Coherent States and Applications in Mathematical Physics (Theoretical and Mathematical Physics) 3030708446, 9783030708443

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Theoretical and Mathematical Physics

Didier Robert Monique Combescure

Coherent States and Applications in Mathematical Physics Second Edition

Coherent States and Applications in Mathematical Physics

Theoretical and Mathematical Physics This series, founded in 1975 and formerly entitled (until 2005) Texts and Monographs in Physics (TMP), publishes high-level monographs in theoretical and mathematical physics. The change of title to Theoretical and Mathematical Physics (TMP) signals that the series is a suitable publication platform for both the mathematical and the theoretical physicist. The wider scope of the series is reflected by the composition of the editorial board, comprising both physicists and mathematicians. The books, written in a didactic style and containing a certain amount of elementary background material, bridge the gap between advanced textbooks and research monographs. They can thus serve as a basis for advanced studies, not only for lectures and seminars at graduate level, but also for scientists entering a field of research.

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Didier Robert Monique Combescure •

Coherent States and Applications in Mathematical Physics Second Edition

123

Didier Robert Departement de mathematiques Laboratoire Jean-Leray Nantes University Nantes, France

Monique Combescure Batiment Paul Dirac IPNL Villeurbanne, France

ISSN 1864-5879 ISSN 1864-5887 (electronic) Theoretical and Mathematical Physics ISBN 978-3-030-70844-3 ISBN 978-3-030-70845-0 (eBook) https://doi.org/10.1007/978-3-030-70845-0 1st edition: © Springer Science+Business Media B.V. 2012 2nd edition: © Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface to the Second Edition

The co-author of the first edition, M. Combescure, agrees that D. Robert would take care of the revisions and extension of the first edition. In this second edition we have revisited the 12 chapters of the first edition and added some minor corrections. These chapters are now gathered together in three parts: I, II and IV. In Part I the basic mathematical structures of coherent states are explained for the harmonic oscillator coherent states. In part II we consider coherent states in various non-Euclidean settings. The 2-torus coherent states are explained with application to quantization of cat maps and quantum ergodicity. Then we consider coherent states in the sense of Gilmore-Perelomov for the groups SUð2Þ and SUð1; 1Þ and we introduce coherent states for the hydrogen atom which are related to the group SOð4Þ and the unit sphere S3 . Part III is new. Here we present more advanced results involving coherent states, in five chapters. In Chaps. 10 and 11 we go more deeply into the connections between coherent states, Weyl quantization and Fourier integral operators. In particular we explain in a general setting the Herman-Kluk formula for the time-dependent Schrödinger propagator. Chapter 12 is concerned with semi-classical measures and quantum chaos: a quantum ergodic theorem is explained. In Chap. 13 we consider open quantum systems. Exact computations are given for quadratic Hamiltonians which are valid for the quantum Brownian motion, by solving a master equation. As a consequence we can estimate the decoherence phenomenon of a coherent state model for the Schrödinger cat. In Chap. 14 we present an adiabatic decoupling result for multi-component systems without crossing eigenvalues, like the Dirac Hamiltonian. As a consequence we compute the time evolution of wave packets for such systems in the semi-classical regime.

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Preface to the Second Edition

The content of the last Part IV is not new. Coherent states are introduced for infinite systems of bosons, with application of the Hepp’s method to the classical limit of large systems. Then we consider finite systems of fermions and we introduce supersymmetric systems with explicit examples using coherent states. Now, I would like to thank my collaborators during these last years for the numerous exchanges with them concerning the matter studied in this book: Clotilde Fermanian-Kammerer, Caroline Lasser, Alberto Maspero and Laurent Thomann. In particular special thanks to Clotilde and Alberto for their careful readings of the new chapters of Part III. This book would not exist without the essential contributions of Monique Combescure to the first edition. At the end, I thank my wife Marie-France for her understanding and patience during the preparation of this new edition and my son Julien for his help for programming pictures with Julia. Nantes, France August 2020

Didier Robert

Preface to the First Edition

The main goal of this book is to give a presentation of various types of coherent states introduced and studied in the physics and mathematics literature during almost a century. We describe their mathematical properties together with application to quantum physics problems. It is intended to serve as a compendium on coherent states and their applications for physicists and mathematicians, stretching from the basic mathematical structures of generalized coherent states in the sense of Gilmore and Perelomov1 via the semi-classical evolution of coherent states to various specific examples of coherent states (hydrogen atom, torus quantization, quantum oscillator). We have tried to show how the field of applications of coherent states is wide, diversified and still alive. Because of our own ability limitations we have not covered all the field. Besides this would be impossible in one book. We have chosen some parts of the subject which are significant for us. Other colleagues may have different opinions. There exist several definitions of coherent states which are not equivalent. Nowadays the most well known is the Gilmore-Perelomov [112, 113, 209] definition: a coherent state system is an orbit for an irreducible group action in an Hilbert space. From a mathematical point of view coherent states appear like a part of group representation theory. In particular canonical coherent states are obtained with the Weyl-Heisenberg 2 group action in L2 ðRÞ and the standard Gaussian u0 ðxÞ ¼ p1=4 ex =2 . Modulo multiplication by a complex number, the orbit of u0 is described by two parameters ðq; pÞ 2 R2 and the L2 -normalized canonical coherent states are uq;p ðxÞ ¼ p1=4 eðxqÞ

2

=2 iððxqÞp þ qp=2Þ

e

:

1

They have discovered independently the relationship with group theory in 1972.

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Preface to the First Edition

Wavelets are included in the group definition of coherent states: they are obtained from the action of the affine group of Rðx 7! ax þ b) on a “mother  func tion” w 2 L2 ðRÞ. The wavelet system has two parameters: wa;b ðxÞ ¼ p1ffiffia w xb a . One of the most useful property of coherent system wz is that they are an “overcomplete” system in the Hilbert space in the sense that we can analyze any g 2 H with its coefficient hwz ; gi and we have a reconstruction formula of g like Z g¼

dz~ gðzÞwz ;

where ~ g is a complex valued function depending on hwz ; gi. Coherent states (with no name) were discovered by Schrödinger (1926) when he searched solutions of the quantum harmonic oscillator being the closest possible to the classical state or minimizing the uncertainty principle. He found that the solutions are exactly the canonical coherent states uz . Glauber (1963) has extended the Schrödinder approach to quantum electro-dynamic and he called these states coherent states because he succeeded to explain coherence phenomena in light propagation using them. After the works of Glauber, coherent states became a very popular subject of research in physics and in mathematics. After the Glauber concept of coherent states several generalizations were elaborated: Gazeau-Klauder coherent states, squeezed coherent states, Perelomov coherent states, wave-packets transform, and may be others, with many applications in different fields in mathematics and physics; some of one is detailed in this book. There exist several other books discussing coherent states. Perelomov’s book [208] plaid an important role in the development of the group aspect of the subject and in its applications in mathematical physics. Several other books brought contributions to the theory of coherent states and worked out their applications in several fields of physics; among them we have [3, 108, 167] but many others could be quoted as well. There is a huge number of original papers and review papers on the subject, we have quoted some of them in the bibliography. We apologize to the authors for forgotten references. In this book we put emphasis on applications of coherent states to semi-classical analysis of Schrödinger type equation (time dependent or time independent). Semi-classical analysis means that we try to understand how solutions of the Schrödinger equations behave as the Planck constant is negligible and how classical mechanics is a limit of quantum mechanics. It is not surprising that semi-classical analysis and coherent states are closely related because coherent states (which are particular quantum states) will be chosen localized close to classical states. Nevertheless we think that in this book we have given more mathematical details concerning these connections than in the other monographs on these subjects.

Preface to the First Edition

ix

Let us give now a quick overview of the content of the book. The first half of the book (Chapter 1 to Chapter 5) is concerned with the canonical (standard) Gaussian Coherent States and their applications in semi-classical analysis of the time-dependent and the time-independent Schrödinger equation. The basic ingredient here is the Weyl-Heisenberg algebra and its irreducible representations. Relationship between coherent states and Weyl quantization is explained in Chapters 2 and 3. In Chapter 4 we compute the quantum time evolution of coherent states in the semi-classical régime; the result is a squeezed coherent state whose shape is deformed, depending on the classical evolution of the system. The main outcome is a proof of the Gutzwiller trace formula given in Chapter 5. The second half of the book (Chapter 6 to Chapter 12) is concerned with extensions to coherent states systems related to other geometry settings. In Chapter 6 we consider quantization of the 2-torus with application to the cat map and an example of “quantum chaos”. Chapter 7 and 8 explain the first examples of non-canonical coherent states where the Weyl-Heisenberg group is replaced successively by the compact group SUð2Þ and the non-compact group SUð1; 1Þ. We shall see that some representations of SUð1; 1Þ are related to squeezed canonical coherent states, with quantum dynamics for singular potentials and with wavelets. We show in Chapter 9 how it is possible to study the hydrogen atom with coherent states related to the group SOð4Þ. In Chapter 10 we considered infinite systems of bosons for which it is possible to extend the definition of canonical coherent states. This is used to prove mean-field limit result for two-body interactions: the linear field equation can be approximated by a non-linear Schrödinger equation in R3 in the semi-classical limit (large number of particles or small Planck constant are mathematically equivalent problems). Chapters 11 and 12 are concerned with extension of coherent states for fermions with applications to supersymmetric systems. Finally in Appendices we have a technical section A around the stationary phase theorem and in section B we recall some basic facts concerning Lie algebras, Lie groups and their representations. We explain how this is used to build generalized coherent systems in the sense of Gilmore-Perelomov. The material covered in these books is designed for an advanced graduate student, or researcher, who wishes to acquaint himself with applications of coherent states in mathematics or in theoretical physics. We have assumed that the reader has a good founding in linear algebra and classical analysis and some familiarity with functional analysis, group theory, linear partial differential equations and quantum mechanics. We would like to thank our colleagues of Lyon, Nantes, and elsewhere, for discussions concerning coherent states. In particular we thank our collaborator Jim Ralston with whom we have given a proof of the trace formula, Stephan Debièvre and Alain Joye.

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Preface to the First Edition

To conclude we wish to express our gratitude to our spouses Alain and Marie-France whose understanding and support have permitted us to spend many hours for the writing of this book. Lyon and Nantes October 2011

Monique Combescure Didier Robert

Contents

Part I 1

2

Basic Euclidian Coherent States and Applications

Introduction to Coherent States . . . . . . . . . . . . . . . . . . . . . . . 1.1 A Very Brief Introduction to the Heisenberg Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Weyl–Heisenberg Group and the Canonical Coherent States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 The Weyl–Heisenberg Translation Operator . . . . . 1.2.2 The Coherent States of Arbitrary Profile . . . . . . . 1.3 The Coherent States of the Harmonic Oscillator . . . . . . . . 1.3.1 Definition and Properties . . . . . . . . . . . . . . . . . . 1.3.2 The Time Evolution of the Coherent State for the Harmonic Oscillator Hamiltonian . . . . . . . 1.3.3 An Overcomplete System . . . . . . . . . . . . . . . . . . 1.4 From Schrödinger to Bargmann–Fock Representation . . . . Weyl Quantization and Coherent States . . . . . . . . . . . . 2.1 Classical and Quantum Observables . . . . . . . . . . . . 2.1.1 Group Invariance of Weyl Quantization . . . 2.2 Wigner Functions . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Coherent States and Operator Norms Estimates . . . 2.4 Product Rule and Applications . . . . . . . . . . . . . . . . 2.4.1 The Moyal Product . . . . . . . . . . . . . . . . . . 2.4.2 Functional Calculus . . . . . . . . . . . . . . . . . 2.4.3 Propagation of Observables . . . . . . . . . . . . 2.4.4 Return to Symplectic Invariance of Weyl Quantization . . . . . . . . . . . . . . . . . . . . . . . 2.5 Husimi Functions, Frequency Sets and Propagation 2.5.1 Frequency Sets . . . . . . . . . . . . . . . . . . . . . 2.5.2 About Frequency Sets of Eigenstates . . . . .

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Wick Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 General Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Application to Semi-classical Measures . . . . . . . . . . . .

Quadratic Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Propagator of Quadratic Quantum Hamiltonians . . 3.2 The Propagation of Coherent States . . . . . . . . . . . . . . 3.3 The Metaplectic Transformations . . . . . . . . . . . . . . . . 3.4 Representation of the Quantum Propagator in Terms of the Generator of Squeezed States . . . . . . . . . . . . . . 3.5 Representation of the Weyl Symbol of the Metaplectic Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 The Classical Motion . . . . . . . . . . . . . . . . . . 3.6.2 The Quantum Evolution . . . . . . . . . . . . . . . .

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The Semiclassical Evolution of Gaussian Coherent States . . . . . . 4.1 General Results and Assumptions . . . . . . . . . . . . . . . . . . . . 4.1.1 Assumptions and Notations . . . . . . . . . . . . . . . . . . . 4.1.2 The Semiclassical Evolution of Generalized Coherent States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Related Works and Other Results . . . . . . . . . . . . . . 4.2 Application to the Spreading of Quantum Wave Packets . . . . 4.3 Evolution of Coherent States and Bargmann Transform . . . . 4.3.1 Formal Computations . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Weighted Estimates and Fourier–Bargmann Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Large Time Estimates and Fourier–Bargmann Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Exponentially Small Estimates . . . . . . . . . . . . . . . . 4.4 Application to the Scattering Theory . . . . . . . . . . . . . . . . . . Trace Formulas and Coherent States . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Semiclassical Gutzwiller Trace Formula . . . . . . . . . . 5.3 Preparations for the Proof . . . . . . . . . . . . . . . . . . . . . . . . 5.4 The Stationary Phase Computation . . . . . . . . . . . . . . . . . . 5.5 A Pointwise Trace Formula and Quasi-Modes . . . . . . . . . 5.5.1 A Pointwise Trace Formula . . . . . . . . . . . . . . . . . 5.5.2 Quasi-Modes and Bohr–Sommerfeld Quantization Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

Part II 6

7

Coherent States in Non Euclidian Geometries

Quantization and Coherent States on the 2-Torus . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The Automorphisms of the 2-Torus . . . . . . . . . . . . . . . . 6.3 The Kinematics Framework and Quantization . . . . . . . . . 6.4 The Coherent States of the Torus . . . . . . . . . . . . . . . . . . 6.5 The Weyl and Anti-Wick Quantizations on the 2-Torus . 6.5.1 The Weyl Quantization on the 2-Torus . . . . . . . 6.5.2 The Anti-Wick Quantization on the 2-Torus . . . 6.6 Quantum Dynamics and Exact Egorov’s Theorem . . . . . 6.6.1 Quantization of SL(2, ZÞ . . . . . . . . . . . . . . . . . 6.6.2 The Egorov Theorem is Exact . . . . . . . . . . . . . . 6.6.3 Propagation of Coherent States . . . . . . . . . . . . . 6.7 Equipartition of the Eigenfunctions of Quantized Ergodic Maps on the 2-Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Spectral Analysis of Hamiltonian Perturbations . . . . . . . .

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Spin 7.1 7.2 7.3

7.4

7.5 7.6

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Coherent States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Groups SOð3Þ and SUð2Þ . . . . . . . . . . . . . . . . . . The Irreducible Representations of SUð2Þ . . . . . . . . . . 7.3.1 The Irreducible Representations of suð2Þ . . . . 7.3.2 The Irreducible Representations of SUð2Þ . . . 7.3.3 Irreducible Representations of SOð3Þ and Spherical Harmonics . . . . . . . . . . . . . . . The Coherent States of SUð2Þ . . . . . . . . . . . . . . . . . . 7.4.1 Definition and First Properties . . . . . . . . . . . . 7.4.2 Some Explicit Formulas . . . . . . . . . . . . . . . . Coherent States on the Riemann Sphere . . . . . . . . . . . Application to High Spin Inequalities . . . . . . . . . . . . . 7.6.1 Berezin-Lieb Inequalities . . . . . . . . . . . . . . . 7.6.2 High Spin Estimates . . . . . . . . . . . . . . . . . . . More on High Spin Limit: From Spin Coherent States to Harmonic-Oscillator Coherent States . . . . . . . . . . .

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Pseudo-Spin Coherent States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction to the Geometry of the Pseudosphere, SOð2; 1Þ and SUð1; 1Þ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Minkowski Model . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Lie Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.3 The Disc and the Half-Plane Poincaré Representations of the Pseudosphere . . . . . . . . . . . . . . . . . . . . . . . .

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8.2

8.3

8.4 8.5

8.6 9

Unitary Representations of SUð1; 1Þ . . . . . . . . . . . . . . . . 8.2.1 Classification of the Possible Representations of SUð1; 1Þ . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Discrete Series Representations of SUð1; 1Þ . . . . 8.2.3 Irreducibility of Discrete Series . . . . . . . . . . . . . 8.2.4 Principal Series . . . . . . . . . . . . . . . . . . . . . . . . 8.2.5 Complementary Series . . . . . . . . . . . . . . . . . . . 8.2.6 Realizations for Bosonic Systems . . . . . . . . . . . Pseudo-Spin-Coherent States for Discrete Series . . . . . . . 8.3.1 Definition of Coherent States for Discrete Series 8.3.2 Some Explicit Formulae . . . . . . . . . . . . . . . . . . 8.3.3 Bargmann Transform and Large k Limit . . . . . . Coherent States for the Principal Series . . . . . . . . . . . . . Generator of Squeezed States. Application . . . . . . . . . . . 8.5.1 The Generator of Squeezed States . . . . . . . . . . . 8.5.2 Application to Quantum Dynamics . . . . . . . . . . Wavelets and Pseudo-Spin Coherent States . . . . . . . . . .

The Coherent States of the Hydrogen Atom . . . . . . . . . . . . 9.1 The S3 Sphere and the Group SOð4Þ . . . . . . . . . . . . . . 9.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Irreducible Representations of SOð4Þ . . . . . . . . 9.1.3 Hyperspherical Harmonics and Spectral Decomposition of DS3 . . . . . . . . . . . . . . . . . . . 9.1.4 The Coherent States for S3 . . . . . . . . . . . . . . . 9.2 The Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 The Fock Transformation: A Map from L2 ðS3 Þ ^ ...... Towards the Pure-Point Subspace of H 9.3 The Coherent States of the Hydrogen Atom . . . . . . . . .

Part III

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More Advanced Results on Harmonic Coherent States and Applications

10 Characterizations of Harmonic Pseudodifferential Operators . . 10.1 Operators Classes Associated with the Harmonic Oscillator . 10.1.1 The Harmonic Classes . . . . . . . . . . . . . . . . . . . . . 10.1.2 The Beals Commutator Classes . . . . . . . . . . . . . . . 10.1.3 Coherent States Classes . . . . . . . . . . . . . . . . . . . . 10.1.4 The Matrix Class . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.5 An Application to Time Dependent Perturbations of Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . 10.2 The Semi-classical Setting . . . . . . . . . . . . . . . . . . . . . . . . .

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313 313 318 324 329 334 337 340 341 341

11 Herman-Kluk Approximation for the Schrödinger Propagator . 11.1 Sub-quadratic Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Semi-classical Fourier-Integral Operators . . . . . . . . . . . . . . 11.3 The Herman-Kluk Semi-classical Approximation . . . . . . . . 11.3.1 Proof of Theorem 57 by Deformation . . . . . . . . . . 11.3.2 A Proof by Solving Transport Equations . . . . . . . . 11.3.3 Error Estimates on Finite Time Intervals . . . . . . . . 11.4 Large Time Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Application to the Aharonov-Bohm Effect . . . . . . . . . . . . . 11.5.1 The Van-Vleck Formula . . . . . . . . . . . . . . . . . . . . 11.5.2 A Setting for the Time-Dependent Aharonov-Bohm Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Application to the Spectrum of Periodic Hamiltonians Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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12 Semi-classical Measures: Stationary and Large Time Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Semi-classical Measures for Bound States . . . . . . . . . . . . 12.1.1 More on the Weyl Law . . . . . . . . . . . . . . . . . . . 12.1.2 More About Semi-classical Measures . . . . . . . . . 12.2 Semi-Classical Quantum Ergodicity . . . . . . . . . . . . . . . . . 12.3 The Quantum Ergodic Birkhoff Theorem . . . . . . . . . . . . . 12.4 Time Dependent Semi-classical Measures . . . . . . . . . . . . . 12.5 Quantum Unique Ergodicity for a Random Hermite Basis .

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349 349 349 351 353 357 362 366

13 Open Quantum Systems and Coherent States . . . . . . . 13.1 Introduction to Open Quantum Systems . . . . . . . . 13.1.1 A General Setting . . . . . . . . . . . . . . . . . . 13.1.2 Coupled Quadratic Hamiltonians . . . . . . . 13.1.3 A Schrödinger Cat Model . . . . . . . . . . . . 13.2 Computation of the Purity . . . . . . . . . . . . . . . . . . 13.2.1 Assumption and Statements . . . . . . . . . . 13.2.2 Proofs of the Statements of Sect. 13.2.1 . 13.3 Separability and Entanglement . . . . . . . . . . . . . . . 13.3.1 Statements . . . . . . . . . . . . . . . . . . . . . . . 13.3.2 Proofs of the Separability Results . . . . . . 13.3.3 Proof of Theorem 72 . . . . . . . . . . . . . . . 13.4 The Master Equation in the Quadratic Case . . . . . 13.4.1 General Quadratic Hamiltonians . . . . . . . 13.4.2 Time Evolution of Reduced Mixed States 13.4.3 About the Schrödinger Cat . . . . . . . . . . .

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373 373 373 377 378 379 379 383 384 384 385 388 392 392 394 399

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xvi

Contents

14 Adiabatic Decoupling and Time Evolution of Coherent States for Multi-component Systems . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Semi-classical Analysis for Multi-components Systems . . . 14.2 Systems with a Scalar Leading Term . . . . . . . . . . . . . . . . 14.3 Spin-Orbit Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Systems with Constant Multiplicities Eigenvalues . . . . . . . 14.5 Figure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part IV

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405 405 408 413 415 427

Coherent States with Infinitely Many Degrees of Freedom . . . . . . . . . . .

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431 431 432 432 435 438 443 443 444 450 453

16 Fermionic Coherent States . . . . . . . . . . . . . . . . . . . . . . . . 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 From Fermionic Fock Spaces to Grassmann Algebras . 16.3 Integration on Grassmann Algebra . . . . . . . . . . . . . . . 16.3.1 More Properties of Grassmann Algebras . . . . 16.3.2 Calculus with Grassmann Numbers . . . . . . . . 16.3.3 Gaussian Integrals . . . . . . . . . . . . . . . . . . . . 16.4 Super-Hilbert Spaces and Operators . . . . . . . . . . . . . . 16.4.1 A Space for Fermionic States . . . . . . . . . . . . 16.4.2 Integral Kernels . . . . . . . . . . . . . . . . . . . . . . 16.4.3 A Fourier Transform . . . . . . . . . . . . . . . . . . 16.5 Coherent States for Fermions . . . . . . . . . . . . . . . . . . . 16.5.1 Weyl Translations . . . . . . . . . . . . . . . . . . . . 16.5.2 Fermionic Coherent States . . . . . . . . . . . . . . 16.6 Representations of Operators . . . . . . . . . . . . . . . . . . . 16.6.1 Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.6.2 Representation by Translations and Weyl Quantization . . . . . . . . . . . . . . . . . . . . . . . . . 16.6.3 Wigner–Weyl Functions . . . . . . . . . . . . . . . . 16.6.4 The Moyal Product for Fermions . . . . . . . . .

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457 457 458 461 462 464 465 467 467 468 470 470 470 472 474 474

15 Bosonic Coherent States . . . . . . . . . . . . . . . . . . . . . . . 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Fock Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.1 Bosons and Fermions . . . . . . . . . . . . . . . 15.2.2 Bosons . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 The Bosons Coherent States . . . . . . . . . . . . . . . . 15.4 The Classical Limit for Large Systems of Bosons . 15.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 15.4.2 The Hepp Method . . . . . . . . . . . . . . . . . 15.4.3 Remainder Estimates in the Hepp Method 15.4.4 Time Evolution of Coherent States . . . . .

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Contents

xvii

16.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.7.1 The Fermi Oscillator . . . . . . . . . . . . . . . . . . 16.7.2 The Fermi-Dirac Statistics . . . . . . . . . . . . . . 16.7.3 Quadratic Hamiltonians and Coherent States 16.7.4 More on Quadratic Propagators . . . . . . . . . . 17 Supercoherent States—An Introduction . . . . . . . . . . . . 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Quantum Supersymmetry . . . . . . . . . . . . . . . . . . . 17.3 Classical Superspaces . . . . . . . . . . . . . . . . . . . . . . 17.3.1 Morphisms and Spaces . . . . . . . . . . . . . . . 17.3.2 Super Algebra Notions . . . . . . . . . . . . . . . 17.3.3 Examples of Morphisms . . . . . . . . . . . . . . 17.4 Super-Lie Algebras and Groups . . . . . . . . . . . . . . . 17.4.1 Super-Lie Algebras . . . . . . . . . . . . . . . . . . 17.4.2 Supermanifolds, A Very Brief Presentation 17.4.3 Super-Lie Groups . . . . . . . . . . . . . . . . . . . 17.5 Classical Supersymmetry . . . . . . . . . . . . . . . . . . . . 17.5.1 A Short Overview of Classical Mechanics . 17.5.2 Supersymmetric Mechanics . . . . . . . . . . . . 17.5.3 Supersymmetric Quantization . . . . . . . . . . 17.6 Supercoherent States . . . . . . . . . . . . . . . . . . . . . . . 17.7 Phase Space Representations of Super Operators . . 17.8 Application to the Dicke Model . . . . . . . . . . . . . . .

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488 488 488 490 495

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499 499 500 502 503 504 505 506 506 507 508 512 512 516 521 523 525 526

Appendix A: Tools for Integral Computations . . . . . . . . . . . . . . . . . . . . . 531 Appendix B: Remainder Estimate for the Moyal Product . . . . . . . . . . . . 539 Appendix C: Semi-classical Functional Calculus . . . . . . . . . . . . . . . . . . . 543 Appendix D: Lie Groups and Coherent States . . . . . . . . . . . . . . . . . . . . . 547 Appendix E: Berezin Quantization and Coherent States . . . . . . . . . . . . . 557 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 Index of Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571 Index of Symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575

Part I

Basic Euclidian Coherent States and Applications

Chapter 1

Introduction to Coherent States

Abstract In this Chapter we study the Weyl–Heisenberg group in the Schrödinger representation in arbitrary dimension n. One shows that it operates in the Hilbert space of quantum states (and on quantum operators) as a phase-space translation. Then applying it to a Schwartz class state of arbitrary profile we get a set of generalized coherent states. When we apply the Weyl–Heisenberg translation operator to the ground state of the n-dimensional Harmonic Oscillator, one gets the standard coherent states introduced by Schrödinger [233] (Naturwissenshaften 14:664–666, 1926) in the early days of quantum mechanics (1926). Later the coherent states have been extensively studied by Glauber [118, 119] (Phys. Rev. 131:2766–2788, 1963; Phys. Rev. 130:2529–2539, 1963) for the purpose of quantum optics and it seems that their name comes back to this work. The standard coherent states have been generalized by Perelomov [208] (Generalised Coherent States and Their Applications, 1986) to more general Lie groups than the Weyl–Heisenberg group. We also introduce the usual creation and annihilation operators in dimension n which are very convenient for the study of coherent states. We show that coherent states constitute a nonorthogonal over-complete system which yields a resolution of the identity operator in the Hilbert space and which allows a computation of the Hilbert–Schmidt norm and of the trace of respectively Hilbert–Schmidt class and trace-class operators. We study their time-evolution for the quantum Harmonic Oscillator hamiltonian and show that a time evolved coherent state located around phase-space point z is up to a phase a coherent state located around the phase-space point z t , where z t is the phase-space point of the classical flow governed by the Harmonic Oscillator. This property was described by Schrödinger as the non-spreading of the time evolution of coherent states under the quantum Harmonic Oscillator dynamics. We also show how to go from the Schrödinger to the Fock–Bargmann representation using the standard coherent states.

© Springer Nature Switzerland AG 2021 D. Robert and M. Combescure, Coherent States and Applications in Mathematical Physics, Theoretical and Mathematical Physics, https://doi.org/10.1007/978-3-030-70845-0_1

3

4

1 Introduction to Coherent States

1.1 A Very Brief Introduction to the Heisenberg Uncertainty Principle One of the main motivation to introduce and study coherent states comes from their close relationship with quantum mechanics. It is well known that a remarkable property is that they are the only states in the Hilbert space L 2 (Rn ) which minimize the uncertainty Heisenberg principle. To fix our notations we start by recalling a mathematical formulation of the uncertainty principle due to H. Weyl. ˆ Bˆ two self-adjoint operLet H be any infinite dimensional Hilbert space and A, ators in H . Recall that when theses operators are not bounded, self-adjointness ˆ D( B) ˆ (see [217]). requires a condition on their domains D( A), The meaning of the hat on operators is to remember that very often an operator Aˆ is obtained after quantization of a classical observable A. ˆ Let be ψ ∈ H , ψ = 1 and define for ψ ∈ D( A): ˆ ˆ ψ := ψ, Aψ, (average).  A ˆ ˆ ˆ ψ A := ( A −  Aψ 1l)2 ψ , (variance).

(1.1) (1.2)

ˆ ∩ D( B), ˆ Then the following Heisenberg inequality holds true for any ψ ∈ D( A) ψ = 1 ˆ ψ Bˆ ≥ 1 |[ A, ˆ B]ψ, ˆ ψ A ψ|, 4 ˆ B] ˆ = Aˆ Bˆ − Bˆ Aˆ is the commutator. The proof is similar to the usual proof where [ A, of the Cauchy–Schwarz inequality [217]. ˆ Bˆ = Q, ˆ then if ψ ∈ D( Q) ˆ ∩ D( P), ˆ ψ = 1, we have Application: Aˆ = P, ˆ ψ Qˆ ≥ ψ P

 . 4

ˆ ψ Qˆ = ψ P

 4

Moreover

(1.3)

if and only if there exist q, p ∈ R, λ > 0, C ∈ C, such that 

λ(x − q)2 ip · x − ψ(x) = C exp  2

 .

(1.4)

We shall see that the r.h.s in (1.4) defines a coherent state localized in the phase space h Rx × Rξ close to (q, p). Recall that  = 2π where h is the Planck constant, h ≈ −34 2 6, 626 × 10 m kg/s, small at the macroscopic scale. Notice the Fourier transform formula:

1.1 A Very Brief Introduction to the Heisenberg Uncertainty Principle



+∞

−∞

x2

e−i x·ξ e− 2 d x =

√

5

ξ2

2πe− 2 ,

˜ ) is localized in p. so it results that the Fourier transform ψ(ξ

1.2 The Weyl–Heisenberg Group and the Canonical Coherent States 1.2.1 The Weyl–Heisenberg Translation Operator Consider quantum mechanics in dimension n (n degrees of freedom). Then the position operator Qˆ has n components Qˆ 1 , . . . Qˆ n where Qˆ j is the multiplication operator in L 2 (Rn ) by the coordinate x j . Similarly the momentum operator Pˆ has n components Pˆ j where ∂ , (1.5) Pˆ j = −i ∂x j  is the Planck constant divided dy 2π . Qˆ and Pˆ are selfadjoint operators with ˆ and D( P). ˆ suitable domains D( Q)   ˆ = u ∈ L 2 (Rn ) | x j u(x) ∈ L 2 (Rn ), ∀ j = 1, . . . , n . D( Q)  ˆ = u ∈ L 2 (Rn ) | D( P)

 ∂u ∈ L 2 (Rn ), ∀ j = 1, . . . , n . ∂x j

The operators Qˆ and Pˆ obey the famous Heisenberg commutation relation [ Pˆ j , Qˆ k ] = −δ j,k i

(1.6)

ˆ The bracket [ A, ˆ B] ˆ is the commutator: on the domain of Qˆ · Pˆ − Pˆ · Q. ˆ B] ˆ = Aˆ Bˆ − Bˆ A. ˆ [ A, ˆ ∩ D( P) ˆ the operator p · Qˆ − q · Pˆ is well On the intersection of the domains D( Q) 2n defined for z = (q, p) ∈ R , where the dot represents the scalar product: p · Qˆ =

n 1

p j Qˆ j .

6

1 Introduction to Coherent States

(z) called the Weyl– It is selfadjoint so it is the generator of a unitary operator T Heisenberg translation operator:

(z) = exp T



 i ˆ ˆ ( p · Q − q · P) . 

(1.7)

Now we use the following Baker–Campbell–Hausdorff formula ˆ Bˆ in the Hilbert space H , Lemma 1 Consider two anti-selfadjoint operators A, ˆ D( B). ˆ We assume the following conditions are satisfied: with domains D( A), (i) There exists a linear subspace space H0 dense in H , which is a core for Aˆ and ˆ B. ˆ B, ˆ et Aˆ , et Bˆ , ∀t ∈ R. (ii) H0 is invariant for A, ˆ B], ˆ well defined in H0 , has a ˆ B] ˆ in H0 and i[ A, (iii) Aˆ and Bˆ commute with [ A, self-adjoint extension in H . Then we have   1 ˆ ˆ ˆ exp( B). ˆ ˆ ˆ (1.8) exp( A + B) = exp − [ A, B] exp( A) 2 Proof Let us introduce F(t)u = e−t

2

ˆ B] ˆ t Aˆ t Bˆ /2[ A,

e e u,

where u ∈ H0 is fixed. Let us compute the time derivative ˆ B]e ˆ −t F (t)u = −t[ A, +e

2

ˆ B] ˆ t Aˆ t Bˆ /2[ A,

e e u ˆ t Bˆ u. e ( Aˆ + B)e

ˆ B] ˆ t Aˆ −t 2 /2[ A,

(1.9)

ˆ The only difficulty is to commute Bˆ with et A . But we have, using the commutations assumptions, d t Aˆ ˆ −t Aˆ ˆ ˆ B]e ˆ −t Aˆ = [ A, ˆ B]. ˆ (e Be ) = et A [ A, dt

So we get that

ˆ F (t) = ( Aˆ + B)F(t)

(1.10) 

and the formula (1.8) follows.

(z): Using this formula, one deduces the multiplication law for the operators T  

(z)T

(z ) = exp − i σ (z, z ) T

(z + z ), T 2

(1.11)

where for z = (q, p), z = (q , p ) , σ (z, z ) is the symplectic product: σ (z, z ) = q · p − p · q ,

(1.12)

1.2 The Weyl–Heisenberg Group and the Canonical Coherent States

7

  i

(z),

(z )T Tˆ (z)Tˆ (z ) = exp − σ (z, z ) T 

and

which is the integral form of the Heisenberg commutation relation. In particular we have: (Tˆ (z))−1 = (Tˆ (z))∗ = Tˆ (−z), since the symplectic product of z by itself is zero. The fact that the Weyl–Heisenberg unitary operator is a translation operator can be seen in the following lemma: Lemma 2 For any z = (q, p) ∈ R2n one has     Qˆ Qˆ − q Tˆ (z) ˆ Tˆ (z)−1 = ˆ . P P−p

(1.13)

ˆ Proof Let us denote L(z) = p · Qˆ − q · Pˆ if z = (q, p). We have easily i

d  it L(z) it ˆ it ˆ ˆ ˆ − L(z) ˆ L(z)]e ˆ ˆ − it L(z) e = e  L(z) [ Q, . Qe dt

ˆ L(z)] ˆ ˆ With the same proof But we have [ Q, = −iq. So we get the formula for Q. ˆ we get the formula for P. Corollary 1

Proof Let

i i ˆ ˆ Tˆ (z) = e−iq· p/2 e  p· Q e−  q· P .

(1.14)

it it 2 ˆ ˆ Uˆ (t) = e−it q· p/2 e  p· Q e−  q· P .

Using Lemma 1 we get d ˆ d i ˆ T (t z) = Uˆ (t) = L(t z)Uˆ (t). dt dt  

Hence the corollary follows. Let us specify the situation in dimension 1. We introduce: i e1 = √ Pˆ  i e2 = √ Qˆ 

8

1 Introduction to Coherent States

e3 = i1l We easily check that [e1 , e2 ] = e3 ,

[e1 , e3 ] = [e2 , e3 ] = 0.

ˆ P, ˆ 1l generate a Lie algebra denoted by h1 which is This means that the operators Q, the Weyl–Heisenberg algebra. The elements of this algebra are defined using triplets of coordinates (s; x, y) ∈ R3 by: W = xe1 + ye2 + se3 .

(1.15)

In quantum mechanics it is more convenient to use the following coordinates: W =−

it i ˆ 1l + ( p Qˆ − q P), 2 

where the real numbers t, q, p are defined as √ √ p = y, q = − x,

t = −2s.

Then we can calculate the commutator of two elements W, W of the Lie algebra h1 : Lemma 3

[W, W ] = (x y − yx )e3 .

(1.16)

Notice here that σ ((x, y), (x , y )) = x y − x y is the symplectic product of (x, y) and (x , y ). Proof We simply use the Lemma 1.



For any W in h1 we can define the unitary operator eW and we get a group using (1.8). This group is denoted H1 . It is a Lie group and its Lie algebra is h1 . The Lie group H1 is simply R3 with the non commutative multiplication (t, z)(t , z ) = (t + t + σ (z, z ), z + z ) where t ∈ R, z ∈ R2 .

(1.17)

We deduce (1.17) from an elementary computation. If W, W ∈ h1 using (1.8) we have 1

eW eW = eW where W

= [W, W ] + W + W . 2 Using the (t, q, p) and (t , q , p ) coordinates for W and W respectively, we get the corresponding coordinates (t

, q

, p

) for W

such that

1.2 The Weyl–Heisenberg Group and the Canonical Coherent States

9

t

= t + t + σ (z, z ), z

= z + z , which is the Weyl–Heisenberg group multiplication (1.17). In the same way we define the Weyl–Heisenberg algebra hn and its Lie Weyl– Heisenberg group Hn . The Weyl–Heisenberg group Hn and Schrödinger representation, n ≥ 1. The Weyl–Heisenberg Lie algebra hn is a real linear space of dimension 2n + 1. Any W ∈ hn has the decomposition W =−

i it ˆ where Qˆ = ( Qˆ 1 , . . . , Qˆ n ), Pˆ = ( Pˆ1 , . . . , Pˆn ), 1l + ( p · Qˆ − q · P), 2 

(t; q, p) = (t; z) ∈ R × R2n is a coordinates system for W . The Lie bracket of W and W , in these coordinates, is [W, W ] =

i σ (z, z )1l. 

This reflects the Heisenberg commutation relations (1.6). As for n = 1 a group multiplication is introduced in R × R2n to reflect multiplication between operators eW . So, Hn is the set R × R2n with the group multiplication (t, z)(t , z ) = (t + t + σ (z, z ), z + z ),

(1.18)

where σ is the symplectic bilinear form in R2n : σ (z, z ) = q · p − q · p, if z = (q, p), z = (q , p ). Hn is a Lie group of dimension 2n + 1. The Schrödinger representation is defined as the following unitary representation of Hn in L 2 (Rn ) : ρ(t, z) = e−it/2 Tˆ (z), (t, z) ∈ Hn . In other words the map (t, z) → ρ(t, z) is a group homomorphism from the Weyl– Heisenberg group Hn into the group of unitary operators in the Hilbert space L 2 (Rn ). By taking the exponential of W one recovers the Weyl–Heisenberg Lie group defined above:   i ˆ = e−it/2 Tˆ (z). ( p Qˆ − q P) e W = e−it/2 exp  Recall that z = (q, p). Remark 1 The Schrödinger representation is irreducible, this will be a consequence of the Schur Lemma 10. According to the celebrated Stone–von Neumann theorem (see [104, 241]) the Schrödinger representation is the unique irreducible representation of Hn , up to conjugation with a unitary operator, for every  > 0.

10

1 Introduction to Coherent States

1.2.2 The Coherent States of Arbitrary Profile The action of the Weyl–Heisenberg translation operator on a state u ∈ L 2 (Rn ) is the following:     i i

x · p u(x − q). (T (z)u)(x) = exp − q · p exp 2 

(1.19)

Physically it translates a state by z = (q, p) in phase space. One has a similar formula for the Fourier transform that we denote F defined as follows:  i F u(ξ ) = (2π )−n/2 e−  x·ξ u(x)d x, Rn

(z)u)(ξ ) = exp (F (T



   i i q · p exp − q · ξ F (u)(ξ − p), 2 

which says that the state is translated both in position and momentum by respectively q and p. Now taking any function u 0 in the Schwartz class S (Rn ) the coherent state associated to it will be simply

(z)u 0 )(x). u z (x) = (T

(1.20)

A useful example for applications is the following generalized Gaussian function. Let us denote Γ ∈ S+ (n) the set of symmetric complex n × n matrices Γ such that its imaginary part Γ is positive-definite. Then we can take u 0 = ϕ (Γ ) , with Γ ∈ S+ (n) and i (1.21) ϕ (Γ ) (x) = (π )−n/4 det 1/4 (Γ )e 2 Γ x·x . Coherent states are closely related with the wave-packet transform. Definition 1 Let z = (q, p) ∈ R2n and u ∈ S (Rn , C) (a profile) . We set WP z u(x) = −n/4 e  p·(x−q) u i

The operator



x−q √ 



.

(1.22)

u → WP z u

is a unitary map in L 2 (Rn , C) which maps continuously Σ1k (Rn ) into Σk (Rn ), where Σk (Rn ) = { f ∈ L 2 (Rn , C), (−2  + |x|2 )k/2 f ∈ L 2 (Rn , C)}, equipped with the -dependent graph-norm. We also denote Σk = Σ1 (Rn ). Gausssian coherent states are the Wave packet transform of Gaussian profile, up to a phase factor:

1.2 The Weyl–Heisenberg Group and the Canonical Coherent States

ϕzΓ = e− 2 q· p WP z (g Γ ), Γ ∈ S+ (n). i

11

(1.23)

Moreover we have the following properties whose proofs are left to the reader i • WP z = e− 2 p·q Tˆ (z) Λ . i • ϕz = e 2 p·q WP z g where g is the standard Gaussian. • For all z, z in R2n , i

 WP z+z = e−  p·q WP z Λ−1  WP z

where Λ u(x) = −n/4 u(x−1/2 ).

1.3 The Coherent States of the Harmonic Oscillator 1.3.1 Definition and Properties They have been introduced by Schrödinger and have been extensively studied and used. They are obtained by taking as reference state u 0 the ground state of the harmonic oscillator  2 x −n/4 (1.24) exp − u 0 (x) = ϕ0 (x) = (π ) 2 Thus ϕz := Tˆ (z)ϕ0 is simply a Gaussian state of the form       i (x − q)2 i x · p exp − . ϕz (x) = (π )−n/4 exp − q · p exp 2  2 (F ϕz )(ξ ) = (π )

−n/4

   q ·ξ (ξ − p)2 iq · p exp −i − . exp 2  2

(1.25)



(1.26)

2n ϕz is a state √ localized in the neighborhood of a phase-space point z = (q, p) ∈ R of size  in all the position and momentum coordinates. Then it is a quantum state which is the analog of a classical state z obtained by the action of the Weyl– Heisenberg group Hn on ϕ0 . They are also called canonical coherent states. They have many interesting and useful properties that we consider now. It is useful to use the standard creation and annihilation operators :

1 ˆ a = √ ( Qˆ + i P), 2

(1.27)

1 ˆ a† = √ ( Qˆ − i P), 2

(1.28)

12

1 Introduction to Coherent States

ˆ ∩ D( P). ˆ Furthermore a simple cona† is simply the adjoint of a defined on D( Q) sequence of the Heisenberg commutation relation is that: [a j , ak† ] = δ j,k .

(1.29)

Then the Hamiltonian of the n-dimensional harmonic oscillator of frequency 1 is

n  1 ˆ2 n  † 2 ˆ ˆ (a†j a j + ) = Hos = ( P + Q ) =  a · a + a · a† 2 2 2 j=1

(1.30)

It is trivial to check that the ground state ϕ0 of Hˆ os is an eigenstate of a with eigenvalue 0. A question is: are the coherent states ϕz also eigenstates of a? The answer is yes and is contained in the following proposition: Proposition 1 Let z = (q, p) ∈ R2n . We define the number α ∈ Cn as

Then the following holds

1 α = √ (q + i p). 2

(1.31)

Tˆ (z)a Tˆ (z)−1 = a − α.

(1.32)

aϕz = αϕz .

(1.33)

Moreover

Proof We simply use Lemma (2) to prove (1.32). Then we remark that: Tˆ (z)a Tˆ (z)−1 ϕz = Tˆ (z)aϕ0 = 0 = (a − α)ϕz .  The Baker–Campbell–Hausforff formula (1.8) is still true for annihilation-creation operators but we need to adapt the proof with the following modifications. Let H0 be the linear space spanned by the products φα (x)eη·x where α ∈ Nn and η ∈ Cn . We can extend the definition of Tˆ (z)u for every u ∈ H0 and z ∈ C2n . Lemma 4 For every u ∈ H0 , z → Tˆ (z)u can be extended analytically to C2n . Moreover Tˆ (z)u ∈ H0 and we have for every z, z ∈ C2n and every u ∈ H0 ,   i





(z + z )u, T (z)T (z )u = exp − σ (z, z ) T 2

(1.34)

where σ is extended as a bilinear form to C2n × C2n . Proof Using formula (1.19), we can extend Tˆ (z)u analytically to C2n . So we can define

1.3 The Coherent States of the Harmonic Oscillator

 exp

13

 i ˆ u := Tˆ (z)u, for z = (q, p) ∈ C2n ( p · Qˆ − q · P) 

Now with the same proof as for (1.8), we get that (1.34) is still true for every z, z ∈  Cn . Using (1.34), in the creation and annihilation operators representation we have that   |α|2 exp(α · a† ) exp(−α¯ · a). Tˆ (z) = exp(α · a† − α¯ · a) = exp − (1.35) 2 Recall that by convention of the scalar product · we have: α¯ · a =

n

α¯ j a j , α · a† =

j=1

n

α j a†j .

j=1

Using (1.35) we have, since exp(−α¯ · a)ϕ0 = ϕ0 :   |α|2 exp(α · a† )ϕ0 . ϕz = exp − 2

(1.36)

Two different coherent states overlap. Their overlapping is given by the scalar product in L 2 (Rn ). We have the following result: Proposition 2     |z − z |2 σ (z, z ) ϕz , ϕz  = exp i exp − . 2 4

(1.37)

Proof: We first establish a useful lemma: Lemma 5

  |z|2 ϕ0 , Tˆ (z)ϕ0  = exp − . 4

(1.38)

Proof We use (1.35). So we get     |α|2 |z|2 † ¯ ¯ ϕ0 , Tˆ (z)ϕ0  = exp − ϕ0 , eα·a e−α·a e−α·a ϕ0  = exp − ϕ0 2 2 4 But since ϕ0 is an eigenstate of a with eigenvalue 0, we simply have ¯ ϕ0  = 1. e−α·a



14

1 Introduction to Coherent States

The operator Tˆ (z) transforms any coherent state in another coherent state up to a phase: Lemma 6

  i

ˆ T (z)ϕz = exp − σ (z, z ) ϕz+z . 2

Proof The proof is immediate using (1.11). The overlap between ϕz and ϕz is given by: ϕz , ϕz  = Tˆ (z)ϕ0 , Tˆ (z )ϕ0  = exp



 i σ (z, z ) ϕ0 , Tˆ (z − z)ϕ0 , 2

where we have used (1.11). Now using the lemma for the last factor we get the result.  In the particular case of the dimension n equals one, the kth eigenstate φk of the harmonic oscillator (the Hermite function, normalized to unity) is generated by (a† )k : φk = (k!)−1/2 (a† )k ϕ0 , so that expanding the exponential, formula (1.36) gives rise to the following wellknown identity: ∞ αk ϕz = exp(−|α|2 /2) √ φk . k! k=0 In arbitrary dimension n, the operator (a†j )k excites the ground state of the harmonic oscillator to the kth excited state of the jth degree of freedom. More precisely let k = (k1 , . . . , kn ) ∈ Nn be a multiindex. The corresponding eigenstate of Hˆ os is: φk (x) = φk1 (x1 ). . .φkn (xn )

(1.39)

and it has eigenvalue E k = (k1 + k2 + · · · + kn + n/2). Note that this eigenvalue is highly degenerate, except E 0 . We have Lemma 7 φk =

n  (a†j )k j  ϕ0 . kj! j=1

(1.40)

The physicists often use the ket notation for the quantum states. Let us define it for completeness: |0 = ϕ0 , |k = φk

1.3 The Coherent States of the Harmonic Oscillator

15

and they also designate the coherent state with the ket notation: |z = ϕz . Then we have: Lemma 8

 ∞  |z|2 α k |z = exp − |k, 4 k =1 k!

(1.41)

j

where α j =

q j +i p j √ 2

and α k = α1k1 α2k2 . . . αnkn , k! = k1 !k2 ! . . . kn !.

1.3.2 The Time Evolution of the Coherent State for the Harmonic Oscillator Hamiltonian A remarkable property of the coherent states is that the Harmonic Oscillator dynamics transforms them into other coherent states up to a phase. This property was anticipated by Schrödinger himself [233] who describes it as the non-spreading of the coherent states wavepackets under the Harmonic Oscillator dynamics. Furthermore the time-evolved coherent state is located around the classical phase-space point of the harmonic oscillator classical dynamics. Let z := (q, p) ∈ R2n be the classical phase-space point at time 0. Then it is trivial to show that the phase-space point at time t is just z t := (qt , pt ) given by z t = Ft z, where Ft is the rotation matrix  Ft =

cos t sin t − sin t cos t



We have the following property: Lemma 9 Define



   ˆ it Hˆ / ˆ Q(t) −it Hˆ os / Q e os . =e ˆ Pˆ P(t)

ˆ ˆ Note that Q(−t), P(−t) are the so-called Heisenberg observables associated to ˆ P. ˆ Then: Q,

16

1 Introduction to Coherent States

(i)



(ii)

   ˆ Qˆ Q(t) = F−t ˆ . ˆ P P(t)

(1.42)

ˆ ˆ e−it Hos / Tˆ (z)eit Hos / = Tˆ (z t ).

Proof (i) One has, using the Schrödinger equation and the commutation property ˆ Pˆ that of Q,     ˆ ˆ d Q(t) − P(t) = . ˆ ˆ Q(t) dt P(t) Then the solution is (1.42). (ii) Then ˆ ˆ it Hˆ os / = p · Q(t) ˆ ˆ ˆ − q · P(t) = pt · Qˆ − qt · P. e−it Hos / ( p · Qˆ − q · P)e



By exponentiation one gets the result.

Proposition 3 The quantum evolution for the harmonic oscillator dynamics of a coherent state ϕz is given by ˆ

e−it Hos / ϕz = e−itn/2 ϕzt . See the picture (1.1), end of this chapter. Proof

ˆ

ˆ

ˆ

ˆ

e−it Hos / ϕz = e−it Hos / Tˆ (z)eit Hos / × e−it Hos / ϕ0 = Tˆ (z t )e−itn/2 ϕ0 = e−itn/2 ϕzt . where we have used that ϕ0 is an eigenstate of Hˆ os with eigenvalue n 2 .



In Chap. 3 we shall see a similar property for any quadratic hamiltonian with possible time-dependent coefficients. Then the quantum time evolution of a coherent state will be a squeezed state instead of a coherent state, located around the phasespace point z t for the associated classical flow which is linear (since the Hamiltonian is quadratic).

1.3 The Coherent States of the Harmonic Oscillator

17

1.3.3 An Overcomplete System We have seen that the coherent states are not orthogonal. So they cannot be considered as a basis of the Hilbert space L 2 (Rn ) of the quantum states, in the naïve sense. Instead they will constitute an overcomplete set of continuous states over which the states and operators of quantum mechanics can be expanded. We can now introduce the Fourier–Bargmann transform that will be studied in more details in Sect. 1.3. We start from u ∈ L 2 (Rn ), u2 := Rn |u 0 (x)|2 d x = 1. Let us define the Fourier–Bargmann transform by the following formula FuB v(z) =: v  (z) = (2π )−n/2 u z , v, z = (q, p) ∈ R2n .

(1.43)

If u is the standard Gaussian ϕ0 , the associated Fourier–Bargmann transform will be denoted F B . Proposition 4 FuB is an isometry from L 2 (Rn ) into L 2 (R2n ). In particular the orbit R2n  z → Tˆ (z)u is dense in L 2 (Rn ). Proof We have i



u z , v = e 2 p·q

Rn

v(x)u 0 (x − q)e−i x· p/ d x.

(1.44)

From Plancherel theorem we get (2π )

−n



 |u (q, p) , v| dp = 2

Rn

Rn

|v(x)u 0 (x − q)|2 d x.

(1.45)

Then we integrate in q variable and change the variables: q = x − q, x = x, so we get that Fu is an isometry. It results easily that the orbit of u is dense in L 2 (Rn ).  Le be U(L 2 (Rn )) the unitary group of the Hilbert space L 2 (Rn ). Corollary 2 The (projective) group representation: R2n  z → Tˆ (z) ∈ U(L 2 (Rn )) is irreducible. Proof Let V be a non trivial subspace of L 2 (Rn ) such that Tˆ (z)V ⊆ V for any z ∈ R2n . Choose u ∈ V , normalized. We know that the orbit of u is dense so V is dense in L 2 (Rn ) which entails that the Schrödinger representation is irreducible.  By polarization we get that the scalar product of two states ψ, ψ ∈ L 2 (Rn ) can be expressed in terms of ψ  , (ψ ) : ψ , ψ =



dz(ψ ) (z)ψ  (z).

(1.46)

We deduce, using Fubini theorem that the function ψ  (z) determines the state ψ completely.We have the reconstruction formula:

18

1 Introduction to Coherent States

ψ = (2π )−n/2

 R2n

dzψ  (z)ϕz .

(1.47)

Another way to prove that the coherent states gives us a resolution of identity is to apply the following Schur’s lemma (see also Appendix). A direct proof uses Corrollary 2: Lemma 10 If Aˆ is a bounded operator in L 2 (Rn ) such that ˆ ∀z ∈ R2n Aˆ Tˆ (z) = Tˆ (z) A, then

Aˆ = C 1ˆl,

for some C ∈ C. We deduce that the coherent states provide a resolution of unity as follows. Define the measure: (1.48) dμ(z) = Cdz = Cdq1 dq2 . . .dqn dp1 dp2 . . .dpn , where and C ∈ C is a constant to be determined later. Let |zz| be the projection operator on the state |z. We consider the operator Aˆ =

 dμ(z)|zz|.

We have the following result: Proposition 5 Aˆ commutes with all the operators Tˆ (z). Proof Using (1.11) we get: ˆ Tˆ (z)] = [ A,



  i

dμ(z ) exp − σ (z, z ) |z z − z| 2    i

− exp( − σ (z, z ) |z + z z | . 2



(1.49) (1.50)

Now using the change of variable z

= z + z in the last term we get zero. Therefore in view of the Schur’s lemma Aˆ must be a multiple of the identity operator: Aˆ = d −1 1l. We determine the constant d by calculating the average of the operator Aˆ in the coherent state |z: d

−1

ˆ = = z| A|z







dμ(z )|z|z | = 2

  |z |2 . dμ(z ) exp − 2

1.3 The Coherent States of the Harmonic Oscillator

19

The constant C can be chosen so that d = 1. Therefore the resolution of the identity takes the form:  dμ(z)|zz| = 1l, (1.51) where dμ(z) is given by (1.48) and the constant C is such that   |z|2 = 1. dz exp − 2 R2n

 C This gives

C = (2π )−n .

(1.52) 

The resolution of identity (1.51) allows to compute the trace of an operator in terms of its expectation value in the coherent states. Let us recall what the trace of an operator is when it exists. Definition 2 An operator Bˆ is said to be of trace class when for some (and then ˆ 1/2 ek  is any) eigenbasis ek of the Hilbert space one has that the series ek , ( Bˆ ∗ B) ˆ convergent. Then the trace of B is defined as ˆ = Tr( B)



ˆ k . ek , Be

(1.53)

k∈N

An operator Bˆ is said to be of Hilbert–Schmidt class if Bˆ ∗ Bˆ is of trace class. Notice that the r.h.s in (1.53) is always defined with values in [0, +∞] if B is any everywhere defined non negative operator. Notations: The space of trace class operators in H is denoted C1 (H ) and the Hilbert–Schmidt class is denoted C2 (H ). They are Banach spaces for their corresponding norms: BTr = Tr((B ∗ B)1/2 ) and BHS = (Tr(B ∗ B))1/2 ). For more details see [217]. Proposition 6 Let Bˆ be an Hilbert–Schmidt operator in L 2 (Rn ) then we have ˆ 2H S = (2π )−n  B

 R2n

ˆ z 2 dz.  Bu

(1.54)

If Bˆ is a trace-class operator in L 2 (Rn ) then we have ˆ = (2π )−n Tr( B)

 R2n

ˆ z dz. u z , Bu

(1.55)

20

1 Introduction to Coherent States

Proof Let {e j } be an orthonormal basis for L 2 (Rn ) (for example the Hermite basis φ j ). We have ˆ 2H S =  B



ˆ j 2  Be

j

=



ˆ j ) 2 . ( Be

(1.56)

j

But we have

ˆ j ) (z) =  Be ˆ j , u z  = e j , Bˆ ∗ u z . ( Be

(1.57)

Using Parseval formula for the basis {e j } we get

ˆ j 2 = (2π )−n  Be

j≥0

 R2n

 Bˆ ∗ u z 2 dz.

(1.58)

ˆ 2H S =  Bˆ ∗ 2H S we get the first part of the corollary. Using that  B For the second part we use that every class trace operator can be writen as Bˆ = Bˆ 2∗ Bˆ 1 where Bˆ 1 , Bˆ 2 are Hilbert–Schmidt. Moreover the Hilbert–Schmidt norm is associated with the scalar product  Bˆ 2 , Bˆ 1  = Tr( Bˆ 2∗ Bˆ 1 ). So we get ˆ = Tr( Bˆ 2∗ Bˆ 1 ) = (2π )−n Tr( B) = (2π )−n

 R

2n

R2n

 Bˆ 2 u z , Bˆ 1 u z dz ˆ z dz. u z , Bu

(1.59) 

These formulas will appear to be very useful in the sequel.

1.4 From Schrödinger to Bargmann–Fock Representation This representation is well adapted to the creation-annihilation operators and to the Harmonic oscillator. It was introduced by Bargmann [20]. In this representation the real phase space R2n is identified to Cn as real spaces: (q, p) → ζ =

q − ip √ 2

and a state ψ is represented by the following entire function on Cn , 

ψHol (ζ ) = ψ  (q, p)e

p 2 +q 2 4

.

1.4 From Schrödinger to Bargmann–Fock Representation

21

Recall that ψ  (z) = (2π )−n/2 ϕz , ψ, z = (q, p).  Proposition 7 The map ψ → ψHol is an isometry from L 2 (Rn ) into the Fock space n n F (C ) of entire functions f on C such that



ζ ·ζ¯

Cn

| f (ζ )|2 e−  |dζ ∧ d ζ¯ | < +∞.

F (Cn ) is an Hilbert space for the scalar product   f2 , f1  =

ζ ·ζ¯

Cn

f 1 (ζ ) f 2 (ζ )e−  |dζ ∧ d ζ¯ |. 

(1.60) 

Proof A direct computation shows that ψHol is holomorphic : ∂ζ¯ ψHol = 0. Recall that the holomorphic and antiholomorphic derivatives are defined as follows. 1 1 ∂ζ = √ (∂q + i∂ p ), ∂ζ¯ = √ (∂q − i∂ p ). 2 2 

We can easily get the following explicit formula for ψHol .    1 x2 √ ζ2 d x. ψ(x) exp − − 2x · ζ +  2 2 Rn (1.61)  The transformation ψ → ψHol is called the Bargmann transform and is denoted by B. Its kernel is the Bargmann kernel: 

ψHol (ζ ) = (π )−3n/4 2−n/2



   1 x2 √ ζ2 − 2x · ζ + , B(x, ζ ) = (π )−3n/4 2−n/2 exp −  2 2 where x ∈ Rn , ζ ∈ Cn .

(1.62)

Recall the notations x 2 = x · x, ζ 2 = ζ · ζ , ζ · ζ¯ = |ζ |2 . Using that ψ → ψ  is an isometry from L 2 (Rn ) into L 2 (R2n ), we easily get that 

ζ ·ζ¯

Cn

 |ψHol (ζ )|2 e−  |dζ ∧ d ζ¯ | = ψ22 .

(1.63)

Hence B is an isometry form L 2 (Rn ) into F (Cn ). ζ ·ζ¯ For convenience let us introduce the Gaussian measure on Cn , dμB = e−  |dζ ∧ d ζ¯ |. It is not difficult to see that F (Cn ) is a complete space. If { f k } is a Cauchy sequence in F (Cn ) then { f k } converges to f in L 2 (Cn , dμB ). So we get in a weak sense that ∂ζ¯ f = 0 so f is holomorphic hence f ∈ F (Cn ). 

22

1 Introduction to Coherent States

Let us now compute the standard harmonic oscillator in the Bargmann representation. We first get the following formula  Rn



∂ζ

∂x ψ(x)B(x, ζ )d x =

Rn

ψ(x)B(x, ζ )d x =

 Rn

ψ(x)

Rn

ψ(x)





x 

√

−

2 x 

√

2 ζ 

−

ζ 





B(x, ζ )d x

(1.64)

B(x, ζ )d x.

(1.65)

Hence B(xψ)(ζ ) = B(∂x ψ)(ζ ) =

√1 (∂ζ 2 √1 (∂ζ 2

+ ζ )Bψ(ζ )

(1.66)

− ζ )Bψ(ζ ).

(1.67)

Then we get the Bargmann representation for the creation and annihilation operators B[a† ψ](ζ ) = ζ B[ψ](ζ ). B[aψ](ζ ) = ∂ζ B[ψ](ζ ).

(1.68) (1.69)

So the standard harmonic oscillator Hˆ os = (a† a + n2 ), has the following Bargmann representation  (1.70) Hˆ os = ζ · ∂ζ + n . 2 Remark 2 It is very easy to solve the time dependent Schrödinger equation for Hˆ os . it it If F ∈ F (Cn ) such that ζ ∂ζ F ∈ F (Cn ), then F(t, ζ ) = e− 2 F(e− 2 ζ ) satisfies i∂t F(t) = Hˆ os F(t), F(0, ζ ) = F(ζ ).

(1.71)

Moreover if  = 1 and if we put e−t/2 in place of e−it/2 we solve the heat equation ∂t F(t) = Hˆ os F(t). We shall see now that the Hermite functions φα have a very simple shape in the Bargmann representation. Let us denote φα# (ζ ) = Bφα . Then we have Proposition 8 For every α ∈ Nn , ζ ∈ Cn , φα# (ζ ) = (2π )−n/2 (α!)−1/2 ζ α .

(1.72)

Moreover {φα# (ζ )}α∈Nn is an orthonormal basis in F (Cn ). Proof Let us first recall the notations in dimension n. For α = (α1 · · · αn ) ∈ N, α! = α1 ! · · · αn ! and for ζ = (ζ1 , · · · , ζn ) ∈ Cn , ζ α = ζ1α1 · · · ζnαn . We get easily that ζ α , ζ β  = 0 if α = β.

1.4 From Schrödinger to Bargmann–Fock Representation

23

It is enough to compute, for n = 1, ζ k 2F (C) and this is an easy computation with the Gamma function. Let us prove now that the system {ζ α }α∈Nn is total in F (Cn ). Let f ∈ F (Cn ) be such that ζ α , f  = 0 for all α ∈ Nn . f is entire so we have f (ζ ) =



fα ζ α ,

α

where f α are the Taylor coefficient of f at 0. The sum is uniformly convergent on every ball of Cn . On the other side from Bessel inequality, we know that the Taylor series α f α ζ α converges in F (Cn ). But we can see that {ζ α }α∈Nn is also an orthogonal system in each ball with center at 0. Then we get that f α = 0 for every α hence f = 0. Let us remark here that we could also prove that the system {φα# (ζ )}α∈Nn is orthogonal using that Hermite functions is an orthonormal system and B is an isometry. Finally, let us prove formula (1.72). It is enough to assume n = 1. We get easily that φ0# = √12π . So for every k ≥ 1, we have, using (1.68),  ζk (a † )k . = B √ φ0 (ζ ) = √ k! 2π k! 

φk# (ζ )

 Then we get the following interesting result. Corollary 3 The Bargmann transform B is an isometry from L 2 (Rn ) onto F (Cn ). The integral kernel of B −1 is    ζ¯ 2 1 x2 √ − 2x · ζ¯ + , B −1 (ζ, x) = (π )−3n/4 2−n/2 exp −  2 2

(1.73)

where x ∈ Rn , ζ ∈ Cn . We also get that the Bargmann kernel is a generating function for the Hermite functions. Corollary 4 For every x ∈ Rn and ζ ∈ Cn we have B(x, ζ ) =

α∈Nn

ζα φα (x). ((2π )n α!)1/2

(1.74)

Proof Compute the Fourier coefficient in the Hermite basis of x → B(x, ζ ).



The standard coherent states also have a simple expression in the Bargmann–Fock space. Let ϕ X be the normalized coherent state at X = (x, ξ ).

24

1 Introduction to Coherent States

Proposition 9 We have the following Bargmann representation for the coherent state ϕ X η¯ η B[ϕ X ](ζ ) = (2π )−n/2 e  (ζ − 2 ) , (1.75) where η =

x−iξ √ . 2

Proof A direct computation gives    √ 1 ζ2 y 2 − y(x + iξ + 2ζ ) − . dy exp −  2 Rn (1.76)

 √ B[ϕ X ](ζ ) = ( 2π )−n

Then we get the result by Fourier transform of the Gaussian e−y . 2



One of the nice properties of the space F (Cn ) is existence of a reproducing kernel. Proposition 10 For every f ∈ F (Cn ) we have f (ζ ) = (2π )−n



η·ζ ¯

Cn

e  f (η)dμB (η), ∀ζ ∈ Cn .

(1.77)

Proof It is enough to assume that f is a polynomial in ζ and that  = 1. So we have f (ζ ) =



ζα cα , with cα = (2π )n/2 (α!)1/2

α

 Cn

η¯ α f (η)dμB (η). (2π )n/2 (α!)1/2 (1.78)

Hence we get −n

f (ζ ) = (2π )

  ζ α η¯ α

 Cn

α

α!

f (η)dμB (η) = (2π )

−n

 Cn

¯ eη·ζ f (η)dμB (η).

(1.79)  η·ζ ¯

Remark 3 The function eζ (η) = (2π )−n e  is a representation of the Dirac delta function in the point ζ . Note that eζ is not in F (Cn ). Moreover we have f (ζ ) = eζ , f  and | f (ζ )| ≤ (2π )−n  f F (Cn ) . Using the Bargmann representation we can give a proof of the well known Mehler formula concerning the Hermite orthonormal basis {φk } in L 2 (R). It is sufficient to assume that  = 1. Theorem 1 For every w ∈ C such that |w| < 1 we have

φk (x)φk (y)w k =

k∈Nn

  1 + w2 2w −n/2 2 −n/2 2 2 π (1 − w ) exp − x·y , (x + y ) + 2(1 − w 2 ) 1 − w2

(1.80)

1.4 From Schrödinger to Bargmann–Fock Representation

25

where k = |k| = k1 + · · · kk , x, y ∈ Rn . Proof The case n ≥ 2 can be easily deduced from the case n = 1. So let n = 1. The left and right side of (1.80) are holomorphic in w in the unit disc {w ∈ C, |w| < 1}. So by analytic continuation principle it is enough to prove it for w = e−t/2 for every t > 0. Hence the right side of (1.80) is the heat kernel denoted K os (t; x, y) of the harmonic oscillator Hˆ os . Using Remark 2 and inverse Bargmann transform we get easily the following integral expression for K os (t; x, y) x 2 +y 2

K os (t; x, y) = 2−1 π −3/2 e− 2    √ 1 exp 2(x · ζ¯ + wy · ζ ) − (w 2 ζ 2 + ζ¯ 2 ) − ζ ζ¯ |dζ ∧ d ζ¯ |. 2 C

(1.81)

The last integral is a Fourier transform of a Gaussian function as it is seen using real √ p , z = (q, p). We have coordinates ζ = q−i 2 −1

K os (t; x, y) = 2 π

−3/2



e− 2 Az·z−i z·Y dY, 1

(1.82)

R2

where     1 3 + w 2 i(1 − w 2 ) i(x + wy) , w = e−t/2 . Y = , A= wy − x 2 i(1 − w 2 ) 1 − w 2

(1.83)

A is a symmetric matrix, its real part is positive definite and det(A) = 1 − w2 . So we have (see [153] or Appendices A, B and C) K os (t; x, y) = π −1/2 (1 − w 2 )−1/2 e−

x 2 +y 2 2

e− 2 A 1

−1

Y ·Y

.

(1.84) 

The Mehler formula follows.

We can deduce in particular the very classical formula for the heat kernel of the harmonic oscillator (n = 1,  = 1). 2 Recall Hˆ os = 1 (− d 2 + x 2 ). 2

dx

Corollary 5 (Mehler formula) Let K τ (x, y) the integral kernel for the operator ˆ e−τ Hos for τ > 0. K τ is in the Schwartz space S (Rx × R y ) and has the following expression K τ (x, y) = √

  2 xy x + y2 + . exp − 2 tanh τ sinh τ 2π sinh τ 1

(1.85)

26

1 Introduction to Coherent States

Motion of Coherent States: Motion of the Schrödinger coherent states driven by the harmonic oscillator. The initial data is 0 = 2,0 and at time t, t = e−it/2 t , t is moving on the circle with center at (0, 0) and radius 2. The localization of the coherent state t is given by the Bargmann transform of (t) which is a Gaussian centered at t .

Chapter 2

Weyl Quantization and Coherent States

Abstract It is well known from the work of Berezin [24] (Commun. Math. Phys. 40:153–174, 1975) in 1975 that the quantization problem of a classical mechanical system is closely related with coherent states. In particular coherent states help to understand the limiting behaviour of a quantum system when the Planck constant  becomes negligible in macroscopic units. This problem is called the semi-classical limit problem. In this chapter we discuss properties of quantum systems when the configuration space is the Euclidean space Rn . So that in the Hamiltonian formalism, the phase space is Rn × Rn with its canonical symplectic form σ . The quantization problem has many solutions, so we choose a convient one, introduced by Weyl [259] (The Classical Groups, 1997) and Wigner [261] (Group Theory and Its Applications to Quantum Mechanics of Atomic Spectra, 1959). We study the symmetries of Weyl quantization, the operationnal calculus and applications to propagation of observables. We show that Wick quantization is a natural bridge between Weyl quantization and coherent states. Applications are given to the semi-classical limit after introducing an efficient modern tool: the semi-classical measures. We illustrate the general results proved in this chapter by explicit computations for the harmonic oscillator. More applications will be given in the following chapters, in particular concerning propagators and trace formulas for a large class of quantum systems.

2.1 Classical and Quantum Observables The quantization problem comes from quantum mechanics and is a mathematical setting for the Bohr correspondence principle between the classical world and the quantum world. Let us consider a system with n degrees of freedom. According the Bohr correspondence principle, it is natural to check a way to associate to every real function A on the phase space R2n (classical observable) a self adjoint operator Aˆ in the Hilbert space L 2 (Rn ) (quantum observable). According the quantum mechanical principles, the map A → Aˆ has to satisfy some properties.

© Springer Nature Switzerland AG 2021 D. Robert and M. Combescure, Coherent States and Applications in Mathematical Physics, Theoretical and Mathematical Physics, https://doi.org/10.1007/978-3-030-70845-0_2

27

28

2 Weyl Quantization and Coherent States

(1) A → Aˆ is linear, Aˆ is self-adjoint if A is real and 1ˆ = 1l L 2 (Rn ) . (2) position observables: x j → xˆ j := Qˆ j where Qˆ j is the multiplication operator by x j (3) momentum observables : ξ j → ξˆ j := Pˆ j where Pˆ j is the differential operator  ∂ . i ∂x j (4) commutation rule and classical limit: for every classical observables A, B we have   i ˆ ˆ  [ A, B] − {A, B} = 0. lim →0  ˆ B] ˆ = Aˆ Bˆ − Bˆ Aˆ is the commutator of Aˆ and B, ˆ {A, B} is the Let us recall that [ A, Poisson bracket defined as follows {A, B}(x, ξ ) = (∂x A · ∂ξ B − ∂x B · ∂ξ A)(x, ξ ), x, ξ ∈ Rn . Let us remark that if we introduce ∇ A = (∂x A, ∂ξ A) then we have {A, B}(x, ξ ) = σ (∇ A(x, ξ ), ∇ B(x, ξ )) (σ is the symplectic bilinear form). If the observables A, B depend only on the position variable (or on the momentum variables) then Aˆ · Bˆ =  A.B, but this is no longer true for a mixed observable. This is related to the non-commutativity for product of quantum observables and the identity: [xˆ j , ξˆ j ] = i so, the quantum observable corresponding to x1 ξ1 is not determined by the rules (1)–(4). We do not want to discuss here the quantization problem in its full generality (see for example [104]). One way to choose a reasonable and convenient quantization procedure is the following, which is called Weyl quantization (see [153] for more details). Let L z be a real linear form on the phase space R2n , where z = ( p, q), L z (x, ξ ) = σ (z, (x, ξ )) (every linear form on R2n is like this). It is not difficult to see that Lˆ z is a well defined quantum Hamiltonian (i.e. an essentially self-adjoint −it  operator in L 2 (Rn )). Its propagator e  L z has been studied in Chap. 1. n For ψ ∈ S (R ), we have explicitly (see Chap. 1) e

−it 

 Lz

ψ(x) = e− 2 t i

2

q· p

it

e  x· p ψ(x − tq).

(2.1)

So, the Weyl prescription is defined by the conditions (1)–(4) and the following (5)

−i L z = Tˆ (z). e−i L z (x,ξ ) → e

In other words Weyl quantization and the exponential commutes on linear forms on the phase space.

2.1 Classical and Quantum Observables

29

We shall use freely the Schwartz space S (Rn )1 and its dual S  (Rn ) (temperate distributions space).  2n ˆ Proposition 11 There  exists a unique continuous map A → A from S (R ) into  L S (Rn ), S  (Rn ) satisfying conditions (1)–(5). Moreover if A ∈ S (R2n ) and ψ ∈ S (Rn ) we have the familiar formula



 x+y −1 A , ξ ei (x−y)·ξ ψ(y)dydξ, 2n 2 R (2.2) and Aˆ is a continuous map from S (Rn ) to S (Rn ). The hermitian conjugate of Aˆ is the quantization of the complex conjugate of A ˆ¯ In particular Aˆ is Hermitian if and only if A is real. ˆ ∗ = A. i.e. ( A) (Opw  )ψ(x)

ˆ := Aψ(x) = (2π )−n



Proof Here it is enough to assume that  = 1. Let us consider the symplectic Fourier transform in S  (R2n ). Assume first that A ∈ S (R2n ) and define  A˜ σ (z) = A(ζ )e−iσ (z,ζ ) dζ. (2.3) R2n

We have the inverse formula −n



A(X ) = (2π )

R2n

A˜ σ (z)eiσ (z,X ) dz.

(2.4)

ˆ A˜ σ (z)ei L z ψ, ηdz.

(2.5)

A˜ σ (z)Tˆ (z)ψdz.

(2.6)

For ψ, η ∈ S (Rn ) we have ˆ = (2π )−n ψ, Aη

 R2n

In other words we must have: ˆ = (2π )−n Aψ

 R2n

 ˆ the function A is called the contravariant symDefinition 3 For a given operator A, ˆ bol of Aˆ and the function A˜ σ is the covariant symbol of A. Let us remark that we have the inverse formula

that f ∈ S (Rn ) means that f is a smooth function in Rn and for every multiindices α, β, β x α ∂x f is bounded in Rn . It has a natural topology. S  (Rn ) is the linear space of continuous linear forms on S (Rn ).

1 Recall

30

2 Weyl Quantization and Coherent States

Proposition 12 If Aˆ is a continuous map from S  (Rn ) to S (Rn ) then we have for every X ∈ R2n , (2.7) A˜ σ (X ) = Tr( Aˆ Tˆ (−X )). Proof For X = 0 the formula is a consequence of the Fourier inversion formula.  For any X we use that the Weyl symbol of Tˆ (−X ) is z → e−iσ (z,X ) As a consequence we have a first norm operator estimate. If A˜ σ ∈ L 1 (R2n ) we have  ˆ ≤ (2π )−n | A˜ σ (z)|dz. (2.8)

A

R2n

The r.h.s in formula (2.2) can be extended by continuity in A to the distribution space S  (R2n ). Let us compute now the Schwartz kernel K A of the operator Aˆ defined in formula (2.6). We have:  K A (x, y) = (2.9) A˜ σ (x − y, p)ei p·(x+y)/2 dp. Rn

Using inverse Fourier transform in p variables, we get K A (x, y) = (2π )

−n



 A Rn

 x+y , ξ ei(x−y)·ξ dξ. 2

(2.10)

and this gives (2.2). The other properties are easy to prove and left to the reader.  Let us first remark that from (2.10) we get a formula to compute the -Weyl symbol of Aˆ if we know its Schwartz kernel K  A(x, ξ ) =

 u u i du. e−  u·ξ K x + , x − 2 2 Rn

(2.11)

Sometimes, we shall use also the notation Aˆ = Opw  A (-Weyl quantization of A). ˆ Hence we shall say that A is an -pseudodifferential operators and that A is its Weyl symbol. For applications it is useful to be able to read properties of the operator Aˆ on its Weyl symbol A. A first example is the Hilbert-Schmidt property.   Proposition 13 Let Aˆ ∈ L S (Rn ), S  (Rn ) . Then Aˆ is Hilbert-Schmidt in L 2 (Rn ) if and only if A ∈ L 2 (R2n ) and we have ˆ 2H S = (2π )−n

A

 R2n

|A(x, ξ )|2 d xdξ.

(2.12)

ˆ Bˆ is a trace In particular if Aˆ and Bˆ are two Hilbert-Schmidt operators then A. operator and we have ˆ B) ˆ = (2π )−n Tr( A.

 R2n

A(x, ξ )B(x, ξ )d xdξ.

(2.13)

2.1 Classical and Quantum Observables

31

Proof We know that ˆ 2H S =

A

 R2n

|K A (x, y)|2 d xd y.

Then we get the proposition using formula (2.10) and Plancherel theorem.



We shall see later many other properties concerning Weyl quantization but most of ˆ like for time we only have sufficient conditions on A to have some property of A, 2 example L continuity or trace-class property. Let us give a first example of computation of a Weyl symbol starting from an ˆ integral kernel. We consider the heat semi-group e−t Hos , of the harmonic oscillator ˆ Hˆ os . Let us denote K w (t; x, ξ ) the Weyl symbol of e−t Hos and K (t; x, y) its integral kernel. From formula (2.11) we get  K w (t; x, ξ ) =

 u u i du. e−  u·ξ K t; x + , x − 2 2 Rn

(2.14)

Using Mehler formula (Chap. 1) we have to compute the Fourier transform of a generalized Gaussian function, so after some computations, we get the following nice formula 2 2 (2.15) K w (t; x, ξ ) = (cosh(t/2))−n/2 e− tanh(t/2)(x +ξ ) . Recall that x 2 = x · x = |x|2 .

2.1.1 Group Invariance of Weyl Quantization Let us first remark that an easy consequence of the definition of Weyl quantization is the invariance by the group of translations in the phase space. More precisely we have, for any classical observable A and any z ∈ R2n , 2n Tˆ (z)−1 Aˆ Tˆ (z) = τ

z A, where τz A(X ) = A(X − z), X ∈ R .

(2.16)

Hamiltonian classical mechanics is invariant by the action of the group Sp(n) of symplectic transformations of the phase space R2n . A natural question to ask is to quantize linear symplectic transformations. We shall see later how it is possible. In this section we state the main results. Recall that the symplectic group Sp(n) is the group of linear transformations of R2n which preserves the symplectic form σ . So F ∈ Sp(n) means that σ (F X, FY ) = σ (X, Y ) for all X, Y ∈ R2n . If we introduce the matrix 

0 1l J= −1l 0



32

2 Weyl Quantization and Coherent States

then F ∈ Sp(n) ⇐⇒ F t J F = J

(2.17)

where F t is the transposed matrix of F. If n = 1 then F is symplectic if and only if det(F) = 1. Linear symplectic transformations can be quantized as unitary operators in L 2 (Rn ): Theorem 2 For every linear symplectic transformation F ∈ Sp(n) and every A ∈ S (R2n ) we have −1 ˆ ˆ ˆ  R(F) A R(F) = A · F. (2.18) ˆ Moreover R(F) is unique up to multiplication by a complex number of modulus 1. ˆ Definition 4 The metaplectic group is the group Met(n) generated by R(F) and λ1l, λ ∈ C, |λ| = 1. Remark 4 A consequence of Theorem 2 is that Rˆ is a projective representation of the symplectic group Sp(n) in the Hilbert space L 2 (Rn ). It is a particular case of a more general setting [255]. More properties of the metaplectic group will be studied in the next chapter. Let us give here some examples of metaplectic transform • The Fourier transform F is associated with the symplectic transformation (x, ξ ) → (ξ, −x). • The partial Fourier transform F j , in variable x j , is associated with the symplectic transform: (x j , ξ j ) → (ξ j , −x j ), (xk , ξk ) → (xk , ξk ), if k = j. • Let A be a linear transformation on Rn , the transformation ψ → |det(A)|1/2 ψ(Ax) is associated with the symplectic transform     x Ax = . FA ξ (At )−1 ξ • Let A be a real symmetric matrix, the transformation ψ → ei Ax·x/2 ψ is associated with the symplectic transform  F=

 1l 0 . A 1l

2.2 Wigner Functions Let ϕ, ψ ∈ L 2 (Rn ). They define a rank one operator Πψ,ϕ η = ψ, ηϕ. Its Weyl symbol can be computed using (2.11).

2.2 Wigner Functions

33

Definition 5 The Wigner function of the pair (ψ, ϕ) is the Weyl symbol of the rank one operator Πψ,ϕ . It will be denoted Wϕ,ψ . More explicitly we have  Wϕ,ψ (x, ξ ) =

 u u  i ψ x− du. e−  u·ξ ϕ x + 2 2 Rn

(2.19)

An equivalent definition of the Wigner function is the following: Wϕ,ψ (z) = (2π )

−n

 R2n

 ϕ, Tˆ (z  )ψe−iσ (z,z )/ dz  ,

(2.20)

ˆ where Tˆ (z) = e−i L z .

We can easily see that (2.19) and (2.20) are equivalent using formula (2.6) and Plancherel formula for symplectic Fourier transform. The Wigner functions are very convenient to use. In particular we have the following nice property: Proposition 14 Let us assume that Aˆ is Hilbert-Schmidt and ψ, ϕ ∈ L 2 (Rn ). Then we have  ˆ = (2π )−n ψ, Aϕ A(X )Wψ,ϕ (X )d X . (2.21) R2n

If A ∈ S  (R2n ) and if ψ, ϕ ∈ S (Rn ), the formula (2.21) is still true in the weak sense of temperate distributions. ˆ = Tr( AΠ ˆ ψ,ϕ ). Hence the first part of the Proof Let us first remark that ψ, Aϕ proposition comes from (2.13). Now if ψ, ϕ ∈ S (Rn ) then we get easily that Wψ,ϕ ∈ S (R2n ). On the other side there exists A j ∈ S (R2n ) such that A j → A in S  (R2n ). So we apply (2.21) to A j and we go to the limit in j.  What Wigner was looking for was a quantum equivalent of the classical probability distribution in the phase space R2n . That is, associated to any quantum state a distribution function in phase space that imitates a classical distribution probability in phase space. Recall that a classical probability distribution is a nonnegative Borel function ρ ; Z → R+ , Z := R2n , normalized to unity:  ρ(z)dz = 1, Z

and such that the average of any observable A ∈ L ∞ (R2n ) is simply given by  ρ(A) =

A(z)ρ(z)dz. Z

From Proposition 14 we see that a possible candidate is

34

2 Weyl Quantization and Coherent States

ρ(z) = (2π )−n Wϕ,ϕ . Actually in the physical literature the expression above (with the factor (2π )−n ) is taken as the definition of the Wigner function but we do not take this convention. In the following we denote by Wϕ the Wigner transform for ϕ, ϕ. What about the expected properties of (2π )−n Wϕ as a possible probability distribution in phase space? Namely: • positivity • normalization to 1 • correct marginal distributions Proposition 15 Let z = (x, ξ ) ∈ R2n and ϕ ∈ L 2 (Rn ) with ϕ = 1. We have: (i)  (2π )−n

Rn

Wϕ (x, ξ )dξ = |ϕ(x)|2 ,

which is the probability amplitude to find the quantum particle at position x. (ii)  (2π )−n

Rn

Wϕ (x, ξ )d x = |ϕ(ξ ˜ )|2 ,

which is the probability amplitude to find the quantum particle at momentum ξ . (iii)  (2π )−n

R2n

Wϕ (x, ξ )d xdξ = 1.

(iv) Wϕ (x, ξ ) ∈ R. Proof (i) Let f ∈ S be an arbitrary test function. We have: 

 y y Wϕ (x, ξ ) f (ξ )dξ = dy ϕ(x ¯ + )ϕ(x − ) dξ e−iξ ·y/ f (ξ ) n 2 2 R  y y = (2π )n dy ϕ(x ¯ + )ϕ(x − )(F f )(y).(2.22) 2 2 Rn 

By taking for the usual Fourier transform F f an approximation of the Dirac distribution at y = 0 we get the result. (ii) is proven similarly. (iii) follows from the normalization to unity of the state ϕ. (iv) We have:   Wϕ (z)∗ = (2π )−n dz  ϕ, Tˆ (−z  )ϕeiσ (z,z )/ . and the result follows by change of the integration variable z  → −z  and by  noting that σ (z, −z  ) = −σ (z, z  ).

2.2 Wigner Functions

35

Let us now compute the Wigner function Wz,z  for a pair (ϕz , ϕz  ) of coherent states. Proposition 16 For every X, z, z  ∈ R2n we have 

n −n

Wz,z  (X ) = 2 

z + z  2 i 1 1   exp − X − − σ (X − z , z − z ) .  2  2

(2.23)

Proof It is enough to consider the case  = 1. Let us apply formula (2.20). Wz,z  (X ) = (2π )−n



ϕz , Tˆ (z  )ϕz  e−iσ (X,z ) dz  . 

R2n

(2.24)

Using a formula from Chap. 1, we have i   ϕz , Tˆ (z  )ϕz   = ϕz , ϕz  +z  e 2 σ (z ,z ) 

 2



= e− 4 |z−z −z | e 2 σ (z,z +z 1

i



)+σ (z  ,z  )

.

(2.25)

Using the change of variables z  = z − z  + u, we have to compute the Fourier 2  transform of the standard Gaussian e−|u| /4 and (2.23) follows. We have the following properties of the Wigner transform: Proposition 17 Let ϕ, ψ ∈ L 2 (Rn ) be two quantum states. Then Wϕ,ψ ∈ L 2 (R2n ) ∩ L ∞ (R2n ) and we have: (i)

Wϕ,ψ L ∞ ≤ 2n ϕ 2 ψ 2 . (ii)

Wϕ,ψ L 2 ≤ (2π )n/2 ϕ 2 ψ 2 . (iii) Let ϕ, ψ ∈ L 2 (Rn ). Then we have: |ϕ, ψ|2 = (2π )−n Wϕ , Wψ  L 2 (R2n ) . Proof (i) is a simple consequence of the definition of the Wigner transform and of the Cauchy–Schwarz inequality. For the proof of (ii) we note that 

 dz|Wϕ,ψ (z)| = 2

 d xdξ |

dyeiξ ·y/ ϕ(x ¯ +

y y )ψ(x − )|2 . 2 2

Using an approximation argument, we can assume that ϕ, ψ ∈ L 1 (Rn ) ∩ L ∞ (Rn ). So we have that ϕ(x ¯ + 2y )ψ(x − 2y ) ∈ L 2 (Rn , dy). According to Plancherel theorem we have:    y y y 2 y −n iξ ·y/ (2π ) dξ | dye ϕ(x ¯ + )ψ(x − )| = dy|ϕ(x ¯ + )ψ(x − )|2 , 2 2 2 2

36

2 Weyl Quantization and Coherent States

so that:  dz|Wϕ,ψ (z)|2 = (2π )n



dx



dy|ϕ(x ¯ + 2y )ψ(x − 2y )|2

= (2π )n ϕ 2 ψ 2

(2.26) 

The Wigner transform operate “as one wishes” in phase space, namely according to the scheme of classical mechanics: ˆ be respectively operators of Proposition 18 Let ϕ, ψ ∈ L 2 (Rn ) and Tˆ (z), R(F) the Weyl–Heisenberg and metaplectic groups, corresponding respectively to – a phase-space translation by vector z ∈ R2n – a symplectic transformation in phase-space We have: WTˆ (z  )ϕ,Tˆ (z  )ψ (z) = Wϕ,ψ (z − z  ) W R(F)ϕ, (z) = Wϕ,ψ (F ˆ ˆ R(F)ψ

−1

(2.27)

z).

(2.28)

Proof We have the nice group property of the Weyl–Heisenberg translation operator:   i  ˆ   ˆ ˆ T (−z )T (X )T (z ) = exp − σ (X, z ) Tˆ (X ),  so that WTˆ (z  )ϕ,Tˆ (z  )ψ (z) = (2π )−n



  i d X exp − σ (z − z  , X ) ϕ, Tˆ (X )ψ 

= Wϕ,ψ (z − z  ).

As a result of the property of the metaplectic transformation we have: −1 ˆ  ˆ ˆ T (z ) R(F) = Tˆ (F −1 z  ). R(F)

Therefore W R(F)ϕ, (z) = (2π )−n ˆ ˆ R(F)ψ (2π )−n





dz  ϕ, Tˆ (F z  )ψe−iσ (z,z )/ =

 dz  ϕ, Tˆ (z  )ψe−iσ (z,F z )/ = (2π )−n





dz  ϕ, Tˆ (z  )ψe−iσ (F

−1 z,z  )/

,

where we have used the change of variable F z  = z  and the fact that a symplectic matrix has determinant one. 

2.2 Wigner Functions

37

Now we get a formula to recover the Weyl symbol of any operator Aˆ in   L S (Rn ), S  (Rn ) :   Proposition 19 Every operator Aˆ ∈ L S (Rn ), S  (Rn ) has a contravariant Weyl symbol A and a covariant Weyl symbol A˜ in S  (R2n ). We have, in the distribution sense in general, in the usual sense if Aˆ is bounded in L 2 (Rn ), A(X ) = (2π )−2n



R4n ϕz



ˆ z Wz  ,z (X )dzdz  . , Aϕ

 ˆ z e− i σ (X,z) dz. A˜ σ (X ) = (2π )−n R2n ϕz+X , Aϕ

(2.29) (2.30)

Proof We compute formally. It is not very difficult to give all the details for a rigorous proof. We apply inverse formula for the Fourier–Bargmann transform (see Chap. 1). So for any ψ ∈ S (Rn ), we have ˆ Aψ(x) = (2π )−2n

 R4n

ˆ z ϕz , ψϕz  (x)dzdz  . ϕz  , Aϕ

(2.31)

ˆ So we get a formula for the Schwartz kernel K A for A, 

K A (x, x ) = (2π )

−2n

 R4n

ˆ z ϕz (x  )ϕz  (x)dzdz  . ϕz  , Aϕ

(2.32)

Then we apply formula (2.11) to get the contravariant symbol A. The formula for the covariant symbol follows from (2.7) and trace computation with coherent states.  The only but important missing property to have a nice probabilistic setting with the Wigner functions is positivity which is unfortunately not satisfied because we have the following result, proved by Hudson [159] for n = 1 then extended to n ≥ 2 by Soto-Claverie [240]. Theorem 3 Wψ (X ) ≥ 0 on R2n if and only if ψ = Cϕz(Γ ) where C is a complex number, Γ a complex, symmetric n × n matrix with a positive non degenerate imaginary part Γ , z ∈ R2n , where we define the Gaussian ϕ (Γ ) (x) = (π )−n/4 det 1/4 Γ exp



 i Γx ·x . 2

(2.33)

Proof We more or less follow the paper of Soto-Claverie [240]. We can check by direct computation that the Wigner density of ϕz (Γ ) is positive (according the definition we have to compute the Fourier transform of the exponent of a quadratic form). We can also give the following more elegant proof. First, it is enough to consider the case z = (0, 0). Second, it is possible to find a metaˆ plectic transformation F such that ϕ0 (Γ ) = R(F)ϕ 0 (see the section on symplectic

38

2 Weyl Quantization and Coherent States

invariance and Chap. 3 for more properties on the metaplectic group). Hence we get (X ) = Wϕ0 (F −1 (X )). But we have computed above Wϕ0 , which is a standard W R(F)ϕ ˆ 0 Gaussian, so it is positive. Conversely, assume now that Wψ (X ) ≥ 0 on R2n . We shall prove that the Fourier– Bargmann transform ψ # (z) is a Gaussian function on the phase space. Hence using the inverse Bargmann transform formula, we shall get that ψ is a Gaussian. Let us first prove the two following properties

|ψ (z)| ≤ Ce #

δ|z|

2

ψ # (z) = 0, ∀z ∈ R2n , and

(2.34)

, ∀z ∈ R , for some C, δ > 0.

(2.35)

2n

We have seen that  |ψ, ϕz |2 = (2π )−n R2n Wψ (X )Wϕz (X )d X  1 2 = 2n R2n Wψ (X )e−  |X −z| d X.

(2.36)

 The last integral is positive because by assumption Wψ (X ) ≥ 0 and Wψ (X )d X = 1. Using again (2.36) we get easily (2.34). The second step is to use a property of entire functions in Cn . Let us recall that in Chap. 1, we have seen that the function  ψa# (ζ )

:= exp

p2 + i p · q 2

 ψ # (q, p)

(2.37)

is an entire function in the variable ζ = q − i p ∈ Cn . Moreover we get easily that ψa# (ζ ) satisfies properties (2.34). To achieve the proof of Theorem 3 we apply the following lemma, which is a particular case of Hadamard factorization theorem for n = 1, extended for n ≥ 2 in [240]. Lemma 11 Let f be an entire function in Cn such that f (ζ ) = 0 for all ζ ∈ Cn and for some C > 0, δ > 0, m (2.38) | f (ζ )| ≤ Ceδ|ζ | , ∀ζ ∈ Cn . Then f (ζ ) = e P(ζ ) , where P is a polynomial of degree ≤ m. 

2.3 Coherent States and Operator Norms Estimates Let us give now a first application of coherent states to Weyl quantization. We assume first that  = 1. Theorem 4 (Calderon–Vaillancourt) There exists a universal constant Cn such that for every symbol A ∈ C ∞ (R2n ) we have

2.3 Coherent States and Operator Norms Estimates

ˆ L (L 2 ,L 2 ) ≤ Cn

A

39 γ

sup |γ |≤2n+1,X ∈R2n

|∂ X A(X )|.

(2.39)

Beginning of the Proof. From (2.32) we get the formula ˆ = (2π )−n ψ, Aη

 R4n

ˆ z ψ # (z  )η# (z)dzdz  . ϕz  , Aϕ

(2.40)

ˆ z  is We shall get (2.39) by proving that the Bargmann kernel K AB (z, z  ) := ϕz  , Aϕ the kernel of a bounded operator in L 2 (R2n ). Let us first recall a classical lemma: Lemma 12 Let (Ω, μ) be a measured (σ -finite) space, K a measurable function on Ω × Ω such that   m K := max{sup |K (z, z  )|dz  , sup |K (z, z  )|dz}. z∈Ω

z  ∈Ω

Ω

Ω

Then K is the integral kernel of a bounded operator TK on L 2 (Ω) and we have

TK ≤ m K . So the Calderon–Vaillancourt will be a consequence of the following Lemma 13 There exists a universal constant Cn such that for every symbol A ∈ C ∞ (R2n ) we have |K AB (z, z  )| ≤ Cn (1 + |z − z  |)−2n−1

sup |γ |≤2n+1,X ∈R2n

γ

|∂ X A(X )|.

(2.41)

Proof We have already seen that K AB (z, z  )

 2n R2n



= A(X )Wz  ,z (X )d X = Rn 

 2 z + z 1   − iσ (X − z , z − z ) d X. A(X ) exp − X − 2 2

(2.42)

First remark that we have ˆ z | ≤ sup |A(X )|. |ϕz  , Aϕ

(2.43)

X ∈R2n

So we only have to consider the case |z  − z| ≥ 1. The estimate is proved by integration by parts (as it is usual for oscillating integral). Let us introduce the phase function

40

2 Weyl Quantization and Coherent States

z + z  2 1 Φ = − X − − iσ (X − z  , z − z  ). 2 2

(2.44)

Computing ∂W Φ, we have |∂ X Φ| ≥ |z − z  | hence ∂X Φ · ∂X Φ e = eΦ . |∂ X Φ|2

(2.45)

So we get the wanted estimates performing 2n + 1 integrations by parts in the integral (2.42) using formula (2.45). This achieves the proof of Calderon–Vaillancourt theorem.  Corollary 6 Aˆ is a compact operator in L 2 (Rn ) if A is C ∞ on R2n and satisfies the following condition: lim |∂zγ A(z)| = 0, ∀γ ∈ N2d , |γ | ≤ 2n + 1.

|z|→+∞

(2.46)

Proof Let us introduce χ ∈ C ∞ (R2n ) such that χ (X ) = 1 if |X | ≤ 21 and χ (X ) = 0 if |X | ≥ 1. Let us define A R (X ) = χ (X/R)A(X ). For every R > 0, Aˆ R is HilbertSchmidt hence compact. Using the Calderon–Vaillancourt estimate, we get R = 0. lim Aˆ − A

|R|→+∞

So Aˆ is compact.



Using the same idea as for proving Calderon–Vaillancourt theorem, we get now a sufficient trace class condition. Theorem 5 There exists a universal constant τn such that for every A ∈ C ∞ (R2n ) we have   γ ˆ Tr ≤ τn |∂ X A(X )|d X. (2.47)

A

2n |γ |≤2n+1 R

In particular if the r.h.s is finite then Aˆ is in the trace class and we have Tr Aˆ = (2π )−n

 R2n

A(X )d X.

(2.48)

Proof Recall that  = 1. From (2.29) we know that Aˆ has the following decomposition into rank one operators Aˆ = (2π )−n

 R4n

ˆ z Πz,z  dzdz  . ϕz  , Aϕ

But we know that Πz,z  T R = 1. So we have

(2.49)

2.3 Coherent States and Operator Norms Estimates

ˆ Tr ≤ (2π )−n

A

 R4n

41

ˆ z |dzdz  . |ϕz  , Aϕ

(2.50)

Using integration by parts as in the proof of Calderon–Vaillancourt, we have ˆ z | ≤ C N (1 + |z − z  |)−N |ϕz  , Aϕ

  |γ |≤N



R2n

γ

e−|X −(z+z )/2| |∂ X A(X )|d X, (2.51) 2

with N = 2n + 1. Now perform the change of variables u = (z + z  )/2, v = z − z  and using Young inequality we get  R4n

ˆ z |dzdz  ≤ τn |ϕz  , Aϕ

  |γ |≤N

γ

R2n

|∂ X A(X )|d X.

(2.52)

Hence (2.47) follows. We can get (2.48) by using approximations with compact support A R like in the proof of Corollary 6. Remark 5 Using interpolation results it is possible to get similar estimates for the ˆ p for 1 < p < +∞ (see [217] for the Schatten classes). Schatten norm A

Let us now compute the action of Weyl quantization on Gaussian coherent states. Lemma 14 Assume that A ∈ S0 (w) (w is a temperate weight, see definition 6). Then for every N ≥ 1, we have ˆ z= Aϕ

 |γ |≤N



|γ | 2

∂ γ A(z) Ψγ ,z + O((N +1)/2 ), γ!

(2.53)

the estimate of the remainder is uniform in L 2 (Rn ) for z in every bounded set of the phase space and γ (2.54) Ψγ ,z = Tˆ (z)Λ Opw 1 (z )g. γ where g(x) = π −n/4 e−|x| /2 , Opw 1 (z ) is the 1-Weyl quantization of the monomial: γ γ  γ    γ (x, ξ ) = x ξ , γ = (γ , γ ) ∈ N2d . In particular Opw 1 (z )g = Pγ g where Pγ is a polynomial of the same parity as |γ |. 2

Proof Let us write ˆ  Tˆ1 (z)g = Λ Tˆ1 (z)(Λ Tˆ1 (z))−1 AΛ ˆ  Tˆ1 (z)g, ˆ z = AΛ Aϕ where Λ is the dilation : Λ ψ = −n/4 ψ(−1/2 x) and Tˆ1 is Tˆ for  = 1. √ ˆ  Tˆ (z) = Opw Let us remark that (Λ Tˆ (z))−1 AΛ 1 [A,z ] where A,z (X ) = A( X + z). So we prove the lemma by expanding A,z in X , around z, with the Taylor formula with integral remainder term to estimate the error term. 

42

2 Weyl Quantization and Coherent States

The following Lemma allows to localized observables acting on coherent states. Lemma 15 Let A be a smooth observable with compact support in the ball B(X 0 , r0 ) of the phase space. Then there exists R > 0 and for all N ≥ 1 there exists C N such that for |z − X 0 | ≥ 2r0 we have ˆ z ≤ C N  N z−N , for |z| ≥ R.

Aϕ

(2.55)

Proof It is convenient here to work on Fourier–Bargmannn side. So we estimate ˆ z   = (2π )−n ϕz , Aϕ

 R2n

A(Y )Wz,z  (Y )dY

(2.56)

As we have already seen, we have 

2



A(Y )Wz,z  (Y )dY =

z + z  2 i 1 1 exp − Y − − σ (Y − z  , z − z  ) A(Y )dY.  2  2 R2n

 n

R2n

Using integrations by parts as above, considering the phase function Ψ (Y ) = −|Y − z+X |2 − iσ (Y − 21 X, z − X ) and the differential operator 2 we get for every M, M  large enough, ˆ z , ϕz  | ≤ C M,M  | Aϕ

 [|Y |≤r0 ]



|Y − z| 1+ √ 

(2.57)

∂Y Ψ ∂ , |∂Y Ψ |2 Y

 −M   |z − z  | M−M 1+ √ dY. (2.58) 

Thefore we easily get the estimate choosing M, M  conveniently and using that the Fourier–Bargmannn transform is an isometry.  We need to introduce some notations for classes of Weyl symbols controlling their smoothness and their growth. Definition 6 A positive function w on Rm is a temperate weight if it satisfies the following property. There exist N , C such that w(X + Y ) ≤ Cw(X )(1 + |X − Y |) N , ∀X, Y ∈ Rm .

(2.59)

A symbol A ∈ C ∞ (R2n ) is a classical observable of weight w and gain ρ ∈ [0, 1] if for every multiindex α there exists Cα such that |∂ Xα A(X )| ≤ Cα w(X )X −ρ|α| , ∀X ∈ R2n The space of symbols of weight w and gain ρ is denoted Sρ (w). A basic example of temperate weight is wμ (X ) = (1 + |X |)μ , μ ∈ R. We shall denote Sμρ = Sρ (wμ ).

2.3 Coherent States and Operator Norms Estimates

43

Remark 6 The product of two temperate weights is a temperate weight and if w is a temperate weight then w −1 is likewise a temperate weight. As proved by Unterberger [245] and rediscovered by Tataru [242], it is possible ˆ z . We state now to characterize the operator class S00 on the matrix element ϕz  , Aϕ a semi-classical version of Unterberger result. Theorem 6 Let Aˆ  be a -dependent family of operators from S (Rn ) to S  (Rn ). 0 2 Then Aˆ = Opw  (A ) with A ∈ S0 with uniform estimate if and only if for every N there exists C N such that we have   − z  | −N ˆ z | ≤ C N 1 + |z √ |ϕz  , Aϕ , ∀ ∈]0, 1), z, z  ∈ R2n . 

(2.60)

0 Proof Suppose that Aˆ = Opw  (A ), with A ∈ S0 is a bounded family. We get estimate (2.60) by integrations by parts as above. Conversely if we have estimates (2.60), using (2.23) and (2.29) we have

A (X ) = (π )

−n



z + z  2 1 ϕz  , Aˆ  ϕz  exp − X −  2 R4n  z + i J (X − ) · (z − z  ) dzdz  . 2



Using the change of variables exists C > 0 such that

z+z  2

= u and z − z  =

√

(2.61)

v we get easily that there

|A (X )| ≤ C, ∀X ∈ R2n ,  ∈]0, 1].

(2.62)

In the same way we can estimate every derivatives of A , after derivation in X in the integral (2.61).  The other main fact in Weyl quantization is existence of an operational calculus. We shall recall its properties in the next section.

2.4 Product Rule and Applications 2.4.1 The Moyal Product One of the most useful properties of Weyl quantization is that we have an operational calculus defined by: 2 This

means that for every γ , sup∈]0,1] ∂ γ A ∞ < +∞.

44

2 Weyl Quantization and Coherent States

The product rule for quantum observables Let us start with A, B ∈ S (R2n ). We ˆ Let us first remark that the look for a classical observable C such that Aˆ · Bˆ = C. integral kernel of Cˆ is  K C (x, y) =

Rn

K A (x, s)K B (s, y)ds.

(2.63)

Using relationship between integral kernels and Weyl symbols, we get −2n



C(X ) = (π )

R4n

e2iσ (Y,Z ) A(X + Z )B(X + Y )dY d Z

(2.64)

where σ is the symplectic bilinear form introduced above. Now let us apply Plancherel formula in R4n and the following Fourier transform formula Lemma 16 Let f (T ) = e 2 B.T,T  , for T ∈ Rm where B is a non degenerate symmetric m × m matrix. Then the Fourier transform f˜ is i

i −1 f˜(ζ ) = (2π )m/2 |det B|−1/2 eiπsgnB e− 2 B ζ,ζ  ,

(2.65)

where sgnB is the signature3 of the matrix B. 

Proof see [153, 217] Hence we get C(x, ξ ) = exp(

i σ (Dx , Dξ ; D y , Dη ))A(x, ξ )B(y, η)|(x,ξ )=(y,η) , 2

(2.66)

We can see easily on formula (B.2) that C ∈ S (R2n ). So that (2.64) defines a noncommutative product on classical observables. We shall denote this product as follows C = A  B (Moyal product). In semi-classical analysis, it is useful to expand the exponent in (B.2), so we get the formal series in :  C j (x, ξ ) j , where C(x, ξ ) = j≥0

C j (x, ξ ) =

1 i ( σ (Dx , Dξ ; D y , Dη )) j A(x, ξ )B(y, η)|(x,ξ )=(y,η) . j! 2

(2.67)

We can easily see that in general C is not a classical observable because of the  dependence. It can be proved that it is a semi-classical observable in the following sense. {b j } be the eigenvalues of B, with their multiplicities. Then sgnB := is the sign of b j . 3 Let

 j

sgnb j where sgnb j

2.4 Product Rule and Applications

45

Definition 7 We say that A is a semi-classical observable of weight w, where w is temperate weight on R2n , if there exist 0 > 0 and a sequence A j ∈ S0 (w), j ∈ N, so that A is a map from ]0, 0 ] into S0 (w)) satisfying the following asymptotic condition : for every N ∈ N and every γ ∈ N2n there exists C N > 0 such that for all  ∈]0, 1[ we have ⎞ ⎛ γ  ∂ −1 j ⎠ ⎝ sup w (z) γ A(, z) −  A j (z) ≤ C N  N +1 , (2.68) R2n ∂z 0≤ j≤N ˆ A0 is called the principal symbol, A1 the sub-principal symbol of A. The set of semi-classical observables of weight w is denoted by Ssc (w). Its range in L (S (Rn ), S  (Rn )) is denoted S sc (w). μ

We may use the notation Ssc = Ssc (wμ ). Now we state the product rule for Weyl quantization. Theorem 7 Let w, w  be two temperate weights in R2n . For every A ∈ S0 (w) and B ∈ S0 (w  ), there exists a unique C ∈ Ssc (ww  )) such that Aˆ · Bˆ = Cˆ with   j C j . The C j are given by C j≥0

C j (x, ξ ) =

1 2j

 (−1)|β| β (Dxβ ∂ξα A).(Dxα ∂ξ B)(x, ξ ). α!β! |α+β|= j

Proof The main technical point is to control the remainder terms uniformly in the semi-classical parameter . This is detailed in the Appendix of the paper [40].  Corollary 7 Under the assumption of the theorem, we have the well known correspondence between the commutator for quantum observables and the Poisson ˆ B] ˆ ∈ Ssc (ww  ) and its principal symbol is bracket for classical observables, i [ A, the Poisson bracket {A, B}. A very useful application of the Moyal product is the possibility to get semiclassical approximations for inverse of elliptic symbol. Definition 8 Let A() be a semiclassical observable in Ssc (w) and X 0 ∈ R2n . We shall say that A is elliptic at X 0 if A0 (X 0 ) = 0. We shall say that A is uniformly elliptic if there exists c > 0 such that |A(X )| ≥ cw(X ), ∀X ∈ R2n .

(2.69)

Theorem 8 Let A ∈ Ssc (w) be an uniformly elliptic semi-classical symbol. Then there exists B ∈ Ssc (w −1 ) such that B  A = 1 (in the sense of asymptotic expansion in Ssc ). Moreover we have Bˆ · Aˆ = 1l + O(∞ ). (2.70)

46

2 Weyl Quantization and Coherent States

where the remainder is estimated in the L 2 norm of operators.   j B j with Moreover the semi-classical symbol B of Bˆ is B = j≥0 −2 B0 = A−1 0 , B1 = −A1 A0 .

(2.71)

Proof Let us denote C j (E, F) the j term in the Moyal product E  F. The method consists to compute by induction B0 , . . . , B N such that (



 j B j )  A(h) = 1 + O( N +1 ).

(2.72)

0≤ j≤N

We start with B0 = A10 . The next step is to compute B1 such that B1 A0 + A1 B0 = 0. Then to compute B2 such that C2 (A0 , B0 ) + C1 (A1 , B1 ) + B2 A0 = 0. So we get all the B j by induction using the asymptotic expansion for the Moyal product. The remainder term in (2.70) is estimated using Calderon Vaillancourt theorem.  We give now a local version of the above theorem, which can be proved by the same method. Theorem 9 Let A ∈ Ssc (m) be an elliptic symbol in an open bounded set Ω of R2n . Then for every χ ∈ C0∞ (Ω) there exists Bχ ∈ S−∞ sc such that Bˆ χ Aˆ = χˆ + O(∞ ).

(2.73)

Remark 7 For application it is useful to note that if A depends in a uniform way of some parameter ε ∈ [0, 1] then B also depends uniformly in ε. In particular ε may depend on .

2.4.2 Functional Calculus An useful consequence of the algebraic properties of symbolic quantization is a functional calculus: under suitable assumptions if Hˆ is an Hermitian semi-classical observable then for every smooth function f , f ( Hˆ ) is also a semi-classical observable. The technical statement is Theorem 10 Let Hˆ be a uniformly elliptic semiclassical Hamiltonian. Let f be a smooth complex valued function such that, for some r ∈ R, we have

2.4 Product Rule and Applications

47

∀k ∈ N, ∃Ck , | f (k) (s)| ≤ Ck sr −k , ∀s ∈ R. Then f ( Hˆ ) is a semiclassical observable with a semiclassical symbol H f (, x, ξ ) such that H f (, ·, ·) ∈ Ssc (wr ) given by the asymptotic expansion: H f (, z) 



 j H f, j (z).

(2.74)

j≥0

In particular we have H f,0 (z) = f (H0 (z)), 

H f,1 (z) = H1 (z) f (H0 (z)),  and for, j ≥ 2, H f, j = d j,k (H ) f (k) (H0 ),

(2.75) (2.76) (2.77)

1≤k≤2 j−1 γ

where d j,k (H ) are universal polynomials in ∂z H (z) with |γ | +  ≤ j. A proof of this theorem can be found in [141, 217] and in [83]. We give a sketch of proof in the Appendix C. In particular we can take f (s) = (λ + s)−1 for λ = 0 (the proof begins with this case) or f with a compact support. From this theorem we can get the following consequences on the spectrum of Hˆ (see [141]). Theorem 11 Let Hˆ be like in Theorem 10. Assume that H0−1 [E − , E + ] is a compact set in Rn × Rn . Consider a closed interval I ⊂ [E − , E + ]. Then we have the following properties. (i) ∀ ∈]0, 0 ], 0 > 0, the spectrum of Hˆ is discrete and is a finite sequence of eigenvalues E 1 () ≤ E 2 () ≤ · · · ≤ E N I () where each eigenvalue is repeated according its multiplicity. Moreover N I = O(−n ) as   0. (ii) For all f ∈ C0∞ (I ), f ( Hˆ ) is a trace class operator and we have Tr[ f ( Hˆ )] 



 j−d τ j ( f ) ,

(2.78)

j≥0

where τ j are distributions supported in H0−1 (I ). In particular we have τ0 ( f ) = (2π )−d τ1 ( f ) = (2π )−d

 R

2n

R2n

f (H0 (z))dz,

(2.79)

f  (H0 (z))H1 (z)dz.

(2.80)

An easy consequence of this is the following Weyl asymptotic formula

48

2 Weyl Quantization and Coherent States

Corollary 8 If I = [λ− , λ+ ] such that λ± are non critical values for H0 4 then we have  n dqdp. (2.81) lim (2π ) N I = →0

[H0 (q, p)∈I ]

Remark 8 Formula (2.81) is very well known and can be proved in many ways, under much weaker assumptions. For a proof using the functional calculus see [217], p. 283–287. Under our assumptions we shall see in a next Chapter that we have a Weyl asymptotic with an accurate remainder estimate: N I = (2π )

−n

 [H (q, p)∈I ]

dqdp + O(1−n ) ,

using a time dependent method due to Hörmander and Levitan ([152] and its bibliography). For more accurate results about spectral asymptotics see [161].

2.4.3 Propagation of Observables Now we come to the main application of the results of this section. We shall give a proof of the correspondence (in the sense of Bohr) between quantum and classical dynamics. As we shall see this theorem is a useful tool for semi-classical analysis although its proof is an easy application of Weyl calculus rules stated above. The microlocal version of the following result is originally due to Egorov [90]. Beals [22] found a nice simple proof. For simplicity we shall assume that the semiclassical Hamiltonian H (t) is subquadratic, H (t) ∈ S2sc , in the following sense |∂zγ H j (t, z)| ≤ Cγ , for |γ | + j ≥ 2; −2 (H (t) − H0 (t) − H1 (t)) ∈ S0sc .

(2.82)

We also assume continuity and uniformity in time t ∈ [−T, T ]. Theorem 12 (The Semiclassical Propagation Theorem) Let us consider a time dependent Hamiltonian satisfying 2.82. Let A ∈ S10 be an observable such that γ ∂ X A ∈ S00 if |γ | ≥ 1. Then we have: (a) For any  > 0 and for every ψ ∈ S (Rn ), the Schrödinger equation i∂t ψt = Hˆ (t)ψt , ψt=s = ψ. 4λ

is a non critical value for H means that ∇ H (z) = 0 if H (z) = λ.

(2.83)

2.4 Product Rule and Applications

49

has a unique solution which we denote ψt = U (t, s)ψ. Moreover U (t, s) can be extended as a unitary operator in L 2 (Rn ). ˆ s) = U (s, t) AU ˆ (t, s) of A, ˆ A(s, ˆ s) = A, ˆ has a semi(b) The time evolution A(t, 1 (t, s) such that A (t, s) ∈ S . More precisely we have classical Weyl symbol A   sc  A(t, s)  j≥0  j A j (t, s), in S0sc , which is uniform in t, s, for t, s ∈ [−T, T ]. Moreover A j (t, s) can be computed by the following formulas: A0 (t, s; z) = A(Φ t,s (z)),  t {A(Φ τ,t ), H1 (τ })(Φ t,τ (z))dτ. A1 (t, s; , z) =

(2.84) (2.85)

s

and for j ≥ 2, A j (t, s; z) can be computed by induction on j. Proof (A sketch). Property (a) will be proved later in this book (see also [191] for  = 1). It is easier to prove it if H is time independent because we can prove in this case that Hˆ is essentially self-adjoint (for a proof see [217]). Then we have U (t) := U (t, 0) = exp(−

it ˆ H ). 

Let us remark that, under the assumption of the theorem, the classical flow for H0 exists globally. Indeed, the Hamiltonian vector field (∂ξ H0 , −∂x H0 ) has a sublinear growing at infinity so, no classical trajectory can blow up in a finite time. Moreover, using usual methods in non linear O.D.E (variation equation) we can prove that A(Φ t,s ) ∈ S00 with semi-norm uniformly bounded for t, s bounded. Now, from the Heisenberg equation and the classical equations of motion we get

 U (s; τ )

∂ U (s, τ ) A0 (t, τ )U (τ, s) = ∂τ 

i ˆ [ H (τ ),  A0 (t, τ )] − {H (τ ), A0 }(Φ t,τ ) U (τ, s), 

(2.86)

where A0 (t, s) = A(Φ t,s ). But, from the corollary of the product rule, the principal symbol of i  ˆ t,τ  [ H (τ ), A 0 (t, τ )] − {H (τ ), A0 } (Φ )  vanishes. So, in the first step, using the product rule formula, we get the approximation 

t s

ˆ (t, s) − A U (s, t) AU 0 (t, s) =  i ˆ ˆ  t,τ U (τ, s)dτ. [ H (τ ), A U (s, τ ) 0 (t, τ )] − { H (τ ), A0 }Φ  

(2.87)

Now, it is not difficult to obtain, by induction, the full asymptotics in . For j ≥ 2,

50

2 Weyl Quantization and Coherent States

A j (t, s; z) =





t

Γ (α, β) s

|(α,β)|+k= j+1 0≤≤ j−1

[(∂ξα ∂xβ Hk (τ )) · (∂ξ α ∂x β A )](Φ t,τ (z))dτ, (2.88)

with Γ (α, β) =

(−1)|β| − (−1)|α| −1−|(α,β)| i . α!β!2|α|+|β|

The main technical point is to estimate the remainder terms. For a proof with more details see [40] where the authors get a uniform estimate up to Ehrenfest time (of order log −1 ).  Remark 9 If H (t) = H0 (t) is a polynomial function of degree ≤ 2 in z on the phase space R2n then the propagation theorem assumes a simpler form : A(t, s) = A(Φ t,s ) and the remainder term is null. This is a consequence of the following exact formula. i ˆ ˆ  [ H , B] = {H, B}, 

(2.89)

where B ∈ S00 . Now we give an application of the propagation theorem and coherent states in semiclassical analysis: we recover the classical evolution from the quantum evolution, in the classical limit   0. Corollary 9 For every observable A ∈ Σ 0 and every z ∈ R2n , we have lim U (t, s)ϕz , Aˆ U (t, s)ϕz  = A(Φ t,s (z))

0

(2.90)

and the limit is uniform in (t, s; z) on every bounded set of Rt × Rs × R2n z . Proof U (t, s)ϕz , Aˆ U (t, s)ϕz  = =

ϕz , U (s, t) Aˆ U (t, s)ϕz   R2n A(t, s; X )Wz,z (X )d X  |X −z|2 = (π )−n R2n A(t, s; X )e−  d X.

(2.91)

So by the propagation theorem we know that A(t, s; X ) = A(Φ t,s (X )) + O(). Hence the corollary follows.  Remark 10 The last result has a long history beginning with Ehrenfest (1930) and continuing with Hepp (1974), Bouzouina–Robert [40]. In this last paper it is proved that the corollary is still valid for times smaller than the Ehrenfest time TE := γ E | log |, for some constant γ E > 0.

2.4 Product Rule and Applications

51

2.4.4 Return to Symplectic Invariance of Weyl Quantization Let us give now a first construction of metaplectic transformations. Other equivalent constructions and more properties will be given later (chapter on quadratic hamiltonians). Lemma 17 For every F ∈ Sp(n) we can find a C 1 - smooth curve Ft , t ∈ [0, 1], in Sp(n), such that F0 = 1l and F1 = F. Proof An explicit way to do that is to use the polar decomposition of F, F = V |F| √ where V is a symplectic orthogonal matrix and |F| = F t F is positive symplectic matrix. Each of these matrices have a logarithm, so F = e K e L with K , L Hamiltonian matrices, and we can choose Ft = et K et L . Ft is clearly the linear flow defined by the  quadratic Hamiltonian Ht (z) = 21 St z · z where St = −J F˙t Ft−1 . Now we use the (exact) propagation theorem. U (t, s) denotes the propagator defined by the quadratic Hamiltonian built in the proof of Lemma 17 and Theorem ˆ 12. Then we define R(F) = U (1, 0). Recall that U (t, 0) is the solution of the Schrödinger equation i

d U (t, 0) = Hˆ (t)U (t, 0), U (0, 0) = 1l. dt

(2.92)

The following theorem translates the symplectic invariance of the Weyl quantization Theorem 13 For every linear symplectic transformation F ∈ Sp(n) and every symbol A ∈ S00 we have −1 ˆ ˆ ˆ  A R(F) = A ·F (2.93) R(F) Proof This is a direct consequence of the exact propagation formula for quadratic Hamiltonians ˆ (t, 0) =  (2.94) U (0, t) AU AΦ t,0  We can get another proof of the following result (see formulas (2.27)). Corollary 10 Let ψ, η ∈ L 2 (Rn ). For every linear symplectic transformation F ∈ Sp(n), we have the following transformation formula for the Wigner function. (z) = Wψ,η (F −1 (z)), ∀z ∈ R2n . W R(F)ψ, ˆ ˆ R(F)η Proof For every A ∈ S (R2n ), we have

(2.95)

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2 Weyl Quantization and Coherent States

ˆ ˆ  R(F)η, Aˆ R(F)ψ =

 R2n

ˆ ˆ A(z)W ( R(F)ψ, R(F)η)(z)dz

−1 ˆ ˆ ˆ = η, R(F) A R(F)ψ  = A(F · z)Wψ,η (z)dz.

(2.96)

R2n



The corollary follows. We have the following uniqueness result.

Proposition 20 Given the linear symplectic transformation F ∈ Sp(n), there exists ˆ a unique transformation R(F), up to a complex number of modulus 1, satisfying (2.18). ˆ  Proof If Vˆ satisfies Vˆ −1 Aˆ Vˆ = A we have that Bˆ com· F then if Bˆ = Vˆ −1 · R(F), 0 ˆ ˆ mutes with every A, A ∈ S0 . In particular B commutes with the Heisenberg-Weyl ˆ But we knows that Tˆ (z)−1 Bˆ Tˆ (z) = translations Tˆ (z), hence Tˆ (z)−1 Bˆ Tˆ (z) = B.  B(· + z). So the Weyl symbol of Bˆ (it is a temperate distribution) is a constant complex number λ. But here Bˆ is unitary, so |λ| = 1. 

2.5 Husimi Functions, Frequency Sets and Propagation 2.5.1 Frequency Sets The Husimi transform of some temperate distributions u ∈ S  (Rn ) is defined as follows Definition 9 The Husimi transform of u ∈ S  (Rn ) is the function Hu (z) defined on the phase space R2n by Hu (z) = (2π )−n |u, ϕz |2 , z ∈ R2n ,

(2.97)

where ϕz is the standard coherent state defined in Chap. 1. The Husimi transform in contrast with the Wigner transform is always nonnegative. We shall see below that the Husimi distribution is a “regularization” of the Wigner distribution. Proposition 21 For every ϕ ∈ L 2 (Rn ) we have Hϕ = Wϕ ∗ G 0 where G 0 is a gaussian function in phase space namely G 0 (z) = (π )−n e−|z|

2

/

.

2.5 Husimi Functions, Frequency Sets and Propagation

53

 One has R2n G 0 (z)dz = 1. This means that the Husimi distribution is a “regularization” of the Wigner distribution. Proof According to the Proposition (17) (iii) we have: Hϕ (z) = (2π )−n Wϕz , Wϕ  L 2 (R2n ) . But we know that Wϕz (X ) = Wϕ0 (X − z). We use Proposition (16):  Wϕz , Wϕ  = 2

n



|X − z|2 exp −  R2n

 Wϕ (X )d X. 

This yields the result.

In semi-classical analysis (or in high frequency analysis) it is important to understand what is the region of the phase space R2n where some states ψ ∈ L 2 (Rn ) depending on , essentially lives when  is small. For that purpose let us introduce the frequency set of ψ Definition 10 Let ψ ∈ L 2 (Rn ), depending on , such that ψ ≤ 1. We say that ψ is negligible near a point X 0 ∈ R2n , if there exists a neighborhood V X 0 such that Hψ (z) = O(∞ ), ∀z ∈ V X 0 .

(2.98)

Let us denote N [ψ ] the set {X ∈ R2n , ψ is negligible near X }. The frequency set FS[ψ ] is defined as the complement of N [ψ ] in R2n . Example 1 • If ψ = ϕz then FS[ϕz ] = {z} i • Let ψ = a(x)e  S(x) where a and S are smooth functions, a ∈ S (Rn ), S real. Then we have the inclusion FS[ψ] ⊆ {(x, ξ )|ξ = ∇ S(x)}.

(2.99)

This is called a Lagrangian or WKB state. There are several equivalent definitions of the frequency set that we now give. Proposition 22 Let ψ be such that ψ ≤ 1 and X 0 = (x0 , ξ0 ) ∈ R2n . The following properties are equivalent (i) Hψ (X ) = O(+∞ ), ∀X ∈ V X 0 . (ii) There exists A ∈ S (R2n ), such that A(X 0 ) = 1 and

54

2 Weyl Quantization and Coherent States

ˆ  = O(+∞ ).

Aψ

(2.100)

(iii) There exists a neighborhood V X 0 of X 0 such that for all A ∈ C0∞ (V X 0 ), ˆ  = O(+∞ ).

Aψ

(2.101)

(iv) There exist χ ∈ C0∞ (Rn ) such that χ (x0 ) = 1 and a neighborhood Vξ0 of ξ0 such that i (2.102) χ (x)e  x·ξ , ψ  = O(+∞ ) for all ξ ∈ Vξ0 . Proof Let us assume (i). Then we have Hψ (z) = O(∞ ), |z − X 0 | < r0 .

(2.103)

Using Lemma 15 we have ˆ z ≤ C N  N /2 < z >−N , if |z − X 0 | > r0 /2.

Aϕ

(2.104)

We have, using linearity of integration, ˆ  = (2π )−n Aψ



ˆ z. dzϕz , ψ  Aϕ

From triangle inequality, we have  ˆ  ≤ (2π )−n dz|ψ , ϕz | Aϕ ˆ z

Aψ    −n ≤ (2π ) dz + dz . [|z−X 0 | 0, r0 > 0 such that |H (X ) − E| ≥ δ, for every X ∈ B(X 0 , r0 ). Let us choose some χ ∈ C0∞ (B(X 0 , r0 )), χ (X 0 ) = 1. Using Theorem 9 and the remark following this theorem (here at the end ε = ), we can find B such that ˆ Hˆ − E  ) = χˆ + O(+∞ ), B( / FS[ψ ]. so we get χˆ ψ = O(+∞ ) hence X 0 ∈

(2.109) 

Assume now that Hˆ satisfies the assumptions of the Propagation theorem and ψ satisfies the Schrödinger equation (2.108). Proposition 24 The frequency set FS[ψ ] is invariant under the classical flow Φ t , for every t ∈ R.

56

2 Weyl Quantization and Coherent States

Proof Let X 0 ∈ / FS[ψ ]. There exists a compact support symbol A elliptic at X 0 ˆ  = O(+∞ ). such that Aψ For every t we have ˆ  = O(+∞ ) = e U (−t) Aψ

it E  

ˆ A(t)ψ .

ˆ Recall that the principal symbol of A(t) is A · Φ t . So we get that if z is near −t +∞ ˆ / FS[ψ ]. So we get that FS[ψ ] Φ (X 0 ), then A(t)ψ = O( ), hence Φ −t X 0 ∈ is invariant.  Remark 11 There are several other names in the literature for the frequency set: “semiclassical wave front set”, “microsupport”. There are also different variants like “analytic-microsupport” [188].

2.6 Wick Quantization 2.6.1 General Properties Following Berezin–Shubin [27] we start with the following general setting. Let M be a locally compact metric space, with a positive Radon measure μ and H an Hilbert space. For each m ∈ M we associate a unit vector em ∈ H such that the map m → em is strongly continuous from M into H . Moreover we assume that the following Plancherel formula is satisfied, for all ψ ∈ H , 

ψ 2 =

|em , ψ|2 dμ(m).

(2.110)

M

Let us denote ψ # (m) = em , ψ. The map ψ → ψ # (m) := I ψ(m) is an isometry from H into L 2 (M). The canonical coherent states introduced in Chap.1 are examples of this setting where M = R2n , H = L 2 (Rn ), z → ϕz , with the measure dμ(z) = (2π )−n dqdp, z = (q, p) ∈ R2n . Definition 11 Let Aˆ ∈ L (H ). (i) The Wick-covariant symbol of Aˆ is the function on M defined by Ac (m) = ˆ m . em , Ae (ii) The contravariant symbol of Aˆ is the function on M, if it exists, such that ˆ = Aψ

 Ac (m)Πm ψdμ(m), ψ ∈ H .

(2.111)

M

For the standard coherent states example, the covariant symbol is called Wick symbol and the contravariant symbol the anti-Wick symbol. ˆ m ). The covariant symbol satisfies the equality Ac (m) = Tr( AΠ

2.6 Wick Quantization

57

Let us compute the anti-Wick symbol of some operator Aˆ with Weyl symbol A. |X −z|2

We know that the -Weyl symbol of the projector Πz is the Gaussian (π )−n e−  . So we get that the Weyl symbol of Aˆ is the convolution of its anti-Wick symbol and a standard Gaussian function:  |X −z|2 −n A(X ) = (π ) Ac (X )e−  dz. (2.112) R2n

This formula shows that if Aˆ has a bounded anti-Wick symbol ( Ac ∈ L ∞ (R2n )) then its Weyl symbol is an entire function in C2n , which is a restriction for a given operator to have an anti-Wick symbol. Let us remark that the Wick symbol is an inverse formula associated with (2.112):  Ac (z) = 2

n R2n

A(X )e−

|X −z|2 

d X.

(2.113)

Now we give another interpretation of the contravariant symbol. Let us first remark that we have I ∗ · I = 1lH . I · I ∗ = ΠH .

(2.114) (2.115)

where ΠH is the orthogonal projector in L 2 (M) on H identified with I (H ). Proposition 25 Let us assume that Aˆ has a contravariant symbol Ac such that Ac ∈ L ∞ (M). Then we have (2.116) Aˆ = I ∗ · Ac · I , where Ac is here the multiplication operator in L 2 (M). Proof For every ψ, η ∈ H we have ˆ η, Aψ =



ˆ η, em em , Aψdμ(m)

(2.117)

Ac (m  )em , Πm  ψdμ(m  ) A (m  )Πm  em , Πm  ψdμ(m  ).

(2.118)

M

and ˆ em , Aψ =

= 



M

M

c

So we get ˆ η, Aψ =



Ac (m  )Πm  em , Πm  ψη, em dμ(m  )dμ(m) M×M

We get the conclusion using the equality

(2.119)

58

2 Weyl Quantization and Coherent States

 η, em   =

em  , em η, em dμ(m)

(2.120)

M

 Estimates on operators with covariant and contravariant symbols are easier to prove than for Weyl symbols. Moreover they can be used as a first step to get estimates in the setting of Weyl quantization as we shall see for positivity. The following proposition is easy to prove. Proposition 26 Let Aˆ be an operator in H with a contravariant symbol Ac . Suppose that Ac ∈ L ∞ (M). Then Aˆ is bounded in H and we have ˆ ≤ Ac ∞ .

Ac ∞ ≤ A

(2.121)

Moreover Aˆ is self-adjoint if and only if Ac is real and Aˆ is non-negative if Ac is μ-almost everywhere non negative on M. For our basic example H = L 2 (Rn ), it is convenient to use the following notation. If A is a classical observable, A ∈ S00 , Opw  (A) denotes the Weyl quantization of A and Opaw  (A) denotes the anti-Wick quantization of A. We shall also use the notation aw Aaw = Opaw  (A).. In other words Op (A) admits A as an anti-Wick symbol. The following proposition is an easy consequence of above results Proposition 27 Let A ∈ S00 (more general symbols could be considered). Then we have 2 w −n − |X| Opaw . (2.122)  (A) = Op (A ∗ G), where G(X ) = (π ) e −n ψ, Opaw  (A)ψ = (2π )

 R2n

A(z)Hψ (z)dz.

(2.123)

where Hψ (z) is the Husimi function of ψ. We get now the following useful consequence for Weyl quantization. Proposition 28 (semiclassical Gårding inequality) Let A ∈ S00 , A ≥ 0 on R2n . Then there exists C ∈ R such that for every  ∈]0, 1] we have ˆ ψ, Aψ ≥ C, ∀ψ ∈ L 2 (Rn ).

(2.124)

Proof We know that Opw  (A ∗ G) is a non negative bounded operator. So the proposition will be proved if (2.125)

Opw  (A ∗ G − A) = O(). Using a standard argument for smoothing with convolution, we get that −1 (A ∗ G − A) ∈ S00 , with uniform estimates in  ∈]0, 1]. Hence we get (2.125) as a consequence of Calderon–Vaillancourt theorem. 

2.6 Wick Quantization

59

ˆ  , for a family These results are useful to study the matrix elements ψ , Aψ {ψ } in the semi-classical regime [140]. This subject is related with an efficient tool introduced by Lions–Paul [181] and Gérard [110] (see also [45]): the semi-classical measures. This is an application of anti-Wick quantization as we shall see now.

2.6.2 Application to Semi-classical Measures Semi-classical measures were introduced to describe localization and oscillations of families of states {ψ } , ψ = 1 (or at least bounded in L 2 (Rn )). Let us first remark that A → ψ, Opaw  Aψ is a probability measure μ in R2n . Moreover this probability measure has a density given by the Husimi function of ψ , dμ = (2π )−n Hψ (z)dz. In particular we have |ψ, Opaw  Aψ| ≤ A ∞ . for every A ∈ Cb (R2n ) (space of continuous, bounded functions on R2n ). Definition 12 A semi-classical measure for the family of normalized states {ψ } is a probability measure μ on the phase space R2n for which there exists at least one sequence {k }, lim k = 0 such that for every A ∈ S00 , we have k→+∞

 lim ψk Opk

k→+∞

aw

Aψk  =

Adμ.

(2.126)

R2n

In other words, the measure sequence μk weakly converges toward the measure μ. Remark 12 Semi-classical measures can also be defined for states ψ ∈ L 2 (Rn , K ) where K is an Hilbert space. By the way in this setting Weyl symbols and anti-Wick symbols are operators in K . We can also define semi-classical measures for statistical mixed states ρ, ˆ where ρˆ is a non negative operator such that Tr ρˆ = 1. For more applications and properties of these extensions see the huge litterature on this subject; for example see [178]. The following proposition is a straightforward application of properties of Husimi function

60

2 Weyl Quantization and Coherent States

Proposition 29 Let μ be a semi-classical measure for {ψ } . Then the support supp(μ) of the measure μ is included in the frequency set FS[ψ ], supp(μ) ⊆ FS[ψ ]. Example 2 (i) Let ψ = ϕz , a standard coherent state. Then this family has one semi-classical measure, μ = δz (Dirac probability) (ii) Let us assume that the states family {ψ } is tight in the following sense. 



lim lim sup

R→+∞

→0

|X |≥R



|ϕ X , ψ | d X 2

= 0.

(2.127)

Then applying the Prokhorov compacity theorem [31] , there exists at least one semi-classical measure for ψ  . One of a challenging problem in quantum mechanics is to compute these semi-classical measures for family of bound states satisfying (2.108). If for some ε > 0, H −1 [E − ε, E + ε] is a bounded set, this family is tight. For classically ergodic systems it is conjectured that there exists only one semi-classical measure which is the Liouville measure [140]. One important property of semi-classical measures is the following propagation result. Let us consider the time dependent Schrödinger equation i∂t ψ (t) = Hˆ ψ (t), ψ (0) = ψ ,

(2.128)

where H is a time independent Hamiltonian. We assume that H is real, subquadratic and  independent (for simplicity). γ

∂ X H ∈ L ∞ (R2n ), for all γ such that |γ | ≥ 2.

(2.129)

Let μ be a semi-classical measure for {ψ }. Theorem 14 For every t ∈ R, {ψ (t)} has a semi-classical measure dμt for the same subsequence k given by the transport of dμ by the classical flow Φ t : μ(t) = (Φ t )∗ μ. Proof For every A ∈ C0 ∞ (R2n ), the semiclassical Egorov theorem and comparaison between anti-Wick and Weyl quantization, give  ψ (t), Opaw  (A)ψ (t) =

R2n

A · Φ t dμψ + O().

(2.130)

Hence we get the result going to the limit for the sequence k . We have the following consequence for the stationary Schrödinger equation.



2.6 Wick Quantization

61

Corollary 12 Let μ be semi-classical measure for a family of bound states {ψ }, satisfying Hˆ ψ = E  ψ . Then μ is invariant by the classical flow Φ t for every t ∈ R. Proof ψ (t) = e−  E ψ satisfies the time dependent Schrödinger equation so using  the Theorem we get (Φ t )∗ μ = μ. it

Now we illustrate Corollary 2.6.2 on Hermite bound states of the harmonic oscillator. We assume n = 1. We can easily compute the Husimi function H j of the Hermite function function φ j . H j (q, p) = |ϕ X , φ j |2 =

(q 2 + p 2 ) j − 1 (q 2 + p2 ) . e 2 2 j j!

(2.131)

We want to study the quantum measures dμ j = (2π )−1 H j (q, p)dqdp when the energies E j = ( j + 21 ) have a limit E > 0. So we have  → 0 and j → +∞. For simplicity we fix E > 0 and choose  =  j = Ej . 2  Let f be in the Schwartz class S (R ). We have to compute the limit of f (X )dμ j (X ) for j → +∞. Using polar coordinates and a change of variables we have to study the large k limit for the Laplace integral 1 I ( j) := ( j + 1)!



∞

√ √ j u j e− E u f ( 2u cos θ, 2u sin θ )du, θ ∈ [0, 2π [.

0

We can asume that f has a bounded support and (0, 0) is not in the support of f . Using the Laplace method we get √ lim I ( j) = f ( 2E(cos θ, sin θ )).

j→+∞

(2.132)

So, we have  lim

j→+∞

1 f (X )dμ j (X ) = √ 2π 2E



2π

√ f ( 2E(cos θ, sin θ ))dθ.

(2.133)

0

On the r.h.s √ of (2.133) we recognize the uniform probability measure on the circle of radius 2E. This measure is a semi-classical measure for the√ quantum harmonic 2E moves on the oscillator. Let us remark that the classical oscillator of energy √ circle of radius 2E in the phase space.

62

2 Weyl Quantization and Coherent States 1.0 0.8 0.6 0.4 0.2 0.0 - 0.2 - 0.4 - 0.6 - 0.8 - 1.0 - 1.0

- 0.8

- 0.6

- 0.4

- 0.2

0.0

0.2

0.4

0.6

0.8

Wigner function of Schrödinger coherent states; on right: horizontal section 2.0 1.8 1.6 1.4 Z

1.2 1.0 0.8 0.6 0.4 0.2

0.0 - 1.5 - 1.0 - 0.5 Y 0.0 0.5 1.0 1.5

1.5

1.0

0.5

- 0.5

0.0

- 1.0

- 1.5

X

Wigner function of squeezed coherent states; on right: horizontal section

1.0

Chapter 3

Quadratic Hamiltonians

Abstract The aim of this chapter is to construct the quantum unitary propagator for Hamiltonians which are quadratic in position and momentum with time-dependent coefficients. We show that the quantum evolution is exactly solvable in terms of the classical flow which is linear. This allows to construct the metaplectic transformations which are unitary operators in L2 (Rn ) corresponding to symplectic transformations. Simple examples of such metaplectic transformations are the Fourier transform, which corresponds to the symplectic matrix J defined in (3.4) and the propagator of the harmonic oscillator, corresponding to rotations in the phase space. The main results of this chapter are computations of the quantum evolution operators for quadratic Hamiltonians acting on coherent states. We show that the time evolved coherent states are still Gaussian states which are recognized to be squeezed states centered at the classical phase space point (see Chap. 8). From these computations we can deduce most of properties concerning quantum quadratic Hamiltonians. In particular we get the explicit form of the Weyl symbols of the metaplectic transformations. These formulae are generalizations of the Mehler formula for the harmonic oscillator. Quadratic Hamiltonians are very important in quantum mechanics because more general Hamiltonians can be considered as non trivial perturbations of time dependent quadratic ones as we shall see in Chap. 5.

3.1 The Propagator of Quadratic Quantum Hamiltonians A classical quadratic Hamiltonian H is a quadratic form defined in the phase space R2n . We assume that this quadratic form is time dependent, so we have H (t, z) =



cj,k (t)zj zk ,

1≤j,k≤n

where z = (q, p) ∈ R2n is the phase space variable and the real coefficients cj,k (t) are continuous functions of time t ∈ R. It can be rewritten as

© Springer Nature Switzerland AG 2021 D. Robert and M. Combescure, Coherent States and Applications in Mathematical Physics, Theoretical and Mathematical Physics, https://doi.org/10.1007/978-3-030-70845-0_3

63

64

3 Quadratic Hamiltonians

H (t, q, p) =

  1 q (q, p)S(t) , p 2

(3.1)

where S(t) is a 2n × 2n real symmetric matrix of the block form  S(t) =

G t LTt Lt Kt

 (3.2)

G t , Lt , Kt are n × n real matrices with G t , Kt being symmetric, and LTt denotes the transpose of Lt . The classical equations of motion for this Hamiltonian are linear and can be written as     q˙ q = JS(t) , (3.3) p˙ p where J is the symplectic matrix  J =

0 1ln −1ln 0

 .

(3.4)

Ler F(t) be the classical flow for the Hamiltonian H (t). It means that it is a symplectic 2n × 2n matrix obeying ˙ F(t) = JS(t)F(t).

(3.5)

with F(0) = 1l. Then the solution of (3.3) with q(0) = q, p(0) = p is simply 

q(t) p(t)



  q = F(t) p

We now consider the quantum Hamiltonian   ˆ Q ˆ ˆ  H (t) = (Q, P) · S(t) . Pˆ

(3.6)

The quantum evolution operator U (t)1 is solution of the Schrödinger equation i

d  (t)U (t), U (t) = H dt

(3.7)

with U (0) = 1l. The following result was already proved in Chap. 2 as a particular case of a more general result. We shall give here a simple direct proof.

1 We

shall explain later why this quantum propagator is a well defined unitary operator in L2 (Rn ).

3.1 The Propagator of Quadratic Quantum Hamiltonians

65

Theorem 15 One has for all times t U (t)



∗

ˆ Q Pˆ





ˆ Q U (t) = F(t) Pˆ

 .

(3.8)

ˆ (t), and similarly for Pˆ (Heisenberg observables). t = U (t)∗ QU Proof Define Q Using the Schrödinger equation one has 

d −i dt But



t Q  Pt



   ˆ Q  = U (t) H (t), U (t). Pˆ ∗

    ˆ ˆ Q Q  = −iJS(t) . H (t), Pˆ Pˆ

t ,  Pt must satisfy the linear equation This means that Q d dt



t Q  Pt



 = JS(t)

t Q  Pt





which is trivially solved by (3.8).

3.2 The Propagation of Coherent States In this section we give the explicit form of the time evolved coherent states in terms of the classical flow F(t) given by the 2n × 2n block matrix form:  At Bt . F(t) = Ct Dt 

(3.9)

It will be shown that the complex n × n matrix At + iBt is always non singular. Then we establish the following result: Ut ϕz = (π )−n/4 T (zt )(det(At + iBt ))−1/2 exp



i (Ct + iDt )(At + iBt )−1 x · x 2



(3.10) where zt = F(t)z is the phase space point of the classical trajectory and  T (z) is the Weyl–Heisenberg translation operator by the vector z = (x, ξ ) ∈ R2n :   i ˆ − x · P) ˆ .  T (z) = exp (ξ · Q 

(3.11)

66

3 Quadratic Hamiltonians

This means that Ut ϕz is a squeezed state centered at the phase space point zt , so the squeezed state moves on the classical trajectory. We take  = 1 for simplicity. A simple example is the harmonic oscillator: 1 d2 1 + x2 . Hˆ os = − 2 2 dx 2

(3.12) ˆ

It is well known that for t = kπ, k ∈ Z the quantum propagator e−it Hos has an explicit Schwartz kernel Kos (t; x, y) (Mehler formula, Chap. 1). It is easier to compute directly with the coherent states ϕz . ϕ0 is the ground state of Hˆ os , so we have ˆ (3.13) e−it Hos ϕ0 = e−it/2 ϕ0 . ˆ

Let us compute e−it Hos ϕz , ∀z ∈ R2 , with the following ansatz ˆ

e−it Hos ϕz = eiδt (z) Tˆ (zt )e−it/2 ϕ0 ,

(3.14)

where zt = (qt , pt ) is the generic point on the classical trajectory (a circle here), coming from z at time t = 0. Let ψt,z be the state equal to the r.h.s in (3.14), and let us compute δt (z) such that ψt,z satisfies the equation i dtd ϕ = Hˆ ϕ ϕ|t=0 = ψ0,z . We have Tˆ (zt )u(x) = ei(pt x−qt pt /2) u(x − qt ) and

ψt,z (x) = ei(δt (z)−t/2+pt x−qt pt /2) ϕ0 (x − qt ).

(3.15)

So, after some computations left to the reader, using properties of the classical trajectories q˙ t = pt p˙ t = −qt , pt2 + qt2 = p2 + q2 , the equation i

1 d ψt,z (x) = (Dx2 + x2 )ψt,z (x) dt 2

(3.16)

is satisfied if and only if δt (z) =

1 (pt qt − pq). 2

(3.17)

 Let us now introduce the following general notations for later use. Ft is the classical flow with initial time t0 = 0 and final time t. It is represented as a 2n × 2n matrix which can be written as four n × n blocks :

3.2 The Propagation of Coherent States

67

 Ft =

At Bt Ct Dt

 .

(3.18)

Let us introduce the following squeezed states: ϕ Γ defined as follows. ϕ Γ (x) = aΓ exp



i Γx·x 2

 (3.19)

where Γ ∈ S+ (n). Recall that S+ (n) is the Siegel space of complex, symmetric matrices Γ such that (Γ ) is positive and non degenerate and aΓ ∈ C is such that the L2 -norm of ϕ Γ is one. T (z)ϕ Γ . We also denote ϕzΓ =  For Γ = i1l, we denote ϕ = ϕ i1l . Theorem 16 We have the following formulae, for every x ∈ Rn and z ∈ R2n , Ut ϕ Γ (x) = ϕ Γt (x), Ut ϕzΓ (x) =  T (Ft z)ϕ Γt (x),

(3.20) (3.21)

where Γt = (Ct + Dt Γ )(At + Bt Γ )−1 and aΓt = aΓ (det(At + Bt Γ ))−1/2 . Beginning of the proof The first formula can be proven by the ansatz  Ut ϕ0 (x) = a(t) exp

 i Γt x · x . 2

where Γt ∈ S+ (n) and a(t) is a complex values time dependent function. We get first a Riccati equation to compute Γt and a linear equation to compute a(t). The second formula is easy to prove from the first, using the Weyl translation operators and the following known property T (Ft z). T (z)Ut ∗ =  Ut  Let us now give the details of the proof for z = 0. We begin by computing the action of a quadratic Hamiltonian on a Gaussian ( = 1). Lemma 18

  i i i Lx · Dx e 2 Γ x·x = LT x · Γ x − TrL e 2 Γ x·x . 2

Proof This is a straightforward computation, using Lx · Dx =

xj Dk + Dk xj 1  Ljk i 2 1≤j,k≤n

68

3 Quadratic Hamiltonians

and, for ω ∈ Rn ,

(ω.Dx )e 2 Γ x·x = (Γ x · ω)e 2 Γ x·x . i

i

 Lemma 19 (GDx · Dx )e 2 Γ x·x = (GΓ x · Γ x − iTr(GΓ )) e 2 Γ x·x . i

i

Proof As above, we get    e 2i Γ x·x = 1 Kx · x + x · LΓ x + 1 GΓ x · Γ x − i Tr(L + GΓ ) e 2i Γ x·x (3.22) H 2 2 2  We are now ready to solve the equation i with

∂ ψ = Hˆ ψ, ∂t

ψ|t=0 (x) = g(x) := π −n/4 e−x

(3.23)

2

/2

.

For simplicity we assume here that Γ = i1l, the proof can be easily generalized to Γ ∈ S+ n. We try the ansatz i ψ(t, x) = a(t)e 2 Γt x·x , (3.24) which gives the equations Γ˙t = −K − 2ΓtT L − Γt GΓt , 1 a˙ (t) = − (Tr(L + GΓt )) a(t) 2

(3.25) (3.26)

with the initial conditions Γ0 = i1l, a(0) = (π )−n/4 . We note that Γ T L et LΓ determine the same quadratic forms. So the first equation is a Ricatti equation and can be written as Γ˙t = −K − Γt LT − LΓt − Γt GΓt ,

(3.27)

where LT denotes the transposed matrix for L. We shall now see that Eq. (3.27) can be solved using Hamilton equation

3.2 The Propagation of Coherent States

69



F˙ t = J

K L LT G

 Ft

F0 = 1l

(3.29) 

We know that

(3.28)

Ft =

At Bt Ct Dt .



is a symplectic matrix ∀t. So using the next lemma, we have det(At + iBt ) = 0 ∀t. Let us denote (3.30) Mt = At + iBt , Nt = Ct + iDt . We shall prove that Γt = Nt Mt−1 . By an easy computation, we get ˙ t = LT Mt + GNt M N˙ t = −KMt − LNt .

(3.31)

Now, compute d ˙ M −1 (Nt Mt−1 ) = N˙ M −1 − NM −1 M dt = −K − LNM −1 − NM −1 (LT M + GN )M −1 = −K − LNM −1 − NM −1 LT − NM −1 GNM −1 ,

(3.32)

which is exactly Eq. (3.27). Now we compute a(t), using the following equality,  ˙ )M −1 = Tr (L + GΓt ) Tr LT + G(C + iD)(A + iB)−1 = Tr(M using TrL = TrLT . Let us recall the Liouville formula

which give directly

d ˙ t Mt−1 ) log(det Mt ) = Tr(M dt

(3.33)

a(t) = (π )−n/4 (det(At + iBt ))−1/2 .

(3.34)

To complete the proof, we need to prove the following Lemma 20 Let F be a symplectic matrix.  F=

A B CD



Then det(A + iB) = 0 and (C + iD)(A + iB)−1 is positive definite.

70

3 Quadratic Hamiltonians

We shall prove a more general result concerning the Siegel space S+ (n). Lemma 21 If

 F=

A B CD

 .

is a symplectic matrix and Z ∈ S+ n then A+BZ et C+DZ are non singular and (C + DZ)(A + BZ)−1 ∈ S+ (n). Proof Let us denote M := A + BZ, N := C + DZ. F is symplectic, so we have F T JF = J . Using     M I =F . N Z we get

 (M , N )J T

T

M N



  I = (I , Z)J = 0. Z

(3.35)

Which gives M TN = NTM . In the same way, we have 1 (M T , N T )J 2i 1 = (I , Z)J 2i



¯ M N¯

 =

1 (I , Z)F T JF 2i

  I Z¯

(3.36)

  1 I = (Z¯ − Z) = −Z Z¯ 2i

We get the following equation ¯ − M T N¯ = 2iZ NTM

(3.37)

Because Z is non degenerate, from (3.37), we get that M and N are injective. If x ∈ Cn , Ex = 0, we have ¯ x¯ = xT M T = 0 M hence xT Z x¯ = 0 then x = 0. So, we can define, α(F)(Z) = (C + DZ)(A + BZ)−1 Let us prove that α(F)Z ∈ S+ (n). We have:

(3.38)

3.2 The Propagation of Coherent States

71

α(F)Z = NM −1 ⇒ (α(F)Z)T = (M −1 )T N T = (M −1 )T M T NM −1 = NM −1 = α(F)Z. We have also: MT

T ¯ T ¯ ¯ −1 NM −1 − N¯ M ¯ = N M − M N = Z. M 2i 2i

and this proves that (α(F)(Z)) is positive and non degenerate. This finishes the proof of the Theorem for z = 0.



The map F → α(F) defines a representation of the symplectic group Sp(n) in the Siegel space S+ (n). For later use it is useful to introduce the determinant: δ(F, Z) = det(A + BZ), F ∈ Sp(n), Z ∈ S+ (n). The following results are easy algebraic computations. Proposition 30 We have, for every F1 , F2 ∈ Sp(n), (i) α(F1 F2 ) = α(F1 )α(F2 ). (ii) δ(F1 F2 , Z) = δ(F1 , α(F2 )(Z))δ(F2 , Z). (iii) For every Z1 , Z2 ∈ S+ (n) there exists F ∈ Sp(n) such that α(F)(Z1 ) = Z2 . In other words the representation α is transitive in S+ (n). Many other properties of the representation α are studied in [104, 183]. For completeness, we state the following Corollary 13 The propagator Ut is well defined and it is a unitary operator in L2 (Rn ). Proof For every coherent state ϕz , Ut ϕz is solution of the Schrödinger equation. As we have seen in Chap. 1, the family {ϕz }z∈R2n is overcomplete in L2 (Rn ). So formula (3.20) wholly determines the unitary group Ut . In a preliminary step we can see that Ut ψ is well defined for ψ ∈ S (Rn ) using inverse Fourier–Bargmann transform, that Ut ψ ∈ S (Rn ) and Ut ψ = ψ. So we can extend Ut in L2 (Rn ). t is a unitary operator and that Hˆ t has a unique selfIn particular it results that U 2 n  adjoint extension in L (R ). It will be useful to compute the Fourier–Bargmann transform of Ut ϕz . ˆ ˆ t ) where R(F) is the metaplectic operator corresponding Recall that U (t) = R(F to the symplectic 2n × 2n matrix F and that Ft has a four blocks decomposition  Ft =

At Bt Ct Dt



Now we define n × n complex matrices Yt , Zt as follows: Yt = At + iBt − i(Ct + iDt ), Zt = At + iBt + i(Ct + iDt ) One has the following property, using the symplectic relations satisfied by the blocks At , Bt , Ct , Dt of Ft :

72

3 Quadratic Hamiltonians

Lemma 22 We have

Z ∗ Z = Y ∗ Y − 41l.

Moreover Yt is invertible. One can define the matrix Wt as follows: Wt = Zt Yt−1 . which satisfies the following property: Lemma 23 (i) W0 = 0. (ii)

Wt∗ Wt < 1l.

(iii) (Γ + i1l)−1 =

1 (1l + W ). 2i

In particular W is a symmetric matrix. Proof (i) is an easy consequence of the fact that F0 = 1l, hence Y0 = 21l, Z0 = 0. (ii) We have W ∗ W = (Y ∗ )−1 Z ∗ ZY −1 = (Y ∗ )−1 (Y ∗ Y − 41l)Y −1 = 1l − 4(YY ∗ )−1 . 

(iii) Is a simple algebraic computation. Theorem 17 The matrix elements ϕz , Ut ϕz  are given by the following formula

ϕz , Ut ϕz  = 2n/2 det −1/2 (At + Dt + i(Bt − Ct ))e−i/2σ (Ft z ,z) 2 2 × e−(x +ξ )/4 e−W (ξ +ix)·(ξ +ix)/4

(3.39)

where z − Ft z = (x, ξ ) and Wt = (At − Dt + i(Bt + Ct ))(At + Dt + i(Bt − Ct ))−1 . Proof For simplicity we forget the time index t everywhere. It is enough to assume that z = 0. From the metaplectic invariance, we get ϕz , Ut ϕz  = ϕz , Tˆ (Fz )ϕ (Γ ) 

= e−iσ (Fz ,z)/2 ϕz−Fz , ϕ (Γ ) . So we have to compute ϕX , ϕ (Γ ) .

(3.40)

3.2 The Propagation of Coherent States

73

We have ϕX , ϕ (Γ )  = π −n/2 det −1/2 (A + iB)e 2 (ip·q−q )

i e 2 (Γ +i)x·x e−ix·(p+iq) dx. 1

2

(3.41)

Rn

So the result follows from computation of the Fourier transform of a generalized Gaussian (or squeezed state). 

3.3 The Metaplectic Transformations Recall that a metaplectic transformation associated with a linear symplectic transR(F) in L2 (Rn ) satisfying one of formation F ∈ Sp(n) in R2n , is a unitary operator  the following equivalent conditions  A R(F) = A ◦ F, ∀A ∈ S (R2n ). R(F)∗  T (X ) R(F) =  T [F −1 (X )], ∀X ∈ R2n . R(F)∗  A R(F) = A ◦ F, R(F)∗ for A(q, p) = qj , 1 ≤ j ≤ n andA(q, p) = pk , 1 ≤ k ≤ n.

(3.42) (3.43) (3.44)

 A is the Weyl quantization of the classical symbol A(q, p) and we recall that the operator  T (X ) is defined by  T (X ) = exp



i ˆ − x · P) ˆ (ξ · Q 

 (3.45)

when X = (x, ξ ) ∈ R2n . We shall prove below that for every F ∈ Sp(2n) there exists a metaplectic transformation  R(F). This transformation is unique up to a multiplication by a complex number of modulus 1. Lemma 24 If R1 (F) and R2 (F) are two metaplectic operators associated to the same symplectic map F then there exists λ ∈ C, |λ| = 1, such that R1 (F) = λR2 (F). Proof Denote Rˆ = R1 (F)R2 (F)−1 . Then we have Rˆ ∗ Tˆ (X )Rˆ = Tˆ (X ) for all X ∈ R2n . Applying Schur lemma (see Chap. 1) we get that Rˆ = λ1l, λ ∈ C. But Rˆ is unitary so that |λ| = 1.  We shall prove here that F →  R(F) defines a projective representation of the real symplectic group Sp(n) with sign indetermination only. More precisely, let us denote by Mp(n) the group of metaplectic transformations and πp the natural projection: Mp → Sp(2n) then the metaplectic representation is a group homomorphism F →

74

3 Quadratic Hamiltonians

ˆ R(F)] = F, ∀F ∈ Sp(2n) For R(F), from Sp(n) onto Mp(n)/{1l, −1l}, such that πp [ more details about the metaplectic transformations see [176]. Proposition 31 For every F ∈ Sp(n) we can find a C 1 —smooth curve Ft , t ∈ [0, 1], in Sp(n), such that F0 = 1l and F1 = F. Proof An explicit way to do that is to use the polar decomposition of F, F = V |F| √ where V is a symplectic orthogonal matrix and |F| = F T F is positive symplectic matrix. Each of these matrices have a logarithm, so F = eK eL with K, L Hamiltonian  matrices, and we can choose Ft = etK etL . Let Ft be as in Proposition 31. Ft is the linear flow defined by the quadratic Hamiltonian Ht (z) = 21 St z · z where St = −J F˙t Ft−1 . So using above results, we define 1 . Here Ut is the solution of the Schrödinger equation  R(F) = U i

d  (t)Ut Ut = H dt

(3.46)

that obeys U0 = 1l. Namely it is the quantum propagator of the quadratic Hamiltonian  (t). That the metaplectic operator so defined satisfies the required properties follows H from Theorem 15. Proposition 32 Let us consider two symplectic paths Ft and Ft joining 1l (t = 0) to  (with obvious notations). 1 = ±U F (t = 1). Then we have U 1 Moreover, if F 1 , F 2 ∈ Sp(2n) then we have ˆ 2 ) = ±  R(F 1 )R(F R(F 1 F 2 ) .

(3.47)

Proof We first remark that the propagator of a quadratic Hamiltonian is determined by its action on squeezed states ϕ Γ and its classical flow. So using (3.20) we see that the phase shift between the two paths comes from variation of argument between 0

and 1 of the complex numbers  det(At + iBt ) and b (t) = det(At + iBt ). ˙b(t) = t b(s) We have arg[b(t)] =  0 b(s) ds and by a complex analysis argument, we have 

 0

with N ∈ Z. So we get

1

 

 1 ˙ ˙ b(s) b (s) ds =  ds + 2π N

b(s) 0 b (s) b(1)−1/2 = eiN π b (1)−1/2 .

The second part of the proposition is an easy consequence of Theorem 16 concerning propagation of squeezed coherent states and Proposition 30. More precisely, the sign indetermination in (3.47) is a consequence of variations for the phase of det(A + iB) concerning F = F 1 and F = F 2 . To compare with F 1 F 2 we apply Proposition 30. 

3.3 The Metaplectic Transformations

75

Remark 13 A geometrical consequence of Proposition 30 is the following. The map ˆ F → R(F) induces a group isomorphism between the symplectic group Sp(n) and the quotient of the metaplectic group Mp(n)/{−1l, 1}. In other words the group Mp(n) is a two-cover of Sp(n). An interesting property of the metaplectic representation is the following. Proposition 33 The metaplectic representation Rˆ has two irreductible non equivalent components in L2 (Rn ). These components are the subspaces L2od (Rn ) of odd states and L2ev (Rn ) of even states. ˆ Proof Let us first remark that the subspaces L2od, ev (Rn ) are invariant for R(F) because quadratic Hamiltonians commute with the parity operator ψ(x) = ψ(−x). ˆ Now we have to prove that L2od, ev (Rn ) are irreducible for R. 2 n ˆ Let us begin by considering the subspace Lev (R ). Let B be a bounded operator in ˆ ˆ L2ev (Rn ) such that R(F) Bˆ = Bˆ R(F) for every F ∈ Sp(n). According to Schur lemma, we have to prove that Bˆ = λ1l, λ ∈ C. In particular Bˆ commutes with the propagator of the harmonic oscillator Ut = it Hˆ os e . We can suppose that Bˆ is Hermitian. So Bˆ is diagonal in the Hermite basis φα ˆ α = λα φα for every α ∈ Nn . (see Chap. 1). We have Bφ To conclude we have to prove that λα = λβ if |α| and |β| are even. Assume for simplicity that n = 1 (the proof is also valid for n ≥ 2). itx2 ˆ Let us consider the metaplectic   transformation Rt = e associated to the sym0t plectic transform Ft = exp . 00 We have  c(t, k, j)φj . Rˆ t φk = j≥0

Using that Rˆ t Bˆ = Bˆ Rˆ t we get c(t, k, j)λj = c(t, k, j)λk .

(3.48)

Now we shall prove that if k − j is even then c(t, k, j) = 0 for some t hence λj = λk . This is a consequence of the following

c(t, k, j) =

2

eitx φk (x)φj (x)dx.

If c(t, k, j) = 0 for every t then R x2m φk (x)φj (x)dx = 0 for every m ∈ N. But this is not possible if k − j is even (see properties of Hermite functions). So we have proved that L2ev (R) is irreducible. With the same proof we also get that L2od (R) is also irreducible. ˆ ˆ Assume now that Bˆ R(F) = R(F) Bˆ and that Bˆ is a linear transformation from ˆ k = λk φk . But an Hermite function is odd or even L2ev (R) in L2odd (R). We still have Bφ  so λk = 0 for all k and Bˆ = 0. So these representations are non equivalent.

76

3 Quadratic Hamiltonians

3.4 Representation of the Quantum Propagator in Terms of the Generator of Squeezed States We have seen in the proof of Proposition 31 that any symplectic matrix has a symplectic polar decomposition. In this section our aim is to analyse more deeply this polar decomposition and its consequence on the metaplectic representation. More precisely let Ut := U (t, 0) be the quantum propagator associated with a time dependent quadratic Hamiltonian H (t, X ) (continuous in time t), we want to analyze the ˆ t ) where Ft = F(t, 0) is the squeezed state Ut ϕz (z ∈ R2n ). Recall that Ut = R(F classical, linear flow. These results were obtained in [61, 65]. For simplicity we forget the time in our notations. Let us start with classical Hamiltonian mechanics in the complex model Cn , √ . As above, F is a classical flow for a quadratic, time dependent Hamiltonian. ζ = q−ip 2 Let us denote by F c the same flow in Cn . We easily get F cζ =

1 (Y ζ + Z¯ ζ¯ ), 2

(3.49)

where Y = A + D + i(B − C), Z = A − D + i(B + C).

(3.50)

Recall that all these matrices are time dependent and Y is invertible. So we have Y = |Y |V (polar decomposition), where |Y |2 = YY ∗ and V is a unitary transformation of Cn (V ∈ SU(n)). We already introduced W = ZY −1 and we know that 0 ≤ W ∗ W < 1l. So we can factorize F c in the following way. F c = Dc · S c ,

(3.51)

S c ζ = V ζ, Dc ζ = (1l − W ∗ W )−1/2 ζ + (1l − W ∗ W )−1/2 W ∗ ζ¯ .

(3.52) (3.53)

Coming back to the real representation in R2n , S is an orthogonal symplectic transˆ formation (a rotation in the phase space). Let us compute R(S). To do that, we write V = eiL (this is possible locally in time). L is an Hermitian complex matrix. V is the flow at time 12 of the quadratic Hamiltonian HSc (ζ, ζ¯ ) =

1 ζ · LT ζ¯ + ζ¯ · Lζ . 2

ˆ ˆ ˆ = e−iHS . So R(S) is a Let H be the real representation of H c , then we have R(S) quantum rotation because we have, for every observable O,

2 This

time is a new time, which has nothing to do with t.

3.4 Representation of the Quantum Propagator in Terms … ˆ

77

ˆ

ˆ −iHS = O  eiHS Oe · S. ˆ It is more difficult to compute R(D) (the dilation or squeezing part). We write down the polar decomposition of W , W = U |W |, |W |2 = W ∗ W , U unitary transformation in Cn . We are looking for a generator at (new) time 1 for the transformation D. Let us introduce a complex transformation B in Cn with the polar decomposition B = U |B| (|B|2 = B∗ B). After standard computations, we get 

0 B∗ exp B 0



 cosh |B| − sinh |B|U ∗ . = U sinh |B| −U cosh |B|U ∗ 

(3.54)

Comparing with previous computation of Dc , we get cosh |B| = (1l − W ∗ W )−1/2 sinh |B|U ∗ = cosh |B|W ∗ sinh |B| = cosh |B||W |.

(3.55) (3.56) (3.57)

We can solve the last equation: |B| = arg tanh |W |. More explicitely we have B=W

 (W ∗ W )n n≥0

2n + 1

(3.58)

.

(3.59)

In particular B is a symmetric matrix (BT = B) because W is a symmetric matrix. As for the rotation part, we can get now a dilation generator, HDc =

i (ζ · Bζ − ζ¯ · B∗ ζ¯ ) 2

such that the (complex) equation of motion is ζ˙ = B∗ ζ¯ . Finally we restore the time t. We have a decomposition Ft = D(Bt )S(Lt ) such that ˆ t ), where λt is a complex number, |λt | = 1 and ˆ t )S(L Ut = λt D(B ˆ t ) = e 21 (a† ·Bt a† −a·Bt ∗ a) D(B

(3.60)

is the generator of squeezed states. More properties of D(B) will will given at the end of this section. We can get now

78

3 Quadratic Hamiltonians

Proposition 34 For every time t we have Ut ϕ0 = (det 1/2 Vt )φBt ,

(3.61)

where φBt is the squeezed state defined by ˆ t )ϕ0 φBt = D(B

(3.62)

and detVt 1/2 is defined by continuity, starting from t = 0 (V0 = 1l). ˆ Proof It is enough to show that ϕ0 is an eigenstate of S(L) with eigenvalue γ = 1 Tr(L). Clearly 2 (a† · Lt a + a · La† )ϕ0 =

  (Lj,i aj† ai + Li,j ai aj† )ϕ0 = Li,i ϕ0 , i,j

i

since ai ϕ0 = 0, ∀i = 1, . . . n. We get the result by exponentiating. Let us remark here that even if L is defined in a small time interval we can conclude because the  prefactor of φBt must be continuous in time t. We can also demonstrate that the quantum evolution of a coherent state ϕz is simply a displaced along the classical motion of a squeezed state: Proposition 35 Let zt = (qt , pt ) be the phase space point of the classical flow at time t starting with initial conditions z = (q, p). Thus     qt q = Ft . pt p One has:

U (t)ϕz = detVt 1/2 Tˆ (zt )Bt .

ˆ t) Proof One uses that due to the fact that U (t) = R(F U (t)ϕz = U (t)Tˆ (z)ϕ0 = Tˆ (zt )U (t)ϕ0 .  More on n-dimensional squeezed states Consider now any complex symmetric n × n matrix W such that W ∗ W < 1l. Take as before the polar decomposition of W to be: W = U |W |, |W | = (W ∗ W )1/2 , U being unitary. We define the n × n complex symmetric matrix B to be:

3.4 Representation of the Quantum Propagator in Terms …

79

B = U arg tanh |W |. More explicitly writing the Taylor expansion of arg tanh u at 0 we find B=W

∞  (W ∗ W )n n=0

2n + 1

and we have |W | = tanh |B|.

(3.63)

Now we construct the unitary operator D(B) in L2 (Rn ) as   1 † ˆ (a · Ba† − a · B∗ a) . D(B) = exp 2 We have Lemma 25 (i) D(B) is unitary with inverse D(−B). (ii) ˆ D(B)



a a†



ˆ D(−B) =



(1l − W W ∗ )−1/2 −W (1l − W ∗ W ))−1/2 −(1l − W ∗ W )−1/2 W ∗ (1l − W ∗ W )−1/2



a a†

 .

Proof Let us denote †ˆ ˆ ˆ ˆ D(−tB), aB† (t) = D(tB)a D(−tB). aB (t) = D(tB)a

Computing   1 † d ˆ ˆ aB (t) = D(tB) a · Ba† − a · B∗ a , a D(−tB) dt 2 we get, using the commuting relations: d aB (t) = −BaB (t). dt and the same for a† :

d † a (t) = B∗ aB∗ (t). dt B   0 B∗ as in (3.54). We get the result by computing exp B 0

We define the n-dimensional squeezed state as ˆ ψ (B) = D(B)ϕ 0 where ϕ0 is the standard Gaussian (ground state of the Harmonic oscillator).



80

3 Quadratic Hamiltonians

We shall first compute ψ (B) in the Fock–Bargmann representation. We recall that in the Fock–Bargmann representation one has B[ϕ0 ](ζ ) = (2π )−n/2 . (independent of ζ ∈ Cn ). Then we try the following ansatz B[ψ

(B)

 1 ζ · Mζ . ](ζ ) = a exp 2 

(3.64)

where M is a complex symmetric matrix that we want to compute. From now on we assume  = 1. Let us consider the antihermitian Hamiltonian HB to be HB =

1 † (a · Ba† − a · B∗ a). 2

Take a fictitious time t to vary between 0 and 1 and consider the evolution operator UB (t) = etHB . Then ψ (B) (t) = UB (t)ϕ0 satisfies the differential equation d (B) ψ (t) = HB ψ (B) (t) dt with the following limiting values ψ (B) (0) = ϕ0 , ψ (B) (1) = ψ (B) . In the Fock–Bargmann representation HB has the following form: HB (ζ, ∂ζ ) =

1 (ζ · Bζ − ∂ζ · B∗ ∂ζ ). 2

Using the ansatz (3.64) for ψ (B) (t) we see that one has       1 1 1 ˙ a˙ + a ζ · M ζ exp ζ · M ζ = a(t)HB (ζ, ∂ζ ) exp ζ · Mζ . 2 2 2 Now we compute the right hand side using the convention of summation over repeated indices; we get:     1 1 1 1 1 ∗ ∗ a(t) ζ · Bζ − ∂ζj B ij Mik ζk − Bij ∂ζj ζk Mki exp ζ · Mζ , 2 2 2 2 2   1  1 ∗ ∗ = a ζ · Bζ − Tr(B M ) − ζ · MB M ζ exp ζ · Mζ 2 2

3.4 Representation of the Quantum Propagator in Terms …

81

Identifying we get: ˙ = B − MB∗ M , a˙ = − 1 aTr(B∗ M ). M 2 Let us solve the differential equation ∂t M = −MB∗ M + B

(3.65)

with Mt=0 = 0. We consider N such as M = UN with U independent of t given by the polar decomposition of B. Then Eq. (3.65) becomes U ∂t N = −U N |B|N + U |B|. Thus it reduces to ∂t N = −N |B|N + |B|. At time t = 0 N = 0 thus the solution at time t = 1 is a function of |B| (thus commuting with |B|) given by N = tanh |B|. This implies that M = U tanh |B| = U |W | = W. Now consider the differential equation satisfied by a(t). We get: 2˙a = −aTr(B∗ M ) = −aTr(|B|U ∗ U N ) = −aTr(|B| tanh |B|). Since a(0) = (2π )−n/2 we get  

1 t a(t) = (2π )−n/2 exp − dsTr(|B| tanh s|B|) 2 0   1 = (2π )−n/2 exp − Tr log cosh t|B| 2 a(1) = (2π )−n/2 (det cosh |B|)−1/2 = (2π )−n/2 (det(1l − |W |2 ))1/4 , where we have used that eTrA = det eA . Thus the squeezed state ψ (B) = D(B)ϕ0 has the following Fock–Bargmann representation

82

3 Quadratic Hamiltonians

ψ (B) (ζ ) = (2π )−n/2 (det cosh |B|)−1/2 exp



1 ζ · Wζ 2

 (3.66)

Now using the inverse Fock–Bargmann transform we go back to the coordinate representation of ψ (B) . One has to compute the following integral:

    2 √ ζ¯ 2 1 x ¯ ¯ − x 2ζ + − ζ · ζ + ζ · Wζ dqdp exp − 2 2 2

(3.67)

The argument of the exponential can be rewritten as: 1 p2 1 x2 ip · q − q2 − + + x · (q + ip) − (q2 + p2 ) 2 4 2 4 2 1 + (q · W q − ip · W q − iq · W p − p · W p) 4 1 1 x2 3 = − + x(q + ip) − q2 − p2 + (q · W q − ip · W q − iq · W p − p · W p). 2 4 4 4 −

(3.68)

Thus it appears a quadratic form in q, p which can be written in matricial form as: 1 − (q, p)M 2 with

  q p

  1 31l − W −i(W + 1l) M = . 2 −i(W + 1l) W + 1l

One has M

−1

  1 1l i . = 2 i (31l − W )(1l + W )−1

Thus the integral over q, p in (3.67) will be the exponential of the following quadratic form   x2 1 x − (3.69) − (x, ix)M −1 ix 2 2 with a phase which is exactly (det(M /2π ))−1/2 = (2π )n/2 (det(1l + W ))−1/2 . It is easy to compute the expression in (3.69). It gives 1 − x · (1l − W )(1l + W )−1 x 2 Thus restoring the  dependence and the factors π we get again a formula proved differently (by solving a Ricatti equation) at the beginning of this chapter.

3.4 Representation of the Quantum Propagator in Terms …

83

Proposition 36 The squeezed state ψ (B) is actually a gaussian in the position representation given by   i (B) x·Γx (3.70) ψ (x) = aΓ exp 2 with

Γ = i(1l − W )(1l + W )−1

and

aΓ = (π )−n/2 (det(1l − |W |2 ))−1/2 (1l + W )−1/2 .

3.5 Representation of the Weyl Symbol of the Metaplectic Operators See Chap. 2 for the definitions of covariant and contravariant Weyl symbols. We ˆ have shown that R(F) = U1 where Ut is the quantum propagator of the quadratic Hamiltonian   ˆ 1 ˆ ˆ Q (3.71) P)Mt H (t) = (Q, Pˆ 2 with Mt = −J F˙ t Ft−1 , Ft being a continuous path in the space Sp(n) joining 1l at t = 0 to F at t = 1. In [66] the authors show the following result Theorem 18 (i) If det(1l + F) = 0 the contravariant Weyl symbol of  R(F) has the following form: R(F, X ) = eiπν | det(1l + F)|−1/2 exp(−iJ (1l − F)(1l + F)−1 X · X )

(3.72)

where ν ∈ Z if det(1l + F) > 0 and ν ∈ Z + 1/2 if det(1l + F) < 0. (ii) If det(1l − F) = 0 the covariant Weyl symbol of  R(F) has the following form:   i R# (F, X ) = eiπμ | det(1l − F)|−1/2 exp − J (1l + F)(1l − F)−1 X · X 4 where μ = ν¯ +

n 2

(3.73)

and ν¯ ∈ Z.

This formula has been heuristically proposed by Mehlig and Wilkinson [193] without the computation of the phase. See also [75]. We can restore the  dependence of R(F, X ) and R# (F, X ) by putting a factor −1 in the argument of the exponentials. Proof Let us state the following proposition which is a direct consequence of (3.39) after algebraic computations.

84

3 Quadratic Hamiltonians

Proposition 37 The matrix elements of  R(F) on coherent states ϕz , are given by the following formula: n −1/2 ˆ ϕz+X |R(F)ϕ z  = 2 (det(1l + F + iJ (1l − F)))   2      X  1 X − iJX X − iJX  × exp − z +  + iσ (X , z) + KF z + · z+ 2 2 2 2

where

KF = (1l + F)(1l + F + iJ (1l − F))−1 .

(3.74)

(3.75)

ˆ Now we can compute the distribution covariant symbol of R(F) by plugging formula (3.74) in formula (2.29). Let us begin with the regular case det(1l − F) = 0 and compute the covariant symbol. Using Proposition (37) and formula (3.39), we have to compute a Gaussian integral with a complex, quadratic, non degenerate covariance matrix (see [153]). This covariance matrix is KF − 1l and we have clearly KF − 1l = −iJ (1l − F)(1l + F + iJ (1l − F))−1 = −(1l − i)−1 where  = (1l + F)(1l − F)−1 J is a real symmetric matrix. So we have (KF − 1l) = −(1l + 2 )−1 , (KF − 1l) = −(1l + 2 )−1 .

(3.76)

So that 1l − KF is in the Siegel space S+ (n) and Theorem (7.6.1) of [153] can be applied. The only serious problem is to compute the index μ. Let us define a path of 2n × 2n symplectic matrices as follows: G t = etπJn if det(1l − F) > 0 and: G t = G 2t ⊗ etπJ2n−2 if det(1l − F) < 0, where  G = 2

η(t) 0 1 0 η(t)



where η is a smooth function on [0, 1] such that η(0) = 1, η(t) > 1 on ]0, 1] and where Jn is the 2n × 2n matrix defining the symplectic matrix on the Euclidean space R2n . G 1 and F are in the same connected component of Sp (n) where Sp (2n) = {F ∈ Sp(n), det(1l − F) = 0}. So we can consider a path s → Fs in Sp (n) such that F0 = G 1 and F1 = F. Let us consider the following “argument of determinant” functions for families of complex matrices. θ [Ft ] = argc [det(1l + Ft + iJ (1l − Ft )] β[F] = arg+ [det(1l − KF )−1 ]

(3.77) (3.78)

3.5 Representation of the Weyl Symbol of the Metaplectic Operators

85

where argc means that t → θ [Ft ] is continuous in t and θ [1l] = 0 (F0 = 1l), and S → arg+ [det(S)] is the analytic determination defined on the Siegel space S+ (n) such that arg+ [det(S)] = 0 if S is real (see [153, vol.1, Sect. (3.4)]). With these notations we have μ=

β[F] − θ [F] . 2π

(3.79)

Let us consider first the case det(1l − F) > 0. Using that J has the spectrum ±i, we get: det(1l + G t + iJ (1l − G t )) = 4n entπi and 1l − KG 1 = 1l. Let us remark that det(1l − KF )−1 = det(1l − F)−1 det(1l − F + iJ (1l + F)). Let us introduce (E, M ) = det(1l − E + M (1l + E)) for E ∈ Sp(2n) and M ∈ sp+ (2n, C). Let consider the closed path C in Sp(2n) defined by adding {G t }0≤t≤1 and {Fs }0≤s≤1 . We denote by 2π ν¯ the variation of the argument for (•, M ) along C . Then we get easily β(F) = θ [F] + 2π ν¯ + nπ, n ∈ Z. (3.80) When det(1l − F) < 0, by an explicit computation, we find arg+ [det(1l − KG 1 )] = 0. So we can conclude as above. The formula for the contravariant symbol can be easily deduced from the covariant formula using a symplectic Fourier transform.  Remark 14 When the quadratic Hamiltonian H is time independent then Ft = etJS , S is a symmetric matrix. So if det(etJS + 1) = 0, then we get applying (3.72),    −1/2        t t JS  JS X · X . exp iσ tanh R(etJS , X ) = eiπν det cosh 2 2 (3.81) This formula was obtained in [154]. In particular the Melher formula for the Harmonic oscillator is obtained with S = 1l. In [75] the author discusses the Maslov index related with the metaplectic representation. Remark 15 In the paper [66] the authors give a different method to compute the contravariant Weyl symbol R(F, X ) inspired by [96]. They consider a smooth family Ft of linear symplectic transformations associated with a family of time dependent quadratic Hamiltonians Ht . After quantization we have a quantum propagator Ut with its contravariant Weyl symbol Utw (X ). We make the ansatz Utw (X ) = αt eX ·Mt X where αt is a complex number, Mt is a symmetric matrix. Using the Schrödinger equation ∂t Utw = Ht  Utw , and the Moyal product, we find for Mt a Riccati equation which is solved with the classical motion. Afterwards, αt is found by solving a Liouville equation hence we recover the previous results (see [66] for details). This approach will be adapted later in this book in the fermionic setting.

86

3 Quadratic Hamiltonians

3.6 Traps We now give an application in physics of our computations concerning quadratic Hamiltonians. The quantum motion of an ion in a quadrupolar radiofrequency trap is solved exactly in terms of the classical trajectories. It is proven that the quantum stability regions coincide with the stability regions of the associated Mathieu equation. By quantum stability we mean that the quantum evolution over one period (the socalled Floquet operator) has only pure-point spectrum. Thus the quantum motion is “trapped” in a suitable sense. We exhibit the set of eigenstates of the Floquet operator.

3.6.1 The Classical Motion Let us consider a three-dimensional Hamiltonian of the following form: p2 e H (t) = + 2m r02

  1 2 2 2 z − (x + y ) (V1 − V0 cos(ωt)). 2

(3.82)

Here p = (px , py , pz ) is the three-dimensional momentum, m the mass of the ion, and r = (x, y, z) is the three-dimensional position. r0 is the size of the trap and e the charge of the electron. V1 (resp. V0 ) is the constant (resp. alternating) voltage. Such time-periodic hamiltonians are realized by using traps that are hyperboloids of revolution along the z-axis that are submitted to a direct current plus an alternating current voltage. Also known as Paul traps they allow to confine isolated ions like Cesium for rather long times, and also a few ions together in the same trap. It is obvious to see that this Hamiltonian is purely quadratic and that it decouples into three one-dimensional Hamiltonians: H (t) = hx (t) ⊗ 1l ⊗ 1l + 1l ⊗ hy (t) ⊗ 1l + 1l ⊗ 1l ⊗ hz (t) p2

2

p2

where hx (t) = hy (t) = 2mx − x2 (α − β cos(ωt)), hz (t) = 2mz + z 2 (α − β cos(ωt)) with α = re2 V1 , β = re2 V0 . 0 0 Thus x(t), y(t), z(t) evolve according to the Mathieu equations: x¨ (t) − x(t)(α/m − cos(ωt)β/m) = 0,

z¨ (t) + z(t)(2α/m − cos(ωt)2β/m) = 0. (3.83)

It is known that each of these equations have stability regions parametrized by (α, β, ω), in which the motion remains bounded. Furthermore it can be shown that it is quasiperiodic with Floquet exponent ρ. See [192]. Theorem 19 There exist ρ, ρ ∈ R and rapidly converging sequences cn , cn ∈ R depending on a = 4α/mω2 , b = 2β/mω2 such that the solution of equations (3.83)

3.6 Traps

87

with x(0) = u, x˙ (t) = v/m, z(0) = u , z˙ (0) = v /m are given by x(t, u, v) =

z(t, u , v ) =

    +∞ +∞ 2v  ωt ωt u + cn cos (2n + ρ) cn sin (2n + ρ) c −∞ 2 mωd −∞ 2     +∞ +∞ 2v  u 

ωt

ωt + c cos (2n + ρ ) c sin (2n + ρ ) c −∞ n 2 mωd −∞ n 2

  where c = Z cn , d = Z (2n + ρ)cn , and similarly for c , d . Note that the Floquet exponent ρ is the same for x, y but ρ = ρ. The proof depends heavily on the linearity of Mathieu’s equations. Note that the stability regions are delimited by the curves Cj for which (3.83) has periodic solutions, ie for which ρ = j ∈ N. For given α, β, there exists ω1 , ω2 ∈ R+ such that for any ω ∈]ω1 , ω2 [ the classical equations of motion (3.83) have stable solutions. ω1 has to be large enough hence the name “radio-frequency traps”.

3.6.2 The Quantum Evolution Since ions are actually quantum objects it is relevant to consider now the quantum problem. As known from the general considerations of this Chapter on quadratic Hamiltonians the quantum evolution for Hamiltonians of the form (3.82) is completely determined by the classical motion. The Hilbert space of quantum states is H = L2 (R3 ). One thus has that the time-periodic Hamiltonian   x2 + y2 2 , Hˆ (t) := − Δ + (α − β cos(ωt)) z 2 − 2m 2 where Δ is the 3-dimensional Laplacian and generates an unitary operator U (t, s) that evolves a quantum state from time s to time t. The Floquet operator is the operator on time evolution over one period T = 2π/ω: U (T , 0) = Ux (T , 0)Uy (T , 0)Uz (T , 0) We shall denote by UF (resp UF ) the operator Ux (T , 0) (resp. Uz (T , 0)). Then one has the following results: Theorem 20 (i) Given α, β ∈ R there exists ω1 , ω2 ∈ R+ as in the previous section such that for any ψ ∈ H and any ε > 0 there exists R ∈ R+ such that sup F(|r | > R)U (t, s)ψ < ε, t

88

3 Quadratic Hamiltonians

F(|r | > R) being the characteristic function of the exterior of the ball |r | ≤ R. (ii) For α, β, ω as above, UF has pure point spectrum of the form {exp (−iρπ(k + 1/2))}k∈N where ρ is as in the preceding section the classical Floquet exponent. Similarly for UF . Remark 16 A complete proof can be seen in [59]. (i) says that the time evolution of any quantum state remains essentially localized along the quantum evolution. Proof (A sketch) To prove (i) it is enough to state the result in one dimension, and to assume m =  = 1. We establish the result for ψ ∈ C0∞ which is a dense set. Let W (−2x, ξ, t) be the Wigner function of the state ψ(t) := U (t, 0)ψ. Then W (x, ., t) ∈ L1 (R) ∀t ∈ R, ∀x ∈ R. Furthermore we have

 x 2  ψ t, −  = d ξ W (x, ξ, t). 2 It follows that

F(|x| > R)ψ(t)2 =

|x|>2R

dxd ξ W (x, ξ, t) =

|x|>2R

dxd ξ W (x(t), x˙ (t), 0),

where x(t) is a solution of (3.83) with x(0) = x, x˙ (0) = ξ . Then we perform a change of variables with uniform (in t) jacobian, the linearity of Mathieu’s equation and the boundedness in time of its solution in the stable region, together with the fact that  W (x, ξ, t) ∈ L1 (R2 ) to conclude for (i). An extension of Theorem 20 to Hermite-like wavefunctions is the following: define k (t, x) = (Lt )

−1/2



L∗t Lt

k/2

  √ iNt x2 , Hk (x/ |Lt |) exp 2Lt

where Hk are the normalized Hermite polynomials and the determination of the square-root is followed by continuity from t = 0. We have U (t, 0)k (0, .) = k (t, .),

∀t ∈ R

This can be proven by using Trotter product formula, a steplike approximation fN (t) of the function f (t) = α − β cos(ωt) and the continuity of solutions of (3.83) when fN → f . See [59, 130, 131] for details. We start now a sketch proof of (ii) of Theorem 20. The normalized eigenstates of UF are generalized squeezed states. Let us assume m = 1 for simplicity. Let F(t) be the symplectic 2 × 2 matrix solution of F˙ = JMF

3.6 Traps

89



where

1 0 0 f (t)

M (t) = f (t) = α − β cos(ωt)



 At Bt . F(t) = Ct Dt 

 We choose suitable initial data F(0) =  g=

 g 0 with 0 g −1

ωd 2c

−1/2

We define Lt = At + iBt , Nt = Ct + iDt . For these initial data we have Lt =

+∞ +∞ g iρωt/2  inωt 1 iρωt/2  inωt e e cn e , Nt = cn e . c gd −∞ −∞

It is clear that |Lt |, NLtt are T -periodic where T = 2π/ω, and furthermore ei(k+1/2)

ρωt 2

(Lt )−1/2



L∗t Lt

k/2

is also T -periodic. Then we introduce the self-adjoint quasi-energy operator K = −i

∂ + hx (t) ∂t

acting in the Hilbert space K = L2 (R) ⊗ L2 (Tω ) of functions depending on both x and t, T -periodic in t. This formalism has been introduced by Howland [157] and Yajima [266]. K is closely related to the quantum evolution operator. We have the following result: Lemma 26 (i) Assume Ψ ∈ K is an eigenstate of K with eigenvalue λ. Then for any t ∈ Tω , Ψ (t, .) ∈ L2 (R) and satisfies U (t + T , t)Ψ (t) = e−iλT / Ψ (t). (ii) Conversely let ψ ∈ L2 (R) satisfy U (T , 0)ψ = e−iT λ/ ψ.

90

3 Quadratic Hamiltonians

Then Ψ = eiλt/ U (t, 0)ψ ∈ K and satisfies KΨ = λΨ. Proof Define

Ψk (t, x) = e(k+1/2)ρωt/2 k (x).

Then using the periodicity properties we have that Ψk ∈ K and    1 k (0, .) U (T , 0)k (0, .) = exp −iπρ k + 2 and thus

  1 ρω KΨk = k + Ψk . 2 2 

Chapter 4

The Semiclassical Evolution of Gaussian Coherent States

Abstract In this chapter we consider semiclassical asymptotics of the quantum evolution of coherent states at any order in the Planck constant . We consider a control in time of the remainder term depending explicitly on  and on the stability matrix of the classical system. We find that the quantum evolved coherent state is in L 2 -norm well approximated by a squeezed state located around the phase-space point z t of the classical flow reached at time t, with a dispersion controlled by the stability matrix at point z t . The idea goes back to Hepp [148] and was further developed by Hagedorn [130, 131]. The method that we develop here follows the paper [65] where we consider general time-dependent Hamiltonians and use the squeezed states formalism and the metaplectic transformation (see Chap. 3). The difference between the exact and the semiclassical evolution is estimated in time t and in the semiclassical parameter  giving in particular the well known Ehrenfest time of order log(−1 ). We then provide two applications of the semiclassical estimates: the first one concerns the semiclassical estimate of the spreading of quantum wave packets, which are coherent states, in terms of the Lyapunov exponents of the classical flow. The second application is to the scattering theory for general short range interactions: then the large time asymptotics can be controlled and the quantum scattering operator acts on coherent states following the classical scattering theory with good estimates in . More accurate estimates can be obtained using the Fourier–Bargmann transform [218]. We consider Gevrey type estimates for the semiclassical coefficients and exponentially small remainder estimates in Sobolev norms for solutions of time dependent Schrödinger equations.

4.1 General Results and Assumptions We shall consider the quantum Hamiltonian Hˆ (t) of a possibly time-dependent problem. We assume that the corresponding time-dependent Schrödinger equation defines a unique quantum unitary propagator U (t, s)1 . Then we consider the canonical gaussian coherent states ϕz where z = (q, p) ∈ R2n is a phase-space point. Then we let shall see later that this condition is fulfilled if H (t, X ) is subquadratic in X . © Springer Nature Switzerland AG 2021 D. Robert and M. Combescure, Coherent States and Applications in Mathematical Physics, Theoretical and Mathematical Physics, https://doi.org/10.1007/978-3-030-70845-0_4 1 We

91

92

4 The Semiclassical Evolution of Gaussian Coherent States

it evolve with the quantum propagator that is we consider the quantum state at time t defined by ψz (t) := U (t, 0)ϕz . Semiclassically (when  is small) ψz (t) is well approximated in L 2 -norm by a superposition of “squeezed states” centered around the phase-space point z t = Φ t,0 z of the classical flow. This study goes back to the pioneering paper by Hepp [148] and was later developed by G. Hagedorn in a series of papers [130, 131]. In [61] an approach is developed connecting the semiclassical propagation of coherent states to the socalled squeezed states. We develop here a generalization of these results, following the paper [65], allowing general time-dependent Hamiltonians and estimating the error term with respect to time t, to z and to . The knowledge of the time evolution for any Gaussian ϕz is a way to get many properties for the full propagator U (t, 0) (we can always assume that the initial time is 0). This is easy to understand using that the family {ϕz , z ∈ R2n } is overcomplete (Chap. 1). Let us denote by K t (x, y) the Schwartz-distribution kernel of U (t, 0). From overcompleteness we get the following formula: K t (x, y) = (2π )−n

 R2n

dz[U (t, 0)ϕz ](x)ϕz (y).

(4.1)

This equality holds as Schwartz distributions on R2n and explains why it is very useful to solve the Schrödinger equation with coherent state ϕz as initial state: i∂t ψz (t) = Hˆ (t)ψz (t), ψz (0) = ϕz .

(4.2)

Several applications will be given later as well for time dependent and time independent Schrödinger equations type.

4.1.1 Assumptions and Notations Let Hˆ (t) be a selfadjoint Schrödinger Hamiltonian in L 2 (Rn ) obtained by quantizing a general time-dependent symbol H (t, X ) which is a classical Hamiltonian. We use the - Weyl quantization (see Chap. 2). H is assumed to be a C ∞ -smooth function for X = (x, ξ ) ∈ Rn × Rn , continuous in t ∈] − T, T [ 0 ≤ T ≤ +∞ and satisfying a global estimate: (A.0) There exist some nonnegative constants m, M, K H,T such that   γ X −M/2 ∂ X H (t, X ) ≤ K H,T ,

4.1 General Results and Assumptions

93

uniformly in X ∈ R2n , t ∈] − T, T [, for |γ | ≥ m.2 So H may be a very general Hamiltonian including time-dependent magnetic fields or non Euclidean metrics. We furthermore assume H (t, x, ξ ) to be such that the classical and quantum evolutions exist from time 0 to time t for t in some interval ] − T, T [ where T < +∞ or T = +∞. More precisely: (A.1) Given some z = (q0 , p0 ) ∈ R2n there exists a positive T such that the Hamilton’s equations q˙t =

∂H (qt , pt , t), ∂p

p˙ t = −

∂H (qt , pt , t) ∂q

have a unique solution for any t ∈] − T, T [ starting from initial data z := (q0 , p0 ). We denote z t = (qt , pt ) := Φ Ht (z) the phase space point reached at time t starting by z at time 0. (A.2) There exists a unique quantum propagator U (t, s), (t, s) ∈ R2 with the following properties: (i) U (t, s) is unitary in L 2 (Rn ) with U (t, s) = U (t, r )U (r, s), ∀(r, s, t) ∈ R3 . (ii) U (., .) is a strongly continuous operator-valued function for the operator norm topology in L 2 (Rn ). The usual norm in L 2 (Rn ) is denoted by . . Remark 17 For general Hamiltonian H (t) the existence time T in (A.1) may depend on z. In Chap. 1 we have described the construction of standard coherent states by applying the Weyl-Heisenberg operator to the ground state ϕ0 the n-dimensional harmonic oscillator with Hamiltonian K 0 :=

1 ˆ2 ( P + Qˆ 2 ) 2

(4.3)

ϕz = Tˆ (z)ϕ0 In Chap. 3 we have computed an explicit formula for the time evolution of coherent states driven by any quadratic Hamiltonian in (q, p). We shall now define generalized coherent states by applying Tˆ (z) to the excited states φν of the n-dimensional harmonic oscillator; given a multi-index ν = (ν1 , . . . νn ), φν is the normalized eigenstate of (4.3) with eigenvalue |ν| + n/2. We recall the notation n  νj |ν| = j=1

2M

= 0 and m = 2 mean that H (t) is subquadratic, (see footnote. 1).

94

4 The Semiclassical Evolution of Gaussian Coherent States

We thus define

φz(ν) = Tˆ (z)φν

and ϕz is simply φz(0) . (ν) Similarly we define generalized squeezed states φz,B centered around the phasespace point z. Let W be a symmetric n × n matrix with polar decomposition W = U |W | where |W | = (W ∗ W )1/2 and U a unitary n × n matrix. We assume that W ∗ W ≤ 1l and define B := U Argtanh|W |.

(4.4)

ˆ Now as in Chap. 3 we construct the unitary operator D(B) in L 2 (Rn ) as   1 ∗ ∗ ∗ ˆ D(B) = exp (a · Ba − a · B a) , 2 a∗ and a being the creation and annihilation operators. We define (ν) ˆ := Tˆ (z) D(B)φ φz,B ν.

Of course we have:

(ν) = ϕ (B) φ0,B

where ϕ (B) is the standard squeezed state, using the notations of Chap. 2 and (0) = ϕz φz,0

using the notation of Chap. 1.

4.1.2 The Semiclassical Evolution of Generalized Coherent States In this section we consider the quantum evolution of superpositions of generalized coherent states of the form φz(ν) and prove under the above assumptions that, up to an error term which can be controlled in t, z, , it is close in L 2 -norm to a superposition where z t := Φ t,0 z is the phase-space point of squeezed states of the form φz(ν) t ,Bt reached at time t by the classical flow Φ t,0 of H (t), and Bt is well-defined through the linear stability problem at point z t . We follow the approach developed in [65] where we use: – the algebra of the generators of coherent and squeezed states

4.1 General Results and Assumptions

95

– the so-called Duhamel’s principle which is nothing but the following identity: 1 U1 (t, s) − U2 (t, s) = i



t

dτ U1 (t, τ )( Hˆ 1 (τ ) − Hˆ 2 (τ ))U2 (τ, s),

(4.5)

s

where Ui (t, s) is the quantum propagator generated by the time-dependent Hamiltonian Hˆ i (t), i = 1, 2. We take Hˆ 1 (t) = Hˆ (t) and Hˆ 2 (t) to be the “Taylor expansion up to order 2” of Hˆ (t) around the classical path z t . More precisely: ∂H ∂H (qt , pt , t) + ( Pˆ − pt ) · (qt , pt , t) Hˆ 2 (t) = H (qt , pt , t) + ( Qˆ − qt ) · ∂q ∂p (4.6)   ˆ − qt 1 ˆ Q , + ( Q − qt , Pˆ − pt )Mt ˆ P − pt 2 Mt being the Hessian of H (t) computed at point z t = (qt , pt ):  Mt =

∂2 H ∂z 2

 |z=zt .

(4.7)

The interesting point is that since Hˆ 2 (t) is at most quadratic, its quantum propagator is written uniquely through the generators of coherent and squeezed states. In Chap. 3 we have shown the link between the quantum propagator of purely quadratic Hamiltonians and the metaplectic transformations. It is shown that the quantum propagator of purely quadratic Hamiltonians can be decomposed into a quantum rotation times a squeezing generator. Let us be more explicit: Let Hˆ Q (t) be a purely quadratic quantum Hamiltonian of the form   Qˆ ˆ ˆ ˆ HQ (t) = ( Q, P)S(t) ˆ P with S(t) a 2n × 2n symmetric matrix of the form 

G t L˜ t S(t) = L t Kt



where G t , K t are symmetric and L˜ denotes the transpose of L. In what follows S(t) will be simply Mt . Let F(t) be the symplectic matrix solution of ˙ F(t) = J Mt F(t) with initial data F0 = 1l, where

(4.8)

96

4 The Semiclassical Evolution of Gaussian Coherent States

 J=

 0 1l . −1l 0

It has the four block decomposition 

 A(t) B(t) F(t) = . C(t) D(t) In Chap. 3 it has been established that the quantum propagator Uq (t, 0) of Hˆ Q (t) ˆ is nothing but the metaplectic operator R(F(t)) associated to the symplectic matrix ˆ ˆ obey the classical NewF(t). It implies that the Heisenberg observables Q(t), P(t) ton’s equations for the Hamiltonian Hˆ Q (t) as expected. Therefore we have: Lemma 27

    Qˆ Qˆ Uq (0, t) ˆ Uq (t, 0) = F(t) ˆ . P P

ˆ P) ˆ representation to the (a∗ , a) representation we easily get Passing from the ( Q, Lemma 28

  ∗   ∗ 1 Yt Z¯ t a a Uq (t, 0) = . Uq (0, t) a a 2 Z t Y¯t

ˆ Furthermore one can show that R(F(t)) decomposes into a product of a rotation part and a “squeezing” part. Defining the complex matrices: Yt = A(t) + i B(t) − i(C(t) + i D(t)),

Z t = A(t) + i B(t) + i(C(t) + i D(t)) (4.9)

we have the following identity: Z ∗ Z = Y ∗ Y − 41l and Yt is invertible. The matrix Wt = Z t Yt−1 is such that Wt∗ Wt < 1l, W0 = 0. Furthermore it has been shown in Chap. 3 (Lemmas 21 and 23) that it is a symmetric matrix. Thus one can define the matrix Bt according to (4.4); note that Bt is not to be confused with the matrix B(t) of the four-block decomposition of F(t). Introducing the polar decomposition of Yt : Yt = |Yt |Vt∗ where

|Y |2 = Y Y ∗

4.1 General Results and Assumptions

97

we see that Vt is a smooth function of t and we can define (at least locally in time) a smooth self-adjoint matrix Γt by Vt = exp(iΓt ). ˆ be the following unitary operator in L 2 (Rn ) (metaplectic transformation): Let R(t)   ∗    i ∗ a 0 Γ˜t ˆ . R(t) = exp (a , a) a Γt 0 2 It has the following property: Lemma 29

 ∗   Vt a∗ a −1 ˆ ˆ R(t) R(t) = . a (V˜t )∗ a

ˆ will be the rotation part of the metaplectic transformation Uq (t, 0) Remark 18 R(t) ˆ t ) will be the squeezing part. while D(B One has the following property: Proposition 38 The quantum propagator solving the time-dependent Schrödinger equation d i Uq (t, s) = Hˆ Q (t)Uq (t, s), Uq (s, s) = 1l, dt is given by

ˆ t ) R(t) ˆ R(s) ˆ −1 D(−B ˆ Uq (t, s) = D(B s ).

Due to the chain rule it is enough to show that ˆ t ) R(t). ˆ Uq (t, 0) = D(B The detailed proof can be found in the paper [65]. Now we can derive an explicit formula for the quantum propagator of Hˆ 2 (t). Let St (z) be the classical action along the trajectory for the classical Hamiltonian H (x, ξ, t) starting at phase-space point z = (q, p) at time 0 and reaching z t = (qt , pt ) at time t: 

t

St (z) =

ds[x˙s · ξs − H (xs , ξs , s)]

0

and define δt = St (z) − The following result holds true:

q t · pt − q · p . 2

(4.10)

98

4 The Semiclassical Evolution of Gaussian Coherent States

Proposition 39 Let U2 (t, s) be the quantum propagator for the hamiltonian Hˆ 2 (t) given by (4.6). Then we have ˆ t ) R(t) ˆ R(s) ˆ −1 D(−B ˆ ˆ U2 (t, s) = exp[i(δt − δs )/]Tˆ (z t ) D(B s ) T (−z s ).

(4.11)

√ p. Proof Here we use the notation z := q+i 2 Using the Baker–Campbell–Hausdorff formula (and omitting the index t in the following formulas) we get:

 1 ˙ d ˆ ∗ ˆ ˙ z · z¯ − z˙ · z¯ . T (z) = T (z) z¯ · a − z˙ · a + dt 2 But  i 1 ˙ 1 ˙ ˙ z · z¯ − z˙ · z¯ = ( p˙ · q − q˙ · p) , and i δ = i S − ( p · q + p˙ · q) , 2 2 2 so that taking the derivative of (4.11) with respect to time t we get i

∂U2 ∂t





 1 1 ( p · q˙ + p˙ · q) − ( p˙ · q − q˙ · p) + q˙ · Pˆ − p − p˙ · Qˆ − q U2 2 2 ˆ ˆ + T (z t )Hq (t)T (−z t )U2    



 1 Qˆ − q ˆ ˆ ˙ ˆ ˆ Q − q, P − p Mt ˆ = q˙ · p − S + q˙ · P − p − p˙ · Q − q + U2 P−p 2

=

− S˙ +

= Hˆ 2 (t)U2 ,

where we have used that q˙ · p − S˙ = H (q, p, t).

(4.12) 

The important fact here will be that U2 propagates coherent states into Weyl translated squeezed states so that using (4.5) we get a comparison between the exact quantum evolution of coherent states and the Weyl-displaced squeezed state centered around the phase-space point z t . Recall the notations of Chap. 1. Let φk be the multidimensional Hermite functions, k ∈ Nn . Let us denote φk,z := Tˆ (z)φk . Consider as an initial state a coherent state ϕz = Tˆ (z)ϕ0 . We get i(δt /+γt ) (Bt ˆ t ) R(t)Ψ ˆ φz t U2 (t, 0)ϕz = eiδt / Tˆ (z t ) D(B 0 =e

using the fact that 1 ˆ R(t)φ 0 = exp(iγt )φ0 , where γt = tr(Γt ). 2 Therefore applying Duhamel’s formula (4.5) to ϕz we get

4.1 General Results and Assumptions

U (t, 0)ϕz − ei(δt /+γt ) φz(Bt t ) =

1 i

99

 0

t

dsU (t, s)[ Hˆ (s) − Hˆ 2 (s)]ei(δs /+γs ) φz(Bs s ) .

(4.13) This will be the starting point of our semiclassical estimate. Taking an arbitrary multiindex k = (k1 , . . . kn ) we denote (Bt ) ˆ t ) R(t)φ ˆ (t) := Tˆ (z t ) D(B φk,z k. t

Starting from (4.13) we deduce that for any integer l ≥ 1 there exist indexed functions cν (t, ) such that     3(l−1)    (Bt ) iδt /  l/2 U (t, 0)ϕz − c (t, )φ (t)e (4.14) k k,z t   ≤ Ct  .   |k|=0 Furhermore the constant Ct can be controlled in the time t and in the center z of the initial state. Let us now explicit the Taylor expansion of the Hamiltonian around the classical phase-space point z t at time t: we denote  1 ∂ ν j f (ζ ) f (ν) (ζ ) νj · (ζ )ν = ν j (ζ j ) , ν = (ν1 , . . . ν2n ) ν! ν ! ∂ζ j j 1 2n

for f being a real function of ζ ∈ R2n . Taking for ζ the phase-space point z we can write the Taylor expansion of H (t, q, p) around the phase-space point z t = (qt , pt ) at time t as: l+1  H (ν) (z t , t) · (ζ − z t )ν ν! |ν|=0   1 H (ν) (t, z t + θ (ζ − z t )) · (ζ − z t )ν (1 − θ )l+1 dθ + (ν − 1)! 0 |ν|=l+2

H (ζ, t) =

=

l+1   H (ν) (t, z t ) · (ζ − z t )ν + rν,t (ζ − z t , t) · (ζ − z t )ν . (4.15) ν! |ν|=0 |ν|=l+2

ˆ P); ˆ we get: Now we perform the -quantization of (4.6) denoting Ω := ( Q, Hˆ (t) − Hˆ 2 (t) = where

l+1   H ν (z t , t) · (Ω − z t )ν + Rν (t), ν! |ν|=3 |ν|=l+2

Rν (t) = Tˆ (−z t )Opw [ζ ν · rν,t (ζ )]Tˆ (z t ).

(4.16)

100

4 The Semiclassical Evolution of Gaussian Coherent States

√ We now insert (4.16) into (4.13) and obtain the semiclassical estimate of order  for the propagation of generalized coherent states, using in particular the CalderonVaillancourt estimate (Chap. 2). Let us introduce some notation to control the estimates for large times: we consider only nonnegative time and define σ (z, t) := sup (1 + |z s |) 0≤s≤t

θ (z, t) = sup [tr(F ∗ (s)F(s))]1/2 0≤s≤t

where F(t) is the symplectic matrix solution of (4.8). Let M1 be any fixed integer not smaller than (M + (m − 2)+ )/2. Let us define ρl (z, t, ) = σ (z, t)

l M1

  |t|  j √ ( θ (z, t))2 j+l .  1≤ j≤l

Let us recall that the constants M, m, K H,T were defined in Assumption (A.0). Note that if (A.0) is satisfied with m = 2 and M = 0, (H (t) is said subquadratic) then we have M1 = 0. The result will be a generalization of (4.14) using as initial state a superposition of generalized coherent states (defined with higher order Hermite functions). Theorem 21 Assume H (t, x, ξ ) and z are such that (A.0)–(A.2) are satisfied. Then for any integers l ≥ 1, J ≥ 1 and any real number κ > 0 there  exist a universal con- stant Γ > 0 such that for every family of complex numbers cμ , μ ∈ Nn , |μ| ≤ J √ there exist cν (t, ) for ν ∈ Nn , |ν| ≤ 3(l − 1) + J such that for 0 <  + θ (t) < κ the following L 2 -estimate holds:   ⎛ ⎞   J +3(l−1) J     (B ) iδt / t U (t, 0) ⎝ ⎠ cj φj,z − e ck (t, )φk,zt (t) ≤    |j|=0 |k|=0 ⎛ ⎞1/2  Γ K H,T ρl (z, , t) ⎝ |cμ |2 ⎠ .

(4.17)

0≤|μ|≤J

Moreover the coefficients cμ (t, ) can be computed by the following formula: cμ (t, ) − cμ = ⎞

⎛ 

 ⎜ ⎜ ⎝

|ν|≤J 1≤ p≤l−1 |μ−ν|≤3l−3



k1 +···+k p ≤2 p+l−1 ki ≥3

⎟ (k1 +···+k p )/2− p a p,μ,ν (t)⎟ ⎠ cν ,

(4.18)

4.1 General Results and Assumptions

101

where the entries a p,μ,ν (t) are given by the evolution of the classical system and are γ universal polynomials in ∂z H (t, z t ) for |γ | ≤ l + 2 satisfying a p,μ,ν (0) = 0. Remark 19 Comments on the error estimate and the Ehrenfest time The error term seems accurate but not very explicit in our general setting. Let us assume for simplicity that T = +∞ and that the classical trajectory z t is bounded and unstable with a Lyapounov exponent λ > 0. So there exists some constant C > 0 such that θ (z, t) ≤ Ceλt , ∀t ≥ 0. Then for every ε > 0 there exists Cε and h ε > 0 such that 0 0 and any integer l ≥ 1 and any f ∈ S (Rn ) there exists Γ > 0 such that the following L 2 -norm estimate holds:     U (t, 0)Tˆ (z)Λ f − Tˆ (z t )U2 (t, 0)Λ Pl ( f, t, ) ≤ Γ K H,t ρl (z, t, ),

(4.23)

where Pl (., t, ) is the (, t)-dependent differential operator defined by Pl ( f, t, ) = f +



k/2− j p jk (t, x, Dx ) f,

(k, j)∈Il

with Il = {(k, j) ∈ N × N : 1 ≤ j ≤ l − 1, k ≥ 3 j, 1 ≤ k − 2 j ≤ l}. Moreover the polynomials pk j (t, x, ξ ) can be computed explicitly in terms of the Weyl symbol of the following differential operators defined above 



t

tl

dtl 0

0



t2

dtl−1 . . .

dt1 Π (t1 , . . . tl ; k1 , . . . kl ).

0

Remark 21 If we consider an -metaplectic transformation V in L 2 (Rn ) we consider the following rescaling V˜ = Λ−1  V Λ (with no confusion with the notation for the transpose of a matrix). By definition since V has a quadratic generator V˜ is -independent. So we have that U2 (t, 0)Λ Pl ( f, t, ) is actually a coherent state ˜ t ) R(t) ˜ f . More explicitly centered at z t with profile D(B ⎛ U2 (t, 0)Λ Pl ( f, t, ) = eiδt / Tˆ (z t )Λ ⎝



⎞ ˜ t ) R(t) ˜ f⎠  j/2 p j (t, x, Dx ) D(B

0≤ j≤l−1

where p j (x, Dx , t) are differential operators with polynomial coefficients depending smoothly on t as long as the classical flow z → z t exists.

4.1.3 Related Works and Other Results In the physics literature the quantum propagation of coherent states has been considered by many authors, in particular by Heller [144, 145] and Littlejohn [182]. In the mathematical literature gaussian wave packets have been introduced and studied in many respects, particularly under the name “gaussian beams” (see [10, 212, 213]). Somehow related to the subject of this Chapter is the study by Paul and Uribe [205, 206] of the -asymptotics of the inner products of the eigenfunctions of a Schrödinger type Hamiltonian with a coherent state and of “semiclassical trace formulas” (see Chap. 5). However their approach differs from the one presented here

104

4 The Semiclassical Evolution of Gaussian Coherent States

by the use of Fourier-integral operators, which were introduced in connection with wave packets propagation in the classical paper by Cordoba and Fefferman [68].

4.2 Application to the Spreading of Quantum Wave Packets In this section we give an application of the estimate of the preceding section to the spreading (in phase space) of a quantum wave packet which is, at time 0 localized in the neighborhood of a fixed point of the corresponding classical motion. Let z = (q, p) be such a fixed point and take as an initial quantum state the coherent state ϕz . The quantum state at time t is ψz (t) = U (t, 0)ϕz . We have seen that we can approximate Ψ (t) by a Gaussian wavepacket again localized around z t = z but with a spreading governed by the stability matrix M0 of the corresponding classical motion (given by (4.7)). A way of measuring the spreading of wave packets around the point z in phase space is to compute  S(t) = Tˆ (−z)ψz (t),

n  (a ∗j a j + a j a ∗j )Tˆ (−z)ψz (t) j=1

= a Tˆ (−z)ψz (t) 2 + a∗ Tˆ (−z)ψz (t) 2 .

(4.24)

The intuition behind this definition is the following. Let Wz,t be the Wigner function of the states ψz (t). According to the properties seen in Chap. 2, (2π )−n Wz,t (X ) is a quasi-probability on the phase space R2n X and we have (2π )−n



ˆ z (t). d X Wz,t (X )A(X ) = Ψ (t), Aψ

T(−z) which has the WeylApplying this relation to Aˆ = Tˆ (z)(a∗ · a + a · a∗ )O symbol A(q, p) = |q − x|2 + | p − ξ |2 , we see that S(t) is the variance of the quasiprobability (2π )−n Wz,t (X ). Let us notice that S(t) is well defined if the estimate (4.14) holds in the Sobolev space Σ2 (Rn ) (defined in Chap. 1) and with some more assumption on the quantum evolution U (t, 0) one can get the estimate (4.14) in Σ2 -norm as we shall see now. More refined estimates in other Sobolev norms will be given in the last section of this chapter. Let us consider some symbol g satisfying assumption (A.0) with m = 0, such that Opw (g) is invertible in the Schwartz space S (Rn ). Let us assume that the following L 2 -operator norm estimate holds:

4.2 Application to the Spreading of Quantum Wave Packets

105

(Opw g(U (t, 0)(Opw g)−1 ≤ Ct,g . Then an estimate analogous to (4.14) holds true: Op w gU (t, 0)Tˆ (z)Λ f − U2 (t, 0)Λ Pl ( f, t, ) ≤ Ct,g Γ K H,tρl (z,t,) (4.25) Obviously S(0) = n and we are interested in the difference ΔS(t) := S(t) − S(0) Let us first calculate  T (t) := Tˆ (−z)φz (t),

n  (a ∗j a j + a j a ∗j )Tˆ (−z)φz (t) , j=1

where φz (t) is the approximant of ψz (t) given by φz (t) = eiδt / Tˆ (z)U0 (t)ψ0 . Then

1 T (t) = n + 2 aU0 (t)ψ0 2 = n + tr(Z t∗ Z (t), 2

where Z t is defined bt (4.9) and using Lemma (28). Therefore T (t) − T (0) =

1 tr(Z t∗ Z t ). 2

We show now that this is the dominant behavior of ΔS(t) up to small correction terms that we can estimate. Let us assume that the classical flow at phase-space point z has finite Lyapunov exponents, with a greatest Lyapunov exponent λ ∈ R. (for notions concerning the stability and Lyapunov exponents for ordinary differential equations we refer to [49]). Then by definition there exists some constant C > 0 such that F(t) ≤ Ceλt , ∀t ≥ 0 where C is independent of t. In what follows we denote by C a generic constant independent of t, . Then under the above assumptions we get √ ψz (t) − ϕz (t) Σ2 (Rn ) ≤ C te3λt , ∀t ≥ 0 . We deduce the following result: Theorem 23 Under the above assumptions we have the long time asymptotics ΔS(t) = ΔT (t) + O(ε ) if one of the two following conditions is fulfilled: (i) λ ≤ 0 (“stable case”) and 0 ≤ t ≤ ε−1/2

106

4 The Semiclassical Evolution of Gaussian Coherent States

(ii) λ > 0 (“unstable case”) and ∃ε > ε such that 0 ≤ t ≤ ((1 − 2ε )/6λ) log(1/). In particular we have Corollary 14 Let us assume that the Hamiltonian H is time independent and that the greatest Lyapunov exponent is λ > 0. Then S(t) − S(0) behaves like e2λt as t → +∞ and  → 0 as long as t[log(1/)]−1 stays small enough. (i) More precisely there exists C > 0 such that e2λt ≤ ΔT (t) ≤ Ce2λt , ∀t ≥ 0. C ΔS(t) = ΔT (t) + O(ε ), for 0 ≤ t ≤

1 1 − 2ε log for some ε > ε. 6λ 

(ii) In particular for n = 1 we have a more explicit result: ΔS(t) =

4b2 + (a − c)2 sinh2 (λt) + O(ε ), 2(b2 − ac)

under the above condition for t where

(4.26)

  ab = M0 is the Hessian matrix of H bc

at z. Proof We get (i) using that the matrix J M0 has at least one eigenvalue with real part λ. To prove (4.26) we compute explicitly √ the exponential of the matrix t J M0 which gives F(t). Its eigenvalues are λ = b2 − ac and 1/λ. So we get the formula  exp(t J M0 ) =

 c sinh(λt) cosh(λt) + λb sinh(λt) λ − aλ sinh(λt) cosh(λt) − λb sinh(λt)

hence we have ΔT (t) = tr(Z t∗ Z t ) =

4b2 + (a − c)2 sinh2 (λt). 2(b2 − ac)



4.3 Evolution of Coherent States and Bargmann Transform In Sect. 4.1 we have studied the evolution of coherent states using the generators of coherent states and the Duhamel formula. Here we present a different approach following [218], working essentially on the Fourier–Bargmann side (see Chap. 1). This

4.3 Evolution of Coherent States and Bargmann Transform

107

approach is useful to get estimates in several norms of Banach spaces of functions and also to get analytic type estimates.

Time evolution of acoherent state We keep the notations of Sect. 4.1. We revisit now the algebraic computations of this section in a different presentation. Recall that we want to solve the Cauchy problem i

∂ψz (t) !(t)ψz (t), ψ(0) = ϕz , =H ∂t

(4.27)

where ϕz is a coherent state localized at a point z ∈ R2n . Our first step is to transform this problem with suitable unitary transformations such that the singular perturbation problem in  becomes a regular perturbation problem. A mathematical statement is in Theorem 24.

4.3.1 Formal Computations We rescale the evolved state ψz (t) by defining f t such that ψz (t) = Tˆ (z t )Λ f t . Then f t satisfies the following equation.

 −1 ! ˆ H (t)Tˆ (z t ) − i∂t Tˆ (z t ) Λ f t i∂t f t = Λ−1  T (z t )

(4.28)

with the initial condition f t=0 = g where g(x) = π −n/4 e− 2 |x| . We get easily the formula √ √ −1 ! w ˆ ˆ (4.29) Λ−1  T (z t ) H (t) T (z t )Λ = O p1 H (t, x + qt , ξ + pt ). 1

Using the Taylor formula we get the formal expansion

2

108

4 The Semiclassical Evolution of Gaussian Coherent States

H (t,

√

x + qt ,

√

√ √ ∂q H (t, z t )x + ∂ p H (t, z t )ξ  + K 2 (t; x, ξ ) +   j/2−1 K j (t; x, ξ ), (4.30)

ξ + pt ) = H (t, z t ) +

j≥3

where K j (t) is the homogeneous Taylor polynomial of degree j in X = (x, ξ ) ∈ R2n . K j (t; X ) =

 1 γ ∂ X H (t; z t )X γ . γ ! |γ |= j

We shall use the following notation for the remainder term of order k ≥ 1, ⎛

⎞  √ Rk (t; X ) = ⎝ H (t, z t + X ) −  j/2 K j (t; X )⎠ .

(4.31)

j a 2 there exists C > 0 such that for all u ∈ S (Rn ) we have, ea|x| u(x) L p (Rnx ) ≤ C eb|X | F B u(X ) L 2 (R2nX ) .

(4.42)

√

More generally, for every a ≥ 0 and every b > a |S|2 there exists C > 0 such that for all u ∈ S (Rn ) and all S ∈ Sp(n) we have " #   ˆ (x) L p (Rnx ) ≤ C eb|X | F B u(X ) L 2 (R2n ) . ea|x| R(S)u

(4.43)

X

We need to control the norms of Hermite functions (see Chap. 1) in some weighted Lebesgue spaces. Let μ be a C ∞ -smooth and positive function on Rm such that lim μ(x) = +∞

(4.44)

|∂ γ μ(x)| ≤ θ |x|2 , ∀x ∈ Rm , |x| ≥ Rγ .

(4.45)

|x|→+∞

for some Rγ > 0 and θ < 1. Lemma 30 For every real p ∈ [1, +∞], for every  ∈ N, there exists C > 0 such that for every α, β ∈ Nm we have: eμ(x) x α ∂xβ (e−|x| ) , p ≤ C |α+β|+1 Γ 2



 |α + β| , 2

(4.46)

4.3 Evolution of Coherent States and Bargmann Transform

111

where • , p is the norm on the Sobolev space3 W , p , Γ is the Euler Gamma function.4 More generally, for every real p ∈ [1, +∞], for every  ∈ N, there exists C > 0 such that    |α + β| −1/2 2 . eμ((Γ ) x) x α ∂ β (e−|x| ) , p ≤ C |α+β|+1 |(Γ )1/2 | + |(Γ )−1/2 |Γ 2 (4.47)

4.3.3 Large Time Estimates and Fourier–Bargmann Analysis In this section we try to control the semi-classical error term Rz(N ) (t, x), for large time, in the Fourier–Bargmann representation. This is also a preparation to control the remainder of order N in , N for analytic Hamiltonians considered in the following sub-section. Let us introduce the Fourier–Bargmann transform of b j (t)g, B j (t, X ) = F B [b j (t)g](X ) = g X , b j (t)g, for X ∈ R2n . The induction Eq. (4.34) becomes for j ≥ 1, ⎛

 ∂t B j (t, X ) =

R

⎞

⎜  ⎜ ⎝ 2n

k+= j+2 ≥3

⎟ < g X , O p1w [K # (t)]g X >⎟ ⎠ Bk (t, X )d X ,

(4.48)



2 with initial condition B j (0, X ) = 0 for j ≥ 1 and with B0 (t, X ) = exp − |X4| . We have seen in the Sect. 4.1 that we have    O p1w [K  (t)]g X , g X  = (2π )−n K  (t, Y )W X,X (Y )dY, (4.49) R2n

where W X,X is the Wigner function of the pair (g X , g X ). Let us now compute the remainder term in the Fourier–Bargmann representation. Using that F B is an isometry we get  R2n

B j (t, X )g X , O p1w [R (t) ◦ Ft,t0 ]g X d X ,

(4.50)

that u ∈ W , p means that ∂xα u ∈ L p for every |α| ≤ . Euler classical Gamma function Γ must not be mixed up with the covariance matrix Γt .

3 Recall 4 The

$ % F B O p1w [R (t) ◦ Ft,t0 (b j (t)g)](X ) =

112

4 The Semiclassical Evolution of Gaussian Coherent States

where R (t) is given by the Taylor integral formula (4.51):  √ /2−1  1 γ ∂ X H (t, z t + θ X )X γ (1 − θ )−1 dθ. R (t, X ) = k! |γ |= 0

(4.51)

We shall use (4.50) to estimate the remainder term Rz(N ) , using estimates (4.42) and (4.43). Now we shall consider long time estimates for B j (t, X ). Lemma 31 For every j ≥ 0, every s ∈ N, r ≥ 1, there exists C( j, α, β) such that for |t| ≤ T , we have    μ(X/4) α β  X ∂ X B j (t, X ) e

3j

s,r

≤ C( j, α, β)σ (t, z) N M1 |F|T (1 + T ) j ,

(4.52)

where |F|T = sup |Ft |. M1 and σ (t, z) were defined in section1, M1 depends on |t|≤T

assumption (A.0) on H (X, t). • s,r is the norm in the Sobolev space W s,r (R2n )5 Proof The main idea of the proof is as follows (see [218] for details). We proceed by induction on j. For j = 0 (4.52) results from (4.46). Let us assume inequality proved up to j − 1. We have the induction formula ( j ≥ 1)   K  (t, X, X )Bk (t, X )d X , (4.53) ∂t B j (t, X ) = 2n k+= j+2 R ≥3

where  1 γ ∂ X H (t, z t )O p1w (Ft Y )γ g X , g X , and γ ! |γ |=  O p1w (Ft Y )γ g X , g X  = 22n (Ft Y )γ W X,X (Y )dY.

K  (t, X, X ) =

(4.54) (4.55)

R2n

By a Fourier transform computation on Gaussian functions, we get the following more explicit expression   γ −β X + X γ Cβ 2−|β| Ft · 2 β≤γ    J (X − X ) 2 e−|X −X | /4 e−(i/2)σ (X ,X ) . ·Hβ Ft 2

O p1w (Ft Y )γ g X , g X  =



(4.56) 

Estimate (4.52) follows easily. 5 The

Sobolev norm is defined here as f s,r =



 |α|≤s

d x| f (x)|r

1/r

for s ∈ N, r ≥ 1.

4.3 Evolution of Coherent States and Bargmann Transform

113

Now we have to estimate the remainder term. Let us compute the Fourier– Bargmann transform of the error term: ⎡

⎤ 

⎢ j/2 (N +1) (t, X ) = F B ⎢ R˜z ⎣ =

 j+k=N +3 k≥3

j+k=N +3 k≥3



R2n

⎥ O p1w [Rk ](t) ◦ Ft ](b j (t)g)⎥ ⎦ (X )

B j (t, X )O p1w [Rk (t) ◦ Ft ]g X , g X d X

Using estimates on the B j we get the following estimate for the error term: Lemma 32 For every κ > 0, for every  ∈ N, s ≥ 0, r ≥ 1, there exists C N , such that for all T and t, |t| ≤ T , we have    α β ,(N +1)  (t, X ) X ∂ X Rz for

s,r

+3 ≤ C N , M N , (T, z)|F|3N (1 + T ) N +1  T

N +3 2

(4.57)

√

|F|T ≤ κ, |α| + |β| ≤ , where M N , (T, z) is a continuous function of γ sup |∂ X H (t, z t )|.

|t−t0 |≤T 3≤|γ |≤N

Proof As above for estimation of the B j (t, X ), let us consider the integral kernels Nk (t, X, X ) = O p1w [Rk (t) ◦ Ft ]g X , g X .

(4.58)

We have

Nk (t, X, X ) =   ×

R2n

(k+1)/2



1 k! |γ |=k+1



1

(1 − θ )k

0

 √ γ γ ∂Y H (t, z t + θ Ft Y )(Ft Y ) .W X ,X (Y )dY dθ.

(4.59)

Let us denote by Nk,t the operator with the kernel Nk (t, X, X ). Using the change and integrations by parts in X as above, we can estimate of variable Z = Y − X +X 2 Nk,t [B j (t, •)](X ). Then using the estimates on the B j (t, X ) we get estimate (4.57).  Now, it is not difficult to convert these results in the configuration space, using 1/2

2 t | +1 (4.42). Let us define λ,t (x) = |x−q . |Ft |2 Theorem 24 Let us assume that (A.0) is satisfied. Then we have for the remainder term, ∂ !(t)ψz(N ) (t, x), Rz(N ) (t, x) = i ψz(N ) (t, x) − H ∂t

114

4 The Semiclassical Evolution of Gaussian Coherent States

the following estimate. For every κ > 0, for every , M ∈ N, r ≥ 1 there exist C N ,M, and N such that for all T and t, |t| ≤ T , we have:  M (N )  λ R (t) ≤ C N , (N +3−)/2 σ (z, t) N M1 |F|3N +3 (1 + T ) N +1 , ,t z T ,r

(4.60)

√ for every  ∈]0, 1], |Ft | ≤ κ. Moreover, if Hˆ (t) admits a unitary propagator (see condition (A.2)) , then under the same conditions as above, we have +3 (1 + T ) N +2 (N +1)/2 . Ut ϕz − ψz(N ) (t) 2 ≤ C N , σ (z, t) N M1 |F|3N T

(4.61)

Proof Using the inverse Fourier–Bargmann transform, we have Rz(N ) (t, x) = Tˆ (z t )Λ

 R2n

 ˆ t ]ϕ X )(x) R ,z(N ) (t, X )d X . ( R[F

Let us remark that using estimates on the b j (t, x), we can assume that N is arbitrary large. We can apply previous results on the Fourier–Bargmann estimates to get (4.60). The second part is a consequence of the first part and of the Duhamel principle.  We see that the estimate (4.60) is much more accurate in norm than estimate (4.61), we have lost much information applying the propagator Ut . The reason is that in general we only know that the propagator is bounded on L 2 and no more. Sometimes it is possible to improve (4.61) if we know that Ut is bounded on some weighted Sobolev spaces. Let us give here the following example. Let Hˆ = −2  + V (x). Assume that V satisfies: V ∈ C ∞ (Rn ), |∂ α V (x)| ≤ Cα V (x), V ≥ 1, |V (x) − V (y)| ≤ C(1 + |x − y|) M , for some M ∈ R. So the time dependent Schrödinger equation for Hˆ has a unitary −1 ˆ propagator Ut = eit H . The domain of Hˆ m can be determined for every m ∈ N (see for example [217]). D( Hˆ m ) = {u ∈ W 2m,2 (Rn ), V m u ∈ L 2 (Rn )}. It is an Hilbert space with the norm defined by u 22m,V =

 |α|≤2m

|α| ∂ α u 2L 2 (Rn ) + V u 2L 2 (Rn )

Using the Sobolev theorem we get the supremum norm estimate for the error: +3 (1 + T ) N +2 (N −n/2)/2 sup |(Ut ϕz )(x) − ψz(N ) (t, x)| ≤ C N , σ (z, t) N M1 |F|3N T

x∈Rn

(4.62)

4.3 Evolution of Coherent States and Bargmann Transform

115

4.3.4 Exponentially Small Estimates Up to now the order N of the semi-classical approximations was fixed, even arbitrary large, but the error term was not controlled for N large. Here we shall give estimates with a control for large N . The method is the same as on the previous section, using systematically the Fourier–Bargmann transform. The proof is not given here, we refer to [218] for more details. For a different approach see [133, 134]. To get exponentially small estimates for asymptotic expansions in small  it is quite natural to assume that the classical Hamiltonian H (t, X ) is analytic in X , where X = (x, ξ ) ∈ R2n . This problem was studied in a different context in [115] concerning Borel summability for semi-classical expansions for systems of bosons. So, in what follows we introduce suitable assumptions on H (t, X ). As before we assume that H (t, X ) is continuous in time t and C ∞ in X and that the quantum and classical dynamics are well defined. Let us define a complex neighborhood of R2n in C2n , Ωρ = {X ∈ C2n , |X | < ρ}

(4.63)

where X = (X 1 , . . . , X 2n ) and | · | is the Euclidean norm in R2n or the Hermitian norm in C2n . Our main assumptions are the following. (Aω ) (Analyticity assumption) There exists ρ > 0, T ∈]0, +∞], C > 0, ν ≥ 0, such that H (t) is holomorphic in Ωρ and for t ∈ IT , X ∈ Ωρ , we have |H (t, X )| ≤ Ceν|X | , and + Y )| ≤ R γ γ !eν|Y | , ∀t ∈ R, Y ∈ R2n

γ |∂ X H (t, z t

(4.64)

for some R > 0 and all γ , |γ | ≥ 3. We begin by giving the results on the Fourier–Bargmann side. It is the main step and gives accurate estimates for the propagation of Gaussian coherent states in the phase space. We have seen that it is not difficult to transfer these estimates in the configuration space to get approximations of the solution of the Schrödinger equation, by applying the inverse Fourier–Bargmann transform as we did in the C ∞ case. The main results are stated in the following theorem. Theorem 25 Let us assume that conditions (A0 ) and (Aω ) are satisfied. Then the following uniform estimates hold.    α β   X ∂ X B j (t, X ) 3 j+1+|α|+|β|

Cλ,T

L 2 (R2d ,eλ|X | d X )

|F|T (1 + |t − t0 |) j j − j (3 j + |α| + |β|) 3j

≤

3 j+|α|+|β| 2

(4.65) .

where Cλ > 0 depends only on λ ≥ 0 and is independent on j ∈ N, α, β ∈ N2n and |t| ≤ T .

116

4 The Semiclassical Evolution of Gaussian Coherent States

Concerning the remainder term estimate we have:    α β ˜ (N )  X ∂ X Rz (t, X ) +3 |F|3N (Cλ ) T

3N +3+|α|+|β|

L 2 (R2n ,eλ|X | d X )

≤ (N +3)/2 (1 + |t|) N +1 (4.66)

(N + 1)−N −1 (3N + 3 + |α| + |β|)

3N +3+|α|+|β| 2

.

√ where λ < ρ, α, β ∈ N2n , N ≥ 1, ν |F|T ≤ 2(ρ − λ), Cλ depends on λ and is independent on the other parameters (, T, N , α, β). From Theorem 25 we easily get weighted estimates for approximate solutions and remainder term for the time dependent Schrödinger equation. Let us recall the Sobolev norms in the Sobolev space W m,r (Rn ). ⎛ u r,m, = ⎝



|α|≤m

|α|/2

 Rn

⎞1/r |∂xα u(x)|r d x ⎠

and a function μ ∈ C ∞ (Rd ) such that μ(x) = |x| for |x| ≥ 1. Proposition 42 For every m ∈ N, r ∈ [1, +∞], λ > 0 and ε ≤ min{1, |F|λ T }, there exists Cr,m,λ,ε > 0 such that for every j ≥ 0 and every t ∈ IT we have ˆ t ]b j (t)geεμ r,m,1 ≤ (Cr,m,λ,ε ) j+1 (1 + |F|T )3 j+2d j j/2 (1 + |t|) j R[F

(4.67)

Theorem 26 With the above notations and under the assumptions of Theorem 25, ψz(N ) (t, x) satisfies the Schrödinger equation !(t)ψz(N ) (t, x) + Rz(N ) (t, x), i∂t ψz(N ) (t, x) = H ⎛ ⎞  ˆ t] ⎝  j/2 b j (t)g ⎠ where ψz(N ) (t, x) = eiδt / Tˆ (z t )Λ R[F

(4.68) (4.69)

0≤ j≤N

is estimated in Proposition 42 and the remainder term is controlled with the following weighted estimates: Rz(N ) (t)eεμ,t r,m, ≤

√  N +3 |F|3T −m (1 + |t|) N +1 , C N +1 (N + 1)(N +1)/2

(4.70)

where C depends only on m, r, ε and not on N ≥ 0, |t| ≤ T and  > 0, with the √ √ t . condition |F|T ≤ κ. The exponential weight is defined by μ,t (x) = μ x−q 

m ≥ 0 and h ε < min{1, |F|ρ T }.

4.3 Evolution of Coherent States and Bargmann Transform

117

Remark 22 We see that the order in j of the coefficient b j (t)g in the asymptotic expansion in  j/2 is C j j j/2 or using Stirling formula C j Γ ( j)1/2 for some constant C > 0, C > 0. So we have obtained that the renormalized evolved state b(t, x)g(x) has a Gevrey-2 asymptotic expansion in 1/2 . Recall that a obtained from ψz (t, x)  formal complex series c j κ j is a Gevrey series of index μ > 0 if there exists j≥0

constants C0 > 0, C > 0 such that |c j | ≤ C0 C j Γ ( j)1/μ , ∀ j ≥ 1. Any holomorphic function f (κ) in a complex neighborhood of 0 has a convergent Gevrey-1 Taylor series. But in many physical examples we have a non-convergent Gevrey asymptotic series f [κ] for a function f holomorphic in some sector with apex 0. Under some technical conditions on f it is possible to define the Borel sum B f (τ ) for the formal power series f [κ] and to recover f (κ) from its Borel sum performing a Laplace transform on B f (τ ) (see [239] for details and bibliography). When it is not possible to apply Borel summability, there exists a well known  method to minimize the error between c j κ j and f (κ). It is called the 1≤ j≤N

astronomers method and consists in stopping the expansion after the smallest term of the series (it is also called “the least term truncation method” for a series). Concerning the semiclassical expansion found for b(t, x)g(x) it is not clear that it is Borel summable or summable in some weaker sense. A sufficient condition would be that the propagator Ut can be extended holomorphically in κ := 1/2 in a (small) sector {r eiθ , 0 < r < r0 , |θ | < ε}. In a different context (quantum field theory for bosons), Borel summability was proved in [115]. Using the astronomers method Theorem 26 we easily get the following consequences. Corollary 15 (Finite Time, Large N ) Let us assume here that T < +∞. There exist c > 0, 0 > 0, a > 0, ε > 0, such that if we choose N = [ a ] − 1 we have

c (4.71) Rz(N ) (t)eεμ,t L 2 ≤ exp − ,  for every |t| ≤ T ,  ∈]0, 0 ]. Moreover, we have

c ψz(N ) (t) − U (t, t0 )ϕz L 2 ≤ exp − . 

(4.72)

And Corollary 16 (Large Time, Large N ) Let us assume that T = +∞ and there exist γ ≥ 0, δ ≥ 0, C1 ≥ 0, such that |Ft,t0 | ≤ exp(γ |t|), |z t | ≤ exp(δ|t|) for every θ ∈ ]0, 1[ there exists aθ > 0 such that if we choose N,θ = [ aθθ ] − 1 there exist cθ > 0, ηθ > 0 such that

118

4 The Semiclassical Evolution of Gaussian Coherent States

c  θ (N,θ +2)/2 Rz(N,θ +1) (t)eεμ,t L 2 ≤ exp − θ  for every |t| ≤

1−θ 6γ

(4.73)

log(−1 ), ∀ ∈  ∈]0, ηθ ]. Moreover we have:

c  θ ψz(N,θ ) (t) − U (t, t0 )ϕz L 2 ≤ exp − θ , 

(4.74)

under the conditions of (4.73). Remark 23 We have considered here standard Gaussian. All the results are true and proved in the same way for Gaussian coherent states defined by g Γ , for any Γ ∈ Σn+ . These results have been proved in [218] and in [133, 134] using different methods. All the results in this subsection can be easily deduced from Theorem 25. Proposition 42 and Theorem 26 are easily proved using the estimates of Sect. 2.2. The proof of the corollaries are consequences of Theorem 26 and Stirling formula for the Euler Gamma function.

4.4 Application to the Scattering Theory In this section we assume that the interaction satisfies a short range assumption and we shall prove results for the action of the scattering operator acting on the squeezed states. One gets a semiclassical asymptotics for the action of the scattering operator on a squeezed state located at point z − in terms of a squeezed state located at point z + where z + = S cl (z − ), S cl being the classical scattering matrix. For the basic classical and quantum scattering theories we refer the reader to [81, 215]. Let us first recall some basic facts on classical and quantum scattering theory. We consider a classical Hamiltonian H for a particle moving in a curved space and in an electromagnetic field; H (q, p) =

1 g(q) p · p + a(q) · p + V (q), q ∈ Rn , p ∈ Rn , 2

(4.75)

g(q) is a smooth positive definite matrix and there exists c > 0, C > 0 such that c| p|2 ≤ g(q) p · p ≤ C| p|2 , ∀(q, p) ∈ R2n , a(q) is a smooth linear form on Rn and V (q) a smooth scalar potential. In what 2 follows it will be assumed that H (q, p) is a short range perturbation of H (0) = p2 in the following sense: there exist ρ > 1, and Cα , for α ∈ Nn such that |∂qα (1l − g(q))| + |∂qα a(q)| + |∂qα V (q)| ≤ Cα q−ρ−|α| , ∀q ∈ Rn .

(4.76)

4.4 Application to the Scattering Theory

119

H and H (0) define two Hamiltonian flows Φ t , Φ0t on the phase space R2n for all t ∈ R. The classical scattering theory establishes a comparison of the two dynamics Φ t , Φ0t in the large time limit. Note that the free dynamics is explicit: Φ0t (q, p) = (q + t p, p). The methods of [81, 215] can be used to prove the existence of the classical wave operators defined by (4.77) Ω±cl X = lim Φ −t (Φ0t X ). t→±∞

  This limit exists for every X ∈ Z0 where Z0 = (q, p) ∈ R2n , p = 0 and is uniform on every compact of Z0 . We also have for all X ∈ Z0 lim

t→±∞

t cl  Φ Ω± (X ) − Φ0t (X ) = 0.

Moreover Ω±cl are C ∞ -smooth symplectic transformations. They intertwine the free and the interacting dynamics: H ◦ Ω±cl X = Ω±cl ◦ H (0) (X ), ∀X ∈ Z0 , and Φ t ◦ Ω±cl = Ω±cl ◦ Φ0t . Then the classical scattering matrix S cl is defined by S cl = (Ω+cl )−1 Ω−cl . This definition makes sense since one can prove (see [215]) that modulo a closed set N0 of Lebesgue measure zero in Z (Z \ Z0 ⊆ N0 ) one has Ω+cl (Z0 ) = Ω−cl (Z0 ) Moreover S cl is smooth in Z \ N0 and commutes with the free evolution: S cl Φ0t = Φ0t S cl . The scattering operator has the following kinematic interpretation: let us consider X − ∈ Z0 and its free evolution Φ0t X − . There exists a unique interacting evolution Φ t (X ) which is close to Φ0t (X − ) for t  −∞. Moreover there exists a unique point X + ∈ Z0 such that Φ t (X ) is close to Φ0t (X + ) for t  +∞. X, X + are given by X = Ω−cl X − and X + = S cl X − . Using [81] we can get a more precise result. Let I be an open interval of R and assume that I is “non trapping” for H which means that for every X such that H (X ) ∈ I we have limt→±∞ |Φ t (X )| = +∞. Then we have:

120

4 The Semiclassical Evolution of Gaussian Coherent States

Proposition 43 If I is a non trapping interval for H then S cl is defined everywhere in H −1 (I ) and is a C ∞ smooth symplectic map. On the quantum side one can define the wave operators and the scattering operator ˆ in a similar way. Let us note that the quantization  H of H is essentially self-adjoint

so that the unitary group U (t) = exp − it Hˆ is well defined in L 2 (Rn ). The free 

evolution U0 (t) := exp − it Hˆ (0) is explicit:  ξ2 i (−t + (x − y) · ξ ) ψ(y) dy dξ. (U0 (t)ψ)(x) = (2π ) exp  2 R2n (4.78) The assumption (4.76) implies that we can define the wave operators Ω± and the scattering operator S () = (Ω+ )∗ Ω− (see [81, 215]). Recall that −n

 



Ω± = lim U (−t)U0 (t). t→±∞

The ranges of Ω± are equal to the absolutely continuous subspace of Hˆ and we have: Ω± U0 (t) = U (t)Ω± , S () U0 (t) = U0 (t)S () .

(4.79)

One wants to obtain a correspondence between lim→0 S () and S cl . There are many works on the subject (see [130, 134, 221, 265]). Here we want to check this classical limit using the coherent states approach like in [130, 134]. We present here a different technical approach extending these results to more general perturbations of the Laplace operator. We recall some notations of Chap. 3: S+ (n) is the Siegel space namely the space of complex symmetric n × n matrices Γ such that Γ is positive and non degenerate. ˆ Given F any 2n × 2n symplectic matrix the unitary operator R(F) is the metaplectic Γ transformation associated to F. g is the gaussian function of L 2 norm 1 defined by 

i Γx ·x g (x) = aΓ exp 2 Γ

and we denote

 (4.80)

ϕzΓ = Tˆ (z)g Γ

Finally Λ is the unitary operator defined in (4.22). The main result of this section states a relationship between the quantum scattering and the classical scattering. Theorem 27 For every N ≥ 1, every z − ∈ Z \ N0 and every Γ− ∈ Σn we have the following semiclassical approximation for the scattering operator S () acting on the Γ Gaussian coherent state ϕz−− :

4.4 Application to the Scattering Theory

121

⎛



ˆ +) ⎝ S () ϕzΓ−− = eiδ+ / Tˆ (z + )Λ R(G

⎞  j/2 b j g Γ− ⎠ + O((N +1)/2 ),

(4.81)

0≤ j≤N

where we define z + = S cl z − , z ± = (q± , p± ). z t = (qt , pt ) is the interacting scattering trajectory z t = Φ t (Ω−cl z − ).  +∞ − p− δ+ = −∞ ( pt qt − H (z t )) dt − q+ p+ −q . 2 ∂z + G + = ∂z− . b j is a polynomial of degree ≤ 3 j, b0 = 1. The error term O((N +1)/2 ) is estimated in the L 2 -norm. Let us denote

ψ− = ϕzΓ−− , and ψ+ := S () ψ− .

Using the definition of S () we have  ψ+ = lim

t→+∞

 lim U0 (t)U (t − s)U0 (s) ψ− .

s→−∞

(4.82)

The strategy of the proof consists in applying the estimate (4.14) at fixed time t to U (t − s) in (4.82) and then to see what happens in the limits s → −∞, t → +∞. Let us denote by Ft0 the Jacobi stability matrix for the free evolution and by Ft (z) the Jacobi stability matrix along the trajectory Φ t (z). We have Ft0

  1ln t1ln . = 0 1ln

We need large time estimates concerning classical scattering trajectories and their Jacobi stability matrices. Proposition 44 Under the assumptions of Theorem 27 there exists a unique scattering solution of the Hamilton’s equation z˙ t = J ∇ H (z t ) such that z˙ t − ∂t Φ0t z + = O(t−ρ ), for t → +∞. z˙ t − ∂t Φ0t z − = O(t−ρ ), for t → −∞. Proposition 45 Let us denote G t,s := Ft−s (Φ0s z − )Fs0 . Then we have: (i) lims→−∞ G t,s = G t exists ∀t ≥ 0.

122

4 The Semiclassical Evolution of Gaussian Coherent States

(ii) limt→+∞ F0−t G t = G + exists. (iii) G t =

∂z t , ∂z −

and G + =

∂z + . ∂z −

These two propositions will be proven later together with the following one. The main step in the proof of Theorem 27 will be to solve the following asymptotic Cauchy problem for the Schrödinger equation with data given at time t = +∞: ) i∂s ψzN−) = Hˆ ψz(N (s) + O((N +3)/2 f N (s)) −

(4.83)

) lim U0 (−s)ψz(N (s) = ϕzΓ−− −

s→−∞

where f N ∈ L 1 (R) ∩ L ∞ (R) is independent of . The following result is an extension for infinite times of results proven in Sect. 1 for finite times. Proposition 46 The problem (4.83) has a solution which can be computed in the following way: ⎛ ) ˆ t) ⎝ (t, x) = eiδ(zt )/ Tˆ (z − )Λ R(G ψz(N −

⎞



 j/2 b j (t, z − )g Γ− ⎠ .

0≤ j≤N

The b j (t, z − , x) are uniquely defined by the following induction formula for j ≥ 1 starting with b0 (t, x) ≡ 1: 

∂t b j (t, z − , x)g(x) =



Opw1 [K l (t)](bk (t, .)g)(x),

(4.84)

k+l= j+2, l≥3

lim b j (t, z − , x) = 0 ,

(4.85)

t→−∞

with 

K j (t, X ) = K j (t, G t (X )) =

 1 γ ∂ X H (z t )(G t X )γ , X ∈ R2n , γ ! |γ |= j

b j (t, z − , x) is a polynomial of degree ≤ 3 j in variable x ∈ Rn with complex time dependent coefficients depending on the scattering trajectory z t starting from z − at time t = − − ∞. Moreover we have the remainder uniform estimate ) ) (t) = Hˆ ψz(N (t) + O((N +3)/2 t−ρ ), i∂t ψz(N − −

(4.86)

uniformly in  ∈]0, 1], and t ≥ 0. Proof (Theorem 27). Without going into the details which are similar to the finite time case, we remark that in the induction formula (4.84) we can use the following estimates to get uniform decrease in time estimates for b j (t, z − , x). First there exist

4.4 Application to the Scattering Theory

123

c > 0 and T0 > 0 such that for t ≥ T0 we have |qt | ≥ ct. Using the short range assumption and conservation of the classical energy we have that for |γ | ≥ 3 there exists Cγ > 0 such that γ (4.87) |∂ X H (z t )| ≤ Cγ t−ρ−1 . Therefore we deduce (4.86) from (4.84) and (4.87). Using Proposition 46 and Duhamel’s formula we get ) ) (s) = ψz(N (t + s) + O((N +1)/2 ), U (t)ψz(N − −

uniformly in t, s ∈ R. But we have: ) (t) − U (t − s)U0 (s)ψ− ≤ ψz(N − ) (t) ψz(N −

We know that

− U (t −

) s)ψz(N (s) −

+ U0 (s)ψ− −

) ψz(N (s) . −

(4.88) (4.89)

) (s) = 0. lim U0 (s)ψ− − ψz(N −

s→−∞

Then going to the limit s → −∞ we get uniformly in t ≥ 0 ) (t) − U (t)Ω− ψ− = O((N +1)/2 ). ψz(N − ) (t) in the limit t → +∞ and we find out that Then we can compute U0 (−t)ψz(N − (N ) () (N +1)/2 ) where S ψ− = ψ+ + O( ) ψ+(N ) = lim U0 (−t)ψz(N (t). − t→+∞



So we have proved Theorem 27. Let us prove now Proposition 44 following the book [215]: Proof Let us denote u(t) := z t − Φt0 z − . We have to solve the integral equation  u(t) = Φt0 (z − ) +

t

−∞

(J ∇ H (u(s)) + Φs0 (z − )) ds

We can choose T1 < 0 such that the map K defined by  K u(t) =

t −∞

(J ∇ H (u(s)) + Φs0 (z − )) ds.

is a contraction in the complete metric space CT1 of continuous functions u from ] − ∞, T1 ] into R2n such that supt≤T1 |u(t)| ≤ 1, with the natural distance. So we

124

4 The Semiclassical Evolution of Gaussian Coherent States

can apply the fixed point theorem to prove Proposition 44 using standard technics.  Proposition 45 can be proved by the same method. Let us now prove Proposition 46. Proof Let us denote z s0 := Φs0 (z − ). Furthermore if S is a symplectic matrix 

A B S= C D



and Γ ∈ S+ (n) (S+ (n) is the Siegel space) we define Σ S (Γ ) = (C + DΓ )(A + BΓ )−1 ∈ S+ (n). Then let Γs0 = Σ Fs0 (Γ− ). One has for every N ≥ 0: i∂t ψz(N ) (t, s, x) = Hˆ (t)ψz(N ) (t, s, x) + Rz(N ) (t, s, x), where ⎛ ˆ t,s Fs0 ) ⎝ ψz(N ) (t, s, x) = eiδs s,t/ Tˆ (z t )Λ R(F



⎞  j/2 b j (t, s)g Γ− ⎠

(4.90)

0≤ j≤N

and ) Rz(N (t, s, x) = eiδs,t / (N +3)/2 × − ⎛ ⎝



⎞ ˆ t,s Fs0 )Opw1 (Rk (t, s) ◦ [Ft,s Fs0 ])(b j (t, s)g Γ− )⎠ . Tˆ (z − )Λ R(F

j+k=N +2, k≥3

(4.91) One denotes Ft,s = Ft−s (Φ0s z − ) the stability matrix at Φt−s (Φs0 (z − )). Moreover the polynomials b j (t, s, x) are uniquely defined by the following induction formula for j ≥ 1 starting with b0 (s, s, x) ≡ 1: ∂t b j (t, s, x) =





Opw1 [K l (t, s)](bk (t, .)g Γ− )(x),

k+l= j+2, l≥3

b j (s, s, x) = 0 where

4.4 Application to the Scattering Theory 

K l (t, s, X ) =

125

 1 γ ∂ X H (Φ t−s (Φs0 z − ))(Ft−s Fs0 X )γ , γ ! |γ |=l

X ∈ R2n .

So using Propositions 44 and 45 we can control the limit s → −∞ in Eqs. (4.90) and (4.91) and we get the proof of the Proposition 46.  The following corollary is an immediate consequence of Theorem 27 and of the properties of the metaplectic transformation: Corollary 17 For every N ∈ N we have S () ϕzΓ−− = eiδ+ /



  j/2 π j

0≤ j≤N

x − q+ √ 



ϕzΓ++ (x) + O(∞ ),

where z + = S cl (z − ), Γ+ = ΣG + (Γ− ), π j (y) are polynomials of degree ≤ 3 j in y ∈ Rn . In particular π0 = 1. Recall that Sm 0 is the space of smooth classical observables L such that for every γ ∈ R2n there exists Cγ ≥ 0 such that γ

|∂ X L(X )| ≤ Cγ X m , ∀X ∈ R2n . The Weyl quantization Lˆ of L is well defined (see Chap. 2). One has the following result: Corollary 18 For any symbol L ∈ Sm 0 , m ∈ R, we have √ S () ϕz− , Lˆ S () ϕz−  = L(S cl (z − )) + O( ) . In particular one recovers the classical scattering operator from the quantum scattering operator in the semiclassical limit. Proof Using Corollary 17 one gets √ ˆ zΓ+  + O( ) S () ϕz− , Lˆ S () ϕz−  = ϕzΓ−+ , Lϕ − and the result follows using results proved in Chap. 2.



Remark 24 A similar result was proven for the time-delay operator in [253]. The proof given here is different and doesn’t use a global non-trapping assumption. Further study concerns the scattering evolution of Lagrangian states (also called WKB states). It was considered by Yajima [265] in the momentum representation and by Robinson [224] for the position representation. The approach developed here provides a more direct and general proof that is detailed in [218].

126

4 The Semiclassical Evolution of Gaussian Coherent States

Note also that under the analytic and Gevrey assumption one can recover the result of [134] for the semiclassical propagation of coherent states with exponentially small estimate.

Chapter 5

Trace Formulas and Coherent States

Abstract The best known trace formula in mathematical physics is certainly the  as Gutzwiller trace formula linking the eigenvalues of the Schrödinger operator H Planck’s constant goes to zero (the semiclassical régime) with the closed orbits of the corresponding classical mechanical system. Gutzwiller gave a heuristic proof of this trace formula, using the Feynman integral representation for the propagator of . In mathematics this kind of trace formula was first known as Poisson formula H d on the circle which gives a relationship between the eigenvalues of the operator idθ and the periods of the rotation group. The generalization to the Laplace operator on a compact manifold, then for more general elliptic operators, was performed using microlocal analysis and the theory of Fourier integral operators. Our goal here is to show how the use of coherent states allows us to give a rather simple and direct rigorous proof of the Gutzwiller trace formula in a rather general setting.

5.1 Introduction A quantum system is described by its Hamiltonian Hˆ and its admissible energies are the eigenvalues E j () (we suppose that the spectrum of the self-adjoint operator Hˆ in the Hilbert space H = L 2 (Rn ) is discrete). The frequency transition between E ()−E () energies E j () and E k () is ω j,k = k  j . If n = 1, or if the system is integrable, it is possible to prove semi-classical expansion for individual eigenvalues E j () when   0. For more general systems it is very difficult and almost impossible to analyze individual eigenvalues. But it is possible to give a statistical description of the energy spectrum in the semi-classical régime by considering mean values    Tr f ( Hˆ ) = f (E j ()).

(5.1)

A first result can be obtained if we suppose that the -Weyl symbol H of Hˆ is smooth and satisfies the assumption of the functional calculus in Chap. 2 (Theorem 10). © Springer Nature Switzerland AG 2021 D. Robert and M. Combescure, Coherent States and Applications in Mathematical Physics, Theoretical and Mathematical Physics, https://doi.org/10.1007/978-3-030-70845-0_5

127

128

5 Trace Formulas and Coherent States

Consider an interval Iε = [λ1 − ε, λ2 + ε] such that H −1 (I ) is compact for ε small enough. The following result is proved in [141] Proposition 47 (i) For every smooth function f supported in Iε we have the asymptotic expansion at any order in  Tr f ( Hˆ )  (2π )−n

 R2n

d X f (H (X )) +



 j−n C j ( f ),

(5.2)

j≥1

where C j ( f ) are computable distributions in the test function f . (ii) If λ1 and λ2 are noncritical values for H 1 and if N I denotes the number of eigenvalues of Hˆ in I := [λ1 , λ2 ] then we have the Weyl asymptotic formula N I = (2π )−n Vol(H −1 (I )) + O(1−n ) .

(5.3)

Remark 25 (i) The first part of the Proposition is an easy application of the functional calculus. Using (i) it is possible to prove a Weyl formula with an error term O(θ−n ) with θ < 23 . The error term with θ = 1 is optimal (in general) and can be obtained using a method initiated by Hörmander [52, 141, 152]. Furthermore using a trick initiated by Duistermaat–Guillemin [89] the remainder term can be improved in o(1−n ) if the measure of closed classical path on H −1 (λi ) is zero, for i = 1, 2 (see [211]). The density of states of a quantum system Hˆ is the sum of delta distributions  D(E) = δ(E − E j ()). The integrated density of states is the spectral repartition function N (E) = { j, E j () < E} where E denotes the number of terms in a series E . So that we have D(E) = ddE N (E). The Gutzwiller trace formula is a semi-classical formula which expresses the density of states of a quantum system in terms of the characteristics of the corresponding classical system (invariant tori for the completely integrable systems, periodic orbits otherwise). Remark that properties of the classical system may have consequence on the error term in the Weyl formula (see the Remark above). The prototype of the trace formula is the Poisson summation formula: +∞  n=−∞

f (n) =

+∞ 

f˜(2π k)

(5.4)

k=−∞

 for any f ∈ S (R). Recall that f˜ is the Fourier transform f˜(τ ) = R dte−itτ f (t). We show in which respect it is a Trace Formula. Consider the quantum momentum operator Pˆ in dimension one Pˆ = −i ddx acting in L 2 ([0, 2π ]) with periodic boundary conditions u(0) = u(2π ). Pˆ is an unbounded operator with purely discrete spectrum and one has ˆ = Z. sp( P) 1 See

definition in Sect. 5.2.

5.1 Introduction

129

Therefore one has ˆ = Tr f ( P)

+∞ 

f (n).

n=−∞

The classical Hamiltonian is H (q, p) = p. Therefore the solutions of the classical equations of motion are q = t mod(2π ). So, the classical trajectories are periodic in phase-space T1 × R and are k -repetitions of the primitive orbit of period 2π, ∀k ∈ Z. Thus the periods of the classical flow are equal to 2kπ, k ∈ Z. Then the Poisson Summation Formula expresses that the trace of a function of a quantum Hamiltonian equals the sum over the periodic orbits of the corresponding classical flow of the Fourier Transform of this function taken at the periods of the classical flow. From the Poisson Summation formula one deduces a Trace Formula for the onedimensional Harmonic Oscillator: take 1 Hˆ os = ( Pˆ 2 + Qˆ 2 ) 2 the Hamiltonian of the Harmonic Oscillator. We assume  = 1 for simplicity. The

spectrum of Hˆ 0 is n + 21 , n ∈ N . Therefore for any f ∈ S ([0, +∞[) one has ∞    1 Tr f ( Hˆ os ) = f (n + ). 2 n=0

Replace f by Tˆ (q, 0) f in Eq. (5.4). One gets 

f (n + q) =

n∈Z

Therefore for q =

1 2



e2iπkq f˜(2kπ ).

k∈Z

one gets +∞    Tr f ( Hˆ os ) = (−1)k f˜(2kπ ). k=−∞

But 2kπ are the periods of the orbits for the classical Harmonic oscillator of frequency 1, which are all repetitions of the primitive orbit of period 2π . One notes the apparition of a factor (−1)k in the trace formula which is the manifestation of the so-called Maslov index of the periodic orbit of period 2kπ . The paper of Gutzwiller appeared in 1971. Between 1973 and 1975 several authors gave rigorous derivations of trace formulas, generalizing the classical Poisson summation formula from d 2 /dθ 2 on the circle to elliptic operators on compact manifolds:

130

5 Trace Formulas and Coherent States

Colin de Verdière [56], Chazarain [51], Duistermaat-Guillemin [89]. The first article is based on a parametrix construction for the associated heat equation, while the two others replace this with a parametrix, constructed as a Fourier integral operator, for the associated wave equation. The pioneering works in quantum physics of Gutzwiller [127, 128] and Balian– ˆ localized Bloch [14, 15] (1972–74) showed that the trace of a quantum observable A, in a spectral neighborhood of size O() of an energy E for the quantum Hamiltonian Hˆ , can be expressed in terms of averages of the classical observable A associated  over invariant sets for the flow of the classical Hamiltonian H associated with with A . This is related to the spectral asymptotics for Hˆ in the semi-classical limit, and it H can be understood as a “correspondence principle” between classical and quantum mechanics as Planck’s constant  goes to zero. More recently, papers by Guillemin–Uribe [126](1989), Paul–Uribe [203, 204] (1991, 1995), Meinrenken [194] (1992) and Dozias [87](1994) have developed the necessary tools from microlocal analysis [153] in a nonhomogeneous (semiclassical) setting to deal with Schrödinger-type Hamiltonians. Extensions and simplifications of these methods have been given by Petkov–Popov [210] and Charbonnel– Popov [50]. In this Chapter we show how to recover the semiclassical Gutzwiller Trace Formula from the coherent states method. The coherent states approach presented here seems particularly suitable when one wishes to compare the phase space quantum picture with the phase space classical flow. Furthermore, it avoids problems with caustics, and the Maslov indices appear naturally. In short, it implies the Gutzwiller trace formula in a very simple and transparent way, without any use of the global theory of Fourier integral operators. In their place we use the coherent states approximation (gaussian beams) and the stationary phase theorem. The use of gaussian wave packets is such a useful idea that one can trace it back to the very beginning of quantum mechanics, for instance, Schrödinger [233] (1926). However, the realization that these approximations are universally applicable, and that they are valid for arbitrarily long times, has developed gradually. In the mathematical literature these approximations have never become textbook material, and this has lead to their repeated rediscovery with a variety of different names, e.g. coherent states and gaussian beams. The first place that we have found where they are used in some generality is Babich [10] (1968) (see also [11]). Since then they have appeared, often as independent discoveries, in the work of Arnaud [7] (1973), Keller and Streifer [165] (1971), Heller [144, 145] (1975, 1987), Ralston [212, 213] (1976,1982), Hagedorn [130, 131] (1980-85), and Littlejohn [182] (1986)—and probably many more that we have not found. Their use in trace formulas was proposed (heuristically) by Wilkinson [262] (1987). The propagation formulas of [130, 131] were extended in Combescure–Robert [66], with a detailed estimate on the error both in time and in Planck’s constant . This propagation formula of coherent states which is described in Chap. 4 allows us to avoïd the whole machinery of Fourier Integral operator theory. The early application of these methods in [10] was for the construction of quasi-modes, and this has been pursued further in [212]

5.1 Introduction

131

and [203]. They have also been applied to the pointwise behaviour of semiclassical measures [206]. The proof we shall present here for the Gutzwiller trace formula is inspired from the paper by Combescure–Ralston–Robert [63].

5.2 The Semiclassical Gutzwiller Trace Formula We consider a quantum system in L 2 (Rn ) with the Schrödinger Hamiltonian  = −2 Δ + V (x), H

(5.5)

where Δ is the Laplacian in L 2 (Rn ) and V (x) a real, C ∞ (Rn ) potential. The corresponding Hamiltonian for the classical motion is H (q, p) = p 2 + V (q), and for a given energy E ∈ R we denote by Σ E the “energy shell”

Σ E := (q, p) ∈ R2n : H (q, p) = E .

(5.6)

 obtained by the -Weyl quantiMore generally we shall consider Hamiltonians H  = Opw (H ), where zation of the classical Hamiltonian H , so that H Opw (H )ψ(x)

= (2π )

−n



 H R2n

i(x−y)·ξ x+y , ξ ψ(y)e  dydξ 2

(5.7)

The Hamiltonian H is assumed to be a smooth, real valued function of z = (x, ξ ) ∈ R2n , and to satisfy the following global estimates • (H.0) there exist non-negative constants C, m, Cγ such that |∂zγ H (z)| ≤ Cγ < H (z) >, ∀z ∈ R2n , ∀γ ∈ N2n





< H (z) > ≤ C < H (z ) > · < z − z > , ∀z, z ∈ R m

2n

(5.8) (5.9)

where we have used the notation < u >= (1 + |u|2 )1/2 for u ∈ Rm . Remark 26 (i) H (q, p) = p 2 + V (q) satisfies (H.0), if V (q) is bounded below by some constant a > 0 and satisfies the property (H.0) in the variable q.  is essentially self(ii) The technical condition (H.0) implies in particular that H 2 n  adjoint on L (R ) for  small enough and that χ ( H ) is a -pseudodifferential operator if χ ∈ C0∞ (R) (see Chap. 2 and [141]) . Let us denote by φ t the classical flow induced by Hamilton’s equations with Hamiltonian H , and by S(q, p; t) the classical action along the trajectory starting at

132

5 Trace Formulas and Coherent States

(q, p) at time t = 0, and evolving during time t  S(q, p; t) =

t

( ps · q˙s − H (q, p)) ds,

(5.10)

0

where (qt , pt ) = φt (q, p), and dot denotes the derivative with respect to time. We shall also use the notation: αt = φ t (α) where α = (q, p) ∈ R2n , is a phase space point. Recall that the Hamiltonian H is constant along the flow φ t . If γ is a periodic trajectory parametrized as t → αt , αTγ∗ = α0 where Tγ∗ is the primitive period (the smallest positive period), the classical action along γ is  Sγ =

Tγ∗

dt pt q˙t :=

0

pdq. γ

An important role in what follows is played by the “linearized flow” around the classical trajectory, which is defined as follows. Let  ∂ 2 H  H (αt ) = ∂α 2 α=αt

(5.11)

be the Hessian of H at point αt = φ t (α) of the classical trajectory. Let J be the symplectic matrix   J = 0 1l − 1l 0

(5.12)

where 0 and 1l are respectively the null and identity n × n matrices. Let Ft be the 2n × 2n real symplectic matrix solution of the linear differential equation F˙t = J H (αt ) Ft ,

(5.13)



1l 0 F0 = = 1l. 0 1l Ft depends on α = (q, p), theinitial point for the classical trajectory, αt . Let γ be a closed orbit on E with period Tγ , and let us denote simply by Fγ the matrix Fγ = F(Tγ ). Fγ is usually called the “monodromy matrix” of the closed orbit γ . Of course, Fγ does depend on α, but its eigenvalues do not, since the monodromy matrix with a different initial point on γ is conjugate to Fγ . Fγ has 1 as eigenvalue of algebraic multiplicity at least equal to 2. In all that follows, we shall use the following definition Definition 13 We say that γ is a nondegenerate orbit if the eigenvalue 1 of Fγ has algebraic multiplicity 2. Let σ denote the usual symplectic form on R2n

5.2 The Semiclassical Gutzwiller Trace Formula

σ (α, α ) = p · q − p · q

133

α = (q, p); α = (q , p )

(5.14)

(· is the usual scalar product in Rn ). We denote by {α1 , α1 } a basis for the eigenspace of Fγ belonging to the eigenvalue 1, and by V its orthogonal complement in the sense of the symplectic form σ

V = α ∈ R2n : σ (α, α1 ) = σ (α, α1 ) = 0 .

(5.15)

Then, the restriction Pγ of Fγ to V is called the (linearized) “Poincaré map” for γ . In more general cases the Hamiltonian flow will contain manifolds of periodic orbits with the same energy. When this happens, the periodic orbits will necessarily be degenerate, but the techniques we use here can still apply. The precise hypothesis for this (“Hypothesis C”) will be given later. Following Duistermaat and Guillemin we call this a “clean intersection hypothesis”, it is more explicit than other versions of this assumption. Since the statement of the trace formula is simpler and more informative when one does assume that the periodic orbits are nondegenerate, we will give only that formula in this case. We  shall now assume the following. Let (Γ E )T be the set of all periodic orbits on E with periods Tγ , 0 < |Tγ | ≤ T (including repetitions of primitive orbits and assigning negative periods to primitive orbits traced in the opposite sense). • (H.1) There exists δ E > 0 such that H −1 ([E − δ E, E + δ E]) is a compact set of R2n and E is a noncritical value of H (i.e. H (z) = E ⇒ ∇ H (z) = 0). • (H.2) All γ in (Γ E )T are nondegenerate, i.e. 1 is not an eigenvalue for the corresponding “Poincaré map”, Pγ . In particular, this implies that for any T > 0, (Γ E )T is a discrete set, with periods −T ≤ Tγ1 < · · · < Tγ N ≤ T .  = Opw (A) be a quantum We can now state the Gutzwiller trace formula. Let A observable, such that A satisfies the following • (H.3) there exists δ ∈ R, Cγ > 0 (γ ∈ N2n ), such that |∂zγ A(z)| ≤ Cγ < H (z) >δ ∀z ∈ R2n , • (H.4) g is a C ∞ function whose Fourier transform g˜ is of compact support with Supp g˜ ⊂ [−T, T ] and let χ be a smooth function with a compact support contained in ]E − δ E, E + δ E[, equal to 1 in a neighborhood of E. Then the following “regularized density of states” ρ A (E) is well defined    ) Aχ  (H )g E − H . ρ A (E) = Tr χ ( H 

(5.16)

 is purely discrete in a neighborhood Note that (H.1) implies that the spectrum of H of E so that ρ A (E) is well defined. We have also, more explicitly, ρ A (E) =

 1≤ j≤N

 g

E − Ej 



ˆ j , ϕ j >, χ 2 (E j ) < Aϕ

(5.17)

134

5 Trace Formulas and Coherent States

where E 1 ≤ · · · ≤ E N are the eigenvalues of Hˆ in ]E − δ E, E + δ E[ (with multiplicities) and ϕ j is the corresponding eigenfunction ( Hˆ ϕ j = E j ϕ j ). Let us remark E−E here that the  scaling:  j is the right one to have a nice semiclassical limit. The first argument is that if n = 1 (and for integrable systems), in regular case, eigenvalues are given by Bohr–Sommerfeld formula [142] and their mutual distance is of order . The second argument is included in the following result [140, 211]. Under assumption (H.1) the Liouville measure d L E is well defined on the energy surface Σ E : dLE =

dΣ E |∇ H |

(dΣ E is the Euclidean measure on Σ E ).

Now we can state the Gutzwiller Trace Formula. Theorem 28 (Gutzwiller Trace Formula) Assume (H.0)–(H.2) are satisfied for H , (H.3) for A and (H.4) for g. Then the following asymptotic expansion holds true, modulo O(∞ ),   ρ A (E) ≡ (π )−n/2 g(0)−(n−1) A(α)dσ E (α) + ck ( g)k + ΣE

k≥−n+2

   n/2−1 i(Sγ /+σγ π/2)  g(Tγ )| det(1l − Pγ )|−1/2 (2π ) e γ ∈(Γ E )T

 j≥1

γ d j ( g ) j

Tγ∗

A(αs )ds +

0

⎫ ⎬

(5.18)

⎭

where Tγ∗ is the primitive period of γ , σγ is the Maslov index of γ ( σγ ∈ Z and is γ ˜ are distributions in g˜ with support in {0}, d j (g) ˜ are computed in the proof), ck (g) distributions in g˜ with support {Tγ }. Remark 27 We can include more general Hamiltonians depending explicitly in , K  H=  j H ( j) such that H (0) satisfies (H.0) and for j ≥ 1, j=1

|∂ γ H ( j) (z)| ≤ Cγ , j < H (0) (z) > .

(5.19)

It is useful for applications to consider Hamiltonians like H (0) + H (1) where H (1) may be, for example, a spin term. In that case the formula (5.18) is true with different coefficients.  the first term in the contribution of Tγ is multiplied by   T ∗ In particular γ (1) exp −i 0 H (αs )ds . Remark 28 For Schrödinger operators we only need smoothness of the potential V . In this case the trace formula (5.18) is still valid without any assumptions at

5.2 The Semiclassical Gutzwiller Trace Formula

135

infinity for V when we restrict ourselves to a compact energy surface, assuming E < lim inf |x|→∞ V (x). Using exponential decrease of the eigenfunctions [143] we can prove that, modulo an error term of order O(+∞ ), the potential V can be replaced by a potential V˜ satisfying the assumptions of the Remark (2.1). It is also possible to get a trace formula for Hamiltonians with symmetries [48].

5.3 Preparations for the Proof We shall make use of the standard coherent states introduced in Chap. 1 and their propagation by the time dependent Schrödiger equation established in Chap. 4. We denote by (5.20) ϕα = Tˆ (α)ψ0 the usual coherent states centered at the point α of the phase space R2n . Then it is known that any operator Bˆ with a symbol decreasing sufficiently rapidly is in trace class [217], and its trace can be computed by (see Chap. 1) Tr Bˆ = (2π )−n

 R2n

ˆ α > dα. < ϕα , Bϕ

(5.21)

The regularized density of states ρ A (E) can now be rewritten as ρ A (E) = (2π )

−n−1 −n



 R×R2n

χ U (t) ϕα > dtdα,  g(t) ei Et/ < ϕα , A

(5.22)

where U (t) is the quantum unitary group :  χ = χ ( H ) Aχ  (H ). U (t) = e−it H / , and A

(5.23)

Our strategy for computing the behavior of ρ A (E) as  goes to zero is first to compute the bracket Aχ U (t)ϕα >, (5.24) m(α, t) =< ϕα ,  where we drop the subscript χ in Aχ for simplicity. First of all we shall use Lemma 14 of Chap. 2, giving the action of an pseudodifferential operator on a Gaussian. Thus m(t, α) is a linear combination of terms like m γ (α, t) =< Ψγ ,α , U (t)ϕα > .

(5.25)

Now we compute U (t)ϕα , using the semiclassical propagation of coherent states result of Chap. 4. We recall that Ft is a time dependent symplectic matrix (Jacobi ˆ matrix) defined by the linear equation (5.13). R(F) denotes the metaplectic repre-

136

5 Trace Formulas and Coherent States

sentation of the linearized flow F (see Chap. 3), and the -dependent metaplectic representation is defined by −1 ˆ Rˆ  (F) = Λ R(F)Λ  .

(5.26)

We will also use the notation  t pt · q t − p · q . ps · q˙s ds − t H (α) − δ(α, t) = 2 0

(5.27)

From Chap. 4 we have the following propagation estimates in the L 2 -norm: for every N ∈ N and every T > 0 it exists C N ,T such that    U (t)ϕα − exp iδ(α, t) Tˆ (αt ) Rˆ  (Ft )Λ PN (x, Dx , t, )ψ 0  ≤ C N ,T  N ,   (5.28) 0 (x) = π −n/4 exp(−|x|2 /2), and PN (t, ) is the (, t)-dependent differential where ψ operator defined by 

PN (x, Dx , t, ) = 1l +

k/2− j pkwj (x, D, t)

(k, j)∈I N

with I N = {(k, j) ∈ N × N, 1 ≤ j ≤ 2N − 1, k ≥ 3 j, 1 ≤ k − 2 j < 2N }, (5.29) where the differential operators pk j (x, Dx , t) are products of j Weyl quantization of homogeous polynomials of degree ks with 1≤s≤ j ks = k (see [66] Theorem (3.5) and its proof). So that we get 0 = Q k j (x)ψ 0 (x), pkwj (x, Dx , t)ψ

(5.30)

where Q k j (x) is a polynomial (with coefficients depending on (α, t)) of degree k having the same parity as k. This is clear from the following facts: homogeneous polynomials have a definite parity, and Weyl quantization behaves well with respect to symmetries: O p w (A) commutes to the parity operator Σ f (x) = f (−x) if and only if A is an even symbol and anticommutes with Σ if and only if A is an odd 0 (x) is an even function. symbol) and ψ So we get iδ(t, α) ·  ( j,k)∈I N ;|γ |≤2N   ˆ t )ψ 0 , Tˆ (αt )Λ Q k, j R(F 0 + O( N ), · Tˆ (α)Λ Q γ ψ

m(α, t) =



ck, j,γ 

k+|γ | 2 −j



exp

(5.31)

5.3 Preparations for the Proof

137

where Q k, j respectively Q γ are polynomials in the x variable with the same parity as k respectively |γ |. This remark will be useful in proving that we have only entire powers in  in (5.18), even though half integer powers appear naturally in the asymptotic propagation of coherent states. By an easy computation we have  ˆ t )ψ 0 , Tˆ (αt )Λ Q k, j R(F 0 = Tˆ (α)Λ Q γ ψ     1 α − αt ˆ t )ψ 0 , Q k, j R(F 0 exp −i σ (α, αt ) Tˆ1 Qγ ψ √ 2  

where Tˆ1 (·) is the Weyl translation operator with  = 1. We set    α − αt ˆ ˆ   Q γ ψ0 , Q k, j R(Ft )ψ0 m k, j,γ (α, t) = T1 √     α − αt ˆ ˆ   m 0 (α, t) = T1 ψ0 , R(Ft )ψ0 . √ 

(5.32)

(5.33) (5.34)

We compute m 0 (α, t) first. We shall use the fact that the metaplectic group transforms Gaussian wave packets into Gaussian wave packets in a very explicit way. If we denote by At , Bt , Ct , Dt the four n × n matrices of the block form of Ft 

At Bt Ft = C t Dt

(5.35)

We have already seen in Chaps. 3 and 4, since Ft is symplectic, that Ut = At + i Bt is invertible. So we have defined Γt = Vt Ut−1 , where Vt = (Ct + i Dt ).

(5.36)

We have from our Chap. 3 ( see also [104], Chap. 4) π −n/2 · m 0 (α, t) = [det Ut ]−1/2 c q − qt i i (Γt + i1l)x · x − √ (x − ) exp · 2 2  Rn ×( p − pt + i(q − qt ))} d x 



(5.37)

But the integration in x is a Fourier transform of a Gaussian and can be performed (see in Appendix Sect. A.1). The complex matrix Γt + i1l is invertible and we have (Γt + i1l)−1 =

1l + Wt 2i

(5.38)

138

5 Trace Formulas and Coherent States

where use the following notations Wt = Z t Yt−1 , Z t = Ut + i Vt , Yt = Ut − i Vt .

(5.39)

It is clear that Y is invertible (see Chap. 3). So we get m 0 (α, t) = 2n π n/2 [det Yt Ut−1 ]−1/2 [det Ut ]−1/2 e  Ψ E (t,α) ∗ c i

−1/2

where [det Yt Ut−1 ]∗ by

−1/2

, [det Ut ]c

(5.40)

are defined below, the phase Ψ E (t, α) is given

Ψ E (t, α) = t (E − H (α)) +

1 2



t

σ (αs − α, α˙ s )ds +

0

i (1l + Wt )(α˘ − α˘ t ) · (α˘ − α˘ t ), 4

(5.41)

with α˘ = q + i p if α = (q, p). −1/2 In (5.40) we have a product of square root of determinant. [det Ut ]c is a branch for [det Ut ]−1/2 with the phase (or argument) obtained by continuity in time from Ut=0 = 1l. For a complex symmetric matrix M with definite-positive real part, −1/2 [det M]∗ is a branch for (det M)−1/2 with the phase obtained by continuity along a path joining M to M, the eigenvalues of M 1/2 having positive real part. Following carefully these phases will give the Maslov correction index. Remark 29 There is here a difference with the paper [63] were the phase was obtained before integration in y ∈ Rn , so computations here will be a little bit more natural and easier. The same phase Ψ E (t, α) appears when computing m k, j,γ (α, t) with non trivial amplitudes. Then the formula for the regularized density of states in (5.22) takes the form   i dt dαa(t, α, )e  Ψ E (t,α) (5.42) ρ A (E) = R

R2n

 where Ψ E is given in (5.41) and a(t, α, )  j∈N a j (t, α) j Our plan is to prove Theorem 28 by expanding (5.42) by the method of stationary phase. The necessary stationary phase lemma for complex phase functions can easily be derived from Theorem 7.7.5 in [153]. There is also an extended discussion of complex phase functions depending on parameters in [153] leading to Theorem 7.7.12, but the form of the stationary manifold here permits us to use the following result proved in Appendix A.2. Theorem 29 (stationary phase expansion) Let O ⊂ Rd be an open set, and let a, f ∈ C ∞ (O) with  f ≥ 0 in O and supp a ⊂ O. We define M = {x ∈ O,  f (x) = 0, f (x) = 0} ,

5.3 Preparations for the Proof

139

and assume that M is a smooth, compact and connected submanifold of Rd of dimension k such that for all x ∈ M the Hessian, f (x), of f is nondegenerate on the normal space N x to M at x.  Under the conditions above, the integral J (ω) = R d eiω f (x) a(x)d x has the following asymptotic expansion as ω → +∞, modulo O(ω−∞ ),  J (ω) ≡

2π ω

d−k 2 

c j ω− j .

j≥0

The coefficient c0 is given by c0 = e

iω f (m 0 )

    f (m)|Nm −1/2 det a(m)d VM (m), i M ∗

(5.43)

where d VM (m) is the canonical Euclidean volume in M, m 0 ∈ M is arbitrary, and −1/2 [det P]∗ denotes the product of the reciprocals of square roots of the eigenvalues of P chosen with positive real parts. Note that, since  f ≥ 0, the eigenvalues of f (m)|Nm lie in the closed right half plane. i

5.4 The Stationary Phase Computation In this section we compute the stationary phase expansion of (5.22) with phase Ψ E given by (5.41). Note that a(t, α, ) is actually, according to (5.31), a polynomial in 1/2 and −1/2 . Hence the stationary phase theorem (with  independent symbol a) applies to each coefficient of this polynomial. We need to compute the first and second order derivatives of Ψ E (t, α). Let us introduce the 2n × 2n complex symmetric matrix 

Wt =

Wt −i Wt . −i Wt −Wt



It is enough to compute first derivatives for Ψ E up to O(|αt − α|2 ). 1 1 ∂t Ψ E (t, α) ≡ E − H (α) − (αt − α) · J α˙ t + (W  − 1l)α˙ t · (αt − α). 2 2i (5.44) 1 1 ∂α Ψ E (t, α) ≡ (1l + F T )J (αt − α) + (F T − 1l)(W  − 1l)(αt − α). (5.45) 2 2i The critical set for the stationary phase theorem is defined as C E = {(α, t) ∈ R2n × R, (Ψ E (α, t)) = 0, ∂t Ψ E (t, α) = 0, ∂α Ψ E (t, α) = 0}.

140

5 Trace Formulas and Coherent States

We have seen in Chap. 3 that since F is symplectic, one has W ∗ W < 1l, so if (Ψ E (α, t)) = 0 then αt = α. Using (5.44) we get C E = {(α, t) ∈ R2n × R, H (α) = E; αt = α}. Hence (t, α) is a critical point means that α is on a periodic path of energy E, for the Hamiltonian H , and period t. The second derivatives of Ψ E restricted on C E can be computed as follows 1  (Wt − 1l)α˙ · α. ˙ (5.46) 2i 1  2 ∂t,α Ψ E (t, α) = −∂α H (α) + (FtT − 1l)(Wt − 1l)(α). ˙ (5.47) 2i 1 1  2 ∂α,α Ψ E (t, α) = (J Ft − FtT J ) + (FtT − 1l)(Wt − 1l)(Ft − 1l). (5.48) 2 2i 2 Ψ E (t, α) = ∂t,t

Let Ψ E (t0 , α0 ) be the Hessian matrix of Ψ E at point (t0 , α0 ) of C E . We have to compute the kernel of Ψ E (α0 , t0 ). Lemma 33 For every (t0 , α0 ) ∈ C E we have ker(Ψ E (t0 , α0 )) = {(τ, v) ∈ R × R2n , v · ∂α H = 0, (Ft0 − 1l)v + τ α˙ = 0}. (5.49) Proof Using the Taylor formula we have (ψ E (t, α)) =

1 Ψ E (t0 , α0 )(t − t0 , α − α0 ) · (t − t0 , α − α0 ) + O(|t − t0 |3 + |α − α0 |3 ). 2

From W ∗ W < 1 we get that, for some c > 0, (ψ E (t, α)) ≥ c|α − αt |2 .

(5.50)

(5.51)

Using that α − αt = (α − α0 ) + (α0 − α0,t0 ) + (α0,t0 − αt0 ) + (αt0 − αt ) we get α − αt = (1l − Ft0 )(α − α0 ) + (t0 − t)α˙ t + O(|t − t0 |2 + |α − α0 |2 ) Then from (5.50) and (5.51) we get, for some c > 0, |Ψ E (t0 , α0 )(t − t0 , α − α0 ) · (t − t0 , α − α0 )| ≥ c(Ft0 − 1l)(α − α0 ) + (t − t0 )α ˙ 2.

So we have proved the part “⊆” in (5.49). The part “⊇” is obvious.

(5.52) 

The first thing to check, in order to apply the stationary phase theorem is that the support of α in (5.42) can be taken as compact, up to an error O(∞ ). We do this in the following way : let us recall some properties of -pseudodifferential calculus proved in [83, 141]. The function m H (z) :=< H (z) > is a weight function. In [83]

5.4 The Stationary Phase Computation

141

it is proved that χ ( Hˆ ) = Hˆ χ where Hχ ∈ S(m −k H ), for every k (χ is like in (H.4)). More precisely, we have in the  asymptotic sense in S(m −k H ), Hχ =



Hχ j  j

j≥0

and support [Hχ, j ] is in a fixed compact set for every j (see (H.4) and [141] for the computations of Hχ, j ). Let us recall that the symbol space S(m H ) is equipped with the family of semi-norms  γ  ∂  −1  sup m H (z)  γ u(z) . ∂z 2n z∈R

Now we can prove the following lemma Lemma 34 There is a compact set K in R2n such that for m(α, t) :=< ϕα , Aˆ χ U (t)ϕα > we have  R2n /K

|m(α, t)|dα = O(+∞ )

uniformly in every bounded interval in t. Proof Let χ˜ ∈ C0∞ (]E − δ E, E + δ E[) such that χ˜ χ = χ . Using (H.3) and the composition rule for -pseudodifferential operators we can see that Aˆ χ ( Hˆ ) is bounded on L 2 (Rn ). So there exists a C > 0 such that |m(α, t)| ≤ C  χ˜ ( Hˆ )ϕα 2

.

But we can write ˜ Hˆ )2 ϕα , ϕα >  χ( ˜ Hˆ )ϕα 2 =< χ(

.

Let us introduce the Wigner function, wα , for ϕα (i.e. the Weyl symbol of the orthogonal projection on ϕα ). We have  < χ( ˜ Hˆ )2 ϕα , ϕα >= (π )−n Hχ 2 (z)wα (z)dz where wα (z) = (π )−n e−

|z−α|2 

.

Using remainder estimates from [141] we have, for every N large enough,

142

5 Trace Formulas and Coherent States

Hˆ χ 2 =



Hχ 2 , j  j +  N +1 R N ()

0≤ j≤N

where the following estimate in Hilbert-Schmidt norm holds sup  R N ()  H S < +∞ . 0 0 such that for every j, we have Supp[Hχ 2 , j ] ⊆ {z, |z| < R}. So the proof of the lemma follows from  2 −n  R N ()  H S = (2π )  R N ()ϕα 2 dα and from the elementary estimate, which holds for some C, c > 0,  |α−r |2 c e−  dzdα ≤ Ce−  . {|z|≤R,|α|≥R+1}

 From (5.49) we see that a sufficient condition to apply the stationary phase theorem is the following “clean intersection condition” for the Hamiltonian flow φ t . Clean Intersection Condition (CI): We assume that C E is a union of smooth compact connected components and on each component, the tangent space T(t0 ,α0 ) C E to C E at (t0 , α0 ) coïncides with the linear space {(τ, v) ∈ R × R2n , v · ∂α H = 0, (Ft0 1l)v + τ α˙ = 0}. So under condition (CI) the kernel of Ψ E (t0 , α0 ) coïncides with the tangent space T(t0 ,α0 ) (C E ). Hence Ψ E (t0 , α0 ) is non-degenerate on the normal space at C E on (t0 , α0 ) as it is required to apply the stationary phase theorem. At this point we have already proved that there exists an asymptotic expansion for the regularized density of states ρ A (E). A more difficult problem is to compute this asymptotics in general. The simpler case is the period 0 of the flow: C E = {0, α), H (α) = E}. Then the property (CI) is satisfied if E is non critical for H . Remark that 0 is not an accumulation point in the periods of classical paths. The Hessian matrix on C E is Ψ E (0, α) =



− 21 ∇ H i∇ H , −∇ H 0

where ∇ H = ∂α H . The normal space Nα to C E has the basis {(1, 0), (0, ∇ H )}. So the determinant of Ψ E (0, α) restricted to N(0,α) is ∇ H 4 . The stationary phase theorem gives us Proposition 48 Let g be such that g˜ is supported in ] − T0 , T0 [ where T0 > 0 and φ t has no periodic trajectory on Σ E with a period in ] − T0 , T0 [\{0}. Then we have

5.4 The Stationary Phase Computation

ρ A (E) = g(0)(2π ˜ )−n

 ΣE

143

χ (H α) A(α)dΣ E (α) 1−n + O(2−n ). ∇ H (α)

(5.53)

Moreover the asymptotics can be extended as a full asymptotics in . As an application of (5.53) we have the following Theorem 30 (Weyl asymptotic formula) Assume that H satisfies condition (H.0) (5.8). Consider λ1 < λ2 such that H −1 [λ1 − ε, λ2 + ε] is compact and are non critical values for H . Let N I be the number of eigenvalues E j () of Hˆ in I = [λ1 , λ2 ]. Then we have N I = (2π )−n Vol{α ∈ R2n , H (α) ∈ I } + O(1−n ).

(5.54)

Proof Using a partition of unity it is enough to consider 

σ (λ) =

χ (E j ())

E j ()≤λ

where χ is supported in a small neighborhood of λ1 or λ2 (between λ1 and λ2 we can apply the functional calculus to get an asymptotic expansion (see [141]). To prove an asymptotic expansion for σ (λ) we use (5.53) choosing A = χ (H ), g˜ even, g(0) ˜ = 1, g ≥ 0 and g(λ) ≥ δ0 for |λ| ≤ ε0 for some δ0 > 0, ε0 > 0. We have   μ−λ 1 σ (μ)dμ = σ (λ) − g    (5.55) (σ (λ) − σ (λ + τ )) g(τ )dτ. From (5.53) we have, after integration, −1



 g

 μ−λ σ (μ)dμ = (2π )−n dαχ (H (α)) + O(−n ).  H (α)≤λ

Using the following estimate, for some C > 0, |σ (λ + τ ) − σ (λ)| ≤ C(1 + |τ |)1−n , we get σ (λ) = (2π )

−n

 H (α)≤λ

dαχ (H (α)) + O(−n ),

then (5.54) follows. Now we prove (5.56). It is enough to consider the case τ ≥ 0. Suppose τ ≤ ε0 . Then

(5.56)

144

5 Trace Formulas and Coherent States

 δ0

λ+τ  λ

 σ (μ) ≤



μ−λ dμg 

= O(1−n ).

For τ = ε0 with  ∈ N, using the triangle inequality, we get |σ (λ + ε0 ) − σ (λ)| ≤ C1−n . Finally for ε0 < τ < ( + 1)ε0 using again the triangle inequality we get (5.56).  Remark 30 Assuming that the set of all periodical trajectories of H in Σ E is of Liouville-measure 0, it is possible to prove by the same method the following result. For every C > 0 we have  lim n−1 { j, E − C ≤ E j () ≤ E + C} =

0

ΣE

d L E =: L E (Σ E ).

(5.57)

This result was already proved in [140, 211] using Fourier Integral operators. Now we come to the proof of the Gutzwiller Trace Formula (5.18). Note that for isolated periodic orbits on Σ E the non-degenerate assumption is equivalent to the condition (CI). So it results from our discussion that in this case the Hessian matrix Ψ E at (t0 , α0 ), where γ is a periodic path with period t0 = kTγ∗ and α0 ∈ γ is non-degenerate on the normal space Nt0 ,α0 at C E . Here Nt0 ,α0 is the linear space {R(1, 0) + {(0, v), v ∈ R2n , σ (v, ∇ H ) = 0}. Our main problem is to (t0 , α0 ) of Ψ E (t0 , α0 ) to Nt0 ,α0 . We compute the determinant of the restriction Ψ E,⊥ 2n shall denote Πα˙ the orthogonal projection in R on J ∇ H (α0 ) := α˙ (tangent vector to γ ). It is convenient to introduce the notations i 1  (Wt0 − 1l), K = (FtT0 − 1l)(G + J ) + i J. 2 2

G=

Using that Ft0 is symplectic we have 2 Ψ E (t0 , α0 ) = K (Ft0 − 1l). ∂α,α

So we have Ψ E (t0 , α0 ) = i −1



G α˙ · α˙ K α˙ . K α˙ K (Ft0 − 1l)

(5.58)

This formula is general. Furthermore we have the very useful result Lemma 35 K is a 2n × 2n invertible matrix and we have  1 U − 1l −i(1l + U ) K −1 = − . 2 i1l + V −(1l + i V ) In particular we have

(5.59)

5.4 The Stationary Phase Computation

145

 −1 Y det K = (−1) det , 2 n

(5.60)

where U = Ut0 , V = Vt0 , Y = Yt0 . Proof We have, using definition of W , W − 1l −i(W − 1l) W − 1l + i J = −i(W + 1l) −(W + 1l   0 2i V 2V (U − i V )−1 = . 0 (U − i V )−1 −2iU −2U 



(5.61) (5.62)

After some algebraic computations, using in particular the symplectic relations, we find   0 −1l − i V i(1l + U ) (U − i V )−1 K = 0 (U − i V )−1 −i1l − V −1l + U 

So we get the Lemma.

Now we begin to use the non-degeneracy condition to compute the determinant of Ψ E,⊥ (t0 , α0 ). We have   det i −1 Ψ E,⊥ (t0 , α0 ) = i −1 det



d K α˙ . K α˙ K (Ft0 − 1l) + iΠα˙

(5.63)

where d = 21 (W  − 1l). Let us introduce now convenient coordinates. We define a Poincaré section S by the equation T (α) = 0 where T is a classical observable such that {T , H }(α) = 1, T (α0 ) = 0, T is defined in an open neighborhood V0 of α0 . The first return Poincaré map P(α) = φ T (α) (α) is defined in V0 ∩ S such that T (φ T (α) (α)) = 0 with T (α0 ) = t0 , T (α) is the first return time. In V0 we can define new symplectic coordinates: (e, τ,  ), where e = H (α), τ = T (α),  (α) ∈ R2(n−1) . The differential P (α0 ) of P at α0 is related with the stability matrix F = (∂α φ t0 )(α0 ): ˙ P (α0 )v = Fv − (F T ∇T · v)α.

(5.64)

For e near E, the Poincaré map Pe is defined in V0 ∩ S ∩ Σe into V1 ∩ S ∩ Σe , where V1 is a neighborhood of α0 , by P(α) = φ T (α) (α), T (α0 ) = t0 . It is a symplectic map and for e = E its differential Pγ is the restriction of P (α0 ) to Nγ := Tα0 (S ∩ Σ E ) (for more details we refer to [135]). Note that Nγ = {v ∈ R2n , v · ∇ H = v · ∇T = 0}. When the energy e is varying around E we have a smooth family of closed trajectories of period T (e) parametrized by α(e) ∈ V0 ∩ S such that α(E) = α0 and T (E) = t0 . T and α are smooth in e. This result is known as the cylinder Theorem

146

5 Trace Formulas and Coherent States

[135]. It is a consequence of the implicit function theorem applied to the equation φ T (e)(α(e)) = α(e). In particular we have ˙ (F − 1l)α (E) = T (E)α.

(5.65)

Note that α (E) · ∇ H = 1 so α (E) = 0 and the non-degeneracy assumption implies that {α, ˙ α (E)} is a basis for the generalized eigenspace E 1 for the eigenvalue 1 of F (we have a non trivial Jordan block if T (E) = 0). ˙ = σ (v, α (E)) = 0}. Let V be the symplectic orthogonal of E 1 : V = {v|σ (v, α) The restriction of F to V is the algebraic linear Poincaré map Pγ (al) . Using (5.64) we can easily proved that Pγ and Pγ (al) are conjugate: M Pγ = Pγ (al) where M is an invertible linear map from Nγ onto V . So we have det(Pγ − 1l) = det(Pγ (al) − 1l). In particular if γ is non-degenerate then Pγ − 1l is invertible. The strategy is to simplify as far as possible the r.h.s in (5.63). To simplify our discussion we shall assume that T (E) = 0. It is not a restriction because if T = 0 we can perturb a little F by F ε , ε > 0, such that F ε α˙ = F on V, F ε α˙ = α, F ε α (E) = α (E) + εα˙ The determinant we have to compute depends only on the symplectic map F, so we can compute with F ε α˙ and take the limit as ε → 0. The first step is to find v ∈ C2n such that ˙ (F − 1l)v + v · α˙ K −1 α˙ = α.

(5.66)

With this v := v0 we get    K α˙ d − v0 · K α˙ (t0 , α0 ) = i −1 det det i −1 Ψ E,⊥ 0 K (F − 1l) + iΠα˙ α) ˙ where Πα˙ = (v· α. ˙ |α| ˙ 2 A direct computation gives

i K −1 α˙ = − (F + 1l)∇ H, 2 so Eq. (5.66) is transformed in (F − 1l)v =

i v · α(F ˙ + 1l)∇ H. 2

Using (F − 1l)T ∇ H = 0 we have v · α˙ = 0, so we have to solve

(5.67)

5.4 The Stationary Phase Computation

147

(F − 1l)v0 = α. ˙

(5.68)

We are looking for v0 = λα˙ + μα (E) and we find 1 v0 = T (E)



α (E) · α˙ α˙ − α (E) . |α| ˙ 2

(5.69)

So our first simplification gives the expression   (t0 , α0 ) = i −1 (d − v0 K˙ α) ˙ det K det(F − 1l + i K −1 Πα˙ ) det i −1 Ψ E,⊥ For the first term we get d − v0 K˙ α˙ = −iv0 · J α˙ = i

˙ σ (α (E), α) T (E)

det K is already computed. We shall compute det(F − 1l + i K −1 Πα˙ ) in a symplectic ˙ v2 = ∇ H ·α1 (E) α (E) where σ (v2 j−1 , v2 j ) = basis {v1 , v2 , v3 , . . . , v2n } where v1 = α, 1 for 1 ≤ j ≤ n and σ (v j , vk ) = 0 if | j − k| = 1. In this basis we have K −1 Πα˙ v j =

v j · α˙ −1 K Πα˙ α˙ |α| ˙ 2

So combining with the first column we can eliminate the terms K −1 Πα˙ v j for j ≥ 2 and using that F − 1l is invertible on V we can assume that in the first column only the two first terms are not zero. Finally we have obtained ⎛

⎞ x1 δ 0 ⎠ = −δx2 det(Pγ − 1l). det F − 1l + i K −1 Πα˙ = det ⎝ x2 0 0 [Pγ − 1l] (5.70) We are left to compute x2 and δ. We have 



δ = σ ((F − 1l)v2 , v2 ) = −

T (E) σ (α, ˙ α (E)) (∇ H · α (E))2

(5.71)

1 x2 = − σ ((F + 1l)∇ H, v1 ) = |∇ H |2 . 2

(5.72)

(t0 , α0 )) = 2n ∇ H 2 det(Y )−1 det(Pγ − 1l). det(i −1 Ψ E,⊥

(5.73)

So we get

Using the expression (5.40) we find the leading term ρ1,γ (E) for the contribution of the periodic path γ in formula (5.18), assuming for simplicity that A = 1,

148

5 Trace Formulas and Coherent States −1/2

−1/2

ρ1,γ (E) = (2π )n/2−1 [det(Y U −1 )]∗ [det U ]c [det Y ]1/2  g(Tγ )[det(Pγ − 1l)]−1/2 eiψ E ∇ H −1

(5.74)

where [u]1/2 denotes a suitable branch for the square root. So we get g(Tγ )Tγ∗ | det(1l − Pγ )|−1/2 ρ1,γ (E) = (2π )n/2−1 ei(Sγ /+σγ π/2)

(5.75)

# with σγ ∈ Z and Sγ = γ pdq. Let us remark that, because Pγ is symplectic and 1 is not eigenvalue of Pγ then we have det(Pγ − 1l) = (−1)σ | det(Pγ − 1l)| where σ is the number of eigenvalues of Pγ smaller than 1. So we see that

eiσγ π/2 = ±eiσ π/2 .

(5.76)

Thus we get that the contribution of the Maslov index in Theorem 28 is to determine the sign in (5.76). We have given here an analytical method for its computation. We do not consider its geometrical interpretation (“Maslov cycle”) for which we refer to the literature on this subject [122, 177, 223] and references in these works. γ The other coefficients, d j are spectral invariants which have been studied by Guillemin and Zelditch. In principle we can compute them using this explicit approach. This completes the proof of Theorem 28.

5.5 A Pointwise Trace Formula and Quasi-Modes From the well known Bohr–Sommerfeld quantization rules it is believed that there exists strong connections between periodic trajectories of a classical system H and bound states of its quantization Hˆ . In this section we discuss some properties of localization for bound states or approximate bound states (quasimodes) near periodic trajectories in the simplest cases. More general results are proved in [206].

5.5.1 A Pointwise Trace Formula The idea of this formula have appeared in [206]. We give here a proof of the main result of [206] for Gaussian coherent states. We assume that properties (H.0) and (H.1) are satisfied. Consider the local density of states defined for every α ∈ R2n by   E − E j ()  2 χ (E j ()) ψα , ψ j  , g ρ E (α) ≡  j

5.5 A Pointwise Trace Formula and Quasi-Modes

149

where ψ j are the normalized eigenfunctions for Hˆ , Hˆ ψ j = E j ()ψ j . Theorem 31 The local density of states ρ E (α) has the following asymptotic behavior as  → 0,  1 k (g, α) 2 +k . (5.77) ρ E (α) ≡ k∈N

˜ Their expressions The coefficients k (g, α) are smooth in α and are distributions in g. depend on the behavior of the path t → φ t α. (i) If the path t → φ t α has no periodic point with period in suppg˜ then k (g, α) are distributions in g˜ supported in {0}. In particular the leading term is 1 1 n+1 g(0) ˜ 0 (g, α) = √ π − 2 ∇ H (α) 2

(5.78)

(ii) If t → φ t α has a primitive period T ∗ , k (g, α) are distributions in g˜ supported in {mT ∗ , m ∈ Z}. In particular the leading term is  1 n+1 ∗ 0 (g, α) = √ π − 2 g(mT ˜ )C(m) 2 m∈Z where

(5.79)

− 21   . C(m) = 1l − WmT ∗ α˙ · α˙

Recall that Wt depends on the monodromy matrix Ft . Proof As for the trace formula (5.18), we first give a time dependent formula for t ˆ ρ E (α) with the propagator U (t) = e−i  H . If Πz is the orthogonal projection on the coherent state ϕz we have $ $ ρ E (α) = Tr g

E − Hˆ 

%

%

χ ( Hˆ )Πα ,

by computing the trace on the basis ψ j . So we get ρ E (α) =

1 2π



iE ϕα , U (t)χ ( Hˆ )ϕα . dt g(t)e ˜ t

(5.80)

In (5.80) the integrand is the same as in the proof of the trace formula. The difference is that here we have only a time integration. So the stationary phase theorem is much simpler to apply: α is fixed such that H (α) = E, the critical set of the phase ψ E is defined by the equation φ t α = α so we have t = mT ∗ where T ∗ is the primitive period (T ∗ = 0 if α is not a periodic point of the flow). ψ¨ E (t, α) is here the second derivative in time of ψ E . So we have

150

5 Trace Formulas and Coherent States

ψ¨ E (t, α) =

i  (1l − Wt )α˙ · α. ˙ 2

For t = 0 we have ψ¨ E (α) = 2i ∇ H 2 . Using that ∇ H = 0, the stationary phase theorem gives the part (i) of the Theorem. For the periodic case we have to recall that Wt is a complex symmetric n × n matrix and that W ∗ W < 1. With this properties we have easily that for every T > 0 there exists cT > 0 such that ˙ 2 for t ∈ [−T, T ]. (1l − Wt )α˙ · α˙ ≥ ct α So the critical points t = mT ∗ , m ∈ Z, are non degenerate and the stationary phase theorem gives the part (ii) of the Theorem. 

5.5.2 Quasi-Modes and Bohr–Sommerfeld Quantization Rules Quasi-modes (or approximated eigenfunctions) can be considered in more general and more interesting cases (see [166, 206, 212, 213]) but for simplicity we shall consider here mainly the fully periodic case. In applications we can have to consider semi-classical H (z) = H0 (z) + H1 (z) + 2 R2 (, z). We always assume that (H.0) 131 and (H.1) 133 are satisfied for H0 , H1 ∈ S(m H0 ) and R(, ·) ∈ Ssc (m H0 ). This class of symbols is defined in Chap. 3. All our results are still valid for these -dependent Hamiltonians. We introduce the condition (H.P) For every E ∈ [E − , E + ], Σ E := H0−1 (E) is connected and the Hamiltonian flow Φ Ht 0 is periodic on Σ E with a period TE . Moreover the average of H1 along any closed trajectory is constant (depends only on the energy E). Remark 31 For n = 1 the periodicity condition is always satisfied. For d > 1 this condition is rather strong. Nevertheless it is satisfied for integrable systems and for systems with a large group of symmetries. Let us first recall a result in classical mechanics (Guillemin–Sternberg, [125]): Proposition 49 Let us assume that above conditions are satisfied. Let γ be a closed path of energy E and period TE . Then the action integral J (E) = γ pdq defines a function of E, C ∞ in ]E − , E + [ and such that J (E) = TE . In particular for one degree of freedom systems we have  J (E) =

dz. H0 (z)≤E

Now we can extend J to an increasing function on R, linear outside a neighborhood of I . Let us introduce the rescaled Hamiltonian Kˆ = (2π )−1 J ( Hˆ ). Using

5.5 A Pointwise Trace Formula and Quasi-Modes

151

properties concerning the functional calculus [141], we can see that Kˆ has all the properties of Hˆ and furthermore the Hamiltonian flow of its principal symbol K 0 has 1 J (E ± ). So a constant period 2π in ΣλK 0 = K 0−1 (λ) for λ ∈ [λ− , λ+ ] where λ± = 2π ˆ ˆ in what follows we replace H by K , its “energy renormalization”. Indeed, the map1 ping 2π J is a bijective correspondance between the spectrum of Hˆ in [E − , E + ] and 1 the spectrum of Kˆ in [λ− , λ+ ], including mutiplicities, such that λ j = 2π J (E j ). K0 Let us denote by a the average of  the action of a periodic path on Σλ and 1 by μ ∈ Z its Maslov index. (a = 2π γ pd x − 2π F). Under the above assumptions the following results were proved in [141], using semi-classical Fourier-integral operators and ideas introduced before by Colin de Verdière [55] and Weinstein [257].

5.5.2.1

Statements of Results Concerning Spectral Asymptotics

Theorem 32 [55, 141, 257] Here we assume that H = H0 (without subprincipal terms).There exist C0 > 0 and 1 > 0 such that spect( Kˆ )

' & Ik (), [λ− , λ+ ] ⊆

(5.81)

k∈Z

with Ik () = [−a + (k −

μ μ ) − C0 2 , −a + (k − ) + C0 2 ]. 4 4

for  ∈]0, 1 ]. Let us remark for  small enough, the intervals Ik () do not intersect and this theorem gives the usual Bohr–Sommerfeld quantization conditions for the energy spectrum, more explicitly, λk =

1 μ J (E k ) = (k − ) − a + O(2 ). 2π 4

Under a stronger assumption on the flow, it is possible to estimate the number of states in each cluster Ik (). (H.F) Φ Kt 0 has no fixed point in Σ FK 0 , ∀λ ∈ [λ− − ε, λ+ + ε] and ∀t ∈]0, 2π [. Let us denote by dk () the number of eigenvalues of Kˆ in the interval Ik (). Theorem 33 [52, 55, 142] Under the above assumptions, for  small enough and −m + (k − μ4 ) ∈ [λ− , λ+ ], we have : dk () ≡



Γ j (−a + (k −

j≥1

with Γ j ∈ C ∞ ([λ− , λ+ ]). In particular

μ )) j−d , 4

(5.82)

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5 Trace Formulas and Coherent States

Γ1 (λ) = (2π )−d

 Σλ

dνλ .

In the particular case n = 1 we have μ = 2 and a = min(H0 ) hence dk () = 1. Furthermore the Bohr–Sommerfeld conditions take the following more accurate form Theorem 34 [141] Let us assume n = 1 and m = 0. Then there exists a sequence f k ∈ C ∞ ([F− , F+ ]), for k ≥ 2, such that λ +



1 h k f k (λ ) = ( + ) + O(∞ ), 2 k≥2

(5.83)

for  ∈ Z such that ( + 21 ) ∈ [λ− , λ+ ]. In particular there exists gk ∈ C ∞ ([λ− , λ+ ]) such that  1 1 λ = ( + ) + h k gk (( + )) + O(∞ ), 2 2 k≥2

(5.84)

where  ∈ Z such that ( + 21 ) ∈ [F− , F+ ]. We can deduce from the above theorem and Taylor formula the Bohr–Sommerfeld quantization rules for the eigenvalues E n at all order in . Corollary 19 There exist λ → b(λ, ) and C ∞ functions b j defined on [λ− , λ+ ] b j (λ) j + O(∞ ) and the spectrum E n of Hˆ is given by such that b(λ, ) = j∈N

 1 E n = b (n + ),  + O(∞ ) , 2

(5.85)

for n such that (n + 21 ) ∈ [λ− , λ+ ]. In particular we have b0 (λ) = J −1 (2π λ) and b1 = 0 . When H −1 (I ) is not connected but such that the M connected components are mutually symmetric, under linear symplectic maps, then the above results still hold [141]. Remark 32 For n = 1, the methods usually used to prove existence of a complete asymptotic expansion for the eigenvalues of Hˆ are not suitable to compute the coefficients b j (λ) for j ≥ 2. This was done recently in [57] using the coefficients d jk appearing in the functional calculus.

5.5 A Pointwise Trace Formula and Quasi-Modes

5.5.2.2

153

A Proof of the Quantization Rules and Quasi-Modes

We shall give here a direct proof for the Bohr–Sommerfeld quantization rules by using coherent states, following [32]. A similar approach, with more restrictive assumptions, was considered before in [204]. The starting point is the following remark. Let r > 0 and suppose that there exists Cr such that for every  ∈]0, 1], there exist E ∈ R and ψ ∈ L 2 (Rd ), such that ( Hˆ − E)ψ ≤ Cr r , and lim inf ψ := c > 0. →0

(5.86)

If these conditions are satisfied, we shall say that Hˆ has a quasi-mode of energy E with an error O(r ). With quasi-modes we can find some points in the spectrum of Hˆ close to the energy E. More precisely, if δ > Ccr , the interval [E − δr , E + δr ] meets the point spectrum of Hˆ . This is easily proved by contradiction, using that Hˆ is self-adjoint. So if the spectrum of Hˆ is discrete in a neighborhood of E, then we know that Hˆ has at least one eigenvalue in [E − δr , E + δr ]. Let us assume that the Hamiltonian Hˆ satisfies conditions (H.0), (H.1), (H.P) 150. Using Proposition 49, we can assume that the Hamiltonian flow Φ Ht has a constant period 2π in H −1 ]E − − ε, E + + ε[, for some ε > 0. Following an old idea in quantum mechanics (A. Einstein), let us try to construct a quasimode for Hˆ with energies E () close to E ∈ [E − , E + ], related with a 2π periodic trajectory γ E ⊂ Σ EH0 , by the Ansatz  ψγ E =

2π

e

it E () 

U (t)ϕz dt,

(5.87)

0

where z ∈ γ E (ψγ E is a state living on γ E ). Let us introduce the real numbers 1 σ () = 2π 



2π

[q(t) ˙ p(t) − H0 (q(t), p(t))]dt +

0

μ , 4

where t → (q(t), p(t)) is a 2π -periodic trajectory γ E in H0 −1 (E), E ∈ [E − , E + ], μ is the Maslov index of γ . In order that the Ansatz (5.87) provides a good quasimode, we must first check that its mass is not too small. Proposition 50 Assume that 2π is the primitive period of γ E . Then there exists a real number m E > 0 such that ψγ E  = m E 1/4 + O(1/2 )

(5.88)

Proof Using the propagation of coherent states and the formula giving √ the action of metaplectic transformations on Gaussians, up to an error term O( ), we have

154

5 Trace Formulas and Coherent States

ψγ E  = (π ) 2

−d/2



2π 2π 0

0

e  Φ(t,s,x) (det(At + i Bt ))−1/2 · i

Rd

−1/2  dtdsd x, · det(As + i Bs )

where the phase Φ is 1 Φ(t, s, x) = (t − s)E + (δt − δs ) + (qs · ps − qt · pt ) + 2  1 x · ( pt − ps ) + Γt (x − qt ) · (x − qt ) − Γs (x − qs ) · (x − qs ) ). 2

(5.89)

Γt is the complex matrix defined in (5.36). Let us show that we can compute an asymptotics for ψγ E 2 with the stationary phase Theorem. Using that (Γt ) is positive non-degenerate, we find that (Φ(t, s, x)) ≥ 0, and {(Φ(t, s, x)) = 0} ⇔ {x = qt = qs }.

(5.90)

On the set {x = qt = qs } we have ∂x Φ(t, s, x) = pt − ps . So if {x = qt = qs } then we have t = s (2π is the primitive period of γ E ) and we get easily that ∂s Φ(t, s, x) = 0. In the variables (s, x) we have found that Φ(t, s, x) has one critical point: (s, x) = (2) Φ at (t, t, qt ). (t, qt ). Let us compute the hessian matrix ∂s,x (2) ∂s,x Φ(t, t, qt ) =

 −(Γt q˙t − p˙t ) · q˙t [Γt (q˙t − p˙t )]T . Γt (q˙t − p˙t ) 2iΓt .

(5.91)

To compute the determinant, we use the identity, for r ∈ C, u ∈ Cd , R ∈ G L(Cd ) 

r uT u R



 1 0 r − u T · R −1 u u T = . 0 R. −R −1 u 1l

(5.92)

Then we get 2 2det[−i∂s,x Φ(t, t, qt )] = Γt q˙t · q˙t + ((Γt ))−1 (Γt q˙t − p˙t ) · (Γt q˙t − p˙t )det[2Γt ]. (5.93)

2 Φ(t, t, qt )] = 0. But E is not critical, so (q˙t , p˙t ) = (0, 0) and we find that det[−i∂s,x The stationary phase Theorem (see Appendix), gives

√ ψγ E 2 = m E 2  + O()

(5.94)

with m 2E

=2

(d+1)/2

√ π



2π 0

2 |det(At + i Bt )|−1/2 |det[−i∂s,x Φ(t, t, qt )]|−1/2 dt. (5.95) 

5.5 A Pointwise Trace Formula and Quasi-Modes

155

We now give one formulation of the Bohr–Sommerfeld quantization rule. This is an alternative proof of Theorem 32 with quasi-modes but the localization of eigenvalues is a little less accurate. Theorem 35 Let us assume that the Hamiltonian Hˆ satisfies conditions (H.0) 131, (H.1) 133, (H.P) 150 with period TE = 2π and that 2π is a primitive period for a periodic trajectory γ E ⊆ Σ E . Then −1/4 ψγ E is a quasi-mode for Hˆ , with an error term O(7/4 ), if E satisfies the quantization condition: σ () :=

μ 1 +b+ 4 2π 

 γE

pdq ∈ Z.

(5.96)

 1 Moreover, the number a := 2π γ E pdq − E is constant on [E − , E + ]. Chosen C > 0 large enough and consider the intervals I (k, ) = [(−

μ μ − b + k) + a − C7/4 , (− − b + k) + a + C7/4 ], 4 4

where b is the average of the subprincipal term H1 along the flow. Then if I (k, ) ∩ [E − , E + ] = ∅ then Hˆ has at least one eigenvalue in I (k, ). Proof We are looking for an energy E  close to E such that Hˆ ψγ E = E  ψγ E + O(2 ). By integration by parts, we get Hˆ ψγ E = i



2π

it E 

∂t U (t)ϕz dt 0 2iπ E  = i e  U (2π )ϕz − ϕz + Eψγ E . e

(5.97)

Using the propagation of coherent states result obtained in Chap. 4 and the 2π periodicity of the flow we get that  2π

U (2π )ϕz = ei(μ/4+b) e  ( i

0

pqdt−2π ˙ E)

ϕz + O().

(5.98)

√ Notice here that the contribution in  vanishes. This can be checked. Using (5.97), (5.98) and (5.88) we get finally a quasi-mode with an error O(7/4 ) and quantization of a series of eigenvalues.  Remark 33 Later, as an application of the Herman–Kluk representation formula (Chap. 11), we shall prove that under the assumptions of Theorem 35 we have Spect[ Hˆ ] ∩ [E − , E + ] ⊆ ∪k∈Z I (k, ).

156

5 Trace Formulas and Coherent States

This is the clustering phenomenon for the eigenvalues when the Hamiltonian flow is periodic. On the contrary if there exits one non periodical trajectory of energy E ∈ [E − , E + ] then the clusters disappear. This will be proved also later. More accurate results on quasi-modes using coherent states are proved in particular in [212], [166] and [218].

Part II

Coherent States in Non Euclidian Geometries

Chapter 6

Quantization and Coherent States on the 2-Torus

Abstract The two dimensional torus T2 is a very simple symplectic space. Nevertheless it gives non trivial examples of chaotic dynamical systems. These systems can be quantized in a natural way. We shall study some dynamical and spectral properties for them.

6.1 Introduction The 2-torus T2 , with its canonical symplectic form, is seen here as a phase space. It is useful to consider classical systems and quantum systems built on T2 for at least two purposes. Dynamical properties of classical non-integrable Hamiltonian systems in the phase space Rd × Rd (d ≥ 2), are quite difficult to study. In particular there are not so many explicit models of chaotic systems. But on the 2-torus it is very easy to get a discrete chaotic system by considering a 2 × 2 matrix F with entries in Z and such that |Tr F| > 2. So we get a discrete flow t → F t z for t ∈ Z, z ∈ T2 ,1 where T2 = R2 /Z2 is the 2-dimensional torus. ˆ In 1980 Hannay–Berry succeeded to construct a “good quantization” R(F) cor2 responding to the “classical system” (T , F). From this starting point many results were obtained concerning consequences of classical chaos on the behavior of the ˆ eigenstates of the unitary family of operators R(F) as well by physicists and mathematicians. In this section we shall explain some of these results and their relationship with periodic coherent states.

1 For

t ∈ N, F t z means that we apply F t-times starting from z and if t > 0 then F t = (F −t )−1 .

© Springer Nature Switzerland AG 2021 D. Robert and M. Combescure, Coherent States and Applications in Mathematical Physics, Theoretical and Mathematical Physics, https://doi.org/10.1007/978-3-030-70845-0_6

159

160

6 Quantization and Coherent States on the 2-Torus

6.2 The Automorphisms of the 2-Torus We have already seen in Chap. 1 that R2 is a symplectic linear space with the canonical symplectic bilinear form σ = dq ∧ dp. T2 is also a symplectic (compact) manifold with the symplectic two-form σ = dq ∧ dp identified with the plane Lebesgue measure. Here we call automorphism of the 2-torus T2 any map F induced by a symplectic matrix F ∈ S L(2, Z). Let F be of the form   ab F= , (6.1) cd with entries in Z safisfying det(F) = ad − bc = 1; the corresponding map of the 2-torus is given by (q, p) ∈ T2  −→ (q  , p ) ∈ T2 , with q  = aq + bp (mod 1),

p  = cq + d p (mod 1)

So F is a symplectic diffeomorphism of T2 . In particular it preserves the Lebesgue measure m L on T2 . We shall consider now the discrete dynamical system in T2 generated by F. Let us first recall the definitions and properties concerning classical chaos (ergodicity, mixing). For more details we refer to the books [69, 163, 187]. A (discrete) dynamical system is a triplet (X, Φ, m) where X is a measurable space, m a probability measure on X and Φ a measurable map on X preserving the measure m: For any mesurable set E ⊂ M one has m(Φ −1 E) = m(E). The orbit (or trajectory) of a point x ∈ X is O(x) := {Φ k (x), k ∈ Z}. The orbit is periodic if Φ T (x) = x for some T ∈ Z, T = 0. Definition 14 For a dynamical system  D = (X, Φ, m) let us consider the time avert age or Birkhoff average E T ( f, x) = T1 t=T t=0 f (Φ (x)), where f is measurable. 1 Φ is ergodic if for any function f ∈ L (X, m) one has lim E T ( f, x) = m( f ) m − everywhere

T →∞

where m( f ) :=

 X

(6.2)

f dm is the spatial average.

Remark 34 If a dynamical system is ergodic its time average (in the sense of the left hand side of (6.2) equals the “space average”, and doesn’t depend on the initial point x ∈ X almost surely. Without dynamical assumptions we have the following Birkhoff Ergodic Theorem (for a proof see [252]). Theorem 36 For any f ∈ L 1 (X, m), for m-almost x ∈ X , the following time average exists:

6.2 The Automorphisms of the 2-Torus

161

lim E T ( f, x) = μxB ( f ).

T →+∞

(6.3)

Moreover f → μxB ( f ) defines a (Birkhoff) Φ-invariant measure on X . Proposition 51 A dynamical system D = (X, Φ, m) is ergodic if and only if one of the following statement is satisfied: (i) Any measurable set E ⊆ X which is Φ-invariant is such that m(E) = 0 or m(X \E) = 0. (ii) If f ∈ L ∞ (X, m) is Φ-invariant ( f ◦ Φ = f ) then it is constant m-everywhere. See [252] for proofs. This means in particular that the periodic orbits of an ergodic dynamical system are rather “rare” from a measurable point of view: Proposition 52 Let Φ be a continuous map on a compact topological space X endowed with a probability measure m which is Φ-invariant and such that m(U ) > 0 for any open set U . If (X, Φ, m) is ergodic, then the set of periodic orbits in X is of measure 0. Although relatively rare, the periodic orbit have a strong importance in the framework of ergodic theory since they allow the construction of invariant measures in the following way. Let x ∈ X and let O(x) be a periodic orbit of period p(x) ∈ R. The following probability measure is clearly invariant: 1  δΦ k (x) p(x) k=1 p(x)

mx =

where δa is the Dirac distribution at point a ∈ R. Given a map Φ in X and m an invariant measure, it is not always true that m is the unique invariant measure. If it is the case the map Φ is said “uniquely ergodic” and the unique invariant measure is ergodic (see [163]). Well known examples are irrational rotations on the circle (or translations on the torus T1 ). If α is an irrational number, Φ(x) = x + α, mod.1 defines a unique ergodic transformation in T1 , the unique invariant measure is the Lebesgue measure (see [163]). In the topological framework one has a characterization of such maps [69]: Proposition 53 Let D = (X, Φ, m) be a dynamical system with X a compact metric space, and Φ a continuous map. D is uniquely ergodic if and only if ∀ f ∈ C (X ): lim 

k→∞

T −1 1  f ◦ Φ l − m( f )∞ = 0, T l=0

where  · ∞ is the norm of the uniform convergence.

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6 Quantization and Coherent States on the 2-Torus

There is a stronger property of dynamical systems which is the “mixing” property: Definition 15 A dynamical system D = (X, Φ, m) is said to be mixing if ∀ f, g ∈ L 2 (X, m) one has  lim f (Φ k (x))g(x)dm(x) = m( f )m(g). k→∞

X

The following result is useful and easy to prove. Proposition 54 D = (X, Φ, m) is mixing if and only if there exists a total set T in L 2 (X, m) such that for every f, g ∈ T we have  f (Φ k (x))g(x)dm(x) = m( f )m(g).

lim

k→∞

X

In the framework where X is a differentiable manifold, one can define the notion of Anosov system (see [163, 183]): Definition 16 A diffeomorphism Φ of a differentiable manifold M is Anosov if ∀x ∈ M, there exists a decomposition of the tangent space at x in direct sum of two subspaces E xu and E xs and constants K > 0, 0 < λ < 1 satisfying s , (Dx Φ)E xs = E Φ(x)

and (Dx Φ n )E xs  ≤ K λn ,

u (Dx Φ)E xu = E Φ(x)

(Dx Φ −n )E xu  ≤ K λn

∀x ∈ M, n ∈ N. We have the following useful stability result (see[183]). Theorem 37 Let M be a compact manifold and Φ an Anosov diffeomorphism on M. There exists ε > 0 small enough such that if Φ − Ψ C 1 (M) < ε then Ψ is an Anosov diffeomorphism on M, where Ψ C 1 (M) = sup (|Φ(x)| + Dx Φ(x)) . x∈M

Theorem 38 If Φ is a diffeomorphism Anosov on T2 then the dynamical system D = (T2 , Φ, μ) is mixing, μ being the normalized Lebesgue measure on T2 . Let F ∈ S L(2, Z). The hyperbolic automorphism of the 2-torus defined by F represents the simplest examples of hyperbolic dynamical systems when |Tr F| > 2. Namely if this is satisfied then F has two eigenvalues λ+ = λ > λ− = λ−1 with λ > 1. Denote Tx (T2 ) the tangent space at point x ∈ T2 , E x+ (resp. E x− ) the eigenspace associated to the eigenvalue λ (resp. λ−1 ) and Dx F : Tx (T2 ) −→ TF x (T2 ) the differential of F. One has

6.2 The Automorphisms of the 2-Torus

163

Dx F(v) = |λ|v if v ∈ E x+ , Dx F(v) = |λ−1 |v if v ∈ E x− , where  ·  is the norm associated to the Riemannian metric ds 2 = dq 2 + dp 2 on T2 . This proves that F is an Anosov diffeomorphism, and is therefore ergodic and mixing. We can also give a more direct proof that F is mixing using Proposition 54. Let us consider the total family in L 2 (T2 ), ek (x) = e2iπk·x , where k ∈ Z2 . We have 

 T2

ek (x)e (Φ n (x))dm(x) =

T2

ek (x)e(Φ T )n (x)dm(x).

If = 0, using that A has eigenvalues λ and λ−1 with λ > 1, we get that (Φ T )n

 is large for ±n large hence we get T2 ek (x)e (Φ n (x))dm(x) = 0. We can conclude using Proposition 54. One can easily identify the periodic points of F: Proposition 55 The periodic points (q, p) ∈ T2 of an hyperbolic automorphism of T2 are exactly points (q, p) such that (q, p) ∈ Q2 /Z2 . Proof Let A be an hyperbolic automorphism of T2 , and n ∈ N∗ . Then the finite set L n = ( nr , ns ), r, s = 1, . . . , n is invariant under A and so all elements of L n are periodic for A. Let m = n ∈ N∗ . Since one has L m,n =

s , r = 1, . . . , m, s = 1, . . . , n ⊂ L mn m n

 r

,

all points of L m,n are also periodic. Thus all points in Q2 /Z2 are periodic for A. No other point can be periodic. Namely a point (q, p) ∈ T2 is periodic of period k ∈ N∗ if and only if there exists (m, n) ∈ Z2 such that (Ak − 1l)

    q m = . p n

But the matrix Ak − 1l is invertible and has only rational entries. Thus       q m r = (Ak − 1l)−1 = k , p sk n with (rk , sk ) ∈ Q2 . This completes the proof.



164

6 Quantization and Coherent States on the 2-Torus

Remark 35 An hyperbolic automorphism of T2 is always mixing (so ergodic) but never uniquely ergodic since every periodic point x gives an invariant probability measure m x .

6.3 The Kinematics Framework and Quantization We follow closely the approaches of [12, 34, 38, 39, 73, 137]. Let us recall that we consider as phase space the 2-torus T2 = R2 /Z2 with its canonical symplectic two form. Using the correspondence principle between classical and quantum mechanics, it seems natural to look for the quantum states ψ having the same periodicity in position and momentum (q, p) as the underlying classical system. The Weyl–Heisenberg translation operators Tˆ (q, p) “translate” the quantum state by a vector z = (q, p) ∈ R2 . So we are looking for some Hilbert space H , included in the Schwartz temperate distribution space S  (R), such that for every ψ ∈ H we have Tˆ (1, 0)ψ = e−iθ1 ψ, Tˆ (0, 1)ψ = eiθ2 ψ,

(6.4) (6.5)

where we allow a phase θ = (θ1 , θ2 ) since two wavefunctions ψ1 , ψ2 satisfying ψ2 = eiα ψ1 define the same quantum state and more importantly, we shall recover the plane model as θ runs over the square [0, 2π [×[0, 2π [. Equation (6.5) means that the -Fourier transform F ψ satisfies F ψ( p + 1) = e−iθ2 F ψ( p).

(6.6)

 Recall that F ψ( p) = (2π )−1/2 R e−iq p/ ψ(q)dq. From Eqs. (6.4), (6.5) we see that ψ must be a joint eigenvector for the Weyl– Heisenberg operators Tˆ (q, p) and we get Tˆ (0, 1)Tˆ (1, 0)ψ = Tˆ (1, 0)Tˆ (0, 1)ψ Since we have that

Tˆ (0, 1)Tˆ (1, 0) = ei/ Tˆ (1, 0)Tˆ (0, 1)

1 = N where conditions (6.4), (6.5) entail the following quantification condition 2π N ∈ N and  is the Planck constant. Moreover, the quantum states ψ live in a N dimensional complex vector space. This result can be obtained using the powerful methods of the geometric quantization [73]. Here we follow a more elementary approach as in [38, 39].

6.3 The Kinematics Framework and Quantization

165

Let us denote by H N (θ ) the linear space of temperate distributions ψ satisfying periodicity conditions (6.4), (6.5) with  = 2π1N (remark that if  = 2π1N and if ψ satisfies (6.4), (6.5) then ψ = 0). So in all this chapter it is assumed that  = 2π1N for some N ∈ N. Proposition 56 H N (θ ) is an N dimensional complex linear sub-space of the temperate distribution space S  (R). Proof Let ψ ∈ H N (θ ). From condition (6.5) we get that the support of ψ is in the 2 , j ∈ Z}. So ψ is a sum of derivatives of Dirac distributions discrete set {q j = 2π2πj+θ N  α (α) ψ = c j δq j . Using uniqueness of this decomposition we can prove that c(α) j =0  (0) for α = 0, so we have ψ = c j δq j where c j = c j . Now, using (6.4) we get a periodicity condition on the coefficient c j . So we have 

c j+N = e c j , ∀ j ∈ Z and ψ = iθ1

cj

0≤ j≤N −1

 

 e

ikθ1

δq j +k .

(6.7)

k∈Z

Conversely it is easy to see that if ψ satisfies (6.7) then ψ ∈ H N (θ ). So the proposition is proven.  From the proof of the proposition, we get a basis of H N (θ ): −1/2 e(θ) j = N



eikθ1 δq j +k , 0 ≤ j ≤ N − 1.

k∈Z

e(θ) j obviously satisfies (6.4). Let us check that it satisfies (6.6) by computing its Fourier transform. As a consequence of the usual Poisson formula: 

e2iπkx =



δ (x),

∈Z

k∈Z

we get after some easy computations −1 −2iπ p( j+θ2 /2π) F e(θ) e j ( p) = N



∈Z

Let us introduce p =

N

+

θ1 2π

and ε (θ) = N −1/2



δ + θ1 . N

(6.8)

2π

e−ikθ2 δ + θ1 +k , for = 0, . . . , N

2π

k∈Z

N − 1. We have now F e(θ) j =

 0≤ ≤N −1

where the matrix element F j, is given by

F j, ε (θ)

(6.9)

166

6 Quantization and Coherent States on the 2-Torus

   i θ1 θ2 2π j + θ2 + θ1 j + . F j, = N −1/2 exp − N 2π We put on H N (θ ) the unique Hilbert space structure such that {e(θ) j }0≤ j≤N −1 is an orthonormal basis. So we see that F is a unitary transformation from H N (θ1 , θ2 ) onto H N (−θ2 , θ1 ). In particular if θ = (0, 0), the matrix {F j, } is the matrix of the discrete Fourier transform. For all ψ ∈ H(θ1 ,θ2 ) we have ψ=

N −1 

c j (ψ)e(θ) j .

j=0

Then the vector

  N −1 c j (ψ) j=0

is interpreted physically as the quantum state of the particle in the position representation. Similarly in the momentum representation one has that ψ˜ θ ∈ H N (−θ2 , θ1 ) is decomposed as N −1  θ ˜ ψ = d j (ψ)ε(θ) j . j=0

One goes from the position to the momentum representation via a generalized Discrete Fourier Transform:      N −1 1  θ1 θ2 θ1 2π k ˆ dk (ψ) = exp −i +k + ) . c j (ψ) exp −i j ( √ N 2π N N N j=0 For θ = (0, 0) we recognize the discrete Fourier operator that we have introduced above. A convenient representation formula for elements of H N (θ ) can be obtained using the following symmetrization operator Σ N(θ) =



(−1) N z1 z2 ei(θ1 z1 −θ2 z2 ) Tˆ (z).

(6.10)

z∈Z2

Let us remark that ψ ∈ H N (θ ) if and only if ψ ∈ S  (R) satisfies Tˆ (z)ψ = (−1) N z1 z2 eiσ ((θ2 ,θ1 ),(z1 ,z2 )) ψ.

(6.11)

Proposition 57 Σ N(θ) defines a linear continuous map from S (R) in S  (R). Its range is H N (θ ). Moreover for every ψ ∈ S (R) we have

6.3 The Kinematics Framework and Quantization (θ) −1/2 e(θ) j , Σ N ψ = N



167

ei θ1 ψ(q j − ) = e(θ) j (ψ)

(6.12)

∈Z



and

R

|ψ(x)|2 d x =

1 4π 2

 [0,2π[2

2 |e(θ) j (ψ)| dθ.

(6.13)

2 The map ψ → {e(θ) 0≤ j≤N −1 can be extended as an isometry from L (R) onto j (ψ)}   dθ the Hilbert space L 2 [0, 2π [2 , C N , 4π 2 .

Proof Recall that we have Tˆ (z)ψ(x) = e−i z1 z2 /2 ei x z2 / ψ(x − z 1 ). So we have

Σ N(θ) ψ =



ei(θ1 z1 −θ2 z2 ) ei x z2 / ψ(x − z 1 ).

z 1 ,z 2 ∈Z

We first compute the z 2 -sum using the Poisson formula: 

ei(x z2 /−θ2 z2 ) =

z 2 ∈Z

we get Σ N(θ) ψ

1  δ k + θ2 N 2π N N k∈Z

   1  ikθ1  i θ1 = e e ψ(q j − ) δq j +k . N 0≤ j≤N −1

(6.14)

∈Z

k∈Z

The equalities (6.12) and (6.13) follow easily from (6.14). (θ) In particular we see that for every j = 0, . . . , N − 1 we have e(θ) j = ΣN ψ j , 1 1 where ψ j (x) = ψ0 (q j − x), ψ0 is C ∞ , with support in [− 4N , 4N ] and ψ0 (0) = 1. (θ) This proved that Σ N (S (R)) = H N (θ ). Let us define the map I (ψ) = {e(θ) j (ψ)}0≤ j≤N −1 . We know that I defines an dθ isometry from L 2 (R) into L 2 ([0, 2π [2 , C N , 4π 2 ). We have to prove now that I is onto. It to prove that the conjugate operator I ∗ is injective on  is enough 2 2 N dθ L [0, 2π [ , C , 4π 2 . To do that we have to compute I ∗ f, ψ where f = ( f 0 , . . . , f N −1 ), f j are periodical functions on the lattice 2π Z × 2π Z and ψ ∈ S (R). This is an exercise left to the reader.  This leads to a direct integral decomposition of L 2 (R): L 2 (R) ∼ =



1 2π

2 

2π 2π 0

0

dθ H N (θ ),

168

6 Quantization and Coherent States on the 2-Torus

ψ∼ =



1 2π

2 

2π 2π

0

0

dθ ψ(θ ), where ψ(θ ) = Σ N(θ) ψ.

This is a Bloch decomposition of L 2 (R) analogous to the description of electrons in a periodic structure. It appears that H N (θ ) is equipped with the natural inner product and the spaces H N (θ ) are the natural quantum Hilbert spaces of states having the torus as phase space. Let us explain now in more details the identification L 2 (R) ∼ =



1 2π

2 

2π 2π 0

dθ H N (θ ).

0

2 ˜ For every ψ ∈ S (R) we define ψ(θ, j) = e(θ) j (ψ) where θ ∈ [0, 2π [ and j ∈ Z. We have seen that ψ → ψ˜ is an isometry from L 2 (R) onto L 2 ([0, 2π [2 ×(Z/N Z), dθ ⊗ dμ N ) where μ N is the uniform probability on Z/N Z. 4π 2 Let Aˆ be some bounded operator in L 2 (R). Assume that Aˆ is a linear continuous operator from S (R) to S (R) and from S  (R) to S  (R) and that Aˆ commutes with ˆ N(θ) = Σ N(θ) A), ˆ for every θ ∈ [0, 2π [2 . Then Aˆ is a decomposable operator Σ N(θ) ( AΣ (see Reed-Simon [215], t.1, p. 281). More precisely we have the following useful result.

Proposition 58 Let us denote by Aˆ N ,θ the restriction of Aˆ to H N (θ ). Then we have, for every ψ1 , ψ2 ∈ L 2 (R), ˆ 1  L 2 (R) = ψ2 , Aψ

 [0,2π[2

ψ2 (θ ), Aˆ N ,θ ψ1 (θ )H N (θ)

dθ , 4π 2

(6.15)

where ψ1 (θ ) = Σ N(θ) ψ1 . Proof This is easily proved using that ψ(θ ) =



˜ and ψ(θ, j)e(θ) j

0≤ j≤N −1

(6.13).



We shall apply the following results proved using Reed-Simon, [215]. Corollary 20 Let Aˆ be a decomposable operator like above. Then we have ˆ L 2 (R) =  A

sup  Aˆ N ,θ H

θ∈[0,2π[2

(θ ) N

(6.16)

and Aˆ is an isometry in L 2 (R) if and only if Aˆ N ,θ is an isometry in H N (θ ) for every θ ∈ [0, 2π [2 . First examples are the Weyl–Heisenberg translations.

6.3 The Kinematics Framework and Quantization

169

Lemma 36 Let z = (z 1 , z 2 ) ∈ R2 . Then Tˆ (z)Σ N(θ) = Σ N(θ) Tˆ (z) if and only if N z ∈ Z2 . Moreover if z 1 = nN1 and z 2 = nN2 we have  n n  n n n 2 (θ) 1 2 1 2 e(θ) exp i(θ e j+n 1 . = exp iπ + 2π j) TˆN ,θ , 2 j N N N N

(6.17)

Proof Exercise. Corollary 21 The unitary (projective) representation (n 1 , n 2 ) → TˆN ,θ ( nN1 , nN2 ) of the group Z2 in H N(θ) is irreducible.  a j e(θ) Proof Let V be an invariant subspace of H N(θ) and v = j , v = 0. 0≤ j≤N −1

If a j = 0 for j = j0 , then using the translation TˆN ,θ ( nN1 , 0) we get that e(θ) j ∈ V n2 ˆ ∀ j. But playing with TN ,θ (0, N ), if m coefficients a j are not 0 there exists a non zero vector of V with m − 1 non null coefficients. So we can conclude that  V = H N (θ ). Remark 36 It has been proved that all irreducible unitary representations of the discrete Heisenberg group are equivalent to (TˆN ,θ ( nN1 , nN2 ), H N (θ )), for some (N , θ ) ∈ N∗ × [0, 2π [2 [77]. For the particular case θ = (0, 0), the states e0j can be identified with the natural basis in C N . Then the translation operators Tˆ (1/N , 0), Tˆ (0, 1/N ) are simply N × N matrices of the following form: Tˆ (0, 1/N ) := Z = diag(1, ω, ω2 , . . . , ω N −1 )

(6.18)

where ω = e2iπ/N is the primitive N th root of unity. ⎛ 0 ⎜1 ⎜ Tˆ (1/N , 0) := X = ⎝ . 0

0 0 . 0

... ... ... ...

0 0 . 1

⎞ 1 0⎟ ⎟. .⎠ 0

(6.19)

These operators (matrices) have been introduced by Schwinger [233] as “generalized Pauli matrices” and are intensively used in quantum information theory for the mutually unbiased bases problem in C N . See [62, 233]. They have the following properties: Proposition 59 (i) X and Z are unitary. (ii) They are idempotent, namely X N = Z N = 1l

170

6 Quantization and Coherent States on the 2-Torus

(the identity matrix in C N ). (iii) They ω-commute:

X Z = ωZ X

(iv) X is diagonalized by the discrete Fourier transform F : F ∗XF = Z where F j,k =

√1 ω jk , N

∀ j, k = 1, . . . , N

Remark 37 A complex N × N matrix is an Hadamard matrix if all its entries have equal modulus. Note that F is an unitary Hadamard matrix of the Vandermonde form, and that X and its powers generate the commutative algebra of the “circulant” matrices. An N × N matrix C is said to be circulant if all its rows and columns are successive circular permutations of the first: ⎛

c1 ⎜c N C = circ(c1 , c2 , . . . , c N ) = ⎜ ⎝ . c2

c2 c1 . c3

⎞ ... c N ... c N −1 ⎟ ⎟ ... . ⎠ ... c1

C = c1 1l + c N X + · · · + c2 X N −1 .

(6.20)

(see [72]) The Discrete Fourier transform in C N is very natural in this context since it transforms any basis vector in the position representation into any basis vector in the momentum representation. Lemma 37 For any circulant matrix C there exists a diagonal matrix D such that F ∗ CF = D. Furthermore D j, j =

√

N cˆ j =

N −1 

ck+1 ω− jk .

0

Proof Use Proposition (59) (iv) and (6.20).



These properties are very useful to construct the N + 1 mutually unbiased bases in Quantum Information Theory for N a prime number. See [62].

6.4 The Coherent States of the Torus

171

6.4 The Coherent States of the Torus Already used in the physical literature in [173] we introduce now the coherent states adapted to the torus structure of the phase-space. They will be the image by the periodisation operator Σ N(θ) of the usual Gaussian coherent states studied in Chap. 1. In dimension 1 one has, for z = (q, p) ∈ R2 . and γ ∈ C, γ > 0,  ϕγ ,z (x) =

γ π

1/4

  (x − q)2 iqp i x p + + iγ , exp − 2  2

(θ) ϕγ(θ) ,z = Σ N ϕγ ,z .

(6.21) (6.22)

It is easily seen from the definition properties of Σ N(θ) and the product rules for Tˆ (z) that  i (−1) N n 1 n 2 ei(θ1 n 1 −θ2 n 2 )+ 2 σ (n,z) Tˆ (n + z)ϕγ ,0 . (6.23) ϕγ(θ) ,z = n 1 ,n 2 ∈Z

For every z  = (z 1 , z 2 ) ∈ Z2 we get N z 1 z 2 i(z 2 θ2 −z 1 θ1 ) iπ N σ (z,z ) (θ) ϕγ(θ) e e ϕγ ,z . ,z+z  = (−1)  







(6.24)

(θ) Thus the states ϕγ(θ) ,z+z  , ϕγ ,z are equal modulo phase factor, so they describe the same physical system and we can identify them. Recall that σ is the symplectic form:

σ ((a, b), (c, d)) = ad − bc.  The set ϕγ(θ) ,z z∈T2 therefore constitutes a coherent states system adapted to the torus. In the basis e(θ) j of the position representation we have (θ)

(θ) c j (q, p) := e j , ϕq, p =



γ π

1/4

  i m qp  1 1 eiθ1 m e  (x j p) exp iγ (x mj − q)2 , √ e−i 2 2 N m∈Z

where x mj = Nj + 2πθ2N − m. Similarly we have in the momentum representation (for γ = i): dk (q, p) = ( with ξmk =

k N

+

(6.25)

  1 1/4 1 − i q p  −imθ2 + i qξmk −1 k  ) √ e 2 (ξm − p)2 , e exp π 2 N m∈Z

θ1 2π N

− m.

An important property which is inherited from the overcompleteness character of the set of coherent states in L 2 (Rn ) is that the ϕγ(θ) ,z z∈T2 form an overcomplete system of H N (θ ) with a resolution of the identity operator H N (θ) :

172

6 Quantization and Coherent States on the 2-Torus

Proposition 60 ∀θ ∈ [0, 2π )2 and ∀ = 1/2π N we have  1lH N (θ) =

T2

dqdp (θ) |ϕ ϕ (θ) | 2π  γ ,q, p γ ,q, p

(θ) where we use the bra-ket notation for the projector on the coherent state ϕq, p.

Proof For simplicity we assume γ = i. Since H N (θ ) is finite dimensional it is enough to prove that ∀( j, k) ∈ [0, N − 1]2 we have  T2

dqdp θ ϕ , eθ eθ , ϕ θ  = δ j,k . 2π  q, p k j q, p

Now using (6.25) together with Fubini’s Theorem we get 

dqd p

T2 2π 

c¯k (q, p)c j (q, p) =



 iθ1 (m−n) 1 d p exp (2πi( j − k − N (m − n)) p) m,n e 0  θ (x m − q)ϕ θ (x m − q). × 01 dq ϕ¯ 0,0 0,0 j j

(6.26)

If j = k then the first integral in the right hand side of (6.26) is zero except for m = n in which case we get 1. Thus we get  (q, p)∈T2

 dqdp |c j (q, p)|2 = 2π  m∈Z

 0

1

θ θ dqϕ0,0 (xmj − q)2 = ϕ0,0 2 = 1.

If j = k the same integral is zero since j − k ∈ N∗ .



We also get the Fourier-Bargmann transform ψ  of any state ψ ∈ H N (θ ) as ψ =

√

θ N ϕq, p , ψ.

√ ) is obviously The map W θ : ψ ∈ H N (θ ) −→ W θ ψ = N ψ  ∈ L 2 (T2 , dqdp 2π isometric. The quantity θ 2 H θ (q, p) = |ϕq, p , ψ| is called the Husimi function of ψ ∈ H N (θ ). We have as a corollary an analogous result as Proposition 6: Corollary 22 Let Aˆ θ ∈ L (Hθ ). Then Tr( Aˆ θ ) =

 T2

θ ˆ θ ϕq, p , Aθ ϕq, p 

dqdp . 2π 

6.4 The Coherent States of the Torus

173

We have the following very useful semiclassical result. Proposition 61 For every complex numbers γ , γ  with positive imaginary part we have: (i) There exist constants C > 0, c > 0 such that for any z, z  ∈ T2 , N ≥ 1, |ϕγθ  ,z  , ϕγθ ,z | ≤ C

√

 2

N e−d(z,z )

cN

,

(6.27)

where d(z, z  ) is the distance between z and z  on the torus T 2 . In particular for γ = γ  = i we can choose c = π . (ii) There exists c > 0 such that ∀θ ∈ [0, 2π )2 we have ∀z = (q, p) ∈ T2 ϕγθ ,z 2 = 1 + O(e−cN ). Proof For simplicity, let assume that γ = γ  = i. The proof is the same for arbitrary γ , γ . We recall that in the continuous case one has   |z  − z|2 |ϕz  , ϕz |2 = (π )−1 exp − 2 so that ϕz  = 1. Thus we shall prove that the analogous properties (i) and (ii) hold for the coherent states of the 2-torus but only in the semiclassical limit N → ∞. A weaker result is given in [38], here we shall give a different proof. We rewrite (6.24): for every z ∈ T 2 , m = (m 1 , m 2 ) ∈ Z2 , (θ) = eiπ N (σ (z,m)+m 1 m 2 ) ei(m 2 θ2 −m 1 θ1 ) eiπ N σ (z,z ) ϕz(θ) . ϕz+m 

(6.28)

Let us denote f z,z  (θ ) = ϕzθ , ϕzθ  and consider f z,z  as a periodic function in θ for the lattice (2π Z)2 . Its Fourier coefficient cm (z, z  ) can be computed using (6.28), 



dθ e−im·θ ϕzθ , ϕzθ  2 4π  dθ θ ϕzθ , ϕz+ = eiπ N (σ (z,m)+m 1 m 2 ) mˇ  2 2 4π [0,2π[

cm (z, z ) =

(6.29)

[0,2π[2

= eiπ N (σ (z,m)+m 1 m 2 ) ϕz  , ϕz+mˇ .

(6.30) (6.31)

where mˇ = (m 1 , −m 2 ). f z,z  being a smooth function in θ , we get | f z,z  (θ )| ≤

 m∈Z2

|cm (z, z  )|.

(6.32)

174

6 Quantization and Coherent States on the 2-Torus

  m| ˇ 2 . So But |cm (z, z  )| = (π )−1 exp − |z −z− 2 

|cm (z, z  )| = (π )−1

m∈Z2

Now we have

  ˇ 2 |z  − z − m| . exp − 2 2

 m∈Z

|z  − z − m| ˇ 2 ≥ |z  − z|2 + |m|2 − 2|m||z − z  |.

So we get  √ m∈Z2 ,|m|≥4 2

   ˇ 2 |z  − z − m|  2 2 ≤ e−|z−z | π N exp − e−|m| π N /2 . 2 2 m∈Z

So for every N ≥ 1 we get  √ m∈Z2 ,|m|≥4 2

  ˇ 2 |z  − z − m|  2 ≤ Ce−|z−z | π N . exp − 2

For the finite sum we have easily  √ m∈Z2 ,|m|≤4 2

  ˇ 2 |z  − z − m|  2 ≤ Ce−d(z,z ) π N . exp − 2

so we get (i). Concerning the proof with any γ , γ  , we have to use the inequality |ϕγ  ,z  , ϕγ ,z | ≤ C−1/2 e−c

|z−z  |2 

.

where C > 0, c > 0 depend on γ , γ  , but not in z, z  . The proof of (ii) uses the same method with z  = z. So we get 

| f z,z (θ ) − 1| ≤

|ϕz , ϕz+m |

m =(0,0)

and | f z,z (θ ) − 1| ≤

√

N



e−|m|

2

πN

.

m =(0,0)

So we have proved (ii).



6.5 The Weyl and Anti-Wick Quantizations on the 2-Torus

175

6.5 The Weyl and Anti-Wick Quantizations on the 2-Torus We will show how a phase space function (classical Hamiltonian) H ∈ C ∞ (T2 ) can be quantized as a selfadjoint operator in the Hilbert space H N (θ ). These functions have to be real.

6.5.1 The Weyl Quantization on the 2-Torus We identify the functions H with the functions C ∞ on R2 of period (1, 1) ∈ R2 . Then we have  Hm,n e2iπσ ((q, p),(m,n)) . H (q, p) = (m,n)∈Z2

So we define, following [78, 137]: Definition 17 OpW (H ) =



Hm,n Tˆ

m,n

m n , . N N

(6.33)

Recall that  = 2π1N , N ∈ N∗ . One has the following property: Proposition 62 Let θ ∈ [0, 2π )2 and  > 0. Then for any function H ∈ C ∞ (T2 ) one has OpW (H )H N (θ ) ⊆ H N (θ ). Proof This follows directly from the definition of H N (θ ) and from Tˆ (m, n)OpW (H )Tˆ (m, n)∗ = OpW (H ), i f m, n ∈ Z. In other words OpW (H ) commutes with Σ N(θ) .



W Thus we define the operator Op,θ (H ) ∈ L (Hθ ) as the restriction of (6.33) to H N (θ ). W (H ) is the fiber at θ of In the decomposition of L 2 (R) as a direct integral, Op,θ W Op (H ).  dθ W Op,θ (H ) 2 . (6.34) OpW (H ) = 4π [0,2π[2

We have also the following formula by restriction to H N (θ ), W (H ) = Op,θ

 n,m∈Z

Hn,m Tˆ

n m , . N N

(6.35)

176

6 Quantization and Coherent States on the 2-Torus

W In particular we see that the map H → Op,θ (H ) cannot be injective, so the Weyl W symbol H of Op,θ (H ) is not unique. It becomes unique by restricting to trigonometric polynomials symbols. Let us denote by T N the linear space spanned by πn,m (q, p) = e2iπ(nq−mp) , for n, m = 0, . . . , N − 1. Then we have W Proposition 63 Op,θ is a unitary map from T N (with the norm of L 2 ([0, 1]2 ) onto L (H N (θ )), equipped with its Hilbert-Schmidt norm. In particular we have, for every H, K ∈ T N ,  W W Tr(Op,θ (H )Op,θ (K )∗ ) = N H (z)K (z)dz. (6.36) T

2

Proof Let us recall the formula Tˆ



k

, N N



iπk /N i(θ2 +2π j) /N (θ) e(θ) e e j+k . j =e

(6.37)

Using that the discrete Fourier transform is unitary, we get       k ˆ k   ∗ ˆ = N δk,k  δ ,  . , , Tr T T N N N N

(6.38)

So the system {N −1/2 Tˆ ( Nk , N )}0≤k, ≤N −1 is an orthonormal basis in L (H N (θ )), equipped with its Hilbert-Schmidt norm and we get the proposition.  Corollary 23 Every linear operator Hˆ in H N (θ ) has a unique Weyl symbol H ∈ TN ,  H (z) = Hm,n e2iπσ (z,(m,n)) , 0≤m,n≤N −1

where

  m n ∗ , . Hm,n = N −1/2 Tr Hˆ Tˆ N N

A semiclassical result for the Weyl quantization is the following: Proposition 64 For all H ∈ C ∞ (T2 ) one has lim

N →∞

 1  W Tr Op,θ (H ) = N

 T2

dz H (z).

Proof Using the orthonormal position basis e(θ) j one has N −1  m n  W   eθj . Tr Op,θ (H ) = Hn,m eθj , Tˆ , N N j=0 m,n

(6.39)

6.5 The Weyl and Anti-Wick Quantizations on the 2-Torus

177

Now we use the property (6.37): N −1   1 W (H )) = 1 Tr(Op,θ Hn, N +k N N j,k=0 ,n∈Z   πn 2π n θ2 ( N + k) − iθ1 + i ( + j + k) eθj , eθj+k  × exp i N N 2π

=



Hn, N (−1) n e−i θ1 +i

,n

nθ2 N

N −1 1  i 2π n j e N . N

(6.40)

j=0

Thus we conclude  1 W Tr(Op,θ (H )) = H0,0 + H N ,n N eiσ (( ,n),(θ1 ,θ2 )) . N ∗

,n∈Z  The last term tends to 0 because of the regularity of H , and the first one is  T2 dz H (z), which completes the proof.

6.5.2 The Anti-Wick Quantization on the 2-Torus As in the continuous case (see Chap. 2) the Anti-Wick quantization is associated to the system of coherent states defined above on the torus T2 . θ Definition 18 Let H ∈ L ∞ (T2 ). Then ϕq, p being the system of coherent states defined in the previous section, we define

 AW (H ) Op,θ

:=

T2

H (z)|ϕzθ ϕzθ |

dz . 2π 

AW Remark 38 Note that Op,θ (H ) for H ∈ C ∞ (T2 ) is simply the restriction of AW Op (H ) (considered as an operator on S  (R)) to H N (θ ). Let us recall that we always assume 2π N = 1.

As for Weyl quantization, anti-Wick quantization on R2 and on T2 are related with a direct integral decomposition Proposition 65 Let H ∈ C ∞ (T2 ). Then we have the direct integral decomposition  OpAW (H ) =

[0,2π[2

AW Op,θ (H )

dθ (2π )2

(6.41)

178

6 Quantization and Coherent States on the 2-Torus

In particular we have the uniform norm estimate AW (H ) ≤ H ∞ . Op,θ

(6.42)

Proof Using periodicity of H and direct integral decomposition of ψ ∈ S (R) (ψ(θ ) = Σ N(θ) ψ), we get OpAW (H )ψ





=



[0,2π[2

n=(n 1 ,n 2 )∈Z2

T

2

θ H (z)ϕz+n , ψ(θ )ϕz+n dz

dθ . (6.43) 4π 2

Using periodicity in z of ϕz and ϕz+n = eiσ (n,z)/2 Tˆ (n)ϕz , we get  OpAW (H )ψ =

 [0,2π[2

T

2

H (z)ϕzθ , ψ(θ )ϕz+n dz

dθ . 4π 2

(6.44)

  AW So we have proved OpAW (H ) θ = Op,θ (H ).



Now we show a link between Anti-Wick quantization and the Husimi function: Proposition 66 One has for any H ∈ C ∞ (T2 ) and for any ψ ∈ Hθ that:  AW ψ, Op,θ H ψ = N

T2

dz H (z)Hψ (z).

And we have the following semiclassical limit: Proposition 67 For any z ∈ T2 , any θ ∈ [0; 2π [2 , and any H ∈ C ∞ (T2 ) we have: AW H ϕzθ  = H (z) lim ϕzθ , Op,θ

N →∞

Proof We denote z = (q, p) ∈ T2 and Bε (z) the ball of center z and radius ε and by Bεc (z) its complementary set. Take ε  0. We have: AW (H )ϕzθ  ϕzθ , Op,θ

 =

Bεc (z)

dz  H (z  )|ϕzθ , ϕzθ |2 + 2π 

 Bε (z)

dz  H (z  )|ϕzθ , ϕzθ |2 2π 

It is clear that the first term in the right hand side tends to 0 as  → 0 because of Proposition (61) (i). For the second term we denote g(z, z  ) = N |ϕzθ , ϕzθ |2 . We have:

6.5 The Weyl and Anti-Wick Quantizations on the 2-Torus

179

   

  dz  H (z  )g(z, z  ) − H (z) ≤ Bε (z)         dz |H (z ) − H (z)|g(z, z ) + |H (z)|  g(z, z  )dz  − 1 ≤ Bε (z) B (z) ε          ε∇ H ∞ g(z, z )dz + |H (z)|  dz g(z, z ) − 1 . (6.45) Bε (z)

Bε (z)

Using the resolution of identity we have that  Bε (z)

dz  g(z, z  ) = ϕz(θ) 2 −

 Bεc (z)

dz  g(z, z  ).

Using Proposition (61) (ii) the first term in the right hand side tends to 1 as N → ∞, and it is clear that the second is small as N → ∞. This completes the proof.  As in the continuous case the Weyl and Anti-Wick quantizations are equivalent in the semiclassical regime: Proposition 68 For any H ∈ C ∞ (T2 ) and any θ ∈ [0, 2π )2 we have W AW (H ) − Op,θ (H )L (H N (θ)) = O(N −1 ), as N → ∞. Op,θ

(6.46)

Proof This result follows from the similar one in the continuous case (see Chap. 2, Proposition 27) using the estimate W AW (H ) − Op,θ (H )L (H N (θ)) ≤ OpW (H ) − OpAW (H )L (L 2 (R) . Op,θ

(6.47) 

6.6 Quantum Dynamics and Exact Egorov’s Theorem 6.6.1 Quantization of SL(2, Z) We now consider a dynamics in phase space induced by symplectic transformations F ∈ S L(2, Z). It creates a discrete time evolution in T2 and the n-step evolution is provided by F n . One wants here to quantize F as a natural operator in H N (θ ). We have seen in Chap. 2 (2.41) that F is quantized in L (L 2 (R)) by the metaplectic ˆ transformation R(F). Let us recall the following property ∗ ˆ ˆ ˆ T (z) R(F) = Tˆ (F −1 z). R(F)

We shall see now how to associate to F an unitary operator in H N (θ ).

(6.48)

180

6 Quantization and Coherent States on the 2-Torus

Proposition 69 Let F ∈ S L(2, Z). Then for any θ ∈ [0, 2π )2 there exists θ  ∈ [0, 2π )2 such that ˆ R(F)H θ ⊆ Hθ  Furthermore θ  is defined as follows:         θ2 ab θ2 mod 2π =F + πN , θ1 θ1 cd mod 2π Moreover we have



(θ) (θ ) ˆ ˆ R(F)Σ N = Σ N R(F).

(6.49)

(6.50)

Proof We use here Proposition 57 and formula (6.10). From (6.48) we get, if ψ ∈ H N (θ ) and z = (z 1 , z 2 ) ∈ Z2 ,    ˆ ˆ ˆ Tˆ (z) R(F)ψ = R(F) Tˆ (F −1 (z))ψ = e−i(σ (z ,(θ2 ,θ1 ))+π N z1 z2 ) R(F)ψ,

where 

z =F

−1

(6.51)

     d −b z1 z1 z=  = z2 z2 −c a

We have σ (z  , (θ2 , θ1 )) = σ (z, F(θ2 , θ1 )) and z 1 z 2 = −cdz 12 + (ad + bc)z 1 z 2 − abz 22 But modulo 2 we have z 12 ≡ −z 1 , z 22 ≡ −z 2 , ad + bc ≡ 1. So we get, modulo 2π , σ (z  , (θ2 , θ1 )) + π N z 1 z 2 ≡ z 1 (dθ1 + cθ2 + π N cd) − z 2 (bθ1 + aθ2 − π N ab) + π N z 1 z 2 .  ˆ So we have R(F)H θ ⊂ Hθ  with θ given by (6.49). Moreover it is easy to check formula (6.50). 

Let us denote θ  := π F (θ ). So π F is a smooth map from the torus R2 /(2π Z)2 into itself. Remark 39 We can easily see that Rˆ θ (J ) = F ∗ for θ = (0, 0) in the basis of

N e(θ) , up to a phase. j j=1

Definition 19 For every F ∈ S L(2, Z), we shall denote Rˆ N ,θ (F) the restriction of ˆ R(F) to H N (θ ). It is the quantization of F in H N (θ ). Rˆ N ,θ (F) is a linear operator from H N (θ ) in H N (θ  ). Proposition 70 Rˆ N ,θ (F) is a one-to-one linear map from H N (θ ) in H N (θ  ). ˆ Furthermore we have the following relationship between Rˆ N ,θ (F) and R(F). ˆ R(F) =



⊕

[0,2π[2

VN ,θ Rˆ N ,θ (F)

dθ (2π )2

(6.52)

6.6 Quantum Dynamics and Exact Egorov’s Theorem

181 

where VN ,θ is the canonical isometry from H N (θ  ) onto H N (θ ) defined by VN ,θ e(θj ) = e(θ) j . In particular for every θ ∈ [0, 2π [2 , Rˆ N ,θ (F) is a unitary transformation from H N (θ ) onto H N (θ  ) ˆ Proof We know that R(F) is an isomorphism from S (R) onto S (R) and from S  (R) onto S  (R). Using that H N (θ ) is finite dimensionnal we get that Rˆ N ,θ (F) is a one-to-one linear map from H N (θ ) in H N (θ  ). We can easily check that VN ,θ Σ N(π F (θ)) = Σ N(θ) for every θ ∈ [0, 2π [2 . So using that π F is an aera preverving transformation we get, for every ψ, η ∈ S (R), ˆ η, R(F)ψ L 2 (R) =

1 4π 2

 [0,2π[2

η(θ ), VN ,θ Rˆ N ,θ ψ(θ )H N (θ) dθ.

So the proposition is proved using standard properties of direct integral decompositions for operators.  One has the following results concerning the interesting case θ  = θ . The proofs are left to the reader or see [38, 39, 137]. Proposition 71 Consider F ∈ S L(2, Z) with |Tr F| > 2. Then ∀N ∈ N∗ there exists θ ∈ [0, 2π )2 so that ˆ R(F)H N (θ ) ⊆ H N (θ ) where θ can be chosen independent of N if and only if F is of the form     even odd odd even , or odd even even odd The case |Tr F| = 3 is the only case where the choice of θ is unique with θ = (π, π ) for N odd and θ = (0, 0) for N even. Moreover in the case   even odd F= odd even the value θ = (0, 0) is a solution of the fixed point equation θ = π F (θ ). Remark 40 It has been shown in [78] that for F ∈ S L(2, Z) of the following form 

2g 1 F= 2g 2 − 1 2g



N −1 ˆ of the form the operator R(F) has matrix elements in the basis e(0) j j=0

  CN 2iπ 2 2 ˆ (g j − jk + gk ) R(F) j,k = √ exp N N

182

6 Quantization and Coherent States on the 2-Torus

where |C N | = 1, so that it is represented as a unitary Hadamard matrix. We don’t know at present whether this property is shared by more general maps F.

6.6.2 The Egorov Theorem is Exact ˆ As in the continuous case the Egorov theorem is exact since R(F) is the metaplectic representation of the linear symplectic map F: Theorem 39 For any H ∈ C ∞ (T2 ) one has W W (H ) Rˆ N ,θ (F) = Op,θ (H ◦ F) Rˆ N ,θ (F)∗ Op,θ

(6.53)

Proof By denoting Tˆθ (z) the restriction of Tˆ (z) to H N (θ ) one has 

W (H ) = Op,θ

Hm,n Tˆθ

m,n∈Z

m n , N N

So W (H ) Rˆ N ,θ (F) = Rˆ N ,θ (F)∗ Op,θ

 m,n

m n , Rˆ N ,θ (F). Hm,n Rˆ N ,θ (F)∗ Tˆθ N N

But we know that Rˆ N ,θ (F)∗ Tˆθ (z) Rˆ N ,θ (F) = Tˆθ (F −1 z), ∀z = (m/N , n/N ) We do the change of variables 

m n



= F −1

  m n

Then we get W (H ) Rˆ N ,θ (F) = Rˆ N ,θ (F)∗ Op,θ

 m  ,n  ∈Z

This completes the proof.

(H ◦ F)m  ,n  Tˆθ



m  n , N N

 W = Op,θ (H ◦ F)



6.6 Quantum Dynamics and Exact Egorov’s Theorem

183

6.6.3 Propagation of Coherent States As in the continuous case the quantum propagation of coherent states is explicit and “imitates” the classical evolution of phase space points. Here the phase space is the 2-torus and ∀z ∈ T2 the time evolution of the point z = (q, p) is given by       q q 1 =F z = , mod . p p 1 

We shall use “generalized coherent states” which are actually “squeezed states”. Take γ ∈ C with γ > 0. The normalized Gaussian ϕγ ∈ L 2 (R) were defined in 6.21.     γ 1/4 iγ x 2 . ϕγ (x) := exp π 2 Then the generalized coherent states in L 2 (R) are ϕγ ,z := Tˆ (z)ϕ γ .

(6.54)

The generalized coherent states on the 2-torus are as above (θ) ϕγ(θ) ,z = Σ N ϕγ ,z ∈ H N (θ ).

We take such coherent state as initial state and apply to it the quantum evolution operator Rˆ θ (F). One has the following result: Proposition 72 Let F ∈ S L(2, Z) be given by   ab F= . cd If π F (θ ) = θ , then Rˆ N ,θ (F)ϕγθ ,z = where F · γ =



|bγ + a| bγ + a

1/2

θ ϕ F·γ ,F z

dγ +c . bγ +a

(θ) 2 ˆ ˆ Proof We know that Σ N(θ) R(F) = R(F)Σ N . Let z = (q, p) ∈ T . Then we have

ˆ Rˆ N ,θ (F)ϕγθ ,z = Rˆ N ,θ (F)Σ N(θ) ϕγ ,z = Σ N(θ) R(F)ϕ γ ,z . The result follows from the propagation of coherent states by metaplectic transformations in the plane (see Chap. 3). 

184

6 Quantization and Coherent States on the 2-Torus

6.7 Equipartition of the Eigenfunctions of Quantized Ergodic Maps on the 2-Torus One of the simplest trace of the ergodicity of a map F on T2 in the quantum world is the equipartition of the eigenfunctions of Rˆ N ,θ (F) in the classical limit N → ∞. It has been established in the literature in different contexts: for the geodesic flow on a compact Riemannian manifold it was proven by [56, 232, 267]. For Hamiltonian flows in Rn it was established in [140], and for smooth convex ergodic billiards in [111]. For the case of the d-torus this problem has been investigated in [37]. Here we restrict ourselves on the case of the 2-torus.

2 Theorem 40 (Quantum ergodicity) Let F be an ergodic area preserving

on T ,

map and Rˆ N ,θ (F) ∈ U (Hθ ) its quantization where θ = π F (θ ). Denote by φ Nj j=1,...,N

the eigenfunctions of Rˆ N ,θ (F). Then there exists E(N ) ⊂ {1, . . . , N } satisfying #E(N ) =1 N →∞ N lim

such that ∀A ∈ C ∞ (T2 ) and all maps j : N ∈ N −→ j (N ) ∈ E(N ) we have:  W (A)φ Nj(N )  = lim φ Nj(N ) , Op,θ

N →∞

T2

A(z)dz ,

(6.55)

A(z)dz

(6.56)

 AW lim φ Nj(N ) , Op,θ (A)φ Nj(N )  N →∞

=

T2

uniformly with respect to the map j (N ). Remark 41 This Theorem says that the Wigner distribution and Husimi distribution (when divided by N ) converge in the sense of distributions to the Liouville distribution along subsequences of density one. We begin with a lemma: Lemma 38 Let us introduce the Radon probability measures μ Nj , μ¯ N as follows: AW μ Nj (A) = φ Nj , Op,θ (A)φ Nj , μ¯ N (A) =

N 1  N μ (A) N j=1 j

These measures are F-invariant. Because φ Nj are eigenstates for Rˆ N ,θ (F). One has ∀A ∈ C ∞ (T2 ):

6.7 Equipartition of the Eigenfunctions of Quantized Ergodic Maps on the 2-Torus

185

lim μ¯ N (A) = μ(A),

N →∞

where μ is the Liouville measure on the 2-torus T 2 . Proof We get that the measures μ Nj are F-invariant, modulo O(N −1 ), using Egorov theorem and that φ Nj are eigenstates for Rˆ N ,θ (F). Clearly we have 1 AW (A)) μ¯ N (A) = Tr(Op,θ N So we have

  1  AW  |μ¯ N (A) − μ(A)| =  Tr(Op,θ (A)) − μ(A) N

  1  AW W W ≤ Op,θ (A) − Op,θ (A)L (H θ ) +  Tr(Op,θ (A)) − μ(A) N We deduce the result using Propositions 64 and 68.



Remark 42 The Lemma is still true for the Schwartz distributions N W N N ν Nj (A) = φ Nj , Op,θ (A)φ Nj  and ν¯ N (A) = N1 j=1 ν j (A). The ν j are exactly Finvariant. Let us prove now Proposition 73 For every A ∈ C ∞ (T2 ) we have lim

N →+∞

1  |μ N (A) − μ(A)|2 = 0 N 0≤ j≤N j

(6.57)

Proof We can replace A by A − μ(A) and assume that μ(A) = 0. Define for n ∈ N∗ the “time-average” of A: An =

k=n 1 A ◦ Fk. n k=1

Using the Remark after Lemma 38, we can replace μ Nj by ν Nj . We have ν j (A) = ν j (An ) for every n ≥ 1. So we get using the Cauchy-Schwarz inequality and F invariance of ν Nj , |ν Nj (A)|2 = |OpW (An )φ Nj , φ Nj |2 ≤ OpW (An )φ Nj 2 ≤ OpW (An )∗ OpW (An )φ Nj , φ Nj  (6.58) But from the composition rule for  Weyl quantization (Chap. 2) we have

186

6 Quantization and Coherent States on the 2-Torus

OpW (An )∗ OpW (An ) = OpW (|An |2 ) + O



1 N

 (6.59)

For every n, consider the limit N → +∞. Using Lemma 38 we get  lim sup |μ Nj (A)|2 N →+∞

=

lim sup |ν Nj (A)|2 N →+∞

≤

T

2

|An |2 dμ

Using the ergodicity assumption and the Lebesgue dominated convergence theorem we have  |An |2 dμ = 0. lim n→+∞ T2

The limit (6.57) follows if μ(A) = 0.



Now, the Bienaymé-Tchebichev inequality gives the following result according which “almost-all” eigenstates are equidistributed on the torus. Proposition 74 For any H ∈ C ∞ (T2 ) and ∀ε > 0

lim



# j : |μ Nj (H ) − μ(H )| < ε

N →∞

N

= 1.

Along the same lines as in [140] one can conclude for the existence of a H independent set E(N ) such that the theorem holds true. Remark 43 A natural question is “is the quantum ergodic theorem true with E(N ) = N ” (unique quantum ergodicity)? The answer is negative. In [74] the following result is proved. Let C =  {τ1 , . . . , τ K }, K periodic orbits for F. Consider  the probability measure μC ,α = α j μτ j , where α j ∈ [0, 1] and α j = 1. 1≤ j≤K

Then there exists a sequence Nk → +∞ such that W (A)φ NNk  = lim φ NNk , Op,θ

k→∞

1 2

 T2

1≤ j≤K

1 A(z)dz + μC ,α (A). 2

(6.60)

This result shows that some eigenstates can concentrate along periodic orbits, this phenomenon is named scarring.

6.8 Spectral Analysis of Hamiltonian Perturbations The previous results can be extended to some perturbations of automorphisms of the torus T2 . Let H be a real periodic Hamiltonian, H ∈ C ∞ (T2 ) and F ∈ S L(2, Z). Let θ ∈ [0, 2π [2 be such that θ = π F (θ ). We consider here the following unitary operator in H N (θ ), where 2π N = 1,

6.8 Spectral Analysis of Hamiltonian Perturbations

187

 ε Uε = exp −i Opw,θ (H ) Rˆ N ,θ (F).  We shall see that if F is hyperbolic and ε small enough the quantum ergodic theorem is still true. To prepare the proof we begin by some useful properties concerning the propat ˆ t ˆ gators V (t) = e−i  H , VN ,θ (t) = e−i  HN ,θ . Let us introduce the following Hilbert spaces Ks = Dom( Hˆ osc + 1)s/2 , where 2 s ≥ 0, with the norm ψ2s = ( Hˆ osc + 1)s/2 ψ2 ( Hˆ osc = −2 ddx 2 + x 2 ). Lemma 39 For every s ≥ 0 and every T > 0, there exists Cs,T > 0 such that V (t)ψs ≤ Cs,T ψs , ∀ψ ∈ Ks , ∀t ∈ [−T, T ], ∀ ∈]0, 2π ]. Proof It is sufficient to assume that s ∈ N (using complex interpolation). For s = 0 we know that V (t) is unitary. Let us denote Λ = ( Hˆ osc + 1)1/2 . Let us assume s = 1. It is enough to prove that ΛV (t)Λ−1 is bounded from L 2 (R) into L 2 (R). We have  d V (−t)ΛV (t) = V (−t)[ Hˆ , Λ]V (t). i dt Using the semi-classical calculus (Chap. 2), and that H is periodic, we know that i ˆ [ H, Λ] is bounded on L 2 (R). So the lemma is proved for s = 1.  Now we will prove the result for every s ∈ N by induction. Assume the lemma is proved for k ≤ s − 1. Compute  d V (−t)Λs V (t) = V (t)[ Hˆ , Λs ]V (t). i dt But i [ Hˆ , Λs ] is an  pseudodifferential operator of order s − 1 for the weight μ(x, ξ ) = (1 + x 2 + ξ 2 )1/2 . In particular the operator i [ Hˆ , Λs ]Λ1−s is bounded  on L 2 (R). So we get the result for s using the induction assumption. The following result will be useful to transform properties from the space L 2 (R) to the spaces H N (θ ). ˜ ) = (ψ(θ, ˜ ˜ 0), . . . , ψ(θ, N− Let ψ ∈ S (R) and for θ ∈ [0, 2π [2 be such that ψ(θ N 1) ∈ C (coefficient of ψ(θ ) in the canonical basis of H N (θ )). Let us denote HNs ([0, 2π [2 ) the periodic Sobolev space of order s ≥ 0 of functions from [0, 2π [2 into C N Its norm is denoted  ·  N ,s . Lemma 40 For every s ≥ 0 there exists Cs such that ˜ N ,s ≤ Cs ψs , ∀ψ ∈ S (R). ψ

(6.61)

In particular we have the following pointwise estimate: for every s > 1 there exists Cs such that ˜ )| ≤ Cs ψs , ∀ψ ∈ Ks . (6.62) |ψ(θ

188

6 Quantization and Coherent States on the 2-Torus

Proof Let us recall that, for every ψ ∈ S (R), θ = (θ1 , θ2 ) ∈]0, 2π ]2 , ˜ ψ(θ, j) = N −1/2



ei θ1 ψ(q j − ), q j =

∈Z

θ2 j + . N 2π N

So we have ˜ ∂θ1 ψ(θ, j) = N −1/2



˜ j), (6.63) ei θ1 i( − q j )ψ(q j − ) + i N −1/2 q j ψ(θ,

∈Z

˜ ∂θ2 ψ(θ, j) = N

−1/2



ei θ1 i

∈Z

d ψ(q j − ). dx

(6.64)

Reasonning by induction on |m| = m 1 + m 2 , m = (m 1 , m 2 ), we get easily  [0,2π[2

˜ |∂θm ψ(θ,

j)| dθ ≤ Cm 2

  k+ ≤|m| R

(|(∂x )k ψ(x)|2 + |x ψ(x)|2 )d x. (6.65)

So estimate (6.61) follows. Estimate (6.62) is a consequence of Sobolev estimate in dimension 2.  Let us now consider the propagation of coherent states ϕγ(θ) ,z under the dynamics VN ,θ (t) in H N (θ ). We shall prove that estimates can be obtained from the corresponding evolution in L 2 (R) (see Chap. 4), using the two previous lemmas. Recall these results. We have checked approximate solutions for the Schrödinger equation: i∂t ψt = Hˆ ψt , ψ0 = ϕγ ,z , γ > 0. (M) such that We have found ψz,t (M) (M) (M) (M) i∂t ψz,t = Hˆ ψz,t + (N +3)/2 Rz,t , ψz,0 = ϕγ ,z

(6.66)

(M) where, for every s ≥ 0, Rz,t K s = O(1) for  → 0. (M) ψz,t has the following expression δt

M ψz,t (x) = ei 



x − qt  j/2 π j (t, √ )ϕzΓt t (x),  0≤ j≤M

(6.67)

where z t = (qt , pt ) is the classical path in the phase space R2 such that z 0 = z satisfying  q˙t = ∂∂Hp (t, qt , pt ) (6.68) p˙t = − ∂∂qH (t, qt , pt ), q0 = q, p0 = p and

6.8 Spectral Analysis of Hamiltonian Perturbations

189

ϕzΓt t = Tˆ (z t )ϕ Γt .

(6.69)

ϕ Γt is the Gaussian state: Γt

ϕ (x) = (π )

−d/4



 i Γt x.x , a(t) exp 2

(6.70)

Γt is a complex number with positive non degenerate imaginary part, δt is a real function, a(t) is a complex function, π j (t, x) is a polynomial in x (of degree ≤ 3 j) with time dependent coefficients. More precisely Γt is given by the Jacobi stability matrix of the Hamiltonian flow z → Φ Ht z := z t . If we denote At =

∂qt ∂ pt ∂qt ∂ pt , Bt = , Ct = , Dt = , ∂q ∂q ∂p ∂p

(6.71)

then we have Γt = (Ct + γ Dt )(At + γ Bt )−1 , Γ0 = γ ,  t q t pt − q 0 p0 , δt (z) = ( ps qs − H (z s ))ds − 2 0 a(t) = [det(At + γ Bt )]−1/2 ,

(6.72) (6.73) (6.74)

where the complex square root is computed by continuity from t = 0. Using the two lemmas and the Duhamel formula, we get, (using the notation ψ (θ) = Σ N(θ) ψ), Proposition 75 For every m ≥ 0 and every θ ∈ [0, 2π [2 we have (m,θ) H N (θ) = O(N −(m+1)/2 ). VN ,θ ϕγ(θ) ,z − ψz,t

(6.75)

(0,θ) In particular we have ψz,t = ϕΓ(θ) with the notation of Sect. 6.4. t ,z t

Let us come back to Hamiltonian perturbations of hyperbolic automorphism F. Let us denote FHε = Φ Hε ◦ F. FHε is symplectic on T 2 (it preserves the area). By the C 1 stability of Anosov dynamical systems, for ε small enough, FHε is Anosov. The quantum analogue of FHε is the unitary operator:  ε Rˆ N ,θ,ε (F) = exp −i Opw,θ (H ) Rˆ N ,θ (F)  We have the following semi-classical correspondence Proposition 76 The following estimates hold true uniformly in z ∈ T 2 and ε ∈ [0, 1] θ (6.76) Rˆ N ,θ,ε (F)ϕγ(θ) ,z = C ε ϕγε ,FHε (z)

190

6 Quantization and Coherent States on the 2-Torus

where γε =

Cε +F·γ Dε Aε +F·γ Bε

Cε = e

and iδε (F(z))/



|γ b + a| γb + a

1/2 

|Aε + F · γ Bε | Aε + F · γ Bε

1/2 .

In particular Cε is a complex number of modulus one. Proof This is a direct consequence of propagation of coherent states (Sect. 4.3).  Now we shall prove some spectral properties for Rˆ N ,θ,ε (F) for ε small enough. Let us denote η Nj , 0 ≤ j ≤ N − 1 the eigenvalues of Rˆ N ,θ,ε (F), so that Rˆ N ,θ,ε (F)ψ jN = η Nj ψ jN where {ψ jN }0≤ j≤N −1 is an orthonormal basis of H N (θ ) and η Nj ∈ S1 , the unit circle of the complex plane. Theorem 41 For ε > 0 small enough, when N → +∞, the eigenvalues {η Nj }0≤ j≤N −1 are uniformly distributed on S1 i.e. for every interval I on S1 ) we have { j; η Nj ∈ I } = μ(I ) (6.77) lim N →+∞ N were μ is the Lebesgue probability measure on S1 . Proof In a first step we will prove that for every f ∈ C 1 (S1 ) we have lim

N →+∞

  1  ˆ Tr f R N ,θ,ε (F) = f (x)dμ(x). N S1

(6.78)

Using Fourier decomposition of f it is enough to prove (6.78) for f (z) = z k , k ∈ Z. Hence we have to prove that for every k = 0, lim

N →+∞

 k 1 Tr( Rˆ N ,θ,ε (F) ) = 0. N

(6.79)

We assume k ≥ 1 (for k ≤ −1 there are obvious modifications). Using that the coherent states are an overcomplete system in H N (θ ), we have  k  Tr( Rˆ N ,θ,ε (F) ) =

T

2

 k ϕzθ , Rˆ N ,θ,ε (F) ϕzθ .

Using the propagation of coherent states we get  k |ϕzθ , Rˆ N ,θ,ε (F) ϕzθ | = |ϕzθ , ϕγ (ε,k),(FHε )k | + O(εN −1/2 ) and using Proposition 61 we have  k √ ε k 2 |ϕzθ , Rˆ N ,θ,ε (F) ϕzθ | ≤ C(ε, k) N e−c(ε,k)d((FH ) z,z) N + O(εN −1/2 ).

6.8 Spectral Analysis of Hamiltonian Perturbations

191

But we know that for ε small enough FHε is Anosov so it is ergodic and its periodic set has zero measure. So for every δ > 0 we have μ{z ∈ T 2 , d((FHε )k z, z) ≥ δ} = 0. Using that O(εN −1/2 ) is uniform in z ∈ T 2 we get (6.79) hence (6.78). 1 1 Now  we get easily the Theorem considering f ± ∈ C (S ) such that f − ≤ 1l I ≤ f + and S1 ( f + − f − )dμ < δ with δ → 0. 

Chapter 7

Spin Coherent States

Abstract In this chapter we consider the unit sphere S2 of the Euclidean space R3 with its canonical symplectic structure is a phase space. Then coherent states are labelled by points on S2 and allow us to build a quantization of the two-sphere S2 . They are defined in each finite dimensional space of an irreducible unitary representation of the symmetry rotation group S O(3) (or its covering SU (2)) of S2 and give a semiclassical interpretation for the spin. As an application we state the Berezin-Lieb inequalities and compute the thermodynamic limit for large spin systems.

7.1 Introduction Up to now we have considered Gaussian coherent states and their relationship with the Heisenberg group, the symplectic group and the harmonic oscillator. These states are used to describe field coherent states (Glauber [118]). For the description of assembly of two-levels atom, physicists have introduced what they have called “atomic coherent states” [6]. These states are defined in Hilbert space irreducible representations of some symmetry Lie group. As we shall see later it is possible to associate coherent states to any irreducible representation of a Lie group. This general construction is due to Perelomov [209]. In this chapter we consider the rotation group S O(3) of the Euclidean space R3 and its companion SU (2). Irreducible representations of these groups are related with the spin of particles as it was discovered by Pauli [207]. In this chapter (and in the rest of the book) we shall use freely some basic notions concerning Lie groups, Lie algebra and their representations. We have recalled most of them in an appendix.

© Springer Nature Switzerland AG 2021 D. Robert and M. Combescure, Coherent States and Applications in Mathematical Physics, Theoretical and Mathematical Physics, https://doi.org/10.1007/978-3-030-70845-0_7

193

194

7 Spin Coherent States

7.2 The Groups S O(3) and SU(2) Let us consider the Euclidean space R3 equipped with the usual scalar product, x = (x1 , x2 , x3 ), y = (y1 , y2 , y3 ), x · y = x1 y1 + x2 y2 + x3 y3 and the Euclidean norm x = (x12 + x22 + x32 )1/2 . The isometry group of R3 is denoted O(3).1 A ∈ O(3) means that Ax = x for every x ∈ R3 . S O(3) is the subgroup of direct isometries i.e. A ∈ S O(3) means that A ∈ O(3) and det A = 1. It is well known that A ∈ S O(3) is a rotation characterized by a unitary vector v ∈ R3 (rotation axis) and an angle θ ∈ [0, 2π [. More precisely, v is an 1-eigenvector for A, Av = v and A is a rotation of angle θ in the plane orthogonal to v. So we have the following formula, for every x ∈ R3 , Ax := R(θ, v)x = (1 − cos θ )(v · x)v + (cos θ )x + sin θ (v ∧ x).

(7.1)

Recall that the wedge product v ∧ x is the unique vector in R3 such that det[v, x, w] = (v ∧ x) · w, ∀w ∈ R3 . It is easy to compute the Lie algebra so(3) of S O(3), so(3) = {A ∈ Mat(3, R), A T + A = 0} where A T is the transposed matrix of A. Considering rotations around vectors of the canonical basis {e1 , e2 , e3 } of R3 we get a basis {E 1 , E 2 , E 3 } of so(3) where Ek =

d R(θ, ek )|θ=0 dθ

It satisfies the commutation relation [E k , E  ] = E m

(7.2)

for every circular permutation (k, , m) of (1, 2, 3). It is well known that any rotation matrix is an exponential. Proposition 77 For every v ∈ R3 , v = 1 and θ ∈ [0, 2π [ we have R(θ, v) = eθ M(v) where M(v) =



(7.3)

vj E j.

1≤k≤3

1 An

isometry in an Euclidean space is automatically linear so that O(3) is a subgroup of G L(R3 ).

7.2 The Groups S O(3) and SU (2)

195

Proof θ → R(θ, v) and θ → eθ M(v) are one parameter groups, so it is enough to see that their derivatives at θ = 0 are the same. This is true because we have v ∧ x = M(v)x.  In the same way as S O(2) (identified to the circle S1 ) S O(3) is connected but not simply connected. To compute irreducible representations of S O(3) it is convenient to consider a simply connected cover of S O(3) which can be realized as the complex Lie group SU (2). SU (2)  is the group of unitary 2 × 2 matrices A with complex a b coefficients, A = , such that |a|2 + |b|2 = 1. −b¯ a¯ The Lie algebra su(2) is the real vector space of dimension 3 of 2 × 2 complex antihermitian matrices of zero trace:   su(2) = X ∈ gl(2, C) | X ∗ + X = 0, Tr X = 0 . The 3 linearly independent matrices:   1 0 1 , A1 = 2 −1 0

  1 0i A2 = , 2 i 0

  1 i 0 A3 = 2 0 −i

form a basis of su(2) on R and satisfy the commutation relations [Ak , A ] = Am ,

(7.4)

provided k, l, m is a circular permutation of 1, 2, 3. Let us consider the adjoint representation of SU (2). This representation is defined in the real vector space su(2) by the formula ρU (A) = U AU −1 , U ∈ SU (2), A ∈ su(2).

(7.5)

In physics the spin is defined by considering the Pauli matrices, which are hermitian 2 × 2 matrices given by σ1 =

      01 0 −i 1 0 , σ2 = , σ3 = 10 i 0 0 −1

(7.6)

which satisfy the commutation relations [σk , σl ] = 2iεk,,m σm

(7.7)

which is equivalent to (7.4) with the correspondance A1 = 2i σ2 , A2 = 2i σ1 , A3 = i σ . Noticing that {σ1 , σ2 , σ3 } is an orthonormal basis for the 3 dimensional real 2 3 linear space H2,0 of Hermitian 2 × 2 matrices with trace 0, which will be identified with R3 . The scalar product in H2,0 is

196

7 Spin Coherent States

A, B =

1 Tr(A∗ B). 2

Let us denote RU the 3 × 3 matrix of ρU in this basis. The following proposition gives the basic relationship between the groups S O(3) and SU (2). Proposition 78 For every U ∈ SU (2) we have: (i) RU has real coefficients. (ii) RU is an isometry in H2,0 (the 3-dimensional linear space defined above). (iii) det RU = 1. (iv) The map U → RU is a surjective group morphism from SU (2) onto S O(3). (v) The kernel of U → RU is ker R = {1l, −1l}. Proof (i) From RU σk , σ  = 21 Tr(U σk U −1 σ ) we get RU σk , σ  = RU σk , σ . (ii) Using commutativity of trace we have RU A, RU B = 21 Tr A A∗ = A, A. (iii) We have det RU = ±1 because RU is an isometry. But det R1l = 1 and SU (2) is connected so det RU = 1. (iv) It is easy to see that R is a group morphism. Let us consider the following generators of SU (2).  U1 (ϕ) =

   e−iϕ/2 0 cos(θ/2) − sin(θ/2) , U (θ ) = . 2 0 eiϕ/2 sin(θ/2) cos(θ/2)

Let us remark that we have, for every ϕ, ϕ , θ , U1 (ϕ)U2 (θ )U1 (ϕ ) =



 cos(θ/2)e−i/2(ϕ+ϕ ) − sin(θ/2)ei/2(ϕ −ϕ) . sin(θ/2)e−i/2(ϕ −ϕ) cos(θ/2)ei/2(ϕ+ϕ )

(7.8)

Then compute the image: ⎛

RU1 (ϕ)

⎞ ⎛ ⎞ cos ϕ sin ϕ 0 cos θ 0 − sin θ = ⎝− sin ϕ cos ϕ 0⎠ , RU2 (θ) = ⎝ 0 1 0 ⎠ 0 0 1 sin θ 0 cos θ

RU1 (ϕ) is the rotation of angle ϕ with axis e3 , RU2 (θ) is the rotation of angle θ with axis e2 . So, if ϕ, θ, η ∈ [0, 2π [ we have RU1 (ϕ)U2 (θ)U1 (η) = RU1 (ϕ) RU2 (θ) RU1 (η) where (ϕ, θ, η) are the Euler angles of the rotation R(ϕ, θ, η) := RU1 (ϕ) RU2 (θ) RU1 (η) . But any rotation can be defined with its Euler angles, so R is surjective. (v) Let U ∈ SU (2) be such that U AU −1 = A for every A ∈ H2,0 . Then we easily  get that U AU −1 = A for every A ∈ Mat(2, C) hence U = λ1l with λ = ±1. We shall see now that every irreducible representation of S O(3) comes from an irreducible representation of its companion SU (2).

7.2 The Groups S O(3) and SU (2)

197

Corollary 24 ρ is an irreducible representation of S O(3) if and only if ρ is an irreducible representation of SU (2) such that ρ RU = ρ R−U , ∀U ∈ SU (2). Proof Let ρ be a represention of SU (2) in a finite dimensional linear space E such that ρ RU = ρ R−U . Then we define a representation ρ˜ of S O(3) in E by the equality ρ(R ˜ U ) = ρ(U ). Conversely every representation of S O(3) comes from a representation of SU (2) like above. In other words the following diagram is commutative R

SU (2) ρ

S O(3) ρ˜

G L(E) where G L(E) is the group of invertible linear maps in E.



Corollary 25 The Lie algebras so(3) and su(2) are isomorph though the isomorphism D R(1l) (differential of R at the unit of the Lie group SU (2)). In particular we have D R(1l)Ak = E k , k = 1, 2, 3. Remark 44 The generators {L k }1≤k≤3 of the rotations with axis ek give a basis of the Lie algebra so(3) as a real linear space. Recall that L = (L 1 , L 2 , L 3 ) is the angular momentum.2 L k belongs to the complex Lie algebra so(3) ⊕ iso(3) and is sometimes denoted Jk , J = i x ∧ ∇x . For example L 3 = i(x2 ∂x1 − x1 ∂x2 ). We have the commutation relations, for every circular permutation (k, , m) of (1, 2, 3), [L k , L  ] = i L m .

(7.9)

This basis can be identified with the matrix basis (i E 1 , i E 2 , i E 3 ) considered before.

7.3 The Irreducible Representations of SU(2) The group SU (2) is simply connected (it has the topology of the sphere S3 ), so we know that all its representations are determined by the representations of its Lie algebra so(2) (see Appendix). Moreover, SU (2) is a compact Lie group so all its irreducible representations are finite dimensional.

2 Multiplication

by i gives self-adjoint generators instead of anti-self-adjoint operators.

198

7 Spin Coherent States

7.3.1 The Irreducible Representations of su(2) We shall first consider the representation of the Lie algebra su(2) and determine all its irreducible representations. Recall that su(2) is a real Lie algebra and it is more convenient to consider its complexification su(2) + isu(2). But any matrix can be decomposed as a sum of an Hermitian and anti-Hermitian part, so we have sl(2, C) = su(2) + isu(2)

(7.10)

  a b , a, b, c ∈ C. It is the Lie algebra of c −a the group of 2 × 2 complex matrices g such that detg = 1. It results from (7.10) that irreducible representations of the real Lie algebra su(2) are determined by irreducible representations of the complex Lie algebra sl(2, C). One considers sl(2, C) endowed with the basis {H, K + , K − } in which the commutation relations are

sl(2, C) is the space of matrices A =

[H, K ± ] = ±K ± , where H=

  1 1 0 , 2 0 −1

[K + , K − ] = 2H, 

K+ =

 01 , 00

K− =

(7.11)

  00 . 10

It is also convenient to introduce Hermitian generators (see footnote 2). K3 = H =

σ1 σ2 σ3 K+ + K− K+ − K− , K1 = = , K2 = = . 2 2 2 2i 2

(7.12)

Let (E, R) be a (finite dimensional) irreducible representation of sl(2, C). For convenience let us denote Hˆ the operator R(H ). Hˆ admits at least an eigenvalue λ and an eigenvector v = 0: Hˆ v = λv. From the commutation relations (7.11) we have Hˆ Kˆ + v = ( Kˆ + Hˆ + Kˆ + )v = (λ + 1) Kˆ + v. Hˆ Kˆ − v = ( Kˆ − Hˆ − Kˆ − )v = (λ − 1) Kˆ − v. Since there must be only a finite number of distinct eigenvalues of Hˆ , there exists an eigenvalue λ0 of Hˆ and an eigenvector v0 such that Hˆ v0 = λ0 v0 ,

Kˆ − v0 = 0,

λ0 is the smallest eigenvalue of Hˆ . One defines then

7.3 The Irreducible Representations of SU (2)

199

vk = ( Kˆ + )k v0 . It must obey

Hˆ vk = (λ0 + k)vk .

One can show by induction on k that Kˆ − vk = ck vk−1 , where ck+1 = ck − 2(λ0 + k), ∀k ∈ N. So we get ck = −k(2λ0 + k − 1), k ∈ N. Since the vectors vk = 0 are linearly independent and the vector space E is finite dimensional there exists an integer n such that v0 = 0, v1 = 0, . . . , vn = 0, vn+1 = 0 Thus from Kˆ − vn+1 = 0 one deduces that 2λ0 + n = 0. Then ∀k ∈ N one has [ Kˆ + , Kˆ − ]vk = 2 Hˆ vk , [ Hˆ , Kˆ ± ]vk = ± Kˆ ± vk .

(7.13)

One deduces that the vectors {vk }nk=0 generate a subspace of E invariant by the representation R and since the representation we look for is irreducible, they generate the complex linear space E which is therefore of finite dimension n + 1. The elements of the basis {vk }nk=0 are called Dicke states in [6]. In conclusion we have found necessary conditions to get an irreducible representation (E (n) , R (n) ) of dimension n + 1 of sl(2, C) with a basis {vk }nk=0 of E (n) such that R (n) (H )vk = (k −

n )vk 2

R (n) (K + )vk = vk+1 R (n) (K − )vk = k(n − k + 1)vk−1

(7.14)

for 0 ≤ k ≤ n and v−1 = vn+1 = 0. We have to check that these conditions can be realized in some concrete linear space. Let E (n) the complex linear space generated by homogeneous polynomials of z 1k z 2n−k degree n in (z 1 , z 2 ) ∈ C with the basis vk = (n−k)! and R (n) (K − ) = z 2

  ∂ ∂ ∂ ∂ 1 z1 , R (n) (K + ) = z 1 , R (n) (H ) = − z2 ∂z 1 ∂z 2 2 ∂z 1 ∂z 2

So we have proved:

200

7 Spin Coherent States

Proposition 79 Every irreducible representation of sl(2, C) of finite dimension is equivalent to (E (n) , R (n) ) for some n ∈ N. In the physics literature one considers j such that n = 2 j; j is thus either integer or half-integer and represents the angular momentum of the particles. We shall see later that representations of S O(3) correspond to j ∈ N so n is even. j is the greatest eigenvalue of R (n) (H ). In quantum mechanics the representation (E (2 j) , R (2 j) ) is denoted (V ( j) , D ( j) ) and one considers the basis of V ( j) indexed by the number m, − j ≤ m ≤ j, where m is integer if j is, and half-integer if j is. States in V ( j) represent spin states and the operators in V ( j) are spin observables. So we introduce the notation Sˆ = R (2 j) (K  ) (it is the spin observable along the axis 0x , 1 ≤  ≤ 3) and Sˆ± = R (2 j) (K ± ). This basis is usually written in the “ket” notation of Dirac as | j, m. The correspondence is the following:  | j, m = (−1)

j+m

( j − m)! v j+m . ( j + m)!

In this basis the representation D j of the elements K 3 , K + , K − (basis of the complex Lie algebra sl(2, C) defined at the begining) act as follows Sˆ3 | j, m = m| j, m Sˆ+ | j, m = ( j − m)( j + m + 1)| j, m + 1 Sˆ− | j, m = ( j + m)( j − m + 1)| j, m − 1. 

Hence Sˆ− | j, − j = 0, | j, m =

( j − m)! ( Sˆ+ ) j+m | j, − j. ( j + m)!(2 j)!

We recall that the two components L 1 , L 2 of the angular momentum are related to the operators L ± as follows: L + = L 1 + i L 2,

L− = L1 − i L2

In the representation space (V ( j) , D j ) of sl(2, C) one can consider the spin operator S = ( Sˆ1 , Sˆ2 , Sˆ3 ) and Sˆ12 + Sˆ22 + Sˆ32 = S2 S2 can be rewritten as

S2 = Sˆ− Sˆ+ + Sˆ3 ( Sˆ3 + 1l).

It is clear that for the representation D j , the vector | j, m is eigenstate of S2 : S2 | j, m = j ( j + 1)| j, m

7.3 The Irreducible Representations of SU (2)

201

Thus S2 acts as a multiple of the identity and is called the Casimir operator of the representation D j . One defines a scalar product on E (2 j) by imposing that the basis | j, m is an orthonormal basis. The representation of the Dicke states | j, m in the polynomials space V ( j) is given by j+m j−m

z1 z2 Dm( j) (z 1 , z 2 ) := | j, m(z 1 , z 2 ) = √ , − j ≤ m ≤ j. ( j + m)!( j − m)!

7.3.2 The Irreducible Representations of SU(2) We shall see now that for every j ∈ N2 we can get a representation T ( j) of SU (2) such that its differential dT ( j) coincides with the representations D ( j) of su(2) that we have studied in the previous section. Furthermore they are the only irreducible representations of SU (2). Since every unitary matrix is diagonalizable with unitary passage matrices we have ∀A ∈ SU (2):   it e 0 g −1 A=g 0 e−it for some t ∈ R. Then we show that the exponential map from su(2) to SU (2) is surjective: From the relation  it  e 0 = exp(itσ3 ) 0 e−it we deduce

A = exp(itgσ3 g −1 )

with igσ3 g −1 ∈ su(2), hence the result. Taking the Pauli matrices as a basis we have that every A ∈ su(2) can be written as A = ia · σ, a = (a1 , a2 , a3 ) ∈ R3 . We deduce easily that: det A = a2 and A2 = −(det A)1l. Therefore we have proven the following lemma

202

7 Spin Coherent States

Lemma 41 For any A ∈ su(2) one has A2 = −(det A)1l. Furthermore one has the following result: Proposition 80 For any A ∈ su(2) such that det A = 1, one has ∀t ∈ R: exp(t A) = cos t1l + (sin t)A

(7.15)

Proof Both members of (7.15) have A as derivative at t = 0. It is therefore enough to show that the map t ∈ R → 1l cos t + A sin t is a one parameter subgroup of G L(2, C). Take s ∈ R. One has (1l cos t + A sin t)(1l cos s + A sin s) = 1l cos t cos s + A2 sin s sin t + (sin s cos t + cos s sin t)A = 1l cos(s + t) + A sin(s + t). (7.16)  As a consequence we see that every element g ∈ SU (2) can be written as g = α1 1l + α2 I + α3 J + α4 K where I =

      0i 0 −1 i 0 , J = , K = i 0 1 0 0 −i

the vector (α1 , α2 , α3 , α4 ) of R4 being of norm 1. Thus SU (2) can be identified with the group of quaternions of norm 1. The group SU (2) acts on C2 by the usual matrix action.  g=

 a b ∈ SU (2) −b¯ a¯

with |a|2 + |b|2 = 1. Then g

    az 1 + bz 2 z1 = ¯ 1 + az z2 −bz ¯ 2

g induces a natural action on functions f : C2 → C: g · f := f ◦ g −1 . Since g has determinant one its inverse g −1 equals

(7.17)

7.3 The Irreducible Representations of SU (2)

g −1 = thus

203

  a¯ −b b¯ a

¯ 1 + az 2 ). ¯ 1 − bz 2 , bz g · f (z 1 , z 2 ) = f (az

The linear V ( j) space of homogeneous polynomials in z 1 , z 2 of degree 2 j (we recall that j ∈ 21 N) is stable by the action of SU (2). Moreover in V ( j) , the map p → g · p is unitary for the scalar product defined above. The action of g ∈ SU (2) on an homogeneous polynomial p of degree 2 j will be also denoted: T j (g) p(z 1 , z 2 ) = g · p(z 1 , z 2 ) = p ◦ g −1 (z 1 , z 2 ). Let us prove the following Lemma. Lemma 42 Consider the homogeneous polynomial p: p(z 1 , z 2 ) =

2j 

2 j−l

cl z l1 z 2

.

l=0

Then the map p →  p2 =

2j 

l!(2 j − l)!|cl |2

l=0

defines an Hilbertian norm of V ( j) that is invariant by the action of SU (2). Proof It is enough to check that  p ◦ g ∗ 2 =  p2 which implies that the representation (V ( j) , T j ) is unitary. Take p(z 1 , z 2 ) = pα,β (z 1 , z 2 ) = (αz 1 + βz 2 )2 j α, β ∈ C which generate V ( j) . Then if p(z 1 , z 2 ) =

2j 

2 j−l cl z l1 z 2 ,

l=0

p ◦ g −1 (z 1 , z 2 ) =

2j  l=0

one has



p (z 1 , z 2 ) =

2j 

cl z l1 z 2

2 j−l

l=0

2 j−l

dl z l1 z 2

,

p ◦ g −1 (z 1 , z 2 ) =

2j  l=0

dl z l1 z 2

2 j−l

204

7 Spin Coherent States

p , p = p ◦ g −1 , p ◦ g −1  =

2j 

cl c¯l l!(2 j − l)!

l=0

 2j

=

dl d¯l l!(2 j − l)! = (2 j)!(α α¯ + β β¯ )2 j .

(7.18)

l=0

Namely the Hermitian scalar product in C2 of (α, β) with (α , β ) is invariant under SU (2).  We shall study now the representation T ( j) of SU (2) in V ( j) . One defines j+m j−m z2 .

f mj (z 1 , z 2 ) = z 1

We first consider diagonal matrices in SU (2). They are of the form   it e 0 gt = exp(−2it K 3 ) = 0 e−it Then (T ( j) (gt ) f mj )(z 1 , z 2 ) = f mj (z 1 e−it , z 2 eit ) = e−2imt f mj (z 1 , z 2 )

(7.19)

Thus every f m is eigenstate of T ( j) (gt ) with eigenvalue e−2imt . We shall now consider the differential dT ( j) (g) for the basis elements K 3 , K ± of sl(2, C): one considers X ∈ sl(2, C), j

 X= and

α β γ −α



  a(t) b(t) gt = exp(t X ) = . c(t) d(t)

Then g(0) = 1l and g (0) = X so that a(0) = d(0) = 1, b(0) = c(0) = 0,

a (0) = α, b (0) = β, c (0) = γ , d (0) = −α.

For any polynomial in two variables f (z 1 , z 2 ) one has: ((dT ( j) )(−X ) f )(z 1 , z 2 ) =

d d (ρ(gt )−1 f )(z 1 , z 2 )|t=0 = ( f ◦ gt )(z 1 , z 2 )|t=0 dt dt

= (αz 1 + βz 2 )∂1 f (z 1 , z 2 ) + (γ z 1 + δz 2 )∂2 f (z 1 , z 2 ). Therefore

7.3 The Irreducible Representations of SU (2)

(dT ( j) )(K 3 ) = (dT ( j) )(K + ) = z 1 ∂2 ,

205

1 (z 1 ∂1 − z 2 ∂2 ) 2 (dρ)(K − ) = z 2 ∂1 .

We shall determine the action of dρ(K 3 ), dT ( j) (K ± ) on the basis vectors f m of V ( j) : dT ( j) (K 3 ) f mj = m f mj j

Furthermore using j+m−1 j−m z2 ,

∂1 f mj = ( j + m)z 1

j+m j−m−1 z2

∂2 f mj = ( j − m)z 1

we get dT ( j) (K + ) f mj = ( j − m) f m+1 , dT ( j) (K − ) f mj = ( j + m) f m−1 . j

j

Denoting 1 fj | j, m = √ ( j − m)!( j + m)! m we get dT ( j) (K 3 )| j, m = m| j, m dT ( j) (K + )| j, m = j ( j + 1) − m(m + 1)| j, m + 1 dT ( j) (K − )| j, m = j ( j + 1) − m(m − 1)| j, m − 1.

(7.20) (7.21) (7.22)

So we deduce the following results: Proposition 81 The differential of the representation T ( j) of SU (2) coincides with the representation D ( j) of su(2). Proposition 82 For any j ∈ 21 N, (V ( j) , T ( j) ) is an irreducible representation of SU (2). It defines an irreducible representation of S O(3) if and only if j ∈ N. / N) then T ( j) is a projective representation of S O(3). If n is odd ( j ∈ N2 , j ∈

Proof It follows from general results about the differential of the representations of Lie groups that the differential of T ( j) is an irreducible representation. The second part comes from the following fact: T ( j) (−g) = T ( j) (g), ∀g ∈ SU (2) if and only if n is even. The proof of the last part is left to the reader.  Corollary 26 Every irreducible representation of SU (2) is equivalent to one of the representations (V ( j) , T ( j) ), j ∈ 21 N.

206

7 Spin Coherent States

Proof Since SU (2) is compact we know that every irreducible representation of SU (2) is finite-dimensional. We have seen that every irreducible representation of finite dimension of su(2) is one of the D j , j ∈ 21 N. This implies the result since SU (2) is connected and simply connected. 

7.3.3 Irreducible Representations of S O(3) and Spherical Harmonics We have seen above that irreducible representations of S O(3) are described by (T ( j) , V ( j) ) for j ∈ N. A more concrete equivalent representation can be obtained with a spectral decomposition of the Laplace operator on S2 . Recall that in spherical coordinates (r, θ, ϕ) we have =

1 ∂2 ∂2 ∂2 2 ∂ ∂2 + 2 S2 + + = + 2 2 2 2 ∂r r ∂r r ∂ x1 ∂ x2 ∂ x3

where S2 is the spherical Laplace operator on S2 , S2 :=

∂2 1 ∂ 1 ∂2 + . + ∂θ 2 tan θ ∂θ sin2 θ ∂ϕ 2

(7.23)

S O(3) has a natural representation  in the function space L 2 (R3 ) (and in L 2 (S2 )) defined as follows. Σ(g) f (x) = f (g −1 x) where g ∈ S O(3), f ∈ L 2 (R3 ) and the Laplace operator commutes with . ( j) Let us introduce the linear space H3 of homogeneous polynomials f in (x1 , x2 , x3 ) of total degree j and satisfying  f = 0 and restricted to the sphere ( j) S2 ; H3 is the space of spherical harmonics. Recall that the Euclidean measure on S2 is dμ2 (θ, ϕ) = sin θ dθ dϕ. ( j)

Theorem 42 H3 is a subspace of C ∞ (S2 ) of dimension 2j+1, invariant for the ( j) action Σ. The representation (Σ, H3 ) is irreducible and is unitary equivalent to the representation (T ( j) , V ( j) ). Recall the following expression for the measure dμ2 on the sphere: dμ2 (θ, ϕ) = sin θ dθ dϕ. Proof We shall prove some properties of spherical harmonics which are proved in more details for example in [171]. ( j) The space H3 is invariant by the generators L 1 , L 2 , L 3 of rotations. In spherical coordinates we have

7.3 The Irreducible Representations of SU (2)

1 ∂ i ∂ϕ   ∂ sin ϕ ∂ 1 cos ϕ − L2 = i ∂θ tan θ ∂ϕ   cos ϕ ∂ ∂ + . L 1 = i sin ϕ ∂θ tan θ ∂ϕ L3 =

207

(7.24) (7.25) (7.26)

We can compute the Casimir operator: L2 := L 21 + L 22 + L 23 = −S2 . In particular we have [L 3 , S2 ] = 0. If L ± := L 1 ± i L 2 then we have [L 3 , L ± ] = ±L ± , [L + , L − ] = 2L 3 L + L − = L2 − L 3 (L 3 − 1l), L − L + = L2 − L 3 (L 3 + 1l).

(7.27) (7.28)

( j)

Let f ∈ H3 . In polar coordinates we have f (r, θ, ϕ) = r j Y (θ, ϕ). So we get that  f = 0 ⇐⇒ −S2 Y = j ( j + 1)Y. ( j)

L 3 can be diagonalized in H3

L 3 Y = λY ⇐⇒ Y (ϕ, θ ) = eimϕ f (θ ), m ∈ Z, − j ≤ m ≤ j. ( j)

So admitting that H3 has dimension 2 j + 1 we see that the representation ( j) (Σ, H3 ) is unitary equivalent to the representation (T j) , V ( j) ).  Remark 45 Using the same method as in Sect. 7.3, for every j ∈ N we have ( j) an orthonormal basis {Y jk }− j≤k≤ j of H3 where Y jk are eigenfunctions of L 3 : L 3 Y jk = kY jk , − j ≤ k ≤ j. In other words Y jk are Dicke states. Moreover they have the following expression:

Y jk (θ, ϕ)

=

2 j + 1 ikϕ e P j (cos θ ) 4π

(7.29)

where P j are the Legendre polynomials P j (u) =

1 dj 2 (u − 1) j . j! du j

2j

For k = 0 we can use the following formula L + Y jk = L − Y jk =

√ √

j ( j + 1) − k(k + 1)Y jk+1 , j ( j + 1) − k(k − 1)Y jk−1 .

Let us prove now two useful properties of the spherical harmonics

(7.30)

208

7 Spin Coherent States ( j)

Proposition 83 (i) For every j ∈ N, H3 has dimension 2 j + 1. (ii) {Y jk , − j ≤ k ≤ j, j ∈ N} is an orthonormal basis of L 2 (S2 ) or, equivalently, ( j)

⊕ j∈N H3

= L 2 (S2 ).

( j)

Proof Let us introduce the space P3 of homogeneous polynomials in (x1 , x2 , x3 ) of total degree j. The dimension of P3, j is ( j+1)(2 j+2) . It easy to prove that  is surjective so we get (i): dim(ker ) =

( j + 1)( j + 2) ( j − 1) j − = 2j + 1 2 2 ( j)

To prove (ii) let us introduce on P3 a scalar product such that we have an orthonork k k x 1x 2x 3 √1 2 3

( j) } . Let us introduce the Hilbert space H := ⊕P3 . So k1 !k2 !k3 ! k1 +k2 +k3 = j the linear operators ∂∂xk and xk are hermitian conjugate. Hence  is conjugate to r 2 = x12 + x22 + x32 . ( j) ( j+2) ( j) ( j−2) We have r 2 P3 ⊆ P3 and P3 = P3 . Using the formula (Fredholm ( j) j ( j−2) ∗ ⊥ 2 property) ker A = (Im A ) , we get P3 = H3 ⊕ r P3 . Step by step we get

mal basis {

( j)

( j)

P3 = H3

( j−2)

⊕ r 2 H3

( j−2)

⊕ · · · ⊕ r 2 H3

,

(7.31)

where j − 1 ≤ 2 ≤ j. Now we can prove that the spherical harmonics is a total system in L 2 (S2 ). Let us remark that the algebra ∪ j∈N P3, j is dense in C(S2 ) for the sup-norm (consequence ( j) of Stone-Weierstrass Theorem). So using (7.31) we get that ∪ j∈N H3 is dense in ( j)  C(S2 ) for the sup-norm, so ∪ j∈N H3 is dense in L 2 (S2 ).

7.4 The Coherent States of SU(2) 7.4.1 Definition and First Properties ( j)

Let us start with a reference (non zero) vector ψ0 ∈ V ( j) and consider elements of ( j) the orbit of ψ0 in V ( j) by the action of the representation T ( j) . We get a family of states of the form ( j) ψg( j) := |g := T ( j) (g)ψ0 . In a more explicit form we have ( j)

ψg( j) (z 1 , z 2 ) = ψ0 (g −1 (z 1 , z 2 )), (z 1 , z 2 ) ∈ C2 .

7.4 The Coherent States of SU (2)

209

In principle any vector ψ0 ∈ V ( j) can be taken as reference state to built coherent states. However the states ψ0 = | j, ± j are better because the total spin operator S = ( Sˆ1 , Sˆ2 , Sˆ3 ), is minimal on these states which are in some sense the closest to ( j) the classical states. In practice we shall choose here ψ0 = | j, j but the choice ( j) ψ0 = | j, − j is also possible with some modifications in the computations. Let us recall the definition of the dispersion for an observable Aˆ for a state ψ, where ψ is a normalized state in an Hilbert space H and Aˆ a self-adjoint operator ˆ and the dispersion ˆ ψ := ψ, Aψ in H . For ψ in the domain of Aˆ the average is A is defined like the variance for a random variable in probability: ˆ 2ψ . ˆ ψ 1l)2 ψ = Aˆ 2 ψ − A ψ Aˆ := ( Aˆ − A ˆ Bˆ are self-adjoint operLet us recall here the Heisenberg uncertainty principle: if A, ators in H and ψ ∈ H , ψ = 1 then we have 1  ˆ ˆ 2 ˆ ˆ

i[ A, B]ψ . (ψ A)( ψ B) ≥ 4

(7.32)

Application to the total spin observable gives ψ S =



ψ Sˆk =

1≤k≤3

For ψ0 = | j, m we get



 Sˆk ψ2 − Sˆk ψ, ψ2 .

1≤k≤3

ψ0 S = j ( j + 1) − m 2 .

So the dispersion is minimal for m = ± j. Heisenberg inequality for spin operators reads (see (7.32))

Sˆ12  Sˆ22  ≥

1 ˆ 2

S3  . 4

(7.33)

It is easy to see that this inequality is an equality for ψ0 = | j, ± j. Now our goal is to study the main properties of the coherent states |g. Let us first remark that the full group G = SU (2) is not a good set to parametrize these coherent states because the map g → |g is not injective. So we introduce the socalled isotropy group H defined as follows H = {g ∈ G, ∃δ ∈ R, T (g)ψ0 = eiδ ψ0 } We find that H is the subgroup of diagonal matrices H=

  α0 , 0 α¯

α = exp(iψ)

210

7 Spin Coherent States

Now the map g˙ → |g is a bijection from the quotient space X := G/H onto the orbit of ψ0 , where G/H is the set of left coset g H of H in G and g → g˙ is the canonical map: G → G/H .  0 1 Let us denote X 0 = X \{ }. −1 0 Lemma 43 X 0 is isomorphic to the set of elements of the form 

  α β 2 2 2 , α ∈ R, α  = 0, β = β + iβ ∈ C, α + β + β = 1 . 1 2 1 2 −β¯ α 

Proof Let us denote g(a, b) a generic element of SU (2), g =

 a b , a, b ∈ C, −b¯ a¯

|a|2 + |b|2 = 1. We first remark that for every |b| = 1, g(0, b) is in the coset of g(0, 1). Now if |a|2 + |b|2 = 1 and a = 0 then we have a unique decomposition g(a, b) = g(a , b )g(α, 0) a , (a )2 + |b |2 = 1. with a > 0, α = |a| So we get the lemma.



Remark 46 Concerning the orbit with our choice of ψ0 the image of X 0 does not 2j contain the monomial z 1 , which are obtained with g(0, 1). Choosing the parametrization θ θ α = cos , β = − sin e−iϕ , 0 ≤ θ < 2π, 0 ≤ ϕ < 2π 2 2 we see that the space X 0 is just a representation of the two-dimensional sphere S2 minus the north pole, namely the set of unit three-dimensional vectors n = (sin θ cos ϕ, sin θ sin ϕ, cos θ ), 0 ≤ θ < π, 0 ≤ ϕ < 2π ; and any element g ∈ X can be written as   θ g = gn = exp i (sin ϕσ1 − cos ϕσ2 ) 2

(7.34)

where σ1 , σ2 are the Pauli matrices. Thus gn describes a rotation by the angle θ around the vector m = (sin ϕ, − cos ϕ, 0) belonging to the equatorial plane of the sphere and perpendicular to n (it is well defined because θ ∈ [0, π [).

7.4 The Coherent States of SU (2)

211

Definition 20 The coherent states of SU (2) are the following states, defined in the representation space V ( j) , |n = T (gn )ψ0 := G(n)ψ0

(7.35)

In the physics literature they are called the spin coherent states because the spin is classified with the irreducible representations of SU (2) (see [207]). These coherent states have several other names: atomic coherent states, Bloch coherent states. Choosing g = gn the coherent state of the SU (2) group can now be written as |n = T (gn )ψ0 = exp(iθ m · K)ψ0 where m = (sin ϕ, − cos ϕ, 0), K = (K 1 , K 2 , K 3 ) m is the unit vector orthogonal to both n and n0 = (0, 0, 1). Note that this definition excludes the south pole n S = (0, 0, −1). Thus a coherent state of SU (2) corresponds to a point of the two-dimensional sphere S2 which may be considered as the phase-space of a classical dynamical system, the “classical spin”. The coherent states associated with the south pole will 2j be the image of (g(0, β)), |β| = 1 giving monomials z 1 . So we have parametrized the spin coherent state by the sphere S2 . Another useful parametrization can be obtained with the complex plane, using the stereographic projection from the south pole of the sphere S2 onto the complex plane C. If n = (n 1 , n 2 , n 3 ) ∈ S2 then the stereographic projection of n from the south 1 +in 2 . So in polar coordinates we have ζ (n) = pole is the complex number ζ (n) = n1+n 3 iϕ tan(θ/2)e . As we have already remarked, the group SU (2) naturally embeds into the complex group S L(2, C) which is the group of complex matrices having determinant one. The following gaussian decomposition in S L(2, C) will be useful. The proof is an easy exercise. Lemma 44 For any g ∈ S L(2, C) of the form  g=

 αβ , with δ = 0 γ δ

one has a unique (Gaussian) decomposition g = t+ · d · t − where d is diagonal and t± are triangular matrices of the form

212

7 Spin Coherent States



 1ζ , 01

t+ = and

d=

t− =

  10 z1

(7.36)

 −1  ε 0 . 0 ε

We have the formulas ε = δ, z =

γ β , ζ = δ δ

(7.37)

Moreover if g ∈ SU (2) then we have |ε|2 = (1 + |ζ |2 )−1 = (1 + |z|2 )−1

(7.38)

This allows to write as consequence of Gauss decomposition, T ( j) (g) = T ( j) (t+ )T ( j) (d)T ( j) (t− ) Let us write ε = r eis with r > 0 and s ∈ R. Taking ψ0 = | j, j as the reference state one gets in the representation (T ( j) , V ( j) ) ˆ

So we have

T ( j) (t− )ψ0 = e z+ S− ψ0 = ψ0 T ( j) (d)ψ0 = e−2 j (log r +is) ψ0

(7.39) (7.40)

T ( j) (g)ψ0 = eiϕ N T (t+ )ψ0 .

(7.41)

From (7.38) we get N = (1 + |ζ |2 )− j . ϕ is a real number (argument of δ). If g = gn then δ is real so we get s = ϕ = 0. In conclusion we have obtained an identification of the coherent states |n with the state |ζ  defined as follows: |ζ  = (1 + |ζ |2 )− j exp(ζ Sˆ+ )| j, j. More precisely we denote |ζ  = |gn  with the following correspondence: θ n = (sin θ cos ϕ, sin θ sin ϕ, cos θ ), ζ = − tan e−iϕ . 2 The geometrical interpretation is that −ζ¯ is the stereographic projection of n. Recall the following expression of gn :  cos θ2 − sin θ2 e−iϕ . sin θ2 eiϕ cos θ2

 gn =

(7.42)

7.4 The Coherent States of SU (2)

213

Another form for |n is given by the following equivalent definition using that gn = ˆ ˆ eiθ(sin ϕ S1 −cos ϕ S2 ) , |n = G(ξ )ψ0 where

G(ξ ) = exp(ξ Sˆ+ − ξ¯ Sˆ− )

  and ξ = tan θ2 eiϕ . The gaussian decomposition also provides a “normal form” of D(ξ ): G(ξ ) = exp(ζ Sˆ+ ) exp(η Sˆ3 ) exp(ζ Sˆ− ) with

η = −2 log |ξ |, ζ = −ζ¯ .

Since ζ, ζ , η do not depend on j it is enough to check this formula in the representation where j = 21 , S = 21 R(σ ), where σ is the three component Pauli matrix. For each n ∈ S2 the coherent state ψn minimizes Heisenberg inequality obtained by translation of (7.33) by G(n) i.e. putting S˜k = G(n) Sˆk G(n)−1 instead of Sˆk , 1 ≤ k ≤ 3.

7.4.2 Some Explicit Formulas Many explicit formulas can be proved for the spin coherent states. These formulas have many similarities with formulas already proved for the Heisenberg coherent states and can be written as well with the coordinates n on the sphere S2 or in the coordinates ζ in the complex plane C. Let us first remark that S2 and C can be identified with a classical phase space. S2 is equipped with the symplectic two form σ = sin θ dθ ∧ dϕ. In the stereographic dζ ∧d ζ¯ projection it is transformed in σ = 2i (1+|ζ . This is an easy computation using |2 )2 ζ = − tan θ2 e−iϕ . Let us consider first some properties of operators G(n).  shall also use the nota We tions D(ξ ) or D(ζ ) where ξ and ζ are given by ξ = tan θ2 eiϕ and ζ = − tan θ2 e−iϕ using polar coordinates for n. The multiplication law for the operators G(n) is given by the following formula Proposition 84 (i) For every n1 , n2 outside the south pole of S2 we have G(n1 )G(n2 ) = G(n3 ) exp(−i(n1 , n2 )J3 )

(7.43)

where (n1 , n2 ) is the oriented area of the geodesic triangle on the sphere with vertices at the points [n0 , n1 , n2 ]. n3 is determined by

214

7 Spin Coherent States

n3 = Rgn1 n2

(7.44)

where Rg is the rotation associated to g ∈ SU (2) as in Proposition 78 and   θ gn = exp i (σ1 sin ϕ − σ2 cos ϕ) . 2

(7.45)

(ii) More generally for every g ∈ SU (2) and every n ∈ S2 such that n and g · n are outside the south pole we have G ( j) (g)ψn = exp(−i jA (g, n))ψg·n ,

(7.46)

where g · n = Rg (n) and A (g, n) is the aera of the spherical triangle [n0 , n, g · n]. Proof We prove the result in the two-dimensional representation of SU (2). The vector n3 is determined only by geometrical rule and is thus independent of the representation. We choose the representation in V 1/2 . Define R(g) to be the rotation in S O(3) induced by any g ∈ SU (2). By the definition (7.45) of gn we have that R(gn )n0 = n, ∀n = (sin θ cos ϕ, sin θ sin ϕ, cos θ ). We need to compute g = gn1 gn2 . We use the following lemma: Lemma 45 ∀g ∈ SU (2) ∃m ∈ S2 and δ ∈ R such that g = gm r3 (δ)   where r3 (δ) = exp i 2δ σ3 . Applying the lemma to g = gn1 gn2 we get g = gm r3 (δ) and we need to identify m with n3 given by (7.44). We have that R(r3 (δ))n0 = n0 . Then R(g) = R(gn1 )R(gn2 ) = R(gm )R(r3 (δ)) Applying this identity to the vector n0 we get R(gn1 )R(gn2 )n0 = R(gn1 )n2 = R(gm )n 0 = m. Thus we have proven that m = n3 = R(gn1 )n2 .

7.4 The Coherent States of SU (2)

215

So we have seen that the displacement operator G(n) transforms any spin coherent state |n1  into another coherent state of the system up to a phase: G(n)|n1  = G(n)D(n1 )ψ0 = G(n2 ) exp(i Sˆ3 (n, n1 ))ψ0 = exp(−i j(n, n1 ))|n2  where n2 = R(gn )n1 . The second factor in the right hand side does depend on the representation. The computation of (n1 , n2 ) will be done later. In the proof of (7.46) the nontrivial part is to compute the phase A (g, n) which also will be done later.  The following lemma shows that the spin is independent of the direction. Lemma 46 One has:

G(n) Sˆ3 G(n)−1 = n · S.

We shall prove the lemma in the representation of the Pauli matrices. First note that G(n) = cos

θ θ + i sin (sin ϕσ1 − cos ϕσ2 ) 2 2

Then   G(n)σ3 G(n)−1 = cos θ2 + i sin θ2 (sin ϕσ1 − cos ϕσ2 ) σ3   cos θ2 − i sin θ2 (sin ϕσ1 − cos ϕσ2 )

(7.47)

We use the properties of the Pauli matrices: σ1 σ3 = −iσ2 , σ2 σ3 = iσ1 , σ2 σ1 = −iσ3 to compute the right hand side. One gets G(n)σ3 G(n)−1 = cos θ σ3 + sin θ (sin ϕσ2 + cos ϕσ1 ) = n · σ The following consequence is that |n is always an eigenvector of the operator n · S: Proposition 85 One has: n · S|n = j|n .

Proof Denote by |n0  the vector

Then

ei S3 θ ψ0 ˆ

S3 |n0  = jeiθ S3 ψ0 = j|n0  since we take ψ0 = | j, j.

(7.48)

216

7 Spin Coherent States

Now we shall use the above lemma: n · S|n = G(n)J3 ψ0 = j|n 

This completes the proof of the proposition. Let us give now a result which is more general than (7.48). ( j)

Proposition 86 With the above notations, in particular ψn being the coherent state located at n ∈ S2 , for any j ∈ N/2 we have Sψn( j) = jnψn( j) +

( j) 2 jv(n)gn · D j−1 ,

(7.49)

where v(n) = (v1 (n), v2 (n), v3 (n)) ∈ C3 is given by     θ θ v1 (n) = cos2 − sin2 e−2iϕ 2 2       θ θ i cos2 + sin2 e−2iϕ v2 (n) = 2 2 2 sin θ −iϕ e . v3 (n) = − 2 In particular we have n · v(n) = 0. ( j) Proof It is enough to compute Sˆk ψg for k = 1, 2, 3 for



 a b g= ∈ SU (2), a ∈ R, b ∈ C. −b¯ a ( j) It is more convenient to compute g −1 · Sˆk g · ψ0 . Elementary computations give the following formulas

√ ( j) ( j) ( j) g −1 · Sˆ3 g · ψ0 = j (a 2 − |b|2 )ψ0 + 2 jabD j−1 , √ ( j) ( j) ( j) = −2 jabψ0 − 2 jb2 D j−1 , g −1 · Sˆ+ g · ψ0 √ ( j) j) ¯ 0( j) − 2 ja 2 D (j−1 = −2 ja bψ . g −1 · Sˆ− g · ψ0 So (7.49) follows directly. Moreover with a little more computation we can check that n · v(n) = 0 which proves again (7.48).  As in the Heisenberg setting, the spin-coherent states family |n is not an orthogonal system. One can compute the scalar product of two coherent states |n, |n :

7.4 The Coherent States of SU (2)

217

Proposition 87 One has

n |n = e

i j(n,n )



1 + n · n 2

j .

(7.50)

where (n, n ) is a real number. If the spherical triangle with vertices {n0 , n, n } is an Euler triangle then (n, n ) is the oriented area of this triangle. Proof To each point n on the sphere S2 we associate its spherical coordinates θ ∈ [0, π ), ϕ ∈ [0, 2π ) as usual: x = sin θ cos ϕ, y = sin θ sin ϕ, z = cos θ. The corresponding element gn ∈ SU (2) is defined as   θ gn = exp i (sin ϕσ1 − cos ϕσ2 ) . 2 The matrix i(sin ϕσ1 − cos ϕσ2 ) can be viewed as a pure quaternion that we denote q. Using (7.15) we have that gn = cos

θ θ + q sin . 2 2

Taking n ∈ S2 with spherical coordinates θ , ϕ we get (using quaternion calculus) θ θ θ θ cos − sin sin cos(ϕ − ϕ ) 2 2 2 2 θ θ θ θ θ θ +σ3 i sin sin sin(ϕ − ϕ) + q cos sin + q sin cos . 2 2 2 2 2 2 gn gn = cos

(7.51)

Therefore gn gn is of the form (7.17) with a = cos

θ θ θ θ cos − sin sin ei(ϕ−ϕ ) . 2 2 2 2

The element gn of SU (2) can be written as  cos θ2 − sin θ2 e−iϕ . gn = sin θ2 eiϕ cos θ2 

Now we turn to the representation T ( j) (g) in the space V ( j) of homogeneous polynomials of degree 2 j in z 1 , z 2 . The coherent state |n is of the form T (gn )ψ0 for some z

2j

n ∈ S2 and ψ0 being a reference state. We choose ψ0 = √(21 j)! in the homogeneous polynomial representation. Note that this is coherent with the choice | j, − j. The

218

7 Spin Coherent States

overlap between two coherent states is given by the scalar product:

T ( j) (gn )ψ0 , T ( j) (gn )ψ0  = 1 θ θ θ θ

(z 2 cos + z 1 sin eiϕ )2 j , (z 2 cos + z 1 sin eiϕ )2 j . (2 j)! 2 2 2 2

(7.52)

We make use of the following result: Lemma 47 Let α,β (z 1 , z 2 ) = (αz 1 + βz 2 )2 j Then the scalar product in V ( j) of two such polynomials equals:

α ,β , α,β  = (2 j)!(α α¯ + β β¯ )2 j . Then we get

2 j  θ θ θ θ

n |n = cos cos + sin sin ei(ϕ−ϕ ) . 2 2 2 2

By an easy calculus we obtain 2    cos θ cos θ + sin θ sin θ ei(ϕ−ϕ )  = 1 + n · n .   2 2 2 2 2 Let us now compute the phase of the overlap n |n. It is a non trivial and interesting computation related with Berry phase as we shall see. It can be extended to a more general setting for coherent states on Kähler manifolds [41]. We follow here the elementary proof of the paper [5]. Let us denote   θ θ θ i(ϕ−ϕ ) θ . η = arg cos cos + sin sin e 2 2 2 2 Using classical trigonometric formula we get tan η =

sin θ sin θ sin(ϕ − ϕ ) . (1 + cos θ )(1 + cos θ ) + sin θ sin θ cos(ϕ − ϕ )

(7.53)

We now compare this formula with the following spherical geometric formula already known by Euler and Lagrange (see [95] for a detailed proof). Let 3 points n1 , n2 , n3 be on the unit sphere S2 , not all on the same great circle and such that the spherical triangle with vertices n1 , n2 , n3 is an Euler triangle i.e. the angles and the sides are all smaller than π . Let ω be the aera of this triangle. Then we have ω |det[n1 , n2 , n3 ]| . (7.54) tan = 2 1 + n1 · n2 + n2 · n3 + n3 · n1

7.4 The Coherent States of SU (2)

219

Notice that this formula takes account of the orientation of the piecewise geodesic curve with vertices n1 , n2 , n3 . The orientation is positive if the frame {On1 , On2 , On3 } is direct. From (7.53) and (7.54) we get directly that tan η2 = tan ω2 . But ω, η ∈] − π, π [ so we can conclude that η = ω.  As it is expected, the spin coherent state system provides a “resolution of the identity” in the Hilbert space V ( j) : Proposition 88 We have the formula 2j + 1 4

 S2

dn|n n| = 1l .

(7.55)

Or using complex coordinates |ζ ,  C

dμ j (ζ )|ζ  ζ | = 1l

(7.56)

where the measure dμ j is dμ j (ζ ) = with d 2 ζ =

d 2ζ 2j + 1 π (1 + |ζ |2 )2

|dζ ∧d ζ¯ | . 2

Proof The two formulas are equivalent by the change of variables ζ = − tan θ2 e−iϕ . So it is sufficient to prove the complex version. Let us recall the analytic expression for |n and |ζ . 1 θ θ (cos z 1 + sin e−iϕ z 2 )2 j . |n = ψn (z 1 , z 2 ) = √ 2 2 (2 j)! |ζ  = ψζ (z 1 , z 2 ) = √

1 1 (z 1 − ζ z 2 )2 j . (2 j)! (1 + |ζ |2 ) j

(7.57)

(7.58)

( j)

Recall that we have an orthonormal basis of Dicke states {Dk }− j≤k≤ j in V ( j) where j+k j−k

z1 z2 ( j) Dk (z 1 , z 2 ) = √ . ( j + k)!( j − k)! So we get ( j)

Dk , ψζ 

 =

(2 j)! ( j + k)!( j − k)!

1/2

(1 + |ζ |2 )− j ζ j−k .

(7.59)

220

7 Spin Coherent States

And using Parseval formula we get ¯ )2 j .

η|ζ  = (1 + |ζ |2 )− j (1 + |η|2 )− j (1 + ηζ

(7.60)

(7.60) is the complex version for the overlap formula of two coherent states (7.52). It is convenient to introduce now the spin Bargmann transform (see Chap. 1 for the Bargmann transform in the Heisenberg setting). For every v ∈ V ( j) we define the following polynomial in the complex variable ζ v j, (ζ ) = ψζ |v(1 + |ζ |2 ) j . If v =



ck ek a direct computation gives

− j≤k≤ j

 C

|v j, (ζ )|2

 d 2ζ π = |ck |2 2 2 j+2 (1 + |ζ | ) 2 j + 1 − j≤k≤ j =

Or equivalently

 C

| ψζ |v|2

π v2 . 2j + 1

d 2ζ π v2 . = (1 + |ζ |2 )2 2j + 1

This formula is equivalent to the overcompleteness formula by polarisation.

(7.61)

(7.62) 

Remark 47 From the proof we have obtained that the spin-Bargmann transform: B j v(ζ ) := v j, (ζ ) is an isometry from V ( j) onto the space P2 j of polynomials of degree at most 2 j equipped with the scalar product

P, Q =

2j + 1 π

 C

¯ )Q(ζ ) P(ζ

d 2ζ (1 + |ζ |2 )2 j+2

In particular we have ψηj, (ζ ) = (1 + |η|2 )− j (1 + ηζ¯ )2 j . We now extend the computation of the phase (n, n ) in the general case by giving for it a different expression related with the well known geometric phase. A similar computation was done in [196]. Proposition 89 For any n, n ∈ S2 we have (n, n ) = −

i j

 [n,n ]

ψn , dψn 

(7.63)

7.4 The Coherent States of SU (2)

221

where the integral is computed on the shortest geodesic arc joining n to n of the one differential form ψn , dψn  To explain the formula we recall here the main idea behind the geometric phase discovered by Berry [28] and Pancharatnam [202] (see also [2]). Let us consider a closed loop n(t) : [0, 1] → S2 which is continuous by part, n(0) = n(1). We define the time-dependent Hamiltonian as Hˆ (t) = n(t) · S. The solution of the time-dependent Schrödinger equation with this Hamiltonian is denoted ψ(t). Let us consider η(t) in the Hilbert space and α(t) ∈ R such that ψ(t) = eiα(t) η(t). We choose α(t) such that η(t), η(t) ˙ = 0. In other terms η(t) describes a parallel transport along the curve. Then the geometrical phase α(t) obeys ˙ i α(t) ˙ + ψ(t), ψ(t) = 0. 

We thus have α(1) := α(γ ) = i

γ

ψ, dψ.

If γ delimitates a portion Γ of S2 we have by Stokes theorem  α(γ ) =

Γ

dψ, dψ

where the product of the differentials is the external product. Let ψn be the coherent state |n obtained at t = 1 from ψ(0) = ψn0 . In the homogeneous polynomial representation we get:  2 j θ 1 θ cos z 1 + sin e−iϕ z 2 ψn = √ . 2 2 (2 j)! Thus  2 j−1  ∂ψn j θ θ θ θ z 1 cos eiϕ − z 2 sin =√ cos z 1 + sin e−iϕ z 2 )2 j . ∂θ 2 2 2 2 (2 j)!   2j ∂ψn θ iϕ θ 2 j−1 θ iϕ =√ i sin e z 1 z 1 sin e + z 2 cos ∂ϕ 2 2 2 (2 j)!

222

7 Spin Coherent States

Using the invariance of the scalar product in V ( j) under SU (2) transformations we perform the change of variables θ θ − z 2 e−iϕ sin , 2 2 θ θ iϕ Z 2 = z 1 e sin + z 2 cos . 2 2 Z 1 = z 1 cos

(7.64)

One thus have using the orthogonality relations in V ( j) :

ψn ,

j iϕ ∂ψn 2 j−1 2j = e Z 1 Z 2 , Z 2  = 0 ∂θ (2 j)!

One can calculate the scalar product of

∂ψn ∂θ

and

(7.65)

∂ψn : ∂ϕ

  2 j 2 −iϕ θ θ iϕ θ ∂ψn ∂ψn 2 j−1 2 j−1 , =i e sin Z 1 Z 2 , Z 1 cos e + Z 2 sin Z2 

∂θ ∂ϕ (2 j)! 2 2 2 1 = i j sin θ, (7.66) 2 since

2 j−1

Z 1 Z 2

2 j−1

, Z1 Z2

 = (2 j − 1).

We shall now calculate the phase of the scalar product ψn , ψn  by calculating the geometric phase along the geodesic triangle T = [n0 , n, n , n0 ]. Denote by  the domain on S2 delimited by T . We have 

 α(T ) = i

T

This yields

ψn , dψn  = i



dψn , dψn .

(7.67)

 α(T ) = − j



sin θ dθ dϕ = − jArea().

We denote by [n1 , n2 ] the portion of great circle on S2 between n1 and n2 . We now integrate (7.67) successively along [n0 , n], [n, n ] and [n , n0 ]. From the fact that [n0 , n], [n , n0 ] lie in verticle planes we have 

 [n0 ,n]

ψn , dψn  =

[n ,n0 ]

ψn , dψn  = 0.

Thus for an Euler triangle we get: − jArea() = − j(n, n ) = i

 [n,n ]

ψn , dψn 

7.4 The Coherent States of SU (2)

223

In the general case we can subdivise the triangle in several Euler triangles with vertex at n0 by adding vertices between n and n . Then we get that (n, n ) = −

i j

 [n,n ]

ψn , dψn .

 As a consequence of our study of the geometric phase, let us now compute the phase in formula (7.43). We already know that G ( j) (n1 )G ( j) (n2 )ψ0 = G ( j) (n3 )eiα and we have to compute α. We have

ψ0 , G ( j) (n1 )D(n2 )ψ0  = eiα ψ0 , G ( j) (n3 )ψ0  = G ( j) (n1 )∗ ψ0 , G ( j) (n2 )ψ0 

(7.68)

From Lemma 48 we know that ψ0 , G ( j) (n3 )ψ0  ≥ 0. But G ( j) (n1 )∗ = G ( j) (n1∗ ) where n1∗ is the symmetric of n1 on the great circle determined by n0 and n1 . Applying the computation of the phase in (7.52) we get α = arg( n1∗ , n2 ). With an elementary geometric argument we get the phase in formula (7.43). As a consequence, we can get the phase A (g, n) in formula (7.46). If g = gm then A (gm , n) = (m, n). For a generic g ∈ SU (2) we have g = gm r3 (δ) for some δ ∈ R. So we have ggn = gm r3 (δ)gn . Let n = (θ, ϕ)(polar coordinates). We have r3 (δ)gn = gn r3 (δ) where n = (θ, ϕ + δ). So using computation of  in (7.43) and elementary geometry we get that A (g, n) is equal to the area of the spherical triangle . [n 0 , n, g · n]. Let us close our discussion concerning the geometric phase for coherent states by the following result: Lemma 48 Let ψ1 and ψ2 be two different states on a great circle of S j (unit sphere on V ( j) ). One parametrizes this by the angle θ in the following way: ψ(θ ) = x1 (θ )ψ1 + x2 (θ )ψ2 where xi (θ ) ∈ R and ψ(0) = ψ1 , ψ(θ0 ) = ψ2 . One assumes that ˙ ) = 0.

ψ(θ ), ψ(θ

224

7 Spin Coherent States

Then ψ1 and ψ2 are in phase, namely

ψ1 , ψ2  > 0

Proof One can assume that a =  ψ1 , ψ2  > 0. Then by an easy calculus we get a 1 sin θ, x2 (θ ) = √ sin θ, a = cos θ0 x1 (θ ) = cos θ − √ 2 1−a 1 − a2 1 ˙ ) = √  ψ(θ ), ψ(θ  ψ1 , ψ2 . 2 1 − a2 One deduces that ψ1 , ψ2  is real therefore positive.



Remark 48 For two coherent states |n and |n  there are two natural geodesics joining them: the geodesic on the two sphere S2 and the geodesic on the sphere S j (sphere (2 j + 1)-dimensional). It is a consequence of results proved above that the geometric phases for these two curves in V ( j) are the same. This was not obvious before computations.

7.5 Coherent States on the Riemann Sphere We have seen that it is convenient to compute on the sphere S2 using complex coordinates given by the stereographic projection. We shall give here more details about this. In particular this gives a semi-classical interpretation for the spin coherent states and a quantization of the sphere S2 . The picture is analogous to the harmonic oscillator coherent states and the associated Wick quantization of the phase space R2d . The stereographic projection of the sphere S2 from its south pole is the transformation πs (n) = ζ , defined by n 1 + in 2 ζ = 1 + n3 where n = (n 1 , n 2 , n 3 ). πs is an homeomorphism from S2 := S2 \{(0, 0, 1)} on the complex plane C. Moreover πs can be extended in an homeomorphism from S2 ˜ := C ∪ {∞} such that πs (0, 0, 1) = ∞. C ˜ is a one dimensional complex and on C compact manifold called the Riemann sphere. πs−1 is determined by the formula πs−1 ζ = (n 1 , n 2 , n 3 ) where n1 =

ζ + ζ¯ ζ − ζ¯ 1 − |ζ |2 , n = = . , n 2 3 1 + |ζ |2 i(1 + |ζ |2 ) 1 + |ζ |2

7.5 Coherent States on the Riemann Sphere

225

˜ (bijective and biholomorphic It is known that the group of automorphisms of C transformations) is the Möbius group, the group of homographic transformations where a, b, c, d ∈ C such that ad − bc = 1. The conventions are: if h(z) = az+b cz+d c = 0 then h(∞) = ∞; if c = 0 then h(∞) = ac , h(− dc ) = ∞.   ab Let us denote g = , g ∈ S L(2, C) and f = h g . Then is not difficult to see cd that h g = 1l if and only if g = ±1l2 . ˜ we get that We have seen that if g ∈ SU (2) then Rg defines a rotation in S2 . In C Rg becomes a Möbius transformation:   a b . Then we have Lemma 49 Let R˜ g = πs Rg πs−1 and g ∈ SU (2), g = −b¯ a¯ aζ + b ˜ R˜ g ζ = , ∀ζ ∈ C. ¯ + a¯ −bζ

(7.69)

Proof We only give a sketch. First it is enough to consider g = g2 (θ ). After some computations we get the result using that Möbius transformations preserve the cross 3 .  ratio ζζ21 −ζ −ζ3 For simplicity we denote R˜ g ζ := g · ζ . Now our aim is to realize the representation (T ( j) , V j ) in a space of holomorphic ˜ This is achieved easily with the spin-Bargmann functions on the Riemann sphere C. j transform B introduced above. Recall that B j v(ζ ) = ψζ , v(1 + |ζ |2 ) j . We get the images of the Dicke basis and of the coherent states: d˜ (ζ ) := B j (dk )(ζ ) =



(2 j)! !(2 j − )!

1/2

ζ  , j + k = .

ψ˜ ζ (z) = ψz , ψζ (1 + |z|2 ) j = (1 + |ζ |2 )− j (1 + ζ¯ z)2 j , z, ζ ∈ C.

(7.70) (7.71)

Let us remark that the Hilbert space V j is transformed in the Hilbert space P2 j (polynomials of degree ≤ 2 j + 1) and that P2 j coïncides with the space of holomorphic functions P on C such that  |P(ζ )|2 (1 + |ζ |2 )−2 j−2 d 2 ζ < +∞. C

So the exponential weight of the usual Bargmann space is replaced here by a polynomial weight. We see now that the action of SU (2) in the Bargmann space P2 j is simple and has a nice semi-classical interpretation. Let us denote T˜ j (g) := B j T j (g)B j . On the Riemann sphere the representation T˜ j has the following expression.

226

7 Spin Coherent States

Proposition 90 For every ψ ∈ P2 j and g ∈ SU (2) we have T˜ j (g)ψ(ζ ) = μ j (g, ζ )ψ( R˜ g−1 (ζ ))  where g =

(7.72)

 a b ¯ )2 j and R˜ g−1 (ζ ) = , μ j (g, ζ ) = (a + bζ −b¯ a¯

aζ ¯ −b ¯ +a . bζ

Proof It is enough to prove the proposition for ψ(ζ ) = ζ  . From definition of B j we get ( j + k)! ¯ 1 + az 2 ) j−k  j . (B j T j (g)dk (ζ ) = (2 j)! √ ¯ 1 − bz 2 ) j+k (bz

(ζ¯ z 1 + z 2 )2 j , (az V ( j − k)!

Using that the scalar product in V j is invariant for the SU (2) action we obtain  (B j T j (g)d )(ζ ) =

(2 j)! !(2 j − )!

1/2

¯ )2 j (a + bζ



aζ ¯ −b ¯ +a bζ

 (7.73) 

Hence we get the proposition

Now, following Onofri [201] we shall give a classical mechanical interpretation of the term μ(g, ζ ). This interpretation can be extended to any semi-simple Lie group, as we shall see later. Let us introduce K (ζ, ζ¯ ) = 2 log(1 + ζ ζ¯ ) (Kähler potential), d the exterior differential, ∂ the exterior differential in ζ , ∂¯ the exterior differential in ζ¯ . ¯ = −∂ ∂. ¯ We have d = ∂ + ∂¯ and d∂ = ∂∂ We introduce the one form θ = −i∂ K and the two form ω = dθ = 2i

dζ ∧ d ζ¯ . (1 + |ζ |2 )2

˜ ω) is a symplectic manω is clearly a non-degenerate antisymmetric two form. So (C, ifold. Moreover it is a Kähler one dimensional complex manifold for the Hermitian metric dζ d ζ¯ ds 2 = 4 . (1 + |ζ |2 )2 It is not difficult to see that ω is invariant by the action of SU (2): g ω = ω. In other ˜ by canonical transformations. words SU (2) acts in C ˜ C is connected and simply connected, so there exists a smooth function S(g, ζ ) such that d S(g, ζ ) = θ − g θ . Now let us compute dμ as follows. From (7.72) with ψ = ψ˜ 0 = 1 and using that ψζ , ψ0  = (1 + |ζ |2 ) j we get: μ j (g, ζ ) =

ψ0 , T j (g −1 gζ ) ψ0  .

ψ0 , ψζ 

7.5 Coherent States on the Riemann Sphere

227

Then we compute dμ = (i j (θ − g θ ))μ = (i jd S)μ. Hence we get the classically mechanical interpretation for μ(g, ζ ) μ(g, ζ ) = μ(g, 0)ei j

ζ 0

(θ−g θ)

.

(7.74) 

7.6 Application to High Spin Inequalities One of the first successful use of spin coherent states was the thermodynamic limit of spin systems as an application of Berezin-Lieb inequalities. Berezin-Lieb inequality holds true for general coherent states.

7.6.1 Berezin-Lieb Inequalities We shall follow here the notations of Sect. 2.6 concerning Wick quantization. We assume here that the Hilbert space H is finite dimensional (this is enough for our application). Let Aˆ ∈ L (H ) with a covariant symbol Ac and contravariant symbol Ac defined on some metric space M with a probability Radon measure dμ(m). It is not difficult to see that these two symbols satisfy the following duality formulas  ˆ = Ac (m)B c (m)dμ(m), Tr( Aˆ B)  M Ac (m ) = M | em |em |2 Ac (m)dμ(m). In particular we have ˆ = Tr( A)

(7.75) (7.76)

 Ac (m)dμ(m).

(7.77)

M

Let us remark while Ac is well defined, Ac is not uniquely defined in general. Theorem 43 Let Hˆ be a self-adjoint operator in H and χ a convex function on R. Then we have the inequalities 

χ (Hc (m))dμ(m) ≤ Tr(χ ( Hˆ )) ≤ M

 χ (H c (m))dμ(m).

(7.78)

M

Proof Let us recall the Jensen inequality [228], which is the main tool for proving the Berezin-Lieb inequalities. For any probability measure ν on M, any convex function χ on R and any f ∈ L 1 (M, dν) we have

228

7 Spin Coherent States



  f dν(m) ≤ χ ( f (m))dν(m).

χ M

(7.79)

M

We start with the formula    Tr χ ( Hˆ ) =

em , χ ( Hˆ )em dμ(m). M

Using the spectral decomposition for self-adjoint operators, we have

em , χ ( Hˆ )em  =

 R

χ (λ)dνm (λ)

where νm is the spectral measure of the state em . It is a probability (discrete) measure. So Jensen inequality gives

em , χ ( Hˆ )em  ≥ χ (Ac (m)). Integrating in m we get the first Berezin-Lieb inequality 

χ (Hc (m))dμ(m) ≤ Tr(χ ( Hˆ )). M

For the second inequality we introduce an orthonormal basis of H : {vn }1≤n≤N of ˆ So we have eigenfunctions of A.

vn |χ ( Hˆ )vn  = χ ( vn | Hˆ vn ) = χ



 H (m)| vn |em | dμ(m) . c

2

M

From Jensen inequality applied with the probability measure | vn |em |2 dμ(m) we get  χ (H c (m))| vn |em |2 dμ(m).

vn |χ ( Hˆ )vn  ≤ M

Summing over n we get the second Berezin-Lieb inequality.



7.6.2 High Spin Estimates We consider a one dimensional Heisenberg chain of N spin Sn = (S1n , S2n , S3n ), 1 ≤ n ≤ N . The Hamiltonian of this system is Hˆ = −

 1≤n≤N −1

Sn Sn+1 .

7.6 Application to High Spin Inequalities

229

Hˆ is an Hermitian operator in the finite dimensional Hilbert space H N = ⊗ N H n where H n = V ( j) for every n. From the coherent states systems in V ( j) we get in a standard way a coherent system in H N parametrized on (S2 ) N (or C N using the complex parametrisation). In the sphere representation, if  = (n1 , . . . , n N ) ∈ S N we define the coherent state ψ = ψn1 ⊗ ψn2 . . . ψn N . We get, as for N = 1, an overcomplete system with a resolution of identity. We denote d N the probability measure (4π )−N dn1 ⊗ · · · ⊗ dn N . Wick and anti-Wick quantization are also well defined as explained in Chap. 1. In particular the Berezin-Lieb inequalities are true in this setting. As usual the following convention is used: if Aˆ ∈ L (H m ) and Bˆ ∈ L (H m ) then Aˆ Bˆ = 1l ⊗ 1l · · · ⊗ Aˆ ⊗ · · · ⊗ Bˆ ⊗ 1l · · · ⊗ 1l with Aˆ at position m, Bˆ at position m . In particular for the covariant and contravariant symbols of Aˆ Bˆ we have the following obvious properties

(AB)c () = Ac (nm )Bc (nm ), m

(AB) () = A (n )B (n ). c

c

m

c

(7.80) (7.81)

We shall prove the following results concerning the symbols of one spin operators (S1 , S2 , S3 ) Proposition 91 The covariant symbols are S1,c (n) = j sin θ cos ϕ, S2,c (n) = j sin θ sin ϕ,

(7.82) (7.83)

S3,c (n) = j cos θ.

(7.84)

The contravariant symbols are S1c (n) = ( j + 1) sin θ cos ϕ,

(7.85)

S2c (n) S3c (n)

(7.86) (7.87)

= ( j + 1) sin θ sin ϕ, = ( j + 1) cos θ.

Proof It seems more convenient to compute in the complex representation ψζ for the coherent states. In the space V ( j) the spin operators are represented by differential operators S1 = S2 = S3 = j

1 (z ∂ + z 2 ∂z1 ), 2 1 z2 1 (z ∂ − z 2 ∂z1 ), 2i 1 z 2 1 (z ∂ 2 1 z1

The coherent state ψζ has the expression

− z 2 ∂z2 ).

(7.88) (7.89) (7.90)

230

7 Spin Coherent States

ψζ (z 1 , z 2 ) = √

1 (1 + |ζ |2 )− j (z 1 − ζ z 2 )2 j . (2 j)!

So a direct computation gives

ζ |S1 |ζ  = j sin θ cos ϕ.

(7.91)

In the same way we can compute S2,c (n) and S3,c (n). Computing contravariant symbols is more difficult because we have no direct formula. The trick is to start with a large enough set of functions in complex variables (ζ, ζ¯ ) and to compute their anti-Wick quantizations. Let us denote Aα,β,τ (ζ, ζ¯ ) = (1 + |ζ |2 )−τ ζ α ζ¯ β and Aˆ α,β,τ (k, ) the matrix elements of the operator Aˆ α,β,τ , − j ≤ k,  ≤ j, in the canonical basis of V ( j) . We shall use the following formula 

ˆ k =

v , Av

Ac (ζ ) v , ζ vk dμ j (ζ )

v , ζ vk  = v , ψζ  ψζ , vk  = ck c ζ¯ j+ ζ j+k where ck = So we get



(2 j)! ( j+k)!( j−k)!

C

(7.92)

1/2 .

Aˆ α,β,τ (k, ) = ck c

 C

(d 2 ζ )ζ j+k+α ζ¯ j++β (1 + |ζ |2 )−2 j−τ −2 .

Using polar coordinates ζ = r eiγ we get Aˆ α,β,τ (k, ) = ck c δα+k,β+ 2(2 j + 1)



∞

dr 0

r 2 j+2α+2k+1 . (1 + r 2 )2 j+τ +2

We compute the one variable integral using beta and gamma functions  0

∞

rs 1 dr = B (1 + r 2 )t 2



s+1 s+1 ,t − 2 2

 =

)Γ (t − 1 Γ ( s+1 2 2 Γ (t)

s+1 ) 2

.

Finally we have the formula Γ ( j + k + α + 1)Γ ( j − α − k + τ + 1) . Aˆ α,β,τ (k, ) = (2 j + 1)ck c δα+k,β+ Γ (2 j + τ + 2) (7.93) For example if τ = 1, α = 0, 1, β = 0, 1, 2 we find

7.6 Application to High Spin Inequalities

231

1 ( j + k + 1)( j − k), 2j + 2 1 Aˆ 0,1,1 (k, k + 1) = ( j + k)( j − k + 1), 2j + 2 1 Aˆ 1,1,1 (k, k + 1) = ( j + k + 1). 2j + 2

Aˆ 1,0,1 (k, k + 1) =

Using these results we easily get a contravariant symbol for S as in (7.85).

(7.94) (7.95) (7.96) 

Now we can compute the covariant and contravariant symbols of the Hamiltonian Hˆ . Applying the above results, translated in the sphere variables, we get 

Hc () = j 2

nm nm+1 ,

1≤m≤N −1

H c () = ( j + 1)2



nm nm+1 .

(7.97) (7.98)

1≤m≤N −1

We are ready now to prove the main result which is a particular case of much more general results proved in [180, 236]. Let us introduce the quantum partition function Z q (β, j) =

1 ˆ Tre−β H , β ∈ R. N (2 j + 1)

The corresponding classical partition function is obtained taking the average of spin operators on coherent states. So we define  Z c (β, j) =

e−β Hc () dμ(). (S2 ) N

Putting together all the necessary results we have proved the following Lieb’s inequalities: Proposition 92 We have the following inequalities, Z c (β, j) ≤ Z q (β, j) ≤ Z c (β, j + 1), ∀ j, integer or half − integer . (7.99) This result can be used to study the thermodynamic limit of large spin systems (see [180]). Corollary 27 With the notations of the previous proposition we have lim

j→+∞

Z q (β, j) = 1. Z c (β, j)

(7.100)

232

7 Spin Coherent States

7.7 More on High Spin Limit: From Spin Coherent States to Harmonic-Oscillator Coherent States We want to give here more details concerning the transition between spin coherent states and Heisenberg coherent states. Let us begin by the following easy connection between spin coherent states and harmonic oscillator coherent state. We compute on the Bargmann side. We start from ¯ )2 j . ψηj, (ζ ) = (1 + |η|2 )− j (1 + ηζ If we replace η and ζ by

√η 2j

and 

j,

lim ψ √η

j→+∞

2j

√ζ 2j

ζ √ 2j



and let j → +∞ then we get ¯ −|η| = eηζ

2

/2

=

√ 2π ϕη (ζ ).

In the right side ϕη (ζ ) is the Bargmann transform of the Gaussian coherent state located in η. In this sense the spin coherent states converge, in the high spin limit, to the “classical” coherent states. In the same way we shall prove that the Dicke states converge to the Hermite basis of the harmonic oscillator. Let us recall that in the Dicke basis of V ( j) the Hamiltonian Hˆ = S3 is diag( j) onal: Hˆ d j,k = kd j,k , − j ≤ k ≤ j. The spin-Bargmann transform of Dk is easily computed: (2 j)! ( j), ζ j+k . Dk (ζ ) = ( j + k)!( j − k)! Recall that (see Chap. 1) the Bargmann transform of the Hermite functions ψ ( ∈ N) is ζ  . ψ (ζ ) = √ 2π ! Then we have the following asymptotic result. Proposition 93 For every  ∈ N and every r > 0 we have lim

j→+∞,k→−∞ j−k→

( j), Dk (ζ )



ζ √ 2j

 =

√



2π ψ (ζ ) .

(7.101)

uniformly in |ζ | ≤ r . Proof The result follows easily using that for j, k large enough such that j − k ≈  we have (2 j)! (2 j)! (2 j) ≈ ≈ .  ( j + k)!( j − k)! !(2 j − )! !

7.7 More on High Spin Limit: From Spin Coherent States …

233

A different approach concerning the high spin limit of spin coherent states is to consider contractions of the Lie group SU (2) in the Heisenberg Lie group H1 (see [160]). We are not going to consider here the theory of Lie group contractions in general but to compute on the example of SU (2) and its representations. Let us consider the irreducible representation of index j of SU (2) in the Bargmann space P2 j . P2 j is clearly a finite dimensional subspace of the Bargmann-Fock space F (C).   Let us compute in this representation the images L ± , L 3 of the generators K ± , K 3 of the Lie algebra sl(C). We get easily ∂ − j, ∂ζ

(7.102)

L + = 2 jζ − ζ 2

(7.103)



L3 = ζ



L− =

∂ . ∂ζ

∂ , ∂ζ

(7.104)

Let us introduce a small parameter ε > 0 and denote 

L ε+ = εL + ,

(7.105)

 εL − ,

(7.106)

1  L ε3 = L 3 + 2 1l. 2ε

(7.107)

L ε−

=

We have the following commutation relations [L ε+ , L ε− ] = 2ε2 L ε3 − 1l, [L ε3 , L ε± ] = ±L ε± .

(7.108)

As ε → 0 equations (7.105) define a family of singular transformations of the Lie algebra su(2) and for ε = 0 we get (formally) [L 0+ , L 0− ] = −1l, [L 03 , L 0± ] = ±L 0±

(7.109)

These commutation relations are those satisfied by the harmonic oscillator Lie algebra: L 0+ ≡ a † , L 0− ≡ a, L 03 ≡ N := a † a. We can give a mathematical proof of this analogy by computing covariant symbols. Proposition 94 Assume that ε → 0 and j → +∞ such that lim(2 jε2 ) = 1. Then we have j j (7.110) lim ψζ /√2 j |L ε3 |ψζ /√2 j  = |ζ |2 = ϕζ |a † a|ϕζ . lim ψζ /√2 j |L ε+ |ψζ /√2 j  = ζ¯ = ϕζ |a † |ϕζ . j

j

(7.111)

234

7 Spin Coherent States

lim ψζ /√2 j |L ε− |ψζ /√2 j  = ζ = ϕζ |a|ϕζ . j

j

(7.112)

Proof From computations of Sect. 7.5 we have j



j

ψζ |L 3 |ψζ  = j So we have

1 − |ζ |2 . 1 + |ζ |2

1 j j

ψζ /√2 j |L ε3 |ψζ /√2 j  = |ζ 2 | + O( ). j

and we get (7.110). The other formulas are proved in the same way using the following relations j



j

ψζ |L + |ψζ  =

2 j ζ¯ 2 jζ j j  , ψζ |L − |ψζ  = . 1 + |ζ |2 1 + |ζ |2



xˆ3 = |0 |ψ θ xˆ2 ϕ xˆ1 −xˆ3 = |1 The sphere S2 as a phase space for the spin is often named the Bloch sphere in quantum optics. (0, 0, 1)

x3 = 0 n(θ, ϕ)

−ζ¯

The stereographic projection.

Chapter 8

Pseudo-Spin Coherent States

Abstract We have seen before that spin coherent states are strongly linked with the algebraic and geometric properties of the Euclidean 2-sphere S2 . We shall now consider analogue setting when the sphere is replaced by the 2-pseudosphere i.e. the Euclidean metric in R3 is replaced by the Minkowski metric and the group SO(3) by the symmetry group SO(2, 1). The main big difference with the Euclidean case is that in the Minkowski case the pseudosphere and its symmetry groupe SO(2, 1) are not compact and that all irreducible unitary representations of SO(2, 1) are infinite dimensional.

8.1 Introduction to the Geometry of the Pseudosphere, SO(2, 1) and SU(1, 1) In this section we shall give a brief introduction to pseudo-euclidean geometry (often named hyperbolic geometry). There exists a huge literature on this subject. For more details we refer to in [13] or to any standard text book on hyperbolic geometry like [46].

8.1.1 Minkowski Model On the linear space R3 we consider the Minkowski metric defined by the symmetric bilinear form x, yM := x  y := x1 y1 + x2 y2 − x0 y0 where x, y ∈ R3 , x = (x0 , x1 , x2 ), y = (y0 , y1 , y2 ). So we get the three dimensional Minkowski space (R3 , ·, ·M ) with its canonical orthonormal basis {e0 , e1 , e2 }. Let us remark that the Minkowski metric is the restriction of the Lorentz relativistic metric, defined by the quadratic form x12 + x22 + x32 − x02 , to the 3 dimensional subspace of R4 defined by x3 = 0. The surface in R3 defined by the equation = {x ∈ R3 , x, xM = −1} is a hyperboloid with two symmetric sheets. The pseudosphere PS2 is one of this sheet. So we © Springer Nature Switzerland AG 2021 D. Robert and M. Combescure, Coherent States and Applications in Mathematical Physics, Theoretical and Mathematical Physics, https://doi.org/10.1007/978-3-030-70845-0_8

235

236

8 Pseudo-Spin Coherent States

can choose the upper sheet: PS2 = {x = (x0 , x1 , x2 ), x12 + x22 − x02 = −1, x0 > 0} PS2 is a surface which can be parametrized with the pseudopolar coordinates (τ, ϕ): x0 = cosh τ, x1 = sinh τ cos ϕ, x2 = sinh τ sin ϕ,

τ ∈ [0, +∞[, ϕ ∈ [0, 2π [.

metric. PS2 is a Riemann surface for the metric induced on PS2 by the Minkowski  In coordinates (x0 , x1 , x2 ) PS2 is defined by the equation x0 =

1 + x12 + x22 . So,

in coordinates {x1 , x2 } on PS2 , the metric ds2 = −dx02 + dx12 + dx22 is given by the following symmetric matrix ⎛ x2 1 − 1+x21+x2 1 2 ⎝ G= x x 1 2

1+x12 +x22

⎞

1

x1 x2 1+x12 +x22 ⎠ . x2 − 1+x22+x2 1 2

Hence we see that ds2 is positive definite on PS2 . In polar coordinates we have a simpler expression: ds2 = d τ 2 + sinh τ 2 d ϕ 2 . The curvature of PS2 is -1 everywhere (compare with the sphere S2 with curvature is +1 everywhere). By analogy with the Euclidean sphere we shall denote n the generic point on PS2 . The Riemannian √ surface measure in pseudopolar coordinates is given by computing the density det G, where G is the matrix of the metric in coordinates (τ, ϕ). So the surface measure on PS2 is d 2 n = sinh τ d τ d ϕ. We can see that the geodesics on the pseudo-sphere PS2 are determined by their intersection with planes through the origin 0. ⎛ ⎞ −1 0 0 We consider now the symmetries of PS2 . Let us denote L = ⎝ 0 1 0⎠ the matrix 0 01 of the Minkowski quadratic form: x, yM = Lx · y (recall that the · denotes the usual scalar product in R3 ). Its invariance group is denoted O(2, 1). So A ∈ O(2, 1) means Ax  Ax = x  x for every x ∈ R3 or equivalently, AT LA = L (AT is the transposed matrix of A). In particular if A ∈ O(2, 1) then det A = ±1. The direct invariance group is the subgroup SO(2, 1) defined by A ∈ O(2, 1) and det A = 1. This group is not connected so we introduce SO0 (2, 1) the component of 1l in SO(2, 1), it is a closed subgroup of SO(2, 1). The pseudosphere PS2 is clearly invariant by SO0 (2, 1). It is not difficult to see that 1 is always an eigenvalue for every A ∈ SO0 (2, 1). Let v ∈ R2 be such that v = 1, Av = v and v, vM = 0. The orthogonal complement

8.1 Introduction to the Geometry of the Pseudosphere, SO(2, 1) and SU (1, 1)

237

of Rv for the Minkowski form ., .M is a two-dimensional plane invariant by A. So A looks like a rotation in Euclidean geometry. Let us give the following examples of transformations in SO0 (2, 1). ⎛

⎞ 1 0 0 Rϕ = ⎝0 cos ϕ − sin ϕ ⎠ : rotation in the plane {e1 , e2 } 0 sin ϕ cos ϕ ⎛ ⎞ cosh τ sinh τ 0 B1,τ = ⎝ sinh τ cosh τ 0⎠ : boost in the direction e1 0 0 1 ⎛ ⎞ cosh τ 0 sinh τ B2,τ = ⎝ 0 1 0 ⎠ : boost in the direction e2 . sinh τ 0 cosh τ

(8.1)

(8.2)

(8.3)

These three transformations generate all the group SO0 (2, 1). This can be easily proved using the following remark: if Av = v and if U is a transformation then AU (U v) = U v where AU = U AU −1 . SO0 (2, 1) is a Lie group. In particular it is a three dimensional manifold.

8.1.2 Lie Algebra We can get a basis for the Lie algebra so(2, 1) of SO(2, 1) by computing the generators of the three one parameter subgroups defined in (8.1). We get ⎛ ⎞ 00 0 d Rϕ |ϕ=0 = ⎝0 0 −1⎠ . E0 : = dϕ 01 0 ⎛ ⎞ 010 d E1 : = B1,τ |τ =0 = ⎝1 0 0⎠ . dτ 000 ⎛ ⎞ 001 d E2 : = B2,τ |τ =0 = ⎝0 0 0⎠ . dτ 100

(8.4)

(8.5)

(8.6)

The commutation relations of the Lie algebra are the following [E0 , E1 ] = E2 , [E2 , E0 ] = E1 , [E1 , E2 ] = −E0 .

(8.7)

If we compare with generators of so(3) we remark the minus sign in the last relation. Let us consider the exponential map exp : so(2, 1) → SO(2, 1). If A = x0 E0 + x1 E1 + x2 E2 where x02 + x12 + x22 = 1, we have the one parameter subgroup of

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8 Pseudo-Spin Coherent States

SO(2, 1), R(θ ) = eθA . R(θ ) is a pseudo-rotation with axis v = (x0 , −x2 , x1 ). Its geometrical properties are classified by the sign of v, vM . • v, vM > 0 (“time-like” axis): R(1) has a unique fixed point on PS2 , each orbit is bounded. R(1) is said to be elliptic • v, vM < 0 (“space-like” axis) : there exists a unique geodesic on PS2 invariant by U (1), R(1) is said hyperbolic • v, vM = 0 (“light-like” axis) : the geodesics asymptotically going to Rv are invariant by R(1). R(1) is said parabolic These classification will be more explicit on other models of PS2 as we shall see. As in the Euclidean case we can realize the Lie algebra relations (8.7) in a Lie algebra of complex 2 × 2 matrices. We replace the group SU (2) by the group SU (1, 1) of pseudo-unitary unimodular matrices of the form:   αβ g= ¯ |α|2 − |β|2 = 1 β α¯ SU (1, 1) is a Lie group of real dimension 3. Let us introduce now a convenient parametrization of SU (1, 1) t t α = cosh ei(ϕ+ψ)/2 , β = sinh ei(ϕ−ψ)/2 2 2 where the triple (ϕ, t, ψ) runs through the domain 0 ≤ ϕ < 2π, 0 < t < ∞, −2π ≤ ψ < 2π . In this way one gets the factorization: g(ϕ, t, ψ) =

  cosh 2t ei(ϕ+ψ)/2 sinh 2t ei(ϕ−ψ)/2 = g(ϕ, 0, 0)g(0, t, 0)g(0, 0, ψ). sinh 2t ei(ψ−ϕ)/2 cosh 2t e−i(ϕ+ψ)/2

In the group SU (1, 1) we choose three one-parameter subgroups consisting of the following matrices  ω1 (t) = g(0, t, 0), ω2 (t) =

 cosh 2t i sinh 2t , ω0 (t) = g(t, 0, 0) t t −i sinh 2 cosh 2

The generators of these one parameter subgroups are       1 01 i 0 1 i 1 0 b1 = , b2 = , b0 = . 2 10 2 −1 0 2 0 −1 These three matrices form a basis of the Lie algebra su(1, 1) with the commutation relations: (8.8) [b1 , b2 ] = −b0 , [b2 , b0 ] = b1 , [b0 , b1 ] = b2

8.1 Introduction to the Geometry of the Pseudosphere, SO(2, 1) and SU (1, 1)

239

We see that the Lie algebras so(2, 1) and su(1, 1) are isomorph and we can identify the basis {E0 , E1 , E2 } and {b0 , b1 , b2 }. Now we shall see that SU (1, 1) is for SO0 (2, 1) what is SU (2) for SO(3). Let us consider the adjoint representation of SU (1, 1). For every g ∈ SU (1, 1) and x = (x0 , x1 , x2 ) we have ⎛ g⎝

0≤k≤2

⎞ xk bk ⎠ g −1 =



yk bk .

0≤k≤2

Let us denote y = ρ(g)x, where y = (y0 , y1 , y2 ). As in the Euclidean case we have the following result. Proposition 95 For every g ∈ SU (1, 1) we have: (i) g → ρ(g) is a group morphism from SU (1, 1) onto SO(2, 1)0 (ii) ρ(g) = 1l if and only if g = ±1l. In particular the groups SO(2, 1)0 and SU (1, 1)/{1l, −1 l} are  isomorph. (iii) If A = v0 b0 + v1 b1 + v2 b2 then for every τ ∈ R, ρ eτ A is a pseudorotation of axis (v0 , −v2 , v1 ). Proof It is easy to see that ρ(g) ∈ SO(2, 1) and ρ is a group morphism. Let us remark that we have ωk (t) = etbk for k = 0, 1, 2. Then we get that ρ(ω0 (t)) is the rotation Rt of axis e0 in the Minkowski space, ρ(ω1 (t)) is the boost B1,t and ρ(ω2 (t)) is the boost B2,t . Then it results that the range of ρ is the full group SO0 (2, 1). (ii) is easy to prove, as in the Euclidean case. We assume v = (v0 , v1 , v2 ) = (0, 0, 0). The kernel of v0 E0 + v1 E1 + v2 E2 is clearly  generated by the vector (v0 , −v2 , v1 )so we get (iii). It will be useful to remark that the group SU (1, 1) is isomorph to the SL(2, R) group of real 2 × 2 matrices A such that detA = 1. The isomorphism is simply the   1 i 1 −1 conjugation A → Λ AΛ where A ∈ SL(2, R), Λ = √2 . i 1 Let us remark that we have su(1, 1) + isu(1, 1) = sl(2, C) so su(1, 1) and su(2) have the same complexification as Lie algebras. We shall see now that this isomorphism has a simple geometry interpretation and this is very useful to get a better insight of the pseudosphere geometry.

8.1.3 The Disc and the Half-Plane Poincaré Representations of the Pseudosphere Considering the stereographic projection from the apex (0, 0, −1) onto the plane {e1 , e2 }, the upper hyperboloïd is transformed into the disc D of radius 1. By this transformation the metric on the pseudosphere becomes a metric on D. Choosing in PS2 polar coordinates (τ, ϕ) and in D polar coordinates (r, −ϕ), we have

240

8 Pseudo-Spin Coherent States

τ x0 = cosh τ, x1 = sinh τ cos ϕ, x2 = sinh τ sin ϕ, r = tanh . 2 In D the metric and the surface element of the pseudosphere have the following expression 4(dr 2 + r 2 d ϕ 2 ) 4rdrd ϕ ds2 = , d 2ζ = 2 2 (1 − r ) (1 − r 2 )2 So we get another model of the pseudosphere named the Poincaré disc model. It is also convenient to introduce a complex representation of D, ζ = re−iϕ . In D geodesics are the straight lines through the origin and arcs circle orthogonal to the boundary (see textbooks on the subject). Let us remark that the metric is conformal so the angles for the metric coincïde with the Euclidean angles. So an isometry of D is a conformal transformation of D. We know that such direct transformations f are homographic: f (ζ ) =

αζ + β , such that |α|2 − |β|2 = 1 ¯ + α¯ βζ

In other words the direct symmetry group of D is given by a representation of  αβ . Moreover we see easily SU (1, 1), g → Hg , where Hg (ζ ) = f (ζ ), g = ¯ β α¯ that Hg = 1lD if and only if g = ±1l. So we recover geometrically the fact that SO(2, 1) ≈ SU (1, 1)/{1l, ±1l}. It is well known that the unit disc can be conformally transformed into the +i or half complex plane H = {z = u + iv, v > 0} by the homography H0 ζ = −iζ ζ +1 −1 −z+i iϕ H0 z = z+i , where ζ ∈ D, z ∈ H. Explicitly, if ζ = re and z = u + iv, we have u=

2r sin ϕ 1 − r2 , v = . 1 + r 2 + 2r cos ϕ 1 + r 2 + 2r cos ϕ

In H the metric and the surface element of the pseudosphere have the following expression du2 + dv2 dudv . , d 2z = ds2 = v2 v So we get another model of the pseudosphere named the Poincaré half-plane model. In this model the boundary of the disc is transformed into the real axis. In H the geodesics are vertical lines and half circles orthogonal to the real axis. As for the disc we can see that the direct symmetry group of H is the group , α, β, γ , δ ∈ R, αδ − γβ = 1 SL(2, R) with the homographic action Hg (z) = αz+β γ z+δ and Hg = 1lH if and only if g = ±1l. We recover the fact that SU (1, 1) and SL(2, R) are isomorph groups.

8.1 Introduction to the Geometry of the Pseudosphere, SO(2, 1) and SU (1, 1)

241

Finally there is another realization of the pseudosphere which is important to define coherent states: PS2 can be seen as a quotient of SU (1, 1) by its compact maximal subgroup U (1). Lemma 50 For every n = (cosh τ, sinh τ sin ϕ, sinh τ cos ϕ) define  gn =

 cosh(τ/2) sinh(τ/2)e−iϕ . sinh(τ/2)eiϕ cosh(τ/2)

Then the map n → {gn ω0 (t), t ∈ [−2π, 2π [} is a bijection from S2 in the right cosets  it/2 of SU(1, 1) modulo U (1) where U (1) is identified with the diagonal matrices 0 e , t ∈ [0, 4π [. 0 e−it/2   αβ , α, β ∈ C, |α|2 − |β|2 = 1. The Lemma is Proof Let us denote g(α, β) = ¯ β α¯ a direct consequence of the following decomposition : there exist α > 0, β ∈ C, t ∈ [0, 4π [, unique such that g(α, β) = g(α , β )ω0 (t) More precisely we have α = |α|, t = 2 arg α, β = eit/2 β. This proves the lemma.  So modulo composition by a rotation of axis e0 on the right, every g ∈ SU (1, 1) is equivalent to a unique gn , n ∈ PS2 . We can see that gn is a pseudorotation with axis v = (0, sin ϕ, cos ϕ) (remark a change of sign). To get that, remark gn−1

d 1 gn = (cos ϕσ1 + sin ϕσ2 ) = (cos ϕb1 − sin ϕb2 ) dτ 2

so gn = eτ (cos ϕb1 −sin ϕb2 ) , this is indeed a pseudorotation of axis direction (0, sin ϕ, cos ϕ).

8.2 Unitary Representations of SU(1, 1) Let us begin by introducing a useful and important invariant operator for a Lie group representation : the Casimir operator. We first define the Killing form on a Lie algebra g, 1 (8.9) X , Y 0 = Tr(ad (X )ad (Y )) 2 where ad is the Lie adjoint-representation: ad (X )Y = [X , Y ], ∀Y ∈ g.

242

g:

8 Pseudo-Spin Coherent States

ad (X ) is the infinitesimal generator of the one parameter group transformation in G θ (Y ) = eθX Y e−θX . We have the antisymmetric property: ad (X )Y , Z0 = −Y , ad (X )Z0 , ∀X , Y , Z ∈ g.

Consider a basis {Xj } of g. The Killing form in this basis has the matrix gj,k = tr(ad (Xj )ad (Xk )). We denote g j,k the inverse of the matrix gj,k . Let us now consider a representation R of a Lie group G in the linear space V and ρ = dR the corresponding representation of its Lie algebra g in L (V ). The Casimir ρ is defined as follows operator Cas ρ Cas =



g j,k ρ(Xj )ρ(Xk ).

Let us remark that if V is infinite dimensional some care is necessary to check the ρ domain of Cas . Nevertheless a standard computation gives the following important property. ρ

ρ

Lemma 51 The Casimir operator Cas commutes with the representation ρ, ρ(X )Cas ρ = Cas ρ(X ), ∀X ∈ g. ρ ρ In particular if the representation is irreducible then Cas = cas 1l, cas ∈ R. Let us now remark that every irreducible unitary representation of SU (1, 1) is infinite dimensional, this is a big difference compare to SU (2). Proposition 96 Let ρ be a unitary representation of SU (1, 1) in a finite dimensional Hilbert space H . Then ρ is trivial, i.e. ρ(g) = 1lH , ∀g ∈ SU (1, 1) Proof Using the isomorphism from SL(2, R) onto SU (1, 1), it is enough to prove the proposition for a unitary representation of SL(2, R). For every x ∈ R, a ∈ R, a = 0, we have    −1  2  a 0 1x a 0 1a x . = 0 1 0 a−1 01 0 a−1 

 1x Then we see that ρ are all conjugate for x > 0. By continuity and compactness 01   10 they are conjugate to 1lH = ρ because the representation is finite dimensional. 01   10 The same property holds true for x < 0 and for matrices . But the group y1     1x 10 SL(2, R) is generated by the two matrices and . So we can conclude 01 y1  that ρ(A) = 1lH for every A ∈ SL(2, R).

8.2 Unitary Representations of SU (1, 1)

243

Unitary irreducible representations of SU (1, 1) have been computed independently by Gelfand-Neumark [109] and by Bargmann [18]. We shall follow here the presentation by Bargmann with some minor modifications.

8.2.1 Classification of the Possible Representations of SU(1, 1) Let ρ be an irreducible unitary representation of SU (1, 1) in some Hilbert space H . From Proposition 96 we know that H is infinite dimensional. In the representation space the generators bj define the operators Bj := i

d ρ(ωj (t))|t=0 , j = 0, 1, 2, dt

with the commutation relations [B1 , B2 ] = −iB0 , [B2 , B0 ] = iB1 , [B0 , B1 ] = iB2 .

(8.10)

Or with the complex notation B± = B2 ± iB1 , we have [B− , B+ ] = 2B0 , [B0 , B± ] = ±B± .

(8.11)

We have ρ(ω0 (t)) = e−itB0 hence e−i4πB0 = 1. So the spectrum of B0 is a subset of { k2 , k ∈ Z}. There exists ψ0 ∈ H , ψ0  = 1 and B0 ψ0 = λψ0 , λ = k20 , k0 ∈ Z. Using the commutation relation we have B0 B+ ψ0 = (λ + 1)B+ ψ0 ,

(8.12)

B0 B− ψ0 = (λ − 1)B− ψ0 .

(8.13)

Reasoning by induction, we get for every k ∈ N, k )ψ0 = (λ + k)(B+ )k ψ0 B0 (B+

(8.14)

B0 (B− ) ψ0 = (λ − k)(B− ) ψ0

(8.15)

k

k

Introduce now the Casimir operator Cas of the representation which is supposed irreducible, so Cas = cas 1l where 1 Cas = B02 − (B− B+ + B+ B− ). 2 Hence we have the equations

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8 Pseudo-Spin Coherent States

(B− B+ + B+ B− )ψ0 = 2(λ2 − cas )ψ0

(8.16)

(B− B+ − B+ B− )ψ0 = 2λψ0 .

(8.17)

So we get B− B+ ψ0 = (λ(λ + 1) − cas )ψ0 B+ B− ψ0 = (λ(λ − 1) − cas )ψ0 .

(8.18) (8.19)

k ψ0 instead of ψ0 we have proved for every k ∈ N, Using now B± k B− B+ ψ0 = ((λ + k − 1)(λ + k) − cas )ψ0

(8.20)

k ψ0 B+ B−

(8.21)

= ((λ − k + 1)(λ − k) − cas )ψ0

Let us denote νk+ = ((λ + k − 1)(λ + k) − cas ) and νk− = ((λ − k + 1)(λ − k) − cas ). Using that B+ = B− ∗ and B− = B+ ∗ we get from (8.20), + k+1 k ψ0 2 = νk+1 B+ ψ0 2 B+ k+1 B− ψ0 2

=

− k νk+1 B− ψ0 2 .

(8.22) (8.23)

From Eq. (8.22) we can start the discussion. k k (I) Suppose that for all k ∈ N, B+ ψ0 = 0 and B− ψ0 = 0. Then for every k ∈ N, λ ± k is an eigenvalue for B0 . (I-1) If 0 is in this family (i.e. λ ∈ Z) then we can suppose that B0 ψ0 = 0 so we can choose λ = 0. From (8.22) we find the necessary condition cas < 0. k we can (I-2) If 0 is not in the family λ ± k , λ = 21 + k0 , k0 ∈ Z and using B± 1 assume that λ = 2 . From (8.22) we get that cas < 1/4. In these two cases we get an orthonormal basis {ϕm }m∈Z for H , such that B0 ϕm = (m + 2ε )ϕm . where ε = 0 in the first case and ε = 1 in the second case. This is really a basis because the linear space span by the ϕm is invariant by the representation which is irreducible. k0 +1 k0 ψ0 = 0 and B+ ψ0 = 0. (II) Suppose now that there exists k0 ∈ N such that B+   k0 Using (8.20) we see that for every  ∈ N, B− ψ0 is proportional to B− B+ ψ0 , so k0 ψ0 . So we have B0 ψ0 = λψ0 , B+ ψ0 = 0. Hence this we can replace ψ0 by B+ gives ν1+ = 0 and cas = λ(λ + 1). If λ = 0 we get B− ψ0 = 0 and the Hilbert space is unidimensional. So we have λ > 0. As above we get an orthonormal basis of H {ϕm }m∈N such that B0 ϕm = (λ − m)ϕm . k0 +1 k0 ψ0 = 0 and B− ψ0 = 0 we have a similar If there exists k0 ∈ N such that B− result with eigenvalues λ + m for B0 .

In case (I) the Casimir parameter cas varies in an interval and we say that the representation belongs to the continuous series; in case (II) the Casimir parameter varies in a discrete set and we say that the representation belongs to the discrete series.

8.2 Unitary Representations of SU (1, 1)

245

8.2.2 Discrete Series Representations of SU(1, 1) In the last section we have found necessary conditions satisfied by any irreducible representation of SU (1, 1). Now we have to prove that this conditions can be realized in some concrete Hilbert spaces.

8.2.2.1

The Hilbert Spaces Hn (D)

Let n be a real number, n ≥ 2, Hn (D) is the Hilbert space of holomorphic functions f on the unit disc D of the complex plane C satisfying f 2H n (D) :=

n−1 π

D

|f (z)|2 (1 − |z|2 )n−2 dxdy < +∞, z = x + iy.

(8.24)

The measure d νn (z) := n−1 (1 − |z|2 )n−2 dxdy is a probability measure on D and π Hn (D) is a complete linear space, with the Hilbert norm defined by (8.24), is a consequence of standard properties of holomorphic functions. It is useful to produce the following characterization of Hn (D) using the series expansion of f ck (f )z k f (z) = k≥0

which is absolutely convergent inside the disc D. In (8.24) let us compute the integral in polar coordinates z = reiθ . From the Parseval formula for Fourier series we get f H n2 (D) = 2(n − 1)π





f 2H n (D) =



r 2k+1 (1 − r 2 )n−2 dr.

(8.25)

0

k≥0

So we have

1

|ck (f )|2

|ck (f )|2

k≥0

Γ (n)Γ (k + 1) Γ (n + k)

(8.26)

This gives a unitary equivalent definition of Hn (D) as an Hilbert space of functions on the unit circle. In particular the scalar product in Hn (D) of f1 and f2 has the following expression f1 , f2 H n (D) =

k≥0

where γn, =

Γ (n)Γ (+1) . Γ (n+)

ck (f1 )ck (f2 )γn,k

(8.27)

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8 Pseudo-Spin Coherent States

From formula (8.26) and (8.27) we get easily that e (z) := orthonormal basis of Hn (D).

8.2.2.2

 √z , γn,

for  ≥ 0, is an

Discrete Series Realization of SU(1, 1) in Hn (D)

These representations can be introduced using Gauss decomposition in the complex Lie group SL(2, C):  β     α 0 1 0 αβ 1 α (8.28) = γ 1 γ δ 0 α1 01 α where α, β, γ , δ are complex numbers such that αδ − βδ = 1, α = 0. Moreover this decomposition as product like     u0 1w 10 0 1 z1 0 u1 is unique. So this allows to define natural of SU (1, 1) in the  actions   disc  D. 10 1z , t (z) the matrices and d (u) = Let us denote by t− (z) the matrices z1 + 01   u 0 . 0 u−1 Let us denote g a generic element of SU (1, 1), 

αβ g= ¯ β α¯



Consider the t− matrix in the Gaussian decomposition of gt− (z). We have t− = ¯ αz ¯ . We get easily that t− (˜z ) where the complex number z˜ is z˜ = M− (g)(z) = β+ α+βz |M− (g)(z)| = 1 if |z| = 1 and from the maximum principle, |M− (g)(z)| < 1 if |z| < 1. So we have defined a right action of SU (1, 1) in D. In the  same  way we get a left 1 z −1 action considering the t+ matrix in the decomposition of g . So we get the 01 action M+ (g)(z) = −β+αz ¯ . α− ¯ βz Now we want to define unitary actions of SU (1, 1) in the space Hn (D) as follows −1 Dn− (g)f (z) = m− g (z)f (M− (g )z)

(8.29)

− where the multiplier m− g (z) is chosen such that Dn defines an unitary representation ¯ − βz)−n . of SU (1, 1). We prove now that it is true with the choice m− g (z) = (α

Theorem 44 For every integer n ≥ 2 we have the following unitary representation of SU (1, 1) in the Hilbert space H (D):

8.2 Unitary Representations of SU (1, 1)

247



 −β¯ + αz (z) = (α¯ − βz) f . α¯ − βz   β + αz ¯ ¯ −n f . Dn+ (g)f (z) = (α + βz) ¯ α + βz

Dn− (g)f

−n

(8.30) (8.31)

Proof It is not difficult to see that Dn± are SU (1, 1) actions in the linear space Hn (D). Let us prove that Dn− is unitary. This follows with the holomorphic change of ¯ αZ ¯ ¯ β+αz , Z = X + iY , z = x + iy. We get z = β+ , dz = (βZ + α)−2 . variables Z = −α−βz ¯ α+βZ dZ We conclude using that for an holomorphic change of variable in the plane, we have from Cauchy conditions,

2



det ∂(x, y) = dz .

∂(X , Y ) dZ So we get for any f ∈ Hn (D),

 

−β¯ + αz 2

(1 − |z|2 )n−2 dxdy = |α¯ − βz|−2n

f |f (Z)|2 (1 − |Z|2 )n−2 dXdY α¯ − βz D D



which says that Dn− is unitary. Let us remark here that the above computations show  that the multiplier m− g is necessary to prove unitarity. The next step is to prove that these representations are irreducible. To do that we first compute the corresponding Lie algebra representation. Let us compute the image of this basis by the representation Dn− . Straightforward computations give d 1 d − D (ω0 (t))f (z)|t=0 = (n + 2z )f (z), dt n 2i dz d − 1 d D (ω1 (t))f (z)|t=0 = (nz + (z 2 − 1) )f (z), dt n 2 dz d − 1 d Dn (ω2 (t))f (z)|t=0 = (nz + (1 + z 2 ) )f (z). dt 2i dz

(8.32) (8.33) (8.34)

So we get the three self-adjoint generators B0 , B1 , B2 , Bj = id Dn− (bj ), where d denotes the differential on the group at 1l, d n +z , 2  dz  i d nz + (z 2 − 1) , B1 = 2 dz   1 d nz + (z 2 + 1) , B2 = 2 dz

B0 =

with the commutation relations

(8.35) (8.36) (8.37)

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8 Pseudo-Spin Coherent States

[B1 , B2 ] = −iB0 , [B2 , B0 ] = −iB1 , [B0 , B1 ] = iB2 .

(8.38)

Using the notation B± = B2 ∓ iB1 , we have B− =

d , dz

B+ = nz + z 2

(8.39) d , dz

(8.40)

and [B− , B+ ] = 2B0 , [B0 , B± ] = ±B± .

(8.41)

Remark 49 The operators Ba for a = 0, 1, 2, ± are non-bounded operators in the Hilbert space Hn (D) so we have to define their domains. Here we know that the representation Dn− is unitary. So Stone’s theorem gives that B0 is essentially selfadjoint and the linear space P ∞ of all polynomials in z is a core for B0 . Moreover the spectrum of B0 is discrete, with simple eigenvalues {n/2 + k, k ∈ N}. B1 and B2 also have a unique closed extension. Moreover it could be possible to characterize their domains (left to the reader!). Operators B± are closable in Hn (D). We keep the same notation B± for their closures. We have the useful property ∗ Lemma 52 B± are adjoint of each other : B± = B∓ . In particular for every ζ ∈ C the operator i(ζ B− − ζ¯ B+ ) is self-adjoint. ∗ Proof It is enough to prove B± = B∓ . This is formally obvious because we know that B1 , B2 are self-adjoint. We left to the reader to check that the domains are the same.  n Let us compute the Casimir operator Cas for the representation Dn− . A direct computation of the coefficients gj,k of the Killing form shows that gj,k = 0 if j = k and g0,0 = −1, g1,1 = g2,2 = 1. So we get

1 n = B02 − B12 − B22 = B02 − (B+ B− + B− B+ ). Cas 2

(8.42)

Let us assume for the moment that Dn− is irreducible. Then by Schur’s lemma we n is a number; this number can be computed using the monomial z 0 . We know that Cas find easily  n n n n − 1 = k(k − 1), k =: . (8.43) = Cas 2 2 2 Let us remark that k := index.

n 2

is the lowest eigenvalue of B0 and is called the Bargmann

8.2 Unitary Representations of SU (1, 1)

249

8.2.3 Irreducibility of Discrete Series Here we prove that for every integer n ≥ 2, the representation Dn− is irreducible. Let E be a closed invariant subspace in Hn (D). The restriction of Dn− to the compact commutative subgroup g(θ, 0, 0) is a sum of one dimensionnal unitary representations. So there exists u ∈ E, ν ∈ R such that Dn− (g(θ, 0, 0))u = eiνθ u, ∀θ ∈ R.  But u has a series expansion u(z) = aj z j . So by identification we have eiνθ aj = e−i(n+2j)θ aj . From this we get that there exists j0 such that aj0 = 0 so we find ν = n + 2j0 . But this entails aj = 0 if j = j0 . Hence we conclude that the monomial z j0 belongs to E. Now playing with B± we conclude easily that E contains all the monomials z j , j ∈ N. We have proved above that the monomials is a total system in Hn (D). So E=Hn (D). The discrete series representations have an important property: they are square integrable (see Appendix). On the group SU (1, 1) we have a left and right invariant Haar measure μ. μ is positive on each non empty open set and unique up to a positive constant (see [169]). The following result is proved in [170]. Proposition 97 For every f ∈ Hn (D) we have SU (1,1)

|Dn− (g)f , f H n (D) |2 d μ(g) < +∞

where d g is the Haar measure on SU (1, 1). There exists other unitary irreducible representations for SU (1, 1): the principal series and the complementary series (see also the book of Knapp [168] for more details). These representations are not square integrable. Up to equivalence, discrete series, principal series and complementary series are the only irreducible unitary representations. Let us first explain now principal series.

8.2.4 Principal Series These representations can also be realized in Hilbert spaces of functions on the unit circle. They are defined in the following way: take a nonnegative number λ and a point z on the unit circle. Then the homographic transformation

250

8 Pseudo-Spin Coherent States

αz + β¯ βz + α¯

z →

obviously maps the unit circle into itself. One defines Piλ (g)f (z) = |βz + α| ¯

−1+2iλ

 f

αz + β¯ βz + α¯

 .

One considers the Hilbert space L2 (S1 ) with the scalar product f1 , f2  =

1 2π



2π

d θ f¯1 (θ )f2 (θ ).

0

To prove the unitarity of the representation Piλ (g) in the Hilbert space we perform the change of variable θ → θ where

eiθ = The jacobian satisfies |

αeiθ + β¯ . βeiθ + α¯

dθ | = |βeiθ + α| ¯ −2 dθ

Thus we have for any real λ:

2π



2π

d θ |(Piλ (g)f )(e )| = iθ

2

0

d θ |f (eiθ )|2 .

0

Now we can prove: Proposition 98 For any λ ∈ R, Piλ is a unitary irreducible representation of 2 1 SU (1, 1) in  the Hilbert space L (S ). Moreover its Casimir operator is: Ciλ = 1 2 − 4 +λ . For the generator ω0 (t) of the Lie group su(1, 1) one gets (Piλ (ω0 (t))f )(eiθ ) = f (ei(θ+t) ). Thus the corresponding generator of SU (1, 1) is simply L0 =

d . dθ

To find the generator associated to ω1 we need to calculate t t | sinh eiθ + cosh |2 = cosh t + cos θ sinh t. 2 2

8.2 Unitary Representations of SU (1, 1)

251

Thus  (Piλ (ω1 (t)f )(e ) = (cosh t + sinh t cos θ ) iθ

−1/2+iλ

f

cosh 2t eiθ + sinh sinh 2t eiθ + cosh

t 2 t 2

 .

Thus the generator L1 of SU (1, 1) is given by   d 1 L1 = − + iλ cos θ − sin θ . 2 dθ Similarly using the third generator ω2 one finds 

 1 d L2 = − − + iλ sin θ − cos θ . 2 dθ Therefore the usual linear combinations B± = ±L1 + iL2 satisfy   d 1 . B+ = − + iλ e−iθ − ie−iθ 2 dθ  d 1 B− = − − + iλ eiθ − ieiθ . 2 dθ 

We define B0 = iL0 = i

d . dθ

Using the same method as for discrete series, we can prove that these representations are irreducible (start with B0 and use B±). To calculate the Casimir operator B02 − 21 (B− B+ + B+ B− ) it is enough to apply it to the constant function. Thus B0 1 = 0, B± 1 = ±(− 21 + iλ)e∓iθ . One finds   1 1 − λ2 1l. C := B02 − (B− B+ + B+ B− ) = − 2 4

8.2.5 Complementary Series When the parameter λ of the principal series is imaginary the representation is not unitary in the space L2 (S1 ). So, following Bargmann [20] we introduce a different Hilbert space. Let us introduce the sesquilinear form depending of the real parameter σ ∈]0, 21 [,

252

8 Pseudo-Spin Coherent States

f1 , f2 σ = c

[0,2π]2

(1 − cos(θ1 − θ2 ))σ −1/2 f1 (θ1 )f2 (θ2 )d θ1 d θ2 .

f1 , f2 σ is well defined if f1 , f2 are continuous on S1 . The constant c is computed such that 1, 1σ = 1, c = 21/2−σ π B(σ, 1/2)−1 .  2π The integral 0 (1 − cos θ )σ −1/2 d θ is computed using the change of variable x = cos(θ ) so we get c. The following properties are useful to build the Hilbert space Hσ . Proposition 99 (i) For every f1 , f2 ∈ C(S1 ) we have [0,2π]2

(1 − cos(θ1 − θ2 ))σ −1/2 |f1 (θ1 )||f2 (θ2 )|d θ1 d θ2 ≤ f1 f2 ,

(ii) ek , e σ = 0, if k = , where ek (θ ) = eikθ . (iii) ek , ek σ = λk (σ ), s where λk (σ ) =

Γ (1/2 + σ )Γ (|k| + 1/2 − σ ) . Γ (1/2 − σ )Γ (|k| + 1/2 + σ )

(8.44)

In particular λ0 = 1 and λk (σ ) > 0 for every k ∈ Z and σ ∈]0, 1/2[. Proof (i) is proved using the change of variable u = θ1 − θ2 and Cauchy-Schwarz inequality. (ii) It is a consequence of the following equality, for k = , ek , e σ =

[0,2π]2

(1 − cos(u))σ −1/2 e−ikθ ei(θ+u) d θ du = 0.

(iii) We have λ−k = λk , so it is enough to consider the case k ≥ 0. Hence we have π (1 − cos(u))σ −1/2 cos(ku)du. ek , ek σ = 2c 0

We compute the integral using the change of variable x = cos u, so cos(ku) = Tk (x), where Tk is the Tchebichev polynomial of order k.

8.2 Unitary Representations of SU (1, 1)

ek , ek σ = 2c

1 −1

253

(1 − x)σ −1/2 (1 − x2 )−1/2 Tk (x)dx.

But we have the following expression known as the Rodrigues formula [70] Tk (x) = (−1)k 2k−1

(k − 1)! dk (1 − x2 )1/2 k ((1 − x2 )k−1/2 ). (2k)! dx

Hence we get the result by integrations by parts and well known formulas for gamma and beta special functions.  j So if f1 , f2 ∈ C(S1 ), fj = ck ek is the Fourier decomposition of fj . Then it results k∈Z

that f1 , f2 σ =



λk (σ )ck1 ck2 .

k∈Z

This shows that (f1 , f2 ) → f1 , f2 σ is a positive-definite sesquilinear form on C(S1 ). Now we can define the complementary series Cσ , 0 < σ < 1/2, as follows. It is realized in the Hilbert space Hσ of functions f on S1 such that λk (σ )|ck |2 < +∞ equipped with the scalar product f1 , f2 σ (see proposition k∈Z

(iii)). So we can define, for f ∈ Hσ , Cσ (g)f (z) = |βz + α| ¯

−1+2σ

 f

αz + β¯ βz + α¯

 .

Proposition 100 For all 0 < σ < 1/2, Cσ is a unitary irreducible representation of SU (1, 1) in Hσ . Moreover its Casimir operator is Cσ = σ 2 − 41 . Proof We use the same methods as for the discrete and continuous series. In particular the computations are the same as for the continuous series with σ in place of iλ. The main difference here is in the definition of the Hilbert space which is necessary to get a unitary representation. 

8.2.6 Realizations for Bosonic Systems Let us start with a one boson system. We consider the usual annihilation and creation operators a, a† in L2 (R) (see Chap. 1). The following operators satisfy the commutation relations (8.41) of the Lie algebra su(1, 1) B+ =

1 † 2 1 1 (a ) , B− = a2 , B0 = (aa† + a† a). 2 2 4

(8.45)

254

8 Pseudo-Spin Coherent States

We have seen in Chap. 3 that the metaplectic representation is a projective representation of the group Sp(1) = SL(2, R) and it is decomposed into two irreducible representations in the Hilbert subspaces of L2 (R), L2ev (R) of even states and L2od (R) of odd states. But the group SL(2, R) is isomorphic to the group SU (1, 1)by the  1i −1 . explicit map g → Fg := M0 gM0 , g ∈ SU (1, 1), Fg ∈ SL(2, R) with M0 = i 1 So the metaplectic representation defines a representation of the group SU (1, 1) in the space L2 (R) with two irreducible components Rˆ ev,od in the space L2ev,od (R). In quantum mechanics it is natural to consider ray-representations (or projective representations) instead of genuine representations. For example the metaplectic representation is a ray-representation. Let us compute the Casimir operators Cev,od for each components. We compute Cev using the bound state ψ0 for the harmonic oscillator (ψ0 ∈ L2ev (R)). 3 Using that Hˆ osc = 21 (aa† + a† a) = aa† − 21 we get Cev ψ0 = − 16 ψ0 and Cod ψ1 = 3 1 3 − 16 ψ1 (B0 ψ0 = 4 ψ0 and B0 ψ1 = 4 ψ1 ). We see that the commutation relations (8.45) define two irreducible “representations” of SU (1, 1) which are neither in the discrete series neither in the continuous series. The reason is that they are ray representations corresponding with the even part and odd part of the metaplectic representation. More details concerning ray representations can be found in [19, 260]. In particular  (1, 1) these ray-representations are genuine representations of the covering group SU (which is simply connected but SU (1, 1) is not). A “double valued representation” ρ in a linear space satisfies ρ(gh) = C(g, h)ρ(g)ρ(h), with C(g, h) = ±1. Let us now consider the two bosons system. We consider two annihilation and † in L2 (R2 ) (see Chap. 1). The following operators satisfies creation operators a± , a± the commutation relations (8.41) of the Lie algebra su(1, 1) B+ = a1† a2† , B− = a1 a2 , B0 =

1 † (a a1 + a2† a2 + 1). 2 1

The Casimir operator is 1 1 Cas = − + (a2† a2 − a1† a1 )2 . 4 4 We know from Chap. 1 that we have an orthonormal basis of L2 (R2 ), {φm1 ,m2 }(m1 ,m2 )∈N2 , where φm1 ,m2 = (a1† )m1 (a2† )m2 φ0,0 such that

(8.46)

8.2 Unitary Representations of SU (1, 1)

m1 + m2 + 1 φm1 ,m2 2 = φm1 +1,m2 +1 = φm1 −1,m2 −1 .

255

B0 φm1 ,m2 =

(8.47)

B+ φm1 ,m2 B− φm1 ,m2

(8.48) (8.49)

A direct computation gives Cas φm1 ,m2

  1 2 = − + (m1 − m2 ) φm1 ,m2 . 4

(8.50)

So if we introduce k = 21 (1 + |n0 |), we get easily (assuming n0 ≥ 0) the following lemma. Lemma 53 For every positive half integer k, the Hilbert space spanned by {φm2 +2k−1,m2 , m2 ∈ N}, is an irreducible space for the representation of the Lie algebra with generators (8.45). We know now that this Lie algebra representation defines a unitary representation of  (1, 1) but only a projective representation of SU (1, 1). SU

8.3 Pseudo-Spin-Coherent States for Discrete Series We can now proceed to the construction of coherent states by analogy with the harmonic oscillator case (Glauber states) and the spin coherent states. We consider here the discrete series representation.

8.3.1 Definition of Coherent States for Discrete Series Let us consider the representation (Dn− , H (D)). It could be possible to work with Dn+ as well. Every g ∈ SU (1, 1) can be decomposed as g = gn h where h ∈ U (1) and n ∈ PS2 . It is convenient to start with ψ0 ∈ H (D) such that hψ0 = ψ0 so we take as a −1/2 fiducial state ψ0 (ζ ) = γn,0 ζ 0 and we define ψn = Dn− (gn )ψ0 . Most of properties of ψn will follow from suitable formula for the operator family D(n) = Dn− (gn ). There are many similarities with the spin setting. We shall explain now these similarities in more details. Using polar coordinates for n we have  cosh(τ/2) sinh(τ/2)e−iϕ . gn = sinh(τ/2)eiϕ cosh(τ/2) 

256

8 Pseudo-Spin Coherent States

So using the definition of the representation Dn− we have the straightforward formula for the pseudospin coherent states. ψζ (z) = (1 − |ζ |2 )n/2 (1 − ζ¯ z)−n , where ζ = sinh(τ/2)eiϕ .

(8.51)

Now we shall give a Lie group interpretation of the coherent states. Let us recall that Bm = id Dn− (1l)bm , m = 0, 1, 2, and B+ = B2 + iB1 , B− = B2 − iB− . Then we have D(n) = exp (−iτ (cos ϕB1 − sin ϕB2 ))

 = exp τ/2(B− eiϕ − B+ e−iϕ ).

(8.52) (8.53)

The second formula reads 

τ D(n) = D(ξ ) = exp ξ¯ B− ξ B+ , with ξ = e−iϕ . 2

(8.54)

We can get a simpler formula using an heuristic following from Gauss decomposition  cosh(τ/2) sinh(τ/2)e−iϕ = gn = sinh(τ/2)eiϕ cosh(τ/2)    cosh(τ/2) 0 0 1 tanh(τ/2)e−iϕ . 1 0 1 0 1 cosh(τ/2) 



1 tanh(τ/2)eiϕ

(8.55)

Recall that B0 ψ0 = 2n ψ0 and |ζ | = tanh(τ/2). Moreover if b ∈ su(1, 1) then we have

 Dn− etb = e−itB , with B = id Dn− (1l)b.

(8.56)

Suppose that (8.56) can be used for b1 ± b2 (which are not in the Lie algebra su(1, 1). Then we get the following representation of pseudocoherent states in the Poincaré disc D, where ζ and n represents the same point on the pseudosphere PS2 , ¯

ψn = ψζ = (1 − |ζ |2 )n/2 eζ B+ ψ0 .

(8.57)

Let us remark that in the spin case this heuristic is rigorous because the representation D j is well defined on SL(2, C) which is not true for Dn− . Nevertheless it is possible to give a rigorous meaning to formula (8.57) as we shall see in the next section.

8.3.2 Some Explicit Formulae We follow more or less the computations done in the spin case. We shall give details only when the proofs are really different.

8.3 Pseudo-Spin-Coherent States for Discrete Series

257

It is convenient to compute in the canonical basis {e }∈N of the representation space Hn (D) (analogue of Dicke states or Hermite basis). We get easily the formula  (n + )( + 1)e+1  B− e = (n +  − 1)e−1 , B− e0 = 0 n B0 e = ( + )e . 2

B+ e =

(8.58) (8.59) (8.60)

Let us remark that the linear space Pj of polynomials of degree ≤ j is stable for B0 and B− but not for B+ . For every  ∈ N we have  e0 = (n(n + 1) · · · (n +  − 1)!)1/2 e . B+ ¯

Following our heuristic argument we expand the exponent eζ B+ as a Taylor series (which is not allowed because B+ is unbounded) and we recover the formula: ψζ (z) = (1 − |ζ |2 )k (1 − ζ¯ z)−2k , k = n/2.

(8.61)

Let us give now a rigorous proof for this. It is enough to explain what is etB+ e for every t ∈ D and and every  ∈ N. For simplicity we assume t ∈] − 1, 1[. Proposition 101 For every m ∈ N, the differential equation φ˙ t = B+ φt , φ0 (z) = z m , has a unique solution holomorphic in (t, z) ∈ D × D given by the following formulas. For m = 0 (8.62) φt (z) = (1 − tz)−n , for m ≥ 1,

φt (z) = (1 − tz)−n − 1 + z m (1 − tz)−n − m.

Proof We check φt (z) =



x (t)z  . So we can compute x (t) using the induction

∈N

formula

(8.63)



t

x+1 (t) = x+1 (0) + (n + )

x (s)ds.

0



The result follows easily. From our computations we get the expansion of ψζ in the canonical basis ψζ = (1 − |ζ |2 )k

 ∈N

Γ (2k + ) Γ ( + 1)Γ (2k)

1/2

ζ¯ e .

(8.64)

258

8 Pseudo-Spin Coherent States

Proposition 102 For every n1 , n2 ∈ PS2 we have D(n1 )D(n2 ) = D(n3 ) exp(−i(n1 , n2 )B0 ),

(8.65)

where (n1 , n2 ) is the oriented area of the geodesic triangle on the pseudosphere with vertices at the points [n0 , n1 , n2 ]. n3 is determined by (8.66) n3 = R(gn1 )n2 , where R(g) is the rotation associated to g ∈ SU (1, 1) and  τ  gn = exp −i (σ1 sin ϕ + σ2 cos ϕ) . 2

(8.67)

Proof Computation of n3 is easy using the following lemma. The phase will be detailed later.  Lemma 54 For all g ∈ SU (1, 1) there exists m ∈ PS2 and δ ∈ R such that g = gm r3 (δ), 

where r3 (δ) = exp i 2δ B0 . The following lemma shows that the pseudospin is also independent of the direction. Lemma 55 One has:

D(n)Bˆ 0 D(n)−1 = −n  B,

(8.68)

where  is the notation for the Minkowski scalar product 235 Proof Let (τ, ϕ) be the pseudopolar coordinates of n, n = n(τ, ϕ). We have D(n(τ )) = exp(−iτ (cos ϕB1 − sin ϕB2 )). Let us use the notation A(τ ) := D(n(τ ))AD(n(τ ))−1 where A is any operator in H (D). Then we have the equalities d B0 (τ ) = − cos ϕB2 (τ ) − sin ϕB1 (τ ), dτ d B1 (τ ) = − sin ϕB0 (τ ), dτ d B2 (τ ) = − cos ϕB0 (τ ). dτ

(8.69) (8.70) (8.71)

We have the following consequences d2 d B0 (0) = − cos ϕB2 − sin ϕB1 , B0 (τ ) = B0 (τ ), 2 dτ dτ

(8.72)

8.3 Pseudo-Spin-Coherent States for Discrete Series

259

hence we get that B0 (τ ) = −n(τ )  B.



The following consequence is that |n is an eigenvector of the operator n  B: Proposition 103 One has: n  B|n = −k|n,

(8.73)

where k = 2n . As in the Heisenberg and spin settings, the pseudospincoherent states family |n is not an orthogonal system in H (D). One can compute the scalar product of two coherent states |n, |n : Proposition 104 One has

n |n = e−ik(n,n )



1 − n  n 2

−k

,

(8.74)

where (n, n ) is the oriented area of the hyperbolic triangle {n0 , n, n }. Proof We use the complex representation of coherent states, starting from the definition, we get ψζ (z) = (1 − |ζ |2 )k (1 − ζ¯ z)−2k . (8.75) Next we use the series expansion (1 − ζ¯ z)−2k =

(2k +  − 1)! ≥0

(2k − 1)!!

(ζ¯ z)

to compute the Fourier coefficient of the coherent state |ζ  in the basis e : 

Γ (2k + ) Γ ( + 1)Γ (2k)

1/2

(1 − |ζ |2 )k ζ  .

(8.76)

n |n = (1 − |ζ |2 )k ((1 − |ζ |2 )k (1 − ζ¯ ζ )−2k .

(8.77)

e |ζ  = The Parseval identity gives

We can translate this equality in pseudopolar cordinates (ζ = tanh(τ/2)e−iϕ ) and we get −2k 

. n |n = cosh(τ/2) cosh(τ /2) − sinh(τ/2) sinh(τ /2)ei(ϕ −ϕ)

(8.78) 

An easy computation now gives the following lemma

260

8 Pseudo-Spin Coherent States

Lemma 56



|n |n| = 2

1 − n  n 2

−2k

.

(8.79)

The computation of the phase  in formula (8.74) can be done as for the spin case using the geometric phase method. As it is expected, the pseudospin coherent state system provides a “resolution of the identity” in the Hilbert space H (D): Proposition 105 We have the formula 2k − 1 4

PS2

d n|nn| = 1l .

(8.80)

Or using complex coordinates |ζ , D

d ν2k (ζ )|ζ ζ | = 1l .

(8.81)

where the measure d νn is d νn (ζ ) = with d 2 ζ =

d 2ζ n−1 , π (1 − |ζ |2 )2

|d ζ ∧d ζ¯ | . 2

Proof The two formulas are equivalent by the change of variables ζ = tanh τ2 e−iϕ . So it is sufficient to prove the complex version. We introduce f˜ (ζ ) = ζ |f H n (D) . Decompose f in the basis of Hn (D), f =



c e , we have

≥0

|f  (ζ )|2 =

≥0

Γ (2k + ) (1 − |ζ |2 )2k |ζ |2 |c |2 . Γ ( + 1)Γ (2k)

After integration in ζ we have 2k − 1 π

D

|f  (ζ )|2

d 2ζ = (1 − |ζ |2 )2

D

|f (z)|2 d 2 z.

So we get the resolution of identity by a polarisation argument.



8.3 Pseudo-Spin-Coherent States for Discrete Series

261

8.3.3 Bargmann Transform and Large k Limit Here we introduce the (pseudospin) Bargmann transform and prove that as k → − contracts to the Harmonic oscillator representation or +∞ the representation D2k Heisenberg-Schrödinger-Weyl representation. Let us denote ϕ k, (ζ ) = ζ |ϕH n (D) (1 − |ζ |2 )−k , ϕ ∈ H2k(D) , ζ ∈ D. In fact this transformation is trivial, here it is identity! But it is convenient to see this as a Bargmann transform. Using Parseval formula we get easily ϕ k, (ζ ) =



−1/2

ζ  γ2k, e , ϕH n (D) =

∈N



e (ζ )e , ϕH n (D) .

∈N

Here we shall note the dependence in the Bargmann index k, so we denote the pseudospin coherent state ψζk (z). Proposition 106 The pseudocoherent states ψζk converge to the Glauber coherent state ϕζ (see Chap. 1) as k → +∞ in the following Bargmann sense and the Dicke states ek converge to the the Hermite function ψ , for every  ∈ N. √ k,  lim ψζ /√2k (ζ / 2k) = ϕζ (ζ ), ∀ζ, ζ ∈ C.

k→+∞

(8.82)

Proof It is an easy exercise, knowing that   |ζ |2  ϕζ (ζ ) = exp ζ¯ ζ − 2 and that

ζ  ψ (ζ ) = √ . 2π ! 

As for the spin case we have analogous results for the generators of the Lie algebras. Let us introduce a small parameter ε > 0 and denote ε B± = εB± ,

B0ε = B0 −

(8.83) 1 1l. 2ε2

We have the following commutation relations

(8.84)

262

8 Pseudo-Spin Coherent States ε ε ε ε [B+ , B− ] = 2ε2 B3ε − 1l, [B3ε , B± ] = ±B± .

(8.85)

As ε → 0 Eq. (8.83) define a family of singular transformations of the Lie algebra su(1, 1) and for ε = 0 we get (formally) 0 0 0 0 , B− ] = −1l, [B30 , B± ] = ±B± . [B+

(8.86)

These commutation relations are those satisfied by the harmonic oscillator Lie algebra 0 0 ≡ a† , B− ≡ a, B00 ≡ N := a† a. : B+ We can give a mathematical proof of this analogy by computing the averages. Proposition 107 Assume that ε → 0 and k → +∞ such that lim 2kε2 = 1. Then we have (8.87) limψζk/√2k |B0ε |ψζk/√2k  = |ζ |2 = ϕζ |a† a|ϕζ . ε |ψζk/√2k  = ζ¯ = ϕζ |a† |ϕζ . limψζk/√2k |B+

(8.88)

ε |ψζk/√2k  = ζ = ϕζ |a|ϕζ . limψζk/√2k |B−

(8.89)

Proof From the proof of Lemma 55 we can compute the following averages. ψnk , Bψnk  = kn. Using the ζ parametrization we get the result as in the spin case.



8.4 Coherent States for the Principal Series As for discrete series we can consider coherent states for the principal and complementary series. The principal series is realized in the Hilbert space L2 (S1 ) with the Haar probability measure on the circle S1 and with its orthonormal basis e (θ ) = eiθ ,  ∈ Z. e0 being invariant by the rotations subgroup of S(1, 1) we define the coherent λ (z), z ∈ S1 , as states ψn,ζ ψniλ (z) = | cosh(τ/2) + sinh(τ/2)e−iϕ z|2iλ−1 , 1 ψζiλ (θ ) = |1 − |ζ |2 | 2 −iλ |1 − ζ¯ z|2iλ−1 .

(8.90) (8.91)

In the first formula coherent states are parametrized by the pseudo-sphere and in the second formula they are parametrized by the complex plane. Properties of these coherent states are analyzed in the book [208] (pp. 77–83).

8.5 Generator of Squeezed States. Application

263

8.5 Generator of Squeezed States. Application We shall prove here that the SU (1, 1) generalized coherent states considered above (introduced by Perelomov [208]) are nothing but the one dimensional squeezed states introduced in Chap. 3. We consider the realization of the Lie algebra su(1, 1) defined by the generators 1 (aa† + a† a), 4 1 B+ = (a† )2 , 2 1 B− = a2 . 2 B0 =

(8.92) (8.93) (8.94)

These generators are defined as closed operators in the Hilbert space L2 (R). They obey the commutation rules (8.41). Furthermore for the Casimir operator we have 3 1l. Cas = − 16 We have already remarked in Sect. 8.2.6 that this representation of the Lie algebra su(1, 1) gives a projective representation of SU (1, 1) (not a genuine group representation).

8.5.1 The Generator of Squeezed States Consider now one dimensional squeezed states. Recall the following definition. Take a complex number ω such that |ω| < 1. We define β(ω) =

ω arg tanh(|ω|), |ω|

D(β) = exp(β Bˆ + − β¯ Bˆ − ). D(β) is also known as the “Bogoliubov transformation” and generates squeezing. For |0 = ϕ0 being the ground state of B0 , let the generalized coherent state ψβ be defined as: ψβ = D(β)|0

ˆ Pˆ of quantum Remark 50 It is not difficult to show that in terms of the operators Q, mechanics one has:   i i 2 2 ˆ ˆ ˆ ˆ ˆ ˆ β(Q − P ) − β(QP + P Q) D(β) = exp 2 2

264

8 Pseudo-Spin Coherent States

We first recall the fundamental property of D(β) proved in Chap. 3 in any dimension. ˆ ∩ D(P) ˆ the following identities holds true: Lemma 57 On D(Q) (i) D(β) is unitary and satisfies: D(β)−1 = D(−β), (ii)

D(β)aD(−β) = (1 − |ω|2 )−1/2 (a − ωa† ),

(iii) D(β)B0 D(−β) = cosh(2r)B0 −

sinh(2r) (B+ eiθ + B− e−iθ ), 2

with β = reiθ being the polar decomposition of β into modulus and phase. The following results are direct consequences of Chap. 3. Proposition 108 (i) Define δ =

1−ω . 1+ω

One has that δ > 0

and

 ψβ (x) =

δ π

1/4 

1+ω |1 + ω|

1/2

  x2 . exp −δ 2

(ii) More generally if φk is the k-th normalized eigenstate of B0 (Hermite function) one has 2−k/2 D(β)φk = √ n!



δ π

1/4 

1+ω |1 + ω|

k+1/2

  √ δx2 . Hk (x δ) exp − 2

where Hk is the normalized k-th Hermite polynomial. Now we address the following question: what is the Wigner function of a squeezed state? It will appear that it is a Gaussian in q, p but with squeezing in some direction and dilatation in the other direction. One has the following result: Proposition 109 The Wigner function Wψβ (q, p) is given by   2 1 q δ − (p + qδ)2 . Wψβ (q, p) = 2 exp −  δ Remark 51 (i) For β = 0, δ = 1, and thus we recover the Wigner function of ϕ0 . (ii) It is clear that 1 dqdpWψβ (q, p) = 1. 2π 

8.5 Generator of Squeezed States. Application

265

The proof is an easy computation of Gaussian integrals.

8.5.2 Application to Quantum Dynamics Consider the time dependent quadratic Hamiltonian Hˆ 2 (t) = λ(t)B+ + λ¯ (t)B− + μ(t)B0 ,

(8.95)

where λ and μ are C 1 functions of t, λ is complex and μ is real. Its propagator is denoted U2 (t, s). This is a particular case of general quadratic Hamiltonian studied in Chap. 1 and in Chap. 3. We revisit here the computation of U2 (t, s) using the su(1, 1) Lie algebra relations satisfied by {B0 , B+ , B− }. It is convenient to formulate the result in an abstract setting. Proposition 110 Assume that B0 , B± are closed operators with a dense domain in ∗ = B− and satisfying the commutation an Hilbert space H such that B0∗ = B0 , B+ relations : [B− , B+ ] = 2B0 , [B0 , B± ] = ±B± . Then Hˆ 2 (t) defined by (8.95) has a propagator given by U2 (t, s) = D(βt ) exp (i(γt − γs )B0 ) D(−βt ),

(8.96)

where the complex function βt and the real function γt satisfy the differential equations iω˙ t = λ¯ ωt2 + μωt + λ, ω0 = 0. γ˙ = −λω¯ − λ¯ ω − μ, γ0 = 0.

(8.97) (8.98)

Proof The first step is to compute the following derivatives i

d D(βt ) = (αt B+ + α¯ t B− + ρt B0 )D(βt ), dt

(8.99)

where ω˙ t , 1 − |ω|2 ωt ω˙¯ t − ω˙ t ω¯ t ρt = i . 1 − |ωt |2

αt = i

Using (8.99) we can compute

(8.100) (8.101)

266

8 Pseudo-Spin Coherent States

i

d U2 (t, s) = (αt B+ + α¯ t B− + ρt B0 − γ˙ D(βt )B0 ) D(−βt ) dt

and we get directly equations (8.97). Let us prove now (8.99). The method is the following. Denote L(t) = βt B+ − β¯t B− . We have L(t + δt) = L(t) + δL(t) ≈ L(t) + δ

d L(t). dt

Applying Duhamel formula we get eL(t+δt) − eL(t) =

1

dsesL(t+δt) δL(t)e(1−s)L(t) .

0

Then as δt → 0 we have d L(t) e = dt



1

(1−s)L(t) ˙ dsesL(t) L(t)e .

0

˙ = β˙t B+ − β˙¯t B− and Now we have L(t) d sL(t) ¯ sL(t) B0 e−sL(t) e B+ e−sL(t) = −2βe ds and

d sL(t) e B− e−sL(t) = −2βesL(t) B0 e−sL(t) . ds

Using Lemma 57 we get formula (8.99).



Remark 52 The differential equation satisfied by ωt is a Ricatti equation. We have seen in Chap. 4 that this equation comes from a classical flow. In particular ωt is defined for every time t and satisfied |ωt | < 1 (ω0 = 0). Let us now consider the time dependent Hamiltonian  2 1 ˆ2 ˆ2 + g , Hˆ g (t) = P + f (t)Q ˆ2 2 2Q where g is a coupling constant and f a function of time t. Properties of this Hamiltonian have been considered by [208] and used in [60] to study the quantum dynamics for ions in a Paul trap. The su(1, 1) Lie algebra relations are satisfied by B0 =

ˆ2 ˆ 2 − Pˆ 2 g2 g2 Pˆ 2 + Q Q QP + PQ + − . , B± = ∓ ˆ2 ˆ2 4 4 4 4Q 4Q

(8.102)

8.5 Generator of Squeezed States. Application

267

So we get 1 1 Hˆ g (t) = (f (t) − 1)B+ + (f (t) − 1)B− + (1 + f (t))B0 . 2 2 2

The algebra is the same as above but here the potential 2gQˆ 2 has a non integrable singularity and we have to take care of the domain of definition for operators B0 , B± . Let us consider the Hilbert space L2 (R+ ). Recall the Hardy inequality

+∞

dx 0

|u(x)|2 ≤4 |x|2



+∞ 0

dx|u (x)|2 , ∀u ∈ H01 (R+ ).

Recall that H01 (R+ ) is the Sobolev space H 1 (R+ ) with the condition u(0) = 0. Denote by L2 1 (R+ ) the space {u ∈ L2 (R+ ), xu ∈ L2 (R+ )}. The sesquilinear form (u, v) → u, B0 v is well defined on V := H01 (R+ ) ∩ L21 (R+ ) and is Hermitian, non negative. So B0 has a self-adjoint extension as a unbounded operator in L2 (R+ ). Furthermore we see that B± are also defined as forms on V and have closed extensions in L2 (R+ ) ∗ = B− . These extensions also satisfy the su(1, 1) Lie algebra relations. such that B+ Hence the unitary operators D(β), eiγ B0 are well defined in L21 (R+ ) with β complex and γ real. So we can apply Proposition 110 to the propagator Ug (t, s) of Hg (t). Corollary 28 The Hamiltonian Hg (t) has a time dependent propagator Ug (t, s) given by (8.96) (8.103) Ug (t, s) = D(βt ) exp (i(γt − γs )B0 ) D(−βt ), where B0 , B± are defined by Eq. (8.102) and βt , γt are determined by (8.97). Moreover they are related to complex solutions of the Newton equation ξ¨t = f (t)ξt , ξ¨0 = ig, ωt =

ξ + iξ˙ , ξ − iξ˙

1 γt = − arg(ξ − iξ˙ ). 2

(8.104)

Remark 53 Note that the solution of the classical equation of motion for H0 (t) solves the quantum evolution problem for Hˆ g for every g ∈ R. The su(1, 1) Lie algebra can also be used to solve the stationary Schrödinger equations for the hydrogen atom. It is nothing else than a group theoretic approach of a method already used by Schrödinger himself [234]. Let us consider the radial Hamiltonian for the hydrogen atom with mass 1,  = 1, charge e, energy E > 0. 

 d2 ( + 1) 2 d2 e2 + +2 − + 2E R(r) = 0. dr 2 r dr 2 r r2

(8.105)

268

8 Pseudo-Spin Coherent States

We transform this equation by the change of variable r = x2 and of function R(r) = x−3/2 f (x). Then we get 

 d2 4( + 1) + 3/4 2 2 − 8Ex − + 8e f = 0. dx2 x2

1 1/4 Another change of variable x = λu with λ = (− 8E ) gives



 d2 4( + 1) + 3/4 2 2 2 − u − + 8λ e ) f = 0. du2 u2

(8.106)

This equation is the eigenvalue equation for the generator B0 with g 2 = 4( + 1) + 3/4. So the negative energies of the radial Schrödinger equation (8.105) are determined by the eigenvalues of B0 . Lemma 58 The self adjoint operator B0 has a compact resolvent. Its spectrum is a discrete set of simple eigenvalues given by 1 2s + 1 + k, k ∈ N, where s = + λk = 4 2



1 + g2 4

1/2 .

(8.107)

Proof The domain of B0 is included in H01 (R+ ) ∩ L21 (R+ ) so we deduce that its resolvent is compact. The computation of the spectrum is standard, using B− and B+ as annihilation and creation operators on the ground state ψ0 . using the results of Sect. 8.2.1. We compute the ground state by solving equation B− ψ0 = 0. This a singular differential equation. We put ψ0 (x) = xs ϕ(x) We can eliminate the singularity   2by

1/2 . Then the equation is satisfied if ϕ(x) = exp −x2 . choosing s = 21 + 41 + g  2 So we have ψ0 (x) = C0 xs exp −x2 where s is like in (8.107) and B0 ψ0 = 2s+1 ψ0 . 4 k ψ0 where Then we get all the spectrum of B0 and all the bounded states ψk = Ck B+ 2  the constants CK are chosen to have an orthonormal basis in L (R+ ).

Applying this lemma and formula (8.106) we get that Eq. (8.105) has non trivial e4 solutions for E = En = − 2 , n ≥ 1, the well known energy levels of the hydrogen 2n atom. More properties will be given in the next chapter. Remark 54 The Casimir operator is here Cas = cas 1l. cas is computed by 1 Cas ψ0 = (B02 − (B+ B− + B+ B− ))ψ0 = k(k − 1)ψ0 , 2 with k = 2s+1 . This is not compatible with a discrete representation of SU (1, 1) 4 except if s is half an integer. What we have considered here is an irreducible repre (1, 1). sentation of the universal cover SU

8.5 Generator of Squeezed States. Application

269

It could be possible to study coherent states ϕβ = D(β)ψ0 in this representation too as we have done for the discrete representations.

8.6 Wavelets and Pseudo-Spin Coherent States As it is well known wavelets are associated with the affine group of transformations of the real axis R: t → at + b where a > 0 and b ∈ R. We denote g(a, b) this affine transformation and AF the group of all affine transformations. Wavelets are real functions defined by the action of the group AF on a given function ψ so we have   t−b 1 . ψa,b (t) = √ ψ a a If ϕa,b is the Fourier transform of ψa,b and ϕ the Fourier transform of ψ, we have ϕa,b (s) =

√ −ibs ae ϕ(as).

Now we shall see, following ideas taken from the paper [29], that wavelets and pseudo-spin coherent states are closely related. This is not surprising using the following facts: the affine group is isomorph to a subgroup of the group SL(2, R)1 which is isomorph to SL(2, R)/SO(2) and this one is isomorph to SU (1, 1)/U (1).2 Recall that the groups SU (1, 1) and SL(2, R) are conjugate   1 1i −1 CSU (1, 1)C = SL(2, R), where C = √ . 2 i 1 Considering consequences of these facts, we shall get that the discrete series representations of SU (1, 1) have realizations with strong connections with the affine group hence relationship between pseudo-coherent states and generalized wavelets will follow. Remark that the affine group can be identify to R∗+ × R with the group law: (a, b) × (a , b ) = (aa , ab + b). Let us consider the mapping √ M : (a, b) →

a 0

√b a √1 a

 .

It is easy to see that M is a group isomorphism from AF into SL(2, R). Its image is denoted Asl .

1 Recall 2 Recall

that SL(2, R) is the group of 2 × 2 real matrices of determinant   −iθ one. 0 e that U (1) is identified here with the group of matrices , θ ∈ R. iθ 0 e

270

8 Pseudo-Spin Coherent States

SO(2) is a compact subgroup of SL(2, R) and it is not difficult to prove that the quotient space SL(2, R)/SO(2) can be identified to Asl :   cos θ sin θ Lemma 59 For every A ∈ SL(2, R) there exits a rotation R(θ ) = − sin θ cos θ and a unique affine transformation (a, b) ∈ R∗+ × R such that A = M (a, b)R(θ ). In particular the left cosets set SL(2, R)/SO(2) is isomorph to Asl . Let us consider the discrete series Dn+ which will be now denoted Dn (n ≥ 2 is an integer, n = 2k where k is the Bargmann index). Using the isomorphism ζ → z(ζ ) = ζ +i from the unit disc D onto the Poincaré half plane H, Dn can be realized in the 1+iζ Hilbert space Hn (H). Hn (H) is the space of holomorphic functions f in H such that f 2H n (H) =

n−1 π

H

|f (X + iY )|2 Y n−2 dXdY < +∞,

with the natural norm. So we have a unitary map f → F from Hn (H) onto Hn (D) where F(ζ ) = 2(1 + iζ )−n f (z(ζ )). In Hn (H) the discrete series Dn gives naturally a unitary representation of SL(2, R)       az − c ab Dn . f (z) = (d − bz)−n f cd −bz + d

(8.108)

To establish a connection with the affine group it is convenient to realize the representation Dn in the space Hˇn (H) of anti-holomorphic functions on H, so that f ∈ Hˇn (H) means that fˇ (z) = f (¯z ) with f ∈ Hn (H). f → fˇ is a unitary map. We denote Iˇn F = fˇ . Then Dn is unitary equivalent to the representation Dˇ n 

     dz − b ab . Dˇ n f (z) = (a − cz)−n f cd −cz + a

(8.109)

Note that the unitary equivalence between Dˇ n and Dn is implemented by the group, isomorphism in SL(2, R),     ab d c → . cd ba The restriction of Dˇ n to the affine group has the following expression 

   z−b . Dˇ n (M (a, b))f (z) = a−n/2 f a

(8.110)

Wavelets are functions of a real variable, so the last step is to find a realization of Dn in the space

8.6 Wavelets and Pseudo-Spin Coherent States

L2n (R+ ) = {ϕ|

+∞

271

t 1−n |ϕ(t)|2 dt < +∞}.

0

with the natural norm. Let us introduce the (anti-holomorphic) Fourier-Laplace transform: (Ln ϕ)(z) = cn

+∞

ϕ(t)e−it¯z dt, z ∈ C, (z) > 0,

0

cn is a normalization constant. Using the Fourier inverse formula and the Plancherel formula we have 1 ty e ϕ(t) = 2π cn 1 2π cn



+∞

−∞

Ln ϕ(x − iy)eitx dx, y > 0, t > 0.

H

|Ln ϕ(x − iy)|2 yn−2 dxdy = Γ (n − 1)21−n

+∞

(8.111)

t 1−n |ϕ(t)|2−n dt. (8.112)

0

So we get isometries between the spaces L2n (R+ ), H (H) and Hˇ (H) choosing cn = 2n−2 . We have obtained the following irreducible unitary representation of the π(n−2)! affine group in the space L2n (R+ ). Wn (a, b) := Ln−1 Dˇ n (M (a, b))Ln . ϕa,b (t) := (Wn (a, b)ϕ) (t) = a1−n/2 e−ibt ϕ(at), which represent wavelets on the Fourier side. Coherent states for SU (1, 1) where defined in the Hilbert space Hn (D) by an action of SU (1, 1), starting from a fiducial states ψ0 invariant by the action of the unit circle U (1) (isomorph to SO(2)). Then SU (1, 1) coherent states are parametrized by the quotient SU (1, 1)/U (1). But from Lemma 59 and the isomorphism between SU (1, 1) and SL(2, R) we get that SU (1, 1)/U (1) can also be parametrized by the affine group: (a, b) → ga,b where we choose one element ga,b in each left coset. We have seen above in the construction of coherent states that SU (1, 1)/U (1) can be parametized by C (or by by pseudosphere) :ξ → gξ . If ga,b and gξ are in the same coset then we have ga,b = gξ h where h ∈ U (1). We have choosen ψ0 rotation invariant so the actions of ga,b and gξ define the same coherent state. Let us move this construction in Hn (H) and in L2n (R+ ). We get respectively fiducial states f0 (z) = dn (1 − iz)n and ϕ0 (t) = en t n−1 e−t where dn and en are suitable constants. Then using properties of the representation Dn we get a bijective correspondence between SU (1, 1) coherent states defined in Hn (D) for Dn+ and wavelets in L2n (R+ ). More precisely we have obtained

272

8 Pseudo-Spin Coherent States

ϕa,b (t) := (Wn (a, b)ϕ0 ) (t) = a1−n/2 e−ibt ϕ(at), which represent wavelets on the Fourier side. Their relationship with the SU (1, 1) coherent states is given by Ln−1 Iˇn ψξ = ϕa(ξ ),b(ξ ) , ∀ξ ∈ C, Iˇn−1 Ln ϕa,b = ψξ(a,b) , ∀(a, b) ∈ R∗+ × R,

(8.113) (8.114)

where ξ → (a(ξ ), b(ξ )) is a bijection from C onto R∗+ × R and (a, b) → ξ(a, b) is a bijection from R∗+ × R onto C. In particular we also have a resolution of identity for wavelets which can be obtained from 8.81 or by a direct computation. n−1 f = 4π



R∗+ ×R

dadb ϕab , f ϕa,b , ∀f ∈ L2n (R+ ). a2

(8.115)

Hn (D), Hn (H) and L2n (R+ ). Finally remark that Wn is a representation in L2n (R+ ) of a subgroup of SU (1, 1) conjugated to the restriction of Dn+ . But all the representations Wn are equivalent contrary to the representations Dn+ which are non-equivalent. If Mn is the unitary map Mn ϕ(t) = t 1−n/2 ϕ(t) from L2n (R+ ) onto L2 (R+ ) then we have clearly Mn Wn = W2 Mn so Wn and W2 are conjugate for every n ≥ 2.

Hyperbolic triangles in the Poincaré disc

Chapter 9

The Coherent States of the Hydrogen Atom

Abstract The aim of this chapter is to present a construction of a set of coherent states for the hydrogen atom proposed by Villegas-Blas (The Laplacian on the n-sphere, the hydrogen atom and the Bargmann space representation. Ph.D. thesis, Mathematics Department, University of Virginia, 1996, [251]), Thomas and Villegas-Blas (Commun Math Phys 187:623–645, 1997, [244]). We show that in a semiclassical sense they concentrate essentially around the Kepler orbits (in configuration space) of the classical motion. A suitable unitary transformation (the Fock operator) maps the pure point subspace of the hydrogen atom Hamitonian onto the Hilbert space L 2 (S3 ) for the S3 sphere. We study the coherent states for the S3 sphere (as introduced by Uribe (J Funct Anal 59:535–556, 1984, [247]) and show that the action of the group S O(4) is irreducible in the space generated by the spherical harmonics of a given degree. Note that coherent states for the hydrogen atom have been extensively studied by J. Klauder and his school. We have chosen not to present them here and refer the interested reader to Klauder and Skagerstam (Coherent states, Worlds Scientific Publishing, Singapore, 1985, [167]).

9.1 The S3 Sphere and the Group S O(4) 9.1.1 Introduction It is well known that the non-relativistic quantum model for the hydrogen atom is 2 1 , p, x ∈ R3 . the quantization Hˆ of the Kepler Hamiltonian H (x, p) = | p|2 − |x| The natural symmetry group for H seems to be the rotation group S O(3). We shall see in Sect. 9.2 that the hydrogen atom has “hidden symmetries” and its symmetry group is the larger group S O(4) which explain the large degeneracies of the energy levels of Hˆ . This is why we start by studying the group S O(4), its irreducible representations and hyperspherical harmonics. Recall that S O(4) is the group of direct isometries of the Euclidean space R4 or its unit sphere S3 ,   S3 = (x = (x1 , . . . , x4 ) |x12 + x22 + x32 + x42 = 1 . © Springer Nature Switzerland AG 2021 D. Robert and M. Combescure, Coherent States and Applications in Mathematical Physics, Theoretical and Mathematical Physics, https://doi.org/10.1007/978-3-030-70845-0_9

273

274

9 The Coherent States of the Hydrogen Atom

Let us introduce the Laplacian ΔS3 for the sphere S3 , as the restriction to the unit  ∂2 . More explicitly computing ΔR4 sphere S3 of the Laplace operator ΔR4 := ∂ x 2j 1≤ j≤4 in hyperspherical coordinates: x1 = r sin χ sin θ cos ϕ x2 = r sin χ sin θ sin ϕ x3 = r sin χ cos θ x4 = r cos χ , where χ , θ ∈ [0, π [, ϕ ∈ [0, 2π [, we get ΔR 4

1 ∂ = 3 r ∂r

  1 3 ∂ r + 2 ΔS3 ∂r r

(9.1)

(9.2)

where 1 ∂2 2 ∂ + ΔS2 , where + 2 ∂χ tan χ ∂χ sin2 χ ∂2 1 ∂ 1 ∂2 + 2+ = . tan θ ∂θ ∂θ sin2 θ ∂ϕ 2

ΔS3 =

(9.3)

ΔS2

(9.4)

Equation (9.2) defines the operator ΔS3 . It is a self adjoint operator in the Hilbert space L 2 (S3 ) for the Euclidean measure dμ3 (θ, ϕ, χ ) = sin2 χ sin θ dθ dϕdχ . The measure dμ3 and the operator ΔS3 are invariant by the group S O(4). The spectrum of ΔS3 can be described has follows: there exists an explicit constant c ∈ R such that if Δ3 := −ΔS3 + c then Δ3 has the discrete spectrum 

 λk = (k + 1)2 |k ∈ N .

Each λk is known to have multiplicity (k + 1)2 (see [197] or what follows), which coïncides with multiplicities of bound states of hydrogen atom as we shall see later.

9.1.2 Irreducible Representations of S O(4) S O(4) is a compact Lie group so we know that all its irreducible representations are finite dimensional. The Lie algebra so(4) of S O(4) is the algebra of antisymmetric 4 × 4 real matrices; so(4) has dimension 6 so the Lie group S O(4) has dimension 6. We shall see that its irreducible representations can be deduced from irreducible representations of SU (2) (computed in the chapter “Spin Coherent States” which will be denoted (SCS)).

9.1 The S3 Sphere and the Group S O(4)

275

It is convenient to use the quaternion field H and its generators {1, I , J , K }. H is a 4-dimensional real   linear space which can be represented as the space of 2 × 2 a b matrices q = where a, b ∈ C. The basis is related with Pauli matrices: −b¯ a¯ 1 = σ0 , I = iσ3 , J = iσ2 , K = iσ1 . The following properties are easy to prove 1. {1l, I , J , K } is an orthonormal basis for the scalar product q, q   := 21 tr(q · q  ). So H can be identified with the Euclidean space R4 . 2. q

· q = q · q = (|a|2 + |b|2 )1l. In particular if q = 0, q is invertible and q −1 = q where |q|2 = |a|2 + |b|2 . |q|2 3. SU (2) = {q ∈ H, |q| = 1} 4. su(2) = {q ∈ H, q + q = 0}. A quaternion q is said pure (or imaginary) if q + q = 0 and real if q = q. 5. g ∈ SU (2) ⇐⇒ g = eθ A = cos θ + (sin θ )A, θ ∈ R, A a pure quaternion. Now we can identify S O(4) with the direct isometries group of the Euclidean space H. In particular if for any g1 , g2 ∈ SU (2) we define τ (g1 , g2 )q = g1 · q · g2−1 then τ (g1 , g2 ) ∈ S O(4). Furthermore we have Proposition 111 τ is a group morphism from SU (2) × SU (2) in S O(4). The kernel of τ is ker τ = {(1l, 1l), (−1l, −1l)} and τ is surjective. In particular the group S O(4) is isomorphic to the quotient group SU (2) × SU (2)/{(1l, 1l), (−1l, −1l)} and its Lie algebra so(4) is isomorphic to the Lie algebra su(2) ⊕ su(2). Proof It is clear that τ is a group morphism. (g1 , g2 ) ∈ ker τ means that g1 · q = q · g2 , ∀q ∈ H. So we get successively :g1 = g2 , g1 = λ1, λ ∈ C, λ = ±1 because g1 ∈ SU (2). To prove that τ is surjective, we use that τ (SU (2) × SU (2)) is a subgroup of S O(4), its action on H is transitive and that we have a surjective group homomorphism g → Rg from SU (2) in S O(3) (acting in pure quaternions). Let A ∈ S O(4). If A1 = 1 then there exists g ∈ SU (2) such that A = τ (g, g). If A1 = q then there exist g1 , g2 ∈ SU (2) such that q = τ (g1 , g2 )1 so we can write A = τ (g1 · g,  g2 · g). Now we use the following classical result to deduce irreducible representations of S O(4) (for a proof see [44]). Proposition 112 Let G 1 , G 2 be two compact Lie groups, (ρ1 , V1 ) and (ρ2 , V2 ) two irreducible representations of G 1 and G 2 respectively. Then (ρ1 ⊗ ρ2 , V1 ⊗ V2 ) is an irreducible representation of G 1 × G 2 . Conversely every irreducible representation of G 1 × G 2 is like this.

276

9 The Coherent States of the Hydrogen Atom

Corollary 29 Let (ρ, V ) be an irreducible unitary representation of S O(4). Then there exists j1 , j2 ∈ N2 such that j1 + j2 ∈ N and such that (ρ, V ) is unitarily equivalent to (T ( j1 ) ⊗ T ( j2 ) , V ( j1 ) ⊗ V ( j2 ) ). Proof Using Proposition 111 we can asume that (ρ, V ) is an irreducible representation of SU (2) × SU (2)/{(1l, 1l), (−1l, −1l)}. So (ρ, V ) is an irreducible representation of SU (2) × SU (2) and from Proposition 112 we have (ρ, V ) ≡ (T ( j1 ) ⊗ T ( j2 ) , V ( j1 ) ⊗ V ( j2 ) ). To get a representation of SU (2) × SU (2)/{(1l, 1l), (−1l, −1l)} it is necessary and sufficient that T ( j1 ) (−1l) ⊗ T ( j2 ) (−1l) = 1l V (1) ⊗V (2) . So we find the condition j1 + j2 ∈ N.



9.1.3 Hyperspherical Harmonics and Spectral Decomposition of ΔS3 As we have already seen in Chap. 7 for S O(3) we shall see now that irreducible representations of S O(4) are closely related with the hyperspherical harmonics and spectral decomposition of ΔS3 . Let us introduce the space P4(k) of homogeneous polynomials f (x1 , x2 , x3 , x4 ) in the four variables (x1 , x2 , x3 , x4 ) of total degree k ∈ N and H˜4(k) the space of f ∈ P4(k) such that ΔR4 f = 0. H˜4(k) is determined by its restriction H4(k) to the sphere S3 which is by definition the space of hyperspherical harmonics of degree k. Let f ∈ P4(k) . For x ∈ R4 we have x = r ω, r > 0, ω ∈ S3 and f (x) = r k ψ(ω). So, using (9.2) we have ΔS3 ψ = −k(k + 2)ψ, ∀ψ ∈ H4(k) .

(9.5)

Let us introduce the modified Laplacian Δ3 = −ΔS3 + 1. Then H4(k) is an eigenspace for Δ3 with eigenvalue λk = (k + 1)2 . The group S O(4) has a natural unitary representation in L 2 (S3 ) defined by the formula ρg ψ(ω) = ψ(g −1 ω), g ∈ S O(4), ω ∈ S3 . ρ commutes with ΔS3 and each H4(k) is invariant by ρ. c The main goal of this sub-section is to prove the following results Theorem 45 (i) The Hilbert space L 2 (S3 ) is the direct Hilbertian sum of hyperspherical subspaces:

9.1 The S3 Sphere and the Group S O(4)

L 2 (S3 ) =



277

H4(k) , (H4(0) = C).

(9.6)

k∈N

Moreover we have

dim(H4(k) ) = (k + 1)2 .

(9.7)

(ii) For every k ∈ N the representation (ρ, H4(k) ) is irreducible and equivalent to the representation (T (k/2) ⊗ T (k/2) , V (k/2) ⊗ V (k/2) ) where (T ( j) , V ( j) ) is the irreducible representation of SU (2) defined in Chap. 7. Proof We shall follow more or less the proof of the similar Proposition in Chap. 7. We first get the following decomposition for homogeneous polynomial spaces in R4 , P4(k) = H˜4(k) ⊕ r 2 P4(k−2) P (k) = H˜ (k) ⊕ r 2 H˜ (k−2) ⊕ · · · ⊕ r 2 H˜ (k−2) 4

4

4

(9.8) (9.9)

4

+ x22 + x32 + x42 . where k − 1 ≤ 2 ≤ k and r 2 is multiplication by r 2 := x12 Using the Stone-Weierstrass theorem it follows that H4(k) is dense in L 2 (S3 ) hence we get the decomposition formula (9.6). Furthermore we have

k∈N

dim(P4(k) ) = dim(H4(k) ) + dim(P4(k−2) ), and dim(P4(k) ) =

  k+3 k

so we get formula (9.7) by the binomial Newton formula. Let us prove now that H4(k) is irreducible for the representation ρ. Denote by Σ4(k) the restriction to S3 of P4(k) . Let us consider a fixed point on S3 , for example n0 = (0, 0, 0, 1). We can identify with S O(3) the subgroup of S O(4) (k) be the subspace of ψ ∈ Σ4(k) such that ρg ψ = ψ, ∀g ∈ S O(3). fixing n0 . Let Σ4,3 Irreducibility of H4(k) will follow from the following lemma. (k) Lemma 60 (i) The dimension of Σ4,3 is [ 2k ] + 1 ([λ] is the greatest integer n such that n ≤ λ). (ii) Let E = {0} be a finite dimensional S O(4)-invariant subspace of the space C(S3 ). Then there exists ψ0 = 0, ψ0 ∈ E such that ρg ψ0 = ψ0 ∀g ∈ S O(3).

Admitting this lemma for a moment let us finish the proof of the theorem. For every k we have the following decomposition of Σ4(k) into irreducible factors: Σ4(k) =



E

1≤≤L 

where each E  is equivalent to some V ( j) ⊗ V ( j ) . Let us apply the lemma (ii) to each E  . We get

278

9 The Coherent States of the Hydrogen Atom



k +1 2

(k) L ≤ dim(Σ4,3 )=

But from (9.8) we get

Σ4(k) =



(k−2 j)

H4

(9.10)

.

2 j≤k (k−2 j)

is reducible then Σ4(k) would have at least [ 2k ] + 2 irreducible So if one of the H4 factors which is not possible because of (9.10). So all the spaces H4(k) are irreducible for ρ.  Let us prove now that (ρ, H4(k) ) is equivalent to the representation (T (k/2) ⊗ T (k/2) , V (k/2) ⊗ V (k/2) ). This is a consequence of the Peter-Weyl theorem whose statement is given now. Theorem 46 ([44]) Let G be a compact Lie group with Haar measure dμ and let (λ) ) be the set of all its irreducible unitary representations (up to unitary (ρλ , Vλ∈Λ equivalence). For each λ ∈ Λ consider an orthonormal basis of V (λ) : {ek(λ) }1≤k≤kλ √ (λ) and the matrix elements ak, (g) = kλ ek(λ) , ρλ (g)(e(λ) ), g ∈ G. (λ) Then {ak, , 1 ≤ k,  ≤ kλ , λ ∈ Λ} is an orthonormal basis of L 2 (G). In the quaternion model we can see that the manifolde S3 is isomorphic to the Lie group SU (2) and up to a normalization constant the Hilbert spaces L 2 (S3 ) and L 2 (SU (2)) coincide. In V ( j) consider the orthonormal basis vk , 0 ≤ k ≤ 2 j and define the linear map ( j) by vk ⊗ v → ak, from V ( j) ⊗ V ( j) in L 2 (SU (2)). We get a unitary map U ( j) from V ( j) ⊗ V ( j) on a subspace E ( j) of L 2 (SU (2)). Now applying the Peter-Weyl theorem we have the following unitary equivalences L 2 (S3 ) ∼ L 2 (SU (2)) ∼

 j∈N/2

E ( j) ∼



V ( j) ⊗ V ( j) .

j∈N/2

Using uniqueness for the decomposition into irreducible representations we can  conclude that H4(k) ∼ V (k/2) ⊗ V (k/2) .

9.1.4 The Coherent States for S3 We want to introduce coherent states defined on S3 . Following Uribe [247] we consider a pair of unit vectors a, b in S3 such that

9.1 The S3 Sphere and the Group S O(4)

279

a·b=0 (· denotes the usual bilinear form in C4 ) Let α = a + ib ∈ C4 The geometrical interpretation is that α represents a tangent vector b at the point a ∈ S3 . So the set A = {α = a + ib | a · b = 0, |a| = |b| = 1} can be identified with the unit tangent bundle S(S3 ) over S3 . Moreover the intersection of S3 and the ˚ real plane containing the vectors a, b is a geodesic on S3 which will be denoted α. We have clearly α˚ = {ω ∈ S3 , |ω · α| = 1}. For every α ∈ A , ω ∈ S3 and k ∈ N we define Ψα,k (ω) = (α · ω)k = (a · ω + ib · ω)k . Uribe ([247]) has defined these coherent states for sphere Sn in any dimension n. We shall call them spherical coherent states. Using definition of A we see that Ψα,k is a spherical harmonics of degree k namely it is an eigenfunction of Δ3 with eigenvalue (k + 1)2 . Let us remark that if ω is not on the geodesic α˚ then we have |α · ω| < 1 hence Ψα,k (ω) is exponentially small as k → +∞, so Ψα,k (ω) lives close to the geodesic α˚ when k is large. The L 2 -norm of Ψα,k in L 2 (S3 ) is given by  Ψα,k 2 =

S3

|α · ω|2k dμ3 (ω)

Using hyperspherical coordinates we get 



π

Ψα,k  = 2π 2

sin

2k+2

π

χ dχ

0

sin2k+1 θ dθ

0

Using well known expression for the Wallis integrals wn = Ψα,k 2 =

2π 2 k+1

π/2 0

sinn (θ )dθ , we get (9.11)

280

9 The Coherent States of the Hydrogen Atom x3

x2

O

• a

b

x1

Coherent states on the sphere S3 Let us check now the completeness of the coherent states Ψα,k . Proposition 113 The coherent states Ψα,k form a complete set on the irreducible eigenspace H4(k) of the modified Laplacian Δ3 associated with the eigenvalue λk :  Pk = C(k)

α∈Γ ˚

|Ψα,k Ψα,k |dμ(α) ˚

(9.12)

where Pk is the projector onto H4(k) of Δ3 belonging to the eigenvalue λk , C(k) is a constant of normalization, Γ is the space of geodesics α, ˚ and dμ(α) ˚ is the S O(4) invariant probability measure on Γ . Moreover we can compute C(k): C(k) =

(k + 1)3 2π 2

Proof Let g ∈ S O(4). One defines gα = ga + igb. It is not difficult to see that S O(4) acts on A transitively so that A = {gα0 , |g ∈ S O(4)} , where α0 ∈ A is fixed. Then the Haar probability measure dμ H on S O(4) induces a pushed forward measure (or image measure) dμ(α) on A . The integral operator in the right hand side of (9.12) commutes with any rotation operator ρg given by a rotation g in S O(4). Therefore by Schur’s lemma this integral operator must be a multiple of the identity on each irreducible subspace H4(k) .

9.1 The S3 Sphere and the Group S O(4)

281

The computation of C(k) is straightforward taking the trace in (9.12) and using (9.11).  As we have already remarked for spin coherent states in Chap. 7, the Ψα,k are not mutually orthogonal. Here it is more difficult to compute Ψα,k , Ψα ,k  for α = α  but it is possible to compute this overlap asymptotically for k → +∞ as it is shown in [244, 251]. Here we only state the result. Proposition 114 For any δ > 0 we have for k → +∞ 2π 2 Ψα,k , Ψα ,k  = k



α · α 2

k  1 1 + O(k δ− 2 ) + O(k −∞ ).

(9.13)

Remark 55 α and α  define the same geodesic if and only if |α · α  | = 1. So (9.13) shows that the overlap is exponentially small if and only if α˚ = α˚  . Remark that on A the geodesic flow is multiplication by eit , t ∈ R.

9.2 The Hydrogen Atom 9.2.1 Generalities We consider the Hydrogen atom Hamiltonian 1 1 Pˆ 2 1 − Hˆ := = − ΔR 3 − ˆ 2 2 |x| | Q|

(9.14)

1 Hˆ is the quantization of the Kepler Hamiltonian H (q, p) = p2 − |q| , 3 3 (q, p) ∈ R \{0} × R . The notations are the same as in Chap. 1. The first expression is more often used in physics the second in mathematics. Hˆ is a selfadjoint operator in L 2 (R3 ) with domain 2

D( Hˆ ) = D( Pˆ 2 ) ∩ D(

1 ) = H 2 (R3 ) (Sobolev space, Kato’s result) ˆ | Q|

For simplicity we have taken m = e =  = 1. It is easy to prove that Hˆ commutes with the angular momentum operator Lˆ = ˆ Q ∧ Pˆ which is a consequence of its S O(3) symmetry property: Hˆ commutes with the three generators Lˆ = ( Lˆ 1 , Lˆ 2 , Lˆ 3 ) of S O(3). Other symmetries were discovered a long time ago for the Kepler problem (Laplace, Runge, Lenz) and give, after quantization, symmetries for the hydrogen atom. Let us consider first the classical Hamiltonian setting and introduce the Laplace–Runge–Lenz vector:

282

9 The Coherent States of the Hydrogen Atom

M := p ∧ L −

q = (M1 , M2 , M3 ) |q|

(9.15)

where the classical angular momentum is L = q ∧ p = (L 1 , L 2 , L 3 ). We have the following properties 1. 2. 3. 4.

{Mk , H } = 0, k = 1, 2, 3. L·M=0 {M j , L k } =  j,k, M {M j , Mk } = −2H  j,k, M

where  j,k, is the usual antisymmetric tensor. In particular we can deduce form these properties that if the energy of H is fixed and negative (H = E < 0) then the six integrals L, M span a Lie algebra (for the Poisson bracket) isomorphic to the Lie algebra so(4) (see Sect. 9.1.2 of this chapter). In these sense the Kepler problem has “hidden symmetries” contained in the Laplace– Runge–Lenz vector M. For an historical point of view about M we refer to [124]. After quantization we get a Laplace–Runge–Lenz operator: X 1 Mˆ = (P ∧ Lˆ − Lˆ ∧ P) − 2 |X | One has the following commutation rules, corresponding to the above classical one (see [241] for detailed computations). 1. 2. 3. 4. 5.

[ Lˆ j , Lˆ k ] = i jkl Lˆ l [ Lˆ j , Mˆ k ] = i jkl Mˆ l [ Mˆ j , Mˆ k ] = −2i jkl Mˆ l Hˆ Lˆ · Mˆ = Mˆ · Lˆ = 0 Mˆ 2 − 1 = 2 Hˆ ( Lˆ 2 + 1l)

It is known that Hˆ has a purely absolutely spectrum on [0, +∞) and a negative point spectrum of the form 1 E n = − 2 , n ∈ N∗ . 2n The degeneracy of the eigenvalue E n is n 2 . We shall see that this is due to the “hidden symmetries” contained in the Laplace–Runge–Lenz operator M commuting with Hˆ ( Lˆ is also commuting with Hˆ but it generates apparent spherical symmetries of Hˆ ). Let us now recall the usual proof for the following result. Lemma 61 The eigenvalue E n = − 2n1 2 of Hˆ has degeneracy n 2 . Proof We give here a sketch of proof, for the details we refer to any text book in quantum mechanics. In spherical coordinates we have 1 1 ∂2 Hˆ = −r 2 r − 2 ΔS2 − . ∂r r r

9.2 The Hydrogen Atom

283

Recall that the spherical harmonics Ym satisfy: Lˆ 2 Ym = ( + 1)Ym (see Chap. 7). Eigenvalues are obtained by solving the radial equation −r

( + 1) 1 ∂2 r+ − f (r ) = E f (r ). 2 ∂r r2 r

So we get the eigenvalues E n for n ≥ 1 and a basis of the eigenspace: En := {ψn,,m , − ≤ m ≤ , 0 ≤  ≤ n}. The degeneracy of E n equals to the dimension of En : dim(Hn ) =

n−1  (2 p + 1). p=0

To this sum of odd numbers up to 2n − 1 we add and subtracts the sum of even numbers up to 2n − 2. This yields dim(Hn ) = n(2n − 1) − n(n − 1) = n 2 .  In the following section we shall recover this result using the S O(4) symmetry in a transparent way.

9.2.2 The Fock Transformation: A Map from L 2 (S3 ) Towards the Pure-Point Subspace of Hˆ We follow a presentation of Bander–Itzykson [17]. Consider the eigenvalue problem of the Hydrogen atom:  Hˆ ψ =

 1 Pˆ 2 − ψ = Eψ ˆ 2 | Q|

In Fourier variable p one gets 

  ˜ ψ(q) 1 p2 ˜ − E ψ(p) = dq 2 2 2π R3 |p − q|2

˜ where we have set  = 1 for simplicity and ψ(p) = (2π )−3/2

(9.16)

R3

dqe−iq·p ψ(q).

284

9 The Coherent States of the Hydrogen Atom

Since the bound states of the hydrogen atom have negative eigenenergies E we define p0 > 0 such that 2E = − p02 Then we define the stereographic projection from the momentum space R3 onto S30 (the sphere S3 with the north pole (0, 0, 0, 1) removed): consider S3 divided into two hemispheres by the momentum space R3 . Given a vector p/ p0 ∈ R3 (homogeneous coordinates), take the line from the north pole to this point. It will intersect the sphere S3 at a point w ∈ S30 . We have: w = (w1 , w2 , w3 , w4 ) 2 p0 pi , i = 1, 2, 3, wi = 2 p + p02 w4 =

(9.17) (9.18)

p2 − p02 . p2 + p02

(9.19)

w := F(p) defines a new parametrization of the sphere S30 . The inverse transformation is simply p0 wi , i = 1, 2, 3. F −1 (w) = pi (w) = 1 − w4 In this parametrization the Euclidean measure dμ3 of S3 can be computed using the formula   ∂F ∂F ,  d 3 p. dμ3 (w) = det  ∂ pk ∂ p So we get

 dμ3 (w) = 2δ(w2 − 1)d 4 w =

2 p02 p2 + p02

3 d 3 p.

When p0 = k we denote by wk (p) the corresponding stereographic transformation (9.17). To the change of variables F is associated the following unitary transform U F from L 2 (R3 ) into L 2 (S3 ): 1 ˜ := Φ(w) = √ U F (ψ)(w) p0



p(w)2 + p02 2 p0

2

˜ ψ(p(w)).

Now we can show that the L 2 norms of ψˆ and Φ are the same:  Φ(w)2L 2 (S3 )

=

 |Φ(w)| dμ3 (w) = 2

S3

R3

p2 + p02 2 ˜ |ψ(p)| dp. 2 p02

(9.20)

9.2 The Hydrogen Atom

285

Due to the virial theorem1 one has  2  p 2 ˜ 2 dp ˜ |ψ(p| |ψ(p)| dp = − E 2 R3 Thus

(9.21)

ˆ Φ(w) L 2 (S3 ) = ψ(p) L 2 (R3 ) .

One has the following remarkable property (just compute): Lemma 62 Given q ∈ R3 we define the point v ∈ S30 by the stereographic equations (9.17) with q instead of p and the same p0 . One has: |p − q|2 =

(p2 + p02 )(q2 + p02 ) |w − v|2 . (2 p0 )2

Therefore from Eq. (9.16) one gets that the equation obeyed by Φ is simply Φ(w) =

1 2π 2 p0

 S3

Φ(v) dμ3 (v). |v − w|2

(9.22)

Consider the operator T in L 2 (S3 ) defined by  (T Φ)(w) =

S3

Φ(v) dμ3 (v). |w − v|2

Note that T commutes with rotations ρR defined by (ρR Φ)(v) = Φ(R −1 v), with R ∈ S O(4). For n ∈ N∗ , H4(n−1) is the finite-dimensional space generated by the harmonic polynomials of degree n − 1 in the variables (v1 , v2 , v3 , v4 ) restricted to S3 . We have shown that the operators ρR restricted to the space H4(n−1) give an irreducible representation of S O(4). Thus due to Schur’s lemma, the operator T acts as a multiple of the identity in H4(n−1) T |H

(n−1) 4

= λn 1l.

We shall now calculate λn . Proposition 115 One has: λn =

2π 2 n

+∇x ·x that the virial theorem says: if ψ is a bound state of Hˆ and Aˆ = x·∇x 2i then 1 −1 ˆ ˆ ˆ ˆ [ H , A]ψ, ψ = 0. For the hydrogen atom we have i [ H , A] = −Δ − |x| . So we have (9.21).

1 Recall

286

9 The Coherent States of the Hydrogen Atom

It is enough to take a particular function in H4(n−1) say: (T G)(v) = (v3 + iv4 )n−1 which satisfies the equation:  G(w) =

S3

G(v) dμ3 (v) = λn G(w). |w − v|2

We introduce the spherical coordinates (χ , θ, φ) in S3 : v1 = sin χ sin θ cos φ v2 = sin χ sin θ sin φ v3 = sin χ cos θ v4 = cos χ and choose w = (0, 0, 0, 1). We get the following equation: 



2π

dχ 0



π

π

dθ 0

dφ 0

(sin χ cos θ + i cos χ )n−1 sin2 χ sin θ = λn i n−1 2(1 − cos χ )

Doing the integration with respect to θ, φ we get 2π n



2π

dχ 0

sin χ sin(nχ ) = λn , 1 − cos χ

which finally yields λn =

2π 2 . n

Comparing with the Eq. (9.22) we obtain p0 = En = −

1 n

and thus

1 2n 2

Thus we recover the point spectrum of the Hydrogen atom with its degeneracy: n 2 = dim(H4(n−1) ). We shall now use Eq. (9.20) to show that the eigenfunctions of Hˆ map onto the spherical harmonics defined on S3 . Recall that Lˆ is the quantum angular momentum operator, and Lˆ 2 = Lˆ 21 + Lˆ 22 + Lˆ 23 . Consider the normalized eigenfunctions of the Hydrogen atom Ψn,,m satisfying:

9.2 The Hydrogen Atom

287

1 Hˆ Ψn,,m = − 2 Ψn,,m n Lˆ 2 Ψn,,m = ( + 1)Ψn,,m ,  = 0, 1, . . . , n − 1 Lˆ 3 Ψn,,m = mΨn,,m , m = −, − + 1, . . . ,  − 1, .

(9.23) (9.24) (9.25)

2 space spanned by the functions  Recall that En is the n -dimensional vector Ψn,,m | 0 ≤ l ≤ n − 1, −l ≤ m ≤ l . Let H pp be the subspace of L 2 (R3 ) spanned by the eigenfunctions of Hˆ . We define the operator U : H pp → L 2 (S3 ) using Eq. (9.20):

1 (U Ψˆ n,,m )(w) = √ p0



p 2 (w) + p02 2 p0

2

Ψˆ n,,m (p(w))

and extend U linearly to all the space H pp . Since U Ψˆ n,,m satisfies Eq. (9.22) U Ψˆ n,,m must belong to the space H4(n−1) , thus one has Δ3 (U Ψˆ n,,m ) = n 2 U Ψˆ n,,m , where Δ3 is the modified Laplacian on S3 with eigenvalue n 2 . Since U preserves the L 2 norm and the spaces En and H4(n−1) have the same finite dimension U is an unitary operator from En onto H4(n−1) . Furthermore it is a unitary   operator from H pp = n∈N En onto L 2 (S3 ) = n∈N H4(n−1) . Let Π pp be the orthogonal projector onto the pure-point spectrum space of Hˆ . One has ∞  1 ˆ −1 − H Π pp = n 2 Πn 2 n=1 where Πn is the projector onto En . One has: Proposition 116

U ( Hˆ −1 Π pp )U −1 = −2Δ3 .

Proof Let Πn be the projector onto H4(n−1) . One has Δ3 =

∞ 

n 2 Πn .

n=1

Since

this yields the result.

U Πn U −1 = Πn 

288

9 The Coherent States of the Hydrogen Atom

9.3 The Coherent States of the Hydrogen Atom Using the unitary operator already introduced in Sect. 9.2, U : H pp → L 2 (S3 ), we define the coherent states of the hydrogen atom as Ψˆ α,k = U −1 Ψα,k .

(9.26)

From now on we define Ψα,k (w) = ck (α · w)k where the constant ck is chosen so that . Ψα,k  L 2 (S3 ) = 1. It was computed in Sect. 9.1: ck2 = k+1 2π 2 Equation (9.26) reads using (9.20): √ Ψˆ α,k (p) = p0



2 p0 2 p + p02

2 Ψα,k (wk (p)) = ck

√

p0

2 p0 2 p + p02

2 (α · wk (p))k

(we recall that wk (p) is the stereographic projection (9.17) for p0 = k). Let us define the dilation operator Dk in L 2 (R3 ) as (Dk Ψ )(x) = k 3/2 Ψ (kx), or in momentum space

p

(Dk Ψˆ )(p) = k −3/2 Ψˆ

k

Defining J to be the multiplication operator by Ψˆ α,k (p) = (Dk J Ψα,k )(p) = ck Dk



2 p2 +1

2 p2 + 1

.

2 we get 

2

(α · w1 (p))k .

(9.27)

Taking the Fourier transform we see that the coherent state for the hydrogen atom in configuration space equals ck Ψα,k (x) = (2π k)3/2





ip · x exp 3 k R



2 p2 + 1

2 (α · w1 (p))k dp.

Now we shall consider the state Ψα,k (x) dilated by k 2 : Φα,k (x) :=(Dk 2 Ψα,k )(x)  3/2   2 k 2 exp(ikp · x + k log(α · w1 (p)))dp. =ck 2π p2 + 1 R3 Note that log(α · w1 (p)) is well defined: if α4 = 0 then |α · w1 (p)| → 0 as |p| → ∞ and one thus gets a decrease outside a compact K of R3 . On K the logarithm is defined locally. If α4 = 0 then

9.3 The Coherent States of the Hydrogen Atom

289

α · w1 (p) = α4 + O(

1 ) |p|

where α4 ∈ C. Then there exists R > 0 such that for |p| > R the logarithm is well defined. The aim is now to show that Φα,k (x) concentrates in the neighborhood of a Kepler orbit when k becomes large. This is a semiclassical result since k plays the role of 1 . For doing this we use complex stationary phase estimates applied to the integral:   Φα,k (x) = ck

k 2π

3/2 

 dp R3

2 p2 + 1

2 exp(k f (x, p))

with f (x, p) = ix · p + log(α · w1 (p)).

(9.28)

The stationary phase condition reads:  f (x · p) = 0,

(9.29)

∇p f (x, p) = 0.

(9.30)

But  f (x, p) = log(|α · w1 (p)|) We have that |α · w1 (p)| ≤ 1 with equality only when w1 (p) is in the plane generated by a = α, b = α. Take for simplicity α = eˆ1 + i(eˆ2 cos γ + eˆ4 sin γ ). . The vectors eˆi are unit vectors in the direction of the comWe assume γ = π2 , 3π 2 ponents wi . Thus the first stationary phase condition (9.29) imposes that w1 (p) must satisfy a parametric equation of the form: w1 (p) = eˆ1 cos β + sin β(eˆ2 sin γ + eˆ4 cos γ ) The corresponding conditions for the momentum components are cos β , 1 − sin β sin γ sin β cos γ p2 = , 1 − sin β sin γ p3 = 0. p1 =

So p describes the circle Cγ in the plane p3 = 0: p12 + ( p2 − tan γ )2 =

1 cos2 γ

.

290

9 The Coherent States of the Hydrogen Atom

One sees easily that α · w1 (p) = eiβ which is a complex number of modulus 1 as required. Now we consider condition (9.30): it reads ixj +

2 [α j + (α4 − α · w1 (p)) p j ] = 0. (p2 + 1)(α · w1 (p))

(9.31)

Thus x must satisfy the parametric equations: x1 (β) = sin β − sin γ x2 (β) = − cos β cos γ x3 = 0 Thus x must belong to the ellipse E (γ ) of equation  (x1 + sin γ ) + 2

of energy −1/2:

x2 cos γ

2 = 1.

(9.32)

1 1 p2 − =− 2 |x| 2

To apply the saddle point method (see Appendix A.4) one needs to show that the Hessian matrix is non singular at the critical point (x(β), p(x(β))). One calculates the Hessian matrix Hβ Hβ = ∂ 2p, p f (x(β), p(x(β))) . First of all we easily see that if x is not on the ellipse E (γ ) then we have Φα,k (x) = O(k −∞ ). A tedious calculation sketched in Appendix A.3 shows that on E (γ ) we have 

| det Hβ | =

(1 − sin β sin γ )4 (sin2 γ + 2 sin β sin γ + 1)(sin2 γ − 2 sin β sin γ + 1). (9.33) So we have det Hβ = 0. From now on it is enough to consider a neighborhood V of a fixed point x0 := x(β) on E (γ ). f (x, p) being holomorphic in a complex neighborhood of (x0 , p(x0 )) in C3 × C3 , the saddle point method (see Appendix A.4) can be applied and gives for every x ∈ V ,

9.3 The Coherent States of the Hydrogen Atom

291

2    2  exp(k f (x, p(x)))  1  |Φα,k (x)| = Cstk  (1 + O(k −1/2 ). 2 1 + p (x)  | det Hβ | (9.34) Let us remark that the exponent in (9.34) is fast decreasing in k for x ∈ / E (γ ). To analyze more carefully the behaviour of |Φα,k (x)| nearby x0 we choose convenient coordinates. Let x(β, t, s) = x(β) + tνβ + s eˆ3 where β ∈ [0, 2π [, νβ is the normal vector to E (γ ) at x(β): νβ = sin β cos γ eˆ1 − cos β eˆ2 . (t, s) are such that t 2 + s 2 < δ 2 with δ > 0 small enough. We use the shorter notations x0 = x(β), xt,s = x(β, t, s), p0 = p(x0 ). Using the Taylor expansion we get 1/2

f (xt,s , p(xt,s )) − f (x0 , p0 ) = ∂x f (x0 , p0 ) · (xt,s − x0 ) 1 2 + ∂x,x f (x0 , p0 )(xt,s − x0 ) · (xt,s − x0 ) 2 1 2 + ∂x,p f (x0 , p0 )(xt,s − x0 ) · (p(xt,s ) − p0 ) + 2 1 2 ∂ f (x0 , p0 )(p(xt,s ) − p0 ) · (p(xt,s ) − p0 ) + O(|xt,s − x0 |3 ) 2 p,p

(9.35)

p(xt,s ) − p0 = ∂x p(x0 )(xt,s − x0 ) + O((t 2 + s 2 )3/2 ).

(9.36)

and

But we have Hβ = −i∂p x so ∂x p(x0 ) = i Hβ−1 and we get, taking the real part in the Taylor expansion,  f (xt,s , p(xt,s )) =

1 Hβ−1 (tνβ + s eˆ3 ) · (tνβ + s eˆ3 ) + O((t 2 + s 2 )3/2 ). 2 (9.37)

2 2 Here we have used that ∂x f is imaginary and ∂x,x (x0 , p0 ) = 0, ∂x, p (x 0 , p0 ) = i. −1 −1 The matrix H ( real part of H ) has the following form (see computations in Appendix A.3):

H −1

⎛ − sin β cos β cos γ sin2 β cos2 γ 1 ⎝ − sin β cos β cos γ cos2 β =− h(β, γ ) 0 0

h(β,γ ) sin2 γ −2 sin β sin γ +1

where h(β, γ ) = (1 − sin β sin γ )2 (sin2 γ + 2 sin β sin γ + 1). The eigenvectors of H −1 are

0 0

⎞ ⎠

292

9 The Coherent States of the Hydrogen Atom

v1 = (cos β, sin β cos γ , 0) v2 = νβ = (sin β cos γ , − cos β, 0) v3 = eˆ3 = (0, 0, 1)

(9.38)

with corresponding eigenvalues: λ1 = 0, sin2 β cos2 γ + cos2 β , h(β, γ ) 1 λ3 = − 2 . sin γ − 2 sin β sin γ + 1

λ2 = −

(9.39)

Since λ2 , λ3 < 0 we see that for k large |Φα,k (x)|2 behaves like a Gaussian highly concentrated around the ellipse at the point x0 . Furthermore it decreases in the direction of the eigenvectors v2 , v3 namely in the direction perpendicular to the plane of the ellipse and in the plane of the ellipse in the direction normal to the ellipse (note that p(x0 ) · v2 = 0). More precisely we have     Φα,k (xt,s )2 = C st   where Q(t, s) =

4  2 1  ek Q(t,s) k + O(k −1 )  2 p(xt,s ) + 1 | det Hβ,t,s |

(9.40)

1 Hβ−1 (tνβ + s eˆ3 ) · (tνβ + s eˆ3 ) + O((t 2 + s 2 )3/2 ). 2

The quadratic form Q 0 (t, s) := 21 Hβ−1 (tνβ + s eˆ3 ) · (tνβ + s eˆ3 ) is definite-negative in the plane (t, s). 2  Now we shall see that as k → +∞ the density probability Φα,k  converges to a probability measure supported in the ellipse E (γ ). The following statement is close to [251] (Thesis, Prop. 4.1). Proposition 117 For every continuous and bounded function ψ in R3 we have  lim

k→+∞ R3

  Φα,k (x)2 ψ(x)dx = C st

 E (γ )

ψ(e)d(e, O)d(e),

(9.41)

where d(e) is the length measure on the ellipse E (γ ), d(e, O) is the distance of e ∈ E (γ ) to its focus O, C st is a normalization constant. Proof From computations already done we have  4   2 1 2 −1/2   (sin2 γ − 2 sin γ sin β + 1)−1/2  p(x )2 + 1  | det H | = (sin γ + 2 sin γ sin β + 1) 0 β

and

9.3 The Coherent States of the Hydrogen Atom

293

det Q 0 = λ2 λ3 =

1 − sin2 γ sin2 β (1 − sin γ sin β)2 (sin γ + 2 sin γ sin β + 1)(sin2 γ − 2 sin γ sin β + 1) 2

So we can apply the stationary phase theorem in variables (t, s) to get  lim

k→+∞ R3

  Φα,k (x)2 ψ(x)dx = Cst



2π

(1 − sin γ sin β)(1 − sin2 γ sin2 β)1/2 ψ(x(β))dβ.

0

But for e = x(β) we have (1 − sin2 γ sin2 β)1/2 ψ(x(β))dβ = d(e) and (1 − sin γ sin β) is the distance between x(β) and O. Let us remark that the speed v(β) of a classical particule travelling on E (γ ) is  1 − sin2 γ sin2 β . v(β) = 1 − sin γ sin β 

Kepler ellipse

x2 , p2 •

pπ/2

Ω•

x3π/2 •

O• • • x7π/4

•

xπ/2 • p7π/4

p3π/2

x1 , p1

Part III

More Advanced Results on Harmonic Coherent States and Applications

Chapter 10

Characterizations of Harmonic Pseudodifferential Operators

Abstract There are many connections between Weyl quantization of classical observables and coherent states as we have already seen in Chap. 2. Here we put emphasis on necessary and sufficient conditions to characterize some classes of classical observables (called symbols). In particular, natural classes of symbols are associated with the Harmonic oscillator. They are defined as follows: for any m ∈ R, ρ ∈ [0, 1], 2n → C, |∂ Xα A(X )| ≤ Cα X m−ρ|α| }. Sm ρ := {A : R These classes were introduced by Shubin [235] (see also [139]). Our aim here is to give several criteria to decide when a linear continuous operator Aˆ : S (Rn ) → S  (Rn ) has its Weyl symbol in some Sm ρ for some m ∈ R. In particular we shall revisit the Beals charaterization [21] and the Chodosh characterization [54]. Some applications are given.

10.1 Operators Classes Associated with the Harmonic Oscillator 10.1.1 The Harmonic Classes These classes are defined by estimates on the size of the derivatives of the Weyl symbols. For any m ∈ R, let us denote by Sm ρ the space of smooth functions A : R2n → C such that for any multiindex α ∈ N2n we have |∂ Xα A(X )| ≤ Cα X m−ρ|α| , ∀X ∈ R2n .

(10.1)

In Chap. 2, we have considered the class Sm 0 . Here we shall consider the case ρ > 0. Another difference with Chap. 2 is that in this chapter the Planck constant is fixed:  = 1 and the orders of operators are measured by powers of the weight λ(X ) := X . So we put emphasis on smoothness and behavior at infinity in the configuration space. So the symbolic calculus rule to compute the Moyal product is formulated as  follows. Let be A ∈ Sρm , B ∈ Sρm . We have seen that Aˆ · Bˆ = Cˆ where © Springer Nature Switzerland AG 2021 D. Robert and M. Combescure, Coherent States and Applications in Mathematical Physics, Theoretical and Mathematical Physics, https://doi.org/10.1007/978-3-030-70845-0_10

297

298

10 Characterizations of Harmonic Pseudodifferential Operators

C(X ) = (π )−2n

 R4n

e2iσ (Y,Z ) A(X + Z )B(X + Y )dY d Z .

(10.2)

Following the same method as in Chap. 2 see also [153, 217], we get that the Weyl  , more precisely we have the following symbol C(x, ξ ) is in Sm+m ρ 



m m+m Theorem 47 For every A ∈ Sm such ρ and B ∈ Sρ , there exists a unique C ∈ Sρ  that Aˆ · Bˆ = Cˆ . Moreover we have an asymptotic formula to compute C: C

Cj j≥0 m+m  −2 jρ

and C j ∈ Sρ

is computed by the formula:

C j (x, ξ ) =

1 2j

 (−1)|β| β (Dxβ ∂ξα A).(Dxα ∂ξ B)(x, ξ ), α!β! |α+β|= j ⎛



and for any N ≥ 1 we have ⎝C −

⎞ C j ⎠ ∈ Sρm+m −2ρ N . 

0≤ j≤N −1 m ˆ ˆ ˆ m+m . Corollary 30 For every A ∈ Sm ρ and B ∈ Sρ the commutator [ A, B] is in S2ρ 



Operators in Sˆρm acts in a nice way in the Harmonic weighted Sobolev spaces H r defined by the powers of the harmonic oscillator Hˆ os = 21 (− + |x|2 ). r/2 For r ≥ 0, H r = Dom( Hˆ os ). H r is an Hilbert space for the graph norm ϕ r 1/2  r/2 := Hˆ os ϕ 2 + ϕ 2 . For r < 0, H r is defined by duality. In other words, f ∈ H r if f is a temperate distribution (i.e. f ∈ S  (Rn )) such that |ϕ, f | ≤ C f ϕ −r, ∀ϕ ∈ S (Rn ) In Chap. 1 of the book we have introduced an orthonormal basis (usually called the tensorial basis) {φk , k ∈ Nn }. φk is a normalized eigenfunction for Hˆ os , with eigenvalue E k = k1 + k2 + · · · + kn + n2 . So for any integer j ∈ N the multiplicity ν j of the eigenvalue j + n2 is the number of multiindex k such that |k| = j. Now we can easily deduce a characterization of the harmonic Sobolev space H r on the above orthornormal basis. Proposition 118 Let ψ ∈ S  (Rn ) and denote ck (ψ) = φk , ψ. Then ψ ∈ H r if |ck |2 (1 + |k|)r < +∞. Moreover the norm and only if k∈Nn



ψ r∗

:=



k∈Nn

is equivalent to the norm ψ r .

1/2 |ck | (1 + |k|) 2

r

10.1 Operators Classes Associated with the Harmonic Oscillator

299

An important property of harmonic pseudodifferential operators is boundness on the scale of harmonic Sobolev spaces. r +m ˆ Theorem 48 Let A ∈ Sm , H r ). ρ , m ∈ R. Then for any r ∈ R, A ∈ L (H

Notice that for m = r = 0 Theorem 48 is the Calderon-Vaillancourt Theorem proved in Chap. 2. We shall deduce the general case from this particular case and a functional calculus result for the harmonic oscillator Hˆ os . Definition 21 A smooth function f from R to C is a symbol of order ν ∈ R if for any j ∈ N we have d j f (λ) | | ≤ C j λν− j , ∀λ ∈ R. dλ j The space of symbols of order ν is denoted S ν . A typical example is f (λ) = (1 + λ2 )ν/2 . Definition 22 A operator A from S  (Rn ) into S (Rn ) is called an N -smoothing operator if for any r ∈ R we have A ∈ L (H r , H r +N ). −j

Example: e−t Hosc is N -smoothing for any N and Hos is 2 j-smoothing. Proposition 119 Let f be a symbol of order ν ∈ R. Then f ( Hˆ os ) is in Sˆ 2ν 1 modulo a smoothing operator. That means here that for any N ≥ 1 there exist A f,N ∈ S2ν H and a smooth function g N such that |g N (λ)| ≤ C N λ−N , and f ( Hˆ os ) = Aˆ f,N + g N ( Hˆ os ).

(10.3)

Proof of Theorem 48 The idea is to apply the Calderon-Vaillancourt Theorem to r ˆ ˆ −(m+r )/2 is bounded on L 2 (Rn ). This would be clear if we had A Hos show that Hˆ os r is a pseudodifferential operator without a smoothing error (we shall proved that Hˆ os prove later that this is true). To tackle here with this technical problem, using the product rule (47), it is not difficult to prove that for any integers M, N we have 1l = Bˆ M,N HosM − Rˆ M,N ,

(10.4)

−N where B M,N ∈ S2M H and R M,N ∈ S H . r ˆ ˆ −r We begin by proving the theorem for m = 0. So we have to prove that Hˆ os A Hos 2 n is bounded on L (R ). Using Proposition 119 we have r ˆ ˆ −r r ˆ ˆ A Hos = Eˆ + Hˆ os A R N + Rˆ N Aˆ Hˆ os−r + Rˆ N Aˆ Rˆ N Hˆ os 2 ˆ ˆ ˆ ˆ where E ∈ S H0 , R N , R N ∈ S−N H . E and R N A R N are bounded on L by the CalderonVaillancourt Theorem.

300

10 Characterizations of Harmonic Pseudodifferential Operators

r ˆ ˆ ˆ Now we have Hˆ os A R N = Aˆ r,N + Rˆ r,N , where Ar,N ∈ S2r H and Rr,N is an N smoothing operator. So it is enough to prove that r > 0 and for any F ∈ S H2r , Fˆ Rˆ r,N is bounded. But we have, using (10.4), is

Fˆ Rˆ r,N = ( Fˆ Bˆ M,N )( Hˆ osM Rˆ N ) − ( Fˆ Rˆ M,N ) Rˆ N . r ˆ ˆ −r Then we get that Hˆ os A Hos is L 2 bounded by choosing M, N large enough. Using the same techniques (we leave the details to the reader) we can extend the result to every m ∈ R. 

Proof of Proposition 119 We shall use an explicit expression for the Weyl symbol of eit Hos which is a particular case of the formulas obtained in Chap. 3 (Sect. 5). Let t ∈] − π, π [. The Weyl symbol W (t, X ) of eit Hos is given by the well known Mehler formula

W (t, X ) = (cos(t/2))−n exp i tan(t/2)|X |2 .

(10.5)

By inverse Fourier transform we have f (Hos ) = (2π )−n

 R

f˜(t)eit Hos dt.

Let χ be a cut-off C ∞ function such that χ (t) = 1 for |t| ≤ π/2 and χ (t) = 0 for |t| ≥ 2π/3. Denote f 0 , f 1 defined by their Fourier transforms: f˜0 (t) = χ (t) f (t) and f˜1 (t) = 1 − f˜0 (t). Using the definition of S ν and standard properties of the Fourier transform, we get that f 1 ∈ S (R) and f 0 ∈ S ν . Notice that we also have f 1 (Hos ) =

 j∈N

 n Πj, f1 j + 2

(10.6)

where Π j denotes the spectral projector on eigenvalue j + n2 . Hence using the characterization given in Proposition 118 we get easily that f 1 (Hos ) is an N -smoothing operator for any N ∈ N. Now we use (10.5) to compute the Weyl symbol of f 0 ( Hˆ os ). Denoting λ = 2|X |2 , we have to compute σ f (λ) = (2π )−1

 R2

χ (t) λ f (λs)eiλ(tan t−ts) dtds. cosn t

(10.7)

In order to simplify the phase we write eiλ(tan t−ts) = eiλ(tan t−t) eiλt (1−s) and we expand eiλ(tan t−t) =



tan t − t + r N (tan t − t) j! 0≤ j≤N −1

10.1 Operators Classes Associated with the Harmonic Oscillator

301

where r N satisfies   dj |s| N − j   , for 0 ≤ j ≤ N .  j r N (s) ≤ ds (N − j)! Now using that tan t − t vanishes at the order 3 at t = 0 and that the critical point of the quadratic form t (1 − s) is at t = 0, s = 1 we get that according the stationary phase principle, σ f (λ) = f (λ) +



γ j,k λk

3k≤ j≤N −1

dj f (λ) + R f,N (λ), dλ j

(10.8)

and for any M there exists N large enough such that   j   k d R (λ)  ≤ C, for k + j ≤ M. λ f,N dλ j In particular we have proved Proposition 119.



Remark 56 The proof given here for Proposition 119 is adapted to the harmonic oscillator. More general methods are known, in particular the Helffer-Sjöstrand formula which is explain in Appendix C.

10.1.2 The Beals Commutator Classes This class was introduced by R. Beals [21] using conditions on commutators Recall ˆ B] ˆ where A, ˆ Bˆ are arbitrary operators. Let be {L j }1≤ j≤J the notation ad Aˆ · Bˆ := [ A, any finite family of linear forms on the phase space R2n . Using Theorem 47 m−ρ J . Hence using we see easily that if A ∈ Sρm then (ad Lˆ 1 ad Lˆ 2 . . . ad Lˆ J ) Aˆ ∈ Sˆ H Theorem 48 we get that (com)ρ (ad Lˆ 1 . . . ad Lˆ J ). Aˆ ∈ L (H r +m−ρ J , H r ), ∀J ≥ 1. for any r ∈ R. For the class S00 of pseudodifferential operators (see the definition in Chap. 3), Beals [21] proved that Aˆ ∈ Sˆ 00 if and only if for any J ≥ 0, ad Lˆ 1 ad Lˆ 2 · · · ad Lˆ J Aˆ is bounded on L 2 (Rn ). J. M. Bony get a simpler proof in [35]. A proof is also given in the book [139] for the harmonic class Sm 1 and in [36] in a more general setting. We shall give here a different proof for the classes Sm ρ with ρ ∈ [0, 1], following [217]. Theorem 49 Let be Aˆ ∈ L (S (Rn ), S  (Rn )). Then Aˆ ∈ Sˆ m ρ if and only if the condition (com)ρ is satisfied.

302

10 Characterizations of Harmonic Pseudodifferential Operators

Proof It is clear that, using the Moyal product rule and Theorem 48, if A ∈ Sm ρ then ˆ A satisfies the condition (com)ρ . Assume now that (com)ρ is satisfied. In a first step we assume that m = ρ = 0. We shall see that the general case will be a consequence of this particular case. We begin by some remarks concerning the duality bracket < ·, · > between L (S  (Rn )) and L (S (Rn )). Ee have, for any A ∈ L (S (Rn ), S  (Rn )) and any B ∈ L (S  (Rn ), S (Rn )), ˆ < A, B >= (2π )n Tr( Aˆ B).

(10.9)

On the l.h.s the bracket is the usual duality between S  (R2n ) and S (R2n ). On the r.h.s the trace is an extension by a regularisation procedure of the usual trace for class trace operators in L 2 (Rn ) (for details see [217]). Using standard properties of the trace (cyclicity), we have the following properties. ˆ B) ˆ = −Tr( A(ad ˆ Lˆ · B)), ˆ Tr((ad Lˆ · A)

(10.10)

for any differential operator Lˆ with polynomials coefficients. In particular if L is a linear form defined by: L(X ) = σ (Z , X ), we have  ˆ ˆ = (2π )−n A(X )D Z B(X )d X, (10.11) Tr( A.ad( Lˆ · B)) R2n

where D Z is the derivative in the direction Z . So for J linear forms L j (X ) = σ (Z j , X ) we have ˆ B). ˆ < D Z 1 . . . D Z J A, B >= Tr(ad Lˆ 1 . . . ad Lˆ J · A).

(10.12)

So using standard estimates of traces, in particular Theorem 5 of Chap. 2, and condition (com)0 , we get ˆ | < D Z 1 . . . ∂ D Z J A, B > | ≤ (2π )n ad Lˆ 1 . . . ad Lˆ J . A

B W 1,2n+1 (R2n )

(10.13)

where W p,m (R2n ) denotes the usual Sobolev space. From (10.13) we get that ∂ γ A ∈ W ∞,−2n−1 (R2n ) for any multiindex γ . To achieve the proof we apply the following result. Let be the Laplace operator on R M and denote G k = (1 − )−k . G k is a convolution operator by the kernel G k (x) = (2π )

−M

 RM

(1 + ξ 2 )−k ei x·ξ dξ.

So G k ∈ L 1 (R M ) if k > M/2 and ∂ γ G k ∈ L 1 (R M ) if k > Hence we have

M+|γ | . 2

10.1 Operators Classes Associated with the Harmonic Oscillator

303

 F(u) = So let N0 ∈ N such that k >

RM

G k (u − v)(1 − )k F(v)dv.

M+N0 . 2

(10.14)

Then we get

|F(u)| ≤ G k W 1,N0

(1 − )k F W ∞,−N0

(10.15)

Applying this estimate to ∂ γ F we get the following Lemma 63 Let be F ∈ S  (R M ) such that for any multiindex γ , ∂ γ F ∈ W ∞,−N0 (R M ) for some N0 > 0. Then for any multiindex β we have ∂ β F ∈ L ∞ (R M ). Applying the above Lemma with M = 2n and N0 = 2n + 1 we have proved the Beals characterization for m = ρ = 0. For later use we notice that we have got the following universal estimates for the symbol: Lemma 64 There exists universal constants Cα,n such that

∂ Xα A L ∞ (R2n ) ≤ Cα,n



γ

Opw1 (∂ X A) L 2 ,L 2

(10.16)

|γ |≤2n+1+|α|

Assume now that m = 0 and ρ ∈]0, 1]. We prove that X ρ|γ | ∂ γ A ∈ S00 by induction on |γ |. ˆ Hˆ 0ρ/2 is bounded on L 2 (Rn ). Using For any linear form L we have that (ad Lˆ · A) ˆ Hˆ 0ρ/2 satisfies (com)0 . the symbolic calculus we get that the operator Bˆ 1 := (ad Lˆ · A) 0 So applying the first part of the proof we get B1 ∈ S0 . Using the symbolic calculus it results that X ρ D Z A ∈ S00 , for any Z ∈ R2n . Assume that for K ≤ J − 1, for any Z 1 , . . . , Z K ∈ R2n , we have X kρ D Z 1 . . . D Z K ∈ S00 . But ˆ Hˆ osJρ/2 = ad Lˆ 1 · (ad Lˆ 2 · · · ad Lˆ J ) · A) ˆ Hˆ os(J −1)ρ/2 Hˆ osρ/2 (ad Lˆ 1 · · · ad Lˆ J ) · A) So we get by induction hypothesis and the first steps, that X  Jρ D Z 1 · · · D Z J A ∈ S00 for any Z 1 , · · · , Z J ∈ R2n . . Corollary 31 [Wiener property] Let be A ∈ S00 and assume that Aˆ −1 is invertible in L 2 (Rn ). Then Aˆ −1 ∈ Sˆ 00 ˆ Aˆ −1 ] = Proof One check the condition (com0 ) for Aˆ −1 starting with the formula [ L, −1 ˆ ˆ ˆ −1 ˆ  − A [ L, A] A .

304

10 Characterizations of Harmonic Pseudodifferential Operators

10.1.3 Coherent States Classes If Aˆ ∈ L (S (Rn ), S  (Rn )) its covariant symbol is defined by Ac (z, z  ) := ˆ z . In this section we assume that  = 1. ϕz , Aϕ Definition 23 Let ρ ∈ [0, 1] and m ∈ R. Ac : R2n × R2n → C is a covariant symbol  2n in the class Sm CS,ρ if for any N ≥ 0, any multiindices α, α ∈ N we have 



|∂zα ∂zα Ac (z, z  )| ≤ C N ,α,α z + z  m−ρ(|α|+|α |) (1 + |z − z  |)−N .

(10.17)

m We shall prove that the classes Sm CS0 and S0 define the same operator classes.

Theorem 50 For any ρ ∈ [0, 1], if Aˆ ∈ Sˆρm then Ac ∈ Sm CS,ρ . m ˆ ˆ Conversely, for ρ = 0, 1, if Ac ∈ Sm then A ∈ S . ρ CS,ρ Proof Let us recall that Wz  ,z (X ) denotes the Wigner function of the pair (ϕz  , ϕz ). We have the following formulas 

Ac (z , z) = (2π ) A(X ) = (2π )−2n

−n

 R2n

 R2n ×R2n

A(X )Wz  ,z (X )d X,

(10.18)

Ac (z  , z)Wz  ,z (X )dzdz  .

(10.19)



Recall that Wz  ,z (X ) = c0 eΦ(z,z ,X ) where c0 is a universal constant and  z  + z 2 z − iσ (X − , z − z  ). Φ(z, z  , X ) = − X − 2 2 It is convenient to change the variables z  , z into u =

z  +z 2

and v = z − z  .

 Assume that A ∈ Sm ρ . We have ∇ X Φ(z, z , X ) = −2(X − u) − i J v. So if v  = 0 we

∇Φ can integrate by parts with the differential operator M := |∇Φ| 2 · ∇ X in the integral −N (10.18). After N iterations we get the estimate in O(v ). To gain the estimate in u, we use the Peetre inequality: for any r ∈ R we have:

u−r ≤ 2−r X −r u − X |r | .

(10.20)

Hence we get estimate (10.17) for α = α  . In the same way we can get (10.17) for any derivatives in α, α  . Assume now that (10.17) is satisfied with ρ = 0. We use here formula (10.19). First using (10.20) we get that X Am ∈ L ∞ (R2n ). So we can assume that m = 0. Then computing ∂ X A, using again (10.20) and estimates on Ac , γ we get that ∂ X A ∈ L ∞ (R2n ). More generally we can prove that ∂ X A ∈ L ∞ (R2n ) for any γ .

10.1 Operators Classes Associated with the Harmonic Oscillator

305

Now consider the case ρ = 1. We have to prove that X |γ | ∂ γ A is bounded in R2n for any γ . We explain the idea of the proof for |γ | = 1. It is easy to extend the the estimate to any γ . In coordinates (u, v) the phase Φ is Φ(z, z  , X ) = Φ(u, v, X ) = −|X − u|2 + i J (X −

u )·v 2

So we have ∂ X Π (u, v, X ) = −∂u Π (u, v, X ) = −2(X − u) − i J v. So integrating by parts we get ∂ X A(X ) = 2

1−n

(2π )

−2n

 R2n ×R2n

∂u Ac (u − v/2, u + v/2)) · ·Wu−v/2,u+v/2 (X )dudv.

(10.21)

Hence using (10.17) and (10.20) we get that X ∂ X A is in L ∞ (R2n ). By iterations γ  we can prove in the same way that, for any γ , X |γ | ∂ X A is in L ∞ (R2n )

10.1.4 The Matrix Class This class was introduced by Chodosh [54] for giving a condition on the matrix ˆ k  to characterize a class of pseudodifferential operators. elements Aj,k := {φj , Aφ Definition 24 Let us introduce the following discrete weighted derivatives on the matrix elements Aj,k .  j + 1Aj+e ,k − k Aj,k−e ,   = j Aj−e ,k − k + 1Aj,k+e .

δ (A)j,k = δ† (A)j,k



(10.22) (10.23)

n Aˆ is in the matrix class Sˆ m ρ,Mat , m ∈ R if for any α, β ∈ N and any N ≥ 1 there exists Cα,β,N such that

|(δ † )β δ α (A)j,k | ≤ Cα,β,N (1 + |j + k|)m−ρ(|α|+|β|)/2 (1 + |j − k|)−N ,

(10.24)

where δ α = δ1α1 · · · δnαn and (δ † )α = (δ1† )α1 · · · (δn† )αn . Theorem 51 Aˆ ∈ Sˆ m 1 if and only if the condition (10.24) is satisfied for ρ = 1.

306

10 Characterizations of Harmonic Pseudodifferential Operators

ˆ −m/4 Aˆ Hˆ os−m/4 . Proof It is enough to assume that m = 0 by applying the result to Hos2

1 0 0 ˆ ∈ Sˆ 1 where Hˆ os, = −∂x + x2 and First assume that Aˆ ∈ Sˆ 1 . Then [ Hˆ os, , A] 2  ˆ k  = ( j − k )A j,k . φj , [ Hˆ os, , A]φ By iteration we get (10.24) for any N ≥ 1 for α = β = 0. So now we can assume that N = 0. Recall the following formulas (see Chap.1) a  φj = a† φj

=

 

j φj−e ,

(10.25)

j + 1 φj+e ,

(10.26)

where {e1 , · · · , en } is the standard basis of Rn . We easily see that we have ˆ k . (δ † )β δ α (A)j,k = φj , adβ1 a1† · · · adβn an† .adα1 a1 · · · adαn an . Aφ

(10.27)

−|α|−|β| so we get (10.24) as explain But adβ1 a1† · · · adβn an† adα1 a1 · · · adαn an . Aˆ is in Sˆ 1 before. Conversely if (10.24) is satisfied using (10.27) and the Beals characterization  Theorem 49 we get that Aˆ ∈ Sˆ m 1.

Remark 57 It is not difficult to prove that if A ∈ Sm ρ , ρ ∈ [0, 1], then (10.24) is satisfied. It is not clear if the reverse claim is true if ρ ∈ [0, 1[. Remark 58 For n = 1 it is not difficult to prove that condition (10.24) is equivalent to the Chodosh condition |  A( j, k)| ≤ C,N (1 + | j − k|)−N (1 + j + k)m− ,

(10.28)

where A( j, k) = A j+1,k+1 − A j,k . Corollary 32 Let be f : R → R a symbol of order ν ∈ R. Then f ( Hˆ os ) ∈ Sˆ 2ν 1 . In other words there exists a symbol Hos f such that f ( Hˆ os ) = Hˆ os f . Moreover H f is a radial function: Hos f (X ) = f  (|X |2 ), where f  is a smooth function on [0, ∞[. Proof Notice that Proposition 119 states that the result is true modulo smoothing operators. Here we get a result without error term using Theorem 51. We have the matrix elements computation:  n . Fj,k = φj ; f ( Hˆ os )φk  = δj,k φj ; f ( Hˆ os )φj  = δj,k f |j| + 2 Hence it is not difficult to check that condition (10.24) is satisfied for the matrix Fj,k . So f ( Hˆ os ) has an exact Weyl symbol in the right class. But we know (see Theorem 2, Chap. 3) that

10.1 Operators Classes Associated with the Harmonic Oscillator

307

ˆ ˆ it Hˆ os = A eit Hos Ae ◦ Fost ,

where Fost is the classical flow of Hos . Hence we get that Hos f (Fost (X )) = Hos f (X )  for every t ∈ R, X ∈ R2 . So Hos f is rotation invariant.

10.1.5 An Application to Time Dependent Perturbations of Harmonic Oscillator Let be Vˆ (t) a time dependent operator such that V (t) ∈ S01 (R2n ), continuous and periodic in t. We consider the Schrödinger equation i∂t ψ(t) = Hˆ os ψ(t) + Vˆ (t)ψ(t), ψ(0) = ψ0 .

(10.29)

The following Lemma claims that the Cauchy problem (10.29) is well posed in L 2 (Rn ). The proof is left to the reader. Proposition 120 (i) For any ψ0 ∈ L 2 (Rn ), (10.29) has a unique solution ψ(t) ∈ L 2 (Rn ) and ψ(t) 0 = ψ0 0 . Moreover this solution solved the fixed point equation  ψ(t) = Uos (t)ψ0 +

t

Uos (t − s)Vˆ (s)ψ(s)ds,

(10.30)

t0

where Uos (t) = e−it Hos . (ii) Moreover the Cauchy problem (10.29) is well posed in H r for any r ∈ R and there exist Cr > 0 such that

ψ(t) r ≤ Cr tr ψ0 r , ∀t ∈ R.

(10.31)

Proof Recall the following Duhamel formula. Denote Hˆ (t) = Hos + Vˆ (t) and ψ(t) = U (t, t0 )ψ0 the solution defined in (10.30). Consider the Cauchy problem i∂t ψ(t) = Hˆ (os )t)ψ(t) + g(t), ψ(0) = ψ0 ,

(10.32)

where g : R → L 2 (R) is continuous. Then we have  ψ(t) = U (t, t0 )ψ0 +

t

U (t, s)g(s)ds.

(10.33)

t0

For simplicity we assume that r = 2k, k ∈ N.We shall prove (ii) by induction on k. Consider the case k = 1. We have i∂t Hos ψ(t) = Hˆ (t) Hˆ 0 ψ(t) + [ Hˆ 0 , Vˆ (t)]ψ(t).

308

10 Characterizations of Harmonic Pseudodifferential Operators

So using (10.33) and that [ Hˆ 0 , Vˆ (t)] ∈ Sˆ10 we have

Hos ψ(t) 0 ≤ Hos ψ0 0 + C|t − t0 ψ0 0

(10.34)

It is not difficult now to achieve the induction by computing i∂t Hosk+1 ψ(t) and using again (10.33). . Now we shall see that the growth estimate is sharp. This is a non trivial result. It was first proved by Delort [80]. We shall prove it here following the rather elementary and illuminating proof given by Maspero [190]. Theorem 52 There exists a time dependent pseudo-differential operator Vˆ (t) ∈ S10 (R2n ), continuous and periodic in time, such that for any r > 0 and any ψ ∈ S (Rn ), ψ = 0, there exists cr > 0 such that

ψ(t) r ≥ cr tr , ∀t ∈ R.

(10.35)

ˆ ˆ −it Hˆ os , Proof We are looking for a time dependent perturbation Vˆ A (t) := eit Hos Ae 0 ˆ ˆ with A ∈ S1 , real. It is clear that ψ(t) is solution of (10.29) for V (t) = V A (t) if and ˆ ˆ ˆ So η(t) = e−it A ψ0 only if η(t) := e−it Hos ψ(t) satisfies the equation i∂t η(t) = Aη(t). and ψ(t) r = η(t) r . So it is enough to prove that

η(t) r ≥ cr tr .

(10.36)

Let r = 2k, k ∈ N. Recall the Liebniz rule for derivatives   m  ˆ j1 · Hˆ · · · (ad A) ˆ jk · Hˆ . ˆ m · Hˆ k = (ad A) (ad A) j1 , · · · jk

(10.37)

j1 +···+ jk =m

We shall use the following Lemma : Lemma 65 There exists a real symbol A ∈ S01 such that for every j ≥ 1 we have ˆ j+1 · Hˆ osj = 0. ˆ j · Hˆ osj = 0 and (ad A) (ad A) This Lemma will be proved later. We have ˆ ˆ

η(t) 2k = eit A Hˆ osk e−it A ψ0 , ψ0  and

ˆ

ˆ

ˆ

ˆ

ˆ  · Hˆ osk e−it A . ∂t (eit A Hˆ osk e−it A ) = eit A (ad A)

Choosing A like in Lemma 65, we get that

10.1 Operators Classes Associated with the Harmonic Oscillator ˆ

ˆ

309

∂t (eit A Hˆ osk e−it A ) = 0 if  ≥ k + 1, and

(10.38)

(it) ˆ ˆ (ad Aˆ · Hˆ os )k . ∂tk (eit A Hˆ osk e−it A ) = k!

(10.39)

k

But there exists ψ0 ∈ S (R) such that (ad Aˆ · Hˆ os )k ψ0 = 0 (notice that i −1 ad Aˆ · Hˆ os ) is self-adjoint). For example we can take ψ0 = φ j0 , for any j0 ≥ 0. So we get

η(t) k 1 ≥ (ad Aˆ · Hˆ os )k ψ0 0 , lim inf t→+∞ tk k! 

and (10.36) follows.

Proof of Lemma 65 Following [190] we define Aˆ on the basis of Hermite functions. ˆ j = δ(φ j+1 + φ j−1 ) (φ j = 0 if j < 0). A direct comLet δ > 0. Define Aˆ by Aφ ˆ 2 · Hˆ os )φ j = 0 for any putation gives (ad Aˆ · Hˆ os )φ j = δ(φ j−1 − φ j+1 ) and ((ad A) ˆ k  satisfies j ∈ N. Moreover it is not difficult to see that the matrix A j,k = φ j , Aφ the condition (10.28). So by the remark 58 we can apply Theorem 51 hence we get that Aˆ ∈ Sˆ 01 . Notice that here (ad Aˆ · Hˆ os ) is injective in L 2 (R) so we can start with any initial data in the Theorem 52.  Remark 59 The lower bound (10.35) shows that as time t is increasing a large part of the energy E(t) := ψ(t), Hˆ os ψ(t) is transferred from low modes to higher modes. This can be observed on the lower bound estimate:  ( j + 1/2)|c j (t)|2 ≥ μt, μ > 0. (10.40) E(t) = j≥0

The contribution of the mode φ j to the energy is given by |c j (t)|2 which can be very large for large time according (10.40). Notice that without time perturbations the energy is conserved or almost conserved. ˆ ˆ ˆ Let us consider a stationary real perturbation V ∈ Sm 1 with m < 2 and H = Hos + V a time independent Hamiltonian. Hˆ is a self-adjoint operator on the domain Dom( Hˆ ) = H 2 . Furthermore we have, for every k ∈ N, H 2k = Dom(H k ) with a norm equivalent to the graph norm of H k . Hence for any k ≥ 0 there exists Ck > 0 such that

ψ(t) 22k ≤ Ck H k ψ0 2 + ψ0 2 ), ∀t ∈ R.

(10.41)

So the upper-bound (10.41) shows that for perturbations of order < 2 of Hos the lower bound (10.40) is a purely dynamical effect, due to the time oscillations of the perturbation Vˆ (t) and not to its size. This is a rather subtle phenomenum analyzed in several papers, [16, 190, 191] and the references in these papers.

310

10 Characterizations of Harmonic Pseudodifferential Operators

10.2 The Semi-classical Setting In quantum mechanics the quantization of a classical observable A depends on an effective small Planck constant  i.e. Aˆ = OpW A) with the Weyl quantization. We consider now the problem of establishing uniform estimates as   0. We are revisiting the characterization of semi-classical pseudo-differential operators with the Planck constant . Let L j linear forms on R2n . Now we denote ˆ P), ˆ 1 ≤ j ≤ J. Lˆ j = L j ( Q, log m Let m be a weight function on R2n . Denote S0,u the class of -dependent symbols m uniformly in S0 which means γ

|∂ X A (X )| ≤ Cγ m(X ), ∀ ∈]0, 1], ∀X ∈ R2n . The constant weight m = 1 corresponds to the pseudo-differential operators of order 0. ˆ z , Recall that the covariant Weyl-symbol of an operator Aˆ is Ac (z, z  ) := ϕz  , Aϕ where ϕz is the coherent state defined in Chap. 1. Proposition 121 Let { Aˆ  }0 0. Remark 60 In Chap. 2 (and Appendix C) we have given asymptotic expansions for f ( Hˆ ) with remainder estimates in L 2 -norms of operators. Here we shall prove that the asymptotic expansions hold true for the semi-norms of the full symbol of f ( Hˆ ) . Theorem 53 Let Hˆ a uniformly elliptic semi-classical Hamiltonian of weight m and f as in Proposition 119 with r = 0 (see Chap. 2). Then f ( Hˆ ) = Opw (H f ) where H f ∈ S00,u . Proof We give here a sketched proof, leaving the details to the reader. The first step is to prove the theorem for the resolvent Bˆ λ = ( Hˆ − λ)−1 and in a second step to use the Helffer-Sjöstrand formula (see Appendix C). We want to apply Proposition 121 to Bˆ λ for λ ∈ C\R. Let L be a linear form on R2n . Using the equality ˆ Bˆ λ ) = − Bˆ λ adL( Hˆ ) Bˆ λ ad L( we get

ˆ Bˆ λ ) L 2 ,L 2 ≤ C1  (1 + |λ|)

ad L( |λ|2

Hence by induction on J ≥ 1 we can prove that for any linear forms L j , j ≤ J ,

(ad Lˆ 1 · · · ad Lˆ J )( Bˆ λ ) L 2 ,L 2 ≤ C J



J |λ J +1

 (1 + |λ| J ), ∀ ∈]0, 1].

Then applying Proposition 121 we get that the full symbol Bλ of ( Hˆ − λ)−1 is in 0 , with control in λ: S0,u γ |∂ X Bλ (X )|

 ≤ Cγ

< λ >|γ | (λ)|γ |+1

 .

(10.43)

312

10 Characterizations of Harmonic Pseudodifferential Operators

Then plugging (10.43) in the Helffer-Sjöstrand formula we get the Theorem. Corollary 33 Let be H a classical symbol of weight m (see Chap. 2) and f like in Theorem 53. Then for λ ∈ C\R, the full symbol Bλ of ( Hˆ − λ)−1 is a semi-classical symbol of weight m −1 . Moreover the symbol H f of f ( Hˆ ) is a semi-classical symbol of order 0 (see Definition 25). Proof For simplicity we give the proof for the resolvent. A similar argument can be use for f ( Hˆ ). In Chap. 2 and Appendix C we have seen that for any N ≥ 1 we have a parametrix Bλ(N ) =  j bλ, j such that 0≤ j≤N

(H − λ)  Bλ(N ) = 1l +  N +1 Rλ(N ) where the symbols bλ, j are computed and Rλ(N ) satisfies for some fixed n 1 , γ |∂ X Rλ(N ) (X )|

   < λ > 2N +|γ |+n 1   ≤ Cγ ,N  . λ 

Hence from the Calderon-Vaillancourt we deduce that there exists c(λ) > 0 (which can be estimated) such that we have, for N large enough,

Rλ(N ) L 2 ,L 2 ≤ c(λ)3(N +1) . So for  small enough: 0 <  ≤ 1/2c(λ)−3 ) we get −1  − 1l =  N +1 Tλ(N ) . 1l +  N +1 Rλ(N ) Now applying Proposition 121 we get that Tλ(N ) ∈ S00,u . So using the composition formula for symbols BΛ = Bλ(N )  (1 +  N +1 Tλ(N ) ). −1

we get that Bλ ∈ Sm 0,u .

Chapter 11

Herman-Kluk Approximation for the Schrödinger Propagator

Abstract In this Chapter our goal is to study the Schrödinger propagator associated with a time dependent sub-quadratic Hamiltonian H (t) on the phase space Rn × Rn = T ∗ (Rn ). The solution ψ(t) of the Schrödinger equation i

∂ψ(t) = Hˆ (t)ψ(t), ψ(t = s) = ψs ∂t

defined a propagator U  (t, s)ψs := ψ(t). When Hˆ (t) = H (t, x, Dx ) (-Weyl quantization of H (t)), we analyze the Schwartz kernel of U  (t, s), describing it by a quadratic phase and an amplitude. At every time t, when  is small enough, we get that it is “essentially supported” in a neighborhood of the graph of the classical flow generated by H (t). This is proved using the Fourier-Bargman representation of the propagator U  (t, s). Furthermore we get that the amplitude has a full uniform asymptotic expansion in  in the phase space. One of the simplest choice of the phase, related with frozen Gaussians, is known in chemical physics as Herman-Kluk formula. Moreover we prove that the semiclassical expansion obtained for the propagator is valid below an Ehrenfest time, |t| 0 is a stability parameter for the classical system. Most of the results can be extended to multicomponent systems with constant multiplicities, like Dirac operators (see Chap. 14). Finally the results are applied to recover the Van-Vleck formula and to revisit the Aharonov-Bohm effect.

11.1 Sub-quadratic Hamiltonians In this Chapter we consider time dependent sub-quadratic Hamiltonians H (t, X ) where t ∈ R, X = (x, ξ ) ∈ Rn × Rn . We assume that H (t, X ) is continuous in time t, C ∞ -smooth in X and sub-quadratic, that means: γ

∂ X H (t, X )| ≤ Cγ (1 + |X |)(2−|γ |)+ , ∀t ∈ R , ∀X ∈ R2n , Cγ > 0,

© Springer Nature Switzerland AG 2021 D. Robert and M. Combescure, Coherent States and Applications in Mathematical Physics, Theoretical and Mathematical Physics, https://doi.org/10.1007/978-3-030-70845-0_11

(11.1)

313

314

11 Herman-Kluk Approximation for the Schrödinger Propagator

Notice that we could consider the case where t is in some bounded real interval. For simplicity we assume that H (t) is defined for t ∈ R even if our estimates will be valid only locally uniformly in time. As before we denote Hˆ (t) the -Weyl quantization of H (t) and the time dependent Schrödinger equation: i

∂ψ(t) = Hˆ (t)ψ(t), ψ(t = s) = ψs . ∂t

(11.2)

A nice property of sub-quadratic Hamiltonians is that classical and quantum dynamics are well defined at any time. The classical motion for the Hamiltonian H (t) is given by the Hamilton equation X˙ (t) = J ∇ X H (t, X (t)), X (t0 ) = X 0 .

(11.3)

Proposition 122 The solution X (t) of (11.3). is well defined for any t ∈ R. Moreover there exits a constant C > 0 (depending on the constants Cγ of (11.1)) such that (11.4) |X (t)|2 ≤ C|X (t0 )|2 eC|t−t0 | , ∀t ∈ R. Proof Let us consider the quadratic Hamiltonian K 0 (X ) := |X |2 . Using (11.1), we have d |X (t)|2 = {K 0 , H (t)}(X (t) ≤ C|X (t0 )|2 . (11.5) dt  Remark 61 The proof of Proposition 122 uses a weaker property than subquadratic. It is enough to assume that |{K 0 , H (t)}(X )| ≤ C|X |2 , ∀t ∈ R, X ∈ R2n . Proposition 123 For any ψs ∈ L 2 (Rn ) (11.2) has a unique solution ψ(t) ∈ L 2 (Rn ) determined by a propagator: ψ  (t) = U  (t, s)ψs where U  (t, s) is a unitary operator in L 2 (Rn ). Moreover U  (t, s) is a bounded operator in the Harmonic weighted Sobolev spaces Σk for any k ∈ N and there exists Ck > 0 such that U  (t, s)ψs k ≤ Ck eCk |t−s| ψs k , ∀t, s ∈ R.

(11.6)

We use the harmonic oscillator K 0 := Hˆ os and the smoothing operators   K 0 −1 R N := 1 + . N The following lemma describes the main properties of the smoothing operator R N . Its proof is left to the reader.

11.1 Sub-quadratic Hamiltonians

315

Let us denote Rk, the operator norm RL (Σk ,Σ ) . Lemma 66 For every N > 0 the following holds true: R N k,k+2 ≤ N .

(11.7)

R N k,k ≤ C. R N − 1lk+2,k ≤ CN .

(11.8) (11.9)

The constant C depending only on k. Now we regularize the operator Hˆ (t) by defining Hˆ N (t) := R N Hˆ (t) R N . For every N the propagator for Hˆ N (t) exists because Hˆ N (t) is a bounded operator. We shall prove that a propagator for Hˆ (t) can be obtained by controlling the limit N → +∞. Lemma 67 For every N ≥ 1 HN (t) is symmetric on L 2 (Rn ) and bounded on Σk for any k > 0. Furthermore there exists a positive C such that for 0 ≤ k ≤ k0 we have: C (11.10)  Hˆ N (t) − Hˆ (t)k+4,k ≤ , N ≥ 1, t ∈ R . N We prove only the estimate. By Lemma 67 one has  Hˆ N (t) − Hˆ (t)k+4,k ≤ R N Hˆ (t)(R N − 1l)k+4,k C . +(R N − 1l) Hˆ (t)(t)k+4,k ≤ N

(11.11)

Consider the regularized Schrödinger equation  i∂t ψ = Hˆ N (t)ψ ψ|t=s = ψs , s ∈ R.

(11.12)

Since the operator Hˆ N (t) is bounded, it generates a flow U N (t, s) ∈ C(R × R, L (Σk )) which is unitary in L 2 (Rn ). Lemma 68 Fix 0 < k. There exists a positive constant Ck , independent of N , such that U N (t, s)k,k ≤ eCk |t−s| , ∀N > 0 . Proof We must show that Hˆ osk U N (t, s) Hˆ os−k is bounded uniformly in N as an operator from L 2 (Rn ) to itself. Remark that, due to the unitarity of U N (t, s) in L 2 (Rn ), one has Hosk U N (t, s) Hos−k 0,0 = U N (t, s)∗ H k U N (t, s) Hˆ os−k 0,0 .

316

11 Herman-Kluk Approximation for the Schrödinger Propagator

Now one has U N (t, s)∗ Hosk U N (t, s) Hos−k = 1l +



t

s



t

= 1l + s −k ×Hos

U N (r, s)∗ [ Hˆ N (r ), Hˆ osk ] U N (r, s) Hˆ os−k dr U N (r, s)∗ R N [ Hˆ (r ), Hˆ osk ]

R N Hosk U N (r, s) Hos−k dr

(11.13)

where we used that [R N , Hosk ] = 0 and [ Hˆ N (t), Hosk ] = R N [H (t), Hosk ] R N . But we can prove easily that [ Hˆ (t), Hˆ osk ] Hˆ os−k 0,0 ≤ Ck for some positive constant Ck , thus it follows (using also Lemma 66 (ii)) that uniformly in N . we have R N [ Hˆ (t), Hˆ osk ] Hˆ os−k R N 0,0 ≤ Ck , ∀N > 0 . Then such estimate combined with the unitarity of U N (t, s) in L (L 2 (Rn )) give  Hˆ k U N (t, s) Hˆ −k 0,0 ≤ 1l +

 s

t

U N (r, s)∗ R N [ Hˆ (r ), Hˆ osk ] Hˆ os−k R N

× Hˆ osk U N (r, s) Hˆ os−k 0,0 dr  t ≤ 1l + Ck  Hˆ k U N (r, s) Hˆ −k 0,0 dr

(11.14)

s

which by Gronwall allows us to conclude that  Hˆ osk U N (t, s) Hˆ os−k 0,0 ≤ eCk |t−s| , thus proving the claim.



Proof of Proposition 123 Fix arbitrary t, s ∈ R. The first step is to show that for every ψ ∈ Σk+4 , the sequence {U N (t, s)ψ} N is a Cauchy sequence in the space Σk . One has  t U N (t, s)ψ − U N  (t, s)ψk =  ∂τ (U N  (t, τ ) U N (τ, s)ψ) dτ k s  t   =  U N  (t, τ ) Hˆ N (τ ) − Hˆ N  (τ ) U N (τ, s)ψ dτ k s   1 1 +  |t − s| e(Ck +Ck+4 )|t−s| ψk+4 . ≤C N N

11.1 Sub-quadratic Hamiltonians

317

For any t, s in a bounded interval, and ψ ∈ Σk+2τ +2m , the sequence {U N (t, s)ψ} N is a Cauchy sequence in Σk+4 . Since Σk+4 is dense in Σk and U N (t, s)k,k ≤ eCk |t−s| uniformly in N , by an easy density argument one shows that {U N (t, s)ψ} N is also a Cauchy sequence in Σk . Thus for every ψ ∈ Σk the limit U  (t, s)ψ := lim U N (t, s)ψ N →∞

exists in Σk . Moreover we have the following error estimate, for C > 0 large enough, C |t − s|eC|t−s| ψk+4 . N

U  (t, s) − U N (t, s)ψk ≤

(11.15)

By the principle of uniform boundedness, U  (t, s) ∈ L (Σk ). Since U N (t, s) is an isometry in L 2 (Rn ), U  (t, s)ψ0 := lim U N (t, s)ψ0 = ψ0 N →∞

which implies that U  (t, s) is an isometry on L 2 (Rn ) satisfying the Schrödinger equation (11.2)). Denote ψ N (t) = U N (t, s)ψs . Then we have ψ (t) = ψs + i N

−1



t

Hˆ N (r )ψ N (r )dr.

(11.16)

s

Using Lemma 67 and estimate (11.15) there exists C > 0, depending on a, b, k such that for a ≤ s ≤ r ≤ t ≤ b we have   1 N N ˆ ˆ  HN (r )ψ (r ) − H (r )ψ(r )k ≤ C ψ(r ) − ψ (r )k+2τ +2m + ψk+2τ +2m . N So we can pass to the limit in (11.16) and we get ψ(t) = ψs + i

−1



t

Hˆ (r )ψ(r )dr.

(11.17)

s

In particular if ψs ∈ Σk+4 then t → ψ(t) is strongly derivable from R into Σk and satisfies the Schrödinger equation ((11.2)). Furthermore U  (t, s)ψ = lim U N (t, s)ψ = lim U N (t, τ )U N (τ, s)ψ = U  (t, τ )U  (τ, s)ψ N →∞

N →∞

where the limits are in the Σk topology. This shows the group property. Estimate (11.6) is deduced from Lemma 68.

318

11 Herman-Kluk Approximation for the Schrödinger Propagator

Finally we have shown that (t, s) → U  (t, s) ∈ L (Σk+4 , Σk ) is strongly continuously differentiable with strong derivatives i∂t U  (t, s) = Hˆ (t)U  (t, s) . With the same proof we get also −i∂s U  (t, s) = U  (t, s) Hˆ (s) .  Remark 62 More general results and applications to growth of Sobolev norms can be found in the paper [191]. More abstract results are also proved in [191] which can be applied to subquadratic multi-component systems in L 2 (Rn , Cm ) considered in Chap. 14. Moreover the conclusion of Proposition 123 is still valid assuming that the commutator [ Kˆ 0 , Hˆ (t)] Kˆ 0−1 is bounded on the spaces Σ1k (Rn ), k ∈ N.

11.2 Semi-classical Fourier-Integral Operators The Schwartz kernel of the Schrödinger propagator U  (t, s) is conveniently describe as a particular case of a class of operators known in P.D.E theory as Fourier Integral operators. They are obtained by a suitable quantization of canonical transformations of the phase space. Recall that a canonical transformation of the phase space is a map which preserves the symplectic two form on Rnx × Rnξ : σ = dx ∧ dξ.

Definition 1 A map χ : R2n → R2n is called a smooth bilipschitz canonical transformation if χ is a smooth transformation of the phase space: (x, ξ ) → χ (x, ξ ) such that • • • •

β

∂xα ∂ξ χ (x, ξ )| ≤ Cα,β , |α| + |β| ≥ 1. χ is a diffeomorphism. Both χ and χ −1 are Lipschitz continuous. χ preserves the symplectic form, i.e. if (y, η) = χ (x, ξ ) then dy ∧ dη = dx ∧ dξ.

To any bilipschitz canonical transformation χ we can associate a class of operators, named semiclassical Fourier integral operators, (F.I.O) by using the FourierBargmann transform. Recall the definition of the Fourier-Bargman transform (Chaps. 1 and 2 for properties) :

11.2 Semi-classical Fourier-Integral Operators

319

F B ψ(X ) = ψ  (X ) = (2π )−n/2 ϕ X , ψ. Definition 2 Let χ : R2n → R2n be a smooth bilipschitz canonical transformation. A family of operators U  : S (Rn ) → S  (Rn ),  ∈ (0, 1], is a semiclassical Fourier integral operator of order1 m ∈ R if the kernel K  of F B U  F B ∗ satisfies the following condition, for every N ≥ 0, there exists C N such that,   |X − χ (Y )| −N |K  (X, Y )| ≤ C N m−n 1 + , ∀X, Y ∈ R2n ,  ∈ (0, 1]. √  (11.18) The above definition describes a natural way to quantize canonical transformations. For example the identity operator is associated with the trivial canonical transformation. Any pseudo-differential operator Opw is also a Fourier integral with the trivial canonical transformation. A metaplectic transformation in L 2 (Rn ) is a F.I.O associated with a linear symplectic transformation χ (see Chap. 2). Conversely to any canonical transformation satisfying Definition 2 one can associate (-uniformly) bounded transformations in L 2 (Rn ) by the following formula. Let us introduce the kernel |X −χ(Y )|2 K χ (X, Y ) = −n a(X, Y )e−  where a ∈ L ∞ (R4n ) and define:  Iχ (u)(x) =

R4n

K χ (X, Y )ϕ X (x)ϕY , udY d X,

where ϕ Z is the normalized coherent state. Using over-completness property of coherent states one can prove that Iχ is a uniformly bounded operator. As an example we have already computed in Chap. 3 (Sect. 3.1) the Schrödinger propagator U (t, 0) when H (t, X ) is a quadratic Hamiltonian. In this case we get using computations of Chap. 3 (see Chap. 2, Wigner functions) |ϕ X , U  (t, 0)ϕY | ≤ C−n e−c

|X −Ft Y |2 

, ∀t ∈ [0, T ],

(11.19)

where Ft is the linear flow defined by H (t) such that F0 = 1l and C, c may depend on 0 ≤ T < +∞. An easy consequence of the definition and Chap. 2, is: Proposition 124 A semiclassical F.I.O of order 0 is uniformly bounded in L 2 (Rn ), as  ∈]0, 1]. We prove now that this class of operators is closed by composition. We have

1 Here

the order is defined according the behavior of operators in .

320

11 Herman-Kluk Approximation for the Schrödinger Propagator

Theorem 54 Let U j be a semiclassical F.I.O of order m j associated to the bilipschitz canonical transformation χ j , j = 1, 2. Then U1 ◦ U2 is a semiclassical FIO of order m 1 + m 2 associated to χ1 ◦ χ2 . Proof Let K j be the kernel of F B U j F B ∗ , j = 1, 2. Then, the kernel K  of F B U1 U2 F B ∗ = F B U1 F B ∗ F B U2 F B ∗ is given by





K (X, Y ) =

R2n

K 1 (X, Z )K 2 (Z , Y )d Z .

Using the definition of semiclassical F.I.O, we have for all N , N  ∈ N, such that N ≥ N, |K  (X, Y )| ≤ C N ,N  m 1 +m 2 −2n

 R2n

     |X − χ1 (Z )| −N |Z − χ2 (Y )| −N dZ 1 + 1+ √ √  

(11.20) On the other hand, one has by the triangle inequality |X − χ1 ◦ χ2 (Y )| ≤ |X − χ1 (Z )| + |χ1 (Z ) − χ1 ◦ χ2 (Y )|, and since χ2 is Lipschitz, we obtain |X − χ1 ◦ χ2 (X )| ≤ C (|X − χ1 (Z )| + |Z − χ2 (Y )|) , C > 0. Thus, Peetre’s inequality yields 1+

  |X − χ1 (Z )| |X − χ1 ◦ χ2 (Y )| ≤C 1+ √ √     |Z − χ2 (Y )| 1+ . √ 

Now, insert (11.21) into (11.20), we obtain   |X − χ1 ◦ χ2 (Y )| −N |K (X, Y )| ≤ C N ,N  m 1 +m 2 −2n 1 + √      |Z − χ2 (Y )| N −N 1+ × dZ . √  R2n Hence, after a change of variables, for N  − N > 2n, we deduce

(11.21)

11.2 Semi-classical Fourier-Integral Operators 

|K (X, Y )| ≤ C N 

m 1 +m 2 −n



321

|X − χ1 ◦ χ2 (Y )| 1+ √ 

−N

, ∀N ≥ 0. 

This achieves the proof of the Theorem. One of the main result of this chapter is the following one [93].

Theorem 55 Let H (t, X ) be a real subquadratic Hamiltonian (2.82), continuous in time for t ∈ R, C ∞ in X . Then for any t, t0 ∈ R, the propagator U H (t, t0 ) for the Schrödinger equation (11.2) is a semiclassical FIO of order 0 associated to the Hamilton flow φ t,t0 . Proof Let us first remark that we know that the propagator U H (t, t0 ) is well defined according Proposition 123 and the classical flow Φ t,t0 is also well defined according Proposition 122. It is clearly uniformly Lipschitz for t, t0 ∈ [−T, T ] and bi-Lipchitz for (Φ t,t0 )−1 = Φ t0 ,t . Step 1. We assume here that the initial data is a coherent state: ψt0 = ϕz , z ∈ R2n . We transform the problem (11.2) into a problem independent of  as in Chap. 4 (Sect. 4.3). Let ψ(t) = T (z t )Λ u(t). Then the function u(t) satisfies the following equation 

  −1 i∂t u(t) = Λ−1 Hˆ (t)T (z t ) − i∂t T (z t ) Λ u(t)  T (z t ) u(t0 ) = g.

(11.22)

We have the useful formula −1 ˆ w Λ−1  T (z t ) H (t)T (z t )Λ = O p1 H (t, z t +

√ X ).

(11.23)

Taylor’s expansion yields H (t, z t +

√

X ) = H (t, z t ) +

√ √ ∂q H (t, z t )x + ∂ p H (t, z t )ξ + P2 (t, X ) + R3 (t, X ),

(11.24) where P2 (t) is a homogeneous polynomial of degree 2 in X : P2 (t, X ) =

1  α ∂ H (t, z t )X α , 2 |α|=2 X

and R3 is the remainder term given by  √ R3 (t, X ) = −1 H (t, z t + X ) − H (t, z t )−  √ ∂q H (t, z t )x − ∂ p H (t, z t )ξ − P2 (t, X ) .

(11.25)

Consider the function v(t) = e−  δt u(t) where δt is a classical action function defined in Chap. 4. By (11.23)–(11.25), the problem (11.22) becomes i

322

11 Herman-Kluk Approximation for the Schrödinger Propagator



i∂t v(t) = O p1w [P2 (t)]v(t) + O p1w [R3 (t)]v(t) v(t0 ) = g.

Since P2 (t) is quadratic, by the metaplectic representation (see Chap. 3), we use another change of function v(t) = R[Ft ]b(t)g. The new unknown function w(t, x) = b(t, x)g(x) satisfies the following Cauchy problem

i∂t w(t, x) = O p1w [R3 (t, Ft (x, ξ ))](w(t, x)) w(t0 , x) = g(x).

(11.26)

Step 2. We claim that problem (11.26) is well defined. This is a consequence of Proposition 123 and of the next lemma. Let us introduce a bounded interval IT = [t0 − T, t0 + T ], for any T > 0. In what follows all our estimates are uniform for t ∈ IT and the constants depend only on T . Lemma 69 Under Hypothesis of Theorem 11.113, R3 (t) is a subquadratic symbol, with uniform estimates with respect to h ∈ (0, 1] and t ∈ IT . Proof Let γ ∈ N2n . If |γ | ≥ 2, one has by (11.25) γ

|∂ X R3 (t, X )| ≤ Cγ ,T , Cγ ,T ≥ 0,

(11.27)

for every  ∈ (0, 1] and t ∈ IT . Then equation (11.27) and the mean value Theorem applied on ∂ X R3 (t) and R3 (t) respectively, show that γ

|∂ X R3 (t, X )| ≤ Cγ ,T |X |2−|γ | , |γ | ≤ 1, 

for all  ∈ (0, 1] and t ∈ IT .

Taking into account that Ft is linear on R2n , it results from Lemma 69 and the chain rule that R3 (t, Ft ) is also subquadratic. Hence, Proposition 123 ensures that problem (11.2) is well defined in weighted Sobolev spaces H k , k ∈ N. Denote by V (t) its quantum propagator. Then, the evolution operator of the Schrödinger equation (11.2) has the following form ψ(t) = U H (t, t0 )ϕz = T (z t )Λ u(t) = T (z t )Λ R[Ft ](V (t)g) e  δt . i

(11.28)

Last step. Using (11.28) we shall deduce that that U H (t, t0 ) is a semiclassical FIO of order 0, associated to the flow φ t,t0 . We have, from the previous computations, denoting z t = φ t,t0 z  , ϕz , U H (t, t0 )ϕz   = ϕz T (z t )Λ u(t)

  = Λ−1  T (−z t )T (z)Λ g, u(t)  

  1 z − z t  σ (z, z t ) T = exp g, u(t) . (11.29) √ 2i 

11.2 Semi-classical Fourier-Integral Operators

323

From Proposition 123 we know that that u(t) ∈ S (Rn ). Hence estimate (11.18) for U H (t, t0 ) is a consequence of the following Lemma: Lemma 70 Let u, v ∈ S (Rn ). Then for all N ∈ N, there exists C N > 0 such that |T (z)u, v| ≤ C N (1 + |z|)−N uH N vH N . Proof The case N = 0 is obvious by Cauchy-Schwarz inequality. For N ≥ 1, we proceed by induction. Let α ∈ Nn such that |α| = N . One has |q α T (z)uv| ≤



|q α u(x − q)v(x)| dx  ≤ Cα |(x − q)β u(x − q)| |x α−β v(x)|dsx β≤α

≤ Cα



x β u L 2 (Rn ) x α−β v L 2 (Rn ) .

(11.30)

β≤α

In the other hand,     | p α T (z)u, v| =  p α ei px u(x − q)v(x) dx      =  ∂xα ei px u(x − q)v(x) dx     ∂ γ u(x − q)| |∂ α−γ v(x) dx ≤ Cα x x γ ≤α

≤ Cα



∂xγ u L 2 (Rn ) ∂xα−γ v L 2 (Rn ) .

γ ≤α

We have |z| N ≤ 2 N −1 (|q| N + | p| N ). Consequently, (11.30) and (11.31) imply for all N ≥ 1: ⎛ |z| N |T (z)uv| ≤ C N ⎝



x β u L 2 (Rn ) x α−β v L 2 (Rn )

β≤α

+





∂xγ u L 2 (Rn ) ∂xα−γ v L 2 (Rn )

γ ≤α

≤ C N uH N vH N . This Lemma simultaneously with (11.29) provide that   |z − z  | −N |K  (z, z  )| ≤ (2π )−n C N 1 + √ t , ∀N ∈ N. 

(11.31)

324

11 Herman-Kluk Approximation for the Schrödinger Propagator

Finally, using the definition of the flow Φ t,t0 we deduce estimate (11.18) and therefore Theorem 11.113.  Remark 63 The above definition given for F.I.O is in agreement with the correspondence principle between classical mechanics and quantum mechanics as it was formulated by N. Bohr.

11.3 The Herman-Kluk Semi-classical Approximation We have seen that the time dependent propagator for a subquadratic Hamiltonian is a semi-classical F.I.O. Notice that the result is easily extended for -dependent (2− j)+ classical Hamiltonians H (t, X )   j H j (t, X ) where H j (t) is in S0 with j≥0

uniformly bounded semi-norms as t ∈ R and     j H j (t, X ) is bounded in S00 , (11.32) ∀N ≥ 1, −N −1 H (t, X ) − 0≤ j≤N

with uniformly bounded semi-norms as t ∈ R and  ∈]0, 1]. In applications the contribution of the sub-principal symbol H1 (t) may have a non trivial effect (this will be more obvious for systems considered later. In this section our aim is to analyze with more accuracy the structure of the Schwartz kernel of the propagator U H (t, t0 ). This can be done using coherent states and the following lemma. Lemma 71 Let Aˆ be a continuous linear operator Aˆ : S (Rn ) → S  (Rn ). Then its Schwartz kernel K A (x, y), as a distribution in S  (Rnx × Rny ), satisfies K (x, y) = (2π ) A

−n

 R2n

Aˆ ∗ ϕz (y) ϕz (x)dz.

(11.33)

In particular let K H (t, t0 ; x, y) be the Schwartz kernel of a quantum propagator U H (t, t0 ) such that in Theorem 56 then we have K H (t, t0 ; x, y) = (2π )−n

 R2n

U H (t, t0 )ϕz (x)ϕz (y)dz .

Proof Recall here that we compute kernels as distributions. ˆ 71 follows from the inverse Bargmann transform applied to Aψ. We get (11.34) using that U H (t, t0 )∗ = U H (t0 , t)−1 .

(11.34)



We wish now to use formula (11.34) in order to apply the results obtained in Chap. 4. concerning semi-classical propagation of coherent states.

11.3 The Herman-Kluk Semi-classical Approximation

325

As it is expected, and suggested by Sect. 11.2, the semi-classical approximation depends stongly on the classical flow generated by the principal term H0 (t, X ) of the Hamiltonian. As in Chap. 4, Let us introduce the flow φ t,t0 for H0 (t, X ) and the stability matrix along this flow: 

At,t0 Bt,t0 = Ct,t0 Dt,t0

Ft,t0



Recall that if φ t,t0 (q, p) = qt,t0 (q, p), pt,t0 (q, p) = z t,t0 (z) then we have At,t0 =

∂qt,t0 ∂qt,t0 ∂ pt,t0 ∂ pt,t0 , Bt,t0 = , Ct,t0 = , Dt,t0 = ∂q ∂p ∂q ∂p

(11.35)

We also introduce the classical action S(t, t0 , z) =



t

( ps,t0 · q˙s,t0 − H (s, z s,t0 ))ds,

(11.36)

t0

and the symmetric complex matrix Γt,t0 = (Ct,t0 + i Dt,t0 )(At,t0 + i Bt,t0 )−1 .

(11.37)

Notice that Γt,t0 depends also on z ∈ R2n the initial data of the classical trajectory z t . Using that the coherent state family {ϕz }z∈R2n is overcomplete in L 2 (Rn ) and the propagation Theorem proved in Chap. 4, we get the following analysis of the Schwartz kernel of the propagator U (t, t0 ). Theorem 56 Let H (t) be a time dependent subquadratic Hamiltonian satisfying (11.32). Let K H (t, t0 ; x, y) be the Schwartz kernel of its quantum propagator U H (t, t0 ).  i K H (t, t0 , x, y)  e  ΦΓ (t,t0 ,z;x,y) c(; t, t0 , z)dz, (11.38) R2n

where c(; t, t0 , ·) is a semi-classical symbol of order 0 and ΦΓt,t0 (t, t0 , z; x, y) = i S(t, t0 , z) + pt · (x − qt ) − p · (y − q) + |y − q|2 + Γt,t0 (x − qt ) · (x − qt ). 2 More precisely, if U H,N (t, t0 ) denotes the operator with the Schwartz kernel K H,N (t, t0 , x, y) = (2π )−3n/2

⎛

 e R2n

i  ΦΓt,t0

(t,z;x,y)

⎝

N  j=0

⎞  j c j (t, t0 , z)⎠ dz,

326

11 Herman-Kluk Approximation for the Schrödinger Propagator

then for every t0 , T , and every N ≥ 1, there exists C N ,T such that U H (t, t0 ) − U H,N (t, t0 ) L 2 →L 2 ≤ C N ,T  N +1 , ∀ ∈ (0, 1], |t − t0 | ≤ T. (11.39) The leading term is given by the following formula c0 (t, t0 ; z) = det

−1/2

(At,t0

  + i Bt,t0 ) exp −i

t

H1 (z s )ds

 (11.40)

t0

where the square root is defined by continuity starting from t = t0 (c0 (t0 ; z) = 2n/2 ). Proof The asymptotic formula (11.38) is a direct application of results proved in Chap. 4 (Sect. 4.3) concerning the time evolution coherent states, using Lemma 71. Hence we get easily the quadratic phase ΦΓ (t, t0 , z; x, y). The only new thing we have to prove is the remainder estimate (11.39) using the sub-quadratic assumption. It is easier to prove this estimate in the Fourier-Bargmann representation. Recall the following formulas obtained in Chap. 4 (4.3.1). (t)ψz(N ) + Rz(N ) (t) i∂t ψz(N ) = H ⎛

where

ˆ t] ⎝ ψz(N ) (t) = eiδt / Tˆ (z t )Λ R[F



(11.41) ⎞

 j/2 b j (t)g ⎠

(11.42)

0≤ j≤N

and ⎛

⎞

⎜ j/2 Rz(N ) (t, x) = eiδt / ⎜ ⎝

 j+k=N +3 k≥3

⎟ ˆ t ]O p1w [Rk (t) ◦ Ft ](b j (t)g)⎟ . Tˆ (z t )Λ R[F ⎠ (11.43)

So have

i∂t U ,(N ) (t, t0 ) = Hˆ (t)U ,(N ) (t, t0 ) + R (N ) (t, t0 )

(11.44)

Denote B R (N ) (t, t0 , z, z  ) = ϕz  , R (N ) (t, t0 )ϕz . We have B R (N ) (t, t0 , z, z  ) = ϕz  , RzN (s). Now, using Lemma 70, we get that for any M > 0. there exists C M,T such that |ϕ , z 

RzN (s)|

≤ C M,T 

(N +3)/2



|z  − z s,t0 | 1+ √ 

−M

.

(11.45)

Notice that (11.45) means that the operator R (N ) (t, t0 ) is semi-classical F.I.O of order N 2+1 + n. We know that U (t0 , t) is a semi-classical F.I.O.

11.3 The Herman-Kluk Semi-classical Approximation

327

Using the Duhamel formula we get for B E (N ) (z, z  ) = ϕz , (U H (t, t0 ) − U ,(N ) (t, t0 ))ϕz , BE

(N )

1 (z, z ) = i 



t t0

ϕz , U H (t, s)Rz(N ) (s, t0 )ϕz  ds

(11.46)

Using Lemma 70 we get that U H (t, s)Rz(N ) (s) is a semi-classical F.I.O of order N +3 + n associated with the canonical transform φ t,t0 . This entails the estimate: 2 U H (t, t0 ) − U ,(M) (t, t0 ) L 2 →L 2 ≤ C M,T 

M+1 2

, ∀  ∈ (0, 1].

(11.47)

But it is not difficult to see that U ,(M) (t, t0 ) is a semiclassical F.I.O of order 0 and for M > N , U ,(M) (t, t0 ) − U ,(N ) (t, t0 ) is of order N + 1. Taking M large enough we get (11.39).  In the following theorem we get that in the phase function ΦΓt,t0 we can replace the matrix Γt,t0 by any matrix Θ ∈ S(n)+ by modifying the amplitude. In particular we can choose Θ = i1l. This property is known as the Hermann-Kluk approximation of the propagator. Theorem 57 ([219]) Let H (t) be a time dependent subquadratic Hamiltonian satisfying (11.32). Let K  (t, t0 ; x, y) be the Schwartz kernel of its quantum propagator U H (t, t0 ). Let be Θ ∈ S(n)+ , a smooth family of complex matrices. dependending smoothly in t, z) ∈ R × R2n . Then we have K  (t, t0 , x, y) 



e  ΦΘ (t,t0 ,z;x,y) a(; t, t0 , z)dz, i

R2n

(11.48)

where a(; t, t0 ) is a semi-classical symbol of order 0 and ΦΘ (t, z; x, y) = S(t, z) + pt · (x − qt ) − p · (y − q) +

 1 Θ(x − qt ) · (x − qt ) + i|y − q|2 . 2

(11.49) More precisely, if U ,(N ) (t, t0 ) denotes the operator with the Schwartz kernel K ,(N ) (t, t0 , x, y) = (2π )−3n/2

 R2n

⎞ ⎛ N  i e  ΦΘ (t,z;x,y) ⎝  j a j (t, t0 , z)⎠ dz, j=0

then for every t, |t − t0 | ≤ T and every N ≥ 1, there exists C N ,T such that U H (t) − U ,(N ) (t) L 2 →L 2 ≤ C N ,T  N +1 , ∀  ∈ (0, 1]. The leading term is given by the following formula

328

11 Herman-Kluk Approximation for the Schrödinger Propagator

  a0 (t, t0 ; z) = det 1/2 (Ct,t0 − i Dt,t0 − Θ(At,t0 − i Bt,t0 )) exp −i

t

H1 (z s )ds



t0

(11.50) where the square root is defined by continuity starting from t = t0 (a0 (t0 , t0 ; z) = 2d/2 ). Moreover, the amplitudes a j are smooth functions defined by transport equations and for every T > 0 they are uniformly bounded in S00 for |t| ≤ T . The previous result is valid on bounded time intervals. Now we give a result with a control on large time intervals. Let us assume that conditions on Hˆ (t) are satisfied uniformly for t ∈ R. Moreover assume that there exists a positive real function μ(T ) ≥ 1, T > 0, such that the classical flow φ t satisfies, for every multiindex γ , |γ | ≥ 1, we have for some Cγ > 0, 

|∂zγ φ t,t (z)| ≤ Cγ μ(T )|γ | , for |t| + |t  | ≤ T, ∀z ∈ R2n .

(11.51)

The condition (11.51) is discussed in [40] . In particular this condition is fulfilled 2 H (t, X ). with μ(T ) = eδT for δ = sup J ∂ X,X X ∈R2n ,t∈R

Theorem 58 Choosing the phase as in theorem 57, for j ≥ 0 the amplitudes a j (t, z) satisfy the following estimates, for every multiindex γ there exist a constant C j,γ such that |∂zν a j (t, z)| ≤ C jγ |det 1/2 M(t, z)|μ(T )8 j+2|ν| , ∀T > 0, |t| ≤ T, ∀z ∈ R2n . (11.52) In particular for μ(T ) = eδT we have the following Ehrenfest type estimate. For every N ≥ 1 and every ε > 0 there exists C N ,ε such that we have U H (t, t0 ) − U ,(N ) (t, t0 ) ≤ C N ,ε ε(N +1) , ∀t, |t| ≤

1−ε | log |, ∀ ∈]0, 1]. 8δ (11.53)

For quadratic Hamiltonians the Herman-Kluk formula is exact (without error term). So assume that H (t, X ) is quadratic in X ∈ R2n then we have the following representation formula for the propagator generated by Hˆ (t). K  (t, t0 , x, y) = (2π )−3n/2 a(t, t0 )



e  ΦΘ (t,z;x,y) dz, i

(11.54)

R2n

where a(t, t0 ) is a number depending only on the coefficient of H (t) and which can be computed. Remark 64 In [219] more explicit and exact computations are given for quadratic Hamiltonians. In previous works an Ehrenfest time TE = c| log |, c > 0, was estimated for propagation of Gaussians in [65] and for propagation of observables in [40]. For

11.3 The Herman-Kluk Semi-classical Approximation

329

Gaussians we got c = 6δ1 , for observables c = 2δ1 and here for the Hermann-Kluk approximation c = 8δ1 . Notice that if μ(T ) ≈ T κ , κ > 0 then we get an algebraic critical time TH K for the 1 validity of the Herman-Kluk approximation with TH K ≈  8κ . We give two different proofs for Theorem 57 and Theorem 58. For Theorem 57 we start from Theorem 56 and we use a deformation argument from the phase ΦΓ (with Γ = Γt,t0 depends also on z ∈ R2n , to the phase Φi1l . For proving Theorem 58 we directly use an ansatz to compute the amplitudes a j (t, t0 , z) solving transport equations.

11.3.1 Proof of Theorem 57 by Deformation Our aim here is to prove Theorem 57 from Theorem 56. For that purpose let us consider a C 1 curve in S(n)+ , s → Θ(s), 0 ≤ s ≤ 1, Θ(0) = Γt,t0 , Θ(1) = Θ. For simplicity we take Θ(s) = sθ + (1 − s)Γt,t0 . We shall prove that we can get a family of semiclassical symbols:  j a (s) a (s) (, t, t0 , z) = j (t, t0 , z) such that j≥0



1

∂s

(s)

a (, t, t0 , z)e

i  ΦΘ(s) (t,z;x,y)

 dz

≈ 0.

(11.55)

0

We have ∂s ΦΘ(s) =

1 (Θ − Γt,t0 )(x − qt ) · (x − qt ) 2

We shall compute (11.55) using integration by parts with the formula M(s)T (x − qt )e  ΦΘ(s) = i

 i (∂q − ∂ p )e  ΦΘ(s) i

(11.56)

where M(s) = Ct,t0 − i Dt,t0 − Θ(s)(At,t0 − i Bt,t0 ). To explain the details it is convenient here to consider a more general setting. Let us consider a class of operators (of Fourier-integral type) defined as follows. χ is a bilipchitz symplectic diffeomorphism in R2n . We denote z = (q, p) ∈ R2n , χ (z) = (Q(z), P(z)) ∈ Rd × Rd and S an action for χ , that means that it is a primitive on R2n of the closed 1-form Pd Q − pdq. We consider the following phases Ψ (χ,Θ,Γ ) (z; x, y) = S(z) + P · (x − Q) − p · (y − q)  1 + Θ(x − Q) · (x − Q) − Γ (y − q).(y − q) . 2

(11.57)

330

11 Herman-Kluk Approximation for the Schrödinger Propagator

Let us denote I (a, Γ, Θ) the operator in S (Rn ) having the following Schwartz kernel  i (χ,Θ,Γ ) −3n/2 (z;x,y) eΨ a(z)dz (11.58) Kχ,a (x, y) = (2π ) R2n

where a ∈ A0 , and Γ, Θ ∈ S+ (n). We assume that the matrix Γ is constant (pratically Γ = i1l) but Θ may depends on z and satisfies Θ(z)v · v ≥ c0 |v|2 , ∀z ∈ R2d , ∀v ∈ Cd , with c0 > 0. |∂zα Θ(z)| ≤ Cα , ∀z ∈ R2d .

(11.59) (11.60)

Pratically if χ is an Hamiltonian flow at time t: z → Φ t (z) then Θ = (C(t, z) + Γ0 D(t, z))(A(t, z) + Γ0 B(t, z))−1 , where we have 

 A(t, z) B(t, z) := ∂z Φ t (z). C(t, z) D(t, z)

Remark 65 We know that (C + DΓ ) and (A + BΓ ) are invertible and that the matrix (C + DΓ )(A + BΓ )−1 ∈ S+ (n). 0 Proposition 125 Let be Γ, Θ as above and a ∈ S0 . Then there exists a semi-classical () j 0 symbol b ≈  b j , in S0 such that we have for the L 2 operator norm, j≥0

I (a, Γ, Θ) = I (b() , Γ, Γ0 ) + O(∞ )

(11.61)

Moreover we have b = b0 + b1 + O(2 ) where b0 (z) = a0 (z) and

det 1/2 (M(1)) det1/2 (M(0))

  M(s) := C + D Γ¯ − (1 − s)Θ + sΓ0 (A + B Γ¯ ).

(11.62)

(11.63)

The proof consists in an evolution argument by considering the path in S+ (n), s → Θ(s) := (1 − s)Θ(z) + sΓ0 , 0 ≤ s ≤ 1 and by looking for s → a(s) ∈ A0 such that I (a, Γ, Θ) = I (a(s), Γ, Θ(s)) + O(∞ ) for any s ∈ [0, 1]. This is done thanks to the matrix M(s) which has good properties according to the next remark.

11.3 The Herman-Kluk Semi-classical Approximation

331

Remark 66 Note that we have M(s) = C + D Γ¯ − Θ(s)(A + B Γ¯ )(C + DΓ ) = (A + BΓ )−1 − Θ(s))(A + BΓ ).

Therefore, by Remark 65, C + DΓ )(A + BΓ )−1 ∈ S+ (n) and so (C + DΓ )(A + ¯ ∈ S+ (n). Therefore, M(s) is invertible for all s ∈ [0, 1]. BΓ )−1 − Θ) We shall also use the following Lemma Lemma 72 For every b ∈ S00 and β ∈ Nn we have the L 2 estimate, uniformly in s ∈ [0, 1],   (11.64) I ((x − Q)β b, Γ, Θ) = O |β|/2 Proof It follows from an elementary Gaussian estimate and the use of the Bargman transform. Indeed, the Schwartz kernel of the operator I ((x − Q)β b, Γ, Θ) is the function  i (x, y) → (2π )−n (x − Q)β e  S(z) gzΓ0 (y)g Θ, Z (x)b(z)dz. R2n

Its Bargman transform is the function |β| 2

kB (X, Y ) =  (2π )

−2n



e  pY ·qY − p X ·q X +  S(z) gY , gzΓ,  ·    x − Q β Θ,  g Z , g X b(z)dz. · √  i

i

R2n

(11.65)

and it is not difficult to see that for all N ∈ N, there exists C N > 0 such that   |β| X  N + Y  N |kB (X, Y )| ≤ C N  2 . The conclusion then follows from Lemma 5.5 of our paper or 3.4 of [219]. Proof We set

Ψ (s) = Ψ (χ,Θ(s),Γ )

and we look for a C 1 family symbol a (s) , 0 ≤ s ≤ 1 such that a (0) = a, a (1) = b and  ∂s

eΨ i

R2d

(s)

(z;x,y) (s)

a (z)dz

We have ∂s Ψ (s) = and we claim that



= O(∞ ), ∀s ∈ [0, 1]

1 ∂s Θ(s)(x − Q) · (x − Q), 2

(11.66)

(11.67)

332

11 Herman-Kluk Approximation for the Schrödinger Propagator

(∂q + Γ¯ ∂ p )Ψ (s) =

 i i  ∂q Θ + Γ¯ ∂ p Θ (x − Q) · (x − Q)Ψ (s) M(s)T (x − Q)Ψ (s) +  2

(11.68) This relation comes from a direct computation. Finally, using that M(s) is invertible according to Remark 66, we can write (∂q + Γ¯ ∂ p )Ψ (s) = i

(x − Q)e  Ψ

(s)

=

 i  T M (s, z)(x − Q) + N (s, z)(x − Q, x − Q) Ψ (s) ,  (11.69)

 i (s)  i (s)  (M T (s))−1 (s, z) ∂q + Γ¯ ∂ p e  Ψ − N0 (s, z)(x − Q, x − Q)e  Ψ i

(11.70) where N0 (s, z) is the smooth tensor of order 2 given by M(s) := C + D Γ¯ − Θ (s) (A + B Γ¯ ) N0 (s, z) :=

  1 T (M (s))−1 ∂q Θ(s) + Γ¯ ∂ p Θ(s) . 2

We are now going to use the latter relation for finding an equation on s → a (s) . Using (11.67), we have ∂s I (a (s) , Γ, Θ(s)) = I (∂s a (s) +

i ∂s Θ(s)(x − Q) · (x − Q)a (s) , Γ, Θ (s) ). 2

We then use (11.70) to write i i (s) ∂s Θ(s)(x − Q) · (x − Q)a (s) e  Ψ 2  i (s)  i (s) = a (s) (M(s)−1 ∂s Θ(s)(x − Q)) · ∂q + Γ¯ ∂ p e  Ψ + N1 (s, z)[x − Q]3 e  Ψ where N1 (s, z) is a smooth tensor of order 3 (independent of ). We then observe   (∂q + Γ ∂ p ) · a (s) M(s)−1 ∂s Θ(s)(x − Q)   = a (s) Tr M(s)−1 ∂s Θ(s)(A + BΓ ) + n 1 (s, z) · (x − Q)   = −a (s) Tr M(s)−1 ∂s M(s) + n 1 (s, z) · (x − Q) for some vector n 1 which is independent from , and where we have used ∂s Θ(s) = Γ0 − Θ(z) = −∂s M(s)(A + BΓ )−1 . Therefore, integration by part in q and p shall generate powers of (x − Q). However, if we stop here the transformations, we shall √ only have a term of order 1 in (x − Q) which only generates a remainder of order . Therefore, we push the transformation one step higher by using again (11.70) and writing

11.3 The Herman-Kluk Semi-classical Approximation

n 1 (s, z) · (x − Q)e  Ψ i

(s)

=

 i (s)   M(s)−1 n 1 (s, z) · ∂q + Γ¯ ∂ p e  Ψ i i (s) −n 1 (s, z) · N0 (s, z)[x − Q]2 e  Ψ .

333

(11.71)

Then, performing a first integration by parts and using Lemma 72, we obtain d I (a (s) , Γ, Θ(s)) = I (∂s a (s) − f (s, z)a (s) , Γ, Θ(s)) + O() ds and

(11.72)

f (s, z) = Tr((M(s))−1 ∂s M(s)) = (det M(s))−1 ∂s (det M(s)).

Note that the second equality is the Liouville formula. At this stage of the proof, by solving the equation ∂s a (s) − f (s, z)a (s) = 0 we get the first term b0 of the asymptotic expansion by the formula (11.62). Note that the structure of the rest term in (11.72) is known, it is of the form I (r  (s), Γ, Θ(s)), where s → r  (s) is a smooth map from [0, 1] to the set S00 and r  admits an asymptotic expansion in  at any order. This opens the way to a recursive argument based on (11.72)  for concluding the proof by constructing the coefficients of the asymptotic series  j b j . Therefore, let us assume that we have constructed C 1 maps t → b j (s) such that ⎛ ⎞ j0  d I⎝  j b j (s), Γ, Θ(s)⎠ =  j0 +1 I (r j0 (s), Γ, Θ(s)) + O( j0 +2 ) ds j=1 and let us look for s → b j0 +1 (s) for pushing the estimate one step further. The analysis above and the recursive assumption leads to ⎛ ⎞ j0 +1  d I⎝  j b j (s), Γ, Θ(s)⎠ =  j0 +1 I (∂s b j0 +1 (s) − f (s, z)b j0 +1 (s), Γ, Θ(s)) ds j=1 +  j0 +1 I (r j0 +1 (s), Γ, Θ(s)) + O( j0 +2 ). (11.73) We then conclude the recursive argument by choosing b j0 +1 (s) such that b j0 +1 (0) = 0 and ∂s b j0 +1 (s) − f (s, z)b j0 +1 (s) = −r j0 (s), whence the result of the proposition.



334

11 Herman-Kluk Approximation for the Schrödinger Propagator

11.3.2 A Proof by Solving Transport Equations This alternative proof seems more convenient to control the estimates in the large time regime. Let us denote I (a, Φ) the formal operator having the Schwartz kernel K a (x, y) = (2π )−3n/2



e  Φ(t,z;x,y) a(t, z)dz. i

(11.74)

R2n

For simplicity we take t0 = 0 in all the proof of Theorem 57. 1 Determine the amplitudes As usual for this kind of problems there are two steps : 2 Estimate the error between a j solving by induction transport differential equations,  the approximated propagator and the exact one. It is convenient to write e  Φ = (π )n/2 ϕzt (x)ϕ¯ z (y)e  (S(t,z)+( p·q− pt ·qt )/2) . i

i

(11.75)

Then we have to compute Hˆ  (t)ϕzt . It is not difficult to add contributions of the lower order terms of the Hamiltonian, so we shall assume for simplicity that H  (t) = H0 (t) := H (t). Lemma 73 For every N ≥ 2 we have Hˆ (t)ϕzt (x) =

x − q   |γ |/2 γ t ∂ X H (t, z t )Πγ √ ϕzt (x) + γ !  |γ |≤N (N +1)/2 T (z t )Λ O p1w [R N (t, z t )]g(x)

(11.76)

where 

1

R N (t, z t , X ) = 0

(1 − s) N N!



√ γ ∂ X H (t, z t + s X )X γ ds.

(11.77)

|γ |=N +1

and Πγ is a universal polynomial of degree ≤ |γ | which is even or odd according |γ | is even or odd. Proof Let us recall that ϕz = Tˆ (z)Λ g. In this proof we put z t = z. An easy property of Weyl quantization gives √ w ˆ ˆ ˆ Λ−1  T (z) H (t) T (z)Λ = O p1 [H ( X + z)],

(11.78)

where the quantization is in X = (x, ξ ) ∈ R2n . So the Lemma follows easily from the Taylor formula with integral remainder.  In this first step we don’t take care of remainder estimates, this will be done in the next step.

11.3 The Herman-Kluk Semi-classical Approximation

335

From the Lemma 73 we can write Hˆ (t)I (a, Φ) ∼ I (b, Φ), where x − q   |γ |/2 γ t ∂ X H (t, z t )Πγ √ a. b∼ γ !  γ We have Πγ (x) =



h γ ,β x β .

(11.79)

(11.80)

β≤γ

A direct computation, using Hamilton equation for the classical flow, gives b ∼ H (t, z t )a + (∂q H (t, z t ) + i∂ p H (t, z t )) · (x − qt )a +   x − q  x − q  t t · √ + Tr(K − i L) a  E √  

(11.81)

2 where we denote ∂ X,X H (t, X ) the Hessian matrix of H (t),



2 ∂ X,X H (t, z t )

 G L = , E = G + 2i L − K . L K

(11.82)

2 2 with G := ∂q,q H (t, z t ), L := ∂q, z ), K := ∂ 2p, p H (t, z t ). p H (t,   t At Bt satisfies Reacall that the stability matrix Ft = C t Dt 2 F˙t = J ∂ X,X H (t, z t )Ft , Ft=0 = 1l. It turns out that the power of (x − qt ) can be transformed into power of , using the following Lemma:

Lemma 74 Let us denote M(t, z) = (Ct − Bt ) − i(At + Dt ). We have (∂q − i∂ p )e  Φ = i M(t, z)T (x − qt )e  Φ , with

(11.83)

  det M(t, z) ≥ 2−n .

(11.84)

i

i

In particular M(t, z) is invertible. Proof For simplicity, let us forget the lower index t. Let us consider the 2d × 2d matrix   1l + A − iC B + i(1l − D) 1l + F + i J (1l − F) = C − i(1l − A) 1l + D + i B   1l + A − iC −i(D + i B) + i = . i(A − iC) 1l + D + i B A linear algebra computation gives

(11.85)

336

11 Herman-Kluk Approximation for the Schrödinger Propagator

  det(1l + F + i J (1l − F)) = 2n det A + D + i(B − C) .

(11.86)

Using that F is symplectic, we get (1l + F + i J (1l − F))∗ (1l + F + i J (1l − F)) = (1l + F τ )(1l + F) + (1 − F τ )(1l − F) ≥ 1l2n

(11.87)

hence (11.84) follows. Let us recall classical computations for the derivatives of the action ∂q S = (∂q qt )T pt − p, ∂ p S = (∂ p qt )T pt .

(11.88) (11.89)

Then we can compute ∂q Φ, ∂ p Φ and we get (2.45). Let us prove now (11.115). We know that Γt (s) is an Hermitian non negative matrix.



Integrating by parts using Lemma 74, we get (i∂t − Hˆ (t))I (a, Φ)  I ( f, Φ)

(11.90)

where   1  f ∼ i ∂t a − Tr M˙ M −1 a 2 x − q   |γ |/2−1 γ t ∂ X H (t, z t )Πγ √ a + γ !  |γ |≥3

(11.91)

Hence using the Liouville formula, we get the first term   a0 (t, z) = 2n/2 det 1/2 i M(t, z)

(11.92)

We shall obtain the next terms a j by successive integrations by parts. This is solved more explicitly with the following Lemma. Lemma 75 For any symbol b ∈ S00 , and every multiindex α ∈ N2d we have   |α| 2 ≤|β|≤|α|

|β|





(x − qt )α e  Φ b(z)dz = i

R2n

2n |σ |≤2|β|−|α| R

f α,β,σ (t, z)e  Φ ∂zσ b(z)dz i

(11.93)

11.3 The Herman-Kluk Semi-classical Approximation

337

where f α,β,σ (t, z) are symbols of order 0, uniformly bounded in S00 on bounded time intervals. They only depend on the classical flow φ t (z) and its derivatives. The Lemma is easily obtained by induction on |α| using Lemma 74  Now, to determine the transport equation, we solve inductively on j ≥ 0, the equations    k ak (t), Φ = O( j+2 ). (11.94) (i∂t − Hˆ (t))I 0≤k≤ j+1

Reasoning by induction on j ≥ 0, we get the transport equation for a j+1 (t) by cancellation of the coefficient of  j+1 in (11.118). ∂t a j+1 (t, z) =

1  ˙ −1  Tr M M a j+1 (t, z) + b j (t, z), a j+1 (0, z) = 0, 2 

where b j (t, z) =

F j,k,α (t, z)∂zα ak (t, z).

(11.95)

(11.96)

|α|+2k≤2( j+2)

Moreover, F j,k,α (t, z) depends only on the classical flow φ t (z) and its derivatives and satisfies, for all z ∈ R2n , |t − t0 | ≤ T , |∂zγ F j,k,α (t, z)| ≤ C j,k,α,γ (T ).

(11.97)

So we get, for every j ≥ 0,  a j+1 (t, z) =

t

  det 1/2 M(t, z)M(s, z)−1 b j (s, z)ds.

(11.98)

0

11.3.3 Error Estimates on Finite Time Intervals Let us denote

where a (N ) (t) =

  R N (t) = (i∂t − Hˆ (t))I a (N ) (t), Φ 

(11.99)

k ak . Using Duhamel formula we have

0≤k≤N

U  (t) − U

N ,

(t) ≤ −1



t

R N (s)ds

(11.100)

0

where t0 = 0, U  (t) = U  (t, 0), U

N ,

  (t) = I a (N ) (t), Φ .

So we have to estimate R N (t). Let us denote K ,(N ) (x, y) the Schwartz kernel

338

11 Herman-Kluk Approximation for the Schrödinger Propagator

of U N , (t) and K˜ R(N ) (X, Y ) the Schwartz kernel of R N (t) in the Fourier-Bargmann representation : K˜ (N ) (X, Y ) =

 Rd ×Rd

K (N ) (x, y)ϕ X (y)ϕY (x)d xd y.

(11.101)

Let R˜ N (t) be the operator with Schwartz kernel K˜ (N ) (X, Y ). The following Lemma is well known. Lemma 76 We have the following L 2 norm estimate R N (t) L 2 (Rn ) ≤ (2π )−n R˜ N (t) L 2 (R2n ) .

(11.102)

In particular we have 

(2π )−n max sup Y



| K˜ R(N ) (X, Y )|d X, sup X



R N (t) L 2 (Rn ) ≤  | K˜ R(N ) (X, Y )|dY . (11.103)

Proof For inequality (11.102) we use that the Fourier-Bargmann transform is an isometry.  Inequality (11.103) is known as Carleman (or Schur) L 2 estimate. Using Lemma 73 we get K˜ (N ) (X, Y ) = 2−3n/2 (π )−n

 R2n



 Tˆ (z t )Λ Opw1 [R N (t)]g, ϕY ϕ X , ϕz  · ·a (N ) (t, z)e  δ(t,z dz, i

where δ(t, z) = S(t, z) + p·q−2pt ·qt . Using Weyl commutation formula we have   |X − z|2 i + σ (X, z) (11.104) ϕ X , ϕz  = exp − 4 2



 Tˆ (z t )Λ O p1w [R N (t)]g, ϕY = O p1w [R N (t)]g, g Y√−zt . (11.105) 

We know the Wigner function W0,Z of the pair (g, g Z ), Z ∈ R2n (see Chap. 2)   Z W0,Z (X ) = 22n exp −|X − |2 − iσ (X, Z ) 2

(11.106)

By a well known property of Weyl quantization [104], for any symbol s, we have O p1w [s]g, g Z 

= (2π )

−n

 R2n

s(X )W0,Z (X )d X

(11.107)

11.3 The Herman-Kluk Semi-classical Approximation

339

We shall use the following Lemma Lemma 77 Let f be a symbol, f ∈ S00 (2n). For every γ ∈ N2n and m > 0 there exists Cγ ,m such that   

R2n

X γ f (X )e−|X −Z |

2

−i J Z ·X

  d X  ≤ Cγ ,m (1 + |Z |)−m

sup |α|≤m+|γ |;

Y ∈R2d

|∂Yα f (Y )| (11.108)

Proof It is enough to assume |Z | ≥ 1. We integrate m times by parts with the differential operator 2(X − Z ) − i J Z · X L = ∂X (11.109) 4|X − z|2 + |J Z |2 using that (L τ )m =



lm,α ∂ Xα , with |lm,α | ≤ Cm,α (|Z | + |X − Z |)−m , where

|α|≤m

θ (X ) = −|X − Z |2 − i J Z · X .  So using Lemma 77 we get the following estimate : for every N , N  there exists C N ,N  (depending only on semi-norms |H (t)|∞,γ , 2 ≤ |γ | ≤ N + N  , such that for X, Y ∈ R2d and |t| ≤ T we have N +1  | K˜ (N ) (X, Y )| ≤ C N ,N  (μ(T )) N +N  2 −n

 R2n

e−

|X −z|2 4

 1+

|Y√ −z t | 

×|a (N ) (t, z)|dz.

−N  (11.110)

Let us denote φ ∗t = φ 0,t = (φ t )−1 . We have the Lipchitz estimate, for |t| ≤ T , |φ ∗,t Y − z| ≤ μ(T )|Y − z t |

(11.111)

So we get   

e−

R2n

|X −z|2 4



1+

 |Y − z t | −N   |φ t ∗ Y − X | −N  dz  ≤ C N  1 + √ √  μ(T ) 

(11.112)

and | K˜ (N ) (X, Y )| ≤ C N ,N  (μ(T )) N +N  

N +1 2

 1+

|φ t ∗ Y −X √ | μ(T ) 

−N 

× supz∈R2n ,|t|≤T |a (N ) (t, z)|

(11.113)

Then using Lemma 76 and choosing N  > 2d, we get the following uniform L 2 estimate for the remainder term, for |t| ≤ T , R N (t) ≤ C N (μ(T )) N +1 (N +1)/2

sup z∈R2n ,|t|≤T

|a (N ) (t, z)|.

(11.114)

340

11 Herman-Kluk Approximation for the Schrödinger Propagator

If T is fixed, pushing the expansion up to 2N instead of N we get easily Theorem 57 using Duhamel formula. 

11.4 Large Time Estimates In this section we revisit the estimates proved in the previous two sections under the assumptions of Theorem 58, in particular using assumption (11.51).  −1 Lemma 78 Let L(t, z) = i M(t, z)T . For any σ ∈ Nn we have ∂zσ L(t, z) ≤ Cσ μ(T )(n+1)|σ |+n−1 , ∀t ∈ [−T, T ], ∀T > 0.

(11.115)

Proof We use Lemma 22 of the book [67]. We have with the notation of [67], M = iY . So we have M ∗ M ≥ 41l. Then we get M −1 (t, z) ≤

1 2

So for any multi-index σ we have ∂zσ M −1  ≤ Cα μ(T )2|σ | , ∀|t| ≤ T.  Lemma 79 For any symbol b ∈ S00 , and every multiindex α ∈ N2n we have  R2n

α

(x − qt ) e

i Φ

b(z)dz =





|α| 2 ≤|β|≤|α|

|β|





2n |σ |≤2|β|−|α| R

f α,β,σ (t, z)e  Φ ∂zσ b(z)dz. i

(11.116) Then we have for any T > 0 and |t| ≤ T , we have |∂zν f α,β,σ (t, z)| ≤ Cα,β,σ,ν μ(T )2(|α|−|σ |+|ν|) .

(11.117)

Proof The Lemma is easily obtained by induction on |α| using Lemma (11.115).  Now, to determine the transport equation, we solve inductively on j ≥ 0, the equation (i∂t − Hˆ (t))I

 

 k ak (t), Φ = O( j+2 )

(11.118)

0≤k≤ j+1

Reasoning by induction on j ≥ 0, we get the transport equation for a j+1 (t) by cancellation of the coefficient of  j+1 in (11.118).

11.4 Large Time Estimates

∂t a j+1 (t, z) =

341

1  ˙ −1  Tr M M a j+1 (t, z) + b j (t, z), a j+1 (0, z) = 0, 2

where b j (t, z) =



F j,k,α (t, z)∂zα ak (t, z).

(11.119)

(11.120)

|α|+2k≤2( j+2)

Moreover, F j,k,α (t, z) depends only on the classical flow φ t (z) and its derivatives and satisfies (11.121) |∂zγ F j,k,α (t, z)| ≤ C j,k,α,γ μ(T )2( j−k+2)+|γ |−|α| where C j,k,α,γ only depends on sup |H (t)|∞,γ , 2 ≤ |γ | ≤ j + 2. So we get, for every j ≥ 0, 

t

a j+1 (t, z) =

|t|≤T

  det 1/2 M(t, z)M(s, z)−1 b j (s, z)ds

(11.122)

0

Moreover, from (11.120) and (11.121), we get, by induction on j, the following estimate, for every j ≥ 0, |t| ≤ T , z ∈ R2n , |∂zν a j (t, z)| ≤ C j,γ |det 1/2 M(t, z)|μ(T )8 j+2|ν|

(11.123)

with the same remark as in (11.121) for the constant C j,γ . Now revisiting Sects. 11.3.2 and 11.4 we get Theorem 58. 

11.5 Application to the Aharonov-Bohm Effect 11.5.1 The Van-Vleck Formula This is one of the oldest mathematical result (1928) in quantum mechanics. It is often deduced using Feynman integral. Here we deduce it using Theorem 56. The goal is to get a point-wise approximation for the Schwartz kernel K  (t, t0 , x, y), as in Theorem 56. In order to apply the stationary phase theorem with complex phase ([153], Theorem 7.7.5) we introduce the following assumption. (V-V). There exist a finite number m of classical trajectories of the Hamilton flow φ t,t0 connecting in the configuration space the initial data (t0 , y) to the final data (t, x) such that (t0 , y) and (t, x) are not conjugate. That means there exist m momenta p ( j) , 1 ≤ j ≤ m such that qt,t0 (y, p ( j) ) = x, and  ∂qt,t0 (y, p ( j) ) = 0 , 1 ≤ j ≤ m. det ∂p 

(11.124)

342

11 Herman-Kluk Approximation for the Schrödinger Propagator

Theorem 59 (i) If qt,t0 (y, p ( j) ) = 0, ∀ p ∈ Rn then K  (t, t0 , x, y) = O(∞ ). If assumption (V-V) is fulfilled then we have the following Van-Vleck formula K (t, t0 , x, y) = (2iπ )−n/2 



π

2 e−i 2 σ j |det∂x,y S(t, t0 ; y, p ( j) )|1/2 +

1≤ j≤m



1+k/2

c j,k (t, t0 ; x, y) + O(∞ ).

(11.125)

1≤ j≤m,k≥0

The asymptotic expansion is in the sense of L 2 -norm of operators. S is the classical action between the data (t0 , y) and (t, x). σ j ∈ Z is the Maslov index of the path ( j). Moreover if Hˆ (t) is an elliptic operator (like for Hˆ (t) = −2  + V (t, x) then the asymptotic expansion is true in the L ∞ -norm in the variables (x, y). Proof This is a direct application of the stationary phase theorem ([153], Theorem 7.7.5) to the oscillating integral (11.38) or (11.48). We have to compute the leading term, which is known as the Van-Vleck semi-classical approximation. In (11.38) we have ΦΓ = 0 ⇐⇒ {q = y, qt = x}. Then we have to compute ∂z ΦΓ and the Hessian matrix ∂z2z ΦΓ at the critical points. We compute the first derivatives of the classical action S in (90).  ∂ p S(t, t0 ; q, p) =  ∂q S(t, t0 ; q, p) =

∂qt,t0 ∂p ∂qt,t0 ∂q

T pt,t0 ,

(11.126)

pt,t0 − p.

(11.127)

T

Then we get ∂q ΦΓ = C T (x − qt,t0 − A T Γ (x − qt,t0 ) − i(y − q)

(11.128)

∂q ΦΓ = D (x − qt,t0 − B Γ (x − qt,t0 ) − (y − q)

(11.129)

T

where A=

T

∂qt,t0 ∂ pt,t0 ∂ pt,t0 ∂qt,t0 , B= ,C = , D= . ∂q ∂p ∂q ∂p

(11.130)

Applying the implicit function theorem in (11.124), we get that the phase ΦΓ has a finite number m of critical points z ( j) = (y, p ( j) ), with q = y, p = p ( j) (t, t0 ; x, y),

11.5 Application to the Aharonov-Bohm Effect

343

1 ≤ j ≤ m. Now we show that these critical points are non degenerate by computing ∂z2z ΦΓ for z = z ( j) . Using (11.126) and that Ft,t0 is symplectic, we get ∂z2z ΦΓ (t, t0 ; z ( j) , x,

 2i1l + (A + i B)−1 B i(A + i B)−1 B . y) = i(A + i B)−1 B −(A + i B)−1 B 

(11.131)

Notice that A, B are computed for (t, t0 , z = z ( j) ). The determinant is easily computed on these points: det(∂z2z ΦΓ ) = det(−2i∂z2z ΦΓ (A + i B)−1 B).

(11.132)

Using assumption (V-V) we get that det(∂z2z ΦΓ ) = 0. Hence the stationary phase theorem can be applied. Moreover we have to compute the correct phase for the square root det −1/2 (∂z2z ΦΓ ) on critical points (analytic continuation and continuity (see [153], section VII). Using the implicit function theorem we get on the critical points 2 S = −B −1 . ∂x,y

 Remark 67 If ∂ p H (q, p) is invertible (like for H (q, p) = | p|2 + V (q)) then for |t − t0 | ≤ δ with δ > 0 small enough then (V-V) is fulfilled with m = 1 and then the Maslov index is 0.

11.5.2 A Setting for the Time-Dependent Aharonov-Bohm Effect The Aharonov-Bohm effect is a purely quantum effect in the sense that a quantum particle may feel the effect of a magnetic potential even if the field vanishes in a neighborhood of the classical trajectory of the particle. This can be checked on the propagator as a consequence of the Van-Vleck formula proved above. Let us consider a two dimensional Schrödinger Hamiltonian 1 Hˆ κ a,V (t) = (Dx − κ a (x))2 + V (t, x), a = (a1 , a2 ). 2 The magnetic field b is perpendicular to the plane R2x , x = (x1 , x2 ) and so is identified with the scalar function b(x) = ∂x2 a1 (x) − ∂x1 a2 (x).

344

11 Herman-Kluk Approximation for the Schrödinger Propagator

For simplicity we assume that, for |α| ≥ 1, ∂x a(x) is bounded on R2 and for |α| ≥ 2, ∂x V (t, x) is bounded on R2 . Under these assumptions the propagator for Hˆ a,V (t) is well defined as a unitary operator on L 2 (R2 ) (see the proof of Proposition 123). Our geometric setting is the following (see also the picture). (GAB). There exist exactly two classical trajectories in the configuration space connecting the initial data (s, y) to the final data (t, x). More precisely, the classical flow (y, p) → q(t, s, y, p), p(t, s, y, p) is such that the equation q(t, s, y, p) = x has exactly two solutions in p = p 1,2 = p (1,2) (t, s, x, y). Thus in the plane configuration space the two paths τ → q(τ, s, y, p 1,2 , s ≤ τ ≤ t are the oriented boundary γ := γ1 ∪ γ2 of a bounded open domain . We assume that the support of the magnetic field b is inside . We denote by ϕ(b) the magnetic flux through the domain . Moreover we put a filter Ψ to prevent any classical particle starting from y to penetrate in supp(b). For that we assume that Ψ ∈ C0∞ (V ) where V is a small open set such that p 1 , p 2 ∈ V and Ψ ( p 1,2 ) = 1. (t, s; x, y) the Schwartz kernel of U Hκa,V (t, s)Ψ (Dx ) and by Let us denote K H,Ψ κa,V Sκa,V (γ ) the classical action along the oriented path γ for the Hamiltonian Hˆ κ a,V (t).

The magnetic Aharonov-Bohm effect

Theorem 60 Under the above setting we have (t, s; x, y)|2 = |2π K H,Ψ κa,V | f 1 (t, s; x, y)e  S0,V (γ1 ) + f 2 (t, s; x, y)e  (S0,V (γ2 )−κϕ(b)) |2 + O() (11.133) i

i

π

2 S(t, s; y, p 1,2 )|1/2 e−i 2 σ1,2 and σ1,2 a Maslov index where f 1,2 (t, s; x, y) = | det ∂x,y like in Sect. 11.5.1.

Proof This is a direct consequence of Theorem 11.5.1 by computing carefully the classical actions. Let us recall here this well known computation. Let us introduce the Lagrangian Lκa,V for the electric potential (κa, V ). We have ˙ = L0,V (q, q) ˙ + κa(q) · q. ˙ Lκa,V (q, q)

11.5 Application to the Aharonov-Bohm Effect

345

The equation of motion of the classical particle is ˙ a¨ = ∂q V (q) + κb(q) · q. Then we have for the actions  Sκa,V (γ ) = S0,V (γ ) + κ

adq. γ

Then using Stokes formula we get the flux: 

 γ1

adq =

γ2

adq − ϕ(b). 

Remark 68 We can compute more explicitly the Aharonov-Bohm effect as follows. Denote ρ1,2 = | f 1,2 | (they are not null). Then we have (t, s; x, y)|2 − |2π K H,Ψ (t, s; x, y)|2 = |2π K H,Ψ κa,V 0,V   κϕ(b) − sin θ ) 2ρ1 ρ2 (sin θ − 

(11.134)

2 where θ = π2 (σ2 − σ1 ) + S1 +−S .  To be more specific assume for example that σ1 = σ2 and S1 = S2 and that the . Then we have magnetic flux is quantized: κϕ(b) = π 2

|2π K H,Ψ (t, s; x, y)|2 = |2π K H,Ψ (t, s; x, y)|2 + 2ρ1 ρ2 . κa,V 0,V So even if the classical particle never meet the magnetic field the motion of corresponding quantum particle is modified by the field. A similar phenomenon can be observed by perturbing a quantum system by a time dependent electric potential. We construct a perturbation of the electric potential V (t, q) as follows. Let W (s, q) = χ1 (s)Y1 (q) + χ2 (s)Y2 (q). Let us consider thin tubular open neighborhoods T1,2 ,  T1,2 , of the pieces T1,2 . Let {q(τ, s, y, p ± ); t1 ≤ τ ≤ t2 }, s < t1 < t2 < t, of path γ± , such that T1,2 ⊆  ∞ ∞  us choose χ1,2 ∈ C0 ]t1 , t2 ], Y1,2 ∈ C0 (T1,2 ), Y1,2 = 1 on T1,2 . Then we have: Theorem 61 Under the above assumptions we have

346

11 Herman-Kluk Approximation for the Schrödinger Propagator

|2π K H,Ψ (t, s; x, y)|2 = V +κ W    f + e i S0,V (γ1 ) + f − e i S0,V (γ2 )−κδ 2 + O(), where δ =

t s

(11.135)

(χ2 (τ ) − χ1 (τ ))dτ .

The electric Aharonov-Bohm effect

Notice that ∂q W (s, q) = 0 in a neighborhood of γ1,2 . So the classical trajectories never see the perturbation of the electric field. Nevertheless we also have here an electric Aharonov-Bohm effect, where the magnetic flux is replaced by the difference of voltage κδ between the tubes T1 and T2 .

11.6 Application to the Spectrum of Periodic Hamiltonians Flow In this section we shall give a sketchy alternative proof of the Theorem 32, Chap. 5., using coherent states and the Herman-Kluk approximation. We assume for simplicity that H is subquadratic (after cut-off it is not a restriction), that H −1 [E − η, E + η] is compact and the Hamiltonian flow Φ Ht is 2π -periodic and 2π is a primitive period on Σe := H −1 (e) for E − η ≤ e ≤ E + η. Moreover Σe is supposed to be connected. Let us choose an energy cutoff χ ∈ C0∞ (]E − η, E + η[), χ = 1 on [E − η/2, E + η/2]. Theorem 62 Let σ () = μ4 + λE (μ is the Maslov index of the periodic orbits γ E ⊂  1 Σ E and λ E = 2π γ E pdq). Then we have ˆ

χ ( Hˆ )e−i  ( H −σ ()) = χ ( Hˆ )(1l + R1 ) 2π

(11.136)

where R1  = O(1) as  → 0. Proof This is a direct application of Theorem 57 using that Φ H2π = 1lR2n . The Maslov index is obtained by computing the argument of a complex determinant associated to the monodromy matrix of the flow.  A s a consequence we deduce the following well known clustering property for the spectrum of periodical classical systems[142].

11.6 Application to the Spectrum of Periodic Hamiltonians Flow

347

Corollary 34 Spect[ Hˆ ] ∩ [E − η/2, E + η/2] ⊆

I (k, ) , k∈Z

where I (k, ) = [(k +

μ μ ) + λ − C2 , (k + ) + λ + C2 ] with C > 0 large enough. 4 4

Proof For  > 0 small enough we can define the bounded operator Lˆ = So we have 2π ˆ ˆ χ ( Hˆ )e−i  ( H −σ ()+ L ) = χ ( Hˆ ).

log(1l+ Rˆ 1 ) . 2iπ

(11.137)

The two compact self adjoint operators χ ( Hˆ ) Hˆ and χ ( Hˆ ) Lˆ commutes so they have a joint basis {ψ j }1≤ j≤M of eigenfunctions in the spectral subspace Range [1l[E−η,E+η] ( Hˆ ): Hˆ ψ j = λ j ψ j ˆ j = νjψj. Lψ ˆ = O() we get for λ j ∈ [E − η/2, E + η/2] that there Now using (11.137) and  L exists k ∈ Z such that λ j − σ () + O(2 ) = k.  Remark 69 If the period depends on the energy e we can apply Proposition 49 of Chap. 5 to reduce to the case of constant period.

Notes Proposition 123 is a particular case of a more general result proved in [191]. Theorem 11.113 was first proved without the small parameter  in [242]. The case with  $ 0 consider here follows [93]. We could easily extent the results of this Chapter to multi-components systems with constant multiplicities (see Chap. 14). This is an exercise for the interested reader. The Herman-Kluk approximation was discovered by chemical-physicists [138, 149, 164]. The first mathematical proof appeared in [227] and in a more general setting in [219].

Chapter 12

Semi-classical Measures: Stationary and Large Time Behavior

Abstract In these chapter our aim is to develop some applications of the theory of semi-classical measures (introduced in Chap. 2) to study quantum properties of ergodic classical system and also for some periodical classical systems. We have already seen that there are several connections between semi-classical measures and coherent states. We first consider quantization of classical ergodic systems and the consequence in the quantum world, a part of a domain much studied in the physicist literature under the name “quantum chaos”. We also consider some quantum properties for quantizations of classical systems with periodic classical flow. We shall see that semi-classical measures are well adapted to study quantum systems in the semi-classical regime at large time, much beyond the well known Ehrenfest time.

12.1 Semi-classical Measures for Bound States In Chap. 2 we have discussed some basic properties of the semi-classical measures associated with a family ψ  of normalized L 2 states on Rn . Recall that a probability measure μ on the phase space R2n is a semi-classical measure (for the family ψ  ) if there exists a sequence  → 0 such that for any A ∈ S00 we have   lim ψ  , opw (A)ψ  = A(X )μ(d X ).  →+∞

R2n

Recall that the mesure μ tell us where the state ψ  is concentrate in the phase space is the semi-classical regime. In particular we shall consider this problem for bound states.

12.1.1 More on the Weyl Law Let H be a classical Hamiltonian satisfying the assumptions (H.0) and (H.1) of Chap. 5. In particular the Liouville measure on Σ E := H −1 {E} is well defined © Springer Nature Switzerland AG 2021 D. Robert and M. Combescure, Coherent States and Applications in Mathematical Physics, Theoretical and Mathematical Physics, https://doi.org/10.1007/978-3-030-70845-0_12

349

350

12 Semi-classical Measures: Stationary and Large Time Behavior

d L E :=

dΣ E . ∇z H (z)

Moreover there exists some 0 > 0, small enough, such that the spectrum of Hˆ in the interval I E (η) := [E − η, E + η] is discrete (only eigenvalues of finite multiplicities). Let N (E, η) be the number (with multiplicities) of eigenvalues of Hˆ in I E (η) and {ψ j }1≤ j≤N (E,η) an orthonormal family of eigenfunctions. So we have Hˆ ψ j = E j ψ j . Notice that E j , ψ j depend on . Moreover we have the following basic result known as the extended semi-classical Weyl law. The proof follows easily from the proof given in Chap. 5 for the Weyl law and the Gutzwiller trace formula. Theorem 63 With the previous notations, for any observable A ∈ S00 , we have: 

ˆ j  = (2π )−n ψ j , Aψ



1≤ j≤N (E,η)

H −1 [E−η,E+η]

A(X )d X + O(1−n ).

(12.1)

In particular the Weyl law read: N (E, η) = (2π )−n VolR2n (H −1 [E − η, E + η]) + O(1−n ).

(12.2)

The following corollary will be useful. Let us consider -dependent energy intervals: I () = [E − δ(), E + δ()], with lim δ() = 0 and δ() ≥ τ , for some →0

τ < 1.

Corollary 35 We have the following consequence: 1 lim 0 N (E, δ())



ˆ j = ψ j , Aψ

1≤ j≤N (E,δ())

 ΣE

A(X )L E (d X ).

(12.3)

 Proof Let FA (λ) = H −1 [E−η0 ,λ] A(X )d X , for η0 > 0, E − η0 < λ < E + η0 . FA is smooth in a neighborhood of E and d FA (λ) = dλ

 Σλ

A(X )L λ (d X ).

Using (12.4), (12.1) and (12.2) we get (12.3).

(12.4) 

Remark 70 In the Cesaro mean we have seen that the quantum averages of the observable Aˆ converges toward the classical average of A for the Liouville probability measure. ˆ j  is much more delicate The behaviour of individual quantum averages ψ j , Aψ ˆ j to understand. A way to do that is to consider the mapping family A → ψ j , Aψ as a distribution, or better if possible, as a Radon measure depending on . This can be done by modifying the Weyl quantization in the positive preserving quantization

12.1 Semi-classical Measures for Bound States

351

i.e. the anti-Wick quantization (see Chap. 2). Afterwards it is possible to study the ˆ j  as we shall see in this Chapter. statistic of the real numbers ψ j , Aψ

12.1.2 More About Semi-classical Measures In Chap. 2 we have introduced the anti-Wick quantization Aˆ aw of an observable A and the Husimi function of a L 2 -normalized family of states ψ  ∈ L 2 (Rn ): Hψ (X ) = (2π )−n |ϕ X , ψ|2 . Recall the two following useful properties (Proposition 27): (1) For any A ∈ S00 we have aw Opw  (A) − Op (A) = O().

(2) For any ψ  ∈ L 2 (Rn ), ψ



 , Opaw  (A)ψ 

= (2π )

−n

 R2n

A(X )Hψ  (X )d X.

Corollary 36 For any ψ ∈ L 2 (Rn ), A → ψ  , Aˆ aw ψ   extends to a positive Radon measure μψ on R2n . Moreover μψ has a density with respect to the Lebesgue measure on R2n equal to the Husimi function Hψ of ψ. Let us denote μj = μψ j where {ψ j }1≤ j≤N (E,δ() . Then for the weak convergence of Radon measures we have  N (E,δ()) lim

→0

j=1

μj

N (E, δ())

= νE ,

where ν E is the Liouville probability measure: ν E =

LE . L E (Σ E )

This Corollary shows that the arithmetical averages of the quantum probability measures μj : A → ψ j , Aˆ aw ψ j  converge towards the Liouville probability on Σ E . We shall see in this chapter that the behavior of the individual measures μj is much more intricate and depends heavily on dynamical properties of the classical flow Φ Ht on Σ E when the frequency set of ψ  is concentrated on Σ E . Existence of semi-classical measures follows from the following statement. Proposition 126 (i) Let {ψ  }0 0. |X |→∞

Then ψ  has a semi-classical probability measure ν. Moreover we have supp(ν) ⊆ H −1 [a, b]. Other examples: Let us consider the coherent state ψ := ϕ X , X ∈ R2n . The only semi-classical mea-

12.1 Semi-classical Measures for Bound States

353

sure is the Dirac mass δ X . S(x) If ψ (x) = f (x)ei  (WKB states), with f ∈ L 2 (Rn ) with compact support and 1 n S ∈ C (R ). The only semi-classical measure is the measure: ν = | f (x)|2 d x ⊗ δ(ξ − ∇ S(x)).

12.2 Semi-Classical Quantum Ergodicity The following semi-classical version of the Schnirelmann theorem was given in [140] following the proof by Colin de Verdière [56] and Zelditch [267] on compact Riemannian manifolds with negative curvature.  Assume: (ERG) The dynamical system Σ E , d L E , Φ Ht is ergodic which means: for every continuous function A on Σ E , we have, for almost all z ∈ Σ E : 1 lim T +∞ T



T 0

 A(Φ Ht 0 (z))dt

=

ΣE

Adν E .

Let us recall that ν E is the Liouville probability measure on Σ E . Remark 72 Well known examples of ergodic dynamical systems are compact manifolds with strictly negative curvature. A. Knauf (1987) has given examples of ergodic systems on the Euclidean space with Hamiltonian H (q, p) = 21 ( p12 + p22 ) + V (q1 , q2 ). Let us consider now -dependent energy intervals: I E () = [E − δ(), E + δ()], with lim δ() = 0, δ() ≥ 2 , for some 2 > 0. Let us denote: Λ() = →0

{ j, E j () ∈ I ()} and N () = #Λ(). Notice that the eigenvalues E j () are repeated according their multiplicity. The quantum signature of classical chaos is given by the following : Theorem 64 (Ergodic Semi-Classical Theorem)[140]) Under the ergodic assumption (ERG), for every  > 0, there exists a subset of density one: M() ⊆ Λ(), depending only on the Hamiltonian Hˆ , such that :  lim

0

#M() #Λ()

 = 1, and

lim

[0, j∈M()]

ˆ j = ψ j , Aψ

 ΣE

Adν E , ∀A ∈ S00 .

(12.7) In other words almost all eigenfunctions are uniformly distributed on the energy shell ΣE . Remark 73 The following question is still open : can we take M() = Λ() in the conclusion of the above lemma, if n ≥ 2?(it is true for n = 1). We shall give two different proofs of Theorem 64. The first one is close to the proof given in [56, 267]. The second one will be given in the next section.

354

12 Semi-classical Measures: Stationary and Large Time Behavior −1

Proof Let us introduce the Schrödinger unitary group : U H (t) = e−it 1 T

ˆ j, ψj =  Aψ



T 0

U H (−t) Aˆ U H (t)ψ j , ψ j dt.

Hˆ

. We have (12.8)

T T Denote A¯ T (X) = T1 0 A(Φ t (X ))dt, Aˆ T = T1 0 U H,∗ (t) Aˆ U H (t)dt, and A E := Σ E Ad L E . From Cauchy-Schwarz inequality we have: ε() :=

1 N (E, δ())



ˆ j  − A E |2 = |ψ j , Aψ

1≤ j≤N (E,δ())

 1 |ψ j , Aˆ T ψ j  − A E |2 ≤ N (E, δ()) 1≤ j≤N (E,δ()) 1 N (E, δ())



ψ j , ( Aˆ T − A E )2 ψ j 

(12.9)

1≤ j≤N (E,δ())

Let BT (X ) = ( A¯ T (X ) − A E )2 . Using using Egorov theorem and the Moyal product rule (see Chap. 2) we get ε() ≤

1 N (E, δ())



ψ j , (Opw  BT )ψ j  + OT ().

(12.10)

1≤ j≤N (E,δ())

 ε() ≤

ΣE

BT (X )L E (d X ) + OT ().

(12.11)

Now using the ergodicity assumption we have  lim

T →+∞ Σ E

BT (X )L E (d X ) = 0.

Hence taking first T large enough then  small enough we get ⎛ lim ⎝

→0



⎞

1 |ψ j , Aˆ T ψ j  − A E |2 ⎠ = 0. N (E, δ()) 1≤ j≤N (E,δ())

(12.12)

(12.7) will be deduced from (12.12) and the Tchebichev inequality following [271]. Denote M() = { j ∈ Λ(), |ψ j , (A − A E )ψ j | ≤ α()} for some suitable 0 < α(). ˆ j  − A E . Then we have Let Δ j := ψ j , Aψ

12.2 Semi-Classical Quantum Ergodicity

355

 1 # (Λ()\M()) α()2 Δ2j ≥ N () 1≤ j≤N () N () and we get

ε() #M() ≥1− N () α()2

Then to conclude we choose α() such that lim α() = 0,

0

() = 0. 0 α()2

and lim

We shall see now some consequences of ergodicity on non diagonal matrices ˆ k . elements: A j,k = ψ j , Aψ Classical ergodicity implies that almost all classical trajectory is dense in Σ E . This implies also that the quantum spectrum is asymptotically dense in R. More precisely we have Theorem 65 Assume that there exists on Σ E a non periodical trajectory for the flow Φ t . Then for every c > 0, δ > 0 and every 0 > 0 the set: T E,δ := {ω jk () : E j (), E k () ∈ [E − c1−δ , E + c1−δ ], 0 <  ≤ 0 } is dense in R, where ω jk () =

E j ()−E k () . 

Proof Let f ∈ C0∞ (R) be such that f = 0 on T E,δ . We have to show that f ≡ 0 on R. Let us introduce χ ∈ C0∞ (] − c, c[) with χ (0) = 1. Following Helton[146] we consider the operator: Aˆ E, f () = ˆ



fˆ(t)U (−t, ) Aˆ E ()U (t, )dt,

(12.13)

ˆ

−E ˆ −E with A E () = χ ( Hh 1−δ ) Aχ ( Hh 1−δ ), A ∈ C0∞ (R2n ). By inverse Fourier transform, we also have:      E j () − E E k () − E ˆ j, ˆ χ Πk AΠ f (ω jk ())χ A E, f () = 2π 1−δ 1−δ   j,k

(12.14) where Πk is the projection on the eigenstate ψk . From (12.13), (12.14) we have: 

fˆ(t)U (−t, ) Aˆ E ()U (t, )dt = 0, ∀A ∈ S00 .

(12.15)

From (12.15) we would like to prove that f ≡ 0 by going to the classical limit   0. To do that we first use the semi-classical propagation theorem (Sect. 1) and functional calculus with parameters for pseudodifferential operators [40]. We test (12.15) by

356

12 Semi-classical Measures: Stationary and Large Time Behavior

ˆ B ∈ S00 . Using (12.15) and computing the trace of the product with any operator B, 0 Egorov theorem we get easily, ∀B ∈ S0 ,     ˆ ˆ fˆ(t)A Φ Ht (z)) B(z)dzdt = 0. lim (2π ) .T r A E, f (). B = 

n

0

So we get



 fˆ(t)A Φ Ht (z)) dt = 0, ∀z ∈ Σ E .

(12.16)

(12.17)

Now, choose z 0 ∈ Σ E such that t → Φ t (z 0 ) is not periodic, we should like to deduce from (12.17) that fˆ ≡ 0. We use now the following result to finish the proof.  F

Lemma 80 For T > 0 we can find ρT > 0 such that the mapping Φ : (t, z) → (t, Φ t (z)) is a diffeomorphism from ] − T, T [×DρT (z 0 ) onto an open neighborhood NT of the curve: {Φ t (z 0 ), −T < t < T }; where DρT (z 0 ) is the euclidean ball with center z 0 and radius ρT in the orthogonal plane to the curve at time 0. Furthermore, for every g in C0∞ (] − T, T [) we can construct some A ∈ C0∞ (R2n ) such that: g(t) = A(Φ t (z 0 )) + h(t), ∀t ∈ R with: (i) Supp(h)∩] − T, T [= ∅, (ii) sup |h| ≤ sup |g|. R

R

Proof Starting with the diffeomorphism F, let us choose u ∈ C0∞ (DρT (z 0 )), u(z 0 ) = 1, 0 ≤ u ≤ 1. We define B(t, z) := g(t)u(z) for (t, z) ∈ DρT (z 0 ) and A(z) := B(F −1 (z)) for z ∈ NT . We have clearly A(Φ t (z 0 )) = g(t) if |t| ≤ T . For |t| ≥ T and Φ t (z 0 ) ∈ NT we have Φ t (z 0 ) = Φ1t (z) with |t1 | ≤ T , z ∈ DρT (z 0 ) so h(t) = −g(t1 )u(z) and we get the announced properties for h. So, with the above notation, using the above lemma and (12.17) we get: 

fˆ(t)g(t)dt =



fˆ(t)h(t)dt ≤ sup |g| R

 |t|≥T

| fˆ(t)|dt,

taking T large, we have clearly fˆ ≡ 0 hence f ≡ 0.



Remark 74 Theorem 65 has to be compared with the clustering property of the spectra when the Hamiltonian flow is periodic (see Chap. 5). It is remarkable that one non periodical trajectory at energy E is enough to perturbed dramatically the spectrum of Hˆ close to E. Concerning the non diagonal matrix elements we have the following result. Theorem 66 Assume that the classical system is ergodic on Σ E . Then for all A ∈ S00 such that A E = 0, we have

12.2 Semi-Classical Quantum Ergodicity

357

⎛ 1 () lim ⎝ →0,δ→0 N



⎞ |A jk |2 ⎠ = 0.

E k ∈I (), |ω jk ()|≤δ

Moreover, the condition () holds if and only if the classical system is ergodic on the energy level E.  Proof Let be θ a bounded with compact support function such that R θ (t)dt = 1.  Denote Aˆ θ = θ (t)U (−t) Aˆ U (t)dt. Let JE be a fix neighborhood of E. Using Parseval equality and localization of eigenfunctions in a neighborhood of Σ E we have 1 N



|ψ j , Aˆ θ ψk |2 =

E j ∈I (), E k ∈JE

 1  |θˆ ω jk |2 |A jk |2 + o(1). N E ∈I ()

(12.18)

j

But we have 1 N

 E j ∈I (), E k ∈JE

1  |ψ j , Aˆ θ ψk |2 = A∗ Aθ ψ j , ψ j  + o(1). N E ∈I () θ

(12.19)

j

Using Corollary 35 we have   2  1  ∗  lim Aθ Aθ ψ j , ψ j  = θ (t)A(Φ Ht (X ))dt  ν E (d X ). →0 N ΣE E ∈I () j

The results follows by choosing θ (t) = enough.

1 2T

1l[−T,T ] , so θˆ (λ) =

sin(T λ ) with Tλ

T large 

Remark 75 We shall see later that quantum ergodicity, in the sense of Theorem 64, can occurs for non classically ergodic systems (even completely integrable systems). For these systems () cannot be satisfied. These probability transitions were considered in [172] in section 51 for n = 1. Classical ergodicity brings information not only on diagonal matrix elements but also on nearby non diagonal matrix elements (see [64] for more details, in particular for considering classical mixing systems).

12.3 The Quantum Ergodic Birkhoff Theorem This section is inspired from a paper by G. Rivière [216]. We adapt it to a semiclassical setting. Let us recall the classical Birkhoff ergodic theorem for the Hamiltonian flow Φ Ht on Σ E with the Liouville invariant measure ν E (it is well defined, only assuming that E is a non-critical energy).

358

12 Semi-classical Measures: Stationary and Large Time Behavior

Theorem 67 (see reference[252]) There exists a Borel subset Γ E ⊆ Σ E , ν E (Γ E ) = 1 such that for all X ∈ Γ E and all A ∈ C 0 (Σ E ) we have 1 lim T →+∞ T

 0

T

A(Φ Ht (X ))dt = μ BX (A)

where μ BX is a Radon invariant (for the flow Φ Ht ) probability on Σ E . We follow the method of [216] to prove the following quantum consequence. ˆ j , A ∈ C ∞ (K E ) where Denote by μ j the Schwartz distribution A → ψ j , Aψ −1 K E := H [E − ε, E + ε]. The Weyl quantization is not positive, so it is more convenient to replace it, as before, by the positive quantization Aˆ aw such that  Aˆ − Aˆ aw  = O() (anti-Wick quantization). Hence A → ψ j , Aˆ aw ψ j  are Radon measures on K E . Let us introduce the Sobolev space W s, p (K E ), s > 0, p > 1. By the Sobolev theorem we have W s, p (K E )) ⊆ C 0 (K E ) if s > 2dp . So we have

μ j ∈ (W s, p (K E )) = W −s, p (K E ) where 1p + p1 = 1 (B denotes the dual space of the Banach space B) . Theorem 68 For any  ∈]0, 1] there exists a subset Λ˜  ⊆ Λ of density one in Λ i.e.: #Λ˜  lim =1 0 N and for any map  → j () ∈ Λ˜  any accumulation point μ of μ j () in the Banach

space W −s, p (Σ E ) is in the closed convex hull of the Birkhoff measures {μ BX , X ∈ Σ E } (for the weak convergence). When the Hamiltonian flow of H is ergodic on Σ E , we have essentially one Birkhoff measure: it is the Liouville measure, so Theorem 68 entails Theorem 64 (Quantum Ergodic Theorem). In the opposite case, if the Hamiltonian flow is periodic on Σ E we have much more Birkhoff measures. Let be T0 a period for the flow. Then for any X ∈ Σ E we have 1 lim T →+∞ T



T

1 A(Φ (X ))dt = T0



t

0

T0

A(Φ t (X ))dt := δγ (A)

0

i.e. the Birkhoff measures are the Dirac measures supported by the periodic trajectories γ of the flow. The following Proposition applies in particular to the Harmonic oscillator: Hˆ = 21 (−2  + |x|2 ) with E > 0. Proposition 127 Let us assume that the Hamiltonian H satisfies the periodicity assumptions of Theorem 35 (Chap. 5). Moreover we assume that Hˆ has only one eigenvalue (with multiplicity) in each cluster I (k, ).Then we have: (i) for any 2π -periodical trajectory γ ⊆ Σ E its Dirac measure δγ is the Wigner measure for eigenstates associated with the eigenvalue E(k, ) ∈ I (k, ) as  → 0.

12.3 The Quantum Ergodic Birkhoff Theorem

359

(ii) More generally any probability measure μ on Σ E , invariant for the flow Φ Ht , is the Wigner measure for a sequence ψ of normalized bound states with Hˆ ψ = E  ψ for  =  and lim E  = E. →+∞

Let us recall here the shape of the clusters in the completely periodic case, where μ is the Maslov index and λ is a geometric constant, I (k, ) = [(k +

μ μ ) + λ − C2 , (k + ) + λ + C2 ] with C > 0 large enough. 4 4

Remark 76 Notice that in part (i) of this proposition the conclusion is valid for any subsequence  → 0. It is not needed to extract a specific subsequence. The proof of the Theorem 68 is a consequence of the following lemma plus the Hahn-Banach Theorem Lemma 81 There exists Λ˜  and  → j () like in Theorem 68 such that essinf X ∈Σ E μ BX (A) ≤ lim inf μ j () (A) ≤ lim sup μ j () (A) ≤ esssup X ∈Σ E μ BX (A) 0

0

(12.20) Proof (Theorem 68 using Lemma 81) By contradiction. Assume that μ is not in the closed convex hull C B of {μ BX , X ∈ Σ E }. From the Hahn-Banach Theorem and reflexivity of the Banach space W s, p (K E ) there exists f ∈ W s, p (K E ) such that ν( f ) ≤ a < μ( f ), ∀ν ∈ C B . In particular sup μ BX ( f ) ≤ a < μ( f )

X ∈Σ E

which cannot be true because we have (12.20).



Proof (Lemma 81) T Let A T (X ) = T1 0 A(Φ t (X ))dt. We have μ j (A) = μ j (A T ) + oT (1) = ψ j , Aˆ aw ψ j  + oT (1). Denote R0 = esssupμ BX (A), A˜ T is a smooth observable such that (i) (ii) (iii)

), ∀X ∈ Σ E , A˜ T (X ) ≤ A T (X√ A˜ T (X ) ≤ R0 + ε, ∀X ∈ Σ E , A˜ T (X ) = A T (X ) if A T (X ) ≤ R0 +

√ ε . 2

We can choose A˜ T = θε (A√T ) where θε is a an increasing function on R such that √ θ (x) = x for |x| ≤ R0 + 2ε and |θε (x)| ≤ R0 + ε for every x ∈ R.

360

12 Semi-classical Measures: Stationary and Large Time Behavior

From the Weyl law we get 1 N



˜ ψ j , Opaw  (A T − A T )ψ j  =

1≤ j≤N

 ΣE

(A T − A˜ T )d L E + oT (1).

(12.21)

To estimate this term we use that   (A T − A˜ T )d L E = (A T − A˜ T )d L E ≤ A L ∞ L E (ΩT,ε ). ΣE

Ωε

where ΩT,ε = {X ∈ Σ E , |A T (X )| ≥ R0 + But from definition of R0 we have

√ ε }. 2

lim L E (ΩT,ε ) = 0.

T →+∞

So, using (12.21), for any η > 0 there exists Tεη and T such that for T ≥ Tεη and 0 <  ≤ T we have 1 N



˜ ψ j , Opaw  (A T − A T )ψ j  ≤ ηε.

1≤ j≤N

Then by Tchebichev inequality we get √ √ 1 ˜ #{ j ∈ Λ , ψ j , Opaw ε} ≤ η ε  (A T − A T )ψ j  ≥ N We also have for  small enough √ ˜ ψ j , Opaw  A T ψ j  ≤ R0 + 2 ε then writing aw ˜ ˜ μ j () = ψ j , Opaw  A T ψ j  + ψ j , Op (A T − A T )ψ j ,

√ and denoting Sε () = { j ∈ Λ , μ j () ≤ R0 + 3 ε}, from the above estimates we get √ #Sε () ≥ 1 − η ε. lim inf 0 N This is true for any η > 0 so S () has density one. ˜ Finally, by a diagonal argument we get, in a subset Λ() of Λ , of density one, (independent on A), such that lim sup μ j () (A) ≤ esssup X ∈Σ E μ BX (A). 0

12.3 The Quantum Ergodic Birkhoff Theorem

361

The other part of inequality (12.20) is obtained by applying the above inequality to −A.  Proof (Proof of Proposition 127) We recall that in Chap. 5 we have obtained quasimodes ψγ E with energy E  such that Hˆ ψγ E = E k ψγ E + O(7/4 ). Moreover ψγ E  = m E 1/4 + O(1/2 ) with m E > 0 and E k ∈ I (k, ). The quasi-mode ψγ E is obtained by propagating coherent states according the formula:  2π t  ˆ ei  (E − H ) ϕz dt, z ∈ γ . (12.22) ψγ E = 0

In particular ψγ E is microlocalized on γ which means FS[ψγ E ] ⊆ γ . Now we defined the L 2 -normalized quasi-mode ψγ := c ψγ E with c > 0, c ≈ −1/4 . Using a stationary phase argument as in Chap. 5 we get for any A ∈ S00 , ˆ γ  = ψγ , Aψ

1 2π



2π

√ A(Φ t (z))dt + O( ).

(12.23)

0

To prove the part (i) of the proposition our strategy consists to prove that we can replace the quasi-mode ψγ by a genuine eigenfunction of Hˆ . Let be η > 0 small enough. Let us denote Πk the spectral projector of Hˆ on the interval I (k, ). We shall always assume that I (k, ) ⊆]E − η, E + η]. Let us choose an energy cutoff χ ∈ C0∞ (]E − η, E + η[), χ = 1 on [E − η/2, E + η/2]. From our assumptions we know that Hˆ Πk = E(k, )Πk for some E(k, ) ∈ I (k, ). Hence we have ˆ

tH χ ( Hˆ )e−i  =



χ (E(k, ))e−i

t E(k,) 

Πk .

k∈Z

Using that E(k, ) = k − μ4  − λ + O(2 ) we get the Fourier series 

t Hˆ

μ

λ

χ (E(k, ))e−itk Πk = χ ( Hˆ )e−i  eit ( 4 +  ) + O(7/4 ).

k∈Z

So we have χ (E(k, ))Πk

1 = 2π



2π

−i χ ( Hˆ )e  t



Hˆ −E 



dt + O(7/4 ).

0

Now we can choose c() of order −1/4 such that c()Ψγ E  = 1 and for z ∈ γ we have

362

12 Semi-classical Measures: Stationary and Large Time Behavior

c()Πk ϕz =

c() ψγ E + O(3/2 ). 2π

(12.24)

So we get the part (i) of the proposition using (12.24) and (12.23). Let us now prove (ii). Let γ j , 1 ≤ j ≤ N different periodical trajectories on Σ E . Let us consider the measure μ = α12 δγ1 + · · · + α 2N δγ N , α j ≥ 0, α12 + · · · + α 2N = 1. We prove that μ is a semi-classical mesure on Σ E . From (12.24) we have obtained quasi-modes {ψ j }1≤ j≤N with frequency set in γ j and such that Πk ψ j = ψ j + O(3/2 ). So they are almost orthogonal.  α j ψ j where ψ j = 2π c()Πk ϕz j , z j ∈ γ j . Let us denote ψα = 1≤ j≤N

To conclude we use that any invariant measure μ on Σ E is the -weak-limit of  finite convex sums of Dirac measures δγ j for periodical orbits γ j .

12.4 Time Dependent Semi-classical Measures Let us consider now a sub-quadratic time dependent semi-classical Hamiltonian  H (t) ≈  j H j (t, X ) satisfying the assumptions of Theorem 12 (Chap. 2). Recall j≥1

that

γ

|∂ X H j (t, X )| ≤ Cγ , for (t, X ) ∈ R × R2n and |γ | + j ≥ 2. We know that Hˆ (t) generates a unitary Schrödinger propagator in L 2 (Rn ) which will be denoted U  (t, t0 ) and Ut  = U  (t, 0). We denote Φ Ht,t00 the classical Hamiltonian flow associated with the principal symbol H0 . Let us consider a normalized initial data ψ0 satisfying (12.5) and let be μ0 a corresponding semi-classical measure. Let {ψ0 }0 0 there exists C T such that for any t ∈ [0, T ] we have Φ Ht,00 X  ≤ C T (X  + 1), ∀X ∈ R2n .

(12.25)

Proof We begin by proving the inequality for T = t0 > 0 small enough. Let us denote X t = Φ Ht,00 X . From the Hamilton equation we get a constant C > 0 such that for all t ≥ 0, X t  ≤ X  + Ct sup X s . 0≤s≤t 1 we get, for t ∈ [0, t0 ], X t  ≤ 2X . Then by iteration, writing So choosing t0 = 2C  for kt0 ≤ t ≤ (k + 1)t0 , Φ t,0 = Φ t,kt0 ◦ Φ kt0 ,(k−1)t0 ◦ · ◦ Φ t0 ,0 , we get (12.25).

Now we consider large time behavior of the measures μt . Following [184, 185] we introduce a time scale τ such that lim sup τ = +∞. →0

Recall that ψt satisfies a semi-classical time dependent Schrödinger equation  = ψ0 . i∂t ψt = Hˆ ψt , ψt=0

(12.26)

{ψ0 } is supposed to be bounded in L 2 (Rn ) and μt is the Husimi measure defined by ψt . We define μ˜ t = μtτ . As above, a compactness argument shows that μ˜ t has accumulation points μ˜ t ∈ L ∞ (Rt , Cb (R2n )). Moreover, for a.e t ∈ R, μ˜ t is a finite measure if ψ0 is uniformly bounded in L 2 (Rn ).  μ˜ t (d X ) ≤ lim sup ψ0 2 . →0

R2n

We shall use the following formula, where θ ∈ L 1 (Rt , dt), for a sequence of  → 0, we have     θ (t) A(X )μ˜ t (d X ) dt/ (12.27) lim θ (t)ψtτ , Aˆ w ψtτ dt = →0





Rt

R2n X

Proposition 129 Recall that lim τ = +∞. →0

(i) a.e t ∈ R, the measure μ˜ t is invariant under the Hamiltonian flow Φ Ht 0 . (ii) Assume that the initial data ψ0 has a semi-classical measure μ0 and that τ = o(−2 ). Let A ∈ S00 invariant under the Hamiltonian flow Φ Ht 0 . Then for a

364

12 Semi-classical Measures: Stationary and Large Time Behavior

subsequence  → 0, for any t ∈ R, we have: 

lim ψtτ , Aˆ w ψtτ  =

→0

R2n

A(X )μ0 (d X ).

(12.28)

Proof We follow [185] (see also [186] for time dependent semi-classical measures for large times). (i) Let As (X ) = A(φ −s (X )). Using the propagation of observables we have  Us Aˆ s Us = Aˆ + O().

Hence we get  R

  θ (t)ψ0 , Utτ, Aˆ w s Utτ ψ0 dt = 

 R

,  θ (t)ψ0 , U(t− ψ0 dt + O(). Aˆ w U(t− s s τ )τ τ )τ 

(12.29)

Hence  R

  θ (t)ψ0 , Utτ, Aˆ w s Utτ ψ0 dt = 

  s ψ0 , Utτ, θ t+ Aˆ w Utτ ψ0 dt + O().  τ R



(12.30) Using that lim τ = +∞ and passing to the limit in  → 0 we get for any θ ∈ L 1 (R), →0



 R

θ (t)

R2n

 As (X )μ˜ t (d X ) =

R

 θ (t)

R2n

A(X )μ˜ t (d X ).

(ii) We have {A, H0 } = 0. So using property of Weyl quantization we get that ˆ = O(3 ) (a little computation shows that the coefficient of 2 vanishes). [ Hˆ , A] We have: d ˆ Utτ ψ0 . ˆ[ Hˆ , A] i ψtτ , Aˆ w ψtτ  = τ ψ0 , Utτ,   dt Then after integration between 0 and t and using that lim 2 τ = 0, we →0

get (12.28).



Corollary 38 Assume that the conditions of Proposition 129 are satisfied and that the Hamiltonian flow of H0 is 2π periodic in the compact set H0−1 [E − η, E + η] (η > 0 small enough) with E non critical for H0 . Then for any semi-classical measure μ˜ t defined by (12.27) and any A ∈ C0∞ (H0−1 ]E − η, E + η[), we have a.e t ∈ R, 

 R2n

where A(X ) =

1 2π

 2π 0

A(X )μ˜ t (d X ) = A(φ s (X ))ds.

R2n

A(X )μ0 (d X ),

(12.31)

12.4 Time Dependent Semi-classical Measures

365

Proof A(X ) is invariant by the Hamiltonian flow so from (12.28) we have 

 R2n

A(X )μ˜ t (d X ) =

R2n

A(X )μ0 (d X ).

Then using that μ˜ t is invariant by the flow and Fubini Theorem we get (12.31).  Let us give now an application of time dependent semi-classical measures to estimate non diagonal matrix elements. Corollary 39 We assume that the assumptions of the Corollary 38 are satisfied. Let ψ j normalized eigenfunctions of Hˆ for eigenvalues E j in clusters I ( j, ). Let ω jk () =

E j ()−E k () . 

Assume that, for a subsequence, lim = 0 for a subsequence  → 0.

→0

ω j,k () ˆ k  = c = 0. Then we have lim ψ j , Aψ →0 

Proof Following [185] again we consider ψ  = ψ j + ψk and its large time evolution. We have ˆ  (s) = ψ j , Aψ ˆ j  + ψ  , Aψ ˆ   + 2(eisωi,k () ψ j , Aψ ˆ k ). ψ  (s), Aψ Using this egality for s = t and using (12.31) with τ = −1 , , there exists a finite real measure μ and and complex finite measure ν on R2n such that for any θ ∈ L 1 (R)s we have     ˜θ(0) A(X )μ(d X ) = 2 θ(c) ˜ A(X )ν(d X ) . Then we get that for a subsequence of , ˆ k  = lim ψ j , Aψ

→0

 A(X )ν(d X ) = 0.



ˆ k  considered here conRemark 77 Notice that the transition amplitudes ψ j , Aψ cerns energies which are in the same cluster I (k, ), so the result is not intuitive. A similar phenomenon was proved for Zoll manifolds [184]. An example of such a situation holds true for a perturbation of the 2-dimensional harmonic oscillator: Hˆ = Hˆ os +  Lˆ where Lˆ is the generator of rotations in the plane: ˆ = 0. So in each specLˆ = (x∂ y − y∂x ). Its spectrum is Z. We know that [ Hˆ os , L] tral subspace E j for the eigenvalue j + 1 of Hˆ os we can find a common basis of ˆ So the eigenvalues of Hˆ in the cluster are eigenfunctions {ψ j, }0≤≤ j for Hˆ os and L. 2 ( j + 1) +  . Then applying the Corollary for this basis we have for  =  , ˆ j,  = 0. lim ψ j, , Aψ

→0

366

12 Semi-classical Measures: Stationary and Large Time Behavior

12.5 Quantum Unique Ergodicity for a Random Hermite Basis We consider now the harmonic oscillator Hˆ os = 21 (− + |x|2 ) in L 2 (Rn ), n ≥ 2. The corresponding classical Hamiltonian flow is surely not ergodic on the energy shell |ξ |2 + |x|2 = 2. Nevertheless we shall see that most of orthonormal basis of eigenfunctions of Hˆ os are quantum ergodic (meaning that in these basis the matrix diagonal elements for a class of observables Opw 1 (A) converge towards the average of A for the Liouville probability. This rather surprising phenomenon was discovered by Zelditch for the Laplace operator on the sphere Sn [268]. In [222] the authors have proved that the same phenomenon holds true for the harmonic oscillator. As we have seen classical ergodicity is related to the semi-classical behaviour of the ˆ k , j = k, close to the diagonal elements. non-diagonal elements ψ j , Aψ The eigenvalues of Hˆ os are λ j = j + n2 , j ∈ N, with mutiplicities m j ≈ j n−1 . Let be E j the corresponding eigenspace , Sm j the unit sphere of E j and {ϕ j,k , 1 ≤ k ≤ m j } a fixed orthonormal basis of E j (Hermite functions). We can identify the space of the orthonormal basis of E j with the unitary group U (m j ). We consider on U (m j ) its Haar probability measure ρ j . Then the set B of the Hilbertian basis of eigenfunctions of Hos in L 2 (Rn ) can be identified with the infinite product B = × j∈N U (Nm j ), which can be endowed with the product of probability measures: dρ = ⊗ j∈N dρ j . Denote by B = (ψ j, ) j∈N, 1≤≤m j ∈ B a typical orthonormal basis of L 2 (Rn ) so that for all j ∈ N, (ψ j, )1≤≤m j ∈ U (m j ) is an orthonormal basis of Em j . Denote S0 the space of symbols on Rn × Rn smooth and homogeneous of degree 0 (outside a neighborhood of 0). Then we have the following result. Theorem 69 For B ∈ B and A ∈ S0 let us denote   ˆ j,  − ν1 (A), D j (B) = max ψ j, , Aψ 1≤≤m j

√ where ν1 is here the uniform probability on the sphere 2S2n−1 . Then we have lim D j (B) = 0, ρ − a.s on B. j→+∞

In other words, a random orthonormal basis of Hermite functions is Quantum Unique Ergodique (QUE). A main step in the proof of Theorem 69 is to prove a semi-classical probabilistic result for the quantum oscillator Hˆ = 21 (−2  + |x|2 ). More generally we assume

12.5 Quantum Unique Ergodicity for a Random Hermite Basis

367

that Hˆ is a rather general quantum Hamiltonian with a discrete spectrum in some small interval I () = [E − δ(), E + δ()]. We assume that H satisfies the assumptions (H.0) and (H.1) of Chap. 5 and δ() is chosen like in Sect. 1 of this Chapter, δ() ≈ 1−τ , with τ > 0. Moreover H satisfies the periodicity conditions of Proposition 35 (Chap. 5) close to the energy E with 0 < c < 1. Let us introduce the following notations. Π  is the spectral projector of Hˆ on the interval I (). {ψ j }1≤ j≤N an orthonormal basis of eigenfunctions of the spectral subspace E := Π  (L 2 (Rn )) and Hˆ ψ j = E j ψ j where E j ∈ I (). S is the unit sphere of E . (Notice that dim(E ) := N ≈ −n+τ .) Moreover if H satisfies the periodicity condition of Proposition 35 (Chap. 5) close to the energy E we shall choose δ() =  and Π  is the spectral projector in [E, E + [ with E > 0. Notice that there is only one eigenvalue in [E, E + [ (with high multiplicity). Lemma 83 Assume that H satisfies the periodicity conditions of Proposition 35 (Chap. 5) close to the energy E. Then choosing I () = [E, E + [ we have (i) N = (2π )1−n L E (Σ E ) + o(1−n ) (ii)  ˆ Tr(Π  A) = A(X )ν E (d X ). lim → N ΣE Proof We can use the same technicalities as in Chap. 5 for proving the Gutzwiller trace formula (see also the Chapter about Herman-Kluk approximation). 2 2 For simplicity we shall assume that Hˆ = Hos := − 2  + |x|2 . So we know from Chap. 1 that 

e−i  Hos ϕz = e−it 2 ϕzt t

n

where z → z t is the classical harmonic flow. Denote Πk the spectral projector for the eigenvalue E j = ( j + n2 ). −1/2

Let us consider a random wave V (ω) = N



X j (ω)ψ j where X j (ω) are

1≤ j≤N

iid random variables of mean 0 and variance 1 (Gaussians or Bernoulli variables). Then we get a Borelian probability measure P on S such that for any continuous function F : S −→ R we have     V (ω) P(dω), F(u)P (du) = F V (ω) S Ω where {Ω, P} is an abstract probability space. For simplicity we assume here that X j follows the complex normal law. This means that (X j , X j ) are independent real random variables and the pair follows the two

368

12 Semi-classical Measures: Stationary and Large Time Behavior

dimensional normal law. Then it can be shown that P (du) is the uniform probability density on the sphere S . Proposition 130 Under the previous assumptions there exist c, C > 0 so that for all r > 0,  ∈]0, 1] and A ∈ S00   2 ˆ Ph u ∈ Sh : |u, A(h)u − ν E (A)| > r ≤ Ce−cN r .

(12.32)

As a consequence, there exists C > 0 so that for all p ≥ 2, h ∈]0, 1] and A ∈ S00 we have   −1/2 √ ˆ p.  u, A(h)u − L E (A) L p ≤ C Nh Ph

Recall that ν E is the Liouville probability on Σ E . It may be surprising that this result is true without any assumption on the dynamical system. Why this is true? This is a consequence of the concentration property of probability measures on spheres of large dimension. For simplicity we assume that P is the uniform probability on S . Proposition 131 (concentration of measure, P. Levy, M. Ledoux [174]) There exist constants K > 0, κ > 0 such that for every Lipschitz function F : S −→ R satisfying |F(u) − F(v)| ≤ F Li p u − v L 2 (Rd ) , ∀u, v ∈ Sh , we have κ N r2   − 2 Pγ u ∈ S : |F − M F | > r ≤ K e FLi p , ∀r > 0, h ∈]0, 1],

(12.33)

where M F is a median for F. Recall that a median M F for F is defined by   1   1 Pγ u ∈ Sh : F ≥ M F ≥ , Pγ u ∈ Sh : F ≤ M F ≥ . 2 2 The factor Nh in the exponential of r.h.s of (12.33) will be crucial in applications to get large deviation estimates. Proof (Proposition 130). ˆ We have Let M A be a median for the map FA (u) := u, Au. ˆ − v. |FA (u) − FA (v)| ≤ 2 Au So we can apply Proposition 131. To go further we have to estimate the median M A by using the expectation E associated with the probability law P . The Proposition 130 is a consequence of the three following lemmas.

12.5 Quantum Unique Ergodicity for a Random Hermite Basis

369

Lemma 84 There exists C > 0 such that −1/2

|E [FA − M A ]| ≤ C N

, ∀ ∈]0, 1].



Lemma 85 lim E [FA ] =

A(X )ν E (d X ).

→0

Lemma 86 Let us denote the random variable v(ω) = V (ω)2 . Then there exists s0 > 0, small enough, such that for any  > 0 we have P[|v − 1| ≥ ε] ≤ e−ε

s0 N 2

, ∀ ∈]0, 1].

We get easily Proposition 130 using Proposition 131 and lemmas 85, 84 Let us prove now the three lemmas. Proof (Lemma 84). We have, using Proposition 131, 

+∞

|E [FA − M A ]| ≤

P [|FA − M A | > r ]dr

0 −1/2

≤ C N

.

(12.34) 

Proof (Lemma 85). Using Corollary 1 of Theorem 63 it is enough to prove 

ˆ ) Tr( AΠ lim E [FA ] − →0 N 

We have E [FA ] =

Ω



 = 0.

V (ω) ˆ V (ω) ,A P(dω). V (ω) V (ω)

Using properties of the random variables, it is clear that we have  Ω

ˆ (ω)P(dω) = V (ω), AV −1/2

Define Ω := {ω : | (ω)2 − 1| ≤ N  I :=

Ω

} and

1 − V 2 ˆ P(dω), I I := V, AV V 2

From Proposition 131 we have

ˆ ) Tr( AΠ . N

 Ωc

1 − V 2 ˆ P(dω) V, AV V 2

370

12 Semi-classical Measures: Stationary and Large Time Behavior −1/2

P(Ωc ) ≤ Ce−cN . And using the Cauchy-Schwarz inequality gives c ˆ I I ≤  AP(Ω )



 (1 + |V ) ) P(dω) . 2 2

Ω

Using that the integral is O(1) as  → 0 we have lim I I = 0. −1/2

We also have I ≤ N

→0

ˆ so the proof of Lemma 85 is completed.  A



Proof (Lemma 86). We reproduce here a usual method to prove a large deviation 2 estimate. Let f (s) = E(es X 1 ) and g(s) = log f (s), for s > 0 close to 0. For τ > 0 we have P[v > τ ] = P[es N v > es N τ ] ≤ E(es N v )e−s N τ . Using the definition of v we have E(es N v ) = e−(sτ −g(s))N . Hence if τ ≥ 1 + ε here s > 0 small enough such that we have for any ε > 0, sτ − g(s) = s(τ − 1) + O(s 2 ) ≥

εs . 2

ε

Then we get P[v > 1 + ε] ≤ e− 2 s N . By the same argument we get ε P[v < 1 − ε] ≤ e− 2 s N .



Now we can prove Theorem 69 as a consequence of Proposition 130. 2 Proof Let Hˆ os = − 2  +

|x|2 . 2

  Every B ∈ B can be identified with B j j≥1 where B j ∈ U (Nh j ). The random variables D j are independent and D j depends only on B j . So for every r > 0 we have     ρ D j (B) > r = ρ j D j (B) > r     ˆ j,  − ν1 (A) > r = ρ j ∃ ∈ N ∩ [1, Nh j ], ϕ j, , Aϕ     ˆ j,  − ν1 (A) > r ≤ ρ j ϕ j, , Aϕ 1≤≤Nh j

   ˆ j )u − ν1 (A) > r . ≤ Nh j Ph j u, A(h In the last line we have used that the probability Ph is the push-forward of the Haar measure of U (Mh ) by the maps: U (Nh ) M → Mv ∈ Sh , for any v ∈ Sh . So using Proposition 130 we get, with positive constants C1 , C2 ,     ρ D j (B) > r ≤ C1 j n−1 exp − C2 j n−1r 2 .

12.5 Quantum Unique Ergodicity for a Random Hermite Basis

371

In particular for any n ≥ 2 we get    ρ D j (B) > r < +∞ j≥1

and the result is a consequence of the Borel-Cantelli Lemma.



Notes There are many papers studying properties of semi-classical mesures in different geometry settings for the Laplace-Beltrami operator on compact Riemannian manifolds [4, 184, 185]. In this chapter we have adapted, in the context of confining Hamiltonians on the Euclidian space Rn , some ideas taken in these papers. Section 5 is taken from a paper by Robert-Thomann [222].

Chapter 13

Open Quantum Systems and Coherent States

Abstract In this chapter we study some properties of systems of coupled quadratic quantum Hamiltonians. The well known example is coupled harmonic oscillators appearing in the Quantum Brownian Motion. An open system is a physical (S) interacting with the environment (E) such that the total system (S ∪ E) is isolated from the rest of the universe. So we have two independent Hamiltonians Hˆ S , Hˆ E and an interaction Vˆ between (S) and (E). The main problem discussed for open systems is to understand the dynamical action of the interaction Vˆ with the environment on the time dependent evolution of (S) which is now irreversible because a part of its energy escapes to the environment. An important physical effect of the interaction of the environment on a quantum system is decoherence which give an explanation for the mechanism driving the quantum-to-classical transition. We restrict ourself to few mathematical problems related with this very active field of research in physics [155, 200, 231, 270]. In a first part we study the purity of a state of (S) (or the linear entropy) in a short time approximation. In a second part we discuss the separability of bipartite systems. In a third part we establish a master equation (or a Fokker-Planck type equation) for the time evolution of the reduced matrix density for coupled quadratic Hamiltonians by a bilinear inter-action. We give an explicit formula for the solution of this master equation so that the time evolution of the reduced density at time t is written as a convolution integral for the reduced density at initial time t0 = 0, with a Gaussian kernel, for 0 ≤ t < tc where tc ∈]0, ∞] is a critical time such that reversibility is lost for t > tc if tc < +∞. Some manifestations of decoherence, for t > 0 small enough, are proved on our simple model which is similar to a representation of the Schrödinger cat.

13.1 Introduction to Open Quantum Systems 13.1.1 A General Setting The general setting considered here is a quantum system (S) interacting with an environment (E). The total system (S ∪ E) is assumed to be an isolated quantum system and we are interested in dynamical properties of (S) alone, which is an open © Springer Nature Switzerland AG 2021 D. Robert and M. Combescure, Coherent States and Applications in Mathematical Physics, Theoretical and Mathematical Physics, https://doi.org/10.1007/978-3-030-70845-0_13

373

374

13 Open Quantum Systems and Coherent States

system because of its interactions with (E). In particular, during the time evolution, the energy of (S) is not preserved and its evolution is not determined by a Schrödinger or Liouville–von Neumann equation unlike for the total system (S ∪ E). A detailed presentation concerning the quantum dynamics of open systems can be found in the book [43]. The Hilbert space of the total system (S ∪ E) is the tensor product  is decomposed as follows: H E ,1 and its Hamiltonian H H = HS ⊗  ⊗ 1l + 1l ⊗ H E + H I (t) = H 0 + V (t). V (t) = H H S  

(13.1)

0 H

A basic example is the quantum Brownian motion, coupling two harmonic oscillators with bilinear potential Vg (t):  E = 1 H (−∂ y22 + ω2j y 2j ), j 2 1≤ j≤N  Vg (t, x, y) = x · g j (t)y j , x, y j , g j (t) ∈ R,

S = 1 (−∂ 22 + ω02 x 2 ), H x 2

(13.2) (13.3)

1≤ j≤N

where H S = L 2 (R), H E = L 2 (R N ). The whole system (S ∪ E) is isolated by assumption and its evolution obeys . the Schrödinger (or Liouville–von Neumann) equation with the Hamiltonian H Assuming that at time t = 0 the system (S) and the environment (E) are decorrelated, the problem is to understand the evolution of the system (S) alone in the Hilbert space H S when an interaction with the environment is switched on.  and All Hamiltonians here are selfadjoint operators on their natural domain, H  are defined in H , H S in H S , H E in H E . Quantum observables are denoted I = V H with a hat accent, the corresponding classical observables (also named Wigner functions or Weyl-symbols) are written without hat accent.  has the following form We shall assume that the interacting potential V g (t) = V



j g j (t) Sj ⊗ E

(13.4)

1≤ j≤N

j are selfadjoint operators in H S and H E respectively. Assumptions where  S j and E on their domains will be given later. Recall that a density matrix ρ  is a positive trace class operator with trace one (Tr( ρ ) = 1), see Chap. 1. Let ρ 0 be a state of the total system (S ∪ E) in the Hilbert space H at the initial time. The time evolution of ρ  obeys the following Liouville– von Neumann equation: 1 Here ⊗  means that the tensor product is the Hilbert space obtained by completion of the algebraic

tensor product.

13.1 Introduction to Open Quantum Systems

, ρ ˙ = i −1 [ H ρ  ], ρ  : t → ρ (t), ρ (0) = ρ 0 .

375

(13.5)

Time derivatives are denoted by a dot, [ · , · ] denotes the commutator of two observables. We assume that for t = 0 the system and the environment are decoupled, which means that E (0), ρ S,E (0) are density matrices in H S,E . ρ (0) = ρ S (0) ⊗ ρ

(13.6)

The time evolution ρ (t) is given by the propagator UV (t, s) generated by the Hamil  tonian HV (t). For V = 0 we have U0 (t, s) = e−i(t−s) HS ⊗ e−i(t−s) HE . So we have ∗ E (t), t ∈ R, where ρ S,E (t) = U S,E ρ S,E (0)U S,E . In particular, if ρ (t) = ρ S (t) ⊗ ρ ρ S (0) is a pure state, then ρ S (t) is pure for any t ∈ R. When the interaction is (t) over the environment degrees switched on, ρ S (t) is defined as the partial trace of ρ of freedom and one question is to check if ρ S (t) is pure or not for t = 0. Let us recall some definitions concerning some properties of density matrix states. Definition 28 Let ρ  be a state in the Hilbert space H , i.e., a nonnegative operator in H with trace 1. (i) ρ  is a pure state if and only if ρ  is the orthogonal projector on a unit vector ψ of H. (ii) ρ  is a mixed state if it is not a pure state.  is a separable state if there exist a probability (iii) Let ρ  be a state in H S ⊗ H E ; ρ space (, P) and a mesurable family of states: ω → ρ ω(S,E) from  into S1 (H S,E )2 3 such that  ρ = ρ ω(S) ⊗ ρ ω(E) P(dω) (13.7) 

(iv) ρ  is entangled if it is not separable. It is easy to see that a density matrix ρ is a mixed state if and only if ρ has an eigenvalue λ ∈]0, 1[. To decide if a state is pure or not we consider its purity function Pρ (t) = Tr( ρ (t)2 ). We clearly have 0 ≤ Pρ (t) ≤ 1. It is not difficult to see that ρ(t) is pure if and only (t) is pure. So 0 is a maximum if Pρ (t) = 1. Suppose that at time t = 0 the state ρ for Pρ , hence we have P¨ ρ (t)(0) ≤ 0. If furthermore P¨ ρ (0) < 0, then for some ε > 0, the state ρ (t) is mixed whenever 0 < t < ε. For isolated (closed) systems, if the initial state is pure, then it stays pure at every time, as it can easily be seen by using Eq. (13.5). us recall that S1 (H ) denotes the space of class-trace operators in the Hilbert space H . In physics literature only discrete probability laws are considered, but it seems necessary to extend the definition to any probability law. 2 Let 3

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13 Open Quantum Systems and Coherent States

Remark 78 The function S (ρ) := 1 − Pρ is called the linear entropy. Recall that in quantum mechanics the Boltzman-von Neumann entropy is defined as S B N (ρ) =: −Tr(ρˆ log(ρ)). ˆ Using the linear approximation log(ρ) ˆ ≈ ρˆ − 1l one get S B N (ρ) ≈ S (ρ). Another quantum entropy, with better properties than S B N , was introduced by ˆ z (see Chap. 2). Wehrl [254] by using the covariant Berezin symbol ρc (z) = ϕz , ρϕ We assume here that H = L 2 (R). The Wehrl entropy is  SW (ρ) = −

R2

ρc (z) log(ρc (z))dz.

For interesting properties of SW (ρ) we refer to [179, 254]. In particular it is proved in [179] that min SW (ρ) ≥ 1 and that SW (ρ) = 1 if and only if ρˆ is the orthogonal ρ

projection on a coherent state ϕz (“Wehrl conjecture” ). So our first step is to compute the second derivative P¨ S (t)(0) for the purity of S (0) is pure. In particular, for a large class of models, the reduced system ρ S when ρ including the Quantum Brownian Motion with N oscillators, we shall prove that for the reduced system we have P¨ S (0) < 0. So for these models there is an indication that the state of the reduced system may be immediately decoherent and is entangled with the environment. Our second goal is to consider the separability (or entanglement) property of the mixed state ρ (t). We explain here results obtained by physicsts: the necessary Horodecki condition [156] which turns out to be sufficient for Gaussian states and this can be applied to the quantum Brownian motion [92, 236]. Our third goal is to compute the time evolution ρ S (t) of the state of the small system (S). We know that ρ S (t) does not satisfy a Liouville–von Neumann equation because the system is open so diffusion terms appear. We shall give a proof that for any time dependent quadratic systems, ρ S (t) obeys an exact “master equation” similar to a Fokker–Planck type equation, with time dependent coefficients, and from this equation we can get an explicit formula for ρ S (t). Similar results were obtained by different methods in [85, 86, 103]. Let us now recall a mathematical definition for the reduced evolution ρ S (t) and related properties. This can be done by introducing partial trace (or relative trace) over the environment for a state ρ  of the global system.  be a trace class operator in H S ⊗ H E . We denote by Tr E A  the Definition 29 Let A  unique trace class operator on H S satisfying, for every bounded operator B on H S ,



  B = Tr A( B ⊗ 1l E ) , Tr S (Tr E A)

(13.8)

We have denoted by “Tr” the trace in the total space H , and by Tr S,E the trace in  = Tr A.  H S,E , respectively. Of course we have the “Fubini property:” Tr S (Tr E A)

13.1 Introduction to Open Quantum Systems

377

 =  if   are trace class operators Moreover, we have Tr E (  B ⊗ C) B · (Tr E C) B and C in H S and H E respectively.  is a density matrix in H S , called the If ρ  is a density matrix in H , then Tr E ρ reduced density matrix or the reduced state. From (13.8) we can compute the matrix  by the formula element of Tr E A    ψ,ϕ ⊗ 1l E ) , ψ, ϕ ∈ H S . = Tr A(Π ψ, (Tr E A)ϕ The partial trace was first introduced by L. Landau (1927) and then more recently in quantum mechanics to explain entanglement and decoherence (see [200, 231] for more details). Let  ψj ⊗ ηj, Ψ ∈ HS ⊗ H E , Ψ = 1≤ j≤N

where Ψ is a pure state in H = H S ⊗ H E . Let us compute the partial trace of ΠΨ in H E . Applying Definition 29, we easily get Tr E (ΠΨ ) =



η j , ηk Πψ j ,ψk

(13.9)

1≤ j,k≤N

where Πψ j ,ψk is a rank one operator in H S : Πψ j ,ψk (ϕ) = ψ j , ϕ ψk . The decoherence phenomenon [162, 270] means that there exists a independent vectors {η j } such that | η j , ηk | becomes negligible for j = k, so that Tr E (ΠΨ ) is very close to 

p j Πψ j ,

p j = η j , η j .

1≤ j≤N

Hen ce quantum interferences between the states ψ j of the system (S) are destroyed (13.9).

13.1.2 Coupled Quadratic Hamiltonians Formula (13.8) is an operator version of the Fubini integration theorem as we can see in the Weyl quantization setting. Let H S = L 2 (Rd ), H E = L 2 (R N ). Denote z = (x, ξ ) ∈ R2d , u = (y, η) ∈ R2N ,  the Weyl symbol of A  in the Schwartz space S (R2(d+N ) ) (see Chap. 2 for A = σw A  is more details); then the Weyl symbol of Tr E A  = (2π )−N σw (Tr E A)(z)

 A(z, u)du. R2N

(13.10)

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13 Open Quantum Systems and Coherent States

Our main applications here concern the Weyl-quadratic case where: • HS and HE are real quadratic Hamiltonians (generalized harmonic oscillators) respectively in the phase spaces R2d , R2N ; E are quantum Hamiltonians (Weyl quantization of HS , respectively HE ) S and H • H 2 d in L (R ), respectively L 2 (R N ); • ρ S is the projector to ψ ∈ S (Rd ); • ρ E is a Gaussian with 0 mean; • S j , respectively E j is a linear form on R2d , respectively R2N (bilinear coupling). In applications one considers thermal equilibrium states for the environment:  ρ E = Z (β)−1 e−β HE , where HE is a quadratic form, positive definite, and β = T1 ,  T > 0, is the temperature, Z (β) = Tr(e−β HE ). In this setting it is possible to find the exact time dependent master equation satisfied by ρ S (t), as we shall see. This model was analyzed in several papers [102, 103].

13.1.3 A Schrödinger Cat Model A coherent state model for the famous Schrödinger cat is built as follows. Let be z ± ∈ R2 , z± = (0, 0). For simplicity we choose z − = −z + . ϕz + is the alive cat state, ϕz − is the dead cat state. χ± is an orthonormal basis of 2 C , for the atomic states (if χ+ the cat is alive, dead if χ− ). The states space of the system “cat+atom” is the Hilbert space L 2 (R) ⊗ C2 and the state of the system is ψcat+at =

1 (ϕz + ⊗ χ+ + ϕz − ⊗ χ− ) N

N is a normalization constant. √ Notice that ϕz ± are almost orthogonal (N ≈ 2 as  → 0). The cat state is the partial trace : ρˆcat = Tr C2 (ρˆcat+at ), where ρˆcat+at is the orthogonal projector on ψcat+at . We find ρˆcat =

1 (Π+ + Π− ) , N

Π± is the orthogonal projector on ϕ±z . So the cat state in entangled. Moreover we get

(13.11)

13.1 Introduction to Open Quantum Systems

379

1 , 2 1 Probability[dead cat] = Tr(ρˆcat Π− ) ≈ . 2

Probability[alive cat] = Tr(ρˆcat Π+ ) ≈

For a finer analysis of the situation we compute the Wigner function of the state ψcat+at using the spin coherent state (Chap. 7). We have seen that coherent 1/2-spin states of an atom are parametrized by the sphere S2 . Let n ∈ S2 , n = (n 1 , n 2 , n 3 ): n 3 = cos θ , n 2 = sin θ cos φ, n 1 = sin θ sin φ. Let the spin coherent state ψn = sin(θ/2)eiφ χ− + cos(θ/2)χ+ . If χ ∈ C2 , norm 1, its Wigner-Husimi function is Wχ (n) = | χ , ψn |2 . Now we have the Wigner function Wcat+at of ψcat+at N2 Wcat+at (X, n) = cos2 (θ/2)Wϕz (X ) + sin2 (θ/2)Wϕ−z (X ) + 2 

2σ (X, z) − φ Wϕ0 (X ) sin(θ ) cos 

(13.12)

X ∈ R2 and n ∈ S2 with coordinates (θ, φ). If n is in the north or south pole, the state of the cat is known. Taking the average on the sphere S2 for Wcat+at (X, n) we find the Wigner function of ρˆcat : N12 (Wϕz+ + Wϕz− ). Choosing n on the equator with φ = 0 we find 

|X −z + |2 |X −z − |2 |X |2 2σ (X, z + ) N2 Wψ cat (X ) = e−  + e−  + 2e−  cos . 2 

(13.13)

corresponding with the state ψ cat = N1 (ϕz + + ϕz − ). This is a superposition of two states usually called “states of the Schrödinger cat”. The first two terms in (13.13) correspond to the Wigner functions of ϕz ± . The third term is an oscillating term between life and death which is responsible of the coherence of the cat state (here is the Schrödinger paradox).

13.2 Computation of the Purity 13.2.1 Assumption and Statements Let us consider the following abstract setting. We fix a selfadjoint and positive opers = Dom(K 0s ) ator K 0 in H and we define the scale of abstract Sobolev space H  s −s ∞ if s ≤ 0. Define H = s∈R H s and if s ≥ 0 and  H sis the dual of H −∞ = s∈R H . Moreover, we assume that K 0−1 is in the Schatten class S p0 (H ). H For definitions and properties of Schatten classes we refer to [120]. We introduce the following definition.

380

13 Open Quantum Systems and Coherent States

Definition 30 A linear operator A : H ∞ → H −∞ is said to be of order m ∈ R if, for any k ∈ R, we have A ∈ L (H k+m , H k ). Now we introduce our assumptions on the Hamiltonian (13.1). • [A0] HS and HE are selfadjoint operators in H S , respectively H E . S ⊗ 1l + 1l ⊗ H E is an operator of order m 0 in H , is a sym0 := H Moreover, H metric operator in H ∞ and is such that [H0 , K 0 ]K 0−1 is of order 0.  ∈ C0 (I0 , L (H k+m 0 , H k )), for every k =V (t) is of order m 0 such that V • [A1] V ∞  is symmetric on H , for t ∈ I0 :=] − T0 , T0 [ for some T0 > 0. Moreover and V (t), K 0 ]K 0−1 ∈ C0 (I0 , L (H k , H k )), for all k. [V • [A2] The initial state ρ (0) is of order r , with r < − p0 − km 0 , for some k ≥ 3. We have the following preliminary results (see [191]). V (t) Proposition 132 Assume that conditions [A0] and [A1] are satisfied. Then H has a unique selfadjoint extension in H . In particular, H0 is essentially selfadV (t) defines a unitary propagator joint. Moreover, the time dependent Hamiltonian H UV (t, s) in H and (t, s) → UV (t, s) is continuous from I0 × I0 into L (H k ), for any k ∈ N. Then the time evolution of an initial matrix density ρ (0) is well defined and is given by (0)U V (t, 0). (13.14) ρ(t) = U V (t, 0)∗ ρ It is useful to compute the second derivative of the purity of the state ρ S of the system. Assume that ρ S (0) is pure and that its purity P S (t) is C 3 in a neighborhood of 0. Then we have P¨ S (0) 2 t + O(t 3 ), t  0. P S (t) = 1 + 2 So if P¨ S (0) < 0, then ρ (t) becomes a mixed state for t close to 0, t = 0. Theorem 70 Assume that conditions [A0], [A1], and [A2] are satisfied. Then t → ρ (t) is C k from I0 into S1 (H ). Assume furthermore that k ≥ 2 and that ρ S (0) is a pure state in H S . Then we have the following formula for the second derivative of P S (t) at 0:  ρ ( (0)V ρ S⊥ (0) ⊗ 1l) , (13.15) P¨ S (0) = −4Tr V S where ρ S⊥ = 1l − ρ Now we get a more explicit formula for the separable interactions (13.4). j E j  ) = δ j, j  d j , 1 ≤ Corollary 40 Assume that we have (13.4) and that Tr E ( ρ E (0) E  j, j ≤ N , then we have P¨ S (0) = −4

   S j ψ S 2 − ψ S ,  S j ψ S 2 d j , 1≤ j≤N

(13.16)

13.2 Computation of the Purity

381

where ρ S (0) is the projection to ψ S . In particular, if there exists j0 such that d j0 > 0 S j0 , then P¨ S (0) < 0. and ψ S is not an eigenfunction of  We can apply this result to the Quantum Brownian Motion (13.2), (13.3). Denote π S (z) := 2π −d ρ S (z) dz and π E (u) := (2π )−N ρ E (u)du (they are quasiprobabilities S (0) is a pure state ψ S ∈ S (Rd ). in R2d , respectively R2N ). Here we assume that ρ We introduce the commutators: j  , E j ], S j ,  S j ], e j  j = [ E s j  j = [ and the classical (symmetric) covariance matrices: Σ (S) j j =

 π S (z)S j  (z)S j (z) dz − R2d



π S (z)S j (z) dz



R2d



R2d



and Σ (E) j j

π S (z)S j  (z) dz

=

π E (u)E j  (u)E j (u) du. R2N

So we get the following statement. Corollary 41 P¨ S (0) = −4Tr R2m (Σ (S) Σ (E) ) +



s j j e j j .

(13.17)

1≤ j, j  ≤m

We remark that on the right-hand side in formula (13.17), the first term is classical and the second term is a quantum correction, which vanishes if the coupling of the system and its environment is only in position (or momentum) variables (as it is usually in the physics literature). Assume now that S j (z) = z j and E j (u) =



G j, u ,

1≤ ≤2N

where 1 ≤ j ≤ 2d. Then G j, is a (2d × 2N )-matrix with real coefficients. With these notations, for the quantum correction we have qc =



s j  j e j  j = Tr R2d (JS G JE G  ),

(13.18)

1≤ j, j  ≤2d

where JS , JE are, respectively, the matrix of the canonical symplectic form in R2d and in R2N : σ S (z, z  ) = z · JS z  , σ E (u, u  ) = u · JE u  (the scalar products are denoted by a dot). For the “classical part” we have

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13 Open Quantum Systems and Coherent States

  − 4Tr R2m Σ (S) Σ (E) = −4tr G  CovρS GCovρ E .

(13.19)

  P¨ S (0) = −4tr R2d CovπS GCovπ E G  + Tr R2d JS G JE G 

(13.20)

Corollary 42

where Covπ is the covariance matrix for the quasiprobability π . Recall here that the Wigner function of a pure state ψ is nonnegative if and only if ψ is a Gaussian by the Hudson theorem (see Chap. 2). We get now a more explicit formula for the quantum Brownian motion or a variant of the quantum Brownian motion. Let us assume that V (x, ξ ; y, η) = x ·



cj yj +

1≤ j≤N

+ξ ·







c j+N η j

1≤ j≤N

dj yj +

1≤ j≤N



 d j+N η j .

1≤ j≤N

Then the quantum correction qc is obtained by the formula  qc = 2(d1 c N +1 + · · · + d N c2N ) − 2(c1 d N +1 + · · · + c N d2N ) . In the quantum Brownian motion, it is assumed that the initial data for the environment is a thermal state:

  (y 2j + η2j ) , 0 < τ < 1. ρτ,E (y, η) = (2τ ) N exp − τ 1≤ j≤N

Corollary 43 Assume that d = 1, c j = 0 for j ≥ N + 2, and d j = 0 for j ≥ 2, ψ S ∈ S (R), and that ρ E (0) = ρτ,E is a thermal state. Then we have 2 2 (c + · · · + c2N + c2N +1 )Covx,x (ψ0 ) P¨ S (0) = τ 1 + 2c1 d1 Covx,ξ (ψ0 ) + d12 Covξ,ξ (ψ0 ) +

c +1 d1  N  

(13.21)

quantum correction

where Cov(ψ S ) is the (2 × 2) covariance matrix of the Wigner function of ψ S and Cov(ψ S ) =

 Covx,x (ψ S ) Covx,ξ (ψ S ) . Covx,ξ (ψ S ) Covξ,ξ (ψ S )

13.2 Computation of the Purity

383

So we see a quantum correction term appearing when the coupling mixes positions and momenta variables. When d1 = 0, we recover a known formula for the quantum Brownian motion.

13.2.2 Proofs of the Statements of Sect. 13.2.1 13.2.2.1

Proof of Theorem 70

It is easier here to consider the interaction formulation of quantum mechanics to  “eliminate” the “free” (noninteracting) evolution: U0 (t) = e−it H0 . We introduce the inter-acting evolution (t)U0 (t), ρ (I ) (t) = U0 (t)∗ ρ We easily see that 



ρ (I ) (t)) = eit HS ρ S (t)e−it HS , ρ S(I ) (t) := Tr E ( ρ SI (t)2 ). We shall use the simpler notation  r (t) = ρ S(I ) (t). Using and P S (t) = Tr S ( cyclicity of the trace, we obtain r (t) r˙ (t)), P˙ S (t) = 2Tr S (  P¨ S (t) = 2 Tr S ( r (t) r¨ (t)) + Tr S ( r˙ (t))2 .

(13.22) (13.23)

We have, at time t = 0, where V = V (0), , ρ  r˙ (0) = i −1 Tr E [V S ⊗ ρ E ], −1 ˙ , ρ , [V , ρ  r¨ (0) = i Tr E [V S ⊗ ρ E ] − Tr E [V S ⊗ ρ E ]]. Notice that



˙ ,ρ  S ⊗ ρ E ] ρ S = 0. Tr S Tr E [V

(13.24) (13.25)

(13.26)

This follows from the definition of the relative trace Tr E and the cyclicity of the trace. Hence we get



˙ ,ρ ˙ ,ρ   S ⊗ ρ S ⊗ ρ E ] ρ S = Tr [V E ] ρ S ⊗ 1 = 0. Tr S Tr E [V Finally we get formula (13.15) using the identity , [V , ρ 2 ρ 2 − 2 V ρ . [V ]] = V + ρ V V



384

13.2.2.2

13 Open Quantum Systems and Coherent States

Proof of the Corollaries

They are easily obtained after standard computations for traces of operators starting with formula (13.15).

13.3 Separability and Entanglement 13.3.1 Statements We assume in this section that H S = L 2 (Rd )

and

H E = L 2 (R N ),

and we use here properties of Weyl quantization (see Chap. 2 for properties used here). We have to consider partial transposition. In the complex Hilbert space L 2 (Rm ), we define the bilinear form  u·v =

u(x)v(x) d x. Rm

 is a bounded operator on L 2 (Rm ), its transpose A  is defined by the formula If A  ·v =u· A  v, u, v ∈ L 2 (Rm ). Au  is given by A (x, ξ ) = A(x, −ξ ) and We easily see that the Weyl symbol A of A  is a quantum state (a density matrix), then A  is also a quantum state. if A ( j)

Now we define the partial transpose in the environment space. Let ψ S ∈ H S and ( j)  (1) (2) in H E such ψ E ∈ H E , j = 1, 2. There exists a unique bounded operator A ψ S ,ψ S that  S(1) ⊗ ψ E(1) ) = ψ E(2) · A  (1) (2) ψ E(1) . (ψ S(2) ⊗ ψ E(2) ) · A(ψ ψ ,ψ S

S

 is a unique bounded operator A E in Definition 31 The partial transposition of A H E satisfying HS ⊗



E ψ S(1) ⊗ ψ E(1) = ψ E(1) · A  (1) (2) ψ E(2) , ψ S(2) ⊗ ψ E(2) · A ψ ,ψ S

( j) ψS

∈ HS ,

( j) ψE

S

∈ HE .

(13.27)

Proposition 133 ([156]) Let ρ  be a separable density matrix in H = H S ⊗ H E . Then the partial transpose ρ E is again a density matrix in H .

13.3 Separability and Entanglement

385

Using anti-Wick quantization (see Chap. 2), we get a sufficient condition for separability.  be a mixed Proposition 134 Assume that H S = L 2 (Rd ) and H E = L 2 (R N ). Let ρ  admits an anti-Wick symbol defining a Radon probstate in L 2 (Rd+N ) such that ρ  is separable. In particular, if ρ  = ρ (Γ,m) is a ability measure in R2(d+N ) . Then ρ Gaussian mixed state with covariance matrix Γ , mean m, and such that Γ ≥ 21 , then ρ (Γ,m) is separable. A remarquable result is the following reverse statement for Proposition 133 in [156, 236].  be a density matrix Theorem 71 Assume that H S = L 2 (R), H E = L 2 (R N ). Let ρ is separable if and only if its partial with a Gaussian Weyl symbol ρ = G (Γ,m) . Then ρ transpose ρ E is a density matrix. Theorem 71 was used in [92] to prove separability results for propagation of Gaussian states in the quantum Brownian motion (13.2), (13.3). We shall give later the details for a mathematical proof of these results. Recall that a thermal state at temperature β1 for the Hamiltonian HE is the mixed state defined as e−β HE Tβ,E = . (13.28) Tr(e−β HE ) ρ (0) Ug (t, 0), the time evolution of the initial density We denote ρ g,β (t) = Ug∗ (t, 0) matrix ρ (0) = ρ S (0) ⊗ Tβ , by the Hamiltonian (13.2), (13.3). Theorem 72 For any quantum Brownian motion (13.2), (13.3) with ω S > 0, ω j > 0, 1 ≤ j ≤ N , we have the following results. (A) Assume that the coupling g is time independent. Then for any quantum Brownian motion model with ω j > 0, 1 ≤ j ≤ N + 1, sufficiently small (see (13.32)), there exists a Gaussian initial density matrix ρ S (0) and a temperature β1 > 0 such that ρ g,β (t) is separable for any t ∈ R. (B) If furthermore g(0) = 0 then, for any pure state ρ S (0) and for any temperature 1 > 0 there exists ε > 0 such that  ρ (t) is not separable for any t ∈]0, ε[. g,β β Remark 79 The results (A) and (B) of Theorem 72 were stated in [92]. A mathematically rigorous proof of (B) was given in [220].

13.3.2 Proofs of the Separability Results First we recall some properties concerning Gaussian density matrices. We recall the definition.  in the Hilbert space L 2 (Rn ) is said to be Gaussian if Definition 32 An operator G its Weyl symbol G is a Gaussian in the phase space R2n , denoted by G Γ,m where Γ

386

13 Open Quantum Systems and Coherent States

is the covariance matrix (a positive definite (2n × 2n)-matrix) and m ∈ R2d is the mean of G. So we have G (Γ,m) (z) = cΓ e− 2 (z−m)·Γ 1

−1

(z−m)

, z = (x, ξ ) ∈ R2n ,

(13.29)

where cΓ = det(Γ )−1/2 . Some condition is needed on the covariance matrix Γ such that (13.29) defines a nonnegative operator in L 2 (Rn ) (see Remark 80). Here J is the matrix of the symplectic form on Rn × Rn and X = (x, ξ ) ∈ R2n . Proposition 135 G Γ,m (X ) defines a density matrix if and only if Γ + i 2J ≥ 0, or equivalently, if and only if all the symplectic eigenvalues of Γ are greater than 21 . Moreover, up to conjugation by a unitary metaplectic transform in L 2 (Rn ), ρ Γ,0 is a product of one degree of freedom thermal states Tβ (see Remark 80). Next, ρ Γ,m (X ) determines a pure state if and only if 2Γ is positive and symplectic or equivalently if 2Γ = F F  , where F is a linear symplectic transformation. For an elementary proof of Proposition 135 we refer to [220]. From these results we can compute the purity (hence the linear entropy): PGΓ,m = (2π )

−n



G Γ,m (z)2 dz = 2−n (detΓ )−1/2 .

(13.30)

R2d

Remark 80 From Proposition 135 we claim that the environment state ρ τ,E is nonnegative if and only if τ ≤ 1. We can give a direct proof of this. It suffices to prove the claim for N = 1. By a holomorphic continuation argument in τ , we can see that, for τ > 1, ρ τ,E is negative on the subspace of L 2 (R) spanned by the odd Hermite functions.

13.3.2.1

Proof of Propositions 133 and 134

To prove Proposition 133 (and the necessity part of Theorem 71), we remark that E )E = ρ S ⊗ ( ρ E ) and that for any ψ E ∈ H E we have ψ E , ρ E ψ E = ( ρS ⊗ ρ E ¯ ¯ E ψ E . So we see that ρ  is a nonnegative operator. It is easy to check that it ψ E , ρ is in trace class so it is a density matrix.  The proof of Proposition 134 results from properties of coherent states. Recall the notation of Chap. 1 for coherent states ϕz . Here we denote z = (z S , z E ), z S ∈ R2d , z E ∈ R2N . So from the definition of coherent states we have ϕz = ϕz S ⊗ ϕz E and Πz = Πz S ⊗ Πz E , where Πz is the orthogonal projection to ϕz . If ρ aw is the antiWick symbol of ρ , then we have  (Πz S ⊗ Πz E )ρ aw (dz S , dz E ).

ρ = R2(d+N )

13.3 Separability and Entanglement

387

Usually it is assumed that an anti-Wick symbol is L ∞ . But the definition can be extended to finite Radon measures on the phase space.  of ρ satisfies Assume now that ρ = ρ Γ . The Fourier transform ρ Γ

ρ (Y ) = e− 2 Y ·Y = ρ aw (Y )e− 2 Y ·Y . 1

So ρ aw defines a Schwartz distribution if and only if Γ ≥ 21 . If this condition is satisfied, ρ aw is a finite Radon measure. Hence we have proved Proposition 134. 

13.3.2.2

Proof of Theorem 71

Here ρ = G (Γ,m) . It suffices to assume that m = 0 and denote G (Γ ) = G (Γ,0) . If G (Γ ) is separable, we saw in Proposition 133 that G (Γ ),E is a density matrix. The reverse property is more difficult to prove. For simplicity, we give the details only for N = 1 following [236]. The general case is proved by induction on N (see [258]) The coordinates in the phase space for X ∈ R4 are denoted as follows: X = (x1 , ξ1 , x2 , ξ2 ), and the matrices are written as 4 blocks of (2 × 2)-matrices:

 A C Γ = , C B

J=

 J1 0 , 0 J1

 0 1 where A, B are symmetric, J1 = . −1, 0 We know that Γ + 2i J ≥ 0 (Proposition 135) so we have A ≥ 12l and B ≥ 12l . To prove that G (Γ ) is separable, we have to prove that Γ ≥ 21 (Proposition 134). From the assumption we know that G (Γ ),E is a mixed Gaussian state with the covariance matrix 



1 0 A Cσ E ; σ = . = Γ 0 −1 σ C  σ Bσ So we also have Γ E + 2i J ≥ 21 . We remark here that separability for states defined in L 2 (R2 ) is preserved by the metaplectic associated with the symplectic transformations of 

transformations S1 0 , where S1 , S2 are (2 × 2)-symplectic matrices. Matrices like the form S = 0 S2 these constitute a subgroup, denoted by Sp1,1 (R), of the symplectic group Sp2 (R) of R2x1 ,ξ1 × R2x2 ,ξ2 . So it suffices to prove that there exists S ∈ Sp1,1 (R) such that S  Γ S ≥ 21 . This is proved in two steps. A, B being real and symmetric, they can be

388

13 Open Quantum Systems and Coherent States

diagonalized by rotations R1 , R2 . So there exits S1 = S1 Γ

 R1 0 such that 0 R2

 a1l2 C  S1 = Γ1 := . C , b1l2

But we have C  = E R3 , where E is symmetric  and R3 is a rotation. Then there exist R1 0   , we have rotations R1 , R2 such that if S2 = 0 R2 ⎛

a ⎜ 0 S2 Γ S2 = Γ1 := ⎜ ⎝c1 0

0 a 0 c2

c1 0 b 0

⎞ 0 c2 ⎟ ⎟ 0⎠ b

with a, b ≥ 21 , c1 , c2 ∈ R. Assume that detC > 0. We may assume that c1 > 0, c2 > 0. symplectic transAt the second step we conjugate Γ1 by a family of 

squeezing r 0 , r, s ∈]0, ∞[. Choosformations Σ(r, s) = Σ(r ) ⊗ Σ(s), where Σ(r ) = 0, r1

1/2 2 +bc2 ing rs = θ := ac , we can find a symplectic rotation R in R4 such that ac1 +bc2 Γ2 := Σ(θ s, s) Γ1 Σ(θ s, s) can be diagonalized. Then we can choose s such that the smallest eigenvalue of Γ2 is greater than 21 . So we get Theorem 71 if detC > 0. A slight modification is needed if detC = 0. Now if detC < 0, we have det(C E ) > 0,  then G (Γ )E is separable so G (Γ ) is separable.

13.3.3 Proof of Theorem 72 (A) Assuming the coupling constants g and β sufficiently small, we have to find γ ,β (t) of the an initial Gaussian density matrix G S such that the time evolution ρ V,β (t) is initial state G S ⊗ Tβ,E stays separable at any time. It is well known that ρ a Gaussian state (for computations see Sect. 13.4). Denote by Γg,β (t) the covariance E E (t) the covariance matrix of ρ g,β (t). matrix of ρ g,β (t) and by Γg,β If Λ E denotes the (2N + 2) × (2N + 2)-matrix defined by the reflection (y1 , y  , η1 , η ) → (y1 , y  , η1 , −η ), for y1 , η1 ∈ R, y  , η ∈ R N , denote M E = Λ E MΛ E ,

13.3 Separability and Entanglement

389

where M is any (2N + 2) × (2N + 2)-matrix. According to Theorem 71 and Proposition 135, we have to prove that E (t) + Γg,β

i i J ≥ 0, or Γg,β (t) + J E ≥ 0. 2 2

(13.31)

Let us introduce the full potential, where y1 = x and y  = (y2 , y3 , . . . , y N +1 ), 1 Wg (y1 , y ) = 2 



 ω2j y 2j

+ y1

1≤ j≤N +1



gj yj.

2≤ j≤N +1

Wg is a quadratic form on R N +1 . Its matrix is denoted by Mg . It is positive definite if the following condition is satisfied: |g j | < 1. 2≤ j≤N +1 2ω1 ω j max

(13.32)

Hence there exists a rotation R in R1+N such that R Mg R  := Dg is a diagonal matrix with eigenvalues  ω2j , 1 ≤ j ≤ N + 1.

 R 0 So if R := , the quadratic form HV is transformed in a sum of independent 0 R oscillators: 

 1 2 2 2  (13.33) H (x, ξ ) := H (R(x, ξ )) = ξ + ωjxj . 2 1≤ j≤N +1 j V (V = Vg ) is denoted by Tδ H and can be The thermal state at temperature 1δ of H computed by using Remark 80. Denote τ (u) = tanh(u/2), u ∈ R. We have Tδ H (x, ξ ) = 2





1 2 2 τ (δ ω j ) exp − τ (δ ωj)  ω j x j + ξ j . (13.34)  ωj 1≤ jk≤N +1 1≤ j≤N +1

−1−N



If Γδ H denotes the covariance matrix of the Gaussian state Tδ H , elementary estimates show that we have ω j ), for 1 ≤ j ≤ 1 + N , 2Γδ H ≥ 1l2+2N if 0 < δ ≤ ( where (u) =

1 u

ln

1+u |1−u|



if u = 1 and (1) = ∞. Using the relation Γδ H = RΓδ H R ,

we get (13.35) for Γδ H .

(13.35)

390

13 Open Quantum Systems and Coherent States

Denote by Φ Ht V the linear flow defined by the quadratic Hamiltonian HV . Then , where we have Γg,β (t) = Φ Ht V Γg,β (0)Φ Ht, V 

S Γ0 0 . Γg,β (0) = 0 Γβ H E So we write Γg,β (t) +

i E i J = Φ Ht V (Γg,β (0) − Γδ HV )Φ Ht, + Γδ HV + J E . V 2 2

(13.36)

To conclude, we see that it is possible to choose the (2 × 2)-symmetric matrix Γ0S and β such that the symmetric matrix E g,β,δ := Γg,β (0) − Γδ H is diagonally dominant [30]. Hence, using the Gersgorin Theorem [30]  and (13.35), we see that E g,β,δ is nonnegative hence we get (13.31). (B) Now we assume that ρ S (0) is a pure state. Then Γ0S is a symplectic symmetric matrix. Here it suffices to assume that N = 1. With the notation of part (A), let us denote Γt := Γg,β (t), we have to prove that the Hermitian matrix 2Γt + i J E has at least one negative eigenvalue for 0 < t < ε if g(0) = 0. Equivalently, we have to prove that the smallest eigenvalue, λmin (t), of 4|i J E Γt |2 is smaller than 1 for t ∈]0, ε[. 

At C t . The eigenvalues of 4|i J E Γt |2 satisfiy Recall that Γt = Ct Bt λ4 − 4λ2 (det At + det Bt − 2 det Ct ) + 16 det Γt = 0. So we have



 λmin (t)2 = 2 dt − dt2 − 4γt =: 2v(t),

(13.37)

where dt = at + bt − 2cc , at = det At , bt = det Bt , ct = det Ct , γt = det Γt . In particular, λmin (0) = 1. The initial data Γ0 is chosen here as the diagonal matrix Γ0 = Diag

1 s , , sω2 2 2r ω2

r

,

where r > 0, s > 1, ω2 > 0. The parameter s is related to the temperature of the thermal state: s = coth(βω2 /2). We are going to prove that v˙ (0) = 0 and v¨ (0) < 0. In [92] it was claimed that v˙ (0) < 0, which is not correct.

13.3 Separability and Entanglement

391

Recall here the Liouville formula to compute the derivative of the determinant for an invertible matrix M(t), smooth in time t:

d d det(M(t)) = det(M(t))Tr M(t)−1 (M(t) . dt dt

(13.38)

The time evolution of Γt is given by the classical flow determined by the quadratic classical Hamiltonian HV (X ) = 21 X · S(t)X , where S(t) is a symmetric matrix. Denote by Φ t this flow. We have Φ˙ t = J S(t)Φ t and Γt = (Φ t ) Γ0 Φ t .

(13.39)

As a consequence of (13.37) and (13.39), we can compute v˙ (0) and v¨ (0). Denote Sˇ := J S, Γ = Γ (0), S = S(0). We compute ˇ − Γ Sˇ  , Γ˙ = − SΓ ˇ Sˇ  + Sˇ 2 Γ + Γ ( Sˇ  )2 . Γ¨ = 2 SΓ

(13.40) (13.41)

For the first derivative we get ⎛

0 ⎜ r ω12 − 2 Γ˙ = ⎜ ⎝ 0 κ r2

1 2r

r 2 ω 2 1

− 0

1 2r

κs 2ω2

0 κ r2

κs 2ω2

0 0

0

⎞

⎟ ⎟. 0⎠ 0

(13.42)

From (13.42) using (13.38) and (13.37), we deduce easily that d˙ = γ˙ = 0, hence v˙ = 0. For the second derivatives similar computations give: a¨ = 2κ 2

rs rs rs , b¨ = κ 2 , c¨ = κ 2 , ω2 2ω2 2ω2

We get finally rs v¨ = κ ω2 2

3 4s 2 − 2s + 2 − 2 s2 − 1

γ¨ = −κ 2  ≤ −3.9

rs (s 2 − 2s − 1). 16ω2 (13.43) rs 2 κ . ω2

(13.44)

Hence t → λmin (t) is strictly concave at t = 0 (see the following figure). Morever, in this figure we see that v(t) oscillates: above 0.5 the state is separable, below 0.5 the state is entangled. 

392

13 Open Quantum Systems and Coherent States

Time dependent oscillations between separability and entanglement.

13.4 The Master Equation in the Quadratic Case 13.4.1 General Quadratic Hamiltonians In this section we get a time dependent partial differential equation (often called the master equation) satisfied by the reduced density matrix ρ S of the open system (S). Moreover, this equation can be solved explicitly using the well-known characteristics method. The quantum Brownian motion model with bilinear position coupling was considered in particular in [103, 106, 136, 158]. Most of papers use the path integral method. In [85, 86] the authors put emphasis on a Fokker–Planck type equation. Our aim here is to give a different mathematical presentation, with straightforward general computations for the time evolution of the reduced density matrix ρ S (t). S and H E , Here we assume that HS and HE are the quadratic Weyl symbols of H respectively. We denote by Φ0t the Hamiltonian flow generated by H0 , and by Φ t the Hamiltonian flow generated by H = H0 + V (t). The symbol V (t) stands for a C 1 -time dependent bilinear coupling between the system and the environment: V (t, z, u) = z · G(t)u, where G(t) is a linear map from R2N to R2d (note that the coupling can mix positions and momenta variables). Recall that R2d (respectively R2N ) is the classical phase space of the system (respectively, the environment). Next, Ψ t = Φ0−t Φ t is the interacting flow in the global phase space R2(d+N ) . The phase space of the global system is identified with the direct sum R2d ⊕ R2N and in this decomposition the flows are represented by 4 matrix blocks

13.4 The Master Equation in the Quadratic Case

Ψt =

Ψiit Ψiet Ψeit Ψeet

 and Ψ0t =

393



t ΦS 0 , with Ψ00 = Ψ 0 = 1lR2d+2N . 0 Φ Et

The interacting classical Hamiltonian is HI (t, z, u) = V (t, z, u) = z · G (I ) (t)u, where G (I ) (t) = (Φ St ) · G(t) · Φ Et . The classical interacting evolution is given by the equation 2 V (t)Ψ t , with J = Ψ˙ t = J ∇z,u

 JS 0 , 0 JE

2 V (t) is the Hessian matrix in variables (z, u) ∈ R2d × R2N . So for the where ∇z,u block components of the interacting dynamics we have

Ψ˙ iit = JS G(t)Ψeit , Ψ˙ iet = JS G(t)Ψeet , Ψ˙ eit = JE G(t) Ψiit , Ψ˙ eet = JE G(t) Ψiet .

(13.45) (13.46)

Since all the Hamiltonians considered here are quadratic (eventually time dependent), they generate well-defined quantum dynamics in Hilbert spaces L 2 (Rn ) where n = d for the system (S), n = N for the environment (E)), and n = d + N for the global system (S ∪ E). Recall the notations 







U (t) = e−it H , U0 (t) = e−it H0 , U S (t) = e−it HS , U E (t) = e−it HE . We have U0 (t) = U S (t) ⊗ U E (t) and the quantum interacting dynamics U I (t) is U0 (t)∗ U (t). At time t = 0 we assume that the states of the system and of the environment are E (0), where ρ S (respectively, ρ E ) is a density matrix decoupled: ρ (0) = ρ S (0) ⊗ ρ in the Hilbert space H S = L 2 (Rd ) (respectively in H E = L 2 (R N )); ρ, ρ S , ρ E are the Weyl symbols (i.e., the Weyl–Wigner functions, with  = 1, of the corresponding density matrices). The coefficients of the quadratic forms HS , HE , V may be time dependent. In this case U (t) means U (t, 0), where U (t, s) is the propagator solving i

∂ (t)U (t, s), U (s, s) = 1lH . U (t, s) = H ∂t

We denote U (t) := U (t, 0). It is well known that for quadratic Hamiltonians, every mixed state ρ  of (S ∪ E) propagates in accordance with the classical evolution (see Chap. 3): ρ (0)U (t), ρ(t, X ) = ρ(0, Φ t X ), ρ (t) := U ∗ (t)

X ∈ R2(d+N ) .

(13.47)

394

13 Open Quantum Systems and Coherent States

Our aim is to compute ρ S (t) = Tr E ( ρ (t)). The Weyl symbol of ρ S (t) is given by the following integral:  ρ S (t, z) =

ρ(t, z, u) du.

(13.48)

R2N

In particular, if ρ(0) is Gaussian in all the variables, X = (z, u) ∈ R2d × R2N , then ρ S (t, z) is Gaussian in z. Nevertheless, a direct computation on the formula (13.48) seems not easy to perform, so a different strategy will be used.

13.4.2 Time Evolution of Reduced Mixed States Here we state and prove the main results of this section. It is convenient to work in the interacting setting. Recall that we have ρ S (0) U S (t)), hence ρ S(I ) (t, z) = ρ S (t, Φ St z). ρ S(I ) (t) = U S∗ (t)

(13.49)

Theorem 73 Let tc > 0 be the largest time tc such that Ψiit is invertible for every E (0) is a Gaussian with t ∈ [0, tc [. Assume that the environment density matrix ρ mean m E . Then there exist two time-dependent (2d × 2d)-matrices A(I ) (t) and B (I ) (t), and a time dependent vector v(I ) (t) ∈ R2d such that for t ∈ [0, tc [ and for S(I ) (t) every ρ S (0) in S (R2d ), the Weyl symbol ρ S(I ) (t, z) of the interacting evolution ρ of the system (S) satisfies the following master equation (a Fokker–Planck type equation): ∂ (I ) ρ (t, z) = (A(I ) (t)∇z ) · zρ S(I ) (t, z) + (B (I ) (t)∇z ) · ∇z ρ S(I ) (t, z) ∂t S + v(I ) (t) · ∇z ρ S(I ) (t, z).

(13.50)

Moreover, we have the following formula to compute A(I ) (t) and B (I ) (t): A(I ) (t) = −JS G(t)Ψeit (Ψiit )−1 , B (I ) (t) =

L(t) + L(t) , 2

(13.51)

where  L(t) = JS · G(t) Ψeet − Ψeit (Ψiit )−1 Ψiet Covρ E (Ψiet ) , (I )

v (t) =

−JS G(t)Ψeet m E .

(13.52) (13.53)

13.4 The Master Equation in the Quadratic Case

395

Proof In this proof (and only here) we shall erase the upper index (I ) for the interacting dynamics. Taking the partial trace in Eq. (13.5), we have  ρ˙ S (t, z) =

{V (t), ρ(t)}(z, u) du. R2N

For simplicity we shall assume that m E = 0. It is not difficult to take this term into account. The Poisson bracket is taken in the variables (z, u) but due to integration in u we have only to consider the Poisson bracket in z. Hence we get  ρ˙ S (t, z) = −

JS G(t)u · ∇z ρ(0, Ψ t (z, u)) du.

(13.54)

R2N

Denote by  f (ζ ) the Fourier transform of f in the variable z. Then we have  ρ˙ S (t, ζ ) = iζ ·



JS G(t)uρ(0, Ψ t (z, u)e−i z·ζ dz du.

R2(d+N )

Now we perform the symplectic change of variables (z  , u  ) = Ψ t (z, u),  ρ˙ S (t, ζ ) = iζ  t  t  × dz  du  JS G(t)(Ψeit z  + Ψeet u  )ρ S (z  )ρ E (u  )e−iζ ·(Ψii z +Ψie u ) .

(13.55)

R2(d+N )

Denoting ϕ = Ψiit z  + Ψiet u  and using the identity i(Ψiit )−1 ∇ζ e−iζ ·ϕ = (z  + (Ψiit )−1 Ψiet u  )e−iζ ·ϕ , we get  S (t, ζ ) ρ˙ S (t, ζ ) = −ζ · JS G(t)Ψeit (Ψiit )−1 ∇ζ ρ  −iζ · dz  du  JS G(t)Ψeit (Ψiit )−1 Ψiet (u  )ρ E (u  )ρ S (z  )e−iζ ·ϕ (13.56)

R2(d+N )



+ iζ ·

dz  du  JS G(t)(Ψeet u  )ρ E (u  )ρ S (z  )e−iζ ·ϕ .

R2(d+N )

To absorb the linear terms in u  , we use the fact that ρ E is Gaussian, ρ E = cΛ e−1/2u·Λu , where Λ is a positive definite (2N × 2N )-matrix and cΛ a normalization constant. So we have Λ−1 ∇u ρ E (u) = uρ E (u), and integrating by parts

396

13 Open Quantum Systems and Coherent States

we have ˙ S (t, ζ ) = −ζ · A(I ) (t) ∇ζ ρ S (t, ζ ) − (ζ · L(t)ζ ) ρ S (t, ζ ). ρ 

(13.57)

We get (13.50) by taking inverse Fourier transforms. Corollary 44 In the notation of Theorem 73, the reduced density matrix for the system satisfies the following master equation: ∂ ρ S (t, z) = {HS , ρ S (t)}(z) ∂t + (A(t)∇z ) · zρ S (t, z) + (B(t)∇z ) · ∇z ρ S (t, z)

(13.58)

+ v(t) · ∇z ρ S (t, z), for t ∈ [0, tc [. Remark 81 The function ρ S (t, z) is of course well defined for every time t ∈ R but the coefficients of the master equation (13.50) may have singularity at t = tc . Denote γ = supt≥0 G(t) and ϕ(t) = Φ St  + Φ Et . Proposition 136 If sup ϕ(t) < +∞ then there exists c > 0 such that tc ≥ gc . t≥0

If there exist C, δ > 0 such that ϕ(t) ≤ Ceδt for every t > 0, then there exists C1 ∈ R such that 1 1 tc ≥ log( ) + C1 . δ g The constants δ and C1 are independant of the coupling strength g. Proof It suffices to work in the interaction representation. Using interacting time evolution Ψ t for the total classical system (S ∪ E), we get t Ψ − 1l ≤ g

t ϕ(s)ds + g

t

0

Denote φ(t) = get

t

ϕ(s)Ψ s − 1lds. 0

ϕ(t)ds. Using the Gronwall lemma and integrating by parts, we

0

Ψ t − 1l ≤ gφ(t) +

g3 φ(t)3 egφ(t) . 6

(13.59)

From inequality (13.59), we easily get the proposition. Remark 82 Explicite computations on the model of coupled harmonic oscillators show that we may have tc < +∞ or tc = ∞. Assume that d = N = 1, V (x, y) = E + V , = H S + H gx y, H

13.4 The Master Equation in the Quadratic Case

S = 1 (−∂ 22 + ω S 2 x 2 ), H x 2

397

E = 1 (−∂ 22 + ω E 2 y 2 ). H y 2

If ω2S = 1, ω2E = −1 and g = 0, then 0 < tc < +∞. If ω2S = 1 = ω2E and 0 ≤ g ≤ √1 , then tc = +∞. 2 Remark 83 The coefficients of equation (13.50) are related to the first and second moments of the reduced density matrix ρ S (t). We denote m (Ij ) (t)

 =

z j ρ S(I ) (t, z) dz,

μ(Ijk) (t)

 =

R2d

z j z k ρ S(I ) (t, z) dz.

R2d

From (13.50), after computations using (13.46), we get m (I ) (t) = Ψiit m (I ) (0).

(13.60)

Computations of the second moments, by using (13.54), (13.46), and (13.50), give μ(I ) (t) =

1 t (I ) Ψii μ (0)Ψiit + Ψiet μ E (0)Ψiet , 2

(13.61)

where μ E (0) is the second moments matrix of ρ E (0). Remark 84 Suppose that the initial state of the total system (S ∪ E) is Gaussian: ρ(0, z, u) = cΓ e−(1/2)Γ

−1

(z,u)·(z,u)

, cγ > 0

is a normalization constant. Then we can compute ρ S (t, z). We get first the Fourier transform: ρ S (t, ζ ) = e−(1/2)Γ (t)(ζ,0)·(ζ,0) , where Γ (t) = (Φ t ) Γ (Φ t ). Using inverse Fourier transforms, we obtain −1

ρ S (t, z) = cγ ,t e−(1/2)ΓS

(t)z·z

(13.62)

where Γ S (t) is the matrix of the positive definite quadratic form Qt (ζ, ζ ) := Γ (t)(ζ, 0) · (ζ, 0) on R2d and cΓ,t = (2π )−d det 1/2 Γt . We can see on this example what is the meaning of the critical times tc . Assume that Γ = Γ S ⊕ Γ E , where the Γ S,E are positive definite quadratic forms on R2d , respectively, on R2N . We have easily: Q0 (ζ, ζ ) = Γ S ζ · ζ and Qt (ζ, ζ ) = Γ S (Φiit )ζ · (Φiit )ζ + Γ E (Φeit )ζ · (Φeit )ζ.

(13.63)

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13 Open Quantum Systems and Coherent States

We see from (13.63) that the initial state ρ S (0) cannot be recovered by its evolution at time tc : only the restriction of Q0 to (ker Φii )⊥ is recovered by Qtc . The physical interpretation is that a part of information contained in ρ S (tc ) has escaped in the environment represented here by Γ E . Now we consider more general initial ρ S ∈ S (Rd ) for the system (S). The master equation (13.50) can easily be solved by the characteristics method after a Fourier transform. As in the proof of Theorem 73, ρ S(I ) (t, ζ ) denotes the Fourier transform of ρ S(I ) (t). Theorem 74 Assume (for simplicity) that the mean of the environment state is 0. With the notation of Theorem 73, we have: 1

ρ S(I ) (t, ζ ) = ρ S (0, (Ψiit )ζ ) exp − ζ · Θ (I ) (t)ζ , t ∈ R, 2

(13.64)

where Θ (I ) (t) = Ψiet Covρ E (0) (Ψiet, ). Remark 85 The interpretation of the right-hand side in formula (13.64) is the following: the first factor is a transport term. The second term is a dissipation term due to the influence of the environment which is controlled by the nonnegative matrix Θ (I ) (t). In formula (74) for the Brownian quantum motion model the matrix Θ (I ) (t) is named the “thermal covariance” when the environment is in a thermal equilibrium state. This term determines a dumping factor for the state of the small system S exponentially t, 2 decreasing in the Fourier variable like e−γ |Ψie ζ | where γ > 0 is the smallest eigenvalue of the covariance matrix of ρ E . As far as Φiit is invertible, from (13.64) we see that the time evolution of ρ S is reversible but this is no longer true for t ≥ tc . From (13.64) we get an explicit representation formula for the reduced density of the system as a convolution integral. Corollary 45 With the notation of Theorem 74, we have 

) ρ S(I )(t, z) = c(I S (t)

1

 ρ S 0, Ψii(t,0) z  exp − (z −z  ) · Θ (I ) (t)−1(z −z  ) dz  , 2

R2d

 1

 ρ S (t, z) = c S (t) ρ S 0, Φii(t,0) z  exp − (z −z  ) · Θ(t)−1(z −z  ) dz  , 2 R2d

where

and

 −1/2 ) −d/2 c(I det(Ψiit )−1 det Θ (I ) (t) S (t) = (2π ) c S (t) = (2π )−d/2 det(Φiit )−1 det (Θ(t))−1/2 .

(13.65)

13.4 The Master Equation in the Quadratic Case

399

If det(Φiit ) = 0 or det((Φiet (Φiet ) ) = 0, the integrals in (13.65) are defined as convolutions of distributions. Formula (13.65) shows clearly the damping influence of the environment on the system because Θ(t) is a non negative matrix under our assumptions. If Θ(t) is  singular, then the Fourier transform of exp − 21 ζ · Θ(t)ζ is a distribution supported in some linear subspace of R2d where the damping occurs.

13.4.3 About the Schrödinger Cat Let us come back to the Schrödinger cat model 13.1.3: ψ cat = N1 (ϕz + + ϕz − ). Here we consider the quantum evolution of the cat state in a system of quantum oscillators like in the quantum Brownian motion: S = 1 (−∂ 22 + ω0 2 x 2 ), H x 2

 E = 1 H (−∂ y22 + ω j 2 y 2j ), j 2 1≤ j≤N

with the bilinear coupling defined by the Weyl symbol V (x, ξ, y, η) = (x, ξ ) · G · (y, η) , where G is a real 2 × 2N matrix, (x, ξ ) ∈ R2 , (y, η) ∈ R2N . The Hilbert spaces are here Hcat = L 2 (R), H E = L 2 (R N ) and for the whole system (S ∪ E), H = L 2 (R × R N ). The initial data for the system (cat ∪ E) is ρ(0) ˆ := ρˆcat ⊗ Tˆω,β for the inverse temperature β > 0 and frequences ω = (ω1 , · · · , ω N ). Notice that the Fourier transform of the Wigner function of the initial cat state is 

2 |ζ −J S z + |2 |ζ +J S z + |2 − |ζ4| + − − 4 4 . cos(ζ · z ) + e +e ρ˜cat (ζ ) = π 2e E has the following Weyl The thermal state Tˆω,β associated with the Hamiltonian H symbol ⎛ Tω,β (Y ) = Z (β, ω)−1 exp ⎝−

j=N 

⎞ tanh(βω j /2)(η2j + ω2j y 2j )⎠ ,

(13.66)

j=1

 where Z (β, ω) is a normalisation constant ( R2N Tβ (Y )dY = (2π ) N ). From (13.66) we see that the covariance matrice Γω,β of the Gaussian state Tω,β the block-diagonal matrix with the 2 × 2 blocks

400

13 Open Quantum Systems and Coherent States

Γω j ,β :=

−2  1 ωj 0 . 0 1 2 tanh(βω j /2)

Using Theorem 74 we can estimate the purity of the quantum superposition of the cat state. Proposition 137 Let us assume for simplicity that the rank of the coupling G is one and N = 1.

 g0 G= , g > 0. 00 Then there exists t1 > 0, C1 > 0 such that for any t ∈]0, t1 ], β = 0, ω1 = 0, g > 0, we have ω1 tanh(βω12 /2) (13.67) Tr(ρˆcat (t)2 ) ≤ C1 tg In particular for t > 0, small enough, the purity of ρcat (t) can be arbitrary small if the coupling strength or the temperature are chosen large enough. Proof Using Theorem 74 we need to compute Ψiet for t close the 0. From (13.45) we have Ψiet = t JS G + O(t 2 ). So the thermal covariance is approximate as Θ (I ) (t) = −t 2 JS G  Γω1 ,β G JS + O(t 3 ). Hence Θ(t) = −



t2 0 0 + O(t 3 ). 2 tanh(βω1 /2) 0 −g 2 ω12

(13.68)

Using Plancherel Theorem we have  Tr(ρˆcat (t) ) = 2

R2

ρ˜cat (t, ζ )2 dζ

Hence, using Theorem 74, the expression of G and that ρˆ S (0) is Gaussian, we get easily (13.67). . Remark 86 (i) From (13.67) we can get an estimate for the rank of ρˆcat (t): Rank[ρˆcat (t)] ≥

log τ −1 log 2

where τ is the r.h.s of (13.67) (using that ρˆcat (t) is a density matrix). In other words the initial coherent state of the cat is transformed into an incoherent mixture of a large number of independent states under the influence of the environment. (ii) It is not difficult to adapt the proof for any coupling G = 0.

13.4 The Master Equation in the Quadratic Case

13.4.3.1

401

Decoherence on the Density Matrix

Now we analyze the integral kernel Kcat (t, x, x  ) of the reduce density matrix ρˆcat (t) outside the diagonal for t > 0 small. A manifestation of decoherence is the fast decreasing of Kcat (t, x, x  ) for x = x  due to the influence of the environment (see many papers in the physics literature [155, 162, 231]). Here we fix  = 1 but we introduce coherent states with a small variance σ 2 > 0 in position x ∈ R. x2 So we introduce the coherent states ϕzσ = Tˆ (z)g σ where g σ (x) = π −1/4 σ −1/2 e− 2σ 2 . We consider here the following superposition of states of the Schrödinger states: ψσ,a (x) :=

σ ϕασ (x) + ϕ−α (x)  , Nσ,a

where α = (a, 0) ∈ R2 and Nσ,a is the normalization constant: 2 2 Nσ,a = 2(1 + e−a /σ ). Denote by Wσ,a the Wigner transform of ψσ,a . We have σ σ + W σ σ . Wσ,a (X ) = Wϕασ ,ϕασ + Wϕ−α ,ϕ−α ϕα ,ϕ−α

We need to compute the Wigner function of the pairs (ϕzσ , ϕzσ ). Using the computations of Chap. 2 (Proposition 16) it is not difficult to prove that, with X = (x, ξ ) ∈ R2 , Wϕzσ ,ϕ σ (X ) = Wz σ ,z σ (X σ ) = z



      1  z q + q  2 p + p  2 2  2 exp − 2 x − − σ ξ − − iσ (X − , z − z ) . σ 2  2  2

(13.69)

where z σ = (σ −1 q, σ p) if z = (q, p). Hence we get 1 Wσ,a (x, ξ ) = Nσ,a

 2 |x+a|2 |x|2 2 2 2 2 2 2 − |x−a| −σ ξ − −σ ξ − −σ ξ 2 2 2 e σ +e σ + 2e σ cos(2aξ ) .

(13.70) And the Fourier transform is:

 (ν+2a)2 (ν−2a)2 ν2 σ2 2 σ2 2 , W˜ σ,a (u, ν) = Cσ,a 2e− 4σ 2 − 4 u cos(au) + e− 4 u e− 4σ 2 + e− 4σ 2 (13.71) where Cσ,a = 2(1+e1−a2 /σ 2 ) . Now we compute the integral kernel of the density matrix. Let ρˆ be an arbitrary density matrix. Using computations of Chap. 2 and playing with the Fourier transform ρ˜ of ρ, we have the following formula for the integral ˆ kernel K ρ (x, x  ) of ρ:

402

13 Open Quantum Systems and Coherent States

K ρ (x, x  ) = (2π )−2

 R



x+x W˜ (u, x − x  )eiu· 2 du.

(13.72)

Plugging (13.64) in (13.72) and using (13.68) we get that for t > 0 the kernel K cat (t, x, x  ) of ρˆcat (t) has now a dumping multiplicative factor : D(t, x, x  ) = e−t

2

A|x−x  |2

g 2 ω2

where A = 2 tanh(βω1 1 /2) . In particular the dumping is stronger at high temperature T = β1 or for large coupling strength. To illustrate the off-diagonal dumping phenomenon let us compute K cat (0, x, x  ) √

1 x+x  2 π − (x−x2 )2 − 12 ( x+x2  +a)2 e 4σ e 4σ + e− 4σ 2 ( 2 −a) √σ

1  2 π − (x+x 2)2 − 12 (x−x  +2a)2 + e 4σ + e− 4σ 2 (x−x −2a) e 16σ σ diag andiag = K cat (0, x, x  ) + K cat (0, x, x  ).

K cat (0, x, x  ) =

(13.73)

So, for σ > 0 small enough, K cat (0, x, x  ) is concentrated in a neighborhood andiag of {(a, a), (−a, −a)} and K cat (0, x, x  ) is concentrated in a neighborhood of {(−a, a), (a, −a)}. So the dumping factor D(t, x, x  ) strongly decreases for t > 0, andiag T large, the anti-diagonal part K cat (0, x, x  ) which is responsible of the quantum interferences for the cat state and modifies weakly the diagonal part. Notice that the diagonal part is the kernel of the decoherent state 21 (Πa,0 + Π−a,0 ) (Πz is the projector on the coherent state ϕz ). This is a toy model for the decoherence phenomenom [198]. diag

13.4.3.2

Decoherence on the Wigner Function

We compute here the Wigner transform Wσ,a,t (x, p) of the density matrix ρˆcat (t), which is nothing else than ρcat (t, x, p). (here p is the momentum, denoted ξ before). We introduce the following notations. • • • •

N (σ, a) = 2(1 + e−a /σ ) f (σ, t) √ = 1 + 4 At 2 σ 2 ) σ (t) = 2σ f (σ, t)−1/2 √ N (σ, a, t) = N (σ, a) f (σ (t).

Then we have

2

2

13.4 The Master Equation in the Quadratic Case

403

σ 2 p2 |x−a|2 |x+a|2 1 (e− σ 2 + e− σ 2 )e− f (σ,t) N (σ, a, t) ⎞

 

 x2 σ (t)2 2 σ (t)2 a2 σ (t)2 ⎟ ⎟ + 2e− σ 2 e− 2 p cos ap exp − 1 − ⎟. ⎠ σ2 σ2 2σ 2    W (σ, a, t, x, p) =

(13.74)

decoherence factor

Let us make more explicit the decoherence factor. We have 

a2 exp − 2 σ

  σ (t)2 2 Aat 2 1− = exp − . 2σ 2 1 + 4 At 2 σ 2

So we see that the decoherence factor is exponentially small as Aat 2 is large and At 2 σ 2 is small. In this regime the oscillations of the cat state are strongly damped as we see on the following pictures.

t=0: full coherence for the density matrix, diagonal and non-diagonal peaks

t>0: the non-diagonal of the density matrix are decreasing when the interaction with the environment is switched on

404

13 Open Quantum Systems and Coherent States

t=0: full coherence for the Wigner quasi-probability with strong oscillations in the middle part

t>0: the quantum coherence for the Wigner quasi-probability in the middle part is decreasing when the interaction with the environment is switched on

Chapter 14

Adiabatic Decoupling and Time Evolution of Coherent States for Multi-component Systems

Abstract Coherent states are well adapted to understand adiabatic decoupling for multi-components systems like the Dirac operator or systems obtained by reduction in the Born-Oppenheimer approximation [58]. Hagedorn [130] has done an accurate analysis of this using a variant of coherent states. The new fact for systems is that now states are vectors valued in Cm for m ≥ 2 and the eigenvalues of the matrix of the system may cross so that one cannot diagonalize it smoothly. In the case when the eigenvalues are not crossing one can construct an adiabatic modification of the spectral projectors on the modes and compute the time evolution of coherent states following the same strategy as in the scalar case. Here we shall consider only non crossing cases where we can prove an adiabatic decoupling between the modes. For the more difficult crossing case we refer to the literature [130] where non-adiabatic transitions can be computed [101, 129] for more recent results.

14.1 Semi-classical Analysis for Multi-components Systems Until now we have assumed that the classical Hamiltonian H (t) is real valued. For systems with spin degrees of freedom we have to consider multi-components systems and so their classical counterparts are Hermitian matrix valued Hamiltoninan, depending on the semi-classical parameter . So we shall consider in this section more general classical Hamiltonians H (t, X ), taking their values in the space Cm,m of m × m complex matrices. | • | denotes a matrix norm on Cm,m . Let us introduce a suitable class of matrix observables of size m. Let recall that Z denotes the phase space Rnx × Rnξ and X = (x, ξ ) a generic point. Sometimes it will be denoted z = (q, p). Our aim in this Chapter is to analyze solutions of the Schrödinger equation i∂t ψ(t) = Hˆ (t)ψ(t), ψ(t0 ) = ϕz0 v0 ,

(14.1)

where ϕz0 is a coherent states living at z 0 ∈ Z , v0 ∈ Cm , Hˆ (t) is the -Weyl quantization of H (t), ψ(t) ∈ L 2 (Rn , Cm ). © Springer Nature Switzerland AG 2021 D. Robert and M. Combescure, Coherent States and Applications in Mathematical Physics, Theoretical and Mathematical Physics, https://doi.org/10.1007/978-3-030-70845-0_14

405

406

14 Adiabatic Decoupling and Time Evolution of Coherent …

Definition 33 We say that A ∈ Sm,ν , ν ∈ R, if and only if A is a C ∞ -smooth function on Z with values in Cm,m such that for every multiindex γ there exists C > 0 such that γ |∂ X A(X )| ≤ CX ν , ∀X ∈ Z .   Let us denote Sm,+∞ = ν Sm,ν . We have obviously ν Sm,ν = S (Z , Cm,m ) (the Schwartz space for matrix valued functions). These symbol spaces are equipped with the natural family of semi-norms (Fréchet spaces).  = (O p w A)ψ by the same For every A ∈ Sm,ν and ψ ∈ S (Z , Cm ) we can define Aψ  , ξ )ψ(y) formula as for the scalar case (m = 1) considered in Chap. 2. Now A( x+y 2 means the action of the matrix A( x+y , ξ ) on the vector ψ(y). Most of general prop2 erties already seen in the scalar case can be easily extended in the matrix case:  is a linear continuous mapping from Sm,ν into S (Rn , Cm ). 1. A → Aψ  and A  is a linear continuous mapping on S (Z , Cm,m ). 2.  A = A We have an operational calculus defined by the product rule for quantum observables. · Let be A, B ∈ S (Z , Cm,m ). We look for a classical observable C such that A   Some computations with the Fourier transform on the phase space give the B = C. following formula (see [153, 217]),  i σ (Dx , Dξ ; D y , Dη ) A(x, ξ )B(y, η)|(x,ξ )=(y,η) , C(x, ξ ) = exp 2 

(14.2)

where σ is the symplectic bilinear form introduced above. By expanding the exponent we get a formal series in : j  j  i σ (Dx , Dξ ; D y , Dη ) A(x, ξ )B(y, η)|(x,ξ )=(y,η) . C(x, ξ ) = j! 2 j≥0

(14.3)

We see that in general C is not a classical observable because of the  dependence of the series. It can be proved that it is a semi-classical observable in the following sense. We say that A is a semi-classical observable of weight ν and size m if there exists A j ∈ Sm,ν , j ∈ N, so that A is a map from ]0, 0 ] into Sm,ν satisfying the following asymptotic condition : for every N ∈ N and every γ ∈ N2n there exists C N > 0 such that for all  ∈]0, 1[ we have ⎞ ⎛ γ  ∂ supX −ν γ ⎝ A(, X ) −  j A j (X )⎠ ≤ C N  N +1 , (14.4) Z ∂ X 0≤ j≤N ˆ A0 is called the principal symbol, A1 the sub-principal symbol of A. The set of semi-classical observables of weight ν and size m is denoted by Sm,ν sc . Now we can state the product rule.

14.1 Semi-classical Analysis for Multi-components Systems

407

m,μ m,ν+μ Theorem 75 For every A ∈ Sm,ν and B ∈ S , there exists a unique C ∈ Ssc ·   with C such that A B=C  j C j . We have the computation rule j≥0

C j (x, ξ ) =

1 2j

 (−1)|β| β (Dxβ ∂ξα A).(Dxα ∂ξ B)(x, ξ ), α!β! |α+β|= j

(14.5)

where Dx = i −1 ∂x . In particular we have C0 = A0 B0 , C1 = A0 B1 + A1 B0 +

1 {A0 , B0 } 2i

(14.6)

where {R, S} is the Poisson bracket of the matrix observables R = {R j,k } and S = {S j,k } defined by the matrices equality    {R j, , S ,k } {R, S} = {R, S} j,k j,k , {R, S} j,k = 1≤ ≤m

Let us recall that the Poisson bracket of two scalar observables F, G is defined by {F, G} = ∂ξ F · ∂x G − ∂x F · ∂ξ G. Notations. We shall denote C := A  B (Moyal product) and A2 = A  A. The Moyal bracket [A, B] = A  B − B  A is the full symbol of the commuˆ B]. ˆ tator operator [ A, The sub-principal term of the Moyal bracket is given by the formula [A, B],1 = [A1 , B0 ] + [A0 , B1 ] +

1 ({A0 , B0 } − {B0 , A0 }). 2i

(14.7)

In particular if the matrices A, B are Hermitian then [A, B],1 is anti-Hermitian. A proof of this theorem in the scalar case, with an accurate remainder estimate, is given in the Appendix of [40]. This rather technical proof could be easily extended to the matrix case considered here. Let us recall also some other useful properties concerning Weyl quantization of observables. Detailed proofs can be found in [153, 217] for the scalar case and the extension to the matricial one is easy.  is bounded in L 2 (Z , Cm ) (Calderon-Vaillancourt theorem). • if A ∈ Sm,0 then A 2 m,m • if A ∈ L (Z , C ) then Aˆ is an Hilbert–Schmidt operator in L 2 (X, Cm ) and its Hilbert–Schmidt norm is  H S = (2π )−n/2  A

 Z

1/2 A(z)2 dz

where A(z) is the Hilbert–Schmidt norm for matrices.

,

408

14 Adiabatic Decoupling and Time Evolution of Coherent …

 is a trace-class operator. Moreover we have • if A ∈ Sm,ν with ν < −2n then A ˆ = (2π )−n tr( A)

 tr(A(z))dz.

(14.8)

Z

• If A, L ∈ L 2 (Z , Cm,m , then Aˆ · Lˆ is a trace class operator in L 2 (X, Cm ) and ˆ = (2π )−n tr( Aˆ · B)

 tr(A(z)B(z))dz. Z

The extension to the matrix case of the propagation of coherent states may be difficult if the principal symbol H0 (t, X ) has crossing eigenvalues. We shall not consider this case here (accurate results have been obtained in [132, 249]). We shall consider here two cases: firstly the principal symbol H0 (t, X ) is scalar and we shall write H0 (t, X ) = H0 (t, X )1lm where we have identified H0 (t, X ) with a scalar Hamiltonian; secondly H0 (t, X ) is a matrix with two distinct eigenvalues of constant multiplicity (like Dirac Hamiltonians). The general case of eigenvalues of constant multiplicities is no more difficult. Our goal in the following sections is to construct asymptotic solutions for the Schrödinger system i

∂ψ(t) (t)ψ(t), ψ(t = t0 ) = ψ0 , =H ∂t

(14.9)

where ψ0 ∈ L 2 (Rn , Cm ). Later we shall give more details when the initial state is a wave-packet.

14.2 Systems with a Scalar Leading Term We shall see that the coherent states analysis of the scalar case can be extended to multicomponent systems as far as the classical Hamiltonian for the principal symbol has eigenvalues of constant multiplicity (without crossing). A state ψ ∈ L 2 (Rn , Cm ) can be seen as a map (x, s) → ψ(x, s) from Rd × {1, . . . , m} into C. The variable s ∈ {1, . . . , m} corresponds to a spin variable in quantum mechanics. The Fourier-Bargmann is defined for u ∈ L 2 (Rn , Cm ), u = (u 1 , . . . , u m ), using the more convenient notation u s (x) = u(x, s), s ∈ {1, 2, . . . , m}, F B [u](z, s) = (2π )−n/2 gz , u s . It defines an isometry from L 2 (Rn , Cm ) into L 2 (R2n , Cm ) (injective but not surjective). Obviously we have the inversion formula

14.2 Systems with a Scalar Leading Term

409

 u s (x) =

Z

FB [u](z, s)ϕz (x)dz, in the L 2 − sense,

(14.10)

where ·, · is the scalar product in L 2 (Rn , Cm ). We shall also use the notation ϕz,v = ϕz v for (z, v) ∈ Z × Cm . Let us introduce the following assumptions. (SQ) H (t) is a semi-classical sub-quadratic matrix  Hamiltonian (continuously H j (t, X ) j and satisfies, for time dependent and -dependent) if H (t, X ) j + |α| ≥ 2 and t ∈ [t0 − T, t0 + T ]:

j≥0

|∂ Xα H j (t, X )| ≤ Cα,T , ∀X ∈ R2n and for any N ≥ 1 and |α| ≥ 2, ⎛ ⎞  α j ≤ C N ,α,T  N +1 , ∀X ∈ R2n ,  ∈ [0, 1]. ∂ ⎝ H (t, X ) − ⎠  H (t, X ) j X j≤N Recall here that H j (t, X ) is a Hermitian complex matrix. It follows that the -Weyl quantization of H (t) has a unique propagator U H (t, t0 ). Moreover in a first step we assume that the principal symbol H0 is a scalar matrix. (SCA) H0 (t, X ) = λ(t, X )1lm such that λ(t, X ) is a scalar subquadratic Hamiltonian in X . Hence the classical flow for λ(t, X ) exists for every initial data (t0 , z 0 ) ∈ R × R2n . We shall denote this flow Φλt,t0 z 0 := z(t, t0 , z 0 ) or with the shorter notation z t := z(t, t0 , z 0 ). The first new fact in this case is the contribution of the subprincipal term H1 (t, X ) in the propagation of states by the unitary propagator U H (t, t0 ) (spin-orbit interaction). The same remark has been done concerning propagation of observables for systems [9]. Let us begin with this last problem. Let us consider a matrix observable of size m, A ∈ Sm,0 and its quantum time ˆ t0 ) := U H (t0 , t) Aˆ U H (t, t0 ). evolution: A(t, Theorem 76 Assume hypothesis (SQ) and (SCA) are satisfied and let A ∈ Sm,0 . Then at any time t the observable ˆ t0 ) := U H (t0 , t) Aˆ U H (t, t0 ), A(t,

(14.11)

has a semi-classical symbol A(t, t0 ) ∈ Sm,0 sc . Its principal symbol A0 (t, t0 ) is given by (14.12) A0 (t, t0 ; X ) = R ∗ (t, t0 , X )A(Φλt,t0 (X ))R(t, t0 , X ), where R(t, t0 , X ) solves the equation

410

14 Adiabatic Decoupling and Time Evolution of Coherent …

∂t R(t, t0 , X ) = −i H1 (t, (Φλt,t0 (X ))R(t, t0 , X ), R(t0 , t0 , X ) = 1lm .

(14.13)

Proof The proof is similar to the proof given in Chap. 2 for the scalar case. We only compute here the principal symbol. Otherwise the details can be reproduced following the Chap. 2 (The semi-classical propagation Theorem). Using (14.6) we get for A0 (t, t0 ) the equation ∂t A0 (t, t0 ) = {A0 (t, t0 ), λ(t)} + i −1 [A0 (t, t0 ), H1 (t)].

(14.14) 

Using (14.13) we get easily that Eq. 14.14 is solved by formula (14.12).

Here the dynamics of the observable is driven by the classical flow distorted by a precession R acting on the components of the states (Larmor precession). This dynamics is written as B(X ) → R ∗ (t, t0 , X )B(Φλt,t0 (X ))R(t, t0 , X ).

(14.15)

Notice that if A is scalar then A(t) is also scalar at any time. Concerning the propagation of coherent states, we also perform the same analysis as for the scalar case (Chap. 4) with natural adaptations. Let us denote by k2 (t, t0 )X · X the quadratic part of the Taylor expansion of λ(t, z) around the trajectory z t . Almost nothing is changed, except that now the coefficients K j (t, t0 ) of the Taylor expansion of H (t, X ) around the classical trajectory z t are time dependent matrices in Cm,m . Moreover here we have to add to the scalar Hamiltonian k2 (t, t0 , X )1lm the matrix H1 (t, z t ). We shall see later that a sub-principal term (of order 1 in ) appears after diagonalizing of non-crossing systems. So the spin of the system will be modified along the time evolution according to the “rotation” matrix R(t, t0 ) defined by (14.13). This phenomenon is related with the Berry phase “factors accompanying adiabatic changes”. (Berry 1984). The following lemma is easy to prove and is standard for differential equations. Lemma 87 Let us denote respectively by U2 (t, t0 ) and U2 (t, t0 ) the quantum prop agators defined by the Hamiltonians k 2 (t, t0 )1lm respectively k2 (t)1lm + H1 (t, z t ). Then we have (14.16) U2 (t, t0 ) = R(t, t0 , z 0 )U2 (t, t0 ). Let us recall that if F(t, t0 ) denotes the linear classical flow associated with k2 (t, t0 )(stability or Jacobi matrix). The corresponding quantum flow U2 (t, t0 ) is also the metaplectic transformations M (F(t, t0 )) (see Chap. 3). The wave packet transform is a convenient way to built coherent states: Definition 34 Let z = (q, p) ∈ R2n and f ∈ S (Rn , Cm ) (a profile) . We set WP z f (x) = −n/4 e  p·(x−q) f i



x−q √ 



.

(14.17)

14.2 Systems with a Scalar Leading Term

The operator

411

f → WP z f

is a unitary map in L 2 (Rn , Cm ) which maps continuously Σ1m,k (Rn ) into Σm,k (Rn ), where Σm,k (Rn ) = { f ∈ L 2 (Rn , Cm ), (−2  + |x|2 )k/2 f ∈ L 2 (Rn , Cm )}, equipped with the -dependent graph-norm. Gaussian coherent states are the Wave Packet transforms of Gaussian profiles, up to a phase factor: ϕzΓ = e− 2 p·q W P z (g Γ ), i

(14.18)

where g Γ (x) = π −n/4 det(Γ )1/4 e 2 Γ x·x , Γ ∈ S+ (n). Moreover we have the following properties whose proofs are left to the reader i

i • WP z = e− 2 p·q Tˆ (z) Λ i • ϕz = e 2 p·q WP z g where g is the standard Gaussian. • For all z, z in R2d ,

 WP z+z = e−  p·q WP z Λ−1  WP z . i

The following Lemma concerns the action of an -Weyl quantized observable on wave packets. Lemma 88 If A is any classical observable with values in Cm,m ,   w opw  (A)WP z = WP z Op1 (A,z ),

(14.19)

√ A,z denotes the Cm,m matrix A,z (X ) = A( X + z), X ∈ R2n . Moreover for any f ∈ S (Rn , Cm ) we have the following expansion at any order:  opw  (A)WP z f =

  |α|2 α N +1 ∂ Xα WP z opw ). 1 (X ) f + O( α! |α|≤N

(14.20)

Proof (14.19) follows from known properties of Weyl quantization. Equation (14.20) follows from (14.19) and the Taylor formula.  As in the scalar case, for every N ≥ 1, we can get approximate solutions ) (t, t0 ) for the exact solution ψz0 ,v (t, t0 ) of the Eq. (14.9) with initial data ψz(N 0 ,v ψz0 ,v (t0 , t0 ) = WP z0 f v, where v ∈ Cm and f ∈ S (Rn )(a wave packet polarized along the direction v). We do not repeat here all the details given in Chap. 4 for the scalar case.

412

14 Adiabatic Decoupling and Time Evolution of Coherent …

First, we get a family of differential operators b j (t, t0 ), polynomials in (x, ∂x ) of degree smaller than 3 j and with coefficients in Cm , uniquely defined by the following induction formula for j ≥ 1, starting with b0 (t, t0 ) f = v f , 

∂t b j (t, t0 ) f (x) =

O p1w [K # (t, t0 )](bk (t, t0 ) f

(14.21)

k+ = j+2, ≥3

b j (t0 , t0 ) f = 0,

(14.22)

where K # (t, t0 , X ) = K (t, t0 , F(t0 , t)X ). Recall that the matrix symbols K (t, t0 ) are obtained by Taylor expanding the full Hamiltonian H (t) around the classical flow Φλt,t0 (z 0 ). Moreover, coming back to the Schrödinger equation, we get, in the same way as for the scalar case, for every N ≥ 0, (N ) (N ) (N ) (t)ψz,v (t, t0 ) = H (t, t0 ) + Rz,v (t, t0 ) i∂t ψz,v

(14.23)

where ⎛ ) ψz(N (t, t0 , ·) = ei S(t,t0 ,z0 )/ WP zt ⎝R(t, t0 )M [F(t, t0 )] 0 ,v



⎞  j/2 b j (t, t0 ) f ⎠

0≤ j≤N

(14.24) and ) Rz(N (t, t0 ) 0 ,v

=e

i S(t,t0 ,z 0 )/





j/2

WP zt

 R(t, t0 )M [F(t, t0 )]

j+k=N +2 k≥3

 N +3 Opw [R (t, t ) ◦ F(t, t )](b (t, t ) f ) = O L 2 (Rn ,Cm ) ( 2 ), k 0 0 j 0 1

(14.25)

where Rk (t, t0 ) is a remainder term in the Taylor expansion of H (t, z) around z t . Putting all these things together we get the following result: Theorem 77 Let us assume that the assumptions (SQ) and (SCA) are satisfied. Then we have, for every (z, v) ∈ Z × Cm , for every t ∈ IT = {t, |t − t0 | ≤ T }, ) (t, t0 ) L 2 (Rd ,Cm ) ≤ C N ,T,z (N +1)/2 . ψz0 ,v (t, t0 ) − ψz(N 0 ,v

(14.26)

In particular we have for the approximation mod O(), in L 2 (Rn , Cm ),   √ i ψz(1) (t, t0 ) = e  S(t,t0 ,z0 ) WP zt R(t, t0 )M [F(t, t0 )] f v + b1 (t, t0 ) f v , 0 ,v (14.27) where

14.2 Systems with a Scalar Leading Term

b1 (t, t0 ) f v =

413

 t  1 α i −1 ∂zα λ(s, z s )Opw 1 ((F(t0 , s)X ) v f ds α! t0 |α|=3  t +i −1 ∂ X H1 (s, z s )Opw 1 (F(t0 , s)X ) f vds.

(14.28)

t0

Remark 87 Using results obtained in [101, 191] we could get estimates in Σk norms. Remark 88 We can apply Theorem 77 with a Gaussian profile f  = g Γ or a finite sum of excited state of the standard harmonic oscillator: f = ck φk1 . (Here |k|≤ν

{φk1 }k∈Nn denotes the usual orthonormal basis of Hermite functions for  = 1). Using creation and annihilation operators we get that ) ψz(N (t, t0 ) = e  S(t,t0 ,z0 ) 0 ,v ⎞  WP zt ⎝R(t, t0 )M [F(t, t0 )] ck (t, t0 , )φk1 ⎠ . i

⎛

(14.29)

|k|≤J +3N

One can compute the vector coefficients ck (t, t0 , ) following the method used in Chap. 4 (Theorem 21). In the next section we can consider the particular case of spin-orbit interaction.

14.3 Spin-Orbit Interaction We have already studied spin-coherent states in Chap. 7. In this section we consider a quantum Hamiltonian where the motion of a particle is coupled with its spin motion.  This is a particular case of Sect. 14.2 with the subprincipal term Ck (t, X ) Sˆk = C(t, X ) · S. H1 (t, X ) = 1≤k≤3

More precisely Sˆk are the spin operators in the complex linear space V ( j) , dimV ( j) = 2 j + 1, j ∈ 21 N is the spin number and Sˆk = D ( j) (σk /2) are defined by the irreducible representation D ( j) of the Lie algebra su(2). Recall that we have the commutation relations (14.30) [Sk , S ] = iεk, ,m Sm . Ck (t) are real scalar symbols satisfying the assumption (SQ). So the full Hamiltonian here is H (t, X ) = H0 (t, X ) + C(t, X ) · S, with the spin operator S = ( Sˆ1 , Sˆ2 , Sˆ3 ). Hence the quantum Hamiltonian Hˆ (t) generates a propagator U (t, t0 ) in the Hilbert space L 2 (Rn , C2 j+1 ).

414

14 Adiabatic Decoupling and Time Evolution of Coherent …

We have seen in Chap. 7 that spin coherent states are labelled by the sphere S2 . So it is natural to study the semi-classical limit of the propagator in the phase space T ∗ (Rn ) × S2 . In particular we can reformulate the propagation for coherent states ϕz ⊗ ψn labelled by (z, n) ∈ R2n × S2 where ϕz was defined in Chap. 1 and ( j) ψn = ψn in Chap. 7. Our goal here is to apply Theorem 77 for f = g Γ0 (Gaussian with covariance matrix Γ0 ), v = ψn0 a spin coherent state where n0 ∈ S2 . So in particular for the leading term we have to compute the time evolution R(t, t0 )ψn0 with H1 = C(t, X ). It is enough to solve the equation for R(t, t0 ) for the spin number j = 21 . Let G(t, t0 ) ∈ SU (2) satisfying i ∂t G(t, t0 ) + (C(t) · σ )G(t, t0 ) = 0, G(t0 , t0 ) = 1. 2 Using the notations of Chap. 7, let ( j)

φ(t) = R(t, t0 )ψn( 0j) = T ( j) (G(t, t0 ))T ( j) (gn0 )ψ0 .

(14.31)

Lemma 89 The adjoint action of G(t, t0 ) has the following property G(t, t0 )(n0 · σ )G(t, t0 )−1 = n(t) · σ where n(t) = RG(t,t0 · n0 satisfies ∂t n(t) = C(t) ∧ n(t). Proof Using the canonical cover of S O(3) by SU (2) we can compute to get the Lemma.  ( j)

Lemma 90 The spin states (14.31) coîncides with the coherent state ψn(t) modulo a phase factor ( j) (14.32) φ(t) = ei jα(t) ψn(t) , where α(t) ∈ R. Proof We have n(t) = RG(t,t0 )gn0 e3 = Rgn (t) e3 , hence there exists α(t) ∈ R such that −1 G(t, t0 )gn0 = e 2 α(t)σ3 . gn(t) i

So we get (14.32) for j = T ( j) .

1 2

hence for any j ∈

In a last step we compute the phase α(t).

N 2

by considering the representation 

14.3 Spin-Orbit Interaction

415

Lemma 91 In polar coordinates (θ, ϕ), n = (sin θ cos ϕ, sin θ sin ϕ, cos θ ) we have  α(t) =

t

˙ )) dτ. (C(τ ) · n(τ ) + (1 − cos θ (τ )ϕ(τ

(14.33)

t0

Proof Applying the equation satisfied by G(t, t0 ) to G(t, t0 ) = e 2 α(t) gn(t) we get i

α(t)g ˙ n(t) = Λ(t) ·

σ (t) − 2i∂t gn(t) . 2

Computing the average of this equation in the initial 21 -spin coherent state ψ0 , a direct computation gives α(t) ˙ = Λ(t) · n(t) + (1 − cos θ (t))ϕ(t). ˙ Here we have used that gn ψ0 , σk gn ψ0  = −nk , 1 ≤ k ≤ 3.  So from Theorem 77 we get the following result (spin-orbit motion) ( j)

Theorem 78 ([33]) For the initial state Ψz0 ,n0 = ϕz0 ⊗ ψn0 we have √ i ( j) Γ (t) ⊗ ψn(t) + O( ) Ψz0 ,n0 (t, t0 ) = e  S(t,t0 )+i jα(t) ϕz(t) where the covariance matrix Γ (t) is computed in Chaps. 3 and 4 for the dynamics generated by the Hamiltonian H0 (t, X ). Remark 89 This Theorem could be extended to an approximation modulo any power of  like in Theorem 77 (see also [33]).

14.4 Systems with Constant Multiplicities Eigenvalues Let us consider now the case where the principal part H0 (t, X ) has several different eigenvalues. For simplicity we assume that H0 (t, X ) has two eigenvalues λ± (t, X ) such that λ± (t, ·) ∈ C ∞ (R2n ). Moreover we assume that λ± (t, X ) have a constant multiplicity μ± for any (t, X ) ∈ [t0 , t0 + T ] × Ω2 , where Ω2 is an open bounded subset of the phase space Z . In the semi-classical limit wave-packets are localized in the phase space and we know that their time evolution follows the classical trajectories, in the scalar case. So we shall see that it is enough to decouple the considered system in a neighborhood of classical trajectories for the different modes.

416

14 Adiabatic Decoupling and Time Evolution of Coherent …

More precisely we introduce the following non-crossing assumptions: (NCC) There exists δ > 0 such that |λ+ (t, X ) − λ− (t, X )| ≥ δ, ∀(t, X ) ∈ [t0 , t0 + T ] × Ω2 . Let us introduce a local existence property for the flows Φ±t,t0 generated by the Hamiltonians λ± (t, X ): for a given T > 0 there exist bounded open set Ω j , 1 ≤ j ≤ 2, such that Ω¯ 1 ⊂ Ω2 and for any X ∈ Ω1 and t ∈ [t0 , t0 + T ], the solution X (t) of the Hamilton equations1 ∂t X ± (t) = J ∇ X λ± (t, X ± (t)), X (t0 ) = X exists and stays in Ω2 ∀t ∈ [t0 , t0 + T ]., where Ω2 is like in (NCC). Let us remark that λ± (t, X ) may collide for some X ∈ R2n \Ω2 which is outside the accessible classical area. Moreover the only global assumption we shall use concerns the quantum propagator for Hˆ (t). In particular this holds true if H (t) is sub-quadratic. Let us denote by π± (t, X ) the spectral projectors on ker(H0 (t, X ) − λ± (t, X )1lm ). All these functions are C ∞ -smooth in X ∈ Ω2 and continuous in t ∈ [t0 , t0 + T ]. To construct asymptotic solutions of equation (14.9) we shall show that the time evolution of the vector states (t, x) → ψ(t, x) ∈ Cm splits into two parts: one part coming from the eigenvalue λ+ (t, X ) the other part coming from λ− (t, X ). We shall see that at the leading order the two quantum evolutions ± are not mixing but they can mix up at the order one or greater in . In physics this phenomenon is called adiabaticity. This will be demonstrated for the propagation of wave-packets (or coherent states) and more generally for initial data with compact frequency set. In a first step we work with formal semi-classical series in  whose coefficients  are Ajj matrix symbols in Cm,m . Let us denote by AΩ1 the set of formal series A = j≥0

where A j is a C ∞ -smooth application from Ω1 in Cm,m . AΩ1 is an algebra for the C j  j with the notations used Moyal product defined by C = A  B where C = j≥0

in (14.5). This product is associative but non commutative. The Moyal bracket will be denoted {A, B} = A  B − B  A where A, B ∈ AΩ1 . Definition  35 A formal self-adjoint observable A is a formal series in Ω1 : A=  j A j A ∈ AΩ1 and A j (X ) is an Hermitian matrix for any X ∈ Ω1 . Moreover j≥0

A is continuous in time t ∈ [t0 , t0 + T ] iff A j (t, ·) is continuous in t ∈ [t0 , t0 + T ]. A is a formal projection in Ω1 if A is self-adjoint and satisfies A  A = A in Ω1 . In a first step we construct a formal semi-classical projector following the quantum evolution driven by Hˆ (t). It will be convenient to write the m × m complex matrices H (t, X ) it is not always true that λ± (t, ·) are subquadratic in R2n . We have easily λ± (t, X ) = O(< X >2 ) and ∂ X λ± (t, X ) = O(< X >) 2 λ (t, ·) is not necessarily bounded in R2n . This is why we introduce a local assumption but ∂ X,X ± which allows us to work with the subquadratic Hamiltonians λ˜ ± (t, X ) = χ(X )λ± (t, X ) for some χ ∈ C0∞ (Ω2 ), χ(X ) = 1 if X ∈ Ω1 .

1 Notice that for a general subquadratic matrix Hamiltonian

14.4 Systems with Constant Multiplicities Eigenvalues

417

symbols in Cm,m in the splitting Cm = π+ (t, X ) ⊕ π− (t, X ) as a 2 × 2 matrix with matrix-valued coefficients. We shall use the following notation. Let be A(t, X ) ∈ Cm,m then   ++ A (t, X ) A+− (t, X ) A(t, X ) = A−+ (t, X ) A−− (t, X ) where A++ (t, X ) = π+ (t, X )A(t, X )π+ (t, X ), A−− (t, X ) = π− (t, X )A(t, X ) π− (t, X ), A+− (t, X ) = π+ (t, X )A(t, X )π− (t, X ), A−+ (t, X ) = π− t, X )A(t, X ) π+ (t, X ). Notice that A(t, X ) is Hermitian if and only if A(t, X )++ , A(t, X )−− are Hermitian and A(t, X )−+ = (A(t, X )+− )∗ . We shall say that A(t, X ) is diagonal if A(t, X )+− = A(t, X )−+ = 0 and that it is off-diagonal if A(t, X )++ = A(t, X )−− = 0. Notice that ∂t π± (t, X ) is off-diagonal (uses that π± (t, X )2 = π± (t, X )) as well as [π± (t, X ), A(t, X )]. In the following Theorem we also denote π(t, X ) = π+ (t, X ) and Π (t, X ) = Π + (t, X ) and π ⊥ (t) = 1lm − π(t) = π− . Theorem 79 (Semiclassical projector-adiabatic evolution) There exists a unique  j self-adjoint formal projection Π (t, X ) =  Π j (t, X ), continuous in t ∈ [t0 , t0 + j≥0

T ], C ∞ -smooth in X ∈ Ω1 , such that Π0 (t, X ) = π(t, X ) and i∂t Π (t) = {H (t), Π (t)}

(14.34)

Moreover the sub-principal term Π1 (t) (Hermitian matrix) is given by the following formulas: −1 π(t){π(t), π(t)}π(t) 2i −1 π(t)⊥ {π(t), π(t)}π(t)⊥ . = 2i

Π1 (t)++ = Π1 (t)−−

Π1 (t)−+ = i(H0 (t) − λ(t))−1 π(t)⊥ S1 (t)−+ ,

(14.35) (14.36)

and Π1 (t)+− = (Π1 (t)−+ )∗ , where 1 S1 (t) = ∂t π(t) + ({H0 (t), π(t)} − {π(t), H0 (t)}) − [H1 (t), π(t)]. 2 Proof Using (14.6) the result will follow by an induction argument. Let us remark that in formula (14.36) (H0 (t) − λ(t))−1 is well defined on the range of π(t)⊥ . We use the following convention: we say that two formal series L , M are equal modulo O( N +1 ) if L j − M j = 0 for 0 ≤ j ≤ N in Ω1 and we shall write L = M + O( N +1 ).

418

14 Adiabatic Decoupling and Time Evolution of Coherent …

Starting with Π0 (t) = π(t) we see that it is a Moyal projector modulo []. For the next term we are looking for Π1 (t) such that Π (1) := π(t) + Π1 (t) is a projector modulo O(2 ) and such that Eq. (14.34) is satisfied modulo O(2 ). So computing the term in  we find the equations Π1 (t) − (π(t)Π1 (t) + Π1 (t)π(t)) =

1 {π(t), π(t)} 2i

(14.37)

and 1 ({H0 (t), π(t)} − {π(t), H0 (t)}) + [H1 (t), π(t)] − i∂t π(t). 2i (14.38) Then the formula (14.35) follows from the first equation. Now we can check that the l.h.s in (14.38) is off-diagonal so we can compute the off-diagonal terms of Π1 (t, X ) and (14.36) follow by elementary algebra. Notice that we have used the gap assumption so that (H0 (t) − λ(t))−1 π(t)⊥ is well defined. Hence we have found that Π1 (t, X ) exists and is unique. Now, reasoning by  induction on N ≥ 1, assume that Π0 (t), . . . , Π N (t) are known (N ) such that Π (t) = k Πk (t) is a projector modulo O( N +1 ) satisfying (14.34) [Π1 (t), H0 (t)] =

0≤k≤N

modulo [ N +1 ] then we have to compute Π N +1 (t) such that Π (N +1) (t) := Π (N ) (t) + Π N +1 (t) is a Hermitian projector and satisfies Eq. (14.34) modulo O( N +2 ). Let us denote R N (t) the coefficient of  N +1 in the -expansion of the matrix symbol Π (N ) (t) − (Π (N ) )2 (t) and S N (t) the coefficient of  N +1 in the -expansion of ∂t Π (N ) (t) − 1i [H (t), Π (N ) (t)] . Hence Π (N +1) (t) must satisfy the equations Π N +1 (t) − (π(t)Π N +1 (t) + Π N +1 (t)π(t)) = R N (t)

(14.39)

[H0 (t), Π N +1 (t)] = i S N (t)

(14.40)

As for the computation of Π1 (t) we have to prove compatibility conditions. We remark first that R N (t, X ) and S N (t, X ) (X ∈ R2d ) are Hermitian matrices. To check (14.39) and (14.40) we have to prove that R N (t) is diagonal and S N (t) is off-diagonal. We have, for the Moyal product,   Π (N ) (t) Π (N ) (t)2 − Π (N ) (t) (1l − Π (N ) (t)) =  (N ) 2   Π (t) − Π (N ) (t) Π (N ) (t) − Π (N ) (t)2 = O(2N +2 ).

(14.41)

Then computing the coefficient of  N +1 in (14.41) we get that π(t)R N (t)π ⊥ (t) = 0, hence taking the adjoint, π ⊥ (t)R N (t)π(t) = 0. Notice that taking the time derivative of

14.4 Systems with Constant Multiplicities Eigenvalues

419

Π (N ) (t)2 − Π (N ) (t) = O( N +1 ) we get

Π (N ) (t)  ∂t Π (N ) (t)  Π (N ) (t) = O( N +1 ).

(14.42)

We have the same properties for 1l − Π (N ) (t). Now let us prove that S N is offdiagonal. Using (14.42) we have   Π (N ) (t) i∂t Π (N ) (t) − [H (t), Π (N ) (t)] Π (N ) (t) =  (N ) 2    Π (t) − Π (N ) (t) H (t)Π (N ) (t) − Π (N ) (t)H (t) Π (N ) (t)2 − Π (N ) (t) . Computing the coefficient of  N +1 in the last equation we get iπ(t)S N (t)π(t) = R N (t)H0 (t)π(t) − π(t)H0 (t)R N (t) = λ(t)[R N (t), π(t)] = 0. By the same argument applied with (1l − Π (N ) (t)) instead of Π (N ) (t) we also have π ⊥ (t)S N (t)π ⊥ (t) = 0. So now we can solve equations (14.39), (14.40) and we get for Π N +1 (t) the following formula Π N +1 (t) = π(t)⊥ R N (t)π(t)⊥ − π(t)R N (t)π(t) +   i π(t)S N (t)π ⊥ (t) − π ⊥ (t)S N (t)π(t) . ⊥ λ(t) − λ (t)

(14.43)

Moreover the proof gives also uniqueness for the series {Π j (t)} j≥0 such that  Π0 (t) = π . Corollary 46 We have the following formal series equality in [t0 , t0 + T ] × Ω2 : Π + (t, X ) + Π − (t, X ) = 1lm . In particular Π + (t)  Π − (t) = Π − (t)  Π + (t) = 0. Corollary 47 Assuming that condition (NCC) is satisfied then in [t0 , t0 + T ] × Ω2 we have for any j ∈ N and α ∈ N2n , the estimates: − j−|α| ∂ Xα Π ± j (t, X ) ≤ C j,α δ

(14.44)

where  ·  is a fixed norm in Cm,m . Proof Consider the case +. We only give an idea of the proof for j = 0, 1. The general case can be checked by an induction argument. For j = 0 we use the integral formula

420

14 Adiabatic Decoupling and Time Evolution of Coherent …

π+ (t, X ) =

i 2π

 γ (t,X )

(H0 (t, X ) − z)−1 dz,

(14.45)

where γ (t, X ) is a loop around λ+ (t, X ) with no other eigenvalue of H (t, X ) inside the loop. But we have (H0 (t, X ) − z)−1 =

π+ (t, X ) π− (t, X ) + . λ+ (t, X ) − z λ− (t, X ) − z

(14.46)

So computing ∂ Xα π+ (t, X ) with (14.45) then plugging (14.46) in (14.45) and using the residue Theorem we get easily (14.44) for j = 0. Then using Theorem 79. we deduce (14.60) for j = 1. For j ≥ 2 we have to use an inductive argument as in the proof of Theorem 79. As above in the following statement we denote λ(t, X ) = λ± (t, X ), π(t, X ) = π± (t, X ) and so on. The assumptions (NCC) are supposed to be satisfied. Theorem 80 (formal adiabatic decoupling) There exists a formal semi-classical time dependent Hermitian Hamiltonian in AΩ2 , H adia (t) =  j H jadia (t), smooth j≥0

in X , continuous in t, such that H0adia (t, X ) = λ(t, X )1lm and Π (t)  H adia (t) = Π (t)  H (t).

(14.47)

Moreover, the subprincipal term H1adia (t) is a Hermitian matrix, such that π(t) satisfies in Ω1 the following adiabatic transport equation along the classical flow for the Hamiltonian λ(t). ∂t π(t) + {λ(t), π(t)} =

1 adia [H (t), π(t)]. i 1

(14.48)

The diagonal part of H1adia satisfies 1 π(t)H1adia (t)π(t) = π(t)H1 (t)π(t) + π(t){π(t), H0 (t)}π(t). 2

(14.49)

Moreover for j ≥ 1, the blocks H jadia+± are unique and the block H jadia−− can be chosen arbitrary. Proof As for the last Theorem we can compute H jadia (t) by induction on j starting with H0adia (t) = λ(t)1lm . Let compute H1adia (t) such that Eq. (14.47) is satisfied modulo O(2 ). This gives π(t)(H1adia (t) − H1 (t)) = Π1 (t) (H0 (t) − λ(t)) +

1 {π(t), H0 (t) − λ(t)}. 2i (14.50)

14.4 Systems with Constant Multiplicities Eigenvalues

421

First of all we have to prove that (14.50) has at least a solution H1adia (t). For that we check that we have   1 (14.51) π(t)⊥ Π1 (t) (H0 (t) − λ(t)) + {π(t), H0 (t) − λ(t)} = 0. 2i Using the following identity for the Poisson bracket of matrix symbols: {A, BC} − {AB, C} = {A, B}C − A{B, C},

(14.52)

we get that {π, λπ} = {π, π }λ + π ⊥ {π, λ}, {π, λ⊥ π ⊥ } = −{π, π }λ + π { π, λ− ⊥}. We apply these formulas to H0 = λπ + λ⊥ π ⊥ to prove (14.51). Then to have uniqueness of H1adia (t) we can impose the (arbitrary) condition (1l − π(t))(H1adia (t) − H1 (t))(1l − π(t)) = 0. So using (14.36), (14.35) and (14.50), H1adia (t) is well determined. adia (t) such that the Eq. (14.50) is Then by induction on j we can compute H j+1 j+2 satisfied modulo O( ). Let us prove now Eq. (14.48). For simplicity let us assume here that H1 (t) = 0. This term is not important. We have   i[H1adia (t), π(t)] = −i π(t)H1adia (t)π(t)⊥ − π(t)⊥ H1adia (t)π(t) . So from (14.50) we get π(t)H1adia (t)π(t)⊥ = Π1 (t)(H0 (t) − λ(t))π(t)⊥ +

1 {π(t), H0 (t) − λ(t)}π(t)⊥ . 2i (14.53)

Using (14.36) we get   1 1 = −π i∂t π − ({H0 , π } − {π, H0 }) π ⊥ + π {π ; H0 − λ}π ⊥ . 2i 2i (14.54) We compute {H0 , λ} = {λπ + λ⊥ π ⊥ , λ} using (14.52) and that {π(t), π(t)} is diagonal and using the adjoint of (14.54) we get (14.48). For the diagonal term using (14.50) we have π(H1adia π ⊥

1 π(t)H1adia (t)π(t) = π(t)H1 (t)π(t) + π(t){π(t), H0 (t)}π(t). 2

(14.55)

422

14 Adiabatic Decoupling and Time Evolution of Coherent …

Then by induction on N ≥ 0, we can check HNadia +1 (t) such that (14.47) is satisfied mod. O( N +2 ). More precisely, assume H jadia well defined for 0 ≤ j ≤ N such that Π (N ) (t)  H adia(N ) (t) = Π (N ) (t)  H (t) + O( N +1 ), where Π (N ) =



k Πk , H adia(N ) (t) =

0≤k≤N



(14.56)

Hkadia (t).

0≤k≤N

We have to determine HNadia +1 (t) such that (14.56) is satisfied for N → N + 1. For simplicity we always assume that H = H0 (the extension with sub-principal terms can be easily done). Let us denote TN (t) the coefficient of  N +1 in Π (N +1) (t)  (H (t) − H adia(N ) (t)). So we have to find HNadia +1 such that π(t)HNadia +1 (t) = TN (t).

(14.57)

Hence we need to check that π ⊥ (t)TN (t) = 0. This is done by computing the coefficient of  N +1 in the expansion of   1lm − Π (N +1) (t) Π (N +1) (t)(H (t) − H adia(N ) (t)) = O( N +2 ). So existence for the H jadia ( j ≥ 1) is proven. From the proof we get that the blocks H jadia,−− can be choose arbitrary, the other blocks are uniquely determined by our algorithm.  From Theorem 80 we get that the time evolution U H (t, t0 ) for the Hamiltonian Hˆ (t) is formally decoupled between the modes −. From Theorem 80 we get two + andadia,± formal quantum Hamiltonians H±adia = j Hj with H0adia,± = λ± . j≥0

Corollary 48 Assume that condition (NCC) is satisfied in [t0 , t0 + T ] × R2n and that Hˆ (t), H±adia (t) generate propagators U H (t, t0 ), U±adia (t, t0 ). Then for any ψ0 ∈ L 2 (Rd , Cm ) we have, at the formal level for Weyl quantization, U H (t, t0 )ψ0 = Πˆ + (t)U+adia (t, t0 )Πˆ + (t0 )ψ0 + Πˆ − (t)U−adia (t, t0 )Πˆ − (t0 )ψ0 . (14.58) ˜ ˜ Proof Let us denote ψ(t) the r.h.s of (14.58) and let us compute i∂t ψ(t). Using Theorems 79 and 80 we get easily ˆ adia ˆ (i∂t − Hˆ (t))Πˆ ± (t) = Π(t)(i∂ t − H± (t)),

(14.59)

˜ ˜ ˜ 0 ) = ψ(t0 ) = ψ0 . Hence by unique= H (t)ψ(t) and ψ(t then we compute i∂t ψ(t) ˜ ness we get that ψ(t) = ψ(t). 

14.4 Systems with Constant Multiplicities Eigenvalues

423

Corollary 49 Assuming that condition (NCC) is satisfied then in [t0 , t0 + T ] × Ω2 we have for any j ∈ N and α ∈ N2n , the estimates: ∂ Xα H jadia,± (t, X ) ≤ C j,α δ 1− j−|α|

(14.60)

where  ·  is a fixed norm in Cm,m . Our aim in the following step is to give an asymptotic meaning as   0 for solutions of (14.58) under the local non-crossing assumption (NCC) concerning the Hamilton flow of λ± (t, X ). To do that we shall assume: (CFS) The frequency set of the initial state ψ0 is localized in a compact set K 0 ⊂ R2n . Recall that the frequency set for U± (t, t0 )ψ0 propagates with the classical flow t,t0 Φ± according to the modified dynamics (14.15). Using this property and assumption (CFS) we shall see that we can give an -asymptotic meaning to (14.58) by localization in bounded set in the phase space. We shall see by applying Theorems (79) and (80) that we can get asymptotic v , z 0 ∈ Ω0 and v is an eigenvector of solutions for (14.9) with ψ0 (x) = WP z0 g(x) H0 (t0 , z 0 ). More generally we can assume that the frequency set FS[ψ0 ] is in a compact K 0 . To do that we have to transform formal expansion in  into semiclassical quantum observables with error estimates. Assuming property (NCC) let us introduce cutoff functions χ j ∈ C0∞ (Ω j ) such that χ1 = 1 in K 0 , χ2 ∈ C0∞ (Ω2 ) such that χ2 = 1 on {Φ±t,t0 X, t ∈ [T0 , t0 + T ], X ∈ K 0 } ⊆ Ω2 . We have to give a meaning as an equality between operators after cutting off. We forget here the sign ±. Proposition 138 Let χ3 (t, X ) be a time dependent cut-off, continuous in t ∈ [t0 , t0 + T ], C ∞ with a compact support in X ∈ Ω2 such that χ2 = 1 on supp(χ3 ). Then we have  (N ) (t)χ (N ) (t)(i∂ − χ H adia,(N ) (t))χ ˆ 3 = χ ˆ 3 + O( N +1 ), (i∂t − Hˆ (t))χ 2Π 2Π t 2 (14.61) Proof It is easier to compute at the symbol level. We start with the equality: (i∂t − H (t))  χ2 Π (N ) (t) = χ2 Π (N ) (t)  (i∂t − χ2 H adia,(N) (t)) +i∂t (χ2 Π (N ) (t)) + χ2 Π (N ) (t)  χ2 H adia,(N ) (t) − H (t)  χ2 Π (N ) (t) where H adia,(N) (t) =



 j H jadia . Using (14.34) we get

0≤ j≤N ) (N ) i∂t (χ2 Π (N ) (t) = χ2 {H (t), Π (N ) (t)}(N i∂t + O( N +1 ).  + χ2 Π (t)

But we have

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14 Adiabatic Decoupling and Time Evolution of Coherent …

H (t)  χ2 Π (N ) (t)) = (χ2 Π (N ) )  H (t) + {H (t), χ2 Π (N ) } . Then using (14.47) (mod O( N +1 )), local properties of the Moyal product and that χ2 = 1 on supp(χ3 ), we get (i∂t − H (t))  χ2 Π (N ) (t)  χ3 = χ2 Π (N (t)  (i∂t − χ2 H adia,(N ) )  χ3 + O( N +1 ).

(14.62)  Let us denote by U±(N ) (t, t0 ) the Schrödinger propagator for the Hamiltonian χ2 H adia,(N ) (t) Let us consider the Schrödinger equation (14.1) with the initial data ψ(t0 ) = ψ0 with a compact frequency set in K 0 . Let us denote   ψ±(N ) (t) = χ2 Π±(N ) (t)U±(N ) (t, t0 )χ2 Π±(N ) (t0 )ψ0 .

(14.63)

Notice that with the previous notations we have χˆ 1 ψ0 = ψ0 + O(+∞ ). Using (14.61) and Egorov theorem we get that (i∂t − Hˆ (t)) (ψ+(N ) (t) + ψ−(N ) (t)) = O L 2 ( N +1 ).   

(14.64)

ψ (N ) (t)

Notice that we have applied (14.62) with χˆ 3 = U±(N ) (t, t0 )χˆ 1 U±(N ) (t0 , t). Using again Theorem 79, and that χ2 = 1 on K 0 , we have 2

2

  ψ (N ) (t0 ) = χ2 Π+(N ) (t0 ) ψ0 + χ2 Π−(N ) (t0 ) ψ0 = ψ0 + O L 2 ( N +1 ). Hence from the Duhamel formula we deduce the asymptotic adiabatic reduction. Theorem 81 Assume that the condition (NCC) is satisfied, that the initial data ψ0 satisfies (CFS) and that the cut-off χ1 , χ2 are fixed as above. Then for t ∈ [t0 , t0 + T ], the Schrödinger equation

satisfies

i∂t ψ(t) = Hˆ (t)ψ(t), ψ(t0 ) = ψ0 + O(∞ ),

(14.65)

ψ(t) = ψ (N ) (t) + O L 2 ( N ),

(14.66)

where ψ (N ) (t) is defined in (14.63) and (14.64). Remark 90 At the beginning of the Chapter we have assumed the H (t) is subquadratic for simplicity. Moreover with this assumption the estimates can be extended to the harmonic spaces. In fact for proving it is enough to assume: (i) Existence of the quantum propagator for Hˆ (t).

14.4 Systems with Constant Multiplicities Eigenvalues

425

(ii) H (t, X ) is in a reasonable class of Hermitian symbols defined for some M ∈ R, by estimates as |∂ Xα H (t, X )| ≤ Cα X  M , ∀t ∈ [t0 , t0 + T ], ∀X ∈ R2n . Remark 91 The adiabatic reduction allows us to apply the results proved in Sect. 14.2. In particular many results known for scalar Hamiltonian can be extended to Hamiltonian systems with constant multiplicity eigenvalues, in the semi-classical regime. We now apply the general adiabatic reduction results when the initial data is a coherent state. Corollary 50 Let v0 be a unitary eigenvector for H (t0 , z 0 ) for the eigenvalue λ+ (t0 , z 0 ). Then the solution ψ(t) for (14.1) satisfies ψ(t) = e  S+ (t,t0 ,z0 v+ (t)ϕzΓt t i

+

  √ (x − qt+ (t)) 1 + u + (t) · + O() √ 

+ where v+ (t) = R+ (t, t0 )v0 and u + (t) = (u + 1 (t), . . . , u n (t)) can be computed using Theorem 77. Moreover v+ (t) solves the differential equation

d v+ (t) + i H1adia (t) v+ (t) = 0, dt and H (t) v+ (t) = λ+ (t) v+ (t),  v+ (t)Cm =  v+ (t0 )Cm . . Proof Using Theorem 79 and (80) we have modulo O(∞ ), + (t)U (t, t )Π + (t )ϕ − −     U (t, t0 )ϕz,v Π + 0 0 z,v + Π (t)U− (t, t0 )Π (t0 )ϕz,v

Therefore it is enough to prove the following formula π± (t, z t± )R ± (t, t0 )π± (t0 , z) = R ± (t, t0 )π± (t0 , z).

(14.67)

Let us denote v(t) = R ± (t, t0 )π± (t0 , z). Computing derivatives in t of v(t) and π± (t, z t± )v(t), and using (14.48), we can see that they satisfy the same differential  equation and we conclude that v(t) = π± (t, z t± )v(t) for every t ∈ IT . Remark 92 It is not difficult to extend the proof of Theorem 77 to systems such that the principal term has any number of constant multiplicities eigenvalues. We left to the reader the study of the remainder term for large T (estimation of the

426

14 Adiabatic Decoupling and Time Evolution of Coherent …

Ehrenfest time) which can be obtained following the method already used for scalar Hamiltonians. To finish this section let us apply the results to the Dirac operator. Let us recall that the Dirac Hamiltonian is defined in the following way. Its symbol is 3  H (t, x, ξ ) = c α j · (ξ j − A j (t, x)) + βmc2 + V (t, x), j=1

where {α j }3j=1 and β are the 4 × 4-matrices of Dirac satisfying the anti-commutation relations α j αk + αk α j = 2δ jk I4 , 1 ≤ j, k ≤ 4, (α4 = β, 1l4 is the 4 × 4 identity matrix). A = (A1 , A2 , A3 ) is the magnetic vector potential and   V+ 1l2 0 V = 0 V− 1l2 where V± , is a scalar potential (12 is the identity matrix on C2 ). The physical constant m (mass) and c (velocity) are fixed so we can assume m = c = 1. We assume for simplicity that the potentials are C ∞ and there exists μ ∈ R such that for very k ∈ N, α ∈ Nd there exists C > 0 such that for all (t, x) ∈ IT × Rd we have |∂tk ∂xα A(t, x)| + |∂tk ∂xα V (t, x) ≤ C < x >μ .  H (t, X ) has two eigenvalues λ± = ± 1 + |ξ − A(t, x)|2 + V± (t, x). To apply our results it could be sufficient to assume that V+ (t, x) − V− (t, x) ≥ −2 + δ for some δ > 0. But to make computations easier we shall assume V+ = V− = V . Then the spectral projections are given by the formula π± (t, x) =

α · (ξ − A(t, x)) + β 1 ±  . 2 2 1 + |ξ − A(t, x)|2

(14.68)

Then we compute the subprincipal terms H ± by the general formula (14.50). After a computation left to the reader, we get the following formula ±i 1 B · S, ( p1l4 − α · H ( p)) · (∂t A + ∂x V ) −  2 2(1 + | p| ) 1 + | p|2 (14.69) where p = ξ − A(t, x), B j,k = ∂x j Ak − ∂xk A j , S j,k = −iα j αk . Let us recall the  notation S · B = B j,k S j,k for the coupling of the spin with the magnetic H1± (t, x, ξ ) =

1≤ j 0 we have  rn n≥0

n!

Φ( f )n ψ (k) < +∞. 

The Weyl operators and their compagnons coherent states are defined as follows: Definition 38 For f ∈ h we define the Weyl translation operators:   T ( f ) = exp a ∗ ( f ) − a( f ) = exp



 d x ( f (x)a ∗ (x) − f¯(x)a(x)) . (15.22)

The coherent state Ψ ( f ) for every f ∈ h is then defined as Ψ ( f ) = T ( f )Ω . The bosonic coherent states have the following expression (analogue of an expression already given in Chap. 1 for finite systems of bosons) Proposition 139 For every f ∈ h we have Ψ ( f ) = e−

f 2 2

 1 √ f ⊗k . k! k≥0

In particular the probability to have k particles in ψ( f ) is equal to e− f f 2k /k!, where we recognize the Poisson law with mean f 2 . 2

Proof We give formal argument from which it is not difficult to supply rigourous proofs. One has the useful formula: T ( f ) = e−

f 2 2

exp(a ∗ ( f )) exp(−a( f ))

(15.23)

since the commutator [a( f ), a ∗ ( f )] = f 2 commutes with a( f ), a ∗ ( f ). We deduce ∗ k ⊗k f 2  (a ( f )) f 2  f Ω = e− 2 Ψ ( f ) = e− 2 √ k! k! k≥0 k≥0   where f ⊗k is the Fock-vector 0, 0, . . . , f ⊗k , 0 . . . . Let πk be the orthogonal projector onto the k-particle space ⊗ks h. We have found

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15 Bosonic Coherent States

πk (Ψ ( f )) = e−

f 2 2

f ⊗k √ k!

so the probability to have k particles in ψ( f ) is equal to πk (Ψ ( f )) 2 = 2  e− f f 2k /k!. The main properties of Weyl operators and coherent states are given in the following proposition. Proposition 140 Let f, g ∈ h. (i) T ( f ) is a unitary operator and one has T ( f )∗ = T ( f )−1 = T (− f ). (ii) The Weyl operator satisfies the commutation relations: T ( f )T (g) = T (g)T ( f ) exp (−2i f, g) = T ( f + g) exp (−i f, g) . In particular we have T (g)Ψ ( f ) = e−ig, f  Ψ (g + f ) (iii) We have T ∗ ( f )a(g)T ( f ) = a(g) + g, f 1l, T ∗ ( f )a ∗ (g)T ( f ) = a ∗ (g) +  f, g1l. (iv) The coherent states are eigenfunctions of the annihilation operators: a(g)Ψ ( f ) = g, f Ψ ( f ). (v) The expectation of the number operator N in the coherent state Ψ ( f ) is Ψ ( f ), NΨ ( f ) = f 2 , also we have for the variance: Ψ ( f ), N2 Ψ ( f ) − Ψ ( f ), NΨ ( f )2 = f 2 . (vi) The coherent states are normalized but not orthogonal to each other:   1 2 Ψ ( f ), Ψ (g) = exp − ( f − g − i f, g) 2 which implies that

15.3 The Bosons Coherent States

441

  1 |Ψ ( f ), Ψ (g)| = exp − f − g 2 . 2 (vii) The set of operators {T ( f ), f ∈ h} is irreducible on F (h): the only bounded operators B in F (h) commuting with T ( f ) for all f ∈ F (h) are the scalar B = λ1l, λ ∈ C. In particular the set of coherent states {Ψ ( f ), f ∈ h} is total in F (h). Proof Properties (i) to (ii) are left to the reader. For (iii) we compute d ∗ T (t f )a(g)T (t f ) = T ∗ (t f )[a(g), a ∗ ( f )]T (t f ) = T ∗ (t f )[g, f ]T (t f ) dt and we get (iii) integrating in t between 0 and 1. (iv) are (v) are consequences of (iii). 1 ∗ For (vi) we write, using (15.23): T ( f )Ω = e− 2 f ea ( f ) Ω. So we get T ( f )Ω, T (g)Ω = Ω, T ∗ ( f )T (g)Ω = Ω, T (g − f )Ωei f,g = e− 2 g− f ei f,g . 1

2

(15.24)

Let us now prove (vii). Remark first that B commutes with Φ( f ) for every f ∈ h hence with a( f ) and a ∗ ( f ). In particular B commutes with the number operator N. Let {ei }i∈I be an orthonormal basis for h and ai = a(ei ). Denote ψk1 ,...,kn = (k1 ! . . . kn !)−1/2 (a1∗ )k1 . . . (an∗ )kn Ω. This is an orthonormal basis for the Fock space F B (h) with the obvious index set. Compute ψk1 ,...,kn , Bψ j1 ,..., jm . This is 0 if the sets {k1 , . . . , kn }, { j1 , . . . , jm }are not equal. Finally we have, using (CCR), ψi1 ,...,in , Bψi1 ,...,in  = Ω, ai1 . . . ain ai∗1 . . . ai∗n Ω = Ω, BΩ hence B = Ω, BΩ1l.



The two following lemmas will be useful later. Let us introduce the operators family Nr = (1l + N2 )r/2 for r ∈ R. Lemma 95 For every r ∈ R and every f ∈ h, Nr T ( f )N−r extends in a bounded operator in the Fock space F B (h). In other words for every r ≥ 0, T ( f ) is bounded on the Hilbert space D(Nr ) for the norm ψ r := Nr ψ F B (h) . Proof From the previous proposition (iii) we have T ∗ ( f )NT ( f ) = N + a( f ) + a ∗ ( f ) + f 2 1l.

(15.25)

We prove the lemma for r = 21 . It is not difficult by iteration and interpolation to prove the result for every r .

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15 Bosonic Coherent States

We have N1/2 T ( f )N−1/2 Ψ 2 = Ψ, N−1/2 T ∗ ( f )NT ( f )N−1/2 Ψ  We can replace N by N, use (15.25) and Cauchy–Schwarz inequality to get N1/2 T ( f )N−1/2 2 ≤ C f 2 .  Lemma 96 Let α ∈ C 1 (R, h), t → α(t). Then T (α(t)) is strongly differentiable in t from D(N1/2 ) to F (h). The derivative is given by d ˙¯ + i(α¯ · α)] ˙ − a(α) ˙ T (α(t)) = T (α(t))[a ∗ (α(t)) dt where α˙ =

(15.26)

d α. dt

Proof It is enough to compute the derivative for t = 0. Using Lemma 3 the computations can be done for ψ ∈ D(N1/2 ). In what follows ψ will be omitted. We have T (α(t)) − T (α(0)) = T (α(0))T ∗ (α(0))(T (α(t)) − 1l) and T ∗ (α(0))T (α(t)) = T (α(t) − α(0))eiα(0),α(t) . Using Duhamel formula on D(N1/2 ) we get ˙ − a(α(0))) ˙ + O(t 2 ) T (α(t) − α(0)) = 1l + t (a ∗ (α(0)) hence d ∗ ˙ − a(α(0)) ˙ + iα(0), α(0). ˙ T (α(0))T (α(t))|t=0 = a ∗ (α(0)) dt 

The formula (15.26) follows.

In the following section we shall study the mean field behavior of large systems of bosons with weak two particles interactions. For that purpose we introduce one particle density operator ΓΨ(1) for every Ψ ∈ F (h), as follows. It is defined as a sesquilinear form in h: ( f, g) →

1 Ψ, a ∗ ( f )a(g)Ψ  :=  f, ΓΨ(1) g, f, g ∈ h. Ψ, NΨ 

If h = L 2 (R3 ) the Schwartz kernel of ΓΨ(1) satisfies ΓΨ(1) (x, y) =

1 Ψ, a ∗ (x)a(y)Ψ  ψ, NΨ 

(15.27)

15.3 The Bosons Coherent States

443

Moreover if Ψ is a k-particle state then ΓΨ(1) is the relative trace in h = L 2 (Rn ) of the projector |Ψ Ψ | and we have ΓΨ(1) (x, y) =

 Rn(k−1) ×Rn(k−1)

d x  dy  Ψ (x  , x)Ψ¯ (y  , y)

We shall be interested to considering the one particle density for Ψ (t) being a time evolution of a coherent state Ψ (ϕ (t)), where ϕ (t) = −1/2 ϕ(t), depending on a (1) . We shall see that, under some small (semi-classical) parameter . We call it Γ,t (1) converges in trace norm operator to |ϕ(t)ϕ(t)| when   0 where conditions, Γ,t ϕ(t) follows a classical evolution.

15.4 The Classical Limit for Large Systems of Bosons 15.4.1 Introduction We have already considered the classical limit problem for systems with a finite number of bosons in Chap. 1. It has been a natural question since the early days of quantum mechanics to compare the classical and quantum mechanical descriptions of physical systems. One of the oldest and by now best known relation between the two theories goes back to Ehrenfest [91]. This has been put on a firm mathematical basis by Hepp [148]. He proved that in the limit where the Planck constant 1 tends to zero, the matrix elements of quantum observables between suitable -dependent coherent states tend to the classical values evolving according to the appropriate equation. Moreover he proved that the quantum mechanical fluctuations evolve according to the equation obtained by linearizing the quantum mechanical evolution around the classical solution. Hepp’s approach covers the case of quantum mechanics (that we have studied in detail in Chap. 4) of boson field theories, both relativistic and nonrelativistic, and more generally of all quantum theories which can be expressed in terms of observables satisfying the Canonical Commutation Relations (CCR). One is led to study a perturbation problem for the evolution of a set of operators satisfying the (CCR) in a suitable representation. The small parameter which characterizes the perturbation theory is 1/2 . In the most favourable cases this evolution is implemented by a unitary group of operators W (t, s). The solution of the unperturbed problem is given by a unitary group U2 (t, s), the infinitesimal generator of which is quadratic in the field operators and depends on the classical solution around which one is considering the classical limit. The operator U2 (t, s) describes the evolution of the quantum fluctuations. Hepp’s result consists in proving strong convergence of W (t, s) towards U2 (t, s) when  goes to zero. We shall explain the results obtained by physics  is a constant equal to 1.055 × 10−34 J.S. As it is usual in quantum mechanics we consider here  as an effective Planck constant obtained by scaling, for example  → √ , where 2m m is the mass and   0 means m  +∞.

1 In

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15 Bosonic Coherent States

Ginibre-Velo [114–116] to estimate the error term in the Hepp’s results. We also explain results obtained more recently by Rodnianski-Schlein [225] using Hepp’s approach to get the convergence of the one particle marginal for evolved coherent states towards a classical field. Note that this result is somehow an extension to the quantum field context of the semi-classical expansion considered before in Chap. 4 for time evolution of Gaussian coherent states for a fixed number of bosons.

15.4.2 The Hepp Method We follow here the presentation given in [115]. We start with the abstract setting of a general Fock space F (h) and an orthonormal basis {ei }i∈I in h. I = {1, 2, . . . , v}, v ≤ +∞. Recall that ai := a(ei ). So the system is described by the family of quantum operators a = (ai )i∈I satisfying the canonical commutation rules: [ai , a j ] = 0, [ai , a ∗j ] = δi, j . The variables a expected to have a classical limit are related to a by a = 1/2 a or a ( f ) = 1/2 a( f ), f ∈ h or ai = 1/2 ai . Consider a self adjoint Hamiltonian H in F B (h), suitably regular (for instance polynomial in a∗ , a), with no explicit -dependence : H=



C(n 1 , . . . , n k |m 1 , . . . , m  )(a1∗ )n 1 . . . (a1∗ )n k a1m 1 . . . am  , C(•|•) ∈ C.

In the Heisenberg picture the time evolution a (t) of a is given by the following equation d (15.28) i a (t) = [a (t), H (a )], a (0) = a . dt We want to relate the time dependent operators a (t) with a family of -independent c-number variables2 : ϕ(t) = {ϕi (t)}i∈I which will appear to be the classical limits. ϕ(t) can be identified with a classical trajectory in the Hilbert space h writing ϕ(t) = ϕi (t)ei . i∈I

2 a c-number here is a family of time dependent complex numbers indexed by

I . It be can identified (1) with a vector in h. In the language of quantum mechanics c-numbers are the opposite of Γ,t , operators in an Hilbert space.

15.4 The Classical Limit for Large Systems of Bosons

445

We thus expand H in power series of ai − ϕi , (ai )∗ − ϕ¯i in a neighborhood of ϕ, ϕ: ¯ H(a ) = H (ϕ) + H1 (a − ϕ) + H2 (a − ϕ) + H≥3 (a − ϕ)

(15.29)

where the functions H1 , H2 , H≥3 are polynomials in a − ϕ, a∗ − ϕ¯ with total degree 1, 2 and ≥ 3 respectively, with time dependent coefficients. Defining Hk (a) = [a, Hk (a)], k = 1, 2, ≥ 3 we see that Hk are -independent, that H1 (a) is a c-number and that H2 is linear in a, a∗ . Equation (15.28) can be rewritten as i

d d  (a − ϕ). ϕ + i (a − ϕ) = H1 (a ) + H2 (a − ϕ) + H≥3 dt dt

(15.30)

But we have H1 (a ) = H1 (a). So we choose ϕ to be a solution of the classical evolution equation associated with Hamiltonian H , namely i We define

d ϕ = H1 (a). dt

(15.31)

ϕ = −1/2 ϕ.

Then Eq. (15.30) becomes i

d  (a − ϕ ) = H2 (a − ϕ ) + −1/2 H≥3 (1/2 (a − ϕ )). dt

Let us now introduce the Weyl operator for any (αi )i∈I T (α) = exp

 

 (αi ai∗

−

αi∗ ai )

= exp(a ∗ (α) − a(α)) where α =

i∈I



αi ei ,

i∈I

(15.32) where we have chosen initial time s = 0 for solving equation (15.28). Recall that T (α) are unitary and obey T (α)∗ aT (α) = a + α. Note that here we have chosen coordinates in h. If f = T (α). We define a new variable b(t) as



αi ei we have T ( f ) =

i∈I

b(t) = T (ϕ (s))∗ (a(t) − ϕ (t))T (ϕ (s)).

446

15 Bosonic Coherent States

The initial value problem for Eq. (15.28) then reduces to finding a family b(t) of operators satisfying the (CCR) and the relations b(0) = a d  (1/2 b) i b = H2 (b) + −1/2 H≥3 dt

(15.33)

Note that the second term in the right hand side of equation (15.33) is O(1/2 ) since  has degree at least two. Therefore b(t) is expected to converge towards the H≥3 solution of the linearized equation i

d  b = H2 (b ), b (0) = a dt

(15.34)

which governs the quantum fluctuations around the classical equation. We introduce the propagator U2 (t, s) defined by the quadratic Hamiltonian Hˆ 2 (t): i

d U2 (t, s) = Hˆ 2 (t)U2 (t, s), U2 (s, s) = 1l. dt

So we have

b (t) = U2 (t, 0)∗ aU2 (t, 0).

In the same way, the time evolution of operators b(t) will be implemented by a unitary group W (t, s) such that: b(t) = W (t, 0)∗ aW (t, 0) where W (t, s) obeys the differential equation i

  d W (t, s) = H2 (a) + −1/2 H≥3 (1/2 a) W (t, s), W (s, s) = 1l. dt

One has the following result, which is easily proved using Lemma 96. Proposition 141 One has: W (t, s) = exp(iω (t, s))Tˆ (ϕ (t))U (t − s)Tˆ (ϕ (s)) with

  U (t − s) = exp −i−1 (t − s)H (a )

and ω (t, s) = 

−1



t s

  dτ H (ϕ(τ )) − ϕ(τ ), H1 (ϕ(τ )) .

Therefore we have proven at least formally the following result.

(15.35)

15.4 The Classical Limit for Large Systems of Bosons

447

Proposition 142 T (ϕ (s))∗ U (t − s)∗ (a(s) − ϕ (t))U (t − s)T (ϕ (s)) = W (t, s)∗ a(s)W (t, s) with U (t) and W (t, s) given by Proposition 141. The difficult mathematical problem is to analyze the unitary propagators U2 (t, s) and W (t, s). One has the following result (see [148]): Let ϕ(t, x) be a solution of the classical Eq. (15.31) with initial data ϕ at t = s. For f ∈ L 2 (Rn ) let us define ϕ  ( f, t) =



d x f (x)ϕ  (t, x)

Similarly consider the solutions b (t) of the linearized problem (15.34). Let (b ) (t, f ) =



f¯i bi (t), if f =

i∈I



f i ei .

i∈I

As usual define the operator valued distributions (b ) (t, x) such that (b ) (t, f ) =



d x f (x)(b ) (t, x).

Here  denotes either nothing or ∗. It is possible to apply the strategy described above to prove semi-classical limit results for boson systems when   0. The Hamiltonian H is defined as H=

 2



d x ∇a ∗ (x) · ∇a(x) +

2 2



d x d y V (x − y)a ∗ (x)a ∗ (y)a(y)a(x).

(15.36) Note that the limit   0 is equivalent to the mean-field limit N  +∞ considered in [225] (N = −1 ) for the Hamiltonian   1 1 HN = d x ∇a ∗ (x) · ∇a(x) + d x d y V (x − y)a ∗ (x)a ∗ (y)a(y)a(x). 2 2N (15.37) Here we have  1 H1 = − Δϕ(x) + ϕ(x) dyV (x − y)|ϕ(y)|2 (15.38) 2 so that one requires that the classical evolution ϕt (x) = ϕ(t, x) is a solution of Hartree equation: 1 d (15.39) i ϕt = − Δϕt + ϕt (V ∗ |ϕt |2 ). dt 2

448

15 Bosonic Coherent States

To state rigorous results some technical assumptions are needed for the potential V . Recall the following definition (see [151] for more details). For simplicity we only consider the case h = L 2 (R3 ). Definition 39 A potential V (x), x ∈ R3 is called a Hardy potential if V is real and there exists C > 0 such that V ϕ L 2 (R3 ) ≤ C ϕ H 1 (R3 ) , ∀ϕ ∈ H 1 (R3 ).

(15.40)

c c ∈ R then V is a Hardy potential (by the usual It is well known that if V (x) = |x| Hardy inequality). Hardy class potentials included the Kato class potentials. (see [151] for details). The following proposition is a particular case of more general ones concerning Hartree equation [116]. The following proposition is sketched in [225], Remark 1.3. In [114–116] more refined results are given concerning solutions for the Hartree equation with singular potentials.

Proposition 143 Let V (x) be a Hardy potential. Let ϕ0 ∈ H 1 (R3 ). Then Eq. (15.39) has a unique solution ϕ ∈ C (R, H 1 (R3 )). Furthermore one has the property of conservation of the L 2 -norm and of the energy: ϕt = ϕ0 , ∀t ∈ R E (ϕt ) = E (ϕ0 ), ∀t ∈ R where E (ϕ) =

1 1 ∇ϕ 2 + 2 2

 d x d y V (x − y)|ϕ(x)|2 |ϕ(y)|2 .

Proposition 144 Let V (x) be an even Hardy potential. Assume that the initial data for the Hartree equation satisfies ϕs ∈ H 1 (R3 ). Then one has for |t − s| < T : (i) The Hamiltonian H is essentially self-adjoint on the dense subspace F0,0 of the finite particle states with compact supported Fourier transform. In particular the time evolutions U (t), U2 (t, s) and W (t, s) are well defined and are unitary operators in the Fock space FB (L 2 (R3 )). (ii) We have the following limit result for the fluctuation operator around the classical solution ϕ(t) (15.41) s − lim W (t, s) = U2 (t, s). →0

(iii) One also has s − lim T (ϕ )∗ U (t − s)∗ exp[(a ∗ ( f ) − ϕ ( f, t)) − h.c.]U (t − s)T (ϕ ) →0

= exp[(b )∗ ( f, t) − b ( f ∗ , t)] (15.42)

15.4 The Classical Limit for Large Systems of Bosons

449

where ϕ = −1/2 ϕ. Proof This proposition is essentially due to Hepp [148]. k

It is well known that for every k ∈ N, H is essentially self-adjoint in ∨L 2 (R3 ) so we get that H is essentially self-adjoint. (ii) is proved with Duhamel formula and the following weight estimate for U2 (t, s) proved in [115]. In this case the generator H2 of U2 (t, s) is given by 1 H2 = 2

where



 1 d x ∇a (x) · ∇a(x) + d xd y V (x − y)|ϕt (y)|2 a ∗ (x)a(x) + 2  1 d xd y ϕt (x)V (x − y)|ϕ¯t (y)|2 a ∗ (x)a(y) + L ∗ + L 2 (15.43) ∗



1 L= 2

d xd y ϕ¯t (x)V (x − y)ϕ¯t (y)a(x)a(y).

(15.44)

Lemma 97 For every δ > 0 and every T > 0 there exists C(δ, T ) such that Nδ U2 (t, s)N−δ ≤ C(δ, T ), for |t − s| ≤ T

(15.45)

The Duhamel formula gives 

t

W (t, s) = U2 (t, s) − i

W (t, τ )(1/2 H3 (τ, a) + H4 (a))U2 (τ, s)dτ

(15.46)

s

where H3 (t, a) = A3 (t) + A3 (t)∗ and  A3 (t) = d yd x V (x − y)ϕ¯t (x)a ∗ (y)a(y)a(x) 

and H4 =

d yd x V (x − y)a ∗ (x)a ∗ (y)a(y)a(x).

(15.47)

(15.48)

Using that F0,0 is dense in F (L 2 (R3 )) and that W (t, s) is unitary in F (L 2 (R3 )), it is enough to prove that lim W (t, s)Ψ = U2 (t, s)Ψ, for every Ψ ∈ F0,0 .

→0

(15.49)

This result is proved using (15.46)–(15.48), assumptions on V and ϕ and standard estimates. (iii) is proven using Proposition 142 and (i). 

450

15 Bosonic Coherent States

Corollary 51 Let A be a smooth and bounded function of a and a (see [23]). Then we have the following semi-classical limit evolution for quantum expectations in coherent states: lim U (t)Ψ (ϕ ), U (t)A(a − ϕ , a∗ − ϕ¯ )U (t)Ψ (ϕ ) =

→0

Ω, A(b (t), (b (t)∗ )Ω

(15.50)

where b (t) = U2 (t, 0)∗ a U2 (t, 0) is the linear evolution at the classical evolution ϕ(t). Remark 96 In [116] the authors proved a full asymptotic expansion in 1/2 for a dense subset of states ψ. We shall see later that we have a better result if V is bounded.

15.4.3 Remainder Estimates in the Hepp Method Hepp’s method was revisited by Ginibre-Velo [116, 117] to extend it to singular potentials and to get quantum correction in  at any order. The spirit of the works of [116, 117] is to exploit the differential equation (15.35) (and the formula (15.46)) to write a Dyson series expansion for W (t, s). The authors obtain a power series in 1/2 and study its analyticity properties in κ := 1/2 . For bounded potentials V this series can be shown to be Borel summable in vector norm when applied to fixed -independent vectors taken from a suitable dense set including coherent states. The potential V is also assumed to be stable, which means that there exists a constant B ≥ 0 such that (15.51) H4 + BN ≥ 0 (N is the number operator (15.16)). Roughly speaking it means that the potential is sufficiently repulsive near the origin if attractive somewhere else. More explicitly we can find in [229] a sufficient condition of stability for a potential V . H4 has the following expression (H4 Ψ )(k) (x1 , . . . , xk ) =



V (xi − x j )Ψ (k) (x1 , . . . , xk ).

i< j

Let V be such that V (x) ≥ ϕ1 (|x|), for |x| ≤ r1 V (x) ≥ −ϕ2 (|x|), for |x| ≥ r2 (15.52) where ϕ1 , ϕ2 are positive decreasing on ]0, r1 [, [r2 , +∞[ respectively and for some ν > 3 we have

15.4 The Classical Limit for Large Systems of Bosons





ϕ1 (t)t ν−1 dt = +∞,

451

ϕ2 (t)t ν−1 dt < +∞.

Then V is stable ([229], Proposition 3.2.8). Remark that the stability condition is a restriction on the negative part of V . Before to state the summability result let us recall some definitions concerning asymptotic series.  αjκ j. Consider a formal power complex series f  (κ) = j∈N 

Definition 40 (i) f is a Gevrey series of order 1/s, s > 0, if there exists C > 0, ρ > 0 such that (15.53) |α| ≤ Cρ j ( j!)1/s , ∀ j ∈ N. (ii) The s-Borel transform of the Gevrey series f  is defined as Bs f (τ ) =



αj

j∈N

Γ (1 + sj )

τj

(15.54)

where the series converges for τ ∈ C, |τ | < ρ −1 . (iii) The Gevrey series f  is said Borel s-summable if (iii)1 its s-Borel transform Bs f has an analytic extension to a neighborhood of the positive real axis, (iii)2 the following integral 

∞

duBs f (u)u s−1 e−( κ )

u s

0

converges for 0 < κ < κ0 . If this holds then we say that f  has a s-Borel sum f (κ) defined by f (κ) = sκ

−s



∞

duBs f (u)u s−1 e−( κ ) . u s

(15.55)

0

Remark 97 (i) For s = 1 the above definition corresponds to the usual Borel summability. (ii) At the formal level, formula (15.55) is easy to check using definition of Γ function and changes of variables. (iii) ( j!) j/s and Γ (1 + sj ) have the same order as j → as a consequence of the Stirling formula Γ (1 + u) =

√

2π u

 u u e

(1 + o(1)) as u → +∞.

452

15 Bosonic Coherent States

We state now a sufficient condition for Borel summability due to Watson, Nevanlinna and Sokal (see [214, 239] and references with extension to any s > 0). Theorem 82 Let f be an holomorphic function in the complex domain Ds,R := {κ ∈ C, κ −s > R −s } for some s > 0. Assume that there exist C > 0, ρ > 0 such that in this domain we have  α j κ j | ≤ Cρ N (N !)1/s |κ| N . (15.56) | f (κ) − 0≤ j 0. δ This is easily seen by stopping the series at the order N ≈ κ s with δ small enough. Adding an analytic condition in a suitable domain as in the above theorem erase this error term so that f (κ) is is uniquely determined. The following result gives an accurate asymptotic description for the quantum fluctuation operator W (t, s), improving Proposition 144. Theorem 83 Let V be stable and bounded potential. Let ϕ ∈ C 1 (R, L 2 (R3 )) be a solution of Hartree equation (15.39). Let β > 0 and Φ ∈ D(exp(β N )). Then there exists θ > 0 such that for all s, t ∈ R, t ≥ s such that t − s ≤ θ , W (t, s)Φ is analytic in κ in the sector − π2 < Argκ < 0, |z| small and has an asymptotic expansion at κ = 0 which is 2-Borel summable to W (t, s)Φ itself. The constant θ depends on V, β, ϕ but can be taken independent of B and uniform in V for V ∞ and B bounded, and uniform in ϕ for ϕ bounded. Proof We only give here the main steps for the proof. We refer to the paper [116] for a detailed proof. A first step is to consider the Duhamel formula (15.46). By iterating it we get a Dyson series. Then by reordering the Dyson series we get a formal power series in 1/2 :   j/2 W j (t, s) (15.57) W  (t, s) = j∈N

where the coefficient W j (t, s) are operators. W0 (t, s) = U2 (t, s). In a second step, 2-Gevrey type estimates 3 are obtained for W j (t, s)Φ where Φ ∈ D(eεN ), ε > 0 (Proposition 3.1 in [116]). In a third step it is proved that W (t, s)Φ has an analytic expansion in κ := 1/2 in a domain like D2,R . Here the stability condition is used. 3 Notations for Borel summability have been defined before for complex valued series, the extension

to vector valued series is straightforward.

15.4 The Classical Limit for Large Systems of Bosons

453

The last step consists in estimating the remainder term of the asymptotic expansion of W (t, s)Ψ in κ small. To do that the Dyson series expansion is performed in two steps, introducing an auxiliary evolution operator U4 (t, s) obeying i

d U4 (t, s) = (H2 (t) + H4 )U4 (t, s). dt

Then if U2 (t, s) is the propagator for H2 (t) (which describes the evolution of the quantum fluctuations), one has  U4 (t, s) = U2 (t, s) − i

t

dτ U2 (t, τ )H4 U4 (τ, s)

s



t

W (t, s) = U4 (t, s) − i

dτ U4 (t, τ )1/2 H3 W (τ, s)

s

Then it can be proved that the series its 2-Borel sum is W (t, s)Φ.

 j∈N

 j/2 W j (t, s)Φ is 2-Borel summable and 

Using similar methods the case of unbounded potentials has been considered in [117]. The authors obtain the same analyticity domain as in [116] of W (t, s) with respect to κ = 1/2 and obtain that the series is still asymptotic (in the Poincaré sense) and Gevrey of order 2. It is not known that the series is still 2-Borel summable for singular potentials. Note that D(exp(β N )) contains the coherent states Ψ ( f ), f ∈ L 2 (R3 ).

15.4.4 Time Evolution of Coherent States We have seen that the Hepp’s method concerns the quantum fluctuations in a neighborhood of a classical trajectory. It does not give informations for the quantum motion itself, in particular when the initial state is a coherent state. In Chap. 4 this problem was considered for finite bosons systems. Our goal here is to explain an extension of this result for large systems of bosons, following the paper [225]. According to the previous section one has to study the evolution operator U(t) = e−iH t associated with the Hamiltonian    1 H= d x ∇a ∗ (x) · ∇a(x) + d x d y V (x − y)a ∗ (x)a ∗ (y)a(y)a(x) 2 2 by the evolution equation i

d U(t) = HU(t) dt

and apply it to the coherent state Ψ (−1/2 ϕ) where ϕ = −1/2 ϕ, ϕ is a solution of the k

Hartree equation (15.39). By definition the Hamiltonian H leaves sectors ∨L 2 (R3 )

454

15 Bosonic Coherent States

with fixed number of particles invariant. One thus have U(t)∗ NU(t) = N

(15.58)

(1) (defined at the end of Sect. 15.3.) is the marginal One has the following result. Γ,t operator in the one particle space L 2 (R3 ) deduced from the quantum evolution U(t)Ψ (ϕ ) of the coherent state Ψ (ϕ ). Recall √ that in an Hilbert space h the trace norm of an operator A is defined as A Tr = Tr(A∗ A) (see Chap. 1).

Theorem 84 Suppose that V is a Hardy potential (see (15.40). Then there exists constants C, K > 0 (only depending on the H 1 (R3 ) norm of ϕ and on C) such that    (1)  Γ,t − |ϕt ϕt | ≤ Ce K t , t ∈ R. Tr

(15.59)

ϕt is the solution of Hartree equation (15.39) at time t with ϕ0 = 1. Proof We shall give here the main ideas of the proof. The details are in the paper [225]. (1) We now write the function Γ,t (x, y) using (15.27). One has to calculate the denominator: U(t)ψ(ϕ )Ω, NU(t)ψ(ϕ )Ω Using equation (15.58) we are left with ψ(ϕ )Ω, Nψ(ϕ )Ω. We now use the translation property of the Weyl operator: ∗

T (ϕ ) NT (ϕ ) =



d x (a ∗ (x) − ϕ¯ (x))(a(x) − ϕ (x))

(15.60)

But using that a(x)Ω = 0 we see that the expectation value of (15.60) in the vacuum simply equals −1 ϕ 2 = −1 . (1) (x, y) has the following decomposition. Therefore Γ,t (1) (x, y) = Ω, T (−1/2 ϕ)∗ U (t)∗ a ∗ (y)a(x)U (t)T (−1/2 ϕ)Ω Γ,t

= ϕ¯t (y)ϕt (x) + 1/2 ϕ(y)Ω, ¯ T (ϕ )∗ U (t)∗ (a(x) − ϕt, (x))U (t)T (ϕ )Ω +1/2 ϕt (x)Ω, T (ϕ )∗ U (t)∗ (a ∗ (x) − ϕt, (y))T (ϕ )Ω +Ω, T (ϕ )∗ U (t)∗ (a ∗ (y) − ϕ¯t, (y))(a(x) − ϕt, (x))U (t)T (ϕ )Ω

15.4 The Classical Limit for Large Systems of Bosons

455

Now we use the fact demonstrated in the previous section that T (ϕs, )∗ U (t)∗ (a(x) − ϕt, (x))U (t)T (ϕs, ) = W (t, s)∗ a(x)W (t, s) Thus we get the equality between the two kernels: (1) (x, y) − ϕt (x)ϕ¯t (y) = Ω, W (t, 0)∗ a ∗ (y)a(x)W (t, 0)Ω + Γ,t

1/2 ϕt (x)Ω, W (t, 0)∗ a ∗ (y)W (t, 0)Ω + 1/2 ϕ¯t (y)Ω, W (t, 0)∗ a(x)W (t, 0)Ω. (15.61) It is remarked in [225] that here it is enough to estimate the Hilbert-Schmidt norm (1) − |ϕt ϕt | instead of its trace-norm. So the main technical part of the paper of Γ,t [225] is to show that the L 2 norm in (x, y) of the right hand side of (15.61) is bounded above by Ce K t , using suitable approximation of the dynamics W (t, s). As in [116, 117] W (t, s) is compared with the dynamics U4 (t, s) generated by H2 (t) + H4 (without the H3 (t) term), namely i

d U4 (t, s) = (H2 (t) + H4 )U4 (t, s). dt

In [225] the following lemmas are proven. Lemma 98 One has (W (t, 0) − U4 (t, 0))Ω ≤ C1/2 e K t . Lemma 99 W (t, 0)Ω, NW (t, 0)Ω ≤ Ce K t . Now, introducing U4 (t, s), the right hand side of equation (15.61) is rewritten as (1) (x, y) − ϕt (x)ϕ¯t (y) = Ω, W (t, 0)∗ a ∗ (y)a(x)W (t, 0)Ω + Γ,t  1/2 ϕt (x) Ω, W (t, 0)∗ a ∗ (y)(W (t, 0) − U4 (t, 0))Ω +  Ω, (W (t, 0)∗ − U4 (t, 0)∗ )a ∗ (y)U4 (t, 0)∗ Ω +  1/2 ϕ¯ t (y) Ω, W (t, s)∗ a(x)(W (t, s) − U4 (t, s))Ω+  Ω, (W (t, s)∗ − U4 (t, s)∗ )a(x)U4 (t, s)Ω (15.62)

where we use that U4 (t, s) preserves the parity of the number of particles: Ω, U4 (t, s)∗ a ∗ (y)U4 (t, s)Ω = Ω, U4 (t, s)∗ a(x)U4 (t, s)Ω = 0.

456

15 Bosonic Coherent States

Then we get from Lemmas 98 and 99 the Hilbert-Schmidt estimate  R3 ×R3

(1) d xd y|Γ,t (x, y) − ϕt (x)ϕ¯t (y)|2 ≤ C2 e2K t , ∀t ≥ 0.



Remark 99 In [53] the authors have extended the previous result to the case of arbitrary factorized initial data.

Chapter 16

Fermionic Coherent States

Abstract This chapter is an introduction to some computation techniques for fermionic states. After defining Grassmann algebras it is possible to get a classical analogue for the fermionic degrees of freedom in a quantum system. Following the basic work of Berezin [23, 26], we show that we can compute with Grassmann numbers as we do with complex numbers: derivation, integration, Fourier transform. After that we show that we have quantization formula for fermionic observables. In particular there exists a Moyal product formula. As an application we consider explicit computations for propagators with quadratic Hamiltonians in annihilation and creation operators.

16.1 Introduction We have seen in the Chapter on Bosons 15 that the canonical commutation relations (CCR) can be realized with real or complex numbers. We shall see here that the anti-commutation relations (CAR) require the introduction of new kind of numbers, nilpotents, called Grassmann numbers. In some sense, nilpotence is classically equivalent to the Pauli exclusion principle which characterizes fermions. This appears to be strange from a physicist point of view because the Pauli exclusion principle is a purely quantum property, without a classical analogue. Nevertheless such a mathematical model exists. Even if “classical fermions” do not exist in Nature, they are a convenient mathematical tool for computations and allow to put bosons and fermions on the same footing. This is important to elaborate supersymmetric models which will be considered in more details in the next Chapter. Our main goal here is to introduce fermionic coherent states and to describe their properties. The main difference from the bosonic case considered in Chap. 1 is that we have to replace complex numbers by Grassmann numbers. So the constructions have to be revisited with some care. So we shall study in more details Grassmann algebras and we shall see that many properties and constructions well known for usual numbers can be extended to Grassmann algebras. Our construction of fermionic

© Springer Nature Switzerland AG 2021 D. Robert and M. Combescure, Coherent States and Applications in Mathematical Physics, Theoretical and Mathematical Physics, https://doi.org/10.1007/978-3-030-70845-0_16

457

458

16 Fermionic Coherent States

coherent states mainly follows the paper [47]. We also consider in more details the propagation of Fermions for quadratic evolutions. In [25] (see the book [208], Chap. 9) the author introduced Fermionic coherent states from another point of view, without Grassmann algebras. He considered an irreducible representation of the rotation group SO(2n, R) (for a system of n fermions) called the spin representation. We shall see that the two points of views are mathematically equivalent.

16.2 From Fermionic Fock Spaces to Grassmann Algebras It follows from the Chapter on bosons that the fermionic Fock space is the Hilbert space FF (h) = ⊕k∈N ∧k h, where h is the one fermion space and ∧k h is the antisymmetric tensor spanned by the states 1  επ ψπ1 ⊗ ψπ2 ⊗ · · · × ψπk , ψj ∈ h. k! π∈Sk

For any f ∈ h, the annihilation and creation operators a(f ), a∗ (f ) are bounded operators in FF (h). This is a consequence of the Canonical Anticommutation Relation a(f1 )a∗ (f2 ) + a∗ (f1 )a(f2 ) = f1 , f2 1l.

(16.1)

Let us begin by an explicit model to realize the CAR relations (16.1) which is called the spin model. We start with HF = C2 and the matrices     01 00 σ+ = , σ− = . 00 10 So we have (CAR) for a = σ− and a∗ = σ+ : [σ+ , σ− ]+ = 1l2 , σ+2 = σ−2 = 0. ∗ This  is a model for the states of one  spin.   We have the number operator N = a a = 10 0 and the ground state is e0 = . 00 1 n We get a model for n spin states in the Hilbert space HH = (C2 )⊗n = C2 . The annihilation and creation operators are

16.2 From Fermionic Fock Spaces to Grassmann Algebras

459

ak = σ3 ⊗ · · · σ3 ⊗a ⊗ 1l2 ⊗ · · · ⊗ 1l2 ,       k−1

n−k

ak∗ = σ3 ⊗ · · · σ3 ⊗a ∗ ⊗ 1l2 ⊗ · · · ⊗ 1l2 .       n−k

The ground state is here Ω0 = e0 ⊗ · · · ⊗ e0 .    n

This model is unitary equivalent to the fermionic Fock model with h = Cn . We have HF = C ⊕ Cn ⊕ (∧2 Cn ) ⊕ · · · ⊕ (∧n Cn ). Notice that dim(HF ) = 2n . Let us give now some specific properties for the general fermionic Fock model (for detailled proofs see [42]). Proposition 145 (1) For every f ∈ h, a(f ) and a∗ (f ) are bounded operators in FF (h): (16.2) a(f ) = a∗ (f ) = f , ∀f ∈ h. (2) Let {ϕj }j∈J be an orthonormal basis for h. The family of states defined as ψj1 ,j2 ,...,jk = a∗ (ϕj1 ) . . . a∗ (ϕjn )Ω, jk ∈ J , n ≥ 0, is an orthonormal basis for FF (h). (3) If T is a bounded operator in FF (h) commuting with all the operators a(f ) and a∗ (g), f , g ∈ h, then T = λ1l for some λ ∈ C. Property (3) of the Proposition means that the Fock representation for Fermion is irreducible. When the system has n identical particles the creation/annihilation operators are denoted aj(∗) := a(∗) (ϕj ), 1 ≤ j ≤ n. A natural problem is to represent the anti-commutation relations (CAR) with d or derivative and multiplication operators as it can be done for (CCR) with q and dq in the Fock–Bargmann representation (see Chap. 1). In other word we would like to represent anticommutation relations like ∂ ∂ θ +θ = 1l. ∂θ ∂θ

(16.3)

This is not possible in the naive sense. This problem is equivalent to build a classical analogue for Fermionic observables. In order to satisfy (16.3) it is necessary to replace the usual real or complex numbers by Grassmann variables as it was discovered by Berezin [26]. Let us define the Grassmann algebras. Definition 41 The Grassmann algebra Gn with n generators {θ1 , . . . , θn } is an algebra, with unit 1, a product and a K-linear space (K = R or K = C) such that

460

16 Fermionic Coherent States

θj θk + θk θj = 0, ∀j, k = 1, . . . , n. and every g ∈ Gn can be written as g = c0 (g) +

 

cj1 ,...,jk (g)θj1 . . . θjk .

(16.4)

k≥1 j1 ,...,jk

where c0 (g) and cj1 ,...,jk (g) are K-numbers. This definition has some direct algebraic consequences given below and easy to prove 1. In equality (16.4) the decomposition is not unique. We get a unique decomposition if we add in the sum the condition j1 < j2 < · · · < jk . So we get a basis of Gn , {θj1 . . . θjk , j1 < j2 < · · · jk ; 1 ≥ k ≥, n} and its dimension is 2n . It is sometimes convenient to introduce E [n] = {0, 1}n , ε = (ε1 , . . . , εn ) where εk = 0, 1, and θ ε = θ1ε1 . . . θnεn . So the above basis can be written as {θ ε , ε ∈ E [n]}. |ε| := ε1 + · · · + εn is the number of fermionic states occupied. In particular (16.4) can be written as g=



cε (g)θ ε

(16.5)

ε∈E [n]

2. g ∈ Gn is invertible if and only if c0 (g) = 0. 3. Equality (16.4) can be interpreted as a generating function for fermionic states where cj1 ,...,jk (g) are the coefficients of a states in the basis {ψj1 ,j2 ,...,jk } of Proposition 145. 4. Derivatives are defined as follows. If g ∈ Gn , for 1 ≤ j ≤ n, we have g = g0 + θj g1 where g0 and g1 are independent on θj . So we define the (left derivative) ∂g = g1 ∂θj We also a right derivative denoted g ∂θ∂ j and defined by writing g = g0 + g2 θj ,

where g0 and g2 are θj -independent, so g2 =: g ∂θ∂ j . 5. We have (CAR) relations

∂ , θˆk = δj,k ∂θj +

(16.6)

where θˆk is left multiplication by θk in Gn and [•]+ is the anticommutator in Gn : [f , g]+ = f g + gf . Let us remark that (16.6) are analogue for fermions of the commutation relation for bosons in the holomorphic representation (see Chap. 1). It is possible to write the relations (CAR) in a real form. For a finite system we introduce the self-adjoint operators:

16.2 From Fermionic Fock Spaces to Grassmann Algebras

 ˆj = Q

461

  ∗ −1  ∗ ˆ (aF,j + aF,j ), Pj = i (aF,j − aF,j ). 2 2

(16.7)

So (16.6) is transformed to ˆ k ]+ = [Pˆ j , Pˆ k ]+ = δj,k 1l [Q ˆ j , Pˆ k ]+ = 0, 1 ≤ j, k ≤ n ˆ j, Q [Q

(16.8)

Equation (16.8) can be compared to the relations (CCR) : anti-commutators replace commutator. We have seen in Chap. 1 that (CCR) is a representation of the Weyl– Heisenberg Lie algebra. (CAR) (16.8) is a representation of the Clifford algebra Cl(Rn ) which is defined below. Definition 42 Let Φ be a symmetric bilinear form on a linear space V (over K = R or C). The Clifford algebra is the associative algebra with unit 1 denoted, Cl(V , Φ), generated by V and such that for every u, v in V we have u · v + v · u = Φ(u, v) · 1.

(16.9)

Taking V = Rn with the canonical basis {ek }1≤k≤n and Φ the usual scalar product the Clifford algebra Cl(Rn ) is the algebra defined by the relations ej · ek + ek · ej = δj,k .

(16.10)

Hence we see that the anti-commutation relations (16.8) for  = 1 define a representations of the Clifford algebra Cl(R2n ) as the commutation relations (CAR) are a representation of the Weyl–Heisenberg algebra. We shall see later that Cl(R2n ) is not a Lie algebra but a graded Lie algebra or super-Lie algebra. On the other side the representation (16.6) of (CAR) is equivalent to the representation given in Proposition 145. Proposition 146 Let us consider a system of n identical fermions ( h = Cn ). The linear map Φ from FF (Cn ) onto Gn is defined by Φ(ψj1 ,...,jk ) = θj1 . . . θjk . We have aj∗ = Φ −1 θˆj Φ, and aj = Φ −1

∂ Φ. ∂θj

(16.11)

16.3 Integration on Grassmann Algebra The construction follows the bosonic construction with the important difference that complex numbers are replaced by Grassmann variables which are anticommuting. In particular the classical-quantum correspondence has many differences from the

462

16 Fermionic Coherent States

bosonic case. The main tool is derivation-integration over Grassmann variables introduced by Berezin and which can appear to be sometimes strange. But it is the right algebra to preserve similarities with Bosons and to have a classical analogue of Fermions.

16.3.1 More Properties of Grassmann Algebras A Grassmann algebra can be defined on a field K = R or K = C. The Grassmann algebra with n generators {θ1 , . . . , θn } will be denoted Gn or K[θ1 , . . . , θn ]. In applications, we have to replace K itself by a Grassmann algebra Gm := K[ζ1 , . . . , ζm ] where [ζj , ζk ]+ = 0. In this case we shall consider the right Gm -module: Gm [θ1 , . . . , θn ] with generators {θ1 , . . . , θn }. It will be sometimes denoted Gnm . More explicitly a generic element g of Gnm can be written as g=



θi1 . . . θik · ci1 ...ik

i1 n. In particular if θ = (γ1 , . . . , γn ) are 2n Grassmann generators then eθ·γ is well defined (θ1 , . . . , θn ), γ =  where θ · γ = θk γk . 1≤k≤n

Lemma 101 If k ≥ 2 we have θ1 θ2 . . . θk = (−1)μ(k) θk θk−1 . . . θ1 .

(16.13)

where μ(k) =

1 if k ≡ 2, 3 mod(4) 0 if k ≡ 0, 1 mod(4)

eθ·γ = 1 + θ · γ +



(−1)ν(ε) θ ε γ ε

(16.14)

(16.15)

ε∈E [n],|ε|≥2

where the integer ν(ε) is defined as follows ν(ε) =

1 if |ε| ≡ 2, 3 mod (4) 0 if |ε| ≡ 0, 1 mod(4)

(16.16)

Proof Equation (16.13) is true for k = 1, 2. We get the general case by induction on k using Lemma 100. Equation (16.15) is proved by induction on n using the identity eθ·γ =

1≤k≤n

e θ k γk =

(1 + θk γk ).

1≤k≤n



464

16 Fermionic Coherent States

16.3.2 Calculus with Grassmann Numbers We have already defined derivatives in Grassmann algebras in the previous section. To preserve analogy with bosons it is useful to define integration in Grassmann variables. Definition 43 Consider a Grassmann algebra Gn with generators {θ1 , . . . , θn }. Let ψ ∈ Gn and 1 ≤ j1 , . . . , jk ≤ n. ψd θj1 . . . d θjk ∈ Gn is defined by the following properties. (i) (ii) (iii) (iv) (v)

ψ → ∫ ψd θj1 . . . d θjk ∈ Gn is linear ∫ d θj = 0, ∀j = 1, . . . , n. ∫ θj d θk = δj,k , ∀j, k. ∫ d θj d θk = 0, ∀j, k.   ∫ ∫ ψ(θj )ϕ(θk )d θj d θk = ψ(θj )d θj ϕ(θk )d θk , ∀j, k

We see that for fermions integration coincides with differentiation; we have easily  ψd θj1 . . . d θjk =

∂ ∂ ... ψ(θ1 , . . . , θn ). ∂θjk ∂θj1

(16.17)

This formula will be used to compute integrals. Lemma 102 (change of variables) Let A be a real invertible n × n matrix (A may have its coefficient in a Grassmann algebra such that A is even and invertible). Then we have:   ψ(Aθ )d θ = det(A) ψ(θ )d θ. (16.18) Proof If θ  = θ A we have



θk =

θj Aj,k .

1≤j≤n

Using definition of determinant we get θn . . . θ1 = θn . . . θ1 (det A).  But we have ψd θ = ψ(1,...,1) where ψ(1,...,1) is the component of ψ in its expansion  ψ(θ ) = θ ε ψε . Hence we deduce (16.18).  ε∈E [n]

From the Leibnitz rule we deduce an integration by parts formula:  

∂ψ ∂θk



∗

ϕd θ d θ =



 Pψ

∂ϕ ∂θk



d θ ∗ d θ.

(16.19)

16.3 Integration on Grassmann Algebra

465

Remark on notation: here d θ ∗ d θ means

d θj∗ d θj . Sometimes it will be

1≤j≤n

denoted d 2 θ .

16.3.3 Gaussian Integrals In order to preserve analogy with the Bargmann-Fock realization with holomorphic states (see Chap. 1) we need to have a complex structure on our Grassmann algebra. So we consider the Grassmann algebra G2n with generators {θ1 , . . . , θn ; θ1∗ , . . . , θn∗ }. In G2n we can define a complex anti- linear involution such that θk → θk∗ . We impose that ψ → ψ ∗ is R-linear and satisfies (i) (zα)∗ = z¯ α ∗ if z ∈ C and α is a generator (ii) (αβ)∗ = β ∗ α ∗ for every α, β ∈ G2n Let us denote θ ∗ = (θ1∗ , . . . , θn∗ ). The Grassmann algebra G2n with its complex structure will be denoted Gnc . Integration on Gnc is defined as follows 

ψd θ d θ ∗ =



ψd θ1 d θ1∗ . . . d θn d θn∗ .

We have the following property.  Lemma 103 The integral ψd θ d θ ∗ is invariant under unitary change of variable: θ = U ζ , U ∈ SU (n). So we have   ψd θ d θ ∗ = ψ(U¯ ζ, U ζ )d ζ d ζ ∗ . Proof From the proof of the change of variable lemma we have: θn . . . θ1 = (det U )ζn . . . ζ1 . Then we get θn∗ θn . . . θ1∗ θ1 = (ζn∗ ζn . . . ζ1∗ ζ1 )det U det U = ζn∗ ζn . . . ζ1∗ ζ1 . 

The lemma follows. Proposition 147 Let B be a n × n Hermitian matrix. Then we have 

e−θ

∗

·Bθ

d θ ∗ · d θ = det B.

(16.20)

Proof There exists a unitary matrix U such that U BU ∗ = D where D is the diagonal matrix D = (b1 , . . . , bn ). Using the change of variables η = U θ we get

466

16 Fermionic Coherent States



e−θ

∗

·Bθ

dθ∗ · dθ =



e−η

∗

·Dη

d η∗ · d η =



∗

e−bj ηj ηj d ηj∗ d ηj = det B.

1≤j≤n

. We can extend the above proposition to more general Gaussian integrals. Proposition 148 Let C[γ , γ ∗ ] be an other copy of the complex Grassmann algebra Gnc and B a n × n Hermitian matrix. Then we have 

e−θ

∗

·Bθ γ ∗ ·θ+θ ∗ ·γ

e

d θ ∗ · d θ = (det B)eγ

∗

·B−1 γ

.

(16.21)

Proof As in the proof of Proposition 147 we begin by a unitary change of variable to diagonalize B. Then we get the result after some computations (reduction to the case n = 1) left to the reader.  Another useful Gaussian integral computation is Proposition 149 Let us consider a quadratic form in (γ , γ ∗ ): Φ(γ , γ ∗ ) =

1 (γ · Kγ + γ ∗ · Lγ ∗ + 2γ ∗ · M γ ) 2

where K, L are antisymmetric matrices. So the following matrix Λ of Φ is antisymmetric, where   K −M T . Λ= M L Assume Λ is non degenerate. Then we have the Fourier transform result 

∗

eΦ(γ ,γ ) eγ

∗

·ξ −ξ ∗ ·γ

d γ ∗ d γ = PfΛeΦ

(−1)

(ξ ∗ ,ξ )

(16.22)

where PfΛ is the Pfaffian of Λ and Φ (−1) is the quadratic form with the matrix Λ−1 . Proof For ξ = 0 it is well known that the integral is equal to PfΛ which is the Pfaffian of Λ (see [269]). Recall that PfΛ2 = det Λ. Then by a usual trick we can eliminate the linear terms in the integral. We do that by performing a change of Grassmann variables γ = θ − αξ , γ ∗ = ζ − βξ . θ and ζ are new Grassmann integration variables (not necessarily conjugated) and αξ , βξ are computed such that the linear terms are eliminated. So we find    ∗ αξ −1 ξ =Λ . βξ ξ The formula (16.22) follows.



16.4 Super-Hilbert Spaces and Operators

467

16.4 Super-Hilbert Spaces and Operators As in the bosonic case we want to find a space to represent fermionic states as functions in some L2 -space.

16.4.1 A Space for Fermionic States Let us define H (n) the subspace of G2n of holomorphic functions in θ = (θ1 , . . . , θn ). ψ ∈ H (n) means ∂θ∂ ∗ ψ = 0 for 1 ≤ k ≤ n. k

H (n) is a complex vector space of dimension n with the basis {θ ε , ε ∈ E [n]}. Moreover H (n) is a Hilbert space for the scalar product  ψ, ϕ :=

eθ

∗

·θ

ψ(θ )∗ ϕ(θ )d θ d θ ∗ .

and {θ ε , ε ∈ E [n]} is an orthonormal basis. We shall denote eε (θ ) = θ ε for ε ∈ E [n]. The space H (n) has dimension 2n over C and is isomorphic to the Fock space HF(n) . Remark 100 The notation ψ ∗ for the complex involution is sometimes replaced by the more suggestive notation ψ¯ which may be sometimes confusing. The creation/annihilation operators in H (n) are (aj∗ ψ)(θ ) = θj ψ(θ ), aj ψ(θ ) =

∂ ψ(θ ). ∂θj

They satisfy anti-commutation rules (CAR) and ak∗ is the hermitian adjoint of ak for every 1 ≤ k ≤ n. The Hilbert space H (n) is not large enough to represent Fermionic states and to compute with them. As usual in Grassmann calculus we need to add new Grassmann generators (γ , γ ∗ ), γ = (γ1 , . . . , γn ). Let us denote Γnc := C[γ , γ ∗ ]. As above we introduce H˜ (n) the sub-Grassmann algebra of Gnc ⊗ Γnc of holomorphic elements in θ . H˜ (n) will be seen as a module1 over the algebra Γnc with basis {θ ε , ; ε ∈ E [n]}. ˜ H (n) is a kind of linear space where the complex field number is replaced by the Grassmann algebra numbers: Γnc . The following operations make sense because they are well defined in Gnc ⊗ Γnc , with usual properties easy to state: if ψ, ϕ ∈ H˜ (n) , λ ∈ Γnc , then ψ + ϕ ∈ H˜ (n) and λψ, ψλ ∈ H˜ (n) . In particular every ψ˜ ∈ H˜ (n) can be decomposed as 1 A module over some ring or algebra is like a linear space where the field number R or C is replaced

by a ring or an algebra (see any textbook in advanced algebra).

468

16 Fermionic Coherent States

ψ=



θ ε cε (ψ)

ε∈E [n]

where cε (ψ) ∈ Γnc . The scalar product in H (n) can be extended as sesquilinear form to H˜ (n) by the formula  ∗ ψ, ϕ := ψ(θ )∗ ϕ(θ )eθ ·θ d θ d θ ∗ but now ψ, ϕ is a Grassmann number in Γnc and not always a complex number. The map (ψ, ϕ) → ψ, ϕ has usual properties: it is sesquilinear and non negative. But it is degenerate and only its restriction to the Hilbert space H (n) is non-degenerate. Here we do not use the general theory of super-space, on our examples direct computations can be done (see the book [76] for more details concerning super-Hilbert spaces). Let us remark that every linear operator in H (n) can be extended as a linear operator in H˜ (n) . In particular the parity operator P in variables θ is well defined. As a first application let us consider the Fermionic Dirac distribution at θ . It is not difficult to establish the following identity for every ψ ∈ H (n) ,  ψ(θ ) =

∗

ψ(γ )e−(θ−γ )·γ d γ ∗ d γ .

(16.23)

In other words the identity is an integral operator with a kernel ∗

E(θ, γ ) := e−(θ−γ )·γ =

(1 − (θk − γk )γk∗ ).

(16.24)

1≤k≤n

It is usual to denote E(θ, γ ) = δ(θ − γ ). We have also the more symmetric form δ(θ − γ ) =

(θk − γk )(θ ∗ − γk∗ )

1≤k≤n

This could be used to prove that every linear operator in H (n) has a kernel.

16.4.2 Integral Kernels Proposition 150 For every linear operator Hˆ : H (n) → H (n)2 there exists KH ∈ C[θ, γ ∗ ] such that

2 As

for bosons, when we consider quantization of observables, it is convenient to denote operators with a hat accent to make a difference between classical and quantum observables. Sometimes this rule is not applied when the context is clear.

16.4 Super-Hilbert Spaces and Operators

Hˆ ψ(θ ) =



469

KH (θ, γ ∗ )ψ(γ )eγ

∗

·γ

d γ d γ ∗ , ∀ψ ∈ H (n) .

(16.25)

Moreover KH ∈ C[θ, γ ∗ ] can be computed as follows: 

KH (θ, γ ∗ ) =

ε ,ε∈E

ˆ ε eε (θ )eε (γ )∗ . eε , He

(16.26)

[n]

Proof Using that {eε } is an orthonormal system it is not difficult to prove that Eq. (16.25) is satisfied for ψ = eε with KH given by (16.26). So get the result for every ψ.  Corollary 52 Every linear operator Hˆ : H (n) → H (n) has a unique decomposition like   Hε,ε a∗ ε aε (16.27) Hˆ = ε ,ε∈E [n]

where Hε,ε ∈ C. These results can be easily extended to linear operators in the super-Hilbert space H˜ (n) . Proposition 151 For every linear operator Hˆ : H˜ (n) → H˜ (n) there exists KH ∈ C[θ, γ , γ ∗ ] such that Hˆ ψ(θ ) =



KH (θ, γ ∗ )ψ(γ )eγ

∗

·γ

d γ d γ ∗ , ∀ψ ∈ H˜ (n) .

(16.28)

Moreover K ∈ C[θ, γ , γ ∗ ] can be computed as follows: K(θ, γ ∗ ) =



eε (θ )eε , Hˆ eε eε (γ )∗ ,

(16.29)

ε ,ε∈E [n]

and every linear operator Hˆ in H˜ (n) has a unique decomposition Hˆ =





Hε,ε a∗ε aε

(16.30)

ε ,ε∈E [n]

where Hε,ε ∈ Γnc . This representation, with annihilation operators, first is call the normal representation of Hˆ . We denote End(H˜ (n) ) the space of linear operators in H˜ (n) . Corollary 53 Every linear operator Hˆ in H˜ (n) has an hermitian conjugate Hˆ ∗ : ψ, Hˆ ϕ = Hˆ ∗ ψ, ϕ, ∀ψ, ϕ ∈ H˜ (n) .

(16.31)

470

16 Fermionic Coherent States

Moreover Hˆ → Hˆ ∗ is a linear complex involution in End(H˜ (n) ), satisfying for every linear operator Hˆ , Fˆ in H˜ (n) and λ ∈ Γnc , ˆ ∗ = Fˆ ∗ Hˆ ∗ . • (Hˆ F) ˆ • (λH )∗ = Hˆ ∗ λ∗ , (Hˆ λ)∗ = λ∗ Hˆ ∗ . • ak∗ is the hermitian conjugate of ak for 1 ≤ k ≤ n. It is convenient to define a fermionic Fourier transform which is a kind of Fouriersymplectic transform. This will be used for fermionic quantization.

16.4.3 A Fourier Transform Definition 44 For any ψ ∈ H˜ (n) the Fourier transform is defined as ψ F (α) =



eξ

∗

·α−α ∗ ·ξ

ψ(ξ )d ξ ∗ d ξ.

The following properties are easy to prove  ∗ ∗ 1. ψ(ξ ) = e(α ·ξ −ξ ·α) ψ F (α)d α ∗ d α (inverse formula) The Fourier transform is idempotent: (ψ F )F = ψ. 2. 1 F (α) = α ∗ · α, ξ F (α) = α, ξ ∗F (α) = −α ∗ . 3.  ψ F (α)(ϕ F (α))∗ d α ∗ d α = ψ(ξ )(ϕ(ξ ))∗ d ξ ∗ d ξ (Parseval’s relation 1) α ∗ d α = ψ(ξ )ϕ(ξ )d ξ ∗ d ξ (Parseval’s relation 2) 4. ψ F (α)ϕ F (−α)d  5. (ψϕ)F (α) = ψ F (α − β)ϕ F (β)d β ∗ d β (Fourier-convolution) ∗ ∗ 6. (τζ ψ)F (α) = ψ F (α)eζ ·α−α ·ζ , (translation-modulation), where τζ ψ(ξ ) = ψ(ξ − ζ ). Let us remark that these properties are also satisfied if ψ, ϕ are replaced by linear operators in End(H˜ (n) ) depending on Grassmann variables.

16.5 Coherent States for Fermions As in the previous section we consider two complex Grassmann algebras Gnc = C[θ, θ ∗ ] and Γnc = C[γ , γ ∗ ]. H (n) (and H˜ (n) ) are spaces of holomorphic states in variables θ .

16.5.1 Weyl Translations The Weyl translations are defined as usual:

16.5 Coherent States for Fermions

471

∗ ∗ Tˆ (γ ) = ea ·γ −γ ·a , γ = (γ1 , . . . , γn ).

(16.32)

Remark that Tˆ (γ ) depends on the 2n independent Grassmann variables (γ , γ ∗ ). Nevertheless we simply note Tˆ (γ ) and sometimes Tˆ (γ , γ ∗ ) if necessary. They are translations in the phase space and in particular we have Tˆ (0, γ ∗ )ψ(θ ) = ψ(θ − γ ∗ ), Recall that γ · a =



Tˆ (γ , 0)ψ(θ ) = eθ·γ ψ(θ ).

γk ak and a∗ · γ =

1≤k≤n



ak∗ γk (beware of the order!).

1≤k≤n

Using anti-commutations relations (CAR) we have the commutation relations: αj aj = −aj αj for αj = γj , γj∗ [aj∗ γj , γk∗ ak ] = γj γk∗ δj,k [aj , ak∗ γk ] = γk δj,k [aj , ak γk ] = 0.

.

(16.33)

We have similar relation inverting a and a∗ and using [A, B]∗ = −[A∗ , B∗ ]. From these relations we get: 

 1 1 + ak∗ γk − γk∗ ak + (ak∗ ak − )γk∗ γk . 2 1≤k≤n 1≤k≤n (16.34) The following properties are easily obtained with little algebraic computations similar to the bosonic case (see Chap. 1). In particular we also have a Baker–Campbell– Hausdorff formula. Tˆ (γ ) =

∗

∗

eak γk −γk ak =

Lemma 104 Let A, B, be Γnc -linear operators in H˜ (n) such that [A, B] commutes with A, B. Then 1 eA eB e− 2 [A,B] = eA+B . Proof Let us remark first that eA is well defined by the Taylor series because dim[H˜ (n) ] < +∞ and eA is a Γnc -linear operator. Hence the result follows as in Lemma 1 of Chap. 1. . Now we state mains properties of the translation operators Tˆ (γ ). 1. Tˆ (γ ) is Γnc -linear in H˜ (n) (on right and left)   2. Tˆ (α + γ ) = Tˆ (α)Tˆ (γ ) exp 21 (α ∗ · γ + α · γ ∗ ) 3. (Tˆ (γ ))−1 = Tˆ (−γ ) = Tˆ (γ )∗ . In particular Tˆ (γ ) is a unitary operator in the superHilbert space, that means that the super-inner product is preserved: Tˆ (γ )ψ, Tˆ (γ )ϕ = ψ, ϕ 4. Translation property:

472

16 Fermionic Coherent States

Tˆ (γ )∗ aTˆ (γ ) = a + γ Tˆ (γ )∗ a∗ Tˆ (γ ) = a∗ + γ ∗ .

(16.35) (16.36)

16.5.2 Fermionic Coherent States Now we can give a definition for Fermionic coherent states. Definition 45 For every Grassmann generator γ we associate a state in the superHilbert space H˜ (n) denoted ψγ = |γ  by the formula ψγ = Tˆ (γ )ψ∅

(16.37)

where ψ∅ (θ ) = e0,...,0 (θ ) is the vacuum state, usually denoted |0. First properties of coherent states ψγ are easily deduced from properties of the translation Tˆ (γ ). 1. ψγ are eigenvectors of annihilation operators a: aψγ = γ ψγ where a = (a1 , . . . , an ) and γ = (γ1 , . . . , γn ). 2. The inner product of two coherent states satisfies

3.

  1 ψγ , ψα  = exp γ ∗ · α − (γ ∗ · γ + α ∗ · α) 2

(16.38)

  1 ∗ (α · γ − γ ∗ · α) ψα+γ . Tˆ (γ )ψα = exp 2

(16.39)

We can give a more explicit formula for ψγ which is very similar to that obtained in the Bargmann representation for bosons (Chap. 1). From above computations we have ψγ (θ ) = e−γ

∗

·γ /2

(1 + ak∗ γk )ψ∅

1≤k≤n

=e

−γ ∗ ·γ /2

=e

−γ ∗ ·γ /2



ak∗1 γk1 . . . ak∗j γkj ψ∅ .

k1 0 the eigenstates for E are pair (ψ B , ψ F ) where ψ F = Qψ ˆ ˆ ˆ If E = 0 then H ψ = 0 if and only if Q 1 ψ = Q 2 ψ = 0 then it is said that supersymmetry is preserved.

17.2 Quantum Supersymmetry

501

ˆ ∩ D( Qˆ ∗ ) such that Qψ ˆ = 0 and Qˆ ∗ ψ = 0 then we But if there exists ψ ∈ D( Q) say that supersymmetry is broken. Then the vacuum energy E satisfies E > 0. Let us now consider the well known Witten example of a quantum mechanical system with supersymmetry. It is a spin 21 exchange model in one degree of freedom.   1 0 2 2 So H = L (R, C ) with the unitary involution σ3 = . The supercharge 0 −1 is     00 . (17.2) Qˆ = 1i ddx + V (x) σ− , σ− = 10     01 Qˆ ∗ = 1i ddx − V (x) σ+ , σ+ = . (17.3) 00 An easy computation gives the following Hamiltonian   d2 σ3 1

2 ˆ − 2 + V (x) 1l2 − V

(x). H= 2 dx 2

(17.4)

Assume for simplicity that V (x) is a polynomial of even degree: V (x) = c0 x 2k + · · · cm x 2k+m with c0 > 0. Then Hˆ is a self-adjoint operator with a discrete spectrum. ˆ = 0 and Qˆ ∗ ψ = 0. We find It is easy to solve equations Qψ ˆ = 0 ⇐⇒ ψ(x) = ψ(x0 )eV (x0 )−V (x) . Qψ Qˆ ∗ ψ = 0 ⇐⇒ ψ(x) = ψ(x0 )eV (x)−V (x0 ) .

(17.5) (17.6)

ˆ = 0 has non zero L 2 solution and Qˆ ∗ ψ = 0 has no non zero L 2 So we see that Qψ solution, hence the supersymmetry is broken. 2 A supersymmetric harmonic oscillator is obtained with V (x) = ωx2 , ω > 0. So we get the Hamiltonian   d2 1 ω 2 2 ˆ − 2 + ω x 1l2 − σ3 Hsos = 2 dx 2 ˆ It is easy to get the spectral decomposition for   basis {ψ  k }k∈N  Hsos using the Hermite ψk (x) 0 (see Chap. 1). Denote ψk (1/2, x) = , with and ψk (−1/2, x) = 0 ψk−1 (x) ψ−1 = 0. For k ≥ 1, {ψk (1/2, ·), ψk (−1/2, ·)} is a basis of eigenvectors for the eigenvalue   ψ0 (x) . k of multiplicity two and 0 is a non degenerate eigenvalue with eigenvector 0 Let us remark that besides the position x we have here another degree of freedom s = 1/2, −1/2 which is discrete and represents the spin of a fermion.

502

17 Supercoherent States—An Introduction

We can rewrite the supersymmetric harmonic oscillator with creation and annihilation operators for bosons and fermions. 1 σ+ 2ω F 1 1 d (ω B q −  dq ), a ∗F = √ σ− . a ∗B = √2ω B 2ω F

aB =

√ 1 (ω B q 2ω B

d +  dq ), a F = √

(17.7) (17.8)

We get the super harmonic oscillator, Hˆ = ω B [a ∗B , a B ]+ + ω F [a ∗F , a F ]

(17.9)

where (a ∗B , a B ) satisfies (CCR) or Weyl–Heisenberg algebra relations and (a ∗F , a F ) satisfies (CAR) or Clifford algebra relations. We shall see that Hˆ is supersymmetric if ω B = ω F . It is possible to consider systems with n bosons generators and m fermions generators as well, writing a B = (a B,1 , . . . , a B,n ) and a F = (a F,1 , . . . , a F,m ) satisfying (CCR) relations for a B , (CAR) relations for a F . The goal of a supersymmetry theory is to put bosons and fermions on the same footing. For example we would like to consider the parameter s for the spin as a classical variable on the same footing as x for the position. In other words the question is to find a kind of classical analogue for the spin. The question seems strange because the spin is purely quantal and disappears in the usual semi-classical limit. Nevertheless Berezin has invented new spaces where this is possible after quantization, he called them superspaces with different super structures (linear spaces, algebras, groups and manifolds (see [24])). We shall give here an introduction to this wide subject, explaining only enough details to understand some semi-classical properties of supercoherent states. Concerning a more complete presentation we refer to the wealth literature [24, 79, 175, 248].

17.3 Classical Superspaces The problem considered here is to find a classical analog for the algebra defined in (17.1). In other words we want to construct classical spaces mixing bosons and fermions. We already know classical spaces for bosons: real or complex vector spaces or manifolds and classical spaces for fermions: Grassmann algebras K[θ1 , . . . , θn ], K = R or C. It is not obvious to understand in geometrical terms what is a classical space for fermions. We shall come back to this point later. Remark first here that it is not the algebra K[θ ] but instead a space mysterious B such that the space of “smooth

17.3 Classical Superspaces

503

functions” on B with values in the Grassmann algebra K is K[θ ]. In particular B is a set of dimension 0. This was more or less implicit in the previous sections.

17.3.1 Morphisms and Spaces The configuration space of a classical system with n bosons and m fermions will be a symbolic space denoted Rn|m such that C ∞ (Rn|m ) := C ∞ (Rn )[θ1 , . . . , θn ] := C ∞ (Rn ) ⊗ Gm where Gm is the Grassmann algebra with m generators. In other words C ∞ (Rn|m ) is a C ∞ (Rn ) module with basis {θ1 , . . . , θn }. (x, θ ) is interpreted as a coordinates system of a point in the symbolic space Rn|m ; x = (x1 , . . . , xn ) are the even coordinates and θ = (θ1 , . . . , θm ) are the odd coordinates. As for Grassmann algebras, C ∞ (Rn|m ) is a super-module, with a parity operator P, C ∞ (Rn )-linear. So C ∞ (Rn|m ) = [C ∞ (Rn|m )]+ ⊕ [C ∞ (Rn|m )]− where [•]+ is the even part and [•]− is the odd part. For applications we need to understand transformation map between different superspaces Rn|m , Rr |s . A first approach to define a map : Rn|m → Rr |s is to use coordinates system:

(y, η) = (x, θ ) where x, y are even coordinates, θ, η are odd coordinates. But the symbolic spaces are defined implicitly by the C ∞ functions defined on them. So we interpret as a change of variables in functions f ∈ C ∞ (Rr |s ),

( f )(x, θ ) = f ( (x, θ ))

(17.10)

where f → ( f ) is a linear map. It is important to underline here that the meaning of the right hand side in (17.10) is given by the left hand side because superspaces are only symbolic and are defined by their algebra of functions and not by a well identified geometrical object in the usual sense. To define supermanifolds it is necessary to define local superspaces in an open set U of Rn . Then U n|m is the superspace defined by its algebra of C ∞ functions : C ∞ (U n|m ) := C ∞ (U ) ⊗ Gm . So we have a natural definition for morphisms from U n|m in V s|r where V is an open set of Rs . Let us give now some more formal definitions following [24, 175, 248].

504

17 Supercoherent States—An Introduction

17.3.2 Super Algebra Notions Definition 50 (super linear spaces) A superlinear space V is a vector space on K = R or C with a decomposition (Z2 -grading) V = V0 ⊕ V1 . V0 is the set of even vector, V1 is the set of odd vectors. Then on V it exists a parity (linear) operator P, defined by Pv = v for v ∈ V0 , Pv = −v for v ∈ V1 . Elements in V0 ∪ V1 are called homogeneous. A linear map F from the superlinear space V into the superlinear space W is said superlinear, or even, if it preserves parities, so F sends V j in W j , j = 0, 1. A linear map from V into W is said odd if F(V j ) ⊆ W j+1 ( j ∈ Z2 ). The parity number π(v) is defined for homogeneous elements: π(v) = 0 if v ∈ V0 ; π(v) = 1 if v ∈ V1 Let us remark that every linear map from V into W is the sum of an even map and an odd map. The following notations will be used: SHom(V, W ) is the linear space of even maps V into W and Hom(V, W ) is the space of all linear maps from V into W . Then Hom(V, W ) is again a super linear space and it even part is the space of even maps from V into W . (Hom(V, W ))0 = SHom(V, W ). Definition 51 (superalgebras) A super algebra is a linear space A with an associative multiplication with unit 1 ( f g) → f · g, such that for every f, g ∈ A+ ∪ A− we have the parity condition: π( f · g) = π( f ) + π(g) The superalgebra A is said (super)commutative if f · g = (−1)π( f )π(g) g · f Examples: Grassmann algebras K[θ1 , . . . , θn ] are super commutative algebras. Definition 52 (superspace) If B is a superspace (for example defined by a Grassmann algebra) the set of B-points of the superspace Rn|m is the set Rn|m (B) defined by the following equality. Rn|m (B) = Hom(C ∞ (Rn|m ), C ∞ (B)). The interpretation is that Rn|m (B) = Hom(B, Rn|m ) so Rn|m (B) can be seen as the “points of Rn|m ” parametrized by B. An other important trick for a correct interpretation of superspace concerns the product of superspaces. We should like to have for example Rn|m = Rn|0 × R0|m (of course Rn|0 is the usual space Rn and R0|m is the space whose C ∞ -functions are R[θ1 , . . . , θm ]).

17.3 Classical Superspaces

505

Let us consider 3 classical super spaces X, Y1 , Y2 . We can understand identification between Hom(X, Y1 × Y2 ) and Hom(X, Y1 ) × Hom(X, Y2 ) through identification between Hom(C ∞ (Y1 × Y2 ), C ∞ (X )) and Hom(C ∞ (Y1 ), C ∞ (X )) × Hom(C ∞ (Y2 ), C ∞ (X )). This is done as follows. If ∗k is a morphism from C ∞ (Yk ) into C ∞ (X ) then we get a unique morphism ∗  from C ∞ (Y1 × Y2 ) into C ∞ (X ) satisfying

∗ ( f 1 ⊗ f 2 ) = ∗1 ( f 1 ) ∗2 ( f 2 ), for f k ∈ C∞ (Yk ).

(17.11)

Conversely if ∗ is given we get ∗1 ( f 1 ) = ∗ ( f 1 ⊗ 1) and ∗2 ( f 2 ) = ∗ (1 ⊗ f 2 ). So the identification is determined by (17.11).

17.3.3 Examples of Morphisms We compute here morphism between some super spaces. (1) is a morphism from Rn in Rr . It is known that we have ( f )(x) = f ( (x)) with smooth function Rn → Rr . (2) is a morphism from R0|1 into R. Using parity we have ( f ) = b0 ( f ) where b0 ( f ) is real, linear and multiplicative in f . So there exists x0 ∈ R such that

( f ) = f (x0 ), which means that is constant, as it must be. (3) is a morphism from R1|1 into R. Using parity, we have ( f ) = c0 ( f ) where c0 is a morphism from C ∞ (R) in C ∞ (R). Hence there exists a smooth function

: R → R such that ( f ) = f ◦

(4) is a morphism from R1|1 into R0|1 . Using parity, we have: (1) = 1 and

(θ ) = c1 θ . (1 is the constant function 1). So we have (b0 + b1 θ ) = b0 + b1 c1 θ . From (3) and (4) we clearly see that (x, θ ) are coordinates for R1|1 . (5) is a morphism from R1|2 into R1 . We have

( f ) = b0 ( f ) + b1 ( f )θ1 θ2 . The super-algebra even morphism conditions give b0 ( f g) = b0 ( f )b0 (g), f, g ∈ C ∞ (R), b1 ( f g) = b0 ( f )b1 (g) + b1 ( f )b0 (g).

b0 (1) = 1.

(17.12) (17.13)

From the first condition, there exists ϕ : R → R such that b0 ( f ) = f ◦ ϕ. Then the second condition means that b1 is a derivation at ϕ(x). So there exists χ (x, t), smooth in a neighborhood of the graph of ϕ such that: b1 ( f )(x) = χ (x, t)

df |t=ϕ(x) . dt

(17.14)

506

17 Supercoherent States—An Introduction

17.4 Super-Lie Algebras and Groups Standard symmetries (at the classical or quantum level as well) are described through group action; Supersymmetries will be described through action of super-groups. Moreover these super-groups are derived by exponentiating super Lie algebras as in the usual case of Lie groups.

17.4.1 Super-Lie Algebras Definition 53 A super Lie algebra s is a super linear space on K = R, C with a bilinear bracket (A, B) → [A, B] satisfying the following identities for homogeneous elements A, B, C, [A, B] = (−1)1+π(A)π(B) [B, A] π([A, B]) = π(A) + π(B)

(17.15) (17.16)

(−1)π(A)π(C) [A, [B, C]] + (−1)π(B)π(A) [B, [C, A]] + (−1)π(C)π(B) [C, [A, B]] = 0 (17.17) Remark 109 The first equality means that [A, B] is anticommutative excepted if A and B are odd. In this case the bracket is commutative. The second identity means that the bracket of two even or two odd elements is even and the bracket of an even and odd element is odd. The third identity is the super-Jacobi identity. As in the standard case of Lie algebras, many examples of super-Lie-algebra are realized as matrix super-algebras. Let us consider a super-vector space on the field K = R, C, V = V0 ⊕ V1 where the even space V0 has dimension n and the odd space V1 has dimension m. Any superlinear operator A in V has a matrix representation  A=

A++ A+− A−+ A−−



Denote End(V ) the space of all super linear operators in V . We have seen that End(V ) is a super vector space with parity π . It is a super Lie algebra for the super-bracket: [A, B] = AB − (−1)π(A)π(B) B A Choosing basis in V0 and V1 the super space V is isomorphic to Kn ⊕ K m . This space is denoted Kn,m (not the same meaning as Kn|m ) and the super-Lie algebra is denoted gl(n|m). We denote sl(n, m) the sub-super-Lie-algebra of gl(n|m) defined by the supertraceless condition: Str A = 0.

17.4 Super-Lie Algebras and Groups

507

su(n, m) is the sub-super-Lie-algebra of matrix A such that: A++ = −A∗++ , A−+ = −i A∗+− . For m = 0 we recover the classical Lie algebras gl(n), sl(n), su(n). Definition 54 Let A be a superalgebra (not necessarily associative). A linear operator D in A is called a super-derivation if it satisfies, for every homogeneous elements f, g in End(A ), D( f · g) = D( f ) · g + (−1)π(D)π( f ) f · D(g).

(17.18)

The basic example is D = ad A where A is a super-Lie algebra and ad A (B) = [A, B] (like in standard Lie algebras). In the super-algebra C ∞ (Rn|m ), X j = ∂∂x and k = ∂∂θ are superderivations (the j

k

first is even, the second is odd). And if h ∈ C ∞ (Rn|m ), h X j and hk are again superderivations.

17.4.2 Supermanifolds, A Very Brief Presentation To define supermanifolds it is necessary to define local superspaces in an open set U of Rn . Then we denote U n|m the superspace defined by its C ∞ functions : C ∞ (U n|m ) := C ∞ (U ) ⊗ Gm . So we have a natural definition for morphisms from U n|m in V s|r where V is an open set of Rs . U n|m is a superspace of dimension n|m. As for a standard manifold, a supermanifold M of dimension n|m is a topological space M0 (called the underlying space) such that for each point m ∈ M0 we have an open neighborhood U and a superalgebra RU isomorphic to C ∞ (V n|m ), where V is an open set of Rn . Moreover a gluing condition has to be satisfied (see [248], p. 135). In particular M0 is a standard manifold of dimension n. We have discussed here the super analogue of real C ∞ manifolds. It is possible to define real analytic or holomorphic supermanifolds. For that we have to replace C ∞ (U ) by the space C ω (U ) of analytic functions in U . In the complex case U is an open set of Cn (recall that complex analytic = holomorphic). Nowadays there exist essentially two kinds of (almost) equivalent definitions of super manifolds. One defines supermanifold by its morphisms [26, 175, 248], it is called the algebro-geometrical approach; the other is closer to the standard definition where a manifold is a set of points but the field of real numbers is replaced by a non commutative and infinite dimensional Banach algebra [76, 226]. Here we shall use the terminology of the algebro-geometrical approach. See [226] for a discussion about comparison of these two definitions. We only give here a very brief and sketchy introduction to this rich subject. Our goal is only to have a better intuition of what is going on some few examples, in particular concerning the Fermi-Bose (or super) oscillator.

508

17 Supercoherent States—An Introduction

Definition 55 Roughly speaking, a C ∞ -super manifold of dimension n|m is a superspace M defined by an underlying manifold of dimension n such that M is locally isomorph to U n|m , where U is a chart (open set in Rn ) of the underlying manifold M0 . There are gluing compatibility conditions between the charts as for manifolds where C ∞ (U n|m ) replaces C ∞ (U ). If U is an open set of Rn , the supermanifold U n|m is defined by the underlying space U and with C ∞ (U n|m ) = C ∞ (U )[θ1 , . . . , θm ], θ1 , . . . , θm are generators of a real Grassmann algebra. Definition 56 A vector field in the super domain U n|m is a derivation in the super algebra C ∞ (U n|m ) As in the standard case the set V (U n|m ) of vector fields in U n|m is a super Lie algebra. Note that it is a module over C ∞ (U n|m ) with the following basis in coordinates (x, θ ): 

 ∂ ∂ ∂ ∂ . ... ; ,..., ∂ x1 ∂ xn ∂θ1 ∂θm

In a supermanifold M the tangent superspace at “P ∈ M” of coordinates (x, θ ) is defined locally in a chart U n|m as the space V (U n|m ) restricted at (x, θ ) (see [248] for a more precise definition).

17.4.3 Super-Lie Groups In short, a super-Lie group is a super-manifold and a super group where the group operations are morphisms. It is not the place here to define rigorously all these terms; we refer to [76, 175, 226, 248] for a detailed study of super-manifolds and super groups. A super Lie-group is defined as a super manifold where the group structure is defined by morphisms μ : G × G → G, ι = G → G, ν0 : R0|0 → G. These morphisms define respectively the multiplication, the inverse element and the unit element. Of course we need conditions to translate the group properties on morphisms (see [248] for details). We have defined before the superspace Rn|m . It is a commutative Lie group for the “natural” addition with the following morphisms. (x, θ ) + (y, ζ ) = (x + y, θ + ζ ). Addition rule has to be defined in terms of morphisms as follows.

(17.19)

17.4 Super-Lie Algebras and Groups

509

Recall that the space Rn|m is defined by its morphisms and for every superspace B, Rn|m (B) is the set of B points of Rn|m , Rn|m (B) = Hom(C ∞ (Rn|m ), C ∞ (B)). Addition in Rn|m (B) is defined as follows. It is first remarked that a morphism ψ ∈ Rn|m (B) is determined by its images on the following generators of Rn|m f j (x, θ ) = x j , gk (x, θ ) = θk , 1 ≤ j ≤ n, 1 ≤ k ≤ m ψ ( f j ) and ψ (gk ) determine ψ . Hence the additive group {Rn|m , +} is defined by the rule, ψ1 , ψ2 ∈ Rn|m (B), (ψ1 + ψ2 )( f ) = ψ1 ( f ) + ψ2 ( f ), for f = f j , gk , for any ψ1 , ψ2 ∈ Rn|m (B) and any superspace B. For simplicity, assume n = m = 1. Practically if f ∈ C ∞ (R1|1 ) we have f (x, θ ) = f 0 (x) + f 1 (x)θ and the meaning of (17.19) is μ ( f )(t, s, θ, ζ ) = f 0 (t + s) + f 1 (t + s)θ + f 1 (t + s)ζ. It is easy to write explicitly ι and ν0 . As in the standard case, it is very important to have connections between super Lie algebras and super groups. Without going into details of the theory of super Lie groups, (see [26, 248]), let us recall here a definition (up to super-isomorphism) of the super Lie algebra of a super group SG of dimension n|m. Definition 57 The super Lie algebra sg of the super group SG is the super linear space of left invariant derivations in the algebra C ∞ (U n|m ) where U is a neighborhood 1 of the underlying Lie group G 0 . The easiest example is the super-linear group GL(n, m) corresponding to the super Lie algebra gl(n, m). The details can be found in [26]. Elements g of GL(n, m) are superlinear isomorphisms in Rn|m . They are parametrized by matrices   A++ A+− A(g) = A−+ A−− where the matrices A++ , A−− have even elements and A+− , A−+ have odd elements. The real component of A(g) is  A0 (g) =

A++ 0 0 A−−



A0 (g) is invertible if and only if det(A++ ) det(A−− ) = 0. Denote G 0 = GL(n) × GL(m). G 0 is a standard manifold of dimension n 2 + m 2 . So we see that the supermanifold GL(n, m) is defined by the sheaf of smooth function C ∞ (GL(n, m)) := C ∞ (G 0 ) ⊗ R2nm

510

17 Supercoherent States—An Introduction

It can be proved that it is a super Lie group [26]. Remark 110 One very useful tool to compute in superanalysis is what Berezin called “the Grassmann analytic continuation principle” [26]. It was sometimes applied implicitly above and we shall often apply it later without more justifications. There exist several ways to explain this principle in a more mathematical rigorous approach (see [79, 138]). The practical rule is the following. Let f be a real function of n real variables and n nilpotent Grassmann numbers ξ1 , . . . , ξn . That means that ξ j ∈ Gn x1 , . . . , xn and ξ j = aj θ  where θ1 , . . . , θn are generators of Gn . Denote x = (x1 , . . . , xn ) ||>0

and ξ = (ξ1 , . . . , ξn ) (using the usual notation in several variable functions), the Grassmann analytic continuation principle says f (x + ξ ) =

 ξ α ∂α f (x). α! ∂ x α α

(17.20)

Recall that α = (α1 , . . . αn ) ∈ Nn , ξ α = ξ1α1 . . . ξnαn , hence we have ξ α = 0 for |α| ≥ 2n so the Taylor expansion (17.20) is finite. We consider now more interesting examples when even and odd coordinates are mixed, that give supersymmetries after quantization. Let us begin by the toy model R1|1 with the law super group: (t, θ ) · (s, ζ ) = (t + s − iθ ζ, θ + ζ ).

(17.21)

Let us denote SG(1|1) this supergroup (it is easy to prove that it is a supergroup with a non commutative product) and sg(1|1) its Lie algebra. If f ∈ C ∞ (R1|1 ), f (t, θ ) = f 0 (t) + θ f 1 (t), and (s, η) the coordinates of a supervector v in R1|1 , the left translation of f by v is defined by 

τ(s,η) f (t, θ ) = f (t − s + iηθ, θ − η) = f 0 (t − s) + iηθ f 0 (t − s) + (θ − η) f 1 (t − s).

(17.22)

Here we have used the Grassmann analytic extension principle. But we have the group property:  τ(s,η) = τ(s,0) ◦ τ(0,η) and 

τ(s,0) f (t, θ ) = f (t − s, θ ), (usual translation by a real number s).(17.23)   ∂ ∂  f (t, θ ) (17.24) τ(0,ζ ) f (t, θ ) = f (t, θ ) + ζ iθ − ∂t ∂θ So we can conclude that the super lie algebra sg(1|1), has the two generators {∂t , Dθ }, where ∂t = ∂t∂ is even, ∂θ = ∂θ∂ , and Dθ = iθ ∂t − ∂θ are odd. We have the commutation rule [Dθ , Dθ ]+ = −2i∂t .

17.4 Super-Lie Algebras and Groups

511

By analogous computations we get a basis for right invariant vector fields. Only the odd parts is modified. We get Qθ = iθ ∂t + ∂θ . Only the last commutation rule is changed: [Qθ , Qθ ]+ = 2i∂t .

(17.25)

Moreover we have the following commutation [Q, D] = 0.

(17.26)

In mechanics the generator of the time translations is the Hamiltonian H . So Eq. (17.25) is a classical analogue for a supersymmetric system. Consider now a more physical example leading to a classical analogue for the Witten model [263]. This kind of model was considered before by several physicists in the seventies (Wess, Zumino, Salam). On R1|2 define the following multiplication (non commutative) rule (t, θ, θ ∗ )(s, ζ, ζ ∗ ) = (t + s − i(θ ζ ∗ − ζ θ ∗ ), θ + ζ, θ ∗ + ζ ∗ ).

(17.27)

For f ∈ C ∞ (R1|2 ) we have f (t, θ, θ ∗ ) = f 0 (t) + f 1 (t)θ + f 2 (t)θ ∗ + f 3 (t)θ θ ∗ . The multiplication morphism μ∗ is determined by μ∗ ( f j ) for 1 ≤ j ≤ 3 (recall that μ∗ is a superalgebra morphism). For every f ∈ C ∞ (R) the meaning of (17.27) is μ ( f )(t, s, θ, θ ∗ , ζ, ζ ∗ ) = f (t + s − i(θ ζ ∗ − ζ θ ∗ ), θ + ζ, θ ∗ + ζ ∗ ) = f (t + s) − i(θ ζ ∗ − ζ θ ∗ ) f (t + s) − θ θ ∗ ζ ζ ∗ f

(t + s) ι and ν0 are easily computed such that {R1|2 , μ} is a super Lie group that we shall denote SG(1|2). As we have done for SG(1|1) we can compute generators for the super Lie algebra sg(1|2) by identifying left invariant vector fields. Denote  τs,η,η∗ the left translation by (s, η, η∗ ),  τη the left translation by (0, η, 0) and  τη∗ the translation by (0, 0, η∗ ). As above we get easily for every f ∈ C ∞ (R1|2 ),   τη f (t, θ ) = f (t, η) + η iθ ∗ ∂t − ∂θ f (t, θ )

(17.28)

τ f (t, θ ) = f (t, η) + η (iθ ∂t − ∂ ) f (t, θ )

(17.29)

 

η∗

θ∗

So we have got the following basis for sg(1|2): ∂t , Dθ = iθ ∗ ∂t − ∂θ , Dθ ∗ = iθ ∂t − ∂θ ∗ , with one even and two odd generators. The commutation rules of this algebra are

512

17 Supercoherent States—An Introduction

Dθ 2 = Dθ ∗ 2 = 0 [Dθ , ∂t ] = [Dθ ∗ , ∂t ] = 0 [Dθ , Dθ ∗ ]+ = −2i∂t

(17.30) (17.31) (17.32)

By analogous computations we get a basis for right invariant vector fields. Only the odd parts is modified. We get Qθ = iθ ∗ ∂t + ∂θ , Qθ ∗ = iθ ∂t − ∂θ ∗ . Only the last commutation rule is changed: [Qθ , Qθ ∗ ]+ = 2i∂t .

(17.33)

Moreover we have the commutation [Q, D] = 0.

(17.34)

So Eq. (17.33) is a classical analogue for the quantum supersymmetric system of Witten. This will be more explicit later.

17.5 Classical Supersymmetry 17.5.1 A Short Overview of Classical Mechanics There are many books concerning this very well known subject. For our purpose we recall here some basic facts concerning Lagrangians and Hamiltonians. We refer for more details to the following books [8, 121, 241]. In physics classical dynamical systems are usually introduced with their Lagrangian L and their action integral S = d xL (x) if x is a coordinates sysn tem for classical paths. For t1 a point particle moving in R with coordinates q = ˙ q˙ is the time derivative of q. The equations (q1 , . . . , qn ) we have S = t0 dtL (q, q), of motion (Euler–Lagrange equations) are deduced from the least action principle d ∂L − ∂q dt



∂L ∂ q˙

 =0

(17.35)

An Euler–Lagrange equation may be very difficult to solve. Many informations about its solutions can be obtained studying its symmetries. According the Noether famous theorem, symmetries give integral of motions I , functions of q, q˙ conserved along the motion dtd I (q, q) ˙ = 0. In particular the Hamiltonian energy function H is conserved:

17.5 Classical Supersymmetry

513

H (q, q) ˙ =

∂L · q˙ − L (q, q). ˙ ∂ q˙

(17.36)

The canonical conjugate momentum p is defined as p=

∂L ∂ q˙

p is a cotangent vector in (Rn )∗ . We consider here for simplicity systems without constraints and that the configuration space is the Euclidean space Rn . (see [8] for manifolds). Let us recall now a statement for the Noether theorem concerning symmetries and integrals of motion. Theorem 87 Let V (q) =



v j (q)

1≤ j≤n

∂ be a vector field and φVs its flow: ∂q j

d s φ (qs ) = v(qs ), q0 = q, where v(q) = (v1 (q), . . . , vn (q)). ds V

(17.37)

(NI) Assume that L is invariant under φVs for s close to 0. Then I V (q, q) ˙ = v(q) ·

∂L ∂ q˙

is an integral of motion: ddtIV = 0. (NII) Assume that there exists a smooth function K on Rn × Rn such that v(q) · then I = v(q) ·

∂L ∂ q˙

∂L d ∂L + v(q) ˙ = K (q, q) ˙ ∂ q˙ ∂q dt

(17.38)

− K (q, q) ˙ is an integral of motion.

Remark 111 It is convenient to state Noether’s theorem with finite small deformations of size . Denote δ q = v(q), δ L = L (q + δ q, q˙ + δ q) ˙ − L (q, q). ˙ The invariance assumptions (17.38) means that δ L = 

d K (q, q) ˙ + O( 2 ). dt

(17.39)

The energy Hamiltonian H was defined as a function of (q, q). ˙ It is convenient to replace the velocity q˙ by the momentum p using the Legendre transform in q˙ → p. H (q, p) = p · q(q, ˙ p) − L (q, q(q, ˙ p)).

(17.40)

514

17 Supercoherent States—An Introduction

L If the matrix ∂∂q∂ is non-degenerate, H is a smooth function of (q, p) only (at least ˙ q˙ locally). In order to quantize the classical system with Lagrangian L it is assumed that H is defined globally. We shall see that for fermions it is not true for interesting examples where the Lagrangian is degenerate: q˙ → p is not onto. But in order to quantize canonically a classical system we have to compute an Hamiltonian and a Poisson bracket. Dirac proposed a method [84] to do that in the degenerate case. Let us describe roughly the Dirac method which will apply to Fermions. We follow here [147] where the reader can find many details. ) When q˙ → p is not onto the range of the mapping (q, q) ˙ → (q, p) ( p = ∂L ∂ q˙ is a non geometrically trivial part of the phase space. In particular it is not possible to recover q˙ from p. In this situation we say that the Lagrangian is singular or degenerate. Following [84], it will be assumed that this is a smooth manifold C described by equations χ j (q, p) = 0, 1 ≤ j ≤ m where the differential dχ j are everywhere independent. The energy Hamiltonian H is always a function of q and p (as in the regular case) but is defined here only under the constraint (q, p) ∈ C . Dirac assumed that H has an extension to the whole phase space Rn × Rn (Rn is identified with (Rn )∗ ) and he computed a modified Poisson bracket {•, •}Di such that for any observable F ∈ C ∞ (Rn × Rn ) the equation of motion is 2

F˙ = {F, H }Di , in particular q˙ = {q, H }Di , p˙ = { p, H }Di .

(17.41)

On C , using definition of p we have d H = qdp ˙ −

∂H ∂H ∂L dq = dq + dp ∂q ∂q ∂p

(17.42)

Then there exists u j (q, q), ˙ 1 ≤ j ≤ m such that q˙ =

 ∂χ j ∂H + uj ∂p ∂p 1≤ j≤m

p˙ = −

 ∂χ j ∂H − uj ∂q ∂q 1≤ j≤m

(17.43) (17.44)

So we have for every observable F F˙ = {F, H } +



u j {F, χ j }

(17.45)

1≤ j≤m

The constraints χ j have to be preserved during the time evolution, which gives the conditions

17.5 Classical Supersymmetry

515



{χk , H } +

u j {χk , χ j } = 0, for 1 ≤ k ≤ m

(17.46)

1≤ j≤m

For simplicity assume now m = 2 and {χ1 , χ2 } = λ = 0. From conditions (17.46) we can compute u 1 , u 2 : u1 =

{χ2 , H } {χ1 , H } , u2 = − . λ λ

Then the Dirac bracket has the following expression: 1 ({F, χ1 }{χ2 , G} − {F, χ2 }{χ1 , G}) λ

{F, G} Di = {F, G} +

(17.47)

It is easy to see that (F, G) → {F, G} Di is a Lie bracket on the linear space C ∞ (Rn × Rn ). Let us consider the following example in the configuration space R2 : L M = x y˙ − y x˙ − V (x, y), V is a smooth potential. This Lagrangian is degenerate. px :=

∂L M ∂L M = −y, p y := =x ∂ x˙ ∂ y˙

(17.48)

The energy Hamiltonian is H = V (x, y). We have the two constraints χ1 = px + y, χ2 = p y − x and {χ1 , χ2 } = 2. Then the Dirac bracket is  ∂ F ∂G ∂ F ∂G − {F, G} Di ∂ px ∂ p y ∂ p y ∂ px (17.49) We can check that the Hamiltonian H and the Dirac bracket (17.49) give the Euler– Lagrange equation for the motion. We find 1 1 = {F, G} − 2 2



∂ F ∂G ∂ F ∂G − ∂x ∂y ∂y ∂x

x˙ = {x, V }Di = −



1 − 2



1 ∂V 1 ∂V ; y˙ = {y, V }Di = 2 ∂y 2 ∂x

(17.50)

To prepare a canonical quantization of the Lagrangian L M we write the commutation relations {x, y}Di = { px , p y }Di = − 1 2 = 0.

1 2

(17.51)

{x, px }Di = {y, p y }Di =

(17.52)

{y, px }Di = {x, p y }Di

(17.53)

A Dirac quantization F → Fˆ satisfies ˆ G] ˆ {F, G}Di → −i[ F,

516

17 Supercoherent States—An Introduction

So we get a representation of the Lie algebra (17.51) in L 2 (R2 ) satisfying     ∂ 1 ∂ 1 x+ , yˆ = y− xˆ = 2 i∂ y 2 i∂ x     ∂ ∂ 1 1 y− , pˆ y = x+ . pˆ x = − 2 i∂ x 2 i∂ y

(17.54)

Notice that this representation in L 2 (R2 ) is not irreducible because we have the constraints pˆ x + yˆ = 0, pˆ y − xˆ = 0 and the relations (17.54) define an Heisenberg Lie algebra of dimension 3. So we can get a quantization equivalent to the Weyl quantization in L 2 (R).

17.5.2 Supersymmetric Mechanics Supermechanics is an extension of classical mechanics. In supermechanics a point has real (or even) coordinates x = (x1 , . . . , xn ) representing the bosonic degrees of freedom and Grassmann coordinates, ξ = (ξ1 , . . . , ξ M ), representing the fermionic degrees of freedom. To encode the two kinds of degree of freedom in the same object one introduces a “real super variables” X living in the configuration space of all the system. A super Lagrangian is a function L S of X and its derivative D X , with values in a Grassmann algebra. In coordinates we can write X = (x, ξ ) where x has real dependent components, ξ have fermionic (real) components. In coordinates we get a pseudo-classical Lagrangian L (x, x, ˙ ξ, ξ˙ ). The conjugate momenta are ∂L ∂L , π= . (17.55) p= ∂ x˙ ∂ ξ˙ Assume for a moment that we can define the Hamiltonian H as the Legendre transform of L in (x, ˙ ξ˙ ). Let F be an observable on the phase space defined by its coordinates (x, ξ, p, π ). Along the motion we want to have as usual F˙ = {F, H } where F is an extension of the usual Poisson bracket. A direct computation, using equations of motion, gives F˙ =



∂H ∂F ∂H ∂F − ∂p ∂x ∂x ∂p



 −

∂H ∂F ∂H ∂F + ∂π ∂ξ ∂ξ ∂π

 .

(17.56)

Recall that H and F are in the superalgebra C ∞ (Rn+n|m+m ). The right side in (17.56) is the super Poisson bracket {F, H }, the first term is the usual antisymmetric Poisson

17.5 Classical Supersymmetry

517

bracket in bosonic variables, inside the second parenthesis we have a symmetric form for the contribution of fermionic variables. For any homogeneous observables F, G the Poisson bracket is a super Lie product in C ∞ (R2n|2m ) satisfying  {F, G} =

∂ F ∂G ∂ F ∂G − ∂x ∂p ∂p ∂x

 + (−1)

π(F)



∂ F ∂G ∂ F ∂G + ∂ξ ∂π ∂π ∂ξ

 .

(17.57)

First consider a simple system with one bosonic state and one fermionic state without interaction. Its states are represented by the superfield: X = x + iθ ξ

(17.58)

x is a real number depending on time t, ξ is an odd number living in a Grassmann algebra. We want that X is even and “real”. We introduce two Grassmann complex ¯ + c(t)η∗ where c(t) is a comconjugate generators {η, η∗ } and choose ξ(t) = c(t)η ∗ plex number. Then we have (iθ ξ ) = iθ ξ so X is even and real. Remark that X is not a C ∞ function on R1|1 because the coefficient ξ is not a complex number but a Grassmann variable. As we have seen the superfields X has to be understood as a morphism from C ∞ (R) in C ∞ (R1|1 ), using the Grassmann analytic extension principle we have: X ∗ ( f )(x, θ ) = f (x) + i f (x)θ ξ. Let us introduce the super Lagrangian LS =

1 D X · D(D X ) 2

where D = Dθ . In coordinates we get the pseudo-classical Lagrangian

L pcl =

dθ L S =

1 2 (x˙ + iξ ξ˙ ) 2

We can easily solve the Euler–Lagrange equations. We get x¨ = 0, ξ˙ = 0. We can consider the same model with 3 superfields X j = x j + iθ ξ j , 1 ≤ j ≤ 3. L pcl = Here we have 3 constraints

1 (x˙ · x˙ − i ξ˙ · ξ ). 2

518

17 Supercoherent States—An Introduction

i χj = πj + ξj. 2 But we have {χ j , χk } = −iδ j,k so the constraints are second order. This system is invariant by any rotation in R3 . Its Dirac canonical quantization gives a spin system [241]. The supergroup SG(1|1) transforms X by right translations, so we have X (t − iθ η, θ + η) = X (t, θ ) + ηQ X (t, θ ) where Q = Qθ . So a variation of X is given by δη X = ηQ X , where δη is a derivation in C ∞ (R)[θ, η, η∗ ] considered as a R(η, η∗ ] module (coefficients of X are in R[η, η∗ ]), δη F = [ηQ, F]. In coordinates we have δη X = δη x + iθ δη ξ where ˙ δη x = iηξ, δη ξ = η x. So we compute the variation of the Lagrangian and the variation of the corresponding action S = dtdθ L . Using that δη is a derivation, we get δη L S = ηQL S = η(iθ ∂t + ∂θ )L . Hence we get





∂η S =

dt∂t

(17.59)

 dθiθ L S .

(17.60)

By extension of the Noether theorem to the fermionic case we see that the vector field Q is the generator of a symmetry for the Lagrangian L S . This symmetry is called supersymmetry because it concerns supervariables mixing usual numbers (real or complex) and Grassmann numbers. For the corresponding variations of the pseudoclassical Lagrangian L pcl we have δη L pcl =

3iη ∂ ξ x˙ 2 ∂t

According to Noether’s results we have the conserved charge I Q = iξ

∂L ∂L 3i ∂ + x˙ ξ x˙ = −2iξ x˙ − ∂ x˙ 2 ∂t ∂ ξ˙

We can compute the Hamiltonian H by Legendre transform. The phase space here is the super linear space R2|1 . We get for the conjugate momentum π=

∂L i = − ξ. ˙ 2 ∂ξ

17.5 Classical Supersymmetry

519 2

So this Lagrangian is degenerate, with the constraint χ = π + 2i ξ . We get H = p2 where p is the conjugate momentum to x. Now we shall consider a more interesting example involving the superharmonic oscillator. We consider the superfields: X = x + θ ψ ∗ + ψθ ∗ + yθ θ ∗ .

(17.61)

This field may depend on time t, so x, y, ψ, ψ are time dependent, where x, y are real numbers, θ, θ are conjugate Grassmann variables, ψ, ψ ∗ are necessarily odd numbers. As above, to compute these numbers we have to introduce an other pair of Grassmann variables {ζ, ζ ∗ }, anticommuting with the pair {θ, θ ∗ }. So we consider that ψ = cζ + dζ ∗ where c, d are complex (time dependent) numbers. As we have already remarked, to avoid mathematical difficulties it is necessary to add extra Grassmann variables. Recall that the superfield X defined in the formula (17.61) is a morphism from R1|4 in R which means that it is defined by a morphism X from C ∞ (R) in C∞ (R)[θ, θ ∗ , ζ, ζ ∗ ] which can be be computed by the the Grassmann analytic continuation principle: (X f )(x, θ ) = f (x) + f (x)(ψθ ∗ + θ ψ ∗ ) + f (x)yθ θ ∗ − f

(x)θ θ ∗ ψψ ∗ . (17.62) There exists other rigorous interpretation about this formula. For a detailed discussion we refer to the paper [138]. Let us introduce a super potential V (X ) where V ∈ C∞ (R) and the super Lagrangian 1 (17.63) L S = Dθ ∗ X Dθ X + V (X ). 2 Assume that V is polynomial for simplicity. Then the Lagrangian has the supersymmetry defined by the odd generators Qθ and Qθ ∗ defined before. The infinitesimal transformations on fields are δη X = η∗ Qθ + ηQθ ∗ X and as above δη is a derivation. η, η∗ are Grassmann variables, independent of the other Grassmann variables. We also have δη V (X ) = η∗ Qθ + ηQθ ∗ V (X ) and δη L S = η∗ Qθ + ηQθ ∗ L S . As for the above toy model, it results that the Lagrangian L S is again supersymmetric with generators Qθ , Qθ ∗ . Now we shall compute in coordinates with the pseudo-classical Lagrangian. Compute Qθ X = ψ ∗ + θ ∗ (i x˙ + y) − iθ θ ∗ ψ˙ ∗ ˙ Qθ ∗ X = −ψ + θ (i x˙ − y) − iθ θ ∗ ψ. So we get

(17.64) (17.65)

520

17 Supercoherent States—An Introduction

δη X = η∗ ψ ∗ − ηψ + θ η(y − i x) + η∗ (y + i x)θ ˙ ∗ ˙ θ ∗. − i(η∗ ψ˙ ∗ + ηψ)θ

(17.66)

Hence in coordinates the supersymmetry has the following infinitesimal representation δη x = η∗ ψ ∗ − ηψ δη ψ = η∗ (y + i x) ˙

(17.67) (17.68)

δη ψ ∗ = η(y − i x) ˙ δη y = −i(η∗ ψ˙ ∗ + ηψ). Now we compute the pseudo-classic Lagrangian: L pcl := computation rules for Berezin integral we get L pcl =

(17.69) (17.70)

dθ dθ ∗ L S . Using the

1 2 i 1 ˙ ∗ ) + V

(x)ψψ ∗ . (x˙ + y 2 ) − V (x)y + (ψ ψ˙ ∗ − ψψ 2 2 2

(17.71)

The real variable can be eliminated because we have ∂∂y˙ L pcl = 0. From Euler– Lagrange equation we get ∂∂y L pcl = 0 so y = V (x). We get now the following Lagrangian Lw =

x˙ 2 1 V (x)2 i ˙ ∗ ) + V

(x)ψψ ∗ . − + (ψ ψ˙ ∗ − ψψ 2 2 2 2

(17.72)

We can compute the two Noether charges associated with the generators Qθ , Qθ ∗ . The results are Q = (x˙ − i V (x))ψ,

Q ∗ = (x˙ + i V (x))ψ ∗ .

(17.73)

An easy exercise is to compute the dynamics for the Fermi oscillator, well known for the Bose (or harmonic) oscillator. The Lagrangian is LF =

i 1 ˙ ∗ ) + ωψψ ∗ . (ψ ψ˙ ∗ − ψψ 2 2

(17.74)

The Euler–Lagrange equation gives ψ˙ = −iωψ hence we get ψ(t) = ηe−iωt , where η is any Grassmann complex variable. It is more suggestive to write down the dynamics in real coordinates:  ψ(t) + ψ ∗ (t) 1  = √ ηe−iωt + η∗ eiωt √ 2 2  ψ ∗ (t) − ψ(t) 1  ∗ iωt ξ2 (t) = = √ η e − ηe−iωt . √ i 2 i 2 ξ1 (t) =

(17.75) (17.76)

17.5 Classical Supersymmetry

521

17.5.3 Supersymmetric Quantization The first step is to compute the Hamiltonian Hw for the Lagrangian Lw . The momenta are defined as usual ∂Lw = x˙ ∂ x˙ ∂Lw i π= = − ψ∗ 2 ∂ ψ˙ ∂L i w = − ψ. π∗ = ∗ 2 ∂ ψ˙ p=

(17.77) (17.78) (17.79)

Hence the Legendre transform is not surjective: the Lagrangian is degenerate. We can extend the Dirac method (see more details in [147]) to fermionic variables with the two constraints     i i χ1 = π + ψ ∗ , χ2 = π ∗ + ψ . 2 2 Recall that here {·} is the super-Poisson-bracket. We have {χ1 , χ2 } = −i, so the constraint is of second order. Denote ˙ + ψ˙ ∗ π ∗ − Lw = H = x˙ p + ψπ

 1 2 p + V (x)2 − V

(x)ψψ ∗ . 2

According to Dirac’s method we compute multipliers u 1 , u 2 such the evolution of any observable F follows the equation F˙ = {F, H } + u 1 {F, χ1 } + u 2 {F, χ2 }. u 1 , u 2 are computed with the compatibility conditions χ˙ k = 0 for k = 1, 2. We get u 1 = −i V

(x)ψ, u 2 = i V

(x)ψ ∗ . The total Dirac Hamiltonian is here, after computation, HDi = H + u 1 χ1 + u 2 χ2 =

 1 2 p + V (x)2 + i V

(x)(ψ ∗ π ∗ − ψπ ). 2

To achieve the quantization of our system we define, as in the bosonic case, the Dirac bracket. Let F, G depending only on fermionic variables. Then the Dirac bracket is {F, G}Di = {F, G} +

1 ({F, χ1 }{χ2 , G} + {F, χ2 }{χ1 , G}) . {χ1 , χ2 }

522

17 Supercoherent States—An Introduction

So we have {ψ, ψ ∗ }Di = −i, {π, π ∗ }Di =

i 1 , {ψ, π } = {ψ ∗ , π ∗ } = . 4 2

(17.80)

A quantization F → Fˆ has to follow the correspondence principle. For F = x, p we consider the usual Weyl–Heisenberg quantization with the commutator rule [x, ˆ p] ˆ = i. If F and G are fermionic variables then ˆ G] ˆ  i{F, G} = [ F,

(17.81)

where the brackets are symmetric (anticommutators). In particular we have ˆ ψˆ ∗ ] = , [π, ˆ πˆ ∗ ] = [ψ,

− −i ˆ πˆ ] = [ψˆ ∗ , πˆ ∗ ] = , [ψ, . 4 2

(17.82)

The Dirac quantum Hamiltonian is  1 2 pˆ + V (x)2 + i V

(x)(ψˆ ∗ πˆ ∗ − ψˆ πˆ ). Hˆ Di = 2

(17.83)

A realization of the commutation relations is obtained with the Pauli matrices: √ √ ψˆ = σ− , ψˆ ∗ = σ+ (17.84) i i πˆ = − σ+ , πˆ ∗ = − σ− . (17.85) 2 2 For  = 1 we get the Witten supersymmetric Hamiltonian considered at the beginning of this Chapter:   d2 σ3 1 − 2 + V (x)2 1l2 − V

(x). (17.86) Hˆ = 2 dx 2 We can also remark that the supercharges Qˆ and Qˆ ∗ are obtained by quantization of the Noether charges Q, Q ∗ . 2 In particular we can consider an harmonic potential V (x) = ωx2 . Choosing a quantization such that πˆ = − 2i ψˆ ∗ and πˆ ∗ = − 2i ψˆ we get the super-harmonic oscillator   d2 1 2 2 ˆ ˆ − 2 + ω x + ωψˆ ∗ ψ. Hsos = (17.87) 2 dx We can realize this Hamiltonian with ψˆ = θ and ψˆ ∗ = ∂θ in the super Hilbert space H S := L 2 (R) ⊗ H˜ (2) . It is nothing but a Grassmann algebra interpretation of the Witten model introduced before with H˜ (2) in place of C2 .

17.5 Classical Supersymmetry

523

The scalar product in the Hilbert space H S is defined as

F, G =

∗

d xdθ dθ ∗ eθ θ F ∗ (x, θ )G(x, θ ),

(17.88)

where x is a real variable, θ is an holomorphic Grassmann variable.

17.6 Supercoherent States As canonical coherent states are built on the Heisenberg-Weyl Lie group, supercoherent states are built on the super Heisenberg-Weyl super Lie group. We first consider the simplest case with one boson and one fermion. So we have creation and annihilation operators a ∗B , a ∗F , a B , a F for bosons and fermions. They satisfy the super-Lie algebra commutation relations: [a B , a ∗B ] = 1l, [a F , a ∗F ]+ = 1l.

(17.89)

All other relations are trivial. We have a supersymmetric harmonic oscillator Hˆ sos with supersymmetry generators Q ∗ , Q, (17.90) Hˆ sos = a ∗B a B + a ∗F a F , Qˆ = a B a ∗F , Qˆ ∗ = a F a ∗B . This definition of Hˆ sos may differ of others by a constant. ˆ = 0 so ( Q, ˆ Qˆ ∗ ) generates a global supersymmetry defined by We have [ Hˆ sos , Q] the following unitary operators ˆ ∗ +η∗ Qˆ

Uη = e η Q

where (η, η∗ ) are any complex conjugate Grassmann numbers. Super translations Tˆ (z, γ ) are parametrized by (z, γ ), z is a complex number, γ is a Grassmann (complex) number Tˆ (z, γ ) = exp(za ∗B − z¯ a B + a ∗F γ − γ ∗ a F ).

(17.91)

Using the Baker–Campbell–Hausdorff formula we have the following useful properties   1 ∗ 2 γ γ − |z| exp(za ∗B ) exp(a ∗F γ ) exp(−¯z a B ) exp(−a F γ ∗ ) T (z, γ ) = exp 2 (17.92)    1 ∗ ∗ z u¯ − z¯ u + δ γ + δγ T (z + u, γ + δ) (17.93) T (z, γ )T (u, δ) = exp 2

524

17 Supercoherent States—An Introduction

In particular T (z, γ )−1 = T (−z, −γ ) is unitary and T (z, γ )−1 a B T (z, γ ) = a B + z T (z, γ )−1 a F T (z, γ ) = a F + γ

(17.94) (17.95)

The ground state of Hˆ sos is the state ψ0,0 := ϕ0 ⊗ ψ0 where ϕ0 and ψ0 are the normalized ground states of the bosonic and fermionic oscillators. So we define super-coherent states by displacement of the ground state of the super-harmonic oscillator by super-translations: ψz,γ = T (z, γ )ψ0,0 .

(17.96)

  γγ∗ ψz,γ = 1 + (|z, 0 + |z, 1γ ) 2

(17.97)

Using (17.92) we have

where we use the notation |z,  = ϕz ⊗ θ  ,  = 0, 1. The coherent states family ψz,γ has the following expected properties. The proofs follow easily from results already established for bosonic and fermionic coherent states. 1. (Normalization) ψz,γ 2 = ψz,γ , ψz,γ  = 1.

(17.98)

2. (Non orthogonality)     1 1 ψz,γ , ψu,δ  = exp γ ∗ δ − (γ ∗ γ + δ ∗ δ) exp − (|z|2 + |u|2 ) + z¯ u . 2 2 (17.99) 3. (Over-completeness) For every ψ ∈ H S we have

ψ(x, θ ) =

ψz,γ , ψψz,γ (x, θ )dzdγ ∗ dγ .

(17.100)

4. (Translation property) T (z, γ )ψu,δ = exp(

 1 z u¯ − z¯ u + δ ∗ γ + δγ ∗ )ψz+u,γ +δ . 2

(17.101)

5. (Eigenfunctions of annihilation operators) a B ψz,γ = zψz,γ , a F ψz,γ = γ ψz,γ . In particular we have the averages for energy and supercharges

(17.102)

17.6 Supercoherent States

525

ψz,γ , Hˆ sos ψz,γ  = |z|2 + γ ∗ γ ∗

(17.103) ∗

ψz,γ , Qψz,γ  = zγ , ψz,γ , Q ψz,γ  = z¯ γ .

(17.104)

Remark 112 Definition and properties of supercoherent states with one boson and one fermion can easily be extended for systems with n bosons and m fermions. Then z = (z 1 , . . . , z n ) ∈ Cn , γ = (γ1 , . . . , γm ) represent a system of m complex Grassetc. The formulas are mann variables, a B = (a B,1 , . . . , a B,n ), a F = (a F,1 , . . . , a F,m ),  the same using the multidimensional notations: γ ∗ · a F = γ j∗ a F, j . Then the 1≤ j≤m

energy Hamiltonian can be Hˆ = ω B · a ∗B a B + ω F a ∗F · a F , ω B,F are real numbers. The Hilbert space of this system is Hn,m = L 2 (Rn ) ⊗ H˜ (m) . To have a supersymmetric system we need that n = m and ω B = ω F .

17.7 Phase Space Representations of Super Operators As it is well known for bosons and as it was studied before for fermions we can extend to mixed systems bosons+fermions representations formulas for operators in the super-Hilbert space H . Phase space representation means that we are looking for a correspondence between functions on the phase space and operators in some Hilbert space. This correspondence is called quantization in quantum mechanics. Our goal here is to revisit Weyl quantizations for observables mixing bosons and fermions. It could be possible to do it also for anti-Wick quantization. To have a better analogy between the bosonic and fermionic variables it is nicer to √ p , ζ ∈ Cn , q, p ∈ write bosonic Weyl quantization in complex coordinates ζ := q+i 2 Rn . The Lebesgue measure in Cn is d 2 ζ = |dζ ∧ dζ ∗ |. Assume for simplicity that Hˆ , Gˆ have smooth Schwartz kernels in S (R2n ). We consider their Weyl symbols Hww , G w w in complex coordinates. Using the same proof as in the fermonic case we have the following Moyal formulas for the Weyl symbols of Gˆ Hˆ :

¯ ¯ d 2 ηG w (ζ − η)Hw (η)e1/2(ζ ·η−ζ ·η) (17.105) (G w  Hw )(ζ ) = Cn

w

← −− → ← −− →

w

(G  H )(ζ ) = G w (ζ )e1/2( ∂ζ · ∂ζ¯ − ∂ζ¯ · ∂ζ ) H w (ζ ).

(17.106)

The first formula gives the covariant symbol, the second formula the contravariant symbol. We have the following relations G w (η) = G w (η)



¯

d 2 ζ eζ ·η−ζ ·η¯ G w (ζ ).

¯ ζ¯ ·η = (2π )−2n d 2 ζ eζ ·η− Gw(ζ ).

(17.107) (17.108)

526

17 Supercoherent States—An Introduction

It is not difficult to get a Moyal formula for classes of operators in the super-space Hn,m = L 2 (Rn ) ⊗ H˜ (m) . As usual it is enough to establish formulas for smoothing operators then the formulas are extended to suitable classes of symbols. So we ˆ Hˆ are linear continuous operators from S (Rn ) ⊗ H˜ (m) into shall assume that G, n (m) S (R ) ⊗ H˜ (that means that their Schwartz kernels are in S (R2n ⊗ Gmc )). The covariant symbol Hw is defined such that Hˆ =



d 2 ζ d 2 γ Hw (ζ, γ )Tˆ (−ζ, −γ )

Hw (ζ, γ ) = Str( Hˆ Tˆ (ζ, γ )),

(17.109) (17.110)

where Str is defined as the usual trace in bosonic variable ζ and the super-trace in the fermionic variable γ . More precisely, for any trace-class operator Hˆ in Hn,m we have Str Hˆ = Tr( Hˆ (1l ⊗ χ )) where χ is the chirality operator in H˜ (m) defined in Chap. 16. The contravariant symbol is the symplectic Fourier transform of the covariant symbol: H w (ζ, γ ) =



¯

¯ −η·ζ +α d 2 ηd 2 α Hw (η, α)eη·ζ

∗

·γ +α·γ ∗

.

(17.111)

So we get the following Moyal formula for super symbols (G w  H w )(ζ, α) = G w (ζ, α)e

← −− → ← −− → ←− − → ← − −→ 1/2 ∂ζ · ∂ζ¯ − ∂ζ¯ · ∂ζ +∂α∗ ·· ∂α + ∂α ·∂α∗

H w (ζ, α). (17.112)

Many results explained before for bosons and fermions could be extended to mixed systems. Instead to do that we now discuss a simple application.

17.8 Application to the Dicke Model This model was studied in [1] as a supersymmetric system. The Hamiltonian for this model is 1 Hˆ = a∗ a + ωσ3 + g(a∗ + a)σ1 2

(17.113)

It concerns a two-level atom interacting with a monochromatic radiation field. a∗ and a are one particle creation/annihilation operators, the Pauli matrices σ = (σ1 , σ2 , σ3 ) denote the radiation field with frequency , ω is the level distance of the states of the atom, g is a coupling constant.

17.8 Application to the Dicke Model

527

The matrices σ± satisfy the (CAR) relation so we can identify them with fermionic creation/annihilation operators. So we can introduce a pair of complex conjugate Grassmann numbers (β, β ∗ ) such that σ+ ≡ βˆ ∗ = ∂β , σ− ≡ βˆ is multiplication by β (see Chap. 16) and σ3 ≡ 2βˆ ∗ βˆ − 1l. But with this choice Hˆ is not an even operator. To overcome this problem we add a new Grassmann pair of complex variables (θ, θ ∗ ) and the real Grassmann number η = θ + θ ∗ . We have ηˆ 2 = 1l (In [1] ηˆ is denoted c and is called a Clifford number, see Chap. 16 for more explanations). Hence we have the two pairs of Grassmann variables (β, β ∗ ), (θ, θ ∗ ) to represent the Pauli matrices: σ+ ←→ ηˆ βˆ ∗ , σ− ←→ ηˆ βˆ ˆ b∗ = ∂β . We have to remark that (bη) ˆ ∗ bηˆ = where we denote ηˆ = θ + ∂θ , b = β, ∗ ˆ b b. With this substitution the Hamiltonian H is transformed into an even Hamiltonian defined in the space L 2 (R) ⊗ H˜ (2) 1 ˆ ∗ + bη) ˆ Hˆ S = a∗ a + ω(b∗ b − 1l) + g(a∗ + a)(ηb 2

(17.114)

Its (contravariant) Weyl symbol is   1 + ωβ ∗ β + g(ζ + ζ¯ )(ηβ ∗ + βη). HS (ζ, β, θ ) =  |ζ |2 − 2 Our aim is to study the dynamics for the Hamiltonian Hˆ S . It is determined by the von Neumann equation i

∂ ρˆt = [ Hˆ S , ρˆt ], ρˆt=0 = ρˆ0 , ∂t

(17.115)

where ρˆ is a density operator (a positive operator of trace 1), ρt is the contravariant Weyl symbol of ρˆt (also called the Weyl–Wigner Distribution Function). Let us remark that Hˆ S depends only on a and γˆ where γ = βη, because we have b∗ b = γˆ ∗ γˆ . Moreover we have [γˆ , γˆ ∗ ]+ = 1l. ρt satisfies a Fokker-Planck type equation which can be computed using the Moyal product formula applied to HS  ρt − ρt  HS . So we get − where we find (see also [1])

∂ρt − → ← − = L ρt + ρt L ∗ ∂t

(17.116)

528

17 Supercoherent States—An Introduction

  1− 1− 1− 1− 1 → → → → − → + −i L = (ζ ∗ − ∂ ζ )(ζ + ∂ ζ ∗ ) + ω (β ∗ + ∂ β )(β + ∂ β ∗ ) − 2 2 2 2 2     1 − 1 − → → − → − → − → g ζ + ζ ∗ + ( ∂ ζ ∗ − ∂ ζ ) + (θ + ∂ θ ) β ∗ − β + ( ∂ ζ − ∂ ζ ) . 2 2 (17.117) Hˆ S being time independent we also have ˆ

ˆ

ρˆt = e−it HS ρˆ0 eit HS . So we start to consider the propagation of the dynamical variables a and γˆ . We first compute the commutators [ Hˆ S , γˆ ] = −γˆ + g(a + a∗ )(2γˆ ∗ γˆ − 1l). [ Hˆ S , γˆ ∗ ] = γˆ ∗ − g(a + a∗ )(2γˆ ∗ γˆ − 1l). [ Hˆ S , a] = −g(γˆ ∗ + γˆ ) − a. [ Hˆ S , a∗ ] = g(γˆ ∗ + γˆ ) + a∗ .

(17.118) (17.119) (17.120) (17.121)

So we find the non-linear differential system ∂ γˆt ∂t ∂ i γˆt∗ ∂t ∂ i at ∂t ∂ i at∗ ∂t i

= −γˆt + g(a + a∗ )(2γˆt∗ γˆt − 1l) = γˆt∗ − g(a + a∗ )(2γˆt∗ γˆt − 1l) = −g(γˆt + γˆt∗ ) − at = −g(γˆ + γˆ ∗ ) + at∗ .

Assume now that the initial Weyl–Wigner distribution ρˆ depends only on a, a∗ , γˆ , γˆ ∗ . Then ρˆt has the following shape ρˆt = ρˆt0 (at , at∗ ) + ρˆt1 (at , at∗ )γˆt + ρˆt1,∗ (at , at∗ )γˆt∗ + ρˆt2 (at , at∗ )γˆt∗ γˆt . Using (17.122) we get that at , at∗ , γˆt∗ γˆt are functions of t and a, a∗ , γˆ , γˆ ∗ . So ρˆt can be written as follows ρˆt = ρˆt(0) (a, a∗ ) + ρˆt(1) (a, a∗ )γˆ + ρˆt(1)∗ (a, a∗ )γˆ ∗ + ρˆt(2) (a, a∗ )γˆ ∗ γˆ .

(17.122)

Now we can give the physical meaning of the bosonic operators ρˆt(•) (at , at∗ ) by taking the average on the fermionic variables (β, θ ). We shall see now that we recover the dynamics generated by the Hamiltonian Hˆ taking the basis {1l2 , σ+ , σ− , σ3 }.

17.8 Application to the Dicke Model

529

 ˆ , where Gˆ is a fermionic We have to compute the relative trace Tr H˜2 ρˆt (1l ⊗ G) operator in H˜2 , using the formula

 ˆ = 1 ρt (a, a∗ , β, θ )G w (β, θ )d 2 βd 2 θ. Tr H˜2 ρˆt (1l ⊗ G) 4 We compute the covariant Weyl symbols of 1l2 , γˆ , γˆ ∗ , γˆ ∗ γˆ . 1w (β, θ ) = β ∗ β + θ ∗ θ. γw (β, θ ) = β(θ − θ ∗ ). γw∗ (β, θ ) = (θ ∗ − θ )β ∗ . (γ γw )(β, θ ) = θ ∗ θ. ∗

Then we get effective distribution functions ρeff (for simplicity the time t is not written explicitly nor the contravariant bosonic variable ζ ) (0) ρeff

=

(1) ρeff = (1)∗ ρeff (2) ρeff



ρ × (θ ∗ θ + β ∗ β)d 2 θ d 2 β = ρ (0) ρ × β(θ + θ ∗ )(θ − θ ∗ )β ∗ d 2 θ d 2 β = −2ρ (1)

ρ × (θ + θ ∗ )β ∗ β(θ − θ ∗ )d 2 θ d 2 β = −2ρ (1)∗

= 2 ρ × θ ∗ θ d 2 θ d 2 β = 2ρ (2) . =

Finally after a direct computation we recover the dynamics for the Hamiltonian Hˆ defined by (17.113) from the dynamics for the super Hamiltonian Hˆ S defined by (17.114): (0) (1) (1),∗ (2) 1l2 + ρeff σ+ ρeff σ− + ρeff σ3 . ρeff = ρeff 

Appendix A

Tools for Integral Computations

A.1

Fourier Transform of Gaussian Functions

This result is the starting point for the stationary phase theorem. 1/2 Let M be a complex matrix such that M is positive-definite. We define [det M ]∗ the analytic branch of (det M )1/2 such that (det M )1/2 > 0 when M is real. Theorem 88 Let A be a symmetric complex symmetric matrix, m × m. We assume that A is non negative and A is non degenerate. Then we have the Fourier transform formula for the Gaussian eiA x·x/2 

−1

Rm

eiA x·x/2 e−ix·ξ d ξ = (2π )m/2 [det (−iA )]−1/2 e(iA ) ∗

ξ ·ξ/2

.

(A.1)

Proof For A real formula (A.1) is well known: first we prove it for m = 1 then for m ≥ 2 by diagonalizing A and using a linear change of variables. For A complex (A.1) is obtained by analytic extension of left and right hand side. 

A.2

Sketch of Proof for Theorem 29

Recall that critical set M of the phase f is M = {x ∈ O, f (x) = 0, f  (x) = 0} . Note that if a is supported outside this set then J (ω) is O(ω−∞ ). Using a partition of unity, we can assume that O is small enough that we have normal, geodesic coordinates in a neighborhood of M . So we have a diffeomorphism, © Springer Nature Switzerland AG 2021 D. Robert and M. Combescure, Coherent States and Applications in Mathematical Physics, Theoretical and Mathematical Physics, https://doi.org/10.1007/978-3-030-70845-0

531

532

Appendix A: Tools for Integral Computations

χ :U →O , where U is an open neighborhood of (0, 0) in Rk × Rd −k , such that χ (x , x ) ∈ M ⇐⇒ x = 0 and if m = χ (x , 0) ∈ M we have χ  (x , 0)(Rkx ) = Tm M χ  (x , 0)(Rdx−k ) = Nm M , (normal space at m ∈ M ) . So the change of variables x = χ (x , x ) gives the integral    J (ω) = eiωf (χ(x ,x )) a(x , x )| det χ  (x , x )|dx dx .

(A.2)

Rd

The phase f˜ (x , x ) := f (χ (x , x )). clearly satisfies {f˜x (x , x ) = 0, f˜ (x , x ) = 0} ⇐⇒ x = 0 Hence, we can apply the stationary phase Theorem 7.7.5 of [153] in the variable x , to the integral (A.2), where x is a parameter (the assumptions of [153] are satisfied, uniformly for x close to 0). We remark that all the coefficients cj of the expansion can be computed using the above local coordinates and Theorem 7.7.5.

A.3

A Determinant Computation

Here we give the details concerning computations of the determinant (9.33) in Chap. 9. We write α = (α  , α4 ) where α  = eˆ 1 + i cos γ eˆ 2 The gradient of the phase (9.28) is ix + G(p) where G(p) is given by G(p) :=

(p2

   2 α + p(α4 − α · w1 (p) = K(p)M (p) + 1)α · w1 (p)

where K(p) =

2 2α  · p + α4 (p2 − 1)

Appendix A: Tools for Integral Computations

and M (p) = α  +

533

2p (α4 − α  · p) 1 + p2

K : R3 → C, M : R3 → C3 . Since x is a constant the hessian of (9.28) is simply DG(p) where DG (resp. DK, DM ) is the first differential of G (resp. K, M ). We want to calculate at the critical point pc with   cos β cos γ sin β c p = , ,0 . 1 − sin β sin γ 1 − sin β sin γ Let δp be an arbitrary increase of p. One has: DG(p)(δp) = (DK(p) · δp) · M (p) + K(p)DM (p) · δp We can write;

  (DK(p) · δp) M (p) = M (p) ⊗ (DK(p))∗ · δp

where DK(p)∗ ∈ (R3 )∗ + i(R3 )∗ C3 . Using the dual structure the identification of (R3 )∗ + i(R3 )∗ with C3 is performed via the isomorphism: u → (v → u · v). We have: K(pc ) = e−iβ (1 − sin β sin γ ) M (pc ) = α  + (i sin γ − eiβ )pc . Thus

DG(pc ) = e−iβ (1 − sin β sin γ )DM (pc ) + M (pc ) ⊗ (DK(pc ))∗

It is convenient to choose as a basis of vectors (pc , qc , eˆ 3 ) where qc = α  + (i sin γ − eiβ )pc . The vectors pc , qc are C-linearly independent for γ = π2 + kπ . A simple calculus yields DM (pc ) = (i sin γ − eiβ )1lR3 − (1 − sin β sin γ )pc ⊗ (qc )∗ DK(pc ) = −e−2iβ (1 − sin β sin γ )2 (α  + α4 pc ) Thus we get

DG(pc ) = e−iβ (i sin γ − eiβ )(1 − sin β sin γ )1lR3

(A.3)

−e−iβ (1 − sin β sin γ )2 pc ⊗ (qc )∗ − e−2iβ (1 − sin β sin γ )2 qc ⊗ (qc + eiβ pc )∗ Let H (p) be the Hessian matrix in the basis (pc , qc , eˆ 3 ) and denote

534

Appendix A: Tools for Integral Computations

H1 (p) =

H (p) . 1 − sin β sin γ

The second line of (A.3) yields a matrix in the plane generated by (pc , qc ) of the form   d1 d2 d3 d4 that we calculate using

 c c c 2 q · p (q ) p ⊗ (q ) = 0 0 c

c ∗



0 0 qq · pc (qc )2

qc ⊗ (qc )∗ = qc ⊗ (pc )∗ = We have (pc )2 =



 0 0 . (pc )2 pc · qc



1 + sin β sin γ 1 − sin β sin γ

qc · pc = sin γ

i − eiβ sin β 1 − sin β sin γ

(qc )2 = −e2iβ We get d1 = −(1 − sin β sin γ ) d2 = −e−iβ (1 − sin β sin γ ) d3 = −e−iβ (1 + i sin γ e−iβ ). d4 = 0. Thus the three-dimensional reduced Hessian matrix H1 equals ⎛

⎞ d1 d2 0 H1 = ⎝d3 0 0 ⎠ 0 0 d5 with d5 = e−iβ (i sin γ − eiβ ). Its determinant equals det H1 = −d2 d3 d5 .

Appendix A: Tools for Integral Computations

535

One has | det H1 |2 = (1 − sin β sin γ )2 [1 − 2 sin β sin γ + sin2 γ ][1 + 2 sin β sin γ + sin2 γ ].

Finaly we get the result:  | det H (pc )| = (1 − sin β sin γ )4 [sin2 γ + 2 sin β sin γ + 1][sin2 γ − 2 sin β sin γ + 1].

We see that it doesn’t vanishes provided sin β sin γ = 1.

A.4 A.4.1

The Saddle Point Method The One Real Variable Case

This result is elementary and very explicit. Let us consider the Laplace integral 

a

I (λ) =

e−λφ(r) F(r)dr

0

where a > 0, φ and F are smooth functions on [0, 1] such that φ  (0) = φ(0) = 0, > 0 if r ∈]0, 1]. Under these conditions we can perform the change φ  (0) > 0, φ(r) √ of variables s = φ(r) where we denote r = r(s) and G(s) = F(r(s))r  (s). So we have Proposition 164 For every N ≥ 1 we have the asymptotic expansion for λ → +∞, I (λ) =



Cj λ−(j+1)/2 + O(λ−(N +1)/2 ),

(A.4)

0≤j≤N −1



where Cj = Γ

 (j) G (0) j +1 . 2 2j!

 −1/2 In particular C0 = F(0) φ  (0) . Proof This is a direct consequence of Taylor expansion applied to G at 0 and using that, for every b > 0 and ε > 0,  0

b

e−λs sj ds = 2

1 Γ 2



 j + 1 λ−(j+1)/2 + O(e−ελ ). 2



536

A.4.2

Appendix A: Tools for Integral Computations

The Complex Variables Case

This is an old subject for one complex variable but there are not so many references for several complex variables. Here we recall a presentation given by Sjöstrand [238] or [237]. Let us consider a complex holomorphic phase function in a open neighborhood A × U of (0, 0) in Ck × Cn , A × U  (a, u) → ϕ(a, u) ∈ C. Assume that • ϕ(0, 0) = 0, ∂u ϕ(0, 0) = 0. 2 ϕ(0, 0) = 0. • det∂(u,u) • ϕ ≥ 0 ∀u ∈ U , ϕ > 0 for all u ∈ ∂UR . where UR := U ∩ Rn and ∂UR is the boundary in Rn of UR . By the implicit function theorem we can choose A × U small enough such that the equation ∂u ϕ(a, z) = 0 has a unique solution z(a) ∈ U , a → z(a) being holomorphic in A . Then we have the following asymptotic result Theorem 89 For every holomorphic and bounded function g in U we have, for k → +∞,  kϕ(a,z(a)) e e−kϕ(a,r) g(r)d r = 

A.5

2π k

n/2

UR

 2 −1/2 det ∂(u,u) ϕ(a, z(a)) g(a) + O(k −n/2−1 ).

(A.5)

Kähler Geometry

Let M be a complex manifold and h an Hermitian form on M : h=



hj,k dzj ⊗ dzk , hj,k (z) = hk,j (z).

j,k

h is a Kähler form if the imaginary part ω = h is closed (d ω = 0) and its real part g = h is positive definite. hj,k (z)dzj ∧ d z¯k . ω=i j,k

(M , h) is said a Kähler manifold if h is a Kähler form on M . Then on M exists a Riemann metric g = h and symplectic two form ω. Locally there exists a real -valued function K, called Kähler potential, such that

Appendix A: Tools for Integral Computations

537

hj,k =

∂2 K. ∂zj ∂zk

The Poisson bracket of two smooth functions φ, ψ on M is defined as follows {φ, ψ}(m) = ω(Xφ , Xψ ) where Xψ is the Hamiltonian vector field at m defined such that d ψ(v) = ω(Xψ , v), for all v ∈ Tm (M ). More explicitly: {ψ, φ}(z) = i



 hj,k (z)

j,k

∂ψ ∂φ ∂φ ∂ψ − ∂zj ∂ z¯k ∂zk ∂ z¯j



where hj,k (z) is the inverse matrix of hj,k (z). The Laplace-Beltrami operator corresponding to the metric g is =

j,k

hj,k (z)

∂ ∂ . ∂zj ∂ z¯k

In particular for the Riemann sphere we have: d ζ d ζ¯ (1 + |ζ |2 )2   ∂φ ∂ψ 2 2 ∂ψ ∂φ − {ψ, φ}(z) = i(1 + |z| ) ∂z ∂ z¯ ∂z ∂ z¯ 2 ∂ .  = (1 + |z|2 )2 ∂z∂ z¯ ds2 = 4

(A.6) (A.7) (A.8)

For the pseudo-sphere we have: d ζ d ζ¯ (1 − |ζ |2 )2   ∂φ ∂ψ 2 2 ∂ψ ∂φ − {ψ, φ}(z) = i(1 − |z| ) ∂z ∂ z¯ ∂z ∂ z¯ 2 ∂ .  = (1 − |z|2 )2 ∂z∂ z¯ ds2 = 4

(A.9) (A.10) (A.11)

Appendix B

Remainder Estimate for the Moyal Product

We give here a remainder estimate for the Moyal product for matrix valued symbols. Let us first recall the formal product rule for quantum observables with Weyl quantization. Let A, B ∈ S (R2n , Cm,m ). The Moyal product C := A  B is the semi-classical ˆ Some computations with the Fourier transform observable C such that Aˆ · Bˆ = C. give the following well known formula [153, 217]  C(x, ξ ) = exp

 i σ (Dq , Dp ; Dq , Dp ) A(q, p)B(q , p )|(q,p)=(q ,p )=(x,ξ ) , 2

(B.1)

where σ is the symplectic bilinear form σ ((q, p), (q , p )) = p · q − p · q and D = i−1 ∇. By expanding the exponential term, we get C(x, ξ ) =

j i ( σ (Dq , Dp ; Dq , Dp ))j A(q, p)B(q , p )|(q,p)=(q ,p )=(x,ξ ) . j! 2 j≥0

(B.2)

So that C(x, ξ ) is a formal power series in  with coefficients given by Cj (x, ξ ) =

1 (−1)|β| β α β (Dx ∂ξ A).(Dxα ∂ξ B)(x, ξ ). 2j |α+β|=j α!β!

(B.3)

It is enough for our purpose to assume that A, B behave like polynomials at infinity. For μ ≥ 0 denote P(m,m) (μ) the linear space of C ∞ symbols A : R2n → Cm,m such that γ |∂X A(X )| ≤ Cγ , for any |γ | ≥ μ. Assuming that A ∈ P(m,m) (μA ), B ∈ P(m,m) (μB ) it is known that, for  fixed, Aˆ Bˆ = Cˆ where C ∈ P(m,m) (μC ), μC ≥ max{μA , μB } (see [153]). Recall the notation C = A  B. © Springer Nature Switzerland AG 2021 D. Robert and M. Combescure, Coherent States and Applications in Mathematical Physics, Theoretical and Mathematical Physics, https://doi.org/10.1007/978-3-030-70845-0

539

540

Appendix B: Remainder Estimate for the Moyal Product

We need here better estimates in small  > 0 and a control in the size of the derivatives of A, B. We introduce the notations X = (x, ξ ) ∈ Rnx × Rnξ and for every N ≥ 1,

A  B(X ) −

j Cj (X ) =: RN (A, B; X ; ).

(B.4)

0≤j≤N

The following estimate is a particular case of Theorem A.1 in [40]. Theorem 90 There exist N0 , N1 such that for all N ≥ N0 and every γ ∈ N2d , there exists a constant K(N , γ ) such that for any A ∈ P(m,m) (μA ), B ∈ P(m,m) (μB ) for every multi-index γ , the following estimate holds, for every X ∈ R2n , ⎛ γ |∂X ⎝A  B(X ) −

⎞



j Cj (X )⎠ | ≤ N +1 K(N , γ )

0≤j≤N



β

∂Xα AL∞ ∂X BL∞

(B.5)

N +1≤|α|,|β|≤N +N1 +|γ |

Proof We follow [40] Appendix A. By Fourier transform computations and Taylor formula, we get the following formula   i N +1 1 (1 − t)N RN ,t (X ; )dt where RN ,t (X ; ) = 2 0 i exp(− σ (u, v))σ N +1 (Du , Dv )A(u + X )B(v + X )dudv. 2t R2n ×R2n

1 RN (A, B; X ; ) = N!   −2n (2π t)



(B.6) Notice that the integral is an oscillating integral as we shall see below. We shall use the following lemma to estimate RN ,t (X ; ). Lemma 109 Let us consider F ∈ S (R2n × R2n ) and the integral   I (λ) = λ2n

R2n ×R2n

exp[−iλσ (u, v)]F(u, v)dudv.

(B.7)

Then for every real number m > 2 there exists C(m, d ) > 0 such that the following estimate holds |I (λ)| ≤ C(m, d ) sup |∂uα ∂vβ F(u, v)|. (B.8) u,v∈R2n |α+β|≤m

Proof The lemma is proved in a standard way, using cut-off and integration by parts as follows. Let us introduce a cut-off χ0 , C ∞ on R, χ0 (x) = 1 for |x| ≤ 1/2 and χ0 (x) = 0 for |x| ≥ 1. We split I (λ) into three pieces

Appendix B: Remainder Estimate for the Moyal Product

541

 I0 (λ) = λ2n

R2n ×R2n

 I1 (λ) = λ2n

exp[−iλσ (u, v), χ0 (λ(u2 + v 2 ))F(u, v)dudv,

exp[−iλσ (u, v)](1 − χ0 (λ(u2 + v 2 ))χ0 (u2 + v 2 )F(u, v)dudv,  I2 (λ) = λ2n exp[−iλσ (u, v)](1 − χ0 (u2 + v 2 ))F(u, v)dudv. R2n ×R2n

R2n ×R2n

For I0 (λ), we easily have |I0 (λ)| ≤ ω4d sup |F(u, v)|,

(B.9)

u2 +v2 ≤1

where ω4d is the volume of the unit ball in R4n . For I1 (λ) and I2 (λ), we integrate by parts with the differential operator i L= 2 u + v2

  ∂ ∂ Ju − Jv , ∂v ∂u

(B.10)

where J is the matrix associated to the symplectic form (σ (u, v) =< Ju, v >). Performing 4n integrations by parts, we can see that it exists a constant cn such that |I1 (λ)| ≤ cn sup |∂uμ ∂vν F(u, v)|.

(B.11)

u2 +v 2 ≤1 |μ|+|ν|≤4n

Similarly, performing m > nd integrations by parts, we get for a constant c(d , m), |I2 (λ)| ≤ cn sup |∂uμ ∂vν F(u, v)|.

(B.12)

u,v∈R2n |μ|+|ν|≤m

 Now we can complete the proof of the theorem by using Lemma 109, the Leibniz formula and the following elementary estimate, using in R2n the coordinates u = (x, ξ ), v = (y, η), |σ N (∂x , ∂ξ ; ∂y , ∂η ))A(x, ξ )B(y, η)| ≤ (2n)N

sup

|α|+|β|=N

β

|∂xα ∂ξ A(x, ξ )∂yβ ∂ηα B(y, η)|.

(B.13)

From the Lemma we get first the Theorem for A, B ∈ S (R2n ) with universal constants. Then the theorem is proved for any A, B ∈ P(m,m) (μ) by smoothing: 2 2 Aη (u) = e−ηu A(u) and Bη (v) = e−ηv B(v) for η > 0 and passing to the limit η → 0.

Appendix C

Semi-classical Functional Calculus

C.1

Introduction

In this appendix we give a proof of Theorem 10. There are many papers or books proving that if Hˆ is a self-adjoint -pseudodifferential operator and f a complex function on R then f (Hˆ ) (defined by the abstract functional calculus) is again a -pseudodifferential operator. Recall that Hˆ is supposed to be a uniformly elliptic semi-classical Hamiltonian. Moreover it is a self-adjoint operator in L2 (Rn ). There are several works concerning functional calculus for pseudodifferential operators and each of them have considered different classes of symbols and different classes of functions f . Notice that if f is holomorphic on a suitable neighborhood of the real axis the problem is solved by proving first that the resolvent (Hˆ − z)−1 is a pseudodifferential operator with a Weyl symbol RH z for z ∈ C\R satisfying suitable estimates with a good in (z). Then the functional calculus result follows by using the Cauchy formula on a suitable contour Γ  1 (z − Hˆ )−1 f (z)dz. (C.1) f (Hˆ ) = 2iπ Γ

In particular the leading symbol of f (Hˆ ) is f (H (x, ξ )). For f only C ∞ smooth on R the Cauchy formula can be replaced by an almost analytic extension of f , following an idea of Helffer-Sjöstrand [143]. The main motivation is to consider smooth function with compact support allowing cut-off in energies. Here we shall show that the same method can be applied for f satisfying (C.2). f is C ∞ -smooth such that there exists r ∈ R such that for all j ∈ N we have  df   (t) ≤ Cj < t >r−j , ∀t ≥ 0. dt © Springer Nature Switzerland AG 2021 D. Robert and M. Combescure, Coherent States and Applications in Mathematical Physics, Theoretical and Mathematical Physics, https://doi.org/10.1007/978-3-030-70845-0

(C.2)

543

544

C.2

Appendix C: Semi-classical Functional Calculus

Functional Calculus and Almost Analytic Extension

Let f be such that (C.2) is statisfied with r < −1. Notice here that this is not restriction because we can write f (t) = (t + i)m g(t) and taking m large enough,g satisfies ((C.2) with < −1. Let χ ∈ C0∞ (R) be a cut-off function such that χ (t) ≡ 1 in a neighborhood of 0. For a given integer N ≥ 1 let us define f˜N (x + iy) =

0≤k≤N

f

(k)

(iy)k χ (x) k!



 |y| .

(C.3)

f˜ is an almost analytic extension of f to C. The following property was considered by Hörmander [153]. We use here the presentation given in [71]. f¯ is almost analytic in the following sense. Let us introduce the Cauchy-Riemann operator  ∂ ∂ 1 +i . ∂¯ = 2 ∂x ∂y Lemma 110 There exists a constant CN such that ¯ N (x + iy)| ≤ CN < x >r−N −1 |y|N , ∀(x + iy) ∈ C. |∂f

(C.4)

with 0 < a < b. The proof of this lemma is elementary using (C.2) and (C.3).  The functional calculus for pseudodifferential operators will be obtained from the following result due to Helffer-Sjöstrand for f with a compact support. Proposition 165 Let f satisfying (C.2) with r < −1 and f˜N an almost analytic extension of f satisfying (C.4) with N ≥ 1 then we have 1 f (Hˆ ) = π

 C

∂¯λ fN (λ)(Hˆ − λ)−1 μL (d λ)

(C.5)

where μL is the Lebesgue measure on the complexe plane C. Proof We follow the proof given in [83, Theorem 8.1]. 1 Remark first that πλ is a fondamental solution for the differential operator ∂¯λ , so we have, for every λ ∈ R, f (ν) = −

1 π

 C

∂¯λ fN (λ)(ν − λ)−1 μL (d λ).

(C.6)

By the general spectral theorem for self-adjoint operators we have < u, f (Hˆ )v >=

 R

f (λ)d (< u, Eλ v >), u, v ∈ H ,

(C.7)

Appendix C: Semi-classical Functional Calculus

545

where Eλ is the spectral projector of Hˆ on ] − ∞, λ]. Hence plugging formula (C.6) in formula (C.7) and using Fubini’s theorem we get the formula (C.5). 

C.2.1

Weyl Calculus and Resolvent Estimates

The main idea is to plug in (C.5) suitable approximations for (Hˆ − z)−1 . These approximations (named parametrices) are obtained using the operational calculus (Moyal product) on semi-classical observables and uniform ellipticity of H . The construction start with bλ,0 (X ) = (H0 (X ) − λ)−1 , X = (x, ξ ) ∈ R2n , λ ∈ C\R, using Weyl calculus we have (Hˆ − λ)bˆ λ,0 = 1l + Rˆ λ,1 with γ |∂X Rλ,1 (X )|

   < λ > n0   ≤C λ 

C, n0 are independent on . The last estimate is established using the following elementary estimates in the complex plane. Assume H0 > 0 and arg λ ∈ [−π/2, π/2] then we have:    λ  |H0 − λ| ≥ |λ|, |H0 − λ| ≥   |H0 |. λ By induction on N , we get bλ,j , for any j ∈ N, such that we have Theorem 91

⎛ (Hˆ − λ) ⎝



⎞ j bˆ λ,j ⎠ = 1l + N +1 Rˆ λ,N +1

0≤j≤N

such that

   < λ > 2N +n0  Rˆ λ,N +1 L2 ,L2 ≤ CN  λ 

Moreover for j ≥ 1, we have, for the symbol of the parametrix, the formula: bλ,j =

1≤k≤2j−1

dj,k bk+1 λ,0 ,

546

Appendix C: Semi-classical Functional Calculus γ

where dj,k are polynomials (independent on λ) in ∂X H for |γ | +  ≤ j. Remark 113 The L2 uniform estimates can be obtained from the constructions of the book [217] (Chap. III, Sect. 3) using improvement in the composition formula obtained in [40].

C.2.2

End of the Proof of Theorem 10

Plugging the parametrix obtained in Theorem 91 in formula (C.5) leads directly to the proof of the Theorem. 

Appendix D

Lie Groups and Coherent States

D.1

Lie Groups and Coherent States

In this appendix we start by a short review of some basic properties of Lie groups and Lie algebras. Then we explain some useful points concerning representation theory of Lie groups and Lie algebras and how they are used to build a general theory of Coherent States systems according to Perelomov [208, 209]. This theory is an extension of the examples already considered in Chaps. 7 and 8.

D.2

On Lie Groups and Lie Algebras

We recall here some basic definitions and properties. More details can be found in [88, 138] or in many other textbooks.

D.2.1

Lie Algebras

A Lie algebra g is a vector space equipped with an anti-symmetric bilinear product: (X , Y ) → [X , Y ] satisfying [X , Y ] = −[Y , X ] and the Jacobi identity [[X , Y ], Z] + [[Y , Z], X ] + [[Z, X ], Y ] = 0. The map (ad X )Y = [X , Y ] is a derivation: (ad X )[Y , Z] = [(ad X )Y , Z] + [Y , (ad X )Z]. If g and h are Lie algebras a Lie homomorphism is a linear map χ : g → h such that [χX , χ Y ] = [X , Y ]. A sub-Lie algebra h in g is a subspace of g such that [h, h] ⊆ h h is an ideal if furthermore we have [h, g] ⊆ h. If χ is a Lie homomorphism then ker χ is an ideal. © Springer Nature Switzerland AG 2021 D. Robert and M. Combescure, Coherent States and Applications in Mathematical Physics, Theoretical and Mathematical Physics, https://doi.org/10.1007/978-3-030-70845-0

547

548

Appendix D: Lie Groups and Coherent States

In the following we assume for simplicity that the Lie algebras g considered are finite-dimensional. g is abelian if [g, g] = 0. g is simple if it is non-abelian and contains only the ideals {0} and g. The centre Z(g) is defined as Z(g) = {X ∈ g, [X , Y ] = 0, ∀Y ∈ g} Z(g) is an abelian ideal. If g has no abelian ideal except {0} then g is said semi-simple. In particular Z(g) = {0}. The Killing form on g is the symmetric bilinear form B(X , Y ) defined as B(X , Y ) = Tr((ad X )(ad X )). g is semi-simple if and only if its Killing form is non-degenerate (Cartan’s criterion).

D.2.2

Lie Groups

A Lie group G is a group equipped with a multiplication (x, y) → x · y and equipped with the structure of a smooth connected manifold (we don’t recall here definitions and properties concerning manifolds, see [138] for details) such that the group operation (x, y) → x · y and x → x−1 are smooth maps. We always assume that Lie groups considered here are analytic. A useful mapping is the conjugation C(x)(y) = xyx−1 , x, y ∈ G. Its tangent mapping at y = e is denoted Ad(x). Ad (x) ∈ GL(g) and x → Ad(x) is a homomorphism from G into GL(g). It is called the adjoint representation of G and Ad(G) is the adjoint group of G. We denote ad the tangent map of x → Ad(x) at x = e. The Lie algebra g associated with the Lie group G is the tangent space Te (G) at the unit e of G. The Lie bracket on g is defined as follows: Let X , Y ∈ g = Te (G) and define [X , Y ] = (ad X )(Y ). g is the Lie algebra associated with the Lie group G. A first example of Lie group is GL(V ) the linear group of a finite dimensional linear space V . Here we have g = L (V , V ) and the Lie bracket is the commutator [X , Y ] = XY − YX . Many informations on Lie groups can be obtained from their Lie algebras through the exponential map exp. Theorem 92 Let G be a Lie group with Lie algebra g. There there exists a unique function exp : →G such that (i) exp(0) = e (ii) dtd exp(tX )|t=0 = X . (iii) exp((t + s)X ) = exp(tX ) exp(sX ), for all t, s ∈ R. (iv) Ad(exp X ) = ead X .

Appendix D: Lie Groups and Coherent States

549

There is an open neighborhood U of 0 in g and an open neighborhood V of e in G such that the exponential mapping is a diffeomorphism form U onto V . This local diffeomorphism can be extended in a global one if moreover the group G is connected and simply connected. Other properties of the exponential mapping are given in [88]. A useful tool on Lie group is integration. Definition 58 Let μ be on the Lie group G. μ is left-invariant  a Radon measure  (left Haar measure) if f (x)d μ(x) = f (yx)d μ(x)  for every y ∈ G and μ is right -invariant (right Haar measure) if f (x)d μ(x) = f (xy)d μ(x) for every y ∈ G. If μ is left and right invariant we say that μ is a bi-invariant Haar measure. If there exists on G a bi-invariant Haar measure, G is said unimodular. Theorem 93 On every connected Lie group G there exits a left Haar measure μ. This measure is unique up to a multiplicative constant. If G is a compact and connected Lie group then a left Haar measure is a right Haar measure and there exists a unique bi-invariant Haar probability measure i.e. every compact Lie group is unimodular. Remark 114 The affine group G aff = {x → ax + b, a, b ∈ R} is not unimodular but the Heisenberg group Hn and SU (1, 1) are. This remark is important to understand the differences between the corresponding coherent states associated with these groups. Coherent states associated with the affine group are called wavelets. In many examples a Lie group G is a closed connected subgroup of the linear group GL(n, R) (or GL(n, C)). Then we can compute a left Haar measure as follows (see [230] for details). Let us consider a smooth system (x1 , . . . , xn ) on an open set U of G and the ∂A matrix of one-forms  = A−1 dxj . Then we have following [230]: j ∂x 1≤j≤n Proposition 166  is a matrix of left-invariant one-forms in U . Moreover the linear space spanned by the elements of  has dimension n. There exist n independent left invariant one-forms ω1 , . . . , ωn and ω1 ∧ ω2 ∧ · · · ∧ ωn defines a left Haar measure on G. Explicit examples of Haar measures: dx (i) On the circle S1 ≡ R/(2π Z) the Haar probability measure is d μ(x) = 2π . (ii) Haar probability measure on SU (2). Consider the parametrization of SU (2) by the Euler angles (see Chap. 7). g(θ, ϕ, ψ) =

  cos(θ/2)e−i/2(ϕ+ψ) − sin(θ/2)ei/2(ψ−ϕ) sin(θ/2)e−i/2(ψ−ϕ) cos(θ/2)ei/2(ϕ+ψ)

A straightforward computation gives

(D.1)

550

Appendix D: Lie Groups and Coherent States

2g −1 d g =



 −i(cos θ d ϕ + d ψ) eiψ (d θ − i sin θ d ϕ) . e−iψ (d θ + i sin θ d ϕ) i(cos θ d ϕ + d ψ)

So we get, after normalization the Haar probability on SU (2): d μ(θ, ϕ, ψ) =

1 sin θ d θ d ϕd ψ. 16π 2

(iii) Haar measure for SU (1, 1). The same method as for SU (2) using the parametrization   cosh 2t ei(ϕ+ψ)/2 sinh 2t ei(ϕ−ψ)/2 . g(ϕ, t, ψ) = sinh 2t ei(ψ−ϕ)/2 cosh 2t e−i(ϕ+ψ)/2 After computations we get 2g −1 d g =



i(cosh td φ + d ψ) e−iψ (dt + i cosh td φ) eiψ (dt − i cosh td φ) i(cosh td φ − d ψ)



and a left Haar measure: d μ(t, φ, ψ) = cosh tdtd φd ψ.

(D.2)

(iii) Let us consider the Heisenberg group Hn (see Chap. 1). These group can also be realized as a linear group as follows. Let ⎛

1 ⎜0 ⎜ ⎜0 ⎜ g(x, y, s) = ⎜ . ⎜ .. ⎜ ⎝0 0

x2 · · · 0 ··· 1 0··· .. .. . . 0 0 ··· 0 0 ···

x1 1 0 .. .

⎞ s y1 ⎟ ⎟ y2 ⎟ ⎟ .. ⎟ .⎟ ⎟ 1 yn ⎠ 0 1

xn 0 0 .. .

where x = (x1 , . . . , xn ) ∈ Rn , y = (y1 , . . . , yn ) ∈ Rn , s ∈ R. We can easily check that ˜ n of the linear group GL(2n + {g(x, y, s), x, y ∈ Rn , s ∈ R} is a closed subgroup H ˜ 1, R). Hn is isomorphic to the Weyl-Heisenberg, group Hn by the isomorphism g(x, y, s) → (s −

x · y x − iy , √ ) 2 2

˜ n i.e. the WeylThe Lebesgue measure dxdyds is a bi-invariant measure on H Heisenberg group is unimodular. We shall see that when considering coherent states on a general Lie group G it is useful to consider a left invariant measure on a quotient space G/H where H is a closed subgroup of G; G/H is the set of left coset, it is a smooth analytic manifold

Appendix D: Lie Groups and Coherent States

551

with an analytic action of G: for every x ∈ G, τ (x)(gH ) = xgH . The following result is proved in [138] Theorem 94 There exists a G-invariant measure d μG/H on G/H if and only if we have | det Ad G (h)| = | det AdH (h)|, ∀h ∈ H , where Ad G is the adjoint representation for the group G. Moreover this measure is unique up to a multiplicative constant and we have for any continuous function f , with compact support in G, 



 f (g)d μG (g) =

G/H

 f (gh)d μH (h) d μG/H (gH )

H

where d μG and d μH are left Haar measure suitably normalized. In this book we have considered the three groups Hn , SU (2) and SU (1, 1) and their related coherent states. In each case the isotropy subgroup H is isomorphic to the unit circle U (1) and we have found the quotient spaces: Hn /U (1) ≡ R2n , SU (2)/U (1) ≡ S2 and SU (1, 1)/U (1) ≡ PS2 with their canonical measure. Each of this space is a symplectic space and can be seen as the phase space of classical systems.

D.3

Representations of Lie Groups

The goal of this section is to recall some basic facts.

D.3.1

General Properties of Representations

G denotes an arbitrary connected Lie group, V1 , V2 , V are complex Hilbert spaces, L (V1 , V2 ) the space of linear continuous mapping from V1 into V2 , L (V ) = L (V , V ), GL(V ) the group of invertible mappings in L (V ), U (V ) the sub-group of GL(V ) of unitary mappings i.e. A ∈ U (V ) if and only if A−1 = A∗ . Definition 59 A representation of G in V is a group homomorphism Rˆ from G in ˆ GL(V ) such that (g, v) → R(g)v is continuous from G × V into V . ˆ If R(g) ∈ U (V ) for every g ∈ G the representation is said unitary. ˆ Definition 60 The subspace E ⊆ V is invariant by the representation Rˆ if R(g)E ⊆E for every g ∈ G. The representation Rˆ is irreducible in V if the only invariant closed subspaces of V are {V , {0}}.

552

Appendix D: Lie Groups and Coherent States

Definition 61 Two representations (Rˆ 1 , V1 ) and (Rˆ 2 , V2 ) are equivalent if there exists an invertible continuous linear map A : V1 → V2 such that R2 (g)A = AR1 (g), ∀g ∈ G. Irreducible representations are important in physics: they are associated to elementary particles (see [256]). Let d μ be a left Haar measure on G. Consider the Hilbert space L2 (G, d μ) and define L(g)f (x) = f (g −1 x) where g, x ∈ G, f ∈ L2 (G, d μ). L is a unitary representation of G called the left regular representation. The Schur lemma is an efficient tool to study irreducible dimensional representations. Lemma 111 (Schur) Suppose Rˆ 1 and Rˆ 2 are finite dimensional irreducible representations of G in V1 and V2 respectively. Suppose that we have a linear mapping A : V1 → V2 such that ARˆ 1 (g) = Rˆ 2 (g)A for every g ∈ G. Then or A is bijective or A = 0. ˆ ˆ = R(g)A for all g ∈ G then A = λ1l for In particular if V1 = V2 = V and AR(g) some λ ∈ C. ˆ V ) is a unitary representation in the Hilbert space H then it is Suppose that (R, irreducible if and only if the only bounded linear operators A in V commuting with ˆ ˆ Rˆ (AR(g) = R(g)A for every g ∈ G) are A = λ1l, λ ∈ C. A useful property of a representation Rˆ is its square integrability (see [123] for details). Definition 62 A vector v ∈ V is said to be admissible if we have  ˆ |R(g)v, v|2 d μ(g) < +∞.

(D.3)

G

The representation Rˆ is said square integrable if Rˆ is irreducible and there exists at least one admissible vector v = 0. If G is compact every irreducible representation is square integrable. The discrete series of SU (1, 1) are square integrable (prove that 1 is admissible using the formula (D.2) for the Haar measure on SU (1, 1)). We have the following result due to Duflo-Moore and Carey (see [123] for a proof). Theorem 95 Let Rˆ be a square integrable representation in V . Then there exists a unique self-adjoint positive operator C in V with a dense domain in V such that: (i) The set of admissible vectors is equal to the domain D(C). (ii) If v1 , v2 are two admissible vectors and w1 , w2 ∈ V then we have 

ˆ ˆ R(g)v 2 , w2 R(g)v1 , w1 d g = Cv1 , Cv2 w1 , w2 . G

(D.4)

Appendix D: Lie Groups and Coherent States

553

(iii) If G is unimodular then C = λ1l, λ ∈ R. Remark 115 If G is unimodular coherent states can be defined as follows. We start from an admissible vector v0 ∈ V , v0  = 1 and an irreducible representation Rˆ in ˆ V . Define the coherent state (or the analyzing wavelet) ϕg = R(g)v 0 . Then the family {λ−1/2 ϕg , |g ∈ G} is overcomplete in V : λ

−1

 ϕg , ψ1 ϕg , ψ2 d μ(g) = ψ1 , ψ2 , ψ1 , ψ2 ∈ V , G

where λ =

D.3.2

 G

2 ˆ |R(g)v 0 , v0 | d μ(g).

The Compact Case

Representation theory for compact group is well known (for a concise presentation see [170] or for more details [171]). Typical examples are SU (2) and SO(3) considered in Chap. 7. Here G is a compact Lie group. The main facts are the following 1. Every finite dimensional representation is equivalent to a unitary representation. 2. Every irreducible unitary representation of G is finite dimensional and every unitary representation of G is a direct sum of irreducible representations 3. If Rˆ 1 , Rˆ 2 are non equivalent irreducible finite representations of G on V1 and V2 then  Rˆ 1 (g)v1 , w1 Rˆ 2 (g)v2 , w2  = 0, for all v1 , w1 ∈ V1 , v2 , w2 ∈ V2 . G

4. If Rˆ is an irreducible unitary representation of G then we have 

ˆ ˆ R(g)v 1 , w1 R(g)v2 , w2  = v1 , v2 w1 , w2 ,

(dimV ) G

for all v1 , w1 ∈ V1 , v2 , w2 ∈ V2 .

(D.5)

5. (Peter-Weyl Theorem) If we denote (Rˆ λ , Vλ ), λ ∈ , the set of all irreducible representations of G and Mλ,v,w (g) = Rˆ λ (g)v, w, then the linear space spanned by {Mλ,v,w (g)|g ∈ G, v, w ∈ Vλ } is dense in L2 (G, d μ).

554

D.3.3

Appendix D: Lie Groups and Coherent States

The Non-compact Case

This case is much more difficult than the compact case and there is so far no a general theory of irreducible unitary representations. The typical example is SU (1, 1) or equivalently SL(2, R) considered in Chap. 7. These groups have the following properties Definition 63 (i) A Lie group G is said reductive if G is a closed connected subgroup of GL(n, R) or GL(n, C) stable under inverse conjugate transpose. (ii) A Lie G is said linear connected semisimple if G is reductive with finite center. Proposition 167 If G is a linear connected semisimple group its Lie algebra g is semisimple. It is known that a compact connected Lie group can be realized as a linear connected reductive Lie group ([168], Theorem 1.15). Let us consider the Lie algebra g of G. The differential of the mapping (A) = A−1,∗ at e = 1l is denoted θ . We have θ 2 = 1l so θ has two eigenvalues ±1. So we have the decomposition g = l ⊕ p where l = ker(θ − 1) and p = ker(θ + 1). Let K = {g ∈ G|θg = g}. The following result is a generalization of the polar decomposition for matrices or operators in Hilbert spaces. Proposition 168 (polar Cartan decomposition) If G is a linear connected reductive group then K is a compact connected group and is a maximal compact subgroup of G. Its Lie algebra is l and the map: (k, X ) → k exp X is a diffeomorphism from K × p onto G.

D.4

Coherent States According Gilmore-Perelomov

Here we describe a general setting for a theory of coherent states in a arbitrary Lie group from the point of view of Perelomov (for more details see [208, 209]). We start from an irreducible unitary representation Rˆ of the Lie group G in the Hilbert space H . Let ψ0 ∈ H be a fixed unit vector (ψ0  = 1) and denote ψg = ˆ R(g)ψ 0 for any g ∈ G. In quantum mechanics states in the Hilbert space H are determined modulo a phase factor so we are mainly interested in the action of G in the projective space P(H ) (space of complex lines in H ). We denote by H the iθ ˆ isotropy group of ψ0 in the projective space: H = {h ∈ G|R(h)ψ 0 = e ψ0 }. So the coherent states system {ψg |g ∈ G} is parametrized by the space G/H of left coset in G modulo H : if π is the natural projection map: G → G/H . Choosing for each x ∈ G/H some g(x) ∈ G we have, with x = π(g), ψg = eiθ(g) ψg(x) . Moreover ψg1 and ψg2 define the same states if and only if π(g1 ) = ψg2 := x hence we have ψg1 = eiθ1 ψg(x) and ψg2 = eiθ2 ψg(x) . So for every x ∈ G/H we have defined the state |x = {eiα ψg } where x = π(g). It is convenient to denote |x = ψg(x) and x(g) = ψ(x). This parametrization of coherent states by the quotient space G/H has the following nice properties.

Appendix D: Lie Groups and Coherent States

555

We have ψg = eiθ(g) |x(g) and θ (gh) = θ (g) + θ (h) if g ∈ G and h ∈ H . The action of G on the coherent state |x satisfies ˆ 1 )|x = eiβ(g1 ,x) |g1 .x R(g

(D.6)

where g1 .x denotes the natural action of G on G/H and β(g1 , x) = θ (g1 g) − θ (g) where π(g) = x (β depends only on x, not in g). Computation of the scalar product of two coherent states gives ˆ 1 −1 g2 )|0 x1 |x2  = ei(θ(g1 )−θ(g2 )) 0|R(g

(D.7)

where x1 = x(g1 ) and x2 = x(g2 ). Moreover if x1 = x2 we have |x1 |x2 | < 1 and g.x1 |g.x2  = ei(β(g,x1 )−β(g,x2 )) x1 |x2 .

(D.8)

Concerning completeness we have Proposition 169 Assume that the Haar measure on G induces a left invariant measure d μ(x) on G/H (see Theorem 94) and that the following square integrability condition is satisfied:  |0|x|2 dx < +∞.

(D.9)

M

Then we have the resolution of identity: 1 d where d =

 M

 x|ψψx d μ(x) = ψ, ∀ψ ∈ H .

(D.10)

M

|0|x|2 dx. Moreover we have a Plancherel identity 1 ψ|ψ = d

 |x|ψ|2 d μ(x).

(D.11)

M

Remark 116 (i) As we have seen in Chap. 2, using formula (D.10) and (D.11), we can consider Wick quantization for symbols defined on M = G/H . (ii) When the square integrability condition (D.9) is not fulfilled (the Poincaré group for example) there exists an extended definition of coherent states. This is explained in [3]. Remark 117 When G is a compact semi-simple Lie group and Rˆ is a unitary irreducible representation of G in a finite dimensional Hilbert space H then it is possible to choose a state ψ0 in H such that if H is the isotropy group of ψ0 then G/H is a Kähler manifold (see [201]). Some results concerning coherent states and quantization have also been obtained for non compact semi-simple Lie groups extending results already seen in Chap. 8 for SU (1, 1).

Appendix E

Berezin Quantization and Coherent States

We have seen in Chap. 12 that canonical coherent states are related to Wick and Weyl quantizations. Berezin in the paper [24] has given a general setting to quantize “classical systems”. Let us explain here very briefly the Berezin construction. Let M be a classical phase space i.e. a symplectic manifold with a Poisson bracket denoted {·, ·} and an Hilbert space H . Assume that for a set of positive numbers , with 0 as limit point, we have a linear mapping A → Aˆ  where A is a smooth function on M and Aˆ  is an operator on H . The inverse mapping is denoted S (Aˆ  ). In general it is difficult to describe in details the definition domain and the range of this quantization mapping. Some example are considered in [246]. Nevertheless for a quantization mapping the two following conditions are required, to preserve the Bohr’s correspondence condition (semi-classical limit), lim S (Aˆ  Bˆ  )(m) = A(m)B(m) ∀m ∈ M , (C1)

→0

1 S ([Aˆ  , Bˆ  ])(m) = {A, B}(m) ∀m ∈ M , (C2). →0 i lim

We have seen in Chap. 1 that these conditions are fulfilled for the Weyl quantization of R2n . In [24] the author have considered the two dimensional sphere and the pseudosphere (Lobachevskii plane). In these two examples the Planck constant  is replaced by 1n where n is an integer parameter depending on the considered representation. The semi-classical limit is the limit n → +∞. For the pseudosphere n = 2k, where k is the Bargmann index. In this section we shall explain some of the Berezin’s ideas concerning quantization on the pseudosphere and we shall prove that the Bohr’s correspondence principle is satisfied using results taken from Chap. 7. The same results could be proved for quantization of the sphere [24], using results of Chap. 7.

© Springer Nature Switzerland AG 2021 D. Robert and M. Combescure, Coherent States and Applications in Mathematical Physics, Theoretical and Mathematical Physics, https://doi.org/10.1007/978-3-030-70845-0

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Nowadays the quantization problem has been solved in much more general settings in particular for Kähler manifolds (the Poincaré disc D or the Riemann sphere S2 are examples of Kähler manifolds) where generalized coherent states are still present. This domain is still very active and is named Geometric Quantization; its study is outside the scope of this book (see the book [264] and the recent review [231]). We have defined before the coherent states family ψζ , ζ ∈ D, for the representations Dn+ of the group SU (1, 1) which is a symmetries group for the Poincaré disc D. Recall that {ψζ }ζ ∈D is an overcomplete system in Hn (D) hence the map ϕ → ϕ  , where ϕ  (z) = ψz , ϕ, is an isometry from Hn (D) into L2 (D). ¯ is defined Let Aˆ be a bounded operator in Hn (D). Its covariant symbol Ac (z, w) as ˆ w ψz , Aψ . ¯ = Ac (z, w) ψz , ψw  It is an holomorphic extension in (z, w) ¯ of the usual covariant symbol Ac (z, z¯ ). Moreover the operator Aˆ is uniquely determined by its covariant symbol and we have  ˆ Aϕ(z) = Ac (z, w)ϕ(w)ψ ¯ (E.1) z , ψw d νn (w). D

From (E.1) we get a formula for the covariant symbol product of the product of two ˆ B. ˆ If (AB)c denotes the covariant symbol of Aˆ Bˆ then we have the formula operators A,  (AB)c (z, z¯ ) =

D

Ac (z, w)B ¯ c (w, z¯ )|ψz , ψw |2 d νn (w).

(E.2)

From our previous computations (Chap. 7) we have  |ψw , ψz |2 =

(1 − |z|2 )(1 − |w|2 ) |1 − z¯ w|2

n .

We have to consider the following operator Tn F(z, z¯ ) =

n−1 4π



 D

F(w, w) ¯

(1 − |z|2 )(1 − |w|2 ) |1 − z¯ w|2

n d μ(w)

for F bounded in D and C 2 -smooth. Proposition 170 We have the following asymptotic expansion, for n → +∞,  Tn F(z, z¯ ) = F(z, z¯ ) 1 −

n (n − 2)2



1 ∂ 2F + (1 − |z|2 )2 (z, z¯ ). n ∂z∂ z¯

(E.3)

Proof Using invariance by isometries of D we show that it is enough to prove formula (E.3) for z = 0.

Appendix E: Berezin Quantization and Coherent States

559

ζ −z Let us consider the change of variable w = 1−¯ . Denote G(ζ, ζ¯ ) = F(w, w) ¯ then zζ we get Tn F(z, z¯ ) = Tn G(0, 0). A direct computation gives

∂2 F ∂2G (0, 0) = (1 − |z¯z |)2 (z, z¯ ) = F(z; z¯ ) ∂z∂ z¯ ∂ζ ∂ ζ¯ where  is the Laplace-Beltrami operator on D. So we have proved (E.3) for any z if it is proved for z = 0. To prove (E.3) for z = 0 we use the real Laplace method for asymptotic expansion of integrals A.4. We write  1  n − 1 2π dθ drF(reiθ , re−iθ )re−(n−2)φ(r) (E.4) Tn F(0, 0) = π 0 0 where φ(r) = log((1 − r 2 )−1 ). Note that φ(r) > 0 for r ∈]0, 1] and φ(0) = 1. So method (see Sect. A.4 for a precise statement) to estimate (E.4) with the large parameter λ = n − 2. Modulo an exponentially small term it is enough to integrate  in r in [0, 1/2]. Using the change of variable φ(r) = s2 , s > 0 we have r(s) = 1 − e−s2 and n−1 Tn F(0, 0) = π

 0

2π

 dθ

c

dse−λs K(r(s))se−s + O(λ−∞ ) 2

0

where K(r) = F(reiθ , re−iθ ). Now to get the result we have to compute the asymp2 totic expansion at s = 0 for L(s) = K(r(s))se−s . Note that L(s) is periodic in θ and we have to consider only the part of the expansion independent in θ . If L0 (s) is this part, we get after computations n−1 n−1 F(0, 0) + L0 (s) = 2π(n − 2) 2π(n − 2)2



 ∂ 2F 1 (0, 0) − F(0, 0) + O( 2 ) ∂z∂ z¯ n 

and formula (E.4) follows.

It is not difficult, using Proposition 170, to check the correspondence principle (C1) and (C2). We get (C1) by applying the Proposition to FAB (w, w) ¯ = Ac (z, w)B ¯ c (w, z¯ ). So we have (AB)c (z, z¯ ) = Tn (z, z¯ ) −→ Ac (z, z¯ )Bc (z, z¯ ). n→+∞

For (C2) we write FAB (z, z¯ ) = Ac (z, z¯ )Bc (z, z¯ )(1 −

n 1 ) + (1 − |z|2 )2 (n − 1)2 n



   1 ∂Ac ∂Bc (z, z¯ ) (z, z¯ ) + O ∂ w¯ ∂w n2

560

Appendix E: Berezin Quantization and Coherent States

So we get n (FAB (z, z¯ ) − FBA (z, z¯ )) = (1 − |z|2 )2

∂Ac ∂Bc ∂Ac ∂Bc (z, z¯ ) (z, z¯ ) − (z, z¯ ) (z, z¯ ) + O ∂ z¯ ∂z ∂z ∂ z¯

  1 n

ˆ B] ˆ and But we know that FAB − FBA is the covariant symbol of the commutator [A, the Poisson bracket {A, B} is  {A, B}(z, z¯ ) = i(1 − |z|2 )2

∂Ac ∂Bc ∂Ac ∂Bc (z, z¯ ) (z, z¯ ) − (z, z¯ ) (z, z¯ ) ∂ z¯ ∂z ∂z ∂ z¯



So we get (C2).  We have seen that the linear symplectic maps and the metaplectic transformations are connected with quantization of the Euclidean space R2n . Here symplectic transformations are replaced by transformations in the group SU (1, 1) and metaplectic ˆ transformations by the representations g → R(g) = Dn− (g). Then we have Proposition 171 (i) For any bounded operator Aˆ in Hn (D) the covariant symbol ˆ Aˆ R(g) ˆ −1 is Ag of R(g) Ag (ζ ) = Ac (g −1 ζ, g −1 ζ ). (E.5) ˆ (ii) The covariant symbol R(g)c of R(g) is given by the formula R(g)c (ζ ) = e

in arg(α+βζ )



1 − |ζ |2 α¯ + β¯ ζ¯ − βζ − α|ζ |2

 (E.6)

 αβ . where g = ¯ β α¯ 

Proof (i) is a direct consequence of definition of Dn− (g) and computations of Chap. 8. For (ii) using formula (8.74) of Chap. 8 in complex variables we get, after computation,     1 − |ζ |2 α + βz n ψζ , ψg−1 ζ  = |α + βz| α¯ + β¯ ζ¯ − βζ − α|ζ |2 which gives (E.6).



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Index of Concepts

A Adiabatic decoupling, 420 Adiabatic evolution, 417 Adjoint representation, 551 Affine group, 269 Aharonov-Bohm effect, 343, 344 Anosov system, 162 Anticommutation relations, 434 Anti-Wick symbol, 56 Automorphism of the 2-torus, 160 B Baker–Campbell–Hausdorff formula, 6 Bargmann–Fock representation, 20 Beals characterization, 301 Berezian, 475 Berezin-Lieb inequality, 228 Berry phase, 218, 410 Birkhoff average, 160 Birkhoff ergodic theorem, 357, 358 Bloch coherent states, 211 Bloch decomposition, 168 Bogoliubov transformation, 263, 493 Bohr correspondence principle, 27 Bohr–Sommerfeld formula, 134 Bohr–Sommerfeld quantization rule, 155 Borel-Cantelli Lemma, 371 Borel summability, 117, 451 Borel transform, 451 Bosons, 431 C Calderon–Vaillancourt, 38 Canonical Anticommutation Relation, 458

Canonical Commutation Relations, 434, 443 Casimir operator, 201, 207, 241 Cayley transform, 498 Chodosh characterization, 305 Circulant matrices, 170 Classical action, 97, 131 Classical scattering matrix, 119 Classical wave operators, 119 Clean intersection condition, 142 Clean intersection hypothesis, 133 Clifford algebra, 461 Clifford algebra relations, 502 Coherent states classes, 304 Completeness, 473 Concentration of measure, 368 Continuous series, 244 Contractions of Lie groups, 233 Contravariant symbol, 29 Covariant symbol, 29 Creation and annihilation operators, 11, 436 Critical set, 531

D Decoherence, 377 Density matrix, 374 Density of states, 128 Density operator, 483 Diagonal peaks, 403 Dicke model, 526 Dicke states, 199 Dirac bracket, 521 Dirac quantum Hamiltonian, 522 Discrete Fourier Transform, 166 Discrete series, 244

© Springer Nature Switzerland AG 2021 D. Robert and M. Combescure, Coherent States and Applications in Mathematical Physics, Theoretical and Mathematical Physics, https://doi.org/10.1007/978-3-030-70845-0

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572 Dispersion, 209 Duhamel formula, 449 Duhamel’s principle, 95 Dyson series, 453

E Egorov theorem, 182 Ehrenfest time, 101 Eigenvalue, 127 Energy shell, 131 Entangled state, 375, 391 Entanglement, 377 Equipartition of eigenfunctions, 184 Equivalent representations, 552 Ergodic system, 353 Euler angles, 196 Euler-Lagrange equations, 512 Euler triangle, 217, 218 Existence of propagators, 314

F Fermions, 431 Field operator, 438 First return Poincaré map, 145 Floquet operator, 86 Fock space of bosons, 433 Fock space of fermions, 433 Fokker-Planck equation, 527 Fourier–Bargmann transform, 17 Fourier-integral operators, 318, 319 Fourier-Laplace transform, 271 Fourier transform, 10, 470 Frequency set, 52 Functional calculus, 127, 543

G Gauss decomposition, 246 Gaussian beams, 103, 130 Gaussian decomposition, 211 Generalized Pauli matrices, 169 Generating function, 460 Generator of squeezed states, 76, 77 Geodesic triangle, 213 Geometric phase, 220 Gevrey series, 117, 451 Gilmore-Perelomov coherent states, 554 Grand canonical ensemble, 489 Grassmann algebra, 459 Grassmann analytic continuation principle, 510 Growth estimates, 308

Index of Concepts Gutzwiller trace formula, 128

H Haar measure, 549 Hadamard matrix, 170 Hamilton’s equations, 93 Hardy potential, 448 Harmonic classes, 297 Harmonic oscillator, 12 Harmonic Sobolev spaces, 299 Hartree equation, 447 Heisenberg commutation relation, 5 Heisenberg picture, 444 Helffer-Sjöstrand formula, 312, 544 Herman-Kluk, 328 Hermann-Kluk approximation, 327 Hilbert–Schmidt class, 19 Horodecki condition, 376 Husimi functions, 52 Hydrogen atom, 281 Hyperbolic geometry, 235 Hyperbolic triangle, 259

I Irreducible representations, 551

K Kähler form, 536 Kepler Hamiltonian, 273

L Laplace integral, 535 Laplace-Runge-Lenz operator, 282 Laplace-Runge-Lenz vector, 281 Larmor precession, 410 Linearized flow, 132 Liouville measure, 134 Lyapunov exponent, 105

M Maslov index, 130, 134 Master equation, 376, 394, 396 Mathieu equations, 86 Mehler formula, 25 Mehlig-Wilkinson formula, 495 Melher formula, 85 Metaplectic group, 32 Metaplectic transformations, 73 Microlocal, 130

Index of Concepts Minkowski metric, 235 Mixed state, 375 Mixing, 162 Möbius transformation, 225 Monodromy matrix, 132 Moyal bracket, 407 Moyal formula, 488 Moyal product, 43, 486, 539

N Nelson criterium, 438 Noether theorem, 512, 513 Non-crossing eigenvalues, 416 Nondegenerate, 133 Nondegenerate orbit, 132 Nondiagonal peaks, 403 Non-oscillating condition, 352 Non trapping, 120 Number operator, 433

O Open quantum systems, 373

P Parametrix, 546 Partial trace, 376, 377 Pauli matrices, 195 Pfaffian, 466 Poincaré disc, 240 Poincaré map, 133 Poisson bracket, 28 Poisson summation formula, 128 Pseudosphere, 235 Pure state, 375 Purity, 376, 379

Q Quantum Brownian motion, 374 Quantum entropy, 376 Quantum ergodicity, 357 Quantum Unique Ergodicity, 366 Quasi-mode of energy E, 153 Quasi-modes, 148, 361 Quaternion field, 275 Quaternions, 202

R Resolution of the identity, 19, 260 Riemann sphere, 224

573 S Scarring, 186 Scattering operator, 120 Scattering theory, 118 Schrödinger cat, 378 Schrödinger cat model, 399 Schrödinger equation, 64 Schrödinger paradox, 379 Schur lemma, 9, 18, 552 Schwartz space, 29 Second quantization, 433 Semi-classical, 128 Semiclassical Garding inequality, 58 Semiclassical Gutzwiller Trace Formula, 131 Semi-classical measures, 59 Semi-classical observable, 406 Semisimple groups, 554 Separability, 385 Separable state, 375 Siegel space, 67, 120 SL(2,R), 239, 269 Space of symbols, 42 Spectrum, 127 Spectrum of periodic Hamiltonians, 346 Spherical harmonics, 206 Spin coherent states, 211 Spin observables, 200 Spin orbit interaction, 413 Spin-orbit motion, 415 Spin representation, 492 Spin states, 200 Square integrable representations, 552 Squeezed state, 67, 78, 263 Stationary phase, 532 Stationary phase expansion, 138 Stereographic projection, 211, 224 Super algebra, 504 Super-coherent states, 499, 523, 524 Super-derivation, 507 Super Lie algebra, 506, 509 Super Lie-group, 508 Superlinear space, 504 Supermanifold, 507 Super-Poisson-bracket, 521 Super-spaces, 499 Supersymmetric harmonic oscillator, 501, 523 Supersymmetry, 499 Supersymmetry generators, 523 Super-trace, 474 Supermechanics, 516 Symplectic matrix, 64

574 Symplectic product, 6

T Temperate distributions, 29 Temperate weight, 42 Thermal covariance, 398 Thermal states, 378 Thermodynamic limit, 231 Tightness-condition, 352 Trace, 474 Trace class, 19 Transport equation, 337

U Uniquely ergodic, 161 Unique quantum ergodicity, 366

Index of Concepts V Vacuum state, 434 Van-Vleck formula, 342 Virial theorem, 285

W Wavelets, 269 Wave operators, 120 Wave packet transform, 411 Wehrl conjecture, 376 Weyl asymptotic, 48 Weyl asymptotic formula, 47, 143 Weyl–Heisenberg algebra, 8 Weyl translation operators, 439 Wigner functions, 32 Witten supersymmetric Hamiltonian, 522

Index of Symbols

Symbols SW (ρ), 376 A  B, 44 Aaw , 59 C ∞ (Rn|m ), 503 ˆ 4 D(A), J , 31 Jn , 84 Kos (t; x, y), 25 O(2, 1), 236 O(3), 194 PS2 , 236 Pγ , 145 SL(2, Z), 160 SO(2, 1), 236 SO(3), 194 SO(4), 273 SO0 (2, 1), 236 SU (1, 1), 238 SU (2), 195 W m,r (Rn ), 116 Ym , 283 ˆ B], ˆ 4 [A, Cm,m , 405 D, 239 S3 , 274  , 11 H, 240 B , 432 α,k (ω), 279 Rn|m , 503 Rn|m (B), 504 k n  (R ), 10 m,k n  (R ), 411 E , 131

BHS , 19 BTr , 19 ˆ D(B), 94 Hˆ os , 12 ˆ 5 P, ˆ 5 Q, ˆ R(F), 32 ˆ 27 A, , 4 ˆ ψ, 4 A a, 12 a† , 12 n, 211 F , 10 Wϕ , 34 sl(2, C), 198 so(2, 1), 237 so(3), 194 so(4), 274 su(2) , 195 μ Sρ , 43 Sm ρ , 297 Ssc (w), 45 μ Ssc , 45 μBx (f ), 161 φk , 14 (ν) φz,B , 94 σ, 9 σ ((x, y), (x , y )), 8 f˜ , 128 ˆ 4 ψ A, (Γ ) ϕ , 10 ϕzΓ , 11 (θ ) ϕγ ,z , 171 |j, m, 200

© Springer Nature Switzerland AG 2021 D. Robert and M. Combescure, Coherent States and Applications in Mathematical Physics, Theoretical and Mathematical Physics, https://doi.org/10.1007/978-3-030-70845-0

575

576 |z, 14  T (z), 6 W P z , 11 {A, B}, 28 {σ1 , σ2 , σ3 }, 195 a(f ), 436 a∗ (f ), 436 x  y, 235 H1 , 8 Hn , 9 N, 436 B , 21 HN (θ), 165 Hn (D), 245 Hu (z), 52 Wϕ,ψ , 33 Wz,z , 35 T2 , 159

Index of Symbols (j)

Dm (z1 , z2 ), 201 FuB , 17 H,4 C1 (H ), 19 C2 (H ), 19

S+ (n), 10 h1 , 8 FS[ψ ], 53 Met(n), 32 Mp(n), 73 Opw  ψ, 29 Opaw  (A), 59 OpAW ,θ , 177 Sp(n), 31 S00,sc , 311 log m

S0,u , 310 ˆ 19 Tr(B), Tr E  A, 376