Inequivalent representations of canonical commutation and anti-commutation relations 9789811521799, 9789811521805

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Mathematical Physics Studies

Asao Arai

Inequivalent Representations of Canonical Commutation and Anti-Commutation Relations Representation-theoretical Viewpoint for Quantum Phenomena

Mathematical Physics Studies Series Editors Giuseppe Dito, Dijon, France Edward Frenkel, Berkeley, CA, USA Sergei Gukov, Pasadena, CA, USA Yasuyuki Kawahigashi, Tokyo, Japan Maxim Kontsevich, Bures-sur-Yvette, France Nicolaas P. Landsman, Nijmegen, The Netherlands Bruno Nachtergaele, Davis, CA, USA

The series publishes original research monographs on contemporary mathematical physics. The focus is on important recent developments at the interface of Mathematics, and Mathematical and Theoretical Physics. These will include, but are not restricted to: application of algebraic geometry, D-modules and symplectic geometry, category theory, number theory, low-dimensional topology, mirror symmetry, string theory, quantum field theory, noncommutative geometry, operator algebras, functional analysis, spectral theory, and probability theory.

More information about this series at http://www.springer.com/series/6316

Asao Arai

Inequivalent Representations of Canonical Commutation and Anti-Commutation Relations Representation-theoretical Viewpoint for Quantum Phenomena

Asao Arai Department of Mathematics Hokkaido University Sapporo Hokkaido, Japan

ISSN 0921-3767 ISSN 2352-3905 (electronic) Mathematical Physics Studies ISBN 978-981-15-2179-9 ISBN 978-981-15-2180-5 (eBook) https://doi.org/10.1007/978-981-15-2180-5 © Springer Nature Singapore Pte Ltd. 2020 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 Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

As is well known, canonical commutation relations (CCR) and canonical anticommutation relations (CAR) are fundamental algebraic relations in quantum physics. Models in quantum mechanics and quantum field theory are constructed based on Hilbert space representations of CCR and/or CAR. There are basically two categories for representations of CCR and CAR, respectively, i.e., reducible and irreducible, and each of them is divided into two classes: equivalent and inequivalent. Quantum theories based on equivalent representations of CCR and/or CAR are physically equivalent, being different only in the framework of the physical picture.1 On the other hand, quantum theories based on inequivalent irreducible representations are essentially different from each other, describing non-comparable physical situations. Roles of equivalent representations and inequivalent irreducible representations of CCR and/or CAR are different. Although equivalent representations are physically equivalent to each other as mentioned above, they may be mathematically important. For example, there may be mathematical problems which are not so easy to solve in a representation, but relatively are very easy to solve in other representations equivalent to the former.2 Concerning inequivalent irreducible

1 For example, the Schrödinger representation of the CCR with d degrees of freedom (see Sect. 2.6.1 for the definition) is within the framework of the physical picture in which the position of a quantum particle is measured in the d-dimensional position space, while the momentum representation of the CCR with the same degree, which is equivalent to the Schrödinger one, is within the framework of the physical picture in which the momentum of a quantum particle is measured in the d-dimensional momentum space. In this case, the d-dimensional Fourier transform is the unitary transformation which maps the Schrödinger representation to the momentum representation, giving the equivalence of them (see Sect. 2.6.1 for more details). 2 A typical example is a one-dimensional quantum harmonic oscillator. The spectrum of the Hamiltonian is easily found in the Born–Heisenberg–Jordan representation of the CCR with one degree of freedom (see Sect. 2.6.2) rather than in the Schrödinger one with the same degree, which is equivalent to the former. In quantum field theory, the Q-space representation (functional Schrödinger representation), which is equivalent to the Fock representation of the CCR over a Hilbert space, is very useful (see Sect. 8.13).

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representations of CCR and/or CAR, the following philosophical point of view is suggested by studies on models in quantum mechanics and quantum field theory: The Universe uses inequivalent irreducible representations of CCR and/or CAR to create or produce “characteristic” quantum phenomena in which macroscopic objects may be involved. Examples of such phenomena include: Aharonov–Bohm effect, Casimir effect, Bose–Einstein condensation, superfluidity, superconductivity, boson masses, and fermion masses (see the text of this book for further details). This book is concerned with mathematical analysis on representations of CCR and CAR, including both finite and infinite degrees of freedom. With a philosophy as stated above, the emphasis of the description is put on inequivalent irreducible representations of CCR and CAR and reading their roles in the context of quantum physics. It may be asked why it still is necessary to consider or study representations of CCR with finite degrees of freedom; is not the von Neumann uniqueness theorem enough? The answer is no, because the von Neumann uniqueness theorem applies only to Weyl representations of CCR with finite degrees of freedom on a separable Hilbert space (unfortunately this point seems not to be understood precisely in general among non-experts of representation theory of CCR). For nonWeyl representations of CCR, even if their degrees of freedom are finite, the von Neumann uniqueness theorem is not valid anymore. But, interestingly enough, the Universe makes use of this invalidity to create or produce “characteristic” quantum phenomena, giving physically important roles to non-Weyl representations of CCR with finite degrees of freedom as well. In the case of CCR with infinite degrees of freedom, even in the category of irreducible Weyl representations, inequivalent ones appear and some of them in quantum field theory correspond to physically interesting quantum phenomena as mentioned above. There may be inequivalent representations of CCR and CAR which are related to spontaneous symmetry breaking.3 In this book, however, we do not discuss this aspect. To do that, one needs approaches using ∗-algebras (C ∗ -algebras, von Neumann algebras, or O ∗ -algebras) and their representations. Representations of CCR and CAR treated in this book are basically the ones that are operator realizations of algebraic relations on Hilbert spaces. This book is organized as follows. Chapter 1 is a summary of mathematical theories which may be used in the following chapters. It is mainly concerned with operator theories on Hilbert spaces. Those who are familiar with the subjects may skip this chapter. In Chap. 2, we describe a general theory of representations of CCR with finite degrees of freedom. For each degree of freedom, three classes of representations of CCR exist basically: ordinary ones in each of which all the representatives are 3 This is seen, e.g., in a Bose–Einstein condensation of an infinite system of free Bose gas at finite temperature.

Preface

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not bounded, weak Weyl ones in each of which half of the representatives are unitary, and Weyl ones in each of which every representative is unitary. In addition to these classes, we consider also weak forms of CCR. We review Schrödinger representations and Born–Heisenberg–Jordan (BHJ) representations of CCR and give a direct proof of the equivalence of them with the same degrees of freedom, which corresponds to the equivalence of Schrödinger’s “wave mechanics” as a quantum theory4 and “matrix mechanics” by BHJ. Concerning the von Neumann uniqueness theorem mentioned above, we only state it without proof. A detailed analysis on weak Weyl representations is given. It is noteworthy that, in weak Weyl representations, each self-adjoint representative is absolutely continuous. Chapter 3 is devoted to the analysis of a representation of the CCR with two degrees of freedom appearing in quantum mechanics of a two-dimensional system of a charged particle moving around a magnetic field concentrated on some points in the plane. It is well known that, in such a physical situation, a characteristic phenomenon, called the Aharonov–Bohm (AB) effect (a shift of the interference pattern of electron beams due to the existence of a solenoidal magnetic field), may occur. We show that the representation is irreducible and formulate a necessary and sufficient condition for the representation to be equivalent to the Schrödinger representation with the same degree. We show that the case where the representation is inequivalent to the Schrödinger one corresponds to the occurrence of the AB effect in the context under consideration. This is an interesting correspondence to note. In Chap. 4, we present a general mathematical theory of time operators. In the first stage of cognition, a time operator T is defined to be a canonical conjugate of a Hamiltonian H , i.e., it is a representative (symmetric, but not necessarily selfadjoint) in a representation of the CCR with one degree of freedom ([T , H ] = i on a suitable subspace of the representation space under consideration). Concerning this definition, the following point may be remarked. There has been a misunderstanding that there exist no time operators as observables (self-adjoint operators). This may be due to Pauli’s statement in the absence of time operators (see Sect. 4.2 for more details) or mathematically non-rigorous arguments which do not care about domains of unbounded operators. In fact, Pauli’s statement is false. This is shown by an explicit construction of a self-adjoint time operator for a class of Hamiltonians (see Sect. 4.4) (but, every Hamiltonian does not have a self-adjoint time operator). We introduce five classes of time operators. A rigorous form of time–energy uncertainty relation is formulated in terms of time operator. If H is semi-bounded (bounded from below or above), then the representation (T , H ) of the CCR with one degree of freedom is obviously inequivalent to the Schrödinger representation with the same degree. Hence, here we have classes of representations inequivalent to Schrödinger representations. We infer that each class of time operators plays different roles in physics. For example, as we shall show, a strong time operator controls the decayin-time rate of transition probabilities of state vectors (see Sect. 4.5.14). There is

4 In a one-body problem in three-dimensional space, Schrödinger’s wave mechanics can be viewed also as a classical field theory.

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a time operator which is related to time delay in a two-body scattering process in non-relativistic quantum mechanics (see Remark 4.15). Chapter 5 is a brief description of part of representation theory of CAR with finite degrees of freedom. Representations of CAR with finite degrees of freedom are physically related to internal degrees of freedom like spin. In Chap. 6 (resp. Chap. 7), we describe elements of the theory of boson (resp. fermion) Fock space, which is used to construct Bose (resp. Fermi) fields in quantum field theory. Only requisites for the following chapters are given. In Chap. 8 (resp. Chap. 9), we analyze representations of CCR (resp. CAR) with infinite degrees of freedom in an abstract framework. The last chapter is devoted to descriptions of various inequivalent representations of CCR or CAR with infinite degrees of freedom and physical correspondences of them, including scaling of quantum fields, infinite volume theories, Bose– Einstein condensation, temperatures, infrared and ultraviolet renormalizations, boson masses, and masses of Dirac particles. For the reader’s convenience, four appendices are added in this book. They are concerned with basic mathematical subjects. This book is written as an introductory text to theories of representations of CCR and CAR in relation to quantum physics with special attention given to their inequivalent representations, being directed to general audiences including graduate students and researchers in mathematics and physics who are interested in mathematical structures of quantum mechanics and quantum field theory. For this reason, in the main text (Chaps. 2–10) of the book, the author tried to make mathematical descriptions as detailed as possible. No completeness is intended for the bibliography in this book, and references in the bibliography are restricted more or less to those which are directly related to the subjects discussed in the text. Articles and books written in Japanese also are included in the bibliography for convenience of readers who can read Japanese. It would be the author’s great pleasure if this book could demonstrate the above-mentioned philosophical point of view to a certain extent and suggest the unfathomable richness and depth of the world of representations of CCR and CAR as fundamental concepts underlying quantum phenomena. Sapporo, Japan October 2019

Asao Arai

Acknowledgments

Theory of representations of CCR has been one of my research subjects since I started my research career in 1978. During past years, I have benefited from conversations or discussions with many researchers on the subject. First of all, I would like to express my hearty thanks to Professor Hiroshi Ezawa (Emeritus Professor, Gakushuin University) who inspired me with the importance of the representation-theoretical point of view for quantum mechanics and quantum field theory, in particular, the importance of inequivalent representations of CCR. I have learned a lot from his excellent book Structures of Quantum Mechanics (IwanamiShoten, 1978, in Japanese) and from his seminars on mathematical physics at Gakushuin University. Professor Ezawa gave me also the opportunity to deliver a series of lectures on representations of CCR and physics in summer schools on mathematical physics of which he was one of the organizers. I would like to thank also Professor Shige-Toshi Kuroda (Emeritus Professor, Gakushuin University), who was my teacher when I was a graduate student at the University of Tokyo, and Professor Huzihiro Araki (Emeritus Professor, Kyoto University) for their kind interest in my work on representations of CCR. My thanks go also to the following researchers who gave me the opportunity to deliver a series of lectures or a seminar on representations of CCR at their institutions (honorific titles omitted): Daisuke Fujiwara, Harald Grosse, Masao Hirokawa, Fumio Hiroshima, Takashi Ichinose, Atsushi Inoue, Hiroshi Isozaki, Tadashi Kawanago, Hideki Kurose, Itaru Mitoma (deceased), Nobuaki Obata, Masanori Ohya (deceased), Izumi Ojima, Tomohiro Sasamoto, Herbert Spohn, Kenji Yajima.

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1

Mathematical Preliminaries .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Linear Operators on Hilbert Spaces . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.1 Definitions and Some Basic Facts. . .. . . . . . . . . . . . . . . . . . . . 1.1.2 Convergences of Bounded Operators . . . . . . . . . . . . . . . . . . . 1.1.3 Spectra.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.4 Unitary Operators and Related Facts . . . . . . . . . . . . . . . . . . . 1.1.5 Symmetric and Self-adjoint Operators .. . . . . . . . . . . . . . . . . 1.1.6 Lowest Energy and Ground State of a Self-adjoint Operator . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.7 Spectral Theorem and Some Facts in Functional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.8 Trace Class and Hilbert–Schmidt Operators.. . . . . . . . . . . 1.1.9 Reduction of Operators .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.10 Strongly Continuous One-Parameter Unitary Groups .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.11 Strongly Continuous One-Parameter Semi-Groups . . . . 1.2 Strong Commutativity of Self-adjoint Operators .. . . . . . . . . . . . . . . . . 1.3 Criteria of Self-adjointness of Symmetric Operators.. . . . . . . . . . . . . 1.4 Basic Properties of Symmetric Operators . . . . .. . . . . . . . . . . . . . . . . . . . 1.5 Tensor Product of Hilbert Spaces . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.6 Symmetric and Anti-symmetric Tensor Product Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.7 Tensor Products of L2 -Spaces . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.8 Tensor Products of Operators . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.9 The Natural Isomorphism Between I2 (H ) and H ⊗ H . . . . . . . 1.10 Convergence of Self-adjoint Operators in Strong Resolvent Sense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.11 Sesquilinear Form and Representation Theorem .. . . . . . . . . . . . . . . . . 1.12 Fourier Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.13 Discrete Fourier Transform .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.14 Momentum Operator with Periodic Boundary Condition.. . . . . . . .

1 1 1 8 8 10 11 12 13 15 19 20 22 24 26 29 31 34 36 38 41 43 44 45 48 50 xi

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1.15 1.16 2

3

Momentum Operators with Other Boundary Conditions . . . . . . . . . Higher-Dimensional Discrete Fourier Transform . . . . . . . . . . . . . . . . .

Representations of Canonical Commutation Relations with Finite Degrees of Freedom .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Definitions and Fundamental Properties .. . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.1 Definition of Representation of Canonical Commutation Relations and Remarks . . . . . . . . . . . . . . . . . . 2.1.2 Some General Properties . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.3 Unitary Equivalence . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Heisenberg Uncertainty Relation.. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Direct Sum Representations .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4 Tensor Product Representations .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5 Irreducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.6 Quantum Mechanics as Representations of CCR . . . . . . . . . . . . . . . . . 2.6.1 Schrödinger Representation . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.6.2 Born–Heisenberg–Jordan Representation . . . . . . . . . . . . . . 2.6.3 Equivalence of Schrödinger Representation—“Wave Mechanics”—and BHJ Representation—“Matrix Mechanics” .. . . . . . . . . . . . . . . . . 2.6.4 Quantum System in a Finite Box . . .. . . . . . . . . . . . . . . . . . . . 2.6.5 Quantum System in a Half-Line . . . .. . . . . . . . . . . . . . . . . . . . 2.7 Cyclic Representations . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.8 Weyl Representations .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.8.1 Heuristics .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.8.2 Weyl Relations and CCR . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.8.3 Weyl Representation of CCR. . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.8.4 Spectral Properties .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.8.5 Irreducibility . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.8.6 Unitary Invariance . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.8.7 Real Linear Combinations of Representatives.. . . . . . . . . 2.8.8 Von Neumann’s Uniqueness Theorem and Remarks .. . 2.9 Weak Form of CCR . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.10 Weak Weyl Representations .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.11 Absolute Continuity of Self-adjoint Representatives in Weak Weyl Representations . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.12 Functional Calculus for a Weak Weyl Relation . . . . . . . . . . . . . . . . . . . Aharonov–Bohm Effect and Inequivalent Representations of CCR .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 Essential Self-adjointness of the Physical Momenta . . . . . . . . . . . . . . 3.3 Strongly Continuous One-Parameter Unitary Groups Generated by the Physical Momenta .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4 Commutation Relations of Exponential Operators .. . . . . . . . . . . . . . . 3.5 Remarks on the Case of Constant Magnetic Fields . . . . . . . . . . . . . . .

54 57 59 59 59 63 66 67 71 72 74 81 83 89

96 100 106 108 114 114 116 120 121 122 123 124 127 130 132 138 143 147 147 149 153 153 155

Contents

3.6 3.7 3.8 3.9 3.10 3.11 4

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Flatness of Vector Potentials and Existence of Representations of CCR . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Inequivalent Representations .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Correspondence to Aharonov–Bohm Effect .. .. . . . . . . . . . . . . . . . . . . . Representations for Different Vector Potentials . . . . . . . . . . . . . . . . . . Charges and Inequivalent Irreducible Representations of CCR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Notes.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

Time Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 An Abstract Definition of Time Operator and Time-Energy Uncertainty Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 A Structure Generating Pairs of a Hamiltonian and a Time Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4 Time Operators of Hamiltonians with Purely Discrete Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4.1 An Operator Associated with a Self-adjoint Operator with Purely Discrete Spectrum . . . . . . . . . . . . . . . 4.4.2 A Time Operator of H . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4.3 A Sufficient Condition for Tmax to be Bounded . . . . . . . . 4.4.4 Unboundedness of TG . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4.5 Concluding Remarks . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5 Strong Time Operators.. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5.1 Definition and Basic Properties . . . . .. . . . . . . . . . . . . . . . . . . . 4.5.2 Self-adjoint Strong Time Operator and Weyl Representation of CCR . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5.3 Non-Self-adjointness of Strong Time Operators . . . . . . . 4.5.4 A Perturbation Theorem .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5.5 Spectral Properties of Strong Time Operators .. . . . . . . . . 4.5.6 Quasi-Weyl Relation .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5.7 A Remark on Uniqueness of Weak Weyl Representations . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5.8 Strong Time Operator of H F . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5.9 Strong Time Operator of an N-body System .. . . . . . . . . . 4.5.10 Aharonov–Bohm Time Operators .. .. . . . . . . . . . . . . . . . . . . . 4.5.11 Strong Time Operators of Free Relativistic Schrödinger Operators .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5.12 Strong Time Operators of a Free Dirac Operator.. . . . . . 4.5.13 A Structure Generating Pairs of a Hamiltonian and a Strong Time Operator .. . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5.14 Decay-in-Time of Transition Probabilities . . . . . . . . . . . . . 4.5.15 Existence of Strong Time Operators . . . . . . . . . . . . . . . . . . . .

157 161 163 165 169 170 171 171 176 178 179 180 182 185 187 188 189 189 192 193 194 195 197 200 201 203 207 210 213 218 220 223

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4.5.16

4.6 4.7 5

6

7

Construction of Strong Time Operators of a Self-adjoint Operator from Those of Another Self-adjoint Operator.. . .. . . . . . . . . . . . . . . . . . . . 227 Other Classes of Time Operators.. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 229 Generalized Time Operators.. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 232

Representations of Canonical Anti-commutation Relations with Finite Degrees of Freedom .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 Representations of the CAR with One Degree of Freedom .. . . . . . 5.3 Representations of the CAR with N Degrees of Freedom .. . . . . . .

235 235 241 244

Elements of the Theory of Boson Fock Spaces . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1 The Boson Fock Space Over a Hilbert Space .. . . . . . . . . . . . . . . . . . . . 6.2 Boson Second Quantization Operators . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3 Boson Γ -Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4 Creation and Annihilation Operators . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.5 Commutation Relations. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.6 Irreducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.7 Relative Boundedness of Creation and Annihilation Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.8 Application to Self-adjointness.. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.9 Representations of Boson Second Quantization Operators in Terms of A(·)# . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.10 Commutation Relations Between A(f )# and db (T ) . . . . . . . . . . . . 6.11 Segal Field Operators .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.11.1 Basic Properties .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.11.2 Self-Adjointness of the Segal Field Operator . . . . . . . . . . 6.11.3 Vacuum Expectation Values . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.11.4 Irreducibility of Segal Field Operators . . . . . . . . . . . . . . . . . 6.11.5 Some Formulae and Spectrum of ΦS (f ) . . . . . . . . . . . . . . . 6.11.6 Properties of Weyl Operators .. . . . . . .. . . . . . . . . . . . . . . . . . . . 6.12 Canonical Free Bose Field and Canonical Conjugate Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.13 Symplectic Spaces and Generalization of Segal Field Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.14 Quadratic Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.15 The Boson Fock Space Over a Direct Sum Hilbert Space . . . . . . . .

247 247 248 252 256 262 267

298 300 306

Elements of the Theory of Fermion Fock Spaces . . .. . . . . . . . . . . . . . . . . . . . 7.1 Definitions and Basic Properties . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2 Fermion Second Quantization Operators . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3 Fermion Γ -Operators .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.4 Fermion Annihilation and Creation Operators . . . . . . . . . . . . . . . . . . . . 7.5 Commutation Relations Between B(·)# and dΓf (·) . . . . . . . . . . . . . . . 7.6 Tranformations of B(·)# by Γf (·). . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

311 311 312 314 316 320 321

267 274 274 281 283 284 287 290 291 292 294 297

Contents

xv

7.7

Representations of Fermion Second Quantization Operators in Terms of B(·)# . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 322 Uniform Differentiability of Basic Operator-Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 323 The Fermion Fock Space Over a Direct Sum Hilbert Space .. . . . . 325

7.8 7.9 8

Representations of CCR with Infinite Degrees of Freedom .. . . . . . . . . . . 8.1 Representation of CCR . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2 Cyclic Representations . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.3 Second Quantization Operator Associated with a Representation of CCR. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.3.1 A General Fact on a Sesquilinear Form.. . . . . . . . . . . . . . . . 8.3.2 A Second Quantization Operator Associated with a Representation of CCR. . . . . . .. . . . . . . . . . . . . . . . . . . . 8.4 Diagonalization of HC (T ) . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.5 Analysis of Bogoliubov Translations . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.6 Bogoliubov Transformations . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.7 Second Quantization Operator Associated with {B(f )|f ∈ H } . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.8 Representations of Heisenberg CCR . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.9 Weyl Representations of CCR Over Real Inner Product Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.10 Translations of Fock Representation of Heisenberg CCR . . . . . . . . 8.11 A Class of Irreducible Weyl Representations of CCR (I) .. . . . . . . . 8.12 A Class of Irreducible Weyl Representations of CCR (II) . . . . . . . . 8.13 Functional Schrödinger Representation.. . . . . . .. . . . . . . . . . . . . . . . . . . .

365 368 370 373 380

Representations of CAR with Infinite Degrees of Freedom .. . . . . . . . . . . 9.1 Definitions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2 Fermionic Bogoliubov Transformations .. . . . . .. . . . . . . . . . . . . . . . . . . . 9.3 A Class of Representations of CAR. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

385 385 387 390

10 Physical Correspondences in Quantum Field Theory . . . . . . . . . . . . . . . . . . 10.1 Quantum Field Models in Hamiltonian Formalism . . . . . . . . . . . . . . . 10.2 Scale Transformations of Time-Zero Fields . . .. . . . . . . . . . . . . . . . . . . . 10.3 Bogoliubov Translations in Finite and Infinite Volume Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.3.1 Finite Volume Case . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.3.2 Infinite Volume Case. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.4 Translations of Heisenberg CCR and BEC . . . .. . . . . . . . . . . . . . . . . . . . 10.5 Improper Bogoliubov Transformation and Renormalization . . . . . 10.5.1 A Model Equivalent to Mqd . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.5.2 A Renormalized Model .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.6 Representations of CCR in a Theory of Weakly Interacting Bosons .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.6.1 Finite Volume Theory . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

395 395 401

9

329 329 333 335 336 338 345 347 348 356 360

402 402 403 404 406 407 408 410 410

xvi

Contents

10.6.2 Infinite Volume Theory .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The Free Hermitian Klein–Gordon Quantum Field . . . . . . . . . . . . . . . 10.7.1 Definition .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.7.2 Boson Masses as Indices of a Family of Mutually Inequivalent Representations of CCR. . . . . . . . . . . . . . . . . . . 10.8 Quantum Fields at Finite Temperatures .. . . . . . .. . . . . . . . . . . . . . . . . . . . 10.9 Van Hove Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.9.1 The Infrared Regular Case . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.9.2 The Infrared Singular Case . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.9.3 Infrared Divergence .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.9.4 Infrared Renormalization .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.9.5 Representations Indexed by Sources .. . . . . . . . . . . . . . . . . . . 10.9.6 Ultraviolet Renormalization .. . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.10 Free Quantum Dirac Fields . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.10.1 Eigenvectors of hD (k) and Some Operators .. . . . . . . . . . . 10.10.2 Construction of a Free Quantum Dirac Field . . . . . . . . . . . 10.10.3 Inequivalence of Free Quantum Dirac Fields of Different Masses . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.7

417 423 423 427 428 431 434 437 438 440 443 444 450 450 453 456

A

Multiplication Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 459

B

Spectral Measures and Functional Calculus . . . . . . . .. . . . . . . . . . . . . . . . . . . . 463

C

Direct Sum Hilbert Spaces and Direct Sum Operators . . . . . . . . . . . . . . . . C.1 Finite Direct Sums of Hilbert Spaces and Operators .. . . . . . . . . . . . . C.2 Infinite Direct Sums of Hilbert Spaces and Operators . . . . . . . . . . . . C.3 A Theorem on Essential Self-adjointness.. . . . .. . . . . . . . . . . . . . . . . . . .

D

Spectra of a Self-adjoint Operator.. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 475

469 469 471 473

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 481 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 489

List of Mathematical Symbols

AD AC[a, b] A∗ {, } B(H , K ) B(H ) C CAR CCR [ , ] CCR(N) CCRsa (N) CCRW (N) CCRWW (N) CONS n Ck C(Rd ) C(D) C n (Rd ) C n (D) C0n (D) C0∞ (D) C ∞ (A) Dc D(A) Dj Δ D⊥ dim V

Restriction of an operator A to a subspace D Absolutely continuous functions on closed interval [a, b] Adjoint of a densely defined linear operator A on a Hilbert space Anti-commutator: {A, B} := AB + BA Everywhere defined bounded linear operators from a Hilbert space H to a Hilbert space K B(H , H ) Complex numbers Canonical anti-commutation relations Canonical commutation relations Commutator: [A, B] := AB − BA Representations of the CCR with N degrees of freedom Self-adjoint representations of the CCR with N degrees of freedom Weyl representations of the CCR with N degrees of freedom Weak Weyl representations of the CCR with N degrees of freedom Complete orthonormal system n! Binomial coefficient with respect to (n, k): n Ck := (n−k)!k! Continuous functions on Rd Continuous functions on an open set D n times continuously differentiable functions on Rd n times continuously differentiable functions on an open set D Elements of C n (D) with compact support in D Elements of C ∞ (D) with compact support in D C ∞ -domain of a linear operator A Complement of a subset D of a set X: D c := X \ D Domain of a linear operator A Generalized partial differential operator in variable xj  Generalized Laplacian: Δ := dj=1 Dj2 Orthogonal complement of a subset D of an inner product space Dimension of a vector space V xvii

xviii

EA Fb (H ) Fd Ff (K ) FL FL,d HC Im z i I IX I2 (H )  , H  ,  k·x K L(H , K ) L(H ) L2 (M) L1 (Rd ) L2 (Rd ) L∞ (Rd ) L1loc (Ω) MF |M| μ(d) L N

H



A

f ∞ ONS ⊕∞ n=0 Hn n ˆ ⊗j =1 Dj ˆ ns D ⊗ ˆ nas D ⊗ ⊗nj=1 Hj ⊗ns H ⊗nas H (d) πS Π+ Π− Re z

List of Mathematical Symbols

Spectral measure of a self-adjoint operator A Boson Fock space over a Hilbert space H d-dimensional Fourier transform on L2 (Rd ) Fermion Fock space over a Hilbert space K Discrete Fourier transform d-dimensional discrete Fourier transform Complexification of a real Hilbert space H The imaginary part of z ∈ C The imaginary unit Identity Identity on a set X Hilbert–Schmidt operators on a Hilbert space H Inner product of an inner product space H Inner product Euclidean inner product of vectors k and x R (real numbers) or C (complex numbers) Linear operators (not necessarily bounded) from a Hilbert space H to a Hilbert space K L(H , H ) Hilbert space of square (Lebesgue) integrable functions on M ⊂ Rd Lebesgue integrable functions on Rd Hilbert space of square (Lebesgue) integrable functions on Rd Essentially bounded functions on Rd Locally integrable functions on an open set Ω ⊂ Rd Multiplication operator by a function F Lebesgue measure of a set M ⊂ Rd d-dimensional Lebesgue measure Natural numbers (positive integers) Norm of an inner product space H Norm Operator norm of a bounded linear operator A Supremum norm of an essentially bounded function f Orthonormal system Infinite direct sum Hilbert space of Hilbert spaces {Hn }∞ n=0 n-fold algebraic tensor product of vector spaces D1 , . . . , Dn n-fold symmetric algebraic tensor product of a vector space D n-fold anti-symmetric algebraic tensor product of a vector space D n-fold tensor product of Hilbert spaces H1 , . . . , Hn n-fold symmetric tensor product of Hilbert space H n-fold anti-symmetric tensor product of Hilbert space H Schrödinger representation of the CCR with degree d The open upper half plane in C The open lower half plane in C The real part of z ∈ C

List of Mathematical Symbols

R R+ ˆ R Rd ˆd R Ran (A) ρ(A) S (Rd ) SR (Rd ) S (Rd ) SR (Rd ) σ (A) σac (A) σc (A) σd (A) σp (A) σsc (A) span D VC Z Z+ z∗

Real numbers Positive real numbers One-dimensional momentum (wave number) space d-dimensional vector space of d-tuples of real numbers: {(x1 , . . . , xd )|xi ∈ R, i = 1, . . . , d} d-dimensional momentum (wave number vector) space Range of a linear operator A Resolvent set of a linear operator A Rapidly decreasing C ∞ -functions on Rd Real-valued rapidly decreasing C ∞ -functions on Rd Tempered distributions on Rd Real tempered distributions on Rd Spectrum of a linear operator A Absolutely continuous spectrum of a self-adjoint operator A Continuous spectrum of a linear operator A Discrete spectrum of a linear operator A Point spectrum of a linear operator A Singular continuous spectrum of a self-adjoint operator A Subspace generated by a subset D of a vector space Complexification of a real vector space V Integers Non-negative integers The complex conjugate of z ∈ C

xix

Chapter 1

Mathematical Preliminaries

Abstract Operator theories on Hilbert spaces including those on tensor product Hilbert spaces are reviewed. We assume that the reader is familiar with elements of theories of Hilbert spaces and Banach spaces. (We will omit proofs for well-known facts.)

1.1 Linear Operators on Hilbert Spaces Throughout this book, we use the term “Hilbert space” to mean “complex Hilbert space” unless otherwise stated. We denote the inner product and the norm of a Hilbert space H by  , H and · H respectively with the convention that the mapping H × H (Ψ, Φ) → Ψ, ΦH ∈ C is linear in Φ and anti-linear in Ψ . We sometimes write  ,  and · instead of  , H and · H respectively, omitting the subscript H , if there is no danger of confusion. In this section, we collect basic definitions and facts in the theory of linear operators on Hilbert spaces.

1.1.1 Definitions and Some Basic Facts Let H and K be Hilbert spaces and D be a subspace of H . A mapping A: D → K ; D Ψ → A(Ψ ) ∈ K is called a linear operator from H to K with domain D if, for all Ψ, Φ ∈ D and α, β ∈ C, A(αΨ + βΦ) = αA(Ψ ) + βA(Φ) (linearity). We denote the domain D of A by D(A). We sometimes write A(Ψ ) = AΨ simply and use the word “operator” to mean “linear operator” unless otherwise stated. If D(A) = H , then A is said to be everywhere defined. If D(A) is dense, then A is said to be densely defined. © Springer Nature Singapore Pte Ltd. 2020 A. Arai, Inequivalent Representations of Canonical Commutation and Anti-Commutation Relations, Mathematical Physics Studies, https://doi.org/10.1007/978-981-15-2180-5_1

1

2

1 Mathematical Preliminaries

There is a notion which is a counterpart to the notion of a linear operator. A mapping A: D → K is called an anti-linear operator with domain D(A) = D if, for all Ψ, Φ ∈ D and α, β ∈ C, A(αΨ + βΦ) = α ∗ A(Ψ ) + β ∗ A(Φ)

(anti-linearity),

where α ∗ denotes the complex conjugate of α. An anti-linear operator is called also a conjugate linear operator. The identity IH on H defined by D(IH ) := H and IH (Ψ ) = Ψ, Ψ ∈ H is a simple example of a linear operator. We sometimes write IH = I if there is no danger of confusion. The zero operator 0H ,K from H to K is defined by D(0H ,K ) := H and 0H ,K (Ψ ) := 0K , Ψ ∈ H , where 0K is the zero vector in K . Usually 0H ,K (resp. 0K ) is simply written 0 (resp. 0) if there is no danger of confusion. For each linear operator A from H to K , the set ∗ DA := {Φ ∈ K |∃ΘΦ ∈ H such that Φ, AΨ K = ΘΦ , Ψ H , ∀Ψ ∈ D(A)}

is a subspace of K . If A is densely defined, then the vector ΘΦ is uniquely ∗ and the correspondence D ∗ Φ → Θ is linear. determined for each Φ ∈ DA Φ A Therefore, under the condition that A is densely defined, one can define a linear operator A∗ from K to H by ∗ , D(A∗ ) := DA

A∗ Φ := ΘΦ ,

Φ ∈ D(A∗ ).

(1.1)

The operator A∗ is called the adjoint of A. We denote by L(H , K ) the set of linear operators from H to K and set L(H ) := L(H , H ). For convenience, we call an element of L(H ) a linear operator on H , even if it is not everywhere defined on H . An operator A ∈ L(H , K ) is said to be equal to B ∈ L(H , K ) if D(A) = D(B) and AΨ = BΨ, ∀Ψ ∈ D(A). In this case we write A = B. For an operator A ∈ L(H , K ), an operator B ∈ L(H , K ) is called an extension of A if D(A) ⊂ D(B) and AΨ = BΨ, Ψ ∈ D(A). In this case we write A ⊂ B. It follows that A = B if and only if A ⊂ B and B ⊂ A. It is easy to see that the relation “⊂” on L(H , K ) is a partial ordering. The adjoint operation makes the order of two densely defined linear operators converse: Proposition 1.1 Let A ∈ L(H , K ) and B ∈ L(H , K ) be densely defined. Suppose that A ⊂ B. Then B ∗ ⊂ A∗ . Proof An easy exercise.



1.1 Linear Operators on Hilbert Spaces

3

For A and B in L(H , K ), the sum A + B ∈ L(H , K ) of A and B is defined as follows: D(A + B) := D(A) ∩ D(B),

(A + B)Ψ := AΨ + BΨ,

Ψ ∈ D(A + B).

Proposition 1.2 Let A ∈ L(H , K ) and B ∈ L(H , K ) be densely defined. (i) If A + B is densely defined, then A∗ + B ∗ ⊂ (A + B)∗ . (ii) If B ∈ B(H , K ), then (A + B)∗ = A∗ + B ∗ . For α ∈ C, αA ∈ L(H , K ) (the scalar multiple of A by α) is defined by D(αA) := D(A),

(αA)Ψ = αAΨ,

Ψ ∈ D(αA).

Let A ∈ L(H , K ) and B ∈ L(K , L ) (L is a Hilbert space). Then the product BA ∈ L(H , L ) of A and B is defined by D(BA) := {Ψ ∈ D(A)|AΨ ∈ D(B)},

(BA)Ψ := B(AΨ ),

Ψ ∈ D(BA).

The next proposition, which shows that the product operation for an operator from the left (resp. right) side preserves the partial ordering of operators, is often useful: Proposition 1.3 Let A, B ∈ L(H , K ) and A ⊂ B. Then: (i) For all C ∈ L(K , L ) (L is a Hilbert space), CA ⊂ CB. (ii) For all D ∈ L(L , H ), AD ⊂ BD. 

Proof An easy exercise.

Concerning the adjoint operation for the product of two operators, the following facts should be kept in mind: Proposition 1.4 Let A ∈ L(H , K ) and B ∈ L(K , L ) be densely defined. Then: (i) If BA is densely defined, then A∗ B ∗ ⊂ (BA)∗ . (ii) If B ∈ B(K , L ), then D(BA) = D(A) and (BA)∗ = A∗ B ∗ . For a subspace D ⊂ D(A), an operator A  D ∈ L(H , K ) is defined as follows: D(A  D) := D,

(A  D)Ψ := AΨ,

Ψ ∈ D.

This operator is called the restriction of A to D. It is obvious that A  D ⊂ A.

4

1 Mathematical Preliminaries

For A, B ∈ L(H , K ) and a subspace D ⊂ D(A) ∩ D(B), we say that “A is equal to B on D” if A  D = B  D, i.e., AΨ = BΨ, ∀Ψ ∈ D. In this case we write A=B

on D.

For A, B ∈ L(H ), the commutator [A, B] ∈ L(H ) of A and B is defined by [A, B] := AB − BA with D([A, B]) = D(AB) ∩ D(BA). Two operators A and B in L(H ) are said to commute if [A, B] ⊂ 0 (i.e., [A, B] = 0 on D([A, B])).1 For any subspace D of D([A, B]), [A, B] has the restriction to D. But the domain of [A, B] cannot be extended in the strict sense of commutator. In this sense, D([A, B]) = D(AB) ∩ D(BA) is called the maximal domain for the commutator [A, B]. If there is a subspace D ⊂ D([A, B]) such that [A, B]Ψ = 0, ∀Ψ ∈ D, then we say that A and B commute on D. In this case, we write “[A, B] = 0 on D”. For each A ∈ L(H , K ), the subset Ran (A) := {AΨ |Ψ ∈ D(A)} is called the range of A, which is a subspace of K . The subset ker A := {Ψ ∈ D(A)|AΨ = 0}

(1.2)

is called the kernel of A, which is a subspace of H . Concepts concerning mapping properties of A ∈ L(H , K ) are defined as follows: A is said to be injective or one-to-one if Ψ1 = Ψ2 (Ψ1 , Ψ2 ∈ D(A)), then AΨ1 = AΨ2 . In this case, the inverse operator A−1 from K to H is defined by D(A−1 ) := Ran (A),

A−1 Φ := ΨΦ ,

Φ ∈ D(A−1 ),

where ΨΦ ∈ D(A) is the unique vector such that AΨΦ = Φ. It is easy to see that A is injective if and only if ker A = {0}. If Ran (A) = K , then A is said to be surjective. If A is injective and surjective, then A is said to be bijective and, in this case, A is called a bijection.

1 Recall

that the domain D(0) of the zero operator 0 on H is H .

1.1 Linear Operators on Hilbert Spaces

5

If there exists a constant C > 0 such that AΨ K ≤ C Ψ H , ∀Ψ ∈ D(A), then A is said to be bounded. In this case, the quantity

A :=

AΨ K Ψ ∈D(A),Ψ =0 Ψ H sup

is called the operator norm of A. It follows that AΨ K ≤ A Ψ , ∀Ψ ∈ D(A). A densely defined bounded operator can be extended uniquely to an everywhere defined bounded operator without change of operator norm: Theorem 1.1 (Extension Theorem) Let A be a densely defined bounded operator  ∈ B(H ) such that A ⊂ A.  on H . Then there exists a unique bounded operator A  = A . Moreover, A

Let B(H , K ) be the set of everywhere defined bounded linear operators from H to K . Then B(H , K ) is a Banach space with operator norm. We set B(H ) := B(H , H ). It holds that, for all A ∈ B(H , K ), the adjoint A∗ of A is in B(K , H ) and

A∗ = A . There is a weaker notion of boundedness of a linear operator. An operator B ∈ L(H , K ) is said to be relatively bounded with respect to an operator A ∈ L(H , K ) or A-bounded if D(A) ⊂ D(B) and there exist constants a, b ≥ 0 such that

BΨ ≤ a AΨ + b Ψ ,

∀Ψ ∈ D(A).

(1.3)

The infimum of the constants a satisfying (1.3) (b may depend on a) is called the relative bound of B with respect to A. We denote it by B A :

B A := inf{a ≥ 0|(1.3) holds}.

(1.4)

An operator A ∈ L(H , K ) is said to be closed if it has the following property: for any sequence {Ψn }∞ n=1 in D(A) (i.e., Ψn ∈ D(A), n ∈ N) such that limn→∞ Ψn = Ψ ∈ H and limn→∞ AΨn = Φ ∈ K , it holds that Ψ ∈ D(A) and AΨ = Φ.

6

1 Mathematical Preliminaries

An operator A ∈ L(H , K ) is said to be closable if it has a closed extension (i.e. there exists a closed operator B ∈ L(H , K ) such that A ⊂ B). In this case, an operator A¯ ∈ L(H , K ), called the closure of A, is associated with A: ¯ = {Ψ ∈ H | ∃{Ψn }∞ D(A) n=1 with Ψn ∈ D(A) such that lim Ψn = Ψ, ∃ lim AΨn }, n→∞

¯ = lim AΨn , AΨ n→∞

n→∞

¯ Ψ ∈ D(A).

It follows that A¯ is closed and it is the smallest closed extension of A (i.e., if B is a closed extension of A, then A¯ ⊂ B). The closure operation A → A¯ for a closable operator A ∈ L(H , K ) preserves the partial ordering of operators: Proposition 1.5 Let A and B be closable operators from H to K satisfying ¯ A ⊂ B. Then A¯ ⊂ B. A subspace D ⊂ D(A) is called a core for A if, for each Ψ ∈ D(A), there exists a sequence {Ψn }∞ n=1 in D such that limn→∞ Ψn = Ψ and limn→∞ AΨn = AΨ . It is easy to see that, if A is closed and D is a core for A, then A = A  D. If A is closable, then a subspace D ⊂ D(A) is a core for A if and only if D is a ¯ In that case, A¯ = A  D. core for A. The following proposition states basic facts on closability of a linear operator: Proposition 1.6 Let A ∈ L(H , K ) be densely defined (hence A∗ exists). Then (i) and (ii) below hold: (i) A∗ is closed. (ii) A is closable if and only if D(A∗ ) is dense in K . In that case, it holds that A = A∗∗ . In particular, if A is closed, then A∗∗ = A. (iii) If A is closable, then (A)∗ = A∗ . Proof See, e.g., [130, Theorem VIII.1] or [23, Proposition 2.20].



Proposition 1.6(ii) is often useful to prove the closability of a densely defined linear operator in applications. The following fact also plays an important role: Lemma 1.1 Let A be a densely defined operator on a Hilbert space. Then ker A∗ A = ker A. Proof It is obvious that ker A ⊂ ker A∗ A. To prove the converse inclusion relation, let Ψ ∈ ker A∗ A. Then Ψ ∈ D(A∗ A) and A∗ AΨ = 0. Taking the inner product of both sides with the vector Ψ and using Ψ ∈ D(A), we obtain AΨ 2 = 0. Hence AΨ = 0, i.e., Ψ ∈ ker A. Thus ker A∗ A ⊂ ker A. 

1.1 Linear Operators on Hilbert Spaces

7

The following decomposition theorem also is often useful: Theorem 1.2 (Hilbert Space Decomposition Theorem) Let A ∈ L(H , K ) be a densely defined closed operator. Then H = ker A ⊕ Ran (A∗ ).

(1.5) 

Proof See, e.g., [23, Lemma 2.28].2 Theorem 1.2 immediately yields:

Corollary 1.1 Let A ∈ L(H , K ) be a densely defined closed operator. Then A is injective if and only if Ran (A∗ ) is dense in H . Proof Suppose that A is injective. Then ker A = {0}. Hence, by (1.5), H = Ran (A∗ ), implying that Ran (A∗ ) is dense in H . Conversely, suppose that Ran (A∗ ) is dense in H . Then H = Ran (A∗ ). Hence, by (1.5), ker A = {0}, implying that A is injective.  For an operator A ∈ L(H , K ) and a subset D ⊂ D(A), we define a subset AD of K by AD := {AΨ |Ψ ∈ D}. If K = H , D is a subspace of H and AD ⊂ D, then we say that A leaves D invariant and D is called an invariant subspace of A. The notions of sum and product for linear operators can be extended to the case of any finite number of linearoperators. For operators A1 , . . . , An ∈ L(H , K ) (n ≥ 2), the operator sum nj=1 Aj = A1 + A2 + · · · + An ∈ L(H , K ) is defined by  D( nj=1 Aj ) := ∩nj=1 D(Aj ),

  ( nj=1 Aj )Ψ := nj=1 Aj Ψ,

 Ψ ∈ D( nj=1 Aj ).

Let H1 , . . . , Hn and Hn+1 be Hilbert spaces and Bj ∈ L(Hj , Hj +1 ) (j = 1, . . . , n). Then the operator product Bn · · · B1 ∈ L(H1 , Hn+1 ) is defined by D(Bn · · · B1 ) := {Ψ ∈ D(B1 )| Bj · · · B1 Ψ ∈ D(Bj +1 ), j = 1, . . . n − 1}, (Bn · · · B1 )Ψ := Bn (Bn−1 (· · · (B2 (B1 Ψ )) · · · )),

Ψ ∈ D(Bn · · · B1 ).

If Bj ∈ B(Hj , Hj +1 ) (j = 1, . . . , n), then Bn · · · B1 ∈ B(H1 , Hn+1 ) and

Bn · · · B1 ≤

n  j =1

2 The

proof of Theorem 1.2 is not so difficult.

Bj .

8

1 Mathematical Preliminaries

In particular, for all A ∈ B(H ),

An ≤ A n .

1.1.2 Convergences of Bounded Operators There are three types of convergence for a sequence of bounded operators. Let H and K be Hilbert spaces and An , A ∈ B(H , K ), n ∈ N. (1) A sequence {An }∞ n=1 is said to uniformly converge to A if limn→∞ An −A = 0. In this case we write u− limn→∞ An = A and A is called the uniform limit of {An }∞ n=1 . (2) A sequence {An }∞ n=1 is said to strongly converge to A if, for all Ψ ∈ H , limn→∞ An Ψ = AΨ . In this case we write s− limn→∞ An = A and A is called the strong limit of {An }∞ n=1 . (3) A sequence {An }∞ is said to weakly converge to A if, for all Ψ ∈ H and Φ ∈ n=1 K , limn→∞ Φ, An Ψ  = Φ, AΨ . In this case we write w− limn→∞ An = A and A is called the weak limit of {An }∞ n=1 .

1.1.3 Spectra For each operator A ∈ L(H ), there exist mutually disjoint four subsets of C corresponding to mapping properties of A − λ := A − λIH (λ ∈ C and IH is the identity on H ): ρ(A) := {λ ∈ C| A − λ is injective, Ran (A − λ) is dense in H and (A − λ)−1 is bounded}, σc (A) := {λ ∈ C| A − λ is injective, Ran (A − λ) is dense in H and (A − λ)−1 is unbounded}, σr (A) := {λ ∈ C| A − λ is injective and Ran (A − λ) is not dense in H }, σp (A) := {λ ∈ C| A − λ is not injective}. The subsets ρ(A), σc (A), σr (A) and σp (A) are called respectively the resolvent set, the continuous spectrum, the residual spectrum and the point spectrum of A. These subsets are mutually disjoint and satisfy ρ(A) ∪ σc (A) ∪ σr (A) ∪ σp (A) = C.

(1.6)

1.1 Linear Operators on Hilbert Spaces

9

The subset σ (A) := ρ(A)c (:= C \ ρ(A))

(1.7)

is called the spectrum of A. We have σ (A) = σc (A) ∪ σr (A) ∪ σp (A)

(1.8)

and σ (A) ∩ ρ(A) = ∅,

σ (A) ∪ ρ(A) = C.

(1.9)

The point spectrum σp (A) is nothing but the set of all eigenvalues3 of A and, for each eigenvalue λ ∈ σp (A), ker(A − λ) is the eigenspace of A with eigenvalue λ. The dimension of the eigenspace (i.e., dim ker(A − λ)) is called the multiplicity of the eigenvalue λ. If the multiplicity of λ is one (resp. more than one), then λ is said to be simple (resp. degenerate). If the multiplicity of λ is finite and λ is an isolated point of σ (A) (i.e., there exists a constant δ > 0 such that {z ∈ C|0 < |z − λ| < δ} ⊂ ρ(A)), then λ is a discrete eigenvalue of A. We denote by σd (A) the set of the discrete eigenvalues of A. It follows that σd (A) ⊂ σp (A). If σ (A) = σd (A), then the spectrum of A is said to be purely discrete or A has a purely discrete spectrum. The next proposition is important, which follows from the closed graph theorem: Proposition 1.7 Let A be a closed linear operator on H . Then λ ∈ ρ(A) if and only if A − λ is bijective. The next proposition states a fundamental fact on the spectrum of a closable operator: Proposition 1.8 For any closable operator A on H , ρ(A) = ρ(A) and σ (A) = σ (A). The next proposition is concerned with an elementary fact on a spectral mapping: Proposition 1.9 Let A ∈ L(H ). Then, for all a ∈ R \ {0} and b ∈ R, σ (aA + b) = {aλ + b|λ ∈ σ (A)}.

(1.10)

σ (−A) = {−λ|λ ∈ σ (A)}.

(1.11)

In particular,

3 Recall that a complex number λ ∈ C is called an eigenvalue of A if there exists a non-zero vector Ψ in D(A) such that AΨ = λΨ ; such a vector Ψ is called an eigenvector of A with eigenvalue λ.

10

1 Mathematical Preliminaries

Proof We first show that ρ(aA + b) = {aλ + b|λ ∈ ρ(A)}.

(1.12)

We denote the set on the right hand side by Ra,b . Let z ∈ ρ(aA + b). Then B := aA + b − z is injective with B −1 bounded and Ran B is dense in H . One can rewrite B = a(A − (z − b)/a), which implies that (z − b)/a ∈ ρ(A). Hence there is an element λ ∈ ρ(A) such that (z − b)/a = λ, i.e., z = aλ + b. Hence ρ(aA + b) ⊂ Ra,b . By tracing conversely the arguments given in the preceding paragraph, one can easily show that Ra,b ⊂ ρ(aA+b). Thus (1.12) holds. This fact implies that σ (aA+ c . Using (1.9), one can show that R c = {aμ + b|μ ∈ σ (A)}. Thus (1.10) b) = Ra,b a,b follows. 

1.1.4 Unitary Operators and Related Facts We recall that a unitary operator U : H → K is an operator such that Ran (U ) = K (surjectivity) and U Ψ, U ΦK = Ψ, ΦH , Ψ, Φ ∈ H (preservation of inner product), which is equivalent to that U ∗ U = IH and U U ∗ = IK . Hence U is bijective with U −1 = U ∗ which also is a unitary operator from K to H . A unitary operator is sometimes called a unitary transformation. For two Hilbert spaces H and K , we write H ∼ = K if there exists a unitary operator U from H to K . It is easy to see that this relation ∼ = is an equivalence relation in the set of Hilbert spaces. Based on this fact, one says that H and K are isomorphic if H ∼ = K . In the case where H ∼ = K under a unitary operator U : U

H → K , we write H ∼ = K (but we sometimes omit U if there is no danger of confusion). Such a unitary operator U is called a Hilbert space isomorphism (or simply an isomorphism) . If an isomorphism U : H → K is independent of the choice of complete orthonormal systems of H and K , then U is called a natural isomorphism. The next theorem is a basic one to find a Hilbert space isomorphism. Theorem 1.3 (Isomorphism Theorem) Let H and K be Hilbert spaces, and {eλ }λ∈Λ and {fλ }λ∈Λ (Λ is an index set not necessarily countable) be complete orthonormal systems of H and K respectively. Then there exists a unitary operator U : H → K such that U eλ = fλ ,

λ ∈ Λ.

For a proof of this theorem, see, e.g., [28, Theorem 2.4], [23, Theorem 2.5]. The spectra of a linear operator A ∈ L(H ) are unitarily invariant: for all unitary operators U : H → K , σ (U AU −1 ) = σ (A),

σ# (U AU −1 ) = σ# (A),

# = c, r, p.

1.1 Linear Operators on Hilbert Spaces

11

A linear operator A on H is said to be unitarily equivalent to a linear operator B on K if there exists a unitary operator U : H → K such that U AU −1 = B. In this case, we write U

A∼ =B

or A ∼ =B

(1.13)

if U is understood. It follows from the above facts that A∼ = B ⇒ σ# (A) = σ# (B),

# = c, r, p.

The next propositions are basic elementary facts often used. Proposition 1.10 Let U be a unitary operator from H to a Hilbert space K and A, B ∈ L(H ). Then U (A + B)U −1 = U AU −1 + U BU −1 .

(1.14)

Proof An easy exercise.



Remark 1.1 The point in Proposition 1.10 is that (1.14) is an operator equality, i.e., D(U (A + B)U −1 ) = D(U AU −1 + U BU −1 ) and, for all Ψ ∈ D(U (A + B)U −1 ), U (A + B)U −1 Ψ = U AU −1 Ψ + U BU −1 Ψ . Proposition 1.11 Let U be a unitary operator on H and D be a subspace of H such that U D ⊂ D and U −1 D ⊂ D. Then U D = D. Proof An easy exercise.



1.1.5 Symmetric and Self-adjoint Operators An operator A ∈ L(H ) is said to be symmetric if D(A) is dense and A ⊂ A∗ . It is easy to see that a densely defined operator A ∈ L(H ) is symmetric if and only if, for all Ψ, Φ ∈ D(A), Ψ, AΦH = AΨ, ΦH . One can easily show that a densely defined operator A ∈ L(H ) is symmetric if and only if Ψ, AΨ H ∈ R, ∀Ψ ∈ D(A). Proposition 1.12 Every symmetric operator A on H is closable with A ⊂ A∗ . Proof By the definition of symmetric operator, A ⊂ A∗ . By Proposition 1.6(i), A∗ is closed. Hence A is closable and A ⊂ A∗ .  For an operator A on H and a dense subspace D of H such that D ⊂ D(A), A is said to be symmetric on D if A  D is a symmetric operator. If A ∈ L(H ) is densely defined and A = A∗ , then A is called a self-adjoint operator.

12

1 Mathematical Preliminaries

It is obvious that a self-adjoint operator is a closed symmetric operator. But one should be careful, in that the converse is not true. A densely defined linear operator A on H is said to be maximally symmetric if it is symmetric and has no non-trivial symmetric extensions (i.e., if B is a symmetric operator such that A ⊂ B, then B = A). It follows that a maximally symmetric operator is closed. It is easy to show that a self-adjoint operator is maximally symmetric. But the converse is not true. A symmetric operator A is said to be essentially self-adjoint if its closure A¯ is self-adjoint. Let A be a symmetric operator on H and D be a dense subspace of H such that D ⊂ D(A). Then it is easy to see that A  D is a symmetric operator. We say that A is essentially self-adjoint on D if A  D is essentially self-adjoint. In that case, it follows that A is essentially self-adjoint and A  D = A¯ (hence D is a core for the ¯ Conversely, if A¯ is self-adjoint and D is a core for A. ¯ then self-adjoint operator A). A is essentially self-adjoint on D. A symmetric operator A on H is said to be bounded from below if there exists a constant γ ∈ R such that Ψ, AΨ  ≥ γ Ψ 2 , ∀Ψ ∈ D(A). In this case we write A ≥ γ . In particular, if A ≥ 0 (the case γ = 0), then A is said to be non-negative. A symmetric operator A satisfying −A ≥ −γ for some γ ∈ R is said to be bounded from above. In this case we write A ≤ γ . In particular, if A ≤ 0, then A is said to be non-positive. A symmetric operator is said to be semi-bounded if it is bounded from below or bounded from above. An operator P ∈ B(H ) is called an orthogonal projection if P = P ∗ (selfadjoint) and P 2 = P (idempotent).

1.1.6 Lowest Energy and Ground State of a Self-adjoint Operator Let H be a self-adjoint operator on H which is bounded from below. Then the infimum of the spectrum of H E0 (H ) := inf σ (H ) is finite. In view of quantum theory and by abuse of terminology, we call E0 (H ) the lowest energy of H . One can show that E0 (H ) =

inf

Ψ ∈D(H ), Ψ =1

Ψ, H Ψ  .

This is called the variational principle for the lowest energy. If E0 (H ) is an eigenvalue of H , then a non-zero vector in the eigenspace ker(H − E0 (H )) is called

1.1 Linear Operators on Hilbert Spaces

13

a ground state of H . In this case, H is said to have a ground state and E0 (H ) is called the ground state energy of H . The ground state of H is said to be unique (resp. degenerate) if dim ker(H − E0 (H )) = 1 (resp. dim ker(H − E0 (H )) ≥ 2). A ground state of H is called a zero-energy ground state if E0 (H ) = 0.

1.1.7 Spectral Theorem and Some Facts in Functional Calculus We recall that, with each self-adjoint operator A on a Hilbert space H , a unique one-dimensional spectral measure4 EA is associated, called the spectral measure of A, and A has the spectral representation  A= λ dEA (λ), R

which means that

   2 2 D(A) = Ψ ∈ H | λ d EA (λ)Ψ < ∞ , R

 Φ, AΨ  =

R

λd Φ, EA (λ)Ψ  ,

Φ ∈ H , Ψ ∈ D(A).

This fact is called the spectral theorem for self-adjoint operators (Theorem B.2 in Appendix B). The spectral theorem allows one to develop the functional calculus based on EA (see Appendix B). Namely, for each Borel measurable function f : R → C ∪ {±∞}, there exists a linear operator f (A) on H such that  f (A) = f (λ)dEA (λ), (1.15) R

which means that    2 2 D(f (A)) = Ψ ∈ H | |f (λ)| d EA (λ)Ψ < ∞ , R

 Φ, f (A)Ψ  =

R

f (λ)d Φ, EA (λ)Ψ  ,

Φ ∈ H , Ψ ∈ D(f (A)).

(1.16) (1.17)

Example 1.1 For a complex number z ∈ C, let fz (λ) := eizλ , λ ∈ R. Then we define eizA by eizA := fz (A).

4 See

Appendix B for the definition of a spectral measure and related facts.

14

1 Mathematical Preliminaries

It follows from Theorem B.1(v) that, for all t ∈ R, eit A is unitary. Example 1.2 Let A be a non-negative self-adjoint operator on H . Then, for all α > 0, the function gα (λ) := χ[0,∞) (λ)|λ|α on R is Borel measurable. Hence one can define gα (A). We denote it by Aα :  Aα :=

R

χ[0,∞) (λ)|λ|α dEA (λ).

Since the function gα is a non-negative continuous function on R, it follows from Theorem B.1(vi) that Aα is a non-negative self-adjoint operator. One can show that (Aα )1/α = A. Example 1.3 Let T ∈ B(H , K ) (K is a Hilbert space). Then T ∗ T is a nonnegative bounded self-adjoint operator on H . Hence, by the preceding example, |T | := (T ∗ T )1/2 is a non-negative bounded self-adjoint operator. The operator |T | is called the absolute value or modulus of T . It follows that |T |2 = T ∗ T . The following proposition is sometimes useful in applications. Proposition 1.13 Let A be a non-negative self-adjoint operator on H and D be a core for A. Then, for all α ∈ (0, 1), D(A) ⊂ D(Aα ) and D is a core for Aα . Proof Let Ψ ∈ D(A). Then (note that supp EA ⊂ [0, ∞)), 

 [0,∞)

λ2α d EA (λ)Ψ 2 =  ≤

 [0,1]

[0,1]

λ2α d EA (λ)Ψ 2 + 

λ2α d EA (λ)Ψ 2 (1,∞)

d EA (λ)Ψ 2 +

λ2 d EA (λ)Ψ 2 (1,∞)

≤ EA ([0, 1])Ψ 2 + AΨ 2 < ∞. Hence Ψ ∈ D(Aα ). Therefore D(A) ⊂ D(Aα ) and

Aα Ψ ≤ Ψ + AΨ ,

Ψ ∈ D(A).

(1.18)

By the assumption on D, for each Ψ ∈ D(A), there exists a sequence {Ψn }n in D such that Ψn → Ψ and AΨn → AΨ as n → ∞. Hence, by (1.18), Aα Ψn → Aα Ψ as n → ∞. This means that Aα  D(A) ⊂ Aα  D.

1.1 Linear Operators on Hilbert Spaces

15

Hence Aα  D(A) ⊂ Aα  D.

(1.19)

Next, let Ψ ∈ D(Aα ) and Ψn := EA ([0, n])Ψ, n ∈ N. Then Ψn ∈ D(A) and Ψn → Ψ as n → ∞. We have  α α 2

A Ψn − A Ψ = λ2α d EA (λ)Ψ 2 . λ>n



2α 2 [0,∞) λ d EA (λ)Ψ < ∞, it follows that the right hand side tends to 0 as α n → ∞. Hence A Ψn → Aα Ψ as n → ∞. Therefore Aα ⊂ Aα  D(A). By this and (1.19), we obtain Aα ⊂ Aα  D. It is obvious that Aα  D ⊂ Aα and hence Aα  D ⊂ Aα . Thus Aα  D = Aα , implying that D is a core for Aα . 

Since

Proposition 1.14 Let A be a self-adjoint operator on H . Then, for all Ψ ∈ D(A),

eiA Ψ − Ψ ≤ AΨ .

(1.20)

Proof By the functional calculus, we have 

eiA Ψ − Ψ 2 = |eiλ − 1|2 d EA (λ)Ψ 2 . R

It is easy to see that |eiλ − 1| ≤ |λ|, Hence

λ ∈ R.



eiA Ψ − Ψ 2 ≤

R

λ2 d EA (λ)Ψ 2 = AΨ 2 . 

Thus (1.20) follows.

1.1.8 Trace Class and Hilbert–Schmidt Operators Let H be a separable Hilbert space and T be a non-negative bounded self-adjoint operator on H . Then, for a complete orthonormal system (CONS) {en }∞ n=1 of H , ∞ e  is a non-negative finite number or infinite, independently of the , T e n n=1 n choice of {en }∞ . Hence one can define n=1 Tr T :=

∞ n=1

It is called the trace of T .

en , T en  .

(1.21)

16

1 Mathematical Preliminaries

A bounded linear operator T ∈ B(H ) is called a trace class operator if Tr |T | < ∞. In this case, T is said to be trace class and the trace Tr T ∈ C of T can be defined by (1.21), independently of the choice of {en }∞ n=1 . Remark 1.2 If H is finite dimensional with dim H = N, then each everywhere defined linear operator T on H is bounded and, for any CONS {en }N n=1 of H , N N Tr T := n=1 en , T en  is in C, independently of the choice of {en }n=1 . Example 1.4 Let f, g ∈ H and define an operator Pf,g on H by Pf,g Ψ := f, Ψ  g,

Ψ ∈H.

(1.22)

Then Pf,g ∈ B(H ) with Pf,g = f g and ∗ Pf,g Ψ = g, Ψ  f,

Ψ ∈H.

Hence Pf,f is self-adjoint and ∗ |Pf,g |2 = Pf,g Pf,g = g 2 Pf,f . 2 = f 2 P In particular, Pf,f f,f and Pf,f ≥ 0. 1/2

Let f = 0. Then it follows that Pf,f = Pf,f / f . Hence |Pf,g | =

g

Pf,f .

f

Let {en }∞ n=1 be a CONS of H . Then ∞

Tr |Pf,g | =

g | en , f  |2 = g f < ∞.

f

n=1

Hence Pf,g is a trace class operator on H and Tr Pf,g = f, g . If f = 0 and g = 0, then Pf,g is an example of one-rank operators. In what follows, we sometimes write Pf,g = f, · g.

(1.23)

A bounded linear operator T ∈ B(H , K ) is said to be Hilbert–Schmidt if T ∗ T is a trace class operator on H . It follows  that T ∈ B(H , K ) is Hilbert–Schmidt if and only if, for a CONS {en }n of H , n T en 2 < ∞. The non-negative number

√ ∗

T en 2 .

T 2 := Tr T T = n

1.1 Linear Operators on Hilbert Spaces

17

is called the Hilbert–Schmidt norm of T . It follows that

T ≤ T 2 Moreover, T ∗ also is Hilbert–Schmidt and

T ∗ 2 = T 2 . Example 1.5 The operator Pf,g in Example 1.4 is Hilbert–Schmidt with

Pf,g 2 = f g . The following representation theorem for Hilbert–Schmidt operators is useful: Theorem 1.4 Let T be a Hilbert–Schmidt operator on a Hilbert space H . Then N (N < ∞ or there exist (not necessarily complete) ONS’s {en }N n=1 and {fn }n=1 of H  2 N = ∞) and positive real numbers μn > 0 (n = 1, . . . , N) such that ∞ n=1 μn < ∞ in the case N = ∞ and T =

N

μn fn , · en .

(1.24)

n=1

Proof It is well known that a Hilbert–Schmidt operator is a compact operator and that a compact operator has a canonical form which, applied to T , takes form (1.24) (see, e.g., [130, Theorems VI.17]). Hence Tfn = μn en . Consider the case N = ∞.  ∞ 2 < ∞. Hence 2 < ∞. Since T is Hilbert–Schmidt, ∞

Tf

μ  n n=1 n=1 n Remark 1.3 The numbers μn (n = 1, 2, . . . , N) are positive eigenvalues of |T | (it is easy to see that T ∗ Tfn = μ2n fn ). They are called the singular values of T . Remark 1.4 In the case N = ∞, the convergence in (1.24) can be taken in operator norm topology. If T in Theorem 1.4 is self-adjoint (i.e., T is a self-adjoint Hilbert–Schmidt operator), then one has a stronger result: Theorem 1.5 Let T be a self-adjoint Hilbert–Schmidt operator on an infinite dimensional Hilbert space H . Then there exist a CONS {en }∞ n=1 of H and real numbers {λn }∞ such that T e = λ e , n ≥ 1 and n n n n=1

T 22 =

∞

λ2n .

n=1

Proof By the Hilbert–Schmidt theorem on a self-adjoint compact operator (e.g., ∞ [130, Theorem VI.16]), there are a CONS {en }∞ n=1 of H and real numbers {λn }n=1

18

1 Mathematical Preliminaries

such that T en = λn en , n ≥1 and λn → 0 as n → ∞. Since T is Hilbert–Schmidt, ∞ 2 2

T 22 = ∞  n=1 T en = n=1 λn . The following proposition formulates a sufficient condition for a densely defined linear operator on H to have an extension to a Hilbert–Schmidt operator on H . Proposition 1.15 Let T ∈ L(H ) be densely defined.  Suppose that there exists a ∞ ∗ ), n ≥ 1 and ∗ 2 CONS {en }∞ of H such that e ∈ D(T n n=1 T en < ∞. Then n=1 T is closable and T is Hilbert–Schmidt. Proof Let Ψ ∈ D(T ) and, for each N ∈ N, define ΦN ∈ H by ΦN :=

N

en , T Ψ  en .

n=1

Then limN→∞ ΦN = T Ψ . In particular, limN→∞ ΦN 2 = T Ψ 2 . We have

ΦN 2 =

N

| en , T Ψ  |2 =

n=1

≤

N



N  | T ∗ en , Ψ | 2 n=1

T ∗ en 2 Ψ 2 ≤ C 2 Ψ 2

n=1

with C :=



∞ ∗ 2 n=1 T en

< ∞, where we have used the Schwarz inequality.

Hence, taking the limit N → ∞, we obtain T Ψ 2 ≤ C 2 Ψ 2 . Hence T is bounded with T ≤ C. It is well known that a densely defined bounded operator is closable and its closure is an everywhere defined bounded operator. Hence T is closable and T ∈ B(H ). This implies also that (T )∗ ∈ B(H ). On the other hand, (T )∗ = T ∗ . Hence T ∗ ∈ B(H ). It follows from this result and the assumption  ∞ ∗ 2 ∗ ∗ ∗ n=1 T en < ∞ that T is Hilbert–Schmidt. Hence (T ) = T also is Hilbert– Schmidt.  Let I2 (H ) be the set of Hilbert–Schmidt operators on H . Then I2 (H ) is a Hilbert space with inner product S, T 2 := Tr S ∗ T ,

S, T ∈ I2 (H )

(see, e.g., [130, Theorem VI.22]). We set

T 2 :=



T , T 2 ,

T ∈ I2 (H ).

1.1 Linear Operators on Hilbert Spaces

19

It is not so difficult to show that, for all S ∈ B(H ) and T ∈ I2 (H ), T S and ST are in I2 (H ) and

T S 2 ≤ T 2 S ,

ST 2 ≤ S T 2 .

(1.25)

The following lemma is a basic fact on I2 (H ): Lemma 1.2 Let {en }∞ n=1 be a CONS of H . Then {Pem ,en }m,n∈H is a CONS of I2 (H ). Proof It is easy to see that {Pem ,en }m,n∈H is an ONS of I 2 (H ). To prove the completeness of it, let T ∈ I2 (H ) be such that T , Pem ,en 2 = 0, m, n ∈ N. It  is easy to see that T , Pem ,en 2 = T em , en . Hence T em , en  = 0, m, n ∈ N, which implies that T = 0. Thus {Pem ,en }m,n∈H is a CONS of I2 (H ). 

1.1.9 Reduction of Operators For a closed subspace M ⊂ H , there exists a unique orthogonal projection PM such that Ran (PM ) = M . The operator PM is called the orthogonal projection onto M . A linear operator A on H is said to be reduced by M if PM A ⊂ APM (i.e., for all Ψ ∈ D(A), PM Ψ ∈ D(A) and APM Ψ = PM AΨ ). In this case, the operator AM on M defined by D(AM ) := D(A) ∩ M ,

AM Ψ = AΨ,

Ψ ∈ D(AM )

is called the reduced part of A to M . We denote by M ⊥ the orthogonal complement of M : M ⊥ := {Φ ∈ H | Φ, Ψ H = 0, ∀Ψ ∈ M }, which is a closed subspace of H . Note that QM := 1 − PM is the orthogonal projection onto M ⊥ (by the projection theorem) and PM A ⊂ APM is equivalent to QM A ⊂ AQM . Hence, if A is reduced by M , then M ⊥ also reduces A. Many properties of a linear operator reduced by a closed subspace are inherited by the reduced part of it: Proposition 1.16 Let M be a closed subspace of H and A be a linear operator on H which is reduced by M . (i) (ii) (iii) (iv)

If A is densely defined, then so is AM . If A is closed, then so is AM . If A is a densely defined closed operator, then so is AM . If A is symmetric, then so is AM .

20

1 Mathematical Preliminaries

(v) If A is self-adjoint, then so is AM . Moreover, for any Borel measurable function f on R, f (A) (see (1.15)) is reduced by M and its reduced part f (A)M is given by f (A)M = f (AM ). (vi) If A is unitary, then so is AM .

1.1.10 Strongly Continuous One-Parameter Unitary Groups Let X be a Banach space with norm · X and I be an interval of R. A mapping F : I → X (I t → F (t) ∈ X ) is called an X -valued function on I. Let a ∈ I. If lim F (a + ε) − F (a) X = 0,

ε→0

then F is said to be strongly continuous at point a. If F is strongly continuous at all points in I, then it is said to be strongly continuous on I. If there exists a vector F (a) ∈ X such that   1   lim  (F (a + ε) − F (a)) − F (a)  = 0, ε→0 ε X then F is said to be strongly differentiable at point a and F (a) is called the strong differential vector of F at point a. If F is strongly differentiable at all points in I, then it is said to be strongly differentiable on I. In this case the mapping F : I → X ; I t → F (t) is called the strong derivative of F . The strong derivative F (t) is written also as F (t) =

d dF (t) = F (t). dt dt

The notion of boundedness of a linear operator on X is defined in the same way as in the case of a linear operator on a Hilbert space.5 We denote by B(X ) the set of everywhere defined bounded linear operators on X . Let I be an interval of R as above and T (·): I → B(X ); I t → T (t) ∈ B(X ) be a mapping from I to B(X ) (i.e., B(X )-valued function on I). For each Ψ ∈ X , we define an X -valued function FΨ on I by FΨ (t) := T (t)Ψ. Let a ∈ I. The mapping T (·) is said to be strongly continuous at point a if, for all Ψ ∈ X , FΨ is strongly continuous at point a. If FΨ is strongly continuous on I, then T (·) is said to be strongly continuous on I. linear operator A on X is said to be bounded if there exists a constant C > 0 such that

AΨ X ≤ C Ψ X , Ψ ∈ D(A). In this case, the operator norm A is defined by A := supΨ ∈D(A),Ψ =0 ( AΨ X / Ψ X ).

5A

1.1 Linear Operators on Hilbert Spaces

21

Let D be a subspace of X and Ψ ∈ D. Then the X -valued function T (·)Ψ : I → X ; I t → T (t)Ψ ∈ X is said to be strongly differentiable at point a if FΨ is strongly differentiable at point a. In this case, we write  dT (t)  FΨ (a) = Ψ or T (a)Ψ. dt t =a If, for all Ψ ∈ D, FΨ is strongly differentiable on I, then T (·) is said to be strongly differentiable on I with domain D. In this case, we write FΨ (t) =

dT (t) Ψ dt

or T (t)Ψ.

The operator dT (t)/dt or T (t) is called the strong derivative of T (t) on D. Let H be a Hilbert space. A family {U (t)}t ∈R of unitary operators on H is called a strongly continuous one-parameter unitary group if the following hold: (i) The mapping R t → U (t) is strongly continuous. (ii) (group property) For all s, t ∈ R, U (t + s) = U (t)U (s). It follows from (ii) and the unitarity of U (0) that U (0) = I (identity). Example 1.6 For each self-adjoint operator H on a Hilbert space H , {e−it H }t ∈R is a strongly continuous one-parameter unitary group. Moreover, for all Ψ ∈ D(H ), ΨH (t) := e−it H Ψ is in D(H ) and strongly differentiable in t, satisfying the differential equation i

d ΨH (t) = H ΨH (t). dt

(1.26)

In general, for an operator A on a Hilbert space H , the differential equation i

d Ψ (t) = AΨ (t) dt

(Ψ (t) ∈ D(A), ∀t ∈ R)

(1.27)

is called the abstract Schrödinger equation with respect to A, where dΨ (t)/dt is taken in the sense of strong derivative. Equation (1.26) shows that, for each Ψ ∈ D(H ), ΨH (t) is a solution of the abstract Schrödinger equation with respect to H such that ΨH (0) = Ψ . Conversely, the initial value problem for the abstract Schrödinger equation with respect to a self-adjoint operator H has a unique solution which is of the form ΨH (t). A fundamental fact on a strongly continuous one-parameter unitary group is stated in the next theorem. Theorem 1.6 (Stone’s Theorem) For each strongly continuous one-parameter unitary group {U (t)}t ∈R on a Hilbert space H , there exists a unique self-adjoint operator S on H such that U (t) = eit S , t ∈ R.

22

1 Mathematical Preliminaries

For a proof of this theorem, see, e.g., [130, Theorem VIII.8] or [23, Theorem 4.5]. The self-adjoint operator S in Theorem 1.6 is called the generator of the unitary group U (t).6 The next theorem may be useful in proving essential self-adjointness for a class of symmetric operators. Theorem 1.7 Let {U (t)}t ∈R be a strongly continuous one-parameter unitary group on H and suppose that there exists a dense subspace D in H such that, for all t ∈ R, U (t)D ⊂ D and, for all Ψ ∈ D, U (t)Ψ is strongly differentiable in t ∈ R. Define an operator S0 on H by D(S0 ) := D and S0 Ψ :=

1 1 lim (U (t) − 1)Ψ, i t →0 t

Ψ ∈ D.

Then S0 is essentially self-adjoint and U (t) = eit S0 , t ∈ R. This theorem immediately implies the following fact: Corollary 1.2 Let S be a self-adjoint operator on H . Suppose that there exists a dense subspace D in H such that eit S D ⊂ D, t ∈ R. Then S is essentially self-adjoint on D. Example 1.7 In quantum theory, the total energy of a quantum system is represented by a self-adjoint operator H on a Hilbert space H , called the Hamiltonian of the system, where non-zero vectors in H describe state vectors of the system. In the case where H is independent of time t ∈ R, the time development of the system is described by the strongly continuous one-parameter unitary group {e−it H/h¯ }t ∈R , where h¯ := h/2π is the reduced Planck constant (h is the Planck constant). If Ψ ∈ H is the initial state (the state vector at t = 0), then the state vector at time t is given by e−it H/h¯ Ψ .

1.1.11 Strongly Continuous One-Parameter Semi-Groups Let X be a Banach space and T (t) ∈ B(X ) with t ≥ 0 be given. A family {T (t)}t ≥0 is called a strongly continuous one-parameter semi-group or (C0 )semi-group if the following hold: (i) The mapping [0, ∞) t → T (t) is strongly continuous: (ii) (semi-group property) For all s, t ∈ [0, ∞), T (t + s) = T (t)T (s). (iii) T (0) = I .

6 There

is a case where iS is called the generator of U (t).

1.1 Linear Operators on Hilbert Spaces

23

Example 1.8 Let H be a self-adjoint operator on a Hilbert space H and bounded below. Then {e−βH }β≥0 is a strongly continuous one-parameter semi-group. Moreover, for all Ψ ∈ D(H ), Ψ (β) := e−βH Ψ is in D(H ) and strongly differentiable in β, satisfying the differential equation d Ψ (β) = −H Ψ (β). dβ

(1.28)

For a (C0 )-semi-group {T (t)}t ≥0, one can define an operator A as follows: D(A) := {Ψ ∈ X |the limit s− lim ε−1 (T (ε)Ψ − Ψ ) exists},

(1.29)

AΨ := lim ε−1 (T (ε)Ψ − Ψ ),

(1.30)

ε↓0

Ψ ∈ D(A).

ε↓0

The operator A is called the generator of the (C0 )-semi-group {T (t)}t ≥0 and symbolically written T (t) = et A ,

t ≥ 0.

The following theorem is fundamental: Theorem 1.8 (Hille–Yosida Theorem) A linear operator A on a Banach space X is the generator of a (C0 )-semi-group {T (t)}t ≥0 on X if and only if the following hold: (i) The operator A is a densely defined closed operator. (ii) There exist real constants M and ω such that the semi-infinite interval (ω, ∞) is included in the resolvent set ρ(A) and

(A − λ)−n ≤

M , (λ − ω)n

λ > ω, n ∈ N.

In that case, {T (t)}t ≥0 is uniquely determined by A and T (t) ≤ Meωt , t ≥ 0. For all Ψ ∈ D(A), T (t)Ψ ∈ D(A), t > 0 and T (t)Ψ is strongly differentiable in t ∈ [0, ∞) with d T (t)Ψ = AT (t)Ψ = T (t)AΨ. dt For a proof of this theorem, see textbooks of functional analysis (e.g., [101, 104, 131, 165]).

24

1 Mathematical Preliminaries

As a special case of the Hille-Yosida theorem, one has the following theorem (the (C0 )-semi-group version of Stone’s theorem): Theorem 1.9 Let {T (t)}t ≥0 be a strongly continuous one-parameter semi-group on a Hilbert space H such that, for all t ≥ 0, T (t) is self-adjoint. Then there exists a unique self-adjoint operator H on H bounded from below such that T (t) = e−t H ,

t ≥ 0.

1.2 Strong Commutativity of Self-adjoint Operators For self-adjoint operators on a Hilbert space, there is a stronger concept of commutativity. Two self-adjoint operators A an B on H are said to strongly commute if the spectral measure EA of A and the spectral measure EB of B commute, i.e., for all Borel sets J, K ⊂ R, EA (J )EB (K) = EB (K)EA (J ). An n-tuple A = (A1 , . . . , An ) of self-adjoint operators on H is said to be strongly commuting if, for all j, k = 1, . . . , n with j = k, Aj and Ak strongly commute. Conditions equivalent to the strong commutativity of two self-adjoint operators on a Hilbert space are summarized in the following proposition7: Proposition 1.17 Let A and B be self-adjoint operators on a Hilbert space H . The following are equivalent: (i) A and B strongly commute. (ii) For all s, t ∈ R, eisA and eit B commute: eisA eit B = eit B eisA . (iii) For all z, w ∈ C \ R, the resolvents (A − z)−1 and (B − w)−1 commute: (A − z)−1 (B − w)−1 = (B − w)−1 (A − z)−1 . There is a simpler criterion for the strong commutativity of two self-adjoint operators. Proposition 1.18 Let A and B be self-adjoint operators on a Hilbert space H . Then A and B strongly commute if and only if, for all t ∈ R, eit B A ⊂ Aeit B . In that case the operator equality eit B A = Aeit B , t ∈ R holds. Proof Suppose that A and B strongly commute. Then part (ii) in Proposition 1.17 holds. This implies that eisA = e−it B eisA eit B = eisA(t )

(1.31)

7 For a proof, see, e.g., [130, Theorem VIII.13], [23, Theorem 4.9] or [143, Propositions 5.27 and 6.15].

1.2 Strong Commutativity of Self-adjoint Operators

25

with A(t) = e−it B Aeit B , where, in the second equality, we have used the unitary covariance of functional calculus (see Theorem B.3 in Appendix B). Hence, for all t ∈ R, A = A(t), i.e., A = e−it B Aeit B . This is equivalent to eit B A = Aeit B ,

t ∈ R.

(1.32)

In particular, eit B A ⊂ Aeit B for all t ∈ R. Conversely, suppose that eit B A ⊂ Aeit B for all t ∈ R. Then, by applying Proposition 1.3(i), we obtain A ⊂ e−it B Aeit B (multiply e−it B from the left). Since the both sides are self-adjoint, they coincide: A = e−it B Aeit B . By the unitary covariance of functional calculus, we obtain the first equation in (1.31), which implies that part (ii) in Proposition 1.17 holds. Thus A and B strongly commute.  The next proposition shows that the concept of strong commutativity of selfadjoint operators is stronger than that of the ordinary commutativity: Proposition 1.19 Let A and B be strongly commuting self-adjoint operators on a Hilbert space H . Then, for all Ψ ∈ D(BA) ∩ D(B), Ψ ∈ D(AB) and [A, B]Ψ = 0. In particular, [A, B] = 0 on D([A, B]), i.e., A and B commute. Proof By Proposition 1.18, we have (1.32). Let Ψ ∈ D(BA) ∩ D(B) and f (t) := eit B AΨ and g(t) := Aeit B Ψ . Then f (t) = g(t). Since AΨ ∈ D(B), it follows that f (t) is strongly differentiable in t with strong derivative f (t) = ieit B BAΨ . Hence g(t) also is strongly differentiable in t. Since Ψ ∈ D(B), h(t) := eit B Ψ is strongly differentiable in t with strong derivative h (t) = ieit B BΨ . Since A is closed, it follows that h (t) ∈ D(A) (hence Ψ ∈ D(Aeit B B), t ∈ R) and g (t) = Ah (t). Therefore eit B BAΨ = Aeit B BΨ . Considering the case t = 0, we see that Ψ ∈ D(AB) and ABΨ = BAΨ . Thus the first half of the present proposition is proved. The second half is a simple consequence of it.  Proposition 1.18 suggests a concept of strong commutativity of a (not necessarily self-adjoint) linear operator with a self-adjoint operator. Definition 1.1 A linear operator A on H (i.e., A ∈ L(H )) is said to strongly commute with a self-adjoint operator B on H if, for all t ∈ R, eit B A ⊂ Aeit B . Proposition 1.20 Let A ∈ L(H ) and B be a self-adjoint operator. Then A strongly commutes with B if and only if eit B Ae−it B = A, ∀t ∈ R.

(1.33)

Proof Suppose that A strongly commutes with B. Then, for all t ∈ R, eit B A ⊂ Aeit B . Hence, by Proposition 1.3, A ⊂ e−it B Aeit B and eit B Ae−it B ⊂ A. Since t ∈ R is arbitrary, we can take −t as t in the former relation to obtain A ⊂ eit B Ae−it B . Hence (1.33) holds.

26

1 Mathematical Preliminaries

Conversely, if (1.33) holds, then eit B A = Aeit B , t ∈ R. Thus A strongly commutes with B.  Proposition 1.21 Let A ∈ L(H ) and B be a self-adjoint operator. Suppose that A strongly commutes with B. Then A and B commute: [A, B] = 0 on D(AB) ∩ D(BA). Proof The proof of this proposition is similar to that of Proposition 1.19.



Remark 1.5 The converse of Proposition 1.21 does not hold.

1.3 Criteria of Self-adjointness of Symmetric Operators In quantum mechanics, physical quantities (observables) are represented by selfadjoint operators. But, in concrete models of quantum mechanics, it is usual that, for unbounded observables, only their symmetry is known a priori. Therefore it is important to have criteria for a symmetric operator to be self-adjoint or essentially self-adjoint. Here, for the reader’s convenience, we recall some theorems on criteria for self-adjointness and essential self-adjointness of symmetric operators. Theorem 1.10 (von Neumann’s Theorem) For any densely defined closed operator A on H , A∗ A is a non-negative self-adjoint operator. For a proof of this theorem, see, e.g., [16, Theorem 2.57] or [131, Theorem X.25]. Theorem 1.11 Let A be a symmetric operator on H . Then the following are equivalent: (i) A is self-adjoint. (ii) A is closed and ker(A∗ ± i) = {0}. (iii) Ran (A ± i) = H . For a proof of this theorem, see, e.g., [130, Theorem VIII.3]. The next theorem immediately follows from Theorem 1.11. Theorem 1.12 Let A be a symmetric operator on H . Then the following are equivalent: (i) A is essentially self-adjoint. (ii) ker(A∗ ± i) = {0}. (iii) Ran (A ± i) are dense. Proof (i)⇒(ii) Suppose that A is essentially self-adjoint. Then A is self-adjoint. Hence, ∗ ∗ by applying Theorem 1.11(ii), ker(A ± i) = {0}. But A = A∗ . Hence (ii) holds.

1.3 Criteria of Self-adjointness of Symmetric Operators

27

∗

(ii)⇒(iii) Condition (ii) implies that ker(A ± i) = {0}. The operator A is a closed symmetric operator. Hence condition (ii) in Theorem 1.11 with A replaced by A is satisfied. Therefore Ran (A ± i) = H . It is easy to see that Ran (A ± i) = Ran (A ± i). Hence Ran (A ± i) are dense. (iii)⇒(i) As just shown, condition (iii) is equivalent to Ran (A ± i) = H . Hence condition (iii) in Theorem 1.11 is satisfied with A replaced by A. Hence A is self-adjoint, i.e., A is essentially self-adjoint.  Theorems 1.11 and 1.12 can be used to find theorems on (essential) self-adjointness of symmetric operators. Theorem 1.13 (Commutator Theorem) Let S be a self-adjoint operator on H such that S ≥ 1 and A be a symmetric operator on H . Suppose that there exists a dense subspace D of H satisfying the following conditions: (i) D ⊂ D(A) ∩ D(S) and D is a core for S. (ii) There exists a constant a > 0 such that, for all Ψ ∈ D, AΨ ≤ a SΨ . (iii) There exists a constant b > 0 such that, for all Ψ ∈ D, | AΨ, SΨ  − SΨ, AΨ  | ≤ b S 1/2 Ψ 2 . Then A is essentially self-adjoint on D and every core for S is a core for the selfadjoint operator A  D. For a proof of this theorem, see, e.g., [16, Theorem 2.32] or [131, Theorem X.37]. For a linear operator A on H , we define n C ∞ (A) := ∩∞ n=1 D(A ),

(1.34)

vector Ψ in C ∞ (A) is said to be analytic if there called the C ∞ -domain of A. A  n n exists a constant t > 0 such that ∞ n=0 A Ψ t /n! < ∞. Theorem 1.14 (Analytic Vector Theorem) Let A be a symmetric operator on H . Suppose that there exists a dense subspace D ⊂ D(A) such that each Ψ ∈ D is an analytic vector of A and AD ⊂ D. Then A is essentially self-adjoint on D. For a proof, see, e.g., [131, p.203, Corollary 2] or [120]. There is a stronger notion of an  analytic vector. A vector Ψ in C ∞ (A) is said to n n be entire analytic if, for all t > 0, ∞ n=0 A Ψ t /n! < ∞. A basic theorem on self-adjointness of perturbations of a self-adjoint operator is given as follows: Theorem 1.15 (The Kato–Rellich Theorem) Let A be a self-adjoint operator on a Hilbert space H and B be a symmetric operator on H such that D(A) ⊂ D(B) and

BΨ ≤ a AΨ + b Ψ ,

Ψ ∈ D(A)

28

1 Mathematical Preliminaries

for constants a and b ≥ 0 with 0 ≤ a < 1. Then A + B is self-adjoint with D(A + B) = D(A) and essentially self-adjoint on any core for A. Moreover, if A is bounded from below, then so is A + B. For proofs of this theorem, we refer the reader to [98, Chapter V, §4] or [131, Theorem X.12]. We want to add a theorem on self-adjointness which is often useful. To prove the theorem, we recall a basic fact in Hilbert space theory: an ONS of a Hilbert space H and α := {αn }∞ Lemma 1.3 Let {en }∞ n=1 be  n=1 be a 2 < ∞. Then the infinite series complex sequence such that ∞ |α | n n=1 Ψ (α) :=

∞

αn en

n=1

converges in H and

Ψ (α) 2 =

∞

|αn |2 .

n=1



Proof An easy exercise.

For a subset E = ∅ of a complex vector space, we denote by span(E ) or span E the subspace algebraically spanned by E : span(E ) :=

⎧ n ⎨ ⎩

j =1

⎫ ⎬ αj Ψj |n ∈ N, αj ∈ C, Ψj ∈ E , j = 1, . . . , n . ⎭

(1.35)

The following theorem also is often useful. Theorem 1.16 Let S be a closed symmetric operator on a Hilbert space. Suppose that there exists a CONS {en }∞ n=1 of H such that, for each n ∈ N, en ∈ D(S) and Sen = λn en with a real constant λn (i.e., λn is an eigenvalue of S). Then S is essentially self-adjoint on span{en |n ∈ N} and hence self-adjoint. Moreover, σ (S) = {λn |n ∈ N}.

(1.36)

Proof Let D := span{en |n ∈ N}. We prove that (S ± i)D are dense in H . Then, by Theorem 1.12, S is essentially self-adjoint on D and hence S  D = S is selfadjoint. Let Ψ be in ((S + i)D)⊥ . Then, for all n ∈ N, (S + i)en , Ψ  = 0. The left hand side is equal to (λn − i) en , Ψ . Hence en , Ψ  = 0. Therefore Ψ = 0. Thus (S + i)D is dense in H . Similarly, one can show that (S − i)D is dense in H . Let us prove (1.36). It is obvious that {λn |n ∈ N} ⊂ σ (S) and hence {λn |n ∈ N} ⊂ σ (S). To prove the converse inclusion relation, let z ∈ ({λn |n ∈ N})c . Then there exists a constant δ > 0 such that |λn − z| ≥ δ, n ∈ N.

1.4 Basic Properties of Symmetric Operators

29

Let Ψ ∈ ker(S − z). Then (S − z)Ψ = 0. Hence en , (S − z)Ψ  = 0. The left hand side is equal to (S − z∗ )en , Ψ  = (λn − z) en , Ψ . Hence en , Ψ  = 0. Therefore Ψ = 0. Thus S − z is injective. To show that S − z is surjective, let Φ ∈ H . Then  ∞  ∞  en , Φ 2 1   ≤ 1 |en , Φ|2 = 2 Φ 2 < ∞. λ −z 2 δ δ n n=1

n=1

Hence, by Lemma 1.3, the infinite series Ψ :=

∞ en , Φ n=1

converges in H . Let ΨN :=

λn − z

en

N

en ,Φ n=1 λn −z en .

Then ΨN → Ψ as N → ∞. It is  easy to see that ΨN ∈ D(S) and (S − z)ΨN = N n=1 en , Φ en → Φ (N → ∞). Hence, by the closedness of S − z, Ψ ∈ D(S − z) = D(S) and (S − z)Ψ = Φ. Hence S is surjective. Thus z ∈ ρ(S), implying that ({λn |n ∈ N})c ⊂ ρ(S). Hence σ (S) ⊂ {λn |n ∈ N}.  Remark 1.6 If {λn }n in Theorem 1.16 has no accumulation points, then σ (S) = σp (S) = {λn |n ∈ N}.

1.4 Basic Properties of Symmetric Operators Non-essential self-adjoint symmetric operators also play important roles in quantum mechanics. For later use, here we state basic properties of a closed symmetric operator. Let Π+ := {z ∈ C|Im z > 0} (the open upper half-plane in C),

(1.37)

Π− := {z ∈ C|Im z < 0} (the open lower half-plane in C),

(1.38)

where Im z denotes the imaginary part of z. Proposition 1.22 Let A be a symmetric operator on H . Then: (i) A is a closed symmetric operator and σ (A) = σ (A),

ρ(A) = ρ(A).

(1.39)

(ii) For all z ∈ C \ R and Ψ ∈ D(A),

(A − z)Ψ ≥ |Im z| Ψ .

(1.40)

In particular, A − z is injective and (A − z)−1 is bounded with D((A − z)−1 ) = Ran (A − z) and (A − z)−1 ≤ 1/|Im z|.

30

1 Mathematical Preliminaries

Proof (i) It is an elementary fact that A is closable and A is a closed symmetric operator. Therefore, by Proposition 1.8, (1.39) holds. (ii) This is a well-known fact.  Proposition 1.23 Let A be a symmetric operator on H . Then: (i) dim ker(A∗ − z) is constant throughout Π+ (z ∈ Π+ ). (ii) dim ker(A∗ − w) is constant throughout Π− (w ∈ Π− ). For a proof of this proposition, see, e.g., [131, Theorem X.1] (note that A∗ = (A)∗ ). For a symmetric operator A, the non-negative integers n+ (A) := dim ker(A∗ − i)

(1.41)

n− (A) := dim ker(A∗ + i)

(1.42)

and

are called the deficiency indices of A. Since A∗ = (A)∗ , we have n± (A) = n± (A). By Proposition 1.23, n+ (A) = dim ker(A∗ − z),

z ∈ Π+

(1.43)

w ∈ Π− .

(1.44)

and n− (A) = dim ker(A∗ − w),

Theorem 1.17 Let A be a symmetric operator on H . Then: (i) If ker(A∗ − z0 ) = {0} for some z0 ∈ Π+ , then Π − ⊂ σ (A). (ii) If ker(A∗ − w0 ) = {0} for some w0 ∈ Π− , then Π + ⊂ σ (A). (iii) If ker(A∗ − z0 ) = {0} and ker(A∗ − w0 ) = {0} for some z0 ∈ Π+ and some w0 ∈ Π− , then σ (A) = C. (iv) On the spectrum σ (A) of A, one of the following holds: (a) (b) (c) (d)

σ (A) = Π + . σ (A) = Π − . σ (A) = C. σ (A) ⊂ R.

(v) A is self-adjoint if and only if case (d) in (iv) holds. In that case, σr (A) = ∅. For a proof of the theorem, see, e.g., [131, Theorem X.1].

1.5 Tensor Product of Hilbert Spaces

31

As for self-adjoint extensions of a symmetric operator, the following theorem holds: Theorem 1.18 Let A be a symmetric operator on H . Then: (i) A is self-adjoint if and only if A is closed and n± (A) = 0. (ii) A has self-adjoint extensions if and only if n+ (A) = n− (A). There exists a one-to-one correspondence between self-adjoint extensions of A and unitary operators from ker(A∗ − i) to ker(A∗ + i). (iii) Let n+ (A) = n− (A) ≥ 1. Then A has uncountably many different self-adjoint extensions. (iv) If n+ (A) = 0 and n− (A) = 0 or n+ (A) = 0 and n− (A) = 0, then A is maximally symmetric. For a proof of this theorem, see, e.g., [131, p. 141, Corollary]. Theorem 1.19 Let A be a symmetric operator on H . (i) If n+ (A) = 0 and n− (A) = 0, then σ (A) = Π + . (ii) If n+ (A) = 0 and n− (A) = 0, then σ (A) = Π − . (iii) If n+ (A) = 0 and n− (A) = 0, then σ (A) = C. Proof (i) The condition n− (A) = 0 and Theorem 1.17(ii) imply that Π + ⊂ σ (A). By condition n+ (A) = 0 and (1.43), ker(A∗ − z) = {0} for all z ∈ Π+ . Hence, by applying Theorem 1.2, we have H = Ran (A − z∗ ) = Ran (A¯ − z∗ ). Hence, ¯ = ρ(A). Therefore Π− ⊂ ρ(A). Thus σ (A) = Π + . for all z ∈ Π+ , z∗ ∈ ρ(A) (ii) The present assumption implies that n+ (−A) = 0 and n− (−A) = 0. Hence, by applying (i) with A replaced by −A, σ (−A) = Π + . Hence, by (1.11), we obtain σ (A) = Π − . (iii) The present assumption and Theorem 1.2 imply that, for all z ∈ Π+ and w ∈ Π− , Ran (A − z∗ ) and Ran (z − w∗ ) are not equal to H . Hence z∗ , w∗ ∈ σ (A), which implies that σ (A) = C. 

1.5 Tensor Product of Hilbert Spaces For complex vector spaces V1 , . . . , Vn with n ∈ N, a mapping f : V1 ×· · ·×Vn → C is called a conjugate n-linear functional if it is separately anti (conjugate)-linear, i.e., for each j = 1, . . . , n and all uj ∈ Vj , (v1 , . . . , vn ) ∈ V1 × · · · × Vn , α ∈ C, j -th 

j -th 

j -th 

f (v1 , . . . , vj + uj , . . . , vn ) = f (v1 , . . . , vj , . . . , vn ) + f (v1 , . . . , uj , . . . , vn ), j -th 

j -th 

f (v1 , . . . , αvj , . . . , vn ) = α ∗ f (v1 , . . . , vj , . . . , vn ).

32

1 Mathematical Preliminaries

The set L∗ (V1 × · · · × Vn ) of conjugate n-linear functionals on V1 × · · · × Vn becomes a complex vector space with the usual operation of addition of functionals and that of scalar multiplication. Let H1 , . . . , Hn be Hilbert spaces (n ≥ 2). Then, for each n-tuple (Ψ1 , . . . , Ψn ) ∈ H1 × · · · × Hn , a conjugate n-linear functional is defined by (⊗nj=1 Ψj )(Φ1 , . . . , Φn ) :=

n   Φj , Ψj ,

(Φ1 , . . . , Φn ) ∈ H1 × · · · × Hn .

j =1

The conjugate n-linear functional ⊗nj=1 Ψj is called the tensor product of Ψ1 , . . . , Ψn . It is also written Ψ1 ⊗ · · · ⊗ Ψn . Let Dj be a subspace of Hj (j = 1, . . . , n). Then the subspace ˆ nj=1 Dj := span{⊗nj=1 Ψj |Ψj ∈ Dj , j = 1, . . . , n} ⊗ of L∗ (H1 × · · · × Hn ) is called the algebraic tensor product of D1 , . . . , Dn . If ˆ nj=1 Dj = ⊗ ˆ n D. Dj = D, j = 1, . . . , n, then we write ⊗ ˆ nj=1 Hj such that It is shown8 that there is an inner product  ,  of ⊗ n     ⊗nj=1 Φj , ⊗nj=1 Ψj = Φj , Ψj ,

Φj , Ψj ∈ Hj , j = 1, . . . , n.

j =1

n

ˆ j =1 Hj with respect to this inner product is called the tensor The completion of ⊗ product Hilbert space (or simply the tensor product) of H1 , . . . , Hn and written ⊗nj=1 Hj . By the definition of completion of an inner product space, the algebraic ˆ nj=1 Hj is identified with a dense subspace of ⊗nj=1 Hj . In the case tensor product ⊗ where Hj = H , j = 1, . . . , n, we write ⊗nj=1 Hj = ⊗n H . It is called the n-fold tensor product of H . The following theorem is a basic theorem in the theory of tensor product Hilbert spaces: Theorem 1.20 Let {ek }∞ k=1 be a CONS of Hj . Then (j )

{ek(1) ⊗ · · · ⊗ ek(n) |kj ∈ N, j = 1, . . . , n} n 1 is a CONS of ⊗nj=1 Hj .

8 See,

e.g., [130, §II.4], [25, Lemma 1.2], [28, Lemma 2.3] or [143, Lemma 7.19].

1.5 Tensor Product of Hilbert Spaces

33

Proof See, e.g., [28, Proposition 2.2(iv)].



In concluding this section, we describe basic facts on continuity and differentiability of a mapping from an interval of R to ⊗nj=1 Hj . Let I be an interval of R and, for each j = 1, . . . , n, Ψj (·): I → Hj ; I t → Ψj (t) ∈ Hj be a mapping from I to Hj . Then a mapping F : I → ⊗nj=1 Hj is defined by F (t) := ⊗nj=1 Ψj (t),

t ∈ I.

(1.45)

Proposition 1.24 (i) If each Ψj (·) is strongly continuous at point a ∈ I, then so is F . (ii) If each Ψj (·) is strongly continuous on I, then so is F . (iii) If each Ψj (·) is strongly differentiable at point a ∈ I, then so is F and

F (a) =

n

Ψ1 (a) ⊗ · · · ⊗ Ψj −1 (a) ⊗ Ψj (a) ⊗ Ψj +1 (a) ⊗ · · · ⊗ Ψn (a),

j =1

(1.46) where Ψj (a) is the strong differential vector of Ψj (·) at point a. (iv) If each Ψj (·) is strongly differentiable on I, then so is F and F (t) is given by the right hand side of (1.46) with a replaced by t. Proof (i) We have for all ε ∈ R such that a + ε ∈ I F (a + ε) − F (a) =

n

Ψ1 (a) ⊗ · · · ⊗ Ψj −1 (a)

j =1

⊗ (Ψj (a + ε) − Ψj (a)) ⊗ Ψj +1 (a + ε) ⊗ · · · ⊗ Ψn (a + ε).

(1.47) Hence

F (a + ε) − F (a) ≤

n j =1

Ψ1 (a) · · · Ψj −1 (a)

× Ψj (a + ε) − Ψj (a) Ψj +1 (a + ε) · · · Ψn (a + ε)

→0

(ε → 0).

Hence F is strongly continuous at point a. (ii) This follows from (i).

34

1 Mathematical Preliminaries

(iii) By (1.47) with ε = 0, we have 1 (F (a + ε) − F (a)) = Ψ1 (a) ⊗ · · · ⊗ Ψj −1 (a) ε n

j =1

1 ⊗ (Ψj (a + ε) − Ψj (a)) ⊗ Ψj +1 (a + ε) ⊗ · · · ⊗ Ψn (a + ε). ε

By the present assumption, limε→0 ε−1 (Ψj (a + ε) − Ψj (a)) = Ψj (a). Hence, by an application of (i), we obtain lim

ε→0

=

1 (F (a + ε) − F (a)) ε

n

Ψ1 (a) ⊗ · · · ⊗ Ψj −1 (a) ⊗ Ψj (a) ⊗ Ψj +1 (a) ⊗ · · · ⊗ Ψn (a).

j =1

Thus F is strongly differentiable at point a and (1.46) holds. (iv) This follows from (iii).



1.6 Symmetric and Anti-symmetric Tensor Product Hilbert Spaces There are special classes of tensor product Hilbert spaces. We denote by Sn the symmetry group of order n (i.e., the set of permutations of 1, . . . , n). Let H be a Hilbert space. Then, for each σ ∈ Sn , there is a unique unitary operator Uσ on ⊗n H such that Uσ (⊗nj=1 Ψj ) = ⊗nj=1 Ψσ (j ) ,

Ψj ∈ H , j = 1, . . . , n.

(1.48)

Using the unitary operators Uσ , σ ∈ Sn , one can define the following operators: Sn :=

1 Uσ , n!

(1.49)

1 sgn(σ )Uσ , n!

(1.50)

σ ∈Sn

An :=

σ ∈Sn

where sgn(σ ) means the signature of σ . One can prove that Sn and An are orthogonal projections: Sn∗ = Sn , A∗n = An ,

Sn2 = Sn , A2n = An .

(1.51) (1.52)

1.6 Symmetric and Anti-symmetric Tensor Product Hilbert Spaces

35

The operator Sn (resp. An ) is called the symmetrizer or the symmetrization operator (resp. the anti-symmetrizer or the anti-symmetrization operator) on ⊗n H . Therefore the ranges of Sn and An are closed and hence Hilbert spaces. We write ⊗ns H := Ran (Sn ),

⊗nas H := Ran (An ).

(1.53)

The Hilbert space ⊗ns H (resp. ⊗nas H ) is called the n-fold symmetric (resp. antisymmetric) tensor product of H . The anti-symmetric tensor product ⊗nas H is sometimes written as ∧n H or ∧n (H ), being called the n-fold wedge product of H . For a subspace D of H , the subspaces ˆ ns D := Sn (⊗ ˆ n D) ⊗ and ˆ nas D := An (⊗ ˆ n D) ⊗ are called respectively the n-fold algebraic symmetric tensor product of D and the n-fold algebraic anti-symmetric tensor product of D. Using (1.51), we have for all Ψj , Φj ∈ H , j = 1, . . . , n Sn (Ψ1 ⊗ · · · ⊗ Ψn ), Sn (Φ1 ⊗ · · · ⊗ Φn ) = Ψ1 ⊗ · · · ⊗ Ψn , Sn (Φ1 ⊗ · · · ⊗ Φn ) =

n 1  Ψj , Φσ (j ) . n!

(1.54)

σ ∈Sn j =1

Let {ej }∞ j =1 be a CONS of H and Ψj1 ...jn := Sn (ej1 ⊗ · · · ⊗ ejn ),

j1 , . . . , jn ∈ N,

j1 ≤ j2 ≤ · · · ≤ jn .

Then, by (1.54), one sees that {Ψj1 ...jn |j1 , . . . , kn ∈ N, j1 ≤ j2 ≤ · · · ≤ jn } is an orthogonal system of ⊗ns H . Note that Ψj1 ...jn = Sn ((⊗n1 ei1 ) ⊗ (⊗n2 ei2 ) ⊗ · · · ⊗ (⊗nr eir ) for some (ei1 , . . . , eir ) with i1 < i2 < · · · ir (i1 , . . . , ir ∈ {j1 , j2 , . . . , jn }, r ∈ N). Hence

Ψj1 ...jn 2 =

n1 ! · · · nr ! . n!

36

1 Mathematical Preliminaries

Using these facts, one obtains the following theorem9: Theorem 1.21 Let

n! . n1 ! · · · nr !

Cj1 ...jn :=

(1.55)

Then {Cj1 ...jn Ψj1 ...jn |j1 , . . . , jn ∈ N, j1 ≤ j2 ≤ · · · ≤ jn } is a CONS of ⊗ns H . For u1 , . . . , un ∈ H , we define the wedge product (or exterior product) of them u1 ∧ · · · ∧ un ∈ ∧n (H ) by u1 ∧ · · · ∧ un :=

√ n!An (⊗nj=1 uj ).

(1.56)

It is easy to see that, for all uj , vj ∈ H (j = 1, . . . , n), u1 ∧ · · · ∧ un , v1 ∧ · · · ∧ vn  =



  sgn(σ ) u1 , vσ (1) · · · un , vσ (n) .

(1.57)

σ ∈Sn

Using this formula, one can prove, in a similar way to above, the following theorem: Theorem 1.22 Let {ej }∞ j =1 be a CONS of H . Then {ei1 ∧ · · · ∧ ein |i1 , . . . , in ∈ N, i1 < i2 < . . . < in } is a CONS of ∧n (H ). 

Proof See [28, Proposition 2.11].

1.7 Tensor Products of L2 -Spaces Let (X, Σ, μ) be a σ -finite measure space and L2 (X, dμ) be the Hilbert space of equivalence classes of Σ-measurable functions on X (see Appendix A). Let n ∈ N, n ≥ 2 and, for j = 1, . . . , n, (Xj , Σj , μj ) be a σ -finite measure space. For each function fj on Xj (j = 1, . . . , n), one can define a function f1 × · · · × fn on the product space X1 × · · · × Xn by (f1 × · · · × fn )(x1 , . . . , xn ) :=

n 

fj (xj ),

(x1 , . . . , xn ) ∈ X1 × · · · × Xn .

j =1

We denote the product measure of μ1 , . . . , μn by μ1 ⊗ · · · ⊗ μn . 9 For

more details, see [28, Proposition 2.9].

1.7 Tensor Products of L2 -Spaces

37

Theorem 1.23 Suppose that each L2 (Xj , dμj ) (j = 1, . . . , n) is separable. Then there exists a unitary operator U : ⊗nj=1 L2 (Xj , dμj ) → L2 (X1 × · · · × Xn , d(μ1 ⊗ · · · × μn )) such that U (⊗nj=1 fj ) = f1 × · · · × fn ,

fj ∈ L2 (Xj , dμj ), j = 1, . . . , n. 

Proof See, e.g., [28, Theorem 2.5].

The unitary operator U in Theorem 1.23 is independent of the choice of complete orthonormal systems of L2 (Xj , dμj ), j = 1, . . . , n. For this reason, we call it the natural isomorphism between ⊗nj=1 L2 (Xj , dμj ) and L2 (X1 × · · · × Xn , d(μ1 ⊗ · · · ⊗ μn )) and write ⊗nj=1 L2 (Xj , dμj ) ∼ = L2 (X1 × · · · × Xn , d(μ1 ⊗ · · · ⊗ μn )).

(1.58)

We now restrict ourselves to the case where X := X1 = X2 = · · · = Xn ,

μ := μ1 = μ2 = · · · = μn

and write μn := μ ⊗ · · · ⊗ μ .    n factors

For an element f of L2 (Xn , dμn ) and a permutation σ ∈ Sn , we define a function fσ on Xn by fσ (x1 , . . . , xn ) := f (xσ (1), . . . , xσ (n) ),

μ-a.e.x = (x1 , . . . , xn ) ∈ Xn .

It follows that fσ ∈ L2 (Xn , dμn ). An element f of L2 (Xn , dμn ) is said to be symmetric (resp. anti-symmetric) if fσ = f (resp. fσ = sgn(σ )f ) for all σ ∈ Sn . Let L2sym (Xn , dμn ) := {f ∈ L2 (Xn , dμn )|∀σ ∈ Sn , fσ = f },

(1.59)

the set of symmetric elements of L2 (Xn , dμn ). It is easy to see that L2sym (Xn , dμn ) is a closed subspace of L2 (Xn , dμn ) and hence it is a Hilbert space. The Hilbert space L2sym (Xn , dμn ) is called the Hilbert space of square integrable symmetric functions on Xn with respect to μn . Contrary to this Hilbert space, one has L2as (Xn , dμn ) := {f ∈ L2 (Xn , dμn )|∀σ ∈ Sn , fσ = sgn(σ )f },

(1.60)

38

1 Mathematical Preliminaries

which is called the Hilbert space of square integrable anti-symmetric functions on Xn with respect to μn . Theorem 1.24 Assume that L2 (X, dμ) is separable. (i) There exists a unitary operator Us : ⊗ns L2 (X, dμ) → L2sym (Xn , dμn ) such that, for all fj ∈ L2 (X, dμ) (j = 1, . . . , n), Us (Sn (⊗nj=1 fj )) =

1 fσ (1) × · · · × fσ (n) . n! σ ∈Sn

(ii) There exists a unitary operator Uas : ∧n (L2 (X, dμ)) → L2as (Xn , dμn ) such that, for all fj ∈ L2 (X, dμ) (j = 1, . . . , n), Uas (An (⊗nj=1 fj )) =

1 sgn(σ )(fσ (1) × · · · × fσ (n) ). n! σ ∈Sn



Proof See, e.g., [28, Theorems 2.10 and 2.11].

We call the unitary operator Us the natural isomorphism between ⊗ns L2 (X, dμ) and L2sym (Xn , dμn ), and the unitary operator Uas the natural isomorphism between ∧n (L2 (X, dμ)) and L2as (Xn , dμn ). These natural isomorphisms are symbolically written ⊗ns L2 (X, dμ) ∼ = L2sym (Xn , dμn ),

∧n (L2 (X, dμ)) ∼ = L2as (Xn , dμn ).

(1.61)

1.8 Tensor Products of Operators We next define basic operators on tensor product Hilbert spaces. Let Aj be a densely defined closable operator from Hj (j = 1, . . . , n) to a Hilbert space Kj . Then it is shown10 that there exists a unique densely defined closed operator ⊗nj=1 Aj from ˆ nj=1 D(Aj ) is a core for it and ⊗nj=1 Hj to ⊗nj=1 Kj such that ⊗ (⊗nj=1 Aj )(⊗nj=1 Ψj ) = ⊗nj=1 Aj Ψj ,

Ψk ∈ D(Ak ),

It follows that ⊗nj=1 Aj = ⊗nj=1 Aj .

10 See

the above-mentioned literature.

k = 1, . . . , n.

1.8 Tensor Products of Operators

39

The operator ⊗nj=1 Aj is called the tensor product operator (or simply the tensor product) of A1 , . . . , An . In the case where Hj = H , Kj = K (j = 1, . . . , n) and Aj = A (j = 1, . . .), we write ⊗nj=1 Aj = ⊗n A ∈ L(⊗n H , ⊗n K ), which is called the n-fold tensor product of A ∈ L(H , K ). The next lemma will be used later: Lemma 1.4 Let A be a densely defined closable operator on a Hilbert space H and B be a densely defined closable operator on a Hilbert space K . Suppose that ˆ A ⊗ I = I ⊗ B on D(A)⊗D(B). Then there is a constant c ∈ C such that A = cI on D(A) and B = cI on D(B). Proof By the present assumption, for all Ψ ∈ D(A) and Φ ∈ D(B), AΨ ⊗ Φ = Ψ ⊗ BΦ. Hence, for all η ∈ H and χ ∈ K , η, AΨ  χ, Φ = η, Ψ  χ, BΦ. Therefore χ, η, AΨ  Φ = χ, η, Ψ  BΦ ,

η, χ, Φ AΨ  = η, χ, BΦ Ψ  .

Thus η, AΨ  Φ = η, Ψ  BΦ, χ, Φ AΨ = χ, BΦ Ψ . Taking Ψ = η with

η = 1 in the first equation, we obtain BΦ = cΦ with c = η, Aη. Putting this into the second equation, we obtain AΨ = cΨ .  We collect some basic facts on tensor product operators.11 Theorem 1.25 Let Aj ∈ B(Hj , Kj ), j = 1, . . . , n.  (i) ⊗nj=1 Aj is bounded with ⊗nj=1 Aj = nj=1 Aj . (ii) If each Aj is unitary, then ⊗nj=1 Aj is unitary. (iii) Let A1 ∈ B(H1 ) and A2 ∈ B(H2 ). Then A1 ⊗ A2 = (A1 ⊗ I )(I ⊗ A2 ) = (I ⊗ A2 )(A1 ⊗ I ). Remark 1.7 Theorem 1.25(iii) can be extended to ⊗nj=1 Aj with Aj ∈ B(Hj ). Theorem 1.26 Consider the case where Kj = Hj (j = 1, . . . , n). (i) If each Aj (j = 1, . . . , n) is symmetric, then ⊗nj=1 Aj is a closed symmetric operator. (ii) If each Aj (j = 1, . . . , n) is self-adjoint, then ⊗nj=1 Aj is self-adjoint. (iii) For a densely defined closable operator Aj on Hj , we define j -th 

A˜ j := I ⊗ · · · ⊗ I ⊗ Aj ⊗I ⊗ · · · ⊗ I,

11 For

7.23].

j = 1, . . . , n.

details of proof, see, e.g., [28, Theorem 3.8], [130, Theorem VIII.33 (a)], [143, Theorem

40

1 Mathematical Preliminaries

Let Aj be a symmetric operator. Then A˜ j is a closed symmetric operator. Moreover, if Aj is self-adjoint, then A˜ j is self-adjoint and (A˜ 1 , . . . , A˜ n ) is strongly commuting. (iv) Let Aj be self-adjoint and Dj be a core for Aj . Then S(A) :=

n

A˜ j

(1.62)

j =1 n

ˆ j =1 Dj . Moreover, for all t ∈ R, is essentially self-adjoint on ⊗ eit S(A) = ⊗nj=1 eit Aj .

(1.63)

(v) If Aj is self-adjoint and bounded from below with Aj ≥ γj (γj is a real constant), then S(A) is self-adjoint and bounded from below with S(A) ≥ n γ . j =1 j Remark 1.8 For a proof of (1.63), Proposition 1.24 can be applied. Concerning the spectrum of S(A), we state only a basic property: Theorem 1.27 Let Aj be a self-adjoint operator on Hj (j = 1, . . . , n) and S(A) be defined by (1.62). Then

σ (S(A)) =

⎧ n ⎨ ⎩

j =1

⎫ ⎬ λj |λj ∈ σ (Aj ), j = 1, . . . , n . ⎭

For a proof of this theorem, see, e.g., [28, Theorem 3.8], [130, Theorem VIII.33 (b)], [143, Corollary 7.25]. The following proposition is about a continuity property of tensor product operator. Proposition 1.25 Let H and K be Hilbert spaces and Tk ∈ B(H ) (k ∈ N). Suppose that s− limk→∞ Tk = T ∈ B(H ) and C := supk≥1 Tk < ∞. Then s− lim Tk ⊗ I = T ⊗ I. k→∞

(1.64)

ˆ , limk→∞ (Tk ⊗ I )Ψ = (T ⊗ I )Ψ . Proof It is easy to see that, for all Ψ ∈ H ⊗K ˆ Let Ψ ∈ H ⊗ K . Then, by the density of H ⊗K in H ⊗ K , for any ε > 0, there ˆ exists a vector Ψε ∈ H ⊗K such that Ψε − Ψ < ε. By the triangle inequality, we have

(Tk ⊗ I )Ψ − (T ⊗ I )Ψ ≤ (Tk ⊗ I )(Ψ − Ψε ) + (Tk ⊗ I )Ψε − (T ⊗ I )Ψε

+ (T ⊗ I )(Ψε − Ψ )

1.9 The Natural Isomorphism Between I2 (H ) and H ⊗ H

41

≤ Tk ⊗ I Ψ − Ψε + (Tk ⊗ I )Ψε − (T ⊗ I )Ψε

+ T ⊗ I Ψ − Ψε

≤ Tk ε + (Tk ⊗ I )Ψε − (T ⊗ I )Ψε + T ε ≤ (C + T )ε + (Tk ⊗ I )Ψε − (T ⊗ I )Ψε . Hence lim supk→∞ (Tk ⊗ I )Ψ − (T ⊗ I )Ψ ≤ (C + T )ε. Taking ε → 0, we obtain limk→∞ (Tk ⊗ I )Ψ − (T ⊗ I )Ψ = 0. Thus (1.64) follows.  Corollary 1.3 Let H and K be Hilbert spaces and Tk ∈ B(H ), Sk ∈ B(K ) (k ∈ N). Suppose that s− limk→∞ Tk = T ∈ B(H ), s− limk→∞ Sk = S ∈ B(K ) with supk≥1 Tk < ∞ and supk≥1 Sk < ∞. Then s− lim Tk ⊗ Sk = T ⊗ S. k→∞

(1.65)

Proof We have Tk ⊗ Sk = (Tk ⊗ I )(I ⊗ Sk ). By Proposition 1.25, s− lim (Tk ⊗ I )(I ⊗ Sk ) = (T ⊗ I )(I ⊗ S). k→∞



Thus (1.65) follows.

1.9 The Natural Isomorphism Between I2 (H ) and H ⊗ H In Sect. 1.1.8, we have seen that the set I2 (H ) of Hilbert–Schmidt operators on a Hilbert space H is a Hilbert space. In this section we prove a theorem on a natural isomorphism of I2 (H ). We recall a basic notion: Definition 1.2 Let V be an inner product space with inner product  , V and norm

· V . A mapping C: V → V is called a conjugation on V if it is an anti-linear mapping satisfying that C 2 = I and

Cf V = f V ,

f ∈V.

(1.66)

Let C be a conjugation on V . Then, using the polarization identity, normpreserving property (1.66) of C and the anti-linearity of C, one can prove the following formula: Cf, CgV = g, f V ,

f, g ∈ V .

(1.67)

It is easy to see that the subset VC := {f ∈ V |Cf = f }

(1.68)

42

1 Mathematical Preliminaries

of V is a real inner product space. For a vector f ∈ V , the vectors Re f :=

1 (f + Cf ) 2

(1.69)

Im f :=

1 (f − Cf ) 2i

(1.70)

and

are in VC . One has f = Re f + i Im f. The vector Re f (resp. Im f ) is called the real (resp. imaginary) part of f with respect to C. We call VC the real part of V with respect to C. The following isomorphism theorem is often useful: Theorem 1.28 Let H be a separable Hilbert space and C be a conjugation on H . Then there exists a unitary operator UC : I2 (H ) → H ⊗ H such that UC (f, · g) = Cf ⊗ g,

f, g ∈ H .

(1.71)

∞ Proof Let {en }∞ n=1 be a CONS of H . Then it is easy to see that {Cen }n=1 is a CONS of H . Hence, by Theorem 1.20, {Cen ⊗ em |n, m ∈ N} is a CONS ofH ⊗ H . Therefore, by Lemma 1.2 and the isomorphism theorem for Hilbert spaces, there exists a unitary operator UC : I2 (H ) → H ⊗ H such that

UC Pem ,en = Cem ⊗ en , By expanding f, g ∈ H as f = obtain Pf,g =

∞

m, n ∈ N.

m=1 em , f  em

∞

and g =

∞

n=1 en , g en ,

we

f, em  en , g Pem ,en

m,n=1

in the Hilbert–Schmidt norm · 2 . Hence UC Pf,g =

∞

f, em  en , g Cem ⊗ en = Cf ⊗ g.

m,n=1

Thus (1.71) holds.



1.10 Convergence of Self-adjoint Operators in Strong Resolvent Sense

43

Theorems 1.4 and 1.28 immediately yield the following corollary: Corollary 1.4 Let T ∈ I2 (H ). Then, with the notation in Theorem 1.4, UC T =

∞

μn Cfn ⊗ en .

(1.72)

n=1

1.10 Convergence of Self-adjoint Operators in Strong Resolvent Sense Let Hn (n ∈ N) and H be self-adjoint operators on a Hilbert space H . Then C\R ⊂ ρ(Hn ) ∩ ρ(H ). Hence, for all z ∈ C \ R, (Hn − z)−1 and (H − z)−1 are in B(H ). Definition 1.3 A sequence {Hn }n of self-adjoint operators is said to converge to H in the strong resolvent sense if, for all z ∈ C \ R, s− lim (Hn − z)−1 = (H − z)−1 , n→∞

where s− lim means strong limit (i.e. limn→∞ (Hn − z)−1 Ψ = (H − z)−1 Ψ, Ψ ∈ H ). A basic criterion on strong resolvent convergence of a sequence of self-adjoint operators is given in the following theorem. Theorem 1.29 Suppose that there exists a core D for H such that D ⊂ ∩∞ n=1 D(Hn ) and limn→∞ Hn Ψ = H Ψ for all Ψ ∈ D. Then {Hn }n converges to H in the strong resolvent sense. Proof Since D is a core for H , for all z ∈ C \ R, (H − z)D := {(H − z)Ψ |Ψ ∈ D} is dense in H . Let Φ ∈ (H − z)D. Then Ψ := (H − z)−1 Φ is in D. We have (Hn − z)−1 Φ − (H − z)−1 Φ = (Hn − z)−1 (H − Hn )Ψ. Hence

(Hn − z)−1 Φ − (H − z)−1 Φ ≤

1

(H − Hn )Ψ . |Im z|

Hence limn→∞ (Hn −z)−1 Φ = (H −z)−1 Φ. Since supn∈N (Hn −z)−1 ≤ 1/|Im z| and (H − z)D is dense, it follows from a limiting argument that s− limn→∞ (Hn − z)−1 = (H − z)−1 . 

44

1 Mathematical Preliminaries

There is an important theorem: Theorem 1.30 A sequence {Hn }n of self-adjoint operators on H converges to H in the strong resolvent sense if and only if s− limn→∞ e−it Hn = e−it H for all t ∈ R. For a proof of this theorem, we refer the reader to [130, Theorem VIII.21].

1.11 Sesquilinear Form and Representation Theorem Let H1 and H2 be Hilbert spaces, and D1 and D2 be subspaces of H1 and H2 respectively. A mapping s: D1 × D2 → C is called a sesquilinear form or a quadratic form with domain D1 × D2 if, for all Φ, Φ ∈ D1 , Ψ, Ψ ∈ D2 and α, β ∈ C, s(Φ, αΨ + βΨ ) = α s(Φ, Ψ ) + β s(Φ, Ψ ), s(αΦ + βΦ , Ψ ) = α ∗ s(Φ, Ψ ) + β ∗ s(Φ , Ψ ). The domain D1 × D2 is called the form domain of s and written D1 × D2 = Q(s). In the case where H = H1 = H2 and D = D1 = D2 , we call a sesquilinear form s with domain D × D a sesquilinear form on H with form domain D and write D = Q(s). Let H be a Hilbert space. A sesquilinear form s on H is said to be Hermitian if s(Ψ, Φ)∗ = s(Φ, Ψ ) for all Ψ, Φ ∈ Q(s). It is easy to see that a sesquilinear form s is Hermitian if and only if s(Ψ, Ψ ) ∈ R for all Ψ ∈ Q(s). A Hermitian sesquilinear form s on H is said to be bounded from below if there exists a constant γ ∈ R such that s(Ψ, Ψ ) ≥ γ Ψ 2 , Ψ ∈ Q(s). If γ can be taken to be zero (i.e., s(Ψ, Ψ ) ≥ 0, Ψ ∈ Q(s)), then s is said to be non-negative. A sesquilinear form s on H is called a symmetric sesquilinear form if it is Hermitian and Q(s) is dense in H . A sesquilinear form s on H is said to be closed if, for any sequence {Ψn }∞ n=1 in Q(s) satisfying the condition that limn→∞ Ψn = Ψ ∈ H and s(Ψn − Ψm , Ψn − Ψm ) → 0 ( n, m → ∞), it holds that Ψ ∈ Q(s) and limn→∞ s(Ψn − Ψ, Ψn − Ψ ) = 0. Example 1.9 Let A be a self-adjoint operator on H . Then, by functional calculus, one can define a sesquilinear form sA with form domain Q(sA ) := D(|A|1/2 ) by  sA (Ψ, Φ) :=

R

λd Ψ, EA (λ)Φ ,

Ψ, Φ ∈ Q(sA ).

1.12 Fourier Analysis

45

This sesquilinear form sA on H is called the sesquilinear form associated with A. It is easy to see that sA is a symmetric form. Moreover, if A is bounded from below with A ≥ γ for a constant γ ∈ R, then sA is closed and one has   sA (Ψ, Φ) = (A − γ )1/2Ψ, (A − γ )1/2 Φ + γ Ψ, Φ ,

Ψ, Φ ∈ Q(sA ). (1.73)

A non-negative closed symmetric form is the sesquilinear form of a non-negative self-adjoint operator: Theorem 1.31 (Representation Theorem) Let s be a non-negative closed symmetric form on H . Then there exists a unique non-negative self-adjoint operator A on H such that Q(s) = D(A1/2 ) and   s(Ψ, Φ) = A1/2Ψ, A1/2 Φ ,

Ψ, Φ ∈ Q(s).

For a proof of this theorem, see, e.g., [98, Chapter VI, Theorem 2.23], [130, Theorem VIII.15] or [16, Theorem 2.56]). In application to quantum mechanics, Theorem 1.31 can be used to define a Hamiltonian.

1.12 Fourier Analysis In this section we recall basic facts in Fourier analysis without proof. For more details, we refer the reader to [131, Chapter IX]. For each natural number d, we denote by Rd := {x = (x1 , . . . , xd )|xj ∈ R, j = 1, . . . , d} the d-dimensional Euclidean space. The dual space of Rd (i.e. the space of linear functionals on Rd ) is naturally identified with the Euclidean space ˆ d := {k = (k1 , . . . , kd )|kj ∈ R, j = 1, . . . , d} R through the Euclidean inner product k · x :=

d

kj xj ,

j =1

which is called also the canonical bilinear form in the variables k and x. In applications to quantum physics below, each point in Rd represents a position in

46

1 Mathematical Preliminaries

ˆ d represents a the space where quantum particles exist, while an element k in R wave number vector of a quantum particle. We denote by L2 (Rd ) the Hilbert space of equivalence classes of square integrable functions on Rd with respect to the d-dimensional Lebesgue measure:   d L (R ) := f : R → C ∪ {±∞}, Borel measurable| 2

 |f (x)| dx < ∞ 2

d

Rd

with inner product  f, gL2 (Rd ) :=

Rd

f (x)∗ g(x)dx,

f, g ∈ L2 (Rd ).

ˆ d ) is defined by The Fourier transform Fd : L2 (Rd ) → L2 (R (Fd f )(k) :=

1 (2π)d/2

 Rd

e−ik·x f (x)dx,

ˆ d , f ∈ L2 (Rd ) a.e.k ∈ R

(1.74)

in the L2 -sense,12 where “a.e.” means “almost everywhere” with respect to the ddimensional Lebesgue measure. One can show that Fd is a unitary transformation ˆ d ) → L2 (Rd ) given by with the inverse Fd−1 : L2 (R (Fd−1 g)(x)

1 = (2π)d/2

 ˆd R

eik·x g(k)dk,

ˆ d) a.e.x, g ∈ L2 (R

(1.75)

in the L2 -sense. We use also the following symbols: fˆ = Fd f,

gˇ = Fd−1 g.

(1.76)

For each j = 1, . . . , d, a linear operator ∂j on L2 (Rd ) is defined as follows: D(∂j ) := C0∞ (Rd ),

∂j f :=

∂f , ∂xj

f ∈ D(∂j ),

(1.77)

where C0∞ (Rd ) is the space of infinitely differentiable functions on Rd with compact support in Rd and ∂f/∂xj is the partial derivative of f in xj ∈ R, the j -th

12 This

means that  lim

Rj →∞,j =1,...,d R ˆd

2       −d/2 −ik·x e f (x)dx  dk = 0. (Fd f )(k) − (2π)   |xj |≤Rj ,j =1,...,d

 But, if f ∈ L2 (Rd ) is integrable, then (Fd f )(k) = (2π)−d/2 Rd e−ik·x f (x)dx (the usual ˆ d. Lebesgue integral) and Fd f is a bounded continuous function on R

1.12 Fourier Analysis

47

component of point x = (x1 , . . . , xd ) ∈ Rd . The operator ∂j is called the partial differential operator in the j -th variable xj . It is shown that ∂j is closable and ∂j∗ = −∂¯j . The closure Dj := ∂¯j

(1.78)

of ∂j is called the generalized partial differential operator in xj , acting in L2 (Rd ). Let pˆ j := −iDj ,

j = 1, . . . , d.

(1.79)

Then it is shown that Fd pˆj Fd−1 = Mkj ,

j = 1, . . . , d,

(1.80)

ˆ d (see Appendix A where Mkj is the multiplication operator by the function kj on R for the definition and basic properties of a multiplication operator). Since Mkj is self-adjoint, it follows that pˆj is self-adjoint with   D(pˆj ) = f ∈ L2 (Rd )|

ˆd R

 |kj fˆ(k)|2 dk < ∞ .

(1.81)

In the context of quantum mechanics, pˆ j is called the j -th momentum operator.13 By (1.80) and the unitary covariance of functional calculus (Theorem B.3 in Appendix B), one has for all t ∈ R Fd e−it pˆ j Fd−1 = e

−it Mkj

.

Hence, for all f ∈ L2 (Rd ), (e−it pˆ j f )(x) = f (x − te j ),

a.e.x ∈ Rd ,

(1.82)

where (e1 , . . . , ed ) ∈ (Rd )d is the standard basis of Rd : j -th 

ej = (0, . . . , 0, 1 , 0, . . . , 0) ∈ Rd ,

j = 1, . . . , d.

(1.83)

13 In applications to quantum physics, we will use the physical unit system where the reduced Planck constant h¯ is equal to 1. In the original physical unit system (SI), pˆ j = −i h¯ Dj .

48

1 Mathematical Preliminaries

Formula (1.82) shows that e−it pˆ j f means the translation of f by the vector tej . It implies also that, for all j, k = 1, . . . , d and t, s ∈ R, e−it pˆ j and e−is pˆ k commute with (e−ia1 pˆ1 · · · e−iad pˆd f )(x) = f (x − a),

∀a = (a1 , . . . , ad ) ∈ Rd , a.e.x ∈ Rd . (1.84)

Hence the d-tuple pˆ := (pˆ 1 , . . . , pˆd )

(1.85)

of self-adjoint operators is strongly commuting (see Proposition 1.17).

1.13 Discrete Fourier Transform In this section, we describe a finite box version of Fourier analysis on Rd . Let us take the bounded closed interval [a, b] ⊂ R and consider the L2 -space    L2 ([a, b]) := f : [a, b] → C ∪ {±∞}, Borel measurable| |f (x)|2 dx < ∞ , [a.b]



which is a Hilbert space with inner product f, g := [a,b] f (x)∗ g(x)dx (f, g ∈ L2 ([a, b])) and with equality f = g in the sense that f (x) = g(x) a.e.x ∈ [a, b] (see Appendix A). We first show that there is a natural isomorphism between L2 ([a, b]) and L2 ([c, d]) ([c, d] is another interval). For this purpose, we define a mapping τ : [a, b] → [c, d] by τ (x) :=

d −c (x − a) + c, b−a

x ∈ [a, b].

Then it is easy to see that τ is bijective. Hence, for all f ∈ L2 ([c, d]), we have   b−a |f (τ (x))|2dx = |f (y)|2 dy < ∞. d − c [c,d] [a,b] Therefore, defining a mapping Uτ : L2 ([c, d]) → L2 ([a, b]) by (Uτ f )(x) :=

d −c f (τ (x)), b−a

f ∈ L2 ([c, d]), a.e.x ∈ [a, b],

(1.86)

we see that Uτ is a unitary operator. We call the unitary operator Uτ the natural isomorphism from L2 ([c, d]) to L2 ([a, b]). Thus, with regard to L2 -spaces on bounded closed intervals, it is sufficient to consider one of them.

1.13 Discrete Fourier Transform

49

Now we consider a closed interval IL := [−L/2, L/2]

(1.87)

with length L > 0 and introduce   2π 2π Z := k = n|n ∈ Z , ΓL := L L

(1.88)

where Z is the set of integers. For each k ∈ ΓL , we define a function φk on IL by 1 φk (x) := √ eikx , L

x ∈ IL .

(1.89)

The following theorem is well known: Theorem 1.32 The set {φk }k∈ΓL is a CONS of L2 (IL ). By Theorem 1.32, every f ∈ L2 (IL ) has the expansion φk , f  φk f = k∈ΓL

in L2 (IL ) and

f 2 =

| φk , f  |2 ,

(1.90)

k∈ΓL

The set 2 (ΓL ) :=

⎧ ⎨ ⎩

u = {uk }k∈ΓL |uk ∈ C, k ∈ ΓL ,

k∈ΓL

|uk |2 < ∞

⎫ ⎬ ⎭

(1.91)

of absolutely square summable sequences on ΓL is an infinite-dimensional complex vector space with the usual addition of two sequences and the usual scalar multiplication for sequences. Moreover, 2 (ΓL ) becomes a Hilbert space with inner product u, v := u∗k vk , u, v ∈ 2 (ΓL ). k∈ΓL

Theorem 1.33 There exists a unique unitary operator FL : L2 (IL ) → 2 (ΓL ) such that  L/2 1 e−ikx f (x)dx, f ∈ L2 (IL ), k ∈ ΓL . (1.92) (FL f )k = φk , f  = √ L −L/2

50

1 Mathematical Preliminaries

Proof By (1.90), one can define a mapping FL : L2 (IL ) → 2 (ΓL ) by (1.92). Then (1.90) implies that FL is isometric14 and hence preserves the inner product. To  show that FL is surjective, let u ∈ 2 (ΓL ). Then k∈ΓL |uk |2 < ∞. Hence, by Lemma 1.3, fu := uk φk k∈ΓL

exists in L2 (IL ). Then φk , fu  = uk . Hence FL fu = u. Therefore FL is surjective. Thus FL is unitary. The uniqueness of FL is easily shown.  We call the unitary operator FL : L2 (IL ) → 2 (ΓL ) the one-dimensional discrete Fourier transform. It follows from (1.92) that (FL φk )k = δkk ,

k, k ∈ ΓL .

(1.93)

By abuse of notation, we write fˆ = FL f,

f ∈ L2 (IL ).

There will be no danger of confusion, provided that one is aware of the space to which f belongs. The inverse discrete Fourier transform FL −1 : 2 (ΓL ) → L2 (IL ) is given by (FL −1 u)(x) =

k∈ΓL

1 uk φk (x) = √ uk eikx , L k∈Γ

u ∈ 2 (ΓL ),

(1.94)

L

in the sense of L2 -convergence.

1.14 Momentum Operator with Periodic Boundary Condition We define a linear operator pˆ L on L2 (IL ) as follows: ⎧ ⎫ ⎨ ⎬ |k fˆk |2 < ∞ , D(pˆL ) := f ∈ L2 (IL )| ⎩ ⎭ pˆ L f := FL−1 (k fˆ) =



(1.95)

k∈ΓL

k fˆk φk ,

f ∈ D(pˆL ),

(1.96)

k∈ΓL

linear operator T from a Hilbert space H to a Hilbert space K is called an isometry if D(T ) = H and T Ψ K = Ψ H , Ψ ∈ H . An isometry T : H → K preserves the inner product: T Ψ, T ΦK = Ψ, ΦH , Ψ, Φ ∈ H . This follows from the polarization identity.

14 A

1.14 Momentum Operator with Periodic Boundary Condition

51

where fˆk := (FL f )k . By (1.93), we have pˆ L φk = kφk ,

k ∈ ΓL .

(1.97)

Namely, φk is an eigenvector of pˆL with eigenvalue k. Theorem 1.34 The operator pˆL is self-adjoint and σ (pˆL ) = σp (pˆ L ) = ΓL ,

(1.98)

where the multiplicity of eigenvalue k of pˆL is one. Proof By the definition of pˆ L , we have FL pˆL FL−1 = Mk ,

(1.99)

where the right hand side is the multiplication operator on 2 (ΓL ) by k. The operator Mk is self-adjoint. Hence pˆL is self-adjoint. As we have seen above, {φk }k∈ΓL is a set of eigenvectors of pˆ L and is a CONS of L2 (IL ). Thus, by Theorem 1.16, (1.98) follows.  One can identify pˆL with an differential operator. To show this, we recall a notion. In general, a function f on the interval [a, b] is said to be absolutely continuous if there  xexists an integrable function g on [a, b] and a constant c ∈ R such that f (x) = a g(y)dy + c. In this case, it is easy to see that f is continuous on [a, b] and differentiable at a.e. x ∈ [a, b] and f (x) = g(x), a.e.x. We denote by AC([a, b]) (or AC[a, b]) the set of absolutely continuous Borel measurable functions on [a, b]. Theorem 1.35   2 2 D(pˆL ) = f ∈ L (IL ) ∩ AC(IL )| f (−L/2) = f (L/2), f ∈ L (IL ) (1.100) pˆ L f = −if ,

f ∈ D(pˆL ).

(1.101)

Proof Let the set on the right hand side of (1.100) be D. Let f ∈ D first. Then, by integration by parts, we have for all k ∈ ΓL 

φk , f = ik φk , f  = ik fˆk .

Hence k∈ΓL

|k fˆk |2 =

 | φk , f |2 = f 2 < ∞. k∈ΓL

(1.102)

52

1 Mathematical Preliminaries

Therefore f ∈ D(pˆL ). Moreover, by using (1.102), we have    φk , pˆL f = FL φk , FL pˆ L f = k fˆk = φk , −if . Since this holds for all k ∈ ΓL and {φk }k∈ΓL is a CONS of L2 (IL ), it follows that pˆ L f = −if . Thus D ⊂ D(pˆL ) and pˆL f = −if for all f ∈ D. Next, let f ∈ D(pˆL ) and v = {vk }k∈ΓL with vk := ik fˆk . Then v ∈ 2 (ΓL ). Hence, by the Cauchy–Schwarz inequality,



|fˆk | =

k∈ΓL \{0}

k∈ΓL \{0}

! " 1 " · |vk | ≤ # |k|

k∈ΓL \{0}

1 k2



|vk |2 < ∞.

k∈ΓL

Hence f (x) =



fˆk φk (x)

k∈ΓL

converges uniformly in x ∈ IL and hence f is a continuous function on IL . It is easy to see that f (−L/2) = f (L/2).  It follows from the condition v ∈ 2 (ΓL ) and Lemma 1.3 that g := k∈ΓL vk φk is in L2 (IL ). In particular, by the Schwarz inequality, g is integrable on IL . Hence one can define an absolutely continuous function  u(x) :=

x

−L/2

 g(y)dy = χ[−L/2,x], g L2 (I ) , L

x ∈ IL ,

where χ[a,b] is the characteristic function of [a, b]. By the unitarity of FL ,  u(x) = FL χ[−L/2,x], FL g 2 (Γ

L)

 = FL χ[−L/2,x], v 2 (Γ ) . L

We have for all k ∈ ΓL  x 1 (FL χ[−L/2,x])∗k vk = ik fˆk √ eiky dy L −L/2  x 1 d iky ˆ = fk √ e dy L −L/2 dy = fˆk (φk (x) − φk (−L/2)). x Hence u(x) = f (x) − f (−L/2). Therefore f (x) = −L/2 g(y)dy + f (−L/2), implying that f ∈ AC(IL ), f (x) = g(x), a.e.x ∈ IL and f ∈ L2 (IL ). Hence f ∈ D. Thus D(pˆL ) ⊂ D. 

1.14 Momentum Operator with Periodic Boundary Condition

53

For a continuous function f on IL , condition f (L/2) = f (−L/2) is called the periodic boundary condition (PBC) on f . As is seen from (1.100), each element of D(pˆL ) obeys the PBC. Based on this property, we call the self-adjoint operator pˆ L the momentum operator with PBC on L2 (IL ) . For later use, we compute the unitary group {e−it pˆ L }t ∈R. We first note that, for each x ∈ R, there exists a unique integer nx ∈ Z such that x −nx L ∈ IL . Concretely, nx is given as follows: ⎧ ⎪ −n ⎪ ⎪ ⎪ ⎪ ⎨ nx = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ n

if −

(2n+1)L 2

≤ x < − (2n−1)L (n ∈ N) 2

if x ∈ IL if

(2n−1)L 2

. 0, k ∈ R √ ⎪ ⎪ ⎨ 2Em (k)(m + Em (k)) 0 1 ⎪ ⎪ 1 k ⎪ ⎪ ˆ 3 \ {0} ; m = 0, k ∈ R ⎩ √ 1 + βα · |k| 2

.

(4.86) By direct computations using the anti-commutation relations (4.77), one can show ˆ 3 (in the case m > 0) or k ∈ R ˆ 3 \ {0} (in the case m = 0), uD (k) that, for each k ∈ R is a unitary matrix with 1 uD (k)−1 = uD (k)∗ = √ (m + Em (k) + α · kβ). 2Em (k)(m + Em (k))

(4.87)

By direct computations again, we obtain uD (k)hD (k)uD (k)−1 = βEm (k).

(4.88)

216

4 Time Operators

Hence, letting UD := uD F3 , we obtain UD HD UD−1 = βEm .

(4.89)

Since βEm is self-adjoint, it follows that HD is self-adjoint with D(HD ) = F3−1 D(Em ) = ∩3j =1 D(pˆj ). We now work with the standard representation for (α, β) so that β is diagonal (see (4.79)). Then (4.89) implies that ⎛

UD HD UD−1

Em ⎜ 0 =⎜ ⎝ 0 0

⎞ 0 0 0 0 ⎟ Em 0 ⎟ = Em ⊕ Em ⊕ (−Em ) ⊕ (−Em ). 0 −Em 0 ⎠ 0 0 −Em (4.90)

ˆ is absolutely continuous with Note that the multiplication operator Em on L2 (R) σ (Em ) = σac (Em ) = [m, ∞). Hence, by spectral properties of direct sum operators, we conclude that HD is absolutely continuous with σ (HD ) = σac (HD ) = (−∞, −m] ∪ [m, ∞). Let VD := F3−1 UD = F3−1 uD F3 . Then VD is unitary on HD . We have F3−1 Em F3 = HR,m , (3)

(3)

where HR,m is defined by (4.72) with d = 3. Hence, by (4.90), we obtain VD HD VD−1 = HR,m ⊕ HR,m ⊕ (−HR,m ) ⊕ (−HR,m ). (3)

(3)

(3)

(3)

(4.91)

4.5 Strong Time Operators

217 (3)

By Theorem 4.25(i), TR,m,j (j = 1, 2, 3) is a strong time operator of HR,m . Hence, for each j = 1, 2, 3, the operator 5 4 TD,j := VD−1 TR,m,j ⊕ TR,m,j ⊕ (−TR,m,j ) ⊕ (−TR,m,j ) VD

(4.92)

is a strong time operator of HD . Moreover, we obtain the following theorem: Theorem 4.26 Let j = 1, 2, 3. (i) The operator TD,j is a non-self-adjoint symmetric operator with σ (TD,j ) = C.

(4.93)

(ii) Write HD,m := HD ,

TD,m,j := TD,j .

Then {(TD,m,j , HD,m )}m≥0 is a family of mutually inequivalent weak Weyl representation of the CCR with one degree of freedom. Proof (i) The first half of this part follows from (4.92) and the fact that a direct sum of non-self-adjoint symmetric operators is a non-self-adjoint symmetric operator. The definition (4.92) implies that σ (TD,m,j ) = σ (TR,m,j ) ∪ σ (−TR,m,j ). By (4.75), we have σ (TR,m,j ) ∪ σ (−TR,m,j ) = Π + ∪ Π − = C. Thus (4.93) holds. (ii) By (4.91), σ (HD,m ) = σ (HD,m ) for m = m (m, m ≥ 0). Hence, if m = m , then (TD,m,j , HD,m ) is inequivalent to (TD,m ,j , HD,m ).  The spectral property (4.93) may be interesting. Theorem 4.26(ii) shows that the set of Dirac particle masses has a meaning as the index set for a family of mutually inequivalent weak Weyl representations with one degree of freedom consisting of pairs of a free Dirac Hamiltonian and a strong time operator of it.

218

4 Time Operators

4.5.13 A Structure Generating Pairs of a Hamiltonian and a Strong Time Operator In Theorem 4.2, we have seen a structure which generates pairs of a Hamiltonian and a time operator from one pair of a Hamiltonian and a time operator. In this subsection, we show that a similar structure exists for strong time operators. 1 (R) the set of real-valued Borel measurable functions f on R We denote by BR satisfying the following: there exists a closed Borel set Nf of Lebesgue measure 0 (i.e. |Nf | = 0) such that f is continuously differentiable on R \ Nf . It follows that, 1 (R), f (λ) is finite and differentiable for a.e. λ ∈ R with respect to for all f ∈ BR the one-dimensional Lebesgue measure. Let H be a self-adjoint operator on a Hilbert space H and T be a strong time operator of H , i.e., T ∈ Ts (H ). Since H is absolutely continuous (Theorem 4.8), it 1 (R) follows that, for all f ∈ BR EH ({λ ∈ R| |f (λ)| = ∞}) = 0,

EH ({λ ∈ R||f (λ)| = ∞}) = 0.

Hence, by functional calculus for H , f (H ) and f (H ) are self-adjoint. 1 (R): We introduce a subset of BR 1 (R)|g (H ) is injective and CT1 ,H (R) := {g ∈ BR

D(T g (H )−1 ) ∩ D(g (H )−1 T ) is dense}.

(4.94)

Then, for each g ∈ CT1 ,H (R), the operator Tg :=

. 1- T g (H )−1 + g (H )−1 T 2

(4.95)

is a symmetric operator. Theorem 4.27 ([34, 92]) Let T ∈ Ts (H ) and suppose that T is closed. Then, for all g ∈ CT1 ,H (R), Tg is a strong time operator of g(H ), i.e., Tg ∈ Ts (g(H )). Proof We have . 1 - itg(H ) e T g (H )−1 e−itg(H ) + eitg(H )g (H )−1 T e−itg(H ) 2 . 1 - itg(H ) −itg(H ) = e Te g (H )−1 + g (H )−1 eitg(H )T e−itg(H ) . 2

eitg(H )Tg e−itg(H ) =

Theorem 2.23(ii) implies that, for all t ∈ R, T e−itg(H ) ⊃ e−itg(H )T + te−itg(H )g (H ), where we have used the assumption that T is closed. Hence eitg(H )T e−itg(H ) ⊃ T + tg (H ).

4.5 Strong Time Operators

219

Hence eitg(H )T e−itg(H )g (H )−1 ⊃ T g (H )−1 + t and g (H )−1 eitg(H )T e−itg(H ) ⊃ g (H )−1 T + t. Therefore eitg(H )Tg e−itg(H ) ⊃ Tg + t. Replacing t with −t, we have e−itg(H )Tg eitg(H ) ⊃ Tg − t. This implies that Tg + t ⊃ eitg(H )Tg e−itg(H ). Hence it follows that eitg(H )Tg e−itg(H ) = T + t. Thus Tg is a strong time operator of g(H ).  Time operators of the Schrödinger operator HNR and the relativistic Schrödinger operator HR can be understood in a natural way in view of Theorem 4.27: ˆ Example 4.9 Consider the case where H = L2 (R) and H = pˆ and T = q. 2

λ (1) Let g(λ) = 2m , λ ∈ R with m > 0. Then g (λ) = λ/m. Hence g (H ) = p/m ˆ is injective. Hence D(T g (H )−1 ) ∩ D(g (H )−1 T ) = D(qˆ pˆ −1 ) ∩ D(pˆ −1 q) ˆ is 1 (R). In this case we have g(H ) = pˆ 2 /2m = dense in L2 (R). Hence g ∈ Cq, ˆ pˆ HNR and

Tg =

m (qˆ pˆ −1 + pˆ −1 q). ˆ 2

This is just the Aharonov–Bohm time operator TAB (see (4.3)). Hence, by Theorem 4.27, √ TAB is a strong time operator of HNR . (2) Let g(λ) = λ2 + m2 , λ ∈ R. Then g (λ) = λg(λ)−1 . Hence g (H ) = pH ˆ R−1 1 and it is injective. It follows that g ∈ Cq, (R). In this case g(H ) = HR . Hence ˆ pˆ Tg =

1 (HR pˆ −1 qˆ + qˆ pˆ −1 HR ) = TR . 2

Hence TR is a strong time operator of HR . To make the dependence of Tg on (T , H ), we write Tg = τ (g, T , H ).

220

4 Time Operators

Using Theorem 4.27, one can define a sequence {(H , (Tn , Hn ))}∞ n=0 of weak Weyl representations with degree one as follows: T0 := T ,

H0 := H,

T1 := τ (g1 , T0 , H0 ),

H1 := g1 (H0 )

(g1 ∈ CT10 ,H0 (R)),

T2 := τ (g2 , T1 , H1 ),

H2 := g2 (H1 )

(g2 ∈ CT11 ,H1 (R)),

.. .

.. .

Tn := τ (gn , Tn−1 , Hn−1 ), .. .

Hn := gn (Hn−1 )

(gn ∈ CT1n−1 ,Hn−1 (R)),

.. .

This may be an interesting structure. The following is an example on an abstract level which may be interesting: Example 4.10 Let T be a closed strong time operator of H . Consider the case where 1 (R) with g (λ) = 1/λ, g(λ) = log |λ|, λ ∈ R \ {0} and g(0) := 0. Then g ∈ BR λ ∈ R \ {0}. Hence g (H ) = H −1 (note that σp (H ) = ∅ by the purely absolute continuity of H and hence H is injective in particular) and g(H ) = log |H |. Hence Tg =

1 (T H + H T ). 2

By Proposition 4.5, D(Tg ) = D(T H ) ∩ D(H T ) is dense. Hence g ∈ CT1 ,H (R). Therefore Tg is a strong time operator of log |H |. For further analysis of this model, see [34].

4.5.14 Decay-in-Time of Transition Probabilities Let H be a self-adjoint operator on H . In the context of quantum mechanics in which H denotes a Hamiltonian, the quantity   PΨ,Φ (t) := | Φ, e−it H Ψ |2

(4.96)

with Ψ = 1 = Φ is called the transition probability from a state Ψ at time 0 to a state Φ at time t (the probability that, when the initial state vector is Ψ , a state vector Φ is found at time t). If H is absolutely continuous, then, by a general theorem (see Theorem D.2),  lim

t →±∞

 Φ, e−it H Ψ = 0,

Ψ, Φ ∈ H .

(4.97)

4.5 Strong Time Operators

221

Hence the transition probability PΨ,Φ (t) decays as the time t tends to infinity. It may be interesting to know the decay order of it. In this subsection, we show that, if H has a strong time operator T , then T controls the decay order (recall that, in this case, by Theorem 4.8, limt →±∞ PΨ,Φ (t) = 0). This shows a physical role of a strong time operator. For each n and Ψ, Φ ∈ D(T n ), we define a finite sequence {d (Φ, Ψ )}n=1 by the following recurrence relations: d1 (Φ, Ψ ) := T Φ Ψ + Φ T Ψ , d (Φ, Ψ ) := T  Φ Ψ + Φ T  Ψ +

−1

 Ck dk (Φ, T

−k

Ψ ),

k=1

 = 2, . . . , n (n ≥ 2),

(4.98)

where  Ck is the binomial coefficient with respect to (, k) with  ≥ k:  Ck

:=

! . ( − k)!k!

(4.99)

Theorem 4.28 Assume that H has a strong time operator T . Let n ∈ N. Then, for all Ψ, Φ ∈ D(T n ) and t ∈ R \ {0},   dn (Φ, Ψ ) | Φ, e−it H Ψ | ≤ . |t|n

(4.100)

Proof We prove (4.100) by induction in n. By (4.39), we have for all t ∈ R te−it H Ψ = (T e−it H − e−it H T )Ψ. Hence

      t Φ, e−it H Ψ = T Φ, e−it H Ψ − eit H Φ, T Ψ .

Therefore, by the Schwarz inequality and the unitarity of e−it H , we obtain   |t| | Φ, e−it H Ψ | ≤ T Φ Ψ + Φ T Ψ . Hence (4.100) with n = 1 follows. We next suppose that (4.100) holds with n replaced by k = 1, . . . , . Then we need only to prove (4.100) with n =  + 1. So let Ψ, Φ ∈ D(T +1 ). It follows from (4.39) and the relation e−it H eit H = I for all t ∈ R that eit H T +1 e−it H = (T + t)+1 = t +1 +

 k=0

+1 Ck T

+1−k k

t .

222

4 Time Operators

Hence        −it H +1−k t +1 Φ, e−it H Ψ = T +1 Φ, e−it H Ψ − T Ψ tk. +1 Ck Φ, e k=0

Hence   |t|+1 | Φ, e−it H Ψ | ≤ T +1 Φ Ψ + Φ T +1 Ψ

+



+1 Ck |

  Φ, e−it H T +1−k Ψ ||t|k .

k=1

Note that Φ, T +1−k Ψ ∈ D(T k ) (k = 1, . . . , ). Hence, by the induction hypothesis,   | Φ, e−it H T +1−k Ψ ||t|k ≤ dk (Φ, T +1−k Ψ ). Therefore    +1−k t +1 | Φ, e−itH Ψ | ≤ T +1 Φ Ψ + Φ T +1 Ψ + Ψ) +1 Ck dk (Φ, T k=1

= d+1 (Φ, Ψ ).

Thus (4.100) with n =  + 1 holds.



Remark 4.17 For all a ∈ R, Ta := T − a also is a strong time operator of H . Hence Theorem 4.28 holds with T replaced by Ta . It may be noteworthy that the constant dn (Φ, Ψ ) in (4.100) depends only on T , Φ and Ψ , not depending on H . We say that a vector Ψ ∈ H has regularity of order n with respect to T if Ψ ∈ D(T n ). Theorem 4.28 tells us that there is a correspondence between the order of regu larity of Φ, Ψ with respect to T and the order of decay-in-time of | Φ, e−it H Ψ |. This is a very interesting structure, showing a role of strong time operators. In the case where Ψ and Φ are unit vectors, we have d1 (Φ, Ψ ) = T Φ + T Ψ , d (Φ, Ψ ) = T  Φ + T  Ψ +

(4.101) −1

 Ck dk (Φ, T

−k

Ψ ),

k=1

 = 2, . . . , n (n ≥ 2).

(4.102)

4.5 Strong Time Operators

223

Theorem 4.28 immediately yields an estimate for PΨ,Φ (t): Corollary 4.7 Assume that H has a strong time operator T . Let n ∈ N. Then, for all Ψ, Φ ∈ D(T n ) with Ψ = Φ = 1and t ∈ R \ {0}, PΨ,Φ (t) ≤

dn (Φ, Ψ )2 . t 2n

(4.103)

In particular, for all Ψ ∈ D(T ) with Ψ = 1,   4(ΔT )2Ψ | Ψ, e−it H Ψ |2 ≤ . t2

(4.104)

Proof Inequality (4.103) follows from (4.100). By Remark 4.17, (4.103) holds with T replaced by T − Ψ, T Ψ . Then (4.103) with n = 1 and Φ = Ψ yields (4.104).   The quantity | Ψ, e−it H Ψ |2 is called the survival probability of the state vector Ψ at time t. It is interesting that the uncertainty (ΔT )Ψ of T appears on the right hand side of (4.104) (cf. [111] also).

4.5.15 Existence of Strong Time Operators As we have already seen in Theorem 4.8, every self-adjoint operator H on a Hilbert space which has a strong time operator is absolutely continuous. A natural question then is: does an absolutely continuous self-adjoint operator H have a strong time operator? In this subsection, we give a partial answer to this question. A vector Ψ ∈ C ∞ (A) (see (1.34)) is called a cyclic vector of A if span{Ψ, An Ψ | n ∈ N} is dense in H , i.e., Ψ is a cyclic vector for {An |n ∈ N} (see Definition 2.4). In this case we say that A has a cyclic vector Ψ . Let H be a self-adjoint operator on a Hilbert space H and EH (·) be the spectral measure of H . Then, for each unit vector Ψ ∈ H , a probability measure μΨ on R is defined by μΨ (B) := EH (B)Ψ 2 ,

B ∈ B,

where B is the family of Borel sets of R. We define a function X on R by X(λ) := λ,

λ ∈ R.

We note the following fact: Lemma 4.10 Assume that H is separable. Suppose that H has a cyclic vector Ψ0 . Then there exists a unitary operator U from H to L2 (R, dμΨ0 ) such that U Ψ0 =

224

4 Time Operators

1 and U H U −1 = MX , the multiplication operator by the function X acting in L2 (R, dμΨ0 ). Moreover, the subspace span{eit X |t ∈ R} is dense in L2 (R, dμΨ0 ). Proof The first half of the lemma follows from an easy extension of Lemma 1 in [130, §VII.2] to the case of unbounded self-adjoint operators [16, Theorem 1.8]. To prove the second half of the lemma, we note that, by the cyclicity of Ψ0 for {H n |n ∈ N}, span{Ψ0 , H n Ψ0 |n ∈ N} is dense in H . By the functional calculus for H , we have 0 lim (−i)n

t →0

eit H − 1 t

1n Ψ0 = H n Ψ0 .

Hence it follows that span{eit H Ψ0 |t ∈ R} is dense in H . By the first half of the lemma, we have U eit H Ψ0 = eit X . Hence span{eit X |t ∈ R} is dense in L2 (R, dμΨ0 ).  Let Ψ ∈ H . If μΨ is absolutely continuous with respect to the Lebesgue measure on R, then  we denote by ρΨ the Radon-Nykodým derivative of μΨ : ρΨ ≥ 0 and μΨ (B) = B ρΨ (λ)dλ, B ∈ B. We introduce a class of self-adjoint operators on H . Definition 4.3 We say that a self-adjoint operator H on H is in the class S0 (H ) if it satisfies the following: (i) H is absolutely continuous. (ii) H has a cyclic vector Ψ0 such that ρΨ0 is differentiable on R and  lim ρΨ0 (λ) = 0,

λ→±∞

ρΨ 0 (λ)2

ρΨ0 (λ)>0

ρΨ0 (λ)

dλ < ∞.

Let H be separable and H ∈ S0 (H ) with a cyclic vector Ψ0 and ⎧ ⎨ ρΨ0 (λ) for ρΨ0 (λ) > 0 W0 (λ) := ρΨ0 (λ) . ⎩ 0 for ρΨ0 (λ) = 0 Then we define an operator Y on L2 (R, dμΨ0 ) as follows: D(Y ) := span{eit X |t ∈ R},

Y := i

i d + W0 . dλ 2

Lemma 4.11 The operator Y is a symmetric operator.

4.5 Strong Time Operators

225

Proof By Lemma 4.10, D(Y ) is dense in L2 (R, dμΨ0 ). Using (ii) in Definition 4.3 and integration by parts, we see that, for all f, g ∈ D(Y ), f, Y gL2 (R,dμΨ ) = Yf, gL2 (R,dμΨ ) . 0

0



Hence Y is a symmetric operator. Lemma 4.12 The operator Y is a strong time operator of MX .

Proof It is obvious that, for all t ∈ R, eit MX D(Y ) ⊂ D(Y ) and hence eit MX D(Y ) = D(Y ). Let f (λ) = eisλ , s ∈ R, λ ∈ R. Then, using the fact that if (λ) = −sf (λ), we see that 0 1 i d + W0 e−i(t −s)λ = tf (λ) + (Yf )(λ). (eit MX Y e−it MX f )(λ) = eit λ i dλ 2 Hence eit MX Y e−it MX = Y + t, implying that Y is a strong time operator of MX .  Theorem 4.29 Assume that H is separable and that H ∈ S0 (H ). Then H has a strong time operator TH given by TH := U −1 Y U.

(4.105)

Proof We have U −1 MX U = H . Then it follows from Proposition 4.7 that TH defined by (4.105) is a strong time operator of H .  It may be instructive to see how the above structure which gives a strong time operator is realized in a simple case: Example 4.11 Consider the case where H = L2 (R) and H = q. ˆ Let f0 (x) :=

- ω .1/4 π

e−ωx

2 /2

with ω > 0. Then f0 = 1. It is well known that span{f0 , qˆ n f0 |n ∈ N} is dense in L2 (R). Hence f0 is a cyclic vector of q. ˆ The spectral measure of qˆ is given by Eqˆ (B) = χB , B ∈ B. Hence 

Eqˆ (B)f =

|f (x)|2 dx.

2

B

Hence qˆ is absolutely continuous and ρf (λ) = |f (λ)|2 ,

λ ∈ R.

226

4 Time Operators

In particular, ρf0 (λ) =

- ω .1/2 π

e−ωλ > 0, 2

ρf 0 (λ) = −2ωλρf0 (λ).

Hence lim ρf0 (λ) = 0,

λ→±∞

 ρ (λ)2 f0 R

ρf0 (λ)

 dλ = 4ω2

R

x 2 f0 (x)2 dx < ∞.

Therefore qˆ ∈ S0 (L2 (R)). In the present example, the unitary operator U : L2 (Rx ) → L2 (Rλ , dμf0 ) takes the form Uf0 = 1,

U qU ˆ −1 = λ.

Let f (x) = x n f0 (x), x ∈ R, n ∈ N. Then (Uf )(λ) = λn , λ ∈ R. Hence (U Dx f )(λ) = (Dλ − ωλ)Uf. In the present case, W0 (λ) = −2ωλ. Hence Y =i

d − iωλ dλ

Therefore it follows that U iDx U −1 = Y on span{λn |n ∈ Z+ }. Thus Tqˆ = iDx with D(Tqˆ ) = span{x n f0 |n ∈ Z+ } is a strong time operator of q. ˆ It is well known that Tqˆ is essentially self-adjoint on span{x n f0 |n ˆ Hence −pˆ is a self-adjoint strong time operator of q. ˆ ∈ Z+ } and Tqˆ = −p. Each element of S0 (H ) is multiplicity free (see [130, p. 232, Definition]). In the case where H is not necessarily multiplicity free, we have the following theorem: Theorem 4.30 Let H be a self-adjoint operator on a separable Hilbert space H . Suppose that there exist mutually orthogonal closed subspaces H1 , . . . , HN of H (N ∈ N or N = ∞) such that H = ⊕N n=1 Hn and H is reduced by each Hn and the reduced part Hn of H to Hn is in the class S0 (Hn ). Then H has a strong time operator T given by T = ⊕N n=1 THn , where THn is defined by (4.105) with H replaced by Hn .

4.5 Strong Time Operators

227

Proof We have H = ⊕N n=1 Hn . By Theorem 4.29, each Hn has a strong time operator THn . Then it is easy to see that T := ⊕N n=1 THn is a strong time operator of H .  Remark 4.18 There is a general structure theorem associated with a self-adjoint operator (see, e.g., [130, p. 226, Lemma 2] for the case of a bounded self-adjoint operator or [16, p. 33, Theorem A.2] for the case of a not necessarily bounded selfadjoint operator). For any self-adjoint operator H on a separable Hilbert space H , there exists a direct sum decomposition H = ⊕N n=1 Hn with N ∈ N or N = ∞ such that (i) H is reduced by each Hn ; (ii) the reduced part of H to Hn has a cyclic vector Ψn . But one cannot apply this structure theorem to show existence of a strong time operator of H , since it is unclear if Ψn can be taken such that Hn is in the class S0 (Hn ).

4.5.16 Construction of Strong Time Operators of a Self-adjoint Operator from Those of Another Self-adjoint Operator We consider two self-adjoint operators H and H acting in Hilbert spaces H and H respectively. If H = H , then H = H + (H − H ) on D(H ) ∩ D(H ) and hence H can be regarded as a perturbation of H . One can use quantum scattering theory to construct a strong time operator of H ac , the absolutely continuous part of H (see Appendix D), from a strong time operator of Hac . This is a basic idea for the subject of this subsection. We denote by Pac (H ) the orthogonal projection onto the absolutely continuous subspace Hac (H ) of H . Lemma 4.13 Assume the following: (A.1) The wave operator

W := s- lim eit H J e−it H Pac (H ) t →∞

exists, where s- lim means strong limit and J : H → H is a bounded linear operator. (A.2) limt →∞ J e−it H Pac (H )Ψ = Pac (H )Ψ , Ψ ∈ H . (A.3) Ran(W ) = Hac (H ). Let U := W  Hac (H ). Then U is a unitary operator from Hac (H ) to Hac (H ) such that Hac = U Hac U −1 .

228

4 Time Operators

Proof This lemma is a well-known fact in quantum scattering theory (see, e.g., [5, 100, 132]).  Theorem 4.31 Assume (A.1)–(A.3) in Lemma 4.13. Suppose that Hac has a strong . time operator T . Then T := U T U −1 is a strong time operators of Hac Proof This follows from Lemma 4.13 and an application of Proposition 4.7.



Remark 4.19 As is easily seen, one can replace condition “ t → ∞ ” in (A.1) and (A.2) with “ t → −∞ ”. Theorem 4.31 can be used to construct strong time operators of H ac from those of Hac . Remark 4.20 To construct a (not necessarily strong) time operator of a self-adjoint operator H itself (not only a time operator of Hac ), one may use the following formulas (see (D.2) and (D.3)): H = Hac (H ) ⊕ Hsc (H ) ⊕ H(H ),

H = Hac ⊕ Hsc ⊕ Hp .

If Hac , Hsc and Hp have time operators Tac , Tsc and Tp respectively, then T := Tac ⊕ Tsc ⊕ Tp is a time operator of H . Example 4.12 We set (d)

H0 := HNR , (d)

where HNR is defined by (4.62). For each r ∈ (0, ∞), we denote by χr the characteristic function of the set {x ∈ Rd | |x| ≥ r}. Let V : Rd → R ∪ {±∞} be a Borel measurable function (potential) satisfying the following: (V.1) V is H0 -bounded (see (1.3)) and V H0 < 1, where V H0 is the relative bound of V with respect to H0 (see (1.4)). By (V.1), one can define a function h on (0, ∞) by h(r) := V (H0 + 1)−1 χr ,

r ∈ (0, ∞),

where χr is regarded as the multiplication operator by the function χr . ∞ (V.2) 0 h(r)dr < ∞. This type of potential V is called an Enss potential.13 Condition (V.1) implies that there exist constants a ∈ [0, 1) and b ≥ 0 such that

Vf ≤ a H0 f + b f ,

f ∈ D(H0 ).

13 More generally, an Enss potential is defined to be a (not necessarily multiplication) symmetric operator V on L2 (Rd ) satisfying (V.1) and (V.2).

4.6 Other Classes of Time Operators

229

Hence, by the Kato–Rellich theorem (Theorem 1.15), the symmetric operator H := H0 + V

(4.106)

is self-adjoint with D(H ) = D(H0 ) and bounded from below. The following theorem is known: Theorem 4.32 (Enss’s Theorem) The wave operators W± := s- limt →±∞ eit H e−it H0 exist and Ran (W± ) = Hac (H ). Moreover, σsc (H ) = ∅, the only possible (finite) accumulation point for σp (H ) is 0 and any non-zero eigenvalue of H has finite multiplicity. Proof See textbooks on quantum scattering theory, e.g., [100, Theorem 4.12] or [132, Theorem XI.112].  By Theorem 4.32, W± are unitary operators from Hac (H0 ) = L2 (Rd ) to Hac (H ). We already know that the d-dimensional Aharonov–Bohm time operator (d) TAB is a strong time operator of H0 (see Theorem 4.21). Hence, by Theorem 4.31, (d) −1 T± := W± TAB W±

(4.107)

acting in Hac (H ) are strong time operators of Hac . Therefore we can apply Theorem 4.28 to obtain estimates for orders of decay-in-time of transition probabilities with respect to H : for all t ∈ R \ {0},   d ± (g, f )2 | g, e−it H f |2 ≤ n 2n , t

f, g ∈ D(T±n ),

n ∈ N,

(4.108)

where dn+ (g, f ) (resp. dn− (g, f )) is the number dn (Φ, Ψ ) defined by (4.98) with (T , Φ, Ψ ) replaced by (T+ , g, f ) (resp. (T− , g, f )).

4.6 Other Classes of Time Operators So far we have considered two kinds of time operators. In fact, there other classes of time operators. In this section we briefly describe them. Let H be a self-adjoint operator on a Hilbert space H . Definition 4.4 A self-adjoint operator T on H is called an ultra-strong time operator of H if (T , H ) is a Weyl representation of the CCR with one degree of freedom, i.e., it satisfies eit T eisH = e−it s eisH eit T ,

s, t ∈ R.

230

4 Time Operators

We denote by Tus (H ) the set of ultra-strong time operators of H . Proposition 4.10 An ultra-strong time operator of H is a strong time operator of H . Proof This follows from the fact that every Weyl representation is a weak Weyl representation (see Theorem 2.20).  By this proposition, we have Tus(H ) ⊂ Ts (H ).

(4.109)

Proposition 4.11 A self-adjoint strong time operator of H is an ultra-strong time operator of H . Proof Let T be a self-adjoint strong time operator of H . Then, by the proof of Theorem 4.10, (T , H ) is a Weyl representation with one degree of freedom. Hence T is an ultra-strong time operator of H .  Example 4.13 The time operator TF in Example 4.1 is an ultra-strong time operator of H F , because TF is a self-adjoint strong time operator of H F . Definition 4.5 A symmetric operator T on H is called a weak time operator of H if there is a non-zero subspace Dw ⊂ D(T ) ∩ D(H ) such that T Ψ, H Φ − H Ψ, T Φ = i Ψ, Φ ,

Ψ, Φ ∈ Dw .

(4.110)

We call Eq. (4.110) the weak CCR for (T , H ) on Dw . The subspace Dw is called a weak CCR-domain for (T , H ). We denote by Tw (H ) the set of weak time operators of H . It is easy to see that a time operator of H is a weak time operator of H : T(H ) ⊂ Tw (H ).

(4.111)

Proposition 4.12 Let T ∈ Tw (H ) with a weak CCR-domain Dw . Then, for all unit vectors Ψ ∈ Dw , time–energy uncertainty relation (4.10) holds. Proof Apply Lemma 2.3.



Finally, we define a notion of a generalized time operator which is weaker than that of a weak time operator. Definition 4.6 ([33]) Let D1 and D2 be non-zero subspaces of H . A sesquilinear form t: D1 × D2 → C with form domain Q(t) = D1 × D2 (see Sect. 1.11) is called an ultra-weak time operator of H if there exists a non-zero subspace D and E of D1 ∩ D2 such that the following hold: (i) E ⊂ D(H ) ∩ D. (ii) (symmetry on D) t(Φ, Ψ )∗ = t(Ψ, Φ), Φ, Ψ ∈ D.

4.6 Other Classes of Time Operators

231

(iii) (ultra-weak CCR) H E ⊂ D1 and, for all Φ, Ψ ∈ E , t(H Ψ, Φ)∗ − t(H Φ, Ψ ) = i Φ, Ψ  . We call E an ultra-weak CCR-domain for (t, H ) and D a symmetry domain of t. We remark that, as is seen in the above definition. An ultra-weak time operator is a sesquilinear form, not an operator. But, by abuse of terminology, we use the name “ultra-weak time operator”. For each T ∈ Tw (H ) with Dw being a weak CCR-domain for (T , H ), one can define a sesquilinear form tT : H × D(T ) → C by tT (Φ, Ψ ) := Φ, T Ψ  ,

Φ ∈ H , Ψ ∈ D(T ).

Proposition 4.13 The sesquilinear form tT is an ultra-weak time operator of H with Dw being an ultra-weak CCR-domain for (tT , H ) and D(T ) a symmetry domain. Proof It is obvious that Dw ⊂ D(H ) ∩ D(T ). We have for all Φ, Ψ ∈ D(T ) tT (Φ, Ψ )∗ = T Ψ, Φ = Ψ, T Φ = tT (Ψ, Φ). Hence the symmetry of tT on D(T ) holds. Moreover, for all Ψ, Φ ∈ Dw tT (H Ψ, Φ)∗ − tT (H Φ, Ψ ) = T Φ, H Ψ  − H Φ, T Ψ  = i Φ, Ψ  . Hence tT is an ultra-weak time operator of H with Dw being an ultra-weak CCRdomain and D(T ) a symmetry domain.  By Proposition 4.13, there is a mapping f from Tw (H ) to Tuw (H ) such that f (T ) = tT ,

T ∈ Tw (H ).

It is easy to see that f is injective. In this sense, we write Tw (H ) ⊂ Tuw. Summing up the above considerations, we have found five classes of time operators with the following inclusion relations: Tus(H ) ⊂ Ts (H ) ⊂ T(H ) ⊂ Tw (H ) ⊂ Tuw (H ). For further discussions of ultra-weak time operators, we refer the reader to [33]. In Sect. 4.5.14, we have seen that a strong time operator is related to orders of decay-in-time of transition probabilities. But we remark that it is still unclear if the other time operators play any proper roles in relation to physics.

232

4 Time Operators

4.7 Generalized Time Operators The notions of time operators introduced in the previous sections can be generalized. Let H be a self-adjoint operator on a Hilbert space H . (i) A symmetric operator T on H is called a generalized time operator of H if there exist a non-zero subspace D ⊂ D(T H )∩D(H T ) and a non-zero bounded self-adjoint operator C on H such that [T , H ] = iC

on D.

(4.112)

(ii) A symmetric operator T on H is called a generalized strong time operator of H if there exists a mapping K: R → B(H ); R t → K(t) ∈ B(H ) such that, for all t ∈ R, K(t) is a bounded self-adjoint operator on H and eit H T e−it H = T + K(t).

(4.113)

Relation (4.113) is called a generalized weak Weyl relation [15]. Commutation relation (4.112) implies a generalized Heisenberg uncertainty relation: Theorem 4.33 Let T be a generalized time operator of H satisfying (4.112). Then, for all unit vectors Ψ ∈ D ∩ (ker C)⊥ , (ΔH )Ψ (ΔT )Ψ ≥

1 δC , 2

(4.114)

where δC :=

inf

Ψ ∈(ker C)⊥ , Ψ =1

| Ψ, CΨ  |.

Proof Let Ψ ∈ D ∩ (ker C)⊥ be a unit vector. Then, applying the Heisenberg– Robertson inequality (2.25) with A = T and B = H , we have (ΔH )Ψ (ΔT )Ψ ≥

1 1 | Ψ, CΨ  | ≥ δC . 2 2 

Hence (4.114) holds. Remark 4.21 Since C = 0 by definition, it follows that

(ker C)⊥

= {0}.

Concerning a generalized strong time operator of H , we first note the following fact: Lemma 4.14 Let T be a generalized strong time operator of H satisfying (4.113). Suppose that K(t) is strongly differentiable at t = 0. Then K (0) is a bounded self-adjoint operator on H .

4.7 Generalized Time Operators

233

Proof By (4.113) and the density of D(T ), we have operator equality K(0) = 0.

(4.115)

Hence K (0) = s- lim

t →0

K(t) . t

We recall that the strong limit of everywhere defined bounded operators is an everywhere defined bounded operator.14 Hence K (0) is in B(H ). The selfadjointness of K (0) follows from that of K(t)/t.  The following proposition shows that, under additional conditions, a generalized strong time operator of H is a generalized time operator of H : Proposition 4.14 Let T be a generalized strong time operator of H satisfying (4.113). Suppose that K(t) is strongly differentiable at t = 0. Then, for all Φ, Ψ ∈ D(T ) ∩ D(H ),  T Φ, H Ψ  − H Φ, T Ψ  = Φ, iK (0)Ψ .

(4.116)

In particular, if D(T ) ∩ D(H ) is dense in H , then [T , H ] = iK (0) on D(T H ) ∩ D(H T ).

(4.117)

Proof Let Φ, Ψ ∈ D(T ) ∩ D(H ). Then, by (4.113), we have       e−it H Φ, T Ψ = T Φ, eit H Ψ + K(t)Φ, eit H Ψ . It follows that both sides are differentiable in t at t = 0 and the differentiations of them at t = 0 give  (−iH )Φ, T Ψ  = T Φ, iH Ψ  + K (0)Φ, Ψ + K(0)Φ, iH Ψ  . By (4.115), the last term on the right hand side vanishes. Hence (4.116) follows.  For further discussions of generalized (strong) time operators, we refer the reader to [15]. Recently, a generalized strong time operator has appeared in a theory of time crystals [118]. This is very interesting.

14 This can be proved by applying the Banach–Steinhaus theorem (principle of uniform boundedness) (see, e.g., [130, Theorem III.9] or [32, Theorem 3.49]).

Chapter 5

Representations of Canonical Anti-commutation Relations with Finite Degrees of Freedom

Abstract Basic facts on representations of canonical anti-commutation relations with finite degrees of freedom are described.

5.1 Introduction Let A be an associative algebra over C and Map(A × A, A) be the set of mappings from A × A to A. The mapping p: A × A → A × A defined by p(X, Y ) := (Y, X),

(X, Y ) ∈ A × A

satisfies p2 = I (identity). The mapping p induces the mapping Rp : Map(A × A, A) → Map(A × A, A) given by (Rp f )(X, Y ) = f (p(X, Y )) = f (Y, X),

f ∈ Map(A×A, A),

(X, Y ) ∈ A×A.

It follows that Rp2 = I. A mapping f ∈ Map(A × A, A) is said to be Rp -symmetric (resp. anti-symmetric) if Rp f = f (resp. −f ). Each f ∈ Map(A × A, A) is uniquely written as a sum of an Rp -symmetric mapping and an Rp -anti-symmetric mapping. Indeed, introducing the mappings f± := f ± Rp f,

© Springer Nature Singapore Pte Ltd. 2020 A. Arai, Inequivalent Representations of Canonical Commutation and Anti-Commutation Relations, Mathematical Physics Studies, https://doi.org/10.1007/978-981-15-2180-5_5

235

236

5 Representations of Canonical Anti-commutation Relations with Finite Degrees. . .

one sees that f+ (resp. f− ) is Rp -symmetric (resp. anti-symmetric) and f =

1 (f+ + f− ). 2

It is easy to see that this representation is unique. This general structure may be applied to a subclass of Map(A×A, A). We define a mapping B: A × A → A by B(X, Y ) := XY,

(X, Y ) ∈ A × A.

Then B is bilinear. We have B+ (X, Y ) = XY + Y X = {X, Y },

B− (X, Y ) = XY − Y X = [X, Y ].

Hence the anti-commutator { , } and the commutator [ , ] can be interpreted as the Rp -symmetric part and the Rp -anti-symmetric part of the bilinear mapping B respectively. Therefore, from symmetry points of view, the commutator and the anticommutator in fact are natural “components” of the one mapping B. Based on the above consideration, one may be led to replace the commutator [ , ] in the Dirac CCR (2.124) by the anti-commutator { , }. Then one obtains anticommutation relations {Xj , Xk∗ } = δj k I,

{Xj , Xk } = 0,

{Xj∗ , Xk∗ } = 0,

j, k = 1, . . . , N

(N ∈ N). (5.1)

These are called the canonical anti-commutation relations (CAR) with N degrees of freedom for {Xj }N j =1 . Remark 5.1 For a real constant q, one can define [X, Y ]q := XY − qY X,

X, Y ∈ A.

This is called the q-commutator. It is obvious that [X, Y ]1 = [X, Y ] and [X, Y ]−1 = {X, Y }. Hence [X, Y ]q interpolates commutator and anti-commutator. Using qcommutator, one can consider q-deformed CCR [Xj , Yk ]qjk = δj k I, [Xj , Xk ]qjk = 0,

(5.2) [Yj , Yk ]qjk = 0, j, k = 1, . . . , N

(N ∈ N),

(5.3)

where qj k ∈ R. As is seen, the case qj k = 1 (resp. qj k = −1) for all j, k = 1, . . . , N in (5.2) and (5.3) with Yj = Xj∗ gives the Dirac CCR (resp. CAR) with N degrees of freedom. Hence the q-deformed CCR interpolate Dirac CCR and CAR. But, in this book, we do not consider representations of q-deformed CCR. For this subject, see, e.g., [48, 66, 88].

5.1 Introduction

237

To consider representations of CAR, we first note the following fact: Lemma 5.1 Let A be a densely defined closed operator on a Hilbert space H . Suppose that there exists a dense subspace D ⊂ D(A∗ A) ∩ D(AA∗ ) such that AA∗ + A∗ A = I

on D.

(5.4)

Then A is bounded with D(A) = H and A ≤ 1. Moreover, operator equality AA∗ + A∗ A = I

(5.5)

holds. Proof For all Ψ ∈ D, we have by (5.4) Ψ, (AA∗ + A∗ A)Ψ  = Ψ 2 . Hence

A∗ Ψ 2 + AΨ 2 = Ψ 2 . This implies that AΨ ≤ Ψ . Hence AD := A  D is bounded with AD ≤ 1. Hence AD ∈ B(H ) with AD ≤ 1. Since AD ⊂ A and A is closed, it follows that AD ⊂ A. Hence A = AD . Thus A ∈ B(H ). Now we know that A∗ A and AA∗ are in B(H ). Hence (5.4) and the uniqueness in the extension theorem (Theorem 1.1) imply (5.5).  Lemma 5.1 shows that, in contrast to the case of the Dirac CCR, AA∗ −A∗ A ⊂ I , a densely defined closed operator A satisfying CAR (5.4) is necessarily everywhere defined and bounded. This is an interesting fact. Based on Lemma 5.1, we define a representation of CAR as follows. Definition 5.1 Let H be a Hilbert space and Bj ∈ B(H ), j = 1, . . . , N. Then ρB := (H , {Bj , Bj∗ |j = 1, . . . , N}) is said to be a representation of the CAR with N degrees of freedom if {Bj , Bk∗ } = δj k I,

{Bj , Bk } = 0,

j, k = 1, . . . , N.

(5.6)

If {Bj , Bj∗ |j = 1, . . . , N} is irreducible, then ρB is said to be irreducible. Remark 5.2 Since each Bj is bounded, the anti-commutativity of Bj and Bk implies that of Bj∗ and Bk∗ (take the adjoint of the second equation in (5.6)): {Bj∗ , Bk∗ } = 0,

j, k = 1, . . . , N.

Example 5.1 Consider the case H = C2 and define b: C2 → C2 by 0 b :=

1 01 . 00

238

5 Representations of Canonical Anti-commutation Relations with Finite Degrees. . .

Then it is easy to see that b∗ =

0

1 00 , 10

b2 = 0,

bb∗ =

0

1 10 , 00

b∗ b =

0

1 00 . 01

Hence (b∗ )2 = 0,

{b, b∗ } = I.

Hence ρb := (C2 , {b, b∗}) is a representation of the CAR with one degree of freedom. Moreover, it is easy to check that {b, b∗ } = CI . Hence ρb is irreducible. This example has a connection with spin, which is an internal degree of freedom of a quantum particle. As is well known, the spin degree of freedom with spin 1/2, is represented by a triple s = (s1 , s2 , s3 ) of 2 × 2 Hermitian matrices given by sj =

1 σj , 2

j = 1, 2, 3

with the Pauli matrices σj , j = 1, 2, 3, defined by (4.78). It is easy to see that the following relations hold: s1 =

1 (b + b ∗ ), 2

s2 =

i ∗ (b − b), 2

s3 =

1 (bb ∗ − b ∗ b). 2

(5.7)

Hence the spin matrices s1 , s2 and s3 are constructed from the representation ρb of CAR. We recall that the Lie algebra su(2) of the two-dimensional special unitary group SU(2) := {U | U is a 2 × 2 unitary matrix with determinant det U = 1} is given by su(2) = {A| A is a 2 × 2 anti-Hermitian matrix with trace zero} and su(2) is a three dimensional real Lie algebra with a basis {e1 , e2 , e3 } of the form e1 :=

1 2

0

1 0i , i 0

e2 :=

1 2

0

1 0 −1 , 1 0

e3 :=

1 2

0

i 0 0 −i

1 .

5.1 Introduction

239

The basis {e1 , e2 , e3 } obeys the following commutation relations: [e2 , e3 ] = e1 ,

[e3 , e1 ] = e2 ,

[e1 , e2 ] = e3 .

It is obvious that e1 = is1 ,

e2 = −is2 ,

e3 = is3 .

(5.8)

Thus su(2) is constructed from the representation ρb of the CAR with one degree of freedom. Conversely, we have by (5.7) b = s1 + is2 . Hence su(2) gives an irreducible representation of the CAR with one degree of freedom. On the other hand, it is well known (e.g., [164, Chapter III, §4]) that possible values {n/2|n ∈ Z+ } = {0, 1/2, 1, 3/2, . . .} of spin angular momentum or orbital angular momentum of a quantum particle appear as the labels of all finitedimensional irreducible representations of su(2), which are mutually inequivalent. In this way, the representation ρb is related to physics of angular momentum of a quantum particle. Example 5.2 Let N ≥ 2 be a natural number and, for k ∈ Z+ , ∧k CN be the k-fold anti-symmetric tensor product (wedge product) of CN (see (1.53)) with convention ∧0 CN := C.1 Then k N (k) N (k) ∈ ∧k CN , k = 0, . . . , N} ∧(CN ) := ⊕N k=0 ∧ C = {ψ = (ψ )k=0 |ψ

is called the fermion Fock space over CN . The name comes from that this Hilbert space serves as a Hilbert space of state vectors of fermions whose oneparticle Hilbert space is CN . The subspace ∧k CN describes the k-particle space of fermions.2 For each z = (z1 , . . . , zN ) ∈ CN , there exists a unique linear operator b(z) on ∧(CN ) such that its adjoint b(z)∗ is of the form (b(z)∗ ψ)(0) = 0, √ (b(z)∗ ψ)(k) := kAk (z ⊗ ψ (k−1) ),

k = 1, . . . , N,

N ψ = (ψ (k) )N k=0 ∈ ∧(C ).

that, if k > N, then ∧k CN = {0}. that a vector state of k identical fermions is anti-symmetric for any permutations of k fermions. In the present context, this corresponds to the anti-symmetry of ∧k CN .

1 Note

2 Recall

240

5 Representations of Canonical Anti-commutation Relations with Finite Degrees. . .

It is easy to see that, for all ψ ∈ ∧(CN ) such that ψ () = 0,  = k and ψ (k) = Ak (u1 ⊗ · · · ⊗ uk )

(u1 , . . . , uk ∈ CN ),

one has (b(z)ψ)() = 0, (b(z)ψ)

(k−1)

 = k − 1,

k  1 = √ (−1)j −1 z, uj CN Ak−1 (u1 ⊗ · · · ⊗ u6j ⊗ · · · ⊗ uk ), k j =1

where u6j indicates the omission of uj . Using these representations, one can show that {b(z), b(w)∗ } = z, wCN ,

{b(z), b(w)} = 0.

(5.9)

It follows that, for each k = 0, 1, . . . , N, b(z) maps ∧k CN to ∧k−1 CN with ∧−1 CN := {0} and b(z)∗ maps ∧k CN to ∧k+1 CN . For this structure, b(z) (resp. b(z)∗ ) is called the fermion annihilation (resp. creation) operator with test vector z. We denote by (e1 , . . . , eN ) the standard basis of CN : (ej )j = δjj , j, j = 1, . . . , N. We define bj := b(ej ),

j = 1, . . . , N.

Then (5.9) implies that {bj , bk∗ } = δj k ,

{bj , bk } = 0,

j, k = 1, . . . , N.

(5.10)

Thus (N)

ρF

:= (∧(CN ), {bj , bj∗ |j = 1, . . . , N})

is a representation of the CAR with N degrees of freedom. Since dim ∧(CN ) = 2N , (N)

this is a finite-dimensional representation. The representation ρF is called the Fock representation of the CAR with N degrees of freedom. (N) The Fock representation ρF has a simple structure. The vector ψ0 ∈ ∧(CN ) with ψ0(k) = δ0k , k = 0, . . . , N

5.2 Representations of the CAR with One Degree of Freedom

241

is called the Fock vacuum in ∧(CN ). It follows that b(z)ψ0 = 0,

z ∈ CN

and, for all uj ∈ CN (j = 1, . . . , N), √ (b(u1 )∗ · · · b(uk )∗ ψ0 )() = δk k!Ak (u1 ⊗ · · · uk ),

k,  = 1, . . . , N.

Hence ∧k (CN ) = span{b(u1 )∗ · · · b(uk )∗ ψ0 |u1 , . . . , uk ∈ CN } = span{bi∗1 · · · bi∗k ψ0 |i1 < · · · < ik , i1 , . . . , ik ∈ {1, . . . , N}}, where we have used that {bj∗ , bk∗ } = 0,

(bj∗ )2 = 0,

j, k = 1, . . . , N.

Thus ∧(CN ) = span{ψ0 , bi∗1 · · · bi∗k ψ0 |k = 1, . . . , N, i1 < · · · < ik , i1 , . . . , ik ∈ {1, . . . , N }}.

Hence ψ0 is a cyclic vector for {bi∗1 · · · bi∗k |k = 1, . . . , N, i1 < · · · < ik , i1 , . . . , ik ∈ {1, . . . , N}}. (N)

In the same way as in the proof of Proposition 2.16, one can show that ρF irreducible.

is

In what follows, we first consider representations of the CAR with one degree of freedom and then those of the CAR with arbitrary finite degrees of freedom.

5.2 Representations of the CAR with One Degree of Freedom Let (H , {B, B ∗ }) be a representation of the CAR with one degree of freedom, i.e., B ∈ B(H ), BB ∗ + B ∗ B = I,

B 2 = 0,

(B ∗ )2 = 0.

(5.11)

We define PB := BB ∗ ,

PB ∗ := B ∗ B.

(5.12)

242

5 Representations of Canonical Anti-commutation Relations with Finite Degrees. . .

Lemma 5.2 The operators PB and PB ∗ are orthogonal projections and PB + PB ∗ = I.

(5.13)

Proof It is obvious that PB and PB ∗ are bounded self-adjoint operators. By the CAR (5.11), we have PB2 = BB ∗ BB ∗ = B(I − BB ∗ )B ∗ = BB ∗ = PB . Hence PB is an orthogonal projection. Similarly, one can show that PB2 ∗ = PB ∗ . Hence PB ∗ is an orthogonal projection.  Let HB := PB H ,

HB ∗ := PB ∗ H .

Then, by (5.13), HB is orthogonal to HB ∗ and H = HB ⊕ HB ∗ .

(5.14)

Lemma 5.3 (i) The operator B is a partial isometry with initial space HB ∗ and final space HB . (ii) The operator B ∗ is a partial isometry with initial space HB and final space HB ∗ . Proof (i) This is just a rephrase of (5.12). (ii) This follows from (5.12) and B = (B ∗ )∗ .



Lemma 5.4 Assume that H is separable. Let d := dim HB (d < ∞ or d = ∞). (0) (1) and {en }dn=1 be a CONS of HB . Then there exists a CONS {en }dn=1 of HB ∗ such that, for all n = 1, . . . , d, B ∗ en(0) = en(1) , Ben(0) = 0,

B ∗ en(1) = 0,

Ben(1) = en(0).

In particular, if d < ∞, then H is even dimensional with dim H = 2d. Proof Let en(1) := B ∗ en(0).

(5.15) (5.16)

5.2 Representations of the CAR with One Degree of Freedom

243

Then Lemma 5.3(ii) implies that {en }dn=1 is a CONS of HB ∗ . Since (B ∗ )2 = 0, B ∗ en(1) = 0. Since BPB = 0, Ben(0) = 0. Moreover, Ben(1) = BB ∗ en(0) = en(0). The last statement follows from the first half of the lemma and (5.14).  (1)

(j )

Let en (j = 0, 1) be as in Lemma 5.4 and En := span{en(0), en(1) }.

(5.17)

H = ⊕dn=1 En .

(5.18)

Then, by (5.14), we have

Lemma 5.4 implies that B and B ∗ are reduced by each En . We denote by Bn the reduced part of B to En . Then Bn∗ is the reduced part of B ∗ to En . Hence we have the direct sum decomposition B = ⊕dn=1 Bn .

(5.19)

We are now ready to prove a basic theorem on representations of CAR: Theorem 5.1 Let ρ := (H , {B, B ∗ }) be a representation of the CAR with one degree of freedom. Assume that H is separable. Then ρ is equivalent to a direct sum representation of ρb . In particular, if ρ is irreducible, then ρ ∼ = ρb . Proof Let 0 1 1 e1 = , 0

0 1 0 e2 = ∈ C2 . 1 (j )

Then {e1 , e2 } be an orthonormal basis of C2 . Let {en }n (j = 0, 1) be as in Lemma 5.4. Then, for each n, there exists a unitary operator Un from En to C2 such that Un en(0) = e1 and Un en(1) = e2 . It is easy to see that Un Bn Un−1 = b,

Un Bn∗ Un−1 = b ∗ .

The operator U := ⊕dn=1 Un is a unitary operator from H to ⊕d C2 . By (5.19), we have U BU −1 = ⊕dn=1 Un Bn Un−1 = ⊕dn=1 b. This implies that U B ∗ U −1 = ⊕dn=1 b∗ . Thus ρ is equivalent to a direct sum representation of ρb . 

244

5 Representations of Canonical Anti-commutation Relations with Finite Degrees. . .

Theorem 5.1 shows that, in contrast to representations of the CCR with one degree of freedom, representations of the CAR with one degree of freedom have a very simple structure.

5.3 Representations of the CAR with N Degrees of Freedom We next go on to analysis of representations of the CAR with N degrees of freedom (N ≥ 2). The following theorem holds: Theorem 5.2 Let ρB := (H , {Bj , Bj∗ |j = 1, . . . , N}) be a representation of the CAR with N degrees of freedom. Assume that H is (N) separable. Then ρB is equivalent to a direct sum of the Fock representation ρF of the CAR with N degrees of freedom: (N) ρB ∼ = ⊕M n=1 ρF

(5.20)

(N) with M ∈ N or M = ∞. In particular, if ρB is irreducible, then ρB ∼ = ρF .

Proof It is obvious that (H , {B1 , B1∗ }) is a representation of the CAR with one degree of freedom. Hence, by (5.14), H = HB1 ⊕ HB1∗ . Moreover, the mappings B1 : HB1∗ → HB1 and B1∗ : HB1 → HB1∗ are unitary operators. By the CAR for {Bj , Bj∗ |j = 1, . . . , N}, we have B2 P# = P# B2 ,

B2∗ P# = P# B2∗ ,

# = B1 , B1∗ .

Hence HB1 and HB1∗ reduce B2 and B2∗ . For notational simplicity, we denote the reduced part of B2 to H# by the same symbol. Hence (H# , {B2 , B2∗ }) is a representation of the CAR with one degree of freedom. Hence HB1 = (HB1 )B2 ⊕ (HB1 )B2∗ ),

HB1∗ = (HB1∗ )B2 ⊕ (HB1∗ )B2∗ .

Therefore we obtain : 9 : 9 H = (HB1 )B2 ⊕ (HB1 )B2∗ ⊕ (HB1∗ )B2 ⊕ (HB1∗ )B2∗ .

5.3 Representations of the CAR with N Degrees of Freedom

245

Let {un }n be a CONS of (HB1 )B2 . Then, by Lemma 5.4, {B2∗ un }n be:a CONS 9 of (HB1 )B2∗ . Hence {un , B2∗ un }n is a CONS of (HB1 )B2 ⊕ (HB1 )B2∗ . Hence, : 9 Lemma 5.4 again, {B1∗ un , B1∗ B2∗ un }n is a CONS of (HB1∗ )B2 ⊕ (HB1∗ )B2∗ . Therefore {un , B2∗ un , B1∗ un , B1∗ B2∗ un }n is a CONS of H . By the CAR for {Bj , Bj∗ |j = 1, . . . , N} again, B3 and B3∗ are reduced by each (H# )! (# = B1 , B1∗ , ! = B2 , B2∗ ). Hence each (H# )! has the orthogonal decomposition (H# )! = ((H# )! )B3 ⊕ ((H# )! )B3∗ . Repeating this process, we finally arrive at the orthogonal decomposition H = (H1 ⊕ H2 ) ⊕ (H3 ⊕ H4 ) ⊕ · · · (H2N −1 ⊕ H2N ) such that the following hold: • BN∗ is a unitary operator from H2k−1 to H2k (k = 1, . . . , 2N−1 ); ∗ • BN−1 is a unitary operator from H4k−3 ⊕ H4k−2 to H4k−1 ⊕ H4k (k = 1, . . . , 2N−2 ); ∗ • BN−2 is a unitary operator from H8k−7 ⊕ H8k−6 ⊕ H8k−5 ⊕ H8k−4 to H8k−3 ⊕ H8k−2 ⊕ H8k−1 ⊕ H8k (k = 1, . . . , 2N−3 ); .. .. . .

• B1∗ is a unitary operator from ⊕2j =1 Hj to ⊕2j =2N−1 +1 Hj . N−1

N

∗ Let {vn }n := {vn }M n=1 be a CONS of H1 with M ∈ N or M = ∞. Then {BN vn }n ∗ is a CONS of H2 . Hence {vn , BN vn }n is a CONS of H1 ⊕ H2 . Then, by the ∗ ∗ algorithm of the construction, {BN−1 vn , BN−1 BN∗ vn }n is a CONS of H3 ⊕ H4 . ∗ ∗ ∗ ∗ Hence {vn , BN vn , BN−1 vn , BN−1 BN vn }n is a CONS of H1 ⊕ H2 ⊕ H3 ⊕ H4 . This implies that ∗ ∗ ∗ ∗ ∗ ∗ vn , BN−2 BN∗ vn , BN−2 BN−1 vn , BN−2 BN−1 BN∗ vn }n {BN−2

is a CONS of H5 ⊕ H6 ⊕ H7 ⊕ H8 . Hence ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ {vn , BN vn , BN −1 vn , BN −1 BN vn , BN −2 vn , BN −2 BN vn , BN −2 BN −1 vn , BN −2 BN −1 BN vn }n

is a CONS of ⊕8j =1 Hj . Repeating this procedure, one can see that {vn , Bi∗1 · · · Bi∗k vn |k = 1, . . . , N, i1 < i2 < · · · < ik , i1 , . . . , ik ∈ {1, . . . , N}}n

246

5 Representations of Canonical Anti-commutation Relations with Finite Degrees. . . N

is a CONS of H = ⊕2j =1 Hj . Therefore, putting Dn := span{vn , Bi∗1 · · · Bi∗k vn |k = 1, . . . , N, i1 < i2 < · · · < ik , i1 , . . . , ik ∈ {1, . . . , N}}.

we have H = ⊕M n=1 Dn . It is easy to see that, for each j = 1, . . . , N and n, Bj and Bj∗ leave Dn invariant. Hence ρn := (Dn , {Bj , Bj∗ |j = 1, . . . , N}) is a representation of the CAR with N degrees of freedom and ρB = ⊕M n=1 ρn . By Example 5.2 and the isomorphism theorem on Hilbert spaces, there exists a unitary operator Wn : Dn → ∧(CN ) such that Wn vn = ψ0 and Wn Bi∗1 · · · Bi∗k vn = bi∗1 · · · bi∗k ψ0 ,

k = 1, . . . , N, i1 < · · · < ik , i1 , . . . , ik ∈ {1, . . . , N}.

Then it is easy to see that Wn Bj Wn−1 = bj ,

j = 1, . . . , N.

Hence each ρn is equivalent to ρF(N) . The operator W := ⊕M n=1 Wn N is a unitary operator from H to ⊕M n=1 ∧ (C ), which gives

W Bj W −1 = ⊕M n=1 bj , Thus (5.20) holds.

j = 1, . . . , N. 

Chapter 6

Elements of the Theory of Boson Fock Spaces

Abstract Basic aspects of the theory of boson Fock space are described in connection with studying representations of CCR with infinite degrees of freedom.

6.1 The Boson Fock Space Over a Hilbert Space Let H be a Hilbert space. Then, for each n ∈ N, one has the n-fold symmetric tensor product Hilbert space ⊗ns H of H (see Sect. 1.5). We set ⊗0s H := C. The infinite direct sum Hilbert space of ⊗ns H , n = 0, 1, 2, . . . Fb (H ) : =

∞ /

⊗ns H

(6.1)

n=0

2

∞ % &∞ = Ψ = Ψ (n) | Ψ (n) ∈ ⊗ns H , n ∈ Z+ ,

Ψ (n) 2 < ∞ n=0

3

n=0

(6.2) is called the boson (or symmetric) Fock space over H . Following the definition of an infinite direct sum Hilbert space, the inner product of Fb (H ) takes the form Ψ, Φ =

∞  n=0

Ψ (n) , Φ (n)

 ⊗ns H

,

Ψ, Φ ∈ Fb (H ).

The Hilbert space Fb (H ) is used to describe state vectors of a Bose field, i.e., a quantum field of identical bosons. In the context of Bose field theories, the Hilbert space ⊗ns H as a closed subspace of Fb (H ) represents the state vector space of a system of n identical bosons. For this reason, ⊗ns H is called the n-boson space or the n-particle space of Fb (H ). In particular, ⊗1s H = H is called the oneparticle Hilbert space of Fb (H ), which describes a Hilbert space of state vectors of one boson. © Springer Nature Singapore Pte Ltd. 2020 A. Arai, Inequivalent Representations of Canonical Commutation and Anti-Commutation Relations, Mathematical Physics Studies, https://doi.org/10.1007/978-981-15-2180-5_6

247

248

6 Elements of the Theory of Boson Fock Spaces

The subspace & % Fb,0 (H ) := Ψ ∈ Fb (H )| ∃n0 ∈ N such that Ψ (n) = 0, ∀n ≥ n0

(6.3)

is called the finite particle subspace of Fb (H ). It is easy to see that Fb,0 (H ) is dense in Fb (H ). For a subspace D of H , we introduce & % ˆ ns D, n ∈ Z+ , (6.4) Fb,fin (D) := Ψ ∈ Fb,0 (H )|Ψ (n) ∈ ⊗ ˆ ns D denotes the n-fold algebraic symmetric tensor product of D. If D is where ⊗ dense in H , then Fb,fin (D) is dense in Fb (H ). Example 6.1 One of the simplest examples of boson Fock space is Fb (C), the boson Fock space over C. Since ⊗ns C ∼ = C, we have 2 Fb (C) ∼ = ⊕∞ n=0 C =  (Z+ ).

Example 6.2 For a measure space (X, Σ, μ), one can consider the Hilbert space L2 (X, dμ) (see Sect. 1.7), which serves as a Hilbert space of state vectors of one boson being in the space X. Then n 2 Fb (L2 (X, dμ)) = ⊕∞ n=0 ⊗s L (X, dμ).

By Theorem 1.24(i), we have the natural isomorphism 2 n n Fb (L2 (X, dμ)) ∼ = ⊕∞ n=0 Lsym (X , dμ ),

where μn := μ ⊗ · · · ⊗ μ, the n-product measure of μ, and L2sym (Xn , dμn )|n=0 := C. ˆd d Typical examples are given by the case (X, Σ, μ) = (Rd , B d , μ(d) L ) or (R , B , (d) (d) μL ), where μL denotes the d-dimensional Lebesgue measure. There are operators acting in the boson Fock space Fb (H ), which play central roles in describing Bose field theories. We next define them.

6.2 Boson Second Quantization Operators Let T be a densely defined closable operator on H . Then, for each natural number n ≥ 2 and j = 1, . . . , n, the operator j -th



Tˆj := I ⊗ · · · ⊗ I ⊗ T ⊗I ⊗ · · · ⊗ I

6.2 Boson Second Quantization Operators

249

is a densely defined closed operator on the n-fold tensor product Hilbert space ⊗n H (see Sect. 1.8). Let (n) T0

⎞ ⎛ n ˆ n D(T ). Tˆj ⎠  ⊗ := ⎝ j =1 n

ˆ D(T ∗ ) and Then T0(n) is densely defined. It is easy to see that D((T0(n) )∗ ) ⊃ ⊗ ⎛ (T0(n) )∗

⊃⎝

n

⎞

j -th

 ∗

ˆ n D(T ∗ ). I ⊗ · · · ⊗ I ⊗ T ⊗I ⊗ · · · ⊗ I ⎠  ⊗

j =1 n

ˆ D(T ∗ ) is dense in ⊗n H , (T0 )∗ is densely defined. Hence, by a general Since ⊗ (n) theorem, T0 is closable. Hence the closure of it (n)

(n)

T (n) := T0

(n ≥ 2)

(6.5)

is a densely defined closed operator on ⊗n H . We set T (1) := T .

(6.6)

Lemma 6.1 For each n ∈ N, T (n) is reduced by ⊗ns H . n

ˆ D(T ), Ψj ∈ D(T ) and Sn be the symmetrization Proof Let Ψ = ⊗nj=1 Ψj ∈ ⊗ operator defined by (1.49). Then 1 n ⊗j =1 Ψσ (j ) . n!

Sn Ψ =

σ ∈Sn

Hence Sn Ψ ∈ D(T (n) ) and T (n) Sn Ψ =

1 (n) n T (⊗j =1 Ψσ (j ) ). n! σ ∈Sn

We have T (n) (⊗nj=1 Ψσ (j ) ) =

n

k -th



Ψσ (1) ⊗ · · · ⊗ T Ψσ (k) ⊗ · · · ⊗ Ψσ (n) .

k=1

= Uσ T (n) Ψ,

250

6 Elements of the Theory of Boson Fock Spaces

where Uσ is defined by (1.48). Hence, by (1.49), we obtain T (n) Sn Ψ = Sn T (n) Ψ. ˆ n D(T ) is a core for ˆ n D(T ). Since ⊗ By linearity, this equation extends to all Ψ ∈ ⊗ T (n) by the definition of T (n) , it follows from a limiting argument that Sn T (n) ⊂ T (n) Sn . Recall that Sn is an orthogonal projection. Hence T (n) is reduced by Ran Sn = ⊗ns H .  We denote by Tb(n) the reduced part of T (n) to ⊗ns H : (n)

D(Tb ) = D(T (n) ) ∩ (⊗ns H ),

(n)

Tb Ψ = T (n) Ψ,

(n)

Ψ ∈ D(Tb ).

(6.7)

The following proposition follows from Proposition 1.16: Proposition 6.1 Let T be a densely defined closable operator on H . (i) The operator Tb(n) is a densely defined closed operator. (n) (ii) If T is self-adjoint, then so is Tb . As a convention, we set (0)

Tb

:= 0

acting in C. The infinite direct sum operator (see Appendix C) (n) db (T ) := ⊕∞ n=0 Tb ,

(6.8)

which is a densely defined closed operator on Fb (H ), is called the boson second quantization operator of T or the boson second quantization of T . Let ΩH ∈ Fb (H ) be the vector defined by (0)

ΩH = 1,

(n)

ΩH = 0,

n ∈ N.

(6.9)

It is called the Fock vacuum in Fb (H ). It is visually written ΩH = {1, 0. 0, . . .}. It follows that db (T )ΩH = 0,

(6.10)

i.e., the Fock vacuum is an eigenvector of db (T ) with eigenvalue 0. The boson second quantization operator Nb := db (I )

(6.11)

6.2 Boson Second Quantization Operators

251

of T = I is called the boson number operator on Fb (H ). The name comes from the easily seen fact Nb  ⊗ns H = n,

n ∈ Z+,

(6.12)

i.e., Nb counts the number of bosons. In many cases in applications to quantum field theory, the operator T is selfadjoint. In this case, we have the following theorem. Theorem 6.1 Let T be a self-adjoint operator on H . Then: (i) db (T ) is a self-adjoint operator. Moreover, for all t ∈ R, (n)

it Tb eit db(T ) = ⊕∞ . n=0 e

(6.13)

(ii) If T is non-negative, then db (T ) is non-negative and, for all β > 0, (n)

−βTb e−βdb (T ) = ⊕∞ . n=0 e

(6.14)

(iii) Let D be a core for T . Then Fb,fin (D) is a core for db (T ). (iv) If σ (T ) = [m, ∞), then σ (db (T )) = {0} ∪ [m, ∞).

(6.15)

Proof (n)

(i) By Theorem 1.26-(iv), T (n) is self-adjoint. Hence Tb is self-adjoint. Therefore, by a general theorem on the direct sum of self-adjoint operators (see Theorem C.2(i) in Appendix C), db (T ) is self-adjoint. Formula (6.13) follows from an application of formula (C.3) in Appendix C. (ii) Let T ≥ 0. Then, by Theorem 1.26(v), T (n) is non-negative and hence so is Tb(n) . Thus, by Theorem C.2(ii), db (T ) is non-negative. Formula 6.14 follows from an application of formula (C.4) in Appendix C. ˆ n D is a core for T (n) . (iii) It follows from a limiting argument that, for all n ∈ N, ⊗ n (n) ˆ s D is a core for Tb . Thus the desired result follows. Hence ⊗ (iv) By Theorem C.2(iii) in Appendix C, . (n) σ (db (T )) = {0} ∪ σ (T ) ∪ ∪∞ )b ) . n=2 σ ((T  ˆ ns D(T ), Ψ, (T (n) )b Ψ ≥ nm Ψ 2 . By a It is easy to see that, for all Ψ ∈ ⊗ limiting argument, this extends to all Ψ ∈ D((T (n) )b ). Hence σ ((T (n) )b ) ⊂ [nm, ∞). Thus (6.15) follows. 

252

6 Elements of the Theory of Boson Fock Spaces

Remark 6.1 More detailed spectral properties of db (T ) are known (see [28, Theorem 5.3]). The following theorem describes a convergent property of a sequence of boson second quantization operators: Theorem 6.2 Let Tn (n ∈ N) and T be self-adjoint operators on H . Suppose that there exists a core D for T such that D ⊂ ∩∞ n=1 D(Tn ) and, for all Ψ ∈ D, limn→∞ Tn Ψ = T Ψ . Then, for all t ∈ R, s- lim eit db(Tn ) = eit db(T ) . n→∞

(6.16)

4 5 Proof It is easy to see that Fb,fin (D) ⊂ D(db (T )) ∩ ∩∞ n=1 D(db (Tn )) and, for all Ψ ∈ Fb,fin (D), lim db (Tn )Ψ = db (T )Ψ.

n→∞

By Theorem 6.1, Fb,fin (D) is a core for db (T ). Hence, by Theorem 1.29, {db (Tn )}n converges to db (T ) in the strong resolvent sense as n → ∞. Then, by Theorem 1.30, we obtain (6.16). 

6.3 Boson Γ -Operators Let T be a densely defined closable operator from a Hilbert space H to a Hilbert space K . Then the n-fold tensor product ⊗n T of T with n ≥ 2 is a unique densely defined closed operator from ⊗n H to ⊗n K such that (⊗n T )(⊗nj=1 Ψj ) = ⊗nj=1 T Ψj ,

Ψj ∈ D(T ), j = 1, . . . , n.

In the same way as in the proof of Lemma 6.1, one can prove that, for each n ≥ 2, Sn (⊗n T ) ⊂ (⊗n T )Sn .

(6.17)

Hence we can define a densely defined closed operator (⊗n T )b from ⊗ns H to ⊗ns K as follows: (⊗0 T )b := 1, (⊗1 T )b := T and, for n ≥ 2, D((⊗n T )b ) := D(⊗n T ) ∩ (⊗ns H ), (⊗n T )b Ψ := (⊗n T )Ψ,

Ψ ∈ D((⊗n T )b ).

In the case H = K , (⊗n T )b is nothing but the reduced part of ⊗n T to ⊗ns H .

6.3 Boson Γ -Operators

253

One can define an infinite direct sum operator from Fb (H ) to Fb (K ) by n Γb (T ) = ⊕∞ n=0 (⊗ T )b .

(6.18)

We call the operator Γb (T ) the boson Γ -operator for T . Theorem 6.3 Let T be a densely defined closable operator from H to K . (i) If T is unbounded or bounded with T > 1, then Γb (T ) is unbounded. (ii) If T is a contraction operator, then Γb (T ) is a contraction operator and Γb (T )∗ = Γb (T ∗ ).

(6.19)

(iii) (product law) Let S ∈ B(H , K ) and T ∈ B(K , X ) (X is a Hilbert space) be contraction operators. Then, Γb (T )Γb (S) = Γb (T S).

(6.20)

(iv) If U : H → K is unitary, then so is Γb (U ) : Fb (H ) → Fb (K ) and Γb (U )−1 = Γb (U −1 ).

(6.21)

Γb (U )Fb,fin (H ) = Fb,fin (K ).

(6.22)

Moreover,

(v) Let H = K and M be a closed subspace of H . Let PM be the orthogonal projection onto M . Then Γb (PM ) is the orthogonal projection onto Fb (M ) ⊂ Fb (H ). Proof (i) Let T be unbounded. Then (⊗1 T )b = T is unbounded. Hence it follows that Γb (T ) is unbounded. Next, let T be bounded with T > 1. Then there exist a constant c > 0 and a unit vector f ∈ H such that Tf ≥ 1 + c. Hence

(⊗n T ) ⊗n f = Tf n ≥ (1 + c)n . Hence, for Ψn := {0, 0, . . . , ⊗n f, 0, . . .}, we have Ψn = 1 and

Γb (T )Ψn = (⊗n T )b ⊗n f ≥ (1 + c)n → ∞ (n → ∞). Hence Γb (T ) is unbounded. (ii) Let T be a contraction operator: T ≤ 1. Then

⊗nb T ≤ ⊗n T = T n ≤ 1.

254

6 Elements of the Theory of Boson Fock Spaces

Hence, for all Ψ ∈ Fb (H ),

Γb (T )Ψ 2 =

∞

(⊗n T )b Ψ (n) 2 ≤

n=0

∞

Ψ (n) 2 = Ψ 2 .

n=0

Hence Γb (T ) ≤ 1. Thus Γb (T ) is a contraction. We have Γb (T )∗ =

∞

((⊗n T )b )∗ .

n=0

Since T is bounded, it follows that ((⊗n T )b )∗ = (⊗n T ∗ )b . Thus (6.19) follows. (iii) This follows from the easily proven fact (⊗n T )b (⊗n S)b = (⊗n T S)b ,

n ∈ N.

(iv) Since U U ∗ = I , it follows from (iii) that Γb (U )Γb (U ∗ ) = Γb (I ) = I . Similarly, U ∗ U = I implies that Γb (U ∗ )Γb (U ) = I . By (6.19), we have Γb (U ∗ ) = Γb (U )∗ . Thus Γb (U ) is unitary. Since U −1 = U ∗ , we have Γb (U −1 ) = Γb (U ∗ ) = Γb (U )−1 . Hence (6.21) holds. ˆ ns K . Hence (6.22) holds. ˆ ns H = ⊗ It is easy to see that (⊗n U )b ⊗ (v) By (ii), Γb (PM ) is a self-adjoint contraction operator. By (iii), 2 Γb (PM )2 = Γb (PM ) = Γb (PM ).

ˆ ns M . ˆ nb H = ⊗ Thus Γb (PM ) is an orthogonal projection. We have (⊗n PM )b ⊗ Hence Γb (PM )Fb,fin (H ) = Fb,fin (M ). Since Fb,fin (H ) is dense in Fb (H ) and Fb,fin (M ) is dense in Fb (M ), it follows that Γb (PM )Fb (H ) = Fb (M ). Thus Γb (PM ) is the orthogonal projection onto Fb (M ).  Basic relations between the boson second quantization operator db (T ) and the boson Γ -operator Γb (T ) are given in the following theorem: Theorem 6.4 Let T be a self-adjoint operator on H . Then: (i) For all t ∈ R, Γb (eit T ) = eit db(T ) .

(6.23)

6.3 Boson Γ -Operators

255

(ii) Let C+ := {z ∈ C|Re z > 0}

(6.24)

and T ≥ 0. Then db (T ) ≥ 0 and, for all z ∈ C+ , Γb (e−zT ) = e−zdb(T ) .

(6.25)

Proof (n)

(i) By Theorem 1.26(iv), for all t ∈ R, ⊗nj=1 eit T = eit T . Hence (⊗nj=1 eit T )b = Sn eit T

(n)

(n)

Sn = eit Tb , (n)

it Tb where we have used Proposition 1.16(v). Hence Γb (eit T ) = ⊕∞ . By n=0 e this fact and formula (6.13), we obtain (6.23). (ii) In the same manner as in (i), one can show that, for all β ≥ 0, Γb (e−βT ) = (n) −βTb ⊕∞ . Hence, by (6.14), we obtain (6.25) with z = β. We next consider n=0 e the general case z = β + it, β ≥ 0, t ∈ R. In this case we have e−zT = e−βT eit T . Hence, by Theorem 6.3(iii),

Γb (e−zT ) = Γb (e−βT )Γb (eit T ) = e−βdb (T ) eit db(T ) = e−zdb (T ) . 

Thus (6.25) follows.

Theorem 6.5 Let U : H → K (a Hilbert space) be a unitary operator and T be a densely defined closable operator on H . Then Γb (U )db (T )Γb (U )−1 = db (U T U −1 ).

(6.26)

Proof One first proves (6.26) on Fb,fin (D(U T U −1 )). Then, by a limiting argument using the fact that Fb,fin (D(U T U −1 )) is a core for db (U T U −1 ), one can show that db (U T U −1 ) ⊂ Γb (U )db (T )Γb (U )−1 . To prove the converse order relation, let D := Γb (U )Fb,fin (D(T )). Then D is a core for Γb (U )db (T )Γ (U )−1 and it is shown that (6.26) holds on D. Hence it follows that db (U T U −1 ) ⊃ Γb (U )db (T )Γ (U )−1 . Thus (6.26) follows. 

256

6 Elements of the Theory of Boson Fock Spaces

6.4 Creation and Annihilation Operators For each f ∈ H , we define an operator A† (f ) on Fb (H ) as follows: 2

3 ∞ √ (n−1) 2 D(A (f )) := Ψ ∈ Fb (H )|

nSn (f ⊗ Ψ ) < ∞ , †

(6.27)

n=1

(A† (f )Ψ )(0) = 0, √ (A† (f )Ψ )(n) := nSn (f ⊗ Ψ (n−1) ),

(6.28) n ≥ 1, Ψ ∈ D(A† (f )).

(6.29)

It is obvious that Fb,0 (H ) ⊂ D(A† (f )).

(6.30)

Hence A† (f ) is densely defined. It follows from (6.29) that A† (f ) maps ⊗ns H to ⊗n+1 H: s A† (f ) ⊗ns H ⊂ ⊗n+1 H, s

n ∈ N.

(6.31)

In particular, A† (f ) leaves Fb,0 (H ) invariant: A† (f )Fb,0 (H ) ⊂ Fb,0 (H ).

(6.32)

It is easy to see that, for all α, β ∈ C and f, g ∈ H , A† (αf + βg) ⊃ αA† (f ) + βA† (g).

(6.33)

Lemma 6.2 For all f ∈ H , A† (f ) is a densely defined closed operator on Fb (H ). Proof We need only to prove the closedness of A† (f ). Let Ψk ∈ D(A† (f )) (k ∈ N) be such that limk→∞ Ψk = Ψ ∈ Fb (H ) and limk→∞ A† (f )Ψk = Φ ∈ Fb (H ). Then, for each n ∈ Z+ , Ψk(n) → Ψ (n) and (A† (f )Ψk )(n) → Φ (n) as k → ∞. The √ latter implies that nSn (f ⊗ Ψk(n−1) ) → Φ (n) as k → ∞. We have f ⊗ Ψk(n−1) → √ it follows that nSn (f ⊗ Ψ (n−1) ) = f ⊗ Ψ (n−1) as k→ ∞. Since Sn is bounded,  √ ∞ (n) 2 (n−1) ) 2 < ∞. Φ (n) . We have ∞ n=0 Φ < ∞. Hence n=1 nSn (f ⊗ Ψ † † (n) (n) † Therefore Ψ ∈ D(A (f ) and (A (f )Ψ ) = Φ , i.e., A (f )Ψ = Φ. Thus A† (f ) is closed.  Since A† (f ) is densely defined, one can define A(f ) := A† (f )∗ ,

(6.34)

6.4 Creation and Annihilation Operators

257

the adjoint of A† (f ). By Lemma 6.2, we have A(f )∗ = A† (f ).

(6.35)

Property (6.31) implies that H, A(f ) ⊗ns H ⊂ ⊗n−1 s

n∈N

(6.36)

with ⊗−1 s H := {0}. In particular, A(f ) leaves Fb,0 (H ) invariant: A(f )Fb,0 (H ) ⊂ Fb,0 (H ).

(6.37)

Relation (6.31) means that A(f )∗ maps the n-particle space into (n + 1)-particle space, i.e., it has the function that creates one particle. On the other hand, (6.36) shows that A(f ) has the function that annihilates one particle. Based on these properties, A(f )∗ and A(f ) are called respectively the boson creation operator and the boson annihilation operator with test vector f . It follows from (6.33) that, for all α, β ∈ C and f, g ∈ H , A(αf + βg) = α ∗ A(f ) + β ∗ A(g)

on Fb,0 (H ).

(6.38)

Note that the mapping H f → A(f )  Fb,0 (H ) is anti-linear, while the mapping H f → A(f )∗  Fb,0 (H ) is linear. Lemma 6.3 The Fock vacuum ΩH is in ∩f ∈H D(A(f )) and A(f )ΩH = 0,

f ∈H.

(6.39)

Proof It follows from (6.28) that, for all Ψ ∈ D(A(f )∗ ), A(f )∗ Ψ, ΩH  = 0 = Ψ, 0. Hence ΩH ∈ D(A(f )∗∗ ) = D(A(f )) and (6.39) holds.  In fact, (6.39) characterizes the Fock vacuum: Proposition 6.2 Let D be a dense subspace of H . Suppose that Ψ ∈ ∩f ∈D D(A(f )) obeys A(f )Ψ = 0,

∀f ∈ D.

Then Ψ = αΩH for some α ∈ C. Proof For all Φ ∈ Fb,0 (H ), ∞  √  n Sn (f ⊗ Φ (n−1) ), Ψ (n) ) . 0 = Φ, A(f )Ψ  = A(f ) Φ, Ψ =



∗



n=1

258

6 Elements of the Theory of Boson Fock Spaces

Let n ∈ N be fixed and take Φ ∈ Fb,0 (H ) such that Φ (m) = 0 for m = n − 1. Then   Sn (f ⊗ Φ (n−1) ), Ψ (n) = 0, ∀Φ (n−1) ∈ ⊗n−1 H. s Since D is dense in H , the subspace span{Sn (f ⊗ Φ (n−1) )|f ∈ D, Φ (n−1) ∈ ⊗n−1 s H} is dense in ⊗ns H . Hence Ψ (n) = 0, n ≥ 1. Therefore Ψ = {Ψ (0) , 0, 0, . . .} = αΩH with α := Ψ (0).



The following proposition may have some interest (see Remark 6.2 below): Proposition 6.3 If A(f ) = A(g)∗ on Fb,0 (H ) (f, g ∈ H ), then f = g = 0. Proof By (6.39), we have A(g)∗ ΩH = 0. On the other hand, it follows from the definition of A(·)∗ that A(g)∗ ΩH = {0, g, 0, 0, . . .}. Hence g = 0. Then A(f ) = 0 on Fb,0 (H ). Hence, for all Ψ ∈ Fb,0 (H ), ΩH , A(f )Ψ  = 0. This implies that A(f )∗ ΩH = 0. Hence f = 0.  Remark 6.2 One can define operator-valued functionals A˜ and A˜ ∗ from H \ {0} to L(Fb (H )) by ˜ ) := A(f ), A(f

A˜ ∗ (f ) := A(f )∗ ,

f ∈ H \ {0}.

˜ ∩ Ran (A˜ ∗ ) = ∅. Proposition 6.3 tells us that Ran (A) For a class of vectors Ψ ∈ D(A(f )), A(f )Ψ is explicitly written: Lemma 6.4 Let n ≥ 1 be fixed. Then, for all Ψ ∈ Fb (H ) of the form Ψ (m) = δmn Sn (⊗n fj ) with fj ∈ H (m ≥ 0), (A(f )Ψ )(m) = 0,

m = n − 1,

1  f, fj Sn−1 (f1 ⊗ · · · ⊗ fˆj ⊗ · · · ⊗ fn ), (A(f )Ψ )(n−1) = √ n

(6.40)

n

j =1

where fˆj indicates the omission of fj .

(6.41)

6.4 Creation and Annihilation Operators

259

Proof Since Ψ (m) = 0 for all m = n, it follows that (A(f )Ψ )(m) = 0 for all m = n − 1. Hence we need only to prove (6.41). Let Φ ∈ Fb,0 (H ) be such that Φ (m) = 0, m = n − 1. Then 

 Φ (n−1) , (A(f )Ψ )(n−1) = Φ, A(f )Ψ   √   = A(f )∗ Φ, Ψ = n Sn (f ⊗ Φ (n−1) ), Ψ (n)  √  = n f ⊗ Φ (n−1) , Sn (⊗nj=1 fj )  1 1 f, fσ (1) = √ n (n − 1)! σ ∈Sn

One can rewrite



σ ∈Sn



as

n

j =1



  · Φ (n−1) , fσ (2) ⊗ · · · ⊗ fσ (n)

σ ∈Sn , σ (1)=j .

Then we obtain

   Φ (n−1) , (A(f )Ψ )(n−1) = Φ (n−1) , χ

with 1  f, fj Sn−1 (f1 ⊗ · · · fˆj ⊗ · · · ⊗ fn ). χ := √ n n

j =1

(n−1) = χ. Thus Since Φ (n−1) is an arbitrary element in ⊗n−1 s H , we have (A(f )Ψ ) we obtain (6.41). 

Proposition 6.4 For all f ∈ H \ {0}, A(f ) and A(f )∗ are unbounded. Proof For each N ∈ N, we define a vector ΨN ∈ Fb (H ) by ΨN(N) := ⊗N f/ f N and ΨN(n) = 0 for n = N. Then ΨN ∈ D(A(f )) and ΨN = 1. Moreover (A(f )ΨN )

(N−1)

√ N = ⊗N−1 f

f N−2

√ and (A(f )ΨN )(n) = 0, n = N − 1. Hence A(f )ΨN = N f → ∞ as N → ∞. Thus A(f ) is unbounded. Hence A(f )∗ also is unbounded.1  Proposition 6.5 Let f ∈ H . Then Fb,0 (H ) is a core for A(f ) and A(f )∗ .

general, if a densely defined closable operator A on a Hilbert space is unbounded, then A∗ is unbounded.

1 In

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6 Elements of the Theory of Boson Fock Spaces

Proof Let Ψ ∈ D(A(f )). For N ∈ N, we define ΨN ∈ Fb,0 (H ) by  (n)

ΨN =

Ψ (n) 0

for 0 ≤ n ≤ N . for n ≥ N + 1

It is easy to see that limN→∞ ΨN = Ψ . It follows from the definition of A(f ) that (A(f )ΨN )(n) = 0 for n ≥ N and (A(f )ΨN )(n) = (A(f )Ψ )(n) for 0 ≤ n ≤ N − 1. Hence

A(f )ΨN − A(f )Ψ 2 =

∞

(A(f )Ψ )(n) 2 → 0 (N → ∞).

n=N

Hence limN→∞ A(f )ΨN = A(f )Ψ . Thus Fb,0 (H ) is a core for A(f ). Similarly, one can show that Fb,0 (H ) is a core for A(f )∗ .  Using (6.29) repeatedly, one can see that, for all n ∈ N, and fj ∈ H (j = 1, · · · , n), √ (A(f1 )∗ · · · A(fn )∗ ΩH )(m) = δnm n!Sn (f1 ⊗ · · · ⊗ fn ), m ≥ 1. (6.42) This means that span{A(f1 )∗ · · · A(fn )∗ ΩH | fj ∈ H , j = 1, . . . , n} is dense in the n boson space ⊗ns H . Formula (6.42) also implies the following inequality: √ (6.43)

A(f1 )∗ · · · A(fn )∗ ΩH ≤ n! f1 · · · fn . It follows from (6.42) that, for each subspace D of H , ' Fb,fin (D) =span ΩH , A(f1 )∗ · · · A(fn )∗ ΩH | n ∈ N, ( fj ∈ D, j = 1, · · · , n .

(6.44)

Hence, in particular, if D is dense in H , then span{ΩH , A(f1 )∗ · · · A(fn )∗ ΩH | n ∈ N, fj ∈ D, j = 1, · · · , n} is dense in Fb (H ). Thus we have the following result: Theorem 6.6 If D is dense in H , ΩH is a cyclic vector of the operator set {A(f1 )∗ · · · A(fn )∗ |n ∈ N, fj ∈ D, j = 1, . . . , n}. This shows a basic role of the creation operators and the Fock vacuum. Moreover, (6.42) implies the following important inner product formula: for all fj , gk ∈ H , j = 1, · · · , n, k = 1, . . . , m (n, m ≥ 1), 

A(f1 )∗ · · · A(fn )∗ ΩH , A(g1 )∗ · · · A(gm )∗ ΩH   = δnm f1 , gσ (1) · · · fn , gσ (n) . σ ∈Sn

(6.45)

6.4 Creation and Annihilation Operators

261

The following corollary follows from (6.45), Theorems 1.21 and 6.6: Corollary 6.1 Let {ej }∞ j =1 be a CONS of H . Then 

Cj ···j ΩH , √1 n A(ej1 )∗ · · · A(ejn )∗ ΩH |n ∈ N, j1 ≤ · · · ≤ jn , jk ∈ N, n!  k = 1, . . . , n

is a CONS of Fb (H ), where Cj1 ···jn is defined by (1.55). Formula (6.42) implies the following proposition: Proposition 6.6 Let T be a densely defined closable operator and n ∈ N. Then, for all f1 , . . . , fn ∈ D(T ), A(f1 )∗ · · · A(fn )∗ ΩH ∈ D(db (T )) ∩ Γb (T ) and db (T )A(f1 )∗ · · · A(fn )∗ ΩH =

n

A(f1 )∗ · · · A(Tfj )∗ · · · A(fn )∗ ΩH ,

j =1

(6.46) ∗

∗

∗

∗

∗

Γb (T )A(f1 ) · · · A(fn ) ΩH = A(Tf1 ) A(Tf2 ) · · · A(Tfn ) ΩH .

(6.47)

Proof One needs only to recall the definitions of db (T ) and Γb (T ) and to apply (6.42).  In what follows, we denote by A(·)# either A(·) or A(·)∗ . A basic relation between A(f )# and Γb (·) is given as follows: Theorem 6.7 Let U be a unitary operator on H . Then, for all f ∈ H , Γb (U )A(f )Γb (U )−1 = A(Uf ),

(6.48)

Γb (U )A(f )∗ Γb (U )−1 = A(Uf )∗ .

(6.49)

Proof Let Ψ ∈ D(A(f )∗ ). Then √ (Γb (U )A(f )∗ Ψ )(n) = (⊗n U ) nSn (f ⊗ Ψ (n−1) ) √ = nSn (Uf ⊗ (⊗n−1 U )Ψ (n−1) ) √ = nSn (Uf ⊗ (Γb (U )Ψ )(n−1) ) = (A(Uf )∗ Γb (U )Ψ )(n) . Hence Ψ ∈ D(A(Uf )∗ Γb (U )) and Γb (U )A(f )∗ Ψ = A(Uf )∗ Γb (U )Ψ . This implies that Γb (U )A(f )∗ ⊂ A(Uf )∗ Γb (U ). Hence Γb (U )A(f )∗ Γ (U )−1 ⊂ A(Uf )∗ . Since f ∈ H is arbitrary, one can replace f with U −1 f to obtain

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6 Elements of the Theory of Boson Fock Spaces

Γb (U )A(U −1 f )∗ Γ (U )−1 ⊂ A(f )∗ . Note that U is an arbitrary unitary operator. Hence one can replace U −1 with U to obtain Γb (U )−1 A(Uf )∗ Γ (U ) ⊂ A(f )∗ , where we have used the fact that Γb (U −1 ) = Γb (U )−1 . Therefore A(Uf )∗ ⊂ Γb (U )A(f )∗ Γb (U )−1 . Thus operator equality (6.49) follows. The adjoint of (6.49) yields (6.48).  Corollary 6.2 Let T be a self-adjoint operator on H . Then, for all f ∈ H and t ∈ R, eit db (T ) A(f )# e−it db(T ) = A(eit T f )# .

(6.50)

Proof One needs only to apply Theorem 6.7 with U = eit T and to use Theorem 6.4(i).  Taking T = I and t = π in (6.50) and using the fact that eiπ = −1, we obtain eiπNb A(f )# e−iπNb = −A(f )# .

(6.51)

This shows that the correspondence A(f )# → −A(f )# is unitarily implementable by eiπNb .

6.5 Commutation Relations In this section we derive commutation relations of A(f )# ’s. For each r ∈ Z+ , we introduce Fb,r (H ) := {Ψ ∈ Fb (H ) | Ψ (n) = 0 for all n ≥ r + 1}. This is a closed subspace of Fb (H ), which is naturally isomorphic to ⊕rn=0 ⊗ns H . In the context of a Bose field theory, Fb,r (H ) is a Hilbert space of state vectors for the subsystem consisting of at most r bosons. Lemma 6.5 Let f ∈ H and r ≥ 0. We set Fb,−1 (H ) = {0}. (i) A(f )∗ is bounded operator from Fb,r (H ) to Fb,r+1 (H ) with

A(f )∗  Fb,r (H ) ≤

√ r + 1 f .

(ii) A(f ) is bounded operator from Fb,r (H ) to Fb,r−1 (H ) with

A(f )  Fb,r (H ) ≤

√ r f .

6.5 Commutation Relations

263

(iii) Let fn , f ∈ H and limn→∞ fn = f . Then, for all Ψ ∈ Fb,0 (H ), lim A(fn )Ψ = A(f )Ψ,

n→∞

lim A(fn )∗ Ψ = A(f )∗ Ψ.

n→∞

Proof Let Ψ ∈ Fb,r (H ). Then it is easy to see that A(f )Ψ ∈ Fb,r−1 (H ) and A(f )∗ Fb,r+1 (H ). (i) We have

A(f )∗ Ψ 2 =

r+1 √

nSn (f ⊗ Ψ (n−1) ) 2 n=1

≤ (r + 1) f 2

r+1

Ψ (n−1) 2 = (r + 1) f 2 Ψ 2 .

n=1

Hence the desired result follows. (ii) By a general fact in Hilbert space theory,2 we have for all Ψ ∈ Fb,r (H )

A(f )Ψ =

sup

Φ∈Fb,r−1 (H ), Φ =1

| Φ, A(f )Ψ  |.

We have  | Φ, A(f )Ψ  | = | A(f )∗ Φ, Ψ | ≤ A(f )∗ Φ Ψ . √ √ By (i), A(f )∗ Φ ≤ r f

Φ . Hence A(f )Ψ ≤ r f Ψ . Thus the desired result follows. (iii) Let Ψ ∈ Fb,0 (H ). Then there √ exists an r ≥ 0 such that Ψ ∈ Fb,r (H ). By (i),

A(fn )∗ Ψ − A(f )∗ Ψ ≤ r + 1 fn − f Ψ . Hence limn→∞ A(fn )∗ Ψ = A(f )∗ Ψ . Similarly, using (ii), we can prove limn→∞ A(fn )Ψ = A(f )Ψ .  As an application of Lemma 6.5, one can prove the following fact: Proposition 6.7 Let D be a dense subspace of H and f ∈ H . Then Fb,fin (D) is a core for A(f ) and A(f )∗ . Proof Let Ψ ∈ D(A(f )). By Proposition 6.5, for any ε > 0, there exists a vector Ψε ∈ Fb,0 (H ) such that Ψε − Ψ < ε and A(f )Ψε − A(f )Ψ < ε. There exists an r ∈ N such that Ψε ∈ Fb,r (H ). Since D is dense in H , there exists a vector

K be a Hilbert space and D be a dense subspace of K . Then, for all Ψ ∈ H , Ψ = supΦ∈D , Φ =1 | Φ, Ψ  |.

2 Let

264

6 Elements of the Theory of Boson Fock Spaces

Φε ∈ Fb,fin (D) such that Φε ∈ Fb,r (H ) and Φε − Ψε < ε. By Lemma 6.5, we have

A(f )Φε − A(f )Ψε ≤

√ √ r + 1 f Φε − Ψε ≤ r + 1 f ε.

Hence, by the triangle inequality, we obtain

Φε − Ψ < 2ε,

√

A(f )Φε − A(f )Ψ < ( r + 1 f + 1)ε.

This implies that Fb,fin (D) is a core for A(f ). Similarly, one can prove that Fb,fin (D) is a core for A(f )∗ .  The operators A(f ) and A(f )∗ (f ∈ H ) satisfy characteristic commutation relations: Theorem 6.8 For all f, g ∈ H , [A(f ), A(g)∗ ] = f, g , [A(f ), A(g)] = 0,

(6.52) ∗

∗

[A(f ) , A(g) ] = 0.

(6.53)

on Fb,0 (H ). Proof For n ≥ 1 and fj ∈ H (j = 1, . . . , n), we define a vector Ψn ∈ Fb,0 (H ) by  Ψn(m) :=

Sn (f1 ⊗ · · · ⊗ fn ) ; m = n . 0 ; m = n

Then

(A(f )Ψn )(m)

⎧ n 1  ⎪ ⎪ f, fj Sn−1 (f1 ⊗ · · · ⊗ fˆj ⊗ · · · ⊗ fn ) ; m = n − 1 √ ⎪ ⎨ n j =1 . = ⎪ ⎪ ⎪ ⎩ 0 ;m=  n−1

By (6.29), (A(g)∗ Ψn )(m) =

√ n + 1Sn+1 (g ⊗ f1 ⊗ · · · ⊗ fn ) ; m = n + 1 . 0 ; m = n + 1

6.5 Commutation Relations

265

Hence (A(f )A(g)∗ Ψn )(n) = f, g Sn (f1 ⊗ · · · ⊗ fn ) +

n 

f, fj Sn (g ⊗ f1 ⊗ · · · ⊗ fˆj ⊗ · · · ⊗ fn ),

j =1

(A(g)∗ A(f )Ψn )(n) =

n



f, fj Sn (g ⊗ f1 ⊗ · · · ⊗ fˆj ⊗ · · · ⊗ fn ).

j =1

Therefore ([A(f ), A(g)∗ ]Ψn )(n) = f, g Ψn(n) . If m = n, then (A(f )A(g)∗ Ψn )(m) = 0, (A(g)∗ A(f )Ψn )(m) = 0. Thus [A(f ), A(g)∗ ]Ψn = f, g Ψn . It is easy to see that [A(f ), A(g)∗ ]ΩH = f, g ΩH . Since any vector in Fb,fin (H ) is written as a finite linear combination of ΩH and vectors of the form Ψn (n ≥ 1, f1 , . . . , fn ∈ H ), it follows that, for all Ψ ∈ Fb,fin (H ), [A(f ), A(g)∗ ]Ψ = f, g Ψ.

(6.54)

We next consider the case where Ψ ∈ Fb,0 (H ). In this case, there exists an r ∈ N such that Ψ ∈ Fb,r (H ). Then there exists a sequence {ΨN }∞ N=1 in Fb,fin (H ) such that, for all N ≥ 1, ΨN ∈ Fb,r (H ) and ΨN → Ψ (N → ∞). By (6.54), A(f )A(g)∗ ΨN − A(g)∗ A(f )ΨN = f, g ΨN . By Lemma 6.5, A(f )A(g)∗ ΨN → A(f )A(g)∗ Ψ,

A(g)∗ A(f )ΨN → A(g)∗ A(f )Ψ

(N → ∞).

Hence, for all Ψ ∈ Fb,0 (H ), (6.54) holds. Namely, (6.52) holds on Fb,0 (H ). Similarly, one can prove (6.53).  Example 6.3 Consider the case where H is finite-dimensional with N = dim H and let {en }N n=1 be a CONS of H . Let An := A(en ),

n = 1, . . . , N.

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6 Elements of the Theory of Boson Fock Spaces

Then it follows that (Fb (H ), Fb,0 (H ), {An , A∗n |n = 1, . . . , N}) is a representation of the Dirac CCR with N degrees of freedom. Hence, in the case where H is infinite dimensional, (Fb (H ), Fb,0 (H ), {A(f ), A(f )∗ |f ∈ H }) gives an infinite-dimensional version of representation of the Dirac CCR with finite degrees of freedom. In view of Example 6.3, we call commutation relations (6.52) and (6.53) the Dirac canonical commutation relations (CCR) over H . But, in what follows, we sometimes call them the CCR over H simply if there is no danger of confusion. We shall discuss in Chap. 8 representation-theoretic aspects of CCR of this type in more details. With regard to the domain of A(f )# , we have in fact the following: Corollary 6.3 For all f ∈ H , D(A(f )) = D(A(f )∗ ) and, for all Ψ ∈ D(A(f )) (= D(A(f )∗ )),

A(f )∗ Ψ 2 = A(f )Ψ 2 + f 2 Ψ 2 .

(6.55)

Proof Using (6.52) with g = f , we have

A(f )∗ Ψ 2 = A(f )Ψ 2 + f 2 Ψ 2 ,

Ψ ∈ Fb,0 (H ).

(6.56)

Let Ψ ∈ D(A(f )). Then there exists a sequence {Ψn }n ⊂ Fb,0 (H ) such that Ψn → Ψ and A(f )Ψn → A(f )Ψ (n → ∞). Replacing Ψ by Ψn −Ψm in (6.56), we see that {A(f )∗ Ψn }n is a Cauchy sequence. Since A(f )∗ is closed, we conclude that Ψ ∈ D(A(f )∗ ) and A(f )∗ Ψ = limn→∞ A(f )∗ Ψn . Hence D(A(f )) ⊂ D(A(f )∗ ) and (6.55) holds. Similarly, one can show that D(A(f )∗ ) ⊂ D(A(f )).  Theorem 6.8 implies a useful formula: Corollary 6.4 Let n ∈ N and f, f1 , . . . , fn ∈ H . Then A(f )A(f1 )∗ · · · A(fn )∗ ΩH =

n 

∗ ∗  f, fj A(f1 )∗ · · · A(f j ) · · · A(fn ) ΩH ,

(6.57)

j =1 ∗ ∗  where A(f j ) indicates the omission of A(fj ) .

Proof By (6.52), we have A(f )A(f1 )∗ · · · A(fn )∗ ΩH = ([A(f ), A(f1 )∗ ] + A(f1 )∗ A(f ))A(f2 )∗ · · · A(fn )∗ ΩH = f, f1  A(f2 )∗ · · · A(fn )∗ ΩH + A(f1 )∗ A(f )A(f2 )∗ · · · A(fn )∗ ΩH . Iterating this equation and using (6.39), we obtain (6.57).



6.7 Relative Boundedness of Creation and Annihilation Operators

267

Corollary 6.5 Let D be a subspace of H . (i) For all f ∈ D, A(f )∗ Fb,fin (D) ⊂ Fb,fin (D). (ii) For all f ∈ H , A(f )Fb,fin (D) ⊂ Fb,fin (D). Proof (i) Since f ∈ D, it follows from (6.44) that A(f )∗ Fb,fin (D) ⊂ Fb,fin (D). (ii) This follows from (6.57).



6.6 Irreducibility In this section, we prove the following theorem: Theorem 6.9 Let D be a dense subspace of H . Then {A(f ), A(f )∗ | f ∈ D} = CI.

(6.58)

In particular, {A(f ), A(f )∗ | f ∈ D} is irreducible. Proof Let T ∈ {A(f ), A(f )∗ | f ∈ D} . Then T ∈ B(Fb (H )) and T A(f )# ⊂ A(f )# T ,

f ∈ D.

In particular, T ΩH ∈ D(A(f )) and A(f )T ΩH = T A(f )ΩH = 0 by (6.39). Hence, by Proposition 6.2, there exists a constant α ∈ C such that T ΩH = αΩH . Let Ψ := A(f1 )∗ · · · A(fn )∗ ΩH ,

fj ∈ D (j = 1, . . . , n).

Then, using the property that T A(f )∗ ⊂ A(f )∗ T , we have T Ψ = αΨ . Hence T = αI on Fb,fin (D). Since Fb,fin (D) is dense in Fb (H ) and T is bounded, it follows that T = αI . Hence (6.58) holds. The irreducibility of {A(f ), A(f )∗ | f ∈ D} follows from Proposition 2.9(i). 

6.7 Relative Boundedness of Creation and Annihilation Operators Let T be a non-negative self-adjoint operator on H and suppose that T is injective. Then, by Theorem 6.1(ii), db (T ) is a non-negative self-adjoint operator. Hence one can define the square root db (T )1/2 of db (T ) via functional calculus. The purpose of this section is to prove that A(f )# with f ∈ D(T −1/2 ) is relatively bounded with respect to db (T )1/2 . But, to prove this fact, we need some preliminaries.

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6 Elements of the Theory of Boson Fock Spaces

For each f ∈ H and n ≥ 0, j = 1, . . . , n + 1, there exists a unique linear operator Kj(n) (f ) from ⊗n H to ⊗n+1 H satisfying 6n H , D(Kj (f )) = ⊗

(6.59)

K1(0)(f )z = zf, z ∈ ⊗0 H = C, . 4 5 - j −1 . Kj(n) (f ) ⊗nk=1 Ψk = ⊗k=1 Ψk ⊗ f ⊗ ⊗nk=j Ψk ,

(6.60)

Ψk ∈ H , n ≥ 1, k = 1, . . . , n,

(6.61)

(n)

5 5 4 4 (n) = ⊗nk=1 Ψk ⊗ f . Indeed, one first where K1(n) (f ) = f ⊗ ⊗nk=1 Ψk and Kn+1 (n)

defines the operation of Kj (f ) to vectors of the form ⊗nk=1 Ψk (Ψk ∈ H , k = 6n H by linearity. This is 1, . . . , n) by (6.61) and then extends it to all vectors Ψ ∈ ⊗ well-defined, i.e., the extension Kj(n) (f )Ψ is independent of the way of representing Ψ as a linear combination of tensors of the form ⊗nk=1 Ψk , Ψk ∈ H , k = 1, . . . , n. 6n H , Lemma 6.6 Let f ∈ H and n ≥ 0, j = 1, . . . , n + 1. Then, for all Ψ ∈ ⊗

Kj(n) (f )Ψ = f

Ψ .

(6.62)

In particular, Kj(n) (f ) is bounded with Kj(n) (f ) = f . 6n H is represented as follows: Proof Each vector Ψ in ⊗ Ψ =

M

(m)

⊗nk=1 Ψk

m=1 (m)

with M ∈ N, Ψk

∈ H , k = 1, . . . , n, m = 1, . . . , M. Let Dk := span{Ψk(m) | m = 1, . . . , M}.

Then dk := dim Dk ≤ M. Hence we can take a CONS {ei(k) | i = 1, . . . , dk } of Dk .  k (km) (k) (m) (km) (k) (m) Hence we have Ψk = di=1 ci ei with ci := ei , Ψk . Therefore Ψ =

d1 i1 =1

where ci1 ···in := (n)

- n

M

Kj (f )Ψ =

m=1 d1 i1 =1

···

···

dn in =1

(km) k=1 cik

dn in =1

ci1 ···in ei(1) ⊗ · · · ⊗ ei(n) , n 1

. . Hence (1)

(j −1)

(j )

(n)

ci1 ···in ei1 ⊗ . . . ⊗ eij−1 ⊗ f ⊗ eij ⊗ · · · ⊗ ein .

6.7 Relative Boundedness of Creation and Annihilation Operators

269

By direct computation, we have d1

(n)

Kj (f )Ψ 2 = f 2

···

i1 =1

dn

|ci1 ···in |2 = f 2 Ψ 2 .

in =1

Thus (6.62) holds and hence Kj(n) (f ) is bounded with Kj(n) (f ) = f .



Lemma 6.7 Let f ∈ H and n ∈ Z+ . Then 1 (n) Kj (f )Ψ, n+1 n+1

Sn+1 (f ⊗ Ψ ) =

6 n H ). Ψ ∈ Sn (⊗

(6.63)

j =1

Proof For each vector η = Sn (Ψ1 ⊗ · · · ⊗ Ψn ) (Ψj ∈ H , j = 1, . . . , n), Sn+1 (f ⊗ η) = Sn+1 (f ⊗ Ψ1 ⊗ · · · ⊗ Ψn ) 1 = (f ⊗ Ψσ (1) ⊗ · · · ⊗ Ψσ (n) (n + 1)! σ ∈Sn

+Ψσ (1) ⊗ f ⊗ Ψσ (2) ⊗ · · · ⊗ Ψσ (n) · · · + Ψσ (1) ⊗ · · · ⊗ Ψσ (n) ⊗ f ) 1 (n) Kj (f )η. n+1 n+1

=

j =1



Hence, by linearity, (6.63) follows. (T )1/2 -boundedness

A(f )#

We are now ready to prove the db of D(T −1/2 ), which plays a basic role in analysis on boson Fock spaces.

with f ∈

Theorem 6.10 Let T be a non-negative and injective self-adjoint operator on H . Then, for all f ∈ D(T −1/2 ), D(db (T )1/2 ) ⊂ D(A(f )) ∩ D(A(f )∗ ) and, for all Ψ ∈ D(db (T )1/2 ),

A(f )Ψ ≤ T −1/2 f db (T )1/2 Ψ , ∗

2

A(f ) Ψ ≤ T

−1/2

2

f db (T )

1/2

(6.64) 2

2

2

Ψ + f Ψ .

(6.65)

270

6 Elements of the Theory of Boson Fock Spaces (1/2)

Proof For each n ∈ N, j = 1, . . . , n, we define a linear operator Tn,j as follows:

on ⊗n H

j th

(1/2) Tn,j



:= I ⊗ · · · ⊗ I ⊗ T 1/2 ⊗I ⊗ · · · ⊗ I.

6 n H ), Then, for all Ψ (n) ∈ Sn (⊗ Kj (f )Ψ (n) = Tn+1,j Kj (T −1/2 f )Ψ (n) . (1/2)

(n)

(n)

(6.66)

Let Φ ∈ Fb,fin (D(T )). Then, by Lemma 6.7 and (6.66), we have 

   √ (A(f )Φ)(n) , Ψ (n) = Φ (n+1) , n + 1Sn (f ⊗ Ψ (n) )   1 (j ) Φ (n+1) , Kn (f )Ψ (n) = √ n + 1 j =1 n+1

  (1/2) 1 = √ Tn+1,j Φ (n+1) , Kj(n) (T −1/2 f )Ψ (n) . n + 1 j =1 n+1

Hence, by the Schwarz inequality and (6.62),      (A(f )Φ)(n) , Ψ (n)  n+1 1 (1/2) (n+1)

T Φ

Kj(n) (T −1/2 f )Ψ (n)

≤ √ n + 1 j =1 n+1,j

1

≤ √ n+1

⎞1/2 ⎛ ⎞1/2 ⎛ n+1 n+1 (1/2) (n+1) 2 ⎠ (n) ⎝ ⎝

T Φ

K (T −1/2 f )Ψ (n) 2 ⎠ j

n+1,j

j =1

⎛ = T −1/2 f ⎝

n+1

j =1

⎞1/2

Tn+1,j Φ (n+1) 2 ⎠ (1/2)

Ψ (n) .

j =1

On the other hand, n+1 j =1

  (1/2)

Tn+1,j Φ (n+1) 2 = Φ (n+1) , Tb(n+1) Φ (n+1) .

6.7 Relative Boundedness of Creation and Annihilation Operators

271

Hence       (n+1) (n+1) 1/2 Φ

Ψ (n) .  (A(f )Φ)(n) , Ψ (n)  ≤ T −1/2 f Φ (n+1) , Tb 6n H ) is dense in ⊗ns H , it follows3 that Since Sn (⊗   (n+1) (n+1) 1/2

(A(f )Φ)(n) ≤ T −1/2 f Φ (n+1) , Tb Φ . Hence

A(f )Φ 2 ≤ T −1/2 f 2 Φ, db (T )Φ = T −1/2 f 2 db (T )1/2 Φ 2 , i.e.,

A(f )Φ ≤ T −1/2 f db (T )1/2Φ .

(6.67)

Since Fb,fin (D(T )) is a core for db (T ) by Theorem 6.1(iii), it is a core for db (T )1/2 (apply Proposition 1.13). Hence, by a limiting argument, (6.67) extends to all Φ ∈ D(db (T )1/2 ) with D(db (T )1/2 ) ⊂ D(A(f )). Thus (6.64) holds. Inequality (6.65) follows from (6.64) and Corollary 6.3.  Remark 6.3 It should be noted that, in Theorem 6.10, separability of H is not assumed. In the case where H is separable, there is another (easier) proof of Theorem 6.10 using the multiplication operator form of spectral theorem (see [28, Remark 5.8]). Taking T = I in Theorem 6.10, we obtain an estimate showing the relative 1/2 boundedness of A(·)# with respect to Nb : Corollary 6.6 For all f ∈ H , D(Nb ) ⊂ D(A(f )) ∩ D(A(f )∗ ) and 1/2

1/2

A(f )Ψ ≤ f Nb Ψ ,

A(f )∗ Ψ ≤ f (Nb + 1)1/2Ψ ,

(6.68) Ψ ∈

1/2 D(Nb ).

(6.69)

Proof Estimate (6.68) is a direct consequence of (6.64). As for (6.69), we need only 1/2 1/2 to note that Nb Ψ 2 + Ψ 2 = (Nb + 1)Ψ 2 , Ψ ∈ D(Nb ).  Corollary 6.7 Let T be a non-negative and injective self-adjoint operator on H .

any vector Φ in a Hilbert space X , Φ = supΨ ∈D ,Ψ =0 | Φ, Ψ  |/ Ψ , where D is a dense subspace of X .

3 For

272

6 Elements of the Theory of Boson Fock Spaces

(i) Let f, fn ∈ D(T −1/2 ) and limn→∞ T −1/2 fn = T −1/2 f . Then, for all Ψ ∈ D(db (T )1/2 ), lim A(fn )Ψ = A(f )Ψ.

n→∞

(6.70)

(ii) Let f, fn ∈ D(T −1/2 ) and limn→∞ fn = f , limn→∞ T −1/2 fn = T −1/2 f . Then, for all Ψ ∈ D(db (T )1/2 ), lim A(fn )∗ Ψ = A(f )∗ Ψ.

n→∞

(6.71)

(iii) Let f ∈ D(T p ) ∩ D(T −1/2 ) with p ∈ N and Ψ ∈ D(db (T )1/2 ). Then the mapping: R t → A(eit T f )# Ψ is p times strongly differentiable and the strong derivative of the p-th order is given by dp A(eit T f )# Ψ = A(i p eit T T p f )# Ψ. dt p

(6.72)

Proof (i) By (6.64), we have

A(fn )Ψ − A(f )Ψ ≤ T −1/2 fn − T −1/2 f db (T )1/2 Ψ → 0 (n → ∞).

Thus (6.70) holds. (ii) By (6.65), we have

A(fn )∗ Ψ − A(f )∗ Ψ ≤ T −1/2 fn − T −1/2 f db (T )1/2 Ψ

+ fn − f Ψ → 0 (n → ∞). Thus (6.71) holds. (iii) Let Ψ ∈ D(db (T )1/2). We first consider the case p = 1. Hence f ∈ D(T ) ∩ D(T −1/2 ). For ε ∈ R \ {0}, we define Bε :=

1 (A(ei(t +ε)T f )# Ψ − A(eit T f )# Ψ ). ε

Then Bε = A(gε )# Ψ with gε := ε−1 (eiεT − 1)eit T f . Since eit T f ∈ D(T ), it follows that limε→0 gε = iT eit T f = ieit T Tf . By the functional calculus for T , gε ∈ D(T −1/2 ) with T −1/2 gε = ε−1 (eiεT − 1)eit T T −1/2 f . It is obvious that T −1/2 f ∈ D(T ). Hence limε→0 T −1/2 gε = iT eit T T −1/2 f = T −1/2 (iT eit T f ). Hence, by (i) and (ii), limε→0 Bε = A(ieit T Tf )# Ψ . Hence A(eit T f )# Ψ is strongly differentiable in t and (6.72) with p = 1 holds. Suppose that the statement of (iii) holds for some p. Then we can apply the preceding argument to f replaced by i p T p f to conclude that the statement of (iii) holds for p replaced by p + 1. 

6.7 Relative Boundedness of Creation and Annihilation Operators

273

Corollary 6.8 1/2

(i) Let f, fn ∈ H , limn→∞ fn = f . Then, for all Ψ ∈ D(Nb ), lim A(fn )# Ψ = A(f )# Ψ.

n→∞

(6.73)

(ii) Let T be a self-adjoint operator on H , f ∈ D(T p ) with p ∈ N and 1/2 Ψ ∈ D(Nb ). Then the mapping: R t → A(eit T f )# Ψ is p times strongly differentiable and the strong derivative of the p-th order is given by (6.72). Proof (i) This follows from (i) and (ii) with T = I in Corollary 6.7. (ii) This follows from (i) and the proof of Corollary 6.7(iii).



Theorem 6.10 also implies infinitesimal smallness of A(f )# (f ∈ D(T −1/2 )) with respect to db (T ): Corollary 6.9 Let T be a non-negative and injective self-adjoint operator on H and f ∈ D(T −1/2 ). Then, for all ε > 0 and Ψ ∈ D(db (T )), 1

T −1/2 f 2 Ψ , 4ε 0 1 1

T −1/2 f 2 + f Ψ .

A(f )∗ Ψ ≤ ε db (T )Ψ + 4ε

A(f )Ψ ≤ ε db (T )Ψ +

(6.74) (6.75)

Proof We estimate the right hand side of (6.64) and (6.65). Let Ψ ∈ D(db (T )). Then, by the Schwarz inequality, we have

db (T )1/2 Ψ =

 Ψ, db (T )Ψ  ≤ Ψ 1/2 db (T )Ψ 1/2 .

Using the elementary inequality ab ≤ εa 2 +

1 2 b 4ε

(a > 0, b > 0, ε > 0),

(6.76)

we obtain

T −1/2 f db (T )1/2 Ψ ≤ ε db (T )Ψ +

1

T −1/2 f 2 Ψ . 4ε

This estimate and (6.64) yield (6.74). Inequality (6.65) implies that

A(f )∗ Ψ ≤ T −1/2 f db (T )1/2 Ψ + f

Ψ . Hence we obtain (6.75).



274

6 Elements of the Theory of Boson Fock Spaces

6.8 Application to Self-adjointness Corollary 6.9 has applications to proving self-adjointness of symmetric operators defined as a perturbation of db (T ). Here we present a basic example. For a selfadjoint operator T on H and a vector g ∈ H , we define 1 Hv := db (T ) + √ (A(g)∗ + A(g)). 2

(6.77)

√ It is easy to see that (A(g) + A(g)∗ )/ 2 is a symmetric operator with D(A(g)) ∩ D(A(g)∗ ) ⊃ Fb,0 (H ) ⊃ Fb,fin (D(T )). Hence Fb,fin (D(T )) ⊂ D(Hv ). Since Fb,fin (D(T )) is dense in Fb (H ), it follows that Hv is a symmetric operator. Theorem 6.11 Assume that T is a non-negative injective self-adjoint operator and g ∈ D(T −1/2 ).

(6.78)

Then Hv is self-adjoint with D(Hv ) = D(db (T )) and bounded from below. Moreover, any core for db (T ) is a core for Hv . √ Proof Let H1 := (A(g)∗ + A(g))/ 2. Then, by Corollary 6.9, D(db (T )1/2 ) ⊂ D(H1 ). Hence D(db (T )) ⊂ D(H1 ) and D(Hv ) = D(db (T )). Let Ψ ∈ D(db (T )). Then, by (6.74) and (6.75), we have 1

H1 Ψ ≤ √ ( A(g)∗ Ψ + A(g)Ψ ) ≤ ε db (T )Ψ + cε Ψ , 2 where ε > 0 is arbitrary and cε > 0 is a constant depending on ε, T −1/2 g and

g . Take ε < 1. Then, by the Kato–Rellich theorem (see Theorem 1.15), we obtain the desired result.  In the context of quantum field theory, the operator Hv denotes the Hamiltonian of an abstract quantum field model, called an abstract van Hove model or an abstract van Hove–Miyatake model (see Sect. 10.9 in the present book).

6.9 Representations of Boson Second Quantization Operators in Terms of A(·)# In this section we derive some formulae for boson second quantization operators in terms of creation and annihilation operators.

6.9 Representations of Boson Second Quantization Operators in Terms of A(·)#

275

Lemma 6.8 Let T ∈ L(H ) be densely defined and S ∈ L(H ). Then, for all f ∈ D(T ∗ ) ∩ D(S) and Ψ ∈ Fb,0 (H ), A(Sf )∗ A(T ∗ f )Ψ = db (PT ∗ f,Sf )Ψ,

(6.79)

where Pf,g (f, g ∈ H ) is defined by (1.22). Proof If Ψ = αΩH (α ∈ C), then (6.79) obviously holds with both sides being the zero vector. Let Ψ be of the form Ψ = A(f1 )∗ · · · A(fp )∗ ΩH

(f1 , . . . , fp ∈ H )

(6.80)

with p ≥ 1. Then we have by CCR A(Sf )∗ A(T ∗ f )Ψ =

p

j th

  A(f1 )∗ · · · A( T ∗ f, fj Sf )∗ · · · A(fp )∗ ΩH

j =1

=

p

j th



∗

A(f1 ) · · · A(PT ∗ f,Sf fj )∗ · · · A(fp )∗ ΩH

j =1

= db (PT ∗ f,Sf )Ψp , where we have used (6.46). Hence (6.79) holds. Then, by linearity, (6.79) holds for all Ψ ∈ Fb,fin (H ). Recall that Fb,fin (H ) is dense in Fb,0 (H ) and p A(Sf )∗ A(T ∗ f ) is a bounded operator on ⊗s H for each p ≥ 1. Moreover PT ∗ f,Sf is a bounded operator. Hence it follows from a limiting argument that (6.79) can be extended to all Ψ ∈ Fb,0 (H ).  Theorem 6.12 Let S, T ∈ B(H ) and {en }∞ n=1 be a CONS of H . Then, for all Ψ ∈ Fb,0 (H ), ∞

A(Sen )∗ A(T ∗ en )Ψ = db (ST )Ψ.

(6.81)

n=1

Proof There exists an integer p0 such that, for all p > p0 , Ψ (p) = 0. Hence we need only to prove (6.81) for Ψ (p) with p ≤ p0 . For each N ∈ N, we define an operator PN by PN :=

N n=1

PT ∗ en ,Sen .

276

6 Elements of the Theory of Boson Fock Spaces

We have by Lemma 6.8 N

A(Sen )∗ A(T ∗ en )Ψ (p) = LN Ψ (p) ,

n=1

where LN = PN ⊗ I ⊗ · · · ⊗ I + I ⊗ PN ⊗ · · · ⊗ I + · · · + I ⊗ · · · ⊗ I ⊗ PN . For all g ∈ H , we have by the Cauchy–Schwarz inequality and the Bessel inequality | g, PN f  | ≤

N

| en , Tf  g, Sen  |

n=1

≤(

N

| en , Tf  |2 )1/2 (

n=1

N  | S ∗ g, en |2 )1/2 n=1

∗

≤ Tf S g ≤ T S ∗ f g . Hence the operator norm PN of PN is uniformly bounded with

PN ≤ T S ∗ . It is easy to see that s- limN→∞ PN = ST . Hence, by an application of Proposition 1.25, we obtain (p)

lim LN Ψ (p) = (ST )b Ψ (p) = (db (ST )Ψ )(p) .

N→∞



Thus (6.81) follows.

Corollary 6.10 Let S be a bounded self-adjoint operator on H and {en }∞ n=1 be a 2 1/2 ∞ CONS of H . Then, for all Ψ ∈ D(db (S ) ), Ψ ∈ ∩n=1 D(A(Sen )) and ∞

A(Sen )Ψ 2 = db (S 2 )1/2 Ψ 2 .

(6.82)

n=1 1/2

In particular, for all Ψ ∈ D(Nb ), ∞ n=1

1/2

A(en )Ψ 2 = Nb Ψ 2 .

(6.83)

6.9 Representations of Boson Second Quantization Operators in Terms of A(·)#

277

Proof We apply (6.81) with T ∗ = S to obtain ∞

A(Sen )Ψ 2 = db (S 2 )1/2 Ψ 2

n=1

for all Ψ ∈ Fb,0 (H ). Let Ψ ∈ D(db (S 2 )1/2) next. For each p ∈ Z+ , we define  Ψp ∈ Fb,0 (H ) by Ψp(k) := δpk Ψ (p) (k ∈ Z+ ). Then we have Ψ = ∞ p=0 Ψp and

db (S 2 )1/2 Ψ 2 =

∞

db (S 2 )1/2 Ψp 2 ,

p=1

where we have used that db (S 2 )1/2 Ψ0 = 0. By the preceding result, we have

db (S 2 )1/2 Ψp 2 =

∞

A(Sen )Ψp 2 .

n=1

Hence

db (S 2 )1/2 Ψ 2 =

∞ ∞

A(Sen )Ψp 2 .

p=1 n=1

This implies that

∞

2 p=1 A(Sen )Ψp

< ∞ and

db (S 2 )1/2 Ψ 2 =

∞ ∞

A(Sen )Ψp 2 .

n=1 p=0

Note that (A(Sen )Ψp )(k) = δk,p−1 (A(Sen )Ψ )(k) . Hence, for all M ∈ N, M p=1 (q)

A(Sen )Ψp = 2

M−1

(A(Sen )Ψ )(q) 2 = A(Sen )ΨM 2 ,

q=0 (q)

where ΨM := Ψ (q) for 0 ≤ q ≤ M and ΨM = 0 for q ≥ M + 1. Hence

db (S 2 )1/2 ΨM 2 ≥ A(Sen )ΨM |2 ,

db (S 2 )1/2 (ΨM − ΨN ) 2 ≥ A(Sen )(ΨM − ΨN ) 2 ,

M > N.

Since db (S 2 )1/2 (ΨM −ΨN ) → 0 as M, N → ∞, it follows that, for each n ∈ N, {A(Sen )ΨM }M is a Cauchy sequence. Since A(Sen ) is closed, one can conclude

278

6 Elements of the Theory of Boson Fock Spaces

that Ψ ∈ D(A(Sen )) and A(Sen )ΨM → A(Sen )Ψ as M → ∞. In particular, Ψ ∈ ∩∞ n=1 D(A(Sen )). Then we have ∞

A(Sen )Ψp 2 = A(Sen )Ψ 2 .

p=0

Thus (6.82) follows. Formula (6.83) is just a special case of (6.82) with S = I .



We next consider the case where S or T is unbounded. Theorem 6.13 Let T be a densely defined closable operator on H and S ∈ B(H ) ∗ such that ST is closable. Let {en }∞ n=1 ⊂ D(T ) be a CONS of H . Then db (ST )Ψ =

∞

A(Sen )∗ A(T ∗ en )Ψ,

Ψ ∈ Fb,fin (D(T )).

(6.84)

n=1

Proof Similar to the proof of Theorem 6.12. But, in the present case, vectors Ψ on which db (ST ) and A(Sen )∗ A(T ∗ en ) act are restricted to those in Fb,fin (D(T )).  in formula (6.84) may be in that the approximating operators NA shortcoming ∗ A(T ∗ e ) (N = 1, 2, . . .) are not necessarily symmetric even if ST A(Se ) n n n=1 is self-adjoint. For a class of non-negative self-adjoint operators T on H , this point can be improved: Theorem 6.14 Let T be a non-negative self-adjoint operator on H such that there 1/2 ) of H satisfying exists a CONS {en }∞ n=1 ⊂ D(T lim

N

N→∞

en , f  T 1/2 en = T 1/2 f,

f ∈ D(T 1/2 ).

(6.85)

A(T 1/2 en )∗ A(T 1/2 en )Ψ.

(6.86)

n=1

Then, for all Ψ ∈ Fb,fin (D(T )), db (T )Ψ =

∞ n=1

Proof Let Ψ be given by (6.80) with fj ∈ D(T ) (j = 1, . . . , p). Then, in the same way as in the proof of Theorem 6.13, one can show that, for each N ∈ N, N

A(T

1/2

∗

en ) A(T

1/2

A(f1 )∗ · · · A(Fj,N )∗ · · · A(fp )∗ ΩH ,

j =1

n=1

where Fj,N := Thus (6.86) holds.

en )Ψp =

p

N n=1



en , T 1/2 fj T 1/2 en . By (6.85), limN→∞ Fj,N = Tfj . 

6.9 Representations of Boson Second Quantization Operators in Terms of A(·)#

279

Example 6.4 Let T be a non-negative unbounded self-adjoint operator on H . Suppose that σ (T ) = σd (T ) (the discrete spectrum of T ) = {λn }∞ n=1 with 0 ≤ λ1 ≤ λ2 ≤ · · · ≤ λn ≤ λn+1 ≤ · · · , counting multiplicities, and λn → ∞ as n → ∞. Then there exists a CONS {ϕn }∞ n=1 of H such that T ϕn = λn ϕn , n ∈ N (ϕn ’s are 1/2 eigenvectors of T ). We have T 1/2 ϕn = λn ϕn . Hence, for all f ∈ D(T 1/2 ), N

ϕn , f  T 1/2 ϕn =

n=1

N 

 ϕn , T 1/2 f ϕn → T 1/2 f.

n=1

Hence (6.85) with en = ϕn holds. Thus, in this case too, (6.86) holds with en = ϕn . A weak form of (6.84) is given as follows: Theorem 6.15 Let S and T be densely defined closable operators on H such that D(S) ∩ D(T ∗ ) is dense in H and ST is a densely defined closable operator. Let ∗ {en }∞ n=1 ⊂ D(S) ∩ D(T ) be a CONS of H . Then, for all Ψ ∈ Fb,fin (D(ST )) and ∗ Φ ∈ Ffin (D(S )), ; Φ, db (ST )Ψ  = lim

N→∞

Φ,

N

< A(Sen )∗ A(T ∗ en )Ψ .

(6.87)

n=1

In particular,

db (T ∗ T )1/2 Ψ 2 =

∞

A(T ∗ en )Ψ 2 ,

Ψ ∈ Fb,fin (D(T ∗ T )).

(6.88)

n=1

Proof Let Ψ = A(f1 )∗ · · · A(fp )∗ ΩH and Φ = A(g1 )∗ · · · A(gp )∗ ΩH with fj ∈ D(ST ) and gk ∈ D(S ∗ ) (j, k = 1, . . . , p). Then, as in the proof of Theorem 6.13, we have ΘN :=

N

A(Sen )∗ A(T ∗ en )Ψ =

A(f1 )∗ · · · A(Fj,N )∗ · · · A(fp )∗ ΩH ,

j =1

n=1

where Fj,N :=

p

N n=1

Φ, ΘN  =



en , Tfj Sen . Hence

p 

  gσ (1) , f1 · · · gσ (j ) , Fj,N · · · gσ (p) , fp .

j =1 σ ∈Sp

We have    lim g, Fj,N = S ∗ g, Tfj = g, STfj ,

N→∞

g ∈ D(S ∗ ).

280

6 Elements of the Theory of Boson Fock Spaces

Hence lim Φ, ΘN  =

N→∞

p 

  gσ (1) , f1 · · · gσ (j ) , STfj · · · gσ (p) , fp

j =1 σ ∈Sp

= Φ, db (ST )Ψ  . By linearity, this extends to all Ψ ∈ Fb,fin (D(ST )) and Φ ∈ Fb,fin (D(S ∗ )). ∗ Hence (6.87) holds. Note that T ∗ T = T T is a non-negative self-adjoint operator on H . Hence db (T ∗ T ) is a non-negative self-adjoint operator on Fb (H ). Therefore db (T ∗ T )1/2 is defined via functional calculus. Taking Φ = Ψ and S = T ∗ in (6.87), we obtain (6.88).  Corollary 6.11 Let T be a non-negative self-adjoint operator on H and {en }∞ n=1 ⊂ D(T 1/2 ) be a CONS of H . Then, for all Ψ ∈ D(db (T )1/2 ), ∞

A(T 1/2 en )Ψ 2 ≤ db (T )1/2 Ψ 2 .

(6.89)

n=1

Proof Since Fb,fin (D(T )) is a core for db (T )1/2 by Theorem 6.1(iii), for each Ψ ∈ D(db (T )1/2 ), there exists a sequence {Ψk }k in Fb,fin (D(T )) such that limk→∞ Ψk = Ψ and limk→∞ db (T )1/2Ψk = db (T )1/2 Ψ . By (6.88) with T replaced by T 1/2 , we have ∞

A(T 1/2 en )Ψk 2 = db (T )1/2 Ψk 2

n=1

and

A(T 1/2 en )Ψk − A(T 1/2 en )Ψ ≤ db (T )1/2 Ψk − db (T )1/2 Ψ → 0 (k → ∞). In particular, limk→∞ A(T 1/2 en )Ψk = A(T 1/2en )Ψ . Hence, by applying Fatou’s theorem, we obtain (6.89).  Remark 6.4 In fact, the equality in (6.89) holds: ∞ n=1

A(T 1/2 en )Ψ 2 = db (T )1/2 Ψ 2 ,

Ψ ∈ D(db (T )1/2).

(6.90)

6.10 Commutation Relations Between A(f )# and db (T )

281

Hence, by the polarization identity, we have 

∞    db (T )1/2 Φ, db (T )1/2 Ψ = A(T 1/2 en )Φ, A(T 1/2 en )Ψ , n=1

Φ, Ψ ∈ D(db (T )1/2 ). (6.91) But, to prove (6.90), we need to pass to another representation of {A(f ), A(f )∗ |f ∈ H } where T is represented by a multiplication operator on an L2 -space. This representation is obtained by employing the multiplication operator form of the spectral theorem. For the details, see [28, Theorem 5.21].

6.10 Commutation Relations Between A(f )# and db (T ) We next consider commutation properties of boson second quantization operators with boson creation and annihilation operators. Proposition 6.8 Let T be a densely defined closable operator on H . (i) For all f ∈ D(T ), Fb,fin (D(T )) ⊂ D([db (T ), A(f )∗ ]) and [db (T ), A(f )∗ ] = A(Tf )∗ on Fb,fin (D(T )). (ii) For all f ∈ D(T ∗ ), Fb,fin (D(T )) ⊂ D([db (T ), A(f )]) and [db (T ), A(f )] = −A(T ∗ f ) on Fb,fin (D(T )). Proof (n) with n ≥ 1 is (i) Let Ψ = {Ψ (n) }∞ n=0 ∈ Fb,fin (D(T )). Then, by (6.44), each Ψ a linear combination of vectors of the form

η = A(f1 )∗ · · · A(fn )∗ ΩH

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6 Elements of the Theory of Boson Fock Spaces

with fj ∈ D(T ), j = 1, . . . , n. By Proposition 6.6, η ∈ D(db (T )) and db (T )η =

n

A(f1 )∗ · · · A(Tfj )∗ · · · A(fn )∗ ΩH .

j =1

Hence db (T )η ∈ D(A(f )∗ ) and A(f )∗ db (T )η =

n

A(f )∗ A(f1 )∗ · · · A(Tfj )∗ · · · A(fn )∗ ΩH .

j =1

Since f is in D(T ), A(f )∗ η ∈ D(db (T )) and db (T )A(f )∗ η = A(Tf )∗ η +

n

A(f )∗ A(f1 )∗ · · · A(Tfj )∗ · · · A(fn )∗ ΩH .

j =1

Hence [db (T ), A(f )∗ ]η = A(Tf )∗ η. It is easy to see that [db (T ), A(f )∗ ]ΩH = A(Tf )∗ ΩH . Thus the desired result follows. (ii) Let f ∈ D(T ∗ ). Let Ψ = A(f1 )∗ · · · A(fn )∗ ΩH with fj ∈ D(T ) (j = 1, . . . , n). Then, by (6.57), we see as in (i) that Ψ ∈ D([db (T ), A(f )]) and [db (T ), A(f )]Ψ = −A(T ∗ f )Ψ Thus the desired result follows.



In the case where T is self-adjoint, non-negative and injective, Proposition 6.8 can be more refined: Theorem 6.16 Let T be a non-negative and injective self-adjoint operator on H . (i) For all Ψ, Φ ∈ D(db (T )) and f ∈ D(T ) ∩ D(T −1/2 ),  db (T )Ψ, A(f )Φ − A(f )∗ Ψ, db (T )Φ = − Ψ, A(Tf )Φ ,   db (T )Ψ, A(f )∗ Φ − A(f )Ψ, db (T )Φ = Ψ, A(Tf )∗ Φ .

(6.92) (6.93)

6.11 Segal Field Operators

283

(ii) For each f ∈ D(T ) ∩ D(T −1/2 ), A(f )# maps D(db (T )3/2 ) to D(db (T )) and satisfies commutation relations [db (T ), A(f )∗ ] = A(Tf )∗ , [db (T ), A(f )] = −A(Tf ) on D(db (T )3/2). Proof (i) By Proposition 6.8, for all Ψ ∈ Fb,fin (D(T )) and Φ ∈ D(db (T )), (6.92) holds. Since Fb,fin (D(T )) is a core for db (T ), for any Ψ ∈ D(db (T )), there exists a sequence {Ψn }n in Fb,fin (D(T )) such that Ψn → Ψ and db (T )Ψn → db (T )Ψ (n → ∞). We have  db (T )Ψn , A(f )Φ − A(f )∗ Ψn , db (T )Φ = − Ψn , A(Tf )Φ . By (6.75), A(f )∗ Ψn → A(f )∗ Ψ (n → ∞). Hence, taking the limit n → ∞ in the above equation, we obtain (6.92). Formula (6.93) follows from taking the complex conjugate of (6.92). (ii) Let Ψ ∈ D(db (T )3/2) and Φ ∈ D(db (T )). Then db (T )Ψ ∈ D(db (T )1/2 ) ⊂ D(A(f )∗ ) ∩ D(A(f )). Hence, by (i), 4 5  A(f )∗ db (T ) + A(Tf )∗ Ψ, Φ = A(f )∗ Ψ, db (T )Φ . This means that A(f )∗ Ψ ∈ D(db (T )) and 5 4 db (T )A(f )∗ Ψ = A(f )∗ db (T ) + A(Tf )∗ Ψ. Hence the statement on A(f )∗ is proved. Similarly, one can prove the statement on A(f ). 

6.11 Segal Field Operators Operators on Fb (H ) can be constructed from A(f ) and A(f )∗ (f ∈ H ). A simple example of such operators is given by A(f ) + A(f )∗ . As already seen in Sect. 6.8, it is a symmetric operator. Hence one can define a closed symmetric operator 1 ΦS (f ) := √ (A(f ) + A(f )∗ ), 2

f ∈H,

(6.94)

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6 Elements of the Theory of Boson Fock Spaces

√ where the constant 1/ 2 is just for convenience. The operator ΦS (f ) is called the Segal field operator with test vector f , which plays a fundamental role in applications to Bose field theory. The purpose of this section is to investigate basic properties of Segal field operators.

6.11.1 Basic Properties Proposition 6.9 Let T be a non-negative and injective self-adjoint operator on H. (i) For all f ∈ D(T −1/2 ), D(db (T )1/2 ) ⊂ D(A(f )) ∩ D(A(f )∗ ) and, for all Ψ ∈ D(db (T )1/2 ), 1 ΦS (f )Ψ = √ (A(f ) + A(f )∗ )Ψ 2

(6.95)

and

ΦS (f )Ψ ≤

√ 1 2 T −1/2 f db (T )1/2 Ψ + √ f Ψ . 2 1/2

(6.96)

1/2

(ii) For all f ∈ H , D(Nb ) ⊂ D(ΦS (f )) and, for all Ψ ∈ D(Nb ), (6.95) holds and 1 0 √ 1 1/2 (6.97)

ΦS (f )Ψ ≤ 2 f Nb Ψ + Ψ . 2 Proof (i) Let f ∈ D(T −1/2 ). Then Theorem 6.10 implies that D(db (T )1/2 ) ⊂ D(ΦS (f )) and (6.95) holds. Hence 1

ΦS (f )Ψ ≤ √ ( A(f )Ψ + A(f )∗ Ψ ). 2 Then, using (6.64) and (6.65), we obtain (6.96). (ii) This follows from (i) in the case T = I .



Let Db := ∩f ∈H D(A(f )) ∩ D(A(f )∗ ).

(6.98)

6.11 Segal Field Operators

285

Then 1/2

D(Nb ) ⊂ Db . The real linearity of A(f )# in f implies that of the correspondence f → ΦS (f )Ψ (Ψ ∈ Db ): ΦS (af + bg)Ψ = (aΦS (f ) + bΦS (g)) Ψ, a, b ∈ R, f, g ∈ H , Ψ ∈ Db . (6.99) Remark 6.5 For any non-real α, if f = 0 (f ∈ H ), then ΦS (αf ) = αΦS (f ). Namely, the correspondence f → ΦS (f )Ψ (Ψ ∈ Db ) is not complex linear. This is because the correspondence f → A(f )Ψ (Ψ ∈ Db ) is anti-linear: A(αf )Ψ = α ∗ A(f )Ψ, α ∈ C (see (6.38)). Proposition 6.10 For all f ∈ H \ {0}, ΦS (f ) is neither bounded from below nor bounded from above. (n)

Proof For each N ∈ N, we define ΨN (f ) ∈ Fb,fin (H ) by ΨN (f ) = √ (n) ( 2 f n )−1 ⊗n f for n = N, N + 1 and ΨN (f ) = 0 for n = N, N + 1. Then

ΨN (f ) = 1 and 

∗

;



1

ΨN , A(f ) ΨN = √ ⊗ 2 f N+1 =

N+1

< √ N + 1 N+1 f, √ ⊗ f 2 f N

f √ N + 1. 2

Hence ΨN , A(f )ΨN  = ΨN , A(f )∗ ΨN ∗ = √ ΨN (f ), ΦS (f )ΨN (f ) =

√

N + 1 f /2. Therefore

N + 1 f

√ → ∞ (N → ∞). 2

Hence ΦS (f ) is not bounded from above. Since ΦS (−f ) = −ΦS (f ), it follows that

ΨN (−f ) = 1 and √ N +1 ΨN (−f ), ΦS (f )ΨN (−f ) = − √

f → −∞ (N → ∞). 2 

Hence ΦS (f ) is not bounded from below. We have for all f ∈ H and Ψ ∈ D(A(f ))

∩ D(A(f )∗ )

i ΦS (if )Ψ = √ (A(f )∗ − A(f ))Ψ. 2

(6.100)

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6 Elements of the Theory of Boson Fock Spaces

By this formula and (6.94), we obtain the following relations: for all Ψ ∈ D(A(f ))∩ D(A(f )∗ ), 1 A(f )∗ Ψ = √ (ΦS (f ) − iΦS (if ))Ψ, 2 1 A(f )Ψ = √ (ΦS (f ) + iΦS (if ))Ψ. 2

(6.101) (6.102)

These are formulae recovering A(·)# from Segal field operators. Proposition 6.11 (i) For each f ∈ H , ΦS (f ) leaves Fb,0 (H ) invariant and obeys commutation relations [ΦS (f ), ΦS (g)] = i Im f, g ,

f, g ∈ H

(6.103)

on Fb,0 (H ). (ii) Let D be a subspace of H . Then, for each f ∈ D, ΦS (f ) leaves Fb,fin (D) invariant. (iii) Let D be a subspace of H and   PΦS (D) := span ΩH , ΦS (f1 ) · · · ΦS (fn )ΩH  n ≥ 1,  fj ∈ D, j = 1, . . . , n . Then Fb,fin (D) = PΦS (D).

(6.104)

In particular, if D is dense, then PΦS (D) is dense in Fb (H ). (iv) (continuity) Let T be a non-negative and injective self-adjoint operator on H . Suppose that fn , f ∈ D(T −1/2 ) and fn → f, T −1/2 fn → T −1/2 f (n → ∞). Then, for all Ψ ∈ D(db (T )1/2 ), ΦS (fn )Ψ → ΦS (f )Ψ (n → ∞). Proof (i) This follows from Theorem 6.8. (ii) By Corollary 6.5, for all f ∈ D, A(f )# Fb,fin (D) ⊂ Fb,fin (D). Hence ΦS (f )Fb,fin (D) ⊂ Fb,fin (D).

6.11 Segal Field Operators

287

(iii) Let fj ∈ D (j = 1, . . . , n). Then, by direct computations, we have ΦS (f1 ) · · · ΦS (fn )ΩH =



1

2n terms

2n/2

A(f1 )# · · · A(fn )# ΩH ,

Using (6.52), (6.53) and (6.39), we see that the vector on the right hand side is in Fb,fin (D). Hence PΦS (D) ⊂ Fb,fin (D). Conversely, let Ψ = A(f1 )∗ · · · A(fn )∗ ΩH ∈ Fb,fin (D) (fj ∈ D, j = 1, . . . , n). Then, using (6.101), we see that the vector on the right hand side is in PΦS (D). Hence Fb,fin (D) ⊂ PΦS (D). Thus Fb,fin (D) = PΦS (D). (iv) This follows from (6.96). 

6.11.2 Self-Adjointness of the Segal Field Operator Lemma 6.9 Let f ∈ H . Then every vector in Fb,0 (H ) is an entire analytic vector of ΦS (f ). Proof Let n ≥ 1. Then we have ΦS (f )n =

1 2n/2

2n terms

A(f )# · · · A(f )#    n factors

on Fb,0 (H ). Let Ψk = {0, · · · , 0, Ψ (k) , 0, · · · } ∈ Fb,0 (H ) (k = 0, 1, 2, · · · ). Then A(f )# · · · A(f )# Ψk ∈ Fb,n+k (H ).    n factors

Hence, by Corollary 6.6,

A(f )# · · · A(f )# Ψk ≤ f (Nb + 1)1/2 A(f )# · · · A(f )# Ψk

      n factors

n−1 factors

≤ (n + k)

1/2

#

f A(f ) · · · A(f )# Ψk

   n−1 factors

≤ (n + k)1/2 (n − 1 + k)1/2 · · · (1 + k)1/2 × f n Ψk .

(6.105)

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6 Elements of the Theory of Boson Fock Spaces

Hence, for all t > 0, ∞

ΦS (f )n Ψk

n=0

≤

n!

∞ n/2 n 2 t f n n=0

n!

tn

(n + k)1/2 (n − 1 + k)1/2 · · · (1 + k)1/2 Ψk .

By d’Alembert’s ratio test, the infinite series on the right hand side converges. Hence Ψk is an entire analytic vector of ΦS (f ). Since any vector Ψ in Fb,0 (H ) is a linear combination of vectors of the form Ψk (k ≥ 0), it follows that Ψ is an entire analytic vector of ΦS (f ).  Theorem 6.17 Let f ∈ H . Then ΦS (f ) is self-adjoint. Moreover, for any dense subspace D of H , Fb,fin (D) is a core for ΦS (f ). Proof It is obvious that ΦS (f )Fb,0 (H ) ⊂ Fb,0 (H ). Hence, by Lemma 6.9 and Theorem 1.14, ΦS (f ) is essentially self-adjoint on Fb,0 (H ) and hence ΦS (f ) is self-adjoint. By the preceding result, for each Ψ ∈ D(ΦS (f )) and any ε > 0, there exists a vector Ψε ∈ Fb,0 (H ) such that Ψε − Ψ < ε and ΦS (f )Ψε − ΦS (f )Ψ < ε. One has Fb,fin (D) ⊂ Fb,0 (H ) and Fb,fin (D) is dense in Fb (H ). There exists an r ∈ N such that Ψε ∈ Fb,r (H ). By Lemma 6.5, ΦS (f ) can be regarded as a bounded operator from Fb,r (H ) to Fb,r+1 (H ). Hence there exists a vector Θε ∈ Fb,fin (D) such that Θε − Ψε < ε and ΦS (f )Θε − ΦS (f )Ψε < ε. Therefore, using the triangle inequality, we obtain

Θε − Ψ < 2ε,

ΦS (f )Θε − ΦS (f )Ψ < 2ε.

This means that Fb,fin (D) is a core for ΦS (f ).



By Theorem 6.17, for all f ∈ H , eiΦS (f ) is a unitary operator on Fb (H ). This unitary operator is called the Weyl operator with test vector f . Theorem 6.18   2 ΩH , eiΦS (f ) ΩH = e− f /4 , 

(6.106)

 in 2 A(f1 )∗ · · · A(fn )∗ ΩH , eiΦS (f ) ΩH = n/2 f1 , f  · · · fn , f  e− f /4 . 2 (6.107)

Proof By Lemma 6.9, we have 

∞ n  i Cn , ΩH , eiΦS (f ) ΩH = n! n=0

6.11 Segal Field Operators

289

where  Cn := ΩH , ΦS (f )n ΩH . Using (6.39) and the commutation relation 1 [A(f ), ΦS (g)] = √ f, g 2

on Fb,0 (H ) (f, g ∈ H ),

(6.108)

we see that Cn =

(n − 1)

f 2 Cn−2 . 2

Note that C1 = 0. Hence C2n−1 = 0, n ∈ N. On the other hand, we have by iteration C2n =

f 2n (2n)! . 22n n!

Hence (6.106) follows. By Lemma 6.9 again, we have ∞ m   i A(f1 )∗ · · · A(fn )∗ ΩH , eiΦS (f ) ΩH = am , m! m=0

where  am := A(f1 )∗ · · · A(fn )∗ ΩH , ΦS (f )m ΩH . It is easy to see that, if m < n, then am = 0. So let m ≥ n. Then, using (6.108) and (6.39), one can show that am =

 m f1 , f   A(f2 )∗ · · · A(fn )∗ ΩH , ΦS (f )m−1 ΩH . √ 2

Iterating the right hand side similarly, we obtain am =

 m! f1 , f  · · · fn , f  ΩH , ΦS (f )m−n ΩH . 2n/2 (m − n)!

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6 Elements of the Theory of Boson Fock Spaces

Hence ⎞ ⎛ ∞ m n ∞ k   i i in ⎝  am = n/2 ΩH , ΦS (f )k ΩH fj , f ⎠ m! 2 k! j =1

m=0

⎛

=

⎞

k=0

  in ⎝  ⎠ ΩH , eiΦS (f ) ΩH . f , f j 2n/2 n

j =1



Thus (6.107) is obtained.

6.11.3 Vacuum Expectation Values In general, for a linear operator X on Fb (H ) such that ΩH ∈ D(X), the number ΩH , XΩH  is called the vacuum expectation value of X. A slight modification of the Segal field operator is given by Ψ (f, g) := A(f ) + A(g)∗ ,

f, g ∈ H .

(6.109)

For each n ∈ N and fj , gj ∈ H (j = 1, . . . , n), we consider the vacuum expectation value Wn ((f1 , g1 ), · · · , (fn , gn )) := ΩH , Ψ (f1 , g1 ) · · · Ψ (fn , gn )ΩH  ,

n ∈ N. (6.110)

of Ψ (f1 , g1 ) · · · Ψ (fn , gn ). Using (6.39) and the CCR for A(·)# , one can easily show that W2 ((f1 , g1 ), (f2 , g2 )) = f1 , g2  .

(6.111)

Theorem 6.19 Let n ∈ N and fj , gj ∈ H , j = 1, . . . , n. Then W2n−1 ((f1 , g1 ), · · · , (f2n−1 , g2n−1 )) = 0, (6.112)    fi1 , gj1 fi2 , gj2 · · · fin , gjn , W2n ((f1 , g1 ), · · · , (f2n , g2n )) = pairings

(6.113)  where pairings means the sum over all (2n)!/2n n! ways of writing 1, . . . , 2n as n distinct (unordered) pairs (i1 , j1 ), . . . , (in , jn ).

6.11 Segal Field Operators

291

Proof By (6.39) and the CCR for A(·)# , we have Wn ((f1 , g1 ), · · · , (fn , gn )) =

n 

   f1 , gj ΩH , Ψ (f2 , g2 ) · · · Ψ (f , g ) · · · Ψ (f , g )Ω j j n n H .

(6.114)

j =2

If n is odd, then a successive iteration of this equation finally gives Wn ((f1 , g1 ), · · · , (fn , gn )) =

n

ci ΩH , Ψ (fi , gi )ΩH  ,

i=1

where ci is a constant. But, by (6.39), ΩH , Ψ (fi , gi )ΩH  = 0. Hence (6.112) follows. Formula (6.113) can be proved by (6.114) and induction in n.  Since 1 ΦS (f ) = √ Ψ (f, f ) 2

on D(A(f )) ∩ D(A(f )∗ ),

Theorem 6.19 implies the following corollary: Corollary 6.12 Let n ∈ N and fj ∈ H , j = 1, . . . , 2n. Then ΩH , ΦS (f1 ) · · · ΦS (f2n−1 )ΩH  = 0, ΩH , ΦS (f1 ) · · · ΦS (f2n )ΩH  =

 1  fi1 , fj1 · · · fin , fjn . n 2 pairings

6.11.4 Irreducibility of Segal Field Operators We next consider irreducibility of Segal field operators. Theorem 6.20 For any dense subspace D of H , {ΦS (f )|f ∈ D} = CI . In particular, {ΦS (f )|f ∈ D} is irreducible. Proof Let T ∈ {ΦS (f )|f ∈ D} . Then, for all f ∈ D, T ΦS (f ) ⊂ ΦS (f )T . Let Df := D(A(f )∗ ) = D(A(f )) (see Corollary 6.3). Hence, for all Ψ, Φ ∈ Df ,   Ψ, T (A(f )∗ + A(f ))Φ = (A(f ) + A(f )∗ )Ψ, T Φ . Replacing f by if , one has   Ψ, T (A(f )∗ − A(f ))Φ = (A(f ) − A(f )∗ )Ψ, T Φ .

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6 Elements of the Theory of Boson Fock Spaces

From these two equations, one obtains  Ψ, T A(f )∗ Φ = A(f )Ψ, T Φ ,

 Ψ, T A(f )Φ = A(f )∗ Ψ, T Φ .

The first equation implies that T Φ ∈ D(A(f )∗ ) and T A(f )∗ Φ = A(f )∗ T Φ, i.e., T A(f )∗ ⊂ A(f )∗ T . Similarly, the second equation implies that T A(f ) ⊂ A(f )T . Hence T ∈ {A(f ), A(f )∗ |f ∈ D} = CI (by Theorem 6.9), i.e., T = cI for some c ∈ C. Thus {ΦS (f )|f ∈ D} = CI and hence {ΦS (f )|f ∈ D} is irreducible. 

6.11.5 Some Formulae and Spectrum of ΦS (f ) Let A be an algebra and A ∈ A. Then, for each n ∈ Z+ , one can define a mapping adn (A): A → A by the following recurrence relations: for all B ∈ A, ad0 (A)B := B,

adn (A)B := [A, adn−1 (A)B],

n ≥ 1.

(6.115)

The mapping adn (A) is called the adjoint operator associated with A. Lemma 6.10 For all A, B ∈ A and N ∈ N, N

N Cn (−1)

n

AN−n BAn = adN (A)B,

(6.116)

n=0

where N Cn is the binomial coefficient with respect to (N, n) (see (4.99)). Proof This can be proved by induction in N, where the following formula is used: n+1 Ck

= n Ck + n Ck−1 ,

k = 1, . . . , n, n ∈ N.

(6.117) 

Theorem 6.21 Let f, g ∈ H . Then the following operator equalities hold: i eiΦS (f ) A(g)e−iΦS (f ) = A(g) − √ g, f  . 2 i eiΦS (f ) A(g)∗ e−iΦS (f ) = A(g)∗ + √ f, g . 2

(6.118) (6.119)

Proof Let Ψ, Φ ∈ Fb,0 (H ). Then, in the same way as in the proof of Lemma 6.9, one can show that e−iΦS (f ) Ψ ∈ D(A(f )# ) and A(g)# e−iΦS (f ) Ψ =

∞ (−i)n n=0

n!

A(g)# ΦS (f )n Ψ.

6.11 Segal Field Operators

293

Hence 

∞ ∞ m  i (−i)n  e−iΦS (f ) Φ, A(g)e−iΦS (f ) Ψ = Φ, ΦS (f )m A(g)ΦS (f )n Ψ . m! n! n=0 m=0

Since the series on the right hand side is absolutely convergent, one can rearrange ∞ ∞  the summation ∞ m=0 n=0 in the form N=0 n+m=N so that 

∞ N  i Φ, AN Ψ  e−iΦS (f ) Φ, A(g)e−iΦS (f ) Ψ = N! N=0

with AN :=

N

N Cn (−1)

n

ΦS (f )N−n A(g)ΦS (f )n .

n=0

By (6.116), we have AN = adN (ΦS (f ))A(g)

on Fb,0 (H ).

We have A0 Ψ = A(g)Ψ, 1 A1 Ψ = [Φ(f ), A(g)]Ψ = − √ g, f  Ψ, 2 AN Ψ = 0,

N ≥ 2.

Hence  e

−iΦS (f )

Φ, A(g)e

−iΦS (f )

=

0

1 > i Ψ = Φ, A(g) − √ g, f  Ψ . 2 

Therefore 1 0 i eiΦS (f ) A(g)e−iΦS (f ) Ψ = A(g) − √ g, f  Ψ. 2 Since Fb,0 (H ) is a core for A(f ) by Proposition 6.5, it follows from a limiting argument that i eiΦS (f ) A(g)e−iΦS (f ) ⊃ A(g) − √ g, f  . 2

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6 Elements of the Theory of Boson Fock Spaces

Since f ∈ H is arbitrary, replacing f with −f gives i e−iΦS (f ) A(g)eiΦS (f ) ⊃ A(g) + √ g, f  2 which is equivalent to i A(g) − √ g, f  ⊃ eiΦS (f ) A(g)e−iΦS (f ) . 2 Thus operator equality (6.118) holds. Operator equality (6.119) follows from taking the adjoint of (6.118).  Corollary 6.13 For all f, g ∈ H , eiΦS (f ) ΦS (g)e−iΦS (f ) = ΦS (g) − Im f, g .

(6.120)

Proof By (6.118) and (6.119), we have eiΦS (f ) ΦS (g)e−iΦS (f ) = ΦS (g) − Im f, g on D(A(g))∩D(A(g)∗ ). Since D(A(g))∩D(A(g)∗ ) is a core for ΦS (g) and ΦS (g) is self-adjoint, operator equality (6.120) follows.  Corollary 6.13 implies the following result on the spectrum of ΦS (f ). Corollary 6.14 For all f ∈ H \ {0}, σ (ΦS (f )) = R.

(6.121)

Proof Let t ∈ R and U (t) := eiΦS (itf ) . Then, by (6.120), we have U (t)ΦS (f )U (t)−1 = ΦS (f ) = ΦS (f ) + t f 2 . By the unitary invariance of spectrum, we have σ (ΦS (f )) = σ (ΦS (f ) + t f 2 ) = {λ + t f 2 |λ ∈ σ (ΦS (f ))}. Since t ∈ R is arbitrary, we obtain (6.121).

6.11.6 Properties of Weyl Operators Theorem 6.22 Let f, g ∈ H .



6.11 Segal Field Operators

295

(i) eiΦS (f ) eiΦS (g) = e−iImf,g eiΦS (g) eiΦS (f ) .

(6.122)

(ii) eiΦS (f +g) = eiImf,g/2 eiΦS (f ) eiΦS (g) ,

(6.123)

(iii) (strong continuity) Let fn ∈ H and fn → f (n → ∞). Then s- lim eiΦS (fn ) = eiΦS (f ) .

(6.124)

n→∞

(iv) (irreducibility) For any dense subspace D ⊂ H , {eiΦS (f ) |f ∈ D} = CI . In particular, {eiΦS (f ) |f ∈ D} is irreducible. Proof (i) By Corollary 6.13 and the unitary covariance of functional calculus, we have eiΦS (f ) eiΦS (g) e−iΦS (f ) = ei(ΦS (g)−Im f,g = e−iIm f,g eiΦS (g). Multiplying eiΦS (f ) from the right side, we obtain (6.122). (ii) Let U (t) := eit

2 Im f,g/2

eit ΦS (f ) eit ΦS (g) ,

t ∈ R.

Then, in the same way as in the proof of Theorem 2.18, one can show that {U (t)}t ∈R is a strongly continuous one-parameter unitary group. Hence, by the Stone theorem, there exists a unique self-adjoint operator S on Fb (H ) such that U (t) = eit S , t ∈ R. It is easy to see that, for all Ψ ∈ D(ΦS (f )) ∩ D(ΦS (g)), 1 lim (U (t) − 1)Ψ = i(ΦS (f ) + ΦS (g))Ψ. t

t →0

Hence Ψ ∈ D(S) and (ΦS (f )+ΦS (g))Ψ = SΨ . Therefore ΦS (f )+ΦS (g)) ⊂ S. Note that ΦS (f ) + ΦS (g) ⊃ ΦS (f + g)  Fb,0 (H ). Since ΦS (f + g) is essentially self-adjoint on Fb,0 (H ), it follows that Φ(f + g) = S. Thus (6.123) holds.

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6 Elements of the Theory of Boson Fock Spaces

(iii) Let Ψ ∈ Fb,0 (H ). Then, by direct computation, we have  

eiΦS (fn ) Ψ − eiΦS (f ) Ψ 2 = 2 Ψ 2 − 2Re Ψ, e−iΦS (fn ) eiΦS (f ) Ψ . By (6.123), e−iΦS (fn ) eiΦS (f ) Ψ = eiIm fn ,f /2 eiΦS (f −fn ) Ψ. Hence  % &

eiΦS (fn ) Ψ − eiΦS (f ) Ψ 2 = 2 Ψ 2 − 2Re eiIm fn ,f /2 Ψ, eiΦS (f −fn ) Ψ . We have limn→∞ Im fn , f  = Im f 2 = 0. By Proposition 1.14 and Proposition 6.11(iv) with T = I ,

eiΦS (f −fn ) Ψ − Ψ ≤ ΦS (f − fn )Ψ → 0 (n → ∞). Hence the right hand side of the above equation converges to 0 as n → ∞. Thus lim eiΦS (fn ) Ψ = eiΦS (f ) Ψ.

n→∞

Since Fb,0 (H ) is dense in Fb (H ) and supn≥1 eiΦS (fn ) = 1 < ∞, a limiting argument gives (6.124). (iv) Let T ∈ {eiΦS (f ) |f ∈ D} . Then, for all f ∈ D, t ∈ R and Ψ ∈ D(ΦS (f )), T eit ΦS (f ) Ψ = eit ΦS (f ) T Ψ . Since eit ΦS (f ) Ψ is strongly differentiable in t and T is bounded, it follows that eit ΦS (f ) T Ψ also is strongly differentiable in t. Hence, considering the strong differentiation of the above equation at t = 0, one obtains that T Ψ ∈ D(ΦS (f )) and T ΦS (f )Ψ = ΦS (f )T Ψ . This means that T ΦS (f ) ⊂ ΦS (f )T . Therefore T ∈ {ΦS (f )|f ∈ D} . Hence, by Theorem 6.20, T = cI for some c ∈ C. Thus {eiΦS (f ) |f ∈ D} is irreducible.  eiΦS (f )

eiΦS (g)

Formula (6.122) for the Weyl operators and is called the Weyl relation for ΦS (f ) and ΦS (g). As a corollary to Theorem 6.22, we obtain an important result: Corollary 6.15 (Strong Commutativity) Let f, g ∈ H . Then ΦS (f ) and ΦS (g) strongly commute if and only if Im f, g = 0. Proof Suppose that ΦS (f ) and ΦS (g) strongly commute. Then, by Proposition 1.17, for all s, t ∈ R, eit ΦS (f ) eisΦS (g) = eisΦS (g)eit ΦS (f ) . Hence, by (6.122), e−it sIm f,g = 1 for all s, t ∈ R. Hence Im f, g = 0.

6.12 Canonical Free Bose Field and Canonical Conjugate Momentum

297

Conversely, suppose that Im f, g = 0. Then Im tf, sg = 0, s, t ∈ R. Using the real linearity of ΦS (f ) in f and (6.122), we obtain eit ΦS (f ) eisΦS (g) = eisΦS (g)eit ΦS (f ) . Hence, by Proposition 1.17 again, ΦS (f ) and ΦS (g) strongly commute. 

6.12 Canonical Free Bose Field and Canonical Conjugate Momentum Let J be a conjugation on H and HJ be the real part of H with respect to J (see (1.68)). For each f ∈ HJ , we define φF (f ) := ΦS (f ),

πF (f ) := ΦS (if ),

(6.125)

where ΦS (f ) is the Segal field operator with test vector f . By Proposition 6.11(i), we have for all f, g ∈ HJ [φF (f ), πF (g)] = i f, g ,

(6.126)

[φF (f ), φF (g)] = 0,

(6.127)

[πF (f ), πF (g)] = 0

on Fb,0 (H ). It follows from (6.99) that, for all f, g ∈ HJ and a, b ∈ R, φF (af + bg) = aφF (f ) + bφF (g), πF (af + bg) = aπF (f ) + bπF (g)

(6.128) on Db .

(6.129)

The mapping f → φF (f ) is called the canonical free Bose field over HJ and the mapping f → πF (f ) is called the canonical conjugate momentum of φF (f ). Theorem 6.22 is rewritten in terms of φF (f ) and πF (f ) as follows: Theorem 6.23 Let f, g ∈ HJ and a, b ∈ R. (i) (Weyl relations) eiφF (f ) eiπF (g) = e−if,g eiπF (g) eiφF (f ) ,

(6.130)

eiφF (f ) eiφF (g) = eiφF (g)eiφF (f ) ,

(6.131)

eiπF (f ) eiπF (g) = eiπF (g) eiπF (f ) .

(6.132)

(ii) eiφF (af +bg) = eiaφF (f ) eibφF (g),

eiπF (af +bg) = eiaπF (f ) eibπF (g) .

(6.133)

298

6 Elements of the Theory of Boson Fock Spaces

(iii) (strong continuity) Let fn ∈ HJ and fn → f ∈ HJ (n → ∞). Then s- lim eiφF (fn ) = eiφF (f ) , n→∞

s- lim eiπF (fn ) = eiπF (f ) . n→∞

(6.134)

(iv) (irreducibility) For any dense subspace W ⊂ HJ , {eiφF (f ) , eiπF (f ) |f ∈ W } = CI . In particular, {eiφF (f ) , eiπF (f ) |f ∈ W } is irreducible. Proof We prove only (iv). Let T ∈ {eiφF (f ) , eiπF (f ) |f ∈ W } . Then, for all f, g ∈ W , T eiφF (f ) = eiφF (f ) T and T eiπF (g) = eiπF (g)T . Hence T eiφF (f ) eiπF (g) = eiφF eiπF (g) T . Using (6.123), we obtain T eiΦS (f +ig) = eiΦS (f +ig) T . Note that {f + ig|f, g ∈ W } is dense in H . Hence, by Theorem 6.22(iv), T = αI for some α ∈ C. Thus the desired result follows. 

6.13 Symplectic Spaces and Generalization of Segal Field Operators As is easily seen, the real number s(f, g) := Im f, g ,

f, g ∈ H

appearing in the commutation relation (6.103) of Segal field operators has the following properties: s(af1 + bf2 , g) = as(f1 , g) + bs(f2 , g), s(f, g) = −s(g, f ),

f1 , f2 , g ∈ H , a, b ∈ R,

f, g ∈ H ,

{f ∈ H |s(f, g) = 0, ∀g ∈ H } = {0}.

(6.135) (6.136) (6.137)

Equations (6.135) and (6.136) imply that the correspondence H × H (f, g) → s(f, g) is a real-bilinear form. Note that the complex Hilbert space H can be viewed as a real Hilbert space by replacing the coefficient field C with R. These properties can be abstracted to obtain a general concept of space. Let X be a vector space over K (K = R or C). Let ω be a mapping from X × X to K satisfying the following: (i) (K-bilinearity) For all f, g, h ∈ X and a, b ∈ K, ω(af + bg, h) = a ω(f, h) + b ω(g, h), ω(h, af + bg) = a ω(h, f ) + b ω(h, g). (ii) (anti-symmetry) For all f, g ∈ X , ω(f, g) = −ω(g, f ). (iii) (non-degeneracy) {f ∈ X |ω(f, g) = 0, ∀g ∈ X } = {0}.

6.13 Symplectic Spaces and Generalization of Segal Field Operators

299

The mapping ω is called a K-symplectic form on X and (X , ω) is called a Ksymplectic space. If K = R (resp. C), then (X , ω) is called a real (resp. complex) symplectic space. Example 6.5 The pair (H , s) described above is a real symplectic space. Example 6.6 Let T be an injective bounded self-adjoint operator on X and ω(f, g) = Im f, T gX , f, g ∈ X . Then ω is an R-symplectic form on X . Example 6.7 Let X be a complex Hilbert space and C be a conjugation on X (see Definition 1.2). Suppose that there exists a bounded linear operator W on X such that Wf = f , f ∈ X and W 2 = −I,

CW = W C.

Let ωC,W (f, g) := Cf, WgX ,

f, g ∈ X .

Then ωC,W is a C-symplectic form on X .4 Hence (X , ωC,W ) is a complex symplectic space. Let (X , ω) be a K-symplectic space. Then a linear operator S: X → X is called an iso-symplectic transformation if ω(Sf, Sg) = ω(f, g),

f, g ∈ X .

It follows from the non-degeneracy of ω that any iso-symplectic transformation is injective. If S is an iso-symplectic transformation and surjective, then S is called a symplectic transformation.

is straightforward to prove the C-bilinearity of ωC,W . With regard to anti-symmetry, one can proceed as follows, where properties (1.67), C 2 = I , CW = W C,Wf, W g = f, g , f, g ∈ X (this follows from the polarization identity and norm-preserving property Wf = f , f ∈ X ) and W 2 = −I are used: 4 It

ωC,W (f, g) = CW g, f  = W Cg, f  = W (W Cg), Wf  = − Cg, Wf  = −ωC,W (g, f ). To prove non-degeneracy of ωC,W , let g ∈ H be such that, for all f ∈ H , ωC,W (f, g) = 0. Then 0 = CW g, f . Hence CW g = 0. Since C 2 = I , it follows that W g = 0, implying g = 0. Thus ωC,W is non-degenerate.

300

6 Elements of the Theory of Boson Fock Spaces

If S is a symplectic transformation, then so is S −1 . Moreover, if S and T are symplectic transformations, then so is ST . It follows that the set Sp(X ) := {S : X → X |S is symplectic} of all symplectic transformations on X forms a group. This group is called the symplectic group on (X , ω). With the above generalization of s(f, g), it may be natural to generalize the concept of Segal field operators. Let (X , ω) be a real symplectic space and {Φω (f )|f ∈ X } be a family of self-adjoint operators on a complex Hilbert space F . Then Φω (f ) is called an ω-Segal field operator with test vector f ∈ X if there exists a dense subspace D of F left invariant by Φω (f ) (f ∈ X ) such that, for all f, g ∈ D and a, b ∈ R, Φω (af + bg) = aΦω (f ) + bΦω (g)

on D

and [Φω (f ), Φω (g)] = iω(f, g),

f, g ∈ X

on D.

6.14 Quadratic Operators For later use (see Sect. 8.6), we introduce operators quadratic in the annihilation operator A(·) and the creation operator A(·)∗ . Let K be a Hilbert–Schmidt operator on a Hilbert space H and J be a conjugation on H . Then, by Theorem 1.4, there NK K exist ONS’s {φn }N n=1 , {ψn }n=1 of H (NK < ∞ or NK = ∞) and λn > 0 (n = 1, . . . , NK ) such that ∞

λ2n < ∞

(in the case NK = ∞)

n=1

and K=

NK

λn ψn , · φn .

(6.138)

n=1

 If NK < ∞, then we define operators A# |K|A# by 

NK A∗ |K|A∗ := λn A(J ψn )∗ A(φn )∗ ,

(6.139)

n=1

A|K|A :=

NK n=1

λn A(ψn )A(J φn ).

(6.140)

6.14 Quadratic Operators

301

In the case NK = ∞, for each N ∈ N, we first define an operator KN on H by KN :=

N

λn ψn , · φn .

(6.141)

n=1

and set 

∗

∗



A |KN |A :=

N

λn A(J ψn )∗ A(φn )∗ ,

(6.142)

n=1

A|KN |A :=

N

(6.143)

λn A(ψn )A(J φn ).

n=1

Lemma 6.11 (i) For all Ψ ∈ Fb,0 (H ), the limit limN→∞ A∗ |KN |A∗  Ψ exists. (ii) For all Ψ ∈ Fb,fin (H ), the limit limN→∞ A|KN |A Ψ exists. Proof For notational simplicity, we set TN A|KN |A.

:= A∗ |KN |A∗  and SN

:=

(i) Let Ψ ∈ Fb,0 (H ). Then (TN Ψ )(0) = 0, (TN Ψ )(1) = 0 and (TN Ψ )(p+2) = where ηN :=



(p + 2)(p + 1)Sp+2 (ηN ⊗ Ψ (p) ),

N

p ≥ 0,

⊗ φn ∈ H ⊗ H . For all M > N, we have

n=1 λn J ψn

ηM − ηN = 2

M

λ2n −→ 0 (M, N → ∞).

n=N+1

Hence {ηN }N is a Cauchy sequence in H ⊗ H . Therefore the limit η(K) := lim ηN = N→∞

∞

λn J ψn ⊗ φn ∈ H ⊗ H

(6.144)

n=1

exists. By the continuity of tensor product operation and the boundedness of Sp+2 , we see that lim (TN Ψ )(p+2) =



N→∞

Thus limN→∞ TN Ψ exists.

(p + 2)(p + 1)Sp+2 (η(K) ⊗ Ψ (p) ).

(6.145)

302

6 Elements of the Theory of Boson Fock Spaces

(ii) It is sufficient to prove that, for all vectors Ψ of the form Ψp = A(f1 )∗ · · · A(fp )∗ ΩH ,

p ≥ 0, fj ∈ H , j = 1, . . . , p,

(6.146)

limN→∞ SN Ψ exists. It is obvious that SN Ψp = 0,

p = 0, 1.

Hence limN→∞ SN Ψp = 0, p = 0, 1. Let p ≥ 2. Then, using the CCR for A(·)# , one can show that SN Ψp =

p

(j,k)

ΦN

∗ ∗ ∗   A(f1 )∗ · · · A(f j ) · · · A(f k ) · · · A(fp ) ΩH ,

j,k=1 j =k (j,k)

where ΦN

:=



N

n=1 λn

J φn , fj ψn , fk . Using (6.138), one can show that (j,k)

lim ΦN

N→∞

 = K ∗ Jfj , fk .

Hence lim SN Ψp =

N→∞

p 

∗ ∗ ∗   K ∗ Jfj , fk A(f1 )∗ · · · A(f j ) · · · A(f k ) · · · A(fp ) ΩH .

j,k=1 j =k

(6.147) 

Thus the desired result follows. Remark 6.6 By Corollary 1.4, η(K) = UJ K.

By Lemma 6.11, one can introduce densely defined operators A|K|A#0 as follows:  D( A∗ |K|A∗ 0 ) := Fb,0 (H ),  ∗  A |K|A∗ 0 Ψ := lim A∗ |KN |A∗ Ψ, N→∞



∗

Ψ ∈ D( A |K|A

∗

0

(6.148) )

(6.149)

and D(A|K|A0 ) := Fb,fin (H ), A|K|A0 Ψ := lim A|KN |A Ψ, N→∞

(6.150) Ψ ∈ D(A|K|A0 ).

(6.151)

6.14 Quadratic Operators

303

Recall that K ∗ also is Hilbert–Schmidt with K∗ =

NK

λn φn , · ψn .

n=1

 Hence A# |K ∗ |A# 0 is defined by replacing K with K ∗ in the above definitions. Then it is easy to see that   ( A∗ |K|A∗ 0 )∗ ⊃ A|K ∗ |A 0 ,  (A|K|A0 )∗ ⊃ A∗ |K ∗ |A∗ 0 .

(6.152) (6.153)

  ∗ Hence A# |K|A# 0 is densely defined. Therefore A# |K|A# 0 is closable. Based on this fact, we define closed operators by 

  A# |K|A# := A# |K|A# 0 ,

(6.154)

 the closure of A# |K|A# 0 . We call operators of the forms A∗ |K|A∗  and A|K|A quadratic operators in A∗ and A respectively. It follows from (6.145) and (6.147) that   ( A∗ |K|A∗ Ψ )(0) = 0, ( A∗ |K|A∗ Ψ )(1) = 0,   ( A∗ |K|A∗ Ψ )(p+2) = (p + 2)(p + 1)Sp+2 (η(K) ⊗ Ψ (p) ),  p ≥ 0, Ψ ∈ D( A∗ |K|A∗ )

(6.155)

(6.156)

and A|K|A = 0 A|K|A Ψp =

on span{ΩH , A(f )∗ ΩH |f ∈ H }, p



(6.157)

∗ ∗ ∗   K ∗ Jfj , fk A(f1 )∗ · · · A(f j ) · · · A(f k ) · · · A(fp ) ΩH

j,k=1 j =k

(6.158) for vectors Ψp of the form (6.146). The following lemma summarizes basic properties of quadratic operators. Lemma 6.12 (i) Fb,0 (H ) ⊂ D(A∗ |K|A∗ ). Moreover, for all Ψ ∈ D(A∗ |K|A∗ ) and p ∈ Z+ ,  

( A∗ |K|A∗ Ψ )(p+2) ≤ (p + 2)(p + 1) η(K) Ψ (p) .

(6.159)

304

6 Elements of the Theory of Boson Fock Spaces

In particular, A∗ |K|A∗  is a bounded operator from the p-particle space p p+2 ⊗s H to the (p + 2)-particle space ⊗s H . ∗ ∗ (ii) D(Nb ) ⊂ D(A |K|A ) and 

A∗ |K|A∗ Ψ ≤ η(K) (Nb + 2)Ψ ,

Ψ ∈ D(Nb ).

(6.160)

(iii) Fb,0 (H ) ⊂ D(A|K|A). Moreover, for each p ∈ Z+ and Ψ ∈ D(A|K|A),

(A|K|A Ψ )(p−2) ≤



(p − 1)p η(K ∗ ) Ψ (p) .

(6.161) p

In particular, A|K|A is a bounded operator from the p-particle space ⊗s H p−2 −1 to the (p − 2)-particle space ⊗s H , where ⊗−2 s H := {0} and ⊗s H := {0}. (iv) D(Nb ) ⊂ D(A|K|A) and

A|K|A Ψ ≤ η(K) Nb Ψ ,

Ψ ∈ D(Nb ).

(6.162)

Proof (i) It is obvious from the definition of A∗ |K|A∗  that Fb,0 (H ) is a subset of D(A∗ |K|A∗ ). Inequality (6.159) follows from (6.156). This implies the desired boundedness property of A∗ |K|A∗ . (ii) By (6.159), we have 

( A∗ |K|A∗ Ψ )(p+2) 2 ≤ (p + 2)(p + 1) η(K) 2 Ψ (p) 2 = η(K) 2 ((Nb + 1)1/2 (Nb + 2)1/2 Ψ )(p) 2 ,

Ψ ∈ D(Nb ).

Taking the summation over p from 0 to ∞, we have ∞



( A∗ |K|A∗ Ψ )(p+2) 2 ≤ η(K) 2 (Nb + 1)1/2(Nb + 2)1/2Ψ 2 .

p=0

Hence Ψ ∈ D(A∗ |K|A∗ ) and (6.160) holds, where we have used the easily 1/2 proved fact that (Nb + 1)1/2Φ ≤ (Nb + 2)1/2 Φ , Φ ∈ D(Nb ). ∗ ∗ ∗ ∗ ∗ (iii) By (6.152), we have A |K |A  ⊃ A|K|A0 . Since A |K ∗ |A∗  is a p p+2 bounded operator from ⊗s H to ⊗s H by (i) with K replaced by K ∗ , it ˆ p+2 ˆ ps H . Hence follows that A|K|A0 is a bounded operator from ⊗ H to ⊗ s Fb,0 (H ) ⊂ D(A|K|A) and 

A∗ |K ∗ |A∗

∗

⊃ A|K|A .

6.14 Quadratic Operators

305

This means that, for each p ∈ Z+ , A|K|A is a bounded operator from p+2 p ⊗s H to ⊗s H with

(A|K|A Ψ )(p) ≤ (p +1)(p +2) η(K ∗) Ψ (p+2) ,

Ψ ∈ D(A|K|A).

Hence (6.161) holds. (iv) Since we have proved (iii), this part can proved in quite the same way as in the proof of (ii).  Lemma 6.13 Let K1 be a Hilbert–Schmidt operator on H and K2 ∈ B(H ). Let {en }∞ n=1 be a CONS of H . Then lim

N→∞

N

 A(K1 en )∗ A(J K2 en )∗ Ψ = A∗ |K1 K2∗ |A∗ Ψ,

Ψ ∈ Fb,0 (H ),

n=1

(6.163) lim

N→∞

N

 A(K1 en )A(J K2 en )Ψ = A|K2 K1∗ |A Ψ,

Ψ ∈ Fb,fin (H ).

(6.164)

n=1

Proof Let ΣN :=

N

n=1 A(K1 en )

ΣN Ψ =

∗ A(J K e )∗ . 2 n

N

Then

A(J K2 en )∗ A(K1 en )∗ Ψ.

n=1

Hence  (ΣN Ψ )(p+2) = (p + 2)(p + 1)Sp+2 (uN ⊗ Ψ (p) )  with uN := ∞ 1.4, we have UJ TN = uN with n=1 J K2 en ⊗ K1 en . By Corollary  N ∗ K e e , · K e . Let P := TN := N 2 n 1 n N n=1 n=1 n , · en . Then TN = K1 PN K2 . It is easy to see that

K1 PN − K1 22 =

∞

K1 en 2 → 0 (N → ∞).

n=N+1

Hence, by (1.25), we obtain

TN − K1 K2∗ 2 ≤ K1 PN − K1 2 K2∗ → 0 (N → ∞). Hence limN→∞ uN = UJ K1 K2∗ in H ⊗ H , implying that lim (ΣN Ψ )(p+2) =

N→∞



(p + 2)(p + 1)Sp+2 (UJ K1 K2∗ ⊗ Ψ (p) ).

306

6 Elements of the Theory of Boson Fock Spaces

By this result and (6.156), we see that (6.163) holds. To prove (6.164), let Ψp be a vector of the form (6.146). Then N

A(K1 en )A(J K2 en )Ψ =

n=1

p

(j,k)

ΨN

∗ ∗ ∗   A(f1 )∗ · · · A(f j ) · · · A(f k ) · · · A(fp ) ΩH ,

j,k=1 j =k

(j,k)

where ΨN

:=

N n=1

 J K2 en , fj K1 en , fk . It is easy to see that (j,k)

lim ΨN

N→∞

  = K1 K2∗ Jfj , fk = (K2 K1∗ )∗ Jfj , fk . 

Thus (6.164) holds.

6.15 The Boson Fock Space Over a Direct Sum Hilbert Space In this section we consider the boson Fock space Fb (H ⊕ K ) over the direct sum Hilbert space H ⊕ K := {(f, g)|f ∈ H , g ∈ K } of two Hilbert spaces H and K . We prove that Fb (H ⊕ K ) is isomorphic to Fb (H ) ⊗ Fb (K ) in a natural way. To avoid confusions, we denote the annihilation operator on Fb (H ) by AH (·). Theorem 6.24 Assume that H and K are separable. Then there exists a unique unitary operator U : Fb (H ⊕ K ) → Fb (H ) ⊗ Fb (K ) which has the following properties: (i) U ΩH ⊕K = ΩH ⊗ ΩK

(6.165)

(ii) 6Fb,fin (K ). U Fb,fin (H ⊕ K ) = Fb,fin (H )⊗

(6.166)

(iii) For all f ∈ H and g ∈ K , U AH ⊕K (f, g)# U −1 = AH (f )# ⊗ I + I ⊗ AK (g)# .

(6.167)

Proof Throughout the proof, we set A(f, g) := AH ⊕K (f, g), (f, g) ∈ ∞ H ⊕ K . Let {en }∞ n=1 and {ξn }n=1 be CONSs of H and K respectively. Then

6.15 The Boson Fock Space Over a Direct Sum Hilbert Space

307

{(en , 0), (0, ξn )|n ∈ N} is a CONS of H ⊕ K . By Corollary 6.1, the set ' Ni1 ···in ,j1 ···jm A(ei1 , 0)∗ · · · A(ein , 0)∗ A(0, ξj1 )∗ · · · A(0, ξjm )∗ ΩH ⊕K | ( i1 ≤ · · · ≤ in , j1 ≤ · · · ≤ jm , ik , j ∈ N, n, m ∈ {0} ∪ N is a CONS of Fb (H ⊕ K ), where Ni1 ···in ,j1 ···jm := A(ei1 , 0)∗ · · · A(ein , 0)∗ A(0, ξj1 )∗ · · · A(0, ξjm )∗ ΩH ⊕K −1 . Similarly, ' Ni1 ···in ;j1 ···jm AH (ei1 )∗ · · · AH (ein )∗ ΩH ⊗ AK (ξj1 )∗ · · · AK (ξjm )∗ ΩK ( | i1 ≤ · · · ≤ in , j1 ≤ · · · ≤ jm , ik , j ∈ N, n, m ∈ {0} ∪ N is a CONS of Fb (H ) ⊗ Fb (K ). Hence, by the isomorphism theorem (Theorem 1.3), there exists a unique unitary operator U : Fb (H ⊕ K ) → Fb (H ) ⊗ Fb (K ) such that, for all m, n ∈ {0} ∪ N, ik , j ∈ N, U A(ei1 , 0)∗ · · · A(ein , 0)∗ A(0, ξj1 )∗ · · · A(0, ξjm )∗ ΩH ⊕K = AH (ei1 )∗ · · · AH (ein )∗ ΩH ⊗ AK (ξj1 )∗ · · · AK (ξjm )∗ ΩK . In particular, (6.165) holds. For all fj ∈ H , gj ∈ K (j ∈ N), we have expansions fj =

∞ n=1

anj en ,

gk =

∞

bnk ξn

n=1

 with anj := en , fj and bnk := ξn , gk . Hence, for all q, r ≥ 0, Φ := A(f1 , 0)∗ · · · A(fr , 0)∗ A(0, g1 )∗ · · · A(0, gq )∗ ΩH ⊕K =

∞

an1 1 · · · anr r bm1 1 · · · bmq q A(en1 , 0)∗ · · · A(enr , 0)∗

n1 ,··· ,nr ≥1 m1 ,··· ,mq ≥1

×A(0, ξm1 )∗ · · · A(0, ξmq )∗ ΩH ⊕K .

308

6 Elements of the Theory of Boson Fock Spaces

By the continuity of U , UΦ =

∞

an1 1 · · · anr r bm1 1 · · · bmq q U A(en1 , 0)∗ · · · A(enr , 0)∗

n1 ,··· ,nr ≥1 m1 ,··· ,mq ≥1

×A(0, ξm1 )∗ · · · A(0, ξmq )∗ ΩH ⊕K =

∞

an1 1 · · · anr r bm1 1 · · · bmq q AH (en1 )∗ · · · AH (enr )∗ ΩH

n1 ,··· ,nr ≥1 m1 ,··· ,mq ≥1

⊗AK (ξm1 )∗ · · · AK (ξmq )∗ ΩK = AH (f1 )∗ · · · AH (fr )∗ ΩH ⊗ AK (g1 )∗ · · · AK (gq )∗ ΩK . On the other hand, for all f ∈ H , g ∈ K , A(f, g)∗ = A(f, 0)∗ + A(0, g)∗ on Fb,fin (H ⊕ K ). Hence Fb,fin (H ⊕ K ) is algebraically spanned by vectors of the form Φ. Thus (6.166) holds. To prove (iii), let Ψ = AH (f1 )∗ · · · AH (fr )∗ ΩH ⊗ AK (g1 )∗ · · · AK (gq )∗ ΩK .

(6.168)

Then U A(f, g)∗ U −1 Ψ = U [A(f, 0) + A(0, g)]∗ A(f1 , 0)∗ · · · A(fr , 0)∗ A(0, g1 )∗ · · · A(0, gq )∗ ΩH ⊕K = U A(f, 0)∗ A(f1 , 0)∗ · · · A(fr , 0)∗ A(0, g1 )∗ · · · A(0, gq )∗ ΩH ⊕K + U A(0, g)∗ A(f1 , 0)∗ · · · A(fr , 0)∗ A(0, g1 )∗ · · · A(0, gq )∗ ΩH ⊕K = AH (f )∗ AH (f1 )∗ · · · AH (fr )∗ ΩH ⊗ AK (g1 )∗ · · · AK (gq )∗ ΩK + AH (f1 )∗ · · · AH (fr )∗ ΩH ⊗ AK (g)∗ AK (g1 )∗ · · · AK (gq )∗ ΩK = [AH (f )∗ ⊗ I + I ⊗ AK (g)∗ ]Ψ. 6Fb,fin (K ). Since Fb,fin (X ) (X Hence (6.167) with # = ∗ holds on Fb,fin (H )⊗ 6Fb,fin (K ) is a is a Hilbert space) is a core for AX (f )∗ , it follows that Fb,fin (H )⊗ core for A(f )∗ ⊗ I + I ⊗ A(g)∗ . Hence operator equality (6.167) with # = ∗ holds. 4 5∗ In general, for a densely defined closable operator A, A∗ = A . Hence 4 5∗ U A(f, g)U −1 = AH (f )∗ ⊗ I + I ⊗ AK (g)∗ ⊃ AH (f ) ⊗ I + I ⊗ AK (g).

6.15 The Boson Fock Space Over a Direct Sum Hilbert Space

309

6Fb,fin (K ), Hence, for all Ψ ∈ Fb,fin (H )⊗ U A(f, g)U −1 Ψ = (AH (f ) ⊗ I + I ⊗ AK (g))Ψ. Then, in the same way as in the case of U A(f, g)∗ U −1 , one can conclude that (6.167) holds for A(f, g)# = A(f, g).  We call the isomorphism U in Theorem 6.24 the natural isomorphism between Fb (H ⊕ K ) and Fb (H ) ⊗ Fb (K ) and write Fb (H ⊕ K ) ∼ = Fb (H ) ⊗ Fb (K ). Example 6.8 We give an important example of the natural isomorphism established above, which is useful in applications to quantum field theory. We consider the boson Fock space Fb (L2 (Rd )) over L2 (Rd ) (d ∈ N). Let σ and κ be constants satisfying 0 ≤ σ < κ and A := {p ∈ Rd | σ ≤ |p| ≤ κ}. Then each element f ∈ L2 (Rd ) has a unique orthogonal decomposition f = f1 + f2 with f1 ∈ L2 (A) and f2 ∈ L2 (Ac ) (explicitly f1 = χA f, f2 = χAc f ). Hence L2 (Rd ) = L2 (A) ⊕ L2 (Ac ). Therefore, by Theorem 6.24, Fb (L2 (Rd )) ∼ = Fb (L2 (A)) ⊗ Fb (L2 (Ac )).

(6.169)

Remark 6.7 As is easily seen, isomorphism (6.169) holds for any Borel set A ∈ B d . Theorem 6.25 Let T and S be densely defined closable operators on H and K respectively. Let U be the unitary operator in Theorem 6.24. Then U db (T ⊕ S)U −1 = db (T ) ⊗ I + I ⊗ db (S).

(6.170)

Proof Let Ψ be a vector of the form (6.168) with fj ∈ D(T ), gk ∈ D(S). Then U db (T ⊕ S)U −1 Ψ =

r

U A(f1 , 0)∗ · · · A(Tfj , 0)∗ · · · A(fr , 0)∗ A(0, g1 )∗ · · · A(0, gq )∗ ΩH ⊕K

j =1

+

q k=1

U A(f1 , 0)∗ · · · A(fr , 0)∗ A(0, g1 )∗ · · · A(0, Sgk ) · · · A(0, gq )∗ ΩH ⊕K

310

=

6 Elements of the Theory of Boson Fock Spaces r

A(f1 , 0)∗ · · · A(Tfj , 0)∗ · · · A(fr , 0)∗ ΩH ⊗ A(0, g1 )∗ · · · A(0, gq )∗ ΩK

j =1

+

q

A(f1 , 0)∗ · · · A(fr , 0)∗ ΩH ⊗ A(0, g1 )∗ · · · A(0, Sgk ) · · · A(0, gq )∗ ΩK

k=1

= [db (T ) ⊗ I + I ⊗ db (S)]Ψ. 6Fb,fin (D(S)), It follows that, for all Ψ ∈ Fb,fin (D(T ))⊗ U db (T ⊕ S)U −1 Ψ = [db (T ) ⊗ I + I ⊗ db (S)]Ψ. Since Fb,fin (D(T )) (resp. Fb,fin (D(S))) is a core for db (T ) (resp. db (S)), it 6Fb,fin (D(S)) is a core for db (T ) ⊗ I + I ⊗ db (S). follows that Fb,fin (D(T ))⊗ We also have 6 Fb,fin (D(S)). U Fb,fin (D(T ) ⊕ D(S)) = Fb,fin (D(T ))⊗ Hence (6.170) holds.



Example 6.9 Let A ⊂ Rd be as in Example 6.8. Let F : Rd → R be a Borel measurable function a.e. finite. Then the multiplication operator by the function F on L2 (Rd ) is self-adjoint. We denote the multiplication operator by the same symbol. Let pA be the orthogonal projection onto L2 (A). Then we have pA = χA , the multiplication operator by the characteristic function of A. It is easy to see that, for all f ∈ D(F ), pA f ∈ D(F ) and FpA f = pA Ff . Hence F is reduced by L2 (A) and hence by L2 (Ac ). We have F = FA ⊕ FAc with FA = χA F and FAc = χAc F . Therefore, by Theorem 6.25, db (F ) ∼ = db (FA ) ⊗ I + I ⊗ db (FAc ) under the natural isomorphism (6.169). Remark 6.8 The contents of this section can be easily extended to the case where H ⊕ K is replaced by the direct sum ⊕N n=1 Hn of N Hilbert spaces H1 , . . . , HN with N ≥ 3.

Chapter 7

Elements of the Theory of Fermion Fock Spaces

Abstract Fundamental aspects of the theory of fermion Fock space are described.

7.1 Definitions and Basic Properties In Example 5.2 in Chap. 5, we mentioned the fermion Fock space ∧(CN ) over the N-dimensional complex Euclidean vector space CN . In this chapter, we consider an abstract (infinite dimensional) version of this space and describe its basic structures. Let K be a complex Hilbert space and, for each p ∈ N, ∧p (K ) be the pfold anti-symmetric tensor product of K (see Sect. 1.5). We set ∧0 (K ) := C. The infinite direct sum Hilbert space of ∧p (K ), p = 0, 1, 2, . . . Ff (K ) : =

∞ /

∧p (K )

p=0

⎧ ⎫ ∞ ⎨ ⎬ (p) p (p) 2 = Ψ = {Ψ (p) }∞ |Ψ ∈ ∧ (K ), p ≥ 0,

Ψ

< ∞ p=0 ⎩ ⎭ p=0

(7.1) is called the fermion Fock space over K or the anti-symmetric Fock space over K . This Hilbert space is used to describe state vectors of a Fermi field, i.e., a quantum field of identical fermions. Remark 7.1 Let K be finite-dimensional with n = dim K . Then, for all p > n, ∧p (K ) = {0}. Hence Ff (K ) = ⊕np=0 ∧p (K ). Therefore Ff (K ) is finiten p n dimensional with dim Ff (K ) = p=0 dim ∧ (K ) = 2 . Hence analysis of Ff (K ) is reduced to finite dimensional analysis. This is a big difference from the case of boson Fock spaces.

© Springer Nature Singapore Pte Ltd. 2020 A. Arai, Inequivalent Representations of Canonical Commutation and Anti-Commutation Relations, Mathematical Physics Studies, https://doi.org/10.1007/978-981-15-2180-5_7

311

312

7 Elements of the Theory of Fermion Fock Spaces

Based on the fact mentioned in Remark 7.1, throughout this chapter, we assume that K is infinite dimensional, unless otherwise stated. In the context of concrete quantum field models, K represents the state vector space of a fermion, called the one-particle space of a fermion or the one-fermion space, and the Hilbert space ∧p (K ) with p ≥ 2 as a closed subspace of Ff (K ) represents the state vector space consisting of p identical fermions. For this reason, ∧p (K ) is called the p-particle space of fermions. The subspace of the Fermion Fock space Ff (K ) corresponding to the finite particle subspace of a boson Fock space is given by % & Ff,0 (K ) := Ψ ∈ Ff (K )| ∃p0 ∈ N such that Ψ (p) = 0, ∀p ≥ p0 .

(7.2)

It is called the finite particle subspace of Ff (K ). It follows that Ff,0 (K ) is dense in Ff (K ). The vector ΩK := {1, 0, 0, . . .} ∈ Ff (K )

(7.3)

is called the fermion Fock vacuum. As in the case of boson Fock space, for a subspace D of K , we introduce a subspace ∧fin (D) of Ff (K ) by 6∞ ˆ ∧fin (D) := ⊕ p=0 Ap (⊗ D), p

(7.4)

p

ˆ D denotes the p-fold algebraic tensor product of D. It follows that, if D where ⊗ is dense in K , then ∧fin (D) is dense in Ff (K ). Example 7.1 Let (X, Σ, μ) be the measure space as in Sect. 1.7 and consider the case K = L2 (X, dμ). Then, by Theorem 1.24(ii), we have p 2 p ∼ ∞ 2 p Ff (L2 (X, dμ)) = ⊕∞ p=0 ∧ (L (X, dμ)) = ⊕p=0 Las (X , dμ ),

We call this isomorphism the natural isomorphism between Ff (L2 (X, dμ)) and 2 p p ⊕∞ p=0 Las (X , dμ ).

7.2 Fermion Second Quantization Operators In the same manner as in the case of second quantization operators on a boson Fock space, one can define second quantization operators on the fermion Fock space Ff (K ).

7.2 Fermion Second Quantization Operators

313

Let T be a densely defined closable operator on K . Then, in the same way as in the proof of Lemma 6.1, one can show that the operator T (p) on ⊗p K (see (6.5)) (p) is reduced by ∧p (K ). We denote the reduced part by Tf : (p)

Tf

:= (T (p) )∧p (K ) ,

(0)

p ≥ 2.

(7.5)

(1)

As a convention, we set Tf = 0 and Tf := T . It follows from Proposi(p) tion 1.16(iii) that Tf is a densely defined closed operator. The infinite direct sum (p) operator of Tf , p = 0, 1, 2, . . . df (T ) := ⊕∞ p=0 Tf

(p)

(7.6)

on Ff (K ) is called the fermion second quantization operator of T . In the same way as in the proof of Theorem 6.1, one can prove the following theorem: Theorem 7.1 Let T be a self-adjoint operator on K . (i) df (T ) is a self-adjoint operator and, for all t ∈ R, (p)

it Tf . eit df(T ) = ⊕∞ p=0 e

(ii) If T is non-negative, then df (T ) is non-negative and, for all z ∈ C+ , (p)

−zTf e−zdf(T ) = ⊕∞ . p=0 e

(iii) Let D be a core for T . Then ∧fin (D) is a core for df (T ). (iv) If σ (T ) = [m, ∞), then σ (df (T )) = {0} ∪ [m, ∞).

(7.7)

It follows from (7.3) that df (T )ΩK = 0.

(7.8)

0 ∈ σp (df (T )).

(7.9)

Hence

The fermion second quantization operator Nf := df (I )

(7.10)

314

7 Elements of the Theory of Fermion Fock Spaces

of the identity I on K is called the fermion number operator. It is obvious that Nf is reduced by each ∧p (K ) and Nf  ∧p (K ) = p

7.3 Fermion Γ -Operators Let T be a densely defined closable operator from a Hilbert space K to a Hilbert space H and p ∈ N. Then the p-fold tensor product ⊗p T of T is a unique densely defined closed operator from ⊗p K to ⊗p H such that p

p

(⊗p T )(⊗j =1 Ψj ) = ⊗j =1 T Ψj ,

Ψj ∈ D(T ), j = 1, . . . , p.

In the same way as in the proof of Lemma 6.1, one can prove that, for each p ≥ 2, AH ,p (⊗p T ) ⊂ (⊗p T )AK ,p ,

(7.11)

where AX ,p (X = K , H ) is the anti-symmetrization operator on ⊗p X . Hence we can define a densely defined closed operator ∧p T from ∧p (K ) to ∧p (H ) as follows: ∧0 T := 1, ∧1 T := T and, for p ≥ 2, D(∧p T ) := D(⊗p T ) ∩ ∧p (K ), (∧ T )Ψ := (⊗ T )Ψ, p

p

(7.12)

Ψ ∈ D(∧ T ). p

(7.13)

In the case H = K , ∧p T is nothing but the reduced part of ⊗p T to ∧p (K ). One can define an infinite direct sum operator Γf (T ) from Ff (K ) to Ff (H ) by p Γf (T ) := ⊕∞ p=0 ∧ T .

(7.14)

We call the operator Γf (T ) the fermion Γ -operator for T . It obvious that Γf (T )ΩK = ΩH .

(7.15)

The following theorems can be proved in quite the same manner as in the proof of Theorem 6.3.

7.3 Fermion Γ -Operators

315

Theorem 7.2 Let T be a densely defined closable operator from K to H . (i) If T is unbounded, then Γf (T ) is unbounded. (ii) If T is a contraction operator, then Γf (T ) is a contraction operator and Γf (T )∗ = Γf (T ∗ ).

(7.16)

(iii) If D is a core for T , then ∧fin (D) is a core for Γf (T ). Theorem 7.3 Let K , H and X be complex Hilbert spaces. (i) If T is self-adjoint on K , then so is Γf (T ). (ii) If U : K → H is unitary, then so is Γf (U ) : Ff (K ) → Ff (H ) and Γf (U )−1 = Γf (U −1 ).

(7.17)

Γf (U ) ∧fin (K ) = ∧fin (H ).

(7.18)

Moreover,

(iii) Let T ∈ B(K , H ) and S ∈ B(H , X ) be contraction operators. Then, Γf (S)Γf (T ) = Γf (ST ). (iv) Let Tn (n ∈ N) and T be contraction operators from K to H satisfying s- limn→∞ Tn = T . Then s- lim Γf (Tn ) = Γf (T ). n→∞

The next theorem gives relations between fermion second quantization operators and fermion Γ -operators: Theorem 7.4 Let T be a self-adjoint operator on K . (i) For all t ∈ R, Γf (eit T ) = eit df(T ) . (ii) If T ≥ 0, then df (T ) ≥ 0 and, for all z ∈ C+ , Γf (e−zT ) = e−zdf(T ) . Proof Similar to the proof of Theorem 6.4.



316

7 Elements of the Theory of Fermion Fock Spaces

7.4 Fermion Annihilation and Creation Operators For each u ∈ K , one can define an operator B † (u) on Ff (K ) as follows: ⎧ ⎫ ∞ ⎨ ⎬  √ (p−1) 2  D(B † (u)) = Ψ = {Ψ (p) }∞ ∈ F (K )

pA (u ⊗ Ψ )

< ∞ , f p p=0 ⎩ ⎭ p=1

(7.19) (B † (u)Ψ )(0) = 0, √ (B † (u)Ψ )(p) = pAp (u ⊗ Ψ (p−1) ),

(7.20) p ≥ 1, Ψ ∈ D(B † (u)).

(7.21)

It is easy to see that B † (u) is a densely defined closed operator with Ff,0 (K ) ⊂ D(B † (u)). We set B(u) := (B † (u))∗ .

(7.22)

Lemma 7.1 For all u ∈ K , ΩK ∈ D(B(u)) and B(u)ΩK = 0.

(7.23)

 Proof Property (7.20) implies that, for all Ψ ∈ D(B † (u)), ΩK , B † (u)Ψ = 0 = 0, Ψ . Hence ΩK ∈ D(B(u)) and B(u)ΩK = 0.  Lemma 7.2 The operator B(u) is a densely defined closed operator with D(B(u)) ⊃ ∧fin (K ) and, for all p ≥ 1 and uj ∈ K , j = 1, . . . , p, (B(u)u1 ∧ · · · ∧ up )(q) = 0, (B(u)u1 ∧ · · · ∧ up )(p−1) =

q = p − 1,

(7.24)

p  (−1)j −1 u, uj u1 ∧ · · · ∧ uˆ j ∧ · · · ∧ up , j =1

(7.25) where uˆ j indicates the omission of uj .

7.4 Fermion Annihilation and Creation Operators

317

Proof Let Ψ = u1 ∧ · · · ∧ up . Since B † (u) maps ∧p (K ) to ∧p+1 (K ), it follows that, for all Φ ∈ D(B † (u)), 

  √  √   Ψ, B † (u)Φ = Ψ, pAp (u ⊗ Φ (p−1) ) = pΨ, u ⊗ Φ (p−1) = Ψp−1 , Φ ,

where (Ψp−1 )(q) = 0 for q = p − 1 and (Ψp−1 )(p−1) = √

 1 sgn(σ ) u, uσ (1) uσ (2) ⊗ · · · ⊗ uσp . (p − 1)! σ ∈S

(7.26)

p

Hence Ψ ∈  D(B(u)) and B(u)Ψ = Ψp−1 . In particular, (7.24) holds. The summation σ ∈Sp on the right hand side of (7.26) is rewritten as σ ∈Sp

=

p



.

j =1 σ ∈Sp , σ (1)=j

Using this structure, one can easily see that the right hand side of (7.26) is equal to that of (7.25).  By Lemma 7.2, the adjoint B(u)∗ of B(u) exists and B(u)∗ = B † (u).

(7.27)

In what follows, we mainly use the symbol B(u)∗ instead of B † (u). As remarked in the proof of Lemma 7.2, B(u)∗ maps ∧p (K ) to ∧p+1 (K ) for all p ≥ 0. This implies that B(u) maps ∧p (K ) to ∧p−1 (K ) with ∧−1 (K ) := {0}. Based on these properties, B(u)∗ and B(u) are called respectively the fermion creation operator and the fermion annihilation operator with test vector u. Proposition 7.1 Let n ∈ N and uj ∈ K , j = 1, . . . , n. Then, ΩK ∈ D(B(u1 )∗ · · · B(un )∗ ) and, for all p ≥ 1, (B(u1 )∗ · · · B(un )∗ ΩK )(p) = δnp u1 ∧ · · · ∧ up . Proof We need only to apply (7.21) repeatedly.

(7.28) 

It follows from Proposition 7.1 that, for every subspace D ⊂ K , ∧fin (D) = span{ΩK , B(u1 )∗ · · · B(up )∗ ΩK |p ≥ 1, uj ∈ D, j = 1, · · · , p}. (7.29)

318

7 Elements of the Theory of Fermion Fock Spaces

Formula (7.28) and Theorem 1.22 imply the following theorem: Theorem 7.5 Let {ej }∞ j =1 be a CONS of K . Then {ΩK , B(ei1 )∗ · · · B(eip )∗ ΩK |p ≥ 1, i1 , . . . , ip ∈ N, i1 < i2 < . . . < ip } is a CONS of Ff (K ). For each u ∈ K , we denote by B(u)# either B(u) or B(u)∗ . Formulae (7.25) and (7.28) give the following result: Corollary 7.1 For all p ∈ N and u1 , . . . , up ∈ K , B(u1 )∗ · · · B(up )∗ ∈ D(B(u)# ) and ∗

∗

B(u)B(u1 ) · · · B(up ) ΩK =

p

∗ ∗  (−1)j −1 B(u1 )∗ · · · B(u j ) · · · B(up ) ΩK .

j =1

(7.30) We now prove a basic theorem: Theorem 7.6 For every u ∈ K , B(u) and B(u)∗ are everywhere defined bounded operators on Ff (K ) with

B(u) = B(u)∗ = u .

(7.31)

Moreover, the following anti-commutation relations hold: {B(u), B(v)∗ } = u, v , {B(u), B(v)} = 0,

{B(u)∗ , B(v)∗ } = 0,

(7.32) u, v ∈ K .

(7.33)

Proof Let u, v ∈ K and Ψp := B(u1 )∗ · · · B(up )∗ ΩK , p ∈ N. Then, by Corollary 7.1, we have B(u)B(v)∗ Ψp = u, v Ψp p  ∗ ∗  − (−1)j −1 u, uj B(v)∗ B(u1 )∗ · · · B(u j ) · · · B(up ) ΩK , j =1

B(v)∗ B(u)Ψp =

p

 ∗ ∗  (−1)j −1 u, uj B(v)∗ B(u1 )∗ · · · B(u j ) · · · B(up ) ΩK .

j =1

Hence (B(u)B(v)∗ + B(v)∗ B(u))Ψp = u, v Ψp .

7.4 Fermion Annihilation and Creation Operators

319

By (7.29), we obtain B(u)B(v)∗ + B(v)∗ B(u) = u, v

on ∧fin (K ).

(7.34)

In particular, B(u)B(u)∗ + B(u)∗ B(u) = u 2

on ∧fin (K ).

Hence, by applying Lemma 5.1 to the case A = u −1 B(u) with u = 0, we conclude that u −1 B(u) is everywhere defined bounded operator with B(u) ≤

u and operator equality B(u)B(u)∗ + B(u)∗ B(u) = u 2 holds (for u = 0, this is trivial). Hence B(u)∗ also is everywhere defined bounded operator and B(u)∗ = B(u) ≤ u . Since ∧fin (K ) is dense in Ff (K ), (7.34) extends to (7.32). Taking u = v in (7.32) and using (7.23), we have

B(u)∗ ΩK = u .

(7.35)

But B(u)∗ ΩH ≤ B(u)∗ ΩK = B(u)∗ . Hence B(u)∗ ≥ u . Thus it follows that B(u)∗ = u . Since B(u) = B(u)∗ , (7.31) holds. We have (B(u)∗ B(v)∗ Ψp )(p+2) = u ∧ v ∧ u1 ∧ · · · ∧ up

(7.36)

= −v ∧ u ∧ u1 ∧ · · · ∧ up

(7.37)

= −(B(v)∗ B(u)∗ Ψp )(p+2).

(7.38)

Hence it follows that {B(u)∗ , B(v)∗ } = 0 on ∧fin (K ). Since ∧fin (K ) is dense in Ff (K ) and B(·)∗ is bounded as already proved, we obtain operator equality {B(u)∗ , B(v)∗ } = 0. Taking the adjoint of this equation, we have operator equality {B(u), B(v)} = 0.  Anti-commutation relations (7.32) and (7.33) are called the canonical anticommutation relations (CAR) over K or indexed by K . Putting u = v in (7.33), we have B(u)2 = 0,

B(u)∗ 2 = 0,

Hence B(u) and B(u)∗ are nilpotent operators.

u∈K.

(7.39)

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7 Elements of the Theory of Fermion Fock Spaces

As we have already seen, for all u ∈ K , B(u) vanishes the fermion Fock vacuum ΩK (see (7.23)). In fact, the converse is true: Theorem 7.7 Let D be a dense subspace of K and Ψ ∈ Ff (K ) be a vector such that, for all u ∈ D, B(u)Ψ = 0. Then there exists a constant α ∈ C such that Ψ = αΩK . Proof For all Φ ∈ Ff (K ), we have  0 = Φ, B(u)Ψ  = B(u)∗ Φ, Ψ . √ Let Φ ∈ ∧p (K ) (p ≥ 0). Then p + 1Ap+1 (u ⊗ Φ), Ψ (p+1) = 0. The subspace algebraically spanned by {Ap+1 (u ⊗ Φ)|u ∈ D, Φ ∈ ∧p (K )} is dense in ∧p+1 (K ). Hence Ψ (p+1) = 0. Since p ≥ 0 is arbitrary, it follows that Ψ = {Ψ (0), 0, 0, . . .} = αΩK with α := Ψ (0).  The following theorem is important: Theorem 7.8 Let D be a dense subspace of K . Then {B(u), B(u)∗ |u ∈ D} = CI . In particular, {B(u), B(u)∗ |u ∈ D} is irreducible. Proof Let T ∈ {B(u), B(u)∗ |u ∈ D} so that, for all u ∈ D, B(u)# T = T B(u)# . Then, by (7.23), B(u)T ΩK = 0. Hence, by Theorem 7.7, there exists a constant α ∈ C such that T ΩK = αΩK . Let Ψ := B(u1 )∗ · · · B(up )∗ ΩK (uj ∈ D, j = 1, . . . , p, p ∈ N). Then it follows from the commutativity of T and B(uj )∗ that T Ψ = αΨ . Since the subspace algebraically spanned by vectors of the form Ψ is dense in Ff (K ) and T is bounded, it follows that T = αI . Thus {B(u), B(u)∗ |u ∈ D} = CI . By this fact and Proposition 2.9(i), {B(u), B(u)∗ |u ∈ D} is irreducible. 

7.5 Commutation Relations Between B(·)# and dΓf (·) Theorem 7.9 Let T be a densely defined closable operator on K . (i) For every u ∈ D(T ), B(u)∗ leaves D(dΓf (T )) invariant and [dΓf (T ), B(u)∗ ] = B(T u)∗

on D(df (T )).

(7.40)

(ii) For every u ∈ D(T ∗ ), B(u) leaves df (T ) invariant and [dΓf (T ), B(u)] = −B(T ∗ u)

on D(df (T )).

(7.41)

7.6 Tranformations of B(·)# by Γf (·)

321

Proof (i) Let Ψ ∈ ∧fin (D(T )). Then Ψ ∈ D(df (T )) and B(u)∗ Ψ ∈ D(df (T )). Moreover, √ √ pAp (T u ⊗ Ψ (p−1) ) + pAp (u ⊗ (df (T )Ψ )p−1 ), √ = pAp (u ⊗ (df (T )Ψ )(p−1) ).

(df (T )B(u)∗ Ψ )(p) = (B(u)∗ df (T )Ψ )(p) Hence

([df (T ), B(u)∗ ]Ψ )(p) =

√ pAp (T u ⊗ Ψ (p−1) ) = (B(T u)∗ Ψ )(p) .

Therefore [df (T ), B(u)∗ ]Ψ = B(T u)∗ Ψ. Let Ψ ∈ D(df (T )). Since ∧fin (D(T )) is a core for dΓf (T ), there exists a sequence {Ψn }n in ∧fin (D(T )) such that Ψn → Ψ, dΓf (T )Ψn → dΓf (T )Ψ (n → ∞). By the result obtained above, we have dΓf (T )B(u)∗ Ψn = B(u)∗ dΓf (T )Ψn + B(T u)∗ Ψn . By the boundedness of B(·)∗ , the right hand side of this equation converges to B(u)∗ dΓf (T )B(u)∗ Ψ + B(T u)∗ Ψ as n → ∞. Hence, by the closedness of dΓf (T ), B(u)∗ Ψ ∈ D(dΓf (T )) and dΓf (T )B(u)∗ Ψ = B(u)∗ dΓf (T )Ψ + B(T u)∗ Ψ . Thus, B(u)∗ leaves D(dΓf (T )) invariant and (7.40) holds on D(df (T )). (ii) Let u ∈ D(T ∗ ). Then, by considering the adjoint of (7.40) with T replaced by T ∗ , we have [df (T ), B(u)] = −B(T ∗ u) on ∧fin (D(T )). Then, by a limiting argument similar to that in (i), one can see that B(u) leaves D(df (T )) invariant and (7.41) holds. 

7.6 Tranformations of B(·)# by Γf (·) Theorem 7.10 Let W be a unitary operator on K . Then Γf (W )B(u)# Γf (W )−1 = B(W u)# ,

u∈K.

(7.42)

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7 Elements of the Theory of Fermion Fock Spaces

Proof Let Ψ = B(u1 )∗ · · · B(un )∗ ΩK (n ≥ 0, uj ∈ K , j = 1, . . . , n). Then it is easy to see that Γf (W )B(u)∗ Ψ = B(W u)∗ Γf (W )Ψ . Since the subspace algebraically spanned by vectors of the form Ψ is dense in Ff (K ) and Γf (W ) and B(u)∗ are bounded, it follows that Γf (W )B(u)∗ = B(W u)∗ Γf (W ). Hence (7.42) with B(u)# = B(u)∗ holds. Similarly (or by taking the adjoint of (7.42) with B(u)# = B(u)∗ ), we obtain (7.42) with B(u)# = B(u).  By this theorem and Theorem 7.4(i), we obtain the following theorem: Theorem 7.11 Let T be a self-adjoint operator on K . Then, for all t ∈ R and u∈K, eit dΓf (T ) B(u)# e−it dΓf (T ) = B(eit T u)# , Corollary 7.2 For all u ∈ K , eiπNf B(u)# e−iπNf = −B(u)# , We introduce a bounded self-adjoint operator (−1)Nf on Ff (K ) by ((−1)Nf Ψ )(p) := (−1)p Ψ (p) ,

Ψ ∈ Ff (K ), p ≥ 0.

(7.43)

By functional calculus, we have (−1)Nf = eiπNf = e−iπNf .

(7.44)

Corollary 7.2 is rephrased as the anti-commutativity of (−1)Nf and B(u)# : {(−1)Nf , B(u)# } = 0.

(7.45)

7.7 Representations of Fermion Second Quantization Operators in Terms of B(·)# As in the case of boson second quantization operators, fermion second quantization operators also have representations in terms of B(·)# . We first derive a basic formula: Lemma 7.3 Let T ∈ L(K ) be densely defined and S ∈ L(K ). Then, for all u ∈ D(S) ∩ D(T ∗ ), B(Su)∗ B(T ∗ u) = df (PT ∗ u,Su ), where Pv,w (v, w ∈ K ) is defined by (1.23).

(7.46)

7.8 Uniform Differentiability of Basic Operator-Valued Functions

323

Proof As in Lemma 6.8, one first proves (7.46) on ∧fin (K ) (see (7.29)). Since B(Su)∗ B(T ∗ u) is bounded (this is different from the bosonic case), one obtains (7.46) as an operator equality.  Theorem 7.12 Let S, T ∈ B(K ) and {en }∞ n=1 be a CONS of K . Then, for all Ψ ∈ Ff,0 (K ), ∞

B(Sen )∗ B(T ∗ en )Ψ = df (ST )Ψ.

(7.47)

n=1

Proof Since we have Lemma 7.3, one can prove (7.47) in the same way as in the proof of Theorem 6.12.  Corollary 7.3 Let S be a bounded self-adjoint operator on K and {en }∞ n=1 be a 2 1/2 CONS of K . Then, for all Ψ ∈ D(df (S ) ), ∞

B(Sen )Ψ 2 = df (S 2 )1/2 Ψ 2 .

(7.48)

n=1 1/2

In particular, for all Ψ ∈ D(Nf ∞

), 1/2

B(en )Ψ 2 = Nf

Ψ 2 .

(7.49)

n=1

Proof We apply (7.47) with T ∗ = S to obtain ∞

B(Sen )Ψ 2 = df (S 2 )1/2 Ψ 2

n=1

for all Ψ ∈ Ff,0 (H ). Then, in the same way as in the proof of Corollary 6.10, one obtains (7.48). Formula (7.49) is just a special case of (7.48) with S = I . 

7.8 Uniform Differentiability of Basic Operator-Valued Functions For a self-adjoint operator T on K and a vector u ∈ K , one has bounded operator-valued functions B(eit T u) and B(eit T u)∗ of t. In this section we consider differentiability of them. To state a theorem on this aspect, we recall a definition of differentiability of an operator-valued function on an interval I of R. Let H be a Hilbert space and a ∈ I. Then an operator-valued function T (·) : I → B(H ); I t → T (t) ∈ B(H ) is

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7 Elements of the Theory of Fermion Fock Spaces

said to be uniformly differentiable at t = a if the uniform limit T (a) := u- lim

t →a

1 (T (t) − T (a)) ∈ B(H ) t −a

exists (i.e., limt →a T (a) − (t − a)−1 (T (t) − T (a)) = 0). If T (·) is uniformly differentiable at any point t in I, then T (·) is said to be uniformly differentiable on I. In this case, we define 1 dT (t) := u- lim (T (t + ε) − T (t)) ∈ B(H ) ε→0 ε dt and call it the uniform derivative of T (·). The uniform derivative dT (t)/dt of T (·) is written also T (t). If dT (t)/dt is uniformly differentiable on I, then T (·) is said to be twice uniformly differentiable on I. In this case, the second uniform derivative of T (·) is defined by dT (t) d 2 T (t) . := dt 2 dt In a similar manner, for any natural number n ≥ 3, one can define n times uniform differentiability of T (·) and the nth uniform derivative T (n) (t) = d n T (t)/dt n inductively: d n T (t) dT (n−1) (t) . := dt n dt Theorem 7.13 Let u ∈ D(T n ) and n ∈ N. Then B(eit T u)# is n times uniformly differentiable in t ∈ R with uniform derivative d n B(eit T u)# = B(i n eit T T n u)# . dt n Proof We first consider the case n = 1. Let t ∈ R, ε ∈ R \ {0}, u ∈ D(T ) and Aε :=

B(ei(t +ε)T u)# − B(eit T u)# − B(ieit T T u)# . ε

Then we have Aε = B(eit T uε )# with uε :=

eiεT u − u − iT u. ε

7.9 The Fermion Fock Space Over a Direct Sum Hilbert Space

325

Hence Aε = eit T uε = uε . It is well known that limε→ uε = 0. Thus the present theorem with n = 1 holds. Similarly, by induction in n, one can prove the theorem with any n. 

7.9 The Fermion Fock Space Over a Direct Sum Hilbert Space Let K1 and K2 be complex Hilbert spaces. Then one can consider the fermion Fock space Ff (K1 ⊕ K2 ) over the direct sum Hilbert space K1 ⊕ K2 . As in the boson Fock space case, there exists a natural isomorphism between Ff (K1 ⊕ K2 ) and Ff (K1 ) ⊗ Ff (K2 ) if K1 and K2 are separable. To be definite, for i = 1, 2, we denote by BKi (u) the annihilation operator on Ff (Ki ) with test vector u ∈ Ki . On the other hand, we simply denote by B(u, v) the annihilation operator on Ff (K1 ⊕ K2 ) with test vector (u, v) ∈ K1 ⊕ K2 . Theorem 7.14 Suppose that K1 and K2 are separable. Then there exists a unique unitary operator U : Ff (K1 ⊕ K2 ) → Ff (K1 ) ⊗ Ff (K2 ) such that U ΩK1 ⊕K2 = ΩK1 ⊗ ΩK2

(7.50)

and, for all n, m ∈ Z+ and ui ∈ K1 , vj ∈ K2 (i = 1, . . . , n, j = 1, . . . , m), U B(u1 , 0)∗ · · · B(un , 0)∗ B(0, v1 )∗ · · · B(0, vm )∗ ΩK1 ⊕K2

(7.51)

= BK1 (u1 )∗ · · · BK1 (un )∗ ΩK1 ⊗ BK2 (v1 )∗ · · · BK2 (vm )∗ ΩK2

(7.52)

Moreover, the following hold: (i) For all u ∈ K1 , v ∈ K2 , U B(u, v)# U −1 = BK1 (u)# ⊗ I + (−1)Nf ⊗ BK2 (v)# .

(7.53)

(ii) Let T1 and T2 be densely defined closable operators on K1 and K2 respectively. Then U df (T1 ⊕ T2 )U −1 = df (T1 ) ⊗ I + I ⊗ df (T2 ).

(7.54)

(iii) Let A1 and A2 be contraction operators on K1 and K2 respectively. Then U Γ (A1 ⊕ A2 )U −1 = Γ (A1 ) ⊗ Γ (A2 ).

(7.55)

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7 Elements of the Theory of Fermion Fock Spaces

∞ Proof Let {en }∞ n=1 and {fn }n=1 be CONS’s of K1 and K2 respectively. Then, by Theorem 1.20,

{B(ei1 )∗ · · · B(eip )∗ ΩK1 ⊗ B(fj1 )∗ · · · B(fjq )∗ ΩK2 | p, q ≥ 0, i1 < . . . < ip , j1 < . . . < jq , ik , j ∈ N, k = 1, . . . , p,  = 1, . . . , q} is a CONS of Ff (K1 ) ⊗ Ff (K2 ). The set {(en , 0), (0, fn )| n ∈ N} is a CONS of K1 ⊕ K2 . Hence, by Theorem 7.5, {B(ei1 , 0)∗ · · · B(eip , 0)∗ B(0, fj1 )∗ · · · B(0, fjq )∗ ΩK1 ⊕K2 | p, q ≥ 0, i1 < . . . < ip , j1 < . . . < jq , ik , j ∈ N, k = 1, . . . , p,  = 1, . . . , q} is a CONS of Ff (K1 ⊕ K2 ). Hence, by the isomorphism theorem, there exists a unique unitary operator U from Ff (K1 ⊕ K2 ) to Ff (K1 ) ⊗ Ff (K2 ) such that, for all p, q ≥ 0 and i1 < . . . < ip , j1 < . . . < jq , ik , j ∈ N, U B(ei1 , 0)∗ · · · B(eip , 0)∗ B(0, fj1 )∗ · · · B(0, fjq )∗ ΩK1 ⊕K2 = B(ei1 )∗ · · · B(eip )∗ ΩK1 ⊗ B(fj1 )∗ · · · B(fjq )∗ ΩK2 Hence, in particular, (7.50) holds. It is easy to see that, for each u ∈ K1 , B(u)∗ =

∞

en , u B(en )∗

(7.56)

n=1

in the sense of operator norm. Using this fact, one can prove (7.52). (i) Let Ψ = B(u1 , 0)∗ · · · B(un , 0)∗ B(0, v1 )∗ · · · B(0, vm )∗ ΩK1 ⊕K2 . Then B(u, v)∗ Ψ = B(u, 0)∗ Ψ + B(0, v)∗ Ψ = B(u, 0)∗ Ψ + (−1)n Ψ , where Ψ := B(u1 , 0)∗ · · · B(un , 0)∗ B(0, v)∗ B(0, v1 )∗ · · · B(0, vm )∗ ΩK1 ⊕K2 . Hence, by the preceding result, we have U B(u, v)∗ Ψ = (B(u)∗ ⊗ I )U Ψ + (−1)n (I ⊗ B(v)∗ )U Ψ = (B(u)∗ ⊗ I )U Ψ + ((−1)Nf ⊗ B(v)∗ )U Ψ. Since the subspace algebraically spanned by vectors of the form Ψ is dense in Ff (K ⊕ K2 ) and B(u, v)∗ , B(u)∗ and (−1)Nf ⊗ B(v)∗ are bounded, it follows that U B(u, v)∗ = (B(u) ⊗ I )U + ((−1)Nf ⊗ B(v)∗ )U.

7.9 The Fermion Fock Space Over a Direct Sum Hilbert Space

327

Thus (7.53) with # = ∗ holds. Taking the adjoint of the operator equality just obtained, we have (7.53) with # = ∅. Propositions (ii) and (iii) can be proved in the same way as in the proof of Theorem 6.25.  Remark 7.2 Note that the operator (−1)Nf appears on the right hand side of (7.53). This is a big difference from the case of boson creation and annihilation operators. Remark 7.3 Theorem 7.14 can be easily extended to the case where K1 ⊕ K2 is replaced by the direct sum ⊕N j =1 Kj of N Hilbert spaces K1 , . . . , KN with N ≥ 3.

Chapter 8

Representations of CCR with Infinite Degrees of Freedom

Abstract A general theory of representations of CCR with infinite degrees of freedom is presented. A basic class of such representations, called Fock representations, is described. Moreover, deformations of Fock representations are discussed.

8.1 Representation of CCR In Chap. 6, we have seen that boson creation and annihilation operators on the boson Fock space Fb (H ) obey the CCR over H (Theorem 6.8) and the Fock vacuum ΩH is a cyclic vector for {A(f1 )∗ A(f2 )∗ · · · A(fn )∗ |n ∈ N, fj ∈ D, j = 1, . . . , n} with D being a dense subspace of H (Theorem 6.6). Moreover, it has been suggested that various operators on the boson Fock space Fb (H ) can be constructed from A(·) and A(·)∗ . One such operator is the Segal field operator ΦS (f ) with test vector f ∈ H , which is a simple linear combination of creation operator A(f )∗ and annihilation operator A(f ) and has interesting properties as shown in Sect. 6.11. Noting these aspects and taking a representation theoretic point of view, one may regard the boson Fock space Fb (H ) as a Hilbert space in which algebraic objects obeying CCR are realized as A(f )# ’s (f ∈ H ). This point of view may lead one to introduce a representation theoretical concept of CCR indexed by a vector space. Definition 8.1 Let F be a Hilbert space and D be a dense subspace in F . Let V be a complex inner product space and C: V → L(F ); V f → C(f ) ∈ L(F ) such that, for all f ∈ V , C(f ) is densely defined and closed. The triplet (F , D, {C(f )|f ∈ V }) (or (F , D, {C(f ), C(f )∗ |f ∈ V })) is called a representation of the CCR over V if the following properties are satisfied: (C.1) For all f ∈ V , D ⊂ D(C(f ))∩D(C(f )∗ ) and C(f )D ⊂ D, C(f )∗ D ⊂ D. (C.2) (anti-linearity in test vectors) For all f, g ∈ V and α, β ∈ C, C(αf + βg) = α ∗ C(f ) + β ∗ C(g) on D.

© Springer Nature Singapore Pte Ltd. 2020 A. Arai, Inequivalent Representations of Canonical Commutation and Anti-Commutation Relations, Mathematical Physics Studies, https://doi.org/10.1007/978-981-15-2180-5_8

329

330

8 Representations of CCR with Infinite Degrees of Freedom

(C.3) (CCR) For all f, g ∈ V , [C(f ), C(g)∗ ] = f, gV , [C(f ), C(g)] = 0,

(8.1) ∗

∗

[C(f ) , C(g) ] = 0

(8.2)

on D. The Hilbert space F is called a representation space of the CCR under consideration. The subspace D is called a CCR-domain for {C(f ), C(f )∗ |f ∈ V }. If {C(f ), C(f )∗ | f ∈ V } is irreducible (resp. reducible), then the representation (F , D, {C(f )|f ∈ V }) is said to be irreducible (resp. reducible). Remark 8.1 The second commutation relation in (8.2) follows from taking the adjoint of the first one. Remark 8.2 Property (C.2) implies the linearity of C(f )∗ in f : for all f, g ∈ V and α, β ∈ C, C(αf + βg)∗ = αC(f )∗ + βC(g)∗ on D. Remark 8.3 By Lemma 2.1(iii), (8.1) with g = f implies that, for f = 0, at least one of C(f ) and C(f )∗ is unbounded. Hence, in fact, both C(f ) and C(f )∗ are unbounded. Example 8.1 Let (F , D, {C(f )|f ∈ V }) be a representation of the CCR over V and U : F → F (F is a Hilbert space) be a unitary operator. Then (F , U D, {U C(f )U −1 | f ∈ V }) is a representation of the CCR over V . Example 8.2 Let H be a Hilbert space and V be a dense subspace of H . Then the triplet ρF (V ) := {Fb (H ), Fb,0 (H ), {A(f )|f ∈ V }} is an irreducible representation of the CCR over V (the irreducibility is due to Theorem 6.9). This representation is called the Fock representation of the CCR over V . Example 8.3 Let V be a dense subspace of a Hilbert space H , L: V → C be a linear functional and CL (f ) := A(f ) + L(f )∗ ,

f ∈V.

(8.3)

Then it is easy to see that {CL (f ), CL (f )∗ |f ∈ V } satisfies the CCR over V on Fb,0 (H ). Let T ∈ {CL (f ), CL (f )∗ |f ∈ V } . Then it follows that T ∈ {A(f ), A(f )∗ |f ∈ V } . Hence T = αI for some α ∈ C. Therefore {CL (f ), CL (f )∗ |f ∈ V } = CI . Thus βL (V ) := (Fb (H ), Fb,0 (H ), {CL (f )|f ∈ V })

(8.4)

8.1 Representation of CCR

331

is an irreducible representation of the CCR over V . The correspondence (A(·), A(·)∗ ) → (CL (·), CL (·)∗ ) of operator-valued functionals on V is called the Bogoliubov translation by the translation functional L(·). Example 8.4 Let W : H → H be an isometry (i.e., W is linear and Wf =

f , f ∈ H ) and AW (f ) := A(Wf ). Let V be a dense subspace of H . Then (Fb (H ), Fb,0 (H ), {AW (f )|f ∈ V }) is a representation of the CCR over V . It may depend on properties of W if this representation is irreducible. (1) The case where W is unitary In this case, W V is dense in H . Hence, by Theorem 6.9, {AW (f ), AW (f )∗ | f ∈ V } is irreducible. (2) The case where W is not unitary In this case, (W V )⊥ = {0}. Hence, for all g ∈ (W V )⊥ \ {0} and f ∈ V , Wf, g = 0. Therefore, by (6.118), i eiΦS (g)AW (f )e−iΦS (g) = AW (f ) − √ Wf, g = AW (f ), 2 i.e., eiΦS (g)AW (f ) = AW (f )eiΦS (g). This implies also that eiΦS (g)AW (f )∗ = AW (f )∗ eiΦS (g) . For every g = 0, eiΦS (g) = I .1 Thus {AW (f ), AW (f )∗ | f ∈ V } = CI . The set {AW (f ), AW (f )∗ |f ∈ V } is ∗-invariant. Hence {AW (f ), AW (f )∗ |f ∈ V } is not irreducible. As in the case of representations of CCR with finite degrees of freedom, we define a concept of equivalence among representations of CCR in the sense defined above. Definition 8.2 Let (F , D, {C(f )|f ∈ V }) and (F , D , {C (f )|f ∈ V }) be two representations of the CCR over V . They are said to be equivalent if there exists a unitary operator U : F → F such that, for all f ∈ V , C (f ) = U C(f )U −1 (operator equality). In that case, we often say simply that {C(f )|f ∈ V } is equivalent to {C (f )|f ∈ V }.

general, for a self-adjoint operator A on a Hilbert space, σ (eiA ) = {eiλ |λ ∈ σ (A)} by the spectral mapping theorem. Recall that, if g = 0, then σ (ΦS (g)) = R by (6.121).

1 In

332

8 Representations of CCR with Infinite Degrees of Freedom

Remark 8.4 The operator equality C (f ) = U C(f )U −1 implies the operator equality C (f )∗ = U C(f )∗ U −1 . Remark 8.5 Suppose that ρ := (F , D, {C(f )|f ∈ V }) is equivalent to ρ := (F , D , {C (f )|f ∈ V }). Then, by Proposition 2.8, ρ is reducible (resp. irreducible) if and only if ρ is reducible (resp. irreducible). Hence, if ρ is reducible and ρ is irreducible, then ρ is inequivalent to ρ . But this type of inequivalence is a trivial inequivalence. Non-trivial inequivalences may occur among irreducible representations. Example 8.5 Let us consider the representation (Fb (H ), Fb,0 (H ), {AW (f )|f ∈ V }) in Example 8.4. Suppose that W is unitary. Then, Γb (W ) is a unitary operator on Fb (H ) and, by Theorem 6.7, Γb (W )A(f )Γb (W )−1 = AW (f ),

f ∈V.

Hence (Fb (H ), Fb,0 (H ), {AW (f )|f ∈ V }) is equivalent to the Fock representation ρF (V ). Example 8.6 We consider the case where W is not unitary in Example 8.4. The reducibility of the representation in this case can be understood structurally as follows. Let M := W V , which is not equal to H in the present case. Hence we have a non-trivial orthogonal decomposition H = M ⊕ M ⊥. Then, by Theorem 6.24, there exists a unitary operator U : Fb (H ) → Fb (M ) ⊗ Fb (M ⊥ ) such that U AW (f )U −1 = AM (Wf ) ⊗ I,

f ∈V,

where AM (·) denotes the annihilation operator on Fb (M ). Let {Ψn }∞ n=1 be a CONS of Fb (M ⊥ ) and Ln := {αΨn |α ∈ C}. Then we have the orthogonal decomposition U Fb (H ) = ⊕∞ n=1 Fb (M ) ⊗ Ln . Since Ln is one dimensional, there exists a unitary operator Vn : Fb (M ) ⊗ Ln → Fb (M ) such that Vn (Ψ ⊗ (αΨn ) = αΨ, α ∈ C, Ψ ∈ Fb (M ). Thus, putting V := ⊕∞ n=1 Vn and Y := V U , we obtain (n) Y Fb (H ) = ⊕∞ n=1 Fb (M )

8.2 Cyclic Representations

333

(n)

(n)

with Fb (M ) := Fb (M ), n ∈ N. We denote by AM (·) the annihilation operator on Fb(n) (M ). Since W V is dense in M , ρn := (Fb(n) (M ), Fb,0 (M ), {A(n) M (Wf ) f ∈ V }) is an irreducible representation of the CCR over V . It is not difficult to show that Y AW (f )Y −1 = ⊕∞ n=1 AM (Wf ). (n)

Thus (Fb (H ), Fb,0 (H ), {AW (f )|f ∈ V }) is equivalent to ⊕∞ n=1 ρn , the direct sum of irreducible representations ρn . As in Example 8.5, we have −1 A(n) M (Wf ) = Γb (W )A(f )Γb (W ) ,

f ∈V,

where Γb (W ) is considered as a unitary operator from Fb (H ) onto Fb (M ). Hence ρn is equivalent to the Fock representation ρF (V ). Thus (Fb (H ), Fb,0 (H ), {AW (f ) |f ∈ V }) is equivalent to an infinite direct sum of the Fock representation ρF (V ). This is a simple example of completely reducible representations.2

8.2 Cyclic Representations We first discuss representations of CCR which can be treated more easily. Definition 8.3 Let ρ := (F , D, {C(f )|f ∈ V })

(8.5)

be a representation of the CCR over V . (i) If there exists a unit vector Ω ∈ D ( Ω = 1) such that C(f )Ω = 0, ∀f ∈ V , then Ω is called a vacuum for ρ or {C(f )|f ∈ V } and ρ is said to have a vacuum Ω. In this case, one can define a subspace: Fρ,fin := span{Ω, C(f1 )∗ · · · C(fn )∗ Ω|n ∈ N, fj ∈ V , j = 1, . . . , n}. (8.6) (ii) Representation ρ is called a cyclic representation if it has a vacuum Ω which is cyclic for {C(f1 )# · · · C(fn )# )|n ∈ N, fj ∈ V , j = 1, . . . , n}. We write this representation as (F , D, {C(f )|f ∈ V }, Ω).

reducible representation of the CCR over an inner product space V is said to be completely reducible if it is equivalent to a direct sum representation of irreducible representations of the CCR over V .

2A

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8 Representations of CCR with Infinite Degrees of Freedom

Example 8.7 The Fock representation ρF (V ) (Example 8.2) is a cyclic representation with a vacuum being equal to the Fock vacuum ΩH . Lemma 8.1 Let ρ be a representation of the CCR over V of the form (8.5) having a vacuum Ω. Then, for all f, g ∈ V , CCR (8.1) and (8.2) hold on Fρ,fin . Proof Let Φ ∈ D and Ψ ∈ Fρ,fin . Then 

 Φ, [C(f ), C(g)∗ ]Ψ = [C(g), C(f )∗ ]Φ, Ψ = g, f  Φ, Ψ  = Φ, f, g Ψ  .

Since D is dense in F , it follows that [C(f ), C(g)∗ ]Ψ = f, g Ψ . Similarly, one can prove (8.2) on Fρ,fin .  Let ρ be a representation of the CCR over V given by (8.5) having a vacuum Ω. Then one can define a linear operator Cfin (f ) (f ∈ V ) on the Hilbert space Fρ := Fρ,fin

(the closure of Fρ,fin )

(8.7)

as follows: D(Cfin (f )) := Fρ,fin ,

Cfin (f )Ψ := C(f )Ψ,

Ψ ∈ Fρ,fin .

Then it easy to see that Cfin (f ) is densely defined and Fρ,fin ⊂ D(Cfin (f )∗ ) with Cfin (f )∗ Ψ = C(f )∗ Ψ, Ψ ∈ Fρ,fin . Hence Cfin (f ) is closable. We denote its closure by CΩ (f ). Lemma 8.1 shows that πΩ (V ) := {Fρ , Fρ,fin , {CΩ (f )|f ∈ V }, Ω} is a cyclic representation of the CCR over V . Theorem 8.1 Let ρ be a representation of the CCR over V given by (8.5) having a vacuum Ω. Then there exists a unitary operator U : Fρ → Fb (H ) such that U Ω = ΩH and U CΩ (f )U −1 = A(f ), f ∈ V . In particular, πΩ (V ) is equivalent to the Fock representation ρF (V ). Proof We define a mapping U0 : Fρ,fin → Fb,0 (H ) by U0 Ω := ΩH , U0 C(f1 )∗ · · · C(fn )∗ Ω

:= A(f1 )∗ · · · A(fn )∗ ΩH

(fj ∈ V , j = 1, · · · , n, n ∈ N). For any element Ψ in Fρ,fin , U0 Ψ is defined by linear extension (this is well-defined). It is easy to see that U0 is isometry and Ran (U0 ) = Fb,fin (V ). Since Fρ,fin is dense in Fρ , by the extension theorem on

8.3 Second Quantization Operator Associated with a Representation of CCR

335

densely defined bounded linear operators, the closure U := U 0 is a unitary operator from Fρ to Fb (H ). Let f ∈ V . Then, by CCR and the fact that C(f )Ω = 0, we have j th

n   CΩ (f )C(f1 ) · · · C(fn ) Ω = f, fj C(f1 )∗ · · · C(fj )∗ · · · C(fn )∗ Ω. ∗

∗

j =1

Hence j th

n   U CΩ (f )C(f1 ) · · · C(fn ) Ω = f, fj A(f1 )∗ · · · A(fj )∗ · · · A(fn )∗ ΩH ∗

∗

j =1

= A(f )A(f1 )∗ · · · A(fn )∗ ΩH . Hence, for all Ψ ∈ Fb,fin (H ), U CΩ (f )U −1 Ψ = A(f )Ψ . Recall that Fb,fin (V ) is a core for A(f ) (see Proposition 6.7). Hence it follows from a limiting argument that A(f ) ⊂ U CΩ (f )U −1 . Exchanging the role of A(f ) and CΩ (f ), we see that CΩ (f ) ⊂ U −1 A(f )U , i.e., U CΩ (f )U −1 ⊂ A(f ). Hence we obtain operator equality U CΩ (f )U −1 = A(f ). Thus the first half of the theorem follows. With regard to the second half of the theorem, we need only to note that U CΩ (f )U −1 = A(f ) implies U CΩ (f )∗ U −1 = A(f )∗ .  Corollary 8.1 Let (F , D, {C(f )|f ∈ V }, Ω) be a cyclic representation of the CCR over V . Suppose that, for all f ∈ V , Fρ,fin is a core for C(f ). Then (F , D, {C(f )|f ∈ V }, Ω) is equivalent to the Fock representation ρF (V ). Proof By the cyclicity of Ω, we have Fρ = F . Hence the operator U in the proof of Theorem 8.1 is a unitary operator from F to Fb (H ) By the assumption on a core of C(f ), CΩ (f ) = C(f ), f ∈ V . Thus, by Theorem 8.1, we obtain U C(f )# U −1 = A(f )# for all f ∈ V . 

8.3 Second Quantization Operator Associated with a Representation of CCR Formula (6.91) may be regarded as a form in which the Fock representation ρF (V ) of the CCR over V (see Example 8.2) yields the boson second quantization operator db (T ) for a class of non-negative self-adjoint operators T . We generalize this structure to introduce a concept of second quantization operator associated with a representation of CCR.

336

8 Representations of CCR with Infinite Degrees of Freedom

8.3.1 A General Fact on a Sesquilinear Form Let A := {An }∞ n=1 be a sequence of densely defined closed operators on a Hilbert space X and 2 DA := Ψ ∈

∩∞ n=1 D(An )|

∞

3 2

An Ψ < ∞ .

n=1

Then one can define a sesquilinear form s: DA × DA (see Sect. 1.11) by s(Φ, Ψ ) :=

∞

An Φ, An Ψ  ,

Φ, Ψ ∈ DA .

(8.8)

n=1

Lemma 8.2 The sesquilinear form s is closed and non-negative. Proof Using the Schwarz inequality, one can easily see that the right hand side of (8.8) is absolutely convergent and that s is a non-negative sesquilinear form on DA . We next show that s is closed. Let Ψk ∈ DA (k ∈ N) and Ψk → Ψ ∈ X (k → ∞), s(Ψk − Ψ , Ψk − Ψ ) → 0 (k,  → ∞). Hence, for any ε >, there exists a number k0 ∈ N such that, for all k,  ≥ k0 , ∞

An Ψk − An Ψ 2 < ε.

(8.9)

n=1

Hence, for all n ∈ N, An Ψk − An Ψ 2 < ε, k,  ≥ k0 . This means that {An Ψk }∞ k=1 is a Cauchy sequence in X . Hence Φn := limk→∞ An Ψk ∈ X exists. Since An is closed, it follows that Ψ ∈ D(An ) and An Ψ = Φn . In particular, Ψ ∈ ∩∞ n=1 D(An ). Taking the limit  → ∞ in (8.9) and using Fatou’s lemma, we obtain ∞

An Ψk − An Ψ 2 ≤ ε,

k ≥ k0 .

(8.10)

n=1

Since

An Ψ 2 ≤ 2( An Ψ − An Ψk 2 + An Ψk 2 ), we have ∞ n=1

An Ψ 2 ≤ 2(ε +

∞ n=1

An Ψk 2 ) < ∞,

k ≥ k0 .

8.3 Second Quantization Operator Associated with a Representation of CCR

337

Hence Ψ ∈ DA . Then (8.10) means that limk→∞ s(Ψk − Ψ, Ψk − Ψ ) = 0. Thus s is closed.  Under an additional condition, we obtain an existence theorem of a self-adjoint operator associated with A : Theorem 8.2 Suppose that DA is dense in X . Then there exists a unique nonnegative self-adjoint operator H on X such that DA = D(H 1/2 ) and, for all Φ, Ψ ∈ DA , 

∞  An Φ, An Ψ  . H 1/2Φ, H 1/2 Ψ =

(8.11)

n=1

Proof Since we have Lemma 8.2, we can apply Theorem 1.31 to conclude that there exists a unique non-negative self-adjoint operator H on X such that DA = D(H 1/2 ) and H 1/2Φ, H 1/2 Ψ = s(Φ, Ψ ) for all Φ, Ψ ∈ DA . Thus (8.11) follows.  By von Neumann’s theorem, A∗n An is a non-negative self-adjoint operator. We introduce a subspace ∗ DA ,2 := {Ψ ∈ ∩∞ n=1 D(An An )|

∞

A∗n An Ψ strongly converges}.

n=1

It follows that DA ,2 ⊂ DA . Corollary 8.2 Suppose that DA is dense in X . Let H be the self-adjoint operator in Theorem 8.2. Then DA ,2 ⊂ D(H ) and HΨ =

∞

A∗n An Ψ,

Ψ ∈ DA ,2 .

(8.12)

n=1

Proof Let Ψ ∈ DA ,2 and Φ ∈ DA = D(H 1/2 ). Then, by (8.11), we have  H

1/2

Φ, H

1/2



Ψ =

∞  n=1

Φ, A∗n An Ψ



; = Φ,

∞

< A∗n An Ψ

,

n=1

 ∗ where we have used the strong convergence of ∞ n=1 An An Ψ . This means that 1/2 1/2 H Ψ ∈ D(H ) (i.e., Ψ ∈ D(H )) and (8.12) holds. 

338

8 Representations of CCR with Infinite Degrees of Freedom

8.3.2 A Second Quantization Operator Associated with a Representation of CCR We fix a representation ρ := (F , D, {C(f )|f ∈ V })

(8.13)

of the CCR over V . Throughout this subsection, we assume that V is a dense subspace of a Hilbert space K . Theorem 8.3 Let T be a non-negative self-adjoint operator on K . Suppose that 1/2 e ∈ V , n ∈ N and D(T 1/2 ) includes a CONS {en }∞ n n=1 of K with T 2 QC := Ψ ∈

 ∞ 

1/2 ∩∞ en )) n=1 D(C(T

3

C(T

1/2

2

en )Ψ < ∞

(8.14)

n=1

is dense in F . Then there exists a unique non-negative self-adjoint operator HC (T ) on F such that D(HC (T )1/2 ) = QC , 

(8.15)

∞    HC (T )1/2 Φ, HC (T )1/2 Ψ = C(T 1/2 en )Φ, C(T 1/2 en )Ψ ,

Φ, Ψ ∈ QC .

n=1

(8.16) Proof We need only apply Theorem 8.2 to the case where X = F and An = C(T 1/2 en ).  We call the non-negative self-adjoint operator HC (T ) a second quantization operator of T associated with {C(f )|f ∈ V }. The operator HC (T ) may depend on the choice of the CONS {en }∞ n=1 . In order to show that QC is dense in concrete representations of CCR on the boson Fock space Fb (H ) over a Hilbert space H , the following lemma may be useful: Lemma 8.3 Let L be a densely defined linear operator on a Hilbert space H and ∗ {en }∞ n=1 ⊂ D(L) be a CONS of H . Let Ψ ∈ Fb,fin (D(L )) ⊂ Fb (H ). Then: (i) ∞ n=1

A(Len )Ψ 2 < ∞.

(8.17)

8.3 Second Quantization Operator Associated with a Representation of CCR

(ii) Suppose that

∞

2 n=1 Len

339

< ∞. Then

∞

A(Len )∗ Ψ 2 < ∞.

(8.18)

n=1

Proof We need only to show (8.17) and (8.18) for Ψ of the form Ψ = A(f1 )∗ · · · A(fp )∗ ΩH ,

p ≥ 0, fj ∈ D(L∗ ), j = 1, . . . , p.

(i) If Ψ = ΩH (the case p = 0), then A(Len )Ψ = 0, n ∈ N. Hence (8.17) trivially holds. Let p ≥ 1 next. Then we have A(Len )Ψ =

p  Len , fj Ψ (j ) , j =1

∗ ∗  where Ψ (j ) := A(f1 )∗ · · · A(f j ) · · · A(fp ) ΩH . Hence ∞

A(Len )Ψ 2 =

n=1

p 

Ψ (j ) , Ψ (k)

∞  

j,k=1

=

p

 L∗ fj , en en , L∗ fk

n=1



 Ψ (j ) , Ψ (k) L∗ fj , L∗ fk .

j,k=1

Therefore (8.17) holds. (ii) By (6.55), we have

A(Len )∗ Ψ 2 = A(Len )Ψ 2 + Len 2 . 

Hence, by (i) and the present assumption, (8.18) holds.

Example 8.8 Let us consider the case ρ = ρF (V ), the Fock representation of the CCR over V ⊂ H . We suppose that V ∩ D(T ) is dense in H . Then there exists a 1/2 e ∈ D(T 1/2 ). Let CONS {en }∞ n n=1 of H with en ∈ V ∩ D(T ) (n ∈ N) so that T 2 QA := Ψ ∈

1/2 ∩∞ en ))| n=1 D(A(T

∞

3

A(T

1/2

en )Ψ < ∞ .

n=1

Then it follows from Lemma 8.3 with L = T 1/2 that Fb,fin (D(T 1/2 )) ⊂ QA .

2

(8.19)

340

8 Representations of CCR with Infinite Degrees of Freedom

Hence QA is dense in Fb (H ). Therefore, by Theorem 8.3, there exists a unique non-negative self-adjoint operator HA (T ) on Fb (H ) such that D(HA (T )1/2 ) = QA and, for all Φ, Ψ ∈ QA , 

∞    HA (T )1/2 Φ, HA (T )1/2Ψ = A(T 1/2 en )Φ, A(T 1/2 en )Ψ .

(8.20)

n=1

There are three cases to be considered separately. (1) The case where T is bounded In this case, (6.85) holds. Hence, by (6.86), for all Ψ ∈ Fb,fin (D(T )) and Φ ∈ D(HA (T )1/2 ), the right hand side of (8.20) is equal to Φ, db (T )Ψ . Hence Ψ ∈ D(HA (T )) and HA (T )Ψ = db (T )Ψ . Since Fb,fin (D(T )) is a core for db (T ), it follows that db (T ) ⊂ HA (T ). Hence HA (T ) = db (T ).

(8.21)

Thus HA (T ) coincides with db (T ). This implies also that HA (T )1/2 = db (T )1/2 . Hence, in the present case, QA = D(db (T )1/2 ).

(8.22)

(2) The case where T is unbounded and satisfies (6.85) In this case too, the arguments in (1) work without alterations. Hence (8.21) and (8.22) hold. (3) The case where T is unbounded and {en }∞ n=1 does not satisfy (6.85) In this case, by Corollary 6.11, D(db (T )1/2 ) ⊂ QA = D(HA (T )1/2 ) and 

   HA (T )1/2 Φ, HA (T )1/2Ψ = db (T )1/2 Φ, db (T )1/2 Ψ , Φ, Ψ ∈ D(db (T )1/2 ).

(8.23)

This means that the sesquilinear form associated with HA (T ) is an extension of that associated with db (T ).3 But it does not mean operator equality (8.21). It follows from (8.23) that, for all Ψ ∈ D(HA (T )) ∩ D(db (T )1/2 ), Ψ ∈ D(db (T )) and HA (T )Ψ = db (T )Ψ .

sesquilinear form s is called an extension of a sesquilinear form s if Q (s) ⊂ Q (s ) and s(Ψ, Φ) = s (Ψ, Φ), Ψ, Φ ∈ Q (s).

3A

8.3 Second Quantization Operator Associated with a Representation of CCR

341

Any CONS {en }∞ n=1 of H satisfies (6.85) with T = I . Hence, by the fact in (1) given above, we obtain the following result: Corollary 8.3 Let {en }∞ n=1 be a CONS of H . Then 2 3  ∞  1/2 ∞ 2 D(Nb ) = Ψ ∈ ∩n=1 D(A(en ))

A(en )Ψ < ∞ ,

(8.24)

n=1



∞  1/2 1/2 A(en )Φ, A(en )Ψ  , Nb Φ, Nb Ψ =

1/2

Φ, Ψ ∈ D(Nb ),

(8.25)

n=1

independently of the choice of {en }∞ n=1 . In what follows, we assume that T obeys the assumption of Theorem 8.3. Proposition 8.1 (i) If ρ has a vacuum Ω, then HC (T ) has a ground state with ground state energy being zero: HC (T )Ω = 0. (ii) Conversely, if HC (T ) has a zero-energy ground state Ω, then C(f )Ω = 0 for all f ∈ span{T 1/2 en |n ∈ N}. Proof (i) The present assumption implies that C(T 1/2 en )Ω = 0, n ∈ N. Hence, by (8.16), Hence HC (T )1/2Ω 2 = 0. Therefore HC (T )1/2Ω = 0. This implies that HC (T )Ω = 0. (ii) By the present assumption, Ω ∈ ker HC (T ). Hence, by Lemma 1.1, Ω ∈  ker HC (T )1/2. Hence HC (T )1/2 Ω, HC (T )1/2 Ω = 0. Then, by (8.16), ∞

C(T 1/2 en )Ω 2 = 0.

n=1

Hence C(T 1/2 en )Ω 2 = 0, n ∈ N, which implies that C(T 1/2 en )Ω = 0, n ∈ N. Hence the desired result follows.  Proposition 8.2 Let N ∈ N and fT ,N :=

N 

 T 1/2 en , f T 1/2 en ,

f ∈V.

n=1

Suppose that the CCR-domain D in the representation ρ is a subset of D(HC (T )1/2 ) and, for all f ∈ V ∩ D(T ) such that Tf ∈ V ,   lim Φ, C(fT ,N )∗ Ψ = Φ, C(Tf )∗ ‘Ψ , Φ, Ψ ∈ D. N→∞

Then, for all Φ ∈ D, Ψ ∈ D ∩ D(HC (T )) and f ∈ V ∩ D(T ) such that Tf ∈ V ,

342

8 Representations of CCR with Infinite Degrees of Freedom

   HC (T )1/2 Φ, HC (T )1/2C(f )∗ Ψ − C(f )Φ, HC (T )Ψ  = Φ, C(Tf )∗ Ψ , (8.26)    HC (T )1/2 Φ, HC (T )1/2C(f )Ψ − C(f )∗ Φ, HC (T )Ψ = − Φ, C(Tf )Ψ  . (8.27) Proof Throughout the proof, we set H := HC (T ). Let Φ ∈ D and Ψ ∈ D ∩ D(H ). Then, by the present assumption, Ψ ∈ D(C(f )∗ ) and C(f )∗ Ψ ∈ D. Hence, by (8.16), we have 

∞    C(T 1/2 en )Φ, C(T 1/2 en )C(f )∗ Ψ . H 1/2Φ, H 1/2 C(f )∗ Ψ = n=1

By CCR on D, we have   C(T 1/2 en )C(f )∗ Ψ = C(f )∗ C(T 1/2 en )Ψ + T 1/2 en , f Ψ and C(f )C(T 1/2 en )Φ = C(T 1/2 en )C(f )Φ. Hence we obtain ∞      H 1/2 Φ, H 1/2 C(f )∗ Ψ − C(f )Φ, H Ψ  = T 1/2 en , f Φ, C(T 1/2 en )∗ Ψ . n=1



The right hand side is equal to limN→∞ Φ, C(fT ,N )∗ Ψ Thus (8.26) holds. Similarly, one can prove (8.27).



= Φ, C(Tf )∗ Ψ . 

Note that (8.26) and (8.27) have a structure similar to the structure of (6.92) and (6.93) apart from domains of commutation relations. Remark 8.6 A slightly different concept of second quantization operator associated with a representation of CCR is introduced in [28, §5.22]. But, in this book, we use the term “second quantization operator associated with {C(f )|f ∈ V }” in the sense defined above. Example 8.9 Let us find a second quantization operator associated with {CL (f )|f ∈ V } in Example 8.3. Let T be a non-negative and injective self-adjoint operator on H and consider the case where V = D(T −1/2 ) and  1  L(f ) = √ T −1/2 g, T −1/2 f , 2

f ∈ D(T −1/2 )

8.3 Second Quantization Operator Associated with a Representation of CCR

343

for some fixed g ∈ D(T −1/2 ). In this case, we write βL (V ) = βL 1/2 ) be a CONS of H . Then T 1/2 e ∈ D(T −1/2 ), n ∈ simply. Let {en }∞ n n=1 ⊂ D(T N. Hence, by Theorem 8.3, there exists a unique non-negative self-adjoint operator HCL on Fb (H ) such that



∞    1/2 1/2 CL (T 1/2 en )Φ, CL (T 1/2 en )Ψ , HCL Φ, HCL Ψ =

Φ, Ψ ∈ QCL .

n=1

For all Ψ, Φ ∈ D(CL (T 1/2 en )) = D(A(T 1/2 en )), we have 

   CL (T 1/2 en )Φ, CL (T 1/2 en )Ψ = A(T 1/2 en )Φ, A(T 1/2en )Ψ   + L(T 1/2 en ) Φ, A(T 1/2 en )Ψ   + L(T 1/2 en )∗ A(T 1/2 en )Φ, Ψ + |L(T 1/2 en )|2 Φ, Ψ  .

(8.28)

In particular,  

CL (T 1/2 en )Ψ 2 = A(T 1/2 en )Ψ 2 + L(T 1/2 en ) Ψ, A(T 1/2 en )Ψ   + L(T 1/2 en )∗ A(T 1/2 en )Ψ, Ψ + |L(T 1/2 en )|2 Ψ 2 .

(8.29)

Let an := A(T 1/2 en )Ψ ,

bn := |L(T 1/2 en )| Ψ .

Then, using the Schwarz inequality, we obtain

CL (T 1/2 en )Ψ 2 ≤ an2 + 2an bn + bn2 ≤ 2(an2 + bn2 ). By the Parseval equality, we have ∞ n=1

bn2 =

1 −1/2 2

T g Ψ 2 < ∞. 2

(8.30)

344

8 Representations of CCR with Infinite Degrees of Freedom

Therefore it follows that QA ⊂ QCL . 1/2 e )) and Conversely, let Ψ ∈ QCL . Then Ψ ∈ ∩∞ n n=1 D(A(T ∞

CL (T 1/2 en )Ψ 2 < ∞.

n=1

By (8.29), we have

CL (T 1/2 en )Ψ 2 ≥ (an − bn )2 . Hence

∞

n=1 (an

− bn )2 < ∞. We have an2 ≤ 2(an − bn )2 + 2bn2 .

Therefore

∞

2 n=1 an

< ∞, implying that Ψ ∈ QA . Thus we obtain QCL = QA .

(8.31)

Let Φ, Ψ ∈ Fb,fin (D(T 1/2 )). Then, in the same way as in the proof of Theorem 6.14, one can show that   1 L(T 1/2 en ) Φ, A(T 1/2 en )Ψ = √ Φ, A(g)Ψ  . N→∞ 2 n=1 lim

N

Using this fact and (8.30), we obtain >   = 1 1/2 1/2 HCL Φ, HCL Ψ = Φ, (Hv + T −1/2 g 2 )Ψ , 2

(8.32)

where Hv is defined by (6.77). By Theorem 6.11, the operator 1 Hˆ v := Hv + T −1/2 g 2 2 is self-adjoint with D(Hˆ v ) = D(db (T )) and bounded from below. The subspace Fb,fin (D(T 1/2 )) is a core for db (T ). Hence, by Theorem 6.11, it is a core for Hˆ v too. Hence, by a limiting argument, one can show that (8.32) extends to all 1/2 Ψ, Φ ∈ D(db (T )) = D(Hˆ v ) with D(db (T )) ⊂ D(HCL ). In particular, Hˆ v ≥ 0

(8.33)

8.4 Diagonalization of HC (T )

345

and 

   1/2 1/2 1/2 1/2 HCL Φ, HCL Ψ = Hˆ v Φ, Hˆ v Ψ ,

Ψ, Φ ∈ D(db (T )).

(8.34)

Since D(db (T )) is a core for db (T )1/2 , it follows that D(db (T )1/2 ) ⊂ 1/2 D(HCL ) = QCL = QA (see (8.31)) and (8.34) extends to all Ψ, Φ ∈ D(db (T )1/2). Let us consider the case where T is bounded. Then, by (8.22), we have 1/2

D(HCL ) = D(db (T )1/2 ). Hence it follows that HCL = Hˆ v .

(8.35)

Thus, in the present case, Hv is a second quantization operator of T associated with {CL (f )|f ∈ D(T −1/2 )} up to an additive constant.

8.4 Diagonalization of HC (T ) We continue to study the self-adjoint operator HC (T ) defined by (8.16). Theorem 8.4 Assume that (6.85) holds with T 1/2 en ∈ V , n ∈ N and suppose that the representation ρ is equivalent to the Fock representation ρF (V ), i.e., there exists a unitary operator U : F → Fb (H ) such that U C(f )U −1 = A(f ),

f ∈V.

(8.36)

Then U HC (T )U −1 = db (T ). Proof By (8.36), for all Ψ ∈ QC , we have ∞ n=1

A(T 1/2 en )U Ψ 2 =

∞

C(T 1/2 en )Ψ 2 < ∞.

n=1

Hence U Ψ ∈ QA . Conversely, if Φ ∈ QA , then ∞ n=1

C(T 1/2 en )U −1 Φ 2 =

∞ n=1

A(T 1/2 en )Φ 2 < ∞.

(8.37)

346

8 Representations of CCR with Infinite Degrees of Freedom

Hence U −1 Φ ∈ QC . Thus U QC = QA . Since (6.85) holds, QA = D(db (T )1/2 ). Therefore, for all Ψ, Φ ∈ D(db (T )1/2 ) = QA (= U QC ), ∞     HC (T )1/2 Φ, HC (T )1/2 Ψ = A(T 1/2 en )U Φ, A(T 1/2 en )U Ψ n=1



 = db (T )1/2 U Φ, db (T )1/2 U Ψ   = U −1 db (T )1/2 U Φ, U −1 db (T )1/2 U Ψ . Note that U −1 db (T )U is a non-negative self-adjoint operator and, by functional calculus, (U −1 db (T )U )1/2 = U −1 db (T )1/2 U. Hence     HC (T )1/2 Φ, HC (T )1/2 Ψ = (U −1 db (T )U )1/2 Φ, (U −1 db (h)U )1/2 Ψ . (8.38) By this fact and the uniqueness of the self-adjoint operator associated with a densely defined non-negative closed sesquilinear form, we obtain that HC (T ) = U −1 db (T )U . This is equivalent to (8.37).  Remark 8.7 As is seen from the proof given above, in the case where T is unbounded and does not satisfy (6.85), one can prove only the following properties unless additional conditions are assumed: D(db (T 1/2 )) ⊂ U QC and (8.38) holds for all Ψ, Φ ∈ D(db (T )1/2). Remark 8.8 For a sufficient condition for the representation ρ to be equivalent to ρF (V ), see Corollary 8.1. In the context of quantum field theory, a Bose field Hamiltonian H , which is a self-adjoint operator on a Hilbert space, is said to be diagonalized if it is unitarily equivalent to a boson second quantization operator db (h) of a self-adjoint operator h (not necessarily bounded) up to an additive constant. In such a case, the spectrum of the Hamiltonian H is completely identified if the spectrum of the one-particle Hamiltonian h is known. Theorem 8.4 shows that, for any non-negative self-adjoint operator T satisfying (6.85) with T 1/2en ∈ V , n ∈ N, the operator HC (T ) can be diagonalized if the representation ρ is equivalent to the Fock representation ρF (V ).

8.5 Analysis of Bogoliubov Translations

347

8.5 Analysis of Bogoliubov Translations In this section, we give a detailed analysis of the Bogoliubov translation (CL , CL∗ ) in Example 8.3. Lemma 8.4 Let D be a dense subspace of H . Suppose that there exists a non-zero vector Ω ∈ ∩f ∈D D(A(f )) such that, for all f ∈ D, CL (f )Ω = 0. Then, there exists a vector h ∈ H such that L(f ) = h, f  , f ∈ D. Proof Without loss of generality, we can assume that Ω = 1. The present assumption implies that A(f )Ω = −L(f )∗ Ω, f ∈ D. There exists an n ∈ Z+ such that Ω (n) = 0. We define Ψn ∈ Fb,0 (H ) by Ψn(n) := Ω (n) and Ψ (m) = 0, m = n and put ωn := Ω, Ψn  = Ω (n) 2 = 0. Then L(f ) = −

1 1  A(f )Ω, Ψn  = − Ω, A(f )∗ Ψn . ωn ωn

By the Schwarz inequality and Lemma 6.5(i), we obtain |L(f )| ≤

1

A(f )∗ Ψn ≤ Cn f , ωn

f ∈ D,

√ with Cn :== (n + 1)/ωn . Hence L is bounded. Since D is dense, it follows from the extension theorem that there exists a unique  L ∈ H ∗ such that  L(f ) = L(f ), f ∈ D. Therefore, by Riesz’s representation theorem, there exists a unique vector h ∈ H such that  L(f ) = h, f  , f ∈ H . Hence we have L(f ) = h, f  , f ∈ D.  Theorem 8.5 There exists a unitary operator U on Fb (H ) such that U CL (f )U −1 = A(f ),

f ∈V

(8.39)

if and only if there exists a vector h ∈ H such that L(f ) = h, f H , f ∈ V . Proof Suppose that there exists a unitary operator U on Fb (H ) such that (8.39) holds. Then, letting Ω := U −1 ΩH , we have CL (f )Ω = 0 and Ω = 1AD ˛ Hence, by Lemma 8.4 with D = V , there exists a vector h ∈ H such that L(f ) = h, f H , f ∈ V . Conversely, suppose that there exists a vector h ∈ H such that L(f ) = h, f  , f ∈ V . Then, by (6.118), we have ei

√ 2 ΦS (ih)

Hence, letting U := e−i

A(f )e−i

√ 2 ΦS (ih) ,

√

2 ΦS (ih)

= CL (f ),

f ∈V.

we obtain U CL (f )U −1 = A(f ), f ∈ V .



348

8 Representations of CCR with Infinite Degrees of Freedom

Corollary 8.4 The representation (Fb (H ), Fb,0 (H ), {CL (f )|f ∈ V }) of CCR is equivalent to the Fock representation ρF (V ) if and only if there exists a vector h ∈ H such that L(f ) = h, f  , f ∈ V .

8.6 Bogoliubov Transformations A more general class of representations of the CCR over a subspace V of a Hilbert space H may be found by considering operators of the form A(W1 f ) + A(W2 f )∗ (f ∈ V ) with suitable mappings W1 and W2 from V to H . This is an idea for what follows in this section. To describe a general structure behind the idea, we first present a complex symplectic space structure of H ⊕ H (see Sect. 6.13). We fix a conjugation J on H (see Definition 1.2). Let γ : H ⊕ H → H ⊕ H be defined by γ (f, g) := (g, −f ),

(f, g) ∈ H ⊕ H .

(8.40)

Then it is easy to see that γ is a bounded linear operator on H ⊕ H with

γ (f, g) = (f, g) ,

(f, g) ∈ H ⊕ H

(8.41)

and γ ∗ = −γ ,

γ 2 = −I.

(8.42)

Moreover J := J ⊕ J

(8.43)

Jγ = γ J.

(8.44)

is a conjugation on H ⊕ H and

Hence, one can apply Example 6.7 to the case X = H ⊕ H , C = J and W = γ to conclude that the mapping ωJ : H ⊕ H → C defined by  ωJ (Ψ, Φ) := (JΨ, γ Φ , Ψ, Φ ∈ H ⊕ H

(8.45)

is a C-symplectic form on (H ⊕H ). Hence (H ⊕H , ωJ ) is a complex symplectic space. Explicitly, we have   ωJ ((f, g), (f , g )) = Jf, g − J g, f ,

(f, g), (f , g ) ∈ H ⊕ H . (8.46)

8.6 Bogoliubov Transformations

349

Note that J and γ have the following operator matrix representations: J =

0

J 0 0J

1

0 ,

γ =

0 I −I 0

1 ,

where an operator matrix 0 A :=

A11 A12 A21 A22

1

with Aij ∈ L(H ) (D(A) := (D(A11 ) ∩ D(A21 )) ⊕ (D(A12 ) ∩ D(A22 ))) acts on (f1 , f2 ) ∈ D(A) as 0 A

f1 f2

1

0 =

A11 f1 + A12 f2 A21 f1 + A22 f2

1 .

Lemma 8.5 Let S be a bounded linear operator on H ⊕ H . Then S is an isosymplectic transformation if and only if (JS∗ J)γ S = γ .

(8.47)

Proof Let Ψ, Φ ∈ H ⊕ H . Then   ωJ (SΨ, SΦ) = ωJ (Ψ, Φ) ⇐⇒ JSΨ, γ SΦ = JΨ, γ Φ   ⇐⇒ S∗ γ JSΨ, Φ = γ JΨ, Φ   ⇐⇒ JΦ, JS∗ γ JSΨ = JΦ, γ Ψ ,   ⇐⇒ JΦ, JS∗ Jγ SΨ = JΦ, γ Ψ , where we have used (8.44). Since Ran J = H ⊕ H , it follows that the last statement is equivalent to (8.47).  For each (f, g) ∈ H ⊕ H , we define an operator Φ(f, g) on Fb (H ) by Φ(f, g) := A(Jf ) + A(g)∗ .

(8.48)

Using CCR of A(·)# and (8.46), one can see that [Φ(f, g), Φ(f , g )] = ωJ ((f, g), (f , g )),

(f, g), (f , g ) ∈ H ⊕ H (8.49)

on Fb,0 (H ). Note that A(f ) = Φ(Jf, 0),

f ∈H.

(8.50)

350

8 Representations of CCR with Infinite Degrees of Freedom

The next proposition immediately follows from (8.49). Proposition 8.3 A linear operator S on H ⊕ H is an iso-symplectic transformation on (H ⊕ H , ωJ ) if and only if [Φ(S(f, g), S(f , g ))] = [Φ(f, g), Φ(f , g )],

(f, g), (f , g ) ∈ H ⊕ H ,

on Fb,0 (H ). In view of Proposition 8.3, it is interesting to ask if Φ(S(·)) is unitarily implementable, i.e., there exists a unitary operator U on Fb (H ) such that Φ(S(f, g)) = U Φ(f, g)U −1 ,

(f, g) ∈ H ⊕ H .

This is a basic motivation of the theory presented below. For each T ∈ B(H ), we define an operator TJ ∈ B(H ) by TJ := J T J.

(8.51)

The operator TJ is called the conjugation of T with respect to J . Note that (TJ )∗ = (T ∗ )J . Let S be an iso-symplectic transformation on (H ⊕ H , ωJ ) and 0 S :=

S1 S2 S3 S4

1

be the operator matrix representation of S (Si ∈ B(H ), i = 1, 2, 3, 4). Then (8.47) is equivalent to the following set of operator equations: (S1J )∗ S4 − (S3J )∗ S2 = I, ∗

(8.52)

∗

(S4J ) S1 − (S2J ) S3 = I,

(8.53)

(S1J )∗ S3 − (S3J )∗ S1 = 0,

(8.54)

∗

∗

(S2J ) S4 − (S4J ) S2 = 0,

(8.55)

where SiJ := (Si )J (i = 1, 2, 3, 4). Note that (8.52) is equivalent to (8.53). To find a subclass of S’s, we consider the subset H (J ) := {(f, Jf ) | f ∈ H } ⊂ H ⊕ H , which may be a natural subset. Indeed, J leaves H JH (J ) = H (J ) .

(J )

invariant bijectively:

8.6 Bogoliubov Transformations

351

We note the following fact: Lemma 8.6 Let S be an iso-symplectic transformation on (H ⊕ H , ωJ ). Then S leaves H (J ) invariant if and only if S2 = S3J ,

S4 = S1J .

(8.56)

Proof We have for all f ∈ H S(f, Jf ) = (S1 f + S2 Jf, S3 f + S4 Jf ). Hence S(f, Jf ) ∈ H (J ) if and only if J S1 f + J S2 Jf = S3 f + S4 Jf.

(8.57)

Suppose that S(f, Jf ) ∈ H (J ) . Then (8.57) holds. Replacing f by if , we obtain −J S1 f + S2J f = S3 f − S4 Jf. Hence it follows that S2J f = S3 f and J S1 f = S4 Jf . Therefore (8.56) holds. Conversely, assume (8.56). Then (8.57) holds. Hence S(f, Jf ) ∈ H (J ).



Let S be an iso-symplectic transformation (H ⊕ H , ωJ ) such that SH (J ) ⊂ H (J ) .

(8.58)

Then, by Lemma 8.6, S is of the form 0 S=

SJ T TJ S

1 (8.59)

with relations S ∗ S − T ∗ T = I,

(8.60)

S ∗ TJ = T ∗ SJ .

(8.61)

We have Φ(S(Jf, 0)) = A(Sf ) + A(J Tf )∗ ,

f ∈H.

This operator is closable. We define B(f ) := A(Sf ) + A(J Tf )∗ ,

f ∈H.

(8.62)

352

8 Representations of CCR with Infinite Degrees of Freedom

It follows that B(f )∗ = (A(Sf ) + A(J Tf )∗ )∗ ⊃ A(J Tf ) + A(Sf )∗ . Lemma 8.7 Let S and T be in B(H ) satisfying (8.60) and (8.61). Then the triple ρS,T (H ) := (Fb (H ), Fb,0 (H ), {B(f ) | f ∈ H })

(8.63)

is a representation of the CCR over H . Proof The anti-linearity of B(f )  Fb,0 (H ) in f is easily seen. It is easy to see that, for all f ∈ H , B(f )# Fb,0 (H ) ⊂ Fb,0 (H ) and, for all Ψ ∈ Fb,0 (H ), [B(f ), B(g)∗ ]Ψ = [A(Sf ) + A(J Tf )∗ , A(Sg)∗ + A(J T g)]Ψ = (Sf, Sg − J T g, J Tf )Ψ  = f, (S ∗ S − T ∗ T )g Ψ = f, g Ψ, where we have used [B(f ), B(g)]Ψ = 0.

(8.60). Similarly, using

(8.61), we can show that 

The correspondence BS,T : (A(·), A(·)∗ ) → (B(·), B(·)∗ ) is called the bosonic Bogoliubov transformation associated with (S, T ). Note that this comes from Φ(S(Jf, 0)) with an iso-symplectic transformation S such that SH (J ) ⊂ H (J ). The transformation BS,T is said to be proper if there exists a unitary operator U on Fb (H ) such that B(f ) = U A(f )U −1 , f ∈ H (hence B(f )∗ = U A(f )∗ U −1 ), while BS,T is said to be improper if it is not proper. Remark 8.9 Conditions (8.60) and (8.61) are weakened. Namely, one can consider the case where S and T are not necessarily bounded operators on H satisfying the weak forms of (8.60) and (8.61): Sf, Sg − Tf, T g = f, g ,

(8.64)

Sf, TJ g = Tf, SJ g

(8.65)

for all f, g in a dense subspace V ⊂ D(T ) ∩ D(S) ∩ D(TJ ) ∩ D(SJ ). In this case, we define B(f ) by (8.62) with f ∈ V . Then it follows that ρS,T (V ) := (Fb (H ), Fb,0 (H ), {B(f )|f ∈ V }) is a representation of the CCR over V . Formula (8.64) implies that

Sf 2 − Tf 2 = f 2 ,

f ∈V.

Hence S is bounded if and only if T is bounded. In this case, (8.64) and (8.65) extend to all f, g ∈ H with S and T replaced by S and T respectively. Then

8.6 Bogoliubov Transformations

353

(8.60) and (8.61) hold with S and T replaced by S and T respectively. But, in the case where S or T is unbounded (then both S and T are unbounded), BS,T may be considered as a new type of bosonic Bogoliubov transformation. We call it a singular bosonic Bogoliubov transformation. For a study on singular bosonic Bogoliubov transformation, see [30]. We first consider a condition under which the representation ρS,T (H ) is not equivalent to any direct sum of ρF (H ). Lemma 8.8 Equation (8.60) implies that S is injective and S ∗ is surjective. Proof Equation (8.60) is equivalent to the equation S ∗ S = I + T ∗ T . Since T ∗ T is a non-negative self-adjoint operator, S ∗ S is bijective. This implies the statement of the present lemma.  The following proposition is a key fact in considering inequivalence of representations of CCR. Proposition 8.4 Assume that H is separable. Let L ∈ L(H ) and D be a dense subspace of H such that D ⊂ D(L). Suppose that there exists a non-zero vector Ω ∈ Fb (H ) such that, for all Ψ ∈ Fb,0 (H ) and f ∈ D,  A(f )∗ Ψ, Ω = A(J Lf )Ψ, Ω .

(8.66)

Then L is bounded and L is Hilbert–Schmidt. Proof For each n ≥ 0, we define a vector Ωn in the n-particle space ⊗ns H by (n) (m) Ωn := Ω (n) and Ωn := 0, m = n. Let Ψ ∈ ⊗ns H . Then, for all f ∈ D, ∗ n+1 A(f ) Ψ ∈ ⊗s H and A(J Lf )Ψ ∈ ⊗n−1 H (⊗−1 s s H := {0}). Hence we have ∗ from (8.66) A(f ) Ψ, Ωn+1  = A(J Lf )Ψ, Ωn−1 . Therefore  Ψ, A(f )Ωn+1  = Ψ, A(J Lf )∗ Ωn−1 . Thus A(f )Ωn+1 = A(J Lf )∗ Ωn−1 ,

n ≥ 0, f ∈ D,

(8.67)

where Ω−1 := 0. In particular, A(f )Ω1 = 0. Hence, by Proposition 6.2, Ω1 = ΩH , Ω1  ΩH = 0. Then, by (8.67), A(f )Ω3 = 0 for all f ∈ D. Hence, by Proposition 6.2 again, Ω3 = 0. By repeating this process with (8.67), we can show by induction in n that Ω2n−1 = 0 for all n ≥ 1. By (8.67), we have A(f )Ω2 = cA(J Lf )∗ ΩH , where c := Ω (0) ∈ C. If c = 0, then A(f )Ω2 = 0 for all f ∈ D. Hence, by Proposition 6.2 again, Ω2 = 0. Then, in the same way as above, we obtain Ω2n = 0 for all n ≥ 0. Thus Ω = 0. But this is a contradiction. Hence c = 0. Therefore A(J Lf )∗ ΩH 2 = A(f )Ω2 2 /|c|2 .

354

8 Representations of CCR with Infinite Degrees of Freedom

Hence Lf 2 = A(f )Ω2 2 /|c|2. Let {en |n ∈ N} ⊂ D be a CONS of H . Then, by (6.83), ∞ n=1

Len 2 =

∞ 1 1 1/2

A(en )Ω2 2 ≤ 2 Nb Ω2 2 < ∞. |c|2 |c| n=1

In particular, Len ≤ C, n ∈ N with C := |c|−1 Nb Ω2 and C is independent of the choice of {en |n ∈ N}. Hence, for each Ψ ∈ D \ {0}, L(Ψ/ Ψ ) ≤ C, i.e.,

LΨ

. Therefore L  D is a bounded operator. Hence L is bounded and ∞ ≤ C Ψ 2  n=1 Len < ∞. Thus L is Hilbert–Schmidt. 1/2

Theorem 8.6 Assume that H is separable. Suppose that Ran S = H and T is not Hilbert–Schmidt. Then ρS,T (H ) is not equivalent to any direct sum of ρF (H ). Proof We prove the contraposition. Hence suppose that there exists a unitary operator U from Fb (H ) to ⊕N p=1 Fb (H ) (N < ∞ or N = ∞) such that A(f ), f ∈ H . Let Ω := U −1 {ΩH , 0, 0, . . .}. Then U B(f )U −1 = ⊕N p=1

Ω = 1 and B(f )Ω = 0, i.e., (A(Sf ) + A(J Tf )∗ )Ω = 0. Hence, for all Ψ ∈ Fb,0 (H ), A(Sf )∗ Ψ, Ω = − A(J Tf )Ψ, Ω. By Lemma 8.8 and the present assumption on S, S is bijective. Hence, putting K := T S −1 , we have for all f ∈ H , A(f )∗ Ψ, Ω = − A(J Kf )Ψ, Ω. Hence, by Proposition 8.4, K is Hilbert–Schmidt. Thus T = KS is Hilbert–Schmidt.4  To see a physical meaning of Theorem 8.6, for a CONS {en }∞ n=1 of H , we introduce a quantity nB :=

∞

B(en )ΩH 2 ,

(8.68)

n=1

which is a finite non-negative number or  +∞. Intuitively, nB is interpreted as an ∗ expectation value of the number operator “ ∞ n=1 B(en ) B(en )” associated with B(·) with respect to the bare vacuum ΩH . By (8.62), we have nB =

∞

T en 2 = Tr T ∗ T .

(8.69)

n=1

Hence nB = +∞ if and only if T is not Hilbert–Schmidt. Therefore the physical phenomenon nB = +∞ is associated with the inequivalence stated in Theorem 8.6.

that, if T is Hilbert–Schmidt on a Hilbert space X , then, for all A ∈ B(X ), AT and T A are Hilbert–Schmidt.

4 Recall

8.6 Bogoliubov Transformations

355

We next consider the Bogoliubov transformation BS,T under a stronger condition for (S, T ). Proposition 8.5 Let S be given by (8.59) satisfying (8.60), (8.61). Then S is symplectic if and only if the following operator equations hold: SS ∗ − TJ TJ∗ = I,

(8.70)

TJ SJ∗ = ST ∗ .

(8.71)

Proof Let S be symplectic. Then S is bijective. Hence the inverse S−1 exists. By (8.47), we have −γ (JS∗ J)γ S = I. Hence S−1 = −γ (JS∗ J)γ =

0

SJ∗ −TJ∗ −T ∗ S ∗

1 .

Then, by direct computations, one can see that SS−1 = I is equivalent to (8.70) and (8.71). Conversely, assume (8.70) and (8.71). Let 0 T :=

SJ∗ −TJ∗ −T ∗ S ∗

1 .

Then, by using (8.70) and (8.71), one can show that ST = I . Hence S is bijective. Therefore S is symplectic.  Lemma 8.9 Assume (8.60), (8.61), (8.70) and (8.71). Then {B(f ), B(f )∗ |f ∈ H } = CI . In particular, {B(f ), B(f )∗ |f ∈ H } is irreducible. Proof Let X ∈ {B(f ), B(f )∗ |f ∈ H } . Then XB(f )# ⊂ B(f )# X, f ∈ H . Using (8.70) and (8.71), one can express A(f ) in terms of B(·)# . Namely, for all f ∈H, A(f ) = B(S ∗ f ) − B(T ∗ Jf )∗ ,

A(f )∗ = B(S ∗ f )∗ − B(T ∗ Jf )

(8.72)

on Fb,0 (H ). Hence, for all Φ, Ψ ∈ Fb,0 (H ), we have  XA(f )Φ, Ψ  = XΦ, A(f )∗ Ψ . Since Fb,0 (H ) is a core for A(f )∗ , it follows that XΦ ∈ D((A(f )∗ )∗ ) = D(A(f )) and A(f )XΦ = XA(f )Φ. Since Fb,0 (H ) is a core for A(f ), it follows from a limiting argument that XA(f ) ⊂ A(f )X. Similarly, one can show that

356

8 Representations of CCR with Infinite Degrees of Freedom

XA(f )∗ ⊂ A(f )∗ X. Hence X ∈ {A(f ), A(f )∗ |f ∈ H } = CI . Therefore X = cI for some c ∈ C. Hence {B(f ), B(f )∗ |f ∈ H } = CI . In particular, {B(f ), B(f )∗ |f ∈ H } is irreducible.  We remark the following fact: Lemma 8.10 Assume (8.60) and (8.70). Then S and S ∗ are bijective. Proof By (8.70), one can apply Lemma 8.8 with S replaced by S ∗ to conclude that S ∗ is injective and S ∗∗ = S is surjective. By this result and Lemma 8.8, we obtain the desired result.  With regard to the converse of Theorem 8.6, the following result is known: Theorem 8.7 Assume (8.60), (8.61), (8.70) and (8.71). Suppose that T is Hilbert– Schmidt. Then ρS,T (H ) is equivalent to ρF (H ). For proofs of this theorem, see, e.g., [123, 135–137, 146], [132, Theorems XI.106 and XI.108], [7] or [93, Chapter 2]. In summary, we have obtained the following characterization on proper Bogoliubov transformations: Corollary 8.5 Assume (8.60), (8.61), (8.70) and (8.71). Then ρS,T (H ) is equivalent to ρF (H ) if and only if T is Hilbert–Schmidt.

8.7 Second Quantization Operator Associated with {B(f )|f ∈ H } 1/2 ) be a Let h be a non-negative self-adjoint operator on H and {en }∞ n=1 ⊂ D(h CONS of H . Then, according to the arguments in Sect. 8.3.2, we define a subspace

2 QB := Ψ ∈

 ∞ 

1/2 ∩∞ en )) n=1 D(B(h

3

B(h

1/2

en )Ψ < ∞ , 2

(8.73)

n=1

where B(f ) is defined by (8.62). In this section, we consider only the case where h is bounded for technical simplicity. We assume the following: Assumption (B) (B.1) Equations (8.60) and (8.61) hold. (B.2) The self-adjoint operator h is bounded and J h ⊂ hJ . (B.3) The operator T is Hilbert–Schmidt. Under Assumption (B), Th := T h1/2

(8.74)

8.7 Second Quantization Operator Associated with {B(f )|f ∈ H }

357

is Hilbert–Schmidt. Hence (Th )J = J Th J is Hilbert–Schmidt. We set Sh := Sh1/2 ,

(8.75)

which also is bounded under Assumption (B). Lemma 8.11 Under Assumption (B), the subspace QB is dense in Fb (H ) with Fb,fin (H )) ⊂ QB .

(8.76)

Proof Let Ψ ∈ Fb,0 (H ). Then an := B(h1/2 en )Ψ 2 ≤ bn := 2( A(Sh en )Ψ 2 + A(J Th en )∗ Ψ 2 ).  By the Hilbert–Schmidtness of Th , we have ∞

2 < ∞. Let Ψ ∈ n=1 J Th en F (H ). Then, by the present assumption and Lemma 8.3, ∞ n=1 bn < ∞. Hence b,fin ∞ a < ∞. Therefore Ψ ∈ Q . Thus (8.76) holds.  B n=1 n By Lemma 8.11, we can apply Theorem 8.3 to obtain the following theorem: Theorem 8.8 Suppose that Assumption (B) holds. Then there exists a unique nonnegative self-adjoint operator HB (h) on Fb (H ) such that D(HB (h)1/2 ) = QB , 

(8.77)

∞    HB (h)1/2 Φ, HB (h)1/2 Ψ = B(h1/2 en )Φ, B(h1/2 en )Ψ , n=1

Φ, Ψ ∈ D(HB (h)1/2 ).

(8.78)

The operator HB (h) is a second quantization operator of h associated with {B(f )|f ∈ H }. To find a concrete form of HB (h), we compute the right hand side of (8.78) for Φ ∈ D(HB (h)1/2 ) and Ψ ∈ Fb,fin (H ). We have B(h1/2 en )∗ B(h1/2 en ) = A(Sh en )∗ A(Sh en ) + A((Th )J J en )∗ A((Th )J J en ) + A(Sh en )∗ A((Th )J J en )∗ + A((Th )J J en )A(Sh en ) + Th en 2 on Fb,0 (H ). Let Ψ ∈ Fb,fin (H ). Then, by Theorem 6.13, we have lim

N→∞

N n=1

A(Sh en )∗ A(Sh en )Ψ = db (Sh Sh∗ )Ψ,

(8.79)

358

8 Representations of CCR with Infinite Degrees of Freedom

lim

N

N→∞

A((Th )J J en )∗ A((Th )J J en )Ψ = db ((Th )J (Th )∗J )Ψ.

(8.80)

n=1

By Lemma 6.13, we have lim

N→∞

N

∗

∗

A(Sh en ) A((Th )J J en ) Ψ = lim

N→∞

n=1

lim

N→∞

N

A((Th )J J en )∗ A(J (Sh )J en )∗ Ψ

n=1

= A |(Th )J (Sh )∗J |A∗ Ψ, 

N

∗

 A((Th )J J en )A(Sh en )Ψ = A|(Sh )J (Th )∗J |A Ψ.

(8.81) (8.82)

n=1

It is obvious that ∞

Th en 2 = Th 22 .

n=1

Hence, introducing the operator Hqd := db (Sh Sh∗ + (Th )J (Th )∗J )   + A∗ |(Th )J (Sh )∗J |A∗ + A|(Sh )J (Th )∗J |A ,

(8.83)

we obtain     HB (h)1/2 Φ, HB (h)1/2 Ψ = Φ, (Hqd + Th 22 )Ψ , Φ ∈ D(HB (h)1/2 ), Ψ ∈ Fb,fin (H ).

(8.84)

It follows from the definition of Hqd and (8.84) that Hqd  Fb,fin (H ) is a symmetric operator bounded from below with Hqd  Fb,fin (H ) ≥ − Th 22 . Theorem 8.9 Suppose that Assumption (B) holds. Then Hqd is essentially selfadjoint on Fb,fin (H ) and H qd ≥ − Th 22 . Proof We write H := Hqd = A + B with A := db (Sh Sh∗ + (Th )J (Th )∗J ),

  B := A∗ |(Th )J (Sh )∗J |A∗ + A|(Sh )J (Th )∗J |A .

Let D := D(A) ∩ Fb,0 (H ) and Ψ ∈ D. Since Fb,fin (H ) is a core for A (note that D(Sh Sh∗ ) + (Th )J (Th )∗J ) = H ), there exists a sequence {Ψn }∞ n=1 in Fb,fin (H ) such

8.7 Second Quantization Operator Associated with {B(f )|f ∈ H }

359

that Ψn → Ψ, AΨn → AΨ as n → ∞. By Lemma 6.12, BΨn → BΨ as n → ∞. Therefore H Ψn → H Ψ as n → ∞. Thus H  D ⊂ H := H  Fb,fin (H ). Hence it is sufficient to show that H is essentially self-adjoint on D. For this purpose, we apply Theorem C.3 in Appendix C to the case where Hn = ⊗ns H , D0 = Fb,0 (H ), Nˆ = Nb and (A, B) given above. By Lemma 6.12, there exist constants ci ≥ 0, i = 1, 2 such that, for all Ψ ∈ Fb,0 (H ),

BΨ ≤ c1 (Nb + 2)Ψ + c2 Nb Ψ ) ≤ (2c1 + c2 ) (Nb + 1)Ψ

≤ (2c1 + c2 ) (Nb + 1)2 Ψ , where we have used the following facts (which are easily proved): for all Ψ ∈ D(Nb ),

Ψ ≤ (Nb + 1)Ψ ,

Nb Ψ ≤ (Nb + 1)Ψ

and, for any non-negative self-adjoint operator X on a Hilbert space, (X + 1) ≤

(X + 1)2 f , f ∈ D(X2 ). Hence condition (i) in Sect. C.3 in Appendix C is satisfied in the present case. Lemma 6.12 implies also that, for |m − n| > 2, Ψ (m) , BΨ (n)  = 0. Hence condition (ii) also in Sect. C.3 in Appendix C is satisfied in the present case. Since H ≥ − Th 22 , it follows that H ≥ − Th 22 on D. Thus we can apply Theorem C.3 to conclude that H is essentially self-adjoint on D. Hence H is a self-adjoint operator. Thus H is essentially self-adjoint on Fb,fin (H ).  Remark 8.10 Let K be a Hilbert–Schmidt operator on H and ξ be a non-negative self-adjoint operator on H such that   Hξ,K := db (ξ ) + A∗ |K|A∗ + A|K ∗ |A is bounded from below on D(db (ξ )) ∩ Fb,0 (H ). Then, in quite the same way as in the proof of Theorem 8.9, one can prove that Hξ,K is essentially self-adjoint on D(db (ξ )) ∩ Fb,0 (H ) and the closure H ξ,K is a self-adjoint operator bounded from below. We now come back to relation (8.84). By Theorem 8.9 and a limiting argument, one can show that (8.84) extends to all Ψ ∈ D(H qd ). Hence it follows that D(H qd ) ⊂ D(HB (h)) and HB (h)Ψ = (H qd + Th 22 )Ψ, Ψ ∈ D(H qd ). This means that H qd + Th 22 ⊂ HB (h). Thus HB (h) = H qd + Th 22 .

(8.85)

360

8 Representations of CCR with Infinite Degrees of Freedom

Theorem 8.10 Assume (8.60), (8.61), (8.70) and (8.71). Suppose, in addition, that Assumption (B) holds. Then there is a unitary operator UB on Fb (H ) such that UB B(f )UB−1 = A(f ),

f ∈H,

(8.86)

UB (H qd + Th 22 )UB−1 = db (h).

(8.87)

Proof Since T is Hilbert–Schmidt, it follows from Corollary 8.5 that there exists a unitary operator UB on Fb (H ) such that (8.86) holds. Hence, by Theorem 8.4, UB HB (h)UB−1 = db (h). By this fact and (8.85), we obtain (8.87).  Remark 8.11 Suppose that S and T are independent of h. Then, by construction, the unitary operator UB is independent of h. Hence, under additional conditions, (8.87) may be extended to the case where h is unbounded through a limiting argument. Remark 8.12 There is another method for proof of (8.87). See, e.g. [7, §IV] or [93, §3.5]. For recent studies on diagonalizations of Hamiltonians quadratic in annihilation and creation operators, see [52, 60, 61, 119]. Theorem 8.10 immediately yields the following corollary: Corollary 8.6 Under the same assumption as in Theorem 8.10, H qd has a unique ground state given by ΨB := UB−1 ΩH , up to constant multiples, with ground state energy − Th 22 : H qd ΨB = − Th 22 ΨB .

8.8 Representations of Heisenberg CCR As in the case of finite degrees of freedom, one can also consider representations of CCR of Heisenberg type. Definition 8.4 Let W be a real inner product space. Let F be a Hilbert space and D be a dense subspace of F . Suppose that, for each f ∈ W , symmetric operators φ(f ) and π(f ) on F are given, satisfying the following conditions: (i) D ⊂ ∩f ∈W D(φ(f )) ∩ D(π(f )) and φ(f )D ⊂ D, π(f )D ⊂ D. (ii) For all f, g ∈ W and a, b ∈ R, φ(af + bg) = aφ(f ) + bφ(g) and π(af + bg) = aπ(f ) + bπ(g) on D.

8.8 Representations of Heisenberg CCR

361

(iii) (Heisenberg CCR over W ) For all f, g ∈ W , [φ(f ), π(g)] = i f, g , [φ(f ), φ(g)] = 0,

[π(f ), π(g)] = 0

on D. Then (F , D, {φ(f ), π(f ) | f ∈ W }) is called a representation of the Heisenberg CCR over W .5 We call D a CCR-domain for {φ(f ), π(f )|f ∈ W }. Two representations (F , D, {φ(f ), π(f ) | f ∈ W }) and (F , D , {φ (f ), π (f ) | f ∈ W }) of the Heisenberg CCR over W are said to be equivalent if there exists a unitary operator U : F → F such that, for all f ∈ W , U φ(f )U −1 = φ (f ) and U π(f )U −1 = π (f ). The representation (F , D, {φ(f ), π(f ) | f ∈ W }) is said to be irreducible (resp. reducible) if {φ(f ), π(f ) | f ∈ W } is irreducible (resp. reducible). Remark 8.13 In this definition, self-adjointness of φ(f ) (resp. π(f )) is not required. The subspace D is not necessarily unique. Remark 8.14 Let f = 0 (f ∈ W ). By Definition 8.4(iii), we have [φ(f ), π(f )] = i f 2

on D.

Hence Q := f −1 φ(f ) and P := f −1 π(f ) obey the CCR with one degree of freedom. Therefore, at least one of φ(f ) and π(f ) is unbounded. Remark 8.15 Remark 8.5 applies also to representations of Heisenberg CCR. Example 8.10 With notation in Sect. 6.12, let W be a subspace of HJ . Then formulas (6.126)–(6.129) show that πF (W ) := (Fb (H ), Fb,0 (H ), {φF (f ), πF (f ) | f ∈ W })

(8.88)

is a representation of the Heisenberg CCR over W . This representation is called the Fock representation of the Heisenberg CCR over W . There is a simple deformation of a given representation of Heisenberg CCR: Proposition 8.6 Let (F , D, {φ(f ), π(f ) | f ∈ W }) be a representation of the Heisenberg CCR over W as above and two linear functionals μ, ν : W → R be given. Define φμ (f ) := φ(f ) + μ(f ), πν (f ) := π(f ) + ν(f ),

5 We

(8.89) f ∈W,

sometimes omit “Heisenberg” if there is no danger of confusion.

(8.90)

362

8 Representations of CCR with Infinite Degrees of Freedom

Then: (i) (F , D, {φμ (f ), πν (f ) | f ∈ W }) is a representation of the Heisenberg CCR over W . (ii) If (F , D, {φ(f ), π(f ) | f ∈ W }) is irreducible, then so is (F , D, {φμ (f ), πν (f ) | f ∈ W }). (iii) If φ(f ) and π(f ) are essentially self-adjoint, then so are φμ (f ) and πν (f ) with φμ (f ) = φ(f ) + μ(f ), πν (f ) = π(f ) + ν(f ),

(8.91) f ∈W.

(8.92) 

Proof An easy exercise.

We call (F , D, {φμ (f ), πν (f ) | f ∈ W }) the translation of (F , D, {φ(f ), π(f ) | f ∈ W }) by (μ, ν). The following proposition shows that a representation of the CCR over a complex inner product space with an additional condition yields a representation of the Heisenberg CCR over a real inner product space. Proposition 8.7 Let V be a complex inner product space and J be a conjugation on V . Suppose that there exists a representation (F , D, {C(f ) | f ∈ V }) of the CCR over V and let 1 φ(f ) := √ (C(f )∗ + C(f )), 2 i π(f ) := √ (C(f )∗ − C(f )), 2

f ∈ VJ ,

where VJ is the real part of V with respect to J (see (1.68)). Then: (i) (F , D, {φ(f ), π(f ) | f ∈ VJ }) is a representation of the Heisenberg CCR over VJ . (ii) If D is a core for C(f ) and C(f )∗ for all f ∈ V and {C(f ), C(f )∗ |f ∈ V } = CI , then {φ(f ), π(f ) | f ∈ VJ } = CI . In particular, {φ(f ), π(f ) | f ∈ VJ } is irreducible. Proof (i) An easy exercise. (ii) Let T ∈ {φ(f ), π(f ) | f ∈ VJ } . Then T φ(f ) ⊂ φ(f )T and T π(f ) ⊂ π(f )T for all f ∈ VJ . It follows that T C(f )∗ = C(f )∗ T ,

T C(f ) = C(f )T

on D.

By the linearity of C(f )∗ in f and the anti-linearity of C(f ) in f , these relations extend to all f ∈ V . Since D is a core for C(f )∗ and C(f )

8.8 Representations of Heisenberg CCR

363

by the present assumption, it follows from a simple limiting argument that T C(f )∗ ⊂ C(f )∗ T and T C(f ) ⊂ C(f )T , f ∈ V . This means that T ∈ {C(f ), C(f )∗ |f ∈ V } = CI . Hence T = αI for some α ∈ C. Therefore {φ(f ), π(f ) | f ∈ VJ } = CI . Hence {φ(f ), π(f ) | f ∈ VJ } is irreducible.  We call (F , D, {φ(f ), π(f ) | f ∈ VJ }) the representation of the Heisenberg CCR induced by (F , D, {C(f ) | f ∈ V }) with conjugation J . Example 8.11 The Fock representation πF (W ) (Example 8.10) is the representation of the Heisenberg CCR induced by (Fb (H ), Fb,0 (H ), {A(f ) | f ∈ WC }) with conjugation J . To consider the converse of Proposition 8.7, we recall a basic concept in Hilbert space theory. Let R be a real Hilbert space with inner product  , R . Then the product space R × R := {(f, g) | f, g ∈ R} becomes a complex vector space with the following operations of addition and scalar multiplication: for all (f, g), (f , g ) ∈ R × R and α = a + ib ∈ C (a, b ∈ R), (f, g) + (f , g ) := (f + g, f + g ), α(f, g) := (af − bg, ag + bf ). This complex vector space is called the complexification of R and denoted by RC . Note that ψ := (f, g) = (f, 0) + i(g, 0) (f, g ∈ R). Hence, identifying (f, 0) with f , we have ψ = f + ig. The vector f (resp. g) is called the real (resp. imaginary) part of ψ and written as f = Re ψ (resp. g = Im ψ). A natural inner product is introduced in RC by 

    f + ig, f + ig R := f, f R + g, g R − i g, f R + i f, g R C

(f, f , g, g ∈ R). Then the inner product space (RC ,  , RC ) becomes a complex Hilbert space. This Hilbert space is called the complexification of the real Hilbert space R and denoted by the same symbol RC for simplicity. Henceforth the subscript RC in  , RC is omitted, provided that there would be no danger of confusion. We define a mapping JR : RC → RC by JR (f + ig) := f − ig,

f, g ∈ R.

Then it is easy to see that JR is a conjugation. This conjugation is called the complex conjugation on RC . It immediately follows that R = (RC )JR . Example 8.12 Let H be a complex Hilbert space and J be a conjugation on H . Then (HJ )C ∼ =H.

364

8 Representations of CCR with Infinite Degrees of Freedom

Proposition 8.8 Let (F , D, {φ(f ), π(f ) | f ∈ W }) be a representation of the Heisenberg CCR over W as in Definition 8.4 and WC be the complexification of W. (i) Let Φ(f ) := φ(Re f ) + π(Im f ),

f ∈ WC .

(8.93)

Then Φ(f ) is symmetric and, for all f, g ∈ WC , [Φ(f ), Φ(g)] = i Im f, g

(8.94)

on D. (ii) Let 1 C(f ) := √ (Φ(f ) + iΦ(if )), 2

f ∈ WC .

Then C(f ) is closable and (F , D, {C(f ) | f ∈ WC }) is a representation of the CCR over WC . 

Proof An easy exercise.

Propositions 8.7 and 8.8 show that considering a representation of the Heisenberg CCR over a real inner product space W is essentially equivalent to considering a representation of the CCR over WC . Remark 8.16 Let (X , ω) be an R-symplectic space. Suppose that a set {Φ(f )|f ∈ (X , ω)} of symmetric (not necessarily self-adjoint) operators on a Hilbert space F is given and D be a dense subspace of F left invariant by each Φ(f ), f ∈ (X , ω). Then the triple (F , D, {Φ(f )|f ∈ (X , ω)}) is called a representation of the generalized Heisenberg CCR over (X , ω) if, for all f, g ∈ X , a, b ∈ R and Ψ ∈ D, Φ(af + bg)Ψ = [a Φ(f ) + b Φ(g)]Ψ and [Φ(f ), Φ(g)]Ψ = iω(f, g)Ψ.

(8.95)

The ω-Segal field operators introduced in Sect. 6.13 give a representation of the generalized Heisenberg CCR over (X , ω).

8.9 Weyl Representations of CCR Over Real Inner Product Spaces

365

8.9 Weyl Representations of CCR Over Real Inner Product Spaces The concept of Weyl representations of CCR with finite degrees of freedom can be naturally extended to the case of infinite degrees of freedom. Definition 8.5 Let W be a real inner product space and F be a complex Hilbert space. Suppose that, for each f ∈ W , two self-adjoint operators φ(f ) and π(f ) on F are defined. The pair (F , {φ(f ), π(f ) | f ∈ W }) is called a Weyl representation of the CCR over W on F if the following hold: (i) For all f, g ∈ W and a, b ∈ R, eiφ(af +bg) = eiaφ(f ) eibφ(g),

(8.96)

eiπ(af +bg) = eiaπ(f ) eibπ(g).

(8.97)

(ii) (Weyl relations over W ) For all f, g ∈ W and t ∈ R eiφ(f ) eiπ(g) = e−if,g eiπ(g)eiφ(f ) , e

iφ(f ) iφ(g)

e

e

iπ(f ) iπ(g)

e

(8.98)

=e

iφ(g) iφ(f )

e

,

(8.99)

=e

iπ(g) iπ(f )

.

(8.100)

e

Remark 8.17 A more general definition of a Weyl representation is possible as given in [37]. A Weyl representation (F , {φ(f ), π(f ) | f ∈ W }) is said to be irreducible (resp. reducible) if {eiφ(f ) , eiπ(f ) | f ∈ W } is irreducible (resp. reducible). Two Weyl representations (F , {φ(f ), π(f ) | f ∈ W }) and (F , {φ (f ), π (f ) | f ∈ W }) are said to be equivalent if there exists a unitary operator U : F → F such that, for all f ∈ W , U φ(f )U −1 = φ (f ) and U π(f )U −1 = π (f ). If dim W = ∞, then (F , {φ(f ), π(f ) | f ∈ W }) is said to be a Weyl representation with infinite degrees of freedom. Remark 8.18 Relations (8.96) and (8.97) imply that, for all a, t ∈ R, eit φ(af ) = eit aφ(f ) ,

eit π(af ) = eit aπ(f ) .

Hence it follows that φ(af ) = aφ(f ),

π(af ) = aπ(f ),

a ∈ R, f ∈ W .

(8.101)

366

8 Representations of CCR with Infinite Degrees of Freedom

Remark 8.19 Relations (8.99) and (8.100) imply that {φ(f ) | f ∈ W } (resp. {π(f ) | f ∈ W }) is a family of strongly commuting self-adjoint operators on F . Hence, for all f, g, φ(f ) + φ(g) is essentially self-adjoint and, for all t ∈ R, e

. it φ(f )+φ(g)

= eit φ(f ) eit φ(g).

Similarly, π(f ) + π(g) is essentially self-adjoint and e

. it π(f )+π(g)

= eit π(f ) eit π(g).

Then, it follows from (i) that, for all t ∈ R, eit φ(f +g) = e

. it φ(f )+φ(g)

,

eit π(f +g) = e

. it π(f )+π(g)

.

Hence φ(f + g) = φ(f ) + φ(g),

π(f + g) = π(f ) + π(g),

f, g ∈ W .

(8.102)

Formulas (8.101) and (8.102) show that φ(f ) and π(f ) are real linear in f with additivity being of the form (8.102). Remark 8.20 Remark 8.5 applies also to Weyl representations of CCR. Remark 8.21 Properties (8.98)–(8.100) imply that, for all f, g ∈ W , φ(f ), φ(g), π(f ) and π(g) obey CCR: [φ(f ), π(g)] = i f, g ,

[φ(f ), φ(g)] = 0,

[π(f ), π(g)] = 0

(8.103)

on the Gårding domain associated with (φ(f ), φ(g), π(f ), π(g)) (see (2.140) and Theorem 2.16). Hence, by Proposition 2.2, each commutation relation in (8.103) holds on its maximal domain. In this sense, {φ(f ), π(f )|f ∈ W } is a representation of the Heisenberg CCR over W . In contrast to the case of Weyl representations of CCR with finite degrees of freedom, a uniqueness theorem similar to the von Neumann uniqueness theorem does not hold anymore for Weyl representations of CCR with infinite degrees of freedom. Namely, there are mutually (non-trivially) inequivalent Weyl representations of the CCR over an infinite dimensional real inner product space. See Corollary 8.7 below and Chap. 10. We say that a representation (F , D, {φ(f ), π(f )|f ∈ W }) of the Heisenberg CCR over W is a Weyl representation of the CCR over W if (F , {φ(f ), π(f )|f ∈ W }) is a Weyl representation of the CCR over W .

8.9 Weyl Representations of CCR Over Real Inner Product Spaces

367

The Fock representation πF (W ) of the Heisenberg CCR over W in Example 8.10 is in fact an irreducible Weyl representation: Theorem 8.11 The Fock representation πF (W ) is an irreducible Weyl representation of the CCR over W . 

Proof This follows from Theorem 6.23.

Proposition 8.9 Let (F , {φ(f ), π(f ) | f ∈ W }) be a Weyl representation of the CCR over a real inner product space W and φμ (f ) and πν (f ) be defined by (8.89) and (8.90) respectively. Then (F , {φμ (f ), πν (f ) | f ∈ W }) is a Weyl representation of the CCR over W . Moreover, if (F , {φ(f ), π(f ) | f ∈ W }) is irreducible, then so is (F , {φμ (f ), πν (f ) | f ∈ W }). Proof It is obvious that φμ (f ) and πν (f ) are self-adjoint and eiφμ (f ) = eiφ(f ) eiμ(f ) ,

eiπν (f ) = eiπ(f ) eiν(f ) .

(8.104)

These relations imply that {eiφμ (f ) , eiπν (f ) |f ∈ W } obeys the Weyl relations over W . Hence (F , {φμ (f ), πν (f ) | f ∈ W }) is a Weyl representation of the CCR over W . Suppose that (F , {φ(f ), π(f ) | f ∈ W }) is irreducible. Let T ∈ {eiφμ (f ) , eiπν (f ) |f ∈ W } . Then, by (8.104), T ∈ {eiφ(f ) , eiπ(f ) | f ∈ W } = CI . Hence T = αI for some α ∈ C. Thus (F , {φμ (f ), πν (f ) | f ∈ W }) is irreducible.  The following proposition is often useful in applications to quantum field theory: Proposition 8.10 Let (F , {φ(f ), π(f ) | f ∈ W }) be a Weyl representation of the CCR over a real inner product space W . Let W be the completion of W and T be an injective linear mapping on W such that, for all f, g ∈ D(T ), f, T gW = Tf, gW and define qT (f ) := φ(T −1 f ),

pT (g) := π(T g)

(8.105)

for f ∈ D(T −1 ) and g ∈ D(T ) such that T −1 f ∈ W and T g ∈ W . Let W0 ⊂ D(T ) ∩ D(T −1 ) be a subspace such that T ±1 (W0 ) ⊂ W . Then: (i) (F , {qT (f ), pT (f )|f ∈ W0 }) is a Weyl representation of the CCR over W0 . (ii) Suppose that the mappings f → eiφ(f ) and f → eiπ(f ) are weakly continuous: if fn , f ∈ W (n ∈ N) and limn→∞ fn = f , then w- limn→∞ eiφ(fn ) = eiφ(f ) and w- limn→∞ eiπ(fn ) = eiπ(f ) , where w- lim means weak limit (see Sect. 1.1.2), and T ±1 (W0 ) are dense in W . Let {φ(f ), π(f )|f ∈ W } = CI . Then {qT (f ), pT (f )|f ∈ W0 } = CI . In particular, (F , {qT (f ), pT (f )|f ∈ W0 }) is irreducible.

368

8 Representations of CCR with Infinite Degrees of Freedom

Proof

 (i) By the assumption for T , we have T −1 f, T g = f, g , f, g ∈ W0 . Hence eiqT (f ) eipT (g) = e−if,g eipT (g)eiqT (f ) . It is easy to show that, for (φ(f ), π(f )) = (qT (f ), pT (f )) (f ∈ W0 ), (8.96), (8.97), (8.99) and (8.100) hold. Thus (F , {qT (f ), pT (f )|f ∈ W0 }) is a Weyl representation of the CCR over W0 . (ii) Let B ∈ {eiqT (f ) , eipT (f ) |f ∈ W0 } . Then, for all f ∈ T −1 (W0 ) and g ∈ T (W0 ), Beiφ(f ) = eiφ(f ) B and Beiπ(g) = eiπ(g)B. Let f ∈ W . Then, by the −1 (W ) such that density of T −1 (W0 ) in W , there exists a sequence {fn }∞ n=1 in T iφ(f ) iφ(f ) n n =e B. Taking the weak limit of fn → f as n → ∞. We have Be both sides as n → ∞, we obtain Beiφ(f ) = eiφ(f ) B. Similarly, one can show that Beiπ(f ) = eiπ(f ) B. Therefore B ∈ {eiφ(f ) , eiπ(f ) } = CI . Hence B = αI for some α ∈ C. Thus {eiqT (f ) , eipT (f ) |f ∈ W0 } = CI . 

8.10 Translations of Fock Representation of Heisenberg CCR Let πF (W ) be the Fock representation of the Heisenberg CCR over W given by (8.88) and consider a translation of it: φμ (f ) := φF (f ) + μ(f ), πν (f ) := πF (f ) + ν(f ),

(8.106) f ∈W,

(8.107)

where μ, ν : W → R are real-valued linear functionals. By Proposition 8.9, πμ,ν (W ) := (Fb (H ), {φμ (f ), πν (f )|f ∈ W })

(8.108)

is an irreducible Weyl representation of the CCR over W . A natural question is: under what conditions is it equivalent or inequivalent to the Fock representation πF (W )? The answer is given in Theorem 8.12 below. We extend μ and ν to linear functionals on WC as follows: for f = f1 +if2 ∈ WC (f1 , f2 ∈ W ), μ(f ) := μ(f1 ) + iμ(f2 ),

ν(f ) := ν(f1 ) + iν(f2 ).

(8.109)

Let 1 L(f ) := √ (μ(f ) + iν(f )), 2

f ∈ WC .

(8.110)

Theorem 8.12 The representation πμ,ν (W ) is equivalent to πF (W ) if and only if there exists a vector h ∈ H such that L(f ) = h, f  , f ∈ WC .

8.10 Translations of Fock Representation of Heisenberg CCR

369

Proof Suppose that πμ,ν (W ) is equivalent to πF (W ). Then there exists a unitary operator U on Fb (H ) such that, for all f ∈ W , U φμ (f )U −1 = φF (f ) and U πν (f )U −1 = πF (f ). Hence U φμ (f ) = φF (f )U and U πν (f ) = πF (f )U . In particular, we have U (φμ (f ) + iπν (f )) = (φF (f ) + iπF (f ))U . Let Φ, Ψ ∈ Fb,0 (H ). Then √

2(A(f ) + L(f )∗ )Ψ, √ (φF (f ) + iπF (f ))∗ Φ = 2A(f )∗ Φ.

(φμ (f ) + iπν (f ))Ψ =

Hence 

 Φ, U (A(f ) + L(f )∗ )Ψ = A(f )∗ Φ, U Ψ .

Since Fb,0 (H ) is a core for A(f )∗ , it follows that U Ψ ∈ D((A(f )∗ )∗ ) = D(A(f )) and A(f )U Ψ = U (A(f ) + L(f )∗ )Ψ , i.e., U −1 A(f )U Ψ = (A(f ) + L(f )∗ )Ψ . It is easy to see that this extends to all f ∈ WC . Recall that Fb,0 (H ) is a core for A(g), g ∈ H . Hence we obtain operator equality U −1 A(f )U = A(f ) + L(f )∗ = CL (f ),

f ∈ WC .

(8.111)

This means that the Bogoliubov translation CL (·) by L is equivalent to the Fock representation of the CCR over WC . Hence, by Theorem 8.5, there exists a vector h ∈ H such that L(f ) = h, f  , f ∈ WC . Conversely, suppose that there exists a vector h ∈ H such that L(f ) = h, f  , f ∈ WC . Then one can define the unitary operator U = e−i

√ 2ΦS (ih)

.

By a general formula, (8.111) holds. Hence, for all f ∈ W , U −1 φF (f )U = φμ (f ),

U −1 πF (f )U = πν (f ),

on D(A(f )) ∩ D(A(f )∗ ). Since φμ (f ) and πν (f ) are essentially self-adjoint on D(A(f )) ∩ D(A(f )∗ ), we obtain operator equalities U −1 φF (f )U = φμ (f ), Thus πμ,ν (W ) is equivalent to πF (W ).

U −1 πF (f )U = πν (f ). 

Corollary 8.7 Suppose that L is not continuous on WC in the strong topology of H . Then πμ,ν (W ) is inequivalent to πF (W ). Proof By Riesz’s representation theorem, there exists a vector h ∈ H such that L(f ) = h, f  , f ∈ WC if and only if L is continuous on WC in the strong topology

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8 Representations of CCR with Infinite Degrees of Freedom

of H . Hence, by the present assumption, there is no such vector h. Therefore the conclusion follows from Theorem 8.12.  There are classes of non-continuous linear functionals on a dense subspace of H . For example, let T be a densely defined closable unbounded linear operator on H . Then D(T ∗ ) = H (if D(T ∗ ) = H , then T ∗ is bounded by the closed graph theorem and hence T = (T ∗ )∗ is bounded, a contradiction). Hence there exists a non-zero vector g ∈ D(T ∗ ). Then the linear functional LT ,g defined by LT ,g (f ) := g, Tf  ,

f ∈ D(T )

is not continuous. (If LT ,g is continuous, then, by Riesz’s representation theorem, there exists a vector h ∈ H such that LT ,g (f ) = h, f  , f ∈ D(T ). Hence g, Tf  = h, f  , f ∈ D(T ). This means that g ∈ D(T ∗ ) and T ∗ g = h. But this is a contradiction.) If H is infinite dimensional, then there are infinitely many densely defined closed unbounded operators. Thus we see that there are infinitely many irreducible Weyl representations which are inequivalent to the Fock representation πF (W ).

8.11 A Class of Irreducible Weyl Representations of CCR (I) In this section we show that one can obtain a simple class of irreducible Weyl representations of the CCR over a real Hilbert space by deforming the Segal field operators. Let H be a Hilbert space and J be a conjugation on H . Suppose that there exist bounded operators S and T on H satisfying the following conditions: for all f, g ∈ HJ , Re Sf, T g = f, g ,

(8.112)

Im Sf, Sg = 0,

(8.113)

Im Tf, T g = 0.

We define the following self-adjoint operators on the boson Fock space Fb (H ) over H : φ (S) (f ) := ΦS (Sf ), π (T ) (f ) := ΦS (iTf ),

(8.114) f ∈ HJ ,

(8.115)

where ΦS (g) (g ∈ H ) is the Segal field operator with test vector g. Using (6.122), (S) (T ) one can show that {eiφ (f ) , eiπ (f ) |f ∈ HJ } obeys the Weyl relations over HJ . Hence we have: Lemma 8.12 The pair (Fb (H ), {φ (S) (f ), π (T ) (f )|f ∈ HJ }) is a Weyl representation of the CCR over HJ .

8.11 A Class of Irreducible Weyl Representations of CCR (I)

One needs an additional condition for {eφ irreducible:

(S) (f )

371

, eiπ

(T ) (f )

|f ∈ HJ } to be

Theorem 8.13 Let W be a dense subspace of HJ . Suppose that S ∗ − iT ∗ is injective. Then (Fb (H ), {φ (S) (f ), π (T ) (f )|f ∈ W }) is an irreducible Weyl representation of the CCR over W . (S)

Proof We need only to show that A := {eiφ (f ) , eiπ To prove this, let B ∈ A . Then, for all f, g ∈ W , BeiΦS (Sf ) = eiΦS (Sf ) B,

(T ) (f )

|f ∈ W } is irreducible.

BeiΦS (iTf ) = eiΦS (iTf ) B.

Let K := S + iT . Then, by (6.123), we have eiΦS (Kf ) = ceiΦS (Sf ) eiΦS (iTf ) ,

f ∈W,

where c := ei f /2 . Hence BeiΦS (Kf ) = eiΦS (Kf ) B for all f ∈ W . Since K ∗ = S ∗ − iT ∗ is injective by the present assumption, Ran ((K ∗ )∗ ) = Ran (K) is dense in H by Corollary 1.1. Hence it follows that KHJ is dense in H and hence KW is dense in H . Hence, by Theorem 6.22(iv), {eiΦS (Kf ) |f ∈ W } is irreducible. It is obvious that the set {eiΦS (Kf ) |f ∈ W } is ∗-invariant. Therefore, by Proposition 2.9, {eiΦS (Kf ) |f ∈ W } = CI . Hence B = αI for some constant (S) (T )  α ∈ C. Thus {eiφ (f ) , eiπ (f ) |f ∈ W } is irreducible. 2

We note the following fact: Lemma 8.13 Equation (8.112) implies that S + T is injective. Proof Let f ∈ ker(S +T ), so that Sf = −Tf . Hence Sf, Tf  = − Tf 2 . Hence, by (8.112), f 2 = − Tf 2 ≤ 0. Therefore f 2 = 0. Thus f = 0.  Concerning equivalence of (Fb (H ), φ (S) (f ), π (T ) (f )|f ∈ HJ }) to the Fock representation πF (HJ ), one has the following theorem: Theorem 8.14 Assume the following: (a) Ran (S + T ) is dense. (b) S − T is not Hilbert–Schmidt. Let W be a dense subspace of HJ . Then (Fb (H ), {φ (S) (f ), π (T ) (f )|f ∈ W }) is inequivalent to the Fock representation πF (W ). Proof Suppose that the representation (Fb (H ), {φ (S) (f ), π (T ) (f )|f ∈ W }) were equivalent to πF (W ). Then there exists a unitary operator U on Fb (H ) such that φF (f ) = U ΦS (Sf )U −1 ,

πF (f ) = U ΦS (iTf )U −1 ,

f ∈W.

(8.116)

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8 Representations of CCR with Infinite Degrees of Freedom

Let Ω := U −1 ΩH . Then it follows from (8.116) and (6.39) that 1 √ A(f )∗ ΩH = U ΦS (Sf )Ω, 2

i √ A(f )∗ ΩH = U ΦS (iTf )Ω, f ∈ W . 2

Hence ΦS (Sf )Ω = −iΦS (iTf )Ω. This implies that, for all Ψ ∈ Fb,0 (H ), 

A((S + T )f )∗ Ψ, Ω = − A((S − T )f )Ψ, Ω ,

f ∈W.

This can be extended as 

A((S + T )f )∗ Ψ, Ω = − A((S − T )Jf )Ψ, Ω ,

f ∈ WC .

Let D := (S + T )WC , which is dense in H by condition (a). Then, for all g ∈ D, 

  A(g)∗ Ψ, Ω = − A(J (S − T )J (S + T )−1 g)Ψ, Ω .

Hence, by Proposition 8.4, W := (S − T )J (S + T )−1 is Hilbert–Schmidt. Therefore W (S +T ) also is Hilbert–Schmidt. But W (S +T ) = (S −T )J . Hence it follows that S − T is Hilbert–Schmidt. But this contradicts (b).  Remark 8.22 Let 1 B0 (f ) := √ (φ (S) (f ) + iπ (T ) (f ))  Fb,0 (H ), 2

f ∈ HJ ,

and B0 (f ) := B0 (f1 )−iB0 (f2 ) for f = f1 +if2 , f1 , f2 ∈ HJ . Since D(B0 (f )∗ ) is dense, B0 (f ) is closable; we denote the closure of it by B(f ). Then the Heisenberg CCR of {φ (S)(f ), π (T ) (f )|f ∈ HJ } imply that, for all f, g ∈ H , [B(f ), B(g)∗ ] = f, g ,

[B(f ), B(g)] = 0 on Fb,0 (H ).

Hence (Fb (H ), Fb,0 (H ), {B(f ), B(f )∗ |f ∈ H }) is a representation of the CCR over H . On the other hand, one has B(f ) =

1 (A((S + T )f ) + A((S − T )Jf )∗ ) 2

on Fb,0 (H ).

Hence (A(·), A(·)∗ ) → (B(·), B(·)∗ ) is a Bogoliubov transformation. Under additional conditions (see Theorem 8.7), one can prove that, if S − T is Hilbert– Schmidt, then (Fb (H ), Fb,0 (H ), {B(f ), B(f )∗ |f ∈ H }) is equivalent to the Fock representation ρF (H ), implying that (Fb (H ), {φ (S) (f ), π (T ) (f )|f ∈ HJ }) is equivalent to πF (HJ ).

8.12 A Class of Irreducible Weyl Representations of CCR (II)

373

8.12 A Class of Irreducible Weyl Representations of CCR (II) Let T be a symmetric operator on H (not necessarily bounded) such that J T ⊂ T J.

(8.117)

Then T is reduced by HJ in the sense that, for all f ∈ D(T ), the real part Re f of f (see (1.69)) is in HJ ∩ D(T ) and Re (Tf ) = T Re f . In particular, D(T ) ∩ HJ is dense in HJ and T (D(T ) ∩ HJ ) ⊂ HJ . Lemma 8.14 Suppose that T is injective. Then J T −1 ⊂ T −1 J . Proof Let f ∈ D(T −1 ) = Ran T . Then g := T −1 f is in D(T ). Hence J g is in D(T ) and T J g = J T g = Jf . Hence Jf ∈ Ran T = D(T −1 ) and J g = T −1 Jf . We have J g = J T −1 f . Thus J T −1 ⊂ T −1 J .  For an injective symmetric operator T on H satisfying (8.117), we define selfadjoint operators φT (f ) := φF (T −1 f ), πT (g) := πF (T g),

f ∈ D(T −1 ) ∩ HJ ,

g ∈ D(T ) ∩ HJ ,

(8.118) (8.119)

acting in Fb (H ). Let W be a dense subspace of HJ and SW (H ) be the set of injective symmetric operators T on H satisfying the following conditions: (i) (8.117) holds. (ii) W ⊂ D(T ) ∩ D(T −1 ) and T ±1 W are dense in HJ . For each T ∈ SW (H ), we define a triple ΠT (W ) := (Fb (H ), Fb,0 (H ), {φT (f ), πT (f )|f ∈ W }).

(8.120)

Proposition 8.11 The triple ΠT (W ) is a representation of the Heisenberg CCR over W . Proof For all f, g ∈ W , the CCR for φT (f ) and πT (g) on Fb,0 (H ) follow from (6.53) and the elementary fact that T −1 f, T g = f, g.  Theorem 8.15 The representation ΠT (W ) is an irreducible Weyl representation of the CCR over W . Proof This follows from an application of Proposition 8.10.



It is interesting to ask under what condition ΠT1 (W ) is equivalent or inequivalent to ΠT2 (W ) for T1 , T2 ∈ SW (H ) with T1 = T2 . To answer this question, we need some preliminaries.

374

8 Representations of CCR with Infinite Degrees of Freedom

For T1 , T2 ∈ SW (H ) such that D(T2−1 T1 )∩D(T2 T1−1 )∩D(T1−1 T2 )∩D(T1 T2−1 ) is dense in H , we define S± :=

1 −1 (T T1 ± T2 T1−1 ). 2 2

(8.121)

Note that D21 := D(S+ ) = D(S− ) = D(T2 T1−1 ) ∩ D(T2−1 T1 ). Hence S± are densely defined and we have ∗ ⊃ S±

1 (T1 T2−1 ± T1−1 T2 ). 2

∗ are densely defined. Therefore S are closable. We set Hence S± ± ∗ ). D12 := D(T1 T2−1 ) ∩ D(T1−1 T2 ) ⊂ D(S±

It follows that D12 ∩ D21 = D(T1−1 T2 ) ∩ D(T1 T2−1 ) ∩ D(T2−1 T1 ) ∩ D(T2 T1−1 ).

(8.122)

Lemma 8.15 The following hold: (i) For all f, g ∈ D21 , S+ f, S+ g − S− f, S− g = f, g ,

(8.123)

S− f, S+ g = S+ f, S− 

(8.124)

(ii) For all f, g ∈ D12 ,  ∗  ∗ ∗ ∗ g − S− f, S− g = f, g , S+ f, S+  ∗  ∗ ∗ ∗ S− f, S+ g = S+ f, S− g .

(8.125) (8.126)

Proof (i) Let f, g ∈ D21 . Then    1  −1 ( T2 T1 f, T2−1 T1 + 2 f, g + T2 T1−1 f, T2 T1−1 g ), 4    1  S− f, S− g = ( T2−1 T1 f, T2−1 T1 − 2 f, g + T2 T1−1 f, T2 T1−1 g ). 4

S+ f, S+ g =

Hence (8.123) holds.. Similarly, one can prove (8.124). (ii) This also can be proved by direct computations as in (i).



8.12 A Class of Irreducible Weyl Representations of CCR (II)

375

Relations (8.123)–(8.126) remind us of the theory of bosonic Bogoliubov transformations (see Remark 8.9). Thus we are led to proceed as follows. We fix T1 , T2 ∈ SW (H ) and set V := D21 .

(8.127)

For each f ∈ V , we introduce a new operator B(f ) by B(f ) := A(S + f ) + A(J S − f )∗ .

(8.128)

B(f )∗ ⊃ A(S + f )∗ + A(J S − f ).

(8.129)

We have

Note that {B(f ), B(f )∗ |f ∈ V } satisfy [B(f ), B(g)∗ ] = f, g , [B(f ), B(g)] = 0,

[B(f )∗ , B(g)∗ ] = 0,

f, g ∈ V ,

on Fb,0 (H ). Hence (Fb (H ), {B(f ), B(f )∗ |f ∈ V }) is a representation of the CCR over V . Remark 8.23 If S+ or S− is unbounded (then both S+ and S− are unbounded), then the correspondence (A(·), A(·)∗ ) → (B(·), B(·)∗ ) is a singular bosonic Bogoliubov transformation in the sense defined in Remark 8.9. An analysis of this singular Bogoliubov transformation is given in [30]. In the present book, we discuss only the case where S± are bounded. Thus, in what follows, we assume the following: (T) The operators T2−1 T1 and T2 T1−1 are bounded. Under this assumption, S± are bounded. Hence, for all f ∈ H , B(f ) is defined by (8.128). Moreover, (8.123)–(8.126) imply the following operator equations: ∗

∗

∗

∗

S + S + − S − S − = I, S−S+ = S +S −, ∗

∗

∗

∗

(8.130) (8.131)

S + S + − S − S − = I, S−S+ = S +S −.

(8.132) (8.133)

Lemma 8.16 Let H be separable and assume (T). Then there exists a unitary operator U on Fb (H ) such that B(f ) = U A(f )U −1 , if and only if S − is Hilbert–Schmidt.

f ∈H

(8.134)

376

8 Representations of CCR with Infinite Degrees of Freedom

Proof This follows from an application of Corollary 8.5.



Theorem 8.16 Let H be separable and assume (T). Suppose that S − is Hilbert– Schmidt. Then there exists a unitary operator U on Fb (H ) such that operator equations U φT1 (f )U −1 = φT2 (f ), U πT1 (f )U −1 = πT2 (f ),

(8.135) f ∈W,

(8.136)

hold. Namely, ΠT1 (W ) ∼ = ΠT2 (W ). Proof By Lemma 8.16, there exists a unitary operator U on Fb (H ) satisfying (8.134). This implies also B(f )∗ = U A(f )∗ U −1 ,

f ∈H.

(8.137)

Hence, for all f ∈ W , we have B(T1±1 f ) = U A(T1±1 f )U −1 ,

B(T1±1 f )∗ = U A(T1±1 f )U −1 .

Hence 1 √ (B(T1−1 f )∗ + B(T1−1 f )) = U φT1 (f )U −1 , 2 i √ (B(T1 f )∗ − B(T1 f )) = U πT1 (f )U −1 . 2 By (8.128) and (8.129), we have B(T1−1 f )∗ + B(T1−1 f ) = A(S + T1−1 f + S − T1−1 f )∗ + A(S − T1−1 f + S + T1−1 f ), B(T1 f )∗ − B(T1 f ) = A(S + T1 f − S − T1 f )∗ + A(S − T1 f − S + T1 f ). on Fb,0 (H ). For all g ∈ D12 , one can show that 

   g, S + T1−1 f + S − T1−1 f = g, T2−1 f ,  g, S + T1 f − S − T1 f = g, T2 f  .

Since D12 is dense by the original assumption, it follows that S + T1−1 f + S − T1−1 f = T2−1 f,

S + T1 f − S − T1 f = T2 f.

8.12 A Class of Irreducible Weyl Representations of CCR (II)

377

Hence B(T1−1 f )∗ + B(T1−1 f ) = A(T2−1 f )∗ + A(T2−1 f ), B(T1 f )∗ − B(T1 f ) = A(T2 f )∗ − A(T2 f ) on Fb,0 (H ). Therefore φT2 (f ) = U φT1 (f )U −1 , πT2 (f ) = U πT1 (f )U −1

on Fb,0 (H ).

(8.138)

This implies that φT2 (f )  Fb,0 (H ) ⊂ U φT1 (f )U −1 , πT2 (f )  Fb,0 (H ) ⊂ U πT1 (f )U −1 . (8.139) Recall that Fb,0 (H ) is a core for φT2 (f ) and πT2 (f ). Moreover, U φT1 (f )U −1 and U πT1 (f )U −1 are self-adjoint. Hence (8.139) yields operator equalities (8.135) and (8.136).  Theorem 8.17 Let H be separable and assume (T). Suppose that there exists a unitary operator U on Fb (H ) satisfying (8.135) and (8.136). Then S − is Hilbert– Schmidt. Proof It is easy to see that, for all Ψ ∈ Fb,0 (H ) and f ∈ W , 1 A(f )Ψ = √ (φT1 (T1 f ) + iπT1 (T1−1 f ))Ψ. 2 Hence 1 A(f )Ψ = √ (U −1 φT2 (T1 f )U Ψ + iU −1 πT2 (T1−1 f )U Ψ ), 2 which implies that 1 U A(f )Ψ = √ (φT2 (T1 f )U Ψ + iπT2 (T1−1 f )U Ψ ). 2 Hence, for all Φ ∈ Fb,0 (H ), 

   U −1 Φ, A(f )Ψ = U −1 (A(S+ f )∗ Φ + A(S− f )Φ), Ψ .

Since Fb,0 (H ) is a core for A(f ), it follows that U −1 Φ ∈ D(A(f )∗ ) and A(f )∗ U −1 Φ = U −1 (A(S+ f )∗ + A(S− f ))Φ.

378

8 Representations of CCR with Infinite Degrees of Freedom

By the density of W in HJ and the strong continuity of the mapping H g → A(g)# Φ, this equation can be extended to all f ∈ HJ with S± replaced by S ± : A(f )∗ U −1 Φ = U −1 (A(S + f )∗ + A(S − f ))Φ,

f ∈ HJ .

 By (6.39), we have A(f )∗ U −1 Φ, ΩH = 0. Hence, putting Ω := U ΩH , we obtain   A(S + f )∗ Φ, Ω = − A(S − f )Φ, Ω ,

f ∈ HJ .

(8.140)

By (8.130), we have ∗

∗

S+S + = 1 + S −S −. ∗

This implies that S + S + is bijective on HJ (note that S ± are reduced by HJ ). In particular, S + is injective on HJ . On the other hand, (8.132) gives ∗

∗

S+S + = 1 + S −S −. ∗

This implies that S + S + is bijective on HJ . In particular, S + is surjective on HJ . −1

Hence S + is bijective on HJ . Hence, putting K := S − S + , we have for all f ∈ HJ , A(f )∗ Φ, Ω = − A(Kf )Φ, Ω. This equation can be extended to all f ∈ H : 

A(f )∗ Φ, Ω = − A(J Kf )Φ, Ω ,

f ∈H.

Hence, by Proposition 8.4, K is Hilbert–Schmidt. Thus S − = KS + is Hilbert– Schmidt.  In summary, we have proved the following theorem: Theorem 8.18 Let H be separable. Let T1 , T2 ∈ SW (H ) and assume (T). Then ΠT1 (W ) ∼ = ΠT2 (W ) if and only if S − is Hilbert–Schmidt. In view of finding inequivalent irreducible Weyl representations of CCR, it may be convenient to rephrase this theorem as follows: Corollary 8.8 Let H is separable. Let T1 , T2 ∈ SW (H ) and assume (T). Then ΠT1 (W ) is inequivalent to ΠT2 (W ) if and only if S − is not Hilbert–Schmidt. Remark 8.24 The conditions for T1 and T2 in Theorem 8.18 are related to an equivalence relation in a subset of SW (H ). Let S× W := {T ∈ SW (H )|T is surjective}. −1 ∈ B(H ). For T , T ∈ S , we write T ∼ T Then, for all T ∈ S× 1 2 1 2 W W, T if the following conditions are satisfied: (1) T2 T1−1 and T2−1 T1 are bounded; (2)

8.12 A Class of Irreducible Weyl Representations of CCR (II)

379

T2 T1−1 − T2−1 T1 is Hilbert–Schmidt. It is easy to see that the relation ∼ is an 6 equivalence relation in S× W . Hence Theorem 8.18 implies the following: Let ∼ T1 , T2 ∈ S × W . Then, if T1 ∼ T2 , then ΠT1 (W ) = ΠT2 (W ). Conversely, if × −1 ∼ ΠT1 (W ) = ΠT2 (W ) for T1 , T2 ∈ SW with T2 T1 and T2−1 T1 being bounded, then T1 ∼ T2 . This may be an interesting structure. Remark 8.25 In the case where at least one of T2−1 T1 and T2 T1−1 is unbounded, we need a separate consideration (see [24, Theorem 5.6]). Such an example appears in relation to Casimir effect in quantum electrodynamics [29, 30]. A simple consequence of Theorem 8.18 is as follows. Corollary 8.9 (Shale [146]) Let S be a bijective bounded self-adjoint operators on H such that SJ = J S. Then the Weyl representation (Fb (H ), {φS (f ), πS (f )|f ∈ HJ }) is equivalent to the Fock representation ρF (H ) if and only if S 2 − I is Hilbert–Schmidt. Proof We need only to apply Theorem 8.18 to the case where T1 = S and T2 = I . In this case, we have S − = S− =

. 1 1S − S −1 = S −1 (S 2 − I ). 2 2

Hence S − is Hilbert–Schmidt if and only S 2 −I is Hilbert–Schmidt. Thus the desired result follows.  Example 8.13 A simple example of (T1 , T2 ) such that S − is not Hilbert–Schmidt is given as follows. Let T ∈ SW (H ) and T1 := T ,

T2 = αT

with α ∈ R \ {0, ±1}. Then, T1 , T2 ∈ SW (H ) and we have S− =

1 2

0

1 1 −α =  0. α

It is easy to see that, in the present example, condition (T) is trivially is satisfied. Suppose that H is infinite dimensional. Then any non-zero scalar operator on H is not Hilbert–Schmidt. Hence S − is not Hilbert–Schmidt. Thus, by Corollary 8.8, ΠT (W ) is inequivalent to ΠαT (W ). Note that, for this inequivalence, the infinite dimensionality of H , which makes ΠT (W ) and ΠαT (W ) to be representations of CCR with infinite degrees of freedom, is essential.

6 Condition

that T −1 ∈ B(H ) (T ∈ S× W ) is needed for ∼ to be transitive.

380

8 Representations of CCR with Infinite Degrees of Freedom

8.13 Functional Schrödinger Representation In concluding this chapter, we briefly mention a representation of the Heisenberg CCR over a real Hilbert space which can be regarded as an infinite-dimensional version of Schrödinger representations of CCR, although it will be not used in the rest of this book (but it is important to know the existence of the representation). For this purpose, we first recall a basic existence theorem on a random process over a real Hilbert space: Theorem 8.19 Let H be a separable real Hilbert space. Then there exist a probability space (Q, Σ, μ) and a family {φ(f )|f ∈ H } of random variables on (Q, Σ) (real-valued Σ-measurable functions on Q which are finite μ-a.e.) such that the following hold: (i) (real linearity in test vectors) For all f, g ∈ H and a, b ∈ R, φ(af + bg) = aφ(f ) + bφ(g),

μ-a.e.

(ii) (fullness of the random variables) The σ -field Σ is the smallest σ -field such that, for all f ∈ H , φ(f ) is measurable. (iii) (Gaussian measure)  2 eiφ(f ) dμ = e− f H /4 , f ∈ H . Q

For a proof of this theorem, see, e.g., [147, §1.1], [148, Theorem 2.3.4], [72, §3.2], [105, §5.4] or [21, §2.4]. Remark 8.26 Theorem 8.19 is concerned with existence of infinite-dimensional ˙ be the one-point Gaussian measures. To explain an idea behind Theorem 8.19, let R compactification of R and Q :=

∞ 

˙ ˙ = {q = (qn )∞ |n ∈ N, qn ∈ R}, R n=1

n=1

˙ and Σ be the σ -algebra generated by the coordinate the infinite direct product of R, ˙ q ∈ Q, n ∈ N. Let μG be the Gaussian functions cn : Q → R; cn (q) := qn ∈ R, 1 measure on (R, B ) given by 1 μG (B) = √ π



e−x dx, 2

B ∈ B 1.

B

Then the measure μ in Theorem 8.19 in the present context is theinfinite product ∞ measure of μG : μ = ⊗∞ n=1 dμG (qn )); n=1 μG (symbolically written dμ(q) = the existence of it can be proved by applying Kolmogorov’s theorem (e.g., [148,

8.13 Functional Schrödinger Representation

381

Theorem 2.1]). This is an infinite-dimensional Gaussian measure. Let {en }∞ n=1 be a CONS of H . Then each f ∈ H is expanded as f =

∞

en f  en .

n=1

One can show that φN (f ) :=

N

en , f  cn

(N ∈ N)

n=1

converges in L2 (Q, dμ) as N → ∞. Hence φ(f ) := lim φN (f ) N→∞

in the L2 -sense is an element of L2 (Q, dμ). Note that φ(en ) = cn .

(8.141)

It is easy to see that 

N

2 n=1 en ,f  /4

eiφN (f ) dμ = e−

,

f ∈H.

Q

The right hand side converges to e− f

2 /4



as N → ∞. Hence it follows that

eiφ(f ) dμ = e− f

2 /4

,

f ∈H.

Q

In this way, one obtains a model for ((Q, Σ, μ), {φ(f )|f ∈ H }). We remark also that there are other models for ((Q, Σ, μ), {φ(f )|f ∈ H }). The family {φ(f )|f ∈ H } of random variables in Theorem 8.19 is called the Gaussian random process indexed by H with probability space (Q, Σ, μ). In the case where H is infinite dimensional, the measure μ is called an infinitedimensional Gaussian measure on (Q, Σ). It follows that, for all n ∈ N and fj ∈ H , j = 1, . . . , n, φ(f1 ) · · · φ(fn ) ∈ L2 (Q, dμ) (for a proof of this fact, see references cited above).

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8 Representations of CCR with Infinite Degrees of Freedom

The following theorem is significant (for a proof, see references cited above): Theorem 8.20 There exists a unitary operator U from the boson Fock space Fb (HC ) over HC to L2 (Q, dμ) such that the following hold: (i) U ΩH = 1 ∈ L2 (Q, dμ). (ii) U Fb,fin (HC ) = span{1, φ(f1 ) · · · φ(fn )|n ∈ N, fj ∈ H , j = 1, . . . , n}. (iii) U φF (f )U −1 = φ(f ), f ∈ H , where φF (f ) is defined by (6.125) with HJ = H and the right hand side is the multiplication operator by the function φ(f ) on Q. The Hilbert space L2 (Q, dμ) is called a Q-space representation of the boson Fock space Fb (HC ). Let H be infinite dimensional and π(f ) := U πF (f )U −1 ,

f ∈H.

(8.142)

Then πS∞ := (L2 (Q, dμ), {φ(f ), π(f )|f ∈ H }) is an irreducible Weyl representation of the CCR over H with infinite degrees of freedom, which is equivalent to the Fock one (Fb (HC ), {φF (f ), πF (f )|f ∈ H }). To write down the form of operation of π(f ), we introduce an operator Df with f ∈ H by Df :=

√ 2U A(f )U −1 .

(8.143)

Let P(H ) := span{1, Pn (φ(f1 ), . . . , φ(fn ))|n ∈ N, Pn is a polynomial of n real variables with complex coefficients, f1 , . . . , fn ∈ H }. Then, by Theorem 8.20(ii), P(H ) is dense in L2 (Q, dμ). One can show that, for all f ∈ H , P(H ) ⊂ D(Df ) and Df Pn (φ(f1 ), . . . , φ(fn )) =

n 

f, fj (∂j Pn )(φ(f1 ), . . . , φ(fn ))

j =1

and π(f ) = −iDf + iφ(f )

on P(H )

(for a proof of these facts, see [21, §6.11], [28, §5.26]).

(8.144)

8.13 Functional Schrödinger Representation

383

Formula (8.144) shows that Df  P(H ) can be regarded as a directional differential operator with the direction of f . We note also that, in the representation πS∞ , φ(f ) is a multiplication operator and each φ(en ) plays a role of a coordinate function in an infinite-dimensional space (see (8.141)). Thus πS∞ can be regarded as an infinite-dimensional version of the Schrödinger representation.7 We call πS∞ the functional Schrödinger representation of the Heisenberg CCR over H on L2 (Q, dμ). For further details of this representation, see [28, §5.26]. The functional Schrödinger representation πS∞ is very useful in constructions and analysis of models of quantum scalar fields. For example, in this representation, an interaction of a self-interacting neutral quantum scalar field with cutoffs can be defined as a multiplication operator on L2 (Q, dμ) with a suitable space Q, and the heat semi-group ((C0 )-semi-group) defined by the Hamiltonian of a neutral quantum scalar field with such an interaction can be represented in terms of functional integrations (Feynman–Kac–Nelson-type formulae). It is known that functional integral methods make it possible to derive detailed properties of quantum field models. For further details of functional integral methods, see, e.g., [21, 62, 72, 80, 105, 147, 148]. We remark also that the representation πS∞ is related to white noise analysis (see, e.g., [122]).

can easily show that, if H is finite-dimensional with dim H = N < ∞, then (L2 (Q, dμ), {φ(f ), π(f ) | f ∈ H }) is equivalent to the Schrödinger representation πS(N) (see [28, §5.26]).

7 One

Chapter 9

Representations of CAR with Infinite Degrees of Freedom

Abstract Some basic representations of CAR with infinite degrees of freedom are described in the framework of fermion Fock spaces.

9.1 Definitions Definition 9.1 Let F and H be complex Hilbert spaces, and ACAR (H ) := {ψ(f ), ψ(f )∗ |f ∈ H } be a subset of B(F), the Banach space of everywhere defined bounded linear operators on F. The pair (F, ACAR (H )) is called a representation of the canonical anti-commutation relations (CAR) over H if the following hold: (i) (anti-linearity in test vectors) For all f, g ∈ H and α, β ∈ C, ψ(αf + βg) = α ∗ ψ(f ) + β ∗ ψ(g). (ii) (CAR) For all f, g ∈ H , {ψ(f ), ψ(g)} = 0,

{ψ(f ), ψ(g)∗ } = f, g .

(9.1)

Remark 9.1 Taking the adjoint of the first equation in (9.1), we have {ψ(f )∗ , ψ(g)∗ } = 0,

f, g ∈ H .

Definition 9.2 A representation (F, ACAR (H )) of the CAR over H is said to be reducible (resp. irreducible) if ACAR (H ) is reducible (resp. irreducible). Definition 9.3 Let (F , A CAR (H )) with A CAR (H ) := {ψ (f ), ψ (f )∗ |f ∈ H }) be another representation of the CAR over H . Then the two representations (F , A CAR (H )) and (F, ACAR (H )) are said to be equivalent if there exists a unitary operator U : F → F such that, for all f ∈ H , ψ (f ) = U ψ(f )U −1 .

© Springer Nature Singapore Pte Ltd. 2020 A. Arai, Inequivalent Representations of Canonical Commutation and Anti-Commutation Relations, Mathematical Physics Studies, https://doi.org/10.1007/978-981-15-2180-5_9

385

386

9 Representations of CAR with Infinite Degrees of Freedom

Example 9.1 Let Ff (K ) be the fermion Fock space over a Hilbert space K (see Chap. 7) and B(u) be the fermion annihilation operator on Ff (K ) with test vector u ∈ K . Then, by Theorems 7.6 and 7.8, (Ff (K ), {B(u), B(u)∗ |u ∈ K })

(9.2)

is an irreducible representation of the CAR over K . This representation is called the Fock representation of the CAR over K . Example 9.2 Let S ∈ B(K ) such that S ∗ S = I , i.e., S is an isometry on K , and let BS (u) := B(Su),

u∈K.

Then it is easy to see that (Ff (K ), {BS (u), BS (u)∗ |u ∈ K }) is a representation of the CAR over K . There are two cases to be considered: (1) The case where S is unitary. In this case, we have S ∗ S = I,

SS ∗ = I.

By Theorem 7.10, BS (u) = Γf (S)B(u)Γf (S)−1 ,

u∈K.

Hence (Ff (K ), {BS (u), BS (u)∗ |u ∈ K }) is equivalent to the Fock representation (Ff (K ), {B(u), B(u)∗ |u ∈ K }). (2) The case where S is not unitary. In this case (Ran S)⊥ = {0}. Let v ∈ (Ran S)⊥ and v = 0. Then, by direct computations using the CAR for B(·)# , we have [BS (u), B(v)∗ B(v)] = Su, v B(v) = 0. Hence B(v)∗ B(v) ∈ {BS (u), BS (u)∗ |u ∈ K } and B(v)∗ B(v) is not a scalar operator. Therefore {BS (u), BS (u)∗ |u ∈ K } is reducible. Thus (Ff (K ), {BS (u), BS (u)∗ |u ∈ K }) is inequivalent to (Ff (K ), {B(u), B(u)∗ |u ∈ K }) (but this is a trivial inequivalence). To analyze the structure of the reducibility of {BS (u), BS (u)∗ |u ∈ K }, we note the orthogonal decomposition K = Ran S ⊕ ker S ∗ . Hence, by Theorem 7.14, Ff (K ) ∼ = Ff (Ran S) ⊗ Ff (ker S ∗ ).

9.2 Fermionic Bogoliubov Transformations

387

Under this isomorphism, BS (u)# is unitarily equivalent to B(Su)# ⊗ I . Let ∗ {Ψn }N n=1 (N ∈ N or N = ∞) be a CONS of Ff (ker S ) and Hn = {αΨn |α ∈ C}. Then Ff (ker S ∗ ) = ⊕N n=1 Hn . Hence ∼ N Ff (K ) ∼ = ⊕N n=1 Ff (Ran S) ⊗ Hn = ⊕n=1 Mn , where Mn := Ff (Ran S). Under this isomorphism, BS (u)# is reduced by Mn and (Mn , {BS (u), BS (u)∗ |u ∈ K }) gives an irreducible representation of the CAR over K . Since Γf (S) is a unitary operator from Ff (K ) to Ff (Ran S), we have Γf (S)B(u)# Γf (S)−1 = BS (u)# on Ff (Ran S). Hence, for each n, (Mn , {BS (u), BS (u)∗ |u ∈ K }) is equivalent to the Fock representation (Ff (K ), {B(u), B(u)∗ |u ∈ K }). Thus (Ff (K ), {BS (u), BS (u)∗ |u ∈ K }) is equivalent to a direct sum representation of the Fock representation of the CAR over K . Remark 9.2 A general theory of representations of CAR is given in [41].

9.2 Fermionic Bogoliubov Transformations For the fermion annihilation and creation operators too, one can consider analogues of Bogoliubov transformations of the boson annihilation and creation operators. Let C be a conjugation on K and S, T ∈ B(K ) satisfying S ∗ S + T ∗ T = I,

S ∗ TC + T ∗ SC = 0,

(9.3)

where, for A ∈ B(K ), AC ∈ B(K ) is the conjugation of A with respect to C (see (8.51)). Then one defines BS,T (u) := B(Su) + B(CT u)∗ ,

u∈K.

(9.4)

It is easy to see that (Ff (K ), {BS,T (u), BS,T (u)∗ |u ∈ K }) is a representation of the CAR over K . The correspondence (B(·), B(·)∗ ) → (BS,T (·), BS,T (·)∗ ) is called the fermionic Bogoliubov transformation associated with (S, T ). For the fermionic Bogoliubov transformation too, one has a result corresponding to Theorem 8.6 on a bosonic Bogoliubov transformation. To state and prove it, we need two lemmas. Lemma 9.1 Assume that S ∈ B(K ) and T ∈ B(K ) satisfy (9.3). Suppose that Ran S is dense in K . Then S is injective.

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9 Representations of CAR with Infinite Degrees of Freedom

Proof Let u ∈ ker S. Then, by (9.3), T ∗ T u = u,

S ∗ CT u = 0.

(9.5)

Since K = ker S ∗ ⊕ Ran S by the Hilbert space decomposition theorem (Theorem 1.2), it follows that ker S ∗ = {0}. Hence, by the second equation in (9.5), CT u = 0. Hence T u = 0. Then the first equation in (9.5) yields u = 0. Hence ker S = {0}, implying that S is injective.  Lemma 9.2 Assume that K is separable. Let D be a dense subspace of K and L ∈ L(K ) with D(L) ⊃ D. Suppose that there exists a non-zero vector Ω ∈ Ff (K ) such that B(CLu)∗ Ω = −B(u)Ω.

u ∈ D,

(9.6)

Then L is bounded and L is Hilbert–Schmidt. (n)

Proof For each n ≥ 0, we define a vector Ωn in the n-particle space by Ωn := ∞ (m) # Ω (n) and Ωn := 0, m = n. Then we have Ω = n=0 Ωn . Since B(·) is bounded, it follows that ∞

B(CLu)∗ Ωn = −

n=0

∞

B(u)Ωn ,

u ∈ D,

n=1

where we have used that B(u)Ω0 = 0. Hence B(u)Ω1 = 0, B(CLu)∗ Ωn−1 = −B(u)Ωn+1 ,

(9.7) n ≥ 1.

(9.8)

It follows from (9.7) and Theorem 7.7 that Ω1 = 0. Then, by (9.8) with n = 2, B(u)Ω3 = 0, u ∈ D. Hence, by Theorem 7.7 again, Ω3 = 0. By induction in n, one can show that Ω2n−1 = 0, n ≥ 1. Taking n = 1 in (9.8), one has B(CLu)∗ Ω0 = −B(u)Ω2 . If Ω0 = 0, then B(u)Ω2 = 0, u ∈ D. Hence, by Theorem 7.7, Ω2 = 0. Then, by induction, one has Ω2n = 0 for all n ∈ Z+ . Thus Ω = 0. But this is a contradiction. Therefore Ω0 = 0. Hence Ω0 = αΩK with some α ∈ C \ {0}. Then B(CLu)∗ ΩK = −α −1 B(u)Ω2 . Taking the norm of both sides, we obtain

Lu = |α|−1 B(u)Ω2 . Let {en }∞ n=1 be a CONS of K with en ∈ D, n ∈ N. Then ∞ n=1

Len 2 = |α|−2

∞ n=1

B(en )Ω2 2 = c0 := |α|−2 Nf

1/2

Ω2 2 < ∞,

9.2 Fermionic Bogoliubov Transformations

389

 2 where we have used (7.49). This implies that ∞ n=1 Len is independent of the √ ∞ choice of {en }n=1 and Len ≤ c0 , n ∈ N. Hence, for all Ψ ∈ D \ {0},

L( Ψ −1 Ψ ) ≤ Therefore LΨ ≤

√ c0 .

√ c0 Ψ . Thus L is bounded and L is Hilbert–Schmidt.



Theorem 9.1 Assume that K is separable. Suppose that Ran S is dense and T is not Hilbert–Schmidt. Then (Ff (K ), {BS,T (u), BS,T (u)∗ | u ∈ K }) is inequivalent to any direct sum representation of the Fock representation (Ff (K ), {B(u), B(u)∗ |u ∈ K }). Proof The method of proof is similar to that of Theorem 8.6. We prove the contraposition of the above statement. Suppose that (Ff (K ), {BS,T (u), BS,T (u)∗ | u ∈ K }) is equivalent to a direct sum representation N N ∗ (⊕N n=1 Ff (K ), {⊕n=1 B(u), ⊕n=1 B(u) |u ∈ K })

of the Fock representation (Ff (K ), {B(u), B(u)∗ |u ∈ K }), where N < ∞ or N = ∞. Then there exists a unitary operator U from Ff (K ) to ⊕N n=1 Ff (K ) such that U BS,T (u)U −1 = ⊕N B(u), u ∈ K . Let n=1 Ω := U −1 (ΩK , 0, 0, . . .). Then Ω = 1 (hence Ω = 0) and BS,T (u)Ω = 0, i.e., B(CT u)∗ Ω = −B(Su)Ω.

(9.9)

Since Ran S is dense, it follows from Lemma 9.1 that S is injective. Hence we have B(CLu)∗ Ω = −B(u)Ω,

u ∈ Ran S,

where L := T S −1 . Then, by Lemma 9.2, L is Hilbert–Schmidt and hence T = LS also is Hilbert–Schmidt.  With regard to irreducibility of {BS,T (u), BS,T (u)∗ |u ∈ K }, the following theorem holds. Theorem 9.2 Assume (9.3) and SS ∗ + TC (TC )∗ = I,

ST ∗ + TC (SC )∗ = 0.

(9.10)

Then (Ff (K ), {BS,T (u), BS,T (u)∗ |u ∈ K }) is an irreducible representation of the CAR over K .

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9 Representations of CAR with Infinite Degrees of Freedom

Proof Using (9.10), one can express B(u) in terms of BS,T (·)# as follows: B(u) = BS,T (S ∗ u) + BS,T (T ∗ Cu)∗ ,

u∈K.

(9.11)

Hence, for all A ∈ {BS,T (u), BS,T (u)∗ |u ∈ K } , A ∈ {B(u), B(u)∗ |u ∈ K } = CI (Example 9.1). Therefore {BS,T (u), BS,T (u)∗ |u ∈ K } = CI . Thus the desired result follows.  The following theorem is a standard result in the theory of fermionic Bogoliubov transformations. Theorem 9.3 Let K be separable. Suppose that (9.3) and (9.10) hold. Then (Ff (K ), {BS,T (u), BS,T (u)∗ |u ∈ K }) is equivalent to the Fock representation (Ff (K ), {B(u), B(u)∗ |u ∈ K }) if and only if T is Hilbert–Schmidt. For a proof of this fact, see, e.g, [137], [123, §2.4] or [156, §10.3]. Cf. also [38] for related aspects.

9.3 A Class of Representations of CAR We now fix an orthogonal decomposition K = K+ ⊕ K− = {f = (f+ , f− )|f+ ∈ K+ , f− ∈ K− } of K with K+ and K− being mutually orthogonal non-trivial closed subspaces (K+ = {0}, K ). We sometimes identify f+ ∈ K+ (resp. f− ∈ K− ) with (f+ , 0) (resp. (0, f− )) so that f is written f = f+ + f− . Let C be a conjugation on K . We denote by B± := B(K , K± ) the Banach space of everywhere defined bounded linear operators from K to K± and introduce a subset of the direct product space B+ × B− : T(K ) := {T = (T+ , T− ) | T± ∈ B± , T+∗ T+ + (T−∗ T− )C = I }, where (T−∗ T− )C := CT−∗ T− C. Let P± be the orthogonal projections onto K± . We assume that P± C = CP± . Then CK± = K± .

(9.12)

9.3 A Class of Representations of CAR

391

Each T ∈ T(K ) defines an element of B(K ) by Tf := (T+ f, T− f ),

f ∈K.

(9.13)

For each T ∈ T(K ), we define an anti-linear mapping ψT : K → B(Ff (K )) by ψT (f ) := B(T+ f, 0) + B(0, T− Cf )∗ ,

f ∈K.

(9.14)

It is obvious that ψT (f )∗ = B(T+ f, 0)∗ + B(0, T− Cf ). Let AT (K ) := {ψT (f ), ψT (f )∗ |f ∈ K }.

(9.15)

Lemma 9.3 For all f, g ∈ K , the following anti-commutation relations hold: {ψT (f ), ψT (g)} = {ψT (f )∗ , ψT (g)∗ } = 0, ∗

{ψT (f ), ψT (g) } = f, g .

(9.16) (9.17)

Proof By direct computations using (9.1), we have   {ψT (f ), ψT (g)∗ } = T+ f, T+ g + T− Cg, T− Cf  = f, T+∗ T+ g + Cg, T−∗ T− f  = f, T+∗ T+ g + (T−∗ T− )C g = f, g . Hence we obtain (9.17). Similarly, one can prove (9.16).



Lemma 9.3 shows that (Ff (K ), AT (K )) is a representation of the CAR over K . Remark 9.3 The standard choice of T = (T+ , T− ) in the literature is given by T+ = P and T− = C(I − P )C, where P is the orthogonal projection onto K+ (see, e.g., [123, pp. 22–23] and [156, §10.1.3]). In this case, the representation is called a quasi-free representation. Hence the representation (Ff (K ), AT (K )) gives a generalization of the quasi-free representation. To find more detailed properties of the representation (Ff (K ), AT (K )), we introduce the following additional conditions for T = (T+ , T− ) ∈ T(K ): (T.1) (T.2) (T.3)

T+ T+∗ = I , T− T−∗ = I , T− CT+∗ = 0.

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9 Representations of CAR with Infinite Degrees of Freedom

Multiplying (T.3) by C from the right and taking the adjoint of the resulting equation, one sees that (T.3) is equivalent to (T.4)

T+ CT−∗ = 0.

We define a subset of T(K ): T∗ (K ) := {T ∈ T(K )|(T.1)–(T.3) hold}.

(9.18)

Lemma 9.4 Let T ∈ T∗ (K ). Then B(f ) = ψT (T+∗ f+ ) + ψT (CT−∗ f− )∗ ,

f = (f+ , f− ), f± ∈ K± .

(9.19)

Proof By (T.1) and (T.3), we have ψT (T+∗ f+ ) = B(f+ , 0),

f+ ∈ K+ .

Property (T.3) implies that T+ CT−∗ = 0. By this fact and (T.2), we obtain ψT (CT−∗ f− ) = B(0, f− )∗ ,

f− ∈ K− .

We have B(f ) = B(f+ , 0) + B(0, f− ). Thus we obtain (9.19).



Lemma 9.5 For all T ∈ T∗ (K ), (Ff (K ), AT (K )) is irreducible. Proof Let A ∈ {ψT (f ), ψT (f )∗ |K } . Then AψT (f ) = ψT (f )A and AψT (f )∗ = ψT (f )∗ A. Hence, by (9.19), AB(f ) = B(f )A and AB(f )∗ = B(f )∗ A for all f ∈ K . Therefore A ∈ {B(f ), B(f )∗ |f ∈ K } . As we have seen in Example 9.1, {B(f ), B(f )∗ |f ∈ K } = CI . Hence A = αI for some α ∈ C. Thus AT (K ) is irreducible.  Lemmas 9.3 and 9.5 immediately yield the following theorem: Theorem 9.4 For each T ∈ T∗ (K ), (Ff (K ), AT (K )) is an irreducible representation of the CAR over K . Thus we have a family {(Ff (K ), AT (K )}T ∈T∗ (K ) of irreducible representations of the CAR over K . We next consider equivalence or inequivalence of two representations (Ff (K ), AT (K )) and (Ff (K ), AS (K )) with S = T (S, T ∈ T∗ (K )). For each pair (S, T ) ∈ T∗ (K ) × T∗ (K ), we define linear operators V and W on K as follows: Vf := (S+ T+∗ f+ , S− T−∗ f− ), Wf := (CS+ CT−∗ f− , CS− CT+∗ f+ ),

(9.20) f = (f+ , f− ) ∈ K .

(9.21)

9.3 A Class of Representations of CAR

393

Let VC := CV C,

WC := CW C.

Lemma 9.6 The following equations hold: V ∗ V + W ∗ W = I, ∗

∗

(9.22)

∗

V WC + W (VC ) = 0,

(9.23)

V V ∗ + WC WC∗ = I,

(9.24)

VW

∗

+ WC VC∗

= 0.

(9.25)

Proof The operators V and W have the following operator matrix representations: 0 V =

S+ T+∗ 0 0 S− T−∗

1

0 ,

W =

0 CS+ CT−∗ CS− CT+∗ 0

1 .

Using these representations and properties of S and T , one can easily prove (9.22)– (9.25) by direct computations.  We define C(f ) := B(Vf ) + B(CWf )∗ ,

f ∈K.

(9.26)

Then it is easy to see that {C(f ), C(f )∗ |f ∈ K } is a representation of the CAR over K . Lemma 9.7 There exists a unitary operator U on Ff (K ) such that B(f ) = UC(f )U∗ ,

f ∈K,

(9.27)

if and only if W is Hilbert–Schmidt. Proof Since we have (9.22)–(9.25), the lemma follows from an application of Theorem 9.3 to the case S = V and T = W .  Now we are ready to state and prove the main theorem in this section: Theorem 9.5 Let T and S be in T∗ (K ) with T = S. Then the two representations (Ff (K ), AT (K )) and (Ff (K ), AS (K )) are equivalent if and only if CS+ CT−∗ and CS− CT+∗ are Hilbert–Schmidt. Proof Suppose that (Ff (K ), AT (K )) and (Ff (K ), AS (K )) are equivalent. Then there exists a unitary operator U on Ff (K ) such that UψS (f )U∗ = ψT (f ),

f ∈K.

(9.28)

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9 Representations of CAR with Infinite Degrees of Freedom

Then, by (T.1)–(T.3), we have 5 4 B(f+ , 0) = U B(S+ T+∗ f+ , 0) + B(0, S− CT+∗ f+ )∗ U∗ ,

f+ ∈ K+ .

(9.29)

f− ∈ K− .

(9.30)

and 4 5 B(0, f− ) = U B(S+ CT−∗ f− , 0)∗ + B(0, S− T−∗ f− ) U∗ ,

Hence, adding the first equation to the second one, we obtain (9.27). Therefore, by Lemma 9.7, W is Hilbert–Schmidt, which is equivalent to that CS+ CT−∗ and CS− CT+∗ are Hilbert–Schmidt. Conversely, suppose that CS+ CT−∗ and CS− CT+∗ are Hilbert–Schmidt. Then W is Hilbert–Schmidt. Hence, by Lemma 9.7, there exists a unitary operator U on Ff (K ) such that (9.27) holds. Then (9.29) and (9.30) hold. Using the property that T+∗ T+ + C(T−∗ T− )C = I , we obtain (9.28).  The contraposition of Theorem 9.5 gives an inequivalence theorem on the two representations (Ff (K ), AT (K )) and (Ff (K ), AS (K )): Theorem 9.6 Let T and S be in T∗ (K ) with T = S. Then the two representations (Ff (K ), AT (K )) and (Ff (K ), AS (K )) are inequivalent if and only if CS+ CT−∗ or CS− CT+∗ is not Hilbert–Schmidt.

Chapter 10

Physical Correspondences in Quantum Field Theory

Abstract It is shown that there are correspondences between characteristic physical situations or phenomena in quantum field theory and inequivalent representations of CCR or CAR.

10.1 Quantum Field Models in Hamiltonian Formalism In what follows, we use the physical unit system where the reduced Planck constant h¯ and the speed of light c are equal to 1. We first give the definition of abstract quantum field models in Hamiltonian formalism.1 Definition 10.1 (Bose Field Models) (1) An abstract relativistic Bose field model is defined to be a triple (F , H , {φ(f ), π(f )|f ∈ W }) consisting of a Hilbert space F , a self-adjoint operator H on F and an irreducible representation (F , D, {φ(f ), π(f )|f ∈ W }) of the Heisenberg CCR over a real inner product space W such that, for all f ∈ W , φ(f ) and π(f ) are closed symmetric operators on F and {φ(f ), π(f )|f ∈ W } = CI .2 The operator-valued functionals f → φ(f ) and f → π(f ) are called the time-zero fields of the model. (2) An abstract non-relativistic Bose field model is defined to be a triple (F , H , {C(f )|f ∈ V }) consisting of a Hilbert space F , a self-adjoint operator H on F and an irreducible representation (F , D, {C(f ), C(f )∗ |f ∈ V }) of the CCR over a complex inner product space V . The operator-valued functional f → C(f ) is called the time-zero field of the model. 1 There

are other ways to define quantum field theories axiomatically. But, in the present book, we do not go into the details (see, e.g., [28, 39, 45, 149, 149, 152]). 2 If φ(f ) or π(f ) is not self-adjoint, then {φ(f ), π(f )|f ∈ W } is not ∗-invariant. For this reason, we assume this stronger irreducible condition. One may replace the condition with that {φ(f ), π(f ), φ(f )∗ , π(f )∗ |f ∈ W }, which is ∗-invariant, is irreducible. © Springer Nature Singapore Pte Ltd. 2020 A. Arai, Inequivalent Representations of Canonical Commutation and Anti-Commutation Relations, Mathematical Physics Studies, https://doi.org/10.1007/978-981-15-2180-5_10

395

396

10 Physical Correspondences in Quantum Field Theory

In each model, the operator H is called the Hamiltonian of the model. In an abstract relativistic Bose field model (F , H , {φ(f ), π(f )|f ∈ W }), the time-t fields φ(t, ·) and π(t, ·) are defined by φ(t, f ) := eit H φ(f )e−it H ,

π(t, f ) := eit H π(f )e−it H ,

t ∈ R, f ∈ W , (10.1)

while, in an abstract non-relativistic Bose field model (F , H , {C(f )|f ∈ V }), the time-t field is defined by C(t, f ) := eit H C(f )e−it H ,

f ∈V.

(10.2)

Remark 10.1 The words “relativistic” and “non-relativistic” here should be understood only as symbolical ones to distinguish two types of Bose field models (only the difference of types of time-zero field(s) is taken into account here). More differences of the two models can be described by taking into account relativistic symmetries and non-relativistic ones as unitary (or projective) representations on F (for more details, see [28, Chapters 8 and 9]). In what follows, we may sometimes omit the word “relativistic” or “non-relativistic”. The irreducibility of the time-zero fields of a Bose field model implies the irreducibility of the time-t fields of it: Proposition 10.1 Let (F , H , {φ(f ), π(f )|f ∈ W }) and (F , H , {C(f )|f ∈ V }) be as in Definition 10.1. Then: (i) For all t ∈ R, {φ(t, f ), π(t, f )|f ∈ W } is irreducible with {φ(t, f ), π(t, f )|f ∈ W } = CI . (ii) For all t ∈ R, {C(t, f ), C(t, f )∗ |f ∈ V } is irreducible. Proof Since {φ(t, f ), π(t, f )|f ∈ W } is unitarily equivalent to {φ(f ), π(f )|f ∈ W } by the unitary operator eit H and the latter is irreducible with {φ(f ), π(f )|f ∈ W } = CI , it follows from an application of Proposition 2.8 that {φ(t, f ), π(t, f )| f ∈ W } is irreducible with {φ(t, f ), π(t, f )|f ∈ W } = CI . Similarly, one can prove the irreducibility of {C(t, f ), C(t, f )∗ |f ∈ V }.  Definition 10.2 (1) An abstract relativistic Bose field model (F , H, {φ(f ), π(f )|f ∈ W }) is said to be equivalent to an abstract relativistic Bose field model (F , H , {φ(f ) , π(f ) |f ∈ W }) if there exist a unitary operator U : F → F and a real constant E such that U H U −1 = H + E,

U φ(f )U −1 = φ(f ) ,

U π(f )U −1 = π(f ) ,

f ∈W.

(10.3) (2) An abstract non-relativistic Bose field model (F , H, {C(f )|f ∈ V }) is said to be equivalent to an abstract non-relativistic Bose field model (F , H , {C(f )

10.1 Quantum Field Models in Hamiltonian Formalism

397

|f ∈ V }) if there exist a unitary operator U : F → F and a real constant E such that U H U −1 = H + E,

U C(f )U −1 = C(f ) ,

f ∈V.

Remark 10.2 One can define a relation (F , H, {φ(f ), π(f )|f ∈ W }) ∼ (F , H , {φ(f ) , π(f ) |f ∈ W }) if there exist a unitary operator U : F → F and a real constant E satisfying (10.3). Then it is easy to see that this relation ∼ is an equivalence relation. Therefore the above definition on equivalence between two Bose field models is well-defined. The same applies to abstract non-relativistic Bose field models. Remark 10.3 A detailed analysis of Hamiltonian formalism using Weyl representations of CCR is given by Araki [36]. Remark 10.4 A concrete relativistic (resp. non-relativistic) Bose field model is defined by giving algebraic relations between H and {φ(f ), π(f )|f ∈ W } (resp. {C(f )|f ∈ V }), which include relations implying field equations for time-t fields. A Hilbert space operator realization of the algebraic relations yields a representation of the model. Therefore, for a model, there may exist inequivalent representations of it. Remark 10.5 Only a unitary equivalence between H and H + E does not imply the equivalence of the models. One can define also an abstract Fermi field model: Definition 10.3 (Fermi Field Model) An abstract Fermi field model is defined to be a triple (F , H , {ψ(f )|f ∈ H }) consisting of a Hilbert space F , a self-adjoint operator H on F and an irreducible representation (F , D, {ψ(f ), ψ(f )∗ |f ∈ H }) of the CAR over a complex Hilbert space H . The operator-valued functional f → ψ(f ) is called the time-zero field of the model. The time t-fields of an abstract Fermi field model (F , H , {ψ(f )|f ∈ H }) are defined by {ψ(t, f ), ψ(t, f )∗ |f ∈ H } with ψ(t, f ) := eit H ψ(f )e−it H ,

f ∈H.

(10.4)

As in the case of Bose field models, the time-t fields of an abstract Fermi field model also are irreducible: Proposition 10.2 Let (F , H , {ψ(f )|f ∈ H }) be an abstract Fermi field model. Then, for all t ∈ R, {ψ(t, f ), ψ(t, f )∗ |f ∈ H } is irreducible. Proof Similar to the proof of Proposition 10.1.



The notion of equivalence between two abstract Fermi field models is defined in the same way as in the case of abstract Bose field models.

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10 Physical Correspondences in Quantum Field Theory

Example 10.1 Let H be a Hilbert space and Fb (H ) be the boson Fock space over H . Let h be a self-adjoint operator on H and db (h) be the boson second quantization operator of h. Then (Fb (H ), db (h), {A(f )|f ∈ H }) is an abstract non-relativistic Bose field model with db (h) being the Hamiltonian of the model. The time-t field is given by A(t, f ) = eit db(h) A(f )e−it db (h) = A(eit h f ), where we have used Corollary 6.2. Using Corollary 6.8(ii), one can show that the operator A(t, f ) obeys the field equation i

d A(t, f )Ψ = A(t, hf )Ψ, dt

1/2

f ∈ D(h), Ψ ∈ D(Nb ),

where the time derivative is taken in the sense of strong derivative. This is an abstract free de Broglie equation [28, Chapter 9].3 Example 10.2 Let T be a non-negative and injective self-adjoint operator on a Hilbert space H such that T 1/2 ∈ SW (H ) (see Sect. 8.12). Let Hb := db (T ) and φT 1/2 (f ) and πT 1/2 (f ) be defined by (8.118) and (8.119) with T replaced by T 1/2 respectively. Then M(T , W ) := (Fb (H ), Hb , {φT 1/2 (f ), πT 1/2 (f )|f ∈ W }) is an abstract relativistic Bose field model. This type of abstract quantum field model is called a free relativistic Bose field model. 3/2 By Theorem 6.16(ii), the commutation relations hold on D(Hb ): [Hb, φT 1/2 (f )] = −iπT 1/2 (f ), [Hb, πT 1/2 (f )] = iφT 1/2 (T 2 f ),

(10.5) f ∈ D(T 2 ) ∩ W .

(10.6)

In view of Remark 10.4, the model M(T , W ) can be regarded as an operator realization of the following algebraic relations: [φ(f ), π(g)] = i f, g , [φ(f ), φ(g)] = 0, [π(f ), π(g)] = 0, [H, φ(f )] = −iπ(f ), [H, π(f )] = iφ(T 2 f ),

f ∈W, f ∈ D(T 2 ) ∩ W

f, g ∈ W , (10.7) (10.8) (10.9)

3 The free bosonic quantum de Broglie field theory described in [28, Chapter 9] is a concrete realization of the present abstract model.

10.1 Quantum Field Models in Hamiltonian Formalism

399

with correspondence (H, φ(f ), π(f )) → (Hb , φT 1/2 (f ), πT 1/2 (f )). For a concrete realization of the model M(T , W ), see Sect. 10.7 below. Let T1 and T2 be non-negative injective self-adjoint operators on H such that (i) 1/2 1/2 −1/2 1/2 1/2 −1/2 T1 , T2 ∈ SW (H ); (ii) T2 T1 and T2 T1 are bounded. Suppose that −1/2

A21 := T2

1/2

T1

1/2

−1/2

− T2 T1

(10.10)

is not Hilbert–Schmidt. Then, by Theorem 8.18, M(T1 , W ) is not equivalent to M(T2 , W ). As a special case, we consider the case where T1 := T 2 and T2 = α 2 T 2 with T ∈ SW (H ) (α > 0, α = 1). Then A21 =

1 − α = 0. α

Suppose that H is infinite dimensional. Then A21 is not Hilbert–Schmidt. Hence, by Theorem 8.18, M(T 2 , W ) is not equivalent to M(α 2 T 2 , W ). If σ (T 2 ) = [0, ∞), then σ (α 2 T 2 ) = [0, ∞). Then we have by Theorem 6.1(iv) σ (db (T 2 )) = σ (db (α 2 T 2 )) = [0, ∞). Hence the spectrum of the Hamiltonian db (T2 ) is equal to that of db (α 2 T 2 ). Therefore we have an example of inequivalent abstract free relativistic Bose field models in which the spectra of Hamiltonians coincide. Example 10.3 Let K be a Hilbert space and Ff (K ) be the fermion Fock space over K . We denote by B(u) the fermion annihilation operator with test vector u ∈ K on Ff (K ). Let h be a self-adjoint operator on K and df (h) be the fermion second quantization operator of h. Then (Ff (K ), df (h), {B(u)|u ∈ K }) is an abstract Fermi field model. The time-t field is given by B(t, u) = eit df(h) B(f )e−idf(h) = B(eit h u), where we have used Theorem 7.11. By Theorem 7.13, B(t, u) obeys the field equation i

d B(t, u) = B(t, hu), dt

u ∈ D(h),

where the time derivative is taken in the sense of uniform derivative. The present model is an abstract form of the free fermionic quantum de Broglie field theory [28, Chapter 9, §9.8].

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10 Physical Correspondences in Quantum Field Theory

Equivalence of quantum field models can be rephrased in terms of time-t fields: Proposition 10.3 Let M = (F , H, {φ(f ), π(f )|f ∈ W }) and M = (F , H , {φ(f ) , π(f ) |f ∈ W }) be abstract relativistic Bose field models. Then M is equivalent to M if and only if there exists a unitary operator U : F → F such that, for all t ∈ R and f ∈ W , U φ(t, f )U −1 = φ (t, f ),

U π(t, f )U −1 = π (t, f ),

(10.11)

where φ (t, f ) and π (t, f ) are the time-zero fields of the model M . Proof Suppose that M is equivalent to M . Then there exist a unitary operator U : F → F and a real constant E satisfying (10.3). Then we have for all t ∈ R and f ∈W −1

U φ(t, f )U −1 = U eit H U −1 U φ(f )U −1 U e−it H U −1 = eit U H U φ (f )e−it U H U = eit (H

+E)

φ (f )e−it (H

+E)



= eit H φ (f )e−it H

−1



= φ (t, f ). Similarly, one can show that U π(t, f )U −1 = π (t, f ). Conversely, suppose that there exists a unitary operator U : F → F such that, for all t ∈ R and f ∈ W , (10.11) holds. Then the case t = 0 gives U φ(f )U −1 = φ (f ),

U π(f )U −1 = π (f ).

Using the first of these relations, we have







U eit H φ(f )e−it H U −1 = eit H φ (f )e−it H = eit H U φ(f )U −1 eit H . This implies that V (t)φ(f )V (t)−1 = φ(f ), where

V (t) := U −1 e−it H U eit H . Similarly, we have V (t)π(f )V (t)−1 = π(f ). Hence V (t) ∈ {φ(f ), π(f )|f ∈ W } = CI . Therefore V (t) = c(t)I for some c(t) ∈ C. It follows that c(t)eit H = −1 eit U H U . It is easy to see that V (t) is a strongly continuous semi-group. Hence it −1 follows that c(t) = eit E for some E ∈ R. Therefore eit (H +E) = eit U H U . This implies that H + E = U H U −1 . Thus M is equivalent to M .  Remark 10.6 Proposition 10.3 holds also for abstract non-relativistic Bose field models and abstract Fermi field models. Remark 10.7 In this book, we do not present a general theory of quantum fields. For this topic, see, e.g., [28, 39, 45, 62, 63, 67, 85, 147, 149, 152].

10.2 Scale Transformations of Time-Zero Fields

401

Remark 10.8 In what follows, we shall consider models whose time-zero fields depend on parameters. In such cases, we are interested in investigating equivalence or inequivalence between time-zero fields of different parameters. In relation to this subject, the following is an elementary fact that one should keep in mind: if timezero fields of different parameters are mutually inequivalent, then the time-t fields of different parameters also are mutually inequivalent. This easily follows from the fact that time-t fields are unitarily equivalent to time-zero fields (see (10.1)–(10.4)). We shall not mention this structure in any of the quantum field models which are discussed below.

10.2 Scale Transformations of Time-Zero Fields Let φT (f ) and πT (f ) be defined by (8.118) and (8.119) respectively with T ∈ SW (H ) and, for λ > 0, define φT ,λ (f ) and πT ,λ (f ) by φT ,λ (f ) := λφT (f ),

πT ,λ (f ) := λ−1 πT (f ),

f ∈W.

(10.12)

The correspondence : λ → φT ,λ (f ) is called the scale transformation of φT (f ) with scaling parameter λ. Note that φT ,λ (f ) = φT /λ (f ),

πT ,λ (f ) = πT /λ (f ).

(10.13)

Hence, by Theorem 8.15, ΠT ,λ (W ) := (Fb (H ), {φT ,λ (f ), πT ,λ (f )|f ∈ W })

(10.14)

is an irreducible Weyl representation of the CCR over W and ΠT ,λ (W ) = ΠT /λ (W ).

(10.15)

Theorem 10.1 Assume that H is separable and infinite dimensional. Let λ1 = λ2 (λ1 , λ2 > 0). Then ΠT ,λ1 (W ) is inequivalent to ΠT ,λ2 (W ). Proof We apply Theorem 8.18 with the following case: T1 =

T , λ1

T2 =

T . λ2

Then it is not so difficult to show that the assumption of Theorem 8.18 is satisfied. We have S := T2−1 T1 − T2 T1−1 =

λ2 λ1 − . λ1 λ2

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10 Physical Correspondences in Quantum Field Theory

Since λ1 = λ2 , S is a non-zero scalar operator. By the present assumption, H is infinite dimensional. Hence S is not Hilbert–Schmidt. Thus, by Theorem 8.18, ΠT ,λ1 (W ) ∼  ΠT ,λ2 (W ).  = We emphasize that, in Theorem 10.1, the infinite dimensionality of H is essential. Let H be infinite dimensional. Then Theorem 10.1 shows that {ΠT ,λ (W )}λ>0 is a family of mutually inequivalent irreducible Weyl representations of the CCR over W . This gives a representation theoretic meaning to scale parameters λ as indices of a family of mutually inequivalent irreducible Weyl representations of CCR.

10.3 Bogoliubov Translations in Finite and Infinite Volume Theories In a theory of Bose gas, a class of Bogoliubov translations (see Example 8.3) appears.4 In this section, we show that, in view of representations of CCR, there is an essential difference between the Bogoliubov translations in a finite volume theory and an infinite volume one.

10.3.1 Finite Volume Case Let M be a bounded open set in the position space Rd (d ∈ N) and Fb (L2 (M)) be the boson Fock space over L2 (M). We denote by aM (·) the annihilation operator on Fb (L2 (M)). We consider a system of Bose gas in M whose particle density is given by a constant ρ > 0. For each f ∈ L2 (M), one has by the Schwarz inequality  |f (x)|dx ≤ f L2 (M) |M|1/2 < ∞, M

where |M| denotes the Lebesgue measure of M. Hence the operator  √ bM (f ) := aM (f ) + ρ f (x)∗ dx, f ∈ L2 (M).

(10.16)

M

is well-defined. This is a Bogoliubov translation whose translation functional is given by  √ LM (f ) := ρ f (x)dx, f ∈ L2 (M). M

4 Here we do not present a systematic description of a theory of Bose gas; we restrict our considerations to the form and properties of CCR only. For more details, see [40, 70]. For physical discussions on the subject, see, e.g., [109, 153, 154].

10.3 Bogoliubov Translations in Finite and Infinite Volume Theories

403

By Example 8.3, ρM := (Fb (L2 (M)), Fb,0 (L2 (M)), {bM (f )|f ∈ L2 (M)}) is an irreducible representation of the CCR over L2 (M). Theorem 10.2 The representation ρM is equivalent to the Fock representation ρF (L2 (M)) = (Fb (L2 (M)), Fb,0 (L2 (M)), {aM (f )|f ∈ L2 (M)}). √ Proof Since M is bounded, ρ ∈ L2 (M). Hence one can rewrite LM (f ) =

√ ρ, f L2 (M) ,

f ∈ L2 (M).

Hence, by Theorem 8.5, bM (f ) is unitarily equivalent to aM (f ).



10.3.2 Infinite Volume Case We next consider the case where a Bose gas is placed in an unbounded open set X ⊂ Rd with |X| = ∞. In this case, the Hilbert space of state vectors of the system under consideration is taken to be the boson Fock space Fb (L2 (X)) over L2 (X). We denote by aX (·) the annihilation operator on Fb (L2 (X)). We assume that the particle density ρ > 0 is a constant. For each f ∈ L1 (X) ∩ L2 (X), one can define an operator bX,ρ (f ) by  √ f (x)∗ dx. bX,ρ (f ) := aX (f ) + ρ X

This is a Bogoliubov translation. Since L1 (X) ∩ L2 (X) is dense in L2 (X), it follows from Example 8.3, (Fb (L2 (X)), Fb,0 (L2 (X)), {bX,ρ (f )|f ∈ L1 (X) ∩ L2 (X)}) is an irreducible representation of the CCR over L1 (X) ∩ L2 (X). Theorem 10.3 The representation (Fb (L2 (X)), Fb,0 (L2 (X)), {bX,ρ (f )|f ∈ L1 (X) ∩ L2 (X)}) is inequivalent to the Fock representation (Fb (L2 (X)), Fb,0 (L2 (X)), {aX (f )|f ∈ L1 (X) ∩ L2 (X)}). √ Proof In the present case, ρ ∈ L2 (X). Hence it follows from the fundamental lemma of the calculus of variations (du Bois–Reymond lemma) that there is no vector h ∈ L2 (X) such that  √ ρ f (x)∗ dx = f, hL2 (X) , f ∈ L1 (X) ∩ L2 (X). X

404

10 Physical Correspondences in Quantum Field Theory



Therefore, by Theorem 8.5, bX,ρ (f ) is not unitarily equivalent to aX (f ).

Physically the inequivalence in Theorem 10.3 is related to the occurrence of a Bose–Einstein condensation (BEC).5 Corollary 10.1 Let ρ1 > 0, ρ2 > 0 and ρ1 = ρ2 . Then {bX,ρ1 (f )|f ∈ L1 (X) ∩ L2 (X)} is inequivalent to {bX,ρ2 (f )|f ∈ L1 (X) ∩ L2 (X)}. Proof Suppose that {bX,ρ1 (f )|f ∈ L1 (X) ∩ L2 (X)} were equivalent to {bX,ρ2 (f )|f ∈ L1 (X) ∩ L2 (X)}. Then there exists a unitary operator U on Fb (L2 (X)) such that bX,ρ2 (f ) = U bX,ρ1 (f )U −1 , f ∈ L1 (X) ∩ L2 (X). Without loss of generality, one can assume that ρ1 > ρ2 . Then the equation √ √ implies that aX (f ) = U bX,ρ (f )U −1 with ρ = ( ρ1 − ρ2 )2 > 0. But this 1 2 contradicts Theorem 10.3. Thus {bX,ρ1 (f )|f ∈ L (X) ∩ L (X)} is not equivalent to {bX,ρ2 (f )|f ∈ L1 (X) ∩ L2 (X)}.  Thus we have a family {{bX,ρ (f )|f ∈ L1 (X) ∩ L2 (X)}|ρ > 0} of mutually inequivalent irreducible representations of the CCR over L1 (X)∩L2 (X). This shows that the particle densities in an infinite volume system have a meaning as the labels of a family of mutually inequivalent irreducible representations of CCR.

10.4 Translations of Heisenberg CCR and BEC We consider the case where X = Rd in the notation in Sect. 10.3.2 and define 1 φˆ F (f ) := √ (aRd (f )∗ + aRd (f )), 2 i πˆ F (f ) := √ (aRd (f )∗ − aRd (f )), 2

(10.17) f ∈ L2R (Rd ),

(10.18)

where L2R (Rd ) denotes the set of real elements in L2 (Rd ): L2R (Rd ) := {f ∈ L2 (Rd )|f ∗ = f }.

(10.19)

Then, for any dense subspace W of L2R (Rd ), πF (W ) := (Fb (L2 (Rd )), Fb,0 (L2 (Rd )), {φˆ F (f ), πˆ F (f )|f ∈ W }) (d)

(10.20)

is an example of Fock representations of Heisenberg CCR (see Example 8.10). Hence it is an irreducible Weyl representation of the CCR over W . In a theory of free

5 For

physical aspects of this phenomenon, see, e.g., [71, 99, 159].

10.4 Translations of Heisenberg CCR and BEC

405

Bose gas in Rd with a constant particle density ρ > 0, a translation of Heisenberg CCR of the following type appears [40, 70]: φˆ θ (f ) := φˆ F (f ) + πˆ θ (f ) := πˆ F (f ) +



 2ρ(cos θ )



 2ρ(sin θ )

Rd

Rd

f (x)dx,

(10.21)

f (x)dx,

(10.22)

where θ ∈ R and f ∈ L1 (Rd ) ∩ L2R (Rd ). We denote by SR (Rd ) the space of real elements in S (Rd ), the space of rapidly decreasing C ∞ -functions on Rd : SR (Rd ) := {f ∈ S (Rd )|f ∗ = f }

(10.23)

By Proposition 8.9, πθ (SR (Rd )) := (Fb (L2 (Rd )), {φˆ θ (f ), πˆ θ (f )|f ∈ SR (Rd )})

(10.24)

is an irreducible Weyl representation of the CCR over SR (Rd ). By the periodicity of cos θ and sin θ , we can assume that θ ∈ (0, 2π] without loss of generality. We first note the following fact: (d)

Theorem 10.4 For all θ ∈ [0, 2π), πθ (SR (Rd )) is inequivalent to πF (SR (Rd )). Proof The representation πθ (SR (Rd )) is the representation πμ,ν (W ) given by (8.108) with the following choice of (H , , W , μ, ν): H = L2 (Rd ),

μ(f ) =

√

W = SR (Rd ),



2(cos θ )

Rd

f (x)dx,

ν(f ) =

 √ 2(sin θ )

Rd

f (x)dx,

f ∈ SR (Rd ).

Hence, in the present case, we have 1 L(f ) = √ (μ(f ) + iν(f )) = eiθ 2

 Rd

f (x)dx.

It is easy to see that there is no vector h ∈ L2 (Rd ) such that, for all f ∈ S (Rd ), L(f ) = h, f L2 (Rd ) . Hence, by Theorem 8.12, the representation πθ (SR (Rd )) is inequivalent to πF(d) (SR (Rd )).



We next consider the family {πθ (SR (Rd ))}θ∈[0,2π), Theorem 10.5 Let θ1 , θ2 ∈ [0, 2π) and θ1 = θ2 . Then πθ1 (SR (Rd )) is inequivalent to πθ2 (SR (Rd )).

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10 Physical Correspondences in Quantum Field Theory

Proof Suppose that πθ1 (SR (Rd )) were equivalent to πθ2 (SR (Rd )). Then there exists a unitary operator U on Fb (L2 (Rd )) such that U φˆ θ1 (f )U −1 = φˆ θ2 (f ),

U πˆ θ1 (f )U −1 = πˆ θ2 (f ),

f ∈ SR (Rd ).

Hence U φˆ F (f )U −1 = φˆ F (f ) + μ12 (f ),

(10.25)

U πˆ F (f )U −1 = πˆ F (f ) + ν12 (f ).

(10.26)

where  √ μ12 (f ) := 2(cos θ2 − cos θ1 ) ν12 (f ) :=

 √ 2(sin θ2 − sin θ1 )

Rd

Rd

f (x)dx,

f (x)dx.

We have 1 L12 (f ) := √ (μ12 (f ) + iν12 (f )) = (eiθ2 − eiθ1 ) 2

 Rd

f (x)dx.

Since θ1 , θ2 ∈ [0, 2π) and θ1 = θ2 , it follows that eiθ2 − eiθ1 = 0. Hence there is no vector h ∈ L2 (Rd ) such that, for all f ∈ S (Rd ), L12 (f ) = h, f L2 (Rd ) . Hence, by Theorem 8.12, (10.25) and (10.26) do not hold. Thus we arrive at a contradiction.  By Theorem 10.5, {πθ (SR (Rd ))}θ∈[0,2π) is a family of inequivalent irreducible Weyl representations of the CCR over SR (Rd ). This family is related to the occurrence of a BEC with a spontaneous symmetry breaking (see [40, 70], [150, Part II, Chapter 13] or [18, Chapter 10] for details).

10.5 Improper Bogoliubov Transformation and Renormalization In this section, we reconsider the representation ρS,T (H ) of the CCR over H defined by (8.63). We take over the notation in Sects. 8.6 and 8.7. As we have seen in Sect. 8.7, a self-adjoint operator H qd associated with (S, T , h) is defined under Assumption (B) (see (8.83)). Hence, if h is injective in addition, then we have a Bose field model Mqd := (Fb (H ), H qd , {φh1/2 (f ), πh1/2 (f )|f ∈ D(h−1/2 ) ∩ HJ }),

10.5 Improper Bogoliubov Transformation and Renormalization

407

where φh1/2 (f ) and πh1/2 (f ) are defined by (8.118) and (8.119) with T = h1/2 respectively. Suppose that Th is not Hilbert–Schmidt. Then Assumption (B) does not hold. In addition, (Th )J (Sh )∗J may not be Hilbert–Schmidt (e.g., if (Sh )∗J is bijective, then ∗ (T If (Th )J (Sh )∗J is not Hilbert–Schmidt, then  h∗)J (Sh )J is ∗not∗Hilbert–Schmidt). A |(Th )J (Sh )J |A cannot be defined and hence Hqd loses mathematical meaning, implying that the model Mqd cannot be defined. The main purpose of this section is to show that, even in the case where Th and (Th )J (Sh )∗J are not Hilbert–Schmidt, there exists a model which may be regarded as a correct or “renormalized” form of Mqd in such a case. We assume the following: Assumption (ST) (ST.1) Relations (8.60), (8.61), (8.70) and (8.71) hold. (ST.2) The operator h is a bounded non-negative injective self-adjoint operator on H and J h ⊂ hJ . We first describe a model equivalent to Mqd in the case where T is Hilbert– Schmidt.

10.5.1 A Model Equivalent to Mqd In this subsection, we assume that T is Hilbert–Schmidt in addition to Assumption (ST). Then we have Theorem 8.10. Therefore the model Mqd is equivalent to the model , {φh 1/2 (f ), πh 1/2 (f )|f ∈ D(h−1/2 ) ∩ HJ }), M qd := (Fb (H ), Hqd

where Hqd := db (h) − Th 22

and φh 1/2 (f ) := UB φh1/2 (f )UB−1 ,

πh 1/2 (f ) := UB πh1/2 (f )UB−1 .

To compute operators φh 1/2 (f ) and πh 1/2 (f ), we use (8.72). By (8.72) and (8.86), we have UB A(f )UB−1 = A(S ∗ f ) − A(T ∗ Jf )∗ , UB A(f )∗ UB−1 = A(S ∗ f )∗ − A(T ∗ Jf )

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10 Physical Correspondences in Quantum Field Theory

on UB Fb,0 (H ). Hence, for all f ∈ D(h−1/2 ) ∩ HJ , φh 1/2 (f ) = ΦS ((S ∗ − T ∗ )h−1/2 f ),

(10.27)

πh 1/2 (f ) = ΦS (i(S ∗ + T ∗ )h1/2 f )

(10.28)

on UB Fb,0 (H ), where ΦS (·) is the Segal field operator (see (6.94)).

10.5.2 A Renormalized Model We next consider the case where Th is not Hilbert–Schmidt. In this case, Hqd may not be defined as we have already discussed. But we note that, in the form of model + T 2 ” (this expression is M qd , the informally renormalized Hamiltonian “Hqd h 2 meaningless, since Th 22 = ∞ by the non-Hilbert–Schmidtness of Th ) may be redefined as db (h). Moreover the right hand sides of (10.27) and (10.28) are welldefined even in the case where Th is not Hilbert–Schmidt (note that, if Th is not Hilbert–Schmidt, then T is not Hilbert–Schmidt, since h1/2 is bounded under the present assumption). Therefore we introduce new operators φS,T (f ) := ΦS ((S ∗ − T ∗ )h−1/2 f ), πS,T (f ) := ΦS (i(S ∗ + T ∗ )h1/2 f ),

(10.29) f ∈ D(h−1/2 ) ∩ HJ .

(10.30)

Theorem 10.6 Suppose that T is not Hilbert–Schmidt in addition to Assumption (ST). Then (Fb (H ), {φS,T (f ), πS,T (f )|f ∈ D(h−1/2 ) ∩ HJ }) is an irreducible Weyl representation of the CCR over D(h−1/2 ) ∩ HJ . Moreover, it is inequivalent to the Fock representation (Fb (f ), {φh1/2 (f ), πh1/2 (f )|f ∈ D(h−1/2 ) ∩ HJ }). Proof Throughout the proof, we set φ(f ) := ΦS ((S ∗ − T ∗ )f ), π(f ) := ΦS (i(S ∗ + T ∗ )f ),

f ∈ HJ .

Then φS,T (f ) = φ(h−1/2 f ),

πS,T (f ) = π(h1/2 f ),

f ∈ D(h−1/2 ) ∩ HJ .

Let Sˆ := S ∗ − T ∗ ,

Tˆ := S ∗ + T ∗ .

10.5 Improper Bogoliubov Transformation and Renormalization

409

Then, using (8.70) and (8.71), one can show that     ˆ Sg ˆ = 0, Im Tˆ f, Tˆ g = 0, Im Sf,   ˆ Tˆ g = f, g , f, g ∈ HJ . Re Sf, We have Sˆ ∗ − i Tˆ ∗ = (1 − i)S − (1 + i)T . Let g ∈ ker(Sˆ ∗ − Tˆ ∗ ). Then (1 − i)Sg = (1 + i)T g. Hence Sg 2 =

T g 2 . By (8.60), Sg 2 = T g 2 + g 2 . Hence g 2 = 0. Hence g = 0. Therefore ker(Sˆ ∗ − Tˆ ∗ ) = {0}, implying that Sˆ ∗ − i Tˆ ∗ is injective. Hence, by an application of Theorem 8.13, (Fb (H ), {φ(f ), π(f )|f ∈ HJ }) is an irreducible Weyl representation of the CCR over HJ . We have Sˆ + Tˆ = 2S ∗ . Hence Ran (Sˆ + Tˆ ) = H . The easily seen equation Sˆ − Tˆ = −2T ∗ implies that Sˆ − Tˆ is not Hilbert–Schmidt. Therefore, by an application of Theorem 8.14, {φ(f ), π(f )|f ∈ W } is inequivalent to {φF (f ), πF (f )|f ∈ W } for any dense subspace W of HJ . This implies that {φ(h−1/2 f ), π(h1/2 f )|f ∈ D(h−1/2 ) ∩ HJ } is inequivalent to {φF (h−1/2 f ), πF (h1/2 f )|f ∈ D(h−1/2 ) ∩ HJ }.  Let Hren (h) := db (h) and define a renormalized model for Mqd as Mqd, ren := (Fb (H ), Hren (h), {φS,T (f ), πS,T (f )|f ∈ HJ }). By Theorem 10.6, the time-zero fields φS,T (f ) and πS,T (f ) in the renormalized model are inequivalent to the Fock ones φh1/2 (f ) and πh1/2 (f ). This is an interesting structure.

410

10 Physical Correspondences in Quantum Field Theory

10.6 Representations of CCR in a Theory of Weakly Interacting Bosons In this section, we consider representations of CCR appearing in a theory of weakly interacting bosons. This theory is related to the superfluidity [44]. For physical discussions of the theory, we refer the reader to [44], [153, §3.6], [154, §4-6] or [109, §7.4].

10.6.1 Finite Volume Theory We first consider a Bose field theory on the cubic box IdL (see (1.114)). We recall that the momentum space corresponding to this configuration space is ΓLd given by (1.115). We work with momentum representation. Then the Hilbert space for this system is taken to be the boson Fock space n 2 d Fb (2 (ΓLd )) = ⊕∞ n=0 ⊗s  (ΓL )

(10.31)

over 2 (ΓLd ). For each p ∈ ΓLd , we define a vector ep ∈ 2 (ΓLd ) by (ep )k := δpk ,

k ∈ ΓLd .

(10.32)

Then {ep }p∈Γ d is a CONS of 2 (ΓLd ). L

We denote by aL (f ) the annihilation operator with test vector f ∈ 2 (ΓLd ), acting in Fb (2 (ΓLd )). For each k ∈ ΓLd , we define an operator ak by ak := aL (ek ).

(10.33)

It follows from the CCR of aL (·) and aL (·)∗ that {ak , ak∗ |k ∈ ΓLd } obeys the CCR indexed by ΓLd : for all k, p ∈ ΓLd , [ak , ap∗ ] = δkp ,

[ak , ak ] = 0

on Fb,0 (2 (ΓLd )).

We consider a system of non-relativistic bosons interacting weakly. For each κ > 0, an approximate Hamiltonian Hκ for such a system is defined by 1 0 1 gρ 2 ∗ k + gρ ak∗ ak + (a−k Hκ := ak∗ + ak a−k ), 2m 2 d d k∈ΓL 0 0 denotes the density of bare bosons with momentum k = 0 and g > 0 is a coupling constant. As is seen from the definition of Hκ , the parameter κ > 0 physically means a momentum for  cutoff ∗ a∗ bosons, called an ultraviolet cutoff for boson momenta6 (note that k∈Γ d a−k k L  and k∈Γ d a−k a−k have no mathematical meaning). It is easy to see that Hκ is a L

symmetric operator with D(Hκ ) ⊃ Fb,0 (2 (ΓLd )). Since Hκ is an operator quadratic in ak and ak∗ , it may be diagonalized through a proper Bogoliubov transformation. Indeed this is the case as shown below. Let 1 2 k + gρ, 2m

%(k) :=

ˆ d. k∈R

(10.35)

Then %(k) ≥ gρ, implying that %(k)2 − (gρ)2 ≥ 0. Hence one can define a function ˆ d by on R η(k) :=



%(k)2 − (gρ)2 ,

ˆ d. k∈R

(10.36)

Note that η(0) = 0. ˆ d \ {0}: Taking this fact into account, we introduce a function on R γ (k) :=

%(k) , η(k)

ˆ d \ {0}. k∈R

(10.37)

It obvious that ˆ d. k∈R

η(k) < %(k),

(10.38)

Hence γ (k) > 1,

ˆ d \ {0}. k∈R

(10.39)

We have γ (k) =

1 1−

(gρ)2

,

ˆ d \ {0}. k∈R

%(k)2

6 A momentum space region in which |k| is very large is called an ultraviolet region, since the wavelength 2π/|k| is very short.

412

10 Physical Correspondences in Quantum Field Theory

For all k ∈ ΓLd \ {0}, %(k) ≥ εL :=

1 2m

0

2π L

12 + gρ.

Hence γ (k) ≤ γL :=

1

1−

(gρ)2

,

k ∈ ΓLd \ {0}.

(10.40)

2 εL

ˆ d \ {0}). Therefore γ (·) is bounded on ΓLd \ {0} (but, unbounded on R By (10.39), one can define two sequences {uk }k∈Γ d and {vk }k∈Γ d by L

uk :=

1 + γ (k) , 2

u0 := 1,

γ (k) − 1 , 2

vk :=

L

k ∈ ΓLd \ {0},

v0 := 0.

(10.41) (10.42)

It is easy to show that the following equations hold: u2k − vk2 = 1, uk vk =

uk = u−k ,

gρ , 2η(k)

vk = v−k ,

k ∈ ΓLd .

k = 0.

(10.43) (10.44)

It follows from the boundedness of γ that {uk }k∈Γ d and {vk }k∈Γ d are bounded. L L We now define ∗ , bk := uk ak + vk a−k

k ∈ ΓLd .

(10.45)

Note that b0 = a0 . Using the CCR for ak# and (10.43), one can show that {bk , bk∗ |k ∈ ΓLd }} obeys the CCR indexed by ΓLd : for all k, p ∈ ΓLd , [bk , bp∗ ] = δkp ,

[bk , bk ] = 0 on Fb,0 (2 (ΓLd )).

(10.46)

Equation (10.45) implies that bk∗ ⊃ uk ak∗ + vk a−k ,

k ∈ ΓLd .

(10.47)

10.6 Representations of CCR in a Theory of Weakly Interacting Bosons

413

Using (10.45) and (10.47), one can represent ak and ak∗ in terms of bk and bk∗ : for all k ∈ ΓLd , ∗ ak = uk bk − vk b−k ,

(10.48)

ak∗ = uk bk∗ − vk b−k

(10.49)

on Fb,0 (2 (ΓLd )). Substituting these representations for ak and ak∗ on the right hand side of (10.34), we obtain

Hκ =

η(k)bk∗ bk + Eκ

on Fb,0 (2 (ΓLd )),

(10.50)

k∈ΓLd ,|k|≤κ

where Eκ := −



η(k)vk2 = −

k∈ΓLd ,|k|≤κ

Note that the operator

(gρ)2 2

 k∈ΓLd ,|k|≤κ

k∈ΓLd ,|k|≤κ

1 < 0. %(k) + η(k)

(10.51)

η(k)bk∗ bk is non-negative on Fb,0 (2 (ΓLd )).

Hence Hκ is bounded from below on Fb,0 (2 (ΓLd )) with Hκ ≥ Eκ on Fb,0 (2 (ΓLd )). Then, in the same way as in the proof of Theorem 8.9, one can prove the following fact: Theorem 10.7 The operator Hκ is essentially self-adjoint on Fb,0 (2 (ΓLd ) and H κ ≥ Eκ . We note that (10.45) defines a Bogoliubov transformation in the following way. We define a mapping jd : 2 (ΓLd ) → 2 (ΓLd ) by (jd f )(k) := f (−k)∗ ,

f ∈ 2 (ΓLd ), k ∈ ΓLd .

(10.52)

Then it is easy to see that jd is a conjugation on 2 (ΓLd ). For each κ > 0, we define functions (sequences) u(κ) and v (κ) on ΓLd as follows:  (u(κ) )k :=

uk for |k| ≤ κ , 1 for |k| > κ

 (v (κ) )k :=

vk for |k| ≤ κ . 0 for |k| > κ

Then u(κ) and v (κ) are real-valued bounded sequences satisfying (u(κ) )2k − (v (κ) )2k = 1, (u(κ) )k = (u(κ) )−k , (v (κ) )k = (v (κ) )−k , k ∈ ΓLd . (10.53)

414

10 Physical Correspondences in Quantum Field Theory

Let b(κ) (f ) := aL (u(κ) f ) + aL (jd (v (κ) f ))∗

f ∈ 2 (ΓLd ),

(10.54)

where, for a sequence w on ΓLd , (wf )k := wk fk , k ∈ ΓLd . Then  b(κ)(ek ) =

bk for |k| ≤ κ . ak for |k| > κ

It is easy to see the correspondence (aL (·), aL (·)∗ ) → (b(κ)(·), b (κ) (·)∗ ) is a Bogoliubov transformation with conjugation J = jd . Note that (8.70) and (8.71) hold for S = u(κ) and T = v (κ) with J = jd . Hence, by Lemma 8.9, we obtain the following: Lemma 10.1 The triple τL := (Fb (2 (ΓLd )), Fb,0 (2 (ΓLd )), {b(κ) (f ) | f ∈ 2 (ΓLd )})

(10.55)

is an irreducible representation of the CCR over 2 (ΓLd ). Let χκ (k) := χ[0,κ] (|k|),

ˆ d. k∈R

(10.56)

Then χκ η can be regarded as a non-negative bounded, multiplication self-adjoint operator on 2 (ΓLd ) satisfying jd (χκ η) ⊂ (χκ η)jd . Theorem 10.8 There exists a unitary operator U on Fb (2 (ΓLd )) such that U b(κ) (f )U −1 = aL (f ),

f ∈ 2 (ΓLd ),

U (H κ − Eκ )U −1 = db (χκ η), (L)

(10.57) (10.58)

(L)

where db (χκ η) denotes the boson second quantization operator of the multiplication operator χκ η, acting in Fb (2 (ΓLd )). Proof Let h = χκ η, S = u(κ) , T = v (κ) and J = jd . Then the pair (h, T ) satisfies Assumption (B) in Sect. 8.7 and (S, T , J ) obeys (8.60), (8.61), (8.70) and (8.71). Hence, by an application of Theorem 8.10, the result follows.  Theorem 10.8 implies the existence of a unique ground state of H κ : Corollary 10.2 The Hamiltonian H κ has a unique ground state Ωκ given by Ωκ := U −1 Ω2 (Γ d ) , L

10.6 Representations of CCR in a Theory of Weakly Interacting Bosons

415

up to constant multiples, with ground state energy Eκ : H κ Ω κ = Eκ Ω κ . Formula (10.57) implies that b(κ)(f )Ωκ = 0,

f ∈ 2 (ΓLd ).

By (10.48) and (10.49), we have aL (f ) = b(κ) (u(κ) f ) − b(κ)(v (κ) jd f )∗ .

(10.59)

on Fb,0 (2 (ΓLd )). Hence

aL (f )Ωκ 2 = b(κ)(v (κ) jd f )∗ Ωκ 2 = v (κ) f 2 . Hence, by Corollary 8.3, we have

1/2

Nb Ωκ 2 =

vk2 ,

(10.60)

k∈ΓLd , |k|≤κ

where Nb is the boson number operator on Fb (2 (ΓLd )). This gives a physical meaning of the sequence {vk }k . By (8.69) with B = b (κ), we have also nb(κ) =





bk(κ) Ω2 (Γ d ) 2 = L

vk2 .

k∈ΓLd , |k|≤κ

k∈ΓLd

We have by (10.38) 1 m 1 > = 2 . %(k) + η(k) 2ε(k) k + 2mgρ It is easy to see that, if d ≥ 2, then

m k + 2mgρ 2

k∈ΓLd

=∞

and hence k∈ΓLd

1 = ∞. %(k) + η(k)

(10.61)

416

10 Physical Correspondences in Quantum Field Theory

By this fact and (10.51), lim Eκ = −∞

κ→∞

for d ≥ 2. This is an example of ultraviolet divergences7 in quantum field models. Hence, to construct a model without ultraviolet cutoff as a suitable limit of the model ML,κ := (Fb (2 (ΓLd )), H κ , {aL (f )|f ∈ 2 (ΓLd )}) with d ≥ 2, one needs an energy renormalization. Remark 10.9 In non-relativistic quantum field theory (quantum theory of de Broglie field), the time-zero field in the Fock representation is given by the annihilation operator (see [28, Chapter 9]). By Theorem 10.8, the model ML,κ is equivalent to the model M L,κ := (Fb (2 (ΓLd )), db(L) (χκ η), {U aL (f )U −1 |f ∈ 2 (ΓLd )}). By (10.59) U aL (f )U −1 = aL (u(κ) f ) − aL (v (κ) jd f )∗

(10.62)

on Fb,0 (2 (ΓLd )). Note that the right hand side is well-defined even if κ = ∞, since u and v are bounded on ΓLd . The operator db(L) (η) is a non-negative self-adjoint operator. Hence we define cL (f ) := aL (uf ) − aL (vjd f )∗ ,

f ∈ 2 (ΓLd ),

(L) Hˆ L := db (η).

(10.63) (10.64)

It is easy to see that, for all f ∈ 2 (ΓLd ), aL (f ) = cL (uf ) + cL (vjd f )∗

(10.65)

on Fb,0 (2 (ΓLd )). Hence it follows that {cL (f )|f ∈ 2 (ΓLd )} is an irreducible representation of the CCR over 2 (ΓLd ). Thus we have a renormalized model Mren,L := (Fb (2 (ΓLd ), Hˆ L , {cL (f )|f ∈ 2 (ΓLd )})

7A

quantity Qκ (a scalar, a vector or an operator) which depends on κ is said to be ultraviolet divergent if Qκ diverges in a suitable sense as κ → ∞ (removal of ultraviolet cutoff).

10.6 Representations of CCR in a Theory of Weakly Interacting Bosons

417

without ultraviolet cutoff, which may be regarded as a genuine form of a fictitious model corresponding to the symbolical expression “ limκ→∞ ML,κ ”. In this model, the ground state (physical vacuum) is given by ΩL := Ω2 (Γ d ) . L

(10.66)

We have k∈ΓLd

vk2 =

(gρ)2 2

k∈ΓLd \{0}

1 . η(k)(%(k) + η(k))

Note that, for all k ∈ ΓLd \ {0}, γ (k)2 1 1 1 γ (k) γL ≤ < < < . 2 2 2 2%(k) η(k)(%(k) + η(k)) 1 + γ (k) %(k) %(k) %(k)2  Since k∈Γ d \{0} 1/%(k)2 < ∞ if and only if d ≤ 3, it follows that the multiplicaL tion operator v is Hilbert–Schmidt if and only d ≤ 3. Therefore the representation {cL (f )|f ∈ 2 (ΓLd )} is equivalent to the Fock representation {aL (f )|f ∈ 2 (ΓLd )} if and only d ≤ 3.

10.6.2 Infinite Volume Theory We next consider an infinite volume theory, corresponding heuristically to the case L → ∞ in the model Mren,L . A standard way to construct such a theory is to apply the idea of the Wightman reconstruction theorem in axiomatic quantum field theory (e.g., [149], [45, Chapter 3, §3]) or Gel’fand–Na˘imark–Segal (GNS) construction in representation theory of ∗-algebras (e.g., [85, Chapter III, §III.2.2], [18, Chapter 2, §2.5]). The vacuum expectation values in the model Mren,L are defined by   Wn(L) (f1 , . . . , fn ) : = ΩL , c˜L (f1 )# · · · c˜L (fn )# ΩL , f1 , . . . , fn ∈ Ed (n ∈ Z+ ), ˆ d ) (the restriction of fj on Γ d is regarded where Ed is a suitable subspace of L2 (R L d 2 as an element of  (ΓL )) and

(2π)d cL (f ), f ∈ Ed . Ld  The multiplicative factor operator (2π)d /Ld is used in order that the infinite (L) volume limit limL→∞ Wn (f1 , . . . , fn ) exists. c˜L (f ) :=

418

10 Physical Correspondences in Quantum Field Theory

Let %, η and γ be as in the preceding subsection. We define functions u and v on ˆ d as follows: R u(k) := u(0) := 1,

1 + γ (k) , 2

γ (k) − 1 , 2

v(k) :=

ˆ d \ {0}, k∈R

v(0) := 0.

(10.67) (10.68)

Then u(k)2 − v(k)2 = 1,

u(k) = u(−k),

v(k) = v(−k),

ˆ d. k∈R

We have gρ , u(k) = √ 2η(k)(%(k) − η(k) gρ v(k) = √ , 2η(k)(%(k) + η(k)

(10.69) k = 0.

(10.70)

Note that lim|k|→0 η(k) = 0. Hence lim|k|→0 u(k) = +∞ and lim|k|→0 v(k) = ˆ d . In the context of +∞. Therefore u and v are unbounded near the origin 0 ∈ R quantum field theory, this kind of singularity is called an infrared singularity.8 To avoid this singularity, we take as Ed the following: ˆ d )|supp f is bounded with supp f ⊂ R ˆ d \ {0}}. Ed := {f ∈ C(R

(10.71)

As is easily seen, for all f ∈ Ed , uf and vf are bounded continuous functions on ˆ d with bounded support and hence uf, vf ∈ L2 (R ˆ d ). R Lemma 10.2 For all f1 , f2 ∈ Ed , 

 lim ΩL , c˜L (f1 )∗ c˜L (f2 )∗ ΩL = −

L→∞



lim ΩL , c˜L (f1 )c˜L (f2 )ΩL  = −

L→∞

 lim ΩL , c˜L (f1 )∗ c˜L (f2 )ΩL =

L→∞



∗



lim ΩL , c˜L (f1 )c˜L (f2 ) ΩL =

L→∞

 

u(k)v(k)f1 (−k)f2 (k)dk,

(10.72)

u(k)v(k)f1 (−k)∗ f2 (k)∗ dk,

(10.73)

ˆd R ˆd R

ˆd R ˆd R

v(k)2 f1 (k)f2 (k)∗ dk,

(10.74)

u(k)2 f1 (k)∗ f2 (k)dk.

(10.75)

ˆ d is very momentum space region in which the modulus |k| of a wave number vector k ∈ R small (hence the wave length 2π/|k| is very long) is called an infrared region. The word “infrared” comes from this correspondence.

8A

10.6 Representations of CCR in a Theory of Weakly Interacting Bosons

419

Proof By (10.63) and (6.39), we have 

(2π)d  ΩL , c˜L (f1 )∗ c˜L (f2 )∗ ΩL = − d ΩL , aL (vjd f1 )aL (uf2 )∗ ΩL L (2π)d =− d u(k)v(k)f1 (−k)f2 (k). L d k∈ΓL

Hence, by a theorem on Riemann integrals, the right hand side converges to that of (10.72) as L → ∞. Formula (10.73) follows from taking the complex conjugate of (10.72). In the same manner as in the proof of (10.72), one can prove (10.74) and (10.75).  Theorem 10.9 For all n ∈ Z+ and f1 , . . . , fn ∈ Ed , the infinite volume limit Wn (f1 , . . . , fn ) := lim Wn(L) (f1 , . . . , fn ) L→∞

exists. (L)

(L)

Proof If n is odd, then, by Theorem 6.19, Wn (·) = 0. Hence limL→∞ Wn (·) = (L) 0. In the case where n is even with n = 2p, by Theorem 6.19 again, Wn is written as a combinatorial sum of     ΩL , cL (fi1 )# cL (fj1 )# ΩL · · · ΩL , cL (fip )# cL (fjp )# ΩL with i1 < · · · < ip , ik < jk . By Lemma 10.2, this quantity converges as L → ∞. Thus the result follows.  We now construct a representation {c(f )|f ∈ Ed } of the CCR over Ed on the boson Fock space ˆ d )) = ⊕∞ ⊗ns L2 (R ˆ d) Fb (L2 (R n=0

(10.76)

ˆ d ) such that, for all n ∈ Z+ and f1 , . . . , fn ∈ Ed , over L2 (R   Wn (f1 , . . . , fn ) = Ω, c(f1 )# · · · c(fn )# Ω

(10.77)

with Ω := ΩL2 (Rˆ d ) . This corresponds to a concrete form of construction by the Wightman reconstruction theorem or the GNS construction in the present context. Intuitively, the infinite space ˆ d ) here is regarded as the infinite volume limit of Id (resp. 2 (Γ d )). Rd (resp. R L L

420

10 Physical Correspondences in Quantum Field Theory

ˆ d ) acting We denote by a(f ) the annihilation operator with test vector f ∈ L2 (R ˆ d )). Let Jd : L2 (R ˆ d ) → L2 (R ˆ d ) be the conjugation defined by in Fb (L2 (R (Jd f )(k) := f (−k)∗ ,

ˆ d ), k ∈ R ˆ d. f ∈ L2 (RL

(10.78)

Corresponding to the case of the renormalized model Mren,L in the finite volume theory, we define a densely defined closed operator c(f ) := a(uf ) − a(Jd (vf ))∗ ,

f ∈ Ed

(10.79)

ˆ d )). It is easy to see that on Fb (L2 (R ˆ d )), Fb,0 (L2 (R ˆ d )), {c(f ) | f ∈ Ed )}) τ∞ := (Fb (L2 (R

(10.80)

is a representation of the CCR over Ed . Remark 10.10 Note that u and v are unbounded self-adjoint multiplication operaˆ d ). Hence the correspondence (a(·), a(·)∗ ) → (c(·), c(·)∗ ) is a singular tors on L2 (R Bogoliubov transformation (see Sect. 8.6). It follows that, for all f ∈ Ed , a(f ) = c(uf ) + c(vJd f )∗

(10.81)

ˆ d ), which, together with the density of Ed in L2 (R ˆ d ), implies that on Fb,0 (L2 (R ∗ {{c(f ), c(f ) | f ∈ Ed } is irreducible. Therefore τ∞ is an irreducible representation of the CCR over Ed . Moreover, by using Theorem 6.19, one can show that (10.77) holds. A big difference between the finite volume theory and the infinite one is stated in the following theorem: Theorem 10.10 The representation τ∞ is inequivalent to the Fock representation ˆ d )), Fb,0 (L2 (R ˆ d )), {a(f ) | f ∈ Ed }) ρF (Ed ) = (Fb (L2 (R over Ed . Proof Suppose that τ∞ were equivalent to the Fock representation ρF (Ed ). Then ˆ d )) such that U c(f )U −1 = a(f ), f ∈ there exists a unitary operator U on Fb (L2 (R −1 Ed . Hence, letting Ω := U ΩL2 (Rˆ d ) , we have that c(f )Ω = 0, ∀f ∈ Ed . Then, ˆ d )) and f ∈ Ed , for all Ψ ∈ Fb,0 (L2 (R  a(uf )∗ Ψ, Ω = a(vJd f )Ψ, Ω .

10.6 Representations of CCR in a Theory of Weakly Interacting Bosons

421

 This implies that a(f )∗ Ψ, Ω = a(Jd vu−1 f )Ψ, Ω , f ∈ Ed . Hence, by Proposition 8.4, vu−1 is a Hilbert–Schmidt operator. But T := vu−1 is the ˆ d \ {0}. Hence multiplication operator by the continuous function v(k)u(k)−1 on R the spectrum σ (T ) of T is not discrete. This is a contradiction.  The model corresponding to the infinite volume limit of the model Mren,L is defined by ˆ d )), Hˆ ∞ , {c(f )|f ∈ Ed }) Mren,∞ := (Fb (L2 (R with Hˆ ∞ := db (η). It is easy to see that the time-t field in the present model is given by ˆ

ˆ

c(t, f ) := eit H∞ c(f )e−it H∞ = c(eit η f ),

f ∈ Ed , t ∈ R.

The Hamiltonian Hˆ ∞ has a unique ground state Ω∞ := ΩL2 (Rˆ d ) with Hˆ ∞ Ω∞ = 0. By Theorem 10.10, the model Mren,∞ is inequivalent to any free Bose field model ˆ d )), db (T ), {a(f )|f ∈ Ed }) with T being a self-adjoint of the form (Fb (L2 (R 2 d ˆ ). operator on L (R In concluding this section, we consider the dependence of c(f ) on the parameters m, g and ρ. We set λ := (m, g, ρ) ∈ R3+ (R+ := (0, ∞))

(10.82)

p(λ) := mgρ.

(10.83)

and define

To indicate the dependence of c(f ) on λ, we write cλ (f ) := c(f ),

τ∞,λ := τ∞ .

(10.84)

Theorem 10.11 If p(λ1 ) = p(λ2 ) (λi := (mi , gi , ρi ) ∈ R3+ , i = 1, 2), then τ∞,λ1 is inequivalent to τ∞,λ2 .

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10 Physical Correspondences in Quantum Field Theory

Proof Suppose that τ∞,λ1 were equivalent to τ∞,λ2 . Then there exists a unitary ˆ d )) such that U cλ1 (f )U −1 = cλ2 (f ), f ∈ Ed . By (10.81), operator U on Fb (L2 (R ˆ d )) we have for all Ψ ∈ Fb,0 (L2 (R a(f )Ψ = cλ1 (u1 f )Ψ − cλ1 (v1 Jd f )∗ Ψ = U −1 cλ2 (u2 f )U Ψ − U −1 cλ2 (v2 Jd f )∗ U Ψ, where ui := u and vi := v with (m, g, ρ) = λi (i = 1, 2). Hence U a(f )Ψ = cλ2 (u2 f )U Ψ − cλ2 (v2 Jd f )∗ U Ψ. ˆ d )), Hence, for all Φ ∈ Fb,0 (L2 (R  Φ, U a(f )Ψ  = (cλ2 (u2 f )∗ − cλ2 (v2 Jd f ))Φ, U Ψ . Using (10.81) again, we see that the right hand side is equal to a(f )∗ Φ, U Ψ . ˆ d )) is a core for a(f ), Hence Φ, U a(f )Ψ  = a(f )∗ Φ, U Ψ . Since Fb,0 (L2 (R ˆ d )) is a core this equation extends to all Ψ ∈ D(a(f )). The subspace Fb,0 (L2 (R ∗ ∗ ∗ for a(f ) too. Hence U Ψ ∈ D((a(f ) ) ) = D(a(f )) and a(f )U Ψ = U a(f )Ψ . Therefore U a(f ) ⊂ a(f )U , which implies that U a(f )∗ ⊂ a(f )∗ U . Hence U ∈ {a(f ), a(f )∗ |f ∈ Ed } . By this fact and Theorem 6.9, we conclude that U = eiθ for some θ ∈ R. Thus cλ1 (f ) = cλ2 (f ), f ∈ Ed . This implies a((u1 − u2 )f ) = a((v1 − v2 )Jd f )∗

ˆ d )). on Fb,0 (L2 (R

Then, by Proposition 6.3, (u1 − u2 )f = 0 and (v1 − v2 )Jd f = 0 for all f ∈ Ed . ˆ d \ {0}. These equations imply that Hence u1 (k) = u2 (k) and v1 (k) = v2 (k), k ∈ R m1 g1 ρ1 = m2 g2 ρ2 .  Theorem 10.11 tells us the following: let two of m, g and ρ be fixed, say, m and g. Then the set of particle densities ρ forms an index set of a family {τ∞,λ }ρ of mutually inequivalent irreducible representations of CCR. The same applies to the parameters m and g. Thus masses, coupling constants and particle densities appear respectively as an index set of a family of mutually inequivalent irreducible representations of CCR. Remark 10.11 There is a theory of interacting non-relativistic electrons with spin 1/2, which is constructed in a framework of fermion Fock space and has a structure similar to the boson theory discussed in this section in the sense that one needs only to replace the bosonic Bogoliubov transformation by a fermionic Bogoliubov

10.7 The Free Hermitian Klein–Gordon Quantum Field

423

transformation (see, e.g., [109, Chapter 5], [154, Chapter 4, §4.4]). In the physics literature, the theory is used to explain superconductivity and called the Bardeen– Cooper–Schrieffer (BCS) theory [42]. In quite the same way as in the theory presented above, one can construct an infinite volume theory of such a system and see that a representation of CAR inequivalent to a Fock representation of CAR appear. But, here, we omit describing the construction (cf. [69, 84]).

10.7 The Free Hermitian Klein–Gordon Quantum Field 10.7.1 Definition We consider a concrete quantum field model describing a quantum field of free relativistic bosons with mass m ≥ 0 in the d-dimensional position space Rd , where d ∈ N is arbitrary. A Hilbert space of state vectors of such a system can be taken to be the boson Fock space ˆ d )) = ⊕∞ ⊗n L2 (R ˆ d) Fb (L2 (R n=0 s ˆ d ). The space R ˆ d here denotes the momentum space of one free over L2 (R ˆd relativistic boson. The energy ωm (k) of one free boson with momentum k ∈ R is given by  ωm (k) =

k 2 + m2 .

(10.85)

We denote by ωm the multiplication operator by the function ωm (·), which is a nonnegative self-adjoint operator. It is easy to see that the operator ωm is injective. The Hamiltonian of the model we consider is defined by Hb := db (ωm ),

(10.86)

the boson second quantization operator of ωm . ˆ d ), we denote by a(f ) the annihilation operator on For each f ∈ L2 (R 2 d ˆ Fb (L (R )) with test vector f . We define a mapping Cd : L2 (Rd ) → L2 (Rd ) by Cd f := f ∗ ,

f ∈ L2 (Rd ).

It is easy to see that Cd is a conjugation. The conjugation Cd is called the complex conjugation on L2 (Rd ). Hence LR (Rd ) := {f ∈ L2 (Rd )|Cd f = f }

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10 Physical Correspondences in Quantum Field Theory

is a real Hilbert space.9 ˆ d ) be the d-dimensional Fourier transform (see (1.74)) Let Fd : L2 (Rd ) → L2 (R and Cˆ d := Fd Cd Fd−1 Then it follows that Cˆ d is a conjugation on L2 (Rˆd ). Concretely, one has (Cˆ d g)(k) = g(−k)∗ ,

ˆ d ), g ∈ L2 (R

ˆ d. a.e.k ∈ R

Hence ˆ d ) := {f ∈ L2 (R ˆ d )|Cˆ d f = f } L2R (R is a real Hilbert space. It follows that ˆ d ). Fd (L2R (Rd )) = L2R (R For a tempered distribution f , we denote by fˆ its Fourier transform and by SR (Rd ) the space of real tempered distributions on Rd . For each s ∈ R, we introduce s ˆ ˆ d )} Ms (Rd ) := {f ∈ SR (Rd )|ωm f ∈ L2 (R

For each f ∈ M−1/2 (Rd ) ∩ M1/2 (Rd ), we define 1 −1/2 −1/2 fˆ) = √ (a(ωm fˆ)∗ + a(ωm fˆ)), 2 i 1/2 1/2 1/2 πm (f ) := πF (ωm fˆ) = √ (a(ωm fˆ)∗ − a(ωm fˆ)), 2 −1/2

φm (f ) := φF (ωm

(10.87) (10.88)

ˆ d )} is the Fock representation of the Heisenberg where {φF (g), πF (g)|g ∈ L2R (R ˆ d ). CCR over L2R (R In what follows, we assume the following: Assumption (d): In the case m = 0, we assume that d ≥ 2. Remark 10.12 If m = 0 and d = 1, then, for all f ∈ SR (R), −1/2

(ω0

9 In

1 fˆ)(k) = √ fˆ(k), |k|

notation in Sect. 1.9, L2R (Rd ) = (L2 (Rd ))Cd .

ˆ \ {0}. k∈R

10.7 The Free Hermitian Klein–Gordon Quantum Field

425

−1/2 ˆ Hence, if infk∈[−δ,δ] |fˆ(k)| > 0 for some δ > 0, then ω0 fˆ ∈ L2 (R). Hence SR (R) is not a subset of M−1/2 (R). This is an example of infrared divergences which may occur in a massless quantum field theory. To avoid this type of divergence in the case d = 1, one can change the test function space M−1/2 (R) ∩ M1/2 (R) to a subspace {f ∈ M−1/2 (R) ∩ ˆ }. But, in this book, we do not M1/2(R)| fˆ vanishes in a neighborhood of 0 ∈ R take such an approach for simplicity.

It is easy to see that SR (Rd ) ⊂ M−1/2 (Rd ) ∩ M1/2(Rd ). Let ˆ d )), Fb,0 (L2 (R ˆ d )), {φm (f ), πm (f )|f ∈ SR (Rd )}). πm := (Fb (L2 (R

(10.89)

Then it is easy to see that πm is a representation of the Heisenberg CCR over SR (Rd ). In fact, we have the following: Theorem 10.12 The representation πm is an irreducible Weyl representation of the CCR over SR (Rd ). Proof Let −1/2 1/2 ˆ d )), Fb,0 (L2 (R ˆ d )), {φF (ωm ˆ d )) ˆ }), πˆ m := (Fb (L2 (R f ), πF (ωm f )|f ∈ (S (R Cd

where ˆ d )|Cˆ d f = f }. ˆ d )) ˆ := {f ∈ S (R (S (R Cd Then πˆ m is an example of ΠT (W ) given by (8.120) with the following choice of (H , W , T ): ˆ d ), H = L2 (R

ˆ d )) ˆ , W = (S (R Cd

1/2

T = ωm .

It is easy to see that Cˆ d ωm ⊂ ωm Cˆ d ,

1/2 −1/2 ˆ d )) ˆ ⊂ D(ωm (S (R ) ∩ D(ωm ). Cd

±1/2 ˆ d )) ˆ ) are Using the du Bois–Reymond lemma, one can show that ωm (S (R Cd ˆ d ). Hence it follows from Theorem 8.15 that πˆ m is an irreducible dense in L2R (R Weyl representation of the CCR over (SR (Rd ))Cˆ d . We have

Fd (SR (Rd )) = (SR (Rd ))Cˆ d .

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10 Physical Correspondences in Quantum Field Theory

Hence πm is obtained by changing the index set (SR (Rd ))Cˆ d of the representation πˆ m to SR (Rd ). Thus πm is an irreducible Weyl representation of the CCR over SR (Rd ).  In conventional models of quantum scalar fields, φm (·) and πm (·) are taken as time-zero fields. A simple example of such models is given by ˆ d )), Hb , {φm (f ), πm (f )|f ∈ SR (Rd )}), (Fb (L2 (R where Hb is defined by (10.86). In this model, the time-t fields are given by φm (t, f ) := eit Hb φm (f )e−it Hb , πm (t, f ) := eit Hb πm (f )e−it Hb ,

t ∈ R, f ∈ SR (Rd ).

Applying Corollary 6.2 with T = ωm , we have −1/2 it ωm

φm (t, f ) = φF (ωm

e

fˆ),

1/2 πm (t, f ) = πF (ωm eit ωm fˆ).

(10.90) (10.91)

ˆ d )). It is easy to We denote by Nb the boson number operator on Fb (L2 (R −1/2 ˆ p d see that, for all f ∈ SR (R ) and p ∈ N, ωm f ∈ D(ωm ). Hence, applying 1/2 Corollary 6.8(ii) with T = ωm , we see that, for all Ψ ∈ D(Nb ), φm (t, f ) is infinitely strongly differentiable and d φm (t, f )Ψ = πm (t, f )Ψ, dt dp p −1/2 φm (t, f )Ψ = φF (i p ωm ωm eit ωm fˆ)Ψ. dt p Let Δ be the d-dimensional generalized Laplacian: Δ :=

d

Dj2 .

(10.92)

j =1

Then, as is well known, we have (Fd Δf )(k) = −k 2 fˆ(k),

ˆ d. f ∈ D(Δ), a.e.k ∈ R

Hence (Fd (−Δ + m2 )f )(k) = ωm (k)2 fˆ(k),

ˆ d. a.e.k ∈ R

10.7 The Free Hermitian Klein–Gordon Quantum Field

427

Therefore, for all f ∈ SR (Rd ), d2 φm (t, f )Ψ + φm (t, (−Δ + m2 )f )Ψ = 0. dt 2

(10.93)

This means that the operator-valued functional10 (t, f ) → φm (t, f ) obeys the free Klein–Gordon equation with mass m. Note that φm (t, f ) is self-adjoint. Hence the expectation value Φ, φm (t, f )Φ with respect to Φ ∈ D(φm (t, f )) is a real number. Based on these facts, the operator-valued functional φm (t, f ) is called the free Hermitian Klein–Gordon quantum field with mass m.

10.7.2 Boson Masses as Indices of a Family of Mutually Inequivalent Representations of CCR As a result in the preceding subsection, we have a family {πm }m>0 of irreducible Weyl representations of the CCR over SR (Rd ), each of which gives the time-zero fields in the theory of the free Hermitian Klein–Gordon quantum field with mass m. An interesting fact in this family is the following: Theorem 10.13 Let m1 , m2 > 0 and m1 = m2 . Then πm1 is inequivalent to πm2 . Proof We apply Theorem 8.18 with the following case: ˆ d ), H = L2 (R 1/2

T1 = ωm1 ,

J = Cˆ d ,

W = Fd SR (Rd ),

1/2

T2 = ωm2 .

Then it is not so difficult to show that the assumption of Theorem 8.18 is satisfied. Hence we need only to show that −1/2 1/2

1/2 −1/2

M := ωm2 ωm1 − ωm2 ωm1

is not Hilbert–Schmidt. It is easy to see that M is a multiplication operator by the function ωm (k) − ωm2 (k) (m1 − m2 )(m1 + m2 )  F (k) =  1 = , ωm1 (k)ωm2 (k) (ωm1 (k) + ωm2 (k)) ωm1 (k)ωm2 (k)

ˆ d. k∈R

The function F is continuous. Without loss of generality, we can assume that m1 > m2 . Then we have ˆ d ) = [0, (m1 − m2 )/√m1 m2 ]. σ (M) = F (R

10 In

fact, the correspondence f → φm (t, f ) is an operator-valued distribution (see [28, §8.2]).

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10 Physical Correspondences in Quantum Field Theory



Hence, M is not Hilbert–Schmidt.11

Theorem 10.13 shows that {πm }m>0 is a family of mutually inequivalent irreducible Weyl representations of the CCR over SR (Rd ). This gives a representation theoretic meaning for boson masses. Namely, the set of positive boson masses forms an index set for a family of mutually inequivalent irreducible Weyl representations of the CCR over SR (Rd ). This is an interesting example of physical correspondences associated with inequivalency of representations of CCR. Remark 10.13 Theorem 10.13 has appeared in [131, Theorem X.46]. But the proof there is completely different from the above one. Our treatment is more general in the sense that it makes clear a general abstract structure behind πm (see Theorem 8.18). It can be shown also that π0 is inequivalent to πm for all m > 0. For the proof of this fact and related aspects, see [24].

10.8 Quantum Fields at Finite Temperatures Let H be an infinite-dimensional separable Hilbert space and hb be a strictly positive self-adjoint operator on H , physically denoting a one-boson Hamiltonian.12 Let J be a conjugation on H and assume that J hb ⊂ hb J. We denote by HJ the real part of H with respect to J as before. We set mb := inf σ (hb ) > 0. Then it follows that, for all β > 0,

e−βhb = e−βmb < 1. Hence, by applying a well-known theorem, we see that 1 ± e−βhb are bijective with bounded inverses (1 ± e−βhb )−1 =

∞

(∓1)n e−nβhb

n=0

in operator norm topology.

that, for any Hilbert–Schmidt operator S, σ (S) \ {0} consists of only isolated eigenvalues with finite multiplicity. 12 We can treat also the case where h is non-negative with inf σ (h ) = 0. But, for simplicity, here b b we restrict ourselves to the case inf σ (hb ) > 0. 11 Recall

10.8 Quantum Fields at Finite Temperatures

429

We consider a system of free bosons described by quantum fields φF (f ) and πF (f ) (f ∈ HJ ) in a canonical equilibrium state at a finite temperature. The Hamiltonian of the system is assumed to be given by db (hb ). We denote by β > 0 the inverse temperature of the system. Let AF be the algebra generated by {φF (f ), πF (f )|f ∈ HJ }. We first consider the case where e−βhb is trace class. Then one can show that, for all A ∈ AF , Ae−βdb (hb ) is trace class (see, e.g., [51, Proposition 5.2.28], [18, Theorem 8.18]). Hence one can define a linear functional ·β on the algebra AF by Aβ :=

Tr (Ae−βdb (hb ) ) , Tr e−βdb (hb )

A ∈ AF .

Then one has φF (f )φF (g)β =

1 f, Sβ g , 2

f, g ∈ HJ ,

(10.94)

where Sβ := (1 + e−βhb )(1 − e−βhb )−1 . Note that Sβ is a bijective bounded operator on H and Sβ−1 is bounded. The linear functional  · β is called the canonical equilibrium state or the Gibbs state with the Hamiltonian db (hb ) at inverse temperature β. In the case where e−βhb is not trace class,  · β is defined to be a quasi free state  with covariance form f, Sβ g (f, g ∈ HJ ) (see [51, p. 40], [18, §8.12]). In this case too, (10.94) holds. To express the right hand side of (10.94) as the expectation value of two quantum fields with respect to the Fock vacuum ΩH , we introduce 1/2

φβ (f ) := φF (Sβ f ).

(10.95)

Then we have  φF (f )φF (g)β = ΩH , φβ (f )φβ (g)ΩH ,

f, g, ∈ HJ .

Let −1/2

πβ (f ) := πF (Sβ

f ),

f ∈ HJ .

(10.96)

−1/2

and W = HJ , one

Πβ := (Fb (H ), {φβ (f ), πβ (f )|f ∈ HJ })

(10.97)

Then, by applying Theorem 8.15 to the case where T = Sβ can conclude that

is an irreducible Weyl representation of the CCR over HJ .

430

10 Physical Correspondences in Quantum Field Theory

With regard to comparison of two representations in {Πβ }β>0 , we have the following: Theorem 10.14 Let β1 = β2 (β1 , β2 > 0). Then Πβ1 ∼ = Πβ2 if and only if e−β2 hb − −β h e 1 b is Hilbert–Schmidt. −1/2

Proof We need only apply Theorem 8.18 to the following case: T = Sβ Πβ1 ∼ = Πβ2 if and only if 1/2 −1/2

S := Sβ2 Sβ1

. Hence

−1/2 1/2 Sβ1

− Sβ2

is Hilbert–Schmidt. By direct computations, we have S = X(e−β2 hb − e−β1 hb ) with −1/2 −1/2 Sβ2 (1 − e−β1 hb )−1 (1 − e−β2 hb )−1 .

X := 2Sβ1

Note that X is a bijective bounded operator. Hence S is Hilbert–Schmidt if and only if so is e−β2 hb − e−β1 hb . Thus the desired result follows.  Corollary 10.3 Suppose that, for some β0 > 0, e−β0 hb is Hilbert–Schmidt. Let β2 > β1 ≥ β0 . Then Πβ1 ∼ = Πβ2 . Proof By the present assumption, for all β > β0 , e−βhb is Hilbert–Schmidt (note that e−2βhb ≤ e−2β0hb ). Hence e−β2 hb − e−β1 hb is Hilbert–Schmidt. Thus, by Theorem 10.14, Πβ1 ∼  = Πβ2 . Corollary 10.4 Suppose that, for some β0 > 0, e−β0 hb is not Hilbert–Schmidt. Let 0 < β2 < β1 ≤ β0 . Then Πβ1 ∼  Πβ2 . = Proof By the present assumption, for all β ∈ (0, β0 ), e−βhb is not Hilbert–Schmidt (note that e−2β0 hb ≤ e−2βhb ). One has e−β2 hb − e−β1 hb = e−β2 hb (1 − e−(β1 −β2 )hb ). If e−β2 hb − e−β1 hb were Hilbert–Schmidt, then e−β2 hb = (e−β2 hb − e−β1 hb )(1 − e−(β1 −β2 )hb )−1 is Hilbert–Schmidt. But this is a contradiction. Hence e−β2 hb −e−β1 hb is not Hilbert– Schmidt. Thus, by Theorem 10.14, Πβ1 ∼  = Πβ2 . Example 10.4 Let d ∈ N and consider a quantum system of free relativistic bosons in the cubic box IdL (see (1.114)). Then the one-boson Hilbert space in the

10.9 Van Hove Model

431

momentum representation is given by 2 (ΓL ) (see (1.117)). In this case, hb is given by (L) , hb = ωm

(10.98)

(L) where ωm is a function on ΓL defined by

 (L) ωm (k) =

k 2 + m2 ,

k ∈ ΓL .

Hence (L) (k)|k ∈ ΓL }. σ (hb ) = σd (hb ) = {ωm

Hence, for all β > 0,

(L)

e−βωm

(k)

< ∞.

k∈ΓL

Therefore e−βhb is trace class and hence Hilbert–Schmidt. Thus, by Corollary 10.3, for all β1 , β2 > 0 with β1 = β2 , Πβ1 ∼ = Πβ2 . Example 10.5 We next consider a quantum system of free relativistic bosons in the ˆ d ) and hb = ωm . Hence σ (hb ) = [m, ∞). infinite space Rd . In this case, H = L2 (R −βh b Therefore, for all β > 0, e is not Hilbert–Schmidt. Thus, by Corollary 10.4, for all β1 , β2 > 0 with β1 = β2 , Πβ1 ∼  Πβ2 . = The above two examples show an essential difference between a finite volume theory and an infinite volume theory. It is noteworthy that, in the infinite volume theory in Example 10.5, the inverse temperatures appear as the indices of the family {Πβ }β>0 of mutually inequivalent irreducible Weyl representations of the CCR over HJ .

10.9 Van Hove Model In quantum field theory, ground states play basic roles. It is an important issue to find criteria for existence or absence of ground states in quantum field models. For some models, it is known that there are correspondences between absence of ground states and inequivalent representations of CCR. In this section we discuss this aspect by using a simple model, called the van Hove model [158] or the van Hove–Miyatake model [112, 113] or a fixed source model [58, 89, 114, 117]. We first give an algebraic structure whose Hilbert space operator realizations yield representations of the van Hove model. Let T be a non-negative and injective

432

10 Physical Correspondences in Quantum Field Theory

self-adjoint operator on a Hilbert space H such that T 1/2 ∈ SW (H ) (see Sect. 8.12) and g ∈ H be a vector satisfying g ∈ D(T −1/2 ) \ {0}.

(10.99)

The following commutation relations for algebraic elements H , φ(f ) and π(f ) (f ∈ D(T 2 ) ∩ W ) are called the van Hove relations associated with (T , g):  [φ(f ), π(f )] = i f, f , (10.100) [φ(f ), φ(f )] = 0,

[π(f ), π(f )] = 0, (10.101)   (10.102) i[H, φ(f )] = π(f ) − Im T −1/2 g, f ,   i[H, π(f )] = −φ(T 2 f ) − Re g, T 1/2 f , f ∈ D(T 2 ) ∩ W , f ∈ W . (10.103) Roughly speaking, the van Hove model for the pair (T , g) is defined to be a Hilbert space operator realization of (H, {φ(f ), π(f )|f ∈ W }) satisfying the van Hove relations associated with (T , g). Remark 10.14 Let W = C ∞ (T ) ∩ D(T −1/2 ) ∩ HJ . Then the van Hove relations associated with (T , g) show that, if H is infinite dimensional, then (H, I, {φ(T 2n f ), π(T 2n f )|n ∈ Z+ , f ∈ W }) forms an infinite-dimensional Lie algebra. By Theorem 6.11, the operator Hv defined by (6.77) is self-adjoint and bounded from below. Hence the following definition is possible: Definition 10.4 The abstract van Hove model or abstract fixed source model associated with (T , g) in the Fock representation is the triple Mv := (Fb (H ), Hv , {φT 1/2 (f ), πT 1/2 (f )|f ∈ W }) (see (8.118) and (8.119)). The vector g is called the fixed source of the model. One may call the model Mv the abstract van Hove–Miyatake model associated with (T , g) [112, 113]. One can easily see that (Hv , {φT 1/2 (f ), πT 1/2 (f )|f ∈ W }) is an operator realization of the van Hove relations associated with (T , g). Example 10.6 The original concrete fixed source model is a model describing an interaction of a fixed source ρ ∈ SR (R3 ) (e.g., fixed nucleons in the threedimensional space R3 ) with a quantum neutral (Hermitian) scalar field φ (e.g., the quantum field of π 0 mesons), which obeys the field equation 0 2 1 ∂ 2 − Δ + m φ(t, x) = ρ(x), (t, x) ∈ R × R3 , ∂t 2

10.9 Van Hove Model

433

in a suitable sense, where Δ is the three-dimensional Laplacian and m > 0 is the mass of one boson. Mathematically, it is a concrete realization of Mv with (H , T , g) given as follows: ˆ 3 ), H = L2 (R

T = ωm ,

ρˆ g = −√ , ωm

where ωm is given by (10.85) with m > 0 (the massive model) and the Fourier √ −1/2 ˆ 3 obeying the condition that ρ/ transform ρˆ of ρ is a function on R ˆ ωm ∈ D(ωm ) −1 (i.e., ρˆ ∈ D(ωm )). The distribution ρ describes a source which generates a quantum field around it. Hence the Hamiltonian of the concrete fixed source model takes the form 1 4 −1/2 ∗ −1/2 5 Hv (ρ) := Hb − √ a(ωm ρ) ˆ + a(ωm ρ) ˆ 2

(10.104)

ˆ 3 )), where Hb is defined by (10.86) and a(·) acting in the boson Fock space Fb (L2 (R 2 ˆ 3 )). Detailed descriptions and analysis of is the annihilation operator on Fb (L (R the model from physics point of view are given in [89, Chapter 9]. A mathematically rigorous comprehensive study of the model Mv including the massless case m = 0 is presented in [26, Chapter 12]; the contents of this section are mainly based on it (see also [28, Chapter 13] and [58, 59, 61]). The purpose of this section is to discuss existence and absence of ground states of the model Mv in relation to representations of CCR. Existence and absence of ground states of Hv depend on regularity of g. There are two cases to be distinguished on regularity of g: (i) The vector g ∈ H is said to be infrared regular with respect to T if g ∈ D(T −1 ).

(10.105)

(ii) The vector g ∈ H is said to be infrared singular with respect to T if g ∈ D(T −1 ).

(10.106)

A remark may be in order. If T −1 is bounded, then D(T −1 ) = H and hence every g ∈ H is infrared regular. Therefore, in this case, the definition above has no substantial meaning. In other words, only in the case where T −1 is unbounded, the definition above is meaningful. It is easy to see that T −1 is unbounded if and only if m(T ) := inf σ (T )

(10.107)

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10 Physical Correspondences in Quantum Field Theory

is zero.13 For convenience, we introduce the following definitions: if m(T ) > 0, then T is said to be massive;14 if m(T ) = 0, then T is said to be massless. Thus, if T is massive, then every g ∈ H is infrared regular with respect to T ; only in the case where T is massless can g be infrared singular with respect to T .

10.9.1 The Infrared Regular Case In this subsection, we assume (10.105). Then one can define a unitary operator by Ug (T ) := e−iΦS (iT

−1 g)

.

(10.108)

The following theorem shows that Hv can be diagonalized: Theorem 10.15 Assume (10.105). Then Ug (T )Hv Ug (T )−1 = db (T ) + Eg (T ),

(10.109)

where 1 Eg (T ) := − T −1/2 g 2 . 2

(10.110)

Proof We first consider the case where T is bounded. Then, by Example 8.9, (8.35) holds and 1/2 QCL = QA = D(db (T ))1/2 ) = D(Hˆ v ).

(10.111)

Hence 

∞    1/2 1/2 CL (T 1/2 en )Φ, CL (T 1/2 )Ψ , Hˆ v Φ, Hˆ v Φ =

1/2

Φ, Ψ ∈ D(Hv ),

n=1

where  1  L(f ) := √ T −1/2 g, T −1/2 f , f ∈ D(T −1/2 ) 2

(10.112)

statement is equivalent to that T −1 is bounded if and only if m(T ) > 0. This statement can be easily proved by using functional calculus. 14 Recall that, in the case where H = L2 (R ˆ d ) and T = ωm with a constant m ≥ 0 (see (10.85)), one has inf σ (ωm ) = m. 13 The

10.9 Van Hove Model

435

and {en }∞ n=1 is a CONS of H . For notational simplicity, we set U := Ug (T ). By Theorem 8.5 and its proof, we have U CL (f )U −1 = A(f ),

f ∈ D(T −1/2 ),

Hence ∞ 

∞    CL (T 1/2 en )Φ, CL (T 1/2 )Ψ = A(T 1/2 en )U Φ, A(T 1/2 en )U Ψ

n=1

n=1

  = db (T )1/2U Φ, db (T )1/2 U Ψ .

Therefore 

   1/2 1/2 Hˆ v Φ, Hˆ v Ψ = db (T )1/2 U Φ, db (T )1/2 U Ψ .

It follows from this fact and (10.111) that operator equality Hˆ v = U −1 db (T )U holds. This is equivalent to (10.109). We next consider the case where T is unbounded. For each n ∈ N, the operator Tn := T (n−1 T + 1)−1 is a non-negative and injective bounded self-adjoint operator on H . It is easy to see that lim Tn f = Tf,

n→∞

Ψ ∈ D(T ).

The domain D(T ) is trivially a core for T . Hence, by Theorem 6.2, s- lim eit db(Tn ) = eit db(T ) . n→∞

By the preceding result, we have Ug (Tn )Hv(n) Ug (Tn )−1 = db (Tn ) + Eg (Tn ), (n)

where Hv

:= db (Tn ) + (n)

√1 (A(g)∗ 2

+ A(g)). Hence, for all t ∈ R,

Ug (Tn )eit Hv Ug (Tn )−1 = eit db(Tn ) eit Eg (Tn ) .

(10.113)

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10 Physical Correspondences in Quantum Field Theory

It is easy to see that lim Hv(n) Ψ = Hv Ψ,

Ψ ∈ Fb,fin (D(T )).

n→∞

Recall that any core for db (T ) is a core for Hv . Hence Fb,fin (D(T )) is a core for Hv . Therefore, by Theorems 1.29 and 1.30, (n)

s- lim eit Hv = eit Hv . n→∞

By functional calculus, one can show that, for all α > 0, −1/2

lim Tn

n→∞

lim T −1 h n→∞ n

f = T −1/2 f, = T −1 h,

f ∈ D(T −1/2 ),

h ∈ D(T −1 ).

The first formula implies that lim Eg (Tn ) = Eg (T ),

n→∞

while the second and Theorem 6.22(iii) imply that s- lim Ug (Tn ) = Ug (T ), n→∞

s- lim Ug (Tn )−1 = Ug (T )−1 . n→∞

Now, taking the limit n → ∞ in (10.113), we obtain Ug (T )eit Hv Ug (T )−1 = eit db(T ) eit Eg (T ) = eit (db(T )+Eg (T )) . By the unitary covariance of functional calculus, the left hand side is equal to exp(itUg (T )Hv Ug (T )−1 ). Thus (10.109) is proved.  Remark 10.15 There is another proof of (10.109) (see [28, Theorem 13.4(i)]). As a consequence of Theorem 10.15 and Corollary 8.4, we obtain the following: Corollary 10.5 Assume (10.105). Then Hv has a unique ground state given by Ωg (T ) := Ug (T )−1 ΩH up to constant multiples with the ground state energy being Eg (T ). Moreover, the representation (Fb (H ), Fb,0 (H ), {CL (f )|f ∈ D(T −1/2 )}) of the CCR over D(T −1/2 ) with L given by (10.112) is equivalent to the Fock representation ρF (D(T −1/2 )).

10.9 Van Hove Model

437

Let H := db (T ) + Eg (T ),

(10.114)

φ (f ) := Ug (T )φT 1/2 (f )Ug (T )−1 ,

(10.115)

π (f ) := Ug (T )πT 1/2 (f )Ug (T )−1 ,

f ∈W.

(10.116)

Then Theorem 10.15 yields the following result: Corollary 10.6 Assume (10.105). Then the abstract fixed source model Mv in the Fock representation is equivalent to the model (Fb (H ), H , {φ (f ), π (f )| f ∈ V }). One can easily prove the following formulae:   φ (f ) = φT 1/2 (f ) − Re T −1 g, T −1/2 f ,   π (f ) = πT 1/2 (f ) + Im T −1/2 g, f .

(10.117) (10.118)

10.9.2 The Infrared Singular Case We next consider the case where g obeys (10.106). Let L be defined by (10.112). We fix a dense subspace V ⊂ D(T −1/2 ). Theorem 10.16 Assume (10.106). Let V be a dense subspace of H which is a core T −1/2 . Then the representation (Fb (H ), Fb,0 (H ), {CL (f )|f ∈ V }) is inequivalent to the Fock representation ρF (V ). Proof Suppose that there existed √a vector  h ∈ H such that L(f ) = h, f  , f ∈  −1/2 −1/2 V . Then T g, T f = 2h, f . Since V is a core for T −1/2 , it follows √ that T −1/2 g ∈ D(T −1/2 ) and T −1/2 T −1/2 g = 2h. The former means that g ∈ D(T −1 ). But this contradicts (10.106). Hence such a vector h does not exist. Thus Corollary 8.4 yields the result.  The inequivalence stated in Theorem 10.16 is related to absence of ground states of Hv : Theorem 10.17 Assume (10.106). Suppose that T is bounded. Then Hv has no ground states. Proof Suppose that Hv had a ground state Ω = 0. Then Hˆ v Ω = 0. Hence, 1/2 1/2 by Example 8.9, Ω ∈ D(HβL ) and HβL Ω = 0. Then, by Proposition 8.1(ii), CL (f )Ω = 0 for all f ∈ W := span{T 1/2 en |n ∈ N}, where {en }∞ n=1 ⊂ D(T 1/2 ) is a CONS of H . Since T 1/2 is bounded, it follows that W is dense

438

10 Physical Correspondences in Quantum Field Theory

in H . Hence, by Lemma 8.4 with D = W , there exists a vector h ∈ H such that L(f ) = h, f H , f ∈ W . Note that W is a core for T −1/2 . Hence it follows that T −1/2 g ∈ D(T −1/2 ), i.e., g ∈ D(T −1 ) and T −1 g = h. But this contradicts (10.106). Thus Hv has no ground states.  Remark 10.16 In Theorem 10.17, the boundedness of T is assumed for simplicity. But the conclusion holds in the theorem also in the case where T is unbounded and absolutely continuous. For details, see [28, Theorem 13.8].

10.9.3 Infrared Divergence In this subsection, we point out that Theorem 10.17 is related to a characteristic phenomenon. We continue to assume (10.106) (the infrared singular case). For each a > 0, we define a vector ga ∈ H by ga := ET ([a, ∞))g. The number a is called an infrared cutoff for g. It is easy to see that ga ∈ D(T −1 ). Hence one can define 1 Ha := db (T ) + √ (A(ga )∗ + A(ga )). 2 We call it the van Hove Hamiltonian with infrared cutoff a. By Corollary 10.5, Ha has a ground state Ωa := Ωga (T ) = eiΦS (iT

−1 g

a)

ΩH .

Theorem 10.18 Assume (10.106). Let Nb be the boson number operator on Fb (H ) (see (6.11)). Then Ωa ∈ D(Nb ) and lim Ωa , Nb Ωa  = ∞.

a→0

(10.119)

Proof Considering the case T = I in Theorem 10.15 with g replaced by −h ∈ H , we have for all h ∈ H 1 1 e−iΦS (ih) Nb eiΦS (ih) = Nb − √ (A(h)∗ + A(h)) + h 2 . 2 2 Taking h = T −1 ga , we obtain e−iΦS (iT

−1 g

a)

Nb eiΦS (iT

−1 g

a)

1 1 = Nb − √ (A(T −1 ga )∗ + A(T −1 ga )) + T −1 ga 2 . 2 2

10.9 Van Hove Model

439

Hence, Ωa ∈ D(Nb ) and, using (6.39) and (6.12), we obtain Ωa , Nb Ωa  =

1 −1

T ga 2 . 2

Condition (10.106) implies that lim T −1 ga 2 = ∞.

(10.120)

a→0



Thus (10.119) follows.

Formula (10.119) physically means that the mean value of bare boson numbers in the ground state Ωa diverges as the infrared cutoff a is removed. This is one of the phenomena called infrared divergences peculiar to theories of massless quantum fields. Thus we see that an infrared divergence is associated with the absence of ground states of the Hamiltonian Hv with g infrared singular with respect to T . There is another interesting phenomenon: Theorem 10.19 Assume (10.106). Then w- lim Ωa = 0,

(10.121)

a→0

where w- lim means weak limit. Proof We have by (6.106) ΩH , Ωa  = e− T

−1 g

2 a /4

.

Hence, by (10.120), lim ΩH , Ωa  = 0.

a→0

Next, let Ψ := A(f1 )∗ · · · A(fn )∗ ΩH ,

f1 , . . . , fn ∈ H .

Then, by (6.107), we have ⎞ ⎛ n  (−1)n   −1 2 Ψ, Ωa  = n/2 ⎝ fj , T −1 ga ⎠ e− T ga /4 . 2 j =1

Hence, by the Schwarz inequality, ⎞ ⎛ n 1 ⎝ −1 2 | Ψ, Ωa  | ≤ n/2

fj ⎠ T −1 ga n e− T ga /4 . 2 j =1

440

10 Physical Correspondences in Quantum Field Theory

It is easy to see that limt →∞ t n e−t all Φ ∈ Fb,fin (H ),

2 /4

= 0. Hence lima→0 Ψ, Ωa  = 0. Thus, for

lim Φ, Ωa  = 0.

a→∞

(10.122)

Finally, let Ψ ∈ Fb (H ). Since Fb,fin (H ) is dense in Fb (H ), there exists a sequence {Ψn }∞ n=1 in Fb,fin (H ) such that limn→∞ Ψn − Ψ = 0. Using the identity Ψ, Ωa  = Ψ − Ψn , Ωa  + Ψn , Ωa  , and the Schwarz inequality with Ωa = 1, we have | Ψ, Ωa  | ≤ Ψ − Ψn + | Ψn , Ωa  |. Hence, by (10.122), lim sup | Ψ, Ωa  | ≤ Ψ − Ψn . a→0

Then, taking the limit n → ∞, we obtain lim supa→0 | Ψ, Ωa  | = 0, which is equivalent to lima→0 Ψ, Ωa  = 0. Thus (10.121) holds.  Formula (10.121) may be regarded as a form of infrared divergence in the present model, suggesting the absence of ground states in the infrared singular case. We call property (10.121) the infrared van Hove–Miyatake phenomenon or the infrared van Hove–Miyatake catastrophe.15

10.9.4 Infrared Renormalization Theorem 10.17 shows that, under the infrared singular condition (10.106), Hv with T bounded has no ground states. But, if one passes to a representation of CCR inequivalent to the Fock representation for the time-zero fields, then one can construct a fixed source model which has a ground state. We show this below. The idea is to note that the operators φ (f ) and π (f ) given by (10.117) and (10.118) respectively can be defined also in the case where (10.106) holds if test vectors f are in a smaller space. In this subsection, we assume (10.106). Let V := D(T −1 ) ∩ D(T 1/2 )

(10.123)

15 Originally the term “van Hove–Miyatake catastrophe” was used in relation to ultraviolet divergence of a massive van Hove–Miyatake model [68]. See Sect. 10.9.6 below.

10.9 Van Hove Model

441

and define Hren := db (T ) + Eg (T ),

(10.124)

  φren (f ) := φT 1/2 (f ) − Re T −1/2 g, T −1 f ,   πren (f ) := πT 1/2 (f ) + Im T −1/2 g, f , f ∈ V .

(10.125) (10.126)

Then it is obvious that Hren is a self-adjoint operator which has a unique ground state given by Ωren := ΩH , up to constant multiples, i.e., Hren Ωren = Eg (T )Ωren . It follows that πren (V ) := (Fb (H ), Fb,0 (H ), {φren (f ), πren (f )|f ∈ V })

(10.127)

is a representation of the Heisenberg CCR over V . Note that this representation is defined even if g is infrared singular with respect to T (i.e., g ∈ D(T −1 )). Lemma 10.3 The representation πren (V ) is an irreducible Weyl representation of the CCR over V . 

Proof This follows from Theorem 8.15 and Proposition 8.9.

One can show that (Hren , {φren (f ), πren (f )|f ∈ V }) satisfies the van Hove relations associated with (T , g). Hence Mv,

ren

:= (Fb (H ), Hren , {φren (f ), πren (f )|f ∈ V })

is a fixed source model in the infrared singular case, which has a ground state. The following theorem shows that Mv, ren is inequivalent to Mv under condition (10.106). Theorem 10.20 Assume (10.106). Then (Fb (H ), {φren (f ), πren (f )|f ∈ V }) is inequivalent to (Fb (H ), {φT 1/2 (f ), πT 1/2 (f )|f ∈ V }). Proof Suppose that the two representations under consideration were equivalent. Then there exists a unitary operator U on Fb (H ) such that U φT 1/2 (f )U −1 = φren (f ),

U πT 1/2 (f )U −1 = πren (f ),

f ∈ V .

442

10 Physical Correspondences in Quantum Field Theory

By change of test vectors f , we have   U φF (f )U −1 = φF (f ) − Re T −1/2 g, T −1/2 f ,   U πF (f )U −1 = πF (f ) + Im T −1/2 g, T −1/2 f ,

f ∈ D(T −3/2 ) ∩ D(T ) ∩ HJ ,

where we have used the fact that (T −1/2 V ) ∩ (T 1/2 V ) = D(T −3/2 ) ∩ D(T ). By the functional calculus for T , D(T −3/2 ) ∩ D(T ) ∩ HJ is dense in HJ . Hence, by Theorem 8.12, there exists a vector h such that   − T −1/2 g, T −1/2 f = h, f  ,

f ∈ D(T −3/2 ) ∩ D(T ).

It follows from the functional calculus for T that D(T −3/2 ) ∩ D(T ) is a core for T −1/2 . Hence T −1/2 g ∈ D(T −1/2 ), i.e., g ∈ D(T −1 ) and T −1 g = −h. But this contradicts (10.106). Thus the desired result follows.  Generally speaking, in a quantum field model which has infrared divergences in a representation of time-zero fields (CCR), a prescription passing to another representation of the model in which there are no infrared divergences is called an infrared renormalization of the model. The theory presented above describes an infrared renormalization of an infrared singular fixed source model. From this point of view, we call Mv, ren the infraredrenormalized van Hove model. A moral of the results obtained above is that, even if a quantum field model has no ground states in a representation of time-zero fields, it may have a ground state in a representation of time-zero fields which is inequivalent to the former. Remark 10.17 The infrared renormalization of the van Hove model described above may be regarded as a mathematically rigorous form for the procedure given by Bloch and Nordsieck [43] which makes a simplified model in quantum electrodynamics (essentially same as the concrete massless fixed source model) free from infrared divergences (cf. also [150, Part II, Example 3.2]). Remark 10.18 As a generalization of a fixed source model, one can consider a model of N non-relativistic quantum particles in a scalar potential interacting with a quantum scalar field. This model is called the Nelson model [74, 121]. It is shown in [106] that the three-dimensional massless Nelson model with N = 1 has no ground states in the infrared singular case. But, in this case, the same idea as above can be applied to construct a massless Nelson model which has a ground state. See [14, 139]. An infrared renormalization of the massless Nelson model using functional integral methods is made in [105, 107].

10.9 Van Hove Model

443

10.9.5 Representations Indexed by Sources It may be interesting to study dependence of the representation (Fb (H ), {φren (f ), πren (f )|f ∈ V }) on the source g. For this purpose, we write Mv, ren = Mren (g), φ (g)(f ) := φren (f ),

Hren = Hren (g), π (g) (f ) := πren (f ),

f ∈ V .

Then πg := (Fb (H ), {φ (g) (f ), π (g) (f )|f ∈ V }) is an irreducible Weyl representation of the CCR over V . We suppose that g varies in the following subset of H : G := {g ∈ D(T −1/2 )|g ∈ D(T −1 }.

(10.128)

Then it may be interesting to examine equivalence or inequivalence of two representations πg1 and πg2 for g1 , g2 ∈ G . One has the following fact: Theorem 10.21 Let g1 , g2 ∈ G . Then πg1 is equivalent to πg2 if and only if g1 − g2 ∈ D(T −1 ). Proof Suppose that πg1 is equivalent to πg2 . Then there exists a unitary operator U on Fb (H ) such that, for all f ∈ V , U φ (g2 ) (f )U −1 = φ (g1 ) (f ),

U π (g2 ) (f )U −1 = π (g1 ) (f ).

This is equivalent to   U φT 1/2 (f )U −1 = φT 1/2 (f ) − Re T −1/2 (g1 − g2 ), T −1 f ,   U πT 1/2 (f )U −1 = πT 1/2 (f ) + Im T −1/2 (g1 − g2 ), f .

(10.129) (10.130)

This means that πg1 −g2 is equivalent to (Fb (H ), {φT 1/2 (f ), πT 1/2 (f )|f ∈ V }). Hence, by Theorem 10.20, g1 − g2 ∈ D(T −1 ). Conversely, suppose that g1 − g2 ∈ D(T −1 ). Then the source g = g1 − g2 is infrared regular. Hence, by Corollary 10.6, there exists a unitary operator U on Fb (H ) such that (10.129) and (10.130) hold. Hence it follows that πg1 is equivalent to πg2 . 

444

10 Physical Correspondences in Quantum Field Theory

Theorem 10.21 implies the following fact on inequivalence of models Mren (g1 ) and Mren (g2 ) (g1 , g2 ∈ G ): Corollary 10.7 Let g1 , g2 ∈ G and g1 − g2 ∈ D(T −1 ). Then Mren (g1 ) is inequivalent to Mren (g2 ). Proof Suppose that Mren (g1 ) is equivalent to Mren (g2 ). Then πg1 is equivalent to πg2 . Hence, by Theorem 10.21, g1 − g2 ∈ D(T −1 ).  Remark 10.19 It is unclear if the converse of Corollary 10.7 holds. To understand Theorem 10.21 in an intrinsic way, we define a relation ∼ in G as follows: g1 , g2 ∈ G , g1 ∼g2 ⇐⇒ g1 − g2 ∈ D(T −1 ). def

(10.131)

It is easy to see that the relation ∼ is an equivalence relation. We denote by [g] the equivalence class of g ∈ G with respect to ∼ and set G / ∼:= {[g]|g ∈ G } (the quotient set by the equivalence relation ∼). Theorem 10.21 is rephrased as follows: πg1 is equivalent to πg2 if and only if [g1 ] = [g2 ]. Therefore, for each equivalence class [g], we define a representation π[g] by π[g] := πg modulo equivalence. Thus we have a family {π[g] |[g] ∈ G / ∼} of mutually inequivalent irreducible Weyl representations of the CCR over V .

10.9.6 Ultraviolet Renormalization The fixed source model in the Fock representation has another type of divergences. Example 10.7 Consider the concrete fixed source model described in Example 10.6 with m > 0 (the massive case). A physical case is given by the case where ρ is a finite linear combination of Dirac’s delta distributions: ρ(x) =

n

λj δ(x − R j ),

j =1

where n ∈ N, δ(·) is the three-dimensional Dirac delta distribution, λj ∈ R \ {0} is a constant and R j ∈ R3 denotes the position of the j th source. In this case ρ is called a point source. Hence ρ(k) ˆ =

n 1 λj e−ikR j . (2π)3/2 j =1

10.9 Van Hove Model

445

It is not difficult to show that ρˆ gm := − √ ωm

(10.132)

ˆ 3 ). Hence, in this case, the interaction term is not in L2 (R 1 4 −1/2 ∗ −1/2 5 − √ a(ωm ρ) ˆ + a(ωm ρ) ˆ 2 in the Hamiltonian Hv (ρ) is meaningless. Therefore the concrete fixed source model in this case cannot be defined in the Fock representation. To avoid this difficulty, one usually introduces a regularized source ρˆκ := χκ ρˆ √ with κ > 0, where χκ is defined by (10.56). It is obvious that ρˆκ / ωm is in ˆ 3 ). As in the case of the model Hκ discussed in Sect. 10.6.1, the parameter L2 (R κ denotes an ultraviolet cutoff. Then one can define a Hamiltonian Hv,κ := H (ρˆκ ) with ultraviolet cutoff κ by replacing ρˆ with ρˆκ in the Hamiltonian Hv (ρ). As we shall show in an abstract setting below, the weak limit of the ground state of Hv,κ as κ → ∞ (the removal of the ultraviolet cutoff) is zero. Moreover, as heuristically seen, one cannot expect that Hv,κ converges in a suitable sense within the Fock ˆ 3 )). These are examples of the so-called ultraviolet divergences. space Fb (L2 (R We now describe ultraviolet divergence in the abstract fixed source model. For simplicity, we assume the following: Assumption (m) The operator T is massive, i.e., m(T ) > 0 and T is unbounded. Then, as already remarked, T −1 ∈ B(H ). Hence conditions (10.105) and (10.106) are trivially satisfied for all g ∈ H . We want to construct a fixed source model with g ∈ H . For this purpose, we introduce a scale of Hilbert spaces. For s > 0, the domain D(T s ) is a Hilbert space with inner product 

 f, f s := T s f, T s f ,

f, f ∈ D(T s ).

We denote this Hilbert space by Hs . We set H0 := H . The sesquilinear form 

f, f

−s

 := T −s f, T −s f

(f, f ∈ H )

on H is an inner product of H . We denote the completion of the inner product space (H , ·, ·−s ) by H−s . We have Hs1 ⊂ Hs2 ⊂ H ⊂ H−s2 ⊂ H−s1 ,

s1 > s2 > 0.

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10 Physical Correspondences in Quantum Field Theory

By the Schwarz inequality, we have  | φ, f H | = | T −s φ, T s f | ≤ φ −s f s ,

φ ∈ H , f ∈ Hs .

Hence, for each f ∈ Hs , the mapping uf : φ → φ, f H is a continuous antilinear functional on H−s with domain D(uf ) = H . Hence it extends uniquely to a continuous anti-linear functional uˆ f on H−s such that uˆ f (φ) = uf (φ), φ ∈ H and |uˆ f (φ)| ≤ φ −s f s ,

φ ∈ H−s , f ∈ Hs .

We write uˆ f (φ) = −s φ, f s ,

φ ∈ H−s , f ∈ Hs .

Throughout this sesquilinear form, one can identify the dual space Hs∗ of Hs with H−s . It follows that, for each φ ∈ H−s , there exists a sequence {φn }∞ n=1 in H such that −s

 φ, f s = lim φn , f H = lim T −s φn , T s f H , n→∞

n→∞

f ∈ Hs .

It is easy to see that, for all s ≥ 0, T is a unitary operator from Hs+1 to Hs . Hence ∗ =H its dual mapping (Banach space adjoint) T from Hs∗ = H−s to Hs+1 −(s+1) , ∗ ∗ 16 defined by (T φ)(f ) := φ(Tf ) (φ ∈ Hs , f ∈ Hs+1 ), is unitary. In what follows we assume the following: Assumption (g) For some p > 0, g ∈ H−(p+1) and (T )−1 g ∈ H . Let np be the natural number such that p + 1 ≤ np < p + 2. Then np ≥ 2. Note that (T )−1 maps H−(s+1) onto H−s . Hence, by Assumption (g), (T )−np g ∈ H−(p+1−np ) . We have 0 ≤ np − (p + 1) < 1. Hence g := (T )−np g is in H . For each κ > 0, ET ([0, κ])g is in C ∞ (T ), where ET is the spectral measure of T . Hence one can define a vector gκ := T np ET ([0, κ])g .

theorem: Let H and K be Hilbert spaces. Then the mapping B(H , K ) T → T ∈ B(K ∗ , H ∗ ) is an isometric isomorphism. In particular, T = T . For a proof, see, e.g., [130, Theorem VI.2].

16 Apply the following

10.9 Van Hove Model

447

The next lemma shows that gκ can be regarded as an ultraviolet cutoff of g: Lemma 10.4 limκ→∞ gκ = g in H−np . Proof By the unitarity of (T )−1 , we have

gκ − g −np = (T )−np gκ − (T )−np g H = ET ([0, κ])g − g H = 0. 

Hence the result follows. Lemma 10.5 limκ→∞ T −1 gκ H = ∞. Proof We have

T −1 gκ H = T np −1 ET ([0, κ])g .

The condition (T )−1 g ∈ H implies that g ∈ D(T np −1 ). Hence it follows that limκ→∞ T np −1 ET ([0, κ])g = ∞. Thus we obtain the result.  We now define a Hamiltonian of the fixed source model with an ultraviolet cutoff source gκ : 1 H (κ) := db (T ) + √ (A(gκ )∗ + A(gκ )). 2

(10.133)

As we already know, Hκ is self-adjoint with D(Hκ ) = D(db (T )) and bounded from below with 1 H (κ) ≥ − T −1/2 gκ 2 . 2 By Corollary 10.5, Hκ has a unique ground state Ω(κ) := eiΦS (iT

−1 g

κ)

ΩH .

Theorem 10.22 (Ultraviolet Divergences) Under Assumption (g), the following hold: (i) limκ→∞ Ω(κ), Nb Ω(κ) = ∞. (ii) w- limκ→∞ Ω(κ) = 0. Proof We have Lemma 10.5. Hence part (i) (resp. (ii) ) can be proved in quite the same way as in the proof Theorem 10.18 (resp. Theorem 10.19).  We call the fact in part (ii) of Theorem 10.22 the ultraviolet van Hove–Miyatake phenomenon (catastrophe) in the cutoff model (Fb (H ), H (κ), {φT 1/2 (f ), πT 1/2 (f ) |f ∈ V }) (cf. [68]). Remark 10.20 In relativistic quantum field theory, there is a model of a neutral quantum scalar field with a self-interaction on the two-dimensional space–time,

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10 Physical Correspondences in Quantum Field Theory

called the P (φ)2 model, where φ denotes the scalar field, P is a real polynomial of one variable bounded from below and P (φ) describes the self-interaction of the quantum field. This model is well-defined in the Fock representation for the time-zero fields without ultraviolet cutoff, but with a volume cutoff  > 0 (for the details, see [147] or [72]). In this volume cutoff model too, the van Hove–Miyatake phenomenon occurs as  → ∞ ([147, Corollary VI.11], [72, Corollary 9.5.6]). Using the same idea as in the infrared renormalization, one can construct a fixed source model without ultraviolet cutoff. Namely, we need only to introduce the following operators: H∞ := db (T ),

(10.134)

φ∞ (f ) := φT 1/2 (f ) − Re

−(p+1) g, T

−3/2

π∞ (f ) := πT 1/2 (f ) + Im

−(p+1) g, T

−1/2

f p+1 , f p+1 ,

(10.135) f ∈ Vp ,

(10.136)

where Vp := Hp+ 1 ∩ HJ . 2

It is obvious that H∞ is a self-adjoint operator and has a unique ground state given by Ω∞ := ΩH , i.e., H∞ Ω∞ = 0. It follows from Theorem 8.15 and Proposition 8.9 that {φ∞ (f ), π∞ (f )|f ∈ Vp } is an irreducible Weyl representation of the CCR over Vp . We call this representation the ultraviolet renormalized representation of the CCR over Vp with respect to (T , g). Moreover, (H∞ , {φ∞ (f ), π∞ (f )|f ∈ Vp }) satisfies the van Hove relations associated with (T , g) with the following replacements: T −1/2 g, f  → −(p+1) g, T −1/2 f p+1 ,

g, T 1/2 f  → −(p+1) g, T 1/2 f p+1 ,

V → Vp . Hence Mv, ∞ := (Fb (H ), H∞ , {φ∞ (f ), π∞ (f )|f ∈ Vp }) is a fixed source model without ultraviolet cutoff, which has a ground state.

10.9 Van Hove Model

449

Remark 10.21 To define the renormalized Hamiltonian H∞ , we have dropped on the right hand side of (10.134) the ground state energy term corresponding to Eg (T ) in (10.124), because, in the present case, T −1/2 g H is ill-defined. This is related to the following fact: Lemma 10.5 implies that limκ→∞ T −1/2 gκ 2 = ∞. Hence limκ→∞ Egκ (T ) = −∞. The operator H∞ heuristically corresponds to the symbolical expression “limκ→∞ (H (κ)−Egκ (T ))”. This kind of procedure is called an energy renormalization. In the same way as in the proof of Theorem 10.20, one can prove the following theorem: Theorem 10.23 Under Assumption (g), the representation (Fb (H ), {φ∞ (f ), π∞ (f ) | f ∈ Vp }) is inequivalent to (Fb (H ), {φT 1/2 (f ), πT 1/2 (f )|f ∈ Vp }). This theorem shows that the model Mv, ∞ is inequivalent to the free field model M(T , Vp ) (see Example 10.2), although the Hamiltonian H∞ of the model Mv, ∞ is the same as that of the latter. The prescription of the infrared and the ultraviolet renormalizations discussed above suggests that, to construct a divergence free quantum field model corresponding to a model which has infrared or ultraviolet divergences in the Fock representation of CCR, one must use a CCR-representation for time-zero fields which is inequivalent to the Fock one. Remark 10.22 Also for the ultraviolet renormalized model M∞ (g) := (Fb (H ), H∞ , {φ∞ (f ), π∞ (f )|f ∈ Vp }), considerations similar to those in Sect. 10.9.5 can be made. Example 10.8 In the original concrete fixed source model without ultraviolet cutoff (see Example 10.7), we have −s ˆ 3) ωm gm ∈ L2 (R −1 g ∈ L2 (R ˆ 3 ). Hence Assumption (g) with H = L2 (R ˆ 3 ), for all s > 1 and ωm m T = ωm and g = gm is satisfied. Therefore we can apply the abstract theory given above to the concrete fixed source model without ultraviolet cutoff to obtain an ultraviolet renormalized model M∞ (gm ).

In concluding descriptions on Bose field theories, we want to make a remark from a point of view of constructive quantum field theory [73, 80, 95]. In the standard method of construction of an interacting model of quantum scalar field, one usually takes as time-zero fields the Fock representation {φm (f ), πm (f )|f ∈ SR (Rd )}. But, to construct a model with the full relativistic symmetry [45, 80, 85, 149, 152], this choice of time-zero fields is not adequate due to Haag’s theorem which states essentially that a relativistic quantum field model whose time-t fields are unitarily equivalent to those of a free quantum field at a fixed time is completely

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10 Physical Correspondences in Quantum Field Theory

equivalent to the free field model [82, 83, 108]. Haag’s theorem suggests that, to construct a non-trivial relativistic quantum field model (a relativistic quantum field model describing an interaction of elementary particles), one must use as time-zero fields ones inequivalent to {φm (f ), πm (f )|f ∈ SR (Rd )}.17 In this sense too, it is important to study irreducible representations of CCR inequivalent to {πm }m≥0 . A more detailed account of this aspect can be found in [150, Part II, Chapter 3]. It should be mentioned that it still is a very important open problem to construct a non-trivial relativistic quantum field model on the four-dimensional Minkowski space–time (see also [95]).

10.10

Free Quantum Dirac Fields

In Sect. 10.7, we have seen that the masses of scalar bosons form the index set of a family of mutually inequivalent Weyl representations of CCR, which consists of time-zero fields of free quantum scalar fields. In this section, we show that, for fermion masses also, there is a fact corresponding to it. For simplicity, we restrict ourselves to the case of Dirac particles, i.e., relativistic charged particles with spin 1/2 and consider free quantum Dirac fields on the four-dimensional space–time. In Sect. 4.5.12, we have already described the free Dirac operator HD of mass m ≥ 0. We follow the notation there.

10.10.1 Eigenvectors of hD (k) and Some Operators For simplicity, we work with the standard representation for the matrices (α, β) given by (4.79). We denote by {er }4r=1 the standard basis of C4 : er = (δrr )4r =1 . For ˆ 3 and s = 1, 2, we define the following vectors in C4 : all k ∈ R um (k, s) := uD (k)−1 es ∈ C4 ,

vm (k, s) := uD (k)−1 es+2 ∈ C4 .

(10.137)

By (4.88), we have hD (k)um (k, s) = Em (k)um (k, s),

hD (k)vm (k, s) = −Em (k)vm (k, s).

Namely, um (k, s) (resp. vm (k, s)) is an eigenvector of hD (k) with positive (resp. negative) energy Em (k) (resp. −Em (k)). Since uD (k)−1 is unitary, it follows that,

17 This is not restricted to relativistic quantum field models. In fact, we have so far seen that this kind of structure exists also in non-relativistic quantum field models discussed in the previous sections (e.g., infinite volume theories of bosons, van Hove model without infrared and/or ultraviolet cutoffs).

10.10 Free Quantum Dirac Fields

451

ˆ d , the set {um (k, s), vm (k, s)|s = 1, 2} is a complete orthonormal for each k ∈ R basis of C4 . Hence 

um (k, s), um (k, s ) C4 = δss ,  vm (k, s), vm (k, s ) C4 = δss ,  um (k, s), vm (k, s ) C4 = 0, s, s = 1, 2.

(10.138) (10.139) (10.140)

and 2 4 5 umr (k, s)umr (k, s)∗ + vmr (k, s)vmr (k, s)∗ = δrr ,

r, r = 1, 2, 3, 4,

s=1

(10.141) where umr (k, s) (resp. vmr (k, s)) is the rth component of the vector um (k, s) (resp. vm (k, s)). Let ˆ 3 ; C4 ), HˆD := F3 HD = L2 (R

(10.142)

where F3 is the three-dimensional Fourier transform. Then one has the orthogonal decomposition HˆD = HˆD+ ⊕ HˆD−

(10.143)

with ˆ 3 ; C2 ). HˆD± := L2 (R We define linear operators Tm± : HˆD → HˆD± by Tm+ f := (um (·, s)∗ f )2s=1 ∈ HˆD+ , Tm− f := (v˜m (·, s)f˜)2s=1 ∈ HˆD− ,

(10.144) f ∈ HˆD ,

ˆ 3 → C4 , where, for w = (wr )4r=1 : R (wf )(k) :=

4

wr (k)fr (k),

ˆ3 f = (fr )4r=1 ∈ HˆD , k ∈ R

r=1

and w(k) ˜ := w(−k),

ˆ 3. k∈R

(10.145)

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10 Physical Correspondences in Quantum Field Theory

It follows that Tm± are bounded with Tm± ≤ It is easy to see that ∗ (Tm+ f+ )r =

2

umr (·, s)f+s ,

√ 2.

f+ = (f+s )2s=1 ∈ HˆD+ ,

(10.146)

s=1 ∗ f− )r = (Tm−

2

vmr (·, s)∗ f˜−s ,

f− = (f−s )2s=1 ∈ HˆD− , r = 1, 2, 3, 4.

s=1

(10.147) We denote by C the complex conjugation on HˆD : Cf := (fr∗ )4r=1 . Throughout this section below, for notational simplicity, the conjugation of a bounded linear operator A ∈ B(HˆD ) with respect to C (see (8.51)) is denoted by A (this is not the closure of A): A := AC = CAC. We regard HˆD± as subspaces of HˆD in the natural way so that C acts also on HˆD± . Important properties of Tm± are summarized in the next lemma: Lemma 10.6 ∗

∗ Tm+ Tm+ + T m− T m− = I,

(10.148)

∗ Tm± Tm±

(10.149)

∗

= I,

Tm+ T m− = 0,

∗

Tm− T m+ = 0.

(10.150)

Proof Throughout the proof, we write T± (resp. u(k, s), v(k, s)) for Tm± (resp. um (k, s), vm (k, s)). Let f, g ∈ HˆD . Then  T+ f, T+ g + T − f, T − g =

=

=

2  3 s=1 R

' ( (u(·, s)f ∗ )(k)(u(·, s)∗ g)(k) + (v(·, s)f ∗ )(k)(v(·, s)∗ g)(k) dk

4  3 r,r =1 R

4  3 r=1 R



2 4 5 ur (k, s)ur (k, s)∗ + vr (k, s)vr (k, s)∗ dk fr (k)∗ gr (k)

fr (k)∗ gr (k)dk,

s=1

10.10 Free Quantum Dirac Fields

453

where we have used (10.141) to obtain the last equality. Hence  T+ f, T+ g + T − f, T − g = f, g . This implies (10.148). We have by (10.146) 

4 2  ur (·, s)fs , ur (·, s )gs T+∗ f, T+∗ g = r=1 s,s =1

 =

ˆ3 R

 u(k, s), u(k, s ) fs (k)∗ gs (k)dk

= f, g , where we have used (10.138). Hence T+ T+∗ = I . Similarly, using (10.147), one can show that T− T−∗ = I . One can see that orthogonality (10.140) implies (10.150).  Lemma 10.6 immediately yields the following result: Lemma 10.7 For all m ≥ 0, Tm := (Tm+ , Tm− ) is an element of T∗ (HˆD ), where T∗ (HˆD ) is defined by (9.18) with K replaced by HˆD . ∗ = 1 and hence It follows from (10.149) that Tm±

Tm± = 1.

(10.151)

10.10.2 Construction of a Free Quantum Dirac Field We construct a free quantum Dirac field on the fermion Fock space Ff (HˆD ) over HˆD . Note that we work with momentum representation. We denote by b(f ) (f ∈ HˆD ) the annihilation operator on Ff (HˆD ). For each f ∈ HˆD , we define ψˆ m (f ) := b(Tm+ f, 0) + b(0, Tm− f ∗ )∗ ,

f ∈ HˆD

(10.152)

and set ρˆm := (Ff (HˆD , {ψˆ m (f ), ψˆ m (f )∗ |f ∈ HˆD }).

(10.153)

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10 Physical Correspondences in Quantum Field Theory

Remark 10.23 In a standard construction of a free quantum Dirac field, the projection method as mentioned in Remark 9.3 is used (see, e.g., [124, 136] or [156, §10.1]). But we find that the projection method is somewhat inconvenient when discussing a family of quantum Dirac fields indexed by mass m, since the orthogonal decomposition of HˆD in the projection method depends on m. Thus we take a slightly different approach in which the orthogonal decomposition (10.143) is fixed independently of m. By Lemma 10.7 and an application of Theorem 9.4, we obtain the following proposition: Proposition 10.4 For each m ≥ 0, ρˆm is an irreducible representation of the CAR over HˆD . Let ψm (t, f ) := ψˆ m (eit hD fˆ),

t ∈ R, f ∈ D(HD ),

(10.154)

and ρm (t) := (Ff (HˆD ), {ψm (t, f ), ψm (t, f )∗ |f ∈ HD }).

(10.155)

Since eit hD is unitary for all t ∈ R, Proposition 10.4 yields the following result: Proposition 10.5 For each m ≥ 0 and t ∈ R, ρm (t) is an irreducible representation of the CAR over HD . Theorem 10.24 Let m ≥ 0 and f ∈ D(HD ). Then the operator-valued function: t → ψm (t, f ) ∈ Ff (HˆD ) on R is differentiable in the operator norm topology and obeys the free functional Dirac equation i

dψm (t, f ) = ψm (t, HD f ). dt

Proof Let ε ∈ R \ {0} and φε := i

ψm (t + ε, f ) − ψm (t, f ) − ψm (t, HD f ). ε

Then we have φε = ψˆ m (eit hD gε ), where 1 0 (eiεhD − 1) − hD fˆ. gε := −i ε

10.10 Free Quantum Dirac Fields

455

We have for all h ∈ HˆD

ψˆ m (h) ≤ Tm+ h + Tm− h∗ ≤ 2 h , where we have used (10.151). Hence φε ≤ 2 gε . Since fˆ ∈ D(hD ), it follows that limε→0 gε = 0. Hence limε→0 φε = 0. Thus the desired result follows.  Based on Theorem 10.24, we call ψm (t, f ) a free quantum Dirac field on the four-dimensional space–time.18 Note that ψˆ m (fˆ) = ψm (0, f ) is the time-zero field of it. Let Hm := df (Em ),

(10.156)

the fermion second quantization operator of the multiplication operator by Em on HˆD , acting in Ff (HˆD ), where Em is defined by (4.85). Theorem 10.25 For all f ∈ HD , ψm (t, f ) = eit Hm ψˆ m (fˆ)e−it Hm ,

t ∈ R.

(10.157)

Proof By (10.152), we have ψˆ m (eit hD fˆ) = b(Tm+ eit hD fˆ, 0) + b(0, Tm− (eit hD fˆ)∗ )∗ . ˆ3 By (10.144), we have for a.e. k ∈ R     (Tm+ eit hD fˆ)s (k) = um (k, s), eit hD (k) fˆ(k) 4 = e−it hD (k) um (k, s), fˆ(k) 4 C C   = eit Em (k) um (k, s), fˆ(k) 4 , C

where we have used the fact that e−it hD (k) um (k, s) = e−it Em (k) um (k, s). Similarly, using the fact that e−it hD (−k) vm (−k, s) = eit Em (−k) vm (−k, s) and Em (−k) = Em (k), one can show that   (Tm− (eit hD fˆ)∗ )s (k) = eit Em (k) v˜m (k, s), f˜ˆ(k)

C4

18 For

a general definition of a free quantum Dirac field, see [28, §12.3.1].

.

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10 Physical Correspondences in Quantum Field Theory

Hence ψˆ m (eit hD fˆ) = b(eit Em Tm+ fˆ, 0) + b(0, eit Em Tm− fˆ ∗ )∗ . = eit Hm b(Tm+ fˆ, 0) + b(0, Tm− fˆ ∗ )∗ e−it Hm = eit Hm ψˆ m (fˆ)e−it Hm , 

where we have used Theorem 7.11. Thus (10.157) is derived.

Formula (10.157) shows that Hm is the Hamiltonian of the free quantum Dirac field ψm (t, f ) of mass m in momentum representation (up to constant additions).

10.10.3 Inequivalence of Free Quantum Dirac Fields of Different Masses We now prove the main result in this chapter: Theorem 10.26 Let m1 = m2 (m1 , m2 ≥ 0). Then the representation ρˆm1 is inequivalent to ρˆm2 . ∗

Proof By Theorem 9.6, we need only to prove that, if m1 = m2 , then Tm1 + T m2 − ∗ or Tm1 − T m2 + is not Hilbert–Schmidt. It is easy to see that ∗

(Tm1 + T m2 − f )r (k) =

2

Krs (k)f˜s (k),

f = (fs )2s=1 ∈ HˆD− , r = 1, 2,

s=1

(10.158) where  Krs (k) := um1 (k, r), vm2 (k, s) C4 ,

ˆ 3 , r, s = 1, 2. k∈R

To make explicit the dependence of uD (k) on m, we write dm (k) := uD (k). Then, by (10.137), we have  Krs (k) = er , dm1 (k)dm2 (k)∗ es+2 .

(10.159)

10.10 Free Quantum Dirac Fields

457

Using (4.86), (4.87), the orthogonality er , es+2  = 0 (r, s = 1, 2) and the fact that βes+2 = −es+2 , we obtain Krs (k) = (m2 − m1 )Fm1 ,m2 (k) er , (α · k)es+2  ,

r, s = 1, 2,

(10.160)

where 1 Fm1 ,m2 (k) :=  2 Em1 (k)Em2 (k)(m1 + Em1 (k))(m2 + Em2 (k)) 0 1 m1 + m2 × 1+ . (10.161) Em1 (k) + Em2 (k) It follows from (9.16) that (α · k)es+2 ∈ ker(β − 1), s = 1, 2. Hence 2

| er , (α · k)es+2  |2 = (α · k)es+2 2 = k 2 ,

r=1

where we have used (4.83). Therefore 2

|Krs (k)|2 = (m2 − m1 )2 Fm1 ,m2 (k)2 k 2 .

(10.162)

r=1

Without loss of generality, one can assume that m1 > m2 ≥ 0. Suppose that ∗ Tm1 + T m2 − were Hilbert–Schmidt. We denote by Kˆ rs the multiplication operator ˆ 3 ). Then, by (10.158), Kˆ rs is Hilbert–Schmidt. by the function on L2 (R 2 Krs (k) ∗ Hence L := r=1 Kˆ rs Kˆ rs is Hilbert–Schmidt. By (10.162), L is the multiplication operator by the function G(k) := (m2 − m1 )2 Fm1 ,m2 (k)2 k 2 , ˆ 3 . Hence, by Theorem A.1(iv) which is bounded, non-negative and continuous on R in Appendix A, L is a bounded self-adjoint operator on HˆD− and ˆ 3 }. σ (L) = {G(k)|k ∈ R It is easy to see that lim|k|→∞ G(k) = 0. Since G is not a constant function, it follows that M := maxk∈Rˆ 3 G(k) > 0. Hence, by the middle value theorem, σ (L) = [0, M]. But this contradicts the Hilbert–Schmidtness of L, because the spectrum of a self-adjoint Hilbert–Schmidt operator is purely discrete in R \ {0}. ∗ Therefore we arrive at a contradiction. Thus Tm1 + T m2 − is not Hilbert–Schmidt. 

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10 Physical Correspondences in Quantum Field Theory

Theorem 10.26 shows that {ρˆm }m≥0 is a family of mutually inequivalent irreducible representations of the CAR over HˆD . This gives a representation theoretical meaning to masses of Dirac particles. Remark 10.24 The contents of this section are taken partly from the paper [27]. It is shown in fact that {ρˆm }m≥0 is an example in a general class of irreducible representations of CAR which are mutually inequivalent. For the details, see [27].

Appendix A

Multiplication Operators

Let (X, Σ, μ) be a measure space, i.e., X is a non-empty set, Σ is a σ -field of subsets of X and μ is a measure on the measurable space (X, Σ). It is well known that the space    |f (x)|2 dμ(x) < ∞ L2 (X, dμ) := f : X → C ∪ {±∞}, Σ-measurable| X

of square integrable functions on (X, Σ, μ) is a Hilbert space with inner product  f, gL2 (X,dμ) :=

f (x)∗ g(x)dμ(x),

f, g ∈ L2 (X, dμ),

X

where, for f, g ∈ L2 (X, dμ), the equality “f = g” is defined to be f (x) = g(x) for μ-a.e.x ∈ X (“μ-a.e.x” means “almost everywhere (a.e.) x with respect to μ”). A Σ-measurable function on X is said to be essentially bounded if there exists a constant C ≥ 0 such that |f (x)| ≤ C, μ-a.e.x ∈ X. In this case we denote the infimum of such constants C by f ∞ , which is called the essential supremum of f and written

f ∞ = ess.supx∈X |f (x)|.

(A.1)

For a Σ-measurable function F on X with |F (x)| < ∞ for μ-a.e.x, one can define a linear operator MF on L2 (X, dμ) as follows:     2 2  |F (x)f (x)| dμ(x) < ∞ , D(MF ) := f ∈ L (X, dμ) X

(MF f )(x) := F (x)f (x),

μ-a.e.x ∈ X, f ∈ D(MF ).

© Springer Nature Singapore Pte Ltd. 2020 A. Arai, Inequivalent Representations of Canonical Commutation and Anti-Commutation Relations, Mathematical Physics Studies, https://doi.org/10.1007/978-981-15-2180-5

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A Multiplication Operators

The essential range of F is defined by ess.ran(F ) := {λ ∈ C|∀ε > 0, μ({x ∈ X| |F (x) − λ| < ε}) > 0}. It is easy to see that ess.ran(F ) is a closed set. Example A.1 For any continuous function F on Rd , ess.ran(F ) = F (Rd ), where F (Rd ) := {F (x)|x ∈ Rd } and, for a subset S ⊂ Rd , S denotes the closure of S. The following theorem states basic properties of the multiplication operator MF : Theorem A.1 Let MF be as above. Then: (i) MF is a closed operator. (ii) If F is essentially bounded, then D(MF ) = L2 (X, dμ) and MF is bounded with MF ≤ F ∞ . (iii) If F is not essentially bounded, then D(MF ) is dense, D(MF ) = L2 (X, dμ) and MF is unbounded. (iv) (MF )∗ = MF ∗ . (v) If F : X → R ∪ {±∞} with F (x) ∈ R for μ-a.e.x, then MF is self-adjoint and σ (MF ) = ess.ran(F ).

(A.2)

Proof 2 (i) Let {fn }∞ n=1 be a sequence in D(MF ) such that fn → f ∈ L (X, dμ) 2 and MF fn → g ∈ L (X, dμ) as n → ∞. Then there exists a subsequence {fnk }∞ k=1 of {fn }n such that limk→∞ fnk (x) = f (x) and limk→∞ F (x)fnk (x) = g(x) for μ-a.e.x ∈ X. Hence F (x)f (x) = g(x) for μ-a.e.x ∈ X. Thus f ∈ D(MF ) and MF f = g. (ii) An easy exercise. (iii) The proof of this part is not so difficult, but takes much space (careful measuretheoretical arguments are needed). Hence we omit it. See, e.g., [23, Example 2.17] or [31, Example 1.33]. (iv) It is obvious that D(MF ) = D(MF ∗ ). Let f ∈ D(MF ∗ ). Then, for all g ∈ D(MF ),  f, MF g = (F ∗ f )(x)g(x)dμ(X) = MF ∗ f, g . X

Hence f ∈ D((MF )∗ ) and (MF )∗ f = MF ∗ f . Therefore MF ∗ ⊂ (MF )∗ . Conversely, let f ∈ D((MF )∗ ). For each n ∈ N, we define χn : X → R as follows: if 0 ≤ |F (x)| ≤ n, then χn (x) := 1 and if |F (x)| > n, then χn (x) := 0. Then, for all h ∈ L2 (X, dμ), χn h ∈ D(MF ) and χn F ∗ f is in L2 (X, dμ). Hence   χn (MF )∗ f, g = f, MF (χn g) = χn F ∗ f, g .

A Multiplication Operators

461

Hence χn (MF )∗ f = χn F ∗ f . Note that limn→∞ χn (x) = 1, μ-a.e.x. Therefore ((MF )∗ f )(x) = F (x)∗ f (x),

μ-a.e.x.

This implies that F ∗ f ∈ L2 (X, dμ) (hence f ∈ D(MF ∗ )) and MF ∗ f = (MF )∗ f . Therefore (MF )∗ ⊂ MF ∗ . (v) The self-adjointness of MF in the present case follows from (iv). For a proof of (A.2), see, e.g., [31, Theorem 2.27]. 

Appendix B

Spectral Measures and Functional Calculus

For each d ∈ N, we denote by B d the Borel field of the d-dimensional Euclidean vector space Rd . Let H be a Hilbert space and P(H ) be the set of all orthogonal projections on H .1 A mapping E: B d → P(H ); B d B → E(B) ∈ P(H ) is called a d-dimensional spectral measure or a d-dimensional resolution of identity2 if (E.1) and (E.2) below hold: (E.1) E(∅) = 0, E(Rd ) = I . (E.2) (complete additivity) If Bn ∈ B d , n ∈ N and Bn ∩ Bm = ∅ (n = m), then E(∪∞ n=1 Bn ) = s- lim

N→∞

N

E(Bn ),

n=1

where s-lim means strong limit (see Sect. 1.1.2). It is not so difficult to prove that (E.1) and (E.2) imply (E.3) For all B1 , B2 ∈ B d , E(B1 )E(B2 ) = E(B1 ∩ B2 ). Hence, in particular, E(B1 ) commutes with E(B2 ): E(B1 )E(B2 ) = E(B2 )E(B1 ),

B1 , B2 ∈ B d .

Thus the set {E(B)|B ∈ B d } is a family of commuting orthogonal projections on H satisfying (E.1)–(E.3). Property (E.2) implies the following: (E.4) (monotonicity) If A ⊂ B, A, B ∈ B d , then, for all Ψ ∈ H , Ψ, E(A)Ψ  ≤ Ψ, E(B)Ψ  . operator P ∈ B(H ) is called an orthogonal projection if P = P ∗ and P = P 2 . is called also a projection-valued measure.

1 An 2 It

© Springer Nature Singapore Pte Ltd. 2020 A. Arai, Inequivalent Representations of Canonical Commutation and Anti-Commutation Relations, Mathematical Physics Studies, https://doi.org/10.1007/978-981-15-2180-5

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464

B Spectral Measures and Functional Calculus

The smallest closed set C0 ∈ B d such that E(C0 ) = I is called the support of E. We write C0 = supp E. Hence E(supp E) = I . For each Ψ ∈ H , there exists a bounded measure μΨ : B d → [0, ∞) defined by μΨ (B) := E(B)Ψ 2 ,

B ∈ Bd .

d For a Borel measurable function f : R → C ∪ {±∞}, the Lebesgue integral f dμΨ ∈ C ∪ {±∞} of f with respect to μΨ (if it exists) is written as Rd  2 Rd f (λ)d Ψ, E(λ)Ψ  or Rd f (λ)d E(λ)Ψ :



 Rd

f (λ)d Ψ, E(λ)Ψ  :=

 Rd

f (λ)d E(λ)Ψ 2 :=

Rd

f (λ)dμΨ (λ).

For each pair (Φ, Ψ ) ∈ H × H , the mapping μΦ,Ψ : B d → C defined by μΦ,Ψ (B) := Φ, E(B)Ψ  ,

B ∈ Bd ,

is  a complex-valued measure. Hence one can define the Lebesgue–Stieltjes integral Rd f (λ)dμΨ,Φ (λ) with respect to μΦ,Ψ under the condition that  Rd

|f (λ)|d|μΨ,Φ |(λ) < ∞,

where |μΨ,Φ |(·) is the total variation of μΨ,Φ (·). We write 

 Rd

f (λ)dμΨ,Φ (λ) =

Rd

f (λ)d Ψ, E(λ)Φ .

For a Borel measurable function f on Rd , the subset   Df := Ψ ∈ H |

Rd

 |f (λ)| d E(λ)Ψ < ∞ 2

2

is a subspace. One can show that, for all Ψ ∈ Df and Φ ∈ H ,    

Rd

 0  f (λ)d Φ, E(λ)Ψ  ≤

Rd

11/2 |f (λ)|2 d E(λ)Ψ 2

Φ .

By this fact and the Riesz representation theorem, there exists a unique linear operator TE (f ) on H such that D(TE (f )) = Df ,  Φ, TE (f )Ψ  =

Rd

f (λ)d Φ, E(λ)Ψ  ,

Ψ ∈ Df , Φ ∈ H .

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465

The operator TE (f ) is written symbolically as  TE (f ) =

f (λ)dE(λ).

One has 

TE (f )Ψ = 2

Rd

|f (λ)|2 d E(λ)Ψ 2 ,

Ψ ∈ D(TE (f )).

The next theorem summarizes some fundamental properties of TE (f ). Theorem B.1 Let f and g be Borel measurable functions on Rd . (i) For all B ∈ B d , E(B)TE (f ) ⊂ TE (f )E(B). (ii) The domain D(TE (f )) is dense in H if and only if E({λ| |f (λ)| = +∞}) = 0. In this case, TE (f ) is a densely defined closed operator and TE (f )∗ = TE (f ∗ ). (iii) If f is continuous on Rd , then TE (f ) is a densely defined closed operator and TE (f )∗ = TE (f ∗ ). (iv) If f is bounded on supp E, i.e., Cf := supλ∈supp E |f (λ)| < ∞, then TE (f ) ∈ B(H ) and TE (f ) ≤ Cf . (v) If |f (λ)| = 1 for all λ ∈ supp E, then TE (f ) is unitary. (vi) If f is real-valued and E({λ ∈ Rd | |f (λ)| = +∞}) = 0, then TE (f ) is selfadjoint. In particular, if f is a real-valued continuous function on Rd , then TE (f ) is self-adjoint. (vii) TE (f ) + TE (g) ⊂ TE (f + g). In particular, if g is bounded on supp E, then TE (f ) + TE (g) = TE (f + g). (viii) D(TE (fg)) ∩ D(TE (g)) = D(TE (f )TE (g)) and TE (f )TE (g) ⊂ TE (fg). In particular, if g is bounded on supp E, then TE (f )TE (g) = TE (fg). (ix) If fn : Rd → C ∪ {±∞} (n ∈ N) is Borel measurable with sup

|fn (λ)| < ∞

n∈N,λ∈supp E

and lim fn (λ) = f (λ),

n→∞

λ ∈ supp E,

then TE (fn ), TE (f ) ∈ B(H ) and s- limn→∞ TE (fn ) = TE (f ). (x) (spectral mapping theorem) If f is continuous on Rd , then σ (TE (f )) = {f (λ)|λ ∈ supp E}, where σ (TE (f )) is the spectrum of TE (f ).

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Calculus of the mapping TE : f → TE (f ) ∈ L(H ) (the set of linear operators on H ) as stated in the above theorem is called functional calculus with respect to the spectral measure E. The following theorem is one of the deepest theorems in Hilbert space theory: Theorem B.2 (Spectral Theorem) For any self-adjoint operator Aon H , there exists a unique one-dimensional spectral measure EA such that A = R λ dEA (λ). Example B.1 Let MF be the multiplication operator discussed in Appendix A. Suppose that F (x) ∈ R μ-a.e.x. Then, by Theorem A.1(v), MF is self-adjoint. For all g ∈ L2 (X, dμ) and f ∈ D(MF ), we have   g, MF f  = g(x)∗ F (x)f (x)dμ(x) = F (x)dμg,f (x), (B.1) X

X



where μg,f (Y ) := Y g(x)∗ f (x)dμ(x) = g, χY f L2 (X,dμ) , Y ∈ Σ (χY is the characteristic function of Y ). For a Borel set B of R, we denote by F −1 (B) the inverse image of F : F −1 (B) := {x ∈ X|F (x) ∈ B} and define  F (B) := μg,f (F −1 (B)) = g, χF −1 (B) f L2 (X,dμ) . νg,f F is a complex measure on (R, B 1 ). It follows from (B.1) that Then νg,f

 g, MF f  =

R

λdμFg,f (λ).

Hence the spectral measure EMF of MF is of the following form: EMF (B)f = χF −1 (B) f,

f ∈ L2 (X, dμ), B ∈ B 1 ,

which means that EMF is the multiplication operator by the characteristic function χF −1 (B) of F −1 (B). It follows from this fact that, for all f, g ∈ L2 (X, dμ) and t ∈ R,       it MF it λ F g, e f = e dνg,f (λ) = eit F (x)g(x)∗ f (x)dμ(x) = g, eit F f . R

X

Hence eit MF f = eit F f,

f ∈ L2 (X, dμ),

i.e., eit MF is the multiplication operator by the function eit F .

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Let A be a self-adjoint operator on H . Then, by an application of the theory of TE (f ) to the case where d = 1 and E = EA , one has an operator TEA (f ) for each Borel measurable function f on R. Usually this operator is written f (A):  f (A) = TEA (f ) =

R

f (λ)dEA (λ).

The correspondence A → f (A) gives an operator-valued function on the set of selfadjoint operators on H . The following theorem is very useful in operator calculus. Theorem B.3 (Unitary Covariance of Functional Calculus) Let A be a selfadjoint operator on a Hilbert space H and U be a unitary operator from H to a Hilbert space K . Then, for all Borel measurable functions f on R, Uf (A)U −1 = f (U AU −1 ). In relation to spectral measures, we need a theorem which is used in the main text of this book. For this purpose, we first present a lemma. Lemma B.1 Let A be a densely defined closed operator on a Hilbert space H and S be a bounded self-adjoint operator on H such that SA ⊂ AS.

(B.2)

Then (S − z)−1 A ⊂ A(S − z)−1 ,

∀z ∈ ρ(S).

(B.3)

Proof It follows from (B.2) that, for all n ∈ N, S n A ⊂ AS n · · · (∗). Let z ∈ C be such that S  < |z|, where S is the operator norm of S. Then z ∈ ρ(S) and, −n−1 S n , we have (S − z)−1 = lim putting SN := − ∞ N→∞ SN in the operator n=0 z norm topology (i.e. limN→∞ SN − (S − z)−1 = 0). By (∗), for all Ψ ∈ D(A), SN AΨ = ASN Ψ . The left hand side converges to (S − z)−1 AΨ as N → ∞. Hence ASN Ψ → (S −z)−1 AΨ as N → ∞. Since A is closed, it follows that (S −z)−1 Ψ ∈ D(A) and A(S − z)−1 Ψ = (S − z)−1 AΨ . Hence (S − z)−1 A ⊂ A(S − z)−1 . Next, let Φ ∈ D(A∗ ), Ψ ∈ D(A) and define functions f and g on ρ(S) by   f (z) := A∗ &, (S − z)−1 ' ,

  g(z) := &, (S − z)−1 A' ,

z ∈ ρ(S).

It is obvious that f and g are analytic on ρ(S) and, by the preceding result, f (z) = g(z) for all z ∈ ρ(S) satisfying |z| > S . Since S is self-adjoint, C \ R ⊂ ρ(S) and σ (S) is a bounded closed subset of R. Hence ρ(S) is arcwise connected. Therefore we can apply the theorem of identity to conclude that f (z) = g(z) for all z ∈ ρ(S).  This implies that (S − z)−1 Ψ ∈ D(A∗∗ ) = D(A) and (B.3) holds.

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B Spectral Measures and Functional Calculus

Remark B.1 If A in Lemma B.1 is bounded, then (B.3) trivially holds. Hence Lemma B.1 is non-trivial only if A is unbounded. Theorem B.4 Let A be a densely defined closed operator on a Hilbert space H and S be a bounded self-adjoint operator on H such that SA ⊂ AS.

(B.4)

Then, for all Borel sets B ⊂ R, ES (B)A ⊂ AES (B), where ES is the spectral measure of S. Proof By the Stone formula, we have 5 14 1 ES ([a, b]) + ES ((a, b)) = s- lim ε↓0 2πi 2



b

-

. (S − λ − iε)−1 − (S − λ + iε)−1 dλ,

a

where the integral on the right hand side is taken in the strong sense and s- lim means strong limit. By Lemma B.1, (S − z)−1 A ⊂ A(S − z)−1 , z ∈ ρ(S). Hence, for all Ψ ∈ D(A), -

. . (S − λ − iε)−1 − (S − λ + iε)−1 AΨ = A (S − λ − iε)−1 − (S − λ + iε)−1 Ψ.

4 5 This equation and the closedness of A imply that ES ([a, b]) + ES ((a, b)) Ψ ∈ D(A) and 5 4 5 4 A ES ([a, b]) + ES ((a, b)) Ψ = ES ([a, b]) + ES ((a, b)) AΨ. Hence it follows that AES ({a})Ψ = ES ({a})AΨ,

AES ((a, b])Ψ = ES ((a, b])AΨ.

Let Φ ∈ D(A∗ ) and  μ1 (B) := A∗ &, ES (B)' ,

μ2 (B) := &, ES (B)A' ,

B ∈ B1.

Then μ1 and μ2 are bounded complex measures satisfying that μ1 ({a}) = μ2 ({a}) and μ1 ((a, b]) = μ2 ((a, b]) for all a, b ∈ R with a < b. Hence, by the uniqueness of Hopf’s extension theorem, μ1 = μ2 . Thus ES (B)Ψ ∈ D(A∗∗ ) = D(A) and AES (B)Ψ = ES (B)AΨ , i.e. ES (B)A ⊂ AES (B). 

Appendix C

Direct Sum Hilbert Spaces and Direct Sum Operators

C.1 Finite Direct Sums of Hilbert Spaces and Operators Let N ≥ 2 (N ∈ N) and H1 , . . . , HN be Hilbert spaces. Then the direct product H1 × · · · × HN = {Ψ = (Ψ1 , . . . , ΨN )|Ψj ∈ Hj , j = 1, . . . , N} becomes a complex vector space with the following operations of addition and scalar multiplication: Ψ + Φ := (Ψ1 + Φ1 , . . . , ΨN + ΦN ), αΨ := (αΨ1 , . . . , αΨN ),

Ψ, Φ ∈ H1 × · · · × HN , α ∈ C.

This vector space becomes a Hilbert space with the following inner product: Ψ, Φ :=

N  j =1

Ψj , Φj H . j

This Hilbert space, denoted by ⊕N j =1 Hj , is called the direct sum Hilbert space of H1 , . . . , HN . Let Aj be a linear operator on Hj . Then the direct sum operator ⊕N j =1 Aj of N A1 , . . . , AN on ⊕j =1 Hj is defined as follows: N

ˆ D(⊕N j =1 Aj ) := ⊕j =1 D(Aj ) = {(Ψ1 , . . . , ΨN )| Ψj ∈ D(Aj ), j = 1, . . . , N} (the algebraic direct sum of D(Aj ), j = 1, . . . , N), (⊕N j =1 Aj )Ψ := (A1 Ψ1 , . . . , AN ΨN ),

Ψ ∈ D(⊕N j =1 Aj ).

© Springer Nature Singapore Pte Ltd. 2020 A. Arai, Inequivalent Representations of Canonical Commutation and Anti-Commutation Relations, Mathematical Physics Studies, https://doi.org/10.1007/978-981-15-2180-5

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With regard to spectral properties of ⊕N j =1 Aj , the following theorem holds: Theorem C.1 Let A := ⊕N j =1 Aj . Then: (i) For all z ∈ C, ker(A − z) = ⊕N j =1 ker(Aj − z). σ (A ). (ii) σ (A) = ∪N j j =1 σ (A (iii) σp (A) = ∪N p j ). j =1 Proof (i) We have A − z = ⊕N j =1 (Aj − z),

(C.1)

which implies the desired result. (ii) The statement of this part is equivalent to that ρ(A) = ∩N j =1 ρ(Aj ). Hence we prove this. Let z ∈ ρ(A). Then A − z is injective and (A − z)−1 is bounded. Hence, by (i), each Aj − z is injective. The boundedness of (A − z)−1 implies that, for a constant C > 0, (A − z)Ψ ≥ C Ψ , Ψ ∈ D(A). Note that

(A − z)Ψ 2 =

N

(Aj − z)Ψj 2 ,

Ψ ∈ D(A).

(C.2)

j =1

Fixing j = 1, . . . , N and taking Ψ ∈ D(A) as Ψk = δj k η (k = 1, . . . , N) with η ∈ D(Aj ) being arbitrary, we obtain (Aj − z)η ≥ C η . Hence (Aj − z)−1 is bounded. Moreover, Ran (A − z) is dense in ⊕N j =1 Hj . By this property and (C.1), for each j = 1, . . . , N, Ran (Aj −z) is dense in Hj . Hence z ∈ ρ(Aj ). Thus ρ(A) ⊂ ∩N j =1 ρ(Aj ). Conversely, let z ∈ ∩N j =1 ρ(Aj ). Then each Aj − z is injective with (Aj − −1 z) being bounded and Ran (Aj − z) is dense in Hj . The former implies that A − z is injective with (A − z)−1 being bounded (use (C.2)), while the latter implies that Ran (A − z) is dense in ⊕N j =1 Hj . Hence z ∈ ρ(A). Thus N ∩j =1 ρ(Aj ) ⊂ ρ(A). (iii) This easily follows from (i).  Let M be a closed subspace of a Hilbert space H and PM be the orthogonal projection onto M . Then, by the projection theorem, for any Ψ ∈ H , there exists a unique pair (ΨM , ΨM ⊥ ) ∈ M ⊕ M ⊥ such that Ψ = ΨM + ΨM ⊥ . It is shown that the correspondence Ψ → (ΨM , ΨM ⊥ ) defines a unitary transformation from H to M ⊕ M ⊥ . In this sense, we identify H with M ⊕ M ⊥ and write H = M ⊕ M ⊥.

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471

Let A ∈ L(H ) be reduced by M . Then it is easy to see that A = AM ⊕ AM ⊥ , where AM is the reduced part of A to M .

C.2 Infinite Direct Sums of Hilbert Spaces and Operators The concept of the direct sum of a finite number of Hilbert spaces is extended in natural way to the case of an infinite number of Hilbert spaces. Let Hn , n ∈ Z+ be Hilbert spaces. Then 2 3 ∞ ∞ (n) ∞ (n) (n) 2 ⊕n=0 Hn := Ψ = {Ψ }n=0 |, Ψ ∈ Hn , n ≥ 0,

Ψ < ∞ n=0

is a Hilbert space with inner product Ψ, Φ :=

∞   Φ (n) , Ψ (n) n=0

Hn

,

Ψ, Φ ∈ ⊕∞ n=0 Hn .

The Hilbert space ⊕∞ n=0 Hn is called the infinite direct sum Hilbert space of {Hn }∞ . n=0 For a subspace Dn of Hn (n ∈ N), one can define a subspace of ⊕∞ n=0 Hn : (n) ∞ ˆ∞ }n=0 | Ψ (n) ∈ Dn , n ≥ 0, ∃n0 such that Ψ (n) = 0, ∀n ≥ n0 }. ⊕ n=0 Dn := {Ψ = {Ψ

This subspace is called the algebraic infinite direct sum of {Dn }∞ n=0 . It is easy to ˆ∞ see that, if Dn is dense in Hn for all n ≥ 1, then ⊕ D is dense in ⊕∞ n=0 n n=0 Hn . Let An be a linear operator on Hn . Then one can define a linear operator ⊕∞ n=0 An on ⊕∞ H as follows: n=0 n 2 3 ∞  (n) ∞ ∞ (n) 2

An Ψ < ∞ , D(⊕ An ) := Ψ ∈ ⊕ Hn Ψ ∈ D(An ), n ≥ 0, n=0

n=0

n=0 (n) ∞ (⊕∞ }n=0 , n=0 An )Ψ := {An Ψ

Ψ ∈ D(⊕∞ n=0 An ).

∞ We call the operator ⊕∞ n=0 An the direct sum of {An }n=0 .

Theorem C.2 Let each An be self-adjoint and A := ⊕∞ n=0 An as above. Then: (i) The operator A is self-adjoint. Moreover, for all t ∈ R, it An eit A = ⊕∞ . n=0 e

(C.3)

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C

Direct Sum Hilbert Spaces and Direct Sum Operators

(ii) If each An is non-negative, then A is non-negative. Moreover, for all β > 0, e−βA is bounded and −βAn e−βA = ⊕∞ . n=0 e

(C.4)

H := ⊕∞ n=0 Hn .

(C.5)

(iii) σ (A) = ∪∞ n=0 σ (An ). (iv) σp (A) = ∪∞ n=0 σp (An ). Proof We set

(i) Let Φ ∈ H . Then, for each n ∈ N, by the self-adjointness of An , there exists a vector Ψ (n) ∈ D(An ) such that i)Ψ (n) = Φ (n) . We have Φ (n) 2 = ∞(An +(n) (n) 2 (n) 2

An Ψ + Ψ . Since n=0 Φ 2 = Φ 2 < ∞, it follows that Ψ := {Ψ (n) }∞ n=0 is in D(A) and (A + i)Ψ = Φ. Hence Ran (A + i) = H . Similarly, one can show that Ran (A − i) = H . Thus, by a general criterion, A is self-adjoint. It is obvious that A is reduced by each Hn identified with {Ψ ∈ H |Ψ (k) = 0, k = n} and its reduced part is An identified with 0 ⊕ · · · ⊕ An ⊕ 0 · · · . Hence, by Proposition 1.16(v), eit A is reduced by each Hn and its reduced part is eit An . Thus (C.3) holds. (ii) For all Ψ ∈ D(A), Ψ, AΨ  =

∞ 

 Ψ (n) , An Ψ (n) .

n=0

 Since each Ψ (n) , An Ψ (n) is non-negative, Ψ, AΨ  ≥ 0. Formula (C.4) can be proved in the same way as in the case of (C.3). (iii) Let z ∈ ρ(A). Then A−z is bijective. This implies that each An −z is bijective. ∞ Hence z ∈ ρ(An ). Therefore ρ(A) ⊂ ∪∞ n=0 ρ(An ). Thus ∪n=0 σ (An ) ⊂ σ (A). Taking the closure of both sides, we obtain ∪∞ n=0 σ (An ) ⊂ σ (A). Let λ ∈ σ (A) \ ∪∞ σ (A ). Then there exists a constant δ > 0 such that n n=0 inf

inf

n≥1 μ∈σ (An )

|λ − μ| ≥ δ.

By functional calculus for An , we have (An − λ)f ≥ δ f , f ∈ D(An ) · · · (∗). Since λ ∈ ρ(An ), An − λ is bijective. Hence, for each Φ ∈ H and n ∈ N, there exists a vector Ψ (n) ∈ D(An ) such that (An −λ)Ψ (n) = Φ (n) . By (∗), we have

Φ (n) 2 ≥ δ 2 Ψ (n) 2 .

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473

Hence ∞

Ψ (n) 2 ≤

n=0

∞ 1 (n) 2

Φ < ∞. δ2 n=0

Therefore Ψ ∈ H . Moreover, ∞ n=0

An Ψ (n) 2 ≤ 2

∞

(An − λ)Ψ (n) 2 + 2λ2

n=0

∞

Ψ (n) 2 < ∞.

n=0

Hence Ψ ∈ D(A) and (A − λ)Ψ = Φ. Thus λ ∈ ρ(A), implying that σ (A) = ∪∞ n=0 σ (An ). (iv) Let λ ∈ σp (A). Then there exists a non-zero vector Ψ ∈ D(A) such that AΨ = λΨ . Hence there exists an n0 ∈ N such that Ψ (n0 ) = 0 and An0 Ψ (n0 ) = λΨ (n0 ) . Hence λ ∈ σp (An ). Thus σ (A) ⊂ ∪∞ n=0 σp (An ). Conversely, let λ ∈ ∪∞ σ (A ). Then there exists an n0 ∈ N such that p n n=0 λ ∈ σp (An0 ). Hence there exists a non-zero vector f ∈ D(An0 ) and An0 f = λf . Let Ψ ∈ H be such that Ψ (n) = 0, n = n0 and Ψ (n) = f . Then Ψ ∈ D(A) \ {0} and AΨ = λΨ . Hence λ ∈ σp (A). Thus ∪∞ n=0 σp (An ) ⊂ σp (A).  Remark C.1 For more detailed spectral properties of infinite direct sum operators, see [28, Theorem 4.2].

C.3 A Theorem on Essential Self-adjointness Let H be the Hilbert space given by (C.5). Then it is easy to show that the subspace D0 := {Ψ ∈ H |∃n0 such that Ψ (n) = 0 for all n ≥ n0 } is dense in H . Let Nˆ be the direct sum of operators nIHn (IX is the identity operator on X ): Nˆ := ⊕∞ n=0 nIHn . Then, by Theorem C.2, Nˆ is a non-negative self-adjoint operator with ˆ = Z+ . σ (Nˆ ) = σp (N)

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Direct Sum Hilbert Spaces and Direct Sum Operators

Let A be a self-adjoint operator on H which is reduced by each Hn (n ∈ Z+ ). Then D(A) ∩ Hn is dense in Hn and hence D := D(A) ∩ D0 is dense in H . Let B be a symmetric operator on H satisfying D0 ⊂ D(B). Then D ⊂ D(A) ∩ D(B) and (A + B)  D is a symmetric operator. We assume the following: (i) There exists a constant c > 0 such that

BΨ ≤ c (Nˆ + 1)2 Ψ ,

Ψ ∈ D.

(ii) There exists an integer p ∈ Z+ such that, for all m, n ∈ Z+ satisfying |m − n| > p, 

 Ψ (m) , BΨ (n) = 0.

Theorem C.3 Let A and B be as above. Suppose that A+B is bounded from below on D. Then A + B is essentially self-adjoint on D. For proofs of this theorem, see [8, 96]. Remark C.2 In [8], a generalized version of Theorem C.3 is proved.

Appendix D

Spectra of a Self-adjoint Operator

Let A be a self-adjoint operator on a Hilbert space H and EA be its spectral measure. For any vector Ψ ∈ H , the mapping μΨ : B 1 → [0, ∞) (B 1 is the Borel field of R) defined by μΨ (B) := EA (B)Ψ 2 ,

B ∈ B1

is a bounded measure on (R, B 1 ). A vector Ψ ∈ H is said to be absolutely continuous with respect to A if the (1) measure μΨ is absolutely continuous with respect to the Lebesgue measure μL on R. In what follows, we write |B| := μ(1) L (B),

B ∈ B 1.

(D.1)

If μΨ is singular with respect to μL , then Ψ is said to be singular with respect to A. We denote by Hac (A) (resp. Hs (A)) the set of vectors absolutely continuous (resp. singular) with respect to A. It is easy to see that Hac (A) and Hs (A) are subspaces of H . The subspace Hac (A) (resp. Hs (A)) is called the absolutely continuous (resp. singular) subspace of A. Theorem D.1 The subspaces Hac (A) and Hs (A) are mutually orthogonal closed subspaces and H = Hac (A) ⊕ Hs (A). Moreover, A is reduced by Hac (A) and Hs (A). Proof See, e.g., [98, p. 518, Theorem 1.5].

© Springer Nature Singapore Pte Ltd. 2020 A. Arai, Inequivalent Representations of Canonical Commutation and Anti-Commutation Relations, Mathematical Physics Studies, https://doi.org/10.1007/978-981-15-2180-5

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D Spectra of a Self-adjoint Operator

We define Hp (A) := Ran (EA (σp (A))), where σp (A) is the point spectrum of A (see Sect. 1.1.3). It is obvious that Hp (A) is closed. It follows that Hp (A) is equal to the closure of the subspace spanned by all eigenvectors of A if σp (A) = ∅. Lemma D.1 Hp (A) ⊂ Hs (A). By Lemma D.1, one can define Hsc (A) := Hp (A)⊥ ∩ Hs (A) ⊂ Hs (A). This closed subspace is called the singular continuous subspace of A. By Theorem D.1, one has the orthogonal decomposition H = Hac (A) ⊕ Hsc (A) ⊕ Hp (A)

(D.2)

and each closed subspace H# (A) (# =ac, sc, p) reduces A. We denote by A# the reduced part of A to H# (A). Hence A is written as the direct sum of them: A = Aac ⊕ Asc ⊕ Ap .

(D.3)

The operator Aac (resp. Asc , Ap ) is called the absolutely continuous part (resp. singular continuous part, point-spectral part) of A. Thus, introducing σac (A) := σ (Aac ),

σsc (A) := σ (Asc ),

σpp (A) := σ (Ap ),

we have by Theorem C.1 σ (A) = σac (A) ∪ σsc (A) ∪ σpp (A). The set σac (A) (resp. σsc (A), σpp (A)) is called the absolutely continuous spectrum (resp. singular continuous spectrum, pure point spectrum) of A. We remark that σac (A), σsc (A) and σpp (A) are not necessarily mutually disjoint. We remark also that σpp (A) = σp (A). Note that σp (A) is not necessarily closed, but σpp (A) is closed. If H = Hac (A), then A is said to be purely absolutely continuous or simply absolutely continuous; in this case, A is said to have a purely absolutely continuous spectrum.

D Spectra of a Self-adjoint Operator

477

If H = Hsc (A), then A is said to be purely singular continuous or simply singular continuous; in this case, A is said to have a purely singular continuous spectrum. If H = Hp (A), then A is said to be purely point spectral or to have a purely point spectrum. As is well known, in the context of quantum theory where −A is a Hamiltonian, the quantity &, eit A ' with unit vectors &, ' ∈ H represents the probability amplitude of a state Ψ at time 0 to a state Φ at time t ∈ R.1 The absolute continuity of A implies the decay in time t of this quantity: Theorem D.2 Suppose that A is absolutely continuous. Then, for all Ψ, Φ ∈ H , lim

t →±∞

  Φ, eit A Ψ = 0.

(D.4)

Proof By the polarization identity, it is sufficient to prove (D.4) in the case where Φ = Ψ . By functional calculus for A, we have    Ψ, eit A Ψ = eit λd EA (λ)Ψ 2 . R

By the absolute continuity of A, there exists a non-negative function f ∈ L1 (R) such that  2

EA (B)Ψ = f (λ)dλ, B ∈ B 1 . B

Hence 

  Ψ, eit A Ψ = eit λ f (λ)dλ. R

Then, by the Riemann–Lebesgue lemma, the right hand side converges to 0 as t → ±∞.  The following proposition is useful for showing that A is absolutely continuous. Proposition D.1 Suppose that there exists a dense subspace D in H such that, for all Ψ ∈ D, the measure Ψ, EA (·)Ψ  is absolutely continuous. Then A is absolutely continuous. Proof Let Ψ ∈ H . Then there exists a sequence {Ψn }∞ n=0 in D (i.e., Ψn ∈ D, ∀n ∈ N) such that Ψn → Ψ (n → ∞). Let B ∈ B 1 be such that |B| = 0. Then, by  quantity | Φ, eitA Ψ |2 represents the transition probability of a state Ψ at time 0 to a state Φ at time t, i.e., the probability that Φ is found at time t under the condition that the initial state (time-zero state) is Ψ .

1 The

478

D Spectra of a Self-adjoint Operator

the present assumption, Ψn , EA (B)Ψn  = 0, n ∈ N. Taking the limit n → ∞, we obtain Ψ, EA (B)Ψ  = 0. Hence Ψ, EA (·)Ψ  is absolutely continuous.  Proposition D.2 Suppose that A is absolutely continuous. Then, for any unitary operator U : H → K (a Hilbert space), U AU −1 is absolutely continuous. Proof The spectral measure EU AU −1 of U AU −1 is given by EU AU −1 (B) = U EA (B)U −1 , B ∈ B 1 . Let |B| = 0. Then, by the absolute continuity of A, EA (B) = 0. Hence EU AU −1 (B) = 0. Thus U AU −1 is absolutely continuous.  By abuse of notation, for each n ∈ N, the n-dimensional Lebesgue measure of B ∈ B n is denoted by |B|. For a subset M ∈ B d with |M| > 0, we define B M := {B ∩ M|B ∈ B d }. The following proposition gives a sufficient condition for a multiplication operator on L2 (M) to be absolutely continuous. Proposition D.3 Let M ∈ B d with |M| > 0 and F be a real-valued function on Suppose that, for all B ⊂ B M with |B| = 0 and f ∈ L2 (M),  M a.e. finite. 2 2 F −1 (B) |f (x)| dx = 0. Then the multiplication operator MF on L (M) (which is self-adjoint by Theorem A.1(v)) is absolutely continuous and σ (MF ) = σac (MF ) = ess.ran(F ),

σsc (MF ) = ∅,

σp (MF ) = ∅.

(D.5)

Proof As we have seen in Example B.1, the spectral measure of MF is given by E(B) = χF −1 (B) , B ∈ B 1 (the multiplication operator by χF −1 (B) ). Hence, for all f ∈ L2 (M), 

E(B)f 2 =

F −1 (B)

|f (x)|2 dx.

If |B| = 0, then the right hand side vanishes by the present assumption and hence

E(B)f 2 = 0. Thus MF is absolutely continuous and hence (D.5) holds.  Example D.1 Let qˆj be the j th position operator in Sect. 2.6.1. Consider the function Fj : Rd → R, Fj (x) := xj , x ∈ Rd . Then qˆj = MFj . It is obvious that Fj is continuous on Rd and Fj−1 (B) = {x ∈ Rd | xj ∈ B} = Rj −1 × B × Rd−j . Hence, for all f ∈ L2 (Rd ) and B ∈ B 1 with |B| = 0,  0

 |f (x)| dx = 2

Fj−1 (B)

1 |f (x)| dx1 · · · dx j −1 dxj +1 · · · dxd dxj = 0, 2

B

Rj−1 ×Rd−j

D Spectra of a Self-adjoint Operator

479

where we have used Fubini’s theorem to rewrite the integral



Fj−1 (B) |f (x)|

2 dx

as an

iterated integral. Hence, by Proposition D.3, MFj , i.e., qˆj is absolutely continuous and σ (qˆj ) = σac (qˆj ) = Fj (Rd ) = R. σsc (qˆj ) = ∅,

σp (qˆj ) = ∅.

Example D.2 Let m > 0. Then the multiplication operator Tˆm := k 2 /2m on ˆ d ) is absolutely continuous (consider the case where F (k) = k 2 /2m, k ∈ R ˆ d) L2 (R and σ (Tˆm ) = σac (Tˆm ) = [0, ∞),

σsc (Tˆm ) = ∅,

σp (Tˆm ) = ∅.

ˆ d ) be the Fourier transform. Then Let Fd : L2 (Rd ) → L2 (R 1 Δ, Fd−1 Tˆm Fd = Tm := − 2m  where Δ := dj=1 Dj2 is the generalized Laplacian. Hence, by Proposition D.2, Tm is absolutely continuous and σ (Tm ) = σac (Tm ) = [0, ∞),

σsc (Tm ) = ∅,

σp (Tm ) = ∅.

 Example D.3 Let m ≥ 0. Then the multiplication operator Rm := k 2 + m2 on L2 (Rdk ) is absolutely continuous (consider the case where F (k) = k 2 + m2 , k ∈ Rd ) and σ (Rm ) = σac (Rm ) = [m, ∞),

σsc (Rm ) = ∅,

σp (Rm ) = ∅.

We have  −Δ + m2 = Fd−1 Rm Fd . Hence

√ −Δ + m2 is absolutely continuous and   σ ( −Δ + m2 ) = σac ( −Δ + m2 ) = [m, ∞),   σsc ( −Δ + m2 ) = ∅, σp ( −Δ + m2 ) = ∅.

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Index

Symbols ∗-invariant, 75 C ∞ -domain, 27 L2R (Rd ), 404 P (φ)2 model, 448 Q-space representation, 382 Rp -anti-symmetric, 235 Rp -symmetric, 235 I2 (H ), 18 SW (H ), 373 su(2), 238 SR (Rd ), 405 CCR(N), 63 CCRsa (N), 63 πF , 361 πμ,ν (W ), 368 n-boson space, 247 n-particle space, 247 p-particle space of fermions, 312 p-representation, 87 q-commutator, 236 q-deformed CCR, 236 q-representation, 83

A Absolutely continuous, 475 Absolutely continuous part, 476 Absolutely continuous self-adjoint operator, 476 Absolutely continuous spectrum, 476 Absolutely continuous subspace, 475 Abstract Fermi field model, 397

Abstract non-relativistic Bose field model, 395 Abstract relativistic Bose field model, 395 Adjoint operator, 292 Aharonov–Bohm effect, 163 Aharonov–Bohm time operator d-dimensional, 209 Algebraic infinite direct sum, 471 Algebraic tensor product, 32 Analytic vector, 27 entire, 27 Angular momentum orbital, 239 spin, 239 Annihilation operator, 98 boson, 95, 257 fermion, 240, 317 Anti-linear operator, 2 Anti-symmetric Fock space, 311 Anti-symmetric tensor product, 35 algebraic, 35 Anti-symmetrization operator, 35 Anti-symmetrizer, 35

B Bardeen–Cooper–Schrieffer (BCS) theory, 423 Bloch–Nordsieck model, 442 Bogoliubov transformation bosonic, 352 fermionic, 387 improper, 352 proper, 352 singular, 420

© Springer Nature Singapore Pte Ltd. 2020 A. Arai, Inequivalent Representations of Canonical Commutation and Anti-Commutation Relations, Mathematical Physics Studies, https://doi.org/10.1007/978-981-15-2180-5

489

490 Bogoliubov translation, 331 Bohr’s complementary principle, 82 Born–Heisenberg–Jordan representation of CCR, 92 Bose–Einstein condensation (BEC), 404 Bose field, 247 Boson Γ -operator, 253 Boson Fock space, 247 over a direct sum Hilbert space, 306 Bosonic Bogoliubov transformation singular, 353 Boson number operator, 251 Bounded from above, 12 Bounded from below, 12 Bounded operator, 5

C Canonical anti-commutation relations (CAR), 236, 319 Canonical commutation relations (CCR), 59, 266 Dirac, 109, 266 weak, 230 Canonical conjugate momentum, 297 Canonical equilibrium state, 429 Canonical free Bose field, 297 CCR-algebra, 62 CCR-domain, 60, 109, 330, 361 ultra-weak, 231 CCR for {Xj }N j =1 , 109 Closable operator, 6 Closed operator, 5 Closure of an operator, 6 Coherent state, 95 Commutant, 75 Commutator, 4 Completely reducible, 333 Complex conjugation, 363, 423 Complexification, 363 Conjugate linear operator, 2 Conjugate n-linear functional, 31 Conjugation, 41, 299, 350, 387 Constructive quantum field theory, 449 Continuous spectrum, 8 Coordinate representation, 83 Core, 6 Creation operator, 98 boson, 95, 257 fermion, 240, 317 Cyclic representation, 111, 333 Cyclic vector, 111, 223

Index D de Broglie equation, 398 Deficiency indices, 30 Degenerate eigenvalue, 9 Densely defined, 1 Diagonalization, 346 Dirac CCR, 109, 266 Dirac particle, 450 Directional differential operator, 383 Direct sum Hilbert space, 71, 469 Direct sum operator, 71, 469 Direct sum representation, 72 Dirichlet boundary condition, 105 Discrete eigenvalue, 9 Discrete Fourier transform, 50, 58 Domain, 1 du Bois–Reymond lemma, 403, 425

E Eigenspace, 9 Eigenvalue, 9 Eigenvector, 9 Energy renormalization, 449 Enss potential, 228 Enss’s theorem, 229 Essentially self-adjoint, 12 Essential range, 460 Everywhere defined, 1 Extension, 2 External time, 171

F Fermi field, 311 Fermion Γ -operator, 314 Fermion annihilation operator, 317 Fermion creation operator, 317 Fermion Fock space, 239, 311 Fermion Fock vacuum, 312 Fermion second quantization operator, 313 Finite particle subspace, 248, 312 Fixed source, 432 Flat, 157 Fock representation, 92, 330 of CAR, 240 Fock vacuum, 241, 250 Form domain, 44 Free Dirac operator, 214 Free Hermitian Klein–Gordon quantum field, 427 Free Klein–Gordon equation, 427 Free quantum Dirac field, 455

Index Free relativistic Bose field model, 398 Functional calculus, 466 Functional Schrödinger representation of Heisenberg CCR, 383

G Gårding domain, 120 Gauge function, 165 Gauge transformation, 165 Gaussian, 85 Gaussian measure, 85 infinite-dimensional, 381 Gaussian random process, 381 Gel’fand–Na˘imark–Segal (GNS) construction, 419 Generalized partial differential operator, 47 Generalized weak Weyl relation, 232 Generator, 22, 23 Gibbs state, 429 Ground state, 13 Ground state energy, 13

H Haag’s theorem, 449 Hamiltonian, 22, 396 Heisenberg CCR, 60, 109, 361 Heisenberg Lie algebra, 61 Heisenberg–Robertson inequality, 71 Heisenberg uncertainty principle, 82 Heisenberg uncertainty relation, 70 Hermite functions, 97 Hilbert–Schmidt, 16, 354 Hilbert–Schmidt norm, 17

I Infinite direct sum Hilbert space, 471 Infinite direct sum representation, 72 Infrared cutoff, 438 Infrared region, 418 Infrared regular, 433 Infrared renormalization, 442 Infrared singularity, 418 Invariant subspace, 7 Inverse operator, 4 Irreducibility, 267 Irreducible, 109, 237, 365 Isomorphic, 10 Isomorphism, 10 natural, 10 Iso-symplectic transformation, 299

491 K Kernel, 4

L Lie algebra, 60 Lie algebra isomorphism, 60 Linear operator, 1 Local quantization of magnetic flux, 161 Lowest energy, 12

M Magnetic field, 147 Magnetic flux, 153 Magnetic momentum, 148 Magnetic translation, 153 Massive, 434 Massless, 434 Maximal domain, 4 Maximally symmetric, 12 Minimal uncertainty vector, 70 Modulus, 14 Momentum operator, 47 with α-boundary condition, 57 with PBC, 53 Momentum representation, 87 Multiplicity, 9

N Natural isomorphism, 37, 38, 48 Nelson model, 442 Non-negative symmetric operator, 12 Non-Weyl representation of CCR, 128 Number operator, 95 boson, 251 fermion, 314 total, 95

O One-fermion space, 312 One-particle Hilbert space, 247 Operator closable, 6 closed, 5 self-adjoint, 11 symmetric, 11 Operator norm, 5 Operator product, 7 Operator sum, 7 Orthogonal projection, 12, 463

492 P Partial differential operator, 47 Pauli matrix, 214 Physical momentum, 148 Point source, 444 Point-spectral part, 476 Point spectrum, 8 Position operator, 83 Probability space, 68 Projection-valued measure, 463 Purely absolutely continuous, 476 Purely discrete, 9 Purely point spectral, 477 Pure point spectrum, 476

Q Quadratic form, 44 Quadratic operator, 303 Quantum electrodynamics, 442 Quasi free state, 429 Quasi-Weyl relation, 198

R Random variable, 68 Gaussian, 85 Range, 4 Reduced part, 19 Reducible completely, 333 Regular, 113 Regularity of a vector, 222 Relative bound, 5 Relatively bounded, 5 Relativistic quantum field theory, 447 Relativistic Schrödinger operator, 212 Representation associated, 109 of CAR, 237, 385 of the CCR, 59 of Dirac CCR, 109 Heisenberg CCR, 361 space, 60, 330 Representatives, 60 Residual spectrum, 8 Resolution of identity, 463 Resolvent set, 8 Restriction, 3

S Scale transformation, 401 Schrödinger equation, 21

Index Schrödinger operator, 86 Schrödinger representation of CCR, 83 Second quantization operator, 338 boson, 250 fermion, 313 Segal field operator, 284 ω-, 300 Self-adjoint operator, 11 Self-adjoint representation, 60 Semi-bounded, 12 Semi-group, 22 Semi-strong self-adjoint representation, 63 Sesquilinear form, 44 closed, 44 Hermitian, 44 symmetric, 44 Simple eigenvalue, 9 Singular, 475 Singular Bogoliubov transformation, 420 Singular bosonic Bogoliubov transformation, 353 Singular continuous part, 476 Singular continuous self-adjoint operator, 477 Singular continuous spectrum, 476 Singular continuous subspace, 476 Singular subspace, 475 Space inversion, 202 Special unitary group, 238 Spectral measure, 463 Spectral theorem, 13 Spectrum, 9 Spin degree of freedom, 238 Standard representation, 214 Stark effect, 173 Strong commutativity, 25, 296 Strong derivative, 20, 21 Strong limit, 8 Strongly commute, 24 Strongly continuous, 20 Strongly differentiable, 20, 21 Strong self-adjoint representation, 63 Strong time operator, 189 generalized, 232 Superconductivity, 423 Support, 464 Survival probability, 223 Symmetric Fock space, 247 Symmetric operator, 11 bounded from above, 12 bounded from below, 12 non-negative, 12 Symmetric tensor product, 35 algebraic, 35 Symmetrization operator, 35

Index Symmetrizer, 35 Symmetry breaking, 406 Symmetry domain, 231 Symplectic form, 299 Symplectic group, 300 Symplectic space, 299 complex, 299 real, 299 Symplectic transformation, 299

T Tensor product, 32 Tensor product Hilbert space, 32 Tensor product operator, 39 Tensor product representation, 73 Time external, 171 laboratory, 171 pragmatic, 171 Time operator, 176 generalized, 232 strong, 189 ultra-strong, 229 ultra-weak, 230 weak, 230 Time-t fields, 396 Time-zero fields, 395 Trace, 15 Trace class operator, 16 Transition probability, 220 Translation of Heisenberg CCR, 362 Trivial reducing subspace, 74

U Ultra-strong time operator, 229 Ultraviolet cutoff, 411, 445 Ultraviolet divergence, 416, 445 Ultraviolet renormalized representation, 448 Ultra-weak CCR-domain, 231 Ultra-weak time operator, 230 Uncertainty, 68 Uniform limit, 8

493 Unitarily equivalent, 11 Unitary covariance, 25, 467 Unitary group, 21

V Vacuum, 111, 333 Vacuum expectation value, 290 van Hove model infrared-renormalized, 442 van Hove–Miyatake catastrophe infrared, 440 van Hove–Miyatake model, 431 van Hove–Miyatake phenomenon infrared, 440 ultraviolet, 447 van Hove relations, 432 Variational principle, 12 Vector potential, 147 trivial, 157 von Neumann’s uniqueness theorem, 127, 156

W Wave mechanics, 86 Wave–particle duality, 87 Weak CCR, 230 Weak CCR-domain, 230 Weak form of CCR, 132 Weak limit, 8 Weak time operator, 230 Weak Weyl relations, 132 generalized, 232 Weak Weyl representation, 134 Wedge product, 35 Weyl operator, 288 Weyl relations, 116, 296, 365 Weyl representation of CCR, 120, 365 Wightman reconstruction theorem, 417, 419

Z Zero-energy ground state, 13