Feynman-Kac-Type Theorems and Gibbs Measures on Path Space: Volume 2 Applications in Rigorous Quantum Field Theory [2nd rev. ed.] 9783110403541, 9783110403503

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Fumio Hiroshima, József Lőrinczi Feynman–Kac-Type Theorems and Gibbs Measures on Path Space

De Gruyter Studies in Mathematics

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Edited by Carsten Carstensen, Berlin, Germany Gavril Farkas, Berlin, Germany Nicola Fusco, Napoli, Italy Fritz Gesztesy, Waco, Texas, USA Niels Jacob, Swansea, United Kingdom Zenghu Li, Beijing, China Karl-Hermann Neeb, Erlangen, Germany

Volume 34/2

Fumio Hiroshima, József Lőrinczi

Feynman–Kac-Type Theorems and Gibbs Measures on Path Space |

Volume 2: Applications in Rigorous Quantum Field Theory 2nd edition

Mathematics Subject Classification 2010 Primary: 47D08, 81Q10, 35J10; Secondary: 47-01, 60-01 Authors Prof. Dr. Fumio Hiroshima Kyushu University Dept. of Mathematics 744 Motooka Nishiku 819-0395 Japan [email protected]

Prof. Dr. József Lőrinczi Loughborough University Department of Mathematical Sciences Schofield Building Loughborough, LE11 3TU United Kingdom [email protected]

ISBN 978-3-11-040350-3 e-ISBN (PDF) 978-3-11-040354-1 e-ISBN (EPUB) 978-3-11-040360-2 ISSN 0179-0986 Library of Congress Control Number: 2019947702 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2020 Walter de Gruyter GmbH, Berlin/Boston Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

Contents Contents of Volume 1 | IX Preface to the second edition | XV 1 1.1 1.2 1.2.1 1.2.2 1.2.3 1.2.4 1.2.5 1.2.6 1.2.7 1.2.8 1.3 1.3.1 1.3.2 1.3.3 1.3.4 1.4 1.4.1 1.4.2 1.5 1.5.1 1.5.2 1.5.3 1.5.4 1.6 1.6.1 1.6.2 1.6.3 1.6.4 1.6.5 1.6.6 1.6.7 1.7 1.7.1

Free Euclidean quantum field and Ornstein–Uhlenbeck processes | 1 Background | 1 Boson Fock space | 3 Second quantization | 3 The case W = L2 (ℝd ) | 9 The case W = ℂ1 | 12 Segal–Bargmann space | 14 Segal fields | 16 Wick product | 20 Exponentials of creation and annihilation operators | 21 The case W = L2 (ℝd ) | 27 Q -spaces | 43 Gaussian random processes | 43 Wiener–Itô–Segal isomorphism and positivity improving | 46 Hypercontractivity | 52 Lorentz covariant quantum fields | 56 Existence of Q -spaces | 57 Countable product spaces | 57 Bochner theorem and Minlos theorem | 59 Functional integral representation of the Euclidean quantum field | 66 Basic results in Euclidean quantum field theory | 66 Markov property of projections | 75 Feynman–Kac–Nelson formula | 78 van Hove Hamiltonian | 80 Infinite dimensional Ornstein–Uhlenbeck processes | 82 Abstract theory of Gaussian measures on Hilbert spaces | 82 Abstract theory of Borel measures on Hilbert spaces | 90 Fock space as a function space | 97 Infinite dimensional Ornstein–Uhlenbeck process | 100 Markov property | 106 Regular conditional Gaussian probability measures | 108 Feynman–Kac–Nelson formula by path measures | 110 Connection with infinite dimensional stochastic analysis and Malliavin calculus | 111 Finite dimensional case | 111

VI | Contents 1.7.2 1.7.3 1.7.4 1.7.5 1.7.6 1.7.7 2 2.1 2.2 2.2.1 2.2.2 2.2.3 2.3 2.3.1

Stochastic derivative and Cameron–Martin space | 115 Malliavin derivative and divergence operator on L2 (X ) | 118 Wiener–Itô chaos expansion | 121 Malliavin derivative and divergence operator on Wiener chaos | 124 Infinite dimensional Ornstein–Uhlenbeck semigroup | 126 Malliavin derivative on white noise space | 127

The Nelson model by path measures | 131 Preliminaries | 131 The Nelson model in Fock space | 132 Definition of the Nelson model | 132 Infrared and ultraviolet divergences | 135 Embedded eigenvalues | 136 The Nelson model in function space | 137 Infinite dimensional Ornstein–Uhlenbeck processes and P(ϕ)1 -processes | 137 2.3.2 Euclidean field and Brownian motion | 143 2.3.3 Extension to general external potential | 148 2.4 Nelson model with Kato-class potential | 151 2.5 Existence and uniqueness of the ground state | 155 2.5.1 Uniqueness | 155 2.5.2 Existence | 157 2.6 Ground state expectations | 164 2.6.1 General expressions | 164 2.6.2 Ground state expectations for second quantized operators | 170 2.6.3 Ground state expectation for fractional powers of the number operator | 176 2.6.4 Ground state expectations of field operators | 180 2.6.5 Gaussian domination | 182 2.7 Infrared divergence | 184 2.8 Gibbs measure associated with the ground state | 189 2.8.1 Local convergence and Gibbs measures | 189 2.8.2 P(ϕ)1 -process associated with the Nelson Hamiltonian | 196 2.8.3 Applications to ground state expectations | 199 2.9 Carmona-type estimates | 207 2.9.1 Exponential decay of bound states: upper bound | 207 2.9.2 Exponential decay of bound states: lower bound | 208 2.10 Martingale properties and applications | 211 2.10.1 Martingale properties | 211 2.10.2 Exponential decay of bound states | 213 2.11 Ultraviolet divergence | 215 2.11.1 Energy renormalization | 215

Contents | VII

2.11.2 2.11.3 2.11.4 2.11.5 2.11.6 2.11.7 2.12 2.12.1 2.12.2 2.12.3 2.12.4 2.12.5 2.12.6 2.12.7 2.12.8 2.13 2.13.1 2.13.2 2.13.3 2.14 2.14.1 2.14.2 2.14.3 3 3.1 3.1.1 3.1.2 3.1.3 3.2 3.2.1 3.2.2 3.2.3 3.3 3.3.1 3.3.2 3.3.3 3.4

Regularized interaction | 219 Removal of ultraviolet cutoff on Fock vacuum | 232 Uniform lower bound and removal of ultraviolet cutoff | 237 Functional integral representation of the ultraviolet renormalized Nelson model | 239 Gibbs measures and applications | 253 Weak coupling limit and removal of ultraviolet cutoff | 262 Translation invariant Nelson model | 267 Definition of translation invariant Nelson model | 267 Functional integral representation | 270 Existence of ground state | 273 Gibbs measure associated with the ground state of Nelson model with zero total momentum | 275 P(ϕ)1 -process associated with Nelson Hamiltonian with zero total momentum | 277 Removal of ultraviolet cutoff | 280 Ground state energy and ultraviolet renormalization term | 285 Gibbs measures and applications | 288 Polaron model | 291 Definition of the polaron model | 291 Functional integral representation | 292 Removal of ultraviolet cutoff | 294 Functional central limit theorem | 296 Gibbs measures with no external potential | 296 Diffusive behavior | 306 Diffusion matrix and effective mass | 314 The Pauli–Fierz model by path measures | 317 Preliminaries | 317 Introduction | 317 Lagrangian QED | 318 Classical variant of nonrelativistic QED | 322 The Pauli–Fierz model in nonrelativistic QED | 324 The Pauli–Fierz model in Fock space | 324 The Pauli–Fierz model in function space | 329 Markov property | 334 Functional integral representation for the Pauli–Fierz Hamiltonian | 337 Hilbert space-valued stochastic integrals | 337 Functional integral representation | 341 Extension to general external potential | 349 The Pauli–Fierz model with Kato-class potential | 351

VIII | Contents 3.5 3.5.1 3.5.2 3.5.3 3.6 3.6.1 3.6.2 3.6.3 3.6.4 3.7 3.8 3.8.1 3.8.2 3.8.3 3.8.4 3.8.5 3.8.6 3.8.7 3.8.8 3.8.9 3.9 3.9.1 3.9.2 3.9.3 3.9.4 3.9.5 3.9.6 3.9.7 3.9.8 3.9.9 3.9.10 3.9.11

4 4.1 4.2 4.2.1

Applications of functional integral representations | 355 Self-adjointness of the Pauli–Fierz Hamiltonian | 355 Positivity improving and uniqueness of the ground state | 364 Spatial decay of bound states | 370 Path measure associated with the ground state of Pauli–Fierz Hamiltonian | 371 Path measure with double stochastic integrals | 371 Expression in terms of iterated stochastic integrals | 375 Weak convergence and Gibbs measures | 378 Gaussian domination of ground states | 381 Translation invariant Pauli–Fierz model | 382 The Pauli–Fierz model with spin | 389 Counting measures | 389 Review of classical cases | 390 Definition of the Pauli–Fierz Hamiltonian with spin | 392 Symmetry and polarization | 395 Scalar representations | 401 Fock representations | 405 Preparation of functional integral representations | 407 Functional integral representations | 421 Translation invariant Pauli–Fierz Hamiltonian with spin | 432 Relativistic Pauli–Fierz model | 439 Definition of relativistic Pauli–Fierz Hamiltonian | 439 Functional integral representations | 442 Self-adjointness | 447 Nonrelativistic limit of relativistic Pauli–Fierz Hamiltonian | 457 Relativistic Pauli–Fierz model with relativistic Kato-class potential | 460 Martingale properties | 465 Spatial decay of bound states | 469 Gaussian domination of ground states | 471 Path measure associated with the ground state of relativistic Pauli–Fierz Hamiltonian | 473 Translation invariant relativistic Pauli–Fierz model | 474 Nonrelativistic limit of translation invariant relativistic Pauli–Fierz Hamiltonian | 478 Spin-boson model by path measures | 479 Definitions | 479 Functional integral representation for the spin-boson Hamiltonian | 483 Preliminaries | 483

Contents | IX

4.2.2 4.2.3 4.3 4.4 4.4.1 4.4.2 4.4.3 4.5 5

Spin process | 484 Functional integral representation | 486 Existence and uniqueness of ground state | 488 Gibbs measure associated with the ground state | 490 Local convergence and Gibbs measures | 490 Ground state properties | 491 van Hove representation | 499 Rabi Hamiltonian | 501 Notes and references | 505 Notes to Chapter 1 | 505 Notes to Chapter 2 | 507 Notes to Chapter 3 | 514 Notes to Chapter 4 | 519

Bibliography | 521 Index | 533

Contents of Volume 1 Preface to the second edition | IX Preface to the first edition | XI 1 1.1 1.2

Heuristics and history | 1 Feynman path integrals and Feynman–Kac formulae | 1 Plan and scope of the second edition | 5

2 2.1 2.1.1 2.1.2 2.1.3 2.1.4

Brownian motion | 9 Concepts and facts of general measure theory and probability | 9 Elements of general measure theory | 9 Probability measures and limit theorems | 16 Random variables | 29 Conditional expectation and regular conditional probability measures | 38 Random processes | 45 Basic concepts and facts | 45 Martingale properties | 50 Stopping times and optional sampling | 53 Markov properties | 67 Feller transition kernels and generators | 72 Invariant measures | 74 Brownian motion and Wiener measure | 77 Construction of Brownian motion | 77 Two-sided Brownian motion | 84 Conditional Wiener measure | 88 Martingale properties of Brownian motion | 89 Markov properties of Brownian motion | 92 Local path properties of Brownian motion | 97 Global path properties of Brownian motion | 103 Stochastic calculus based on Brownian motion | 107 The classical integral and its extensions | 107 Stochastic integrals | 108 Extension of stochastic integrals | 115 Itô formula | 119 Stochastic differential equations | 128 Brownian bridge | 134 Weak solution and time change | 136 Girsanov theorem and Cameron–Martin formula | 140

2.2 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.2.6 2.3 2.3.1 2.3.2 2.3.3 2.3.4 2.3.5 2.3.6 2.3.7 2.4 2.4.1 2.4.2 2.4.3 2.4.4 2.4.5 2.4.6 2.4.7 2.4.8

XII | Contents of Volume 1 3 3.1 3.1.1 3.1.2 3.1.3 3.1.4 3.1.5 3.2 3.2.1 3.2.2 3.3 3.3.1 3.3.2 3.4 3.4.1 3.4.2 3.5 3.5.1 3.5.2 3.5.3 3.6 3.6.1 3.6.2

Lévy processes | 143 Lévy processes and the Lévy–Khintchine formula | 143 Infinitely divisible random variables | 143 Lévy–Khintchine formula | 149 Lévy processes | 154 Martingale properties of Lévy processes | 160 Markov properties of Lévy processes | 161 Sample path properties of Lévy processes | 165 Càdlàg version | 165 Two-sided Lévy processes | 169 Random measures and Lévy–Itô decomposition | 178 Poisson random measures | 178 Lévy–Itô decomposition | 186 Itô formula for semimartingales | 188 Point processes | 188 Itô formula for semimartingales | 194 Exponentials of Lévy processes and recurrence properties | 201 Exponential functionals of Lévy processes | 201 Capacitary measures | 203 Recurrence properties of Lévy processes | 204 Subordinators and Bernstein functions | 206 Subordinators and subordinate Brownian motion | 206 Bernstein functions | 209

4 4.1 4.1.1 4.1.2 4.1.3 4.1.4 4.1.5 4.1.6 4.1.7 4.1.8 4.2 4.2.1 4.2.2 4.2.3 4.2.4 4.3 4.3.1 4.3.2

Feynman–Kac formulae | 217 Schrödinger semigroups | 217 Schrödinger equation and path integral solutions | 217 Linear operators and their spectra | 218 Spectral resolution | 223 Compact operators and trace ideals | 227 Schrödinger operators | 232 Schrödinger operators through quadratic forms | 236 Confining potentials and decaying potentials | 239 Strongly continuous operator semigroups | 243 Feynman–Kac formula for Schrödinger operators | 246 Bounded smooth external potentials | 246 Derivation through the Trotter product formula | 249 Kato-class potentials | 251 Feynman–Kac formula for Kato-decomposable potentials | 264 Properties of Schrödinger operators and semigroups | 270 Kernel of the Schrödinger semigroup | 270 Positivity improving and uniqueness of ground state | 271

Contents of Volume 1

4.3.3 4.3.4 4.3.5 4.3.6 4.3.7 4.3.8 4.4 4.4.1 4.4.2 4.4.3 4.4.4 4.5 4.6 4.6.1 4.6.2 4.6.3 4.6.4 4.7 4.7.1 4.7.2 4.7.3 4.7.4 4.7.5 4.8 4.8.1 4.8.2 4.8.3 4.9 4.9.1 4.9.2 4.9.3 4.9.4 4.9.5 4.9.6 4.9.7 4.9.8 4.9.9 4.9.10

| XIII

Degenerate ground state and Klauder phenomenon | 275 Existence and non-existence of ground states | 277 Sojourn times and existence of bound states | 282 The number of eigenfunctions with negative eigenvalues | 289 Application to canonical commutation relations | 307 Exponential decay of eigenfunctions | 314 Feynman–Kac formula for Schrödinger operators with vector potentials | 320 Feynman–Kac–Itô formula | 320 Alternative proof of the Feynman–Kac–Itô formula | 324 Extension to singular external and vector potentials | 327 Kato-class potentials and Lp -Lq boundedness | 333 Feynman–Kac formula for unbounded semigroups and Stark effect | 335 Feynman–Kac formula for relativistic Schrödinger operators | 339 Relativistic Schrödinger operator | 339 Relativistic Kato-class potentials | 344 Decay of eigenfunctions | 351 Non-relativistic limit | 356 Feynman–Kac formula for Schrödinger operators with spin | 359 Schrödinger operators with spin 21 | 359 A jump process | 361 Feynman–Kac formula for the jump process | 363 Extension to singular external potentials and singular vector potentials | 367 Decay of eigenfunctions and martingale properties | 371 Feynman–Kac formula for relativistic Schrödinger operators with spin | 375 Relativistic Schrödinger operator with spin 21 | 375 Martingale properties | 381 Decay of eigenfunctions | 384 Feynman–Kac formula for nonlocal Schrödinger operators | 388 Nonlocal Schrödinger operators | 388 Vector potentials | 389 Ψ-Kato-class potentials | 392 Fractional Kato-class potentials | 401 Generalized spin | 406 Recurrence properties and existence of bound states | 412 The number of eigenfunctions with negative eigenvalues | 413 Decay of eigenfunctions | 423 Massless relativistic harmonic oscillator | 432 Embedded eigenvalues | 436

XIV | Contents of Volume 1 5 5.1 5.1.1 5.1.2 5.1.3 5.1.4 5.1.5 5.1.6 5.2 5.2.1 5.2.2 5.2.3 5.3 5.3.1 5.3.2 5.4 5.5 5.5.1 5.5.2 5.5.3 6

Gibbs measures associated with Feynman–Kac semigroups | 449 Ground state transform and related processes | 449 Ground state transform and the intrinsic semigroup | 449 Ground state-transformed processes as solutions of SDE | 454 P(ϕ)1 -processes with continuous paths | 458 Dirichlet principle | 464 Mehler’s formula | 467 P(ϕ)1 -processes with càdlàg paths | 476 Gibbs measures on path space | 481 From Feynman–Kac formulae to Gibbs measures | 481 Gibbs measures on Brownian paths | 485 Gibbs measures on càdlàg paths | 492 Gibbs measures for external potentials | 494 Existence | 494 Uniqueness | 497 Gibbs measures for external and pair interaction potentials: direct method | 503 Gibbs measures for external and pair interaction potentials: cluster expansion | 511 Cluster representation | 511 Basic estimates and convergence of cluster expansion | 516 Further properties of the Gibbs measure | 518 Notes and references | 521 Notes to the Preface | 521 Notes to Chapter 1 | 522 Notes to Chapter 2 | 522 Notes to Chapter 3 | 525 Notes to Chapter 4 | 526 Notes to Chapter 5 | 535

Bibliography | 539 Index | 553

Preface to the second edition In the present edition of this monograph, first published in 2011, the material covered has doubled and thus it was natural to split the presentation in two volumes. This second part is a companion volume to “Feynman–Kac-Type Formulae and Gibbs Measures”, where we laid the foundations to the applications of Feynman–Kac representations to a rigorous study of models of quantum field theory presented in this volume. Our goal in the present edition was to make the material more self-contained and up-to-date, reflecting recent work by the authors and their collaborators on developing new methods of functional integration to ground state problems of fundamental models of quantum field theory. After a detailed introduction to a mathematical treatment of the free Euclidean quantum field and infinite dimensional Ornstein– Uhlenbeck processes, we devote separate chapters to the Nelson, the Pauli–Fierz, and the spin-boson models. While our approach is via path integrals, there is much interaction between functional analysis, operator theory and stochastic analysis at work, which is presented systematically in the hope that newcomers to the field will find it useful, while it will be a resource also to experts. In the presentation we typically follow the pattern of first introducing and discussing a model Hamiltonian on Fock space (suitable for the operator language) and a unitary equivalent L2 space (suitable for the stochastic language), then derive a functional integral representation, next describe the related path measure as a Gibbs measure, and finally study ground state properties of the initial Hamiltonian by making use of this Gibbs measure. We recommend to use this book with reference to the material of Volume 1, and we provide further commentary in Chapter 5 to a selection of the vast bibliography. We are grateful to our collaborators Massimiliano Gubinelli, Masao Hirokawa, Oliver Matte, Toshimitsu Takaesu for recent joint work on the problems discussed in this book, and express our appreciation to Gonzalo A. Bley, Thomas Norman Dam, Jacob Schach Møller, Yuma Takabayashi, for comments on the previous edition. The first named author thanks the kind hospitality of Aarhus University and the International Network Program of the Danish Agency for Science, Technology and Innovation, where part of this work has been done. His work was also financially supported by JSPS KAKENHI Grant Number JP16H03942 and JSPS KAKENHI Grant Number JP16K17612. The second named author thanks IHES, Bures-sur-Yvette, for visiting fellowships through several years, where part of this book has been written, and specifically Professor David Ruelle for sustained interest and discussions. Last but not least, we also express our appreciation to Sabina Dabrowski of Walter de Gruyter, and Ina Talandienė and her team at VTeX UAB, Lithuania, for their professionalism and support.

https://doi.org/10.1515/9783110403541-201

1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes 1.1 Background A free quantum field describes a large collection of quantum particles of the same type in which there is no interaction between any of these particles nor with any other particles in the environment. In this section, we introduce basic notions of the mathematical theory of free quantum fields. There are various formulations of this theory of which we will discuss two. One uses field operators acting on a Hilbert space, providing a framework in which problems of spectral theory, scattering theory and others are addressed directly. Another is the so-called Euclidean quantum field theory which uses an L2 space-valued random process to address similar problems by using probabilistic methods. Our goal here is to explain the equivalence of the theory of free boson fields with the theory of infinite dimensional Gaussian random processes. The essential link between the two descriptions is a Feynman–Kac-type formula. First, we define the quantum field in boson Fock space. While there are many advantages to the Fock space picture, what is needed for an approach using a Feynman– Kac-type formula is that the Hamiltonian is given in Schrödinger representation, i. e., acting on a L2 (Q , dμ) space. Fock space is not of this form, and thus the Gaussian space introduced in the previous section and the Wiener–Itô–Segal isomorphism are needed. Secondly, we discuss the quantum field in terms of Euclidean quantum field theory. In order to obtain the integral representation of (F, e−tHf G) for F, G ∈ L2 (Q , dμ) with its Hamiltonian Hf one way is to changing eitH to e−tH regarding it as the analytic continuation of t to it. In this procedure, the Minkowski metric turns into a Euclidean metric: −t 2 + x2 → t 2 + x2 , (t, x) ∈ ℝ × ℝ3 . In the Lorentz covariant formulation of scalar quantum field theory, the so-called two-point function is given by (1, ϕ(f )ϕ(g)1) =

∫ f (x)g(y)W(x − y)dxdy ℝ4 ×ℝ4

with the Wightman distribution W(x − y) = (1, ϕ(x)ϕ(y)1). Here, x = (t, x) and ϕ(x) = e−itH ϕ((0, x))eitH . The Wightman distribution of the free field Hamiltonian is W((t, x)) =

2

2

1 1 eit |k| +ν −ik⋅x dk. ∫ 2 (2π)3 √|k|2 + ν2 3 √

ℝ

The field at time zero is given by ∫ ϕ(f )ϕ(g)dμ = Q https://doi.org/10.1515/9783110403541-001

1 dk ̂ . ∫ f ̂(k)g(k) 3 2 √ (2π) |k|2 + ν2 ℝ3

2 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes Due to the singularity of the so-called Feynman propagator 1 , k02 − |k|2 − ν2 it is natural to make an analytic continuation into the region k02 < 0. The depth and utility of this point of view has been long appreciated, and Nelson discovered a method of recovering a Minkowski regional field theory from Euclidean field theory. The constructive quantum field theory developed initially by Glimm and Jaffe made use of the Euclidean method. The analytic continuation t 󳨃→ it of W((t, x)) yields the Schwinger function W((it, x)) =

eitk0 −ik⋅x 1 1 dk, ∫ 2 (2π)4 |k|2 + ν2 + k02

k = (k0 , k) ∈ ℝ × ℝ3 .

ℝ4

The Euclidean time-zero quantum field ϕE (f ) has covariance ∫ ϕE (f )ϕE (g)dμE =

1 dk ̂ . ∫ f ̂(k)g(k) 2 |k|2 + ν2 + k02 ℝ4

QE

To establish the connection between L2 (Q ) and the Euclidean space L2 (QE , dμE ), a family of operators {It }t∈ℝ is introduced. The functional integral representation of e−t(Hf +HI ) with some perturbation HI can be also constructed by using the Trotter product formula, the Markov property of Is I∗s and the identity I∗t Is = e−|s−t|Hf , we have (F, e−tH G) = ∫ F0 Gt eK dμE ,

(1.1.1)

QE

where F0 = I0 F, Gt = It G and eK denotes an integral kernel. The right-hand side of 󸀠 (1.1.1) is the integral over QE , e. g., Sreal (ℝd+1 ), which we call functional integral in distinction to path integral. An alternative approach is based on a path measure constructed on C(ℝ; M−2 ), i. e., the set of M−2 -valued continuous paths, where M−2 is the dual of a Hilbert space M+2 . For path integral representations of the heat semigroup (f , e−tH g) in quantum mechanics, a path measure is given on C([0, ∞); ℝd ). In quantum field theory, the same procedure yields M−2 -valued paths. In the finite mode ak = (1/√2)(xk + 𝜕k ) and a∗k = (1/√2)(xk − 𝜕k ), k = 1, . . . , n, and the free field Hamiltonian Hf becomes n

n

√|k|2 + ν2 (−Δk + |xk |2 − 1). 2 k=1

Hf → ∑ √|k|2 + ν2 a∗k ak = ∑ k=1

In this representation, Hf has the form of an infinite dimensional harmonic oscillator. It is known that the generator of the d-dimensional Ornstein–Uhlenbeck process

1.2 Boson Fock space |

3

on C(ℝ; ℝd ) is the harmonic oscillator on L2 (ℝd ). This suggests that it might be possible to construct an infinite dimensional Ornstein–Uhlenbeck process (ξt )t∈ℝ on Y = C(ℝ, M−2 ) whose generator is the free field Hamiltonian Hf . This can be done by way of Kolmogorov’s consistency theorem through the finite dimensional distributions giving (F, e−tH G) = ∫ F(ξ0 )G(ξt )eK(ξ ) d𝒢 Y

with a suitable kernel K(ξ ).

1.2 Boson Fock space 1.2.1 Second quantization In this section we start the above outlined program by explaining the Fock spacebased formulation of quantum field theory. Instead of physical space-time dimension, we work in general d dimension. Let W be a separable Hilbert space over ℂ. Consider the operation ⊗nsym of n-fold symmetric tensor product defined through the symmetrization operator Sn (f1 ⊗ ⋅ ⋅ ⋅ ⊗ fn ) =

1 ⊗ ⋅ ⋅ ⋅ ⊗ fπ(n) , ∑ f n! π∈℘n π(1)

n ≥ 1,

where f1 , . . . , fn ∈ W and ℘n denotes the permutation group of order n. Define n

n

ℱb = ℱb (W ) = ⊗sym W = Sn (⊗ W ), (n)

(n)

where ⊗0sym W = ℂ. The space ∞

(n)

ℱb = ℱb (W ) = ⨁ℱb (W ), n=0

where ⨁∞ n=0 is understood to be completed (rather than simple algebraic) direct sum, is called boson Fock space over W . The Fock space ℱb can be identified with the space of ℓ2 -sequences (Ψ(n) )n∈ℕ such that Ψ(n) ∈ ℱb(n) and ∞

‖Ψ‖2ℱb = ∑ ‖Ψ(n) ‖2ℱ (n) < ∞. n=0

b

ℱb is a Hilbert space endowed with the scalar product ∞

(Ψ, Φ)ℱb = ∑ (Ψ(n) , Φ(n) )ℱ (n) . n=0

b

(1.2.1)

4 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes The vector Ωb = (1, 0, 0, . . . ) is called Fock vacuum. Write Pn for the projection from ℱb onto ℱb(n) . The subspace Pn ℱb can be interpreted as consisting of the states of the quantum field having exactly n boson particles. Functions in Pn ℱb then determine the exact behavior of these particles, while the permutation symmetry corresponds to the fact that the particles are indistinguishable. In a model of an electron coupled to its radiation field which will be studied in Chapter 3, the particles will be photons. If ‖Φ‖ℱb = 1, then (Φ, Pn Φ)ℱb represents the probability of having exactly n bosons in the state described by Φ. Note that by (1.2.1) this probability must decay faster than 1/√n for large n in order to have Φ ∈ ℱb . The reason for this constraint is mathematical convenience rather than physical necessity, a fact that we will encounter when discussing the infrared divergence of a specific quantum field model. In the description of the free quantum field, the following operators acting in ℱb are used. There are two fundamental boson particle operators, the creation operator denoted by a∗ (f ), f ∈ W , and the annihilation operator by a(f ), f ∈ W , both acting on ℱb , defined by (a∗ (f )Ψ)(n) = √nSn (f ⊗ Ψ(n−1) ),

n ≥ 1,

(a∗ (f )Ψ)(0) = 0 with domain

󵄨󵄨 ∞ D(a∗ (f )) = {(Ψ(n) )n≥0 ∈ ℱb 󵄨󵄨󵄨 ∑ n‖Sn (f ⊗ Ψ(n−1) )‖2ℱ (n) < ∞}, 󵄨 n=1 b

and

a(f ) = (a∗ (f ̄))∗ .

As the terminology suggests, the action of a∗ (f ) increases the number of bosons by one, while a(f ) decreases it by one. Since one is the adjoint operator of the other, the relation (Φ, a(f )Ψ) = (a∗ (f ̄)Φ, Ψ) holds. Furthermore, since both operators are closable by the dense definition of their adjoints, we will use and denote their closed extensions by the same symbols. Let D ⊂ W be a dense subset. It is known that L. H.{Ωb , a∗ (f1 ) ⋅ ⋅ ⋅ a∗ (fn )Ωb | fj ∈ D, j = 1, . . . , n, n ≥ 1} is dense in ℱb (W ), where L. H. is a shorthand for the linear hull. The space (n)

ℱb,fin = {(Ψ

)n≥0 ∈ ℱb | Ψ(m) = 0 for all m ≥ M with some M}

is called finite boson subspace. The field operators a, a∗ leave ℱb,fin invariant and satisfy the canonical commutation relations [a(f ), a∗ (g)] = (f ̄, g)1, on ℱb,fin .

[a(f ), a(g)] = 0,

[a∗ (f ), a∗ (g)] = 0

1.2 Boson Fock space |

5

Given a bounded operator T on W , the second quantization of T is the operator Γ(T) on ℱb defined by ∞

Γ(T) = ⨁(⊗n T). n=0

Here it is understood that ⊗0 T = 1. In most cases Γ(T) is an unbounded operator resulting from the fact that it is given by a countable direct sum. However, for a contraction operator T, the second quantization Γ(T) is also a contraction on ℱb , or equivalently, Γ is a functor Γ : C (W → W ) → C (ℱb → ℱb ), of the set C (X → Y) of contraction operators from X to Y. The functor Γ has the semigroup property, while C (W → W ) is a ∗-algebra with respect to operator multiplication and conjugation ∗ (i. e., taking adjoints). The map Γ pulls this structure over to ℱb so that Γ(S)Γ(T) = Γ(ST),

Γ(S)∗ = Γ(S∗ ),

Γ(1) = 1,

(1.2.2)

for S, T ∈ C (W → W ). The above definitions imply the intertwining properties Γ(T)a∗ (f ) = a∗ (Tf )Γ(T),

a(f )Γ(T) = Γ(T)a(jT jf ), ∗

(1.2.3) (1.2.4)

which can be checked directly. From this the commutation relations [Γ(T), a∗ (f )] = a∗ ((T − 1)f )Γ(T),

[Γ(T), a(f )] = Γ(T)a(j(1 − T )jf ), ∗

(1.2.5) (1.2.6)

follow, where jf = f ̄ denotes the complex conjugate of f . If, in particular, T satisfies T ∗ T = 1, then (1.2.3) and (1.2.4) yield Γ(T)a∗ (f )Γ(T ∗ ) = a∗ (Tf ), Γ(T)a(f )Γ(T ) = a(jTjf ). ∗

(1.2.7) (1.2.8)

For a self-adjoint operator h on W , the structure relations (1.2.2) imply in particular that {Γ(eith ) : t ∈ ℝ} is a strongly continuous one-parameter unitary group on ℱb . By Stone’s theorem there exists a unique self-adjoint operator dΓ(h) on ℱb such that Γ(eith ) = eitdΓ(h) ,

t ∈ ℝ.

6 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes The operator dΓ(h) is called the differential second quantization of h or simply second quantization of h. Thus we have eitdΓ(h) a∗ (f )e−itdΓ(h) = a∗ (eith f ), eitdΓ(h) a(f )e−itdΓ(h) = a(jeith jf ).

(1.2.9) (1.2.10)

Since dΓ(h) = −i

d Γ(eith )⌈t=0 dt

on a suitable domain, we have ∞

n

n=1

j=1

j

dΓ(h) = 0 ⊕ {⨁ (∑ 1 ⊗ ⋅ ⋅ ⋅ ⊗ h ⊗ ⋅ ⋅ ⋅ ⊗ 1)} ,

(1.2.11)

where the bar denotes closure, and j on top of h indicates its position in the product. Thus the action of dΓ(h) on each ℱb(n) is given by dΓ(h)Ωb = 0 and n

dΓ(h)a∗ (f1 ) ⋅ ⋅ ⋅ a∗ (fn )Ωb = ∑ a∗ (f1 ) ⋅ ⋅ ⋅ a∗ (hfj ) ⋅ ⋅ ⋅ a∗ (fn )Ωb . j=1

It can be also seen by (1.2.11) that Spec(dΓ(h)) = {λ1 + ⋅ ⋅ ⋅ + λn | λj ∈ Spec(h), j = 1, . . . , n, n ≥ 1} ∪ {0},

Specp (dΓ(h)) = {λ1 + ⋅ ⋅ ⋅ + λn | λj ∈ Specp (h), j = 1, . . . , n, n ≥ 1} ∪ {0}. If 0 ∉ Specp (h), the multiplicity of 0 in Specp (dΓ(h)) is one. A crucial operator in quantum field theory is the boson number operator N = dΓ(1) defined as the second quantization of the identity operator on W . Since NΩb = 0,

Na∗ (f1 ) ⋅ ⋅ ⋅ a∗ (fn )Ωb = na∗ (f1 ) ⋅ ⋅ ⋅ a∗ (fn )Ωb , it follows that Spec(N) = Specp (N) = ℕ ∪ {0}.

1.2 Boson Fock space |

7

Let ρ be a real-valued measurable function on ℝ. The operator ρ(N) is self-adjoint on the dense domain 󵄨󵄨 ∞ D(ρ(N)) = {(Φ(n) )n≥0 ∈ ℱb 󵄨󵄨󵄨 ∑ ρ(n)2 ‖Φ(n) ‖2ℱ (n) < ∞}. 󵄨 n=0 b In the case of ρ(x) = eβx with β > 0 (note the positive exponent), Ψ ∈ D(eβN ) implies that the probability of finding n bosons decays exponentially as n → ∞. We will use the following facts below. The superscript in a♯ indicates that any of the creation or annihilation operators is meant. Proposition 1.1 (Relative bound for a♯ (f )). Let h be a positive self-adjoint operator, f ∈ D(h−1/2 ) and Ψ ∈ D(dΓ(h)1/2 ). Then Ψ ∈ D(a♯ (f )) and ‖a(f )Ψ‖ ≤ ‖h−1/2 f ‖‖dΓ(h)1/2 Ψ‖, ∗

−1/2

‖a (f )Ψ‖ ≤ ‖h

1/2

f ‖‖dΓ(h) Ψ‖ + ‖f ‖‖Ψ‖.

(1.2.12) (1.2.13)

In particular, D(dΓ(h)1/2 ) ⊂ D(a♯ (f )), whenever f ∈ D(h−1/2 ). While we do not give a proof of this proposition, we will prove it for a special case below. Example 1.2. (1) A first application of the above estimates is that a∗ (f ) and a(f ) are well-defined on D(N 1/2 ) for all f ∈ W . For f ∈ W , ‖a(f )Ψ‖ ≤ ‖f ‖‖N 1/2 Ψ‖, ∗

‖a (f )Ψ‖ ≤ ‖f ‖(‖N

1/2

Ψ‖ + ‖Ψ‖).

(1.2.14) (1.2.15)

(2) Another application is considering a perturbation of dΓ(h) by adding the operator Φ(f ) =

1 ∗ (a (f ) + a(f ̄)). √2

Let f ∈ D(h−1/2 ). Then dΓ(h) + αΦ(f ) is self-adjoint on D(dΓ(h)) for all α ∈ ℝ. This follows from (1.2.12), (1.2.13) and the Kato–Rellich theorem. To obtain the commutation relations between a♯ (f ) and dΓ(h), suppose that f ∈ D(h−1/2 ) ∩ D(h). Then [dΓ(h), a∗ (f )]Ψ = a∗ (hf )Ψ,

[dΓ(h), a(f )]Ψ = −a(hf ̄)Ψ,

(1.2.16)

for Ψ ∈ D(dΓ(h)3/2 ) ∩ ℱb,fin . By a limiting argument, (1.2.16) can be extended to Ψ ∈ D(dΓ(h)3/2 ), and it is seen that a♯ (f ) maps D(dΓ(h)3/2 ) into D(dΓ(h)).

8 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes n n−1/2 Lemma 1.3. Let f ∈ ⋂∞ )). Then n=1 (D(h ) ∩ D(h

a♯ (f ) : D(dΓ(h)n+1/2 ) → D(dΓ(h)n ) n for all n ≥ 1. In particular, a♯ (f ) maps ⋂∞ n=1 D(dΓ(h) ) into itself.

Next we discuss the tensor product of boson Fock spaces. Proposition 1.4. (1) Let W1 and W2 be Hilbert spaces. Then ℱb (W1 ⊕ W2 ) ≅ ℱb (W1 ) ⊗ ℱb (W2 ).

(2) Let A and B be self-adjoint operator in W1 and W2 , respectively. Then dΓ(A ⊕ B) ≅ dΓ(A) ⊗ 1 + 1 ⊗ dΓ(B). Proof. Let Ω1 (resp. Ω2 ) be the Fock vacuum in ℱb (W1 ) (resp. ℱb (W2 )). Define the operator U : ℱb (W1 ⊕ W2 ) → ℱb (W1 ) ⊗ ℱb (W2 ) by n

n

j=1

j=1

U ∏ a∗ (fj ⊕ gj )Ωb = ∏ (a∗ (fj ) ⊗ 1 + 1 ⊗ a∗ (gj )) Ω1 ⊗ Ω2 . Here fj ∈ W1 , j = 1, . . . , n and gi ∈ W2 , i = 1, . . . , n We see that 󵄩󵄩 n 󵄩󵄩 󵄩󵄩 n 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 ∗ = 󵄩󵄩󵄩 ∏ (a∗ (fj ) ⊗ 1 + 1 ⊗ a∗ (gj )) Ω1 ⊗ Ω2 󵄩󵄩󵄩 󵄩󵄩U ∏ a (fj ⊕ gj )Ωb 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩ℱb (W1 ⊕W2 ) 󵄩󵄩 󵄩󵄩ℱb (W1 )⊗ℱb (W2 ) j=1 j=1 From this equality U can be extended to the unitary operator from ℱb (W1 ⊕ W2 ) to ℱb (W1 ) ⊗ ℱb (W2 ), thus statement (1) follows. Next we prove statement (2). We have Γ(eitA⊕itB ) = Γ(eitA ⊕ eitB ) and n

n

j=1

j=1

UΓ(eitA ⊕ eitB ) ∏ a∗ (fj ⊕ gj )Ωb = U ∏ a∗ (eitA fj ⊕ eitB gj )Ωb n

= ∏ (a∗ (eitA fj ) ⊗ 1 + 1 ⊗ a∗ (eitB gj )) Ω1 ⊗ Ω2 j=1

n

= (Γ(eitA ) ⊗ 1 + 1 ⊗ Γ(eitB )) ∏ (a∗ (fj ) ⊗ 1 + 1 ⊗ a∗ (gj )) Ω1 ⊗ Ω2 j=1

n

= (Γ(eitA ) ⊗ 1 + 1 ⊗ Γ(eitB )) U ∏ a∗ (fj ⊕ gj )Ωb . j=1

Hence UΓ(eitA ⊕ eitB )U −1 = Γ(eitA ) ⊗ 1 + 1 ⊗ Γ(eitB ) follows. Taking the derivative at t = 0, yields statement (2).

1.2 Boson Fock space |

9

Inductively we can also see that n

n

(1.2.17)

ℱb (⨁ Wj ) ≅ ⊗ ℱb (Wj ). j=1

j=1

1.2.2 The case W = L2 (ℝd ) Take now W = L2 (ℝd ) and consider the boson Fock space ℱb (L2 (ℝd )). In this case, for n ∈ ℕ the space ℱb(n) can be identified with the set of symmetric functions on L2 (ℝdn ) through ⊗nsym L2 (ℝd ) ≅ {f ∈ L2 (ℝdn ) | f (k1 , . . . , kn ) = f (kπ(1) , . . . , kπ(n) ), ∀π ∈ ℘n }. The creation and annihilation operators are realized as (a(f )Ψ)(n) (k1 , . . . , kn ) = √n + 1 ∫ f (k)Ψ(n+1) (k, k1 , . . . , kn )dk, n ≥ 0, (a∗ (f )Ψ)(n) (k1 , . . . , kn ) =

1 { √n

0,

ℝd n ∑j=1 f (kj )Ψ(n−1) (k1 , . . . , k̂j , . . . , kn ),

n ≥ 1, n = 0.

Here Ψ ∈ ℱb(n) is denoted as a pointwise defined function for convenience, however, all of these expressions are to be understood in L2 -sense. Formally, we also use a notation of kernels a(k) and a∗ (k) by writing (a(k)Ψ)(n) (k1 , . . . , kn ) = √n + 1Ψ(n+1) (k, k1 , . . . , kn ), (a∗ (k)Ψ)(n) (k1 , . . . , kn ) = and

1 n ∑ δ(k − kj )Ψ(n−1) (k1 , . . . , k̂j , . . . , kn ), √n j=1

[a(k), a∗ (k 󸀠 )] = δ(k − k 󸀠 ),

[a(k), a(k 󸀠 )] = 0 = [a∗ (k), a∗ (k 󸀠 )].

Let ων : L2 (ℝd ) → L2 (ℝd ) be the multiplication operator called dispersion relation given by ων (k) = √|k|2 + ν2 ,

k ∈ ℝd ,

with ν ≥ 0. Here ν describes the boson mass. The second quantization of the dispersion relation is n

(dΓ(ων )Ψ)(n) (k1 , . . . , kn ) = (∑ ων (kj )) Ψ(n) (k1 , . . . , kn ). j=1

10 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes Definition 1.5 (Free field Hamiltonian). The self-adjoint operator dΓ(ων ) is called free field Hamiltonian on ℱb (L2 (ℝd )) and we use the notation Hf = dΓ(ων ). The spectrum of the free field Hamiltonian is Spec(Hf ) = [0, ∞), with component Specp (Hf ) = {0}, which is of single multiplicity with Hf Ωb = 0. Formally we may write the free field Hamiltonian as Hf = ∫ ων (k)a∗ (k)a(k)dk. ℝd

Physically, this describes the total energy of the free field since a∗ (k)a(k) gives the number of bosons carrying momentum k, multiplied with the energy ων (k) of a single boson and integrated over all momenta. The commutation relations are [Hf , a(f )] = −a(ων f ),

[Hf , a∗ (f )] = a∗ (ων f )

and in formal expression they can be written as [dΓ(ων ), a(k)] = −ων (k)a(k),

[dΓ(ων ), a∗ (k)] = ων (k)a∗ (k).

The relative bounds of a♯ (f ) with respect to the free Hamiltonian Hf can be seen from (1.2.18) and (1.2.19). If f /√ων ∈ L2 (ℝd ), then ‖a(f )Ψ‖ ≤ ‖f /√ων ‖‖Hf1/2 Ψ‖, ∗

‖a (f )Ψ‖ ≤

‖f /√ων ‖‖Hf1/2 Ψ‖

(1.2.18) (1.2.19)

+ ‖f ‖‖Ψ‖

hold, following directly from the lemma below. Lemma 1.6. Let h : ℝd → ℂ be a measurable function, and g ∈ D(h). Then for every Ψ ∈ D(dΓ(|h|2 )1/2 ) we have Ψ ∈ D(a♯ (hg)), and it follows that ‖a(hg)Ψ‖2 ≤ ‖g‖2 ‖dΓ(|h|2 )1/2 Ψ‖2 , 2

2

2 1/2

2

(1.2.20) 2

2

‖a (hg)Ψ‖ ≤ ‖g‖ ‖dΓ(|h| ) Ψ‖ + ‖hg‖ ‖Ψ‖ . ∗

Proof. Let Ψ ∈ D(dΓ(|h|2 )). First note that n

(Ψ(n) , (dΓ(|h|2 )Ψ)(n) ) = ∑ ∫ |Ψ(n) (k1 , . . . , kn )|2 |h(ki )|2 dk1 ⋅ ⋅ ⋅ dkn i=1

ℝnd

= n ∫ |Ψ(n) (k1 , . . . , kn )|2 |h(k1 )|2 dk1 ⋅ ⋅ ⋅ dkn . ℝnd

(1.2.21)

1.2 Boson Fock space |

11

Thus 󵄨󵄨 󵄨󵄨2 ‖(a(hg)Ψ)(n−1) ‖2 = n ∫ 󵄨󵄨󵄨 ∫ g(k1 )h(k1 )Ψ(n) (k1 , . . . , kn )dk1 󵄨󵄨󵄨 dk2 ⋅ ⋅ ⋅ dkn 󵄨 󵄨 ℝ(n−1)d ℝd

≤ n‖g‖2 ∫ |h(k1 )|2 |Ψ(n) (k1 , . . . , kn )|2 dk1 ⋅ ⋅ ⋅ dkn ℝnd 2 (n)

= ‖g‖ (Ψ , (dΓ(|h|2 )Ψ)(n) ), and summing over n gives (1.2.20). By the closedness of both dΓ(|h|2 ) and a(hg), we can extend to Ψ ∈ D(dΓ(|h|2 )1/2 ). Equality (1.2.21) can be derived from the canonical commutation relation [a(hg), a∗ (hg)] = ‖hg‖2 and (1.2.20). We can show the next result similarly to Lemma 1.6. Lemma 1.7. Let h : ℝd → ℂ be measurable, and gj ∈ D(h) for j = 1, . . . , m. Then for every Ψ ∈ D(dΓ(|h|2 )m/2 ) we have Ψ ∈ D(∏m j=1 a(hgj )), and it follows that m 󵄩󵄩 m 󵄩 󵄩󵄩∏ a(hgj )Ψ󵄩󵄩󵄩 ≤ (∏ ‖gj ‖) ‖dΓ(|h|2 )m/2 Ψ‖. 󵄩󵄩 󵄩󵄩 j=1

(1.2.22)

j=1

Proof. Let Ψ ∈ D(dΓ(|h|2 )m ). First note that m

n

(Ψ(n) , (dΓ(|h|2 )m Ψ)(n) ) = ∫ |Ψ(n) (k1 , . . . , kn )|2 (∑ |h(ki )|2 ) dk1 ⋅ ⋅ ⋅ dkn . i=1

ℝnd

By permutation symmetry of Ψ(n) (k1 , . . . , kn ) we can replace m

n

(∑ |h(ki )|2 ) = i=1

m

∏ |h(kij )|2

∑

{i1 ,...,im }⊂{1,...,n} j=1

with n

m

m

(∑ |h(ki )|2 ) = C(n, m) ∏ |h(kj )|2 . i=1

j=1

Here C(n, m) = n(n − 1) ⋅ ⋅ ⋅ (n − m + 1). We have m

(Ψ(n) , (dΓ(|h|2 )m Ψ)(n) ) = C(n, m) ∫ |Ψ(n) (k1 , . . . , kn )|2 ∏ |h(kj )|2 dk1 ⋅ ⋅ ⋅ dkn . ℝnd

j=1

On the other hand, by the definition of the annihilation operators we have

12 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes (n−m) 2 󵄩󵄩 󵄩󵄩 m 󵄩󵄩 󵄩󵄩(∏ a(hgj )Ψ) 󵄩󵄩 󵄩󵄩 j=1

m

󵄨󵄨2 󵄨󵄨 = C(n, m) ∫ 󵄨󵄨󵄨 ∫ (∏ gj (kj )h(kj )) Ψ(n) (k1 , . . . , kn )dk1 ⋅ ⋅ ⋅ dkm 󵄨󵄨󵄨 dkm+1 ⋅ ⋅ ⋅ dkn 󵄨 󵄨 j=1

ℝ(n−m)d ℝmd

m

m

≤ C(n, m) (∏ ‖gj ‖2 ) ∫ (∏ |h(kj )|2 ) |Ψ(n) (k1 , . . . , kn )|2 dk1 ⋅ ⋅ ⋅ dkn j=1

ℝnd

j=1

m

= (∏ ‖gj ‖2 ) (Ψ(n) , (dΓ(|h|2 )m Ψ)(n) ) j=1

and summing over n gives (1.2.22). By closedness of the operators dΓ(|h|2 )m/2 and 2 m/2 ). ∏m j=1 a(hgj ), we can extend to Ψ ∈ D(dΓ(|h| ) Corollary 1.8. If fj ∈ L2 (ℝd ) and gj ∈ D(1/√ων ) for j = 1, . . . , n, then n 󵄩󵄩 n 󵄩 󵄩󵄩 ∏ a(fj )Ψ󵄩󵄩󵄩 ≤ (∏ ‖fj ‖) ‖N n/2 Ψ‖, 󵄩󵄩 󵄩󵄩 j=1

j=1

Ψ ∈ D(N n/2 ),

n

n 󵄩󵄩 󵄩 󵄩󵄩 ∏ a(gj )Φ󵄩󵄩󵄩 ≤ (∏ ‖gj /√ων ‖) ‖H n/2 Φ‖, 󵄩󵄩 󵄩󵄩 f j=1

j=1

Φ ∈ D(Hfn/2 ).

Proof. This is obtained directly from Lemma 1.7 by putting h = 1 and h = √ων . 1.2.3 The case W = ℂ1 Now we consider the boson Fock space over the one-dimensional vector space W = ℂ1 over ℂ, and write for simplicity ℂ for ℂ1. We have ̇ ℱb (ℂ) = {a∗ (1)n Ωb | n ∈ ℕ}. We see that a(1)Ωb = 0 and the canonical commutation relation [a(1), a∗ (1)] = 1 is satisfied. On the other hand, we consider the operators a=

d 1 (x + ), √2 dx

a∗ =

1 d (x − ), √2 dx

2

acting in L2 (ℝ), and Φ0 (x) = π −1/4 e−x /2 . We see that ‖Φ0 ‖ = 1, aΦ0 = 0 and the canonical commutation relation [a, a∗ ] = 1 holds. It can be also seen that Φn =

1 (a∗ )n Φ0 √n!

1.2 Boson Fock space |

13

is the nth Hermite function and {Φn }n∈ℕ̇ is a complete orthonormal system of L2 (ℝ), i. e., ̇ L2 (ℝ) = L. H.{(a∗ )n Ωb | n ∈ ℕ}. Now we construct a unitary operator between ℱb (ℂ) and L2 (ℝ). Let U : ℱb (W ) → L2 (ℝ) be defined by Ua∗ (1)n Ωb = (a∗ )n Φ0 ,

n ∈ ℕ.̇

The operator U maps a dense subspace to a dense subspace, and furthermore we see that ‖Ua∗ (1)n Ωb ‖ℱb (W ) = ‖(a∗ )n Φ0 ‖L2 (ℝ) . From this equality U can be extended to the unitary operator from ℱb (W ) to L2 (ℝ). Then 2

ℱb (ℂ) ≅ L (ℝ)

follows. By the discussion above we also see that ℱb (ℂ) ≅ L. H.{Φn },

n ∈ ℕ̇

(n)

and dimℱb(n) (ℂ) = 1 for n ∈ ℕ.̇ We summarize this in the proposition below. Proposition 1.9. Let d ≥ 1. It follows that d

2

d

ℱb (ℂ ) ≅ L (ℝ )

and d

ℱb (ℂ ) ≅ L. H.{Φn1 ⊗ ⋅ ⋅ ⋅ ⊗ Φnk | n1 + ⋅ ⋅ ⋅ + nk = n, nj ∈ ℕ,̇ k ≥ 0}. (n)

Proof. For d = 1 we already showed it. In the case of d ≥ 2, inductively we have d

d

d

d 2

2

d

ℱb (ℂ ) = ℱb (⊕ ℂ) ≅ ⊗ ℱb (ℂ) ≅ ⊗ L (ℝ) ≅ L (ℝ ),

thus the first statement follows. Since d

d

d

∞

∞

(n)

(n)

ℱb (ℂ ) ≅ ⊗ ℱb (ℂ) = ⊗ ⨁ ℱb (ℂ) = ⨁ 𝒦b n=0

n=0

with (n)

𝒦b =

⨁

n1 +⋅⋅⋅+nk =n,k≥0

the second statement also follows.

(n )

(nk )

ℱb 1 (ℂ) ⊗ ⋅ ⋅ ⋅ ⊗ ℱb

(ℂ),

14 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes The number operator N in ℱb (ℂ) acts as UNa∗ (1)n Ωb = nUa∗ (1)n Ωb = n(a∗ )n Φ0 = a∗ a(a∗ )n Φ0 . Hence the number operator in ℱb (ℂ) corresponds to the Hamiltonian of the harmonic oscillator 1 1 1 N ≅ a∗ a = − Δ + x 2 − 2 2 2 and Spec(N) = ℕ.̇ Example 1.10 (Mehler’s formula). Let 0 < c < 1. The second quantization Γ(c) is the integral operator on L2 (ℝ) and its integral kernel is given by Γ(c)(x, y) = (

1/2 (1 + c2 )(x 2 + y2 ) − 4xyc 1 ) exp (− ). π(1 − c2 ) 2(1 − c2 )

Example 1.11. (1) Let W be a Hilbert space. Then ℱb (ℂd ⊕ W ) ≅ L2 (ℝd ) ⊗ ℱb (W ) and the relation dΓ(1ℂd ⊕ 1W ) ≅ a∗ a ⊗ 1 + 1 ⊗ N holds. (2) Let W = L2 (ℝd ). Then ℱb (ℂd ⊕ L2 (ℝd )) ≅ L2 (ℝd ) ⊗ ℱb (L2 (ℝd )) and the relation dΓ(1ℂd ⊕ ω) ≅ a∗ a ⊗ 1 + 1 ⊗ Hf holds. 1.2.4 Segal–Bargmann space In Section 1.2.3 we showed that L2 (ℝ) ≅ ℱb (ℂ) and we established that L2 (ℝ) can be regarded as a special case of the boson Fock space. Here we discuss an alternative representation of L2 (ℝ) with an annihilation operator and a creation operator. Definition 1.12 (Segal–Bargmann space). The Segal–Bargmann space ℬ is the space of holomorphic functions f on ℂ satisfying the condition ‖f ‖2ℬ =

1 ∫ |f (z)|2 exp(−|z|2 ) dz < ∞, π ℂ

where dz = dxdy under the identification ℂ ≅ ℝ ⊕ ℝ by z = x + iy. ℬ is a Hilbert space with respect to the associated inner product

(f , g)ℬ =

1 ∫ f (z)g(z) exp(−|z|2 ) dz. π ℂ

Let Fa (z) = exp(az). The function Fa is called a coherent state with parameter a, and the function κ(a, z) = Fa (z) is called reproducing kernel for the Segal–Bargmann space. Note that 1 f (a) = ∫ κ(a, z)f (z) exp(−|z|2 ) dz. π ℂ

1.2 Boson Fock space |

15

n n m Any function f ∈ ℬ can be represented as f (z) = ∑∞ n=0 cn z and we see that (z , z )ℬ = 0 unless n = m. Then

{z n }n∈ℕ̇ d . We see that makes a complete orthonormal system of ℬ. We define b∗ = z and b = dz ∗ the adjoint of b is b under the Segal–Bargmann inner product, and it satisfies that [b, b∗ ] = 1. An important fact is that there exists a unitary operator U from L2 (ℝ) to ℬ defined by

1 (Uf )(z) = ∫ exp (− (z 2 − 2√2zx + x 2 )) f (x)dx 2 ℝ

and the inverse formula is given by 2 1 ̄ + x2 )) (Uf )(z)e−|z| dz. f (x) = ∫ exp (− (z̄2 − 2√2zx 2

ℂ

This leads to the identification ℬ ≅ L2 (ℝ). Proposition 1.13. It follows that ∞

L2 (ℝ) ≅ ℬ = ⨁ L. H.{z n }. n=0

Under this identification we have z≅

1 d ), (x − √2 dx

d 1 d ≅ ) (x + dz √2 dx

and it follows that z

1 x2 1 d ≅− Δ+ − . dz 2 2 2

Example 1.14. The eigenvalue equation 1 x2 1 (− Δ + − )Φ = αΦ 2 2 2 can be reduced to finding a holomorphic function Φ such that z

d Φ(z) = αΦ(z). dz

The solution is Φ(z) = z α which is holomorphic. Hence α ∈ ℕ̇ and Φ(z) = z n ,

n ∈ ℕ,

and we conclude that z n ∈ ℬ corresponds to the nth Hermite function ( √12 (x − 2

in L2 (ℝ), where Φ0 (x) = π −1/4 e−|x| /2 .

n d )) Φ0 dx

16 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes 1.2.5 Segal fields The creation and annihilation operators are not symmetric and do not commute. Roughly speaking, the creation operator corresponds to √12 (x − 𝜕x ) and the annihi-

lation operator to √12 (x + 𝜕x ) in L2 (ℝd ). We can, however, construct symmetric and commutative operators by combining the two field operators leading to so-called Segal fields. As it will be seen below, Segal fields linearly span Fock space, which suggests that ℱb may be realized conveniently as a space L2 (Q ) with infinitely many variables on a suitable Q instead of necessarily using operators. This is due to the fact that Segal fields are commutative. In this representation, Segal fields translate into real-valued multiplication operators. Definition 1.15 (Segal field). The Segal field Φ(f ) on the boson Fock space ℱb (W ) is defined by Φ(f ) =

1 ∗ (a (f ) + a(f ̄)), √2

f ∈W,

i ∗ (a (f ) − a(f ̄)), √2

f ∈W.

and its conjugate momentum by Π(f ) =

As above, f ̄ denotes the complex conjugate of f . By the above definition both Φ(f ) and Π(g) are symmetric, however, they are not linear in f and g over ℂ. Note that, in contrast, they are linear operators over ℝ. It is straightforward to check that [Φ(f ), Π(g)] = i Re(f , g), and [Φ(f ), Φ(g)] = i Im(f , g),

[Π(f ), Π(g)] = i Im(f , g).

In particular, for real-valued f and g the canonical commutation relations become [Φ(f ), Π(g)] = i(f , g),

[Φ(f ), Φ(g)] = [Π(f ), Π(g)] = 0.

Applying the bounds (1.2.18)–(1.2.19) to h = 1, we see that ℱb,fin is the set of analytic vectors of Φ(f ), i. e., m

‖Φ(f )n Ψ‖t n 0. Then K is essentially self-adjoint on 𝒟 . By Nelson’s analytic vector theorem, Φ(f ) and Π(g) are essentially self-adjoint on ℱb,fin . We keep denoting the closures of Φ(f )⌈ℱb,fin and Π(g)⌈ℱb,fin by the same symbols. Next we review the algebraic calculus involving commutators, which are useful in a rigorous treatment of commutation relations. Suppose that [A, B] = C and [C, B] = D, ignoring domain issues for simplicity. Let n ∈ ℕ, and suppose that D = 0, i. e., C and B are commutative. Then we have [A, Bn ] = nBn−1 C. We define eB by the formal series ∑∞ n=0

Bn . n!

(1.2.23)

By (1.2.23) it can be directly seen that

[A, eB ] = eB C. In particular e−B AeB = A + C and e−B An eB = (A + C)n follow. Next consider the case where [D, B] = 0, giving [A, Bn ] = nBn−1 C +

n(n − 1) n−2 B D. 2

From this it follows that 1 [A, eB ] = eB (C + D) 2 and, in particular, we have 1 n e−B An eB = (A + C + D) . 2

(1.2.24)

These relations can be generalized. Define adA B = [A, B]. Hence ad2A B = [[A, B], B] and ad3A B = [[[A, B], B], B] etc, follow. Thus (1.2.24) can be written as n 1 e−B An eB = (ad0A B + ad1A B + ad2A B) 2

(1.2.25)

with ad0A B = A and ad1A B = adA B. In general, it can be seen that m

1 n adA B) . n! n=0 ∞

e−B Am eB = ( ∑

These expressions are formal and require a check of the convergence of ∑∞ n=0 and specifying the domain where both sides of (1.2.26) are well-defined.

(1.2.26) 1 adnA B n!

18 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes Example 1.17. (1) Since [Φ(f ), iΠ(g)] = − Re(f , g), we have Φ(f )eiΠ(g) = eiΠ(g) (Φ(f ) − Re(f , g)) and e−iΠ(g) Φ(f )eiΠ(g) = Φ(f ) − Re(f , g).

(1.2.27)

(2) Let h be self-adjoint and g ∈ D(h). It is seen that [dΓ(h), iΠ(g)] = −Φ(hg) and [−Φ(hg), iΠ(g)] = (hg, g). Hence 1 dΓ(h)eiΠ(g) = eiΠ(g) (dΓ(h) − Φ(hg) + (hg, g)). 2 We also see that

1 e−iΠ(g) dΓ(h)eiΠ(g) = dΓ(h) − Φ(hg) + (hg, g). 2

(1.2.28)

Equalities (1.2.27)–(1.2.28) can rigorously be verified by a limiting argument as in the proposition below. Proposition 1.18. (1) The operator e−iΠ(g) maps D(Φ(f )) onto itself, and (1.2.27) holds on D(Φ(f )). (2) If h is self-adjoint and g ∈ D(h), then e−iΠ(g) maps D(dΓ(h)) onto itself and (1.2.28) is satisfied on D(dΓ(h)). Proof. Let F ∈ ℱb,fin . We have n

n n−1 (iΠ(g))m (iΠ(g))m (iΠ(g))m F= ∑ Φ(f )F − ∑ Re(f , g)F. m! m! m! m=0 m=0 m=0

Φ(f ) ∑

By Nelson’s analytic vector theorem, we see that n

(iΠ(g))m F → eiΠ(g) F, m! m=0 ∑

n

(iΠ(g))m F → eiΠ(g) (Φ(f ) − Re(f , g))F m! m=0

Φ(f ) ∑

as n → ∞. Then eiΠ(g) F ∈ D(Φ(f )) and (1.2.27) follows on ℱb,fin by the closedness of Φ(f ). Let F ∈ D(Φ(f )). Since ℱb,fin is a core of Φ(f ), there exists a sequence (Fn )n∈ℕ ⊂ ℱb,fin such that Fn → F and Φ(f )Fn → Φ(f )F as n → ∞. It follows that e−iΠ(g) Φ(f )eiΠ(g) Fn = (Φ(f ) − Re(f , g))Fn and eiΠ(g) Fn → eiΠ(g) F as n → ∞. Hence the closedness of Φ(f ) yields again that eiΠ(g) F ∈ D(Φ(f )) and (1.2.27) is true on D(Φ(f )), and (1) is proven. Part (2) can be shown similarly.

1.2 Boson Fock space |

Corollary 1.19. (1) If f ≢ 0, then

Spec(Φ(f )) = ℝ,

19

Specp (Φ(f )) = 0.

(2) Let h be self-adjoint and g ∈ D(h). Then

1 inf Spec(dΓ(h) + Φ(hg)) = − (hg, g), 2 1 (dΓ(h) + Φ(hg))eiΠ(g) Ωb = − (hg, g)eiΠ(g) Ωb . 2

Proof. For every α ∈ ℝ the operator eiαΠ(f ) is unitary and we have e−iαΠ(f ) Φ(f )eiαΠ(f ) = Φ(f ) − α‖f ‖2 . Since α ∈ ℝ is arbitrary, Spec(Φ(f )) = ℝ follows. Let Ψ be an eigenvector of Φ(f ) such that Φ(f )Ψ = EΨ. Then Φ(f )eiαΠ(f ) Ψ = (E − α‖f ‖2 )eiαΠ(f ) Ψ. This yields that eiαΠ(f ) Ψ is an eigenvector of Φ(f ) associated with eigenvalue E − α‖f ‖2 . This contradicts the separability of ℱb , and (1) follows. By 1 e−iΠ(g) (dΓ(h) + Φ(hg))eiΠ(g) = dΓ(h) − (hg, g) 2 and inf Spec(dΓ(h)) = 0, implying (2). Example 1.20. Let h be self-adjoint. Choosing any n ∈ ℕ, it is important to identify 2n which space is left by e−Φ(f ) invariant. We have η = [dΓ(h), −Φ(f )2n ] = 2niΦ2n−1 (f )Π(hf ) + and

2n(2n − 1) 2n−2 Φ (f )(hf , f ) 2

ξ = [η, −Φ(f )2n ] = −(2n)2 Φ(f )4n−2 (hf , f ). Thus we can derive that

In particular,

2n 2n 1 [dΓ(h), e−Φ(f ) ] = e−Φ(f ) (η + ξ ). 2

2n 2n 1 m dΓ(h)m e−Φ(f ) = e−Φ(f ) (dΓ(h) + η + ξ ) . 2

m Set h = 1, for which dΓ(1) = N is the number operator. Let C ∞ (N) = ⋂∞ m=0 D(N ). For ∞ F ∈ C (N), a computation gives that for arbitrary m ∈ ℕ, 2n

N m e−Φ(f ) F 2n

= e−Φ(f ) ( N + 2niΦ(f )2n−1 Π(f ) + 2n

m 2n(2n − 1) (2n)2 Φ(f )2n−2 ‖f ‖2 − Φ(f )4n−2 ‖f ‖2 ) F. 2 2

This implies that e−Φ(f ) leaves C ∞ (N) invariant.

20 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes 1.2.6 Wick product Loosely speaking, the so-called Wick product :a♯ (f1 ) ⋅ ⋅ ⋅ a♯ (fn ): is defined in a product of creation and annihilation operators by moving the creation operators to the left and the annihilation operators to the right without taking the commutation relations into account. For example, :a(f1 )a∗ (f2 )a(f3 )a∗ (f4 ): = a∗ (f2 )a∗ (f4 )a(f1 )a(f3 ). Definition 1.21 (Wick product). The Wick product :∏ni=1 Φ(gi ): of ∏ni=1 Φ(gi ) is recursively defined by the equalities :Φ(f ): = Φ(f ), n

n

i=1

i=1

:Φ(f ) ∏ Φ(fi ): = Φ(f ):∏ Φ(fi ): −

1 n ∑(f , fj ):∏ Φ(fi ):. 2 j=1 i=j̸

By the above definition, we have n n :Φ(f )n : = 2−n/2 ∑ ( ) a∗ (f )m a(f )n−m m m=0

or

k 1 n! :Φ(f ) : = ∑ Φ(f )n−2k (− ‖f ‖2 ) = n! ∑ k!(n − 2k)! 4 k=0 k=0 n

[n/2]

[n/2]

k

1 2 Φ(f )m (− 4 ‖f ‖ ) . ∑ m! k! m+2k=n

Here [m] denotes the intger part of m. Note that :Φ(f1 ) ⋅ ⋅ ⋅ Φ(fn ):Ωb = 2−n/2 a∗ (f1 ) ⋅ ⋅ ⋅ a∗ (fn )Ωb . From this, the orthogonality property n

m

i=1

i=1

n

(:∏ Φ(fi ):Ωb , :∏ Φ(gi ):Ωb ) = δnm 2−n/2 ∑ ∏(gi , fπ(i) ) π∈℘n i=1

follows. Hence n

󵄨󵄨 L. H.{:∏ Φ(fi ):Ωb 󵄨󵄨󵄨 fj ∈ W , j = 1, . . . , n} = ℱb(n) . 󵄨 i=1

The Wick product of the exponential can be computed directly to yield m

2 2 αn :Φ(f )n :Ωb = e−(α /4)‖f ‖ eαΦ(f ) Ωb . m→∞ n! n=0

:eαΦ(f ) :Ωb = s-lim ∑ Hence for real-valued f and g,

(Ωb , Φ(f )Ωb ) = 0, and

1 (Ωb , Φ(f )Φ(g)Ωb ) = (f , g), 2 2

2

(Ωb , eαΦ(f ) Ωb ) = e(α /4)‖f ‖ .

(1.2.29)

(1.2.30)

(1.2.31)

Since the commutator [Φ(f ), Φ(g)] = 0, equalities (1.2.30)–(1.2.31) suggest that with a real-valued test function f the expression Φ(f ) can be realized as a Gaussian random variable, as it will be further explored in the next section.

1.2 Boson Fock space |

21

1.2.7 Exponentials of creation and annihilation operators The field operator Φ(f ), f ∈ W , is a self-adjoint operator, while a∗ (f ) and a(f ) are not. ∗ Nevertheless, we can define ea (f ) and ea(f ) by the series ea

♯

1 ♯ n a (f ) , n! n=0 ∞

(f )

= ∑

which are closed operators and play an important role in quantum field theory. Fur∗ thermore, ea (f ) Ωb becomes an eigenvector of the annihilation operator a(g) with g ≠ 0 ∗ ∗ such that a(g)ea (f ) Ωb = (g,̄ f )ea (f ) Ωb . Hence we see that the spectrum of the closed operator a(g) coincides with ℂ and Specp (a(g)) = ℂ follows. We discuss now these operators rigorously. Let f ∈ W and define the exponential of the creation operators Ff by 1 ∗ n a (f ) n! n=0 ∞

Ff = ∑ with the domain

∞ ∞ 1 󵄨 D(Ff ) = {Φ ∈ ⋂ D(a∗ (f )n ) 󵄨󵄨󵄨 ∑ ‖a∗ (f )n Φ‖ < ∞}. n! n=1 n=0

Let Φ ∈ ℱb (m) . Then we have √m + n − 1 ⋅ ⋅ ⋅ √m n ‖f ‖ ‖Φ‖ < ∞, n! n=1 ∞

‖Ff Φ‖ ≤ ‖Φ‖ + ∑

and ℱb,fin ⊂ D(Ff ) follows. We also define the exponential of annihilation operators by 1 a(f )n n! n=0 ∞

Gf = ∑ with the domain

∞ ∞ 1 󵄨 D(Gf ) = {Φ ∈ ⋂ D(a(f )n ) 󵄨󵄨󵄨 ∑ ‖a(f )n Φ‖ < ∞}. n! n=1 n=0

For convenience, we write Ff = e a

∗

(f )

Gf = ea(f ) ,

,

unless any confusion may arise. It is seen that (ea (f ) )∗ ⊃ ea(f ) , which implies that ∗ ∗ ea (f ) is closable. We will denote the closure of ea (f ) by the same symbol. Similarly, we denote the closure of ea(f ) by the same symbol. The vector defined by ∗

C(f ) = ea

∗

(f )

Ωb

̄

22 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes is called a coherent vector. By the definition of the creation operators, C(f ) is formally written as ∞

C(f ) = ∑

n=0

⊗nsym f √n!

.

Let D be a subspace of W and define the linear hull of coherent vectors with test functions in D by C (D) = L. H.{C(g) | g ∈ D}.

Proposition 1.22 (Denseness of coherent vectors). (1) If f1 , . . . , fn ∈ W are linearly independent vectors, then C(f1 ), . . . , C(fn ) are also linearly independent. (2) If D ⊂ W is a dense subset, then C (D) is also dense in ℱb (W ). Proof. Suppose that ∑nj=1 aj C(fj ) = 0 with aj ∈ ℂ for j = 1, . . . , n. In particular we have ∑nj=1 aj a∗ (fj )Ωb = 0, which implies that ∑nj=1 aj fj = 0. Then a1 = . . . = an = 0 and statement (1) follows. In order to prove (2), it is enough to show that C (D)⊥ = {0}. Let Φ ∈ ℱb (W ) such that (Φ, C(f )) = 0 for any f ∈ D. Hence (Φ(n) , a∗ (f )n Ωb ) = 0 for any n. Putting f = ∑m j=1 aj fj with aj ∈ ℂ, we see that m

j

n

0 = (Φ(n) , a∗ ( ∑ aj fj ) Ωb ) = j=1

∑

j1 +⋅⋅⋅+jm =n, ji ≥0,i=1,....,m

j

m a11 ⋅ ⋅ ⋅ amm (Φ(n) , ∏ a∗ (fi )ji Ωb ). j1 ! ⋅ ⋅ ⋅ jm ! i=1

The right-hand side is a polynomial of degree n in a1 , . . . , an , identically zero. In particular, (Φ(n) , a∗ (f1 ) ⋅ ⋅ ⋅ a∗ (fn )Ωb ) = 0. Since L. H.{a∗ (f1 ) ⋅ ⋅ ⋅ a∗ (fn )Ωb | fj ∈ D} is dense in ℱb (n) (W ), Φ(n) = 0 follows. Since n is arbitrary, we conclude that Φ = 0. ∗ n It can be seen that C(g) ∈ ⋂∞ n=0 D(a (f ) ) and M ∗ n 󵄩󵄩 󵄩 󵄩󵄩C(f + g) − ∑ a (f ) C(g)󵄩󵄩󵄩 → 0 󵄩󵄩 󵄩󵄩 n! n=0

as M → ∞. Also, C(g) ∈ D(ea

∗

(f )

) and ea

∗

follows by the closedness of ea

∗

a(g)ea

∗

(f )

(f )

(f )

C(g) = C(f + g)

. We also see that

Φ = ea

∗

(f )

a(g)Φ + (g,̄ f )ea

∗

(f )

Φ

for Φ ∈ ℱb,fin . In particular, a(g)C(f ) = (g,̄ f )C(f )

(1.2.32)

1.2 Boson Fock space |

23

and recursively we get P(a(g))C(f ) = P((g,̄ f ))C(f ) for any polynomial P. We summarize these facts in the proposition below. Proposition 1.23 (Algebraic properties). Let f , g ∈ W , and P be a polynomial. Then ∗ ∗ ∗ (1) ea (g) ea (f ) Ωb = ea (f +g) Ωb ; ∗ ∗ (2) P(a(g))ea (f ) Ωb = P((g,̄ f ))ea (f ) Ωb ; ∗ ∗ (3) ea(g) ea (f ) Ωb = e(g,f̄ ) ea (f ) Ωb . Proof. (1) and (2) are already shown. Since M

M a(g)n a∗ (f ) (g,̄ f )n a∗ (f ) e Ωb = ∑ e Ωb n! n! n=0 n=0

∑

and the right-hand side converges to e(g,f̄ ) ea the closedness of ea(g) .

∗

(f )

Ωb as M → ∞, statement (3) follows by

Corollary 1.24. Let f ∈ W and α ∈ ℂ. Then it follows that on D(ea

♯

(f )

)

♯ (a♯ (f ) + α)n = eα ea (f ) . n! n=0

∞

∑

Proof. Let Φ ∈ D(ea

♯

(f )

(1.2.33)

). We have m a♯ (f )k m−k αl (a♯ (f ) + α)n Φ= ∑ ∑ Φ. n! k! l=0 l! n=0 k=0 m

∑

It follows that ♯ 󵄩󵄩 m a♯ (f )k m−k αl 󵄩󵄩 󵄩󵄩 ∑ ∑ Φ − eα ea (f ) Φ󵄩󵄩󵄩 󵄩󵄩 󵄩 k! l=0 l! k=0

m ♯ 󵄩󵄩 󵄩󵄩 m a♯ (f )k 󵄩󵄩 m a♯ (f )k m−k αl a♯ (f )k α 󵄩󵄩󵄩 e Φ󵄩󵄩 + eα 󵄩󵄩󵄩 ∑ Φ − ea (f ) Φ󵄩󵄩󵄩. ≤ 󵄩󵄩󵄩 ∑ ∑ Φ− ∑ 󵄩 󵄩 󵄩 󵄩 k! l=0 l! k! k! k=0 k=0 k=0

By a simple limiting argument it is seen that the right-hand side above converges to zero as m → ∞, which yields (1.2.33). We can also determine the spectrum of a♯ (f ) by using their algebraic properties. Recall that Specp (A) and Specr (A) describe the point spectrum of A and the residual spectrum of A, respectively.

24 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes Proposition 1.25 (Spectral properties). Let f ≠ 0. Then it follows that Spec(a(f )) = Specp (a(f )) = ℂ, Spec(a∗ (f )) = Specr (a∗ (f )) = ℂ. In particular Specp (a∗ (f )) = 0. Proof. Since a(f )C(g) = (f ̄, g)C(g), we have a(f )C(αf /‖f ‖2 ) = αC(αf /‖f ‖2 ) for every α ∈ ℂ. This implies that C(αf /‖f ‖2 ) is an eigenfunction associated with eigenvalue α. This shows the first statement. To prove the second, suppose that a∗ (f )Φ = 0. By the identity ‖a∗ (f )Φ‖2 = ‖a(f )Φ‖2 + ‖f ‖2 ‖Φ‖2 , we have Φ = 0, and hence 0 ∈ ̸ Specp (a∗ (f )). Let a∗ (f )Φ = zΦ with z ≠ 0. Since zΦ(0) = 0 and √nSn (f ⊗ Φ(n−1) ) = zΦ(n) for n ≥ 1, by induction Φ(n) = 0 for all n ≥ 0 and Φ = 0. Thus z ∈ ̸ Specp (a∗ (f )) and a∗ (f ) − z is injective for every z ∈ ℂ. By the decomposition ℱb = ker(a(f ̄)−z ∗ )⊕Ran(a∗ (f ) − z) and dim ker(a(f ̄) − z ∗ ) ≥ 1, we see that Ran(a∗ (f ) − z) ≠ ℱb . This implies that the domain of (a∗ (f ) − z)−1 is not dense, and the second statement follows. Proposition 1.26 (Continuity). Let Φ ∈ ℱb,fin . The map W ∋ f 󳨃→ ea strongly continuous.

♯

(f )

Φ ∈ ℱb is

Proof. Suppose that fm → f strongly in W as m → ∞. Let Φ ∈ ℱb(N) . We see that

ea(fm ) Φ = ∑Nn=0

a(fm )n Φ. n!

Since ∑Nn=0

a(fm )n Φ n!

→ ∑Nn=0

a(f )n Φ n!

as m → ∞,

lim ea(fm ) Φ = ea(f ) Φ

m→∞

follows. Next consider the continuity of f 󳨃→ ea (f ) Φ. Let ε > 0 and ‖fm − f ‖ < ε for sufficiently large m. We fix c > 0 such that ‖fm ‖ < c for all m. It is seen that ∗

n−1 ∞ 󵄩󵄩 a∗ (fm ) a∗ (f ) 󵄩 󵄩󵄩 ≤ ∑ √(N + n − 1) ⋅ ⋅ ⋅ √N ∑ ‖f ‖k ‖f − f ‖‖f ‖n−k−1 ‖Φ‖ 󵄩󵄩e Φ − e Φ 󵄩 m m 󵄩 󵄩 n! n=1 k=0

cn−1 √(N + n − 1) ⋅ ⋅ ⋅ √N ‖Φ‖. (n − 1)! n=1 ∞

≤ε∑ This implies that ‖ea

∗

(fn )

Φ − ea

∗

(f )

Φ‖ → 0 as n → ∞.

Proposition 1.27 (Differentiability). Let h be a self-adjoint operator in W , f ∈ D(h), and ♯ ith Φ ∈ ℱb,fin . Then the map ℝ ∋ t 󳨃→ ea (e f ) Φ ∈ ℱb is strongly differentiable and ♯ ith d a♯ (eith f ) e Φ = a♯ (iheith f )ea (e f ) Φ. dt

Proof. The proof is straightforward. Let ε ∈ ℝ and Φ ∈ ℱb(N) . We show the statement only for a∗ (f ), the proof for a(f ) is similar. We write a∗ (eith f ) = a∗ (0) and a∗ (ei(t+ε)h f ) = a∗ (ε) for notational simplicity. We have ∗ ∗ 1 a∗ (ε) (e − ea (0) ) Φ − a∗ (iheith f )ea (0) Φ = Aε + Bε , ε

1.2 Boson Fock space |

25

where Aε = a∗ ((

∞ eiεh − 1 1 n−1 ∗ n−k−1 ∗ k − ih) eith f ) ∑ a (0) Φ, ∑ a (ε) ε n=1 n! k=0

1 n−1 ∗ n−k−1 ∗ k ∞ 1 ∗ n a (0) − ∑ a (0) ) Φ. ∑ a (ε) n! n=1 n! k=0 n=0 ∞

Bε = a∗ (iheith f ) ( ∑ We see that

󵄩󵄩 ∞ √N + n − 1 ⋅ ⋅ ⋅ √N n−1 󵄩󵄩 eiεh − 1 f − ihf 󵄩󵄩󵄩 ∑ ‖f ‖ ‖Φ‖ ‖Aε ‖ ≤ 󵄩󵄩󵄩 󵄩 n=1 󵄩 ε n! and n−1 1 ∑ ‖(a∗ (ε)n−k−1 − a∗ (0)n−k−1 )a∗ (0)k a(iheith f )Φ‖ (n − 1)! n=1 k=0 ∞

‖Bε ‖ ≤ ∑

√N + n − 1 ⋅ ⋅ ⋅ √N n−1 ∑ (n − k − 1)‖f ‖n−2 ‖hf ‖‖(eiεh − 1)f ‖‖Φ‖ (n − 1)! n=1 k=0 ∞

≤∑

√N + n − 1 ⋅ ⋅ ⋅ √N n(n − 1) n−2 ‖f ‖ ‖Φ‖. (n − 1)! 2 n=1 ∞

≤ ‖hf ‖‖(eiεh − 1)f ‖ ∑

Hence limε→0 ‖Aε ‖ = 0 and limε→0 ‖Bε ‖ = 0. Next we discuss the relationship between ea Let h be a self-adjoint operator in W and define

∗

and the second quantization Γ(T).

(f )

D = L. H.{a (f1 ) ⋅ ⋅ ⋅ a (fn )Ωb , Ωb | fj ∈ W , j = 1, . . . , n, n ≥ 1}, ∗

∗

Dh = L. H.{a (f1 ) ⋅ ⋅ ⋅ a (fn )Ωb , Ωb | fj ∈ D(h), j = 1, . . . , n, n ≥ 1}. ∗

∗

Recall that jf = f ̄ denotes the complex conjugate of f . Proposition 1.28 (Intertwining properties). (1) Let T be a contraction operator on W . Then the following hold on ℱb,fin : Γ(T)ea

∗

(f )

Γ(T)ea(jT

∗

= ea

∗

jf )

(Tf )

Γ(T),

= ea(f ) Γ(T).

(2) Let h be self-adjoint in W and f ∈ D(h). Then the following hold on Dh : dΓ(h)ea

∗

dΓ(h)e

(f )

a(f )

= a∗ (hf )ea

= −a(jhjf )e

∗

(f )

a(f )

+ ea

∗

+e

(f )

a(f )

dΓ(h),

dΓ(h).

26 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes ∗ a Proof. Let Φ = ∏m j=1 a (gj )Ωb ∈ D . It follows that Φ ∈ D(e have

∗

Γ(T)ea

∗

(f )

(f )

) and Γ(T)D ⊂ D . We

∗ a∗ (Tf )n m ∗ ∏ a (Tgj )Ωb = ea (Tf ) Γ(T)Φ, n! n=0 j=1

∞

Φ= ∑

and the first statement of (1) is proven on D . Let Φ ∈ ℱb(N) . Then there exists Φn ∈ D such that Φn ∈ ℱb(N) and ‖Φn − Φ‖ → 0 as n → ∞. We can also see that ea

∗

a (f )

a (f )

∗

(f )

Φn →

a∗ (f )

∗

e Φ as n → ∞. The limit of Γ(T)e Φn exists, and the identity Γ(T)e Φn = a∗ (Tf ) a∗ (f ) a∗ (Tf ) e Γ(T)Φn implies that Γ(T)e Φ=e Γ(T)Φ. Hence the first statement of (1) is proven on ℱb,fin . The second statement of (1) can be shown by taking the adjoint of both sides of the first statement. To show (2), let Φ ∈ Dh and T = eith . It follows that Γ(eith )ea

∗

(f )

Φ = ea

∗

(eith f )

Γ(eith )Φ.

(1.2.34)

In a similar way to Proposition 1.27, it can be seen that the right-hand side above is differentiable with respect to t at t = 0. The result is ∗ ∗ d a∗ (eith f ) ith e Γ(e )Φ = ia∗ (hf )ea (f ) Φ + iea (f ) dΓ(h)Φ. dt

(1.2.35)

This implies that the left-hand side of (1.2.34) is also differentiable with respect to t, ∗ thus ea (f ) Φ ∈ D(dΓ(h)), and the derivative of the left-hand side at t = 0 is ∗ ∗ d Γ(eith )ea (f ) Φ = idΓ(h)ea (f ) Φ. dt

(1.2.36)

Comparing (1.2.35) with (1.2.36), the first statement of (2) follows. The second statement can be shown by taking the adjoint of both sides in the first statement. Finally, we discuss the representation of eΦ(f ) in terms of ea known as Baker–Campbell–Hausdorff formula. Define

∗

(f )

and ea(f ) which is

Db = L. H.{C(g), Φ | g ∈ W , Φ ∈ ℱb,fin }.

Proposition 1.29 (Baker–Campbell–Hausdorff formula). Let f ∈ W and α ∈ ℂ. Then eα(a

∗

(f )+a(f ̄))

= eαa

∗

(f ) αa(f ̄)

e

1

2

2

e 2 α ‖f ‖

(1.2.37)

holds on Db . Proof. We show (1.2.37) on C(g), the proof on ℱb,fin is similar. We have eαa

∗

(f ) αa(f ̄)

e

C(g) = eα(f ,g) C(αf + g).

(1.2.38)

1.2 Boson Fock space |

27

Let ψ(f ) = a∗ (f ) + a(f ̄), which is self-adjoint and we have αn ψ(f )n n! n=0 ∞

eαψ(f ) = ∑ on ℱb,fin . Let Cm (g) = ∑m n=0 pression

a∗ (g)n Ωb . n!

(1.2.39)

By using the expansion (1.2.39), we have the ex-

m

(a∗ (g) + α(f , g))n αψ(f ) e Ωb . n! n=0

eαψ(f ) Cm (g) = ∑ 1

2

2

Together with eαψ(f ) Ωb = e 2 α ‖f ‖ eαa

∗

(f )

Ωb this implies

m

(a∗ (g) + α(f , g))n 21 α2 ‖f ‖2 αa∗ (f ) e Ωb . e n! n=0

eαψ(f ) Cm (g) = ∑ By Corollary 1.24 we have

1

2

2

eαψ(f ) C(g) = eα(f ,g) e 2 α ‖f ‖ C(αf + g).

(1.2.40)

Using (1.2.38) and (1.2.40), the claim follows. Example 1.30. Choosing α = i in the Baker–Campbell–Hausdorff formula, we obtain ∗ ̄ for the unitary operator ei(a (f )+a(f )) that ei(a

∗

Since ‖ei(a

∗

(f )+a(f ̄))

(f )+a(f ̄))

= eia

∗

‖ = 1, the operator eia

∗

‖eia

∗

e

(f ) ia(f ̄)

e

(f ) ia(f ̄)

e

(f ) ia(f ̄) − 21 ‖f ‖2

e

.

is bounded and it follows that 1

2

‖ = e 2 ‖f ‖ .

From the discussions above we also obtain the following expressions of the action of the operators on coherent vectors: 2

e−iΦ(f ) C(g) = e−‖f ‖ /4 e−i(f ,g)/ 2 C(g − if /√2), 2

√

e−iΠ(f ) C(g) = e−‖f ‖ /4 e−(f ,g)/ 2 C(g + f /√2), √

e−idΓ(A) C(g) = C(e−iA g). 1.2.8 The case W = L2 (ℝd ) From now on, we consider the case when W = L2 (ℝd ), and let ων be the multiplication operator by ων = ων (k) = √|k|2 + ν2 , ν ≥ 0, in L2 (ℝd ). Thus Hf = dΓ(ων ).

28 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes ∗ We estimate the norm of the product ‖ ∏m j=1 a (fj )Φ‖ of creation operators. Since we already have a bound on the product of annihilation operators ‖ ∏m j=1 a(fj )Φ‖ by ∗ Corollary 1.8 when W = L2 (ℝd ), we may also estimate ‖ ∏m j=1 a (fj )Φ‖, which is though technically more complicated than for annihilation operators. To obtain this, we use the fact m m 󵄩2 󵄩󵄩 m ∗ 󵄩󵄩 ∏ a (fj )Φ󵄩󵄩󵄩 = (Φ, ∏ a(fj̄ ) ∏ a∗ (fj )Φ) 󵄩󵄩 󵄩󵄩 j=1

j=1

j=1

and compute the commutator m

m

j=1

j=1

[ ∏ a(fj̄ ), ∏ a∗ (fj )]. We find that m m 󵄩󵄩 m ∗ 󵄩2 󵄩 m 󵄩2 󵄩󵄩 ∏ a (fj )Φ󵄩󵄩󵄩 = 󵄩󵄩󵄩 ∏ a(fj̄ )Φ󵄩󵄩󵄩 + (Φ, [ ∏ a(fj̄ ), ∏ a∗ (fj )]Φ). 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 j=1

j=1

j=1

j=1

It remains to estimate the second term at the right-hand side. Lemma 1.31. Let Φ ∈ ℱb,fin and fi , gj ∈ W for i, j = 1, . . . , m. Then n

n

∏ a(ḡj ) ∏ a∗ (fj )Φ j=1

j=1

n

= ∑ ∑

∑

m=0 Cm ∋A Cn−m ∋B

∑ (∏ (gl , fσ(l) )) ( ∏ a∗ (fp )) (∏ a(ḡq )) Φ.

σ:Ac →B bijection

(1.2.41)

q∈A

p∈Bc

l∈Ac

Here Ck = {A ⊂ {1, . . . , n} | #A = k}, C0 = 0, and in ∑ σ:Ac →B the summation is extended bijection

over all bijections from Ac to B. In particular, n

n

n−1

[ ∏ a(ḡj ), ∏ a∗ (fj )] = ∑ ∑ j=1

j=1

∑

m=0 Cm ∋A Cn−m ∋B

∑ (∏ (gl , fσ(l) )) ( ∏ a∗ (fp )) ∏ a(ḡq ).

σ:Ac →B bijection

l∈Ac

p∈Bc

q∈A

̄ ∗ (f )Φ = (g, f )Φ + a∗ (f )a(g)Φ. ̄ Proof. For m = 1 we have a(g)a For m ≥ 2, the proof can be completed by induction. From this lemma we can estimate ‖ ∏nj=1 a∗ (fj )Φ‖. Lemma 1.32. Let fi , gj ∈ D(1/√ων ) for i, j = 1, . . . , n and Φ ∈ D(Hfn/2 ). Then n n n 󵄨󵄨 n ∗ 󵄨 󵄨󵄨( ∏ a (gj )Φ, ∏ a∗ (fj )Φ)󵄨󵄨󵄨 ≤ n!2n (∏ ‖fl ‖ω ‖gl ‖ω ) ∑ 1 ‖H m/2 Φ‖2 , 󵄨󵄨 󵄨󵄨 m! f m=0 j=1

j=1

where ‖f ‖ω = ‖f ‖ + ‖f /√ων ‖.

l=1

1.2 Boson Fock space |

29

Proof. By Lemma 1.31 we have n

n

n

n

j=1

j=1

j=1

( ∏ a∗ (gj )Φ, ∏ a∗ (fj )Φ) = (Φ, ∏ a(ḡj ) ∏ a∗ (fj )Φ) j=1

n

= ∑ ∑

∑

m=0 Cm ∋A Cn−m ∋B

∑ (Φ, ∏ (gl , fσ(l) ) ∏ a∗ (fp ) ∏ a(ḡq )Φ).

σ:Ac →B bijection

q∈A

p∈Bc

l∈Ac

(1.2.42)

By ‖ ∏nj=1 a(hj )Φ‖ ≤ ∏nj=1 ‖hj ‖‖Hfn/2 Φ‖ and #Bc = m = #A, the right-hand side of (1.2.42) can be estimated as 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨(Φ, ∏ a∗ (fp ) ∏ a(ḡq )Φ)󵄨󵄨󵄨 = 󵄨󵄨󵄨( ∏ a(fp̄ )Φ, ∏ a(ḡq )Φ)󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 c c q∈A

p∈B

q∈A

p∈B

≤ ( ∏ ‖fp /√ων ‖) (∏ ‖gq /√ων ‖) ‖Hfm/2 Φ‖2 . q∈A

p∈Bc

Since ‖f ‖ ≤ ‖f ‖ω and ‖f /√ω‖ ≤ ‖f ‖ω , we have n 󵄨󵄨 󵄨 󵄨󵄨(Φ, ∏ (gl , fσ(l) ) ∏ a∗ (fp ) ∏ a(ḡq )Φ)󵄨󵄨󵄨 ≤ (∏ ‖gl ‖ω ‖fl ‖ω ) ‖H m/2 Φ‖2 . 󵄨󵄨 󵄨󵄨 f c c q∈A

p∈B

l∈A

l=1

(1.2.43)

Hence by (1.2.42) and (1.2.43) we have n n n 󵄨󵄨 󵄨 󵄨󵄨(Φ, ∏ a(ḡj ) ∏ a∗ (fj )Φ)󵄨󵄨󵄨 ≤ ∑ ∑ 󵄨󵄨 󵄨󵄨 m=0 j=1

n

∑

Cm ∋A Cn−m ∋B

j=1

∑ (∏ ‖gl ‖ω ‖fl ‖ω ) ‖Hfm/2 Φ‖2

σ:Ac →B bijection

l=1

n

n n! 1 (∏ ‖gl ‖ω ‖fl ‖ω ) ‖Hfm/2 Φ‖2 (n − m)!m! m! l=1 m=0

= n! ∑

n 1 (∏ ‖gl ‖ω ‖fl ‖ω ) ‖Hfm/2 Φ‖2 . m! l=1 m=0 n

≤ n!2n ∑

By Lemma 1.32 and Corollary 1.8, we have useful bounds for products of annihilation operators and creation operators. We summarize them below. Suppose that fj ∈ D(1/√ων ) for j = 1, . . . , n. Then for Φ ∈ D(Hfn/2 ) we have n 󵄩󵄩 n 󵄩 󵄩󵄩 ∏ a(fj )Φ󵄩󵄩󵄩 ≤ (∏ ‖fj /√ων ‖) ‖H n/2 Φ‖, 󵄩󵄩 󵄩󵄩 f j=1

(1.2.44)

j=1

n n 󵄩󵄩 n ∗ 󵄩 󵄩󵄩 ∏ a (fj )Φ󵄩󵄩󵄩 ≤ √n!2n/2 (∏ ‖fl ‖ω ) ( ∑ 1 ‖H m/2 Φ‖2 ) 󵄩󵄩 󵄩󵄩 m! f m=0 j=1

l=1

1/2

.

(1.2.45)

By introducing a scaling parameter 0 < s < 1, we also have n 󵄩󵄩 n 󵄩 󵄩󵄩 ∏ a(fj )Φ󵄩󵄩󵄩 ≤ s−n/2 (∏ ‖fj /√ων ‖) ‖(sHf )n/2 Φ‖, 󵄩󵄩 󵄩󵄩 j=1

j=1

(1.2.46)

30 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes 1/2 󵄩󵄩 󵄩󵄩 n n n 󵄩 󵄩󵄩 󵄩󵄩∏ a∗ (fj )Φ󵄩󵄩󵄩 ≤ √n!2n/2 s−n/2 (∏ ‖fl ‖ω ) ( ∑ 1 ‖(sHf )m/2 Φ‖2 ) . 󵄩󵄩 󵄩󵄩 m! 󵄩󵄩 󵄩󵄩 j=1 m=0 l=1

Although the exponential operator ea is bounded for every t > 0.

∗

(f )

is unbounded, it can be seen that ea

∗

Proposition 1.33 (Boundedness). Let t > 0. (1) If f ∈ D(1/√ων ), then both ea

∗

(f ) − 2t Hf

e

(1.2.47) (f ) − 2t Hf

e

t

and e− 2 Hf ea(f ) are bounded.

(2) Let N be the number operator and f ∈ L2 (ℝd ). Then both ea are bounded.

∗

(f ) − 2t N

e

t

and e− 2 N ea(f )

Proof. Choose t < 1. By (1.2.47) for any s < t, we have for Φ ∈ ℱb , 󵄩󵄩 m 1 ∗ n − t Hf 󵄩󵄩 m 1 ∗ n − t Hf 󵄩󵄩 ∑ a (f ) e 2 Φ󵄩󵄩 ≤ ∑ ‖a (f ) e 2 Φ‖ 󵄩󵄩 󵄩󵄩 n! n! n=0 n=0

1/2

t 1 n/2 −n/2 n n 1 2 s ‖f ‖ω ( ∑ ‖(sHf )k/2 e− 2 Hf Φ‖2 ) √ k! n=0 n! k=0

m

.

≤ ∑ We see that (∑m n=0 D(ea

∗

(f )

t 1 ∗ a (f )n e− 2 Hf Φ)m∈ℕ n!

t

is a Cauchy sequence in ℱb . Hence e− 2 Hf Φ ∈

) and on both sides above we have ‖ea

∗

(f ) − 2t Hf

as m → ∞, where

e

1

Φ‖ ≤ A(f , s)‖e− 2 (t−s)Hf Φ‖, 1 n/2 −n/2 n 2 s ‖f ‖ω . √ n=0 n! ∞

A(f , s) = ∑

(1.2.48)

1

Choosing s such that s < t, we see that ‖e− 2 (t−s)Hf Φ‖ ≤ ‖Φ‖ and ea bounded. Let now t ≥ 1. Choosing s = 1 above, we have ‖ea

∗

(f ) − 2t Hf

e

∗

(f ) − 2t Hf

e

for t < 1 is

1

Φ‖ ≤ A(f , 1)‖e− 2 (t−1)Hf Φ‖ ≤ A(f , 1)‖Φ‖.

t

t

t

Thus ea (f ) e− 2 Hf for t ≥ 1 is bounded. Finally, since (e− 2 Hf ea(f ) )∗ ⊃ ea (f ) e− 2 Hf , the second statement follows, and (1) is proven. Part (2) is similarly proven by replacing ων with the identity 1. ∗

∗

̄

We can also estimate the norms of these operators, which can be obtained from the estimates in the proof of Proposition 1.33. Corollary 1.34. (1) If f ∈ D(1/√ων ), then ‖ea

∗

(f ) − 2t Hf

e

2

1

‖ ≤ √2e(2/s)‖f ‖ω ‖e− 2 (t−s)Hf ‖, 1 2 a∗ (f ) − 2t Hf ‖e e ‖ ≤ √2e2‖f ‖ω ‖e− 2 (t−1)Hf ‖,

0 < s < t < 1,

t≥1

1.2 Boson Fock space |

31

In particular, we have ‖ea

∗

‖e

(f ) −tHf a(f ̄)

e

e

a∗ (f ) −tHf a(f ̄)

e

e

2

‖ ≤ 2e(4/s)‖f ‖ω ,

‖ ≤ 2e

4‖f ‖2ω

,

0 < s < t < 1,

t ≥ 1.

(2) If f ∈ L2 (ℝd ), then ‖ea

∗

(f ) − 2t N

1

2

e

0 < s < t < 1,

‖ ≤ √2e(2/s)(2‖f ‖) ‖e− 2 (t−s)N ‖, t 1 ∗ 2 ‖ea (f ) e− 2 N ‖ ≤ √2e2(2‖f ‖) ‖e− 2 (t−1)N ‖,

t ≥ 1.

In particular, ‖ea

∗

‖e

(f ) −tN a(f ̄)

e

e

a (f ) −tN a(f ̄) ∗

e

e

2

‖ ≤ 2e(16/s)‖f ‖ ,

‖ ≤ 2e

2

16‖f ‖

,

0 < s < t < 1, t ≥ 1.

Proof. We can estimate A(f , s) defined in (1.2.48) as ∞ 1 n/2 n −n/2 n −n 1 n/2 −n/2 n 2 s ‖f ‖ω = ∑ 2 ‖f ‖ω s 2 2 √ √ n! n=0 n! n=0 ∞

A(f , s) = ∑

1/2

1 n 2n −n −n 8 ‖f ‖ω s 2 ) n! n=0 ∞

≤ (∑

∞

1/2

( ∑ 12 ⋅ 2−n ) n=0

2

≤ √2e(2/s)‖f ‖ω .

(1) follows from Proposition 1.33, and (2) can be derived by replacing ων with 1. We have already obtained strong continuity of the map L2 (ℝd ) ∋ f 󳨃→ ea t ∗ can furthermore prove uniform continuity of f 󳨃→ ea (f ) e− 2 Hf for t > 0.

♯

(f )

Φ. We

Corollary 1.35 (Uniform continuity). (1) Let f , g ∈ D(1/√ων ). Then ‖ea

∗

‖ea

∗

(f ) − 2t Hf

e

(f ) − 2t Hf

e

− ea

∗

(g) − 2t Hf

2

e

‖ ≤ √2‖f − g‖ω e(2/s)(‖f ‖ω +‖g‖ω +1) , 2 a∗ (g) − 2t Hf −e e ‖ ≤ √2‖f − g‖ω e2(‖f ‖ω +‖g‖ω +1) ,

0 < s < t < 1, t ≥ 1.

(1.2.49)

In particular, let f , fn ∈ D(1/√ων ) for n ≥ 1 such that ‖f − fn ‖ω → 0 as n → ∞. Then ea

∗

(fn ) − 2t Hf

e

is uniformly convergent to ea

∗

(f ) − 2t Hf

e

as n → ∞.

(2) Let f , g ∈ L2 (ℝd ). Then ‖ea

∗

‖e

(f ) − 2t N

e

a∗ (f ) − 2t N

e

− ea

∗

(g) − 2t N

e

2

‖ ≤ 2√2‖f − g‖e(2/s)(2‖f ‖+2‖g‖+1) , t ∗ 2 − ea (g) e− 2 N ‖ ≤ 2√2‖f − g‖e2(2‖f ‖+2‖g‖+1) ,

0 < s < t < 1,

t ≥ 1.

(1.2.50)

In particular, let f , fn ∈ L2 (ℝd ) for n ≥ 1 such that ‖f − fn ‖ → 0 as n → ∞. Then t t ∗ ∗ ea (fn ) e− 2 N is uniformly convergent to ea (f ) e− 2 N as n → ∞.

32 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes Proof. It is seen by inspection that n 󵄩 ∗ j−1 ∗ 󵄩 󵄩󵄩 ∗ n ∗ n−j 󵄩 ∗ n 󵄩󵄩(a (f ) − a (g) )Ψ󵄩󵄩󵄩 ≤ ∑ 󵄩󵄩󵄩󵄩a (f ) a (f − g)a (g) Ψ󵄩󵄩󵄩󵄩 j=1

n

n

1/2 1 ‖(sHf )m/2 Ψ‖2 ) m! m=0

n−j ≤ ∑ √n!2n/2 s−n/2 ‖f ‖j−1 ω ‖f − g‖ω ‖g‖ω ( ∑ j=1

n

1/2 1 ‖(sHf )m/2 Ψ‖2 ) ‖f − g‖ω . m! m=0

≤ √n!2n/2 s−n/2 (‖f ‖ω + ‖g‖ω + 1)n ( ∑ Hence from ‖(ea

∗

(f ) − 2t Hf

− ea

∗

e

(g) − 2t Hf

e

t 1 ‖(a∗ (f )n − a∗ (g)n )e− 2 Hf Φ‖, n! n=0

∞

)Φ‖ ≤ ∑

(1.2.49) follows. The bounds (1.2.50) can be derived by replacing ων by the identity map 1. We can slightly improve Theorem 1.33 and Corollary 1.34 above. In the corollary t t ∗ ∗ we gave bounds on ea (f ) e− 2 Hf and ea (f ) e− 2 N by estimating A(f , s). This estimate, however, can be modified by introducing a parameter q > 1 as 1/2

1 n 2n 2n −n −n 2 q ‖f ‖ω s q ) n! n=0 ∞

A(f , s) ≤ ( ∑

∞

( ∑ 12 ⋅ q−n )

1/2

≤√

n=0

q (q/s)‖f ‖2ω e . q−1

q An advantage of this estimate is to get the coefficient √ q−1 as close to 1 as convenient. This will be used in Lemma 2.17 in the next chapter. Here we show a modified version of Corollary 1.34.

Corollary 1.36. Let q > 1. (1) Let f ∈ D(1/√ων ). Then ‖ea

∗

‖e

(f ) − 2t Hf

e

a∗ (f ) − 2t Hf

e

1

2

q (q/s)‖f ‖ω − 2 (t−s)Hf ‖ ≤ √ q−1 e ‖e ‖,

‖≤

1 2 q √ q−1 eq‖f ‖ω ‖e− 2 (t−1)Hf ‖,

0 < s < t < 1, t ≥ 1.

In particular, we have ‖ea

∗

‖e

(f ) −tHf a(f ̄)

e

e

e

e

a∗ (f ) −tHf a(f ̄)

‖≤ ‖≤

q (2q/s)‖f ‖2ω e , q−1 q 2q‖f ‖2ω e , q−1

0 < s < t < 1,

t ≥ 1.

(2) Let f ∈ L2 (ℝd ). Then ‖ea

∗

‖e

(f ) − 2t N

e

a∗ (f ) − 2t N

e

2

1

q (q/s)(2‖f ‖) e ‖e− 2 (t−s)N ‖, ‖ ≤ √ q−1

‖≤

1 2 q √ q−1 eq(2‖f ‖) ‖e− 2 (t−1)N ‖,

0 < s < t < 1, t ≥ 1.

1.2 Boson Fock space |

33

In particular, we have ‖ea

∗

‖e

(f ) −tN a(f ̄)

e

e

e

e

a∗ (f ) −tN a(f ̄)

‖≤ ‖≤

q (8q/s)‖f ‖2 e , q−1 q 8q‖f ‖2 e , q−1

0 < s < t < 1,

t ≥ 1.

In later developments discussed in the chapters below we often encounter exponentials of creation and annihilation operators, which will require some estimates. Next we discuss some technical estimates in this direction. Let Φ(f ) = √12 (a∗ (f )+a(f ̄)). Products such as m

∏ Φ(fj ) j=1

can be represented in terms of sums of Wick-ordered operators. For simplicity we apply the shorthand ̂) ⋅ ⋅ ⋅ Φ(f ̂) ⋅ ⋅ ⋅ Φ(f ):, cjm1 ,...,jk = :Φ(f1 ) ⋅ ⋅ ⋅ Φ(f j1 jk m

(1.2.51)

̂) means skipping Φ(f ). We also write below, where Φ(f cm = :Φ(f1 ) ⋅ ⋅ ⋅ Φ(fm ):.

(1.2.52)

Lemma 1.37 (Wick’s theorem). Let fj ∈ L2 (ℝd ), j = 1, . . . , n. Then ∏m j=1 Φ(fj ) can be represented in terms of Wick-ordered operators as 2n

n

pair

k 1 ( (fj2i−1 , fj2i )) cj2n , ∑ ∏ k 1 ,...,j2k 2 j1 ,...,j2k i=1 k=1

∏ Φ(fj ) = c2n + ∑ j=1

m = 2n,

(1.2.53)

where ∑pair is run over all k-pairs formed of the elements of {1, . . . , 2n}, and j ,...,j 1

2k

pair

k 1 , ∑ (∏(fj2i−1 , fj2i )) cj2n+1 k 1 ,...,j2k k=1 2 j1 ,...,j2k i=1

2n+1

n

∏ Φ(fj ) = c2n+1 + ∑ j=1

m = 2n + 1,

(1.2.54)

where ∑pair is run over all k-pairs formed of the elements of {1, . . . , 2n + 1}. j ,...,j 1

2k

Proof. The lemma directly follows by the definition of Wick product. Lemma 1.38. If f ∈ L2 (ℝd ), then n

Φ(f ) = n! ∑

∑

k=0 l+m+2k=n

k

l m ( ‖f4‖ ) ( √1 a∗ (f )) ( √1 a(f ̄)) 2 2 2

[n/2]

k!

l!

m!

.

Here [m] denotes the integer part of m, i. e., [2n/2] = n and [(2n + 1)/2] = n.

(1.2.55)

34 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes Proof. The number of all k-pairs chosen from {1, . . . , n} is 2k 2k − 2 2 1 n n! 1 ( )( )( )⋅⋅⋅( ) = k . 2 2 2 k! 2k 2 k!(n − 2k)! We then have from (1.2.53) [n/2]

Φ(f )n = ∑

k=0

k

n! ‖f ‖2 ( ) :Φ(f )n−2k :, k!(n − 2k)! 4

where m m :Φ(f )m : = 2−m/2 ∑ ( ) a∗ (f )k a(f ̄)m−k . k k=0

From here we obtain k

n−2k n−2k n − 2k ∗ l ̄ n−2k−l ‖f ‖2 n! ( ) 2− 2 ∑ ( ) a (f ) a(f ) l k!(n − 2k)! 4 l=0

[n/2]

Φ(f )n = ∑

k=0

[n/2] n−2k

= n! ∑ ∑

2 1 ∗ l 1 n−2k−l ( ‖f4‖ )k ( √2 a (f )) ( √2 a(f ̄))

k!

k=0 l=0

l!

(n − 2k − l)!

= (1.2.55).

Remark 1.39 (Baker–Campbell–Hausdorff formula: formal derivation). We can derive the Baker–Campbell–Hausdorff formula from (1.2.55) by the formal expansion e

Φ(f )

1 ∗ l 1 m ∞ 2 1 ∗ ( ‖f4‖ )k ( √2 a (f )) ( √2 a(f ̄)) Φ(f )n a (f ) √12 a(f ̄) ‖f4‖ = ∑ = ∑ ∑ = e √2 e e . n! k! l! m! n=0 n=0 l+m+2k=n 2

∞

This is suggestive, however, a rigorous derivation will also control the operator domains. Consider X = ea (g) ea(g)̄ for g ∈ L2 (ℝd ). We have already seen above that the opert ∗ ator ea (g) e− 2 Hf ea(g)̄ is bounded for every t > 0. This can be extended to more general cases which will be useful later on. By equality (1.2.55) we have the explicit form of the Wick ordering of Φ(f )n , and see that ∗

2

XΦ(f ) Ψ = n! ∑

∑

2

l m ( √12 a∗ (f )) ( √12 a(f ̄))

l!

m!

k

( ‖f4‖ ) ( √1 a∗ (f ) + y) 2

[n/2]

∑

X

k!

k=0 l+m+2k=n

= n! ∑

k

( ‖f4‖ )

[n/2]

n

k!

k=0 l+m+2k=n

l!

l

X

Ψ

m ( √12 a(f ̄))

m!

Ψ.

(1.2.56)

Here we write y = (g, f )/√2. The operator XΦ(f )n is unbounded, but our interest is in estimating the operator Z(t, n), which is defined by inserting e−tHf into XΦ(f )n as 2

[n/2]

Z(t, n) = n! ∑

∑

k=0 l+m+2k=n

k

( ‖f4‖ ) ( √1 a∗ (f ) + y) 2 k!

l!

l

e

a∗ (g)̄ −tHf a(g)̄

e

e

m ( √12 a(f ̄))

m!

.

1.2 Boson Fock space |

35

Furthermore, below we will be interested also in estimates of tn Z(t, n) n! n=0 ∞

Z(t) = ∑ and its norm.

Remark 1.40. Similarly to the formal derivation of the Baker–Campbell–Hausdorff formula in Remark 1.39, we can proceed to obtain further expressions of interest. Note that for any real-valued functions f , g we have [Φ(f ), Φ(g)] = 0 and eΦ(g) eΦ(f ) = eΦ(f +g) . The Baker–Campbell–Hausdorff formula gives 1

2

eΦ(g) eΦ(f ) = e 4 ‖f +g‖ e

a∗ (f +g) √2

e

a(f +g) √2

(1.2.57)

.

This can also be obtained from (1.2.56) by a formal computation. First of all, 1 41 ‖g‖2 a∗√(g) a(√g)̄ e e 2 e 2 Φ(f )n . n! n=0 ∞

eΦ(g) eΦ(f ) = ∑

On replacing g by g/√2 in equality (1.2.56), we then obtain 1 41 ‖g‖2 a∗√(g) a(√g)̄ e e 2 e 2 Φ(f )n n! n=0 ∞

∑

=e

1 ‖g‖2 4

2

∑

∑

1

2

1

2

l

k!

k=0 l+m+2k=n

= e 4 ‖g‖ e 4 ‖f ‖ e

k

( ‖f4‖ ) ( √1 a∗ (f ) + 21 (g, f )) 2

[n/2]

a∗ (f ) 1 + 2 (f ,g) √2

e

l!

a∗ (g) √2

e

a(g)̄ √2

e

a(f ̄) √2

1

2

e

= e 4 ‖f +g‖ e

a∗ (g) √2

e

a∗ (f +g) √2

a(g)̄ √2

e

m ( √12 a(f ̄))

a(f +g) √2

m!

.

Here we used that f and g are real, thus ‖f ‖2 + ‖g‖2 + 2(f , g) = ‖f + g‖2 . Lemma 1.41. Let f ∈ D(1/√ων ). Then Z(t, n) is bounded for every t > 0, with the following bounds: (1) if t ≥ 1, then 󵄩 1 󵄩2 ‖Z(t, n)Ψ‖ ≤ zt (n)󵄩󵄩󵄩e− 2 (t−1)Hf 󵄩󵄩󵄩 ‖Ψ‖,

where 2

[n/2]

zt (n) = n! ∑

∑

k

( ‖f4‖ ) (‖f ‖ω + |y|)l ‖f ‖m ω k!l!m!

k=0 l+m+2k=n

ξ (l, m),

with y = (g, f )/√2 and (√2‖g‖ω )j+j √(j + l)!√(j󸀠 + m)! ; j!j󸀠 ! j,j󸀠 =0 ∞

ξ (l, m) = ∑

󸀠󸀠

36 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes (2) if t < 1, then for every 0 < s < t, 󵄩 󵄩2 ‖Z(t, n)Ψ‖ ≤ zt (s, n)󵄩󵄩󵄩e− 2 (t−s)Hf 󵄩󵄩󵄩 ‖Ψ‖, 1

where 2

[n/2]

zt (s, n) = n! ∑

∑

k

( ‖f4‖ ) (‖f ‖ω + |y|)l ‖f ‖m ω k!l!m!

k=0 l+m+2k=n

ξs (l, m)

and ∞

ξs (l, m) = ∑

(√2‖g‖ω )j+j √(j + l)!√(j󸀠 + m)!

j,j󸀠 =0

󸀠󸀠

s(j+l)/2 s(j +m)/2 j!j󸀠 ! 󸀠

.

Proof. In this proof we set s = 1 for t ≥ 1 and 0 < s < t for t < 1. Since N √ N 󵄩󵄩 N a∗ (f )n 󵄩󵄩 ( 2‖f ‖ω )n s−n/2 󵄩󵄩 21 sHf 󵄩󵄩 1 󵄩󵄩 ∗ n 󵄩󵄩 󵄩󵄩 ∑ Φ󵄩󵄩󵄩 ≤ ∑ Φ󵄩󵄩, 󵄩󵄩e 󵄩󵄩a (f ) Φ󵄩󵄩 ≤ ∑ 󵄩󵄩 󵄩 n=0 n! √n! n! n=0 n=0

and m m ‖(a∗ (f ) + y)m a∗ (g)n Ψ‖ ≤ ∑ ( ) |y|k ‖a∗ (f )m−k a∗ (g)n Ψ‖ k k=0

m 1 m ≤ ∑ ( ) |y|k (√2‖f ‖)m−k (√2‖g‖ω )n √(n + m − k)!s−(n+m)/2 ‖e 2 sHf Ψ‖ k k=0 1

≤ s−(n+m)/2 (√2‖f ‖ + |y|)m (√2‖g‖ω )n √(n + m)!‖e 2 sHf Ψ‖, we have for nonnegative integers m ≥ 0, ∗ 󵄩󵄩 ∗ 󵄩 ∞ 1 󵄩󵄩 ∗ m ∗ n 󵄩 󵄩󵄩(a (g) + y)m ea (f ) Φ󵄩󵄩󵄩 ≤ ∑ 󵄩(a (g) + y) a (f ) Φ󵄩󵄩󵄩 󵄩 󵄩 n! 󵄩 n=0

(√2‖f ‖ω + |y|)m (√2‖g‖ω )n √(n + m)! 󵄩󵄩 21 sHf 󵄩󵄩 Φ󵄩󵄩. 󵄩󵄩e s(n+m)/2 n! n=0 ∞

≤ ∑ Also, [n/2]

‖Z(t, n)Ψ‖ ≤ ∑

∑

k=0 l+m+2k=n

[n/2]

≤ ∑

∑

k=0 l+m+2k=n

2

k

2

k

n! ( ‖f4‖ )

∗ l ∗ ̄ m 󵄩 t ̄ a(f ) 󵄩󵄩 a (f ) + y) ea (g) e− 2 Hf ea(g) ( ) Ψ󵄩󵄩󵄩 󵄩󵄩( √2 √2 k!l!m!

n! ( ‖f4‖ ) 󵄩 a∗ (f ) l ∗ ̄ m 󵄩󵄩 t 󵄩󵄩󵄩󵄩 t ̄ a(f ) 󵄩󵄩 + y) ea (g) e− 2 Hf 󵄩󵄩󵄩󵄩󵄩󵄩e− 2 Hf ea(g) ( ) 󵄩󵄩󵄩‖Ψ‖. 󵄩󵄩( 󵄩󵄩 󵄩 √2 k!l!m! 󵄩 √2

(1.2.58)

1.2 Boson Fock space |

37

Applying the estimates ∞ √ l ∗ 󵄩󵄩 a∗ (f ) ( 2‖g‖ω )j (‖f ‖ω + |y|)l √(j + l)! 󵄩󵄩 a (g) − 2t Hf 󵄩 󵄩󵄩( e , 󵄩󵄩 ≤ A ∑ 󵄩󵄩 √ + |y|) e 󵄩 s(j+l)/2 j! 2 j=0

j m √ 󵄩󵄩 − t Hf a(g)̄ a(f ̄) m 󵄩󵄩 √(j + m)! 󵄩󵄩e 2 e ( 󵄩󵄩 ≤ A ∑ ( 2‖g‖ω ) ‖f ‖ω ) 󵄩󵄩 󵄩󵄩 (j+m)/2 √2 s j! ∞

j=0

1

in (1.2.58) with A = ‖e− 2 (t−s)Hf ‖, the claim follows. Next we estimate the operator Z(t) = ∑∞ n=0

tn Z(t, n). n!

Lemma 1.42. If f , g ∈ D(1/√ων ), then Z(t) is bounded for t > 0 with ‖Z(t)‖ ≤

(1 −

h √ ( 2‖g‖ω √s

e

‖f ‖2 4

‖e−

t−s H 2 2 f‖

C(h)2

+ ‖f ‖ω + y)) (1 −

h √ ( 2‖g‖ω √s

+ ‖f ‖ω ))

.

(1.2.59)

Here s = 1 for t ≥ 1 and 0 < s < t for t < 1, and C(h) is a constant dependent on h > 0 h √ ( 2‖g‖ω + ‖f ‖ω + y) > 0 for y = (g, f )/√2. such that 1 − √s Proof. We set again s = 1 for t ≥ 1 and 0 < s < t for t < 1. By Lemma 1.41 and the definition of Z(t) we have 󵄩󵄩2 󵄩󵄩 1 ‖Z(t)Ψ‖ ≤ zt 󵄩󵄩󵄩e− 2 (t−1)Hf 󵄩󵄩󵄩 ‖Ψ‖, 󵄩 󵄩

where ∞

2

zt = ∑ t

n

n=0

∑

k!l!m!

l+m+2k=n 2

∞

= ∑

∑

k

( ‖f4‖ ) (‖f ‖ω + |y|)l ‖f ‖m ω

ξs (l, m)

k

( ‖tf4‖ ) (‖tf ‖ω + t|y|)l ‖tf ‖m ω k!l!m!

n=0 l+m+2k=n

ξs (l, m).

Replacing tf and t|y| by f and |y|, respectively, we get 2

k

( ‖f4‖ ) (‖f ‖ + |y|)l ‖f ‖m 2 ω ω zt = ∑ ∑ ξs (l, m) = e‖f ‖ /4 ξs (y)ξs (0), k! l! m! n=0 l+m+2k=n ∞

(1.2.60)

where (√2‖g‖ω )j (‖f ‖ω + y)l √(j + l)! . s(j+l)/2 l!j! j=0 l=0 ∞ ∞

ξs (y) = ∑ ∑

Let a = √2‖g‖ω /√s and b = (‖f ‖ω + y)/√s. We rewrite ξs (y) as aj bl √(j + l)! . l!j! j=0 l=0 ∞ ∞

ξs (y) = ∑ ∑

(1.2.61)

38 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes Let 0 < h be such that h < 1/(a + b). Then aj bl hj+l (j + l)! (1/h)j (1/h)l C < ∞. ≤ l!j! 1 − h(a + b) √(j + l)! j,l=0 ∞

ξs (y) = ∑

Here C = C(h) = sup(j,l)∈ℕ×ℕ

(1/h)j (1/h)l . √(j+l)!

we have (1.2.59), and zt is finite.

It can be also shown that ξ (0) is finite. Hence

In the next step we extend Lemma 1.42. The operator Z(t, n) is defined through the ∗ commutator [X, Φ(f )n ], where X = ea (g) ea(g)̄ . We consider the operator W(t, n) defined through the commutator n

[X, ∏ Φ(fj )].

(1.2.62)

j=1

To determine this commutator explicitly, we apply Wick’s theorem to the product n

[n/2]

j=1

k=1

∏ Φ(fj ) = cn + ∑

pair

k 1 ( (fj , fj )) cjn1 ,...,j2k . ∑ ∏ 2k j1 ,...,j2k i=1 2i−1 2i

Recall that ∑pair denotes summation over all k-pairs chosen from {1, . . . , n}, and cn j1 ,...,j2k and cjn1 ,...,j2k are defined by (1.2.52) and (1.2.51), respectively. Since n

n

cn = :∏ Φ(fj ): = 2−n/2 ( ∑ j=1

∑

∏

k=1 {j1 ,...,jk }⊂{1,...,n} j∈{j1 ,...,jk }

a∗ (fj )

n

∏

i∈{j1 ,...,jk }c

a(fī ) + ∏ a(fī )) , i=1

we can compute cjn1 ,...,j2k as n−2k

cjn1 ,...,j2k = 2−(n−2k)/2 ( ∑

∑

l=1 {i1 ,...,il }⊂{j1 ,...,j2k

}c

∏

j∈{i1 ,...,il }

a∗ (fj )

i∈{i1 ,...,il

∏

}c ∩{j

1 ,...,j2k

}c

a(fī ) + anj1 ,...,j2k ) ,

where ̂ ̂ anj1 ,...,j2k = a(f1̄ ) ⋅ ⋅ ⋅ a(f j1 ) ⋅ ⋅ ⋅ a(f j2k ) ⋅ ⋅ ⋅ a(fn̄ ), and we note that {j1 , . . . , j2k }c = {1, . . . , n} \ {j1 , . . . , j2k }. Hence n

[n/2]

j=1

k=1

X ∏ Φ(fj ) = Xcn + ∑ n

pair

k 1 ∑ (∏(fj2i−1 , fj2i )) Xcjn1 ,...,j2k k 2 j1 ,...,j2k i=1 n

= 2−n/2 X ∏ a(fī ) + ∑ i=1

[n/2]

+ ∑

k=1

∑

l=1 {i1 ,...,il }⊂{1,...,n}

̄ B̄ AX

pair

k n−2k 1 n ( (f , f )) (Xa + ∑ ∏ ∑ ∑ j j j1 ,...,j2k 2k j1 ,...,j2k i=1 2i−1 2i l=1 {i ,...,i }⊂{j ,...,j 1

l

1

c

2k }

AXB) ,

1.2 Boson Fock space |

39

where A = A({i1 , . . . , il }) =

∏ (

a∗ (fj ) √2

j∈{i1 ,...,il }

B = B({i1 , . . . , il }c ∩ {j1 , . . . , j2k }c ) =

+ yj ),

(1.2.63) ∏

i∈{i1 ,...,il }c ∩{j1 ,...,j2k }c

a(fī ) , √2

(1.2.64)

Ā = A({i1 , . . . , il }), B̄ = B({i1 , . . . , il }c ) =

a(fī ) , √2

∏

i∈{i1 ,...,il }c

with yj = (g, fj )/√2. It is simply written as n

[n/2]

j=1

k=0

X ∏ Φ(fj ) = ∑

pair

k n−2k 1 ( (f , f )) ∑ ∏ ∑ ∑ j2i−1 j2i 2k j ,...,j i=1 l=0 {i ,...,i }⊂{j ,...,j 1

2k

1

l

c

AXB.

2k }

1

Here and in what follows it is understood that {i1 , . . . , il } = 0 if l = 0, {j1 , . . . , j2k }c = {1, . . . , n} if k = 0 and pair

k

n−2k

n

∑ (∏(fj2i−1 , fj2i )) ∑

j1 ,...,j2k

∑

l=0 {i1 ,...,il }⊂{j1 ,...,j2k }c

i=1

AXB = ∑

∑

l=0 {i1 ,...,il }⊂{1,...,n}

AXB

−tHf if k = 0. Since X ∏m j=1 Φ(fj ) is not bounded, we then define W(t, n) by inserting e m into X ∏j=1 Φ(fj ) as [n/2]

W(t, n) = ∑

k=0

pair

n−2k 1 ∑ ∑ ∑ 2k j1 ,...,j2k l=0 {i ,...,i }⊂{j ,...,j 1

l

1

k

c

2k }

(∏(fj2i−1 , fj2i )) Aea i=1

∗

(g) −tHf a(g)̄

e

e

B.

(1.2.65)

Furthermore, consider tn W(t, n). n! n=0 ∞

W(t) = ∑

(1.2.66)

The formula proved in Theorem 1.43 below will be used in estimating integral kernels of semigroups generated by the so-called Pauli–Fierz model with spin 1/2 discussed in Chapter 3 below, or the spin-boson model in Chapter 4. The integral kernels of both are of the form given by (1.2.66). Next we show that these integral kernels are bounded. Theorem 1.43. Let fj ∈ D(1/√ων ), j ∈ ℕ. Suppose that there exists α > 0 such that ‖fj ‖ω ≤ α for every j. Then W(t, n) and W(t) are bounded for every t > 0, with norm 2

α 󵄩 t−s 󵄩2 e 4 󵄩󵄩󵄩e− 2 Hf 󵄩󵄩󵄩 C(h)2 ‖W(t)‖ ≤ . h √ h √ (1 − √s ( 2‖g‖ω + α + y)) (1 − √s ( 2‖g‖ω + α))

Here s = 1 for t ≥ 1, 0 < s < t for t < 1, y = α‖g‖/√2, and C(h) is a constant dependent h √ on h > 0 such that 1 − √s ( 2‖g‖ω + α + y) > 0.

40 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes Proof. We set s = 1 for t ≥ 1 and 0 < s < t for t < 1. Let A and B be as given in (1.2.63) and (1.2.64), respectively. We estimate 󵄩󵄩 a∗ (g) −tHf a(g)̄ 󵄩󵄩 󵄩󵄩󵄩 a∗ (g) − 2t Hf 󵄩󵄩󵄩󵄩󵄩󵄩 − 2t Hf a(g)̄ 󵄩󵄩󵄩 󵄩󵄩Ae e e e B󵄩󵄩󵄩 ≤ 󵄩󵄩Ae e B󵄩󵄩, 󵄩󵄩󵄩󵄩e 󵄩 󵄩 󵄩󵄩 󵄩 with 󵄩󵄩 a∗ (g) − t Hf 󵄩󵄩 ∞ (α + y)l (√2‖g‖ω )j √(l + j)! 󵄩󵄩 − 1 (t−s)Hf 󵄩󵄩 󵄩󵄩Ae 󵄩󵄩e 2 󵄩󵄩 , e 2 Φ󵄩󵄩󵄩 ≤ ∑ 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 s(j+l)/2 j! j=0

󵄩󵄩 − t Hf a(g)̄ 󵄩󵄩 ∞ αn−2k−l (√2‖g‖ω )j √(n − 2k − l + j)! 󵄩󵄩 − 1 (t−s)Hf 󵄩󵄩 󵄩󵄩 . 󵄩󵄩e 2 󵄩󵄩e 2 e B󵄩󵄩 ≤ ∑ 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 s(j+n−2k−l)/2 j! j=0

Let y = α‖g‖/√2, then |yj | ≤ y. Similarly to the proof of Lemma 1.41 we obtain the bounds 󵄩󵄩 1 󵄩󵄩2 ‖W(t, n)Ψ‖ ≤ wt (n) 󵄩󵄩󵄩e− 2 (t−1)Hf 󵄩󵄩󵄩 ‖Ψ‖, 󵄩 󵄩

t ≥ 1,

where wt (n) = n! ∑

k

2

[n/2]

∑

( α4 ) (α + y)l αm k!l!m!

k=0 l+m+2k=n

ξ (l, m),

and 󵄩󵄩 1 󵄩󵄩2 ‖W(t, n)Ψ‖ ≤ wt (s, n) 󵄩󵄩󵄩e− 2 (t−s)Hf 󵄩󵄩󵄩 ‖Ψ‖, 󵄩 󵄩

0 < s < t < 1,

where 2

[n/2]

wt (s, n) = n! ∑

∑

k

( α4 ) (α + y)l αm k!l!m!

k=0 l+m+2k=n

ξs (l, m).

Here ξ (l, m) and ξs (l, m) are defined as in Lemma 1.41. This shows that W(t, n) is bounded. To show boundedness of W(t) we proceed similarly to the proof of Lemma 1.42. We have 󵄩󵄩 1 󵄩󵄩2 ‖W(t)Ψ‖ ≤ wt 󵄩󵄩󵄩e− 2 (t−1)Hf 󵄩󵄩󵄩 ‖Ψ‖, 󵄩 󵄩 where 2 2

∞

wt = ∑

∑

n=0 l+m+2k=n

k

( t 4α ) (tα + ty)l (tα)m k!l!m!

We replace tα and ty by α and y, respectively, and get

ξs (l, m).

1.2 Boson Fock space

| 41

k

2

( α4 ) (α + y)l αm 2 wt = ∑ ∑ ξs (l, m) = eα /4 ξs (y)ξs (0), k! l! m! n=0 l+m+2k=n ∞

where (√2‖g‖ω )j (α + y)l √(j + l)! . s(j+l)/2 l!j! j=0 l=0 ∞ ∞

ξs (y) = ∑ ∑

The right-hand side above is finite as shown in Lemma 1.42. Finally, we prove continuity of W(t, n) and W(t). Here W(t, n) and W(t) are defined by (1.2.65) and (1.2.66), respectively. Let fj , g ∈ L2 (ℝd ) for j ∈ ℕ. Then W(t, n) is a map from L2 (ℝd ) × (×n L2 (ℝd )) ∋ g × (×nj=1 fj ) 󳨃→ W(t, n). Lemma 1.44. (1) The map L2 (ℝd ) × (×n L2 (ℝd )) ∋ g × (×nj=1 fj ) 󳨃→ W(t, n) is continuous in the uniform norm. (2) Suppose that there exists α > 0 such that fjh → fj as h → 0 is uniformly in j and

α = sup{‖fjh ‖ω | j ∈ ℕ, h > 0}. Then the map ×∞ L2 (ℝd ) ∋ ×∞ j=1 fj 󳨃→ W(t, n) is continuous in the uniform norm.

Proof. We set W(t, n) = W 0 (t, n) + W + (t, n), where [n/2]

W + (t, n) = ∑

k=1 n

pair

n−2k 1 ∑ ∑ ∑ 2k j1 ,...,j2k l=0 {i ,...,i }⊂{j ,...,j 1

W 0 (t, n) = ∑ Aea l=0

∗

(g) −tHf a(g)̄

e

e

l

1

k

2k

}c

(∏(fj2i−1 , fj2i )) Aea

∗

(g) −tHf a(g)̄

e

i=1

e

B,

B.

We show the statements (1) and (2) for W 0 (t, n) and W + (t, n). Suppose that g h → g and fjh → fj as h → ∞ in ‖ ⋅ ‖ω . Let Wh+ (t, n) be defined by

W + (t, n) with fj and g replaced by fjh for j = 1, . . . , n, and g h , respectively. Define the

finite constants α = sup {‖fjh ‖ω | j = 1, . . . , n, h > 0} and β = sup {‖g h ‖ω | h > 0}. We estimate ‖Wh+ (t, n) − W + (t, n)‖. Let (√2β)j+j √(j + l)!√(j󸀠 + m)! . ξ (l, m) = ∑ j!j󸀠 ! j,j󸀠 =0 󸀠󸀠

∞

We see that [n/2]

Wh+ (t, n) − W + (t, n) = ∑

k=1

pair

n−2k 1 ∑ ∑ ∑ 2k j1 ,...,j2k l=0 {i ,...,i }⊂{j ,...,j 1

l

1

5

c

2k }

∑ Jp ,

p=1

42 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes where ∑4p=1 Jp0 and ∑5p=1 Jp are obtained by telescoping Wh+ (t, n) − W + (t, n) defined by k

k

i=1

i=1

J1 = (∏(fjh2i−1 , fjh2i ) − ∏(fj2i−1 , fj2i )) Ah ea

∗

k

J2 = ∏(fj2i−1 , fj2i )(Ah − A)ea

∗

k

∗

(g h )

e

− ea

∗

i=1 k

J4 = ∏(fj2i−1 , fj2i )Aea

∗

e

(g) −tHf

i=1 k

J5 = ∏(fj2i−1 , fj2i )Aea

∗

e

e

(g)

e

Bh ,

Bh , ̄h

)e−tHf ea(g ) Bh ,

̄h

(ea(g ) − ea(g) )Bh ,

(g) −tHf a(g)̄

i=1

e

(g h ) −tHf a(ḡ h )

i=1

J3 = ∏(fj2i−1 , fj2i )A(ea

(g h ) −tHf a(ḡ h )

e

̄

(Bh − B).

Suppose that there exists δ > 0 such that ‖fjh − fj ‖ω ≤ ε for any j and ‖g h − g‖ ≤ ε for every h < δ. Then pair

n−2k 1 ∑ k ∑ ∑ ∑ k=1 2 j1 ,...,j2k l=0 {i ,...,i }⊂{j ,...,j

[n/2]

1

l

1

[n/2]

c

‖J1 ‖ ≤ εn! ∑

∑

k=1 l+m+2k=n

2k }

α 2

2

k−1

( α4 )

(α + y)l αm ξ (l, m). (k − 1)! l!m!

Similar estimates can be obtained for J2 and J4 , and the net result is [n/2]

∑

k=1

pair

n−2k 1 ∑ ∑ ∑ 2k j1 ,...,j2k l=0 {i ,...,i }⊂{j ,...,j 1

l

1

2

≤ εn! ∑

k

( α4 )

[n/2]

∑

k=1 l+m+2k=n

c

(

k!

(‖J2 ‖ + ‖J4 ‖)

2k }

(α + y)l−1 αm (α + y)l αm−1 + ) ξ (l, m). (l − 1)!m! l!(m − 1)!

Finally, J3 and J5 are estimated as [n/2]

∑

k=1

pair

n−2k 1 ∑ ∑ ∑ 2k j ,...,j l=0 {i ,...,i }⊂{j ,...,j 1

2k

1

l

1

2

k

c

(‖J3 ‖ + ‖J5 ‖)

2k }

( α4 ) (α + y)l αm ≤ εn! ∑ (ξ (l + 1, m) + ξ (l, m + 1)). ∑ k! l!m! k=1 l+m+2k=n [n/2]

Together with them (1) follows. By the estimates above, it is seen that

where

󵄩󵄩 1 󵄩󵄩2 ‖(W + (t) − Wh+ (t))Ψ‖ ≤ εwt+ 󵄩󵄩󵄩e− 2 (t−1)Hf 󵄩󵄩󵄩 ‖Ψ‖, 󵄩 󵄩 ∞ [n/2]

wt+ = ∑ ∑

∑

(aklm + bklm + cklm )

n=1 k=1 l+m+2k=n

1.3 Q-spaces |

with aklm =

α 2

2

(α + y)l αm ξ (l, m), (k − 1)! l!m! k

( α4 ) k! 2

cklm

k−1

( α4 ) 2

bklm =

43

( k

(α + y)l−1 αm (α + y)l αm−1 + ) ξ (l, m), (l − 1)!m! l!(m − 1)!

( α4 ) (α + y)l αm = (ξ (l + 1, m) + ξ (l, m + 1)). k! l!m!

Thus (2) follows. Using the decomposition

n

4

Wh0 (t, n) − W 0 (t, n) = ∑

∑ Jp0 ,

∑

l=0 {i1 ,...,il }⊂{1,...,n} p=1

where

J10 = (Ah − A)ea

∗

J20 = A(ea

∗

J30 = Aea

∗

J40 = Aea

∗

h

(g )

e

(g h ) −tHf a(ḡ h )

e

− ea

(g) −tHf

∗

(g)

Bh , ̄h

)e−tHf ea(g ) Bh ,

̄h

(ea(g ) − ea(g) )Bh ,

(g) −tHf a(g)̄

e

e

e

̄

(Bh − B),

we can similarly prove the statements (1) and (2) for W 0 (t, n) and W 0 (t).

1.3 Q-spaces 1.3.1 Gaussian random processes We define a family of Gaussian random variables indexed by a given real vector space h on a probability space (Q , Σ, μ), and study L2 (Q ) = L2 (Q , Σ, μ). Note that L2 (Q ) is a vector space over ℂ. Our main goal here is to show unitary equivalence of L2 (Q ) and ℱb (hℂ ), where hℂ is the complexification of h. Definition 1.45 (Gaussian random variables indexed by h). We say that (ϕ(f ), f ∈ h) is a family of Gaussian random variables on a probability space (Q , Σ, μ) indexed by a real inner product space h whenever (1) ϕ : h ∋ f 󳨃→ ϕ(f ) is a map from h to a Gaussian random variable on (Q , Σ, μ) with zero mean and covariance 1 ∫ ϕ(f )ϕ(g)dμ = (f , g)h , 2 Q

(2) ϕ(αf + βg) = αϕ(f ) + βϕ(g), α, β ∈ ℝ, (3) Σ is the completion of the minimal σ-field generated by {ϕ(f ) | f ∈ h}.

44 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes Property (3) in Definition 1.45 is expressed by saying that the σ-field Σ is full. The existence of such a family of Gaussian random variables will be discussed in Section 1.4. Here we first look at some properties. Define n

S (Q ) = {F(ϕ(f1 ), . . . , ϕ(fn )) 󵄨󵄨󵄨 F ∈ S (ℝ ), fj ∈ h, j = 1, . . . , n, n ≥ 1}.

󵄨

Theorem 1.46. Let (Q , Σ, μ) be a probability space and (ϕ(f ), f ∈ h) a family of Gaussian random variables indexed by h on it. Then the following hold. (1) If Σ is a full σ-field, then S (Q ) is dense in L2 (Q ). (2) If Σ is a complete σ-field and S (Q ) is dense in L2 (Q ), then Σ is a full σ-field. Proof. To prove (1), we regard S (Q ) as a set of bounded multiplication operators on L2 (Q ). Hence the closure S (Q ) taken in strong topology is a von Neumann algebra. It follows that S (Q ) ⊂ L∞ (Q ), since a strongly convergent bounded operator is bounded. Let 𝒫 be the set of all projections in S (Q ). Every projection is the characteristic function of a measurable subset in Q . Let Σ𝒫 denote the family of measurable subsets of Q associated with 𝒫 , satisfying Σ𝒫 ⊂ Σ. It is easily seen by a limiting argument that ϕ(f ) is measurable with respect to Σ𝒫 . Hence Σ = Σ𝒫 , as Σ is the minimal σ-field generated by {ϕ(f ) | f ∈ h}. Since a function in L∞ (Q ) can be approximated by a finite linear sum of characteristic functions and Σ = Σ𝒫 , S (Q ) = L∞ (Q ) follows ‖⋅‖L2 (Q)

as sets of bounded operators. It is clear that L∞ (Q ) = S (Q )1 ⊂ S (Q ) 2

‖⋅‖L2 (Q)

2

. Since

= L (Q ). This shows (1). L (Q ) is dense in L (Q ), it follows that S (Q ) Next we prove (2). Let Σ0 be the completion of the minimal σ-field generated by {ϕ(f ) | f ∈ h}, and then Σ0 ⊂ Σ. We shall prove Σ0 ⊃ Σ. Let M be the L2 -closure of the set of all Σ0 -measurable functions in L2 (Q ). Then for any f ∈ M there exists a sequence of Σ0 -measurable functions fn such that fn converges to f a. e. Since Σ0 is complete, f is also Σ0 -measurable. Thus any f ∈ M is Σ0 -measurable. Let PM be the projection on M. Let A ∈ Σ. By the assumption, there exists a sequence Φn ∈ S (Q ) such that ‖Φn − 1A ‖L2 (Q) → 0 as n → ∞. Since Φn ∈ M, we can see that ∞

‖Φn − 1A ‖2 = ‖1A PM c ‖2 + ‖(Φn − 1A )PM ‖2 and ‖(Φn − 1A )PM ‖2 → 0 as n → ∞. This implies that ‖1A PM c ‖ = 0 and then 1A PM = 1A . Hence 1A is a Σ0 -measurable function. Thus A is Σ0 measurable and Σ0 ⊃ Σ. This completes (2). The following is a useful consequence saying that every nonnegative function in L (Q ) can be approximated by nonnegative functions in S (Q ). 2

Corollary 1.47. Let (Q , Σ, μ) be a probability space such that Σ is full. Let F ∈ L2 (Q ) be nonnegative. Then there exists a sequence (Fn )n∈ℕ ⊂ S (Q ) such that Fn ≥ 0 for n ∈ ℕ and ‖Fn − F‖L2 (Q) → 0 as n → ∞.

1.3 Q-spaces | 45

Proof. Notice that S (Q ) is dense in L2 (Q ). There exists (Gn )∞ n∈ℕ ⊂ S (Q ) such that Gn → F as n → ∞. Choosing a subsequence n󸀠 , we see that Gn󸀠 → F as n󸀠 → ∞ a. s. Thus |Gn󸀠 | → F a. s. We approximate |Gn |(ϕ(f1 ), . . . , ϕ(fm )) by functions in S (Q ). Let ρ ∈ C0∞ (ℝd ) such that ≥0 ρ(x) { =0

|x| < 1 |x| ≥ 1

and

∫ ρ(x)dx = 1. ℝd

Define ρε (x) = ε−d ρ(x/ε) and Fnε (x) = ∫ℝd ρε (x − y)|Gn |(y)dy. It then follows for every 1 ≤ p < ∞ that ‖Fnε − |Gn |‖Lp (ℝd ) → 0 as ε ↓ 0. This yields ‖Fnε (ϕ(f1 ), . . . , ϕ(fm )) − |Gn |(ϕ(f1 ), . . . , ϕ(fm ))‖L2 (Q) m 󵄩󵄩 1 󵄩󵄩 = 󵄩󵄩󵄩 ∫ (F̌nε (k) − |Gň |(k))e−i ∑j=1 kj ϕ(fj ) dk 󵄩󵄩󵄩 2 󵄩 (2π)m/2 󵄩L (Q) ℝm

󵄨󵄨 󵄨󵄨 1 ≤ ∫ 󵄨󵄨F̌ ε (k) − |Gň |(k)󵄨󵄨󵄨dk → 0, 󵄨 (2π)m/2 m 󵄨󵄨 n

as ε ↓ 0.

ℝ

We define Wick products in L2 (Q ) in the same way as for boson Fock space. Definition 1.48 (Wick product). The Wick product of ∏ni=1 ϕ(fi ) is recursively defined by :ϕ(f ): = ϕ(f ), n

n

i=1

i=1

:ϕ(f ) ∏ ϕ(fi ): = ϕ(f ):∏ ϕ(fi ): −

1 n ∑(f , fj ):∏ ϕ(fi ):. 2 j=1 i=j̸

(1.3.1)

Since μ is a Gaussian measure, any Wick-ordered polynomial satisfies n

:∏ ϕ(fi ): ∈ L2 (Q , dμ). i=1

Definition 1.48 has the virtue of allowing explicit calculations, and we will need especially equality (1.3.1) later on. In particular, for fi , gj ∈ h, n

m

i=1

i=1

n

(:∏ ϕ(fi ):, :∏ ϕ(gi ):) = δmn ∑ 2−n ∏(fi , gπ(i) ). π∈℘n

i=1

Furthermore, m

α2 ‖f ‖2 αn :ϕ(f )n : = e 4 eαϕ(f ) . m→∞ n! n=0

:eαϕ(f ) : = s-lim ∑ Write

n 󵄨󵄨 L2n (Q ) = L. H. {:∏ ϕ(fi ): 󵄨󵄨󵄨 fi ∈ h, i = 1, . . . , n}. 󵄨 i=1

We see that

L2n (Q )

2

⊂ L (Q ) and

L2m (Q )

⊥ L2n (Q ) if n ≠ m.

(1.3.2)

46 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes Proposition 1.49 (Wiener–Itô decomposition). We have that ∞

L2 (Q ) = ⨁L2n (Q ). n=0

(1.3.3)

2 2 iϕ(f ) Proof. Denote G = ⨁∞ ∈ G , which n=0 Ln (Q ). It is clear that G ⊂ L (Q ). By (1.3.2), e n means that F(ϕ(f1 ), . . . , ϕ(fn )) ∈ G for F ∈ S (ℝ ). Since S (Q ) is dense in L2 (Q ) by Theorem 1.46, L2 (Q ) = G follows.

Equality (1.3.3) is called the Wiener–Itô decomposition. The following result says that the family of Gaussian random variables (ϕ(f ), f ∈ h) is unique up to unitary equivalence. Proposition 1.50 (Uniqueness of Gaussian random variables). Let (ϕ(f ), f ∈ h) and (ϕ󸀠 (f ), f ∈ h) be Gaussian random variables indexed by a real vector space h on (Q , Σ, μ) and (Q 󸀠 , Σ󸀠 , μ󸀠 ), respectively. Then there exists a unitary operator U : L2 (Q ) → L2 (Q 󸀠 ) such that U1 = 1 and Uϕ(f )U −1 = ϕ󸀠 (f ). Proof. Define U : L2 (Q ) → L2 (Q 󸀠 ) by U:ϕ(f1 ) ⋅ ⋅ ⋅ ϕ(fn ): = :ϕ󸀠 (f1 ) ⋅ ⋅ ⋅ ϕ󸀠 (fn ):. By Proposition 1.49, U can be extended to a unitary operator.

1.3.2 Wiener–Itô–Segal isomorphism and positivity improving Define the complexification of the real Hilbert space h by hℂ = {{f , g} | f , g ∈ h} such that for λ ∈ ℝ, λ{f , g} = {λf , λg},

iλ{f , g} = {−λg, λf },

{f , g} + {f 󸀠 , g 󸀠 } = {f + f 󸀠 , g + g 󸀠 }.

The scalar product on hℂ is defined by ({f , g}, {f 󸀠 , g 󸀠 })hℂ = (f , f 󸀠 ) + (g, g 󸀠 ) + i((f , g 󸀠 ) − (g, f 󸀠 )). With these definitions, (hℂ , (⋅, ⋅)hℂ ) is a Hilbert space over ℂ. Proposition 1.51 (Wiener–Itô–Segal isomorphism). There exists a unitary operator θW : ℱb (hℂ ) → L2 (Q ) such that (1) θW Ωb = 1; (2) θW ℱb(n) (hℂ ) = L2n (Q ); −1 (3) θW Φ(f )θW = ϕ(f ), where Φ(f ), f ∈ h, is a Segal field.

1.3 Q-spaces |

47

Proof. Define θW : ℱb (hℂ ) → L2 (Q ) by n

n

i=1

i=1

θW :∏ Φ(fi ):Ωb = :∏ ϕ(fi ):,

f1 , . . . , fn ∈ h,

θW Ωb = 1. Notice that ‖:∏ni=1 Φ(fi ):Ωb ‖ℱb (hℂ ) = ‖:∏ni=1 ϕ(fi ):‖L2 (Q) can be checked and subspace L. H.{:∏ni=1 Φ(fi ):Ωb , Ωb | fi ∈ h, i = 1, . . . , n, n ≥ 1} is dense in the Fock space ℱb (hℂ ). It is straightforward to check that θW can be extended to a unitary operator from ℱb (hℂ ) to L2 (Q ) by Proposition 1.49, and the proposition follows. The unitary operator θW is called Wiener–Itô–Segal isomorphism, and it allows to identify ℱb (hℂ ) with L2 (Q ). Let T : h → h be a contraction operator. T can be linearly extended to a contraction −1 on hℂ . The operator θW Γ(T)θW : L2 (Q ) → L2 (Q ) is called the second quantization of T and we will denote it by the same symbol Γ(T). It is seen directly that Γ(T) acts as n

n

i=1

i=1

Γ(T):∏ ϕ(fi ): = :∏ ϕ(Tfi ): and Γ(T)1 = 1. −1 For a self-adjoint operator h, the differential second quantization θW dΓ(h)θW will also be denoted by the same symbol dΓ(h). We have n

dΓ(h):ϕ(f1 ) ⋅ ⋅ ⋅ ϕ(fn ): = ∑:ϕ(f1 ) ⋅ ⋅ ⋅ ϕ(hfj ) ⋅ ⋅ ⋅ ϕ(fn ): j=1

with dΓ(h)1 = 0 for fj ∈ D(h), j = 1, . . . , n. Using Wiener–Itô–Segal isomorphism allows to identify ℱ (hℂ ) with L2 (Q ), and discuss the fundamental question of positivity of eigenfunctions. In particular, one expects the same positivity properties of ground states of self-adjoint operators acting in L2 (Q ) as of operators acting on L2 (ℝd ). Now we turn to discussing the relationship between positivity preserving operators and second quantization. Let (M, μ) be a σ-finite measure space. Recall that a bounded operator A on L2 (M, dμ) is called a positivity preserving operator if (f , Ag) ≥ 0 for all nonnegative f , g ∈ L2 (M, dμ). A is also called a positivity improving operator if (f , Ag) > 0 for all nonnegative f , g ∈ L2 (M, dμ).

48 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes Proposition 1.52 (Positivity preserving). Let T be a contraction operator on a real Hilbert space h. Then Γ(T) is positivity preserving on L2 (Q ). Proof. Note that Γ(T):exp(αϕ(f )): = :exp(αϕ(Tf )): for every α ∈ ℂ. Thus 1

2

2

1

2

Γ(T)eαϕ(f ) = :eαϕ(Tf ) :e 4 α ‖f ‖ = eαϕ(Tf ) e 4 α (f ,(1−T

∗

T)f )

.

Hence for F(ϕ(f1 ), . . . , ϕ(fn )) ∈ S (Q ), Γ(T)F(ϕ(f1 ), . . . , ϕ(fn )) n n 1 ∗ 1 ̂ 1 , . . . , kn )e− 4 ∑i,j=1 (fi ,(1−T T)fj )ki kj ei ∑j=1 kj ϕ(Tfj ) dk1 ⋅ ⋅ ⋅ dkn . = ∫ F(k n/2 (2π) n

(1.3.4)

ℝ

Since ‖T‖ ≤ 1, {(fi , (1 − T ∗ T)fj )}i,j is positive semi-definite. The right-hand side of (1.3.4) is expressed by the convolution of F and the Gaussian kernel DT , i. e., Γ(T)F(ϕ(f1 ), . . . , ϕ(fn )) = (2π)−n/2 (F ∗ DT )(ϕ(Tf1 ), . . . , ϕ(Tfn )). Thus F ≥ 0 implies that Γ(T)F ≥ 0. Let Ψ ∈ L2 (Q ) be a nonnegative function. By Corollary 1.47, we can construct a sequence (Fn )n∈ℕ ⊂ S (Q ) such that Fn → Ψ as n → ∞ in L2 (Q ) and Fn ≥ 0. Hence Γ(T)Ψ ≥ 0 follows by a limiting argument. Next we show a sufficient condition under which a positivity preserving operator Γ(T) becomes a positivity improving operator. First we give an abstract necessary and sufficient condition for a semigroup to be positivity improving. Proposition 1.53 (Positivity improving semigroup). Let (M, μ) be a σ-finite measure space, and A a self-adjoint operator on L2 (M), bounded from below. Suppose that e−tA is positivity preserving for all t > 0 and E = inf Spec(A) is an eigenvalue. Then the following properties are equivalent: (1) E is a simple eigenvalue with a strictly positive eigenvector. (2) e−tA is positivity improving for all t > 0. Proof. (1) → (2): Fix t > 0. Let U = e−tA and write S = U/‖U‖. Let ES (λ) be the spectral projection of S. We have lim (f , Sn g) = lim ∫ λn d(f , ES (λ)g)

n→∞

n→∞

[0,1]

= lim ∫ λn d(f , ES (λ)g) + (f , ES ({1})g) = (f , ES ({1})g). n→∞

[0,1)

Thus Sn weakly converges to the projection ES ({1}) and s-limn→∞ Sn = ES ({1}) follows. Note that ES ({1}) = (v, ⋅)v, where v is the unique and strictly positive eigenvector of T such that Tv = ‖T‖v. Let f , g ∈ L2 (M) and f , g ≥ 0. Note that {t > 0 | (f , e−tA g) ≥ 0} =

1.3 Q-spaces |

49

[0, ∞). Hence we see that s-limn→∞ (f , Sn g) = (f , v)(v, g) > 0, and (f , Sn g) > 0 for some n > 0. This implies that B = {t > 0 | (f , e−tA g) > 0} ≠ 0. We see that F(z) = (f , e−zA g) is an analytic function in a neighborhood of the positive axis. This implies that (0, ∞) \ B is a discrete set, in particular, B contains arbitrarily small positive numbers. Fix s ∈ B. We see that (f , e−sA g) = ∫ f (m)(e−sA g)(m)dμ > 0 M

and f (m)(e

−sA

g)(m) is not identically zero. Define the function η(m) = min{f (m), (e−sA g)(m)},

and note that it is not identically zero. Hence for arbitrary t > 0 we have (f , e−(t+s)A g) = (f , e−tA (e−sA g)) ≥ (f , e−tA η) = (e−tA f , η) ≥ (e−tA η, η) > 0. This implies that e−(t+s)A is positivity improving for arbitrary t > 0. Hence e−tA is positivity improving for any t ≥ s, and we can chose arbitrary small positive s. This implies that e−tA is positivity improving for all t > 0. (2) → (1): Fix t > 0 and write U = e−tA . We prove that there is no nontrivial closed subspace invariant under U or any other bounded multiplication operator on L2 (M). Suppose that S ⊂ L2 (M) be a nontrivial closed subspace invariant under U and every bounded multiplication operator. Let f ∈ S and write h = f ̄/|f |. The function h defines a bounded multiplication operator. Since S is invariant under every bounded multiplication operator, |f | = hf ∈ S. Similarly, we have |g| ∈ S⊥ for g ∈ S⊥ . Let f ∈ S and g ∈ S⊥ such that f and g are not identically zero. Since U leaves S invariant, U|f | ∈ S. Hence we have (|g|, U|f |) = 0, which is in contradiction with the positivity improving property of U. This implies that there is no nontrivial closed subspace invariant under U and any bounded multiplication operator. By assumption ‖U‖ is an eigenvalue. Let f be an eigenvector of U associated with ‖U‖ and suppose that f is real-valued. Since |f |±f ≥ 0 and U is positivity improving, it follows that either U(|f |±f ) > 0 or identically U(|f | ± f ) = 0. Thus either U|f | > ±Uf or identically U|f | = ±Uf . We have U|f | > |Uf | or

U|f | = |Uf |,

however, ‖U‖‖f ‖2 ≥ (|f |, U|f |) ≥ (|f |, |Uf |) ≥ (f , Uf ) = ‖U‖‖f ‖2 . Hence 0 = (|f |, U|f | − |Uf |) = ∫ |f (m)||(U|f | − |Uf |)(m)|dμ. M

(1.3.5)

50 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes From (1.3.5), it follows that U|f | = |Uf |, which implies that U|f | = ‖U‖|f |. In particular, |f | is also an eigenvector of U. Let S = {g ∈ L2 (M) | gf = 0 a. e.}. We see that S is a subspace of L2 (M) invariant under every bounded multiplication operator. We show that US ⊂ S. Let S+ = {g ∈ S | g ≥ 0}. For g ∈ S+ we have (Ug, |f |) = (g, U|f |) = ‖U‖(g, |f |) = 0, and hence Ug ∈ S+ , i. e., U leaves S+ invariant. Since S = S+ − S+ + i(S+ − S+ ), U leaves S invariant. Thus S is invariant under U and every bounded multiplication operator, and thus S = M or S = {0}, but f ∈ ̸ S. We have S = {0} and f ≠ 0 a. e. which implies that |f | > 0 a. e. As a result it follows that every real-valued eigenvector with eigenvalue ‖U‖ is strictly positive and U|f | = ‖U‖|f |. Thus |f | − f is a nonnegative eigenvector such that U(|f | − f ) = ‖U‖(|f | − f ). Since U is positivity improving, it is either |f | − f > 0 a. e. or |f | − f = 0 a. e. This means that f is either strictly positive or strictly negative. Let f and g be two real-valued eigenvectors such that Uf = ‖U‖f and Ug = ‖U‖g. It is impossible to choose f and g such that (f , g) = 0. We see that a real-valued eigenvector is unique up to multiple constants. Now let f = ℜf + iℑf be an arbitrary eigenvector with eigenvalue ‖U‖. Since ℜ(Uf ) = Uℜf , ℜf and ℑf are eigenvectors of U. The uniqueness of real-valued eigenvectors implies that ℜf = aℑf with some a ∈ ℝ. Hence f is the complex multiple of the unique real-valued eigenvector, i. e., f = (1 + ia)ℜf . Thus g = f /(1 + ia) is the strictly positive eigenvectors with Ug = ‖U‖g. Corollary 1.54 (Positivity improving). Let hℂ = L2 (ℝd ) and f : ℝd → ℝ be a nonnegative multiplication operator on L2 (ℝd ) such that the Lebesgue measure of the set {k ∈ ℝd | f (k) = 0} is zero. Let dΓ(f ) be the differential second quantization on L2 (Q ). Then the semigroup {e−tdΓ(f ) : t > 0} is positivity improving. Proof. The semigroup e−tdΓ(f ) is positivity preserving and 1 is a simple eigenvector of dΓ(f ) with eigenvalue zero, since Specp (f ) ∋ 0. Hence the corollary follows from Proposition 1.53. Finally, we show in an abstract setting an expression of the ground state eigenvalue of a self-adjoint operator K acting in L2 (M). Let κ = inf Spec(K) and suppose that K has a ground state ϕ, i. e., Kϕ = κϕ, and (ϕ, f )L2 (M) ≠ 0.

(1.3.6)

By spectral theory we can directly see that κ = − lim

t→∞

1 log(f , e−tK f ). t

(1.3.7)

Lemma 1.55. Let (M, F , μ) be a measure space, and K a self-adjoint operator on L2 (M), bounded from below. Suppose that K has a unique strictly positive ground state. Let f ∈ L2 (M) be a nonnegative function, not identically zero. Then (1.3.7) holds.

1.3 Q-spaces | 51

Proof. Let E denote the spectral projection of the self-adjoint operator K. Write νf (⋅) = (f , E(⋅)f ). We have (f , e−tK f ) = e−tκ ( ∫ e−t(λ−κ) dνf (λ) + νf ({κ})). (κ,∞)

Note that ∫(κ,∞) e−t(λ−κ) dνf (λ) + νf ({κ}) → νf ({κ}) > 0 as t → ∞. This implies 1 1 − log(f , e−tK f ) = κ − log ( ∫ e−t(λ−κ) dνf (λ) + νf ({κ})) → κ t t (κ,∞)

as t → ∞. As seen in Proposition 1.53, if e−tK is positivity improving and K has a ground state, then the ground state is unique and strictly positive. Hence (1.3.7) follows for any nonnegative f , e. g., f (x) = 1{|x|≤1} . However, we can prove (1.3.7) under weaker conditions without assuming the existence of any ground state. Lemma 1.56. Let M be a metric space, (M, F , μ) a σ-finite measure space, and K a selfadjoint operator on L2 (M), bounded from below. Suppose that e−tK is positivity preserving for t ≥ 0, and f ∈ L2 (M) satisfies infm∈D f (m) > 0 for every compact set D ⊂ M. Then (1.3.7) holds, in particular, if μ(M) < ∞, then κ = − lim

t→∞

1 log(1, e−tK 1). t

(1.3.8)

Proof. Denote by E the spectral projection of the self-adjoint operator K, and let Ef be the infimum of the support of measure νf (⋅) = (f , E(⋅)f ). We have ∞

1 1 − log(f , e−tK f ) = − log ∫ e−λt dνf (λ) t t Ef

∞

∞

Ef

Ef

1 1 = − log e−Ef t ∫ e−(λ−Ef )t dνf (λ) = Ef − log ∫ e−(λ−Ef )t dνf (λ). t t Note that νf ([Ef , Ef + ε)) ≠ 0 for every ε > 0 by the definition of Ef . We have ∞

1 1 log ∫ e−(λ−Ef )t dνf (λ) ≤ log (νf ([Ef , Ef + ε)) + e−εt ) → 0, t t Ef

since νf ([Ef , Ef + ε)) + e−εt → νf ([Ef , Ef + ε)) ≠ 0. Thus − lim

t→∞

1 log(f , e−tK f ) = Ef . t

52 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes It remains to prove κ = Ef . Let O = {g ∈ L2 (M) | supp g ⊂ ⋃N>0 BN }, where BN is a ball of radius N centered at the origin. Notice that O is dense in L2 (M). We show that inf Eg = κ.

g∈O

(1.3.9)

Since trivially infg∈O Eg ≥ κ, it suffices to prove that infg∈O Eg ≤ κ. Suppose, to the contrary, that infg∈O Eg > κ. Then there exists ε > 0 such that E([κ, κ + ε))L2 (M) ≠ 0. Take a nonzero h ∈ E([κ, κ + ε))L2 (M). Thus for every g ∈ O , (g, h) = (g, E([κ, κ + ε))h) = (E([κ, κ + ε))g, h) = (0, h) = 0 holds. This is in contradiction with the denseness of O , and hence (1.3.9) holds. Let g ∈ O . We have |g| ≤ Cf on M, where C=

ess supm∈M |g(m)| < ∞, ess infm∈supp g f (m)

and notice that ess infm∈supp g f (m) > 0 since supp g is compact. Since e−tK is positivity preserving we see that (g, e−tK g) ≤ (|g|, e−tK |g|) ≤ C 2 (f , e−tK f ), and hence 1 1 − lim log(g, e−tK g) ≥ − lim log C 2 (f , e−tK f ). t→0 t t→0 t This means that Eg ≥ Ef for every g ∈ O , and κ = infg∈O Eg ≥ Ef . Since Ef ≥ κ trivially holds, Ef = κ and the lemma follows. 1.3.3 Hypercontractivity Let Δ be the d-dimensional Laplacian. It is known that etΔ , t > 0, is a contraction from Lp (ℝd ) to Lq (ℝd ) for 1 ≤ p ≤ q ≤ ∞. In this section we discuss the hypercontractivity of Γ(T), as a counterpart of the Lp − Lq bound of etΔ . Let T be a contraction operator on a real Hilbert space h. It was shown that Γ(T) : L2 (Q ) → L2 (Q ) is positivity preserving and Γ(T)1 = Γ(T)∗ 1 = 1. Operators with this property are called doubly Markovian. In the definition below we use a more abstract setting. Definition 1.57 (Doubly Markovian operator). Let (M, ℬM , P) be a probability space. A bounded operator S on L2 (M) is doubly Markovian if (1) S is positivity preserving; (2) S1 = S∗ 1 = 1.

1.3 Q-spaces | 53

Lemma 1.58. Suppose that S is a doubly Markovian operator on L2 (M). Then its adjoint S∗ is also doubly Markovian on it. Proof. To say that S is positivity preserving, is equivalent to (f , Sg) ≥ 0 for all positive f , g ∈ L2 (M). Thus if S is positivity preserving, so is S∗ . Hence S∗ is also doubly Markovian. Every doubly Markovian operator has the following property. Theorem 1.59 (Lp -contraction). Let (M, ℬM , P) be a probability space, and suppose that S is doubly Markovian on L2 (M). Then S is a contraction on Lp (M) for every 1 ≤ p ≤ ∞, i. e., ‖Sf ‖p ≤ ‖f ‖p for f ∈ L2 (M) ∩ Lp (M). Proof. Let f ∈ L1 (M) ∩ L2 (M) and f ≥ 0. We have ‖Sf ‖L1 = (1, |Sf |)L2 = (1, Sf )L2 = (S∗ 1, f )L2 = (1, f )L2 = ‖f ‖L1 . For general f ∈ L1 (M) ∩ L2 (M), we have ‖Sf ‖L1 = (1, |Sf |)L2 ≤ (1, S|f |)L2 = ‖S|f |‖L1 = ‖f ‖L1 . Hence S is a contraction on L1 (M). By Lemma 1.58 S∗ is also doubly Markovian and thus a contraction on L1 (M), i. e., ‖S∗ f ‖1 ≤ ‖f ‖1 for f ∈ L1 (M) ∩ L2 (M). By duality, ‖Sf ‖∞ ≤ ‖f ‖∞ . The theorem follows by the Riesz–Thorin interpolation theorem. Now we investigate the second quantization Γ(T) of a contraction T. We denote the norm on the Banach space Lp (Q ) by ‖ ⋅ ‖p for 1 ≤ p ≤ ∞. Lemma 1.60. Let T be a contraction. Then Γ(T) is doubly Markovian and a contraction on Lp (Q ), for 1 ≤ p ≤ ∞. Proof. We have already seen that Γ(T) is doubly Markovian. Hence Γ(T) is a contraction on Lp (Q ) by Theorem 1.59. Lemma 1.61. Let Φ ∈ L2n (Q ). Then it follows that ‖Φ‖4 ≤ 2n ‖Φ‖2 . Proof. Let ℤn+ denote the set of sequences of length n of nonnegative integers. Let Φ ∈ L2n (Q ) and {ei }i∈ℕ be an orthogonal system in h such that (ei , ej )2 = 2δij . Since n

{:∏ ϕ(eil ): | i1 ≤ i2 ≤ . . . ≤ in } l=1

54 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes form the orthogonal system in L2n (Q ), we have Φ=

n

∑

i=(i1 ,...,in )∈ℤn+

ai :∏ ϕ(eil ): l=1

and n

n

l=1

l=1

(:∏ ϕ(eil ):, :∏ ϕ(ejl ):) = ∑ δi1 jπ(1) ⋅ ⋅ ⋅ δin jπ(n) . 2

π∈℘n

Thus we see that n

n

l=1

l=1

‖Φ‖22 = ∑ aī aj (:∏ ϕ(eil ):, :∏ ϕ(ejl ):) i,j∈ℤn+

2

= ∑ ∑ aī aπ(i) = n! ∑ |ai |2 . π∈℘n i∈ℤn+

(1.3.10)

i∈ℤn+

Here we used that ai = aπ(i) for every π ∈ ℘n , where π(i) = (iπ(1) , . . . , iπ(n) ). Let Pn be the set of pairings of 4n distinct objects i = {i1 , . . . , in }, j = {j1 , . . . , jn }, k = {k1 , . . . , kn }, l = {l1 , . . . , ln } ∈ ℤn+ in such a way that no two elements of i, no two elements of j, no two elements of k, and no two elements of l are paired together. In other words, let m m m Pn = {X1 , . . . , XN } with Xm = {{am 1 , b1 }, . . . , {a2n , b2n }}, m = 1, . . . , N, where N denotes the m m m m m m m number of elements of Pn , and {am p , bp } ⊄ i, {ap , bp } ⊄ j, {ap , bp } ⊄ k and {ap , bp } ⊄ l hold for all 1 ≤ m ≤ N and 1 ≤ p ≤ 2n. Let |Pn | denote the number of elements in Pn . We see that ‖Φ‖44 =

n

∑

i,j,k,l∈ℤn+

n

n

n

l=1

l=1

l=1

aī aj̄ ak al ∫ :∏ ϕ(eil ): :∏ ϕ(ejl ): :∏ ϕ(ekl ): :∏ ϕ(eml ):dμ Q l=1

is a sum of |Pn | terms each of which is δp1 q1 ⋅ ⋅ ⋅ δp2n q2n with {{p1 , q1 }, . . . , {p2n , q2n }} ∈ Pn . After computations we finally obtain 2

‖Φ‖44 ≤ |Pn |( ∑ |ai |2 ) . i∈ℤn+

From this and (1.3.10) we have ‖Φ‖44 ≤

|Pn | ‖Φ‖42 . (n!)2

|Pn | is less than the number of ways of pairing 4n objects into 2n pairs without any restriction. Hence |Pn | ≤ (4n)!/(2n)!22n . Thus |Pn | 1 4n 2n 1 ≤ 2n ( ) ( ) ≤ 2n 24n 22n ≤ 24n . 2 2n n (n!) 2 2 Thus we have ‖Φ‖44 ≤ 24n ‖Φ‖42 .

1.3 Q-spaces | 55

Lemma 1.62. Let 2 ≤ p ≤ 4 and Φ ∈ L2n (Q ). Then 1− p2 n

) ‖Φ‖2 .

‖Φ‖p ≤ (4 Proof. By the Hölder inequality we have

‖Φ‖pp ≤ ‖Φ‖2p−4 ‖Φ‖4−p 4 2 . Thus the claim follows from Lemma 1.61. Lemma 1.63. Let 2 < p ≤ 4 and 0 < c < ‖Γ(c)Φ‖p ≤ Proof. Let Φ =

(n) ⨁∞ n=0 Φ

∈

1

2

41− p

1

1− p2

1 − c4

2 ⨁∞ n=0 Ln (Q ).

. Then ‖Φ‖2 ,

Φ ∈ L2 (Q ).

Then we have

∞

∞

1− p2 n

‖Γ(c)Φ‖p ≤ ∑ ‖Γ(c)Φ(n) ‖p ≤ ∑ ‖Γ(c)Φ(n) ‖2 (4 n=0 ∞

)

n=0

∞

1− p2 n

1− p2 n

) ‖Φ(n) ‖2 ≤ ‖Φ‖2 ∑ (c4

≤ ∑ (c4

n=0

n=0

)

We can make this lemma sharper by using the identities Lp (Q ⊕ Q ) ≅ Lp (Q )⊗Lp (Q ) and Γ(c ⊕ c) ≅ Γ(c) ⊗ Γ(c). Lemma 1.64. Let 2 < p ≤ 4 and 0 < c
log 4. We have an even sharper result, which we quote without proof. Theorem 1.67 (Hypercontractivity). Let 1 < p ≤ q < ∞ and ‖T‖ ≤ √(p − 1)/(q − 1). Then Γ(T) is a contraction from Lp to Lq , i. e., ‖Γ(T)Φ‖q ≤ ‖Φ‖p ,

Φ ∈ Lp (Q ).

1.3.4 Lorentz covariant quantum fields Following the conventions in physics, a quantum field is required to have the so-called Lorentz covariance property, i. e., to transform under the Lorentz group. The measure δ(k02 − k 2 − ν2 )dk, k = (k0 , k) ∈ ℝ × ℝ3 , on the 4-dimensional space-time ℝ4 supported on the light cone with mass ν is Lorentz-covariant and ∫ ℝ3

dk 1 dk = ∫ (2π)δ(k02 − |k|2 − ν2 )⌈{k0 >0} 2ων (k) (2π)3 (2π)4 ℝ4

is a Lorentz invariant integral. On normalization, the Lorentz covariant scalar field is formally given by ϕ(t, x) = ∫ ℝ3

1 (a(k)e−itων (k)+ik⋅x + a∗ (k)eitων (k)−ik⋅x ) dk. √2ων (k)

Instead of Segal fields, the conventional time-zero scalar field in ℱb (L2 (ℝ3 )) is obtained, 1 ∗ ̂ ̃ (a (f /√ων ) + a(f ̂/√ων )). √2

(1.3.11)

1.4 Existence of Q-spaces | 57

̃ ̃ ̄ Here f ̂(k) = f ̂(−k). Note that f ̂ = f ̂ for real f , and (1.3.11) is linear in f . In Chapter 2, we will discuss the Nelson model of a scalar quantum field given by (1.3.11). Note that (1.3.11) can be generalized by using a covariance operator C. Suppose that (ϕ(f ), f ∈ h) is a family of Gaussian random variables on (Q , Σ, μ). Let C : h → h be self-adjoint and ker C = {0}. Define h󸀠 by the closure of D(C) with respect to the norm ‖C⋅‖h

and (f , g)h󸀠 = (Cf , Cg)h . Then a quantum field ϕC (f ) ‖ ⋅ ‖h󸀠 = ‖C ⋅ ‖h , i. e., h󸀠 = D(C) is defined by ϕC (f ) = ϕ(Cf ) for f ∈ D(C), and 1 (ϕ(Cf ), ϕ(Cg))L2 (Q) = (Cf , Cg)h , 2

f , g ∈ D(C).

Let f ∈ h󸀠 . Then there exists a sequence fn ∈ D(C) such that ‖fn − f ‖h󸀠 → 0 as n → ∞. Thus we can see that ‖ϕC (fn ) − ϕC (fm )‖L2 (Q) = ‖fn − fm ‖h󸀠 → 0 as n, m → ∞ and (ϕC (fn ))n∈ℕ is a Cauchy sequence in L2 (Q ). Define ϕC (f ) for f ∈ h󸀠 by the limit s-limn→∞ ϕC (fn ). For ϕC (f ) the Wick product can be defined in the same way as for ϕ(f ), i. e., :ϕC (f ): = ϕC (f ), n

n

i=1

i=1

:ϕC (f ) ∏ ϕC (fi ): = ϕC (f ):∏ ϕC (fi ): −

1 n ∑(f , fj )h󸀠 :∏ ϕC (fi ):. 2 j=1 i=j̸

The Wiener–Itô–Segal isomorphism θW : L2 (Q , dμ) → ℱb (h󸀠ℂ ) acts as n

θW :∏ ϕC (fi ): = a∗ (f1 ) ⋅ ⋅ ⋅ a∗ (fn )Ωb , i=1

θW 1 = Ωb .

Here a∗ and a are the creation and annihilation operators in ℱb (h󸀠ℂ ), respectively, satisfying the canonical commutation relations [a(f ), a∗ (g)] = (f ̄, g)h󸀠 ,

[a(f ), a(g)] = 0 = [a∗ (f ), a∗ (g)].

In particular, [a(f ), a∗ (g)] = (Cf , Cg)h for f , g ∈ D(C). The map θW can be extended to a unitary operator between L2 (Q ) and ℱb (h󸀠ℂ ) by linearity and using the fact that L. H. {∏nj=1 a∗ (fj )Ωb , Ωb | fj ∈ D(C), j = 1, . . . , n, n ∈ ℕ} is dense in ℱb (h󸀠ℂ ). Thus the unitary equivalence L2 (Q ) ≅ ℱb (h󸀠 ) is implemented by θW .

1.4 Existence of Q-spaces 1.4.1 Countable product spaces In the previous section we have seen that a Q -space is unique up to unitary equivalence. On the other hand, the Q -space is not canonically given. In this section, we consider explicit choices of a measure space (Q , Σ, μ) on which we will be able to construct a family of Gaussian random variables indexed by a real Hilbert space h.

58 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes Theorem 1.68. Let h be a real Hilbert space. Then there exists a probability space (Q , Σ, μ) and a family of Gaussian random variables (ϕ(f ), f ∈ h) with mean zero and covariance 1 ∫ ϕ(f )ϕ(g)μ(dϕ) = (f , g)h . 2

Q

Proof. We show that Q can be obtained as a countable product space and, roughly, the measure μ may be regarded as the infinite product of Gaussian distributions ∞

2

dμ = ∏ π −1/2 e−xi dxi i=1

on

∞

∏ ℝ. i=1

We establish this next. ̇ Let ℝ̇ = ℝ∪{∞} be the one-point compactification of ℝ, and put Q = ×∞ n=1 ℝ. By the Tychonoff theorem, Q is a compact Hausdorff space. Let C(Q ) be the set of continuous functions on Q , and Cfin (Q ) ⊂ C(Q ) the set of continuous functions on Q depending on a finite number of variables. For F = F(x1 , . . . , xn ) ∈ Cfin (Q ) define ℓ : Cfin (Q ) → ℂ by ℓ(F) =

1

n

π n/2

2

∫ F(x1 , . . . , xn )e− ∑i=1 xi dx1 ⋅ ⋅ ⋅ dxn . ℝn

Since |ℓ(F)| ≤ ‖F‖∞ , ℓ is a bounded linear functional on Cfin (Q ). On the other hand, Cfin (Q ) is a dense subset in C(Q ) by the Stone–Weierstrass theorem, thus ℓ can be uniquely extended to a bounded linear functional ℓ̃ on the Banach space (C(Q ), ‖⋅‖∞ ). ̃ It is clear that ℓ(F) ≥ 0, for F ≥ 0. The Riesz–Markov theorem furthermore implies that ̃ there exists a probability space (Q , Σ, μ) such that ℓ(F) = ∫Q F(q)dμ(q). Let {en }∞ n=1 be a complete orthonormal system of a real Hilbert space h. For each en we define en (q) = xn for q = {xk }∞ k=1 ∈ Q . Note that ̃ 2) = ∫ |en (q)|2 dμ(q) = ℓ(e n Q

2 1 1 ∫ x 2 e−x dx = . 2 π 1/2

ℝ

Since C(Q ) is dense in L2 (Q ) in the ‖ ⋅ ‖L2 (Q) norm, Cfin (Q ) is dense in L2 (Q ). By Theorem 1.46 the minimal σ-field generated by {en (⋅)}∞ n=1 is Σ. On expanding f ∈ h as n f = ∑∞ α e and writing ϕ (f ) = α e (⋅), it is seen that ∑ n n=1 n n k=1 k k ‖ϕm (f ) − ϕn (f )‖2L2 (Q) =

1 n ∑ α2 → 0 m, n → ∞. 2 k=m+1 k 2

Thus ϕ(f ) = s-limn→∞ ϕn (f ) exists in L2 (Q ) and ∫Q eiϕ(f ) dμ = e−(1/4)‖f ‖ easily follows. From this we obtain ∫Q ϕ(f )dμ = 0 and 1 ∫ ϕ(f )ϕ(g)dμ = (f , g), 2

Q

1.4 Existence of Q-spaces | 59

and thereby we have constructed the desired family of Gaussian random variables (ϕ(f ), f ∈ h) indexed by h on (Q , Σ, μ). 1.4.2 Bochner theorem and Minlos theorem Next we construct the Q space in Theorem 1.68. Like seen above, this is given by the ̇ countable product ×∞ n=1 ℝ, however, the family of Gaussian random variables (ϕ(f ), f ∈ h), depends on the base of the given Hilbert space h, therefore, the realization is not unique. The Q -space introduced in this section is obtained through a generalization of Bochner’s theorem, known as Minlos’ theorem. In this construction, the measure on Q is given on an abstract nuclear space such as S 󸀠 (ℝd ). Let X be a random variable on (Ω, F , P), and consider its characteristic function C(z) = 𝔼P [eiz⋅X ],

z ∈ ℝd .

Clearly, C(z) has the properties n

(1) ∑ αi ᾱ j C(zi − zj ) ≥ 0, i,j=1

(2) C is uniformly continuous,

(3) C(0) = 1.

(1.4.1)

The converse statement is known as Bochner’s theorem. Theorem 1.69 (Bochner). Suppose that the function C : ℝd → ℂ satisfies condition (1.4.1). Then there exists a probability space (ℝd , ℬ(ℝd ), μB ) such that C(z) = ∫ eiz⋅x μB (dx). ℝd 2

Proof. Let Ct (z) = e−t|z| /2 . It is seen that Ct (z)C(z) is positive definite, which implies that ρt (x) =

1 ∫ e−iz⋅x Ct (z)C(z)dz ≥ 0 (2π)d/2 ℝd

and ∫ ρt (x)dx = (2π)d/2 Ct (0)C(0) = (2π)d/2 . ℝd

Put μt (dx) = (2π)−d/2 ρt (x)dx. Thus μt (dx) is a probability measure on ℝd with characteristic function Ct (z)C(z). Since limt↓0 Ct (z)C(z) = C(z) pointwise and C(z) is continuous, there exists a measure μ such that μt is weakly convergent to μ and its characteristic function is C(z). Setting μ = μB proves the claim.

60 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes A function C satisfying (1) of (1.4.1) for all zi , zj ∈ ℂ, αi , αj ∈ ℂ and n ∈ ℕ is called positive definite. Bochner’s theorem can be extended to infinite dimension, however, as it will be illustrated in Remark 1.75 below, the Hilbert space h is too small to support the corresponding measure μ. To replace h, we will introduce a suitable nuclear space. Let h be a complete and separable topological vector space with a countable family of norms {‖ ⋅ ‖n }∞ n=0 such that ‖ ⋅ ‖0 < ‖ ⋅ ‖1 < ‖ ⋅ ‖2 < . . ., and there exists a scalar product (⋅, ⋅)n generating these norms so that ‖f ‖2n = (f , f )n for each n. Define ∞

h = ⋂ En , n=0

‖⋅‖n

where En = h and En is a Hilbert space with the scalar product (⋅, ⋅)n . Identifying E0 with E0∗ gives rise to the nested inclusions . . . ⊂ E2 ⊂ E1 ⊂ E0 ≅ E0∗ ⊂ E1∗ ⊂ E2∗ ⊂ . . . . ∗ Furthermore, let h∗ = ⋃∞ n=0 En , and denote its norm by ‖x‖−n = supξ ∈En |x(ξ )|.

Definition 1.70 (Nuclear space). h is a called a nuclear space whenever for every m > 0 there exists n > m such that the injection ιnm : En → Em is a Hilbert–Schmidt operator. Example 1.71. Let Hosc = 21 (−Δ + |x|2 − d) be the harmonic oscillator on L2 (ℝd ) with purely discrete spectrum {n1 + ⋅ ⋅ ⋅ + nd | nj ∈ ℕ ∪ {0}, j = 1, . . . , d}. Using a suitable orthonormal basis {gi1 ,...,id}(i1 ,...,id )∈(ℕ∪{0})d , Hosc is diagonalized and (gi1 ,...,id,Hgi1 ,...,id) =

∑dj=1 ij . It is easily seen that (Hosc + 1)−(d+ε)/2 is a Hilbert–Schmidt operator for every ε > 0 and ∞

∞

d

i1 =0

id =0

ν=1

−(d+ε)

Tr[(Hosc + 1)−(d+ε) ] = ∑ ⋅ ⋅ ⋅ ∑ ( ∑ (iν + 1))

.

Let (Ψ, Φ)n = (Ψ, (Hosc + 1)n Φ). Then ‖Ψ‖n < ‖Ψ‖n+1 and denote the completion of L2 (ℝd ) with respect to ‖ ⋅ ‖n by ℋn . We have . . . ⊂ ℋn+1 ⊂ ℋn ⊂ . . ., and it can be proven that ∞

S = ⋂ ℋn , n=−∞

󸀠

∞

S = ⋃ ℋn . n=−∞

Let 1n+1 = 1 : ℋn+1 → ℋn be the identity map. Thus (Hosc + 1)1/2 1n+1 is unitary and so 1 = 1n+1 ⋅ ⋅ ⋅ 1n+D+1 : ℋn+D+1 → ℋn is a Hilbert–Schmidt operator for sufficiently large D since 1 = (Hosc + 1)−D/2 (Hosc + 1)D/2 1 is a product of a Hilbert–Schmidt operator with a unitary operator. Theorem 1.72 (Minlos). Let h be a nuclear space, and suppose that C : h → ℂ satisfies n

(1) ∑ αi ᾱ j C(ξi − ξj ) ≥ 0, i,j=1

(2) C is uniformly continuous,

(3) C(0) = 1.

(1.4.2)

1.4 Existence of Q-spaces |

61

Then there exists a probability space (h∗ , ℬ, μ) such that C(ξ ) = ∫ ei⟨⟨x,ξ ⟩⟩ dμ(x),

ξ ∈ h,

(1.4.3)

h∗

where ⟨⟨x, ξ ⟩⟩ denotes the pairing between x ∈ h∗ and ξ ∈ h. Proof. Suppose that h = ⋂∞ n=0 En . Let F ⊂ h be a finite dimensional subspace, and for ξ1 , . . . , ξn ∈ F and B ∈ ℬ(ℝn ) define Aξ1 ,...,ξn ,B = {x ∈ h∗ | (⟨⟨x, ξ1 ⟩⟩, . . . , ⟨⟨x, ξn ⟩⟩) ∈ B}. Let AF = σ({Aξ1 ,...,ξn ,B | ξ1 , . . . , ξn ∈ F, B ∈ ℬ(ℝn ), n ∈ ℕ}) be the minimal σ-field generated by {Aξ1 ,...,ξn ,B | ξ1 , . . . , ξn ∈ F, B ∈ ℬ(ℝn ), n ∈ ℕ} and A∞ =

⋃

F⊂h,dim F 0, such that m(A) < ε holds for every A ∈ A with A ∩ Sn = 0. Proof. (󳨐⇒) Suppose that m can be extended to a measure μ on h∗ . Assume that γn → ∗ c ∞ as n → ∞. Then ⋃∞ n=1 Sn = h , and μ(Sn ) < ε for a sufficiently large n. Hence (A) ≤ μ(Snc ) < ε. ∗ (⇐󳨐) Assume that ⋃∞ n=1 An = h and Am ∩ An = 0 for m ≠ n. Since m is a finitely ∞ additive set function, ∑n=1 m(An ) ≤ 1. Suppose that ∑∞ n=1 m(An ) = 1 − 3ε < 1. There exists an open set A󸀠n such that A󸀠n ⊃ An and m(A󸀠n \ An ) < ε/2n . Since Sn is weakly compact, there exists k such that A󸀠 = ⋃kn=1 A󸀠n is an open covering of Sn . Thus k

1 = m(A󸀠 ∪ A󸀠 c ) = m(A󸀠 ) + m(A󸀠 c ) ≤ ∑ m(An ) + ε + ε ≤ 1 − ε < 1, n=1

which is a contradiction. Hence we have ∑∞ n=1 m(An ) = 1. This implies that m is completely additive, and the existence of the extension of m follows from Hopf’s extension theorem.

64 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes Using the two lemmas above, we can complete the proof of Theorem 1.72. By (2) of (1.4.1), C(⋅) is continuous for every ‖ ⋅ ‖p . Fix a p. By (3) of (1.4.1) for every ε > 0 there exists Bγ ⊂ Ep such that |C(ξ ) − 1| ≤

ε , 2β2

ξ ∈ Bγ ,

(1.4.7)

where β = 1/(1 − e−1/2 ). There exists n > p such that the injection ιnp : En → Ep is a Hilbert–Schmidt operator. Also, there exists a neighborhood V of 0 in En such that ιnp V ⊂ Bγ . Take Bt ⊂ En∗ ⊂ h∗ , where 2β‖ιnp ‖HS

t=

√εγ 2

.

Let F ⊂ En be a finite dimensional subspace and A ⊂ h∗ \ Bt with A ∈ AF ⊂ A. Note that πF (Bt )∩πF (A) = 0. Since ιnp is a Hilbert–Schmidt operator, there exists a complete orthonormal system {em }∞ m=1 ⊂ En such that ιnp em = λm em . Since dim F = n, y ∈ F ∩ V can be expressed as y = z1 e1 + ⋅ ⋅ ⋅ + zn en with some e1 , . . . , en and z1 , . . . , zn ∈ ℝ. For y = z1 e1 + ⋅ ⋅ ⋅ + zn en ∈ F ∩ V, ιnp y ∈ Bγ implies that ‖ιnp y‖2p = z12 λ12 + ⋅ ⋅ ⋅ + zn2 λn2 ≤ γ 2 . We identify ιnp (F ∪ V) with a subspace of ℝn . Set D = ιnp (F ∩ V) = {z ∈ ℝn | ∑ni=1 λi2 zi2 ≤ γ 2 } ⊂ Ep . The restriction of C(ξ ) to ξ ∈ F, CF (ξ ), satisfies |CF (ξ ) − 1| ≤ ε/(2β2 ) for ξ ∈ D by (1.4.7). By Lemma 1.73, we estimate as mF ([πF (Bt )]c ) ≤ β2 (

2 2 ε 2 n 2 ε 2β ‖ιnp ‖HS + + λ ) ≤ = ε. ∑ i 2 2β2 γ 2 t 2 i=1 γ2 t 2

Hence m(A) = m̃ F (A) = mF (πF (A)) ≤ mF ([πF (Bt )]c ) < ε. By a limiting argument, we conclude that m(A) < ε for every A ⊂ h∗ such that A ∩ Bt = 0. Hence there exists an extended measure μ ⊂ m on (h∗ , A) by Lemma 1.74. Thus we constructed the probability space (h∗ , A, μ). Step 4: By Step 3 we can write (1.4.4) as C(ξ ) = ∫ ei⟨⟨x,ξ ⟩⟩ dμ(x) h∗

for ξ ∈ E . Remark 1.75. Let h be a Hilbert space and C(ξ ) = exp(−(1/4)‖ξ ‖2 ) for ξ ∈ h. By Minlos’ theorem C(ξ ) satisfies conditions (1.4.2). Let {ξn }∞ n=1 be a complete orthonormal system of h. If a measure μ on h satisfying (1.4.3) exists, then 2

∫ eiz(x,ξn ) dμ(x) = e−(1/4)z . h

(1.4.8)

1.4 Existence of Q-spaces |

65

Since (x, ξn ) → 0 as n → ∞, the left-hand side of (1.4.8) converges to 1, in contradiction with the right-hand side. Moreover, from C(∑nk=1 zk ξk ) = exp (−(1/4) ∑nk=1 zk2 ) and the law of large numbers, it would follow that 1 n ∑ ⟨⟨x, ξk ⟩⟩2 = 1 n→∞ n k=1 lim

which is also a contradiction. We further discuss Gaussian measures on Hilbert spaces in Section 1.6. Let Sreal (ℝd ) denote the set of real-valued, rapidly decreasing and infinitely differentiable functions on ℝd . It is known that Sreal (ℝd ) is a nuclear space. Corollary 1.76. Suppose that Sreal (ℝd ) is dense in a real Hilbert space ℋ. Then there exists a family of Gaussian random variables (ϕ(f ), f ∈ ℋ) indexed by ℋ on a probability 󸀠 space (Sreal (ℝd ), ℬ, μ). Proof. Write C(ξ ) = exp(−(1/4)‖ξ ‖2 ) for ξ ∈ Sreal (ℝd ). Since C(ξ ) = ∫ eiϕ(f ) μ(dϕ) Q

by Theorem 1.68, C satisfies (1.4.2). Thus Minlos’ theorem implies that a probability 󸀠 space (Sreal (ℝd ), ℬ, μ) exists such that C(ξ ) =

∫

ei⟨⟨ϕ,ξ ⟩⟩ dμ(ϕ).

󸀠 (ℝd ) Sreal

󸀠 Define ϕ(ξ ) = ⟨⟨ϕ, ξ ⟩⟩ and ϕ ∈ Sreal (ℝd ) and ξ ∈ Sreal (ℝd ). We have

∫

ϕ(ξ )dμ = 0,

󸀠 (ℝd ) Sreal

∫ 󸀠 (ℝd ) Sreal

1 ϕ(ξ )ϕ(η)dμ = (ξ , η), 2

ξ , η ∈ Sreal (ℝd ).

Since ℬ is the minimal σ-field generated by cylinder sets, ℬ is the minimal σ-field generated by {ϕ(ξ ), ξ ∈ Sreal (ℝd )}. For f ∈ ℋ, there is a sequence (ξn )n∈ℕ ⊂ Sreal (ℝd ) such that ξn → f as n → ∞ in ℋ. Thus ∫ 󸀠 (ℝd ) Sreal

1 |ϕ(ξn ) − ϕ(ξm )|2 dμ = ‖ξn − ξm ‖2ℋ → 0 2

󸀠 as n, m → ∞. Hence (ϕ(ξn ))n∈ℕ is a Cauchy sequence in L2 (Sreal (ℝd )). Define ϕ(f ) = s-limn→∞ ϕ(ξn ). Then {ϕ(f ) | f ∈ ℋ} is a family of Gaussian random variables on 󸀠 (Sreal (ℝd ), ℬ, μ) indexed by ℋ.

66 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes Example 1.77. Suppose that B(f , g) is a positive semi-definite quadratic form on Sreal (ℝd ). Denote the completion of Sreal (ℝd ) with respect to the norm ‖ ⋅ ‖ = √B(⋅, ⋅) by h. Thus h is the real Hilbert space with inner product B(⋅, ⋅). Hence by Theorem 1.68 there exists a probability space (Q , Σ, μ) with a Gaussian measure μ such that ∫Q eiϕ(f ) μ(dϕ) = e−B(f ,f )/4 . From this we see that e−B(f ,f )/4 satisfies (1.4.2). By 󸀠 Minlos’ theorem we can construct a measure μ on Sreal (ℝd ) such that

∫

ei⟨⟨f ,x⟩⟩ dμ(x) = e−B(f ,f )/4 .

󸀠 (ℝd ) Sreal

1.5 Functional integral representation of the Euclidean quantum field 1.5.1 Basic results in Euclidean quantum field theory In this section we define the state space of a single boson and study Euclidean quantum fields. Let S 󸀠 (ℝd ) be the space of tempered distributions on ℝd . For s ∈ ℝ consider the inhomogeneous Sobolev space H s (ℝd ) i. e., H s (ℝd ) = {u ∈ S 󸀠 (ℝd ) | û ∈ L1loc (ℝd ), (1 + |k|2 )s/2 û ∈ L2 (ℝd )}. Here û describes the Fourier transform of u on S 󸀠 (ℝd ). In particular, H 0 (ℝd ) = L2 (ℝd ) and H s (ℝd ) ⊂ L2 (ℝd ) for every s > 0. The scalar product on H s (ℝd ) is defined by (f , g)H s (ℝd ) = ((1 + |k|2 )s/2 f ̂, (1 + |k|2 )s/2 g)̂ L2 (ℝd ) . Hence H s (ℝd ) is a Hilbert space equipped with the scalar product defined above. Furthermore, we can regard H −s (ℝd ) as the dual of H s (ℝd ), for every s ∈ ℝ. We can also define homogeneous Sobolev spaces. For s ∈ ℝ, the homogeneous Sobolev space Ḣ s (ℝd ) is defined by H s (ℝd ) with (1 + |k|2 )s/2 replaced by |k|s , i. e., Ḣ s (ℝd ) = {u ∈ S 󸀠 (ℝd ) | û ∈ L1loc (ℝd ), |k|s û ∈ L2 (ℝd )}. The scalar product on Ḣ s (ℝd ) is given by (f , g)H s (ℝd ) = (|k|s f ̂, |k|s g)̂ L2 (ℝd ) . It is known that Ḣ s (ℝd ) is a Hilbert space if and only if s < d/2. Let S0 (ℝd ) be the set of u ∈ S (ℝd ) such that the Fourier transform û vanishes near k = 0. It is also known that S0 (ℝd ) is dense in Ḣ s (ℝd ) for s < d/2. Moreover, Ḣ −s (ℝd ) can be regarded as the dual of Ḣ s (ℝd ) for |s| < d/2.

1.5 Functional integral representation of the Euclidean quantum field | 67

Example 1.78. (1) Let δx be the delta distribution with mass at x = 0 in d-dimensions. Then δ̂ x =

1 (2π)d/2

and δx ∈ H ε (ℝd ) for ε < −d/2. (2) Let f ∈ Ḣ −1/2 (ℝd ) and δx be the delta distribution with mass at x = 0 in onedimension. Then δx ⊗ f ∈ Ḣ −1 (ℝd+1 ). We can actually see that 1/2 ̂ δ̂ x ⊗ f (k0 , k) = f (k)/(2π)

and 󵄩󵄩 󵄩󵄩2 |f ̂(k)|2 |f ̂(k)|2 󵄩󵄩 δ̂ 󵄩󵄩 x ⊗f 󵄩󵄩 󵄩󵄩 dk < ∞. dk dk = π = ∫ ∫ 0 2 2 󵄩󵄩 (k + |k|2 )1/2 󵄩󵄩 2 d+1 |k| k0 + |k|2 󵄩 0 󵄩L (ℝ ) d d+1 ℝ

ℝ

In view of applications to quantum field theory, we make some modifications to these spaces. Let ων (k) = √|k|2 + ν2 , where ν ≥ 0. We define Hνs (ℝd ) by H s (ℝd ) with (1 + |k|2 )1/2 replaced by ων . Hence H s (ℝd ) and Hνs (ℝd ) are equivalent for ν > 0, and Hνs (ℝd ) = Ḣ s (ℝd ) for ν = 0. We write H s (ℝd ),

d

ℋs (ℝ ) = { ̇ νs d H (ℝ ),

ν > 0, ν = 0.

Two specific Hilbert spaces will play a special role, as singled out next. Definition 1.79 (ℋM , ℋ̂ M , ℋE and ℋ̂ E ). (1) We write ℋM = ℋ−1/2 (ℝd ) and ℋE = ℋ−1 (ℝd+1 ). (2) We define the Fourier transform (in the sense of tempered distributions) of ℋM and ℋE by ℋ̂ M and ℋ̂ E , respectively. 2 d+1 ̂ ̂ ν , G/ω We note that F,̂ Ĝ ∈ ℋ̂ E if and only if F/ω ) and define ν ∈ L (ℝ

̂ ̂ ν , G/ω (F,̂ G)̂ ℋ̂ = 2(F/ω ν )L2 (ℝd+1 ) . E

(1.5.1)

Here the coefficient 2 is added for a normalization which will be used in Lemma 1.81 below. We also see that f ̂, ĝ ∈ ℋ̂ M if and only if f ̂/√ων , g/̂ √ων ∈ L2 (ℝd ) and ̂ (f ̂, g)̂ ℋ̂ = (f ̂/√ων , g/√ω ν )L2 (ℝd ) . M

(1.5.2)

For notational simplicity, we write (F,̂ G)̂ ℋ̂ = (F,̂ G)̂ −1 = (F, G)−1 , E

(f ̂, g)̂ ℋ̂ = (f ̂, g)̂ −1/2 = (f , g)−1/2 . M

Although ℋM , ℋ̂ M , ℋE and ℋ̂ E depend on the space dimension and ν ≥ 0, we do not indicate this explicitly. We also define two specific real Hilbert spaces below.

68 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes Definition 1.80 (M and E ). We define (1) M = {f ∈ ℋM | f is real-valued}, (2) E = {f ∈ ℋE | f is real-valued}. Both M and E are Hilbert spaces over ℝ. Note that Mℂ ≅ ℋM and Eℂ ≅ ℋE . In summary, we have Mℂ ≅ ℋM ≅ ℋ̂ M and Eℂ ≅ ℋE ≅ ℋ̂ E . Let (ϕ(f ), f ∈ M ) be a family of Gaussian random variables indexed by f ∈ M on a probability space (Q , Σ, μ) and (ϕE (F), F ∈ E ) on (QE , ΣE , μE ). The family of Gaussian random variables (ϕE (F), F ∈ E ) is called a Euclidean field. One possible choice of Q 󸀠 󸀠 and QE is Sreal (ℝd ) and Sreal (ℝd+1 ), respectively, however, we do not fix any particular space. Note that ϕ(f ) and ϕE (F) are Gaussian random variables with mean zero and covariance 1 ∫ ϕ(f )ϕ(g)dμ = (f , g)−1/2 , 2

1 ∫ ϕE (F)ϕE (G)dμE = (F, G)−1 . 2

Q

QE

Under the identification Mℂ ≅ ℋ̂ M we have L2 (Q ) ≅ ℱb (ℋ̂ M )

(1.5.3)

by the Wiener–Itô–Segal isomorphism θW : ℱb (ℋ̂ M ) → L2 (Q ), as well as −1 θW ϕ(f )θW =

1 ∗ ̂ ̃ (a (f ) + aM (f ̂)). √2 M

Here a∗M and aM denote the annihilation operator and the creation operator in ℱb (ℋ̂ M ), which satisfy the canonical commutation relations ̂ = (f ̄̂, g)̂ −1/2 . [aM (f ̂), a∗M (g)]

̂ = 0 = [a∗M (f ̂), a∗M (g)], ̂ [aM (f ̂), aM (g)]

(1.5.4)

We introduce a family of transformations It from L2 (Q ) to L2 (QE ) through the second quantization of a specific transformation τt from M to E . Define τt : M → E ,

τt : f 󳨃→ δt ⊗ f .

Here δt (x) = δ(x − t) is the delta function with mass at t. Note that for f ∈ M , ̂ ̂ δt ⊗ f (k, k0 ) = δ̂ t ⊗ f (−k, −k0 ) = δt ⊗ f (k, k0 ). Thus δt ⊗ f = δt ⊗ f , which implies that τt preserves realness. Define ω̂ ν = ων (−i∇) = √−Δ + ν2 . Lemma 1.81. It follows that τt∗ τs = e−|t−s|ων , ̂

s, t ∈ ℝ.

In particular, τt is an isometry between M and E for every t ∈ ℝ.

1.5 Functional integral representation of the Euclidean quantum field | 69

Proof. We see that (δt ⊗ f , δs ⊗ g)−1 =

1 1 ̂ dk ∫ f ̂(k)g(k)dk ∫ e−i(s−t)k0 π ων (k)2 + |k0 |2 0 ℝd

ℝ

̂ = ∫ f ̂(k)g(k) ℝd

e

−|s−t|ων (k)

ων (k)

dk = (f , e−|s−t|ων g)−1/2 . ̂

(1.5.5)

Definition 1.82 (Second quantization of contraction operators). (1) Let T ∈ C (M → M ). Define Γ(T) ∈ C (L2 (Q ) → L2 (Q )) by Γ(T):ϕ(f1 ) ⋅ ⋅ ⋅ ϕ(fn ): = :ϕ(Tf1 ) ⋅ ⋅ ⋅ ϕ(Tfn ):

(1.5.6)

with Γ(T)1M = 1M , where 1M denotes the identity function in L2 (Q ). (2) Let T ∈ C (E → E ). Define ΓE (T) ∈ C (L2 (QE ) → L2 (QE )) by (1.5.6) with ϕ replaced by ϕE . (3) Let T ∈ C (M → E ). Define ΓInt (T) ∈ C (L2 (Q ) → L2 (QE )) by ΓInt (T):ϕ(f1 ) ⋅ ⋅ ⋅ ϕ(fn ): = :ϕE (Tf1 ) ⋅ ⋅ ⋅ ϕE (Tfn ): with ΓInt (T)1M = 1E , where 1E denotes the identity function in L2 (QE ). Definition 1.83. Let It = ΓInt (τt ) : L2 (Q ) → L2 (QE ), t ∈ ℝ, be a family of isometries, i. e., It 1M = 1E ,

It :ϕ(f1 ) ⋅ ⋅ ⋅ ϕ(fn ): = :ϕE (δt ⊗ f1 ) ⋅ ⋅ ⋅ ϕE (δt ⊗ fn ):.

Definition 1.84 (Free field Hamiltonian). The self-adjoint operator Ĥ f = dΓ(ω̂ ν ) is called the free field Hamiltonian in L2 (Q ). Note the following relationship between Hf on ℱb (ℋ̂ M ) and Ĥ f . We have −1 Ĥ f = θW Hf θW ,

where θW : ℱb (ℋ̂ M ) → L2 (Q ) is the Wiener–Itô–Segal isomorphism. From the identity τs∗ τt = e−|t−s|ω̂ ν , it follows that I∗t Is = e−|t−s|Hf , ̂

s, t ∈ ℝ.

From equality (1.5.7) we obtain the following representation.

(1.5.7)

70 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes Proposition 1.85 (Functional integral representation for free field Hamiltonian). Let F, G ∈ L2 (Q ) and t ≥ 0. Then (F, e−t Hf G)L2 (Q) = ∫ F0 (ϕ)Gt (ϕ)dμE (ϕ), ̂

(1.5.8)

QE

where F0 = I0 F and Gt = It G. Proof. Noting the factorization property (1.5.7), we see directly that (F, e−t Hf G)L2 (Q) = (F, I∗0 It G)L2 (Q) = (I0 F, It G)L2 (QE ) . ̂

Hence the proposition follows. In order to extend equality (1.5.8) to Hamiltonians of the form Ĥ f + perturbation, we will use the Markov property of projections Et = It I∗t ,

t ∈ ℝ,

(1.5.9)

which will be proven below. The following argument shows why the Markov property is useful. Let Ĥ = Ĥ f + HI with interaction term HI . By the Trotter product formula we have e−t H = s-lim(e−(t/n)Hf e−(t/n)HI )n . ̂

̂

n→∞

(1.5.10)

Substituting e−|t−s|Hf = I∗t Is into (1.5.10), we have ̂

n

e−t H = s-lim I∗0 (∏ Itj/n e−(tj/n)HI I∗tj/n ) It , ̂

n→∞

j=1

where ∏nj=1 Tj = T1 T2 ⋅ ⋅ ⋅ Tn . Thus we obtain n

e−t H = s-lim I∗0 ( ∏ Etj/n e−(tj/n)HI (tj/n) Etj/n )It ̂

n→∞

j=1

(1.5.11)

by the definition of the projection Es . Here HI (t/n) denotes an operator acting on L2 (QE ). Applying the Markov property of Es implies that all Es in equation (1.5.11) can be disregarded, thus the net result is n

(F, e−t H G) = lim (F0 , ( ∏ e−(tj/n)HI (tj/n) )Gt ) ̂

n→∞

j=1

1.5 Functional integral representation of the Euclidean quantum field | 71

∞

= lim (F0 , e− ∑j=1 (tj/n)HI (tj/n) Gt ) n→∞

t

= ∫ F0 Gt e− ∫0 HI (s)ds dμE . QE

We investigate the Markov property of Et and establish the functional integral representation of (F, e−t H G) in the next section. We finally consider estimates of the operator I∗a eϕE (f ) Ib in L2 (Q ). Formally, ̂

t

(F, e−t H G) = ∫ F0 Gt e− ∫0 HI (s)ds dμE ̂

QE

and the right-hand side above can be rewritten as t

t

∫ F0 Gt e− ∫0 HI (s)ds dμE = (F, I∗0 e− ∫0 HI (s)ds It G). QE t

We are interested in I∗0 e− ∫0 HI (s)ds It as an operator in L2 (Q ), in particular, operators of the form Ia eϕE (f ) Ib are of special interest. As it will be seen below, eϕE (f ) is an unbounded operator but Ia eϕE (f ) Ib is bounded when a ≠ b. In the massive case, when ω(k) = √|k|2 + ν2 and ν > 0, it can be seen that eϕE (f ) ∈ ⋂p≥1 Lp (QE ) and Ia eϕE (f ) Ib becomes a bounded operator by using the so-called hypercontractivity of e−tHf . This will be discussed in Theorem 1.87 below. In this section we discuss the boundedness of I∗a eϕE (f ) Ib also for the massless case. First we consider the intertwining properties of It and τt . We can identify ϕE (f ) ̃ ̃ with 1 (a∗ (f ̂) + a (f ̂)), and ϕ(f ) with 1 (a∗ (f ̂) + a f ̂))), respectively. Under this cor√2

E

E

√2

M

M

respondence It can be identified as a map from ℱb (ℋ̂ M ) to ℱb (ℋ̂ E ), i. e., n

n

j=1

j=1

It ∏ a∗M (fĵ )Ωb = ∏ a∗E (τ̂t fj )ΩEb and I∗t from ℱb (ℋ̂ E ) to ℱb (ℋ̂ M ). Here ΩEb denotes the Fock vacuum in ℱb (ℋ̂ E ). Lemma 1.86 (Intertwining properties). On the finite particle subspace the relations It a∗M (f ̂) = a∗E (τ̂ t f )It ,

(1.5.12)

It aM (f ̂) = aE (τ̂ t f )It ,

(1.5.13)

∗ ∗ I∗t a∗E (f ̂) = a∗M (τ̂ t f )It , ∗ ∗ ̂ I∗t aE (ê t f ) = aM (τt f )It

(1.5.14) (1.5.15)

hold, where et = τt τt∗ . In particular, ∗ ̂ ∗ It a∗M (τ̂ t f ) = aE (et f )It ,

(1.5.16)

72 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes ∗ ̂ ̂ It aM (τ̂ t f ) = aE (et f )It = aE (f )It ,

(1.5.17)

I∗t a∗E (τ̂ tf ) ∗ It aE (τ̂ tf )

(1.5.18)

= =

a∗M (f ̂)I∗t , aM (f ̂)I∗t .

(1.5.19)

Proof. These intertwining properties can be checked by applying vectors of the form ∏nj=1 a∗M (fĵ )Ωb for (1.5.12), (1.5.13), (1.5.16) and (1.5.17), and ∏nj=1 a∗E (fĵ )ΩEb for (1.5.14), (1.5.15), (1.5.18) and (1.5.19). Note that n

n

n

n

∗ ̂ ∗ ̂ E ∗ ̂ E ̂ ̄ aE (ê t f )It ∏ aM (fj )Ωb = aE (et f ) ∏ aE (τt fj )Ωb = ∑(et f , τt fj ) ∏ aE (τt fi )Ωb j=1

j=1

n

j=1

n

i=j̸

n

E ∗ ̂ ̂ = ∑(f ̄, τt fj ) ∏ a∗E (τ̂ t fi )Ωb = aE (f )It ∏ aM (fj )Ωb . j=1

i=j̸

j=1

Then also the second equality of (1.5.17) follows. ∗ ̂ Theorem 1.87. Suppose that f ̂ ∈ ℋ̂ E and τ̂ t f /√ω ∈ ℋM for t = a, b with a ≠ b. Then

1 1 ∗ 2 ∗ 2 ̂ ‖I∗a eϕE (f ) Ib ‖ ≤ 2 exp ( ‖f ̂‖ℋ̂ + (1 ∨ )(‖τ̂ a f ‖ω + ‖τb f ‖ω )) . E 4 |a − b| Here x ∨ y = max{x, y} and ∗ ∗ ∗ √ ∗ √ ∗ ̂ ̂ ̂ ̂ ‖τ̂ a f ‖ω = ‖τa f ‖ℋ̂ + ‖τa f / ω‖ℋ̂ = ‖τa f / ω‖L2 (ℝd ) + ‖τa f /ω‖L2 (ℝd ) . M

M

̃ ♯ Proof. We identify ϕE (f ) with √12 (a∗E (f ̂)+aE (f ̂)), where aE denotes the annihilation and creation operators in ℱb (ℋ̂ E ). By the Baker–Campbell–Hausdorff formula we have 1

eϕE (f ) = e √2

a∗E (f ̂)

1

e √2

̃ aE (f ̂)

1

e4

‖f ̂‖2ℋ̂

E

.

We compute τb∗ by (τb f , G)−1 = ∫ ℝd

̄ ̂ f ̂(k) g(k) ∗ G) dk = (f ̂, ĝ τ̂ −1/2 , b √|k| √|k|

where ̂ g(k) =√

2 |k| ̂ , k)dk . eibk0 G(k ∫ 0 0 π k02 + |k|2 ℝ

∗ G = g.̂ The intertwining properties in Lemma 1.86 yield We have τ̂ b 1

I∗a eϕE (f ) Ib = e √2

∗ a∗M (τ̂ a f ) −|a−b|Hf

e

1

e √2

̃ ∗f ) aM (τ̂ b

1

e4

‖f ̂‖2ℋ̂

E

.

1.5 Functional integral representation of the Euclidean quantum field | 73

1

The operators e √2

|a−b| ∗ a∗M (τ̂ a f ) − 2 Hf

e

and e−

|a−b| Hf 2

1

e √2

̃ ∗f ) aM (τ̂ b

are bounded with

1 2 ∗ 2 ∗ √ ̂ ) (‖τ̂ a f ‖ℋ̂ M + ‖τa f / ω‖ℋ̂ M )) , |a − b| ̃ 1 |a−b| ∗f ) 1 a (τ̂ ∗ f ‖2 ∗ f /√ω‖2 )) . ‖e− 2 Hf e √2 M b ‖ ≤ √2 exp ((1 ∨ ) (‖τ̂ + ‖τ̂ b ℋ̂ M b ℋ̂ M |a − b| 1

‖e √2

|a−b| ∗ a∗M (τ̂ a f ) − 2 Hf

e

‖ ≤ √2 exp ((1 ∨

Hence we have 1

‖I∗a eϕE (f ) Ib ‖ ≤ ‖e √2

|a−b| ∗ a∗M (τ̂ a f ) − 2 Hf

e

‖‖e−

|a−b| Hf 2

1

e √2

̃ ∗f ) aM (τ̂ b

1

‖e 4

‖f ̂‖2ℋ̂

E

1 1 ∗ 2 ∗ 2 ̂ ) (‖τ̂ ≤ 2 exp ( ‖f ̂‖2ℋ̂ + (1 ∨ a f ‖ω + ‖τb f ‖ω )) . E 4 |a − b|

We single out some special cases of Theorem 1.87 which will be used below in the study of the Nelson Hamiltonian. Corollary 1.88. Let T ≥ 0, and X : ℝ → ℝd be a measurable function. Let f ∈ ℋM , i. e., f ̂/√ω ∈ L2 (ℝd ). Define T

ΦX = ϕE ( ∫(τs f )(⋅ − X(s))ds). 0

The following hold: (1) If f ̂/ω, f ̂/√ω3 ∈ L2 (ℝd ), then T ‖I∗0 eΦX IT ‖ ≤ 2 exp ( ‖f ̂/ω‖2 + 2(T ∨ 1)(‖f ̂/ω‖2 + ‖f ̂/√ω3 ‖2 )) . 2

(1.5.20)

(2) If f ̂/ω ∈ L2 (ℝd ), then T ‖I∗0 eΦX IT ‖ ≤ 2 exp ( ‖f ̂/ω‖2 + 2T(T ∨ 1)(‖f ̂/√ω‖2 + ‖f ̂/ω‖2 )) . 2

(1.5.21)

Proof. We see that T

T

T

T

0

0

0

0

󵄩󵄩 󵄩2 󵄩󵄩 ∫ τs fds󵄩󵄩󵄩 = ∫ ds ∫(e−|s−t|ω f ̂, f ̂) ̂ dt = 2 ∫ ((1 − e−sω )f ̂/√ω, f ̂/√ω) ̂ ds ≤ 2T‖f ̂/ω‖2 . 󵄩󵄩 󵄩󵄩ℋ̂ E ℋM ℋM We also see that T

󵄩󵄩 ∗ 󵄩2 2 ̂ 󵄩󵄩τ ∫ τs fds󵄩󵄩󵄩 󵄩󵄩 T 󵄩󵄩ℋM ≤ T‖f /ω‖ , 0

T

󵄩󵄩 ∗ 󵄩2 2 ̂ 󵄩󵄩τ ∫ τs fds󵄩󵄩󵄩 󵄩󵄩 0 󵄩󵄩ℋM ≤ T‖f /ω‖ , 0

T

󵄩󵄩 ∗ 󵄩2 ̂ √ 3 2 󵄩󵄩τ ∫ τs fds/√ω󵄩󵄩󵄩 󵄩󵄩 T 󵄩󵄩ℋM ≤ T‖f / ω ‖ , 0

T

󵄩󵄩 ∗ 󵄩2 ̂ √ 3 2 󵄩󵄩τ ∫ τs fds/√ω󵄩󵄩󵄩 󵄩󵄩 0 󵄩󵄩ℋM ≤ T‖f / ω ‖ . 0

74 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes Hence (1) follows by Theorem 1.87. For (2) we estimate as T

T

󵄩2 󵄩󵄩 ∗ 2 ̂ 2 󵄩󵄩τ ∫ τs fds󵄩󵄩󵄩 󵄩󵄩ℋM ≤ T ‖f /√ω‖ , 󵄩󵄩 T

󵄩󵄩 ∗ 󵄩2 2 ̂ 2 󵄩󵄩τ ∫ τs fds/√ω󵄩󵄩󵄩 󵄩󵄩 T 󵄩󵄩ℋM ≤ T ‖f /ω‖ , 0

0

T

T

󵄩2 󵄩󵄩 ∗ 2 ̂ 2 󵄩󵄩τ ∫ τs fds󵄩󵄩󵄩 󵄩󵄩ℋM ≤ T ‖f /√ω‖ , 󵄩󵄩 0

󵄩󵄩 ∗ 󵄩2 2 ̂ 2 󵄩󵄩τ ∫ τs fds/√ω󵄩󵄩󵄩 󵄩󵄩 0 󵄩󵄩ℋM ≤ T ‖f /ω‖ .

0

0

Corollary 1.89. Let ΦX be as in Corollary 1.88. Suppose that f ∈ ℋM , i. e., f ̂/√ω ∈ L2 (ℝd ), and f ̂/ω, f ̂/√ω3 ∈ L2 (ℝd ). Let 1

‖f ̂/ω‖2 + 2(‖f ̂/ω‖2 + ‖f ̂/√ω3 ‖2 ) }. 1 ̂ ‖f /ω‖2 + 2(‖f ̂/√ω‖2 + ‖f ̂/ω‖2 ) 2

E(f ̂) = max { 2

(1.5.22)

Then ‖I∗0 eΦX IT ‖ ≤ 2eTE(f ) , ̂

T ≥ 0.

(1.5.23)

Proof. We have T ‖I∗0 eΦX IT ‖ ≤ 2 exp ( ‖f ̂/ω‖2 + 2T(‖f ̂/ω‖2 + ‖f ̂/√ω3 ‖2 )) 2 for T ≥ 1, and T ‖I∗0 eΦX IT ‖ ≤ 2 exp ( ‖f ̂/ω‖2 + 2T(‖f ̂/√ω‖2 + ‖f ̂/ω‖2 )) 2 for T ≤ 1. By Corollary 1.88 above it can be seen that if f ̂/√ω, f ̂/ω ∈ L2 (ℝd ), then I∗0 eΦX IT is a bounded operator, and supT≥0 T12 log ‖I∗0 eΦX IT ‖ < ∞. Furthermore, the extra condition 1 log ‖I∗ eΦX I ‖ ≤ E(f ̂) < ∞. f ̂/√ω3 ∈ L2 (ℝd ) gives that sup T≥0 T

0

T

In Corollary 1.36 we introduced a parameter q > 1 and obtained an alternative ∗ bound on ea (f ) e−tHf . Similarly we can also modify Theorem 1.87 and Corollary 1.88.

∗ ̂ Theorem 1.90. Let q > 1. Suppose that f ̂ ∈ ℋ̂ E and τ̂ t f /√ω ∈ ℋM for t = a, b with a ≠ b. Then

‖I∗a eϕE (f ) Ib ‖ ≤ √

q q 1 1 ∗ 2 ∗ 2 ̂ exp ( ‖f ̂‖ℋ̂ + (1 ∨ )(‖τ̂ a f ‖ω + ‖τb f ‖ω )) . E q−1 4 2 |a − b|

Corollary 1.91. Let q > 1 and ΦX be as in Corollary 1.88. Suppose that f ∈ ℋM , i. e., f ̂/√ω ∈ L2 (ℝd ). Then the following hold:

1.5 Functional integral representation of the Euclidean quantum field | 75

(1) If f ̂/ω, f ̂/√ω3 ∈ L2 (ℝd ), then ‖I∗0 eΦX IT ‖ ≤ √

q T exp ( ‖f ̂/ω‖2 + q(T ∨ 1)(‖f ̂/ω‖2 + ‖f ̂/√ω3 ‖2 )) . q−1 2

(1.5.24)

In particular, let 1

‖f ̂/ω‖2 + q(‖f ̂/ω‖2 + ‖f ̂/√ω3 ‖2 ) }. 1 ̂ ‖f /ω‖2 + q(‖f ̂/√ω‖2 + ‖f ̂/ω‖2 ) 2

Eq (f ̂) = max { 2

(1.5.25)

Then it follows that ‖I∗0 eΦX IT ‖ ≤ √

q TEq (f ̂) e , q−1

T ≥ 0.

(1.5.26)

(2) If f ̂/ω ∈ L2 (ℝd ), then ‖I∗0 eΦX IT ‖ ≤ √

q T exp ( ‖f ̂/ω‖2 + qT(T ∨ 1)(‖f ̂/√ω‖2 + ‖f ̂/ω‖2 )) . q−1 2

(1.5.27)

1.5.2 Markov property of projections Let O ⊂ ℝ and define d

E (O ) = {f ∈ E | supp f ⊂ O × ℝ }.

We denote the projection E → E (O ) by eO . Let ΣO be the minimal σ-field generated by {ϕE (f ) | f ∈ E (O )}, and define 2

EO = {Φ ∈ L (QE ) | Φ is ΣO -measurable}.

Also, let et = τt τt∗ , t ∈ ℝ. The collection {et }t∈ℝ is a family of projections from E to Ran(τt ). Let Σt , t ∈ ℝ, be the minimal σ-field generated by {ϕE (f ) | f ∈ Ran(et )}, and define 2

Et = {Φ ∈ L (QE ) | Φis Σt -measurable},

t ∈ ℝ.

We will see below that F ∈ E[a,b] can be characterized by supp F ⊂ [a, b] × ℝd . Lemma 1.92. (1) E ({t}) = Ran(et ) and every f ∈ Ran(et ) can be expressed as f = δt ⊗ g for some g ∈ M . In particular, e{t} = et . ‖⋅‖−1

(2) E ([a, b]) = L. H. {f ∈ E | f ∈ Ran(et ), a ≤ t ≤ b}

holds.

76 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes 󸀠 Proof. (1) The key fact is that any f ∈ Sreal (ℝd+1 ) with support {t}×ℝd can be expressed as n

f (x0 , x) = ∑ δ(j) (x0 − t) ⊗ gj (x), j=0

(x0 , x) ∈ ℝ × ℝd ,

󸀠 where gj ∈ Sreal (ℝd ) and δ(j) is the jth derivative of the delta distribution δ. We have

f ̂(k0 , k) =

1 n ∑ (ik )j e−ik0 t ĝj (k), √2π j=0 0

(k0 , k) ∈ ℝ × ℝd ,

and hence n = 0 and g0 ∈ M is necessary and sufficient for f ∈ E . Then f ∈ E ({t}) if and only if there exists g ∈ M such that f = δt ⊗ g, which gives that f ∈ Ran(et ). In particular, E ({t}) ⊂ Ran(et ). The inclusion Ran(et ) ⊂ E ({t}) is trivial. (2) Let A1 = E ((a, b)) and take f ∈ E ([a, b]). For λ ∈ (1, ∞), we have fλ (x0 , x) = f (λx0 + (1 − λ)(a + b)/2, x) ∈ A1 , and ‖fλ − f ‖−1 → 0 as λ → 1. Hence A1 is dense in E ([a, b]). Let A2 = C ∞ (ℝd ) ∩ A1 , and define fε = ρε ∗ f , where ρε (x) = ρ(x/ε)/εd , ρ ∈ C0∞ (ℝd ), ρ(x) ≥ 0, ∫ ρ(x)dx = 1 and supp ρ ⊂ {x ∈ ℝd | |x| ≤ 1}. Then fε ∈ A2 for sufficiently small ε and ‖fε − f ‖−1 = 0 as ε → 0. Hence A2 is dense in E ([a, b]). Next we show that A3 = C0∞ (ℝd+1 ) ∩ A1 is dense in E ([a, b]). For ν > 0 we define 1, χn (x) = { 0,

|x| ≤ n,

|x| > n + 1,

χn ∈ C0∞ (ℝd ) and 0 ≤ χn ≤ 1.

For f ∈ A2 we have fn = χn f ∈ A2 and ‖fn − f ‖−1 = 0 as n → ∞. Since A2 is dense in A3 , A3 is dense in E ([a, b]). If ν = 0, let A4 = {f ∈ C0∞ (ℝd+1 ) ∩ A1 | supp f ̂ ⊂ ℝd+1 \ {0}}. It is similar to show that A4 is dense in E ([a, b]). Finally, we show that any f ∈ A3 can be represented as a limit of finite linear sums of vectors in Ran(et ), a < t < b. Take f ∈ A3 and define j j 1 δ (x0 − ) f ( , x) . n n n j=−∞ ∞

fn (x0 , x) = ∑ We have

j e−ijk0 /n 1 g ( , k) , √2π n n j=−∞ ∞

fn̂ (k0 , k) = ∑

where g(X, k) = (2π)−d/2 ∫ℝd f (X, x)e−ikx dx. Note that |fn̂ (k0 , k)| ≤ cN /(|k|2 + 1)N for some N and fn̂ (k0 , k) → f ̂(k0 , k) as n → ∞ for every k. Hence ‖fn − f ‖−1 → 0 as n → ∞ by the Lebesgue dominated convergence theorem. This completes the proof of (2) with ν ≠ 0. In the case of ν = 0, we use A3 replaced by A4 .

1.5 Functional integral representation of the Euclidean quantum field | 77

Lemma 1.93. Let a ≤ b ≤ t ≤ c ≤ d. Then: (1) ea eb ec = ea ec ; (2) e[a,b] et e[c,d] = e[a,b] e[c,d] ; (3) ec eb ea = ec ea ; (4) e[c,d] et e[a,b] = e[c,d] e[a,b] . Proof. Since ea eb ec = τa τa∗ τb τb∗ τc τc∗ = τa e−|a−b|ων e−|b−c|ων τc∗ = τa e−|a−c|ων τc∗ = τa τa∗ τc τc∗ = ea ec , ̂

̂

̂

(1) readily follows. Take f , g ∈ E . By Lemma 1.92 (2), it is clear that Nn

e[c,d] f = s-lim ∑ fnα , n→∞

α=1 Nm

e[a,b] g = s-lim ∑ fmβ , m→∞

β=1

fnα ∈ Ran(etn ), α

tnα ∈ [c, d],

fmβ ∈ Ran(etm ), β

tmβ ∈ [a, b].

Hence by (1), we have (e[a,b] et e[c,d] f , g) = lim

m,n→∞

Nn ,Mm

∑ (et fnα , gmα ) = lim

Nn ,Mm

m,n→∞

α,β=1

∑ (fnα , gmα ) = (e[a,b] e[c,d] f , g).

α,β=1

This gives (2). Statements (3) and (4) are similarly proven. Define Et = It I∗t = ΓE (et ),

t∈ℝ

and

EO = ΓE (eO ),

O ⊂ ℝ.

Notice that E{t} = Et for t ∈ ℝ by Lemma 1.92. The following relationship holds between the projections E[a,b] and the set of Σ[a,b] -measurable functions. Proposition 1.94. (1) Ran(E[a,b] ) = E[a,b] ; (2) E[a,b] Et E[c,d] = E[a,b] E[c,d] and E[c,d] Et E[a,b] = E[c,d] E[a,b] hold for a, b, c, d such that a ≤ b ≤ t ≤ c ≤ d; (3) E[a,b] E[c,d] = E[c,d] E[a,b] = E[a,b] if [a, b] ⊂ [c, d]. Proof. Let {gn }∞ n=1 be a complete orthonormal system of E ([a, b]). We have Ran(E[a,b] ) = L. H.{:ϕE (g1 )n1 ⋅ ⋅ ⋅ ϕE (gk )nk : | gj ∈ {gn }∞ n=1 , nj ≥ 0, j = 1, . . . , k, k ≥ 0}, which implies that Ran(E[a,b] ) ⊂ E[a,b] . Noticing that :exp(iϕE (f )): ∈ Ran(E[a,b] ), we obtain exp(iϕE (f )) ∈ Ran(E[a,b] ). Thus F(ϕE (f1 ), . . . , ϕE (fn )) ∈ Ran(E[a,b] ), for F ∈ S (ℝn ) and f1 , . . . , fn ∈ E ([a, b]). Since {F(ϕE (f1 ) ⋅ ⋅ ⋅ ϕE (fn )) | F ∈ S (ℝn ), f1 , . . . , fn ∈ E ([a, b])}

78 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes is dense in E[a,b] , we conclude that E[a,b] ⊂ Ran(E[a,b] ). Hence (1) is obtained. (2) follows from (2) and (4) of Lemma 1.93. Since a Σ[a,b] -measurable function is also Σ[c,d] -measurable, (3) also follows. Remark 1.95. Part (1) of Lemma 1.94 implies that E[a,b] is a projection to the set of Σ[a,b] measurable functions in L2 (QE ). This is a conditional expectation, and we can write E[a,b] F as 𝔼[F|Σ[a,b] ]. We also denote Et F by 𝔼[F|Σt ]. Part (2) of Lemma 1.94 is called the Markov property of the family Es , s ∈ ℝ. Proposition 1.96 (Markov property). Let F ∈ Es+t . Then 𝔼[F|Σ(−∞,s] ] = 𝔼[F|Σs ]. Proof. It follows that 𝔼[F|Σ(−∞,s] ] = E(−∞,s] Es+t F = E(−∞,s] Es Es+t F from the Markov property. Since E(−∞,s] Es = E(−∞,s] E{s} = E{s} = Es by Proposition 1.94, we see that E(−∞,s] Es Es+t F = Es Es+t F = Es F = 𝔼[F|Σs ]. 1.5.3 Feynman–Kac–Nelson formula Consider the polynomial P(X) = a2n X 2n + a2n−1 X 2n−1 + ⋅ ⋅ ⋅ + a1 X + a0 with a2n > 0. For f ∈ M , define HI = :P(ϕ(f )):. Note that HI is a polynomial of ϕ(f ) of degree 2n and there exists M > −∞ such that HI > M. Define the self-adjoint operator HP = Ĥ f +̇ HI

(1.5.28)

acting in L2 (Q ). As was already mentioned in this section, a key ingredient in constructing a functional integral representation of (F, e−tHP G) is the Trotter product formula and the Markov property of the family Es , s ∈ ℝ. Before going to state the functional integral representation of e−tHP , we give a corollary to Lemma 1.94. Corollary 1.97. Let F ∈ E[a,b] , G ∈ E[c,d] with a ≤ b ≤ t ≤ c ≤ d. Then it follows that (F, Et G) = (F, G). Proof. By Proposition 1.94, it is immediate to see that (F, Et G) = (E[a,b] F, Et E[c,d] G) = (E[a,b] F, E[c,d] G) = (F, G). Now we can prove the functional integral representation of HP given by (1.5.28).

1.5 Functional integral representation of the Euclidean quantum field | 79

Theorem 1.98 (Feynman–Kac–Nelson formula). Let F, G ∈ L2 (Q ). Then t

(F, e

−tHP

G) = ∫ F0 Gt exp ( − ∫:P(ϕE (δs ⊗ f )):ds)dμE , 0

QE

where F0 = It F and Gt = It G. Proof. By the Trotter product formula and the fact e−|s−t|Hf = I∗t Is , we have (F, e−tHP G) = lim (F, (e−(t/n)Hf e−(t/n)HI )n G) n→∞

n

= lim (F0 , ∏(Iti/n e−(t/n)HI I∗ti/n )Gt ) n→∞

i=1 n

= lim (F0 , ∏(Eti/n Ri Eti/n )Gt ) , n→∞

i=1

(1.5.29)

where Ri = exp(−(t/n):P(ϕE (δit/n f )):) and we used that Is exp(−tHI )I∗s = Es exp(−t:P(ϕE (δs ⊗ f )):)Es

(1.5.30)

as operators. In fact, we can check that n

n

i=1

i=1

It ϕ(f ):∏ ϕ(fi ): = It (:ϕ(f ) ∏ ϕ(fi ): + n

n 1 n ∑(f , fj ):∏ ϕ(fi ):) 2 j=1 i=j̸

= :ϕE (δt ⊗ f ) ∏ ϕE (δt ⊗ fi ): + i=1

n

n 1 n ∑(δt ⊗ f , δt ⊗ fj ):∏ ϕ(δt ⊗ fi ): 2 j=1 i=j̸

= ϕE (δt ⊗ f )It :∏ ϕ(fi ):. i=1

Thus It ϕ(f )I∗t = ϕE (δt ⊗ f )Et = Et ϕE (δt ⊗ f )Et . Inductively, it follows that n

n

j=1

j=1

It (∏ ϕ(fj )) I∗t = Et (∏ ϕE (δt ⊗ fj )) Et . A limiting argument gives then (1.5.30). Notice that n

(F0 , ∏(Eti/n Ri Eti/n )Gt ) = ( ⏟⏟⏟F⏟⏟0⏟⏟ , Et/n R 1 Et/n (E2t/n R2 E2t/n ) ⋅ ⋅ ⋅ (Et Rn Et )Gt ) . ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ i=1

∈E{0}

(1.5.31)

∈E[t/n,t]

By Corollary 1.97, we can remove Et/n in (1.5.31). Furthermore, note that n

(F0 , ∏(Eti/n Ri Eti/n )Gt ) = ( R ⏟⏟⏟⏟⏟⏟⏟ 1 F0 , Et/n (E 2t/n R2 E2t/n ) ⋅ ⋅ ⋅ (Et Rn Et )Gt ) . ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ i=1

∈E[0,t/n]

∈E[2t/n,t]

(1.5.32)

80 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes Similarly, Et/n can be removed in (1.5.32), and since n

(F0 , ∏(Eti/n Ri Eti/n )Gt ) = ( R ⏟⏟⏟⏟⏟⏟⏟ 1 F0 , E2t/n R 2 E2t/n ⋅ ⋅ ⋅ (Et Rn Et )Gt ) , ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ i=1

∈E[0,t/n]

∈E[2t/n,t]

also E2t/n can be removed. Recursively, we can delete all Es in the rightmost side of (1.5.29). Thus (F, e−tHP G) = lim (F0 , R1 ⋅ ⋅ ⋅ Rn Gt ) n→∞

n

= lim (F0 , exp (− ∑(t/n):P(ϕE (δjt/n ⊗ f )):) Gt ) n→∞

j=1

t

= ∫ F0 Gt exp (− ∫:P(ϕE (δs ⊗ f )):ds) dμE . QE

0

1.5.4 van Hove Hamiltonian We consider the so-called van Hove Hamiltonian HvH , which is the simplest model in quantum field theory. It is given by HP by choosing P(x) = x, i. e., HvH = Ĥ f + ϕ(f ).

(1.5.33)

Alternatively, HvH can be also defined by Hf + Φ(f ̂) on ℱb (ℋ̂ M ), where Φ(f ̂) =

1 ∗ ̂ ̃ (a (f ) + aM (f ̂)), √2 M

f ̂ ∈ ℋ̂ M .

Here aM and a∗M are the annihilation operator and the creation operator in ℱb (ℋ̂ M ), respectively, and they satisfy canonical commutation relations (1.5.4). Suppose that f ̂/ω, f ̂/√ω ∈ L2 (ℝd ) and f is a real-valued function. Then HvH is self-adjoint on D(Ĥ f ) and bounded from below by the Kato–Rellich theorem. Corollary 1.99 (Functional integral representation of van Hove Hamiltonian). We have t

(Φ, e−tHvH Ψ)L2 (Q) = (I0 Φ, e−ϕE (∫0 δs ⊗fds) It Ψ)L2 (QE ) .

(1.5.34)

ε ε Proof. Let HvH = Ĥ f + yε (ϕ(f )), where yε (x) = x + εx2 for ε ≥ 0. HvH is self-adjoint on D(Ĥ f ) and essentially self-adjoint on any core of Ĥ f for sufficiently small 0 ≤ ε by ε the Kato–Rellich theorem. In particular, e−tHvH → e−tHvH strongly as ε ↓ 0. By the ε Feynman–Kac–Nelson formula it can be seen that the semigroup generated by HvH is represented as ε

t

t

(Φ, e−tHvH Ψ)L2 (Q) = (I0 Φ, e− ∫0 yε (ϕE (∫0 δs ⊗f ))ds It Ψ)L2 (QE ) . Taking ε ↓ 0 on both sides, the result follows.

(1.5.35)

1.5 Functional integral representation of the Euclidean quantum field | 81 t

In order to factorize e−ϕE (∫0 δs ⊗fds) in (1.5.34) in a suitable form, we recall that 1 ∗ ̂ ̃̂ (a (F) + aE (F)), F̂ ∈ ℋ̂ E , √2 E 1 ∗ ̂ ̃ ϕ(f ) ≅ (a (f ) + aM (f ̂)), f ̂ ∈ ℋ̂ M , √2 M

ϕE (F) ≅

(1.5.36) (1.5.37)

where a∗E (F)̂ and aE (F)̂ are the creation operator and the annihilation operator on boson Fock space ℱb (ℋ̂ E ), i. e., they satisfy that ̂ = (F,̄̂ G)̂ ̂ . ̂ a∗ (G)] [aE (F), E ℋ

̂ = 0 = [a∗ (F), ̂ ̂ aE (G)] ̂ a∗ (G)], [aE (F), E E

E

We use the identifications (1.5.36) and (1.5.37). By the Baker–Campbell–Hausdorff formula we see that t

e−ϕE (∫0 δs ⊗fds) = e

t t? 1 ̂ − √12 a∗E (∫0 δ̂ s ⊗f ds) − √2 aE (∫0 δs ⊗f ds)

e

t

1

2

e 4 ‖ ∫0 δs ⊗fds‖ℋE .

Recall that τs f = δs ⊗ f and τs∗ τt = e−|s−t|ω̂ . The intertwining properties ̃ ∗ ̂

I∗0 eaE (F) = eaM (τ0 F) I∗0 , ∗

̂

∗

̃̂

It eaM (τt F) = eaE (F) It

∗ ̂

are satisfied. Thus we have t

I∗0 e−ϕE (∫0 δs ⊗fds) It = e

t t ̃̂ − √12 a∗M (∫0 e−sω f ̂ds) −tHf − √12 aM (∫0 e−(t−s)ω fds)

e

e

1

e 2 Wt ,

(1.5.38)

where t

t

t

0

0

0

|f ̂(k)|2 1 1 󵄩󵄩 󵄩󵄩2 dk. Wt = 󵄩󵄩󵄩 ∫ δs ⊗ fds󵄩󵄩󵄩 = ∫ ds ∫ dr ∫ e−|s−r|ω(k) 󵄩ℋE 2 2󵄩 ω(k) ℝd

The following result gives an expression of the van Hove Hamiltonian in terms of exponentials of the creation and annihilation operators. Proposition 1.100. Denote U(f , t) = −

1 1 (1 − e−tω )f ̂, √2 ω

̃ , t) = − 1 1 (1 − e−tω )f ̃̂. U(f √2 ω

̃ , t) ∈ ℋ̂ M and it follows that Then U(f , t), U(f 1

(Φ, e−tHvH Ψ)L2 (Q) = (Φ, eaM (U(f ,t)) e−t Hf eaM (U(f ,t)) Ψ) ∗

̂

̃

L2 (Q)

e 2 Wt .

Proof. From (1.5.34) and (1.5.38) it follows that (Φ, e−tHvH Ψ)L2 (Q) = (Φ, eaM (S(f ,t)) e−t Hf eaM (S(f ,t)) Ψ) ∗

̂

̃

L2 (Q)

1

e 2 Wt ,

(1.5.39)

82 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes where

t

S(f , t) = −

t

̃̂ ̃ , t) = − 1 ∫ e−sω fds, S(f √2

1 ∫ e−sω f ̂ds, √2 0

0

̃ , t) = U(f ̃ , t). and S(f , t) = U(f , t), S(f We note that the operator eaM (U(f ,t)) e−t Hf eaM (U(f ,t)) given in (1.5.39) is bounded, ∗ ̂ ̃ since eaM (U(f ,t)) e−(t/2)Hf and e−(t/2)Hf eaM (U(f ,t)) are bounded for all t > 0. ∗

̂

̃

1.6 Infinite dimensional Ornstein–Uhlenbeck processes 1.6.1 Abstract theory of Gaussian measures on Hilbert spaces As was explained in Remark 1.75, it is not possible to construct a Gaussian measure on any Hilbert space 𝒦 such that the covariance form is defined by the scalar product on 𝒦. Nevertheless, if the covariance is given in the form (A−1 ⋅, A−1 ⋅)𝒦 with a suitable self-adjoint operator A having a Hilbert–Schmidt inverse, i. e., such that Tr(A−2 ) < ∞, then there exists a Gaussian measure on an enlarged Hilbert space. ∞ We define ℝ∞ by the infinite direct product ∏∞ n=1 ℝ, and define a topology on ℝ by the product topology. Let 𝒦 be a separable Hilbert space over ℝ, L+1 (𝒦) be the set of symmetric positive trace class operators on 𝒦, and consider Q ∈ L+1 (𝒦),

a ∈ 𝒦.

Let {en }∞ n=1 be a complete orthonormal system of 𝒦 such that Qen = λn en . Hence Tr Q = λ . Consider ∑∞ j=1 j ∞ ∞ 󵄨󵄨󵄨 2 ℓ2 = {x = (xk )∞ k=1 ∈ ℝ 󵄨󵄨󵄨 ∑ |xk | < ∞}. k=1

Define the map γ : 𝒦 → ℓ2 by γ(x) = (xk )∞ k=1 , where xk = (ek , x)𝒦 . Then γ is a unitary operator and we write γ(a) = (ak )∞ . Let μ k = Nak ,λk be a Gaussian measure on ℝ with k=1 mean ak and covariance λk . Also, let ∞

μ̄ = ∏ μk

on (ℝ∞ , ℬ(ℝ∞ )).

k=1

Due to the expressions n

n

n

n

∫ ∑ xj2 ∏ μj (dxj ) = ∑ λj + ∑ a2j ,

ℝn j=1 n

j=1

j=1

n

n

j=1

j=1

j=1

∫ ∑ xj hj ∏ μj (dxj ) = ∑ aj hj ,

ℝn j=1

1.6 Infinite dimensional Ornstein–Uhlenbeck processes | 83 n

n

n

n

j=1

i=1

j=1

j=1

∫ (∑(xj − aj )hj ) (∑(xi − ai )ki ) ∏ μj (dxj ) = ∑ λj hj kj , ℝn

n

n

n

n

1

2

∫ ei ∑i=1 xj hj ∏ μj (dxj ) = ei ∑i=1 aj hj − 2 ∑i=1 λj hj , j=1

ℝn

we find that ̄ = Tr Q + ‖a‖2𝒦 , ∫ ‖x‖2𝒦 μ(dx)

̄ = (a, h)𝒦 , ∫ (x, h)𝒦 μ(dx)

ℝ∞

ℝ∞

̄ = (Qh, k)𝒦 , ∫ (x − a, h)𝒦 (x − a, k)𝒦 μ(dx) ℝ∞ 1

1/2

̄ = ei(a,h)𝒦 − 2 ‖Q ∫ ei(x,h)𝒦 μ(dx)

h‖2𝒦

.

ℝ∞

Notice that ℓ2 ⊂ ℝ∞ . ̄ Lemma 1.101. Let Q ∈ L1+ (𝒦) with Ker Q = {0} and a ∈ 𝒦. Then ℓ2 is μ-measurable and μ(ℓ2 ) = 1. Proof. Define the projection Pn : ℝ∞ → ℝ∞ by Pn ((xk )∞ k=1 ) = (x1 , . . . , xn , 0, . . .). The map ∞ 2 ℝ ∋ x 󳨃→ fn (x) = ‖Pn x‖ℓ2 is continuous, and fn is measurable. Thus limn→∞ fn (x) = f (x) is also measurable, i. e., f (x) = ‖x‖2ℓ2 is a measurable function. This means that {x ∈ ℝ∞ | f (x) < ∞} is a measurable set. We also see that n

∫ f (x)dμ = lim ∫ fn (x)dμ = lim ∑ ∫ ‖xk ‖2 dμ n→∞

ℝ∞

n→∞

ℝ∞ n

k=1 ℝ∞

= lim ∑ (λk + a2k ) = Tr Q + ‖a‖2 < ∞. n→∞

k=1

This implies that μ({x ∈ ℝ∞ | f (x) < ∞}) = 1 and the lemma is proven. Theorem 1.102. Let Q ∈ L+1 (𝒦) and a ∈ 𝒦. Suppose that Ker Q = {0}. Then there exists a probability measure μ∞ on (𝒦, ℬ(𝒦)) such that ∫ ‖x‖2𝒦 μ∞ (dx) = Tr Q + ‖a‖2𝒦 ,

(1.6.1)

𝒦

∫(x, h)𝒦 μ∞ (dx) = (a, h)𝒦 ,

(1.6.2)

𝒦

∫(x − a, h)𝒦 (x − a, k)𝒦 μ∞ (dx) = (Qh, k)𝒦 ,

(1.6.3)

𝒦 1

1/2

∫ ei(x,h)𝒦 μ(dx) = ei(a,h)𝒦 − 2 ‖Q 𝒦

h‖2𝒦

.

(1.6.4)

84 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes Proof. It is known that 𝒦 ≅ ℓ2 since 𝒦 is separable by the assumption. Then it suffices to construct a Gaussian measure on ℓ2 . By Lemma 1.101 it follows that μ∞ = μ⌈̄ ℓ2 is a probability measure satisfying (1.6.1)–(1.6.4). Definition 1.103 (Gaussian measures on a Hilbert space). The probability measure μ∞ on (𝒦, ℬ(𝒦)) defined in Theorem 1.102 is called a Gaussian measure with mean a ∈ 𝒦 and covariance Q ∈ L1+ (𝒦). By Theorem 1.102 we can construct a Gaussian measure on a Hilbert space, which satisfies (1.6.4). Let a = 0. Then we have ∫ ei(Q

−1/2

x,h)

1

2

μ∞ (dx) = e− 2 ‖h‖ .

(1.6.5)

𝒦

In (1.6.5) a requirement is that x ∈ D(Q−1/2 ) = Q1/2 𝒦, thus one may expect ∫ ei(Q

−1/2

x,h)

1

2

μ∞ (dx) = e− 2 ‖h‖

(1.6.6)

Q1/2 𝒦

to hold. Putting Xh (x) = (Q−1/2 x, h), the expression (1.6.6) would appear to give a Gaussian random variable Xh (⋅) with covariance ‖h‖2𝒦 . The next result, however, says that this cannot be the case. Lemma 1.104. Let Q ∈ L1+ (𝒦) and Ker Q = {0}. If μ∞ is the probability measure on (𝒦, ℬ(𝒦)) given in Theorem 1.102, then μ(Q1/2 𝒦) = 0. Proof. We identify 𝒦 with ℓ2 . Notice that ∞ ∞ ∞ 1 2 ∞ 󵄨󵄨󵄨 yj ≤ n2 } = ⋃ Un . Q1/2 𝒦 = ⋃ {y = (yk )∞ k=1 ∈ ℝ 󵄨󵄨󵄨 ∑ λ n=1 n=1 j=1 j

Moreover, define 2k 1 2 ∞ 󵄨󵄨󵄨 Unk = {y = (yk )∞ yj ≤ n2 }. k=1 ∈ ℝ 󵄨󵄨󵄨 ∑ λ j=1 j

Since Unk ↓ Un as k → ∞, we have μ∞ (Un ) = limk→∞ μ∞ (Unk ). By a direct computation μ∞ (Unk ) = where S2k−1 =

−1/2 (∏2n j=1 λj )

(2π)n/2

k

2π . (k−1)!

∫ 1{∑2k ℝ2k

1 j=1 λ j

e y2 ≤n2 } j

− 21 ∑2k j=1

1 λj

|yj |2

n

dy =

1 2 S2k−1 ∫ e− 2 r r 2k−1 dr, k (2π)

0

Hence we have n

μ(Unk ) ≤

S2k−1 n2k 2k−1 r dr ≤ →0 ∫ (2π)k 2k k! 0

as k → ∞. Hence μ(Un ) = 0 for each n and the proposition follows.

1.6 Infinite dimensional Ornstein–Uhlenbeck processes | 85

1

The dense linear space Q 2 𝒦 is called Cameron–Martin space. This space is crucial in defining Gaussian measures on a Hilbert space. Recall that, in general, when a probability measure P is absolutely continuous with respect to a probability measure dP . Q, the notation P ≪ Q is used and the Radon–Nikodým derivative is denoted by dQ Thus P(A) = ∫ A

dP (x)dQ(x). dQ

When this is not the case, P is said to be singular with respect to Q, denoted P ⊥ Q, i. e., in this case there exists a measurable set A such that P(A) = 0 and Q(A) = 1. If P ≪ Q and Q ≪ P hold simultaneously, denoted P ∼ Q, then P and Q are called equivalent. Let (Ω, F ) be a measurable space, and μ, ν probability measures on this space. Take ξ = 21 (μ + ν); then we have μ ≪ ξ and ν ≪ ξ . The expression H(μ, ν) = ∫ √ Ω

dμ dν dξ dξ dξ

is called Hellinger integral. Lemma 1.105. Let μ and ν be probability measures on a measurable space (Ω, F ). If H(μ, ν) = 0, then μ ⊥ ν. dμ dν . The equality H(μ, ν) = ∫Ω √f (x)g(x)dξ = 0 implies Proof. Denote f = dξ and g = dξ that fg = 0 a. e. Define the sets A = {x ∈ Ω | f (x) = 0}, B = {x ∈ Ω | g(x) = 0} and C = A ∪ B = {x ∈ Ω | f (x)g(x) = 0}. Thus μ(A) = ∫Ω f (x)dξ = 0 and ν(B) = ∫Ω g(x)dξ = 0. Since μ(C) = ν(C) = 1, we can conclude that μ(B \ A) = 1 and ν(A \ B) = 1 and (B \ A) ∩ (A \ B) = 0.

Lemma 1.106. Let μ and ν be probability measures on a measurable space (Ω, F ). If dμ μ ∼ ν, then H(μ, ν) = ∫ √ dν dμ = ∫ √ dν. Ω

Proof. We have

dμ dν dξ dξ

=

dμ

dμ dν dμ . dξ dμ dξ

dν

Ω

Hence

H(μ, ν) = ∫ √ Ω

dμ dν dμ dν dξ = ∫ √ dμ. dξ dμ dξ dμ Ω

The second expression is shown similarly. Example 1.107. Choose μ = N0,λ and ν = Na,λ to be Gaussian measures on ℝ. We have μ ∼ ν and a2 ax dν = e− λ + λ . dμ

86 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes In this case the Hellinger integral H(μ, ν) is a2

1 a2 λ

ax

H(μ, ν) = ∫ √e− λ + λ dμ = e− 8

.

ℝ

We extend this to n-dimensional case. Example 1.108. Let μn = N0,(λ1 ,...,λn ) and νn = N(a1 ,...,an ),(λ1 ,...,λn ) be Gaussian measures on ℝn . By similar computations as above, it is seen that H(μn , νn ) = e

a2j

− 81 ∑nj=1

λj

.

From Example 1.108 above, formally we see that 1

−1/2

lim H(μn , νn ) = e− 8 ‖Q

n→∞

a‖2

.

In particular, a ∈ ̸ Q1/2 𝒦 implies that the right-hand side is zero, and then μ∞ ⊥ ν∞ may be expected. In general, it can be seen that n

n

n

j=1

j=1

j=1

H(∏ μj , ∏ νj ) = ∏ H(μj , νj ). This identity can be extended to infinite product measures and applied to investigating the absolute continuity of infinite product measures. Proposition 1.109 (Kakutani). Let μk and νk be probability measures on (ℝ, ℬ(ℝ)) and ∞ suppose that μk ∼ νk , for each k ∈ ℕ. Let μ = ∏∞ k=1 μk and ν = ∏k=1 νk . Then ∞

H(μ, ν) = ∏ H(μj , νj ).

(1.6.7)

j=1

Furthermore, if H(μ, ν) > 0, then μ ∼ ν, and if H(μ, ν) = 0, then μ ⊥ ν. When μ ∼ ν, the Radon–Nikodým derivative is given by ∞ dμ dμ j (x) = ∏ (xj ). dν dν j j=1

(1.6.8)

Proof. Write n

Ψn (x) = Ψn (x1 , . . . , xn ) = ∏ √ j=1

dνj

dμj

(xj )

for x ∈ ℝ∞ . We show that (Ψn )n∈ℕ is a Cauchy sequence in L2 (ℝ∞ , dμ). It is straightforward to see that

1.6 Infinite dimensional Ornstein–Uhlenbeck processes | 87

n

‖Ψn − Ψm ‖2 = ∫ ∏ ℝ∞ j=1

m 󵄨󵄨 󵄨󵄨2 dνj 󵄨 󵄨 (xj )󵄨󵄨󵄨1 − ∏ √ (xj )󵄨󵄨󵄨 μ(dx) 󵄨󵄨 󵄨󵄨 dμj dμ j j=n+1

dνj

m

= 2 (1 − ∏ ∫ √ j=n+1 ℝ

m

dνj

dμj

(xj )μj (dxj )) = 2 (1 − ∏ H(μj , νj )) . j=n+1

(1.6.9)

We consider two cases. First suppose that n

lim ∏ H(μj , νj ) > 0.

n→∞

j=1

We have − ∑∞ n=1 log H(μj , νj ) < ∞ and (− log H(μj , νj ))n∈ℕ is a Cauchy sequence. Thus lim

m,n→∞

m

∏ H(μj , νj ) = 1

j=n+1

which implies by (1.6.9) that (Ψn )n∈ℕ is a Cauchy sequence. Hence there exists Ψ ∈ L2 (ℝ∞ , dμ) such that Ψn → Ψ as n → ∞. We also see that (Ψ2n )n∈ℕ is a Cauchy sequence in L1 (ℝ∞ , dμ) by the bound 󵄨󵄨 󵄨2 󵄨󵄨 ∫ |Ψ2 (x) − Ψ2 (x)|dμ(x)󵄨󵄨󵄨 ≤ 4 ∫ |Ψm (x) − Ψn (x)|2 dμ(x). m n 󵄨󵄨 󵄨󵄨 ℝ∞

ℝ∞

Let A ∈ ℬ(ℝ∞ ) and χn (x) = 1A (Pn x), where Pn x = (x1 , . . . , xn , 0, . . .) for x ∈ ℝ∞ . We have n

∫ χn (x)ν(dx) = ∫ χn (x) ∏ ℝ∞

ℝn

j=1

dνj

dμj

n

∏ μj (dxj ) = ∫ χn (x)Ψ2n (x)μ(dx). j=1

ℝ∞

Hence ν(A) = ∫ 1A (x)ν(dx) = lim ∫ χn (x)Ψ2n (x)μ(dx). ℝ∞

n→∞

ℝ∞

Since 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 2 2 󵄨󵄨 ∫ 1A (x)Ψ (x)μ(dx) − ∫ χn (x)Ψn (x)μ(dx)󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨ℝ∞ 󵄨󵄨 ℝ∞ 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 2 2 ≤ 󵄨󵄨 ∫ 1A (x)Ψ (x)μ(dx) − ∫ χn (x)Ψ (x)μ(dx)󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨ℝ∞ 󵄨󵄨 ℝ∞ 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 + 󵄨󵄨󵄨 ∫ χn (x)Ψ2 (x)μ(dx) − ∫ χn (x)Ψ2n (x)μ(dx)󵄨󵄨󵄨 → 0 󵄨󵄨 󵄨󵄨 󵄨󵄨ℝ∞ 󵄨󵄨 ℝ∞

88 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes as n → ∞ by the L1 -convergence of Ψ2n , we conclude that ν(A) = lim ∫ χn (x)Ψ2n (x)μ(dx) = ∫ 1A Ψ2 (x)μ(dx). n→∞

ℝ∞

ℝ∞

Hence ν ≪ μ and the Radon–Nikodým derivative is given by dν (x) = Ψ2 (x). dμ This implies (1.6.8). Furthermore, it follows that H(μ, ν) = ∫ √ ℝ∞

dν (x)μ(dx) = ∫ Ψ(x)μ(dx) = lim ∫ Ψn (x)μ(dx) n→∞ dμ ∞ ∞ ℝ

n

= lim ∏ ∫ √ n→∞

j=1 ℝ

dνj

dμj

ℝ

n

(xj )μj (dxj ) = lim ∏ H(μj , νj ). n→∞

j=1

Hence (1.6.7) is obtained. Next suppose that n

lim ∏ H(μj , νj ) = 0.

n→∞

j=1

Let ε > 0 and An = {x = (x1 , . . . , xn ) ∈ ℝn | Ψ2n (x, 0, . . .) = ∏nj=1 n

n

n

j=1

j=1

dνj (x ) dμj j

> 1}. We have

(∏ μj )(An ) ≤ ∫ Ψn (x) ∏ μj (dxj ) ≤ ∏ H(μj , νj ) < ε j=1

An

for sufficiently large n. Since ∏nj=1

dνj (x ) dμj j

n

(∏ j=1

we have ∏nj=1

that

dμj (x ) dνj j

dνj

dμj

≤ 1 on ℝn \ An and the identity n

) (∏ j=1

dνj

) = 1,

> 1 on ℝn \ An . In the same way as the estimate above, we obtain

n

n

(∏ νj )(ℝn \ An ) ≤ ∫ √∏ j=1

dμj

ℝn \An

j=1

dμj dνj

n

n

j=1

j=1

∏ νj (dxj ) ≤ ∏ H(μj , νj ) < ε

∞ ̃ ̃ for sufficiently large n. Define à n = An × ∏∞ j=n+1 ℝ. Then μ(An ) ≤ ε and ν(ℝ \ An ) ≤ ε. ∞ ε ε ∞ Hence there exists Bn such that μ(Bn ) ≤ 2n and ν(ℝ \Bn ) ≤ 2n . Let C = ⋂n=1 Bn . We have μ(C) ≤ 2εn for all n ∈ ℕ, and this implies that μ(C) = 0. We also see that ν(ℝ∞ \ C) ≤ ε.

1.6 Infinite dimensional Ornstein–Uhlenbeck processes | 89

Hence for each n ∈ ℕ there exists Cn such that μ(Cn ) = 0 and ν(ℝ∞ \ Cn ) ≤ 1/n. Let D = ⋃∞ n=1 Cn . We have μ(D) = 0

and

ν(ℝ∞ \ D) = 0.

From this μ ⊥ ν follows. Finally, we see that H(μ, ν) = ∫ √ D

dμ dν dμ dν dξ + ∫ √ dξ dξ dξ dξ dξ 1/2

dμ dξ ) ≤ (∫ dξ D

1/2

= (μ(D)ν(D))

Dc

dν (∫ dξ ) dξ D

c

1/2

c

dμ + (∫ dξ ) dξ

1/2

+ (μ(D )ν(D ))

1/2

Dc

1/2

dν (∫ dξ ) dξ Dc

= 0.

Hence H(μ, ν) = 0 = limn→∞ ∏nj=1 H(μj , νj ) follows. There are various applications of Proposition 1.109. One allows to compare Gaussian measures with zero and nonzero means. Corollary 1.110. Let Q ∈ L1+ (𝒦), Ker Q = {0} and a ∈ 𝒦. (1) If a ∈ ̸ Q1/2 𝒦, then Na,Q ⊥ N0,Q . (2) If a ∈ Q1/2 𝒦, then Na,Q ∼ N0,Q and 1 −1/2 2 −1/2 dN0,Q = e− 2 ‖Q a‖ +(Q a,x) . dNa,Q

Proof. We have n

n

H(μ, ν) = lim ∏ H(μj , νj ) = lim ∏ e n→∞

n→∞

j=1

j=1

− 81

a2j λj

.

By Proposition 1.109, if a ∈ Q1/2 𝒦, then H(μ, ν) > 0 and μ ∼ ν, if a ∈ ̸ Q1/2 𝒦, then H(μ, ν) = 0 and μ ⊥ ν. The Radon–Nikodým derivative is also given by dN0,Q − 1 ∑n = lim e 2 j=1 dNa,Q n→∞

a2j λj

+∑nj=1

aj xj λj

1

−1/2

= e− 2 ‖Q

a‖2 +(Q−1/2 a,x)

.

From this corollary it follows that either Na,Q ∼ N0,Q or Na,Q ⊥ N0,Q , and clearly Na,Q ∼ N0,Q if and only if a is in the Cameron–Martin space Q1/2 𝒦, and if a is not contained in the Cameron–Martin space, then Na,Q ⊥ N0,Q . In the next corollary we compare the Gaussian measures N0,Q and N0,αQ , α ≥ 0. Corollary 1.111. Let Q, R ∈ L1+ (𝒦) and Ker Q = Ker R = {0}. Suppose that [Q, R] = 0. Let {en }∞ n=1 be a complete orthonormal system such that Qen = λn en and Ren = rn en for each n ∈ ℕ. Then

90 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes λ −r

2

k k (1) N0,Q ∼ N0,R if and only if ∑∞ k=1 ( λk +rk ) < ∞. (2) If N0,Q ≁ N0,R , then N0,Q ⊥ H0,R .

Proof. A computation gives ∞

H(N0,Q , N0,R )4 = ∏ (1 − ( j=1

λ −r

λk − rk 2 ) ). λk + rk

2

k k We see that ∑∞ k=1 ( λ +r ) < ∞ if and only if H(N0,Q , N0,R ) > 0. k

k

Corollary 1.112. Let α > 0, α ≠ 1, Q ∈ L1+ (𝒦) and Ker Q = {0}. Then N0,Q ⊥ N0,αQ . λ −r

2

2

∞ 1−α k k Proof. Let rj = αλj . Since ∑∞ k=1 ( λk +rk ) = ∑k=1 ( 1+α ) = ∞, the claim follows by Corollary 1.111.

Using the above corollary, we can construct an uncountable family of Gaussian measures {N0,αQ }α≥0 such that N0,αQ ⊥ N0,βQ for any pair α ≠ β. This indicates that Gaussian measures on an infinite dimensional Hilbert space generally have a more complicated structure than in finite dimensions.

1.6.2 Abstract theory of Borel measures on Hilbert spaces Consider the Borel measure space (𝒦, ℬ(𝒦), μ) on a separable Hilbert space 𝒦 over ℝ, and define ̂ μ(x) = ∫ ei(x,y) μ(dy). 𝒦

It is seen that ̂ (1) μ(0) = μ(𝒦); ̂ is continuous; (2) x 󳨃→ μ(x) (3) μ̂ is positive definite. A natural converse question is if it is possible to construct a Borel measure on (𝒦, ℬ(𝒦)) such that properties (1)–(3) are satisfied. We summarize the properties of positive definite functions next. Lemma 1.113. Let ϕ : 𝒦 → ℂ be positive definite. Then the following hold: (1) |ϕ(x)| ≤ |ϕ(0)|, ϕ(x) = ϕ(−x). (2) |ϕ(x) − ϕ(y)| ≤ 2√ϕ(0)√ϕ(0) − ϕ(x − y). (3) |ϕ(0) − ϕ(x)| ≤ √2ϕ(0)√ϕ(0) − ℜϕ(x).

1.6 Infinite dimensional Ornstein–Uhlenbeck processes | 91

Proof. For x, y ∈ 𝒦 define ϕ(0) A=( ϕ(−x)

ϕ(x) ) ϕ(0)

ϕ(0) and B = (ϕ(−x) ϕ(−y)

ϕ(x) ϕ(0) ϕ(x − y)

ϕ(y) ϕ(y − x)) . ϕ(0)

Since ϕ is positive definite, both A and B are positive definite matrices. We have A∗ = A. Hence ϕ(x) = ϕ(−x) follows, and det A ≥ 0 implies that |ϕ(x)| ≤ ϕ(0). It is straightforward to see that det B = ϕ(0)3 − ϕ(0)|ϕ(x − y)|2 − ϕ(0)|ϕ(x) − ϕ(y)|2 + 2ℜ (ϕ(y)ϕ(x)ϕ(x − y) − ϕ(0)) .

(1.6.10)

Using the inequality a3 − ab2 ≤ 2a2 |a − b| for |b| < a we see that ϕ(0)3 − ϕ(0)|ϕ(x − y)|2 ≤ 2ϕ(0)2 |ϕ(0) − ϕ(x − y)|. Inserting this into (1.6.10), we obtain 0 ≤ det B ≤ 4ϕ(0)2 |ϕ(0) − ϕ(x − y)| − ϕ(0)|ϕ(x) − ϕ(y)|2 , which proves (2). Part (3) follows from the estimate |ϕ(0) − ϕ(x)|2 ≤ 2ϕ(0)2 − 2ϕ(0)ℜϕ(x). Proposition 1.114 (Minlos–Sazonov). Let 𝒦 be a separable Hilbert space over ℝ and ϕ : 𝒦 → ℂ be positive definite. The following properties are equivalent: (1) There exists a finite Borel measure μ on (𝒦, ℬ(𝒦)) such that ϕ(x) = ∫ ei(x,y) μ(dy). 𝒦

(2) For every ε > 0, there is a symmetric positive trace class operator Qε such that (Qε x, x) < 1 implies that ℜ(ϕ(0) − ϕ(x)) < ε. (3) There exists Q ∈ L1+ (𝒦) such that ϕ is continuous with respect to ‖ ⋅ ‖Q , where ‖x‖Q = √(Qx, x). Proof. (1) 󳨐⇒ (2): Suppose (1). For every γ > 0 we obtain ℜ(ϕ(0) − ϕ(x)) = ∫ (1 − cos(x, z))μ(dz) + ∫ (1 − cos(x, z))μ(dz) ‖z‖≤γ

‖z‖>γ

1 ≤ ∫ (z, x)2 μ(dz) + 2μ({z ∈ 𝒦 | ‖z‖ > γ}). 2 ‖z‖≤γ

92 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes Define the sesquilinear form B : 𝒦 × 𝒦 → ℂ by B(f , g) = ∫‖z‖≤γ (z, f )(z, g)μ(dz). By the Riesz representation theorem, there exists a bounded operator B̄ such that B(f , g) = ̄ Furthermore, for any complete orthonormal system {ej }∞ of 𝒦 we see that (f , Bg). j=1

∞

̄ j ) = ∫ ‖z‖2 μ(dz) ≤ γ 2 < ∞, ∑(ej , Be j=1

‖z‖≤γ

which implies that B̄ is a trace class operator. On the other hand, there exists γ such that μ({z ∈ 𝒦 | ‖z‖ > γ}) ≤ ε/4. Thus we have ε ε ℜ(ϕ(0) − ϕ(x)) ≤ (x, Qε x) + , 2 2 ̄ Hence (2) follows. where we set Qε = B/ε. (2) 󳨐⇒ (1): Suppose (2). Let ‖x‖2 ≤ 1/‖Qε ‖. We have |(Qε x, x)| ≤ 1, and furthermore ℜ(ϕ(0) − ϕ(x)) ≤ ε, which implies that ϕ is continuous at x = 0 and then continuous at all x ∈ 𝒦. Let {ej }∞ j=1 be a complete orthonormal system in 𝒦. Let Λ = ℕ and for {i1 , . . . , in } ⊂ Λ define fi1 ,...,in (x1 , . . . , xn ) = ϕ(x1 ei1 + ⋅ ⋅ ⋅ + xn ein ),

fi1 ,...,in : ℝn → ℂ,

which is positive definite. Thus by the Bochner theorem there exists a finite Borel measure μi1 ,...,in on (ℝn , ℬ(ℝn )) such that fi1 ,...,in (x) = ∫ eix⋅y μi1 ,...,in (dy).

(1.6.11)

ℝn

We define the family of probability spaces (ℝA , ℬ(ℝA ), μA ), A ⊂ Λ, #A < ∞. Let F((x0 , x)) = f (x0 ) for x0 ∈ ℝn and x ∈ ℝm with f ∈ S (ℝn ). We have ∫ F((x0 , x))μi1 ,...,in ,in+1 ,...,in+m (dx0 dx)

ℝn+m

= ∫ f (x0 )μi1 ,...,in ,in+1 ,...,in+m (dx0 dx) ℝn+m

=

1 ∫ f ̌(k0 )dk0 ∫ e−ik0 ⋅x0 μi1 ,...,in ,in+1 ,...,in+m (dx0 dx) (2π)n/2 n n+m ℝ

ℝ

ℝ

ℝ

1 = ∫ f ̌(k0 )dk0 ∫ e−ik0 ⋅x0 μi1 ,...,in (dx0 ) = ∫ f (x0 )μi1 ,...,in (dx0 ). (2π)n/2 n n n ℝ

Let E = Ei1 × ⋅ ⋅ ⋅ × Ein with Eij ∈ ℬ(ℝ). By a limiting argument we see that ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ μi1 ,...,in ,in+1 ,...,in+m (E × ℝ × ⋅ ⋅ ⋅ × ℝ) = ∫ 1E×ℝ×⋅⋅⋅×ℝ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ (x0 , x)μi1 ,...,in ,in+1 ,...,in+m (dx0 dx) m

ℝn+m

m

1.6 Infinite dimensional Ornstein–Uhlenbeck processes | 93

= ∫ 1E (x0 )μi1 ,...,in (dx0 ) = μi1 ,...,in (E). ℝn

The Kolmogorov consistency condition μi1 ,...,in ,in+1 ,...,in+m (E × ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ℝ × ⋅ ⋅ ⋅ × ℝ) = μi1 ,...,in (E) m

then follows. By the Kolmogorov extension theorem, there exists a measure γ on a measurable space (ℝ∞ , ℬ(ℝ∞ )) such that γ ∘ πi−1 (Ei1 × ⋅ ⋅ ⋅ × Ein ) = μi1 ,...,in (Ei1 × ⋅ ⋅ ⋅ × Ein ), 1 ,...,in

(1.6.12)

where πi1 ,...,in : ℝ∞ → ℝn is the projection defined by πi1 ,...,in (x) = (xi1 , . . . , xin ). Define X : ℝ∞ → ℝ by Xi (x) = π{i} (x) = xi , and X : ℝ∞ → 𝒦 by X(x) = ∑∞ j=1 Xj (x)ej . We show that ∞

∑ Xj2 (x) < ∞

(1.6.13)

j=1

for γ-a. e. in Lemma 1.115 below. Hence X is measurable and defines the image measure ∞ ̃ =x on (𝒦, ℬ(𝒦)) by μ = γ ∘ X −1 . For x ∈ 𝒦, we define x̃ = ((ej , x))∞ j=1 ∈ ℝ . We have X(x) and n

∫ ei(x,y) μ(dy) = ∫ ei(X(x),X(y)) γ(dy)̃ = lim ∫ ei ∑j=1 xj yj γ(dy)̃ ̃

̃

𝒦

n→∞

ℝ∞ n

ℝ∞

̃ ̃

n

= lim ∫ ei ∑j=1 xj yj μ1,...,n (dy)̃ = lim ϕ(∑ x̃j ej ) = ϕ(x). n→∞

̃ ̃

n→∞

ℝn

j=1

The first equality follows by a change of variables, the second by the Lebesgue dominated convergence theorem, the third by Kolmogorov’s extension theorem (1.6.12), the fourth by the Bochner theorem (1.6.11), and the fifth is derived from the continuity of ϕ. This completes the proof of (1). (2) ⇒ (3): Suppose (2). Take ε = 1/k in (2) and λk such that ∞

∑ λk Tr Q1/k < ∞.

k=1

Define Q = ∑∞ k=1 λk Q1/k . Q is a positive and symmetric trace class operator. Let (Qx, x) < λk . We have (Q1/k x, x) < 1, which by the assumption implies that ℜ(ϕ(0) − ϕ(x)) < 1/k. Hence ϕ(x) is continuous at x = 0 by ‖ ⋅ ‖Q , and continuous at all x ∈ 𝒦. (3) ⇒ (2): Suppose (3). For every ε > 0 there exists δ > 0 such that ‖x‖Q < δ implies ℜ(ϕ(0)−ϕ(x)) < ε. Set Qε = Q/δ. We see that ℜ(ϕ(0)−ϕ(x)) < ε for x satisfying (Qε x, x) < 1 and Qε satisfies (2).

94 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes Lemma 1.115. The property (1.6.13) holds. 1

2

Proof. Let N be a Gaussian measure on ℝn such that ∫ℝn e−ikx N(dx) = e− 2 |k| . We have n

1

n

2

ϕ(0) − ∫ e− 2 ∑j=1 Xk+j γ(dx) = ϕ(0) − ∫ γ(dx) ∫ e−i ∑j=1 yj Xk+j N(dy) ℝn

ℝ∞

ℝ∞

= ϕ(0) − ∫ N(dy) ∫ e ℝn

−i ∑nj=1 yj Xk+j

n

γ(dx) = ϕ(0) − ∫ ϕ(∑ yj ek+j )N(dy) j=1

ℝn

ℝ∞ n

= ∫ (ϕ(0) − ϕ(∑ yj ek+j )) N(dy). j=1

ℝn

Since (Qε x, x) < 1 gives ℜ(ϕ(0) − ϕ(x)) < ε, we have (ϕ(0) − ℜϕ(x)) ≤ ε + 2ϕ(0)(Qε x, x) for x ∈ 𝒦. Hence 1

n

2

n

n

j=1

j=1

ϕ(0) − ∫ e− 2 ∑j=1 Xk+j γ(dx) ≤ ε + 2ϕ(0) ∫ (Qε ∑ yj ek+j , ∑ yj ek+j )N(dy) ℝ∞

ℝn

n

= ε + 2ϕ(0) ∑ (Qε ek+j , ek+i ) ∫ yi yj N(dy) i,j=1

ℝn

n

n

= ε + 2ϕ(0) ∑(Qε ek+j , ek+j ) ∫ yj2 N(dy) = ε + 2ϕ(0) ∑(Qε ek+j , ek+j ). j=1

j=1

ℝn

We obtain ∞ n 1 2 󵄨󵄨 󵄨 󵄨󵄨ϕ(0) − ∫ e− 2 ∑j=1 Xk+j γ(dx)󵄨󵄨󵄨 ≤ ε + 2ϕ(0) ∑ (Qε ej , ej ). 󵄨󵄨 󵄨󵄨 j=k+1

ℝ∞

Taking k → ∞ and ε → 0 gives 1

2

lim ∫ e− 2 ∑j=k+1 Xj γ(dx) = ϕ(0).

k→∞ 1

∞

ℝ∞

2

This implies that e− 2 ∑j=k+1 Xj converges to the constant function 1 in L1 (ℝ∞ , γ). Hence ∞

there exists a subsequence k 󸀠 such that e 2 ∑∞ j=1 Xj < ∞ γ-a. e.

− 21 ∑∞ X2 j=k 󸀠 +1 j

converges to 1 γ-a. e., which shows

Definition 1.116 (Sazonov topology). Let 𝒦 be a real separable Hilbert space. A nonnegative trace class operator B defines the seminorm pB (f ) = √(f , Bf )𝒦 . The Sazonov topology on 𝒦 is the locally convex topology defined by the family of seminorms {pB | B : nonnegative trace class operator on 𝒦}.

1.6 Infinite dimensional Ornstein–Uhlenbeck processes | 95

Proposition 1.117 (Borel measures on a Hilbert space). Let 𝒦 be a separable Hilbert space over ℝ. Suppose that C : 𝒦 → ℂ is positive definite. There exists a finite Borel measure μ on 𝒦 such that C(f ) = ∫ ei(ψ,f )𝒦 dμ(ψ) 𝒦

if and only if C(⋅) is continuous in the Sazonov topology. Proof. This follows from Proposition 1.114. Proposition 1.117 is called the Minlos–Sazonov theorem. Example 1.118. Let B ≥ 0 be a self-adjoint bounded operator and CB (f ) = e−(f ,Bf ) . It is well known that CB is continuous in the Sazonov topology if and only if B is trace-class. Therefore, whenever B is trace-class, there is a finite Borel measure μB on 𝒦 such that ∫𝒦 ei(ψ,f )𝒦 dμB (ψ) = e−(f ,Bf )𝒦 . Let A ≥ 0 be a self-adjoint operator on 𝒦 such that A−1 is a Hilbert–Schmidt operator. Define ∞

C ∞ (A) = ⋂ D(An ) n=0

and write ‖f ‖n = ‖An/2 f ‖𝒦 . Also, the completion 𝒦n = C ∞ (A)

‖⋅‖n

defines a Hilbert space with scalar product (f , g)n = (An/2 f , An/2 g)𝒦 ,

f , g ∈ D(An/2 ).

We have the sequence of Hilbert spaces . . . ⊂ 𝒦+2 ⊂ 𝒦+1 ⊂ 𝒦0 ⊂ 𝒦−1 ⊂ 𝒦−2 ⊂ . . . . Under the identification 𝒦0 ≅ 𝒦0∗ we also have ∗

𝒦n ≅ 𝒦−n ,

n ≥ 1.

Denote the dual pair between 𝒦−2 and 𝒦+2 by ⟨⟨ψ, f ⟩⟩. An immediate corollary of the Minlos–Sazonov theorem is the following. Lemma 1.119. If L ≥ 0 is a bounded self-adjoint operator on 𝒦 such that [A−1 , L] = 0, then there exists a Borel probability measure mL on (𝒦−2 , ℬ(𝒦−2 )) such that 1

∫ ei⟨⟨ψ,f ⟩⟩ dmL (ψ) = e− 2 (f ,Lf )0 . 𝒦−2

(1.6.14)

96 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes In particular, if L = 1, then there exists a probability measure m1 on 𝒦−2 such that 1

2

∫ ei⟨⟨ψ,f ⟩⟩ dm1 (ψ) = e− 2 ‖f ‖0 𝒦−2

Proof. We show that A−1 is a Hilbert–Schmidt operator also on 𝒦−2 . To see this, note that for a complete orthonormal system {ei }∞ i=1 of 𝒦 consisting of eigenvectors of A, the set {Aei }∞ i=1 is also a complete orthonormal system of 𝒦−2 , and we put ẽ i = Aei . We have ∞

∞

i=1

i=1

∑(ẽi , A−2 ẽi )−2 = ∑(ei , A−2 ei )0 = Tr(A−2 ) < ∞. Hence A−1 is a Hilbert–Schmidt operator on 𝒦−2 . Moreover, since ‖Lf ‖−2 = ‖A−1 Lf ‖0 = ‖LA−1 f ‖0 ≤ ‖L‖‖f ‖−2 , L is bounded on 𝒦−2 , so that LA−1 is a Hilbert–Schmidt operator on 𝒦−2 . By Proposition 1.117 and Example 1.118 there exists a probability measure mL on 𝒦−2 such that 1

∫ ei(ψ,f )−2 dmL (ψ) = e− 2 (f ,LA

−2

f )−2

,

f ∈ C ∞ (A).

𝒦−2

Inserting A2 f into f above, we have 1

∫ eiρ(ψ,f ) dmL (ψ) = e− 2 (f ,Lf )0 ,

(1.6.15)

𝒦−2

where we put ρ(ψ, f ) = (ψ, A2 f )−2 ,

f ∈ C ∞ (A),

ψ ∈ 𝒦−2 .

Thus we have the bound |ρ(ψ, f )| ≤ ‖ψ‖−2 ‖f ‖2 . Using this, we can extend ρ(ψ, f ) to f ∈ 𝒦+2 by ρ(ψ, f ) = lim ρ(ψ, fn ), n→∞

where fn → f in 𝒦+2 as n → ∞. ρ(ψ, f ) with f ∈ 𝒦+2 also satisfies |ρ(ψ, f )| ≤ ‖ψ‖−2 ‖f ‖2 , hence ρ(ψ, f ) = ⟨⟨ψ, f ⟩⟩. Moreover, for f ∈ 𝒦2 there exists a sequence (fn )n∈ℕ ⊂ C ∞ (A) such that fn → f in both 𝒦0 and 𝒦2 as n → ∞, since A is closed and C ∞ (A) is a core of A. Hence (1.6.15) can be extended to f ∈ 𝒦+2 by a limiting argument, and the lemma follows. We can extend Lemma 1.119 to f ∈ 𝒦. Theorem 1.120 (Gaussian measure mL on 𝒦−2 ). Suppose that L ≥ 0 is a bounded selfadjoint operator on 𝒦 and [A−1 , L] = 0. Then there exists a Borel probability measure mL on (𝒦−2 , ℬ(𝒦−2 )) and a family of Gaussian random variables ψ(f ) indexed by f ∈ 𝒦 such that

1.6 Infinite dimensional Ornstein–Uhlenbeck processes | 97

(1) 𝒦 ∋ f 󳨃→ ψ(f ) ∈ ℝ is linear; (2) ψ(f ) = ⟨⟨ψ, f ⟩⟩ for f ∈ 𝒦+2 ; (3) it satisfies 1

∫ eiψ(f ) dmL (ψ) = e− 2 (f ,Lf )0 .

(1.6.16)

𝒦−2

Proof. For f ∈ 𝒦+2 we write ψ(f ) = ⟨⟨ψ, f ⟩⟩. Equality (1.6.16) follows by Lemma 1.119. For f ∈ 𝒦 \ 𝒦+2 , ψ(f ) can be defined through a limiting procedure. By (1.6.14) it can be seen that ∫ ψ(f )2 dmL (ψ) = (f , Lf )0 ,

f ∈ 𝒦+2 .

𝒦−2

Since for f ∈ 𝒦 there exists a sequence (fn )n∈ℕ ⊂ 𝒦+2 such that fn → f strongly in 𝒦 as n → ∞, we can define ψ(f ) by ψ(f ) = s-lim ψ(fn ) n→∞

in L2 (𝒦−2 , dmL ). The theorem follows by a limiting argument. Remark 1.121. The map 𝒦+2 ∋ f 󳨃→ ψ(f ) ∈ ℝ is continuous with |ψ(f )| ≤ ‖ψ‖−2 ‖f ‖2 , but not as a map 𝒦 ∋ f 󳨃→ ψ(f ) ∈ ℝ. Example 1.122. It is impossible to construct a Gaussian measure on a Hilbert space 𝒦0 with covariance (⋅, ⋅)𝒦0 . By Theorem 1.120 however we can construct a Gaussian measure with covariance (⋅, ⋅)𝒦0 on Banach space 𝒦−2 by taking the identity as L. 1.6.3 Fock space as a function space In order to construct functional integral representation of quantum field Hamiltonians, it will be convenient to define boson Fock space as an L2 -space over a function space with a Gaussian measure. From now on, we will work in the following setup: (1) Dispersion relation. We choose ν = 0 and ω0 (k) = ω(k) = |k|. (2) Lorentz-covariant space. The real Hilbert space M is the real part of ℋM for ων replaced by ω. (3) Hilbert–Schmidt operator. A given positive self-adjoint operator 𝒟 with Hilbert– Schmidt inverse on M and such that √ω𝒟−1 is bounded on M . Let Spec(𝒟) = ∞ {λi }∞ i=1 and {ei }i=1 be the normalized eigenvectors such that 𝒟 ei = λi ei . In this case ∞ −2 −2 Tr(𝒟 ) = ∑i=1 λi < ∞. We define a real vector space by ‖𝒟 n/2 ⋅‖M

Mn = C ∞ (𝒟)

.

98 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes Remark 1.123. The dispersion relation ω can be chosen to be more generally a measurable function ω : ℝd → ℝ, with 0 ≤ ω(k) = ω(−k), and ω(k) = 0 only on a set of Lebesgue-measure zero. The results in this section will equally apply to ω. Remark 1.124. (1) The positive self-adjoint operator 1 1 (d+1)/2 1 T = (− Δ + |k|2 + ) 2 2 2 is a specific case of 𝒟 above, see Example 1.71. Indeed T −1 is a Hilbert–Schmidt operator and √ωT −1 is bounded due to ‖√ω(−Δ + |k|2 + 1)−(d+1)/2 ‖ ≤ 1. (2) The condition that √ω𝒟−1 is bounded on M will be used in Lemma 1.134 below, where we will prove path continuity of the infinite dimensional Ornstein– Uhlenbeck process. Moreover, this condition yields that the embedding ι : M+2 ∋ f 󳨃→ f ∈ L2 (ℝd ) is bounded. It can indeed be seen that ‖f ‖L2 (ℝd ) = ‖√ω𝒟−1 𝒟f ‖M ≤ ‖√ω𝒟−1 ‖‖f ‖M+2 .

(1.6.17)

∗ We identify M+2 with the topological dual of M−2 by M−2 ≅ M+2 , and denote the pairing between M+2 and M−2 by ⟨⟨⋅, ⋅⟩⟩.

Theorem 1.125 (Gaussian measure G on M−2 ). In the above set-up there exists a probability measure G on M−2 and a family of Gaussian random variables ξ (f ) indexed by f ∈ M such that (1) M ∋ f 󳨃→ ξ (f ) ∈ ℝ is linear; (2) ξ (f ) = ⟨⟨ξ , f ⟩⟩ when f ∈ M+2 ; (3) the expression 1

2

∫ eiξ (f ) dG = e− 2 ‖f ‖ℋM M−2

holds. Proof. This is immediate by Theorem 1.120 with A = 𝒟, L = 1, 𝒦 = M , ψ(f ) = ξ (f ) and mL = G. Now we discuss the unitary equivalence of the boson Fock space ℱb (ℋ̂ M ) and L2 (M−2 , dG). Let Φ(f ) =

1 ∗ ̂ ̃ (a (f ) + aM (f ̂)), √2 M

f ̂ ∈ ℋ̂ M ,

1.6 Infinite dimensional Ornstein–Uhlenbeck processes | 99

where aM and a∗M denote the creation operator and the annihilation operator in the boson Fock space ℱb (ℋ̂ M ), respectively. The field operator Φ(f ) is formally written as Φ(f ) = ∫ ℝd

1 (a∗ (k)f ̂(k) + a(k)f ̂(−k)) dk. √2ω(k)

Recall that Ωb denotes the Fock vacuum. For f , g ∈ M , we see that (Φ(f )Ωb , Φ(g)Ωb )ℱ

̂

b (ℋM )

1 = (f ̂, g)̂ ℋ̂ = 𝔼G [ξ (f )ξ (g)]. M 2

(1.6.18)

For f ∈ ℋ̂ M , f = g + ih with g, h ∈ M , define ξ (f ) = ξ (g) + iξ (h). The Wick product of ∏nj=1 ξ (fj ) is inductively defined by :ξ (f ): = ξ (f ), n

n

j=1

j=1

:ξ (f ) ∏ ξ (fj ): = ξ (f ):∏ ξ (fj ): −

n 1 n ∑(f , fi )ℋM :∏ ξ (fj ):. 2 i=1 j=i̸

Proposition 1.126 (Unitary equivalence). There exists a unitary operator θM : ℱb (ℋ̂ M ) → L2 (M−2 , dG) satisfying (1)–(3) below: (1) θM Ωb = 1; (2) θM :∏ni=1 Φ(fî ):Ωb = :∏ni=1 ξ (fi ):; −1 (3) θM Φ(f ̂)θM = ξ (f ) for f ∈ M as an operator. In particular, it follows that 2

ℱb (ℋ̂ M ) ≅ L (M−2 , dG).

(1.6.19)

Proof. The proof is similar to that of Proposition 1.51. We define the linear operator θM : ℱb (ℋ̂ M ) → L2 (M−2 , dG) by θM Ωb = 1,

n

n

i=1

i=1

θM :∏ Φ(fî ):Ωb = :∏ ξ (fi ):,

fi ∈ M .

(1.6.20)

By the commutation relations, it follows that 󵄩󵄩 n 󵄩 󵄩󵄩 n 󵄩󵄩 󵄩󵄩:∏ Φ(fî ):Ωb 󵄩󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩ℱb (ℋ̂ M ) = 󵄩󵄩󵄩:∏ ξ (fi ):󵄩󵄩󵄩L2 (M−2 ,dG) i=1

i=1

and the linear hull of :∏ni=1 Φ(fî ):Ωb and :∏ni=1 ξ (fi ): is a dense subspace in ℱb (ℋ̂ M ) and L2 (M−2 , dG), respectively. Hence θM can be extended to the unitary operator from ℱb (ℋ̂ M ) to L2 (M−2 , dG) and the proposition follows.

100 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes Definition 1.127 (Second quantization of contraction operators). ̃ (1) For T ∈ C (M → M ) its second quantization Γ(T) ∈ C (L2 (M−2 , dG) → L2 (M−2 , dG)) is defined by ̃ Γ(T):ξ (f1 ) ⋅ ⋅ ⋅ ξ (fn ): = :ξ (Tf1 ) ⋅ ⋅ ⋅ ξ (Tfn ):

(1.6.21)

̃ with Γ(T)1 = 1, where 1 denotes the identity function in L2 (M−2 , dG). ̃ (2) Let h be a self-adjoint operator in Mℂ . Its differential second quantization dΓ(h) ̃ −ith −itd Γ(h) ̃ is defined by the self-adjoint operator such that Γ(e )=e for t ∈ ℝ. Definition 1.128 (Free field Hamiltonian). The free field Hamiltonian in L2 (M−2 , dG) is defined by ̃ Hf̃ = dΓ(ω(−i∇)).

(1.6.22)

−1 From this definition it can be seen that Hf̃ = θM Hf θM , where Hf is the free field ̂ Hamiltonian of ℱb (ℋM ). In the previous section we defined the self-adjoint operator HP = Ĥ f + :P(ϕ(f )): in L2 (Q ) and have obtained a functional integral representation of the semigroup e−tHP by making use of the Euclidean field. The operator

H̃ P = Hf̃ + :P(ξ (f )):

(1.6.23)

−1 is the operator associated with HP in L2 (M−2 , dG) by θM HP θM = H̃ P . We can also coñ −t HP struct a functional integral representation of e in terms of an infinite dimensional Ornstein–Uhlenbeck process which will be investigated next.

1.6.4 Infinite dimensional Ornstein–Uhlenbeck process In this section we define an infinite dimensional Ornstein–Uhlenbeck process as a Hilbert space-valued random process (ξs )s∈ℝ on the set of continuous paths C(ℝ; M−2 ). The one-dimensional Ornstein–Uhlenbeck process (Xt )t∈ℝ has been discussed in Sections 2.4.5 and 5.1.5 in Volume 1, and we have seen that it is a stationary Gaussian Markov process defined on the probability space X = C(ℝ; ℝ) with a measure characterized by having mean and covariance given by 𝔼μ [Xt ] = 0

and

𝔼μ [Xs Xt ] =

σ 2 −|s−t|ω e . 2ω

The construction of an infinite dimensional Ornstein–Uhlenbeck process is similar to this. Denote the set of M−2 -valued continuous functions on the real line ℝ by Y = C(ℝ; M−2 ).

(1.6.24)

1.6 Infinite dimensional Ornstein–Uhlenbeck processes | 101

On Y we introduce the topology derived from the metric 1 [( sup ‖f (x) − g(x)‖M−2 ) ∧ 1] . k 0≤|x|≤k k=0 2 ∞

dist(f , g) = ∑

(1.6.25)

This metric induces the locally uniform topology, giving rise to the Borel σ-field ℬ(Y). In order to construct (ξs )s∈ℝ , we introduce the Euclidean version of M , E and a self-adjoint operator D. Let E be given by Definition 1.79 with ων replaced by ω. Let D be a positive self-adjoint operator with Hilbert–Schmidt inverse on E , and define ‖Dn/2 ⋅‖E

En = C ∞ (D)

.

The main theorem in this section is the following. Theorem 1.129. (1) There exists a probability measure 𝒢 on (Y, ℬ(Y)) and an M−2 -valued continuous random process ℝ ∋ s 󳨃→ ξs ∈ M−2 on the probability space (Y, ℬ(Y), 𝒢 ) such that 𝔼𝒢 [(ξs , ξt )M−2 ] =

1 ∞ 1 ̂ (e , e−|s−t|ω ej )ℋM , ∑ 4 i,j=1 λi λj i

∞ where Spec(𝒟) = {λj }∞ j=1 , and {ej }j=1 are the normalized eigenvectors 𝒟 ej = λj ej . (2) There exists a family of random processes (ξs (f ))s∈ℝ indexed by f ∈ M on the probability space (Y, ℬ(Y), 𝒢 ) such that the following properties hold: (a) linearity: M ∋ f 󳨃→ ξs (f ) ∈ ℝ is a linear map; (b) boundedness: ξs (f ) = ⟨⟨ξs , f ⟩⟩ for f ∈ M+2 , in particular, |ξs (f )| ≤ ‖ξs ‖M−2 ‖f ‖M+2 for f ∈ M+2 ; (c) path continuity: (i) if f ∈ M+2 , then ℝ ∋ s 󳨃→ ξs (f ) ∈ ℝ is almost surely continuous; (ii) if f ∈ M , then ℝ ∋ s 󳨃→ ξs (f ) ∈ L2 (E−2 , d𝒢 ) is strongly continuous; (d) characteristic function: ξs (f ) is a Gaussian random process with respect to 𝒢 with mean zero and

𝔼𝒢 [e

i ∑nj=1 ξsj (fj )

] = exp (−

1 n ̂ ∑ (f , e−|si −sj |ω fj )ℋM ) . 4 i,j=1 i

We will prove Theorem 1.129 below. Definition 1.130 (Infinite dimensional Ornstein–Uhlenbeck process). The M−2 -valued random process (ξs )s∈ℝ on the probability space (Y, ℬ(Y), 𝒢 ) given in Theorem 1.129 is called infinite dimensional Ornstein–Uhlenbeck process. The next lemma follows from Theorem 1.120. Lemma 1.131. There exists a Borel probability measure γ on (E−2 , ℬ(E−2 )) and a family of Gaussian random variables ξ ̃ (f ) indexed by f ∈ E with mean zero such that

102 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes (1) E ∋ f 󳨃→ ξ ̃ (f ) ∈ ℝ is linear; (2) ξ ̃ (f ) = ⟨⟨ξ ̃ , f ⟩⟩E when f ∈ E+2 , where ⟨⟨ξ ̃ , f ⟩⟩E denotes the pairing between E+2 and E−2 ; (3) it satisfies 1

2

̃ ∫ eiξ (f ) dγ(ξ ̃ ) = e− 2 ‖f ‖E . E−2

Take g ∈ M . We have already checked in (1.5.5) that δt ⊗ g ∈ E and (δs ⊗ f , δt ⊗ g)E = (f , e−|s−t|ω g)ℋM ̂

for all f , g ∈ M . For every t ∈ ℝ and g ∈ M , put ξt̃ (g) = ξ ̃ (δt ⊗ g). In particular, it follows that ∫ ξs̃ (f )dγ(ξ ̃ ) = 0, E−2

1 ̂ ∫ ξs̃ (f )ξt̃ (g)dγ(ξ ̃ ) = (f , e−|s−t|ω g)ℋM . 2

E−2

Lemma 1.132. For every fixed t1 < t2 < . . . < tn ∈ ℝ, there exists a Gaussian measure

n 𝒢t1 ,...,tn on M−2 = ×nk=1 M−2 and a family of random variables ξt̃ 1 ,...,tn (g) indexed by g ∈ M n

such that (1) M n ∋ g 󳨃→ ξt̃ 1 ,...,tn (g) ∈ ℝ is linear; n (2) ξt̃ 1 ,...,tn (g) = ⟨⟨ξt̃ 1 ,...,tn , g⟩⟩ when g ∈ M+2 , where ⟨⟨⋅, ⋅⟩⟩ denotes the pairing between n n M+2 and M−2 ; (3) ̃ i ∑n ξ ̃ (g ) (1.6.26) ∫ eiξt1 ,...,tn (g) d𝒢t1 ,...,tn = ∫ e j=1 tj j dγ(ξ ̃ ) n M−2

E−2

n for all g = (g1 , . . . , gn ) ∈ M+2 .

Proof. By the definition of ξt̃ we have ∫e E−2

i ∑nj=1 ξt̃ j (gj )

dγ(ξ ̃ ) = exp ( −

1 n ∑ (δ ⊗ gj , δtl ⊗ gl )E ). 4 j,l=1 tj

The quadratic form n

(f , g) 󳨃→ Qt1 ,...,tn (f , g) = ∑ (δtj ⊗ fj , δtl ⊗ gl )E j,l=1

1.6 Infinite dimensional Ornstein–Uhlenbeck processes | 103

on M n × M n satisfies |Qt1 ,...,tn (f , g)| ≤ ‖f ‖M n ‖g‖M n . Thus there exists a bounded operator Lt1 ,...,tn = L on M n such that Qt1 ,...,tn (f , g) = (f , Lg)M n . Hence we can write ∫e

i ∑nj=1 ξt̃ j (gj )

1

dγ(ξ ̃ ) = e− 4 (g,Lg)M n .

E−2

Let 𝒟n = ⊕n 𝒟. Notice that n

n

(f , L𝒟n−1 g)M n = ∑ (δtj ⊗ fj , δtl ⊗ 𝒟−1 gl )E = ∑ (δtj ⊗ 𝒟−1 fj , δtl ⊗ gl )E i,j=1

i,j=1

=

(𝒟n−1 f , Lg)M n

=

(f , 𝒟n−1 Lg)M n .

Thus [L, 𝒟n−1 ] = 0. By Theorem 1.120, there exist a measure 𝒢t1 ,...,tn and a random variable ξt̃ 1 ,...,tn (g) such that 1

∫ eiξt1 ,...,tn (g) d𝒢t1 ,...,tn = e− 4 (g,Lg)M n , ̃

n M−2

and hence (1.6.26) is obtained. Let J = ℝ×ℕ be the index set and Λ ⊂ J be such that |Λ| < ∞, where |Λ| = #Λ. Write

J Λ Λ M−2 = {f : Λ → M−2 }, let πΛ : M−2 → M−2 be the projection defined by πΛ f = f ⌈Λ , and

define

J

A = {πΛ (E) ∈ M−2 | Λ ⊂ J, |Λ| < ∞, E ∈ ℬ(M−2 )}. −1

|J|

J Now we construct a probability measure 𝒢 on (M−2 , σ(A )) by using Kolmogorov’s extension theorem. J , σ(A )), an M−2 -valued Lemma 1.133. There exists a probability measure 𝒢 on (M−2 random process (ξt )t∈ℝ , and an ℝ-valued random process (ξt (f ))t∈ℝ with f ∈ M on J (M−2 , σ(A ), 𝒢 ) such that the following hold. (1) Its covariance is

𝔼𝒢 [(ξs , ξt )M−2 ] =

1 ∞ 1 ̂ ∑ (e , e−|s−t|ω ej )ℋM , 4 j=1 λj2 j

∞ where {λi }∞ i=1 and {ei }i=1 are given in Section 1.6.3. In particular,

𝔼𝒢 [‖ξt ‖2M−2 ] =

1 Tr(𝒟−2 ). 4

(2) M ∋ f 󳨃→ ξs (f ) ∈ ℝ is linear, and ξs (f ) = ⟨⟨ξs , f ⟩⟩ whenever f ∈ M−2 .

(1.6.27)

104 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes (3) (ξt (f ))t∈ℝ is a Gaussian random process with zero mean and 𝔼𝒢 [e

i ∑nj=1 ξsj (fj )

] = exp (−

1 n ̂ ∑ (f , e−|si −sj |ω fj )ℋM ). 4 i,j=1 i

Proof. Write ẽj = 𝒟−1 ej = λj−1 ej . The set {ẽj }∞ j=1 forms a complete orthonormal system in M+2 . For every k1 , . . . , kn ∈ J with kj = (tj , ij ), and E1 × ⋅ ⋅ ⋅ × En ∈ ℬ(ℝn ) put n

μk1 ,...,kn (E1 × ⋅ ⋅ ⋅ × En ) = 𝔼γ [ ∏ 1Ej (ξt̃ j (ẽij ))]. j=1

(1.6.28)

n By Lemma 1.132 there exists a probability measure 𝒢t1 ,...,tn on M−2 such that n

μk1 ,...,kn (E1 × ⋅ ⋅ ⋅ × En ) = ∫ ∏ 1Ej (ξĩ j )d𝒢t1 ,...,tn , n j=1 M−2

j

where ξĩ j = ξt̃ 1 ,...,tn (0 ⊕ ⋅ ⋅ ⋅ ẽij ⋅ ⋅ ⋅ ⊕ 0). Since the family of set functions {μΛ }Λ⊂J,|Λ| 0. Let g ∈ M+2 . A direct computation gives ̂ 𝔼𝒢 [|ξs (g) − ξt (g)|4 ] = 𝔼γ [|ξs̃ (g) − ξt̃ (g)|4 ] = 3(g, (1 − e−|s−t|ω )g)2ℋ̂

= 3( ∫ ℝd

1−e ω(k)

−|s−t|ω(k)

2

M

2 ̂ |g(k)| dk) ≤ 3|s − t|2 ‖g‖4L2 (ℝd ) .

106 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes Notice that the embedding ι : M+2 → L2 (ℝd ) is bounded with the bound ‖ιg‖L2 (ℝd ) ≤ ‖√ω𝒟−1 ‖‖g‖M+2 . From this it follows that 𝔼𝒢 [|ξs (g) − ξt (g)|4 ] ≤ C‖g‖4M+2 with a constant C > 0. Since ‖ξs ‖M−2 =

sup

f =0,f ̸ ∈M+2

|ξs (f )| , ‖f ‖M+2

there exists fε ∈ M+2 such that ‖ξs − ξt ‖M−2 − ε ≤

|ξs (fε ) − ξt (fε )| , ‖fε ‖M+2

and we obtain that 𝔼𝒢 [‖ξs − ξt ‖4M−2 ] ≤ ε + C|s − t|2 for every ε > 0. Hence (1.6.32) follows. Proof of Theorem 1.129. We denote the continuous version of (ξt )t∈ℝ by ξ ̄ = (ξt̄ )t∈ℝ , and the image measure 𝒢 ∘ ξ ̄ −1 on Y by the same symbol 𝒢 . Thus we have constructed above the probability space (Y, ℬ(Y), 𝒢 ). Parts (1) and (2) of the theorem follow by Lemma 1.133 except for path continuity, which we show now. Whenever f ∈ M+2 , we see that ξs (f ) = ⟨⟨ξs , f ⟩⟩. Hence s 󳨃→ ξs (f ) is continuous almost surely since ℝ ∋ s 󳨃→ ξs ∈ M−2 is continuous. For f ∈ M , lim 𝔼𝒢 [|ξs (f ) − ξt (f )|2 ] = s→t

1 ̂ lim(f , (1 − e−|s−t|ω )f )ℋM = 0. 2 s→t

This gives the desired result.

1.6.5 Markov property The random process (ξt )t∈ℝ is stationary due to the form of its covariance as given by (1.6.27). In Proposition 1.96 we have shown that the Euclidean quantum field has the Markov property. In this section we show that (ξt )t∈ℝ also is a Markov process. Recall that δs ⊗ f ∈ E for f ∈ M , and (ϕE (F), F ∈ E ) is a Euclidean field on (QE , ΣE , μE ). First we show the relationship of ξs̃ (f ) = ξ ̃ (δs ⊗ f ), (ξs )s∈ℝ , and the Euclidean field ϕs (f ) = ϕE (δs ⊗ f ).

1.6 Infinite dimensional Ornstein–Uhlenbeck processes | 107

Proposition 1.135. Let Fj be Borel measurable bounded functions on ℝ, and fj ∈ M . Then n

n

n

j=1

j=1

j=1

𝔼𝒢 [ ∏ Fj (ξtj (fj ))] = 𝔼γ [ ∏ Fj (ξt̃ j (fj ))] = 𝔼μE [ ∏ Fj (ϕtj (fj ))]. Proof. Let F ∈ S (ℝn ). From (1.6.31), it follows that 𝔼𝒢 [F(ξs1 (f1 ), . . . , ξsn (fn ))] n 1 ̌ 1 , . . . , kn )𝔼𝒢 [ei ∑j=1 kj ξsj (fj ) ]dk1 . . . dkn = F(k ∫ (2π)n/2 ℝnd

n ̃ 1 ̌ 1 , . . . , kn )𝔼γ [ei ∑j=1 kj ξsj (fj ) ]dk1 . . . dkn = F(k ∫ (2π)n/2

ℝnd

= 𝔼γ [F(ξs̃ 1 (f1 ), . . . , ξs̃ n (fn ))]. The first equality can be completed by a limiting argument, the second is proven similarly. Lemma 1.136. Let E 󸀠 ⊂ E be a closed subspace, and P : E → E 󸀠 be the corresponding orthogonal projection. Consider GP = σ(ξ ̃ (f ) | f ∈ E 󸀠 ). Then for every α ∈ ℂ and f ∈ E it follows that α2

2

𝔼γ [eαξ (f ) |GP ] = eαξ (Pf ) e 2 ‖f −Pf ‖E . ̃

̃

Proof. We have that 𝔼γ [eαξ (f ) |GP ] = 𝔼γ [eαξ (Pf ) eαξ (f −Pf ) |GP ] = eαξ (Pf ) 𝔼γ [eαξ (f −Pf ) |GP ], ̃

̃

̃

̃

̃

(1.6.33)

since ξ ̃ (Pf ) is GP -measurable. Moreover, since ξ ̃ (f − Pf ) is independent of GP , we have α2

2

(1.6.33) = eαξ (Pf ) 𝔼γ [eαξ (f −Pf ) ] = eαξ (Pf ) e 2 ‖f −Pf ‖E . ̃

̃

̃

For any interval I ⊂ ℝ, write FI = σ(L. H. {ξt (f ) | t ∈ I, f ∈ M }).

Theorem 1.137 (Markov property). Let F be a bounded F[s,∞) -measurable function. Then 𝔼𝒢 [F|F(−∞,s] ] = 𝔼𝒢 [F|F{s} ].

(1.6.34)

Proof. Denote by E(−∞,s] the closed subspace of E generated by the linear hull of the set {δr ⊗ f ∈ E | r ≤ s, f ∈ M }, and similarly by E{s} the closed subspace generated

108 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes by{δs ⊗ f ∈ E | f ∈ M }. Write P(−∞,s] and P{s} for the corresponding projections. We claim that for every t ≥ s and every g ∈ M , P(−∞,s] (δt ⊗ g) = δs ⊗ e−(t−s)ω g = P{s} (δs ⊗ e−(t−s)ω g). ̂

̂

(1.6.35)

Indeed, δs ⊗ e−(t−s)ω̂ g ∈ E(−∞,s] , and for all r ≤ s, dk ̂ ̂ (δr ⊗ f , δt ⊗ g)E = ∫ e−(s−r)ω(k) f ̂(k)(e−(t−s)ω g) ̂ (k) = (δr ⊗ f , δs ⊗ e−(t−s)ω g)E . 2ω(k) ℝd

From this we get (F, δt ⊗ g)E = (F, δs ⊗ e−(t−s)ω g)E , ̂

F ∈ E(−∞,s] ,

by linearity and approximation, and thus the first equality in (1.6.35) is shown. The second equality now follows from the fact that δs ⊗ e−(t−s)ω̂ g is not only in E(−∞,s] but also in E{s} . By Lemma 1.136, we furthermore have 𝔼γ [eαξt (g) |GP(−∞,s] ] = eαξ (P(−∞,s] δt ⊗g) exp ( ̃

̃

α2 ‖(1 − P(−∞,s] )δt ⊗ g‖2E ) , 2

and Proposition 1.135 and (1.6.33) give 𝔼𝒢 [eαξt (g) |F(−∞,s] ] = eαξs (e

−(t−s)ω̂

g)

exp (

α2 ̂ (g, (1 − e−2(t−s)ω )g)M ) ∈ F{s} 4

for t ≥ 0. Again by approximation, we find 𝔼𝒢 [F|F(−∞,s] ] ∈ F{s} for all bounded F[s,∞) -measurable functions. Thus (1.6.34) follows.

1.6.6 Regular conditional Gaussian probability measures The Gaussian measure 𝒢 is a probability measure on (Y, ℬ(Y)) and ξ0 : Y → M−2 is a random variable. Since both Y = C(ℝ; M−2 ) and M−2 are Polish spaces, we can define the regular conditional probability measure 𝒢 ξ with ξ ∈ M−2 by ξ

𝒢 (⋅) = 𝒢 (⋅ | ξ0 = ξ ).

In particular, we may assume that ξ

𝒢 (ξ0 ∈ B) = 1ξ (B),

B ∈ ℬ(M−2 ),

and ξ

𝒢 (ξ0 = ξ ) = 1,

𝒢 ∘ ξ0 -a. s. ξ . −1

1.6 Infinite dimensional Ornstein–Uhlenbeck processes | 109

ξ

ξ

We compute 𝔼𝒢 [ξt (f )] and 𝔼𝒢 [ξt (f )ξs (g)] for later use. Note that 𝔼𝒢 [e

∑nj=1 ξtj (fj )

|F{0} ] = 𝔼𝒢 [e

∑nj=1 ξtj (fj )

(1.6.36)

|σ(ξ0 )].

It is seen that the right-hand side of (1.6.36) is a function of ξ0 and then we write h(ξ0 ). ξ

𝔼𝒢 [e

∑nj=1 ξtj (fj )

] is defined by h(ξ0 ) with ξ0 replaced by ξ .

Theorem 1.138. If f , g ∈ M and ξ ∈ M−2 , then ξ

𝔼𝒢 [ξt (f )] = ξ (e−|t|ω f ),

(1.6.37)

̂

1 1 ̂ ̂ ̂ ̂ ξ 𝔼𝒢 [ξt (f )ξs (g)] = (f , e−|s−t|ω g)ℋM − (f , e−(|t|+|s|)ω g)ℋM + ξ (e−|t|ω f )ξ (e−|s|ω g). 2 2 (1.6.38)

Proof. In a similar way to the proof of Theorem 1.137, we see that 𝔼𝒢 [e where

∑nj=1 ξtj (fj )

n

n

|F{r} ] = e∑j=1 ξr (e

−|tj −r|ω̂ f

j)

eQ/4 ,

(1.6.39)

Q = ∑ ((fi , e−|ti −tj |ω fj )ℋM − (fi , e−|ti −r|ω e−|tj −r|ω fj )ℋM ) . ̂

̂

̂

i,j=1

Let α, β ∈ ℝ. Inserting αf and βg into f1 and f2 in (1.6.39), respectively, and setting n = 2 and r = 0, we have 𝔼𝒢 [eαξt (f )+βξs (g) |σ(ξ0 )] = eαξ0 (e

−|t|ω̂

f )+βξ0 (e−|s|ω̂ g) Q/4

e

,

where Q = α2 (‖f ‖2ℋM − (f , e−2|t|ω f )ℋM ) + β2 (‖g‖2ℋM − (f , e−2|s|ω g)ℋM ) ̂

̂

+ 2αβ((f , e−|s−t|ω g)ℋM − (f , e−(|t|+|s|)ω g)ℋM ). ̂

̂

Hence ξ

𝔼𝒢 [eαξt (f )+βξs (g) ] = eαξ (e

−|t|ω̂

f )+βξ (e−|s|ω̂ g) Q/4

e

.

ξ

Equality (1.6.37) is derived through 𝜕α 𝔼𝒢 [eαξt (f )+βξs (g) ]α=0 , while formula (1.6.38) can be ξ

obtained via 𝜕α 𝜕β 𝔼𝒢 [eαξt (f )+βξs (g) ]α=β=0 .

Using Theorem 1.138 we can also compute the covariance to be ξ

ξ

ξ

cov(ξt (f ); ξt (g)) = 𝔼𝒢 [ξt (f )ξt (g)] − 𝔼𝒢 [ξt (f )]𝔼𝒢 [ξt (g)]

1 1 ̂ ̂ = (f , e−|s−t|ω g)ℋM − (f , e−(|t|+|s|)ω g)ℋM . 2 2

In particular, cov(ξt (f ); ξt (g)) is independent of ξ .

110 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes 1.6.7 Feynman–Kac–Nelson formula by path measures By making use of 𝒢 , an alternative functional integral representation of (F, e−tHP G) in Theorem 1.98 can be given. Define H̃ P = Hf̃ +̇ :P(ξ (f )):,

f ∈ M,

where we recall that Hf̃ denotes the free field Hamiltonian on L2 (M−2 , dG). Theorem 1.139 (Feynman–Kac–Nelson formula). Let F, G ∈ L2 (M−2 , dG). Then (F, e−t HP G) = 𝔼𝒢 [F(ξ0 )G(ξt )e− ∫:P(ξs (f )):ds ]. ̃

Proof. Let F(ξ ) = P1 (ξ (f1 ), . . . , ξ (fn )) and G(ξ ) = P2 (ξ (f1 ), . . . , ξ (fm )) with a polynomial Pj . In the proof of Theorem 1.98, we have n−1

̃ (f )): −1 −1 −1 −1 F(ξ ̃ )e− ∑j=1 (tj/n):P(ξtj/n (θW F, e−tHP θW G)ℱb = lim 𝔼γ [θW θW G(ξt̃ )]. 0 n→∞

−1 Thus by the unitary equivalence θW e−tHP θW = e−t HP and Proposition 1.135 we see that ̃

n−1

(F, e−t HP G)L2 (M−2 ,dG) = lim 𝔼𝒢 [F(ξ0 )e− ∑j=1 (tj/n):P(ξtj/n (f )): G(ξt )] ̃

n→∞

= 𝔼𝒢 [F(ξ0 )e− ∫:P(ξs (f )):ds G(ξt )]. Here we used the strong continuity of s 󳨃→ ξs (f ). By a limiting argument F, G can be extended to F, G ∈ L2 (M−2 , dG). Corollary 1.140 (Generator and invariant measure of (ξt )t∈ℝ ). The generator of (ξt )t∈ℝ is −H̃ f and its invariant measure is G. Proof. Let F, G ∈ L2 (M−2 , dG). By Theorem 1.139 we have (F, e−t Hf G) = 𝔼𝒢 [F(ξ0 )G(ξt )]. ̃

Hence the generator of (ξt )t∈ℝ is −H̃ f . Let p(t, ξ , A) be the probability transition kernel of (ξt )t∈ℝ . Since H̃ f 1 = 0, we have ∫ p(t, ξ , A)dG = (1, e−t Hf 1A )L2 (M−2 ) = G(A). ̃

M−2

This implies that G is an invariant measure for (ξt )t∈ℝ . Making use of the regular conditional probability measure 𝒢 ξ , we can derive the ̃ functional integral representation of e−t HP .

1.7 Connection with infinite dimensional stochastic analysis and Malliavin calculus | 111

Corollary 1.141. Let G ∈ L2 (M−2 , dG). Then ξ

(e−t HP G)(ξ ) = 𝔼𝒢 [e− ∫:P(ξs (f )):ds G(ξt )], ̃

a. e. ξ ∈ M−2 .

Proof. The proof is straightforward. By Theorem 1.139 we have ξ

(F, e−t HP G) = 𝔼G [F(ξ )𝔼𝒢 [G(ξt )e− ∫:P(ξs (f )):ds ]]. ̃

Since F is arbitrary, the corollary follows.

1.7 Connection with infinite dimensional stochastic analysis and Malliavin calculus 1.7.1 Finite dimensional case While the subject of this brief section differs from the main theme of this book, here we point out some important connections with infinite dimensional analysis, which is a language that some readers may find more familiar. Euclidean quantum field theory and commutative or non-commutative infinite dimensional stochastic analysis have a deep relationship, which would deserve and require a book-length discussion, therefore we need to refer the interested reader to a well-established literature for details. Here we will only highlight some basic concepts and facts, and mostly will have to skip proofs. First we make our observations in one dimension. Let (ℝ, ℬ(ℝ), P) be a probability space, and a standard Gaussian random variable be given on it. Denote by P(E) =

x2 1 ∫ e− 2 dx, √2π

E ∈ ℬ(ℝ),

E

the related standard Gaussian measure, and 𝔼[f ] = ∫ℝ f (x)P(dx). Denote ℕ̇ = {0} ∪ ℕ. Consider L2 (ℝ, dP) and its dense subspace n

̇ ℒ = L. H. {x | n ∈ ℕ}. Also, consider the differentiation operator df (x) =

df (x) dx

with the domain Dom(d) = ℒ. (To avoid confusion with operators, in this section we denote operator domain by Dom.) Its adjoint d∗ in L2 (ℝ, dP) is obtained by a straightforward calculation.

112 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes Proposition 1.142. For every f ∈ ℒ we have d∗ f (x) = −df (x) + xf (x),

x ∈ ℝ.

Proof. Integration by parts gives ∫ df (x) ⋅ h(x)dP(x) = − ∫ f (x)d∗ h(x)dP(x), ℝ

h ∈ ℒ.

ℝ

A similar calculation and an extension argument give the corollary below. Corollary 1.143. On ℒ we have [d, d∗ ] = 1, where the right-hand side denotes the identity operator in L2 (ℝ, dP). For higher derivatives we write dn to mean d repeated n times, and similarly we write (d∗ )n to mean n iterations of d∗ . It is understood that d0 = 1 = (d∗ )0 . Consider the operator N = d∗ d

(1.7.1)

on L2 (ℝ, dP). On ℒ we have Nf (x) = −

df (x) d2 f (x) +x . dx dx 2

This means that N is the generator of an Ornstein–Uhlenbeck process with state space ℝ and path measure given by P. As seen in Section 5.1.5 in Volume 1, its eigenfunctions are given by the Hermite polynomials x2

Hn = (−1)n e 2

dn − x22 e , dxn

n ∈ ℕ,̇

(1.7.2)

̇ holds. It can be obtained directly from (1.7.2) that so that NHn = nHn , n ∈ ℕ, Hn = (d∗ )n 1,

n ∈ ℕ,̇

(1.7.3)

and it is also direct to see that the set of functions {

1 H } √n! n n∈ℕ̇

forms a complete orthonormal system in L2 (ℝ, dP).

(1.7.4)

1.7 Connection with infinite dimensional stochastic analysis and Malliavin calculus | 113

Remark 1.144 (Hermite polynomial). In Section 5.1.5 in Volume 1 we used Hermite polynomials defined as n 2 d 2 H̄ n = (−1)n ex e−x , n dx

n ∈ ℕ.̇

(1.7.5)

See also (2.6.26) below. Although the two definitions (1.7.2) and (1.7.5) are not exactly identical, each is a rescaling of the other as n

H̄ n (x) = 2 2 Hn (√2 x) ,

n x Hn (x) = 2− 2 H̄ n ( ). √2

Hermite polynomials defined by (1.7.5) are orthogonal with respect to a Gaussian weight function so that ∞

2 1 ∫ H̄ m (x)H̄ n (x)e−x dx = 2n n!δmn √π

−∞

holds, hence {

1 H̄ } √n!2n n n∈ℕ̇

(1.7.6)

2

forms a complete orthonormal system in L2 (ℝ, e−x /√πdx). They also satisfy (−

1 d2 d + x ) H̄ n (x) = nH̄ n (x). 2 2 dx dx

The functions defined by (1.7.5) are sometimes called physicists’ Hermite polynomials, and those defined by (1.7.2) are called probabilists’ Hermite polynomials. The above lead to the following expansion formula. Proposition 1.145. For every f ∈ C ∞ (ℝ) with

dn f dxn

∈ L2 (ℝ, dP), n ∈ ℕ,̇ we have

1 𝔼P [dn f ]Hn . n! n=0 ∞

f = ∑

(1.7.7)

Proof. Since the functions (1.7.4) also belong to C ∞ (ℝ), we have for every f ∈ C ∞ (ℝ) that there exists a sequence of numbers c0 , c1 , . . . ∈ ℝ such that ∞

f (x) = ∑ cn Hn (x), n=0

a.e. x ∈ ℝ.

Using orthogonality, it follows that cn =

1 1 ∫ f (x)Hn (x)dP(x) = 𝔼P [fHn ]. n! n! ℝ

114 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes By Proposition 1.142 and equality (1.7.3) we furthermore have 𝔼P [fHn ] = 𝔼P [f (d∗ )n 1] = 𝔼P [dn f ], proving the claim. Next we consider d ≥ 1 and an ℝd -valued Gaussian random variable with related probability measure P(E) =

|x|2 1 ∫ e− 2 dx, d/2 (2π)

E ∈ ℬ(ℝd ),

E

and again write 𝔼P [f ] = ∫ℝd f (x)dP(x) to keep the notation simple. Recall that, in general, given an open subset U ⊂ ℝn , a function f : U → ℝm is said to have a directional derivative at x ∈ U in direction y ∈ ℝn if the limit Dy f (x) = lim ε↓0

f (x + εy) − f (x) ∈ ℝm ε

(1.7.8)

exists. With the canonical basis vectors e1 , . . . , en of ℝn the derivative Dej f (x) = 𝜕xj f (x) = 𝜕j f (x) is the partial derivative of f with respect to xj . Furthermore, f is called differentiable at x ∈ U whenever there exists an m × n-matrix D(x) such that lim z→0 z∈ℝn

|f (x + z) − f (x) − D(x)z| = 0. |z|

(1.7.9)

Then the classical fact is that differentiability of f at x ∈ U implies that Dy f exists for all y ∈ ℝn and Dy f (x) = D(x)y holds, and conversely, if all 𝜕j f (x) for x ∈ U exist and are continuous functions of x, then f is differentiable at x and 𝜕j fi (x) = Dij (x), where fi (x) is the ith component of f (x). Returning to our setting, with the canonical basis vectors e1 , . . . , ed of ℝd we thus define the d-dimensional counterpart of d to be the partial derivatives 𝜕j in direction ej . Its adjoint in L2 (ℝd , dP) is now 𝜕j∗ f (x) = −𝜕j f (x) + xj f (x). Corollary 1.143 becomes [𝜕j , 𝜕k∗ ] = δjk 1,

j, k = 1, . . . , d.

To define higher order derivatives and their adjoints, consider the map ℓ : {1, . . . , d} → ̇ Then for every ℓ ∈ ℳd we define ℕ̇ and the set ℳd = {ℓ | ℓ : {1, . . . , d} → ℕ}. 𝜕ℓ = 𝜕1ℓ(1) ⋅ ⋅ ⋅ 𝜕dℓ(d)

and

(𝜕∗ )ℓ = (𝜕1∗ )ℓ(1) ⋅ ⋅ ⋅ (𝜕d∗ )ℓ(d) .

1.7 Connection with infinite dimensional stochastic analysis and Malliavin calculus | 115

The operator corresponding to (1.7.1) is now d

N = ∑ 𝜕j∗ 𝜕j , j=1

and the eigenvalue equation NHℓ = |ℓ|Hℓ ,

d

|ℓ| = ∑ ℓ(j), j=1

ℓ ∈ ℳd ,

is satisfied by the d-dimensional Hermite polynomials Hℓ (x) = Πdj=1 Hℓ(j) (xj ) = (𝜕∗ )ℓ 1ℝd , where 1ℝd denotes the identity function on ℝd . A complete orthonormal system of L2 (ℝd , dP) is given by {

1 Hℓ } . Πdj=1 ℓ(j)! ℳd

Finally, Proposition 1.145 generalises now as below. Proposition 1.146. For every f ∈ C ∞ (ℝd ) with f = ∑ ℓ∈ℳd

dn f dxn

∈ L2 (ℝd , dP), n ∈ ℕ,̇ we have

1 𝔼P [𝜕ℓ f ]Hℓ . Πdj=1 ℓ(j)!

(1.7.10)

From the above it is clear that the operators 𝜕∗ , 𝜕, and N are reminiscent of the creation, annihilation, and number operators, respectively. Next we discuss their counterparts on infinite dimensional spaces, and show that the analogies observed in finite dimensions are indeed valid.

1.7.2 Stochastic derivative and Cameron–Martin space We consider the counterparts of (1.7.8)–(1.7.9) in Banach spaces. Let B be a general Banach space over ℝ with norm ‖⋅‖B , U ⊂ B an open subset in the topology generated by the given norm, and f : U → ℝm . Then the function f is said to have a Gâteaux (directional) derivative Dy f (x) in direction y ∈ B at x ∈ U if the similar expression at the right-hand side of (1.7.8) exists, and f is called Fréchet differentiable at x if there exists a bounded linear map D : B → ℝm with components Dj , j = 1, . . . , m, in the dual space B∗ of B such that the expression as the right-hand side of (1.7.9) exists with |z| replaced by ‖z‖B . Furthermore, Gâteaux and Fréchet differentiability are related similarly as in finite dimensional Euclidean spaces.

116 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes In what follows we restrict to a specific Banach space by choosing B = X = C(ℝ+ ; ℝ), and consider the standard Wiener space (X , ℬ(X ), 𝒲 ), where 𝒲 is Wiener measure defined in Definition 2.95 in Volume 1, and used throughout in this book. We note that the constructions that we will present are possible to do for the more general class of so-called isonormal Gaussian processes, but we only discuss it for one-dimensional Brownian motion (Bt )t≥0 on standard Wiener space. Also, we note that, since there exists no Lebesgue type measure on C(ℝ+ ; ℝ) or in other infinite dimensional spaces, the role of a suitable (Gaussian) measure on the given space is now clear, which does have an infinite dimensional extension. We denote expectation with respect to 𝒲 by 𝔼. First we consider the case when Brownian motion is run in a bounded timeinterval only. Let T > 0 be arbitrary, and take the interval [0, T]. We denote the projection of X to this interval by XT = C([0, T]; ℝ). From the construction of standard Wiener space we know that the Borel σ-field ℬ(XT ) coincides with the σ-field generated by cylinder sets. It will then suffice to consider derivatives of functions F in n

𝒞 (XT ) = ⋃ {F : XT → ℝ | ∃ f ∈ Cpoly (ℝ ), ∃ 0 < t1 < . . . < tn < T such that ∞

n∈ℕ

F(ω) = f (ω(t1 ), . . . , ω(tn )), ω ∈ XT } called smooth cylinder functions, where ∞ Cpoly (ℝn ) = {f ∈ C ∞ (ℝn ) | ∀k = (k1 , . . . , kn ) ∈ ℕ̇ n , ∃ polynomial pk with

̄ x = (x1 , . . . , xn ), x̄ = (|x1 |, . . . , |xn |)}, |f (k) (x)| ≤ pk (x), and f (k) denote usual derivatives. It is known that 𝒞 (XT ) is dense in Lp (XT , d𝒲 ) for every p ≥ 1. In order to construct a directional derivative, we need translations of paths ω ∈ XT . Define the operator θϖ : XT → XT to be θϖ (ω) = ω + ϖ, i. e., a pathwise translation by ϖ: θϖ (ω)(t) = ω(t) + ϖ(t) for 0 ≤ t ≤ T. The map θϖ is measurable for every ϖ and it is an XT -valued random variable with induced probability measure −1 𝒲 ϖ = 𝒲 ∘ θϖ . Consider the subset of paths t

2

CT = {h ∈ XT | ∃ gh ∈ L ([0, T], dt) such that h(t) = ∫ gh (s)ds, t ∈ [0, T]}. 0

Denoting gh corresponding to h by h,̇ under the inner product T

(h1 , h2 ) = ∫ ḣ 1 (t)ḣ 2 (t)dt, 0

1.7 Connection with infinite dimensional stochastic analysis and Malliavin calculus | 117

the set CT becomes a Hilbert space, called Cameron–Martin space. An application of the Girsanov theorem then implies that whenever ϖ ∈ CT , the measure 𝒲 ϖ is absolutely continuous with respect to Wiener measure, with Radon–Nikodým derivative T

T

d𝒲 ϖ 1 ̇ ̇ 2 ds). = exp ( ∫ ϖ(s)dB s − ∫ ϖ(s) d𝒲 2 0

(1.7.11)

0

Furthermore, it is known that that 𝒲 ϖ is absolutely continuous with respect to 𝒲 if and only if ϖ ∈ CT . The implication is then that there is a natural limitation in taking Gâteaux derivatives of Brownian paths in arbitrary directions, and to keep absolute continuity the directions need to be selected from Cameron–Martin space. Then for every ϖ ∈ CT we can consider the directional derivative (F ∘ θεϖ )(ω) − F(ω) . ε→0 ε

Dϖ F(ω) = lim

Definition 1.147 (Stochastic differentiability). Let F : XT → ℝ, and suppose that Dϖ F exists as a limit in L2 (XT ), for every ϖ ∈ CT . Also, assume that there exists a random T process (ut )t≥0 with the property ∫0 𝔼[u2t ]dt < ∞ such that ∞

̇ Dϖ F(ω) = ∫ ut (ω)ϖ(t)dt,

for every ϖ ∈ CT .

0

Then F is said to be stochastically differentiable, and its stochastic derivative is Dt F(ω) = ut (ω),

𝒲 − a. s.

(1.7.12)

The set of stochastically differentiable random variables is denoted by 𝒟1,2 . Note that DF = Dt F(ω) ∈ L2 (XT × [0, T]). The above concept can be applied to all F = F(ω) = f (ω(t1 ), . . . , ω(tn )) ∈ 𝒞 (XT ), and we get n

Dϖ F(ω) = ∑ j=1

𝜕f (ω(t1 ), . . . , ω(tn ))1[0,tj ] . 𝜕xj

(1.7.13)

To obtain the adjoint of the directional derivative Dϖ , we use the following integration by parts formula. Proposition 1.148. For every ϖ ∈ CT and F, G ∈ 𝒞 (XT ) we have 𝔼[Dϖ F ⋅ G] = 𝔼[FD∗ϖ G] with

T

D∗ϖ G

̇ = −Dϖ G + ∫ ϖ(t)dB t. 0

118 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes Proof. By the Cameron–Martin formula we have 𝔼[F ∘ θεϖ G] − 𝔼[FG] = ∫ F(ω + εϖ)G(ω)d𝒲 (ω) − ∫ F(ω)G(ω)d𝒲 (ω) XT

XT

= ∫ F(ω)G(ω − εϖ)d𝒲 εϖ (ω) − ∫ F(ω)G(ω)d𝒲 (ω) XT

XT

= − ∫ F(ω) (G(ω) − G(ω − εϖ) XT

d𝒲 εϖ (ω)) d𝒲 (ω). d𝒲

Then on dividing by ε, taking the limit ε → 0, using (1.7.11), dominated convergence, and the properties of 𝒞 (XT ), the claim follows. 1.7.3 Malliavin derivative and divergence operator on L2 (X ) In this section we take ℝ+ instead of bounded intervals, and consider Brownian motion on (X , ℬ(X ), 𝒲 ). The space L2 (ℝ+ ) is a Hilbert space with inner product ∞

⟨f , g⟩ = ∫ f ̄(t)g(t)dt. 0

For f ∈ L2 (ℝ+ ) consider the stochastic integral ∞

I(f ) = ∫ f (t)dBt 0

of the non-random function f . Using the stochastic integral, Brownian motion Bt can be re-expressed by choosing f (t) = 1[0,t] . Due to the Itô isometry, for each f the linear functional I(f ) is a Gaussian random variable with mean zero and variance ‖f ‖2L2 (ℝ+ ) = ⟨f , f ⟩. Also, consider n

𝒮 = ⋃ {F : X → ℝ | ∃ h ∈ Cpoly (ℝ ), ∃ 0 < t1 < . . . < tn such that ∞

n∈ℕ

F = h(I(f1 ), . . . , I(fn )), f1 , . . . , fn ∈ L2 (ℝ+ )}. The set 𝒮 is dense in L2 (ℝ+ ). Inspired by (1.7.13) we have the following concept. Definition 1.149 (Malliavin derivative on L2 (X )). For F = h(I(f1 ), . . . , I(fn )) ∈ 𝒮 the L2 (ℝ+ )-valued random variable n

Dt F = ∑ j=1

𝜕h (I(f1 ), . . . , I(fn ))fj (t) 𝜕xj

is called the Malliavin derivative of F.

1.7 Connection with infinite dimensional stochastic analysis and Malliavin calculus | 119

Since 𝔼[‖Dt F‖2L2 (ℝ+ ) ] < ∞, it is seen that Dt is a map Dt : L2 (X ) → L2 (X ; L2 (ℝ+ )). We use the notation ∞

Dg F = ⟨DF, g⟩ = ∫ (Dt F)(t)g(t)dt. 0

Higher order derivatives can similarly be defined. For F ∈ 𝒮 we have for k ∈ ℕ k

n

𝜕k h (I(f1 ), . . . , I(fn ))fj1 (t1 ) ⋅ ⋅ ⋅ fjk (tk ) 𝜕xj1 ⋅ ⋅ ⋅ 𝜕xjk i=1 j =1

Dkt1 ,...,tk F = Dt1 ⋅ ⋅ ⋅ Dtk F = ∑ ∑ i

and it is clear that Dk : L2 (X ) → L2 (X ; ⊗k L2 (ℝ+ )). We use the identifications L2 (X ; ⊗k L2 (ℝ+ )) ≅ L2 (X ) ⊗ (⊗k L2 (ℝ+ )) ≅ L2 (X ) ⊗ L2 (ℝk+ ) ≅ L2 (X × ℝk+ ). We also define the adjoint of the Malliavin derivative. Consider the family of smooth cylinder processes 𝒮 = ⋃ {u : X × ℝ → ℝ | ∃ F1 , . . . , Fn ∈ 𝒮 , ∃ f1 , . . . , fn ∈ L2 (ℝ+ ) 󸀠

+

n∈ℕ

such that ut = ∑nj=1 Fj fj (t)}. Definition 1.150 (Divergence operator on L2 (X )). The linear operator δ : L2 (X × ℝ+ ) → L2 (X ) called divergence operator is defined by the equality 𝔼[Fδ(u)] = 𝔼[⟨DF, u⟩],

F ∈ 𝒮 , u ∈ 𝒮 󸀠.

Integration by parts applied to this definition gives the following expression. Proposition 1.151. If F = h(I(f1 ), . . . , I(fn )) ∈ 𝒮 and u = ∑nj=1 Fj fj (t) ∈ 𝒮 󸀠 , then n

n

j=1

j=1

δ(u) = ∑ Fj I(fj ) − ∑⟨DFj , fj ⟩. Example 1.152. We have Dt F = f (t) for F = I(f ), and δ(f ) = I(f ) for f ∈ L2 (ℝ+ ).

(1.7.14)

120 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes Proposition 1.153. Let F ∈ 𝒮 , u ∈ 𝒮 󸀠 and g ∈ L2 (ℝ+ ). Then (1) Dg δ(u) = δ(Dg u) + ⟨g, u⟩; (2) δ(Fu) = Fδ(u) − ⟨DF, u⟩; (3) for every v ∈ 𝒮 󸀠 ∞∞

∞

𝔼[δ(u)δ(v)] = 𝔼[ ∫ ut vt dt] + 𝔼[ ∫ ∫ (Ds ut )(Dt vs )dsdt]. 0 0

0

The following observation allows to extend the Malliavin operators to more singular functions. Lemma 1.154. The Malliavin derivative Dk ⌈𝒮 is a closable operator from Lp (X ) to Lp (X ; ⊗k L2 (ℝ+ )), for every p ≥ 1. We denote the closed extension Dk ⌈𝒮 by the same label Dk . Define the family of norms k

p/2

j

‖F‖k,p = (𝔼[|F|p ] + 𝔼[ ∑ ( ∫ |Dt1 ,...,tj F|2 dt1 ⋅ ⋅ ⋅ dtj ) j=1

1/p

])

,

k ∈ ℕ, p ≥ 1,

(R+ )n

and denote 𝔻k,p = 𝒮 k,p . Then the domain of the extended kth order Malliavin derivative Dk is Dom(Dk ) = 𝔻k,p , in particular, since 𝒮 is a core of Dk , we see that F ∈ 𝔻k,p if ‖⋅‖

Lp (X )

Lp (X ;⊗k L2 (ℝ+ ))

󳨀→ and only if there exists (Fn )n∈ℕ ⊂ 𝒮 such that Fn 󳨀→ F and Dk Fn as n → ∞. Let p = 2 and k = 1. Then 𝔻1,2 is a Hilbert space with inner product

Dk F

(F, G)𝔻1,2 = 𝔼[FG] + 𝔼[⟨DF, DG⟩]. A random variable F ∈ 𝔻1,2 is called Malliavin differentiable. The divergence operator δ can also be extended by requiring (1.7.14) to hold for every F ∈ 𝔻1,2 . Since δ is the adjoint of D, it is a closed operator so that if (un )n∈ℕ ⊂ 𝒮 󸀠 is such that un

L2 (X ;L2 (ℝ+ ))

L2 (X )

u and δ(un ) 󳨀→ G as n → ∞, then u ∈ Dom(δ) and ‖⋅‖ ̃k,p = 𝒮 󸀠 k,p . δ(u) = G. We also consider the closure 𝔻 󳨀→

Proposition 1.155. The formulae in Proposition 1.153 can be shown to hold also for the extended operators in the following sense. ̃2,2 and g ∈ L2 (ℝ+ ), then Dg u ∈ Dom(δ), δ(u) ∈ 𝔻1,2 , and (1) If u ∈ 𝔻 Dg δ(u) = δ(Dg u) + ⟨g, u⟩ holds. (2) If F ∈ 𝔻1,2 and u ∈ Dom(δ) such that Fu ∈ L2 (X ; L2 (ℝ+ )), then δ(Fu) = Fδ(u) − ⟨DF, u⟩ whenever the right-hand side is square-integrable.

1.7 Connection with infinite dimensional stochastic analysis and Malliavin calculus | 121

̃1,2 , then (3) If u, v ∈ 𝔻 ∞∞

∞

𝔼[δ(u)δ(v)] = 𝔼[ ∫ ut vt dt] + 𝔼[ ∫ ∫ (Ds ut )(Dt vs )dsdt]. 0 0

0

Remark 1.156. The following chain rule applies for the Malliavin derivative. Let ϕ ∈ C 1 (ℝ) with the property that there exist λ ≥ 0, C > 0 such that |ϕ󸀠 (x)| ≤ C(1 + |x|λ ). If p F ∈ 𝔻1,p with p ≥ λ + 1, then ϕ(F) ∈ 𝔻1,q with q = λ+1 and Dϕ(F) = ϕ󸀠 (F)DF. Remark 1.157 (Malliavin derivative and stochastic derivative). The Malliavin derivative on 𝔻1,2 is consistent with the stochastic derivative given in Definition 1.147, and on the space 𝔻1,2 ∩ 𝒟1,2 the two operators coincide. Remark 1.158 (Divergence operator and Skorokhod integral). If u ∈ Dom(δ) is a progressively measurable random process, then ∞

δ(u) = ∫ ut dBt , 0

i. e., it is a stochastic integral. When (ut )t≥0 is not adapted, δ(u) can be obtained as a more general stochastic integral called Skorokhod integral. This can be expressed as ∞

δ(u) = ∫ ut δBt . 0

For details we refer to the literature.

1.7.4 Wiener–Itô chaos expansion Recall the class M 2 (S, T) of square integrable processes, adapted to the augmented filtration (FtBM )t≥0 of Brownian motion, introduced in Definition 2.94 in Volume 1. A basic property of Itô integrals with integrands in M 2 (S, T) is that they are martingales with respect to the same filtration (Theorem 2.132 in Volume 1). The following converse statements also hold. Theorem 1.159 (Martingale representation theorem). If (Mt )t≥0 is a martingale with respect to (FtBM )t≥0 such that Mt ∈ L2 (X ), t ≥ 0, then there exists a unique random process (ft )t∈[0,T] ∈ M 2 (0, T), for all T > 0, such that t

Mt = 𝔼[Mt ] + ∫ fs dBs . 0

122 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes In particular, for a random variable X ∈ L2 (X ), the random process (Mt )t≥0 with Mt = 𝔼[X|FtBM ], is a martingale, on which applying the martingale representation theorem gives the theorem below. Theorem 1.160 (Itô representation theorem). Let T > 0. If F ∈ L2 (X ) is an FTBM -measurable random variable, then there exists a unique random process (ft )t∈[0,T] ∈ M 2 (0, T) such that T

F = 𝔼[F] + ∫ ft dBt .

(1.7.15)

0

At each fixed t, we can apply the Itô representation theorem to the random variable ft in (1.7.15) again. This gives that for each t ∈ [0, T] there exists a unique random process (gs,t )s∈[0,t] ∈ M 2 (0, t) such that t

ft = 𝔼[ft ] + ∫ gs,t dBs . 0

Inserting this in (1.7.15) furthermore gives T

T

t

F = 𝔼[F] + ∫ 𝔼[ft ]dBt + ∫ dBt ∫ gs,t dBs . 0

0

0

Iterating this procedure indefinitely leads to a canonical representation of an arbitrary random variable in L2 (X ), which we discuss next. Take an arbitrary n ∈ ℕ and consider a function f : (ℝ+ )n → ℝ. The function f is called symmetric if f (tπ(1) , . . . , tπ(n) ) = f (t1 , . . . , tn ),

for all π ∈ ℘n , t1 , . . . , tn ∈ ℝ+ ,

where ℘n denotes the permutation group of order n. The symmetrization of a function f is defined by f ̃(t1 , . . . , tn ) =

1 ∑ f (t , . . . , tπ(n) ). n! π∈℘n π(1)

Clearly, f is symmetric if and only if f ̃ = f . We denote the space of square-integrable symmetric functions with respect to Lebesgue measure on (ℝ+ )n by L2sym (ℝn+ ). For f ∈ L2 (ℝ+ ) consider again the stochastic integral ∞

I(f ) = ∫ f (t)dBt . 0

1.7 Connection with infinite dimensional stochastic analysis and Malliavin calculus | 123

Definition 1.161 (Iterated stochastic integral). Let n ∈ ℕ and f ∈ L2sym (ℝn+ ). The n-fold iterated stochastic integral of f is defined by ∞

tn

t2

In (f ) = n! ∫ dBtn ∫ dBtn−1 ⋅ ⋅ ⋅ ∫ f (t1 , . . . , tn )dBt1 . 0

0

0

Let f ∈ L2 (ℝn+ ). The n-fold iterated stochastic integral of f is defined by I(f ) = I(f ̃). Note that for each iterate the integrands are adapted, and due to Itô isometry the iterated stochastic integrals are well-defined to any order n. Also, from the properties of the stochastic integral, it is straightforward to deduce the orthogonality relation 𝔼[Im (f )In (g)] = n!(f ̃, g)̃ L2 (ℝn+ ) δmn ,

m, n ∈ ℕ.

(1.7.16)

In the special case when f (t1 , . . . , tn ) = h(t1 ) ⋅ ⋅ ⋅ h(tn ), with a suitable h ∈ L2 (ℝ+ ), there is the more explicit expression n

In ( ∏ h(tj )) = ‖h‖nL2 (ℝ+ ) Hn ( j=1

I(h) ), ‖h‖L2 (ℝ+ )

(1.7.17)

where Hn is the Hermite polynomial of n defined by (1.7.2). From these properties the following fundamental fact can be obtained. Theorem 1.162 (Wiener–Itô chaos expansion on L2 (X )). Let T > 0 and suppose that F ∈ L2 (X ) is FTBM -measurable. Then there exists a unique sequence of non-random functions (fn )n∈ℕ with fn ∈ L2sym (ℝn+ ), n ∈ ℕ, such that ∞

F = ∑ In (fn ) n=0

in L2 (X ), where I0 (f0 ) = 𝔼[F]. Furthermore, the isometry ∞

‖F‖2L2 (X ) = ∑ n!‖fn ‖2L2

n sym (ℝ+ )

n=0

holds. Define ℋ0 = ℝ and ℋn = L. H.{Hn (I(f )) | f ∈ L2 (ℝn+ ), ‖f ‖L2 (ℝn+ ) = 1} for n ∈ ℕ, where the

closure is taken in the L2 (X ) norm. The subspace ℋn is called the nth Wiener chaos. Due to (1.7.17) the subspaces ℋm and ℋn are orthogonal unless m = n. Theorem 1.162 then amounts to saying that the orthogonal decomposition ∞

L2 (X ) = ⨁ ℋn n=0

holds.

(1.7.18)

124 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes 1.7.5 Malliavin derivative and divergence operator on Wiener chaos From Theorem 1.162 we know that a random variable included in L2 (X ) can be canonically represented through its Wiener–Itô chaos expansion, which makes it natural to see how the Malliavin operators D and δ act on functions in this representation. Proposition 1.163. Let F ∈ L2 (X ) and consider its nth Wiener chaos ℋn , for arbitrary n ∈ ℕ,̇ and In (fn ), where fn ∈ L2sym (ℝn+ ) is uniquely determined. Then In (fn ) ∈ 𝔻1,2 and Dt In (fn ) = nIn−1 (fn (⋅, t)),

n ∈ ℕ,

(1.7.19)

where tn−1

∞

t1

In−1 (fn (⋅, t)) = ∫ dBtn−1 ∫ dBtn−2 ⋅ ⋅ ⋅ ∫ f (t1 , . . . , tn−1 , t)dBt1 . 0

0

0

Proof. First consider the case fn (t1 , . . . , tn ) = ∏nj=1 hn (tj ), with hn ∈ L2 (ℝ+ ) such that ‖h‖L2 (ℝ+ ) = 1. Then by using (1.7.17) we obtain Dt In (fn ) = Dt Hn (I(hn )) = Hn󸀠 (I(hn ))Dt (I(hn ))

n−1

= nHn−1 (I(hn ))hn (t) = nhn (t)In−1 ( ∏ hn (tj )) = nIn−1 (fn (⋅, t)). j=1

The general case can be obtained by using that the Hermite polynomials make a complete orthonormal system in L2 (ℝ+ ). Furthermore, by using the isometry (1.7.16) we obtain ∞

∞

∞

𝔼[ ∫ (Dt In (fn ))2 dt] = n2 ∫ 𝔼[In−1 (fn (⋅, t))2 ]dt = n2 (n − 1) ∫ ‖fn (⋅, t)‖2L2 0

0

= nn!‖fn ‖2L2

n sym (ℝ+ )

= n𝔼[In (fn )2 ],

0

n−1 sym (ℝ+ )

dt

which shows that In (fn ) ∈ 𝔻1,2 . The above observation leads then to the following result. Theorem 1.164 (Malliavin derivative on Wiener chaos in L2 (X )). Let F ∈ L2 (X ) with 1,2 Wiener–Itô chaos expansion F = ∑∞ n=0 In (fn ). We have that F ∈ 𝔻 if and only if ∞

𝔼[‖DF‖2L2 (X ) ] = ∑ nn!‖fn ‖2L2 n=0

n sym (ℝ+ )

and then ∞

Dt F = ∑ nIn−1 (fn (⋅, t)). n=0

0 and F ∈ 𝔻1,2 be an FT -measurable random variable. Then T

F = 𝔼[F] + ∫ 𝔼[Dt F|Ft ]dBt . 0

1.7.6 Infinite dimensional Ornstein–Uhlenbeck semigroup Similarly to Mehler’s formula in the finite dimensional case (see Section 5.1.5 in Volume 1), we can construct the following semigroup. Definition 1.167 (Infinite dimensional Ornstein–Uhlenbeck semigroup). Let F ∈ L2 (X ) with Wiener–Itô chaos expansion F = ∑∞ n=0 In (fn ). Also, let (Wt )t≥0 be an independent Brownian motion (Bt )t≥0 on (X , ℬ(X ), 𝒲 ), and define Tt (F) = 𝔼W [F(e−t B + √1 − e−2t W)] ,

t ≥ 0,

where 𝔼W denotes expectation with respect to the Brownian motion (Wt )t≥0 . The oneparameter semigroup {Tt : t ≥ 0} on L2 (X ) is called the infinite dimensional Ornstein– Uhlenbeck semigroup. The following representation of the infinite dimensional Ornstein–Uhlenbeck semigroup on chaos expansions holds. Proposition 1.168. Let F ∈ L2 (X ) with Wiener–Itô chaos expansion F = ∑∞ n=0 In (fn ). Then ∞

Tt (F) = ∑ e−nt In (fn ). n=0

1.7 Connection with infinite dimensional stochastic analysis and Malliavin calculus | 127

The generator of the infinite dimensional Ornstein–Uhlenbeck semigroup is defined by 1 LF = lim (Tt − 1)F, ε↓0 t for F ∈ L2 (X ) for which the limit exists in L2 (X ). The following property holds. Proposition 1.169 (Generator of infinite dimensional Ornstein–Uhlenbeck semigroup). Let F ∈ L2 (X ) with Wiener–Itô chaos expansion F = ∑∞ n=0 In (fn ). 2 (1) The property F ∈ Dom(L) holds if and only if ∑∞ n ‖fn ‖L2sym (ℝn+ ) < ∞ and then n=1 ∞

LF = − ∑ nIn (fn ). n=1

(2) The property F ∈ Dom(L) holds if and only if F ∈ 𝔻1,2 and DF ∈ Dom(δ), and then δDF = −LF. From Theorem 1.169 we also obtain that [D, δ] = 1L2 (X ) as a counterpart of Corollary 1.143. 1.7.7 Malliavin derivative on white noise space In this section we briefly sketch an alternative way of introducing the Malliavin operators, when instead of Wiener space the space of tempered distributions is used. For simplicity, we only consider the one-dimensional case, as before. Denote the 󸀠 duality product for f ∈ Sreal (ℝ) and ϕ ∈ Sreal (ℝ) by ⟨⟨f , ϕ⟩⟩. Consider the measurable 󸀠 space (Sreal (ℝ), F ), where F is its Borel σ-field. By the Bochner–Minlos–Sazonov theorem (Proposition 1.114 above) there exists a Gaussian probability measure μ on 󸀠 Sreal (ℝ) such that ∫

1

2

ei⟨⟨ω,ϕ⟩⟩ dμ(ω) = e− 2 ‖ϕ‖ ,

ϕ ∈ Sreal (ℝ).

󸀠 (ℝ) Sreal

We denote

󸀠 (L2 ) = L2 (Sreal (ℝ), dμ).

(1.7.20)

Definition 1.170 (White noise process). The random process (wϕ )ϕ∈Sreal (ℝ) on the 󸀠 probability space (Sreal (ℝ), F , μ) indexed by Sreal (ℝ) and given by the measurable 󸀠 map w : Sreal (ℝ) × Sreal (ℝ) → ℝ, wϕ (ω) = ⟨⟨ω, ϕ⟩⟩, is called white noise process.

󸀠 ω ∈ Sreal (ℝ), ϕ ∈ Sreal (ℝ),

128 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes Remark 1.171. By Wiener–Itô–Segal isomorphism (Proposition 1.51) we have the equivalence (L2 ) ≅ ℱb (L2 (ℝ)). Under this equivalence the Segal field Φ(ϕ) is equivalent to wϕ for ϕ ∈ L2real (ℝ). Below we construct the triplet (𝒮 ) ⊂ (L2 ) ⊂ (𝒮 )∗ in (1.7.21), which is a counterpart of the triplet S (ℝ) ⊂ L2 (ℝ) ⊂ S 󸀠 (ℝ), and thereby we show that white noise analysis can be interpreted as a distribution theory on boson Fock space. Using the isometry 𝔼μ [wϕ2 ] =

∫

⟨⟨ω, ϕ⟩⟩2 dμ(ω) = ‖ϕ‖2 ,

󸀠 (ℝ) Sreal

the white noise process can be extended to ϕ ∈ L2real (ℝ). Writing Bt (ω) = {

⟨⟨ω, 1[0,t) ⟩⟩, ⟨⟨ω, 1(t,0]) ⟩⟩,

t ≥ 0, t < 0,

we can define a two-sided Brownian motion (Bt )t∈ℝ indexed by ℝ, and by further using the Kolmogorov–Čentsov theorem it is possible to show that it has a continuous version. We have that ∞

wϕ (ω) = I(ϕ)(ω) = ∫ ϕ(t)dBt (ω) −∞

holds as a stochastic integral, and then 𝔼μ [wϕ2 ] = ‖ϕ‖2 is usual Itô isometry. The white noise variant of Wiener–Itô chaos expansion can be obtained in the following way. Recall the Hermite polynomials as given in (1.7.2) and consider the related family of Hermite functions given by en (t) =

1

t2

√√π(n − 1)!

Hn−1 (√2t)e− 2 ,

n ∈ ℕ,

which are Schwartz functions and make a complete orthonormal system in L2 (ℝ). One can check that 2 1 ∫ en (t)em (t)dt = ∫ Hn−1 (√2t)Hm−1 (√2t)e−t dt √π(n − 1)!(m − 1)! ℝ

ℝ

=

1

√(n − 1)!(m − 1)!

2

∫ Hn−1 (t)Hm−1 (t) ℝ

e−t /2 dt = δnm √2π

Define Jn (ω) = wen (ω). ̇ Also, let ℐ = ⋃∞ n=1 {ℓ = (ℓ1 , . . . , ℓn ) | ℓ1 , . . . , ℓn ∈ ℕ} be the family of nonnegative integer multi-indices, write L0 = 1, and for ℓ ∈ ℐ define n

Lℓ (ω) = ∏ Hℓk (Jk (ω)), k=1

ℓ = (ℓ1 , . . . , ℓn ) ≠ 0.

1.7 Connection with infinite dimensional stochastic analysis and Malliavin calculus | 129

Theorem 1.172 (Wiener–Itô chaos expansion on white noise). Choose T ∈ ℝ and consider an FT -measurable random variable F ∈ (L2 ). There exists a unique sequence of numbers (cℓ )ℓ∈ℐ ⊂ ℝ such that F = ∑ cℓ Lℓ . ℓ∈ℐ

Moreover, ‖F‖2(L2 ) = ∑ℓ∈ℐ ℓ! cℓ2 holds, where ℓ! = ℓ1 ! ⋅ ⋅ ⋅ ℓn ! for ℓ = (ℓ1 , . . . , ℓn ) ∈ ℐ . We note that the equivalence with the expansion given in Theorem 1.162 is secured by the following relationships. We can show by using (1.7.17) that n

In (⊗ℓsym en ) = ∏ Hℓk (Jk ) = Lℓ , k=1

ℓ = (ℓ1 , . . . , ℓn ) ∈ ℐ , n = |ℓ|.

On the other hand, fn in the expansion in Theorem 1.162 can be expressed like fn =

∑

ℓ=(ℓ1 ,...,ℓm )∈ℐ |ℓ|=n

cℓ (⊗ℓ1 e1 ) ⊗sym ⋅ ⋅ ⋅ ⊗sym (⊗ℓm em ).

These relationships allow to introduce the following spaces. For m ∈ ℕ and ℓ = ℓk (ℓ1 , . . . , ℓm ) denote (m, ℓ) = ∏m k=1 (2k) . Definition 1.173 (Hida space). Consider ϕ = ∑ℓ∈ℐ aℓ Lℓ and F = ∑ℓ∈ℐ bℓ Lℓ . (1) Let p > 0. ϕ is said to belong to Hida test function Hilbert space (𝒮 )p if ‖ϕ‖2(𝒮)p = ∑ (m, ℓ)pℓ ℓ! a2ℓ < ∞. ℓ∈ℐ

We define Hida test function space by (𝒮 ) = ⋂p>0 (𝒮 )p . (2) Let q > 0. F is said to belong to Hida distribution Hilbert space (𝒮 )−q if ‖F‖2(𝒮)−q = ∑ (m, ℓ)−qℓ ℓ! b2ℓ < ∞. ℓ∈ℐ

We define Hida distribution space by (𝒮 )∗ = ⋃q>0 (𝒮 )−q . We note that (𝒮 )∗ is the dual space of (𝒮 ), and for F ∈ (𝒮 )∗ and ϕ ∈ (𝒮 ) of the form in the definition above the formula ⟨⟨F, ϕ⟩⟩ = ∑ ℓ! aℓ bℓ ℓ∈ℐ

applies. Here we also used ⟨⟨⋅, ⋅⟩⟩ for the distribution product between (𝒮 )∗ and (𝒮 ). Also, the inclusions (𝒮 ) ⊂ (𝒮 )p ⊂ (L2 ) ⊂ (𝒮 )−q ⊂ (𝒮 )∗

(1.7.21)

130 | 1 Free Euclidean quantum field and Ornstein–Uhlenbeck processes hold for every 0 < p, q. The topology on (𝒮 ) is given by requiring that (ϕn )n∈ℕ ⊂ (𝒮 ) is convergent to ϕ ∈ (𝒮 ) if ‖ϕn − ϕ‖(𝒮)p → 0 as n → ∞, for every p > 0. Similarly, the topology on (𝒮 )∗ is defined by saying that (Fn )n∈ℕ ⊂ (𝒮 )∗ is convergent to F ∈ (𝒮 )∗ if there exists q > 0 such that ‖Fn − F‖(𝒮)−q → 0 as n → ∞. Using these spaces, now we can introduce Malliavin derivative on white noise. Definition 1.174 (Malliavin derivative on white noise). Let F ∈ (L2 ), and ϖ ∈ L2 (ℝ) be a nonrandom function. The directional derivative of F in (𝒮 )∗ is given by Dϖ F(ω) = lim

ε→0

F(ω + εϖ) − F(ω) , ε

ϖ ∈ S 󸀠 (ℝ),

if the limit exists in (𝒮 )∗ . If, moreover, there exists a function u : ℝ → (𝒮 )∗ such that ∞ ∫−∞ u(t)ϖ(t)dt is a convergent integral in (𝒮 )∗ and ∞

Dϖ F = ∫ u(t)ϖ(t)dt,

ϖ ∈ L2 (ℝ),

−∞

then F is called Malliavin differentiable in (𝒮 )∗ and u(t) = Dt F,

t ∈ ℝ,

is called its Malliavin derivative in (𝒮 )∗ at t. Remark 1.175 (Malliavin derivative on Wiener chaos in (L2 )). (1) We can also express the Malliavin derivative on the Wiener chaos in white noise variables. A calculation gives m

m

k=1

j=1 j=k̸

Dt Lℓ = ∑ ( ∏ Hℓj (Jj ))ℓk Hℓk −1 (Jk )ek (t),

ℓ = (ℓ1 , . . . , ℓm ) ∈ ℐ .

In fact, this can be used for a definition of Dt on the expansion series. For F = ∑ℓ∈ℐ cℓ Lℓ ∈ (𝒮 )∗ the Malliavin derivative at t ∈ ℝ is then defined to be ∞

m

k=1

j=1 j=k̸

Dt F = ∑ cℓ ∑ ( ∏ Hℓj (Jj ))ℓk Hℓk −1 (Jk )ek (t) ℓ∈ℐ

and Dom(Dt ) contains all the F ∈ (𝒮 )∗ for which this series is convergent in the topology of (𝒮 )∗ . It can be shown that (L2 ) ⊂ Dom(Dt ) ⊂ (𝒮 )∗ . (2) Higher order derivatives and a divergence operator as the adjoint of the Malliavin derivative can similarly be defined on white noise than previously seen, for details we refer to the literature.

2 The Nelson model by path measures 2.1 Preliminaries In this section we consider a model of an electrically charged, spinless and nonrelativistic particle interacting with a scalar boson field, which we call Nelson model. The coupling between particle and field is assumed to be linear. First we explain how the Nelson model can be derived from a Lagrangian, and then continue with a rigorous definition and analysis in the following sections. The Lagrangian density LN = LN (x, t), (x, t) ∈ ℝ3 × ℝ of the Nelson model is described by LN = iΨ Ψ̇ + ∗

1 1 1 𝜕 Ψ∗ 𝜕j Ψ + 𝜕μ ϕ𝜕μ ϕ − ν2 ϕ2 + Ψ∗ Ψϕ, 2m j 2 2

where Ψ = Ψ(x, t) is a complex scalar field describing a nonrelativistic, spinless electron, and ϕ(x, t) is a neutral scalar field describing scalar bosons, ν ≥ 0 is the mass of bosons, and m > 0 is the mass of the electron. Here 𝜕μ ϕ𝜕μ ϕ = ϕ̇ ϕ̇ − 𝜕j ϕ𝜕j ϕ, 𝜕j = 𝜕xj , the dots denote time derivative, and the stars denote complex conjugate. The dynamics of this system is given by the Euler–Lagrange equation leading to the system of nonlinear field equations 2 ∗ { {(◻ + ν )ϕ(x, t) = Ψ (x, t)Ψ(x, t), { {(i𝜕 + 1 Δ) Ψ(x, t) = ϕ(x, t)Ψ(x, t). { t 2m

The Hamiltonian density is derived from the Legendre transform of LN . The conjugate momenta are defined by Φ=

𝜕LN = iΨ∗ , 𝜕Ψ̇

(2.1.1)

π=

𝜕LN = ϕ.̇ 𝜕ϕ̇

(2.1.2)

Thus the Hamiltonian density HN = HN (x, t) is given by 1 1 |𝜕x Ψ|2 + (ϕ̇ 2 + (𝜕x ϕ)2 + ν2 ϕ2 ) − Ψ∗ Ψϕ HN = ΦΨ̇ + π ϕ̇ − LN = 2m 2 and the Hamiltonian HN = ∫ [Ψ∗ (− ℝ3

1 1 Δ) Ψ + (ϕ̇ 2 + (𝜕x ϕ)2 + ν2 ϕ2 ) − Ψ∗ Ψϕ] dx. 2m 2

Since the Nelson model describes a charged particle interacting with a scalar field, and the particle is supposed to carry low energy, there is no annihilation and creation https://doi.org/10.1515/9783110403541-002

132 | 2 The Nelson model by path measures of particles and the number of particles is conserved. In order to obtain the quantized model, the kinetic part is replaced by ∫ Ψ∗ (− ℝ3

1 1 Δ) Ψdx → − Δ 2m 2m

and the interaction by − ∫ Ψ∗ Ψϕdx → ϕ(x). ℝ3

Adding an external potential V, the formal expression of the Nelson model is − where Hf =

1 Δ + V + Hf + ϕ(x), 2m

1 ∫ (ϕ̇ 2 + (𝜕x ϕ)2 + ν2 ϕ2 ) dx. 2 ℝ3

2.2 The Nelson model in Fock space 2.2.1 Definition of the Nelson model We give a rigorous definition of the Nelson model. In the previous section we introduced the Hilbert spaces ℱb (ℋ̂ M ), L2 (Q ) and L2 (M−2 , dG), serving as state spaces of the quantum field. We have seen the unitary equivalence 2

2

ℱb (ℋ̂ M ) ≅ L (Q ) ≅ L (M−2 , dG)

(2.2.1)

in (1.5.3) and (1.6.19), following the identifications 1 ∗ ̂ ̃ (a (f ) + aM (f ̂)) ≅ ϕ(f ) ≅ ξ (f ), √2 M

f ∈ M.

(2.2.2)

Here a∗M and aM denote the creation operator and the annihilation operator in the boson Fock space ℱb (ℋ̂ M ), respectively, satisfying the canonical commutation relations ̂ = 0 = [a∗M (f ̂), a∗M (g)], ̂ [aM (f ̂), aM (g)]

̂ = (f ̄̂, g)̂ ℋ̂ . [aM (f ̂), a∗M (g)] M

(2.2.3)

The Hilbert space L2 (ℝd ) describes the state space of the nonrelativistic particle, and the scalar boson field is defined by ℱN = ℱb (ℋ̂ M ),

2.2 The Nelson model in Fock space

| 133

i. e., the boson Fock space over ℋ̂ M . The joint state space is described by the tensor product 2

d

ℋN = L (ℝ ) ⊗ ℱN .

The free particle Hamiltonian is described by the Schrödinger operator 1 Hp = − Δ + V 2 acting in L2 (ℝd ). We denote the ground state energy of Hp by Ep = inf Spec(Hp ). We introduce a class of potentials. Definition 2.1 (Definition of ℛKato ). Let V be relatively bounded with respect to − 21 Δ with a relative bound strictly smaller than one, i. e., D(V) ⊂ D(−(1/2)Δ) and 󵄩󵄩 1 󵄩󵄩 ‖Vf ‖ ≤ a󵄩󵄩󵄩− Δf 󵄩󵄩󵄩 + b‖f ‖ 󵄩 2 󵄩 for f ∈ D(V) with some a < 1 and b ≥ 0. Then we say that V ∈ ℛKato . The following are standing assumptions throughout Section 2.2. Assumption 2.2. The following conditions hold: (1) Dispersion relation: ω(k) = ων (k) = √|k|2 + ν2 , ν ≥ 0. ̂ ̂ ̂ ̂ = φ(−k) and φ/ω, φ/√ω (2) Charge distribution: φ ∈ S 󸀠 (ℝd ), φ(k) ∈ L2 (ℝd ). (3) External potential: V ∈ ℛKato . Note that we use the notation ω instead of ων for notational simplicity unless any ̂ ̂ = φ(−k) is necessary and sufficonfusion may arise. We also note that condition φ(k) cient for φ being a real tempered distribution. The free field Hamiltonian Hf = dΓ(ω) on ℱN accounts for the energy carried by the field configuration. The particle-field interaction Hamiltonian HI acting on the Hilbert space ℋN describes then the interaction energy between the boson field and the particle. To give a definition of this operator, we identify ℋN as the space of ℱN -valued L2 -functions on ℝd : ⊕

d

󵄨󵄨 󵄨

2

ℋN ≅ ∫ ℱN dx = {F : ℝ → ℱN 󵄨󵄨󵄨 ∫ ‖F(x)‖ℱN dx < ∞}. ℝd

ℝd

For every x ∈ ℝd the operator HI (x) is defined by the time-zero field HI (x) =

1 ̃̂ ik⋅x )) . ̂ −ik⋅x ) + aM (φe (a∗ (φe √2 M

134 | 2 The Nelson model by path measures We will use the formal notation HI (x) = ∫ ℝd

1

√2ω(k)

−ik⋅x ∗ ik⋅x ̂ ̂ (φ(k)e a (k) + φ(−k)e a(k)) dk

̂ ̂ = φ(−k), for convenience. Here we assume that [a(k), a∗ (k 󸀠 )] = δ(k − k 󸀠 ). Since φ(k) HI (x) is symmetric, and it can be shown by using Nelson’s analytic vector theorem that HI (x) is essentially self-adjoint on the finite particle subspace ℱN,fin = ℱb,fin (ℋ̂ M ).

We denote the self-adjoint extension of HI (x) by HI (x). The interaction HI is then defined by the self-adjoint operator ⊕

HI = ∫ HI (x)dx ℝd

acting as (HI Ψ)(x) = HI (x)Ψ(x) with domain D(HI ) = {Ψ ∈ ℋN | Ψ(x) ∈ D(HI (x)) a. e. x and ∫ ‖HI (x)Ψ(x)‖2ℱN dx < ∞}. ℝd

Definition 2.3 (Nelson Hamiltonian in Fock space). The operator HN = Hp ⊗ 1 + 1 ⊗ Hf + HI

(2.2.4)

acting in ℋN is called the Nelson Hamiltonian. Let H0 = Hp ⊗ 1 + 1 ⊗ Hf ,

D(H0 ) = D(Hp ⊗ 1) ∩ D(1 ⊗ Hf ).

A first natural question about the Nelson Hamiltonian is whether it is self-adjoint. Proposition 2.4 (Self-adjointness). We have the following properties. (1) H0 is self-adjoint on D(H0 ) and bounded from below. (2) HN is self-adjoint on D(H0 ) and bounded from below. Furthermore, it is essentially self-adjoint on any core of H0 . Proof. H0 is self-adjoint and bounded from below as it is the sum of two commuting self-adjoint operators bounded from below. By Lemma 1.6, D(HI ) ⊃ D(H0 ), and Proposition 1.6 gives ̂ ̂ √ω‖)‖(Hf + 1)1/2 Ψ‖ℱN ‖HI (x)Ψ‖ℱN ≤ (2‖φ/ω‖ + ‖φ/

2.2 The Nelson model in Fock space

| 135

for Ψ ∈ D(Hf ) and every x ∈ ℝd . Thus for Φ ∈ D(1 ⊗ Hf ), ̂ ̂ √ω‖)‖1 ⊗ (Hf + 1)1/2 Φ‖ℋN . ‖HI Φ‖ℋN ≤ (2‖φ/ω‖ + ‖φ/ Since ‖1 ⊗ (Hf + 1)1/2 Ψ‖ ≤ ‖(H0 − Ep + 1)1/2 Ψ‖ ≤ ε‖H0 Ψ‖ + bε ‖Ψ‖ with some bε ≥ 0, the Kato–Rellich theorem yields that HN is self-adjoint on D(H0 ) and bounded from below.

2.2.2 Infrared and ultraviolet divergences It is a standard fact in quantum field theory that a rigorous description runs into mathematical difficulties caused by divergent integrals at the two ends of the spectrum. Since in quantum theory the electron is regarded as a point particle, the physical form factor corresponds to φph (x) = δ(x). Due to φ̂ ph (k) = (2π)−d/2 , this implies ∫ ℝd

|φ̂ ph (k)|2 ω(k)

dk = ∞.

This situation is called ultraviolet divergence. In a mathematical description therefore a regularized form factor φ̂ is imposed so that the above integral is finite and HI (x) is well-defined as an operator on ℱN . At the other end of the energy spectrum, an infrared divergence is encountered. ̂ Suppose that φ(k) = (2π)−d/2 for |k| < ε, with some ε > 0. Since for massless bosons the physical dispersion relation is ω(k) = |k|, in this case ∫ |k| 0. Moreover, under condition (2.2.5) it can be proven that there is no ground state in ℋN , which justifies the expectation that in the ground state

136 | 2 The Nelson model by path measures only a finite number of bosons bind to the particle. To cope with this difficulty, we impose the infrared regularity condition ∫ ℝd

2 ̂ |φ(k)| dk < ∞ 3 ω(k)

(2.2.6)

so that whenever the above integral is infinite, we speak of infrared singularity. Note that, in contrast, for massive bosons where ω(k) = √|k|2 + ν2 with ν > 0, no infrared divergence occurs. Although the cutoffs are mathematical artifacts and not intrinsically physical properties, we impose both an ultraviolet and an infrared cutoff in order that the model is well-defined, and at a stage of discussing this theory we will ask whether in a proper limit these cutoffs can be removed. Two specific choices of ω and φ̂ will be particularly important and we single them out. Choosing ω(k) = |k| and

̂ φ(k) = g1{κ 0. Moreover, HNε = Hp ⊗1+1⊗ Ĥ f + Ĥ Iε is self-adjoint on D(Hp ⊗1+1⊗ Ĥ f ) and essentially self-adjoint on any core of Hp ⊗ 1 + 1 ⊗ Ĥ f for 0 ≤ ε < c with some c. In particular, ε

e−tHN → e−tHN strongly as ε ↓ 0. First we construct a functional integral representation ε of (F, e−tHN G) for ε > 0 and by a limiting argument we obtain the extension (F, e−tHN G) to ε = 0. For simplicity, first we assume that V ∈ C0∞ (ℝd ). Let h = − 21 Δ. By the Trotter

product formula and the factorization formula e−|s−t|Hf = I∗s It , we have ̂

t

ε

t

̂ε

t

t

n−1

t

t

̂ε

t

e−tHN = s-lim(e− n HI e− n h e− n V e− n Hf )n = s-lim I∗0 (∏ I jt e− n HI e− n h e− n V I∗jt ) It . (2.3.13) n→∞

̂

n→∞

j=0

n

̂ε

t

n

ε

Inserting (2.3.13) in (F, e−tHN G) gives n−1

ε

t

t

(F, e−tHN G) = lim (I0 F, ( ∏ I jt e− n HI e− n h e− n V I∗jt )It G). n→∞

j=0

n

n

̂ε

̂ε

Here ∏nj=1 Tj = T1 ⋅ ⋅ ⋅ Tn . Using the identity Is e−HI Is = Es e−HI (s) Es for s ∈ ℝ, where ⊕

Ĥ Iε (s) = ∫ yε (ϕE (δs ⊗ φ(⋅ − x))) dx ℝd

and Es = Is I∗s is a projection, we see that n−1

ε

t

̂ε

jt

t

t

(F, e−tHN G) = lim (I0 F, (∏ E jt e− n HI ( n ) e− n h e− n V E jt ) It G) . n→∞

j=0

n

n

By the Markov property we may drop Es on the right-hand side above. We obtain n−1

ε

t

̂ε

jt

t

t

(F, e−tHN G) = lim (I0 F, (∏ e− n HI ( n ) e− n h e− n V ) It G) . n→∞

j=0

2.3 The Nelson model in function space

| 145

The right-hand side above can be expressed in terms of Wiener measure. Since for any time-division t0 ≤ t1 ≤ . . . ≤ tn n

n

(f0 , ∏ e−(tj −tj−1 )h fj ) = ∫ 𝔼x [f0 (B0 )( ∏ fj (Btj ))]dx, j=1

j=1

ℝd

it follows that ε

(F, e−tHN G) = lim ∫ 𝔼x [e n→∞

− ∑n−1 j=0 V(B jt ) n

n−1 t ̂ ε jt H ( ) n I n

(I0 F(B0 )e− ∑j=0

It G(Bt )) ]dx.

ℝd

Note that s 󳨃→ δs ⊗ φ(⋅ − Bs ) is strongly continuous as a map ℝ → E , almost surely. Hence s 󳨃→ ϕE (δs ⊗ φ(⋅ − Bs )) is also strongly continuous as ℝ → L2 (QE ). We can compute the limit and the result is t

ε

t

(F, e−tHN G) = ∫ 𝔼x [e− ∫0 V(Bs )ds (I0 F(B0 ), e−εQt −ϕE (∫0 δs ⊗φ(⋅−Bs )ds) It G(Bt ))] dx. ℝd

Here

t

Qt = ∫ ϕE (δs ⊗ φ(⋅ − Bs ))2 ds. 0 ε

We see that HNε → HN on D(Hp ⊗ 1 + 1 ⊗ Hrad ), and e−tHN G → e−tHN G follows as ε ↓. On the other hand, we have the bound t 󵄨󵄨 󵄨 󵄨󵄨 (I0 F(B0 ), e−εQt −ϕE (∫0 δs ⊗φ(⋅−Bs )ds) It G(Bt )) 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 t

2

2

2

≤ ‖F(B0 )‖‖G(Bt )‖e 2 ‖φ/√ω‖ +2t(t∨1)(‖φ/√ω‖ +‖φ/ω‖ ) ̂

̂

̂

and t

∫ 𝔼x [e− ∫0 V(Bs )ds ‖F(B0 )‖‖G(Bt )‖] dx < ∞. ℝd

Hence the Lebesgue dominated convergence theorem yields that t

t

lim ∫ 𝔼x [e− ∫0 V(Bs )ds (I0 F(B0 ), e−εQt −ϕE (∫0 δs ⊗φ(⋅−Bs )ds) It G(Bt ))] dx ε↓0

ℝd

t

t

= ∫ 𝔼x [e− ∫0 V(Bs )ds (I0 F(B0 ), e−ϕE (∫0 δs ⊗φ(⋅−Bs )ds) It G(Bt ))] dx, ℝd

implying t

t

(F, e−tHN G) = ∫ 𝔼x [e− ∫0 V(Bs )ds (I0 F(B0 ), e−ϕE (∫0 δs ⊗φ(⋅−Bs )ds) It G(Bt ))] dx.

(2.3.14)

ℝd

Thus the theorem follows for V ∈ C0∞ (ℝd ). By a limiting argument similar to the proof of Theorem 4.88 in Volume 1, we can extend (2.3.12) to V ∈ ℛKato .

146 | 2 The Nelson model by path measures Remark 2.13. We can also derive the self-adjoint generator HN from Theorem 2.12 by taking the derivative of the functional integral representation with respect to t at t = 0. Suppose that V and φ are sufficiently smooth and bounded. Also assume that F and G are sufficiently smooth and F, G ∈ D(HN ) for simplicity. Let t

t

Rt = ∫ 𝔼x [e− ∫0 V(Bs )ds (I0 F(B0 ), e−ϕE (∫0 δs ⊗φ(⋅−Bs )ds) It G(Bt ))L2 (QE ) ] dx. ℝd

We have Rt − R0 = at + bt + ct , where t

t

at = ∫ 𝔼x [(e− ∫0 V(Bs )ds − 1)(I0 F(B0 ), e−ϕE (∫0 δs ⊗φ(⋅−Bs )ds) It G(Bt ))L2 (QE ) ] dx, ℝd

t

bt = ∫ 𝔼x [(I0 F(B0 ), (e−ϕE (∫0 δs ⊗φ(⋅−Bs )ds) − 1)It G(Bt ))L2 (QE ) ] dx, ℝd

ct = ∫ 𝔼x [(I0 F(B0 ), (It G(Bt ) − I0 G(B0 )))L2 (QE ) ] dx. ℝd

It is straightforward to see that at = ∫ 𝔼x [−V(B0 )(I0 F(B0 ), I0 G(B0 ))L2 (QE ) ] dx = (F, −VG), t→0 t lim

ℝd

lim

t→0

bt = ∫ 𝔼x [(I0 F(B0 ), −ϕE (δ0 ⊗ φ(⋅ − B0 ))I0 G(B0 ))L2 (QE ) ] dx = (F, −ϕ(φ(⋅ − x))G). t ℝd

By I∗0 It = e−tHf we can also see that lim

t→0

ct 1 1 = lim (F, (e−tHf etΔ/2 − 1)G)L2 (QE ) = (F, (−Hf + Δ)G). t→0 t t 2

Together with them we have 1 lim (Rt − R0 ) = (F, −HN G). t

t→0

Hence we can derive HN as a generator. Conversely in a similar way to the Schrödinger operator − 21 Δ + V with singular external potentials we can also define self-adjoint operator HN with singular external potentials V through the functional integral reprsentation in Sections 2.3.3 and 2.4 below. 2 ̂ Suppose that ∫ℝd |φ(k)| /ω(k)3 dk < ∞, and define 1

2 2 ̂ ̂ ̂ √ω3 ‖2 ) ‖φ/ω‖ + 2(‖φ/ω‖ + ‖φ/ }. 2 2 2 ̂ ̂ ̂ ‖φ/ω‖ + 2(‖φ/√ω‖ + ‖φ/ω‖ ) 2

E(φ)̂ = max { 21

(2.3.15)

2.3 The Nelson model in function space

| 147

From (1.5.23) it follows that t

‖I∗0 e−ϕE (∫0 δs ⊗φ(⋅−Bs )ds) It ‖ ≤ 2etE(φ) , ̂

t ≥ 0.

(2.3.16)

More generally, for q > 1 define 1

2 2 ̂ ̂ ̂ √ω3 ‖2 ) ‖φ/ω‖ + q(‖φ/ω‖ + ‖φ/ }. 2 2 2 ̂ ̂ ̂ + ‖φ/ω‖ ) ‖φ/ω‖ + q(‖φ/√ω‖ 2

Eq (φ)̂ = max { 21

(2.3.17)

By (1.5.26) we then have t

‖I∗0 e−ϕE (∫0 δs ⊗φ(⋅−Bs )ds) It ‖ ≤

q tEq (φ)̂ e , q−1

Corollary 2.14 (Diamagnetic inequality). Assume ∫ℝd Then |(F, e−tHN G)ℋN | ≤

t ≥ 0.

2 ̂ |φ(k)| dk ω(k)3

(2.3.18)

< ∞, and take q > 1.

q ̂ (‖F‖L2 (Q) , e−t(Hp −Eq (φ)) ‖G‖L2 (Q) )L2 (ℝd ) . q−1

Proof. By the functional integral representation of (F, e−tHN G) and the bound (2.3.18) we see that |(F, e−tHN G)| ≤

t q tEq (φ)̂ e ∫ 𝔼x [e− ∫0 V(Bs )ds ‖F(B0 )‖ ⋅ ‖G(Bt )‖] dx. q−1

ℝd

Similarly to Theorem 2.8, we can also construct a functional integral representation for other types of Green functions. j

Theorem 2.15 (Euclidean Green functions). Let F0 , Fn ∈ ℋN , Fj = fj ⊗Φj (ϕ(g1 ), . . . , ϕ(gnj j )), j

where fj ∈ L∞ (ℝd ), Φj ∈ L∞ (ℝnj ), gi ∈ M , i = 1, . . . , nj , j = 1, . . . , n − 1. Take arbitrary 0 = t0 < t1 < . . . < tn = t, n ∈ ℕ. Then n

(F0 , ( ∏ e−(tj −tj−1 )HN Fj ))

ℋN

j=1

t

t

n−1

= ∫ 𝔼x [e− ∫0 V(Bs )ds (I0 F0 (B0 ), e−ϕE (∫0 δs ⊗φ(⋅−Bs )ds) ( ∏ F̃j (Btj ))It Fn (Bt ))]dx, (2.3.19) j=1

ℝd j

where F̃j (x) = fj (x)Φj (ϕE (δtj ⊗ g1 ), . . . , ϕE (δtj ⊗ gnj j )). Proof. For smooth functions with compact support fj and Φj , j = 1, . . . , n, we have Is Fj I∗s = fj Es Φj (ϕE (δs ⊗ gnj 1 ), . . . , ϕE (δs ⊗ gnj 1 )) Es as a bounded operator. Similarly to the proof of Theorem 2.12 the Trotter product formula implies (2.3.19). For general Fj , a limiting argument can be used.

148 | 2 The Nelson model by path measures 2.3.3 Extension to general external potential In Assumption 2.2 we required the external potential to belong to ℛKato . In Section 2.3.3 we extend the functional integral representation to a wider class of external potentials, but we use Assumption 2.16 on dispersion relations and charge distributions below. Assumption 2.16. The following conditions hold: (1) Dispersion relation: ω(k) = ων (k) = √|k|2 + ν2 , ν ≥ 0. ̂ ̂ ̂ ̂ (2) Charge distribution: φ ∈ S 󸀠 (ℝd ), φ(k) ∈ L2 (ℝd ). = φ(−k) and φ/ω, φ/√ω For convenience, we denote HN with V ≡ 0 by 1 K = − Δ ⊗ 1 + 1 ⊗ Ĥ f + HÎ . 2 Let z > 0 and suppose that ‖|V|(−(1/2)Δ + z)−1 ‖ ≤ a

(2.3.20)

with an a > 0. We see that −1 󵄩 󵄩 ‖|V|(K + z)−1 F‖ ≤ a󵄩󵄩󵄩 (1 + HÎ (−(1/2)Δ + Ĥ f + z)−1 ) F 󵄩󵄩󵄩

and we also know that ‖HÎ (−(1/2)Δ + Ĥ f + z)−1 F‖ ≤ b‖F‖ with some b. Whenever b < 1 we furthermore have 1 󵄩󵄩 −1 −1 󵄩 󵄩󵄩 (1 + HÎ (−(1/2)Δ + Ĥ f + z) ) 󵄩󵄩󵄩 ≤ 1−b and ‖|V|(K + z)−1 ‖ ≤

a . 1−b

It is, however, not trivial to see b < 1 in general, and the right-hand side above is a/(1 − b) > a. Surprisingly, in the lemma below we can nevertheless show that (2.3.20) implies ‖|V|(K + z)−1 ‖ ≤ a 2 ̂ under the infrared regular condition ∫ℝd |φ(k)| /ω(k)3 dk < ∞. Denote

F(x)/‖F(x)‖L2 (Q) sgn F(x) = { 0

if ‖F(x)‖L2 (Q) ≠ 0,

if ‖F(x)‖L2 (Q) = 0.

2.3 The Nelson model in function space

| 149

2 ̂ Lemma 2.17. Let ∫ℝd |φ(k)| /ω(k)3 dk < ∞ hold. 1 (1) If V is − 2 Δ-form bounded with a relative bound a, then |V| is also K-form bounded with a relative bound smaller than a. (2) If V is − 21 Δ-bounded with a relative bound a, then |V| is also K-bounded with a relative bound smaller than a.

Proof. Let q > 1, and ψ ∈ C0∞ (ℝd ) such that ψ > 0. Applying F = sgn(e−tK G) ψ in the diamagnetic inequality we obtain |(F, e−tK G)| ≤

q tEq (φ)̂ e (‖F‖, e−t(−(1/2)Δ) ‖G‖), q−1

1 where Eq (φ)̂ is given by (2.3.17). By formula (T + z)−1/2 = √π ∫0 t −1/2 e−t(T+z) dt for any self-adjoint operator T and z < inf Spec(T), we furthermore get ∞

(ψ, ‖(K + z)−1 G‖L2 (Q) )L2 (ℝd ) ≤

1 q −1 (ψ, ( − Δ − Eq (φ)̂ + z) ‖G‖L2 (Q) ) 2 d L (ℝ ) q−1 2

̂ Since ψ ∈ C0∞ (ℝd ) is arbitrary, we for z < inf Spec(K) and z < inf Spec (− 21 Δ − Eq (φ)). have ‖(K + z)−1 G(x)‖L2 (Q) ≤

−1 q 1 ( − Δ − Eq (φ)̂ + z) ‖G(x)‖L2 (Q) q−1 2

for a. e. x ∈ ℝd . We obtain that ‖|V|1/2 (K + z)−1/2 G‖ℋN ‖G‖ℋN

󵄩󵄩 1/2 1 󵄩󵄩 −1/2 q 󵄩󵄩󵄩|V| (− 2 Δ − Eq (φ)̂ + z) ‖G(⋅)‖L2 (Q) 󵄩󵄩󵄩L2 (ℝd ) ≤ . q−1 ‖G‖ℋN

Taking the supremum on both sides above with respect to ‖G‖ℋN ≠ 0, we furthermore obtain ‖|V|1/2 (K + z)−1/2 ‖ ≤

q 󵄩󵄩 1/2 󵄩 󵄩󵄩|V| (−(1/2)Δ − Eq (φ)̂ + z)−1/2 󵄩󵄩󵄩 . 󵄩 q−1 󵄩

̂ Since V is − 21 Δ-form bounded with relative bound a, V is |−(1/2)Δ−E(φ)|-form bounded with relative bound smaller than a. This gives ‖|V|1/2 (−(1/2)Δ − E(φ)̂ + z)−1/2 ‖ ≤ a and ‖|V|1/2 (K + z)−1/2 ‖ ≤

q a, q−1

where q > 1 is arbitrary. Then ‖|V|1/2 (K + z)−1/2 ‖ ≤ a follows and (1) is proven. Part (2) can be shown similarly. Theorem 2.18 (Functional integral representation of Nelson Hamiltonian with singular 2 ̂ external potential). Let ∫ℝd |φ(k)| /ω(k)3 dk < ∞ hold. Take K +̇ V+ −̇ V− with V+ ∈ L1loc (ℝd ) and − 21 Δ-relatively form bounded V− with relative bound strictly less than 1. Then (2.3.12) holds for K +̇ V+ −̇ V− .

150 | 2 The Nelson model by path measures Proof. The proof is similar to the Feynman–Kac formula for Schrödinger operator with singular external potentials in Theorem 4.88 in Volume 1, with h = − 21 Δ replaced by K. If V ∈ L∞ (ℝd ), then (2.3.12) holds. Assume that V+ ∈ L1loc (ℝd ) and V− is relatively form bounded with respect to − 21 Δ. Define V+ (x),

V+n (x) = {

n,

V+ (x) < n,

V− (x), V− (x) < m, V−m (x) = { m, V− (x) ≥ m,

V+ (x) ≥ n,

and write Vn,m = V+n − V−m . We have t

(F, e−t(K+Vn,m ) G) = ∫ 𝔼x [e− ∫0 Vn,m (Bs )ds (F(B0 ), K G(Bt ))]dx.

(2.3.21)

ℝd t

Here K = I∗0 e−ϕE (∫0 δs ⊗φ(⋅−Bs )ds) It . Define the closed quadratic forms 1/2 1/2 1/2 1/2 qn,m (F, F) = (K 1/2 F, K 1/2 F) + (V+n F, V+n F) − (V−m F, V−m F),

1/2 1/2 qn,∞ (F, F) = (K 1/2 F, K 1/2 F) + (V+n F, V+n F) − (V−1/2 F, V−1/2 F),

q∞,∞ (F, F) = (K 1/2 F, K 1/2 F) + (V+1/2 F, V+1/2 F) − (V−1/2 F, V−1/2 F), whose form domains respectively are Q(qn,m ) = Q(K), Q(qn,∞ ) = Q(K) and Q(q∞,∞ ) = Q(K) ∩ Q(V+ ). Clearly, qn,m ≥ qn,m+1 ≥ qn,m+2 ≥ . . . ≥ qn,∞ and qn,m → qn,∞ in the sense of quadratic forms on ⋃∞ m=1 Q(qn,m ) = Q(K). Since qn,∞ is closed on Q(K), by the monotone convergence theorem for forms, the associated positive self-adjoint operators satisfy K + V+n − V−m → K + V+n −̇ V− in strong resolvent sense, which implies that for all t ≥ 0, exp (−t (K + V+n − V−m )) → exp (−t (K + V+n −̇ V− ))

(2.3.22)

strongly as m → ∞. Similarly, we have qn,∞ ≤ qn+1,∞ ≤ qn+2,∞ ≤ . . . ≤ q∞ and qn,∞ → q∞,∞ in form sense on {F ∈ ⋂∞ n=1 Q(qn,∞ ) | supn∈ℕ qn,∞ (F, F) < ∞} = Q(K) ∩ Q(V+ ). Hence by the monotone convergence theorem for forms again we obtain exp(−t(K +̇ V+n −̇ V− )) → exp(−t(K +̇ V+ −̇ V− ))

(2.3.23)

strongly as n → ∞. By taking first m → ∞ and then n → ∞, it can be proven that both sides of (2.3.21) converge, thus the left-hand is convergent by (2.3.22)–(2.3.23). On the other hand, let F, G ∈ ℋN and F, G ≥ 0. Note that (F(B0 ), K G(Bt )) ≥ 0. It follows that t

lim lim ∫ 𝔼x [e− ∫0 Vn,m (Bs )ds (F(B0 ), K G(Bt ))]dx

n→∞ m→∞

ℝd

t

= ∫ 𝔼x [e− ∫0 V(Bs )ds (F(B0 ), K G(Bt ))]dx ℝd

2.4 Nelson model with Kato-class potential | 151

by the monotone convergence theorem for integrals for the limit of m → ∞ and the dominated convergence theorem for n → ∞. Hence the right-hand side of (2.3.21) cont

verges to ∫ℝd 𝔼x [e− ∫0 V(Bs )ds (F(B0 ), K G(Bt ))]dx for every F, G ∈ ℋN by the decomposition F = ℜF + ℑG, ℜF = ℜF+ − ℜF− , ℑF = ℑF+ − ℑF− , G = ℜG + ℑG, ℜG = ℜG+ − ℜG− and ℑG = ℑG+ − ℑG− . The functional integral representation (2.3.12) in Theorem 2.12 can further be extended to Kato-decomposable V, which will be done in the next section.

2.4 Nelson model with Kato-class potential Theorem 2.12 gives a functional integral representation of (F, e−tHN G) in terms of Brownian motion (Bt )t≥0 and the Euclidean field ϕE . In Theorem 2.18 we extended this formula to singular external potentials. In this section we extend it to the more singular class of Kato-decomposable potentials. When V is Kato-decomposable a functional integral representation of (F, e−tLN G) can be actually constructed by using the P(ϕ)1 -process (Xt )t≥0 and the infinite dimensional Ornstein–Uhlenbeck process (ξt )t≥0 , but we need to assume that a ground state of Hp exists in order to define (Xt )t∈ℝ . In Section 2.4, we do not assume existence of a ground state of Hp , and only use Assumption 2.19 below. Assumption 2.19. The following conditions hold. (1) Dispersion relation: ω(k) = ων (k) = √|k|2 + ν2 , ν ≥ 0. ̂ ̂ ̂ ̂ (2) Charge distribution: φ ∈ S 󸀠 (ℝd ), φ(k) ∈ L2 (ℝd ). = φ(−k) and φ/ω, φ/√ω (3) External potential: V is Kato-decomposable. First we define HN with a Kato-decomposable potential as a self-adjoint operator in L2 (ℝd ) ⊗ L2 (Q ), via the functional integral representation given by t

(F, e−tHN G) = ∫ 𝔼x [e− ∫0 V(Bs )ds (I0 F(x), e−ϕE (Kt ) It G(Bt ))]dx, ℝd

where

t

Kt = ∫ δs ⊗ φ(⋅ − Bs )ds.

(2.4.1)

0

Thus we see that t

(e−tHN G)(x) = 𝔼x [e− ∫0 V(Bs )ds I∗0 e−ϕE (Kt ) It G(Bt )].

(2.4.2)

Conversely, we show that a sufficient condition to define the right-hand side of (2.4.2) is that V is of Kato-decomposable. The idea used for Schrödinger operators with Katoclass potentials will be extended to the Nelson Hamiltonian.

152 | 2 The Nelson model by path measures Let V be a Kato-decomposable potential and define the family of operators t

(Tt F) (x) = 𝔼x [e− ∫0 V(Br )dr I∗0 e−ϕE (Kt ) It F(Bt )]. Lemma 2.20. Operator Tt is bounded on ℋN , for every t > 0. Proof. Let F ∈ ℋN . Since ‖I0 e−ϕE (Kt ) It F(Bt )‖ ≤ 2eE(t) ‖F(Bt )‖, where t E(t) = ‖f ̂/ω‖2 + 2t(t ∨ 1)(‖f ̂/√ω‖2 + ‖f ̂/ω‖2 ), 2

(2.4.3)

is given by (1.5.21), we get from t

‖Tt F‖2ℋN = ∫ ‖𝔼x [e− ∫0 V(Br )dr I∗0 e−ϕE (Kt ) It F(Bt )]‖2L2 (Q) dx ℝd

that t

‖Tt F‖2ℋN ≤ 2 ∫ 𝔼0 [e−2 ∫0 V(Br +x)dr ]𝔼0 [‖F(Bt + x)‖2 ]e2E(t) dx. ℝd t

Since V is Kato-decomposable, we have supx∈ℝd 𝔼0 [e−2 ∫0 V(Br +x)dr ] = C < ∞, and thus ‖Tt F‖2ℋN ≤ 4Ce2E(t) ‖F‖2ℋN results. In what follows we show that {Tt : t ≥ 0} is a symmetric C0 -semigroup. To obtain this, we introduce the time shift operator ut f (x) = f (x0 − t, x),

x = (x0 , x) ∈ ℝ × ℝd

on L2 (ℝd ). It is straightforward that u∗t = u−t and u∗t ut = 1. We denote the second quantization of ut by Ut = ΓE (ut ), which acts on L2 (QE ) and is a unitary map. We see that Ut Is = Is+t . Lemma 2.21. Ts Tt = Ts+t holds for s, t ≥ 0. Proof. By the definition of Tt we have s

t

Ts Tt F = 𝔼x [e− ∫0 V(Br )dr I∗0 e−ϕE (Ks ) Is 𝔼Bs [e− ∫0 V(Br )dr I∗0 e−ϕE (Kt ) It F(Bt )]].

(2.4.4)

∗ ∗ By the formulae Is I∗0 = Is I∗s U−s = Es U−s and It = U−s It+s , equality (2.4.4) can be written as s

t

∗ −ϕE (Kt ) 𝔼x [e− ∫0 V(Br )dr I∗0 e−ϕE (Ks ) Es 𝔼Bs [e− ∫0 V(Br )dr U−s e U−s It+s F(Bt )]] .

(2.4.5)

∗ −ϕE (Kt ) Since Us is unitary, we have U−s e U−s = e−ϕE (u−s Kt ) as an operator. The test func∗ tion of the exponent u−s Kt is given by ∗

t

u∗−s Kt

= ∫ δr+s ⊗ φ(⋅ − Br )dr. 0

2.4 Nelson model with Kato-class potential | 153

Moreover, by the Markov property of Et , t ∈ ℝ, we may drop Es in (2.4.5), and by the Markov property of (Bt )t≥0 we have s

s+t

Ts Tt F = 𝔼x [e− ∫0 V(Br )dr I∗0 e−ϕE (Ks ) 𝔼x [e− ∫s s+t

= 𝔼x [e− ∫0 s+t

where Kss+t = ∫s

V(Br )dr −ϕE (Kss+t )

e

V(Br )dr ∗ −ϕE (Ks+t ) I0 e Is+t F(Bs+t )]

Is+t F(Bs+t )|FsBM ]]

= Ts+t F,

δr ⊗ φ(⋅ − Br )dr and (FtBM )t≥0 denotes the natural filtration of (Bt )t≥0 .

Strong continuity of the map t 󳨃→ Tt on ℋN can also be checked, while T0 = 1 is trivial. Theorem 2.22. The semigroup {Tt : t ≥ 0} is a symmetric C0 -semigroup. Proof. Since we already know that {Tt : t ≥ 0} is a C0 -semigroup, it is enough to show that Tt is symmetric, i. e., (F, Tt G) = (Tt F, G). Let R = ΓE (r) be the second quantization of the reflection r, and Ut = ΓE (ut ). We have t

(F, Tt G) = ∫ 𝔼x [e− ∫0 V(Bs )ds (Ut RIt F(B0 ), Ut Re−ϕE (Kt ) I0 G(Bt ))] dx ℝd

t

= ∫ 𝔼x [e− ∫0 V(Bs )ds (It F(B0 ), e−ϕE (ut rKt ) I0 G(Bt ))] dx, ℝd

where

t

ut rKt = ∫ δt−s ⊗ φ(⋅ − Bs )ds. 0 d Noticing that Ḃ s = Bt−s − Bt = Bs for 0 ≤ s ≤ t, we can replace Bs with Ḃ s and obtain t

̇ ̇ (F, Tt G) = ∫ 𝔼0 [e− ∫0 V(Bs +x)ds (It F(Ḃ 0 + x), e−ϕE (ut rKt ) I0 G(Ḃ t + x))]dx. ℝd t Here K̇ t = ∫0 δs ⊗ φ(⋅ − Ḃ s )ds. Swapping ∫ℝd dx and ∫X d𝒲 0 , and changing the variable −Bt + x to y, we have t

(F, Tt G) = ∫ 𝔼0 [e− ∫0 V(Bt−s +y)ds (It F(Bt + y), e−ϕE (Kt ) I0 G(B0 + y))]dy. ̃

ℝd t

Here K̃ t = ∫0 δt−s ⊗ φ(⋅ − Bt−s − y)ds. Thus we conclude that (F, Tt G) = (Tt F, G) and the theorem follows. By Theorem 2.22 there exists a self-adjoint operator HNKato such that Kato

Tt = e−tHN ,

t ≥ 0.

154 | 2 The Nelson model by path measures Definition 2.23 (Nelson Hamiltonian for Kato-class potential). We call the self-adjoint operator HNKato Nelson Hamiltonian for Kato-class potential V. In what follows we write again HN for HNKato , for notational simplicity. The semigroup generated by − 21 Δ + V with Kato-decomposable potential V has t

a number of regularity properties. Let Kt f (x) = 𝔼x [e− ∫0 V(Bs )ds f (Bt )] and suppose that f ∈ Lp (ℝd ) for some 1 ≤ p ≤ ∞. Then x 󳨃→ Kt f (x) is continuous for every t > 0 by Proposition 4.109 in Volume 1. We can also have a counterpart of this fact for HN with Kato-decomposable potential V. Write t

I[0,t] = I∗0 e−ϕE (∫0 δs ⊗φ(⋅−Bs )ds) It . The estimate ‖I[0,t] ‖ ≤ 2eE(t)

(2.4.6)

is established in (2.4.3). By the functional integral representation of the semigroup generated by HN we have t

(F, e−tHN G)ℋN = ∫ 𝔼x [e− ∫0 V(Bs )ds (F(B0 ), I[0,t] G(Bt ))L2 (Q) ]dx.

(2.4.7)

ℝd

From this we obtain t

Φt (x) = (e−tHN Φ)(x) = 𝔼x [e− ∫0 V(Bs )ds I[0,t] Φ(Bt )]

(2.4.8)

for almost every x ∈ ℝd . We remark that Φt is a vector in L2 (ℝd × Q ). Hence it is meaningless to consider Φt (x) for each x ∈ ℝd . In the theorem below we can show that the right-hand side of (2.4.8) gives the continuous version of (Ψ, Φt (x))L2 (Q) . Theorem 2.24. Let Ψ ∈ L2 (Q ) and Φ ∈ L2 (ℝd ) ⊗ L2 (Q ). Then for every t > 0, the map t

ℝd ∋ x 󳨃→ 𝔼x [e− ∫0 V(Bs )ds (Ψ, I[0,t] Φ(Bt ))] is continuous, i. e., (Ψ, e−tHN Φ(x))L2 (Q) has a continuous version. Proof. For arbitrary t ≥ 0 it follows that t

(Ψ, Φt (x)) = 𝔼x [e− ∫0 V(Bs )ds (Ψ, I[0,t] Φ(Bt ))]. t

Write ρt (x) = 𝔼x [e− ∫0 V(Bs )ds (Ψ, I[0,t] Φ(Bt ))]. Since V is a Kato-decomposable potential, ρ ∈ L1 (ℝd ). We apply the smoothing effect of the heat semigroup discussed in Proposition 4.109 in Volume 1. Let f ∈ L1 (ℝd ) and define (Pr f )(x) = ∫ℝd Πr (x − y)f (y)dy. The map x 󳨃→ Pr f (x) is continuous, and furthermore t

𝔼x [e− ∫r V(Bs )ds f (Bt )] = Pr e−(t−r)Hp f (x).

2.5 Existence and uniqueness of the ground state

| 155

Using the Markov property of Brownian motion we have t

t

Pr ρt−r (x) = 𝔼x [e− ∫r V(Bs )ds (Ψ, I∗0 e−ϕ(∫r δs−r ⊗φ(⋅−Bs )ds) It−r Φ(Bt ))] and (Pr ρt−r )(x) is continuous in x. We have ρt (x) − ρt (y) = ρt (x) − (Pr ρt−r )(x) + (Pr ρt−r )(x) − (Pr ρt−r )(y) + (Pr ρt−r )(y) − ρt (y). Since V is a Kato-decomposable potential and (2.4.6) holds, it is straightforward to see that |ρt (x) − (Pr ρt−r )(x)| → 0 as r → 0 uniformly in x. Similarly |ρt (y) − (Pr ρt−r )(y)| → 0 as r → 0 uniformly in y. On the other hand for each r and t, (Pr ρt−r )(x)−(Pr ρt−r )(y) → 0 as |x − y| → 0. From here we conclude that ρ(x) is continuous in x.

2.5 Existence and uniqueness of the ground state 2.5.1 Uniqueness First we discuss the uniqueness of the ground state of HN by means of the functional integral representation derived in the previous sections. Proposition 2.25 (Positivity improving). The semigroup {e−t Hf : t > 0} is positivity improving. ̂

Proof. The proposition follows from Corollary 1.54. Uniqueness of the ground state follows directly from functional integral representation of the Nelson Hamiltonian and the Perron–Frobenius theorem. Theorem 2.26 (Uniqueness of ground state). Let either Assumption 2.2 or Assumption 2.19 hold. If HN has a ground state, then it is unique. Proof. By Theorem 2.12 it suffices to show that (F, e−tHN G) ≠ 0 for every non-negative F and G such that F ≢ 0 and G ≢ 0, since e−tHN is positivity preserving. Using (2.3.12) we have t

(F, e−tHN G) = ∫ 𝔼x [e− ∫0 V(Bs )ds (I0 F(B0 ), e−ϕE (K) It G(Bt ))L2 (QE ) ]dx, ℝd t

where K = ∫0 δs ⊗φ(⋅−Bs )ds. Suppose (I0 Ψ, e−ϕE (K) It Φ)L2 (QE ) = 0 for some nonnegative Ψ and Φ. This means that μE (supp[(e−ϕE (K) )It Φ] ∩ supp[I0 Φ]) = 0.

(2.5.1)

Since ϕE (K) ∈ L2 (QE ), we see that μE ({ϕE ∈ QE | eϕE (K) = 0}) = 0. Hence (2.5.1) implies ̂ ̂ that μE (supp[It Ψ] ∩ supp[I0 Φ]) = 0, and thus 0 = (It Ψ, I0 Φ) = (Ψ, e−t Hf Φ). Since e−t Hf

156 | 2 The Nelson model by path measures is positivity improving, this is a contradiction. Hence (I0 Ψ, e−ϕE (K) It Φ)L2 (QE ) > 0 and (I0 F(B0 ), e−ϕE (K) It G(Bt ))L2 (QE ) > 0. Thus (F, e−tHN G) > 0 follows for any nonnegative F and G. If the ground state Ψg of HN exists, by the proof of Theorem 2.26 we see that Ψg = Ψg (x, ϕ) is strictly positive for almost every (x, ϕ) ∈ ℝd × Q . Thus we see that (Ψ, Ψg (x))L2 (Q) > 0 for almost every x ∈ ℝd for any Ψ ≥ 0 but Ψ ≢ 0. We can also get to an improved statement. Corollary 2.27. Let either Assumption 2.2 with V+ ∈ L1loc (ℝd ) or Assumption 2.19 hold. Let Ψ ∈ L2 (Q ) be non-negative and Ψ ≢ 0. Then (Ψ, Ψg (x))L2 (Q) > 0 for all x ∈ ℝd . Proof. It has been established that (Ψ, Ψg (x))L2 (Q) ≥ 0 for any x ∈ ℝd . We show that (Ψ, Ψg (x))L2 (Q) ≠ 0 for all x ∈ ℝd . Suppose that (Ψ, Ψg (x))L2 (Q) = 0 for some x ∈ ℝd . Since Ψg = e−t(HN −E) Ψg , for every fixed x we have the functional integral representation t

(Ψ, Ψg (x))L2 (Q) = etE 𝔼𝒲 [e− ∫0 V(Bs +x)ds (Ψ, I[0,t] (x)Ψg (Bt + x))L2 (Q) ]. t

Here I[0,t] (x) = I0 e−ϕE (∫0 δs ⊗φ(⋅−Bs −x)ds) It . Since t

e− ∫0 V(Bs +x)ds (Ψ, I[0,t] (x)Ψg (Bt + x))L2 (Q) ≥ 0,

a. s.,

equality (Ψ, Ψg (x))L2 (Q) = 0 implies that t

e− ∫0 V(Bs +x)ds (Ψ, I[0,t] (x)Ψg (Bt + x))L2 (Q) = 0

a. s.

t

By Lemma 4.95 in Volume 1, V+ ∈ L1loc (ℝd ) implies that e− ∫0 V(Bs +x)ds > 0 a. s. This gives (Ψ, I[0,t] (x)Ψg (Bt + x))L2 (Q) = 0

a. s.

t

Furthermore, e−ϕE (∫0 δs ⊗φ(⋅−Bs )ds) > 0 a. s. on X × Q and we have (Ψ, I∗0 It Ψg (Bt + x))L2 (Q) = 0,

a. s.

(Ψ, e−t Hf Ψg (Bt + x))L2 (Q) = 0

a. s.

Thus we conclude that ̂

Since e−t Hf is positivity improving, we see that Ψg (Bt + x) = 0 a. s. on X × Q . Hence ̂

0 = 𝔼𝒲 [(Ψ, Ψg (Bt + x))L2 (Q) ] = ∫ Πt (x − y)(Ψ, Ψg (y))L2 (Q) dy. ℝd

Πt (x − y) > 0 implies that (Ψ, Ψg (y))L2 (Q) = 0 for almost every y ∈ ℝd . Since Ψ is an arbitrary nonnegative vector, it follows that Ψg (y) = 0 for almost every y ∈ ℝd which is a contradiction. Hence (Ψ, Ψg (y))L2 (Q) > 0 for all y ∈ ℝd .

2.5 Existence and uniqueness of the ground state

| 157

2.5.2 Existence We turn to the existence problem of a ground state of the Nelson Hamiltonian. Theorem 2.28 (Existence of ground state). In addition to Assumption 2.5, suppose that 2 ̂ /ω(k)3 dk < ∞ and ∫ℝd |φ(k)| Σp − Ep > ∫ ℝd

2 ̂ |φ(k)| |k|2 dk 2 2ω(k) 2ω(k) + |k|2

(2.5.2)

hold, where Σp denotes the infimum of the essential spectrum of Hp . Then HN has a ground state. 2 ̂ Under ∫ℝd |φ(k)| /ω(k)3 dk < ∞, we have the uniform bound 0

∞

∫ ds ∫ |W(Xs − Xt , s − t)|dt ≤ ∫ −∞

0

ℝd

2 ̂ |φ(k)| dk < ∞ 2ω(k)3

(2.5.3)

2 ̂ on paths. In Section 2.7 we will also see that ∫ℝd |φ(k)| /ω(k)3 dk < ∞ is crucial for a ground state to exist, and will show that in the physically important cases the existence of a ground state fails if the infrared regularity condition does not hold. Condition (2.5.2) is a restriction on the coupling constant, which in fact only comes to effect in the case where the particle Hamiltonian Hp has absolutely continuous spectrum. On physical grounds, (2.5.2) is not expected to be necessary. This is because the coupling to the quantum field should make the particle more heavy, enhancing the binding under V.

Lemma 2.29. Let f ∈ L2 (ℝd ) and be a real-valued function. If ess infk∈K f (k) > 0 for every compact set K ⊂ ℝd , then (f ⊗ 1, e−(T+t)HN f ⊗ 1) = e−tEN . T→∞ (f ⊗ 1, e−THN f ⊗ 1) lim

Proof. Recall that if μ is a measure on ℝ with inf supp(μ) = E(μ), then E(μ) = − lim

T→∞

1 ln (∫ e−Tx μ(dx)) , T

(2.5.4)

ℝ

and lim

T→∞

∫ℝ e−(T+t)x μ(dx) ∫ℝ e−Tx μ(dx)

= e−tE(μ) .

(2.5.5)

158 | 2 The Nelson model by path measures Applying (2.5.5) to the spectral measure μf ⊗1 of HN associated with vector f ⊗ 1 yields the claim with EN replaced by inf supp(μf ⊗1 ) = E(μf ⊗1 ). Thus it only remains to prove E(μf ⊗1 ) = EN . For this, define the class of functions 󵄨󵄨 󵄨

2

d

O = {F ∈ L (P0 ) 󵄨󵄨󵄨 supp F ⊂ ⋃ BN (ℝ ) × BM (M−2 )}, N,M>0

where BN (ℝd ) and BM (M−2 ) denote the balls centered in the origin in ℝd and M−2 of radius N and M, respectively. O is dense in L2 (P0 ). Pick g ∈ O . Since e−tHN is positivity preserving, we get (g, e−THN g) ≤ (|g|, e−THN |g|) ≤ C 2 (f ⊗ 1, e−THN f ⊗ 1) with C=

ess sup(x,ξ )∈ℝd ×M−2 |g(x, ξ )| ess inf(x,ξ )∈supp |g| f (x)

.

Equality (2.5.4) shows that E(μf ⊗1 ) ≤ E(μg ), for all g ∈ O . However, since O is a dense subset in D(HN ), we have E(μf ⊗1 ) ≥ EN = inf {E(μg ) | g ∈ O } ≥ E(μf ⊗1 ), and thus EN = E(μf ⊗1 ). Recall that Ψp is the normalized ground state of Hp , and for T > 0 define ΨTg =

e−THN (Ψp ⊗ 1)

‖e−THN (Ψp ⊗ 1)‖

(2.5.6)

.

Since ‖ΨTg ‖ = 1, ΨTg has a subsequence ΨTg such that ΨTg is weakly convergent to a 󸀠 vector Ψ∞ g . We reset T to T, and defining 󸀠

󸀠

γ(T) = (Ψp ⊗ 1, ΨTg )2 consider the limit of γT and more general cases for later use as T → ∞ in what follows. Thus we choose ρ ∈ L2 (ℝd ) and define γρ (T) =

(ρ ⊗ 1, e−THN (ρ ⊗ 1))2 ‖e−THN (ρ ⊗ 1)‖2

(2.5.7)

replacing Ψp by ρ in γ(T). The next criterion is useful for showing the existence of a ground state of HN .

2.5 Existence and uniqueness of the ground state

| 159

Proposition 2.30 (Criterion for existence of ground state). Suppose that ρ ∈ L2 (ℝd ) and ess infx∈K ρ(x) > 0 for every compact set K ⊂ ℝd . Denote lim γρ (T) = a.

T→∞

If a > 0, then HN has a ground state. Proof. Since a > 0, for sufficiently large T it follows that b < √γρ (T) with some b > 0. Let E(⋅) be the spectral measure of HN associated with ρ ⊗ 1, and δ = inf suppE. Then δ = EN follows from the proof of Lemma 2.29. By the definition of δ, we see that E([δ, δ + ε)) ≠ 0 for every ε > 0, and E([δ, δ + ε)) is strongly convergent to E({δ}) as ε ↓ 0. We have √γρ (T) =

∫[δ,δ+ε) e−T(λ−δ) dE + ∫[δ+ε,∞) e−T(λ−δ) dE (∫[δ,∞) e−2T(λ−δ) dE)

1/2

.

By the Schwarz inequality we have ∫[δ,δ+ε) e−T(λ−δ) dE (∫[δ,∞)

1/2 e−2T(λ−δ) dE)

Next consider timated as

1/2

≤ E([δ, δ + ε))

∫[δ+ε,∞) e−T(λ−δ) dE

1/2

(∫[δ,∞) e−2T(λ−δ) dE)

∫

(∫[δ,δ+ε) e−2T(λ−δ) dE) (∫[δ,∞)

e−2T(λ−δ) dE)

1/2

1/2

≤ E([δ, δ + ε))1/2 .

. Since the denominator and the numerator can be es-

e−T(λ−δ) dE ≤ e−Tε ,

[δ+ε,∞)

∫ e−2T(λ−δ) dE ≥ [δ,∞)

∫

e−2T(λ−δ) dE ≥ e−εT E([δ, δ + ε/2)),

[δ,δ+ε/2)

we see that ∫[δ+ε,∞) e−T(λ−δ) dE

1/2

(∫[δ,∞) e−2T(λ−δ) dE)

≤

e−Tε/2 e−Tε ≤ . 1/2 + ε/2)) E([δ, δ + ε/2))1/2

e−Tε/2 E([δ, δ

It follows that √γρ (T) ≤ E([δ, δ + ε))1/2 +

e−Tε/2 . E([δ, δ + ε/2))1/2

Take T → ∞ on both sides above. We obtain b ≤ E([δ, δ + ε)) for every ε > 0, and b ≤ E({δ}) is obtained by taking the limit ε ↓ 0. Hence 0 ≠ E({δ}) = E({EN }) and thus a ground state of HN exists.

160 | 2 The Nelson model by path measures Proposition 2.30 gives a sufficient condition for HN to have a ground state. A similar argument can also be used to show absence of a ground states of HN . Proposition 2.31 (Criterion for absence of ground state). Let ρ ∈ L2 (ℝd ) be a nonnegative function and not identically zero. If a = limT→∞ γρ (T) = 0, then HN has no ground state. Proof. We may suppose that inf Spec(HN ) = 0, so that limT→∞ e−THN = 1{0} (HN ) in strong sense. If 0 is an eigenvalue of HN , i. e., HN has a ground state Ψg , then a = (ρ ⊗ 1, Ψg ) > 0, since Ψg is strictly positive by Theorem 2.26. Thus a = 0 implies that HN has no ground state. Corollary 2.32. Suppose that ess infx∈K Ψp (x) > 0 for every compact set K ⊂ ℝd . Then HN has a ground state if and only if Ψ∞ g ≠ 0. Proof. Since ΨTg is nonnegative, the weak limit Ψ∞ g is also nonnegative. Hence ∞ 2 ∞ limT→∞ γ(T) = (Ψp ⊗ 1, Ψg ) > 0 if Ψg ≠ 0 and = 0 if Ψ∞ g = 0. The corollary follows then by Proposition 2.30. Remark 2.33. Note that since V is Kato-decomposable, by Lemma 4.179 in Volume 1, the ground state Ψp (x) is continuous and strictly positive. Hence ess infx∈K Ψp (x) > 0 for every compact set. Proof of Theorem 2.28. Write Ψp instead of Ψp ⊗ 1. By Corollary 2.32, we need to prove that ΨTg is not weakly convergent to zero. Let S[a,b] =

b

b

a

a

1 ∫ ds ∫ W(Xs − Xt , s − t)dt 2

and put f (T, t) = (ΨTg , (e−tHp ⊗ P0 )ΨTg ), where P0 denotes the projection onto the one-dimensional subspace generated by the constant function 1 ∈ L2 (G). We claim that lim inf f (T, t) ≥ exp(−t(EN + ∫ T→∞

ℝd

2 2 ̂ ̂ |φ(k)| (1 + e−tω(k) ) |φ(k)| dk) − dk). ∫ 2ω(k)3 2ω(k)2

(2.5.8)

ℝd

To prove this, write f (T, t) =

(Ψp , e−THN (e−tHp ⊗ P0 )e−THN Ψp ) (Ψp , e−(2T+t)HN Ψp ) (Ψp , e−(2T+t)HN Ψp )

(Ψp , e−2THN Ψp )

.

The second ratio above converges to e−EN t as T → ∞ by Lemma 2.29 since Ψp > 0. The first ratio, denoted g(t, T), can be written in terms of a functional integral. The

2.5 Existence and uniqueness of the ground state

| 161

denominator is 𝔼𝒩0 [eS[−T,T+t] ]e−(2T+t)Ep

(2.5.9)

due to (2.3.9) and shift invariance of (Xt )t≥0 . For the numerator of g(t, T), notice that (Ψp , e−THN (e−tHp ⊗ P0 )e−THN Ψp ) = (hT , e−tHp hT )L2 (ℝd ) , where hT (x) = (1, e−THN Ψp )L2 (G) (x). Also, notice that t

∫ hT (x)f (x)Ψp (x)dx = (f Ψp ⊗ 1, e−THN Ψp ⊗ 1) = 𝔼𝒫0 [f (X0 )e− ∫0 ξs (φXs )ds ]e−TEp ℝd

t

= 𝔼𝒩0 [f (X0 )𝔼𝒢 [e− ∫0 ξs (φXs )ds ]]e−TEp = ∫ Ψp (x)2 f (x)𝔼x𝒩0 [eS[0,T] ]e−TEp dx. ℝd

Comparing the leftmost and right-hand side expressions, we have hT (x) = Ψp (x)F(x)e−TEp with F(x) = 𝔼x𝒩0 [eS[0,T] ]. Thus (hT , e−tHp hT ) = (F, e−tLp F)e−(2T+t)Ep = 𝔼𝒩0 [F(X0 )F(Xt )]e−(2T+t)Ep X

= ∫ 𝔼x𝒩0 [𝔼x𝒩0 [eS[0,T] ]𝔼𝒩t [eS[0,T] ]]e−(2T+t)Ep dN0 . ℝd

0

By reflection symmetry, X

(hT , e−tHp hT ) = ∫ 𝔼x𝒩0 [𝔼x𝒩0 [eS[−T,0] ]𝔼𝒩t [eS[0,T] ]]e−(2T+t)Ep dN0 ℝd

0

and by the Markov property (hT , e−tHp hT ) = ∫ 𝔼x𝒩0 [𝔼x𝒩0 [eS[−T,0] ]𝔼x𝒩0 [eS[t,T+t] |σ(Xt )]]e−(2T+t)Ep dN0 . ℝd

Independence of X−t , t ≥ 0 and Xs , s ≥ 0, implies (hT , e−tHp hT ) = ∫ 𝔼x𝒩0 [𝔼x𝒩0 [eS[−T,0]+S[t,T+t] |σ(Xt )]]e−(2T+t)Ep dN0 ℝd

= ∫ 𝔼x𝒩0 [eS[−T,0]+S[t,T+t] ]e−(2T+t)Ep dN0 ℝd

= 𝔼𝒩0 [eS[−T,0]+S[t,T+t] ]e−(2T+t)Ep .

162 | 2 The Nelson model by path measures Finally, using (2.5.9) it follows that g(T, t) =

𝔼𝒩0 [eS[−T,0]+S[t,T+t] ]

=

𝔼𝒩0 [eS[−T,T+t] ]

𝔼𝒩0 [eSΔ +S[−T,T+t] ] 𝔼𝒩0 [eS[−T,T+t] ]

,

where SΔ = S[−T, 0] + S[t, T + t] − S[−T, T + t]. Making use of the uniform pathwise estimate 0 ∞

|SΔ | ≤

t ∞

t t

2 ̂ |φ(k)| 1 dk (2 ∫ ∫ +2 ∫ ∫ + ∫ ∫) e−|s−r|ω(k) dsdr ∫ 2 2ω(k) ℝd

0 t

−∞ 0

0 0

2

2 ̂ ̂ |φ(k)| |φ(k)| (1 + 2e−tω(k) ) ≤t∫ dk, dk + ∫ 2ω(k)3 2ω(k)2 ℝd

ℝd

we can compare the numerator and denominator of g(t, T) to find that g(t, T) ≥ exp (−t ∫ ℝd

2 2 ̂ ̂ |φ(k)| |φ(k)| (1 + 2e−tω(k) ) dk − dk). ∫ 2ω(k)3 2ω(k)2 ℝd

This proves (2.5.8), and we see that lim infT→∞ ‖e−tHp /2 ⊗ P0 ΨTg ‖ is nonzero. In order to show that the family ΨTg does not converge to zero, we replace e−tHp /2 ⊗P0 by a compact operator, which we will choose to be a spectral projection of e−tHp /2 . Let 1[a,b] (Hp ) denote the projection onto the spectral subspace of Hp corresponding to Spec(Hp )∩[a, b]. By the definition of Σp and by standard properties of Schrödinger operators, Hp has only finitely many eigenvalues of finite multiplicity below Σp − δ for every δ > 0, and thus 1[Ep ,Σp −δ] (Hp ) ⊗ P0 is a finite rank operator. We have (1[Ep ,Σp −δ] (Hp ) ⊗ P0 )ΨTg → (1[Ep ,Σp −δ] (Hp ) ⊗ P0 )Ψ∞ g in strong sense as T → ∞. On the other hand, 󵄩󵄩 −tHp 󵄩 󵄩󵄩e 1(Σp −δ,∞) 󵄩󵄩󵄩󵄩 ≤ e−t(Σp −δ) . 󵄩 Hence −tHp (Ψ∞ 1[Ep ,Σp −δ] (Hp ) ⊗ P0 )Ψ∞ g , (e g )

= lim (ΨTg , (e−tHp ⊗ P0 )ΨTg ) − (ΨTg , (e−tHp 1(Σp −δ,∞) (Hp ) ⊗ P0 )ΨTg ) T→∞

and we estimate −tHp −(EN +C)t−C(t) (Ψ∞ 1[Ep ,Σp −δ] (Hp ) ⊗ P0 )Ψ∞ − e−t(Σp −δ)t g , (e g )≥e

= e−(EN +C)t (e−C(t) − e−(Σp −δ−EN −C)t ),

(2.5.10)

2.5 Existence and uniqueness of the ground state

| 163

where we used (2.5.8) and (2.5.10), and the notations C=∫ ℝd

2 ̂ |φ(k)| dk, 2ω(k)2

C(t) = ∫ ℝd

2 ̂ |φ(k)| (1 + e−tω(k) ) dk. 2ω(k)3

By choosing δ small enough and t large enough, we find that Ψ∞ g is not zero provided EN < Σp − ∫ ℝd

2 ̂ |φ(k)| dk. 2 2ω(k)

(2.5.11)

The final step is to convert (2.5.11) to a condition involving the bottom Ep of the spectrum of Hp instead of that of HN . This is achieved by the estimate EN ≤ Ep − ∫ ℝd

2 ̂ |φ(k)| dk 2ω(k)(ω(k) + |k|2 /2)

(2.5.12)

which will be proved in Lemma 2.34 below. Combining (2.5.12) and (2.5.11) completes the proof. A concluding lemma provides the missing piece in the proof above. Lemma 2.34. The estimate (2.5.12) holds. Proof. Instead of L2 (P0 ), we show (2.5.12) on ℋN = L2 (ℝd ) ⊗ ℱN . Let Pf = dΓ(k) be the momentum operator of ℱN , and define ψf = eix⊗Pf Ψp ⊗ e−iΠ(f ) ΩN , where ΩN is the Fock vacuum in ℱN , Π(f ) = i(a∗ (f ) − a(f ̄)) is the conjugate momentum, and f will be determined below. By a direct computation, 2 ̂ 2φ(k)ℜf (k) |k|2 1 ]dk. )|f (k)|2 + EN ≤ (ψf , HN ψf ) = Ep + ( ∫ k|f (k)|2 dk) + ∫ [(ω(k) + 2 2 √2ω(k) d d ℝ

ℝ

(2.5.13)

Let f be such that f (−k) = f (k). The second term at the right-hand side of (2.5.13) is zero, and the last term is 1 ∫ (ω(k) + |k|2 ) (|f (k) + Φ(k)|2 − Φ2 (k)) dk, 2

ℝd

where Φ(k) = −

̂ φ(k)

√2ω(k)(ω(k) + |k|2 /2)

The minimizer f (k) = Φ(k) yields the bound (2.5.12).

.

164 | 2 The Nelson model by path measures Corollary 2.35 (Existence of ground state for any coupling strength). Let ω(k) = |k| ̂ and φ(k) = g1{κ≤|k|≤Λ} with coupling constant g ∈ ℝ. Suppose also that Spec(Hp ) is purely discrete and ess infx∈K Ψp (x) > 0 for every compact set. Then for all 0 < κ < Λ and g ∈ ℝ the operator HN has a unique ground state. 2 ̂ Proof. Since Σp − Ep = ∞ and ∫ℝd |φ(k)| /ω(k)3 dk < ∞, the corollary follows by Theorem 2.28.

For confining potentials V the operator H has a purely discrete spectrum, which gives Σp − Ep = ∞.

2.6 Ground state expectations 2.6.1 General expressions Throughout Section 2.6 we use Assumption 2.36 below. Assumption 2.36. The following conditions hold: (1) Dispersion relation: ω(k) = ων (k) = √|k|2 + ν2 , ν ≥ 0. ̂ ̂ ̂ ̂ (2) Charge distribution: φ ∈ S 󸀠 (ℝd ), φ(k) = φ(−k) and φ/ω, φ/√ω ∈ L2 (ℝd ). 2 3 ̂ (3) Infrared regularity: ∫ℝd |φ(k)| /ω(k) dk < ∞. (4) External potential: V is Kato-decomposable and Hp has a ground state Ψp with Hp Ψp = Ep Ψp . (5) Ground state: HN has the unique ground state Ψg . We write d𝒫T =

T 1 ∫−T e ξs (φXs )ds d𝒫0 ZT

with normalizing constant ZT . With these ingredients, the representation (ΨTg , (f ⊗ 1)ΨTg ) = 𝔼𝒫T [f (X0 )] holds. Due to linearity of the coupling, the field variables ξ can be integrated out in 𝔼𝒫T [f (X0 )] and this gives the functional integral representation (ΨTg , (f ⊗ 1)ΨTg ) = 𝔼𝒩T [f (X0 )] , where d 𝒩T =

T T 1 21 ∫−T ds ∫−T e ZT

W(Xs −Xt ,s−t)dt

d𝒩0 .

Then formally, (Ψg , (f ⊗ 1)Ψg ) = lim (ΨTg , (f ⊗ 1)ΨTg ) = 𝔼𝒩 [f (X0 )] T→∞

(2.6.1)

2.6 Ground state expectations | 165

holds with a probability measure 𝒩 . In the next theorem we establish the existence of this measure. Theorem 2.37 (Tightness). The family of probability measures {𝒩T }T≥0 is tight. In particular, there exists a subsequence (Tn󸀠 )n∈ℕ such that 𝒩Tn󸀠 is weakly convergent to a probability measure 𝒩 on X. Proof. By the Prokhorov theorem it suffices to show that (1) limξ →∞ supT>0 𝒩T (|X0 |2 > ξ ) = 0, (2) limδ↓0 supT>0 𝒩T (max|s−t| ε) = 0 for any ε > 0. We have 2

T

T

𝒩T (|X0 | > ξ ) = (Ψg , 1{|x|2 >ξ } Ψg ).

Let ε > 0 be arbitrary. Since ΨTg → Ψg strongly as T → ∞, there exists T0 > 0 such that for all T > T0 , (ΨTg , 1{|x|2 >ξ } ΨTg ) ≤ (Ψg , 1{|x|2 >ξ } Ψg ) + ε. Hence 2

T

T

𝒩T (|X0 | > ξ ) ≤ sup (Ψg , 1{|x|2 >ξ } Ψg ) + (Ψg , 1{|x|2 >ξ } Ψg ) + ε. 0≤T≤T0

We estimate the first factor in the right-hand side above. Note that 1

(ΨTg , 1{|x|2 >ξ } ΨTg )

=

T

T

𝔼𝒩0 [1{|X0 |2 >ξ } e 2 ∫−T ds ∫−T W(Xs −Xt ,s−t)dt ] 1

T

T

𝔼𝒩0 [e 2 ∫−T ds ∫−T W(Xs −Xt ,s−t)dt ]

.

(2.6.2)

Since T

T

−T

−T

2 ̂ |φ(k)| 1 (e−2Tω(k) − 1 + 2Tω(k)) dk ≤ aT + b ∫ ds ∫ W(Xs − Xt , s − t)dt ≤ ∫ 3 2 2ω(k) ℝd

with some a, b, together with (2.6.2) it gives (ΨTg , 1{|x|2 >ξ } ΨTg ) ≤

𝔼𝒩0 [1{|X0 |2 >ξ } ]eaT+b e−(aT+b)

= (Ψp , 1{|x|2 >ξ } Ψp )e2(aT+b) .

Thus sup (ΨTg , 1{|x|2 >ξ } ΨTg ) → 0

0≤T≤T0

as ξ → ∞ and (1) follows.

166 | 2 The Nelson model by path measures To prove (2), it suffices to show that 𝔼𝒩T [|Xs − Xt |2n ] ≤ D|s − t|n for some n ≥ 1 with a constant D. Let t ≥ s. We have d

2n

2n μ )(−1)k 𝔼𝒩T [(Xsμ )2n−k (Xt )k ] k

𝔼𝒩T [|Xs − Xt |2n ] = ∑ ∑ ( μ=1 k=0 d

2n

2n )(−1)k (e−sHN ΨTg , xμ2n−k e−(t−s)HN xμk e+tHN ΨTg ). k

=∑∑( μ=1 k=0

Here we note that HN = HNKato , and write ΔBt−s = |B0 − Bt−s |. The last term can be represented by functional integration given by Theorem 2.12 in terms of Brownian motion (Bt )t≥0 and the Euclidean field ϕE as 𝔼𝒩T [|Xs − Xt |2n ]

t−s

V(Br )dr

(I0 e−sHN ΨTg (B0 ), e−ϕE (∫0

t−s

V(Br )dr

‖e−sHN ΨTg (B0 )‖‖I∗0 e−ϕE (∫0

− ∫0 = ∫ 𝔼x [ΔB2n t−s e ℝd

− ∫0 ≤ ∫ 𝔼x [ΔB2n t−s e

t−s

δr ⊗φ(⋅−Br )dr)

t−s

It−s e+tHN ΨTg (Bt−s ))]dx

δr ⊗φ(⋅−Br )dr)

It−s ‖‖e+tHN ΨTg (Bt−s )‖]dx,

ℝd

where the norms with no subscript are L2 (Q )-norms. We also note that t−s

‖I∗0 e−ϕE (∫0

δr ⊗φ(⋅−Br )dr)

It−s ‖ ≤ 2e(t−s)E(φ) = C. ̂

(2.6.3)

Here E(φ)̂ is given by (2.3.15). By the Schwarz inequality, 𝔼𝒩T [|Xs − Xt |2n ] t−s

− ∫0 ≤ C𝔼[ΔB2n t−s ∫ e

V(Br +x)dr

‖e−sHN ΨTg (x)‖‖etHN ΨTg (Bt−s + x)‖]dx

ℝd t−s

−2 ∫0 ≤ C𝔼[ΔB2n t−s ( ∫ e

V(Br +x)dr

1/2

ℝd

ℝd t−s

1/2 −2 ∫0 ≤ C(𝔼[ΔB4n t−s ]) (𝔼[ ∫ e

V(Br +x)dr

ℝd

≤ CD|s − t|n ( sup 𝔼x [e x∈ℝd

1/2

‖e−sHN ΨTg (x)‖2 dx) ( ∫ ‖etHN ΨTg (Bt−s + x)‖2 dx) ]

t−s −2 ∫0

V(Br )dr

1/2

‖e−sHN ΨTg (x)‖2 dx ∫ ‖etHN ΨTg (Bt−s + x)‖2 dx]) ℝd

]) ‖e−sHN ΨTg ‖ℋN ‖etHN ΨTg ‖ℋN .

Finally, noticing that ‖e−sHN ΨTg ‖‖etHN ΨTg ‖ → ‖Ψg ‖2 as T → ∞, we complete (2).

1/2

2.6 Ground state expectations | 167

We now establish an explicit formula for expectations (Ψg , (g ⊗ L)Ψg ) of operators g ⊗ L as averages with respect to the probability measure 𝒩 . In order to state our main theorem, we introduce some special elements of ℋM . For every T ∈ [0, ∞] and every path, define T

φ+T,X (x)

= −∫e

−|s|ω̂

0

T

φXs (x)ds i. e.,

−ik⋅Xs −|s|ω(k) + (k) = − ∫ φ(k)e ̂ ̂ φ e ds T,X 0

0

φ−T,X (x) = − ∫ e−|s|ω φXs (x)ds

i. e.,

̂

0

−ik⋅Xs −|s|ω(k) − (k) = − ∫ φ(k)e ̂ ̂ φ e ds, T,X

−T

−T

where ω̂ = ω(−i∇). We also define 0

φ−X (x)

=− ∫ e

−|s|ω̂

φXs (x)ds

and

φ+X (x)

∞

= − ∫ e−|s|ω φXs (x)ds. ̂

0

−∞

It follows that φ±T,X ∈ ℋM and 0

(φ−T,X , φ+T,X )ℋM

T

= 2 ∫ ds ∫ W(Xs − Xt , s − t)dt, 0

−T

2 ̂ |φ(k)| dk < ∞. 3 ω(k)

‖φ±T,X ‖2ℋM ≤ ∫ ℝd

Write x,ξ

x

ξ

(x, ξ ) ∈ ℝd × M−2 .

𝒫0 = 𝒩0 ⊗ 𝒢 ,

Theorem 2.38 (Ground state expectations for bounded operators). Let L be a bounded operator on L2 (G), and g ∈ L∞ (ℝd ) viewed as a multiplication operator. Then 0

(Ψg , (g ⊗ L)Ψg ) = 𝔼𝒩 [(:eξ (φX ) :, L:eξ (φX ) :) −

+

L2 (G)

g(X0 )e− ∫−∞ ds ∫0

∞

W(Xs −Xt ,s−t)dt

].

(2.6.4)

Proof. By Theorem 2.7 we have (F, ΨTg ) =

1

‖e−TLN 1‖

(F, e−TLN 1) =

T 1 x,ξ 𝔼G×N0 [F(x, ξ )𝔼𝒫 [e− ∫0 ξs (φXs )ds ]] . 0 √ZT

Hence we arrive at ΨTg (x, ξ ) =

T 1 x,ξ 𝔼𝒫 [e− ∫0 ξs (φXs )ds ] 0 √ZT

(2.6.5)

168 | 2 The Nelson model by path measures x,ξ

for almost every (x, ξ ) ∈ ℝd × M−2 . Notice that 𝔼𝒫 ξ

𝔼𝒢 [e

T − ∫0 ξs (φXs )ds

ξ

0

ξ

= 𝔼x𝒩0 𝔼𝒢 . We can compute

] by using that (see Theorem 1.138) T

𝔼𝒢 [ ∫ ξs (φXs )ds] = ξ (φ+T,X ), 0

T 2 ξ 𝔼𝒢 [( ∫ ξs (φXs )ds) ] 0 T

T

0

0

−

T 2 ξ (𝔼𝒢 [ ∫ ξs (φXs )ds]) 0

2 ̂ |φ(k)| 1 = ∫ ds ∫ dt ∫ e−ik⋅(Xs −Xt ) (e−|s−t|ω(k) − e−(|t|+|s|)ω(k) )dk. 2 2ω(k) ℝd

Now the integration with respect to 𝒢 ξ in (2.6.5) can be carried out with the result ΨTg (x, ξ ) =

T 1 T + 1 𝔼x𝒩0 [eξ (φT,X ) e 2 ∫0 ds ∫0 dt ∫ℝd √ZT

2 ̂ |φ(k)| 2ω(k)

e−ik⋅(Xs −Xt ) (e−|s−t|ω(k) −e−(|t|+|s|)ω(k) )dk

] . (2.6.6)

2

By the Wick ordering :eξ (f ) : = e−(1/4)‖f ‖ℋM eξ (f ) from (1.2.29), and then 1

T

eξ (φT,X ) = :eξ (φT,X ) :e 2 ∫0 +

+

T

ds ∫0 dt ∫ℝd

2 ̂ |φ(k)| 2ω(k)

e−ik⋅(Xs −Xt ) e−(|t|+|s|)ω(k) dk

.

(2.6.7)

Hence, by combining (2.6.6) and (2.6.7), ΨTg (x, ξ ) =

T 1 T + 1 𝔼x𝒩0 [:eξ (φT,X ) :e 2 ∫0 ds ∫0 W(Xs −Xt ,s−t)dt ]. √ZT

(2.6.8)

By the reflection symmetry of Xt , we also have ΨTg (x, ξ ) =

0 1 0 − 1 𝔼x𝒩0 [:eξ (φT,X ) :e 2 ∫−T ds ∫−T W(Xs −Xt ,s−t)dt ]. √ZT

(2.6.9)

Now we write (2.6.9) for the left and (2.6.8) for the right entry of the scalar product (ΨTg , (1 ⊗ L)ΨTg ), use independence of Xs , s ≤ 0 and Xt , t ≥ 0, and we add and subtract 0

T

the term − ∫−T ds ∫0 W(Xs − Xt , s − t)dt in the exponent to obtain 0

T

(ΨTg , (g ⊗ L)ΨTg )L2 (P0 ) = 𝔼𝒩T [(:eξ (φT,X ) :, L:eξ (φT,X ) :)g(X0 )e− ∫−T ds ∫0 −

+

W(Xs −Xt ,s−t)dt

]. (2.6.10)

This is the finite T version of (2.6.4). It remains to take the limit T → ∞. On the lefthand side of (2.6.10), this is immediate since ΨTg → Ψg in L2 (P0 ) and L is continuous. On the right-hand side, we already know that 𝒩T 󸀠 → 𝒩 weakly with some subsequence. Reset T 󸀠 to T. Thus it only remains to show that the integrand of (2.6.10) is uni± (k)| ≤ |φ(k)|/ω(k) ̂ ̂ formly convergent in the paths. For the first factor we find that |φ T,X ± ± ̂ ̂ uniformly in T and the paths, and φ → φ strongly in ℋ as T → ∞ uniformly T,X

X

M

2.6 Ground state expectations | 169

in paths. Thus :eξ (φT,X ) : → :eξ (φX ) : in L2 (G) uniformly in paths. Since the same argu− + ment applies to φ−T,X and L is continuous, (:eξ (φT,X ) :, L:eξ (φT,X ) :) is uniformly convergent +

+

to (:eξ (φX ) :, L:eξ (φX ) :) as T → ∞. Moreover, the infrared regularity condition implies −

+

0

T

−T

0

2 󵄨 󵄨󵄨 ̂ 󵄨󵄨 ∫ ds ∫ W(Xs − Xt , s − t)dt 󵄨󵄨󵄨 ≤ ∫ |φ(k)| dk < ∞ 󵄨󵄨 󵄨󵄨 3 2ω(k) ℝd

and hence 0

T

0

∞

∫ ds ∫ W(Xs − Xt , s − t)dt → ∫ ds ∫ W(Xs − Xt , s − t)dt −T

0

−∞

0

as T → ∞ uniformly in paths. Hence the integrand in (2.6.10) is uniformly convergent as T → ∞. Most operators of physical interest are, however, not bounded. Therefore, we need to extend formula (2.6.4) to unbounded operators. Theorem 2.39 (Ground state expectations for unbounded operators). Let g be a measurable function on ℝd and L a self-adjoint operator in L2 (G) such that 𝔼𝒩 [|g(X0 )|2 ‖L:eξ (φX ) :‖2L2 (G) ] < ∞.

(2.6.11)

±

Then Ψg ∈ D(g ⊗ L), and (2.6.4) holds. Proof. Without loss of generality, we can suppose that g is real-valued and g(x), |g(x)| < M, gM (x) = { M, |g| ≥ M. Let LN = 1[−N,N] (L). The operator gM ⊗ LN is bounded, hence (2.6.4) holds for gM ⊗ LN . 2 ̂ Let I = ∫ℝd |φ(k)| /ω(k)3 dk. Using the Schwarz inequality, we have 0

‖(gM ⊗ LN )Ψg ‖2 = 𝔼𝒩 [(:eξ (φX ) :, L2N :eξ (φX ) :)|gM (X0 )|2 e− ∫−∞ ds ∫0 −

+

≤ eI 𝔼𝒩 [|gM (X0 )|2 ‖LN :e ≤ eI 𝔼𝒩 [|g(X0 )|2 ‖L:e

ξ (φ−X )

ξ (φ−X )

:‖‖LN :e

:‖‖L:e

ξ (φ+X )

ξ (φ+X )

∞

W(Xs −Xt ,s−t)dt

]

:‖]

:‖] ,

which is bounded according to (2.6.11). By the monotone convergence theorem, this shows that Ψg ∈ D(g ⊗ L), and then we obtain that (gM ⊗ LN )Ψg → (g ⊗ L)Ψg as M, N → ∞. Hence (Ψg , (gM ⊗ LN )Ψg ) → (Ψg , (g ⊗ L)Ψg )

(2.6.12)

170 | 2 The Nelson model by path measures as N, M → ∞. On the other hand, it follows from (2.6.11) that :eξ (φX ) : ∈ D(L) for 𝒩 -almost all paths. From this we conclude ±

(:eξ (φX ) :, LN :eξ (φX ) :) → (:eξ (φX ) :, L:eξ (φX ) :) −

+

−

+

for 𝒩 almost all paths as N → ∞. We also have (:eξ (φX ) :, LN :eξ (φX ) :) ≤ ‖:eξ (φX ) :‖‖LN :eξ (φX ) :‖ ≤ eI/2 ‖L:eξ (φX ) :‖ −

+

−

+

+

for all paths, and the right-hand side is 𝒩 -integrable. Thus the Lebesgue dominated convergence theorem implies 0

𝔼𝒩 [(:eξ (φX ) :, LN :eξ (φX ) :)gM (X0 )e− ∫−∞ ds ∫0 −

+

→ 𝔼𝒩 [(:e

ξ (φ−X )

:, L:e

ξ (φ+X )

:)g(X0 )e

0 − ∫−∞

∞

W(Xs −Xt ,s−t)dt

∞ ds ∫0

]

W(Xs −Xt ,s−t)dt

(2.6.13)

]

as M, N → ∞. By (2.6.12) and (2.6.13), the theorem follows. 2.6.2 Ground state expectations for second quantized operators Our next goal is to find similar expressions for the ground state expectation of second quantized operators. In Section 1.2.1 we introduced the second quantization Γ(T) and the differential second quantization dΓ(T) on a boson Fock space ℱN , and in Section 1.6.3, further on the function space L2 (G). Here we use the notation Γ(T), dΓ(T) on ̃ ̃ ℱN , and Γ(T), dΓ(T) on L2 (G). It is seen that ̃ −1 TF), Γ(T) ≅ Γ(F

̃ −1 TF), dΓ(T) ≅ dΓ(F

where F : L2 (ℝd ) → L2 (ℝd ) denotes Fourier transform. We have already seen that this unitary equivalence is derived from the Wiener–Itô–Segal isomorphism. Theorem 2.40. Suppose that L is a bounded self-adjoint operator on ℋM . Then ̃ ̃ Ψg ∈ D(Γ(L)) ∩ D(dΓ(L)) and 1

0

(φX ,LφX )ℋM − ∫−∞ ds ∫0 ̃ 2 (Ψg , Γ(L)Ψ e g )L2 (P0 ) = 𝔼𝒩 [e −

+

∞

W(Xs −Xt ,s−t)dt

1 − + ̃ (Ψg , dΓ(L)Ψ g )L2 (P0 ) = 𝔼𝒩 [(φX , LφX )ℋM ]. 2

]

Proof. Note that 1

+ 2

ξ (φX ) 2 ̃ ‖Γ(L):e :‖L2 (G) = e 2 ‖LφX ‖ℋM , ±

1 ± 2 1 ξ (φ±X ) 2 ̃ ‖dΓ(L):e :‖L2 (G) = (‖Lφ±X ‖2ℋM + (φ±X , Lφ±X )2ℋM )e 2 ‖φX ‖ℋM , 2

2 ̂ and ‖φ±X ‖ℋM ≤ ∫ℝd |φ(k)| /ω(k)3 dk < ∞. Since ‖Lf ‖ℋM ≤ ‖L‖‖f ‖ℋM for all f ∈ ℋM , (2.6.11) is satisfied. By Theorem 2.39 the statement follows.

2.6 Ground state expectations | 171

̃ = Ñ Now we turn to specific cases of particular interest. Let dΓ(1) = N and dΓ(I) be the number operators. Corollary 2.41 (Super-exponential decay of boson number). For every α ∈ ℂ we have ̃ that Ψg ∈ D(eαN ) and (Ψg , eαN Ψg ) = 𝔼𝒩 [e−(1−e ̃

α

0

) ∫−∞ ds ∫0 W(Xs −Xt ,s−t)dt ∞

].

̃ ̃ α 1), Theorem 2.40 gives the result. Proof. Since eαN = Γ(e

By Corollary 2.41, ̃ dn (Ψ , eαN Ψg )⌈α=0 . (Ψg , Ñ n Ψg ) = dαn g

In particular, 0

∞

̃ g ) = 𝔼𝒩 [ ∫ ds ∫ W(Xs − Xt , s − t)dt ] . (Ψg , NΨ ] [−∞ 0

(2.6.14)

Corollary 2.42 (Pull-through formula). We have 2 ̂ 󵄩󵄩 󵄩2 −1 −ik⋅x ̃ g ) = ∫ |φ(k)| Ψg 󵄩󵄩󵄩 dk. (Ψg , NΨ 󵄩󵄩(HN − EN + ω(k)) e 2ω(k)

(2.6.15)

ℝd

Proof. Computing (2.6.14), we obtain 0

∞

−∞

0

2 ̂ ̃ g ) = ∫ ds ∫ dt ∫ |φ(k)| (Ψg , NΨ e−|s−t|ω(k) 𝔼𝒩 [e−ik⋅(Xs −Xt ) ]dk. 2ω(k) ℝd

Since 0

∞

0

0

−∞

∞

∫ ds ∫ e−|s−t|ω(k) 𝔼𝒩 [e−ik⋅(Xs −Xt ) ]dt = ∫ ds ∫ (e−ik⋅x Ψg , e−(t−s)(HN −EN +ω(k)) e−ik⋅x Ψg )dt −∞

0

󵄩 󵄩2 = 󵄩󵄩󵄩(HN − EN + ω(k))−1 e−ik⋅x Ψg 󵄩󵄩󵄩 ,

the corollary follows. Equality (2.6.15) is called pull-through formula. It can also be obtained by the formal computation (HN − EN )a(k)Ψg = [(HN − EN ), a(k)]Ψg = (−ω(k) + [HI , a(k)])Ψg and ̃ g ) = ∫ ‖a(k)Ψg ‖2 dk = ∫ ‖(HN − EN + ω(k))−1 [HI , a(k)]Ψg ‖2 dk. (Ψg , NΨ ℝd

ℝd

172 | 2 The Nelson model by path measures −ik⋅x √ ̂ Note that [HI , a(k)] = φ(k)e / 2ω(k). The formula is useful in studying the spectral properties of the Nelson model. This formal derivation above is justified, for instance, by Corollary 2.42. ̃ can be extended to more genThe expectation for the number operator Ñ = dΓ(1) ̃ ̂ eral operators of the form dΓ(g).

Corollary 2.43. Let 0 ≤ g be a measurable function such that ∫ |g(k)| ℝd

2 ̂ |φ(k)| dk < ∞. ω(k)3

̂ where ĝ = F −1 gF = g(−i∇), and Then Ψg ∈ D(dΓ(̃ g)), 0

∞

−∞

0

2 ̂ |φ(k)| ̂ g) = ∫ g(k)dk ∫ ds ∫ e−|s−t|ω(k) 𝔼𝒩 [e−ik⋅(Xs −Xt ) ]dt. (Ψg , dΓ(̃ g)Ψ 2ω(k) ℝd

(2.6.16)

In particular, the following hold. (1) Pull-through formula: ̂ g ) = ∫ |g(k)| (Ψg , dΓ(̃ g)Ψ ℝd

2 ̂ |φ(k)| ‖(HN − EN + ω(k))−1 e−ik⋅x Ψg ‖2 dk. 2ω(k)

(2) Upper bound: ̂ g) ≤ (Ψg , dΓ(̃ g)Ψ

2 ̂ |φ(k)| 1 dk. ∫ |g(k)| 2 ω(k)3

(2.6.17)

ℝd

(3) Lower bound: if Ψg ∈ D(|x| ⊗ 1), then there exists C > 0 such that 2 ̂ |φ(k)| 1 ̂ g ). (1 − C|k|2 )dk ≤ (Ψg , dΓ(̃ g)Ψ ∫ |g(k)| 3 2 ω(k) ℝd

Proof. It can be checked that ̂ ξ (φX ) :‖2L2 (G) ≤ ‖dΓ(̃ g):e ±

2 1 ± 2 ̂ |φ(k)| 1 ̂ ±X ‖2ℋ + (φ±X , gφ ̂ ±X )2ℋ ) e 2 ‖φX ‖ℋM (1 + I) eI ∫ |g(k)| dk (‖gφ 3 M M 2 ω(k) ℝd

2 ̂ with I = ∫ℝd |φ(k)| /ω(k)3 dk, and the right-hand side above is finite. Equality (2.6.16) follows from Theorem 2.40. The pull-through formula can be proven in a similar way to the case involving N. The upper bound (2.6.17) follows directly from the representation (2.6.16). From the estimate 1 − (|k|2 |x|2 )/2 ≤ cos k ⋅ x we get

𝔼𝒩 [cos(k ⋅ (Xs − Xt ))] ≥ 1 −

|k|2 𝔼 [X 2 + Xs2 − 2Xt ⋅ Xs ] ≥ 1 − |k|2 ‖|x|Ψg ‖2 . 2 𝒩 t

2.6 Ground state expectations | 173

The last bound above follows from 𝔼𝒩 [Xs ⋅ Xt ] = ‖e−(HN −EN )(|s−t|/2) |x|Ψg ‖2 ≥ 0. Writing C = ‖|x|Ψg ‖2 , we have 2 ̂ |φ(k)| 1 ̂ g ) ≥ ∫ |g(k)| (Ψg , dΓ(̃ g)Ψ (1 − C|k|2 )dk. 2 ω(k)3 ℝd

This proves the lower bound. Corollary 2.44 (Infrared divergence). Let M̃ ⊂ ℝd be a compact set and write M = ℝd \ M.̃ If d ≤ 3, Ψg ∈ D(|x| ⊗ 1), 0 < κ < Λ < ∞, and 0, |k| < κ, { { { ̂ φ(k) = {1, κ ≤ |k| ≤ Λ, { { {0, |k| > Λ. Then < ∞, 0 ∉ M,

̂ )Ψg ) { lim (Ψg , dΓ(̃ 1M

κ→0

= ∞, 0 ∈ M.

In particular, ̃ g ) = ∞. lim (Ψg , NΨ

κ→0

(2.6.18)

Proof. By Corollary 2.43, we have 2

̂ ̂ )Ψg ) ≥ ∫ |φ(k)| (1 − C|k|2 )dk. (Ψg , dΓ(̃ 1M 2ω(k)3 M

The corollary follows from ∫ M

2 = ∞, 0 ∈ M, ̂ |φ(k)| dk { 2ω(k)3 < ∞, 0 ∉ M.

Remark 2.45. Since the informal description of dΓ(g) is ̃ dΓ(g(−i∇)) ≅ dΓ(g) = ∫ g(k)a∗ (k)a(k)dk, ℝd

the above results can be compactly written as 2 2 ̂ ̂ |φ(k)| |φ(k)| 2 2 (1 − C|k| ) ≤ ‖a(k)Ψ ‖ ≤ . g 2ω(k)3 2ω(k)3

(2.6.19)

The quantity in the middle of (2.6.19) is the density at momentum k of the expected number of bosons.

174 | 2 The Nelson model by path measures We can also estimate the boson number distribution of the ground state Ψg from both sides. Corollary 2.46 (Boson number distribution). Let Pn be the projection onto the nth Fock space component. Then (Ψg , Pn Ψg )L2 (P0 ) =

1 𝔼 [I n e−IX ], n! 𝒩 X

where 0

∞

−∞

0

1 IX = (φ+X , φ−X )ℋM = ∫ ds ∫ W(Xs − Xt , s − t)dt. 2 Proof. By the definition of Wick exponential, we have (:eξ (φX ) :, Pn :eξ (φX ) :)L2 (G) = −

+

1 (φ+ , φ− )n . 2n n! X X ℋM

The corollary follows from Theorem 2.38. Denote by pn = (Ψg , Pn Ψg ) the probability of n bosons occurring in the ground state of HN . Recall that ω(k) = √|k|2 + ν2 . Corollary 2.47. Suppose that φ ≥ 0 and ν > 0. Then there exists 0 < D such that (I/2)n I/2 Dn −I/2 e ≤ pn ≤ e , n! n!

(2.6.20)

2 ̂ where I = ∫ℝd |φ(k)| /ω(k)3 dk.

Proof. The right-hand side of (2.6.20) is obvious by Corollary 2.46. Note that W(X − Y, t) =

∞

1 ̂ ∫ (φX , e−rω φY )dr 2 |t|

and (φX , e

−r ω̂

∞

2

1 re−r /(4p) ̂2 φY ) = (φX , e−pω φY )dp. ∫ 2√π p3/2 0

2

Since e−rω̂ is positivity preserving and φ ≥ 0, we see that W(Xs − Xt , s − t) ≥ 0, and hence IX ≥ 0. The left-hand side follows from pn ≥

1 −I/2 1 e 𝔼𝒩 [IXn ] ≥ e−I/2 (𝔼𝒩 [IX ])n . n! n!

D is the expectation of the double integral above.

2.6 Ground state expectations | 175

Let g be the multiplication operator by g. Although we mentioned the unitary ̂ we are interested in investigating dΓ(g), ̂ i. e., dΓ(g(−i∇)) equivalence dΓ(g) ≅ dΓ(̃ g), ̃ in ℋN and dΓ(g) in L2 (P0 ). ̂ 3/2 ∈ L1 (ℝd ). Then Corollary 2.48. Let g ∈ L∞ (ℝd ) ∩ L1 (ℝd ) be real-valued, and φ/ω 0

∞

1 ∫ ds ∫ dt 2(2π)d/2

̃ (Ψg , dΓ(g)Ψ g) =

−∞

where J(k, k 󸀠 ) =

0

∫ J(k, k 󸀠 )𝔼𝒩 [ei(k⋅Xs −k ⋅Xt ) ]dkdk 󸀠 , 󸀠

ℝd ×ℝd

̂ φ(k ̂ 󸀠 ) −ω(k)|s|−ω(k󸀠 )|t| φ(k) ̌ − k 󸀠 ). e g(k √ω(k)ω(k 󸀠 )

In particular, we have the upper bound ̃ (Ψg , dΓ(g)Ψ g) ≤

1 ̂ 3/2 ‖2L1 ‖g‖L1 . ‖φ/ω 2(2π)d

(2.6.21)

Proof. In order to apply Theorem 2.40, we calculate (φ−X , gφ+X )ℋM . The result is 0

(φ−X , gφ+X )ℋM

∞

1 = ∫ ds ∫ dt (2π)d/2 0

−∞

∫ J(k, k 󸀠 )dkdk 󸀠 . ℝd ×ℝd

Remark 2.49. The informal description of dΓ(g(−i∇)) is ̃ dΓ(g) ≅ dΓ(g(−i∇)) = ∫ g(x)a∗ (x)a(x)dx. ℝd

Let M ⊂ ℝd be a measurable set. The expression (Ψg , dΓ(1M (−i∇))Ψg ) counts the expected number of bosons with position inside M. It is common to write dΓ(1M (−i∇)) = ∫ a∗ (x)a(x)dx. M

A special case of Corollary 2.48 is as follows. Corollary 2.50. Let M ⊂ ℝd be a compact set and write Vol M = ∫M dx. Suppose that ̂ 3/2 ∈ L1 (ℝd ). Then φ/ω (Ψg , dΓ(1M (−i∇))Ψg ) ≤

1 ̂ 3/2 ‖2L1 Vol M. ‖φ/ω 2(2π)d

In particular, lim

Vol M→∞

(Ψg , dΓ(1M (−i∇))Ψg ) Vol M

≤

1 ̂ 3/2 ‖2L1 < ∞. ‖φ/ω 2(2π)d

In what follows, for notational simplicity we use the notations dΓ(g) and Γ(g) in ̂ respectively. the function spaces L2 (M−2 , dG) for dΓ(̃ g)̂ and Γ(̃ g),

176 | 2 The Nelson model by path measures 2.6.3 Ground state expectation for fractional powers of the number operator For more refined information, we also consider the functional (Ψg , Ñ α Ψg ) for an arbitrary number α ∈ ℝ and derive an expression in terms of the measure 𝒩 . Let (Nt )t≥0 be a Poisson process with intensity 1 on a probability space (Ω, F , P), and recall that 𝔼P [e−uNt ] = et(e

−u

−1)

,

u ≥ 0.

(2.6.22)

Let f ∈ C ∞ ((0, ∞)) with f ≥ 0, and recall from Section 3.6.2 in Volume 1, that f is dn f called a Bernstein function whenever (−1)n dx n (x) ≤ 0 for all n ∈ ℕ. We denote the set of Bernstein functions by ℬ, and by ℬ0 the set of Bernstein functions with vanishing right limits at zero. For simplicity, in this section we denote 0

∞

W = ∫ ds ∫ dt ∫ −∞

0

ℝd

2 ̂ |φ(k)| e−|s−t|ω(k) e−ik⋅(Xs −Xt ) dk 2ω(k)

and

I= ∫ ℝd

2 ̂ |φ(k)| dk. ω(k)3

Also, throughout this section we assume that W ≥ 0,

(2.6.23)

almost surely. Hence together with (2.6.23) we have in fact 0 ≤ W ≤ I/2 a. s. Example 2.51. A case of φ ∈ S 󸀠 (ℝd ) such that W ≥ 0 is the following. Suppose that φ ∈ L2 (ℝd ) and φ(x) ≥ 0 for all x ∈ ℝd . By Fourier transform, ∫ ℝd

2 ̂ |φ(k)| e−|t|ω(k) e−ik(x−y) dk = (φ(⋅ − x), ω(−i∇)−1 e−|t|ω(−i∇) φ(⋅ − y)) ω(k) ∞

= ∫ (φ(⋅ − x), e−sω(−i∇) φ(⋅ − y)) ds |t|

for every x, y ∈ ℝd and t ∈ ℝ. Since e−sω(−i∇) is positivity improving and φ ≥ 0 by assumption, we have (2.6.23). Corollary 2.52. Let m ∈ ℤ and G ∈ ℬ0 . Then m

̃ g ) = ∑ S(m, r)𝔼𝒩 𝔼P [W r G(NW + r)] , m ≥ 1 󳨐⇒ (Ψg , Ñ m G(N)Ψ r=1

̃ g ) = 𝔼𝒩 𝔼P [G(NW )] , m = 0 󳨐⇒ (Ψg , G(N)Ψ m ≤ −1 󳨐⇒ (Ψg , (Ñ + 1)m G(Ñ + 1)Ψg ) = 𝔼𝒩 𝔼P [(NW + 1)m G(NW + 1)] , where S(m, r) =

r (−1)r r ∑(−1)s ( ) sm s r! s=1

2.6 Ground state expectations | 177

are the Stirling numbers of the second kind, and NW is Nt evaluated at the random time t = W. Proof. We first prove the case m ≥ 0. Let ρ(β) = 𝔼𝒩 [e−(1−e

−β

)W

] = (Ψg , e−βN Ψg ) ̃

for β ∈ ℂ. It suffices to consider only m ≥ 1. There exists a Lévy measure λ such that ∞

G(u) = ∫ (1 − e−uy )λ(dy),

(2.6.24)

0

and

̃ dm Ñ m = (−1)m m e−βN ⌈ . β=0 dβ

(2.6.25)

Let ρ(m) = dm ρ/dβm . By Corollary 2.41 and a combination of (2.6.24)–(2.6.25) we have ∞

̃ g ) = (−1)m ∫ (ρ(m) (0) − ρ(m) (β))λ(dβ). (Ψg , Ñ m G(N)Ψ 0

We note that in general that the mth derivative of g(x) = f (ex ) is m

g (m) (x) = ∑ erx S(m, r)f (r) (ex ). r=1

Since ρ(β) = f (e−β ) with f (x) = 𝔼𝒩 [e−(1−x)W ], we have m

𝜕m ρ(β) = (−1)m ∑ S(m, r)e−rβ 𝔼𝒩 [W r e−(1−e

−β

)W

r=1

].

This yields ∞ m

̃ g ) = 𝔼𝒩 [ ∫ ∑ S(m, r)W r (1 − e−rβ e−(1−e (Ψg , Ñ m G(N)Ψ 0 r=1

−β

)W

)λ(dβ)].

Using (2.6.22) gives 1 − e−rβ e−(1−e

−β

)W

= 𝔼P [1 − e−β(NW +r) ] .

Hence m

∞

r=1

0

̃ g ) = 𝔼𝒩 [ ∑ S(m, r)W r 𝔼P [ ∫ (1 − e−β(NW +r) )λ(dβ)]], (Ψg , Ñ m G(N)Ψ

178 | 2 The Nelson model by path measures and the corollary follows for m > 0. Next we consider the case m < 0 by using a similar strategy. By a combination of the Laplace transform, m

m

̃ e− ∑j=1 βj (N+1) ∏ dβj = (Ñ + 1)−m

∫

j=1

[0,∞)m

and ∞

̃ G(Ñ + 1) = ∫ (1 − e−β(N+1) )λ(dβ), 0

we obtain (Ψg , (Ñ + 1)−m G(Ñ + 1)Ψg ) m

=

=

∫

∞

[0,∞)m j=1

∫ [0,∞)m

m

m

∏ dβj ∫ (Ψg , (e− ∑j=1 βj (N+1) − e−(∑j=1 βj +β)(N+1) )Ψg )λ(dβ) ̃

̃

0

m

∞

j=1

0

m

∏ dβj ∫ e− ∑j=1 βj 𝔼𝒩 [e−(1−e

− ∑m j=1 βj

)W

− e−(1−e

− ∑m j=1 βj +β

e ] λ(dβ).

)W −β

The second equality follows by Corollary 2.41. In terms of the Poisson process (Nt )t≥0 , we can further rewrite as m

=

∫

[0,∞)m j=1 m

=

∫

∞

m

m

∏ dβj ∫ 𝔼𝒩 𝔼P [e− ∑j=1 βj (NW +1) − e−(∑j=1 βj +β)(NW +1) ] λ(dβ) 0 ∞

m

∏ dβj ∫ 𝔼𝒩 𝔼P [e− ∑j=1 βj (NW +1) (1 − e−β(NW +1) )] λ(dβ).

[0,∞)m j=1

0

Integrating with respect to λ(dβ) and then with ∏m j=1 dβj , we finally obtain m

=

∫

m

∏ 𝔼𝒩 𝔼P [e− ∑j=1 βj (NW +1) G(NW + 1)] dβj = 𝔼𝒩 𝔼P [(NW + 1)−m G(NW + 1)] .

[0,∞)m j=1

In order to avoid singularities, in Corollary 2.52 we replaced Ñ with Ñ + 1 for the case m < 0, however, any positive multiple of 1 can be used. By the same corollary we also have the result below. Corollary 2.53 (Fractional powers of boson number). Let m ∈ ℤ and 0 ≤ α < 2. Then m

α

m ≥ 1 󳨐⇒ (Ψg , Ñ m+ 2 Ψg ) = ∑ S(m, r)𝔼𝒩 𝔼P [W r (NW + r)α/2 ] , r=1

α/2 m = 0 󳨐⇒ (Ψg , Ñ α/2 Ψg ) = 𝔼𝒩 𝔼P [NW ], α

m ≤ −1 󳨐⇒ (Ψg , (Ñ + 1)m+ 2 Ψg ) = 𝔼𝒩 𝔼P [(NW + 1)m+α/2 ] .

2.6 Ground state expectations | 179

The above corollary allows to derive the asymptotic behavior of the ground state expectation of noninteger powers of the boson number operator in the strong coupling limit. We introduce a coupling constant g in the Nelson Hamiltonian. It gives in ℋN , HN = Hp ⊗ 1 + 1 ⊗ Hf + gHI or in L2 (𝒫0 ), LN = Lp ⊗ 1 + 1 ⊗ Hf̃ + g HĨ . Corollary 2.54. If G ∈ ℬ0 is a Bernstein function with G(0) = 0, then ̃ g ) ≤ G(g 2 I/2). (Ψg , G(N)Ψ Proof. Since any Bernstein function G is increasing and concave, by Jensen’s inequality and assumption (2.6.23) we obtain more generally that ̃ g ) ≤ G((Ψg , NΨ ̃ g )) ≤ G(I/2), (Ψg , G(N)Ψ ̃ g ) = 𝔼𝒩 [W]. since (Ψg , NΨ ̃ g )/G(g 2 ) in terms By Corollary 2.54 we have an upper bound of limg→∞ (Ψg , G(N)Ψ 2 2 of limg→∞ G(g I/2)/G(g ). Corollary 2.55. Let 0 < α < 2. Then lim

g→∞

(Ψg , Ñ α/2 Ψg ) gα

≤ (I/2)α/2

and lim

g→∞

(Ψg , log(1 + Ñ α/2 )Ψg ) α log g

≤ 1.

Proof. Set G(u) = uα/2 to obtain the first statement follows, and G(u) = log(1 + uα/2 ) to get the second. For the special case (Ψg , Ñ k Ψg ), k ≥ 1, we have both upper and lower bounds. Corollary 2.56. Let k = m+ α2 ≥ 1, m ∈ ℕ and 0 ≤ α < 2. Then there exists A independent of g such that (I/2 − A)k ≤ lim

g→∞

(Ψg , Ñ k Ψg ) g 2k

≤ (I/2)k .

Proof. By Jensen’s inequality we have 𝔼P [(NW + r)α/2 ] ≤ (𝔼P [NW + r])α/2 = (I/2 + r)α/2 .

180 | 2 The Nelson model by path measures In particular, it follows that m

α

(Ψg , Ñ m+ 2 Ψg ) ≤ ∑ S(m, r)(I/2 + r)α/2 (I/2)r , r=1

m+ α2

implying limg→∞ (Ψg , Ñ Ψg )/g 2m+α ≤ S(m, m)(I/2)k = (I/2)k . This gives the upper bound. Again, from Jensen’s inequality it follows that ̃ g )k . (Ψg , Ñ k Ψg ) ≥ (Ψg , NΨ Hence the lower bound can be derived from (3) of Corollary 2.43. 2.6.4 Ground state expectations of field operators For β ∈ ℝ and g ∈ ℋM next we consider (Ψg , eβξ (g) Ψg )L2 (P0 ) , i. e., the moment generating function of the random variable ξ 󳨃→ ξ0 (g). Corollary 2.57 (Exponential moments). Let β ∈ ℂ. Then (Ψg , eβξ (g) Ψg )L2 (P0 ) is finite and β2

(Ψg , eβξ (g) Ψg )L2 (P0 ) = 𝔼𝒩 [e 2 Ig −βIX ], where 2 ̂ |g(k)| dk Ig = ∫ 2ω(k)

∞

̂ ̂ g(k) φ(k) IX = ∫ dk ∫ e−ω(k)|s| e−ik⋅Xs ds. 2ω(k)

and

ℝd

ℝd

−∞

Proof. Note that β2

0

(:eξ (φX ) :, eβξ0 (g) :eξ (φX ) :) = e 2 Ig −βIX e∫−∞ ds ∫0 −

+

∞

W(Xs −Xt ,s−t)dt

.

The statement is then a direct consequence of Theorem 2.39. From this corollary, we immediately get (Ψg , ξ (g)n Ψg )L2 (P0 ) =

dn (Ψ , eβξ (g) Ψg )L2 (P0 ) ⌈β=0 dβn g

for all n ∈ ℕ.

Let Hn be the Hermite polynomials of order n defined by Hn (x) = (−1)n ex

2

dn −x2 e . dxn

(2.6.26)

The first few examples are H0 (x) = 1,

H1 (x) = 2x,

H2 (x) = 4x2 − 2,

H3 (x) = 8x3 − 12x.

2.6 Ground state expectations | 181

We use the well-known generating function to write e Hence we obtain

β2 I −βIX 2 g

n

∞

= ∑ Hn (iIX /√Ig ) n=0

(iβ√Ig ) n!

.

dn β22 Ig −βIX e ⌈β=0 = in Hn (iIX /√Ig ) Ign/2 . dβn

In the next corollary we look at the mean and variance of the random variable ξ 󳨃→ ξ0 (g) for g ∈ ℋM by making use of Corollary 2.57. Corollary 2.58 (Average field strength and field fluctuations). The following hold. (1) Average field strength: for n = 1 ∞

̂ ̂ g(k) φ(k) (Ψg , ξ (g)Ψg ) = − ∫ dk ∫ e−ω(k)|s| 𝔼𝒩 [e−ik⋅Xs ]ds. 2ω(k) ℝd

−∞

(2) Field fluctuations: for n = 2 2

2 ̂ ̂ g(k) ̂ φ(k) |g(k)| (Ψg , ξ (g) Ψg ) = ∫ dk + 𝔼𝒩 [( ∫ dk ∫ e−ω(k)|s| e−ik⋅Xs ds) ]. 2ω(k) 2ω(k) ∞

2

ℝd

ℝd

−∞

Remark 2.59. (1) By using the previous result and the Schwarz inequality, we find that (Ψg , ξ (g)2 Ψg ) − (Ψg , ξ (g)Ψg )2 ≥ ∫ ℝd

2 ̂ |g(k)| dk = (1, ξ (g)2 1) − (1, ξ (g)1)2 . 2ω(k)

The latter term represents the fluctuations of the free field. It is then seen that fluctuations increase by coupling the field to the particle. (2) Note that 𝔼𝒩 [e−ik⋅Xs ] = ∫ Ψp (x)2 λ(x)2 e−ik⋅x dx, ℝd

where λ(x)2 = ∫M Ψ2g (ξ , x)dG is the density of 𝒩 with respect to N0 , and Ψ2p is −2 the density of N0 with respect to Lebesgue measure and the square of the ground state of Hp . (3) Writing χ = Ψ2p λ2 for the position density of the particle, and taking g to be a delta function in momentum space and in position space, respectively, we find (Ψg , ξ (k)Ψg )L2 (P0 ) = −

̂ χ(k) ̂ φ(k) d/2 (2π) ω2 (k)

k ∈ ℝd ,

182 | 2 The Nelson model by path measures and (Ψg , ξ (x)Ψg )L2 (P0 ) = (χ ∗ Vω ∗ φ)(x) x ∈ ℝd ,

(2.6.27)

respectively. Here Vω denotes the Fourier transform of −1/ω2 and is the Coulomb potential for massless bosons, i. e. for ω(k) = |k|. Equality (2.6.27) gives the classical field generated by a particle with position distribution χ(x)dx. 2.6.5 Gaussian domination A further important application of Corollary 2.57 is a Gaussian domination property of the ground state Ψg . It is well-known that the nth eigenfunction hn of the harmonic 2

oscillator − 21 Δ+ 21 |x|2 is given by Hn (x)e−|x| /2 , where Hn denotes the Hermite polynomial 2

of order n. Hence hn (x) ∼ e−|x| /2 as |x| → ∞, and 2

lim ‖eβ|x| hn ‖ = ∞.

β↑1/2

The ground state of the Nelson model has a similar property, which we can show by using the probability measure 𝒩 . ̂ ̂ ∈ L2 (ℝd ). Then Lemma 2.60. Let β > 0 and g ∈ S 󸀠 (ℝd ) such that g/ω, g/√ω 1

2

(Ψg , e−βξ (g) Ψg ) =

βK(g)2

2 √1 + β‖g/√ω‖ ̂

𝔼𝒩 [e

− 1+β‖g/√ω‖ ̂

],

where K(g) denotes the random variable defined by ∞

1 ̂ √ω, g/̂ √ω)dr. K(g) = ∫ (e−|r|ω e−ik⋅Br φ/ 2 −∞

Proof. For convenience, we use the same notations Ig and IX as in Corollary 2.57. First, ̂ ̂ ̂ ̂ since φ/ω ∈ L2 (ℝd ) and g/ω ∈ L2 (ℝd ), it follows that |K(g)| ≤ ‖φ/ω‖‖ g/ω‖ < ∞. By Corollary 2.57 we see that (Ψg , eikβξ (g) Ψg ) = 𝔼𝒩 [e−k Thus 2

2

(Ψg , e−β ξ (g) /2 Ψg ) =

2 2

β Ig /2 −ikβIX

e

].

2 1 ∫ e−k /2 (Ψg , eikβξ (g) Ψg )dk √2π

ℝ

2

2

β K(g) /2 2 2 2 − 1 1 2 𝔼𝒩 [ ∫ e−k /2 e−k β Ig /2 e−ikβIX dk] = 𝔼𝒩 [e 1+β Ig ]. = √2π √1 + β2 Ig ℝ

Replacing β2 /2 by β > 0, the claim follows.

2.6 Ground state expectations | 183

Corollary 2.61 (Gaussian domination of the ground state). Let g ∈ S 󸀠 (ℝd ) such that 2 2 ̂ ̂ ̂ ∈ L2 (ℝd ), and β < 1/‖g/√ω‖ . Then Ψg ∈ D(e(β/2)ξ (g) ) and g/ω, g/√ω βK(g)2

1

2

‖e(β/2)ξ (g) Ψg ‖2 =

2 √1 − β‖g/√ω‖ ̂

2 ̂ 𝔼𝒩 [e 1−β‖g/√ω‖ ].

In particular, 2

lim

2 )−1 ̂ β↑(2‖g/√ω‖

‖eβξ (g) Ψg ‖ = ∞.

Proof. Write ℂ+ = {z ∈ ℂ | ℜz > 0} and ℂ− = {z ∈ ℂ | ℜz < 0}, and take the open disc B = {z ∈ ℂ | |z| < 1/(2Ig )}. Define ρ(z) =

2

1 √1 + 2zIg

𝔼𝒩 [e

− zK(g) 1+2zI

g

],

z > 0.

̂ ̂ Since |K(g)| ≤ ‖g/√ω‖‖ φ/√ω‖ uniformly in the paths, the function ρ can be analytically continued to a function ρ̃ on ℂ+ ∪B. Let Bδ (w) ⊂ ℂ be the ball of radius δ centered at w. Suppose that δ < 2Ig and choose w ∈ ℝ ∩ B such that (Bδ (w) ∪ ℂ− ) ∩ B ≠ 0. ̃ can be expanded as The analytic function ρ(z) ∞

̃ = ∑ (z − w)n bn (w), ρ(z)

z ∈ Bδ (w) ∩ B.

n=0

(2.6.28) 2

On the other hand, since Ψg ∈ D(ξ 2 (g)), the function ℂ ∋ z 󳨃→ (Ψg , e−zξ (g) Ψg ) ∈ ℂ is differentiable on ℂ+ . Hence (Ψg , e

−zξ (g)2

∞

1 Ψg ) = ∑ (z − w) ∫ (−λ)n e−wλ dEλ , n! n=0 ∞

n

(2.6.29)

0

where Eλ denotes the spectral measure of the positive self-adjoint operator ξ (g)2 associated with Ψg . Comparing (2.6.28) and (2.6.29), we obtain ∞

1 bn (w) = ∫ (−λ)n e−wλ dEλ , n!

n = 0, 1, 2, . . . .

0

Inserting (2.6.30) into (2.6.28), we have ∞

1 ̃ = ∑ (z − w) ρ(z) ∫ (−λ)n e−wλ dEλ , n! n=0 ∞

n

0

z ∈ Bδ (w) ∩ B.

(2.6.30)

184 | 2 The Nelson model by path measures Moreover, the right-hand side is absolutely convergent for every z ∈ Bδ (w) ∩ B. For z ∈ Bδ (w) ∩ B ∩ ℝ we have M

∫e 0

−zλ

M

M

󵄨󵄨 ∞ 󵄨󵄨 1 󵄨󵄨󵄨 n 1 󵄨󵄨󵄨 n −wλ n −wλ dEλ ≤ ∑ |z − w| 󵄨󵄨 ∫(−λ) e dEλ 󵄨󵄨󵄨 = ∑ |z − w| 󵄨󵄨 ∫ λ e dEλ 󵄨󵄨󵄨 󵄨 󵄨 󵄨 󵄨 n! n! n=0 n=0 ∞

∞

n

≤ ∑ |z − w|n n=0

0 ∞

0

∞

󵄨󵄨 󵄨󵄨 1 󵄨󵄨󵄨 n 1 󵄨󵄨󵄨 n −wλ n −wλ 󵄨󵄨 ∫ λ e dEλ 󵄨󵄨󵄨 = ∑ |z − w| 󵄨󵄨 ∫ (−λ) e dEλ 󵄨󵄨󵄨 < ∞. 󵄨 n=0 󵄨 n! 󵄨 n! 󵄨 ∞

0

0

M

This implies that limM→∞ ∫0 e−zλ dEλ < ∞ for z ∈ Bδ (w)∩B∩ℝ. Thus we conclude that 2

2

̃ follows for z ∈ Bδ ∩ B ∩ ℝ. Here we notice Ψg ∈ D(e−(z/2)ξ (g) ) and ‖e−(z/2)ξ (g) Ψg ‖2 = ρ(z) that for every δ < 1/(2Ig ) we can choose w ∈ ℝ ∩ B such that ℂ− ∩ B ∩ Bδ (w) ≠ 0.

2.7 Infrared divergence As seen above, in order to give a rigorous mathematical definition of the Nelson model, an infrared and an ultraviolet cut-off was necessary to make sure that divergences do not occur. To control the ultraviolet part, we introduced a charge distribution φ making the coupling to the field sufficiently regular, and to control the infrared part we 2 ̂ |φ(k)|

required the integral ∫ℝd ω(k)3 dk to be finite. Now we discuss the model in the conditions of infrared singularity, while keeping the ultraviolet cutoff in place. In Section 2.7 we use Assumption 2.62 below. Assumption 2.62. The following conditions hold. (1) Dispersion relation: ω(k) = |k|. ̂ ̂ ̂ ̂ = φ(−k) and φ/ω, φ/√ω (2) Charge distribution: φ ∈ S 󸀠 (ℝd ), φ(k) ∈ L2 (ℝd ). (3) External potential: V satisfies the bound V(x) ≥ C|x|2β with constant C > 0 and exponent β > 0. In particular, we consider here the massless Nelson model. Under the conditions above, Hp has a unique, strictly positive ground state Ψp at eigenvalue Ep , and β+1

Ψp (x) ≤ e−C|x|

holds with a constant C. Under the infrared regularity condition ∫ ℝd

2 ̂ |φ(k)| dk < ∞ 3 ω(k)

2.7 Infrared divergence

| 185

we have seen in the previous sections that HN has a unique, strictly positive ground state, and we have obtained detailed information on its properties. What we discuss here is whether there is any ground state in the same space when the infrared regularity condition is not guaranteed. Theorem 2.63 (Absence of ground state). Let d = 3, φ ∈ L1 (ℝ3 ), φ ≥ 0 such that φ ≢ 0. Then HN has no ground state. Remark 2.64. Under the assumptions in Theorem 2.63 we are in the situation of in2 ̂ |φ(k)| ̂ frared singularity. Indeed, ∫ℝ3 ω(k)3 dk = ∞, since φ(0) > 0 and φ̂ is continuous. that

Recall that γ(T) = (Ψp ⊗ 1, ΨTg )2 . In order to prove Theorem 2.63, it suffices to show lim γ(T) = 0

T→∞

b

b

by Proposition 2.31. We use the notation S[a, b] = ∫a ds ∫a W(Xt − Xs , t − s)dt for the pair potential W of the Nelson model. Since γ(T) =

(Ψp ⊗ 1, e−TLN Ψp ⊗ 1)2

(Ψp ⊗ 1, e−2TLN Ψp ⊗ 1)

,

we can express γ(T) in terms of the P(ϕ)1 -process (Xt )t∈ℝ associated with the selfadjoint operator Lp = UΨp (Hp − Ep )UΨ−1p as γ(T) =

(𝔼𝒩0 [eS[0,T] ])2 𝔼𝒩0 [eS[−T,T] ]

(2.7.1)

.

Here we used the shift invariance of the process, i. e., 𝔼𝒩0 [eS[−T,T] ] = 𝔼𝒩0 [eS[0,2T] ]. Let 𝒩T be the probability measure given in (2.6.1). Lemma 2.65. It follows that 0

T

γ(T) ≤ 𝔼𝒩T [e− ∫−T ds ∫0

W(Xs −Xt ,s−t)dt

].

(2.7.2)

Proof. The numerator of (2.7.1) can be estimated by the Schwarz inequality with respect to dN0 and the reflection symmetry of the process as (𝔼𝒩0 [eS[0,T] ])2 ≤ ∫ (𝔼x𝒩0 [eS[0,T] ])2 dN0 = ∫ (𝔼x𝒩0 [eS[0,T] ])(𝔼x𝒩0 [eS[−T,0] ])dN0 . ℝ3

ℝ3

186 | 2 The Nelson model by path measures Since X−s , s ≥ 0, and Xt , t ≥ 0, are independent, we also see that (𝔼𝒩0 [eS[0,T] ])2 ≤ ∫ 𝔼x𝒩0 [eS[0,T]+S[−T,0] ]dN0 = 𝔼𝒩0 [eS[0,T]+S[−T,0] ]. ℝ3

T

0

Since, moreover, S[0, T] + S[−T, 0] = S[−T, T] − ∫−T ds ∫0 W(Xs − Xt , s − t)dt, we have 0

T

(𝔼𝒩0 [eS[0,T] ])2 ≤ 𝔼𝒩0 [eS[−T,T]−∫−T ds ∫0

W(Xs −Xt ,s−t)dt

].

Thus 0

γ(T) ≤

T

𝔼𝒩0 [eS[−T,T]−∫−T ds ∫0 𝔼𝒩0

W(Xs −Xt ,s−t)dt

[eS[−T,T] ]

]

0

T

= 𝔼𝒩T [e− ∫−T ds ∫0

W(Xs −Xt ,s−t)dt

],

and the lemma follows. The pair potential W(x, t) can be computed explicitly. Using the integral kernel e−|t|

√−Δ

(x, y) =

|t| 1 , π 2 (|x − y|2 + |t|2 )2

we obtain W(x − y, t − s) =

∞

1 ̂ φ(k)dk ̂ ∫ d|T| ∫ e−ik⋅(y−x) e−|T|ω(k) φ(k) 2 ℝ3

|s−t|

φ(u)φ(v)dv 1 = > 0. ∫ du ∫ 2 4π |x − y + u − v|2 + |s − t|2 ℝ3

(2.7.3)

ℝ3

To show that γ(T) → 0, we first restrict to the set AT = {ω ∈ X | |Xt (ω)| ≤ T λ , |t| ≤ T} with some λ < 1. We split up the right-hand side of (2.7.2) by restricting to AT and X\AT , and show that the corresponding expectations converge to zero separately. Lemma 2.66. It follows that 0

T

lim 𝔼𝒩T [1AT e− ∫−T ds ∫0

T→∞

W(Xs −Xt ,s−t)dt

] = 0.

Proof. By using the estimate |Xs − Xt + x − y|2 + |s − t|2 ≤ 8T 2λ + 2|x − y|2 + |s − t|2 on AT , and 0

T

∫ ds ∫ −T

0

a2 + T 2 /2 dt ≥ log ( ), a2 + |s − t|2 a2

(2.7.4)

2.7 Infrared divergence

| 187

we obtain by (2.7.3) that 0

T

∫ ds ∫ W(Xs − Xt , s − t)dt −T

0 T

0

= ∫ ds ∫ dt ∫ dx ∫ 0

−T

ℝ3

T

0

ℝ3

≥ ∫ ds ∫ dt ∫ dx ∫ 0

−T

ℝ3

ℝ3

φ(x)φ(y) dy |Xs − Xt + x − y|2 + |s − t|2 φ(x)φ(y) dy 8T 2λ + 2|x − y|2 + |s − t|2

= ∫ dx ∫ φ(x)φ(y) log ( ℝ3

ℝ3

8T 2λ + 2|x − y|2 + T 2 /2 ) dy. 8T 2λ + 2|x − y|2

The right-hand side in the formula above diverges as T → ∞, since λ < 1. This gives (2.7.4). Lemma 2.67. It follows that 0

T

lim 𝔼𝒩T [1X\AT e− ∫−T ds ∫0

T→∞

W(Xs −Xt ,s−t)dt

] = 0.

Proof. Note that 0

T

∫ ds ∫ W(Xs − Xt , s − t)dt ≤ −T

0

We have 2

γ(T) ≤ e(T/4)‖φ/ω‖ ̂

2

≤ e(T/4)‖φ/ω‖ ̂

T 2 ̂ ‖φ/ω‖ . 2

𝔼𝒩0 [1X\AT eS[−T,T] ] 𝔼𝒩0 [eS[−T,T] ]

(𝔼𝒩0 [eS[−T,T] ])1/2 𝔼𝒩0 [eS[−T,T] ]

(𝔼𝒩0 [1X\AT ])1/2 .

(2.7.5)

Moreover, there exists a constant δ > 0 such that 2 2 ̂ ̂ −Tδ‖φ/ω‖ ≤ |S[−T, T]| ≤ Tδ‖φ/ω‖ .

Thus (𝔼𝒩0 [e2S[−T,T] ])1/2 𝔼𝒩0 [eS[−T,T] ]

2

≤ e2Tδ‖φ/ω‖ . ̂

To complete the proof, we argue that this exponential growth is balanced by the second factor of (2.7.5). The estimate 𝔼𝒩0 [1X\AT ] ≤ T −λ √a + Tb exp(−cT λ(β+1) )

(2.7.6)

188 | 2 The Nelson model by path measures is obtained in Lemma 2.68. Here a, b, c > 0. By choosing 1/(β+1) < λ < 1, the right-hand side of (2.7.6) goes to zero as T → ∞. Now we are ready to prove the absence of a ground state. Proof of Theorem 2.63. We have 0

T

γ(T) ≤ 𝔼𝒩T [e− ∫−T ds ∫0 0

W(Xs −Xt ,s−t)dt

] = 𝔼𝒩T [1AT F] + 𝔼𝒩T [1X\AT F],

T

where F = e− ∫−T ds ∫0 W(Xs −Xt ,s−t)dt . Since limT→∞ 𝔼μT [1AT ] = 0 by Lemma 2.65 and limT→∞ 𝔼μT [1X\AT ] = 0 by Lemma 2.66, the theorem follows. To prove (2.7.6), we make use of the path properties of the P(ϕ)1 -process (Xt )t≥0 . Note that 𝔼𝒩0 [1X\AT ] = 𝒩0 ( sup |Xt | ≥ T λ ) . t∈[−T,T]

Lemma 2.68. Estimate (2.7.6) holds. Proof. Suppose that f ∈ C ∞ (ℝd ) is an even function, and = |x|, { { { f (x) {≤ |x|, { { {= 0,

|x| ≥ T λ ,

T λ − 1 < |x| < T λ , |x| ≤ T λ − 1.

We have 𝔼𝒩0 [1X\AT ] = 𝔼𝒩0 [1{sup|s|T λ } ] = 𝔼𝒩0 [1{sup|s|T λ } ]. By reflection symmetry of (Xt )t∈ℝ , 𝔼𝒩0 [1{sup|s|T λ } ] = 2𝔼𝒩0 [1{sup0≤s≤T |f (Xs )|>T λ } ] and by the Dirichlet principle, 𝔼𝒩0 [ sup |f (Xs )| > T λ ] ≤ |s| 0. We consider d-dimensional two-sided Brownian motion (Bt )t∈ℝ on the probability space (X, ℬ(X), 𝒲 x ). Fix a nonnegative function f ∈ L2 (ℝd ) and let 1

LT = f (B−T )f (BT )e 2

T

T

T

∫−T ds ∫−T W(Bs −Bt ,s−t)dt − ∫−T V(Bs )ds

e

.

(2.8.1)

190 | 2 The Nelson model by path measures Define the probability measure μT on (X, ℬ(X)) by ℬ(X) ∋ A 󳨃→ μT (A) =

1 ∫ 𝔼x [1A LT ]dx, ZT

(2.8.2)

ℝd

where ZT is the normalizing constant such that 1 ∫ 𝔼x [LT ]dx = 1. ZT ℝd

We call the measure μT the finite volume Gibbs measure associated with the Nelson model. Using the functional integral representation we see that (f ⊗ 1, e−2THN f ⊗ 1) = ∫ 𝔼x [LT ]dx. ℝd

Our goal in this section is to show that μT converges in local sense as T → ∞ and (an infinite volume) Gibbs measure exists. This method will also be useful in constructing a limit measure on a path space with jumps to discuss the relativistic Pauli–Fierz model in Section 3.9 and the spin-boson model in Section 4.4 below. BM Let F[−T,T] = σ(Br , −t ≤ r ≤ t) be the natural filtration of (Bt )t∈ℝ . The sets BM

GT = ⋃ F[−s,s] , 0≤s≤T

BM

G = ⋃ F[−s,s] s≥0

are finitely additive families of sets. We consider the collection of probability spaces (X, σ(G ), μT ), T > 0, where the probability measure μT is given by (2.8.2). Definition 2.70 (Local convergence). Let μ∞ be a probability measure on (X, σ(G )). The family of probability measures {μT }T≥0 is said to converge to μ∞ in local sense if μT (A) → μ∞ (A) as T → ∞, for every A ∈ G . We begin with an outline of showing local convergence. Let ΨTg =

e−T(HN −EN ) f ⊗ 1 ‖e−T(HN −EN ) f ⊗ 1‖

(2.8.3)

and fT = e−T(HN −EN ) f ⊗ 1. Note that in (2.5.6) we used ΨTg given by (2.8.3) with f replaced by Ψp , in this section, however, ΨTg is defined by (2.8.3). First, by using ΨTg we define a family of finitely additive set functions ρT on (X, GT ), T > 0, and obtain its extension ρ̄ T to a probability measure on a measurable space (X, σ(GT )). Thus we arrive at the probability space (X, σ(GT ), ρ̄ T ).

2.8 Gibbs measure associated with the ground state

| 191

Using functional integration we will see that ρ̄ T (A) = ρT (A) = μT (A),

A ∈ Gt , t ≤ T.

(2.8.4)

Next, by using the ground state Ψg we define a finitely additive set function μ on (X, G ), and with its extension to a probability measure μ∞ on (X, σ(G )), we get the probability space (X, σ(G ), μ∞ ). Making use of the strong convergence of ΨTg to Ψg as T → ∞, we prove that ρT (A) → μ(A),

T → ∞,

for A ∈ G , which together with (2.8.4) will imply that μT (A) → μ∞ (A),

A ∈ G.

Through the construction of μ∞ we will also obtain an explicit form of this probability measure for A ∈ G . Define T

T

I[−T,T] = e− ∫−T V(Bs )ds I∗−T e−ϕE (∫−T δs ⊗φ(⋅−Bs )ds) IT . Note that I[−T,T] : L2 (Q ) → L2 (Q ) is an almost surely bounded linear operator. Define an additive set function μ : G → [0, ∞) by μ(A) = e2EN T ∫ 𝔼x [1A (Ψg (B−T ), I[−T,T] Ψg (BT ))] dx,

BM A ∈ F[−T,T] .

ℝd BM BM Lemma 2.71. The set function μ is well-defined, i. e., for A ∈ F[−T,T] ⊂ F[−S,S] ,

μ(A) = e2EN T ∫ 𝔼x [1A (Ψg (B−T ), I[−T,T] Ψg (BT ))] dx ℝd

= e2EN S ∫ 𝔼x [1A (Ψg (B−S ), I[−S,S] Ψg (BS ))] dx. ℝd

Proof. Let μ(T) = μ⌈F BM , which is also a probability measure on the measurable [−T,T]

BM space (X, F[−T,T] ). Let −S < −T = t0 < t1 < . . . < tn = T < S, n ∈ ℕ, be an arbitrary set of time-points. By Theorem 2.15, the family of finite dimensional distributions is given by t ,...,tn

0 μ(T)

(A0 × ⋅ ⋅ ⋅ × An ) = μ(T) (Bt0 ∈ A0 , . . . , Btn ∈ An ) n

= e2EN T ∫ 𝔼x [( ∏ 1Aj (Btj )) (Ψg (B−T ), I[−T,T] Ψg (BT )) ]dx ℝd

j=0

192 | 2 The Nelson model by path measures = (Ψg , 1A0 e−(t1 −t0 )(HN −EN ) ⋅ ⋅ ⋅ e−(tn −tn−1 )(HN −EN ) 1An Ψg ) , where Aj ∈ ℬ(ℝ), j = 1, . . . , An . By the identity e−(t0 +S)(HN −EN ) Ψg = Ψg we have = (Ψg , e−(t0 +S)(HN −EN ) 1A0 ⋅ ⋅ ⋅ 1An e−(S−tn )(HN −EN ) Ψg ) n

= e2EN S ∫ 𝔼x [( ∏ 1Aj (Btj )) (Ψg (B−S ), I[−S,S] Ψg (BS )) dx] =

ℝd t0 ,...,tn μ(S) (A0

j=0

× ⋅ ⋅ ⋅ × An ).

Furthermore, the family of finite dimensional distributions {μΛ(T) , Λ ⊂ [−T, T], #Λ < ∞} satisfies the Kolmogorov consistency condition t ,...,tn

0 μ(T)

t ,...,tn ,tn+1 ,...,tn+l

0 (A0 × ⋅ ⋅ ⋅ × An ) = μ(T)

l

(A0 × ⋅ ⋅ ⋅ × An × ∏ ℝd ).

Let πΛ : ℝ[−T,T] → ℝ#Λ be a projection, where Λ ⊂ [−T, T] and #Λ < ∞. Define −1

𝒜 = {πΛ (E) ∈ ℝ

[−T,T]

| E ∈ ℬ(ℝ#Λ ), Λ ⊂ [−T, T], #Λ < ∞}.

By the Kolmogorov extension theorem, there exists a unique measure m on the measurable space (ℝ[−T,T] , σ(𝒜)) such that m ({ω ∈ ℝ[−T,T] | ω(t0 ) ∈ A0 , . . . , ω(tn ) ∈ Atn }) = μΛ(T) (A0 × ⋅ ⋅ ⋅ × Atn ) = μΛ(S) (A0 × ⋅ ⋅ ⋅ × Atn ), E ,...,E

where Λ = {t0 , . . . , tn } ⊂ [−T, T] ⊂ [−S, S]. Note that 𝒜 ∋ At11,...,tn n , with the cylinder sets n

E ,...,E

At11,...,tn n = {ω ∈ X | (ω(t0 ), . . . , ω(tn )) ∈ ∏ Ej , Ej ∈ ℬ(ℝ), j = 1, . . . , n, n ≥ 1} j=1

BM and the Borel σ-field generated by the cylinder sets as above coincides with F[−T,T] . BM BM Also, σ(𝒜) ⊃ F[−T,T] and it follows that μ(S) (A) = m(A) = μ(T) (A) for all A ∈ F[−T,T] , by uniqueness of the extension.

Lemma 2.72. The set function μ is completely additive on (X, G ). Proof. Let Aj ∈ G and Ai ∩ Aj = 0 for i ≠ j. Suppose that A = ⋃∞ j=1 Aj ∈ G . Hence there exists s > 0 such that A ∈ Gs . We have ∞

μ(A) = e2EN s ∫ 𝔼x [ ∑ 1Aj (Ψg (B−s ), I[−s,s] Ψg (Bs )) ]dx j=1

ℝd ∞

∞

= ∑ e2EN s ∫ 𝔼x [1Aj (Ψg (B−s ), I[−s,s] Ψg (Bs ))] dx = ∑ μ(Aj ) j=1

ℝd

by the monotone convergence theorem.

j=1

2.8 Gibbs measure associated with the ground state

| 193

By Lemma 2.72 above, there exists a unique probability measure μ∞ on (X, σ(G )) such that μ∞ (A) = μ(A) for A ∈ G , by Hopf’s extension theorem. We define the additive set function ρT (A) = e2EN s ∫ 𝔼x [1A ( ℝd

fT−s (B−s ) f (B ) , I[−s,s] T−s s )]dx, ‖fT ‖ ‖fT ‖

ρT : GT → ℝ,

BM for A ∈ F[−s,s] with s ≤ T.

Lemma 2.73. The set function ρT is well-defined, and ρT (A) = e2EN r ∫ 𝔼x [1A ( ℝd

fT−r (B−r ) f (B ) , I[−r,r] T−r r )]dx ‖fT ‖ ‖fT ‖

= e2EN s ∫ 𝔼x [1A ( ℝd

fT−s (B−s ) f (B ) , I[−s,s] T−s s )] dx, ‖fT ‖ ‖fT ‖

(2.8.5)

BM holds for all r ≤ s ≤ T and A ∈ F[−r,r] .

Proof. The proof of the second statement is similar to the proof of Lemma 2.71. Denote the left-hand side of (2.8.5) by ρ(r) (A), and the right-hand side by ρ(s) (A). The finite dimensional distributions of ρ(r) are given by t ,...,tn

0 ρ(r)

(A0 × ⋅ ⋅ ⋅ × An ) = ρ(r) (Bt0 ∈ A0 , . . . , Btn ∈ An ) =

n e2EN r x 𝔼 [( 1Aj (Btj )) (fT−r (B−r ), I[−r,r] fT−r (Br )) ]dx. ∏ ∫ ‖fT ‖2 j=0 ℝd

By Theorem 2.15 the right-hand side above can be represented as = = = = =

(fT−r , e−(t0 +r)(HN −EN ) 1A0 e−(t1 −t0 )(HN −EN ) ⋅ ⋅ ⋅ 1An e−(r−tn )(HN −EN ) fT−r ) ‖fT ‖2

(f ⊗ 1, e−(T+t0 )(HN −EN ) 1A0 e−(t1 −t0 )(HN −EN ) ⋅ ⋅ ⋅ 1An e−(T−tn )(HN −EN ) f ⊗ 1) (fT−s , e

−(t0 +s)(HN −EN )

1A0 e

‖fT ‖2 −(t1 −t0 )(HN −EN ) ‖fT ‖2

⋅ ⋅ ⋅ 1An e−(s−tn )(HN −EN ) fT−s )

n e2EN s x 𝔼 [( 1Aj (Btj ))(fT−s (B−s ), I[−s,s] fT−s (Bs ))]dx ∏ ∫ ‖fT ‖2 j=0

ℝd t0 ,...,tn ρ(s) (A0

× ⋅ ⋅ ⋅ × An ).

Note that both ρΛ(r) and ρΛ(s) , Λ ⊂ [−T, T], #Λ < ∞, satisfy the consistency condition. BM Note also that ρ(r) ⌈F BM and ρ(s) ⌈F BM are probability measures on (X, F[−r,r] ). By the [−r,r]

[−r,r]

BM Kolmogorov extension theorem we see that ρ(r) (A) = ρ(s) (A) for every A ∈ F[−r,r] ⊂ BM F[−s,s] , which proves the lemma.

194 | 2 The Nelson model by path measures By Hopf’s extension theorem there exists a probability measure ρ̄ T on (X, σ(GT )) such that ρT = ρ̄ T ⌈GT . Lemma 2.74. Let s ≤ T and A ∈ Gs . Then ρ̄ T (A) = μT (A). Proof. For Λ = {t0 , t1 , . . . , tn } ⊂ [−s, s] and A0 × ⋅ ⋅ ⋅ × An ∈ ×nj=0 ℬ(ℝd ), we define ρΛT (A0 × ⋅ ⋅ ⋅ × An ) = ρT (Bt0 ∈ A0 , . . . , Btn ∈ An ) =

n

e2EN s

‖fT ‖2

∫ 𝔼x [( ∏ 1Aj (Btj )) (fT−s (B−s ), I[−s,s] fT−s (Bs )) ]dx ℝd

j=0

and μΛT (A0 × ⋅ ⋅ ⋅ × An ) = μT (Bt0 ∈ A0 , . . . , Btn ∈ An ) =

n 1 ∫ 𝔼x [( ∏ 1Aj (Btj ))LT ]dx. ZT j=0 ℝd

Both ρΛT and μΛT are probability measures on ((ℝd )#Λ , ℬ(ℝd )#Λ ). We have μΛT (A0 × ⋅ ⋅ ⋅ × An ) = =

(f ⊗ 1, e−(t0 +T)HN 1A0 e−(t1 −t0 )HN 1A1 ⋅ ⋅ ⋅ 1An e−(T−tn )HN f ⊗ 1) e

2EN s

(fT−s , e

−(t0 +s)HN

1A0 e

‖fT ‖2 −(t1 −t0 )HN ‖fT ‖2

1A1 ⋅ ⋅ ⋅ 1An e−(s−tn )HN fT−s )

by the definition of fT−s . The right-hand side above can be expressed as n

= e2EN s ∫ 𝔼x [( ∏ 1Aj (Bj )) ( j=0

ℝd

f (B ) fT−s (B0 ) , I[−s,s] T−s s ) ]dx. ‖fT ‖ ‖fT ‖

It follows that ρΛT (A0 × ⋅ ⋅ ⋅ × An ) = μΛT (A0 × ⋅ ⋅ ⋅ × An ). Note that t ,...,tn

μT ⌈GT (Bt0 ∈ A0 × ⋅ ⋅ ⋅ × Btn ∈ An ) = μT0 =

t ,...,t ρT0 n (A0

(A0 × ⋅ ⋅ ⋅ × An )

× ⋅ ⋅ ⋅ × An ) = ρ̄ T (Bt0 ∈ A0 × ⋅ ⋅ ⋅ × Btn ∈ An ).

The probability measures μΛT and ρΛT satisfy the Kolmogorov consistency condition. By the Kolmogorov extension theorem, ρ̄ T = μT ⌈GT follows from the uniqueness of the extension. Lemma 2.75. If A ∈ G , then limT→∞ μT (A) = μ∞ (A). Proof. Suppose that A ∈ Gs for some s. By Lemma 2.74, we have lim μT (A) = lim ρ̄ T (A) = lim e2EN s ∫ 𝔼x [1A (

T→∞

T→∞

T→∞

ℝd

fT−s (B−s ) f (B ) , I[−s,s] T−s s )] dx. ‖fT ‖ ‖fT ‖

Since fT /‖fT ‖ → Ψg strongly as T → ∞, we have lim μT (A) = e2EN s ∫ 𝔼x [1A (Ψg (B−s ), I[−s,s] Ψg (Bs ))] dx = μ∞ (A).

T→∞

ℝd

(2.8.6)

2.8 Gibbs measure associated with the ground state

| 195

BM BM Remark 2.76. In a similar way to Lemma 2.71 it is seen that for A ∈ F[a,b] ⊂ F[−S,S] ,

μ(A) = eEN (b−a) ∫ 𝔼x [1A (Ψg (Ba ), I[a,b] Ψg (Bb ))] dx ℝd

= e2EN S ∫ 𝔼x [1A (Ψg (B−S ), I[−S,S] Ψg (BS ))] dx. ℝd

In addition to this we can also show that ρT (A) = eEN (b−a) ∫ 𝔼x [1A ( ℝd

fT+a (Ba ) f (B ) , I[a,b] T−b b )]dx ‖fT ‖ ‖fT ‖

BM holds for all −T < a < b < T and A ∈ F[a,b] . From these facts for A ∈ F[a,b] we get limT→∞ μT (A) = μ∞ (A) and

μ∞ (A) = eEN (b−a) ∫ 𝔼x [1A (Ψg (Ba ), I[a,b] Ψg (Bb ))] dx.

(2.8.7)

ℝd

Finally, we obtain local convergence of {μT }T>0 . Theorem 2.77 (Local convergence). The family of probability measures {μT }T≥0 converges to μ∞ on (X, σ(G )) in local sense, i. e., μT (A) → μ∞ (A) as T → ∞ for each A ∈ G , and μ∞ is independent of f . Proof. By Lemma 2.75, it follows that μT (A) → μ∞ (A) for A ∈ G . To show that μ∞ is independent of the choice of f , suppose that μ󸀠∞ is a local limit of μ󸀠T defined by μT with f replaced by f 󸀠 such that 0 ≤ f 󸀠 ∈ L2 (ℝd ). By the construction of μ∞ we have μ∞ (A) = μ󸀠∞ (A) for A ∈ G by (2.8.6). Uniqueness of the extension implies that μ∞ = μ󸀠∞ , thus μ∞ is independent of f . The probability measure μ∞ is the Gibbs measure associated with the Nelson model. Corollary 2.78. Let f , g be bounded measurable functions on ℝd . Then it follows that for a ≤ b, 𝔼μ∞ [f (Ba )g(Bb )] = (f Ψ, e−(b−a)(HN −EN ) gΨ).

(2.8.8)

−1 Proof. Let G1 , G2 ∈ ℬ(ℝd ). We have 1G1 (Ba )1G2 (Bb ) = 1B−1 and by Remark 2.76 a (G1 )∩Bb (G2 ) we find that

−1 𝔼μ∞ [1G1 (Ba )1G2 (Bb )] = μ∞ (B−1 a (G1 ) ∩ Bb (G2 )) −1 = eEN (b−a) ∫ 𝔼x [1B−1 (Ψg (Ba ), I[a,b] Ψg (Bb ))] dx a (G1 )∩B (G2 )

ℝd

b

196 | 2 The Nelson model by path measures = eEN (b−a) ∫ 𝔼x [(1G1 (Ba )Ψg (Ba ), I[a,b] 1G2 (Bb )Ψg (Bb ))] dx = (1G1 Ψ, e

ℝd −(b−a)(HN −EN )

1G2 Ψ).

Hence a limiting argument yields the corollary. By a limiting argument (2.8.8) can be extended to unbounded functions. Corollary 2.79. Suppose that Ψ ∈ D(|x|). Then it follows that for a ≤ b, 𝔼μ∞ [|Ba ||Bb |] = (|x|Ψ, e−(b−a)(HN −EN ) |x|Ψ). Furthermore it follows that 𝔼μ∞ [Ba ⋅ Bb ] = (xΨ, e−(b−a)(HN −EN ) xΨ). Proof. Let fn (x) = |x| for |x| ≤ n and fn (x) = n for |x| ≥ n. Since fn is bounded, 𝔼μ∞ [fn (Ba )fn (Bb )] = (fn Ψ, e−(b−a)(HN −EN ) fn Ψ). Since e−(b−a)(HN −EN ) is positivity improving and (fn Ψ, e−(b−a)(HN −EN ) fn Ψ) ≤ (|x|Ψ, e−(b−a)(HN −EN ) |x|Ψ), the monotone convergence theorem yields that lim (f Ψ, e n→∞ n

−(b−a)(HN −EN )

fn Ψ) = (|x|Ψ, e−(b−a)(HN −EN ) |x|Ψ).

Hence the monotone convergence theorem yields again that lim 𝔼μ∞ [fn (Ba )fn (Bb )] = 𝔼μ∞ [|Ba ||Bb |]

n→∞

and the first statement follows. The second statement can be also show in a similar manner.

2.8.2 P(ϕ)1 -process associated with the Nelson Hamiltonian In Section 2.8.1 we defined the probability measure μ∞ on (X, σ(G )) by using the ground state Ψg of HN . This can be extended to a probability measure μ̂ on X × Y. Recall that (ξt )t∈ℝ is the infinite dimensional Ornstein–Uhlenbeck process on (Y, ℬ(Y)). We write M = ℝd × M−2

2.8 Gibbs measure associated with the ground state

| 197

and define the measure dxdG = dM on M. Also, we define an M-valued random process on (X × Y, ℬ(X) × ℬ(Y), 𝒲 0 × 𝒢 ) by Xt = (Bt , ξt ). Denote

b

b

I[a,b] = e− ∫a V(Bs )ds e− ∫a ξs (φ(⋅−Bs ))ds . Using I[a,b] we have the functional integral representation x,ξ

(F, e−(b−a)HN G)L2 (M) = ∫ 𝔼𝒲×𝒢 [(F(ξa , Ba ), I[a,b] G(ξb , Bb ))] dM M

of the semigroup defined on L2 (M). Let F[a,b] = σ(Xr , r ∈ [a, b]).

We define μ̂ by x,ξ ̂ μ(A) = eEN (b−a) ∫ 𝔼𝒲×𝒢 [1A Ψg (ξa , Ba )I[a,b] Ψg (ξb , Bb )] dM M

for A ∈ F[a,b] . It is seen that μ̂ is well defined in the sense that x,ξ

eEN (b−a) ∫ 𝔼𝒲×𝒢 [1A Ψg (ξa , Ba ), I[a,b] Ψg (ξb , Bb )] dM M

x,ξ

= eEN (b −a ) ∫ 𝔼𝒲×𝒢 [1A Ψg (Ba󸀠 )I[a󸀠 ,b󸀠 ] Ψg (ξb󸀠 , Bb󸀠 )] dM 󸀠

󸀠

M

for a󸀠 , b󸀠 such that [a, b] ⊂ [a󸀠 , b󸀠 ]. This can be proven in the same way as Lemma 2.73. Consider G ̂ = ⋃ F[−T,T] T≥0

̂ which we The measure μ̂ can be extended to a probability measure on (X × Y, σ(G )), denote by the same label μ.̂ Let x,ξ μ̂ T (A) = eEN (b−a) ∫ 𝔼𝒲×𝒢 [1A M

fT+a (ξa , Ba ) f (ξ , B ) , I[a,b] T−b b b ] dM, ‖fT ‖ ‖fT ‖

where fT = e−T(HN −EN ) f ⊗ 1 with 0 ≤ f ∈ L2 (ℝd ). We note that fT+a /‖fT ‖ → eaEN Ψg as T → ∞. We have the following result.

198 | 2 The Nelson model by path measures Theorem 2.80 (Local convergence). The family of probability measures {μ̂ T }T≥0 con̂ in local sense, i. e., μ̂ T (A) → μ(A) ̂ verges to μ on (X × Y, σ(G )) as T → ∞ for every A ∈ G ,̂ and μ is independent of f . Proof. The proof is similar to that of Lemma 2.75 and is left to the reader. Note that (f Ψg , e−|s−t|(HN −EN ) gΨg )L2 (M) = (f , e−|s−t|L g)L2 (M,Ψ2 dM) , where L=

1 (H − EN )Ψg . Ψg N

Hence by the definition of μ̂ we see that for 0 ≤ t0 ≤ t1 ≤ . . . ≤ tn and bounded functions f1 , . . . , fn and f , n

𝔼μ̂ [ ∏ fj (Xtj )] = (f0 , e−(t1 −t0 )L f1 ⋅ ⋅ ⋅ fn−1 e−(tn −tn−1 )L fn )L2 (M,Ψ2g dM) , j=0

𝔼μ̂ [f (Xs )] = (1, f )L2 (M,Ψ2g dM) ,

s ≥ 0.

(2.8.9) (2.8.10)

̂ μ)̂ is a Theorem 2.81 (P(ϕ)1 -process). The random process (Xt )t≥0 on (X × Y, σ(G ), 2 Markov process under the natural filtration, with invariant measure Ψg dM. Proof. Define p(t, X, A) = (e−tL 1A )(X) for all t ≥ 0, X ∈ M and A ∈ ℬ(M). Clearly, p(t, X, M) = 1 and p(t, ⋅, A) is Borel measurable. Writing p(t, X, A) = ∫M 1A (Y)p(t, X, dY) gives ∫M f (Y)p(t, X, dY) = (e−tL f )(X) for every bounded function f . With semigroup property e−sL e−tL 1A (X) = e−(s+t)L 1A (X) we have p(s + t, X, A) = e−(s+t)L 1A (X) = e−sL e−tL 1A (X) = ∫ e−tL 1A (Y)p(s, X, dY) = ∫ p(t, Y, A)p(s, X, dY), M

M

i. e., the Chapman–Kolmogorov identity is verified. Hence p : [0, ∞) × M × ℬ(M) → ℝ is a probability transition kernel. By (2.8.9) we also see that 𝔼μ̂ [1A0 (X0 ) ⋅ ⋅ ⋅ 1An (Xtn )] = (1A0 , e−(t1 −t0 )L 1A1 ⋅ ⋅ ⋅ 1An−1 e−(tn −tn−1 )L 1An )L2 (M,Ψ2g dM) n

n

j=0

j=1

= ∫ ( ∏ 1Aj (Xj ))( ∏ p(tj − tj−1 , Xj−1 , dXj ))1A0 (X0 )Ψ2g (X0 )dM(X0 ). M (n+1)

Thus (Xt )t≥0 is a Markov process with probability transition kernel p(t, X, A). Moreover, we have

2.8 Gibbs measure associated with the ground state

| 199

∫ p(t, X, A)Ψ2g (X)dM = (1, e−tL 1A )L2 (M,Ψ2g dM) = (Ψg , e−t(HN −EN ) 1A Ψg )L2 (M)

M

= (Ψg , 1A Ψg )L2 (M) = ∫ 1A (X)Ψ2g (X)dM M

and thus Ψ2g dM is an invariant measure for (Xt )t≥0 . The random process (Xt )t≥0 is called the P(ϕ)1 -process associated with the Nelson Hamiltonian.

2.8.3 Applications to ground state expectations Likewise with 𝒩 , the Gibbs measure μ∞ can be used to derive ground state expectations. As seen in (1.5.7) in Chapter 1, in order to factorize e−t Hf we needed an extra Hilbert space ℋE in addition to ℋM , and the map τt : ℋM → ℋE so that τs∗ τt = e−|t−s|ω̂ . ̂

This proved to be a key formula allowing to write It = Γ(τt ) and I∗s It = e−|t−s|Hf , which were useful in deriving a functional representation. Using the same idea, we construct a functional integral representation of (e−THN F, e−βdΓ(ρ) e−THN F) here, with a nonnegative measurable function ρ. To achieve this, we use the extra Hilbert space ̂

2

d+2

2

d+2

ℋρ = L (ℝ

)

and its Fourier image ℋ̂ ρ = FL (ℝ

),

where F denotes Fourier transform on ℋρ . We will denote the scalar products on ℋ̂ ρ and ℋρ by (⋅, ⋅)ρ̂ and (⋅, ⋅)ρ , respectively. We define a family of Gaussian random variables {ϕρ (f ), f ∈ L2real (ℝd+2 )} on a probability space (Qρ , Σρ , μρ ) indexed by L2real (ℝd+2 ). For f ̂ ∈ ℋ̂ , the variables of f ̂ will be denoted by (k, k , k ) ∈ ℝd × ℝ × ℝ. Also, we define ρ

a family of isometries qs : ℋE → ℋρ by

0

1

√2 ρ(k) e−isk1 √ q̂ f ̂(k, k0 ). s f (k, k0 , k1 ) = √π √ω(k)2 + |k |2 ρ(k)2 + |k1 |2 0 Lemma 2.82. It follows that qs∗ qt = e−|s−t|ρ(−i∇)⊗1 ,

s, t ∈ ℝ.

Here we used the identification L2 (ℝd+1 ) ≅ L2 (ℝd ) ⊗ L2 (ℝ).

200 | 2 The Nelson model by path measures Proof. We have ρ(k) 2 ei(s−t)k1 ̂ k0 ) dkdk0 dk1 . ∫ f ̂(k, k0 )g(k, π ω(k)2 + |k0 |2 ρ(k)2 + |k1 |2

̂ (qs f , qt g)ℋρ = (q̂ sf , q t g)ℋ̂ = ρ

ℝd+2

(2.8.11)

Using the formula e−|s−t|ρ(k) 1 ei(s−t)k1 dk1 = ∫ 2 2 π ρ(k) + |k1 | ρ(k) ℝ

and integrating (2.8.11) with respect to k1 , we have e−|s−t|ρ(k) ̂ k0 ) dkdk0 = (f ̂, e−|s−t|ρ⊗1 g)̂ ℋ̂ . (qs f , qt g)ℋρ = 2 ∫ f ̂(k, k0 )g(k, E ω(k)2 + |k0 |2 ℝd+1

Next we define a family of second quantizations Ξs by Ξt = Γint (qt ) : L2 (Q ) → L2 (Qρ ),

t ∈ ℝ.

Let ρ̂ = ρ(−i∇). We see that Ξs is an isometry for every t and, furthermore, it factorizes ̂ e−tdΓ(ρ⊗1) . Lemma 2.83. It follows that Ξ∗s Ξt = e−|t−s|dΓ(ρ⊗1) , ̂

s, t ∈ ℝ,

(2.8.12)

s ≥ 0, t ∈ ℝ

(2.8.13)

and the intertwining property It e−sdΓ(ρ) = e−sdΓ(ρ⊗1) It , ̂

̂

holds. ̂ Proof. The factorization formula (2.8.12) follows from the identity qs∗ qt = e−|s−t|ρ⊗1 . It is seen that

(f , τt e−sρ g)ℋE = ̂

√2 ̂ k0 ) e−sρ(k) e−itk0 f ̂(k, k0 )g(k, dkdk0 ∫ √π ω(k)2 + |k0 |2 ℝd+1

= (f ̂, e

−sρ⊗1

̂ −sρ⊗1 τ̂ τt g)ℋE . t g)ℋ̂ = (f , e E

Hence τt e−sρ = e−sρ⊗1 τt , and from this (2.8.13) follows.

2.8 Gibbs measure associated with the ground state

| 201

Theorem 2.84. Let ρ be a positive function on ℝd . For every F, G ∈ L2 (Q ) and β > 0 we have (e−THN F, e−βdΓ(ρ) e−THN G) ̂

T

= ∫ 𝔼x [e− ∫−T V(Bs )ds (e−ϕE (L−T ) I−T F(B−T ), e−βdΓ(ρ⊗1) e−ϕE (L+T ) IT G(BT )) ]dx ̂

(2.8.14)

ℝd

or, equivalently, (e−THN F, e−βdΓ(ρ) e−THN G) ̂

T

ρ

= ∫ 𝔼x [e− ∫−T V(Bs )ds (Ξ0 I−T F(B−T ), e−ϕρ (KT ) Ξβ IT G(BT ))L2 (Qρ ) ] dx,

(2.8.15)

ℝd

where T

0

L−T = ∫ τs φ(⋅ − Bs )ds,

L+T = ∫ τs φ(⋅ − Bs )ds, 0

−T

and T

0

ρ

KT = ∫ q0 τs φ(⋅ − Bs )ds + ∫ qβ τs φ(⋅ − Bs )ds. 0

−T

Proof. Let V be Kato-decomposable or V ∈ C0∞ (ℝd ). It follows that T

e−THN F(x) = 𝔼x [e− ∫0

T

V(Bs )ds ∗ −ϕE (∫0 τs φ(⋅−Bs )ds) I0 e IT F(BT )]

almost every x ∈ ℝd . Hence we have (e−THN F, e−βdΓ(ρ) e−THN G) = (F, e−THN e−βdΓ(ρ) e−THN G) ̂

̂

T

= ∫ 𝔼μ 𝔼x [F(B0 )e− ∫0

T

V(Bs )ds ∗ −ϕE (∫0 τs φ(⋅−Bs )ds) I0 e IT

ℝd

T

𝔼BT [e−βdΓ(ρ) e− ∫0 ̂

T

V(Bs )ds ∗ −ϕE (∫0 τs φ(⋅−Bs )ds) I0 e IT G(BT )]] dx.

∗ Since IT I∗0 = ET U−T and IT = U−T I2T , by the Markov property of Brownian motion and ̂ the intertwining property e−βdΓ(ρ⊗1) IT = IT e−βdΓ(ρ)̂ , we have

(e−THN F, e−βdΓ(ρ) e−THN G) ̂

T

= ∫ 𝔼μ 𝔼x [F(B0 )e− ∫0 ℝd

T

𝔼x [e− ∫0

T

̂ V(Bs )ds ∗ −ϕE (∫0 τs φ(⋅−Bs )ds) −βdΓ(ρ⊗1) I0 e e

V(Bs+T )ds

T

∗ −ϕE (∫0 ET U−T e

τs φ(⋅−Bs+T )ds)

󵄨 U−T I2T G(B2T )󵄨󵄨󵄨FTBM ]] dx.

202 | 2 The Nelson model by path measures T

∗ −ϕE (∫0 Since U−T e we have

τs φ(⋅−Bs+T )ds)

T

U−T = e−ϕE (∫0

τs+T φ(⋅−Bs+T )ds)

, and the Markov property of Es

(e−THN F, e−βdΓ(ρ) e−THN G) ̂

T

= ∫ 𝔼μ 𝔼x [F(B0 )aV I∗0 e−ϕE (∫0 ℝd

T

= ∫ 𝔼x [aV (e−ϕE (∫0

2T

̂ τs φ(⋅−Bs )ds) −βdΓ(ρ⊗1) −ϕE (∫T τs φ(⋅−Bs )ds)

τs φ(⋅−Bs )ds)

e

e

2T

Φ0 , e−βdΓ(ρ⊗1) e−ϕE (∫T ̂

τs φ(⋅−Bs )ds)

I2T G(B2T )] dx

Φ2T )] dx.

(2.8.16)

ℝd 2T

Here we set aV = e− ∫0 V(Bs )ds , Φ0 = I0 F(B0 ) and Φ2T = I2T G(B2T ) for a notational ̂ ̂ simplicity. We factorize e−βdΓ(ρ⊗1) as e−βdΓ(ρ⊗1) = Ξ∗0 Ξβ and insert this into the above identity (2.8.16) to see that (e−THN F(B0 ), e−βdΓ(ρ) e−THN G) ̂

T

τs φ(⋅−Bs )ds) ∗ Ξ0 Ξ0 Φ0 , Ξβ e−ϕE (∫T τs φ(⋅−Bs )ds) Ξ∗β Ξβ Φ2T )] dx.

T

τs φ(⋅−Bs )ds) ∗ Ξt

= ∫ 𝔼x [aV (Ξ0 e−ϕE (∫0

2T

ℝd

Again using the identity Ξt e−ϕE (∫0

T

= Etq e−ϕE (∫0

qt τs φ(⋅−Bs )ds) q Et ,

where Etq = Ξt Ξ∗t is a projection, we have (e−THN F, e−βdΓ(ρ) e−THN G) ̂

T

= ∫ 𝔼x [aV (E0q e−ϕρ (∫0

2T

q0 τs φ(⋅−Bs )ds) q E0 Ξ0 Φ0 , Eβq e−ϕρ (∫T qβ τs φ(⋅−Bs )ds) Eβq Ξβ Φ2T )] dx,

ℝd

and, furthermore, by the Markov property of Etq which can be obtained similarly as for Et , we drop Etq in the above identity. Thus (e−THN F(B0 ), e−βdΓ(ρ) e−THN G) ̂

T

= ∫ 𝔼x [aV (e−ϕρ (∫0

q0 τs φ(⋅−Bs )ds)

2T

Ξ0 Φ0 , e−ϕρ (∫T

qβ τs φ(⋅−Bs )ds)

Ξβ Φ2T )] dx.

(2.8.17)

ℝd

From a time shift by −T of (2.8.17) equality (2.8.15) follows, and (2.8.14) is obtained from (2.8.16). For general V ∈ ℛKato one can apply a limiting argument. Proposition 2.85. Let ρ be a positive function on ℝd . Define fT = e−THN f ⊗ 1. Then it follows that 0 T (fT , e−βdΓ(ρ) fT ) = 𝔼μT [e− ∫−T ds ∫0 Wβ (Bs −Bt ,s−t)dt ] , 2 ‖fT ‖

2.8 Gibbs measure associated with the ground state

where

| 203

2 ̂ |φ(k)| e−|t|ω(k) e−ik⋅x (1 − e−βρ(k) )dk. 2ω(k)

Wβ (x, t) = ∫ ℝd

Proof. By Theorem 2.84 we have T

ρ

(fT , e−βdΓ(ρ) fT ) = ∫ 𝔼x [e− ∫−T V(Bs )ds (1, e−ϕρ (KT ) 1)L2 (Qρ ) ] dx. ̂

ℝd ρ

ρ 2

1

A computation gives (1, e−ϕρ (KT ) 1) = e 2 ‖KT ‖ρ and ρ ‖KT ‖2ρ

0

T

󵄩󵄩 󵄩󵄩2 󵄩󵄩 󵄩󵄩2 = 󵄩󵄩󵄩 ∫ q0 τs φ(⋅ − Bs )ds󵄩󵄩󵄩 + 󵄩󵄩󵄩 ∫ qβ τs φ(⋅ − Bs )ds󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 0

−T

0

T

+ 2ℜ( ∫ q0 τs φ(⋅ − Bs )ds, ∫ qβ τs φ(⋅ − Bs )ds). 0

−T −βρ(k) ̂ ∗ Since q? f (k, k0 ), we have 0 qβ f (k, k0 ) = e ρ ‖KT ‖2ρ

0

0

2 ̂ |φ(k)| e−|s−t|ω(k) e−ik⋅(Bs −Bt ) dk 2ω(k)

= ∫ ds ∫ dt ∫ −T

ℝd

−T T

T

+ ∫ ds ∫ dt ∫ 0

0 0

T

ℝd

2 ̂ |φ(k)| e−|s−t|ω(k) e−ik⋅(Bs −Bt ) dk 2ω(k)

+ 2 ∫ ds ∫ dt ∫ 0

−T T

T

ℝd

2 ̂ |φ(k)| e−|s−t|ω(k) e−βρ(k) e−ik⋅(Bs −Bt ) dk 2ω(k) 0

T

= ∫ ds ∫ W(Bs − Bt , s − t)dt − 2 ∫ ds ∫ Wβ (Bs − Bt , s − t)dt. −T

−T

0

−T

On the other hand, T

1

T

T

‖fT ‖2 = ∫ 𝔼x [e− ∫−T V(Bs )ds e 2 ∫−T ds ∫−T W(Bs −Bt ,s−t)dt ] dx. ℝd

It follows that T

1

T

T

0

T

x − ∫ V(Bs )ds 2 ∫−T ds ∫−T Wdt − ∫−T ds ∫0 Wβ dt e e ] dx (fT , e−βdΓ(ρ) fT ) ∫ℝd 𝔼 [e −T = T T 2 1 T ‖fT ‖ ∫ 𝔼x [e− ∫−T V(Bs )ds e 2 ∫−T ds ∫−T Wdt ] dx ℝd 0

T

= 𝔼μT [e− ∫−T ds ∫0 Hence the proposition follows.

Wβ (Bs −Bt ,s−t)dt

].

(2.8.18)

204 | 2 The Nelson model by path measures By this proposition, in particular, we have 0 T −β (fT , e−βN fT ) = 𝔼μT [e−(1−e ) ∫−T ds ∫0 W(Bs −Bt ,s−t)dt ] . 2 ‖fT ‖

The expectation (Ψg , e−βN Ψg ) expressed in terms of 𝒩 in Corollary 2.41, can also be computed in terms of μ∞ . Corollary 2.86 (Super-exponential decay of boson number). For every β ∈ ℂ we have Ψg ∈ D(e−βN ) and 0

∞

(Ψg , e−βN Ψg ) = 𝔼μ∞ [exp (−(1 − e−β ) ∫ ds ∫ W(Bs − Bt , s − t)dt)] . −∞ 0 ] [

(2.8.19)

Proof. Let β > 0. For easing the notation we write 0

T

WT = −(1 − e−β ) ∫ ds ∫ W(Bs − Bt , s − t)dt. −T

0

Note that for every δ > 0 there is Sδ such that |WT − W∞ | ≤ δ for all T > Sδ , uniformly in the paths. We have 𝔼μT [eWT ] − 𝔼μ∞ [eW∞ ] = 𝔼μT [eWT ] − 𝔼μT [eW∞ ] + 𝔼μT [eW∞ ] − 𝔼μ∞ [eW∞ ].

(2.8.20)

Also, 󵄨󵄨 W W 󵄨 󵄨󵄨𝔼μT [e T ] − 𝔼μT [e ∞ ]󵄨󵄨󵄨 ≤ Cδ

(2.8.21)

with a constant C. The second term at the right-hand side of (2.8.20) can be estimated as 󵄨󵄨 󵄨 W 󵄨 W W 󵄨 W 󵄨󵄨𝔼μT [e ∞ ] − 𝔼μ∞ [e ∞ ]󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨𝔼μT [e ∞ ] − 𝔼μT [e Sδ ]󵄨󵄨󵄨 󵄨 󵄨 + 󵄨󵄨󵄨𝔼μT [eWSδ ] − 𝔼μ∞ [eWSδ ]󵄨󵄨󵄨 󵄨 󵄨 + 󵄨󵄨󵄨𝔼μ∞ [eWSδ ] − 𝔼μ∞ [eW∞ ]󵄨󵄨󵄨.

(2.8.22) (2.8.23) (2.8.24)

For (2.8.22) and (2.8.24) we have the same upper bound as in (2.8.21). The term (2.8.23) converges to zero since μT → μ∞ as T → ∞ in local sense. Thus the corollary follows for β > 0. We can extend this to all β ∈ ℂ in the same way as in Corollary 2.61. Let ℂ+ = {z ∈ ℂ | ℜz > 0} and ℂ− = {z ∈ ℂ | ℜz < 0}. Denote the right-hand side of (2.8.19) by g(β), and the left-hand side by f (β). Informally, g can be extended to the whole complex plane ℂ with g = f on ℂ+ and then f can be also extended to ℂ, and thus 2 ̂ g = f holds on ℂ by the identity theorem. Since |W∞ | ≤ 21 ∫ℝd |φ(k)| /ω(k)3 dk < ∞, the function g can be analytically continued in ℂ. We denote its analytic continuation

2.8 Gibbs measure associated with the ground state

| 205

by g.̃ Let β0 > 0, and consider the ball Br (z0 ) = {z ∈ ℂ | |z − z0 | < r}. Fix an arbitrary R such that R > β0 . We see that ∞

̃ g(β) = ∑ (β − β0 )n bn (β0 )

(2.8.25)

n=0

for β ∈ BR (β0 ), and (2.8.25) is absolutely convergent. Denote by ν(dρ) the spectral measure of N associated with Ψg . We have ∞

f (β) = ∫ e−βρ ν(dρ), 0

β ∈ ℂ+ .

Note that f (β) is analytic on ℂ+ . We see that (β − β0 )n f (β) = ∑ ∫ (−ρ)n e−β0 ρ ν(dρ), n! n=0 ∞

∞

0

β ∈ Bβ0 (β0 ).

Since f = g̃ on Bβ0 (β0 ), we also see that bn (β0 ) =

∞

1 ∫ (−ρ)n e−β0 ρ ν(dρ). n! 0

It follows that (β − β0 )n ∫ ρn e−β0 ρ ν(dρ), n! n=0 ∞

∞

̃ g(β) = ∑

β ∈ BR (β0 ).

(2.8.26)

0

Notice that the right-hand side of (2.8.26) is absolutely convergent and BR (β0 )∩ℂ− ≠ 0. In particular, for β ∈ ℝ ∩ BR (β0 ) ∩ ℂ− , i. e., for −(R − β0 ) < β < 0,

(2.8.27)

by Fatou’s lemma we have for every M > 0, M

M

∞

0

0

∞ (β − β0 )n (β − β0 )n ∫ ρn e−β0 ρ ν(dρ) ≤ ∑ ∫ ρn e−β0 ρ ν(dρ) < ∞. n! n! n=0 n=0 ∞

∫ e−βρ ν(dρ) ≤ ∑ 0

∞ ∫0 e−βρ ν(dρ)

Thus < ∞ for β with (2.8.27), which implies that Ψg ∈ D(e−(β/2)N ) and (2.8.19) holds for β with (2.8.27). Since R is an arbitrary number, we get (2.8.19) for ∞ all β ∈ ℝ ∪ ℂ+ . From this it follows that |(Ψg , e−βN Ψg )| < ∫0 e−ℜβρ ν(dρ) < ∞ for all β ∈ ℂ. The function f can be extended in ℂ to a function f ̃. In particular, f ̃ = g̃ on ∞ D = ℝ ∪ ℂ+ and f ̃ has a convergent series expansion in D. That is, ∫0 e−zρ ν(dρ) = n n ∞ −βρ n ρ ν(dρ) for z ∈ BK (β), for every K and β < 0. Hence f ̃ is also ∑∞ n=0 (−1) (z − β) ∫0 e ̃ analytic on ℂ, and g̃ = f on ℂ follows.

206 | 2 The Nelson model by path measures We can also estimate (Ψg , NΨg ) by using the probability measure μ∞ which is the same as in Section 2.6.2. For m ≥ 0 we have (Ψg , N m Ψg ) = (−1)m

0 ∞ −β dm 𝔼μ∞ [e−(1−e ) ∫−∞ ds ∫0 Wdt ]⌈ . m dβ β=0

From this we see that there exist positive constants a and b such that −a + bI ≤ (Ψg , NΨg ) ≤

g2 I, 2

(2.8.28)

2 ̂ where I = ∫ℝd |φ(k)| /ω(k)3 dk, and it can be seen that limI→∞ (Ψg , NΨg ) = ∞. In Section 2.6.5 we showed a Gaussian domination of the ground state Ψg using the path measure 𝒩 . We can also see this by using the Gibbs measure μ∞ .

Corollary 2.87 (Gaussian domination of the ground state). Take g ∈ S 󸀠 (ℝd ) such that 2 2 ̂ ∈ L2 (ℝd ), and β < 1/‖g/√ω‖ ̂ ̂ g/ω . Then Ψg ∈ D(e(β/2)ϕ(g) ) and g/√ω, 1

2

‖e(β/2)ϕ(g) Ψg ‖2 =

2 √1 − β‖g/√ω‖ ̂

𝔼μ∞ [ exp (

βK(g)2 ) ], 2 ̂ 1 − β‖g/√ω‖

where K(g) denotes the random variable defined by ∞

1 ̂ √ω, g/̂ √ω)dr. K(g) = ∫ (e−|r|ω e−ik⋅Br φ/ 2 −∞

In particular, it follows that 2

lim

2 )−1 ̂ β↑(2‖g/√ω‖

‖eβϕ(g) Ψg ‖ = ∞.

Proof. Let f ∈ L2 (ℝd ) and f ≥ 0, and write fT = e−THN f ⊗ 1. For every β ∈ ℂ we have T (fT , eikβϕ(g) fT ) 1 x − ∫−T V(Bs )ds = 𝔼 [f (B )f (B )e ST ] dx, ∫ −T T ‖fT ‖2 ‖fT ‖2

ℝd

where 0

T

ST = (1, e−ϕE (∫−T τs φ(⋅−Bs )ds) eikβϕE (τ0 g) e−ϕE (∫0

τs φ(⋅−Bs )ds)

1)

L2 (QE )

.

A computation gives ST = e−

k 2 β2 4

2 ̂ −ikβ ∫ℝd ( ‖g/√ω‖

e

̂ ̂ g(k) φ(k) 2ω(k)

T

∫−T e−ω(k)|s| e−ik⋅Bs ds)dk

1

T

T

e 2 ∫−T ds ∫−T W(Bs −Bt ,s−t)dt .

Hence it follows that T ̂ ̂ g(k) φ(k) k 2 β2 2 (fT , eikβϕ(g) fT ) ̂ −ikβ ∫ℝd ( 2ω(k) ∫−T e−ω(k)|s| e−ik⋅Bs ds)dk − 4 ‖g/√ω‖ = 𝔼 [e e ]. μT 2 ‖fT ‖

2.9 Carmona-type estimates | 207

Taking the limit T → ∞ we obtain (Ψg , eikβϕ(g) Ψg ) = 𝔼μ∞ [e−k

2 2

β Ig /2 −ikβIB

e

],

(2.8.29)

̂ ̂ g(k) φ(k) 2 ̂ where Ig = ‖g/√ω‖ /2 and IB = ∫ℝd ( 2ω(k) ∫−∞ e−ω(k)|s| e−ik⋅Bs ds) dk. Thus using (2.8.29), in the same way as Lemma 2.60 we can show that for β < 0, ∞

2

(Ψg , eβϕ(g) Ψg ) =

1 2 √1 − β‖g/√ω‖ ̂

𝔼μ∞ [exp (

βK(g)2 )] . ̂ 1 − β‖g/√ω‖

2 ̂ in the same way as in Corollary 2.61. Furthermore, we can extend this to β < 1/‖g/√ω‖

2.9 Carmona-type estimates 2.9.1 Exponential decay of bound states: upper bound In this section we briefly discuss the decay of bound states for potentials in the Carmona classes discussed in Section 4.3.8 in Volume 1, for Schrödinger operators. We consider the following class of potentials. See also Definition 4.175 in Volume 1. Definition 2.88. The potential classes 𝕍upper and 𝕍lower are defined by the following properties. 𝕍upper : V ∈ 𝕍upper if and only if V = W − U such that (1) U ≥ 0 and U ∈ Lp (ℝd ) for p > d/2 and 1 ≤ p < ∞; (2) W ∈ L1loc (ℝd ) and W∞ = infx∈ℝd W(x) > −∞. 𝕍lower : V ∈ 𝕍lower if and only if V = W − U such that (1) U ≥ 0 and U ∈ Lp (ℝd ) for p > d/2 and 1 ≤ p < ∞; (2) W ≥ 0 and W ∈ 𝒦loc (ℝd ). 2 ̂ Lemma 2.89 (Carmona-type estimate). Suppose ∫ℝd |φ(k)| /ω(k)3 dk < ∞ and let Asupper sumption 2.16 hold. Let V = U − W ∈ 𝕍 and HN Φb = EΦb . Then for every t, a > 0 and every 0 < α < 1/2, there exist constants D1 , D2 , D3 > 0 such that α a2 t

‖Φb (x)‖L2 (Q) ≤ t −d/2 D1 eD2 ‖U‖p t eEt (D3 e− 4 ̃

e−tW∞ + e−tWa (x) )‖Φb ‖ℋN ,

where Ẽ = E + E(φ)̂ and Wa (x) = inf {W(y) | |x − y| < a}. Proof. Writing Φb = etE e−tHN Φb , the functional integral representation gives t

t

Φb (x) = etE 𝔼x [I∗0 e−ϕE (∫0 δs ⊗φ(⋅−Bs )ds) It e− ∫0 V(Bs )ds Φb (Bt )] .

208 | 2 The Nelson model by path measures From this, it follows directly that t

‖Φb (x)‖L2 (Q) ≤ et(E+E(φ)) 𝔼x [e− ∫0 V(Bs )ds ‖Φb (Bt )‖L2 (Q) ] . ̂

The lemma follows in a similar way to Carmona’s estimate for Schrödinger operators in Lemma 4.176 in Volume 1. From the above lemma, we can draw decay results similarly to the proof of eigenfunction decay for Schrödinger operators. For V = W − U ∈ 𝕍upper , define Σ = lim inf|x|→∞ V(x). Notice that Σ = lim inf|x|→∞ W(x), and Σ ≥ W∞ holds. Since the proof is similar to the proofs of Corollaries 4.177 and 4.178 in Volume 1, we only state the results. 2 ̂ Corollary 2.90 (Upper bound of exponential decay). Suppose ∫ℝd |φ(k)| /ω(k)3 dk < ∞ and let Assumption 2.16 hold. Let V = W − U ∈ 𝕍upper and HN Φb = EΦb . (1) Suppose that W(x) ≥ γ|x|2n outside a compact set K, for some n > 0 and γ > 0. Take 0 < α < 1/2. Then there exists a constant C1 > 0 such that

‖Φb (x)‖L2 (Q) ≤ C1 exp (−

αc n+1 |x| ) ‖Φb ‖ℋN , 16

where c = infx∈ℝd \K W|x|/2 (x)/|x|2n . ̂ Σ > W∞ , and let 0 < β < 1. Then (2) Decaying potential: Suppose that Σ − E > E(φ), there exists a constant C2 > 0 such that ‖Φb (x)‖L2 (Q) ≤ C2 exp (−

β Σ − E − E(φ)̂ |x|) ‖Φb ‖ℋN . 8√2 √Σ − W∞

(3) Confining potential: Suppose that lim|x|→∞ W(x) = ∞. Then there exist constants C, δ > 0 such that ‖Φb (x)‖L2 (Q) ≤ C exp (−δ|x|) ‖Φb ‖ℋN . 2.9.2 Exponential decay of bound states: lower bound By the Carmona-type estimate in the previous section we can show not only upper bounds but also lower bounds of positive eigenfunctions. In Section 2.9.2 we suppose Assumption 2.16 and the existence of the unique ground Ψg . Here we derive an exponential lower bound of the ground state of HN . First we recall some estimates of probabilities of sets of paths discussed in (4.3.85) of Section 4.3.8 in Volume 1. Let P(a, [c, d], t) = { sup |Bs | ≤ a} ∩ {Bt ∈ [c, d]}. 0≤s≤t

2.9 Carmona-type estimates | 209

The following statement has been established in Lemma 4.180 in Volume 1: Suppose that a > 0, α > 0 and t > 0 satisfy that α < a/2 and a2 /t > β. Here β is the unique positive solution of F(ξ ) = 1 − (e−21ξ /8 + e−5ξ /8 + e−165ξ /8 ) = 0. Then for every x ∈ [−(a − α), a − α] we have 𝒲 (P(a, [x − α, x + α], t)) ≥

a2 α a2 F( )e− 2t . √2πt t

(2.9.1)

t

We recall that I[0,t] (x) = I∗0 e−ϕE (∫0 δs ⊗φ(⋅−Bs −x)ds) It . Lemma 2.91. Suppose that V ∈ 𝕍lower and let K ⊂ ℝd be compact. Then there exists εK > 0 such that inf (1, I[0,t] (x)Ψg (Bt + x))

L2 (Q)

Bt +x∈K

≥ εK exp (−

t

t

0

0

1/2

1 ( ∫ ds ∫ W(s − r, Bs − Br )dr) Mg ) . εK

Here Mg = supx∈ℝd ‖Ψg (x)‖L2 (Q) . Proof. Fix x and let P(A) = (QE , ΣE ), where

1 𝔼 [1 I Ψ (B N μE A t g t

+ x)], A ∈ ΣE , be a probability measure on

N = 𝔼μE [It Ψg (Bt + x)] is the normalising constant. Note that It Ψg (Bt + x) > 0. Since V is Kato-decomposable, by Theorem 2.24 and Corollary 2.27 we get that 𝔼μE [It Ψg (y)] = (1, Ψg (y)) is continuous in y and (1, Ψg (y)) > 0. Hence there exists εK > 0 such that inf (1, Ψg (y)) ≥ εK

(2.9.2)

N ≥ inf (1, Ψg (Bt + x)) ≥ εK .

(2.9.3)

y∈K

and Bt +x∈K

By Jensen’s inequality we have t

(1, I[0,t] (x)Ψg (Bt + x)) = N𝔼P [e−ϕE (∫0 δs ⊗φ(⋅−Bs −x)ds) ] t

t

1

≥ Ne−𝔼P [ϕE (∫0 δs ⊗φ(⋅−Bs −x)ds)] = Ne− N (ϕE (∫0 δs ⊗φ(⋅−Bs −x)ds),It Ψg (Bt )) .

(2.9.4)

Since 1/2

t t t 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 (ϕ ( δ ⊗ φ(⋅ − B − x)ds), I Ψ (B )) ≤ ( ds W(s − r, Bs − Br )dr) ∫ ∫ ∫ 󵄨󵄨 E s s t g t 󵄨󵄨󵄨 󵄨󵄨 󵄨 0

together with (2.9.3)–(2.9.4) the result follows.

0

0

Mg ,

210 | 2 The Nelson model by path measures Lemma 2.92. Let V = W − U ∈ 𝕍lower . Suppose that a1 . . . , ad , α1 , . . . , αd , b1 . . . , bd and t satisfy that a2j /t > β, aj /2 > αj and |[−aj , aj ] ∩ [−xj − bj , −xj + bj ]| > 2αj . Then (1, Ψg (x)) ≥ e−tWa (x) etE εK e−

√t‖φ/ω‖M ̂ g /εK

d

∏ j=1

αj

√2πt

F(

a2j t

a2j

(2.9.5)

)e− 2t ,

where Wa (x) = sup{W(y) | |yj − xj | < aj , j = 1, . . . , d} and εK is given in Lemma 2.91 with the compact set K = [−b1 , b1 ] × ⋅ ⋅ ⋅ × [−bd , bd ]. Proof. Let A = ⋂dj=1 P(aj , [−bj − xj , bj − xj ], t). By (2.9.1) we see that for αj < aj /2 and a2j /t > β d

d

j=1

j=1

𝒲 ( ⋂ P(aj , [kj , kj + 2α], t)) ≥ ∏

αj

√2πt

F(

a2j t

a2j

)e− 2t ,

for any kj ∈ [−aj , aj ]. By the assumption there exists kj ∈ [−aj , aj ] such that [kj , kj +2α] ⊂ [−bj − xj , bj − xj ]. We have d

d

d

j=1

j=1

j=1

𝒲 ( ⋂ P(aj , [−bj − xj , bj − xj ], t)) ≥ 𝒲 ( ⋂ P(aj , [kj , kj + 2α], t)) ≥ ∏

t

αj

√2πt

F(

a2j t

a2j

)e− 2t

(2.9.6)

t

2 ̂ uniformly in paths. By Lemma 2.91 we see and ∫0 ds ∫0 W(s − r, Bs − Br )dr ≤ t‖φ/ω‖ that t

(1, Ψg (x)) = etE 𝔼[e− ∫0 V(x+Bs )ds (1, I[0,t] (x)Ψg (x + Bt ))] t

≥ etE 𝔼[1A e− ∫0 W(x+Bs )ds (1, I[0,t] (x)Ψg (x + Bt ))] t

√t‖φ/ω‖M ̂ g /εK

≥ etE 𝔼[1A e− ∫0 Wa (x)ds εK e− ≥ etE e−tWa (x) εK e−

√t‖φ/ω‖M ̂ g /εK

]

d

𝒲 ( ⋂ P(aj , [kj , kj + 2α], t)). j=1

By (2.9.6) the lemma follows. Theorem 2.93 (Lower bound of exponential decay). Let V = W − U ∈ 𝕍lower . Suppose that W(x) ≤ γ|x|2m outside a compact set with γ > 0 and m > 1. Then there exist conm+1 stants δ, D > 0 such that Ψg (x) ≥ De−δ|x| . Proof. The proof is a minor modification of Theorem 4.182 in Volume 1. We fix a sufficiently large compact set K ⊂ ℝd . Since supx∈ℝd ‖Ψg (x)‖L2 (Q) < ∞ it suffices to show for large enough |x|. We also have ∞

‖Ψg (x)‖2 = ∑ ‖Ψg (x)(n) ‖2 ≥ ‖Ψg (x)(0) ‖2 = (1, Ψg (x))2 . n=0

2.10 Martingale properties and applications | 211

Thus it is sufficient to estimate (1, Ψg (x)) from below. We set t = |x|−(m−1) , aj = 1 + |xj |, αj = 1/2, and bj = 1 for j = 1, . . . , d. Hence the assumptions in Lemma 2.92 are satisfied. We also see that tWa (x) ≤ γ24m |x|m+1 . Inserting this into (2.9.5), we have 4m

(1, Ψg (x)) ≥ εK e−γ2 4m

≥ εK e−(γ2

̂ |x|m+1 E|x|−(m−1) −|x|−(m−1)/2 ‖φ/ω‖M g /εK

e

m+1

+2)|x|

e

eE|x|

−(m−1)

d

∏( j=1

e−|x|

−(m−1)/2

̂ ‖φ/ω‖M g /εK

|x|(m−1)/2 − 21 (1+|xj |)2 |x|m−1 )e 4√2π

4m m+1 ε ≥ K e−(γ2 +2)|x| . 2

Here a large enough |x| is taken so that both inequalities F((1 + |xj |)2 |x|m−1 ) > e

E|x|

−(m−1)

e

−|x|

−(m−1)/2

̂ ‖φ/ω‖M g /εK

>

1 2

hold. From this we obtain (1, Ψg (x)) ≥ De

−δ|x|

m+1

.

1 2

and

Remark 2.94. (1) In Theorem 2.93 we do not need to assume ∫ℝd |φ(k)|2 /ω(k)3 dk < ∞. (2) Estimate (2.9.2) can be also shown for any strictly positive eigenfunction Φ of HN , i. e., infy∈K (1, Φ(y))L2 (Q) ≥ cK with a strictly positive constant cK . Hence under the m+1

same assumptions in Theorem 2.93 it can be shown that ‖Φ(x)‖L2 (Q) ≥ De−δ|x|

.

2.10 Martingale properties and applications 2.10.1 Martingale properties In Sections 4.6.3 and 4.8.2 in Volume 1, we have obtained a martingale related to the semigroup and process associated with the relativistic Schrödinger operators. In a similar way we can construct a martingale related to the semigroup and process associated with the Schrödinger operator Hp = − 21 Δ + V. For an eigenfunction φ of Hp we can define the random process t

ht (x) = etEp e− ∫0 V(Br +x)dr φ(Bt + x),

t ≥ 0.

As it was shown there, 𝔼[ht (x)] = f (x) for all t ≥ 0 and the random process (ht (x))t≥0 is a martingale with respect to the filtration (FtBM )t≥0 . This construction can be extended to the Nelson Hamiltonian, which we present now. In Section 2.10.1 we work under Assumption 2.19, so that the potential V is Katodecomposable. Consider an eigenfunction and eigenvalue satisfying HN Φ = EΦ, and define t

t

Ht (x) = etE e− ∫0 V(Br +x)dr e−ϕE (∫0 δr ⊗φ(⋅−x−Br )dr) It Φ(Bt + x).

(2.10.1)

212 | 2 The Nelson model by path measures Here (Ht (x))t≥0 is a random process on (X × QE , ℬ(X) × ΣE , 𝒲 × μE ) parametrized by x ∈ ℝd . By the functional integral representation we have ∗ ̄ (Ψ, Φ) = (Ψ, e−t(HN −E) Φ) = ∫ 𝔼μE [Ψ(x)𝔼 𝒲 [I0 Ht (x)]]dx, ℝd

implying Φ(x) = (e−t(HN −E) Φ)(x) = 𝔼𝒲 [I∗0 Ht (x)]. Define the filtration (Mt )t≥0 by BM

Mt = Ft

× Σ(−∞,t] ,

t ≥ 0.

Theorem 2.95 (Martingale property). The random process (Ht (x))t≥0 is a martingale with respect to (Mt )t≥0 . Proof. We have s

s

𝔼μE 𝔼𝒲 [Ht (x)|Ms ] = etE e− ∫0 V(Br +x)dr e−ϕE (∫0 δr ⊗φ(⋅−x−Br )dr) t

t

× 𝔼μE 𝔼𝒲 [e− ∫s V(Br +x)dr e−ϕE (∫s δr ⊗φ(⋅−x−Br )dr) It Φ(Bt + x)|Ms ] . We compute the conditional expectation at the right-hand side. Using the Markov property of (Bt )t≥0 we obtain t

t

𝔼μE 𝔼𝒲 [e− ∫s V(Br +x)dr e−ϕE (∫s δr ⊗φ(⋅−x−Br )dr) It Φ(Bt + x)|Ms ] t−s

B

= 𝔼μE [𝔼𝒲s [e− ∫0

t

V(Br +x)dr −ϕE (∫s δr ⊗φ(⋅−x−Br−s )dr)

e

It Φ(Bt−s + x)|Σ(−∞,s] ]] .

From the Markov property of projection Es = Is I∗s it furthermore follows that t−s

B

= 𝔼μE [𝔼𝒲s [e− ∫0 B

= Es 𝔼𝒲s [e

t−s − ∫0

t

V(Br +x)dr −ϕE (∫s δr ⊗φ(⋅−x−Br−s )dr)

e

V(Br +x)dr

e

t −ϕE (∫s

δr ⊗φ(⋅−x−Br−s )dr)

It Φ(Bt−s + x)|Σs ]]

It Φ(Bt−s + x)] .

Denote Us = ΓE (us ), where us : E → E denotes the time-shift operator defined by us f (t, x) = f (t + s, x). Since Es = Is I∗0 U−s , we have t−s

B

= Is I∗0 U−s 𝔼𝒲s [e− ∫0 B

t−s − ∫0

B

t−s − ∫0

= Is I∗0 𝔼𝒲s [e = Is I∗0 𝔼𝒲s [e

t

V(Br +x)dr −ϕE (∫s δr ⊗φ(⋅−x−Br−s )dr)

V(Br +x)dr V(Br +x)dr

e

e e

t −ϕE (∫s

δr−s ⊗φ(⋅−x−Br−s )dr)

t−s −ϕE (∫0

δr ⊗φ(⋅−x−Br )dr)

It Φ(Bt−s + x)]

It−s Φ(Bt−s + x)]

It−s Φ(Bt−s + x)] .

We conclude that s

s

𝔼μE 𝔼𝒲 [Ht (x)|Ms ] = esE e− ∫0 V(Br +x)dr e−ϕE (∫0 δr ⊗φ(⋅−x−Br )dr) e−(t−s)(HN −E) Φ(Bs + x) = Hs (x). Hence (Ht (x))t≥0 is a martingale.

2.10 Martingale properties and applications | 213

2.10.2 Exponential decay of bound states Let τ be a stopping time with respect to Mt . Then Ht∧τ (x) is also a martingale, in particular, 𝔼μE 𝔼𝒲 [Ht (x)] = 𝔼μE 𝔼𝒲 [Ht∧τ (x)]. Lemma 2.96. Let HN Φ = EΦ. Then ‖Φ(⋅)‖L2 (Q) ∈ L∞ (ℝd ). Proof. It follows that Φ(x) = 𝔼μE 𝔼𝒲 [I∗0 Ht (x)] for every t > 0. Since t

‖I∗0 e−ϕE (∫0 δs ⊗φ(⋅−Bs )ds) It ‖ ≤ 2eE(t) ,

(2.10.2)

2 2 2 ̂ ̂ ̂ where E(t) = 2t ‖φ/√ω‖ + 2t(1 ∨ t)(‖φ/√ω‖ + ‖φ/ω‖ ), we obtain t

‖Φ(x)‖ ≤ 2etE eE(t) (𝔼x𝒲 [e−2 ∫0 V(Br )dr ])

1/2

1/2

(𝔼x𝒲 [‖Φ(Bt )‖2 ])

.

t

Since supx∈ℝd 𝔼x𝒲 [e−2 ∫0 V(Br )dr ] < ∞ and 𝔼x𝒲 [‖Φ(Bt )‖2 ] ≤ C‖Φ‖, the lemma follows. 2 ̂ Remark 2.97. We note that if ∫ℝd |φ(k)| /ω(k)3 dk < ∞, then (2.10.2) can be replaced with t

‖I∗0 e−ϕE (∫0 δs ⊗φ(⋅−Bs )ds) It ‖ ≤ 2etE(φ) , ̂

(2.10.3)

where E(φ)̂ is given by (2.3.15). 2 ̂ Theorem 2.98 (Upper bound of exponential decay). Assume ∫ℝd |φ(k)| /ω(k)3 dk < ∞. Let HN Φ = EΦ. If one of the conditions (1) lim|x|→∞ V(x) = ∞; (2) lim|x|→∞ V− (x) + E + E(φ)̂ < 0,

holds, then there exist constants c, C > 0 such that ‖Φ(x)‖L2 (Q) ≤ Ce−c|x| . Proof. Suppose (1). Let τR = inf{t | |Bt | > R}, which is a stopping time with respect to the natural filtration of Brownian motion (Bt )t≥0 . Consider WR (x) = inf {V(y) | |x − y| < R}. Note that WR (x) ≤ V(x + y) for |y| < R, and WR (x) − E − E(φ)̂ → ∞

as |x| → ∞.

Let Ψ ∈ L2 (Q). The process (I0 Ψ ⋅ Ht (x))t≥0 is also a martingale since 𝔼μE 𝔼𝒲 [I0 Ψ ⋅ Ht (x)|Ms ] = I0 Ψ ⋅ 𝔼μE 𝔼𝒲 [Ht (x)|Ms ] = I0 Ψ ⋅ Hs (x),

0 < s < t.

214 | 2 The Nelson model by path measures Hence it follows that (I0 Ψ, Φ(x))L2 (Q) = 𝔼μE 𝔼𝒲 [I0 Ψ ⋅ H0 (x)] = 𝔼μE 𝔼𝒲 [I0 Ψ ⋅ Ht∧τR (x)]. Using estimate (2.10.3) we get sup

‖Φ(x)‖ =

|(I0 Ψ, Φ(x))| =

Ψ∈L2 (Q),‖Ψ‖=1

sup

Ψ∈L2 (Q),‖Ψ‖=1 t∧τR

≤ ‖𝔼𝒲 [I∗0 Ht∧τR (x)]‖ ≤ 2𝔼𝒲 [e− ∫0

󵄨󵄨 󵄨 󵄨󵄨𝔼μE 𝔼𝒲 [I0 Ψ ⋅ Ht∧τR (x)]󵄨󵄨󵄨

̂ (V(Br +x)−E−E(φ))dr

] sup ‖Ψg (x)‖. x∈ℝd

We estimate the above expectation as t∧τR

𝔼𝒲 [e− ∫0

̂ (V(Br +x)−E−E(φ))dr t∧τR

= 𝔼𝒲 [1{τR 0, and that s-limΛ→∞ φ̂ κ,Λ β exists in ℋ̂ M . The conjugation map with respect to the unitary operator eiGΛ is called Gross transform. A direct calculation shows that the field operators transform as f ̂(k) φ̂ κ,Λ (k) g β(k)e−ik⋅x dk, e−iGΛ aM (f ̂)eiGΛ = aM (f ̂) − ∫ √2 √ω(k) √ω(k) 3 ℝ

e−iGΛ a∗M (f ̂)eiGΛ

f ̂(k) φ̂ κ,Λ (k) g β(k)e+ik⋅x dk. = a∗M (f ̂) − ∫ √2 √ω(k) √ω(k) 3 ℝ

Hence it follows that 󵄩󵄩 φ̂ κ,Λ 󵄩󵄩2 √β󵄩󵄩󵄩 . e−iGΛ HI eiGΛ = HI − g 󵄩󵄩󵄩 󵄩 √ω 󵄩 For the momentum operator we obtain e−iGΛ peiGΛ = p − gAΛ − gA∗Λ , where AΛ = −

φ̂ κ,Λ (k) φ̂ κ,Λ (k) 1 1 dk and A∗Λ = − dk, ∫ a(k)kβ(k)eik⋅x ∫ a∗ (k)kβ(k)e−ik⋅x √2 √2 √ω(k) √ω(k) 3 3 ℝ

ℝ

and e−iGΛ p2 eiGΛ 2

= p2 − g(2pAΛ + 2A∗Λ p) + g 2 (A2Λ + 2A∗Λ AΛ + A∗Λ ) − g[p, A∗Λ ] − g[AΛ , p] + g 2 [AΛ , A∗Λ ]. A calculation gives [p, A∗Λ ] = CΛ∗ =

φ̂ κ,Λ (k) 1 dk, ∫ a∗ (k)e−ik⋅x |k|2 β(k) √2 √ω(k) 3 ℝ

[p, AΛ ] = CΛ =

φ̂ κ,Λ (k) 1 dk, ∫ a(k)e−ik⋅x |k|2 β(k) √2 √ω(k) 3 ℝ

φ̂ κ,Λ 󵄩󵄩2 1 󵄩󵄩 󵄩󵄩 . [AΛ , A∗Λ ] = 󵄩󵄩󵄩|k|β 󵄩 √ω 󵄩󵄩 2

2.11 Ultraviolet divergence

| 217

Furthermore, we have e−iGΛ Hf eiGΛ = Hf + gBΛ + gB∗Λ +

g2 ‖βφ̂ κ,Λ ‖2 , 2

where BΛ = −

φ̂ κ,Λ (k) 1 dk, ∫ a(k)ω(k)β(k)eik⋅x √2 √ω(k) 3 ℝ

φ̂ κ,Λ (k) 1 B∗Λ = − dk. ∫ a∗ (k)ω(k)β(k)e−ik⋅x √2 √ω(k) 3 ℝ

Hence, with H0 = Hp ⊗ 1 + 1 ⊗ Hf , e−iGΛ HN eiGΛ = H0 − g(pAΛ + A∗Λ p) +

g2 2 2 (A + A∗Λ + 2AΛ A∗Λ ) + gHIκ + EΛ , 2 Λ

(2.11.3)

where HIκ = HI + BΛ + B∗Λ − 21 CΛ − 21 CΛ∗ . We have the expression HIκ =

1 (a∗ ({1 e−ik⋅x ) + aM ({1|k| 0 is regarded as the ultraviolet cutoff parameter. The Nelson Hamiltonian is also defined in L2 (ℝ3N ) ⊗ L2 (Q ) by Ĥ Nε = Hp ⊗ 1 + 1 ⊗ Ĥ f + g Ĥ Iε , where N

⊕

Ĥ Iε = ∑ ∫ ϕ(φ(⋅ − xj ))dxj , j=1

ℝ3

and ϕ(f ) can be extended by ϕ(f ) = ϕ(ℜf )+iϕ(ℑf ) for f ∈ ℋM . We also write HNε for Ĥ Nε . The main purpose is to consider the limit ε ↓ 0 of HNε . We show that this limit can be sensibly defined by an energy renormalization. Define 2

Eε = −

g2 e−ε|k| N∫ β(k)dk. 2 ω(k) ℝ3

(2.11.10)

220 | 2 The Nelson model by path measures Notice that Eε → −∞ as ε ↓ 0. In the following we will fix a time interval [−T, T] and track the dependence on T of estimates. ε A functional integral representation of (F, e−2THN G) is given in terms of Brownian motion and Euclidean field by Theorem 2.12. In particular, for F = f ⊗ 1 and G = h ⊗ 1, it can be described in terms of two-sided 3N-dimensional Brownian motion (Bt )t∈ℝ , Bt = (B1t , . . . , BNt ), as T

ε

(f ⊗ 1, e−2THN h ⊗ 1) = ∫ 𝔼x [f (B−T )h(BT )e− ∫−T V(Bs )ds e

g2 2

SεT

] dx,

ℝ3N

where N

T

T

j

SεT = ∑ ∫ ds ∫ Wε (Bis − Bt , s − t)dt i,j=1 −T

and Wε (x, t) = ∫ ℝ3

(2.11.11)

−T 2 1 e−ε|k| e−ik⋅x e−|t|ω(k) dk. 2ω(k)

Definition 2.101 (Regularized pair potential and pair interaction). We call regularized pair potential, and g 2 SεT in (2.11.11) regularized pair interaction.

Wε (x, t)

Consider the function 2

e−ε|k| e−ik⋅x−|t|ω(k) φε (x, t) = ∫ β(k)dk, 2ω(k)

ε ≥ 0,

(2.11.12)

ℝ3

where β(k) is given by (2.11.2). Lemma 2.102. There exists a functional S0ren such that T

lim 𝔼x [e− ∫−T V(Bs )ds e ε↓0

g2 2

(SεT −4NTφε (0,0))

T

] = 𝔼x [e− ∫−T V(Bs )ds e

g2 2

S0ren

].

(2.11.13)

Notice that Wε (x, t) is smooth and Wε (x, t) → W0 (x, t) as ε ↓ 0 for every (x, t) ≠ (0, 0), where W0 (x, t) = ∫ ℝ3

1 e−ik⋅x e−|t|ω(k) dk. 2ω(k)

It is seen, however, that Wε (0, 0) → ∞ as ε ↓ 0, i. e., W0 (x, t) has a power-like singularity at (0, 0), thus (2.11.13) is nontrivial to obtain. We will prove Lemma 2.102 through a sequence of lemmas below. As ε ↓ 0, only the diagonal part of the interaction has a singular term. Fix 0 < τ ≤ T and denote T, { { ⟨t⟩ = {t, { {−T,

t ≥ T, |t| < T, t ≤ −T.

2.11 Ultraviolet divergence

| 221

We decompose the regularized interaction into diagonal and off-diagonal parts as SεT = SεD,T + SεOD,T , where SεD,T

N

T

⟨s+τ⟩

j

= 2 ∑ ∫ ds ∫ Wε (Bis − Bt , s − t)dt i,j=1 −T

s

and SεOD,T

N

T

T

j

= 2 ∑ ∫ ds ∫ Wε (Bis − Bt , s − t)dt. i,j=1 −T

⟨s+τ⟩

Here the quantity SεD,T is the integral over a neighborhood of the diagonal region {(t, t) ∈ ℝ2 | |t| ≤ T}, and SεOD on its complement. Notice that for τ = T we have SεOD,T = 0. It is easy to show that the pathwise limit N

T

⟨s+τ⟩

j

lim SεOD,T = S0OD,T = 2 ∑ ∫ ds ∫ W0 (Bis − Bt , s − t)dt ε↓0

i,j=1 −T

s

exists. A representation in terms of a stochastic integral will help us deal with the more difficult term SεD,T . We give an upper bound of φε (x, t) in the lemma below. Lemma 2.103. There exists a constant c > 0 such that the following bounds on φε hold uniformly in ε, x, t: |φε (x, t)| ≤ c(1 + | log(|t| ∧ 1)|), |∇x φε (x, t)| ≤ c|t|−1 ,

t ≠ 0,

|∇x φε (x, t)| ≤ c|x| ,

|x| ≠ 0,

−1

where ∇φε = (𝜕1 φε , 𝜕2 φε , 𝜕3 φε ) denotes the gradient of φε . Furthermore, similar bounds hold for the function φ0 − φε with a constant c󸀠 (ε) > 0 such that limε→0 c󸀠 (ε) = 0: |φε (x, t) − φ0 (x, t)| ≤ c󸀠 (ε)(1 + | log(|t| ∧ 1)|), |∇x φε (x, t) − ∇x φ0 (x, t)| ≤ c󸀠 (ε)|t|−1 ,

t ≠ 0,

|∇x φε (x, t) − ∇x φ0 (x, t)| ≤ c󸀠 (ε)|x|−1 ,

|x| ≠ 0.

Proof. After integration over the angular variables, we obtain ∞

φε (x, t) = 2π ∫ 0

2

e−εr −r|t| sin(r|x|) dr. 2+r r|x|

(2.11.14)

222 | 2 The Nelson model by path measures Hence ∞

∞

e−r|t| e−r dr = 2πC ∫ dr, 2+r 2|t| + r

|φε (x, t)| ≤ 2πC ∫ 0

0

where we used that sin a/a ≤ C for some C > 0. Thus 1

∞

0

1

1 e−r |φε (x, t)| ≤ 2πC ∫ dr + 2πC ∫ dr 2|t| + r r which gives the bound |φε (x, t)| ≤ C(1 + | log(|t| ∧ 1)|). The first bound on the gradient follows directly by |∇x φε (x, t)| ≤ ∫ ℝ3

2 r2 1 e−ε|k| e−|k||t| dk ≤ 2π ∫ e−rt dr ≤ c ∫ e−rt dr. 2 2(|k| + |k| /2) r + r 2 /2

∞

∞

0

0

Next consider the second bound on the gradient. Differentiation in (2.11.14) gives ∞

∇x φε (x, t) = 2π ∫ 0

2

e−εr −r|t| cos(r|x|)rx sin(r|x|)x ( ) dr − r(2 + r) |x|3 |x|2 2

∞

2

2πx e−εr /|x| −|t|r/|x| = 2 ∫ (r cos r − sin r) dr, r(2|x| + r) |x| 0

where we made the change of variable r 󳨃→ r|x|. The integral is uniformly bounded in ε and t; indeed 2

∞

2

󵄨󵄨 󵄨󵄨 e−εr /|x| −|t|r/|x| 󵄨󵄨 ∫ (r cos r − sin r) dr 󵄨󵄨󵄨 󵄨󵄨 󵄨 r(2|x| + r) 0

1

2

∞

2

∞

2

2

󵄨󵄨 󵄨󵄨 e−εr /|x| −|t|r/|x| 󵄨󵄨 󵄨󵄨 e−εr /|x| −|t|r/|x| |r cos r − sin r| 󵄨󵄨 + 󵄨󵄨 ∫ 󵄨󵄨 ∫ cos r dr sin rdr 󵄨󵄨󵄨 dr + ≤∫ 󵄨 󵄨 󵄨 2 󵄨 󵄨 󵄨 󵄨 (2|x| + r) r(2|x| + r) r 1

≤∫ 0

1

1

0

2

2

󵄨󵄨 󵄨󵄨 e−εr /|x| −|t|r/|x| Cr 3 1 cos rdr 󵄨󵄨󵄨 + ∫ 2 dr dr + 󵄨󵄨󵄨 ∫ 2 󵄨 󵄨 (2|x| + r) r r ∞

∞ 1

1

using |r cos r − sin r| ≤ Cr 3 for a constant C and all r ∈ [0, 1]. It remains to show that the integral ∞

2

2

e−εr /|x| −|t|r/|x| cos rdr J=∫ (2|x| + r) 1

2.11 Ultraviolet divergence

| 223

is bounded. We have 2

2

∞

2

e−ε/|x| −|t|/|x| d e−εr /|x| −|t|r/|x| J=− sin 1 − ∫ ( ) sin rdr. (2|x| + 1) dr (2|x| + r) 1

Thus 2

∞

2

∞

󵄨󵄨 d e−εr /|x| −|t|r/|x| 󵄨󵄨 dr |J| ≤ sin 1 + ∫ 󵄨󵄨󵄨 ( ) 󵄨󵄨󵄨dr ≤ sin 1 + ∫ 2 < ∞. 󵄨 dr 󵄨 (2|x| + r) r 1

1

In order to subtract the singular part from

SεD,T ,

we define

T

N

XεT = 2 ∑ ∫ φε (Bis − Bjs , 0)ds, i=j̸ −T

T

N

⟨s+τ⟩

j

j

YεT = 2 ∑ ∫ ds ∫ ∇φε (Bis − Bt , s − t)dBt , i,j=1 −T N

s

T

j

ZεT = −2 ∑ ∫ φε (Bis − B⟨s+τ⟩ , s − ⟨s + τ⟩)ds. i,j=1 −T

A heuristic idea to subtract the singular part is to apply the Itô formula to the random j variable φε (Bis − Bt , s − t). Fix s, i and j with i ≠ j. We formally have 1 d dφε = ∇φε dBit + ( Δ + )φε dt. 2 dt Since ( 21 Δ +

d )φε dt

j

= −Wε = −Wε (Bis − Bt , s − t), we see that j

dφε = ∇φε dBt − Wε dt, and hence it follows that T

T

s

s

j

j

∫ Wε dt = ∫ ∇φε dBt + φε (Bis − Bjs , 0) − φε (Bis − BT , s − T). By the symmetry we can also see that T

T

−T

s

j

j

∫ Wε dt = 2 ∫ ∇φε dBt − 2φε (Bis − BT , s − T) + 2φε (Bis − Bjs , 0) and finally T

T

∫ ds ∫ Wε dt = −T

−T

T

T

j 2 ∫ ds ∫ ∇φε dBt s −T

T

−2 ∫ −T

φε (Bis

−

j BT , s

T

− T)ds + 2 ∫ φε (Bis − Bjs , 0)ds. −T

224 | 2 The Nelson model by path measures Hence we have T

N

T

∑ ∫ ds ∫ Wε dt =

i,j=1 −T

−T

T

N

T

j 2 ∑ ∫ ds ∫ ∇φε dBt i,j=1 −T s N

N

T

j

− 2 ∑ ∫ φε (Bis − BT , s − T)ds i,j=1 −T

T

+ 2 ∑ ∫ φε (Bis − Bjs , 0)ds + +4Tφε (0, 0). i=j̸ −T

In the lemma below we rigorously derive the formula above. Lemma 2.104. We have SεD,T = 4TNφε (0, 0) + XεT + YεT + ZεT for all ε > 0. Proof. Note that φε (x, t) solves the equation 1 (𝜕t + Δx ) φε (x, t) = −Wε (x, t), 2 Fix i and j. We have φε (x, t) = ∫ℝ3

x ∈ ℝ3 , t ≥ 0.

2

e−ε|k| β(k)ρ(x, |t|)dk, 2ω(k)

(2.11.15)

where

ρ(x, t) = ρ(x, t, k) = e−ik⋅x e−tω(k) . Notice that 2

2

e−ε|k| e−ε|k| β(k)ρ(x, |t|, k)dk = ∫ β(k)ρ(−x, |t|, k)dk. 2ω(k) 2ω(k)

∫ ℝ3

ℝ3

Let t ≥ s. The Itô formula yields j ρ(Bt , t)

−

ρ(Bjs , s)

t

=

∫ ∇ρ(Bjr , r)dBjr s t

=

t

1 + ∫ (𝜕t + Δx ) ρ(Bjr , r)dr 2

∫ −ikρ(Bjr , r)dBjr s

s

t

+ ∫(−ω(k) − |k|2 /2)ρ(Bjr , r)dr. s

i

Multiplying by eik⋅Bs esω(k) on both sides above, we get j

ρ(Bt − Bis , t − s) − ρ(Bjs − Bis , 0) =e

ik⋅Bis

t

∫ −iρ(Bjr , r s

− s)k ⋅

dBjr

t

+ ∫(−ω(k) − |k|2 /2)ρ(Bjr − Bis , r − s)dr s

(2.11.16)

2.11 Ultraviolet divergence t

=

∫ −iρ(Bjr s

Bis , r

−

− s)k ⋅

dBjr

| 225

t

+ ∫(−ω(k) − |k|2 /2)ρ(Bjr − Bis , r − s)dr. s

t

i

Here we used that eik⋅Bs and ∫s −iρ(Bjr , r − s)k ⋅ dBjr are independent as t ≥ s. Multiplying by

2

e−ε|k| β(k) 2ω(k)

on both sides and integrating with respect to k gives 2

∫ ℝ3

e−ε|k| j β(k) (ρ(Bt − Bis , |t − s|) − ρ(Bjs − Bis , 0)) dk 2ω(k) t

2

= ∫ (∫ s

t

ℝ3

e−ε|k| β(k)(−ik)ρ(Bjr − Bis , |r − s|)dk) ⋅ dBjr 2ω(k) 2

+ ∫ (∫ s

ℝ3

t

=

e−ε|k| β(k)(−ω(k) − |k|2 /2)ρ(Bjr − Bis , |r − s|)dk) dr 2ω(k)

∫ ∇φε (Bjr s

−

Bis , |r

−

s|)dBjr

t

1 + ∫ (𝜕t + Δx ) φε (Bjr − Bis , r − s)dr. 2 s

Hence by using reflection symmetry (2.11.16) we obtain j

φε (Bis − Bt , s − t) − φε (Bis − Bjs , 0) t

t

s

s

1 = ∫ ∇φε (Bis − Bjr , s − r) ⋅ dBjr + ∫ (𝜕t + Δ) φε (Bis − Bjr , s − r)dr. 2 Denote t = ⟨s + τ⟩. We have j

φε (Bis − B⟨s+τ⟩ , s − ⟨s + τ⟩) − φε (Bis − Bjs , 0) ⟨s+τ⟩

⟨s+τ⟩

s

s

1 j j j = ∫ ∇φε (Bis − Bt , s − t)dBt + ∫ (𝜕t + Δ) φε (Bis − Bt , s − t)dt. 2 By (2.11.15), for each i, j, ⟨s+τ⟩

j

∫ Wε (Bis − Bt , s − t)dt s

j

⟨s+τ⟩

j

j

= φε (Bis − Bjs , 0) − φε (Bis − B⟨s+τ⟩ , s − ⟨s + τ⟩) + ∫ ∇φε (Bis − Bt , s − t)dBt s

follows. Inserting this expression into SεD,T , proves the claim.

226 | 2 The Nelson model by path measures Lemma 2.104 suggests the definition Sεren = SεT − 4NTφε (0, 0), as a renormalized action, and this can be expressed as Sεren = SεOD,T + XεT + YεT + ZεT .

(2.11.17)

We will estimate each term of (2.11.17). The estimates of SεOD,T and ZεT are straightforward. Lemma 2.105. Let ε ≥ 0. There exist constants cZ and cS such that |SεOD,T | ≤ cS (T + 1), |ZεT | ≤ cZ T uniformly in the paths and in ε ≥ 0. Proof. We see that T−τ

∞

−T

0

|ZεT | ≤ 4πN 2 ( ∫ ds ∫

T

∞

T−τ

0

e−rτ e−r(T−s) dr + ∫ ds ∫ dr) ≤ cz T 1 + r/2 1 + r/2

with some cz > 0. It can be also seen that |SεOD,T |

T−τ

T

≤ ∫ ds ∫ dt ∫ −T

τ+s

ℝ3

1 e−(t−s)ω(k) dk. 2ω(k)

Since ∫ ℝ3

1 c e−(t−s)ω(k) dk ≤ 2ω(k) (t − s)2

with a constant c. It follows that |SεOD,T |

T−τ

T

≤ ∫ ds ∫ dt −T

τ+s

τ c 2T − 1) + log ( )) = c (( τ 2T (t − s)2

and the bound |SεOD,T | ≤ cs (T + 1) follows. Next we estimate XεT . Lemma 2.106. Let ε ≥ 0. There exists a constant cX independent of ε, such that for all α > 0 and T > 0, T

2

sup 𝔼x [eα|Xε | ] ≤ ecX (α +α)(T+

x∈ℝ3N

√T)

.

2.11 Ultraviolet divergence

| 227

Proof. Notice that XεT

N

T

=∑∫

∞

4π

j

i=j̸ −T

|Bis − Bs |

N

T

1

i=j̸

−T

0

We have

XεT ≤ ∑ 4π ∫ ( ∫

ds ∫ 0

sin(r|Bis − Bjs |) −εr2 e dr, r + r 2 /2

ε ≥ 0.

∞

1 1 dr + ∫ dr)ds. 2 /2)|Bi − Bj | 1 + r/2 (r + r s s 1

Hence by ∞

a = 2π ∫ 1

1 dr, r + r 2 /2

we have |XεT |

b = 8N(N − 1)π log(3/2),

T

N

≤ a∑ ∫

i i=j̸ −T |Bs

1

j

− Bs |

ds + bT.

The lemma follows from Corollary 2.110 below. Finally, we estimate the crucial YεT term. By the stochastic Fubini theorem we can interchange the stochastic and Lebesgue integrals when ε > 0. Thus YεT has the representation T

N

j

j

YεT = ∑ ∫ Φε,t dBt , j=1 −T

ε > 0,

where Φε,t = (Φ1ε,t , . . . , ΦNε,t ) is a process with values in ℝ3N given by N

j

t

j

Φε,t = 2 ∑ ∫ ∇φε (Bis − Bt , s − t)ds. i=1 ⟨t−τ⟩

For ε = 0, define

T

N

j

j

Y0T = ∑ ∫ Φ0,t dBt . j=1 −T

Lemma 2.107. Let ε ≥ 0 and independent of ε, such that

1 2

< θ < 1. Then there exists a constant cY = cY (θ), T

2

sup 𝔼x [eαYε ] ≤ ecY (α +α

x∈ℝ3N

2 1−θ

)(T+1)

for all α > 0,

lim 𝔼x [|YεT − Y0T |2 ] = 0 for all x ∈ ℝ3N . ε↓0

228 | 2 The Nelson model by path measures Proof. We have T

T

N

2

N

2

t

|∇φε (Bis

∫ |Φε,t | dt ≤ 4 ∑ ∫ [∑ ∫ i=1 −T [ j=1 ⟨t−τ⟩ −T t

T

N

≤ 4c2 N ∑ ∫ [ ∫ |Bis − i,j=1 −T [⟨t−τ⟩

−

j Bt , s

j Bt |−θ |s

− t)|ds] dt ]

2

− t|−(1−θ) ds] dt, ]

where we used Jensen’s inequality, Lemma 2.103, and an interpolation to obtain the bound |∇φε (x, t)| ≤ c|x|−θ |t|−(1−θ) ,

θ ∈ [0, 1],

which is uniform in ε ∈ [0, 1]. With a suitable 21 < θ < 1, the Schwarz inequality applied to the latter integral gives T

N

T

t

t

j ∫ |Φε,t | dt ≤ 4c N ∑ ∫ [ ∫ |Bis − Bt |−2θ ds] ( ∫ |s − t|−2(1−θ) ds) dt i,j=1 −T −T [⟨t−τ⟩ ] ⟨t−τ⟩ 2

2

2 2θ−1

≤ 4c τ

t

T

N

j N ∑ ∫ [ ∫ |Bis − Bt |−2θ ds] dt i,j=1 −T [⟨t−τ⟩ ]

≤ 4c2 τ2θ−1 NQ,

where c is the constant in Lemma 2.103, which is independent of ε, and T

N

⟨s+τ⟩

j

Q = ∑ ∫ ds ∫ |Bis − Bt |−2θ dt. i,j=1 −T

s

By the Girsanov theorem, T

2

T

1

2

(𝔼x [eαYε ]) ≤ 𝔼x [e2α ∫−T Φε,t dBt − 2 (2α) = 𝔼x [e

T 2α2 ∫−T

|Φε,t |2 dt

T

∫−T |Φε,t |2 dt

] 𝔼x [e2α

] ≤ 𝔼x [eγQ ] ,

where γ = 8c2 Nα2 τ2θ−1 , and we recall that we have chosen tion 2.108 and Corollary 2.109 below, we obtain that sup 𝔼x [eγQ ] ≤ e(γ+γ

1 1−θ

)(a+bT)

x∈ℝ3N

with nonnegative constants a and b, and then it follows that T

2

sup sup 𝔼x [e2αYε ] ≤ e(α +α

ε∈(0,1] x∈ℝ3N

2 1−θ

)(c1 +c2 T)

2

T

∫−T |Φε,t |2 dt

] (2.11.18)

1 2

< θ < 1. By Proposi(2.11.19)

2.11 Ultraviolet divergence

| 229

with nonnegative constants c1 and c2 , for all α ∈ ℝ. For all ε > 0, we see that T

2

∫ |Φε,t − Φ0,t | dt ≤

4c2ε N

−T

N

T

t

∑ ∫[ ∫

i,j=1 −T

|Bis

⟨t−τ⟩ T

N

−

t

j Bt |−2θ ds](

∫ |s − t|−2(1−θ) ds)dt ⟨t−τ⟩

t

j

≤ 4c2ε τ2θ−1 N ∑ ∫ [ ∫ |Bis − Bt |−2θ ds]dt i,j=1 −T

≤

4c2ε τ2θ−1 NQ,

⟨t−τ⟩

where, as above, we used Lemma 2.103 and an interpolation to obtain the bound |∇x φε (x, t) − ∇x φ0 (x, t)| ≤ cε |x|−θ |t|−(1−θ) ,

θ ∈ [0, 1]

for all ε > 0, and where the constant cε is such that cε → 0 as ε ↓ 0. The process Φε converges to Φ0 in L2 ([−T, T], ℝ3N ) almost surely for all x ∈ ℝ3N . The almost sure convergence is then a consequence of the fact that Q < +∞ almost surely under 𝒲 x for all x ∈ ℝ3N , which we have already shown. The convergence of Φε implies also the convergence of YεT to Y0T , at least in L2 (X, 𝒲 x ), We show a general proposition concerning an estimate of pair potentials. j

Proposition 2.108. Suppose that (Bt )t≥0 , j = 1, . . . , N, are d-dimensional Brownian moj tions, and (Bit )t≥0 and (Bt )t≥0 are independent for i ≠ j. Let f be a nonnegative function, write f∗ (t) = ess supt≤s≤T f (s) for t ≤ T, and choose 1 ≤ α < 2. Then the following hold. (1) For each j there exist nonnegative constants aα and bα such that s

T

𝔼[ exp ( ∫ ds ∫ 0

0

T

f (s − t) j

j

|Bs − Bt |α

dt)]

s

≤ exp (aα ∫ ds( ∫ f∗ (t)dt) 0

T

2 2−α

s

+ bα ∫ ds ∫ f∗ (t)t −α/2 dt).

0

0

0

(2) Let i ≠ j. Then there exist nonnegative constants a󸀠α and b󸀠α such that T

s

sup 𝔼[ exp ( ∫ ds ∫

(x,y)∈ℝ2d

0

T

|Bis 0

≤ exp (a󸀠α T( ∫ f (t)dt)

f (s − t) j

+ x − Bt − y|α T

2 2−α

+ b󸀠α T 1−α/2 ∫ f (t)dt).

0

0

(3) For each j, there exist nonnegative constants x

T

sup 𝔼 [ exp ( ∫

x∈ℝd

0

f (s) j

|Bs |α

ds)] ≤

dt)]

a󸀠󸀠 α

and

b󸀠󸀠 α

T 2 󸀠󸀠 exp (aα ∫ f∗ (t) 2−α dt 0

such that +

T 󸀠󸀠 bα ∫ f∗ (t)t −α/2 dt). 0

230 | 2 The Nelson model by path measures We

can x

supx∈ℝ3N 𝔼 [e

apply

α|XεT |

Proposition

2.108

to

estimate

supx∈ℝ3N 𝔼x [eγQ ]

and

].

Corollary 2.109. Let 1 ≤ α < 2 and γ ≥ 0. Then there exist nonnegative constants a and b such that T

N

s

sup 𝔼x [ exp (γ ∑ ∫ ds ∫

x∈ℝdN

|Bis s−τ

i,j=1 0

1 −

2

j Bt |α

dt)] ≤ exp ((γ 2−α + γ)(a + bT)) .

(2.11.20)

In particular, estimate (2.11.19) follows with nonnegative constants a and b dependent on τ, and limτ→0 a = limτ→0 b = 0. Proof. We use notation as in Proposition 2.108. Let f (t) = 1[0,τ] (t). We have T

s

∫ ds ∫

f (s − t)

|Bis 0

0

−

j Bt |α

T

s

dt = ∫ ds ∫

|Bis s−τ

0

1

j

− Bt |α

dt.

Since f∗ (t) = f (t), we see that T

s

2/(2−α)

∫ ds( ∫ f∗ (t)dt) 0

0

τ

2/(2−α)ds

= ∫s

T

+ ∫ τ2/(2−α) ds τ

0

2 − α (4−α)/(2−α) =( τ − τ(4−α)/(2−α) ) + τ2/(2−α) T = a1 (τ) + a2 (τ)T 4−α

and T

∫ ds ∫ f∗ (t)t 0

T

τ

s −α/2

0

1 1 s1−α/2 ds + ∫ τ1−α/2 ds dt = ∫ 1 − α/2 1 − α/2

=(

τ

0

2−α/2

2−α/2

1−α/2

τ τ τ − )+ T = b1 (τ) + b2 (τ)T. (1 − α/2)(2 − α/2) 1 − α/2 1 − α/2

By Proposition 2.108 for each j we have T

s

𝔼[ exp γ( ∫ ds ∫ dt 0

s−τ

j |Bs

1 −

2

j Bt |α

)] ≤ exp (aα γ 2−α (a1 (τ) + a2 (τ)T) + bα γ(b1 (τ) + b2 (τ)T)) 2

≤ exp ((γ 2−α + γ)(d1 + d2 T)) ,

(2.11.21)

with nonnegative constants d1 and d2 independent of γ. We can also estimate the case when i ≠ j. Since 2 T 2−α 󸀠 aα T( ∫ f (t)dt)

0

+

T 󸀠 1−α/2 bα T ∫ f (t)dt 0

= a󸀠α Tτ2/(2−α) + b󸀠α T 1−α/2 τ,

2.11 Ultraviolet divergence

| 231

it follows by Proposition 2.108 again that T

s

dt

sup 𝔼[ exp γ( ∫ ds ∫

(x,y)∈ℝ2d

j

|Bis + x − Bt − y|α

s−τ

0

)] 2

≤ exp (a󸀠α γ 2/(2−α) Tτ2/(2−α) + b󸀠α γT 1−α/2 τ) ≤ exp ((γ 2−α + γ)(d1󸀠 + d2󸀠 T))

(2.11.22)

with nonnegative constants d1󸀠 and d2󸀠 independent of γ. Here we used that T 1−α/2 ≤ s

T

T +1. Let Xi,j = ∫0 ds ∫s−τ

dt |Bis −Bjt |α

and write Y1 = X1,1 , . . . , YN = X1,N , YN+1 = X2,1 , . . . , Y2N =

X2,N , . . . , YN 2 = XN,N for convenience. Applying the Schwarz inequality, we have N

x

T

s

sup 𝔼 [ exp (γ ∑ ∫ ds ∫

x∈ℝdN

|Bis s−τ

i,j=1 0

dt

j

− Bt |α

)]

N2

N2

k=1

x∈ℝdN k=1

1/2k

= sup 𝔼x [exp (γ ∑ Yk )] ≤ sup ∏ (𝔼x [exp(2k γYk )]) x∈ℝdN

.

(2.11.23)

By (2.11.21)–(2.11.22), each 𝔼x [exp(2k γYk )] can be estimated and we have the bound (2.11.20). By the definitions of d1 , d1󸀠 , d2 and d2󸀠 , it can be seen that limτ→0 d1 = limτ→0 d1󸀠 = limτ→0 d2 = limτ→0 d2󸀠 = 0, and then limτ→0 a = limτ→0 b = 0 follows. Finally, we have T

N

𝔼x [eγQ ] = 𝔼x [eγ ∑i,j=1 ∫−T ds ∫s 2T

N

= 𝔼x [e∑i,j=1 ∫0

⟨s+τ⟩

ds ∫s

|Bis −Bjt |−2θ dt

(s+τ)∧2T

]

|Bis −Bjt |−2θ dt

N

2T

] ≤ 𝔼x [e∑i,j=1 ∫0

s+τ

ds ∫s

|Bis −Bjt |−2θ dt

]

and the right-hand side above can be estimated as in (2.11.20) with T and α replaced by 2T and 2θ, respectively. Corollary 2.110. There exist nonnegative constants a and b such that N T

sup 𝔼x [ exp (γ ∑ ∫

i i=j̸ 0 |Bs

x∈ℝdN

ds −

j Bs |

)] ≤ exp (aγ 2 T + bγ √T) .

T

In particular, supx∈ℝ3N 𝔼x [eα|Xε | ] ≤ exp (cX (α2 + α)(T + √T)). j

ij

Proof. Let i ≠ j and fix i and j. Since (Bt )t≥0 and (Bit )t≥0 are independent, (Wt )t∈ℝ = j (Bit − Bt )t∈ℝ is a Brownian motion. By (3) of Proposition 2.108, we have x

T

𝔼 [ exp (γ ∫ 0

ds ij

|Ws |α

)] ≤ exp (aα γ 2/(2−α) T + bα γT 1−α/2 ) .

Putting α = 1, in a similar way to (2.11.23) we prove the corollary.

232 | 2 The Nelson model by path measures Now we can estimate Sεren for ε ≥ 0. Note that S0ren = S0OD,T + X0T + Y0T + Z0T . Lemma 2.111. Let 1/2 < θ < 1. Then there exists a constant cren independent of ε ≥ 0 such that for all α ∈ ℝ and f , h ∈ L2 (ℝ3N ) we have for all ε > 0 that T

ren

∫ 𝔼x [f (B−T )h(BT )e− ∫−T V(Bs )ds eαSε ]dx ≤ ‖f ‖‖h‖e4T‖V‖∞ e

2

cren (α 1−θ +α2 +α)(T+1)

.

ℝ3N

Proof. Recall the decomposition Sεren = SεOD,T + XεT + YεT + ZεT . By Schwarz inequality we have T

ren

∫ 𝔼x [|f (B−T )h(BT )|e− ∫−T V(Bs )ds eαSε ]dx ℝ3N T

OD,T

≤ ‖f ‖‖h‖ sup (𝔼x [e−2 ∫−T V(Bs )ds e2α(Sε

+XεT +YεT +ZεT )

x∈ℝ3N

1/2

])

.

By Lemmas 2.105, 2.106 and 2.107, and the fact that V is bounded, we see that there exists a constant cren such that T

OD,T

sup 𝔼x [e−2 ∫−T V(Bs )ds e2α(Sε

+XεT +YεT +ZεT )

x∈ℝ3N

] ≤ e4T‖V‖∞ e2cren (α

2 1−θ

+α2 +α)(T+1)

,

(2.11.24)

and the lemma follows. 2.11.3 Removal of ultraviolet cutoff on Fock vacuum In this section we show that HNε + g 2 Nφε (0, 0) converges to a self-adjoint operator HNren as ε ↓ 0. Lemma 2.112. If α ∈ ℝ, then for every x ∈ ℝ3N , T

T

lim 𝔼x [|eαUε − eαU0 |] = 0,

U = SOD , X, Y, Z.

ε↓0

T

(2.11.25)

Proof. Let U = X. We obtain that |XεT | ≤ bT + ∫−T VCoul (Bs )ds with b > 0, where N

VCoul (x) = C ∑ i=j̸

|x i

1 − xj |

with a constant C, and the fact that T

T

T

𝔼x [|eαXε − eαX0 |] ≤ 2 sup 𝔼x [eα(bT+∫−T VCoul (Bs )ds) ] < ∞. x∈ℝd

2.11 Ultraviolet divergence

| 233

Since XεT → X0T a. s. with respect to 𝒲 x for every x ∈ ℝ3N , the Lebesgue dominated convergence theorem implies (2.11.25). T T Let U = Y. It suffices to show that supx∈ℝ3N 𝔼x [|eα(Yε −Y0 ) − 1|] → 0. We have T

2

T

T

T

T

T

𝔼x [(eα(Yε −Y0 ) − 1) ] = 𝔼x [e2α(Yε −Y0 ) ] + 1 − 2𝔼x [eα(Yε −Y0 ) ] . T

T

We will show below that limε↓0 𝔼x [eα(Yε −Y0 ) ] = 1. Define the random process (δΦt )t∈ℝ = (Φε,t − Φ0,t )t∈ℝ so that YεT

−

Y0T

T

= ∫ δΦt dBt . −T

By the Girsanov theorem, 2T

𝔼x [eα ∫0

2

2T

δΦt dBt − α2 ∫0 |δΦt |2 dt

]=1

(2.11.26)

for every α ∈ ℝ, hence it follows that T

2

T

α2

T

(𝔼x [eα(Yε −Y0 ) ] − 1) ≤ 𝔼x [e2α ∫−T δΦt dBt ] 𝔼x [(1 − e− 2

2

T

∫−T |δΦt |2 dt

) ].

(2.11.27)

We see that by (2.11.26) again T

sup 𝔼x [e2α ∫−T δΦt dBt ] ≤ sup (𝔼x [e4α

2

1/2

T

∫−T |δΦt |2 dt

])

(2.11.28)

󵄨󵄨2 󵄨󵄨 α ) ] ≤ 𝔼x [󵄨󵄨󵄨 ∫ |δΦt |2 dt 󵄨󵄨󵄨 ] → 0 󵄨 󵄨2 −T ] [

(2.11.29)

x∈ℝ3N

x∈ℝ3N

and, furthermore, α2

𝔼x [(1 − e− 2

T

∫−T |δΦt |2 dt

2

2

T

as ε ↓ 0. Here (2.11.29) can be shown by Lemma 2.107. The right-hand side of (2.11.28) is uniformly bounded in ε, which can be proven in the same way as in the proof of Lemma 2.111. Hence (2.11.27) converges to zero as ε ↓ 0 and (2.11.25) for U = Y follows. T T Let U = Z. It suffices to show that supx∈ℝ3N 𝔼x [|eα(Zε −Z0 ) − 1|] → 0. We have N

T

i

i

j

j

ZεT − Z0T = 2 ∑ ∫ ds ∫ e−ik⋅(Bs +x −B⟨s+τ⟩ −x ) e−(⟨s+τ⟩−s)ω(k) i,j=1 −T

ℝ3

2

1 − e−ε|k| β(k)dk. ω(k)

Let ηε = α(ZεT − Z0T ). It can be directly seen that |ηε |n ≤ cn αn T n εn for a constant c. We have ∞ 1 1 x 𝔼 [|ηε |n ] ≤ ∑ cn T n εn → 0 n! n! n=1 n=1 ∞

|𝔼x [eηε ] − 1| ≤ ∑

as ε ↓ 0, uniformly in x ∈ ℝ3N . Thus (2.11.25) for U = Z follows. For U = SOD , we obtain (2.11.25) in a similar way.

234 | 2 The Nelson model by path measures Lemma 2.113. Let α ∈ ℝ and f , h ∈ L2 (ℝ3N ). Then T

T

ren

ren

lim ∫ 𝔼x [f (B−T )h(BT )e− ∫−T V(Bs )ds eαSε ]dx = ∫ 𝔼x [f (B−T )h(BT )e− ∫−T V(Bs )ds eαS0 ]dx. ε↓0

ℝ3N

ℝ3N ren

ren

Proof. Writing θε (x) = 𝔼x [(eαSε − eαS0 )2 ]dx, by telescoping we obtain T ren ren 󵄨 󵄨󵄨 󵄨󵄨 ∫ 𝔼x [f (B−T )h(BT )e− ∫−T V(Bs )ds (eαSε − eαS0 )] dx󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨

ℝ3N

1/2

≤ e2T‖V‖∞ ∫ |f (x)| (𝔼x [|h(BT )|2 ])

θε (x)1/2 dx.

ℝ3N

Note that from Lemma 2.111 it follows that supx∈ℝ3N θε (x) < ∞, and by Lemma 2.112 we see that limε↓0 θε (x) = 0 for each x ∈ ℝ3 . Hence the Lebesgue dominated convergence theorem implies that the second term converges to zero, and the lemma follows. Lemma 2.114. It follows that ε

lim (f ⊗ 1, e−2T(HN +g ε↓0

2

Nφε (0,0))

T

h ⊗ 1) = ∫ 𝔼x [f (B−T )h(BT )e− ∫−T V(Bs )ds e

g2 2

ℝ3N

S0ren

] dx, (2.11.30)

where S0ren

N

T

= 2∑ ∫ i=j̸ −T

φ0 (Bis

−

Bjs , 0)ds T

t

i,j=1 −T

−T

N

N

T

j

− 2 ∑ ∫ φ0 (Bis − BT , s − T)ds i,j=1 −T

j

+ 2 ∑ ∫ ( ∫ ∇φ0 (Bis − Bt , s − t)ds) dBt ,

(2.11.31)

and the integrands are given by φ0 (X, t) = ∫ ℝ3

e−ik⋅x e−|t|ω(k) β(k)dk, 2ω(k)

∇φ0 (X, t) = ∫ ℝ3

−ie−ik⋅x e−|t|ω(k) β(k)kdk. 2ω(k)

Proof. We have ε

(f ⊗ 1, e−2T(HN +g

2

Nφε (0,0))

T

h ⊗ 1) = ∫ 𝔼x [f (B−T )h(BT )e− ∫−T V(Bs )ds e

g2 2

Sεren

] dx.

ℝ3 T

The right-hand side converges to ∫ℝ3 𝔼x [f (B−T )h(BT )e− ∫−T V(Bs )ds e Thus (2.11.30) follows. We also see that

g2 2

S0ren

]dx as ε ↓ 0.

2.11 Ultraviolet divergence

S0ren

N

T

= 2∑ ∫ i=j̸ −T

φ0 (Bis

−

Bjs , 0)ds

T

N

N

t

T

| 235

j

+ 2 ∑ ∫ ( ∫ ∇φ0 (Bis − Bt , s − t)ds)dBt i,j=1 −T

(2.11.32)

⟨t−τ⟩

j

− 2 ∑ ∫ φ0 (Bis − B⟨s+τ⟩ , s − ⟨s + τ⟩)ds + S0OD,T .

(2.11.33)

i,j=1 −T

Taking τ = T, we obtain S0OD,T = 0 and (2.11.31) follows. Now we extend the result above from functions of the form f ⊗ 1 to functions of the form f ⊗ F(ϕ(f1 ), . . . , ϕ(fn )) with F ∈ S (ℝn ). Lemma 2.115. Let ρj ∈ M for j = 1, 2, f , h ∈ L2 (ℝ3N ) and α, β ∈ ℂ. Then ε

lim (f ⊗ eαϕ(ρ1 ) 1, e−2T(HN +g

2

ε↓0

Nφε (0,0))

h ⊗ eβϕ(ρ2 ) 1)

T

= ∫ 𝔼x [f (B−T )h(BT )e− ∫−T V(Bs )ds e

g2 2

S0ren + 41 ξ

] dx,

ℝ3N

where ̄ ρ̂ 1 /√ω, e−2Tω ρ̂ 2 /√ω) ξ = ξ (g) = ᾱ 2 ‖ρ̂ 1 /√ω‖2 + β2 ‖ρ̂ 2 /√ω‖2 + 2αβ( T

N

̄ ∑ ∫ ds ∫ + 2αg j=1 −T

ℝ3

T

N

+ 2βg ∑ ∫ ds ∫ j=1 −T

ℝ3

ρ̂1 (k) −|s−T|ω(k) −ik⋅Bjs e e dk √ω(k) ρ̂2 (k) −|s+T|ω(k) −ik⋅Bjs e e dk. √ω(k)

Proof. By the functional integral representation in Theorem 2.12 we have ε

(f ⊗ eαϕ(ρ1 ) 1, e−2T(HN +g

2

Nφε (0,0))

h ⊗ eβϕ(ρ2 ) 1)

T

= ∫ 𝔼x [f (B−T )h(BT )e− ∫−T V(Bs )ds ℝ3N

N

T

j

× 𝔼μE [eαϕE (δ−T ⊗ρ1 ) eβϕE (δT ⊗ρ2 ) e−gϕE (∑j=1 ∫−T δs ⊗φε (⋅−Bs )ds) ]] e−2Tg ̄

2

Nφε (0,0)

dx.

It can be directly seen that N

T

j

𝔼μE [eαϕE (δ−T ⊗ρ1 ) eβϕE (δT ⊗ρ2 ) e−gϕE (∑j=1 ∫−T δs ⊗φε (⋅−Bs )ds) ] e−2Tg ̄

j

2

j

Nφε (0,0)

=e

2

g2 2

Sεren + 41 ξε

,

where ξε is defined by ξ with e−ik⋅Bs /√ω(k) replaced by e−ik⋅Bs e−ε|k| /2 /√ω(k). Thus ε

(f ⊗ eαϕ(ρ1 ) 1, e−2T(HN +g

2

Nφε (0,0))

h ⊗ eβϕ(ρ2 ) 1) T

= ∫ 𝔼x [f (B−T )h(BT )e− ∫−T V(Bs )ds e ℝ3N

g2 2

Sεren + 41 ξε

] dx.

236 | 2 The Nelson model by path measures Notice that with a constant C we have ξε ≤ C uniformly in the paths and ε ≥ 0. Hence we can complete the proof of the lemma in a similar way to Lemma 2.114. Consider the dense subspace of ℋN given by 2

3N

𝒟 = L. H.{f ⊗ F(ϕ(f1 ), . . . , ϕ(fn )), f ⊗ 1 | f ∈ L (ℝ

), F ∈ S (ℝn ),

fj ∈ C0∞ (ℝ3 ), 1 ≤ j ≤ n, n ≥ 1}.

(2.11.34)

Lemma 2.116. For Φ = f ⊗ F(ϕ(u1 ), . . . , ϕ(un )), Ψ = h ⊗ G(ϕ(v1 ), . . . , ϕ(vm )) ∈ 𝒟 ε

lim(Φ, e−2T(HN +g ε↓0

=

1

2

Nφε (0,0))

Ψ) T

(2π)(n+m)/2

̂ )dk dk ∫ 𝔼x [f (B )h(B )e− ∫−T V(Bs )ds e ̂ 1 )G(k ∫ F(k 2 1 2 −T T

ℝn+m

g2 2

S0ren + 41 ξ (k1 ,k2 )

ℝ3N

] dx

(2.11.35)

holds, where ξ (k1 , k2 ) = −‖k1 ⋅ u/√ω‖2 − ‖k2 ⋅ v/√ω‖2 − 2(k1 ⋅ u/√ω, e−2Tω k2 ⋅ v/√ω) N

T

− 2ig ∑ ∫ ds ∫ j=1 −T N

T

ℝ3

+ 2ig ∑ ∫ ds ∫ j=1 −T

ℝ3

̂ (k) −|s−T|ω(k) −ik⋅Bjs k1 ⋅ u e e dk √ω(k) k2 ⋅ v̂(k) −|s+T|ω(k) −ik⋅Bjs e e dk, √ω(k)

with u = (u1 , . . . , un ), v = (v1 , . . . , vm ), and k1 ∈ ℝn , k2 ∈ ℝm . Proof. We have F(ϕ(u1 ), . . . , ϕ(un )) =

1 iϕ(l⋅u) ̂ dl, ∫ F(l)e (2π)n/2 n ℝ

1 ̂ iϕ(l⋅v) dl. G(ϕ(v1 ), . . . , ϕ(vm )) = ∫ G(l)e (2π)m/2 m ℝ

Hence ε

2

(Φ, e−2T(HN +g Nφε (0,0)) Ψ) ε 2 1 ̂ )(f ⊗ e−iϕ(k1 ⋅u) , e−2T(HN +g Nφε (0,0)) h ⊗ e−iϕ(k2 ⋅v) )dk dk . ̂ 1 )G(k = ∫ F(k 2 1 2 (n+m)/2 (2π) m+n ℝ

Thus the statement follows by Lemma 2.115. ε

2

The functional integral representation of limε↓0 (Φ, e−2T(HN +g Nφε (0,0)) Ψ) for arbitrary Φ, Ψ ∈ L2 (ℝd ) ⊗ L2 (Q ) will be given in Section 2.11.5 in terms of exponentials of creation and annihilation operators.

2.11 Ultraviolet divergence

| 237

2.11.4 Uniform lower bound and removal of ultraviolet cutoff In this section we show a crucial lower bound on HNε + g 2 Nφε (0, 0), uniform in ε > 0, and complete the removal of the ultraviolet cutoff. Corollary 2.117. There exists a constant C ∈ ℝ such that HNε +g 2 Nφε (0, 0) > C uniformly in ε > 0. Proof. Consider the quadratic function N

v(x1 , . . . , xN ) = ∑ |x j |2 . j=1

We denote HNε with V replaced by αv + V by HNε (α) for α ≥ 0. Since V is bounded, 1 N − ∑ Δj + αv + V, 2 j=1

α > 0,

has a compact resolvent, which implies that HNε (α) for α > 0 has a unique ground state Ψg (α) by Theorem 2.28. Since Ψg (α) > 0 almost surely due to the positivity improving ε

property of e−tHN (α) , it can be shown that (f ⊗ 1, Ψg (α)) ≠ 0 for every 0 ≤ f ∈ L2 (ℝ3N ), where f ≢ 0. Thus ε 2 1 log(f ⊗ 1, e−T(HN (α)+g Nφε (0,0)) f ⊗ 1). T→∞ T (2.11.36)

inf Spec (HNε (α) + g 2 Nφε (0, 0)) = − lim

Using the functional integral representation we have ε

(f ⊗ 1, e−2T(HN (α)+g

2

Nφε (0,0))

T

ren

f ⊗ 1) = ∫ 𝔼x [f (B−T )f (BT )e− ∫−T (αv+V)(Bs )ds eSε ]dx. ℝ3N

For

1 2

< θ < 1, by Lemma 2.111 there exists a constant cren such that ε

(f ⊗ 1, e−2T(HN (α)+g

2

Nφε (0,0))

T

ren

f ⊗ 1) ≤ ∫ 𝔼x [|f (B−T )||f (BT )|e− ∫−T (αv+V)(Bs )ds eSε ] dx ℝ3N

≤ ‖f ‖2 e2T‖V‖∞ e

4

cren (g 1−θ +g 4 +g 2 )(T+1)

,

which implies, together with (2.11.36), that inf Spec (HNε (α) + g 2 Nφε (0, 0)) + b ≥ 0,

α > 0.

(2.11.37)

4

Here b = cren (g 1−θ + g 4 + g 2 ) + ‖V‖∞ and b is independent of α. Thus ε

|(F, e−2T(HN (α)+g

2

Nφε (0,0))

G)| ≤ ‖F‖‖G‖e2Tb

(2.11.38)

238 | 2 The Nelson model by path measures follows for every α > 0. Let F, G ∈ ℋN . By the functional integral representation again, we have T

ε

T

N

j

(F, e−2THN (α) G) = ∫ 𝔼x [e− ∫−T (αv+V)(Bs )ds (F−T (B−T ), e−ϕE (∫−T ∑j=1 δs ⊗φ(⋅−Bs )ds) GT (BT ))] dx, ℝ3

where F−T = I−T F and GT = IT G. The Lebesgue dominated convergence theorem furthermore implies that ε

lim (F, e−2T(HN (α)+g α↓0

2

Nφε (0,0))

ε

G) = (F, e−2T(HN (0)+g

2

Nφε (0,0))

G).

Taking the limit α ↓ 0 on both sides of (2.11.38), we have ε

|(F, e−2T(HN (0)+g

2

Nφε (0,0))

G)| ≤ ‖F‖‖G‖e2Tb .

This implies that (2.11.37) also holds for α = 0. Since HNε = HNε (0) and V is bounded, we obtain inf Spec(HNε + g 2 Nφε (0, 0)) + b ≥ 0. Setting C = −b yields the corollary. We arrived at the main theorem in this section. Theorem 2.118 (Removal of ultraviolet cutoff). There exists a self-adjoint operator HNren such that ε

ren

s-lim e−t(HN −Eε ) = e−tHN , ε↓0

t ≥ 0,

where the renormalization term Eε is given by 2

Eε = −g N φ̂ ε (0, 0) = −g 2 N ∫ ℝ3 ε

2

e−ε|k| β(k)dk. 2ω(k)

2

Proof. Let F, G ∈ ℋN and Cε (F, G) = (F, e−t(HN +g Nφε (0,0)) G). By Lemma 2.115 we obtain that Cε (F, G) is convergent as ε ↓ 0, for every F, G ∈ 𝒟, where 𝒟 is given by (2.11.34). By the uniform bound ε

‖e−t(HN +g

2

Nφε (0,0))

‖ < e−tC

obtained from Corollary 2.117, and since 𝒟 is dense in ℋN , we see that (Cε (F, G))ε>0 is a Cauchy sequence for F, G ∈ ℋN . Let C0 (F, G) = limε↓0 Cε (F, G). Hence we get |C0 (F, G)| ≤ e−tC ‖F‖‖G‖. The Riesz representation theorem implies that there exists a bounded operator Tt such that C0 (F, G) = (F, Tt G),

F, G ∈ ℋN .

2.11 Ultraviolet divergence

ε

Thus s-limε↓0 e−t(HN +g ε

s-lim e−t(HN +g ε↓0

2

2

Nφε (0,0))

| 239

= Tt follows. Furthermore, we also have that

Nφε (0,0)) −s(HNε +g 2 Nφε (0,0))

e

ε

= s-lim e−(t+s)(HN +g ε↓0

2

Nφε (0,0))

= Tt+s .

Since the left-hand side above is Tt Ts , the semigroup property of Tt follows. Since ε 2 e−t(HN +g Nφε (0,0)) is a symmetric semigroup, Tt is also symmetric. By the functional integral representation (2.11.35), the functional (F, Tt G) is continuous at t = 0 for every F, G ∈ 𝒟. Since 𝒟 is dense in ℋN and ‖Tt ‖ is uniformly bounded in a neighborhood of t = 0, it also follows that Tt is strongly continuous at t = 0. The semigroup version of Stone’s theorem implies that there exists a self-adjoint operator HNren , ren bounded from below, such that Tt = e−tHN , t ≥ 0. The proof is completed by setting Eε = −g 2 Nφε (0, 0). We can also determine the explicit form of the pair interaction associated with HNren . Corollary 2.119 (Pair interaction associated with renormalized Nelson Hamiltonian). The pair interaction associated with HNren is given by g 2 S0ren . Proof. By Lemma 2.114 we see that T

ren

(f ⊗ 1, e−2THN h ⊗ 1) = ∫ 𝔼x [f (B−T )h(BT )e− ∫−T V(Bs )ds e

g2 2

S0ren

] dx.

ℝ3N

2.11.5 Functional integral representation of the ultraviolet renormalized Nelson model ren

We apply Proposition 1.100 to construct a functional integral representation of e−tHN in terms of exponentials of creation and annihilation operators. We recall that (ϕ(f ), f ∈ M ) and (ϕE (F), F ∈ E ) are families of Gaussian random variables indexed by M and E , respectively, and 1 ∗ ̂ ̃̂ (a (F) + aE (F)), F̂ ∈ ℋ̂ E , √2 E 1 ∗ ̂ ̃ (a (f ) + aM (f ̂)), f ̂ ∈ ℋ̂ M , ϕ(f ) ≅ √2 M ϕE (F) ≅

(2.11.39) (2.11.40)

̂ a∗ (f ̂) and aE (F), ̂ aM (f ̂) are creation and annihilation operators on hold. Here a∗E (F), M ̂ the boson Fock spaces ℱb (ℋE ) and ℱb (ℋ̂ M ), respectively, satisfying ̂ = (f ̄̂, g)̂ ℋ̂ , [a∗M (f ̂), aM (g)] M

̂ = (F,̄̂ G)̂ ̂ . ̂ aE (G)] [a∗E (F), ℋ E

ren

We have already given a functional integral representation of (F, e−THN G) for F, G ∈ 𝒟, where 𝒟 is given in (2.11.34). This will now be extended to ℋN .

240 | 2 The Nelson model by path measures Theorem 2.120. Let F, G ∈ ℋN and T

g UT (k) = − ∫ e−|s+T|ω(k) e−ik⋅Bs ds, √2 −T

(2.11.41)

T

g Ũ T (k) = −Ū −T (k) = − ∫ e−|s−T|ω(k) eik⋅Bs ds. √2

(2.11.42)

−T

We have UT , Ũ T , UT /√ω, Ũ T /√ω ∈ ℋ̂ M a. s., i. e., UT /√ω, Ũ T /√ω, UT /ω, Ũ T /ω ∈ L2 (ℝ3 ) a. s., and it follows that T

ren

(F, e−2THN G)ℋN = ∫ 𝔼x [e− ∫−T V(Bs )ds e

g2 2

S0ren

ℝ3

̃ (F(B−T ), AAG(B T ))L2 (Q) ] dx,

(2.11.43)

where A and à are bounded operators on L2 (Q ) a. s. defined by A = eaM (UT ) e−THf , ∗

2

̃ Ã = e−THf eaM (UT ) .

(2.11.44)

2

Proof. Let ϱε = (e−ε|k| /2 ) ̌ = e−|x| /2ε /ε for ε > 0. We have ε

(F, e−2T(HN −Eε ) G) T

= ∫ 𝔼x [e− ∫−T V(Bs )ds e

g2 2

Sεren

ℝ3

T

(F(B−T ), I∗−T e−gϕE (∫−T δs ⊗ϱε (⋅−Bs )ds) IT G(BT ))] dx.

Let 2

e−isk0 e−ε|k| /2 . us,ε (k0 , k) = δ? s ⊗ ϱε (k0 , k) = √2π By the identification T

ϕE ( ∫ δs ⊗ ϱε (⋅ − Bs )ds) ≅ −T

T

T

−T

−T

1 [a∗ ( ∫ us,ε e−ik⋅Bs ds) + aE ( ∫ ũ s,ε eik⋅Bs ds)] √2 E

and the identity T T 󵄩2 󵄩󵄩2 󵄩󵄩 1 󵄩󵄩󵄩󵄩 1 󵄩󵄩 󵄩󵄩 −ik⋅Bs 󵄩 −ik⋅Bs 󵄩 󵄩 u e ds = u e ds ∫ ∫ 󵄩 󵄩󵄩 󵄩󵄩 󵄩 s,ε s,ε 󵄩󵄩ℋ̂ E 󵄩󵄩󵄩 k 2 + |k|2 󵄩󵄩L2 (ℝ4 ) 2 󵄩󵄩󵄩 0 −T

T

T

−T

= ∫ ds ∫ dt ∫ −T

−T

ℝ4

ut,ε us,ε e−ik⋅(Bs −Bt ) k02 + |k|2

dk0 dk = SεT ,

| 241

2.11 Ultraviolet divergence

the Baker–Campbell–Hausdorff formula yields T

e−gϕE (∫−T δs ⊗ϱε (⋅−Bs )ds) = e

g2 2

T

T

g g SεT −a∗E ( √2 ∫−T us,ε e−ik⋅Bs ds) −aE ( √2 ∫−T ũ s,ε eik⋅Bs ds)

e

e

.

We can furthermore compute by using the intertwining property as T

I∗−T e−gϕE (∫−T us,ε (⋅−Bs )ds) IT = e

g2 2

SεT a∗M (UTε ) −2THf aM (Ũ Tε )

e

e

e

,

where UTε (k)

T

2 g =− ∫ e−|s+T|ω(k) e−ε|k| /2 e−ik⋅Bs ds, √2

−T

(2.11.45)

T

2 g ε Ũ Tε (k) = −Ū −T (k) = − ∫ e−|s−T|ω(k) e−ε|k| /2 eik⋅Bs ds. √2

(2.11.46)

−T

ε

Thus we have a functional integral representation of e−2T(HN −Eε ) in terms of exponen∗ ε ̃ε tials of annihilation operators and creation operators, eaM (UT ) e−2THf eaM (UT ) by T

ε

(F, e−2T(HN −Eε ) G) = ∫ 𝔼x [e− ∫−T V(Bs )ds e

g2 2

Sεren

(F(B−T ), Aε Ã ε G(BT ))] dx,

ℝ3

where Aε is defined by A with UT replaced by UTε , and à ε by à with Ũ T replaced by Ũ Tε . Write ψ(Bs , s) = e−|s+T|ω(k) e−ik⋅Bs . By the Itô formula we have T

T

ψ(BT , T) − ψ(B−T , −T) = ∫ ∇ψ(Bs , s)dBs −

1 ∫ e−|s+T|ω(k) e−ik⋅Bs ds. β(k) −T

−T

Here we recall that β = β(k) = 1/(ω(k) + |k|2 /2). We have T

β∇ψ(Bs , s) UT g βα(T) g = − dBs . ∫ √2 ω √2 ω ω

(2.11.47)

−T

Here α(T) = ψ(BT , T) − ψ(B−T , −T) = e−2|T|ω(k) e−ik⋅BT − e−ik⋅B−T . Notice that |α(T)| ≤ Ω(k),

(2.11.48)

242 | 2 The Nelson model by path measures where 2, 2|k| (|T| +

Ω(k) = 2 ∧ (2|T|ω(k) + |k||BT − B−T |) ≤ {

|BT −B−T | ). 2

For every k ∈ ℝ3 , Ω(k) is a random variable. We show that the right-hand side of (2.11.47) is an L2 -function with respect to k almost surely. It is trivial to see that 󵄨󵄨 β(k)α(T) 󵄨󵄨2 Ω(k)2 󵄨󵄨 dk ≤ ∫ dk < ∞. ∫ 󵄨󵄨󵄨 󵄨 2 󵄨 ω(k) 󵄨 ω(k) (|k|2 /2 + ω(k))2

ℝ3

ℝ3

T β(k)∇ψ(Bs ,s) dBs ω(k)

Next we prove that ∫−T

∈ L2 (ℝ3 ). By the Itô isometry,

T

T

󵄨󵄨2 󵄨󵄨 β(k)∇ψ(Bs , s) 󵄨󵄨 β(k)∇ψ(Bs , s) 󵄨󵄨2 𝔼 [ ∫ 󵄨󵄨󵄨 ∫ dBs 󵄨󵄨󵄨 dk] = ∫ 𝔼x [󵄨󵄨󵄨 ∫ ds󵄨󵄨󵄨 ]dk 󵄨 󵄨 󵄨 󵄨 ω(k) ω(k) x

ℝ3 −T

T

= ∫ ds ∫ −T

ℝ3

ℝ3

−T

(1 − e−4Tω(k) )|k|2 e−2|s+T|ω(k) |k|2 dk = dk < ∞. ∫ ω(k)2 (|k|2 /2 + ω(k))2 2ω(k)3 (|k|2 /2 + ω(k))2 ℝ3

T (k) 2 Thus it follows that 𝔼x [∫ℝ3 | Uω(k) | dk] < ∞, which implies UT /ω ∈ L2 (ℝ3 ) and Ũ T /ω ∈ 2 3 L (ℝ ) a. s. Similarly we can also show UT /√ω ∈ L2 (ℝ3 ) and Ũ T /√ω ∈ L2 (ℝ3 ) a. s.

Hence the first statement is proven. Let Qε = Aε Ã ε for ε ≥ 0 with Q0 = AA,̃ and T

Pε (x) = 𝔼x [e− ∫−T V(Bs )ds e

g2 2

Sεren

(F(B−T ), Qε G(BT ))] .

In Lemmas 2.125–2.126 below we prove the crucial facts (1) Pε ∈ L1 (ℝ3x ) (2)

ε ≥ 0,

s-lim Pε = P0 ε↓0

(2.11.49) 1

in L .

(2.11.50)

By (2.11.49)–(2.11.50) we see that ren

ε

(F, e−2THN G) = lim(F, e−2T(HN −Eε ) G) = lim ∫ Pε (x)dx = ∫ P0 (x) ε↓0

= ∫ 𝔼x [e

ε↓0

T − ∫−T

V(Bs )ds

e

g2 2

S0ren

ℝ3

ℝ3

̃ (F(B−T ), AAG(B T ))] dx.

ℝ3

We give a remark on conditions UT , Ũ T , UT /√ω, Ũ T /√ω ∈ ℋ̂ M a. s. Condition UT , Ũ T ∈ ℋ̂ M implies that both a∗M (UT ) and aM (Ũ T ) are well-defined, and condition UT /√ω, Ũ T /√ω ∈ ℋ̂ M implies that both A and à are bounded. We proceed through a sequence of lemmas to prove (2.11.49)–(2.11.50).

2.11 Ultraviolet divergence

| 243

Lemma 2.121. Let n ∈ ℕ and T > 0. Then there exist Dj = Dj (ξ ), j = 1, . . . , 5, independent of (x, g, n) ∈ ℝ3 × ℝ × ℕ, such that limξ →0 Dj (ξ ) = ∞ for j = 1, 2 and limξ →0 Dj (ξ ) = 0 for j = 3, 4, 5, and ε

̃ε

𝔼x [‖eaM (UT ) e−2THf eaM (UT ) ‖n ] ≤ 2n Deg ∗

with D=

(1 −

g 2 nD3 )3/2 (1

1

−

g 4 n2 D

2

nD1 +g 4 n2 D2

3/8 (1

4)

ε≥0

,

− g 4 n2 D5 )3/8

.

Here we chose ξ such that (1 − g 2 nD3 )(1 − g 4 n2 D4 )(1 − g 4 n2 D5 ) > 0. 2

Proof. We prove the case ε = 0 and T ≥ 1. Noticing that e−ε|k| /2 < 1, we can show the lemma for ε > 0 in the same way as for ε = 0. For T < 1, by using the estimate for t < 1 in Corollary 1.34, we can prove this in a similar way to the case T ≥ 1. We have the estimate 2

𝔼x [‖eaM (UT ) e−2THf eaM (UT ) ‖n ] ≤ 2n 𝔼x [e4n‖UT /√ω‖ω ] ∗

̃

2

by Corollary 1.34. Here ‖ ⋅ ‖ω = ‖ ⋅ ‖ + ‖ ⋅ /√ω‖. We estimate 𝔼x [eδ‖UT /√ω‖ω ] for δ ≥ 0. Denote g 2 /2 = a. By the Itô formula (2.11.47), we have 󵄩󵄩 U 󵄩󵄩2 󵄩󵄩2 󵄩󵄩 βα(T) 󵄩󵄩2 󵄩󵄩 T β∇ψ(B , s) 󵄩󵄩 T 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 󵄩 s dBs 󵄩󵄩󵄩 = 2aAT + 2aBT . 󵄩󵄩 󵄩󵄩 ≤ 2a󵄩󵄩 󵄩󵄩 + 2a󵄩󵄩󵄩 ∫ 󵄩󵄩 √ω 󵄩󵄩ω 󵄩󵄩ω 󵄩󵄩 √ω 󵄩󵄩ω 󵄩󵄩 √ω −T

Thus 2

𝔼x [eδ‖UT /√ω‖ω ] ≤ 𝔼x [e2aδ(AT +BT ) ] ≤ (𝔼x [e4aδAT ])1/2 (𝔼x [e4aδBT ])1/2 . By Lemmas 2.123–2.124 below we have the bounds 𝔼x [e4aδAT ] ≤ 𝔼x [e4aδCA ],

(2.11.51)

2 2

2 2

1/4

𝔼x [e4aδBT ] ≤ (𝔼x [ea δ C1 ]𝔼x [ea δ C2 ]) with CA = ∫ ℝ3

,

2Ω(k)2 1 1 ( + ) dk, 2 2 (ω(k) + |k| /2) ω(k) ω(k)2 ∞

C1 =C[( ∫

0 ∞

∞ 2 (T √r) ∧ √r 2 2 Ω(r) dr) + ( dr) ], ∫ 2 √r(1 + r/2) √r(1 + r/2) 0 ∞

2 (T √r) ∧ √r 2 Ω(r) C2 =C[( ∫ dr) ] dr) + ( ∫ r(1 + r/2) r(1 + r/2)2 0

2

0

(2.11.52)

244 | 2 The Nelson model by path measures with a positive constant C. Here Ω(r) = 2 ∧ (2|T| + |BT − B−T |)r. We notice that CA , C1 and C2 depend on w ∈ X due to Ω(⋅). Hence 1/2

2

𝔼x [eδ‖UT /√ω‖ω ] ≤ (𝔼x [e4aδCA ]) 2 2

2 2

2 2

(𝔼x [ea δ C1 ]𝔼x [ea δ C2 ])

1/8

.

2 2

We estimate 𝔼x [e4aδCA ] and 𝔼x [ea δ C1 ]𝔼x [ea δ C2 ] separately. Let ξ > 0. First note that CA ≤ ∫ |k|>ξ

1 8 1 + ( ) dk (ω(k) + |k|2 /2)2 ω(k) ω(k)2

+ ∫ |k|≤ξ

2(2|T| + |BT − B−T |)2 |k|2 1 1 ( ) dk + ω(k) ω(k)2 (ω(k) + |k|2 /2)2

and ∫ |k|≤ξ

2(2|T| + |BT − B−T |)2 |k|2 1 1 ( ) dk + 2 2 ω(k) ω(k)2 (ω(k) + |k| /2)

≤ ∫ |k|≤ξ

1 16|T|2 |k|2 1 ( ) dk + 2 2 (ω(k) + |k| /2) ω(k) ω(k)2

+ |BT − B−T |2 ∫ |k|≤ξ

4|k|2 1 1 + ( ) dk. 2 2 (ω(k) + |k| /2) ω(k) ω(k)2

Hence we conclude that CA ≤ CA1 + |BT − B−T |2 CA2 ,

(2.11.53)

where CA1 = CA1 (ξ ) = ∫ |k|>ξ

1 1 8 + ( ) dk (ω(k) + |k|2 /2)2 ω(k) ω(k)2

+ ∫ |k|≤ξ

CA2 = CA2 (ξ ) = ∫ |k|≤ξ

16|T|2 |k|2 1 1 + ( ) dk, (ω(k) + |k|2 /2)2 ω(k) ω(k)2

1 1 4|k|2 ( + ) dk. (ω(k) + |k|2 /2)2 ω(k) ω(k)2

Note that CA1 < ∞ and CA2 < ∞, and limξ →0 CA2 (ξ ) = 0 and limξ →0 CA1 (ξ ) = ∞. In general, if 1/4T > c > 0 then we have

2.11 Ultraviolet divergence

2

𝔼x [ec|BT −B−T | ] =

1 (2πT)3

1 ≤ (2πT)3

2

2

| 245

2

∫ ec|x−y| e−|x| /2T e−|y| /2T dxdy ℝ3 ×ℝ3 1

2

2

∫ e−( 2T −2c)(|x| +|y| ) dxdy = ℝ3 ×ℝ3

1

1 . (1 − 4cT)3

(2.11.54)

2 2

Since 𝔼x [e4aδCA ] ≤ e4aδCA 𝔼x [e4aδ|B−T −BT | CA ], 𝔼x [e4aδCA ] < ∞ if 4aδCA2 < 1/4T. We can control CA2 by varying ξ , leading to 1

2 2

𝔼x [e4aδCA ] ≤ e4aδCA 𝔼x [e4aδ|B−T −BT | CA ] ≤

1

e4aδCA , (1 − 16aδCA2 T)3

for ξ such that 1 − 16aδCA2 T > 0. Hence 𝔼x [e8ng

2

CA

2 1

e8ng CA (1 − 32ng 2 TCA2 )3

]≤

(2.11.55)

follows for ξ such that 1 − 32ng 2 TCA2 > 0. Next we estimate C1 and C2 through similar computations as applied to CA . We need to estimate ( ∫0

∞

ξ

∞ Ω(rdr) 2 2 Ω(r)dr ) and ( ∫0 r(1+r/2) 2) . √r(1+r/2)2

Splitting up the integral ∫0 dr like ∫0 dr + ∫ξ dr, we have ∞

∞

(∫ 0 ∞

(∫ 0

∞

2

Ω(rdr) ) ≤ (2|T| + |B−T − BT |)2 CB1 + CB3 , √r(1 + r/2)2 2

Ω(r)dr dr) ≤ (2|T| + |B−T − BT |)2 CB2 + CB4 , r(1 + r/2)2

where CB1

ξ

2

√rdr = 2( ∫ ), (1 + r/2)2 0 ∞

CB3 = 2( ∫ ξ

CB2 2

2dr ), √r(1 + r/2)2

ξ

= 2( ∫ 0

2

dr ), (1 + r/2)2 ∞

CB4 = 2( ∫ ξ

2

2dr ). r(1 + r/2)2

Hence 2 12 C1 ≤ D11 B + |B−T − BT | DB ,

C2 ≤

D21 B

+ |B−T −

BT |2 D22 B,

where we denoted 1 D12 B = 2CB ,

2 D22 B = 2CB ,

2 1 3 15 D11 B = 8|T| CB + CB + CB ,

2 2 4 25 D21 B = 8|T| CB + CB + CB ,

(2.11.56) (2.11.57)

246 | 2 The Nelson model by path measures 2

2

2

2

√r with CB15 = ( ∫0 √r(1+r/2) dr) and CB25 = ( ∫0 r(1+r/2)√r dr) . The computation at the right-hand side of (2.11.56) is similar to (2.11.53). The result is

∞ (T√r)∧

x

𝔼 [e

∞ (T√r)∧

a2 δ2 Cj

2 2

]≤

j1

e a δ DB

j2

(1 − 4a2 δ2 TDB )3

,

j = 1, 2,

j2

for ξ such that 1 − 4a2 δ2 TDB > 0. We obtain 𝔼[e4aδBT ] ≤ (

2 2

11

1/4

21

ea δ (DB +DB ) ) 22 3 3 2 2 (1 − 4a2 δ2 TD12 B ) (1 − 4a δ TDB )

.

(2.11.58)

Together with (2.11.55) and (2.11.58), we have 𝔼x [‖eaM (UT ) e−2THf eaM (UT ) ‖n ] ∗

≤ 2n

̃

2 1

1

2 4

11

21

e4ng CA e 2 n g (DB +DB ) . 3/8 (1 − 16n2 g 4 TD22 )3/8 (1 − 32ng 2 TCA2 )3/2 (1 − 16n2 g 4 TD12 B) B

The proof of Lemma 2.121 also gives the corollary below. Corollary 2.122. For every T > 0 there exist constants D1 and D2 independent of x such 2 ε that for every ε ≥ 0 it follows that 𝔼x [e‖UT /√ω‖ω ] ≤ D1 eD2 . Next we prove the bounds (2.11.51) and (2.11.52) used in the proof of Lemma 2.121. Lemma 2.123. It follows that 𝔼x [e4aδAT ] ≤ 𝔼x [e4aδCA ]. Proof. By direct inspection, we have 󵄩2 󵄩 󵄩2 󵄩󵄩 βα(T) 󵄩󵄩2 󵄩 󵄩󵄩 ≤ 2 󵄩󵄩󵄩 βα(T) 󵄩󵄩󵄩 + 2 󵄩󵄩󵄩 βα(T) 󵄩󵄩󵄩 AT = 󵄩󵄩󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩󵄩 ω 󵄩󵄩󵄩 󵄩 √ω 󵄩󵄩ω 󵄩 √ω 󵄩󵄩 ≤∫ ℝ3

2Ω(k)2 1 1 ( + ) dk = CA . (ω(k) + |k|2 /2)2 ω(k) ω(k)2

Next we estimate the expectation of the exponent of the random process 󵄩󵄩 T β∇ψ(B , s) 󵄩󵄩2 󵄩 󵄩 s BT = 󵄩󵄩󵄩 ∫ dBs 󵄩󵄩󵄩 , 󵄩󵄩 󵄩󵄩ω √ω

T ≥ 0,

−T

which is defined by the absolute value of Hilbert space-valued stochastic integrals. The estimate is not straightforward. 2 2

2 2

Lemma 2.124. It follows that 𝔼x [e4aδBT ] ≤ (𝔼x [ea δ C1 ]𝔼x [ea δ C2 ])1/4 .

| 247

2.11 Ultraviolet divergence

Proof. We have BT ≤ 2B1T + 2B2T , where 󵄩󵄩 T β∇ψ(B , s) 󵄩󵄩2 󵄩 󵄩 s = 󵄩󵄩󵄩 ∫ dBs 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 √ω

B1T

B2T

and

−T

󵄩󵄩 T β∇ψ(B , s) 󵄩󵄩2 󵄩 󵄩 s = 󵄩󵄩󵄩 ∫ dBs 󵄩󵄩󵄩 . 󵄩󵄩 󵄩󵄩 ω −T

We see that 1

1/2

𝔼x [e4aδBT ] ≤ (𝔼x [e16aδBT ])

1/2

2

(𝔼x [e16aδBT ])

.

We only consider B1T , the case B2T can be dealt with similarly. Denote α = 16aδ. Note T

that B1T = ∫−T Φs dBs , where (Φs )s∈ℝ is an ℝ3 -valued random process defined by T

Φs = ( ∫ −T

β∇ψ(Bs , s) β∇ψ(Br , r) dBr , ) , √ω √ω L2 (ℝ3 )

s ∈ ℝ.

By the Girsanov theorem, T

1

T

𝔼x [eαBT ] = 𝔼x [eα ∫−T Φs dBs ] = 𝔼x [eα ∫−T Φs dBs −α T

≤ (𝔼x [e2α ∫−T Φs dBs −2α

2

1/2

T

∫−T |Φs |2 ds

])

2

T

T

∫−T |Φs |2 ds+α2 ∫−T |Φs |2 ds

(𝔼x [e2α

2

T

∫−T |Φs |2 ds

]

1/2

])

,

thus it is enough to estimate the second factor since the first equals one. Inserting T formula (2.11.47) into ∫−T Φs dBs gives T

T

2 󵄨󵄨 T β∇ψ(B , r) β∇ψ(Bs , s) 󵄨󵄨󵄨󵄨 󵄨 r dBr , )󵄨󵄨 ds ∫ |Φs | ds = ∫ 󵄨󵄨󵄨( ∫ 󵄨󵄨 󵄨󵄨 √ω √ω 2

−T

−T T

−T

2 󵄨󵄨 √2 U βα(T) β∇ψ(Bs , s) 󵄨󵄨󵄨󵄨 󵄨 T = ∫ 󵄨󵄨󵄨 ( − , ) 󵄨󵄨 ds 󵄨󵄨 g √ω 󵄨󵄨 √ω √ω −T

≤

T

T

−T

−T

󵄨󵄨 βα(T) β∇ψ(Bs , s) 󵄨󵄨2 4 󵄨󵄨 U β∇ψ(Bs , s) 󵄨󵄨󵄨2 ) 󵄨󵄨 ds + 2 ∫ 󵄨󵄨󵄨 ( , ) 󵄨󵄨󵄨 ds. (2.11.59) ∫ 󵄨󵄨󵄨 ( T , 2 󵄨 󵄨 󵄨 √ω 󵄨 √ω √ω √ω g

We estimate both terms at the right-hand side separately. In the calculations below C will denote a positive constant which is independent of g and may differ from line to line. The integrand is estimated directly as ∞

󵄨󵄨 βα(T) β∇ψ(Bs , s) 󵄨󵄨 Ω(k)e−sω(k) |k| e−sr Ω(r) 󵄨󵄨 ( ) 󵄨󵄨󵄨 ≤ ∫ dk ≤ C ∫ dr 󵄨󵄨 √ω , 2 2 󵄨 √ω ω(k)(ω(k) + |k| /2) (1 + r/2)2 ℝ3

0

248 | 2 The Nelson model by path measures and then T

T

∞

∞

−T

0

0

󵄨󵄨 βα(T) β∇ψ(Bs , s) 󵄨󵄨2 e−sr Ω(r) e−sr Ω(r 󸀠 ) 󸀠 , ) 󵄨󵄨󵄨 ds ≤ C ∫ ds ∫ dr ∫ dr ∫ 󵄨󵄨󵄨 ( 󵄨 √ω 󵄨 √ω (1 + r/2)2 (1 + r 󸀠 /2)2

−T

∞

∞

0

0

󸀠

∞

∞

0

0

Ω(r)Ω(r 󸀠 ) Ω(r)Ω(r ) 󸀠 dr ≤ C dr dr 󸀠 < ∞. ≤ C ∫ dr ∫ ∫ ∫ (1 + r 󸀠 /2)2 (1 + r/2)2 (r + r 󸀠 ) (1 + r 󸀠 /2)2 (1 + r/2)2 √rr 󸀠 󸀠

Thus T

∞

−T

0

2 󵄨󵄨 βα(T) β∇ψ(Bs , s) 󵄨󵄨2 Ω(r) dr) . , ) 󵄨󵄨󵄨 ds ≤ C( ∫ ∫ 󵄨󵄨󵄨 ( 󵄨 󵄨 √ω √ω √r(1 + r/2)2

(2.11.60)

Next we estimate T

T

T

󵄨󵄨 U β∇ψ(Bs , s) 󵄨󵄨2 󵄨󵄨2 ψ(Bt , t) β∇ψ(Bs , s) g 2 󵄨󵄨󵄨 ) 󵄨󵄨󵄨 ds = , ) dt 󵄨󵄨󵄨 ds. ∫ 󵄨󵄨󵄨 ( T , ∫ 󵄨󵄨 ∫ ( 󵄨 √ω 󵄨 󵄨 󵄨 √ω √ω √ω 2

−T

(2.11.61)

−T −T

The integrand is estimated as ∞

󵄨󵄨 ψ(Bt , t) β∇ψ(Bs , s) 󵄨󵄨 |k|e−tω(k) e−sω(k) re−tr e−sr 󵄨󵄨 ( , ) 󵄨󵄨󵄨 ≤ ∫ dk ≤ C ∫ dr. 󵄨󵄨 2 󵄨 √ω √ω (1 + r/2) ω(k)(ω(k) + |k| /2) 0

ℝ3

Hence it follows that T

T

󵄨󵄨2 󵄨󵄨 ψ(Bt , t) β∇ψ(Bs , s) , ) dt 󵄨󵄨󵄨 ds ∫ 󵄨󵄨󵄨 ∫ ( 󵄨 󵄨 √ω √ω

−T −T

T

T

T

∞

∞

0

0

≤ C ∫ ds ∫ dt ∫ dp ∫ dr ∫ −T ∞

−T ∞

−T ∞

−∞ ∞

0 ∞

0

0

0

≤ C ∫ ds ∫ dr ∫ 10 = C ∫ dr ∫

re−|t|r e−sr r 󸀠 e−|p|r e−sr 󸀠 dr (1 + r/2) (1 + r 󸀠 /2) 󸀠

󸀠

e−sr (1 − e−Tr ) e−sr (1 − e−Tr ) 󸀠 dr (1 + r/2) (1 + r 󸀠 /2) 󸀠

󸀠

(1 − e−Tr )(1 − e−Tr ) dr 󸀠 (r + r 󸀠 )(1 + r/2)(1 + r 󸀠 /2) 󸀠

2

2

∞ 2 (T √r) ∧ √r 1 − e−Tr ≤ C (∫ dr) ≤ C ( ∫ dr) < ∞. √r(1 + r/2) (1 + r/2) ∞

0

0

By (2.11.60) and (2.11.62) we see that 1

2

𝔼x [eαBT ] ≤ (𝔼x [e2α CC1 ])

1/2

,

(2.11.62)

2.11 Ultraviolet divergence

| 249

where C is independent of g by (2.11.59) and (2.11.61), and 2

2

∞ 2 (T √r) ∧ √r Ω(r) C1 = ( ∫ dr) < ∞. dr) + ( ∫ √r(1 + r/2)2 (1 + r/2) ∞

0

0

In the same way as above, we can also see that 2

2

𝔼x [eαBT ] ≤ (𝔼x [e2α CC2 ])

1/2

,

where 2

2

∞ 2 (T √r) ∧ √r Ω(r) C2 = ( ∫ dr) + ( ∫ dr) < ∞. √r(1 + r/2) r(1 + r/2)2 ∞

0

0

Hence 2

1/4

2

𝔼x [e4aδBT ] ≤ (𝔼x [e2α CC1 ]𝔼x [e2α CC2 ])

,

where α = 16aδ. The time shift defined by Bs → Bs+T allows to write 2T

ε

(F, e−2T(HN −Eε ) G) = ∫ Pε (x)dx = ∫ 𝔼x [e− ∫0 ℝ3

V(Bs )ds

e

g2 2

S̃εren

(F(B0 ), Q̃ ε G(B2T ))] dx,

ℝ3

where S̃εren and Q̃ ε are defined by Sεren and Qε shifted by T, i. e., 2T

2T

2T

0

s

S̃εren = −2 ∫ φε (Bs − B2T , s − 2T)ds + 2 ∫ ds ∫ ∇φε (Bs − Bt , s − t)dBt , 0

Q̃ ε = e

a∗M (ST )

STε = −

e−2THf e

aM (S̃T )

,

2T

2 g ∫ e−|s|ω(k) e−ε|k| /2 e−ik⋅Bs ds, √2

0

2T

2 g S̃Tε = − ∫ e−|s−2T|ω(k) e−ε|k| /2 eik⋅Bs ds. √2

0

We also define ST and S̃T by STε and S̃Tε with ε = 0, respectively. Using this formula, we prove (2.11.49) and (2.11.50). The proof is in some sense straightforward, using the properties of exponentials of annihilation operators and creation operators discussed ∗ in Section 1.2.7. In particular, the uniform continuity of the map f 󳨃→ ea (f ) e−THf for any t > 0 is a key ingredient. It results in the uniform bound of 𝔼x [‖Qε − Q0 ‖n ] for every n ∈ ℕ, and we can show that ∫ℝ3 |Pε (x) − P0 (x)|dx → 0 as ε ↓ 0. We show now the claim announced in (2.11.49).

250 | 2 The Nelson model by path measures Lemma 2.125. Pε ∈ L1 (ℝ3x ), for every ε ≥ 0. Proof. By Lemma 2.121 and the bound (2.11.24) there exist C1 and C2 , independent of x ∈ ℝ, such that 𝔼x [eg

2 ̃ ren Sε

‖Q̃ ε ‖2 ] ≤ (𝔼x [e2g

2 ̃ ren Sε

1/2

])

1/2 (𝔼x [‖Q̃ ε ‖4 ]) ≤ C1 eC2 .

We find that 2

∫ |Pε (x)|dx ≤ e2T‖V‖∞ ∫ ‖F(x)‖𝔼x [eg S̃εren ‖Q̃ ε ‖‖G(BT )‖] dx ℝ3

ℝ3

≤ e2T‖V‖∞ ∫ ‖F(x)‖ (𝔼x [eg ≤ C1 eC2 e

ℝ3 2T‖V‖∞

2 ̃ ren Sε

‖Q̃ ε ‖2 ])

1/2

1/2

(𝔼x [‖G(B2T )‖2 ])

dx

‖F‖‖G‖.

Next we also show (2.11.50). Lemma 2.126. s-limε↓0 Pε = P0 in L1 . Proof. We suppose that T ≥ 1; for T < 1 the proof is similar by a scaling argument. We have ‖Pε − P0 ‖L1 (ℝ3 ) = e2T‖V‖∞ ∫ ‖F(x)‖ (𝔼x [‖e

g2 2

S̃εren

Q̃ ε − e

g2 2

S̃0ren

Q̃ 0 ‖2 ])

1/2

1/2

(𝔼x [‖G(BT )‖2 ])

dx. (2.11.63)

ℝ3

We show that the first expectation above converges to zero as ε ↓ 0 uniformly with respect to x ∈ ℝ3 . We have the estimate 󵄩󵄩 Sε S 󵄩󵄩󵄩 󵄩 ‖Q̃ ε − Q̃ 0 ‖ ≤ 2J 󵄩󵄩󵄩 T − T 󵄩󵄩󵄩 , 󵄩󵄩 √ω √ω 󵄩󵄩ω 2

ε

2

ε

2

where J = 2e2(‖ST /√ω‖ω +‖ST /√ω‖ω +1) (e2‖ST /√ω‖ω +e2‖ST /√ω‖ω ). This bound follows by Corollaries 1.34–1.35. Hence 8 1/2 󵄩󵄩 Sε 1/2 S 󵄩󵄩󵄩 󵄩 𝔼x [‖Q̃ ε − Q̃ 0 ‖4 ] ≤ 24 (𝔼x [󵄩󵄩󵄩 T − T 󵄩󵄩󵄩 ]) (𝔼x [J 8 ]) 󵄩󵄩 √ω √ω 󵄩󵄩ω 8 1/2 󵄩󵄩 Sε S 󵄩󵄩󵄩 󵄩 ≤ 24 C1 eC2 (𝔼x [󵄩󵄩󵄩 T − T 󵄩󵄩󵄩 ]) , 󵄩󵄩 √ω √ω 󵄩󵄩ω

with constants C1 and C2 . The second estimate above follows by Lemma 2.121 and the STε ST 8 bound (2.11.24). We show that limε↓0 supx∈ℝ3 𝔼x [‖ √ω − √ω ‖ω ] = 0, i. e., lim sup 𝔼x [( ∫ ε↓0 x∈ℝ3

ℝ3

4

|ST (k) − STε (k)|2 dk) ] = 0, ω(k)n

n = 1, 2.

2.11 Ultraviolet divergence

| 251

Take the case n = 1, and denote 2

ψ̃ ε (Bs , s) = e−ε|k| /2 e−|s|ω(k) e−ik⋅Bs for ε ≥ 0. The Itô formula gives 2T

STε β∇ψ̃ ε (Bs , s) g β(ψ̃ ε (B2T , 2T) − ψ̃ ε (B0 , 0)) g = − ⋅ dBs . ∫ √ω √2 √ω √2 √ω

(2.11.64)

0

Let Ψε (Bs , s) = ψ̃ ε (Bs , s) − ψ̃ ε (B0 , 0) and αε (T) = Ψε (B2T , 2T) − Ψ0 (B2T , 2T). Notice that 2 |αε (T)| ≤ 2(1 − e−ε|k| /2 ). Hence we have 2T

4 4 󵄨󵄨2 󵄨󵄨 g βαε (T) |S (k) − STε (k)|2 β∇Ψε (Bs , s) g − dBs 󵄨󵄨󵄨 dk) ] 𝔼 [( ∫ T dk) ] = 𝔼x [( ∫ 󵄨󵄨󵄨 ∫ 󵄨 󵄨 √2 √ω √2 √ω ω(k) 3 3 x

ℝ

0

ℝ

2T

4 󵄨2 󵄨󵄨 βα (T) 󵄨󵄨2 󵄨 󵄨󵄨 dk + ∫ 󵄨󵄨󵄨 ∫ β∇Ψε (Bs , s) dBs 󵄨󵄨󵄨 dk) ] ≤ C𝔼 [( ∫ 󵄨󵄨󵄨 ε 󵄨󵄨 󵄨 √ω 󵄨󵄨 󵄨󵄨 √ω x

ℝ3

ℝ3

2T β∇Ψε (Bs ,s) dBs . By √ω

with a constant C > 0. Denote XT (k) = ∫0 have 4

0

the Schwarz inequality we

4

󵄨 󵄨2 𝔼x [( ∫ 󵄨󵄨󵄨XT (k)󵄨󵄨󵄨 dk) ] ≤ ∫ ∏(𝔼x [|XT (kn )|8 ])1/4 dk1 dk2 dk3 dk4 ℝ3

n=1

ℝ3×4

󵄨 󵄨8 = ( ∫ (𝔼x [󵄨󵄨󵄨XT (k)󵄨󵄨󵄨 ])

1/4

4

dk)

ℝ3

Hence by the Burkholder–Davis–Gundy inequality, 2T

󵄨󵄨 󵄨󵄨8 󵄨󵄨 β∇Ψε (Bs , s) 󵄨󵄨8 󵄨󵄨 ds] 𝔼 [󵄨󵄨󵄨XT (k)󵄨󵄨󵄨 ] ≤ 284 (2T)3 𝔼x [ ∫ 󵄨󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 √ω x

2T

≤ 284 (2T)3 ∫ 0 4

≤

3

0

2

|k|8 e−8|s|ω(k) (1 − e−ε|k| /2 )8 ds ω(k)4 (ω(k) + |k|2 /2)8 2

28 (2T) |k|3 (1 − e−ε|k| /2 )8 . 8 (ω(k) + |k|2 /2)8

Thus we obtain 1/4

󵄨󵄨 󵄨󵄨8 1/4 (1 − e−ε|k| /2 )8 ) ∫ (𝔼x [󵄨󵄨󵄨XT (k)󵄨󵄨󵄨 ]) dk ≤ C ∫ ( 󵄨 󵄨 (1 + |k|/2)3 (ω(k) + |k|2 /2)5 2

ℝ3

ℝ3

2

(1 − e−εr /2 )2 r 2 dr → 0 ≤C∫ (1 + r/2)3/4 (r + r 2 /2)5/4 ∞

0

dk

252 | 2 The Nelson model by path measures as ε ↓ 0. This gives 2T

4 󵄨󵄨 β∇Ψε (Bs , s) 󵄨󵄨2 lim sup 𝔼 [( ∫ 󵄨󵄨󵄨 ∫ dBs 󵄨󵄨󵄨 dk) ] = 0. 󵄨 󵄨 ε↓0 x∈ℝ3 √ω x

ℝ3

0

On the other hand, we also have 4 4 󵄨󵄨 βα (T) 󵄨󵄨2 󵄨 󵄨8 1/4 󵄨󵄨 dk) ] ≤ ( ∫ (𝔼x [󵄨󵄨󵄨 βαε (T) 󵄨󵄨󵄨 ]) dk) 𝔼x [( ∫ 󵄨󵄨󵄨 ε 󵄨󵄨 √ω 󵄨󵄨 󵄨 √ω 󵄨󵄨 ℝ3

ℝ3

and 2

󵄨󵄨 βα (T) 󵄨󵄨8 (1 − e−ε|k| /2 )8 󵄨󵄨 ] ≤ 𝔼x [󵄨󵄨󵄨 ε . 󵄨 √ω 󵄨󵄨 ω(k)4 (ω(k) + |k|2 /2)8 It follows that 4 4 −εr 2 󵄨󵄨 βα (T) 󵄨󵄨2 󵄨󵄨 dk) ] ≤ ( ∫ (1 − e ) dr) → 0 𝔼 [( ∫ 󵄨󵄨󵄨 ε 󵄨 󵄨 √ω 󵄨 r(1 + r/2)2 ∞

x

2

0

ℝ3

as ε ↓ 0. Hence for a given δ > 0, there exists ε0 such that for all ε ≤ ε0 , sup 𝔼x [eg

x∈ℝ3

2 ̃ ren S0

‖Q̃ ε − Q̃ 0 ‖2 ] ≤ δeC .

(2.11.65)

Furthermore, for every δ > 0 there exists ε0󸀠 such that for all ε ≤ ε0󸀠 , 1/2 g 2 ̃ ren 󵄨2 g 2 ̃ ren 󵄨4 󵄨󵄨 g2 ̃ ren 󵄨󵄨 g2 ̃ ren 1/2 󵄨 󵄨 𝔼x [󵄨󵄨󵄨e 2 Sε − e 2 S0 󵄨󵄨󵄨 ‖Q̃ 0 ‖2 ] ≤ (𝔼x [󵄨󵄨󵄨e 2 Sε − e 2 S0 󵄨󵄨󵄨 ]) (𝔼x [‖Q̃ 0 ‖4 ]) ≤ δC1 eC2 . 󵄨 󵄨 󵄨 󵄨

These estimates follow by Corollary 2.122 and (2.11.24). Hence for a given δ > 0 we obtain 2 g 2 ̃ ren 󵄨2 󵄨 󵄨󵄨 g ̃ ren sup 𝔼x [󵄨󵄨󵄨e 2 Sε − e 2 S0 󵄨󵄨󵄨 ‖Q̃ 0 ‖2 ] ≤ δC1 eC2 (2.11.66) 󵄨 󵄨 3 x∈ℝ for sufficiently small ε. Due to (2.11.65)–(2.11.66) we see that g 2 ̃ ren 󵄨󵄨2 󵄨󵄨 g2 ̃ ren sup 𝔼x [󵄨󵄨󵄨e 2 Sε Q̃ ε − e 2 S0 Q̃ 0 󵄨󵄨󵄨 ] 󵄨 󵄨 x∈ℝ3

≤ sup 𝔼x [eg x∈ℝ3

2 ̃ ren S0

g 2 ̃ ren 󵄨2 󵄨󵄨 g2 ̃ ren 󵄨 ‖Q̃ ε − Q̃ 0 ‖2 ] + sup 𝔼x [󵄨󵄨󵄨e 2 Sε − e 2 S0 󵄨󵄨󵄨 ‖Q̃ 0 ‖2 ] ≤ δD 󵄨 󵄨 x∈ℝ3

(2.11.67)

with a constant D > 0 for sufficiently small ε > 0. Together with (2.11.67), we conclude from (2.11.63) that ‖Pε − P0 ‖L1 (ℝ3 ) ≤ δDe2T‖V‖∞ ‖F‖‖G‖.

2.11 Ultraviolet divergence

| 253

ren

ε

We see that e−tHN is the strong limit of positivity preserving operators e−t(HN −Eε ) . ren Hence e−tHN is at least positivity preserving. By using formula (2.11.43) we can furthermore show that in fact the semigroup is positivity improving. ren

Corollary 2.127. For every t > 0 the operator e−tHN is positivity improving. In particular, if a ground state of HNren exists, then it is unique. Proof. Let t = 2T and Φ ∈ L2 (Q ) be nonnegative. Φ can be approximated by functions (Φn )n∈ℕ ⊂ S (Q ) such that Φn = Fn (ϕ(f1n ), . . . , ϕ(fmn n )), where Fn ∈ S (ℝmn ) is a j

nonnegative function and fi ∈ M . Suppose that Ψ = Fn (ϕ(f1 ), . . . , ϕ(fm )) ∈ S (Q ). For g ∈ M we have eaM (g) Ψ = Fn (ϕ(f1 ) + (g, f1 )ℋM , . . . , ϕ(fm ) + (g, fm )ℋM ) ≥ 0. We conclude that e−THf eaM (g) is positivity improving for every T > 0. In particular, the ∗ ̃ bounded operator AÃ = eaM (UT ) e−2THf eaM (UT ) is also positivity improving for all T > 0. 2 3 Let F, G ∈ L (ℝ × Q ) be nonnegative functions. By formula (2.11.43) we have T

ren

(F, e−2THN G) = ∫ 𝔼x [e− ∫−T V(Bs )ds e

g2 2

S0ren

̃ (F(B−T ), AAG(B T ))] dx > 0.

ℝ3

2.11.6 Gibbs measures and applications In this section we show that the family of probability measures {μren T }T>0 defined below ren has a limit μren as T → ∞ in local sense. By using μ we can derive further ground ∞ ∞ state expectations. In Section 2.11.6 we use Assumption 2.128 below. Assumption 2.128. The following conditions hold. (1) Dispersion relation: ω(k) = |k|; (2) External potential: V is a continuous bounded function; (3) Number of particle: N = 1; (4) Infrared cutoff: function ξλ : ℝ3 → ℝ satisfies that (i) ξλ=0 (k) = 1; (ii) ∫{|k|≤1} |ξλ (k)|2 /ω(k)3 dk < ∞ for λ > 0; 1 ∗ ̂ ik⋅x ξλ ) + aM (φe ̂̃ −ik⋅x ξλ )); (iii) HIε (x) = (a (φe √2 M ren (5) Ground state: For λ > 0, Hren = HN=1 has the normalized ground state Ψren . The examples of infrared cutoff functions are (1) ξλ (k) = 1{|k|≥λ} , (2) ξλ (k) = √ω(k)/ωλ (k) with ωλ (k) = √|k|2 + λ2 . We regard λ ≥ 0 as a parameter, i. e., in the case of λ > 0, infrared cutoff is introduced, and in the case of λ = 0, no infrared cutoff is introduced.

254 | 2 The Nelson model by path measures Remark 2.129 (Existence of ground state and infrared cutoff). The existence of the ground state is significantly dependent on infrared cutoffs. It is established that Hren has the ground state for λ > 0 but no ground state for λ = 0. Let Hε = Hp ⊗ 1 + 1 ⊗ Hf + gHIε , and the renormalization term with the infrared cutoff is given by 2

Eε = −

g 2 e−ε|k| |ξλ (k)|2 β(k)dk. ∫ 2 ω(k) ℝ3

For λ = 0 it was shown that Hε − Eε → Hren as ε → 0 in the sense of strong semigroup in Theorem 2.118. In the same way as this for λ > 0 we can also show Hε − Eε converges as ε → 0, and we denote the limit by the same notation Hren . Note also that Ψren > 0 and the ground state is unique. We consider two-sided Brownian motion on the probability space (X, ℬ(X), 𝒲 x ). Fix a nonnegative function f ∈ L2 (ℝ3 ) and let ren

LT

= LTren (λ) = f (B−T )f (BT )e

g2 2

T

S0ren (T) − ∫−T V(Bs )ds

e

.

Here S0ren (T) = S0ren (λ, T) is given by (2.11.32) with infrared cutoff λ ≥ 0, i. e., T

t

T

S0ren (T) = 2 ∫ ( ∫ ∇φ0 (Bs − Bt , s − t)ds) dBt − 2 ∫ φ0 (Bs − Bt , s − T)ds −T

−T

(2.11.68)

−T

with φ0 (x, t) = φ0 (λ, x, t) = ∫ ℝ3

e−ik⋅x−|t|ω(k) |ξλ (k)|2 β(k)dk. 2ω(k)

Define the probability measure μren T on (X, ℬ (X)) by ren

ℬ(X) ∋ A 󳨃→ μT (A) =

1 ∫ 𝔼x [1A LTren ]dx, ZT ℝ3

where ZT is the normalizing constant such that Z1 ∫ℝ3 𝔼x [LTren ]dx = 1. We call the T measure μren T the finite volume Gibbs measure associated with the renormalized Nelson model. Using the functional integral representation we see that (f ⊗ 1, e−2THren f ⊗ 1) = ∫ 𝔼x [LTren ]dx. ℝ3

2.11 Ultraviolet divergence

| 255

ren We shall show that μren T converges to μ∞ in local sense as T → ∞ and Gibbs measure exists. The procedure of the proof of the convergence is however similar to the case of the Nelson Hamiltonian discussed in Section 2.8, and hence we show only an outline of the proof. BM Let F[−T,T] = σ(Br , −t ≤ r ≤ t) and we set BM

GT = ⋃ F[−s,s] , 0≤s≤T

BM

G = ⋃ F[−s,s] . s≥0

We consider the collection of probability spaces (X, σ(G ), μren T ), T > 0. As seen below, in a similar way to the series of Lemmas 2.71–2.74 we can show the convergence of μren T as T → ∞. Let fT = e−T(Hren −Eren ) f ⊗ 1, where Eren = inf Spec(Hren ). Define T

− ∫−T V(Bs )ds Iren e [−T,T] = e

g2 2

S0ren (T) a∗M (UT ) −2THf aM (Ũ T )

e

e

e

.

Here T

g UT (k) = − ∫ ξλ (k)e−|s+T|ω(k) e−ik⋅Bs ds, √2 −T T

g Ũ T (k) = − ∫ ξλ (k)e−|s−T|ω(k) eik⋅Bs ds. √2 −T

Define an additive set function μren : G → [0, ∞) by μren (A) = e2Eren T ∫ 𝔼x [1A (Ψren (B−T ), Iren [−T,T] Ψren (BT ))]dx, ℝ3

BM A ∈ F[−T,T] .

BM BM The set function μren is well-defined, i. e., for A ∈ F[−T,T] ⊂ F[−S,S] ,

μren (A) = e2Eren T ∫ 𝔼x [1A (Ψren (B−T ), Iren [−T,T] Ψren (BT ))] dx ℝ3

= e2Eren S ∫ 𝔼x [1A (Ψren (B−S ), Iren [−S,S] Ψren (BS ))] dx. ℝ3

The set function μren is also completely additive on (X, G ). By this, there exists a unique probability measure μren ∞ on (X, σ(G )) such that μren = μren ∞ ⌈G

256 | 2 The Nelson model by path measures by Hopf’s extension theorem. We define the additive set function ρT,ren : GT → ℝ by ρT,ren (A) = e2Eren s ∫ 𝔼x [1A ( ℝ3

fT−s (B−s ) ren fT−s (Bs ) , I[−s,s] )]dx ‖fT ‖ ‖fT ‖

BM for A ∈ F[−s,s] with s ≤ T. The set function ρT,ren is well-defined, and

ρT,ren (A) = e2Eren r ∫ 𝔼x [1A ( ℝ3

fT−r (B−r ) ren fT−r (Br ) , I[−r,r] )]dx ‖fT ‖ ‖fT ‖

= e2Eren s ∫ 𝔼x [1A ( ℝ3

fT−s (B−s ) ren fT−s (Bs ) , I[−s,s] )] dx ‖fT ‖ ‖fT ‖

BM holds for all r ≤ s ≤ T and A ∈ F[−r,r] . By Hopf’s extension theorem again there exists a probability measure ρ̄ T,ren on (X, σ(GT )) such that

ρT,ren = ρ̄ T,ren ⌈GT . We can also see the relationship between ρ̄ T,ren and μren T . Let s ≤ T and A ∈ Gs . We have ρ̄ T,ren (A) = μren T (A). Finally, we obtain local convergence of {μren T }T>0 below. Theorem 2.130 (Local convergence). The family of probability measures {μren T }T≥0 converges to μren on (X, σ( G )) in local sense for arbitrary λ > 0. ∞ Proof. Suppose that A ∈ Gs for some s. We have 2Eren s μren ∫ 𝔼x [1A ( T (A) = ρ̄ T,ren (A) = e ℝ3

fT−s (B−s ) ren fT−s (Bs ) , I[−s,s] )] dx. ‖fT ‖ ‖fT ‖

Since λ > 0, Hren has the ground state. Hence fT /‖fT ‖ → Ψren strongly as T → ∞ and we have 2Eren s ren lim μren ∫ 𝔼x [1A (Ψren (B−s ), Iren T (A) = e [−s,s] Ψren (Bs ))] dx = μ∞ (A).

T→∞

ℝ3

The probability measure μren ∞ is the Gibbs measure associated with the renormalized Nelson model. In a similar way to Corollaries 2.78 and 2.79 we have the corollaries below. Corollary 2.131. Let f and g be bounded measurable functions on ℝd . Then it follows that for a ≤ b, 𝔼μ∞ [f (Ba )g(Bb )] = (f Ψren , e−(b−a)(Hren −Eren ) gΨren ).

| 257

2.11 Ultraviolet divergence

Proof. The proof is the same as that of Corollary 2.78. Corollary 2.132. Suppose that Ψren ∈ D(|x|). Then it follows that for a ≤ b, 𝔼μ∞ [|Ba ||Bb |] = (|x|Ψren , e−(b−a)(Hren −Eren ) |x|Ψren ). Furthermore it follows that 𝔼μ∞ [Ba ⋅ Bb ] = (xΨren , e−(b−a)(Hren −Eren ) xΨren ). Proof. The proof is same as that of Corollary 2.79. Now we shall investigate expectation values of self-adjoint operator O with respect to Ψren by the Gibbs measure we constructed. In order to investigate (Ψren , OΨren ) we discuss a functional integral representation of (e−THren F, e−dΓ(ρ) e−THren F) with F = f ⊗ 1, since e−THren f ⊗ 1/‖e−THren f ⊗ 1‖ → Ψren as T → ∞ for 0 < f ∈ L2 (ℝ3 ). Lemma 2.133. Suppose that ρ is a positive function on ℝ3 such that ∫ ℝ3

1 − e−ρ(k) |ξλ (k)|2 dk < ∞. ω(k)3

(2.11.69)

Then it follows that 0

T

(fT , e−dΓ(ρ) fT ) = 𝔼μren [exp (−g 2 ∫ ds ∫ Wρ (Bs − Bt , s − t)dt)] , T ‖fT ‖2 0 −T [ ] where Wρ (x, t) = ∫ ℝ3

e−|t|ω(k) −ik⋅x e (1 − e−ρ(k) )|ξλ (k)|2 dk. 2ω(k)

Proof. By Proposition 2.85 we have (e−T(Hε −Eε ) f ⊗ 1, e−dΓ(ρ) e−T(Hε −Eε ) f ⊗ 1) ‖e−T(Hε −Eε ) f ⊗ 1‖2 T

=

∫ℝ3 𝔼x [e− ∫−T V(Bs )ds e

g2 2

∫ℝ3 𝔼x [e

0

T

(SεT −4φε (0,0)) −g 2 ∫−T ds ∫0 Wρε (Bs −Bt ,s−t)dt

T − ∫−T

e

V(Bs )ds

e

g2 2

(SεT −4φε (0,0))

] dx

where φε (x, t) is given by (2.11.12) with infrared cutoff λ ≥ 0; 2

φε (x, t) = ∫ ℝ3

e−ε|k| e−ik⋅x−|t|ω(k) β(k)|ξλ (k)|2 dk 2ω(k)

] dx

,

258 | 2 The Nelson model by path measures and SεT = SεD,T + SεOD,T , where SεD,T

T

⟨s+τ⟩

= 2 ∫ ds ∫ Wρε (Bs − Bt , s − t)dt, −T

s

T

T

−T

⟨s+τ⟩

SεOD,T = 2 ∫ ds ∫ Wρε (Bs − Bt , s − t)dt with Wρε (X, t) = ∫ ℝ3

2

e−ε|k| −ik⋅x −|t|ω(k) e e (1 − e−ρ(k) )|ξλ (k)|2 dk. 2ω(k)

Since 󵄨󵄨 0 󵄨󵄨 T 󵄨󵄨 󵄨󵄨 1 − e−ρ(k) 󵄨󵄨 󵄨 ε |ξλ (k)|2 dk < ∞ 󵄨󵄨 ∫ ds ∫ Wρ (Bs − Bt , s − t)dt 󵄨󵄨󵄨 ≤ ∫ 3 󵄨󵄨 󵄨󵄨 ω(k) 󵄨󵄨 ℝ3 󵄨󵄨−T 0 uniformly in paths, we have (fT , e−dΓ(ρ) fT ) (e−T(Hε −Eε ) f ⊗ 1, e−dΓ(ρ) e−T(Hε −Eε ) f ⊗ 1) = lim ε→0 ‖fT ‖2 ‖e−T(Hε −Eε ) f ⊗ 1‖2 T

=

∫ℝ3 𝔼x [e− ∫−T V(Bs )ds e

= 𝔼μren [e−g T

2

0

g2 2

∫ℝ3 𝔼x [e

0

T

S0ren (T) −g 2 ∫−T ds ∫0 Wρ (Bs −Bt ,s−t)dt

e

T − ∫−T

V(Bs )ds

e

g2 2

S0ren (T)

]

]

T

∫−T ds ∫0 Wρ (Bs −Bt ,s−t)dt

].

Then the desired result is obtained. Theorem 2.134. Suppose that ρ is a positive function on ℝ3 and (2.11.69) holds. Let λ > 0. Then 0

(Ψren , e

−dΓ(ρ)

∞

[exp (−g 2 ∫ ds ∫ Wρ (Bs − Bt , s − t)dt)] . Ψren ) = 𝔼μren ∞ −∞ 0 ] [

Proof. By Lemma 2.133 we see that (fT , e−dΓ(ρ) fT ) T→∞ ‖fT ‖2

(Ψren , e−dΓ(ρ) Ψren ) = lim = lim 𝔼μren [e−g T T→∞

2

0

T

∫−T ds ∫0 Wρ (Bs −Bt ,s−t)dt

Then we show the theorem.

] = 𝔼μren [e−g ∞

2

0

∫−∞ ds ∫0 Wρ (Bs −Bt ,s−t)dt ∞

].

2.11 Ultraviolet divergence

| 259

Using Theorem 2.134 and the functional integral representation of (F, e−THren G) due to Theorem 2.120, we can show the super-exponential decay of the boson number in the ground state. The number operator N can be decomposed as N = N0 + N∞ ,

(2.11.70)

where N0 = dΓ(1{|k|≤1} ) denotes the number operator for lower momentum and N∞ = dΓ(1{|k|>1} ) for higher momentum. Lemma 2.135. Let λ ≥ 0. Then eβN∞ e−2THren is bounded for T > β. Proof. We have (F, eβN∞ e−2THren G) T

= ∫ 𝔼x [e− ∫−T V(Bs )ds e

g2 2

S0ren

(F(B−T ), eβN∞ eaM (UT ) e−2THf eaM (UT ) G(BT ))] . ∗

̃

ℝ3

It can be seen that eβN∞ eaM (UT ) e−2THf eaM (UT ) = eaM (β1{|k|>1} UT ) eβN∞ e−2THf eaM (UT ) ∗

∗

̃

̃

and T

T

T

T

eβN∞ e−2THf = e− 2 Hf eβN∞ e−THf e− 2 Hf = e− 2 Hf Γ(e−T|k|+β1{|k|>1} )e− 2 Hf T

T

= e− 2 Hf Γ(e1{|k|>1} (−T|k|+β) )Γ(e−T|k|1{|k| β a. e. We can also show that ∗

̃

T

T

𝔼x [‖eaM (β1{|k|>1} UT ) e− 2 Hf eβN∞ e−THf e− 2 Hf eaM (UT ) ‖n ] < ∞ ∗

̃

for T > β in the similar manner to Lemma 2.121, and hence |(F, eβN∞ e−2THren G)| ≤ C‖F‖‖G‖ is proven in the same way as Lemma 2.125, where C is a constant. This implies that eβN∞ e−2THren is bounded for T > β. In the case of λ = 0 we expect that (Ψren , N0 Ψren ) = ∞ by an infrared divergence. We shall show this in Corollary 2.137 below. We can however show the super-exponential decay of boson number of Ψren for λ > 0. Corollary 2.136 (Super-exponential decay of boson number). Let λ > 0. Then Ψren ∈ D(eβN ) for arbitrary β ∈ ℂ.

260 | 2 The Nelson model by path measures Proof. We have Ψren = e2TEren e−2THren Ψren for T > 0. Let β > 0. Hence ‖eβN∞ Ψren ‖ = e2TEren ‖eβN∞ e−2THren Ψren ‖ < ∞ by Lemma 2.135. Hence Ψren ∈ D(eβN∞ ),

(2.11.71)

∀β ∈ ℝ.

Next we estimate eβN0 Ψren . By Theorem 2.134 we see that (Ψren , e−βN0 Ψren ) = 𝔼μren [e−(1−e ∞

−β

0

)g 2 ∫−∞ ds ∫0 W0 (Bs −Bt ,s−t)dt ∞

(2.11.72)

],

where W0 (x, t) = Wβ1{|k|≤1} (x, t) = ∫ |k|≤1

e−|t|ω(k) −ik⋅x e |ξλ (k)|2 dk. ω(k)

Since λ > 0, ∞ 󵄨󵄨 0 󵄨󵄨 |ξ (k)|2 󵄨󵄨 󵄨 󵄨󵄨 ∫ ds ∫ W0 (Bs − Bt , s − t)dt 󵄨󵄨󵄨 ≤ ∫ λ 3 dk < ∞. 󵄨󵄨 󵄨󵄨 ω(k) −∞ 0

|k|≤1

By an analytic continuation with respect to β we can see that 0 ∞ 󵄨󵄨 󵄨 −(1−eβ )g 2 ∫−∞ ds ∫0 W0 (Bs −Bt ,s−t)dt 󵄨󵄨 󵄨󵄨 < ∞ |(Ψren , eβN0 Ψren )| = 󵄨󵄨󵄨󵄨𝔼μren [e ] 󵄨󵄨 󵄨 ∞

for β ∈ ℂ. In particular Ψren ∈ D(eβN0 ),

∀β ∈ ℂ.

(2.11.73)

By (2.11.71) and (2.11.73) we see that |(Ψren , eβN Ψren )| = |(Ψren , eβN0 eβN∞ Ψren )| ≤ ‖eβN0 Ψren ‖‖eβN∞ Ψren ‖ < ∞. Then the proof is complete. The expectation of the number operator with respect to the ground state for the Nelson Hamiltonian HN with ultraviolet cutoff diverges as infrared cutoff is removed. This is shown in Corollary 2.44. In the corollary below we can also show that Ψren has the same property. Corollary 2.137 (Infrared divergence). Suppose that Ψren ∈ D(|x| ⊗ 1). Then lim(Ψren , NΨren ) = ∞. λ↓0

Proof. We have (Ψren , NΨren ) ≥ (Ψren , N0 Ψren ). Taking the derivative of (2.11.72) at β = 0 we have

2.11 Ultraviolet divergence 0

| 261

∞

(Ψren , N0 Ψren ) = 𝔼μren [ ∫ ds ∫ W0 (Bs − Bt , s − t)dt] ∞ 0

0

−∞

|ξλ (k)|2 −|s−t|ω(k) e 𝔼μren [e−ik⋅(Bs −Bt ) ]dk ∞ 2ω(k)

∞

= ∫ ds ∫ dt ∫ −∞

0

0

∞

{|k|≤1}

|ξλ (k)|2 −|s−t|ω(k) e 𝔼μren [cos(k ⋅ (Bs − Bt ))]dk. ∞ 2ω(k)

= ∫ ds ∫ dt ∫ −∞

0

{|k|≤1}

Inequality 𝔼μren [cos(k ⋅ (Bs − Bt ))] ≥ 1 − ∞

|k|2 𝔼 ren [B2 + B2s − 2Bt Bs ] ≥ 1 − |k|2 ‖|x|Ψren ‖2 2 μ∞ t

follows from Corollary 2.132: [Bs ⋅ Bt ] = (xΨren , e−|s−t|(Hren −Eren ) xΨren ) ≥ 0. 𝔼μren ∞ Writing C = ‖|x|Ψren ‖2 , we have (Ψren , N0 Ψren ) ≥

|ξλ (k)|2 1 (1 − C|k|2 )dk. ∫ 2 ω(k)3 {|k|≤1}

Since ξλ (k) → 1 as λ → 0, the right-hand side above diverges as λ → 0. This proves the corollary. We can also see Gaussian domination of Ψren by using the Gibbs measure μren ∞ . Corollary 2.138 (Gaussian domination of the ground state). Let λ > 0. Take h ∈ S 󸀠 (ℝ3 ) 2 2 ̂ ̂ 2 ∈ L1 (ℝ) and β < 1/‖h/√ω‖ ̂ such that h/√ω ∈ L2 (ℝ3 ), ξλ h/ω . Then Ψren ∈ D(e(β/2)ϕ(h) ) and 1

2

‖e(β/2)ϕ(h) Ψren ‖2 =

2 ̂ √1 − β‖h/√ω‖

𝔼μren [ exp ( ∞

βg 2 K(h)2 ) ], 2 ̂ 1 − β‖h/√ω‖

where K(h) denotes the random variable defined by K(h) =

∞

1 ∫ (ξλ e−|r|ω e−ik⋅Br /√ω, h/̂ √ω)dr. 2 −∞

In particular, it follows that lim

2 )−1 ̂ β↑(2‖h/√ω‖

2

‖eβϕ(h) Ψren ‖ = ∞.

262 | 2 The Nelson model by path measures Proof. For every β ∈ ℝ we have (fT , eikβϕ(g) fT ) (e−T(Hε −Eε ) f ⊗ 1, eikβϕ(g) e−T(Hε −Eε ) f ⊗ 1) = lim 2 ε→0 ‖fT ‖ ‖e−T(Hε −Eε ) f ⊗ 1‖2 T 1 x − ∫−T V(Bs )ds = 𝔼 [f (B )f (B )e ST ] dx, ∫ −T T ‖fT ‖2 ℝ3

where ST = e −

k 2 β2 4

̂ T h(k) ξ (k)(∫−T 2ω(k) λ

2 ̂ −ikβg ∫ℝ3 ‖h/√ω‖

e

e−ω(k)|s| e−ik⋅Bs ds)dk

e

g2 2

S0ren

.

Hence it follows that k 2 β2 2 ̂ (fT , eikβϕ(h) fT ) −ikβg ∫ℝ3 [e− 4 ‖h/√ω‖ e = 𝔼μren 2 T ‖fT ‖

̂ T h(k) ξ (k)(∫−T 2ω(k) λ

e−ω(k)|s| e−ik⋅Bs ds)dk

].

Taking the limit T → ∞ we obtain (Ψren , eikβϕ(h) Ψren ) = 𝔼μ∞ [e−k

2 2

β Ih /2 −ikβgIB

e

],

(2.11.74)

h(k) 2 ̂ /2 and IB = ∫ℝ3 2ω(k) where Ih = ‖h/√ω‖ ξλ (k) (∫−∞ e−ω(k)|s| e−ik⋅Bs ds) dk. Thus using (2.11.74), in the same way as Lemma 2.60 we can show that for β < 0, ̂

1

2

(Ψren , eβϕ(h) Ψren ) =

2 ̂ √1 − β‖h/√ω‖

∞

𝔼μ∞ [exp (

βg 2 K(h)2 )] . ̂ 1 − β‖h/√ω‖

2 ̂ Furthermore, we can extend this to β < 1/‖h/√ω‖ in the same way as in Corollary 2.61.

2.11.7 Weak coupling limit and removal of ultraviolet cutoff We consider the weak coupling limit for the massless Nelson model with many particles. For simplicity, we assume d = 3 and no external potential. The N-particle Nelson Hamiltonian is defined by 1 N HN = − ∑ Δj ⊗ 1 + 1 ⊗ Hf + gHI , 2 j=1 where HI is defined in (2.11.9) with φ̂ replaced with (2.11.75) below. With given parameter Λ > 0, we define the ultraviolet cutoff function of HN to be ̂ φ(k) =

1 1|k|≤Λ (k). (2π)3/2

(2.11.75)

2.11 Ultraviolet divergence

| 263

Also, we write HI (Λ) for HI with cutoff function (2.11.75). We also introduce the scaling parameter κ > 0 such that aM (f ) → κaM (f ) and a∗M (f ) → κa∗M (f ). The scaled Nelson Hamiltonian is then defined by HN (κ, Λ) = Hp + gκHI (Λ) + κ2 Hf .

(2.11.76)

We call the limit of HN (κ, Λ) as Λ → ∞ weak coupling limit. Here we consider the asymptotic behavior of e−THN (κ,Λ) as Λ → ∞ and κ → ∞ simultaneously. Before going to a rigorous computation, we give a heuristic description of the 2 asymptotic behavior. First note that e−Tκ Hf → P0 as κ → ∞, with P0 being the projection onto the subspace spanned by the Fock vacuum. One can expect that e−THN (κ,Λ) → e−TH∞ (Λ) ⊗ P0 as κ → ∞ with a self-adjoint operator H∞ (Λ) in L2 (ℝ3N ). It can indeed be seen that (f ⊗ 1, e−THN (κ,Λ) g ⊗ 1) = ∫ 𝔼x [f (B0 )g(BT )e(g

2

T

T

/2) ∫0 ds ∫0 W(Bs −Bt ,s−t)dt

] dx,

ℝ3

where N

1 −ik⋅(Bis −Bjt ) 2 −κ2 |s−t|ω(k) 1 e κ e dk. ∫ 3 ω(k) 2(2π) i,j=1

W(Bs − Bt , s − t) = ∑

|k|≤Λ

Since as κ → ∞, κ 2 e−κ

2

|s−t|ω

→

2 δ(t − s), ω

it follows that T

T

T

0

0

0

g2 g2 N dt lim lim [ ∫ ds ∫ W(Bs − Bt , s − t)dt − g 2 TNE(Λ)] = , ∑∫ Λ→∞ κ→∞ 2 4π i 0. Proof. This is proven in the same way as Propositions 2.30 and 2.31. Using Lemma 2.150, we can show the existence of a ground state of HN (0). By the functional integral representation we have (1, e−THN (0) 1) = 𝔼 [e

g2 2

T

T

∫0 ds ∫0 Wdt

],

(2.12.14)

where W = W(Bs − Bt , s − t) = ∫ ℝd

2 ̂ |φ(k)| e−ik⋅(Bs −Bt ) e−|s−t|ω(k) dk. 3 2ω(k)

2 ̂ Theorem 2.151 (Existence of ground state). If ∫ℝd |φ(k)| /ω(k)3 dk < ∞, then HN (0) has a ground state and it is unique.

2.12 Translation invariant Nelson model | 275

Proof. Uniqueness follows from Corollary 2.148. By (2.12.14) we have

γ(T) =

(𝔼 [e 𝔼 [e

2

g2 2

∫0 ds ∫0 Wdt

g2 2

∫−T ds ∫−T Wdt

T

T

T

])

T

.

]

By reflection symmetry of Brownian motion 𝔼 [e

γ(T) =

g2 2

T

T

∫0 ds ∫0 Wdt

𝔼 [e

] 𝔼 [e

g2

T ∫ 2 −T

g2 2

T ds ∫−T

0

0

∫−T ds ∫−T Wdt

Wdt

]

,

]

and by using the independence of Bs and Bt for s < 0 < t, the expression can be rearranged into γ(T) =

𝔼 [e

g2 2

T

𝔼 [e 0

T

2

T

0

0

∫0 ds ∫0 Wdt+ g2 ∫−T ds ∫−T Wdt

Since ∫−T ds ∫0 Wdt ≤

g2 2

T

T

∫−T ds ∫−T Wdt

1 ∫ 2 ℝd

]

=

𝔼 [e

g2 2

T

𝔼 [e

]

T

0

T

∫−T ds ∫−T Wdt−g 2 ∫−T ds ∫0 Wdt g2 2

T

T

∫−T ds ∫−T Wdt

]

.

]

2 ̂ |φ(k)| /ω(k)3 dk, we obtain

γ(T) ≥ exp ( −

2 ̂ |φ(k)| g2 dk) > 0 ∫ 2 ω(k)3

(2.12.15)

ℝd

for all T > 0. Hence the theorem follows. In the proof of Theorem 2.151 above W(Bs − Bt , s − t) is the pair potential of the Nelson Hamiltonian with a fixed total momentum. 2.12.4 Gibbs measure associated with the ground state of Nelson model with zero total momentum A Gibbs measure can be defined to the process describing the Nelson model with zero total momentum in a similar way to (2.8.2). Let p = 0, EN (0) = E, and denote the ground state of HN (0) simply by Ψg . Define the probability measure νT on (X, ℬ(X)) by ℬ(X) ∋ A 󳨃→ νT (A) = 𝔼[1A e

g2 2

T

T

∫−T ds ∫−T W(Bs −Bt ,s−t)dt

],

which we call the finite volume Gibbs measure associated with the Nelson model with zero total momentum. Recall that we proved that the similar family of measures {μT }T≥0 in (2.8.2) associated with the Nelson model is tight. For {νT }T≥0 , however, it is not straightforward to show tightness. The strategy to show the local convergence of

276 | 2 The Nelson model by path measures {νT }T≥0 is, however, a minor modification of the arguments in Section 2.8. We show an the outline of this argument. BM Let F[−T,T] = σ(Br , −t ≤ r ≤ t) be the natural filtration of (Bt )t∈ℝ , and define BM BM GT = ⋃0≤s≤T F[−s,s] and G = ⋃0≤s F[−s,s] . Also, define the collection of probability spaces (X, σ(G ), νT ), T > 0, and the additive set function ν : G → [0, ∞), ν(A) = e2ET 𝔼 [1A (Ψg , I[−T,T] Ψg )] , Here

BM A ∈ F[−T,T] .

T

I[−T,T] = I∗−T e−gϕE (∫−T δs ⊗φ(⋅−Bs )ds) IT . BM BM The set function ν is well-defined in the sense that for A ∈ F[−T,T] ⊂ F[−S,S] , T ≤ S,

ν(A) = e2ET 𝔼 [1A (Ψg , I[−T,T] Ψg )] = e2ES 𝔼 [1A (Ψg , I[−S,S] Ψg )] holds. This can be checked similarly to Lemma 2.71. By Hopf’s extension theorem, ν can be uniquely extended to a measure ν∞ on (X, σ(G )). Furthermore, define the additive set function ϱT : GT → ℝ, ϱT (A) = e2Es 𝔼 [1A (

1T−s 1T−s ,I )] ‖1T ‖ [−s,s] ‖1T ‖

BM for A ∈ F[−s,s] with s ≤ T. Here 1t = e−t(HN (0)−E) 1. Similarly to Lemma 2.73, it can be also seen that ϱT is well-defined and can be uniquely extended to a probability measure ϱ̄ T on (X, σ(GT )). Hence it will follow that

ϱ̄ T (A) = νT (A)

(2.12.16)

for all A ∈ Gs with s ≤ T. Theorem 2.152 (Local convergence). The family of probability measures {νT }T≥0 on (X, σ(G )) converges to the probability measure ν∞ as T → ∞ on (X, σ(G )) in local sense, i. e., νT (A) → ν∞ (A) as T → ∞, for every A ∈ G . Proof. Suppose that A ∈ Gs with some s. We have lim νT (A) = lim ϱ̄ T (A) = lim e2Es 𝔼 [1A (

T→∞

T→∞

T→∞

1T−s (B−s ) 1 (B ) , I[−s,s] T−s s )] . ‖1T ‖ ‖1T ‖

Since 1T /‖1T ‖ → Ψg strongly as T → ∞, we have lim νT (A) = e2Es 𝔼 [1A (Ψg , I[−s,s] Ψg )] = ν∞ (A).

T→∞

We call ν∞ the Gibbs measure associated with the Nelson model with zero total momentum. Using ν∞ we can also prove the super-exponential decay of the ground state Ψg of HN (0).

2.12 Translation invariant Nelson model | 277

Corollary 2.153 (Super-exponential decay of boson number). Let β ∈ ℂ. Then Ψg ∈ D(e−βN ) and (Ψg , e−βN Ψg ) = 𝔼ν∞ [e−(1−e

−β

0

) ∫−∞ ds ∫0 W(Bs −Bt ,s−t)dt ∞

],

β ∈ ℂ.

In particular, Ψg ∈ D(eβN ) for all β > 0. Proof. The proof is similar to Corollary 2.86. We can also prove Gaussian domination of the ground state Ψg of HN (0). Corollary 2.154 (Gaussian domination of the ground state). Take g ∈ S 󸀠 (ℝd ) such 2 2 ̂ ̂ ∈ L2 (ℝd ), and β < 1/‖g/√ω‖ ̂ that g/√ω, g/ω . Then Ψg ∈ D(e(β/2)ϕ(g) ) and 1

2

‖e(β/2)ϕ(g) Ψg ‖2 =

2 √1 − β‖g/√ω‖ ̂

𝔼ν∞ [ exp (

βK(g)2 ) ], 2 ̂ 1 − β‖g/√ω‖

where K(g) denotes the random variable defined by ∞

1 ̂ √ω, g/̂ √ω)dr. K(g) = ∫ (e−|r|ω e−ik⋅Br φ/ 2 −∞

In particular, it follows that lim

2

2 )−1 ̂ β↑(2‖g/√ω‖

‖eβϕ(g) Ψg ‖ = ∞.

Proof. The proof is similar to Corollary 2.87. 2.12.5 P(ϕ)1 -process associated with Nelson Hamiltonian with zero total momentum We consider the Nelson Hamiltonian with zero total momentum HN (0) in this section. We denote its ground state by Ψg = Ψg (0). In Section 2.12.4 we constructed a probability measure ν∞ defined on (X, σ(G )). In this section we extend this measure to a probability measure ν̂ on X × Y. This is a similar procedure to Section 2.8.2 for the Nelson Hamiltonian. In terms of infinite dimensional Ornstein–Uhlenbeck process (ξt )t∈ℝ we have 0,ξ

t

(Ψ, e−tHN (0) Φ)L2 (G) = ∫ 𝔼𝒲×G [τ̂B0 Ψ(ξ0 )e−g ∫0 ξs (φ(⋅−Bs ))ds τ̂Bt Φ(ξt )] dG, M−2

(2.12.17)

278 | 2 The Nelson model by path measures where we write τ̂x = e−iPf ⋅x . ̂

We note that (τ̂x F)(ξ (f1 ), . . . , ξ (fn )) = F(ξ (f1 (⋅ − x)), . . . , ξ (fn (⋅ − x))) for F ∈ L , where n

d

L = {F(ξ (f1 ), . . . , ξ (fn )) | F ∈ S (ℝ ), fj ∈ C0 (ℝ ), j = 1, . . . , n, n ≥ 1}. ∞

Since τ̂x ξ (f ) = ξ (f (⋅ − x)), we can regard τ̂x as a map from M−2 → M−2 . Thus (τ̂x ξ )(f ) = ξ (f (⋅ − x)), and for F ∈ L we see that τ̂x F(ξ ) = F(τ̂x ξ ). For every F ∈ L2 (G) there exists a sequence (Fn )n∈ℕ ⊂ L such that Fn → F in L2 (G) as n → ∞. We have (τ̂x Fn )(ξ ) = Fn (τ̂x ξ ). Taking a subsequence nk we have (τ̂x Fnk )(ξ ) → (τ̂x F)(ξ ) for almost every ξ ∈ M−2 and Fnk (τ̂x ξ ) → F(τ̂x ξ ) a. s. ξ ∈ M−2 , implying τ̂x F(ξ ) = F(τ̂x ξ ) a. s. Hence (2.12.17) can be rewritten as t

0,ξ

(Ψ, e−tHN (0) Φ)L2 (G) = ∫ 𝔼𝒲×G [Ψ(τ̂B0 ξ0 )e−g ∫0 τBs ξs (φ)ds Φ(τ̂Bt ξt )] dG. ̂

(2.12.18)

M−2

We now define an M−2 -valued random process (Yt )t∈ℝ on (X × Y, ℬ(X) × ℬ(Y)) by Yt = τ̂Bt ξt .

(2.12.19)

Hence we also have t

0,ξ

(Ψ, e−tHN (0) Φ)L2 (G) = ∫ 𝔼𝒲×G [Ψ(Y0 )e−g ∫0 τBs Ys (φ)ds Φ(Yt )] dG. ̂

(2.12.20)

M−2

Let F[−T,T] = σ(Yr , −t ≤ r ≤ t) and define GT̂ = ⋃0≤s≤T F[−s,s] and G ̂ = ⋃0≤s F[−s,s] . Also, define the additive set function ν̂ : G ̂ → [0, ∞), 0,ξ

ν(A) = eE(b−a) ∫ 𝔼𝒲×G [1A Ψg (Y0 )I[−T,T] Ψg (Yt )] dG,

A ∈ F[a,b] .

M−2

Here

b

I[a,b] = e−g ∫a Ys (φ)ds . The set function ν is well-defined in the sense that for A ∈ F[a,b] ⊂ F[a󸀠 ,b󸀠 ] , 0,ξ

ν(A) = eE(b−a) ∫ 𝔼𝒲×G [1A Ψg (Y0 )I[a,b] Ψg (Yt )] dG M−2 0,ξ

= eE(b −a ) ∫ 𝔼𝒲×G [1A Ψg (Y0 )I[a󸀠 ,b󸀠 ] Ψg (Yt )] dG 󸀠

󸀠

M−2

2.12 Translation invariant Nelson model | 279

holds. This can be checked similarly to Lemma 2.71. By Hopf’s extension theorem, ν̂ ̂ We denote the can be uniquely extended to a probability measure on (X × Y, σ(G )). extension by the same symbol. Moreover, define the additive set function ν̂T : GT → ℝ, 0,ξ ν̂T (A) = eE(b−a) ∫ 𝔼𝒲×G [1A M−2

1T+a (Ya ) 1T−b (Yb ) I ] dG ‖1T ‖ [−T,T] ‖1T ‖

for A ∈ F[a,b] with s ≤ T. Here 1t = e−t(HN (0)−EN (0)) 1. Since 1t+a /‖1t ‖ → eaEN (0) Ψ as t → ∞, we can prove the local convergence of νT as T → ∞ in a similar way to Theorem 2.80 for the Nelson model. Theorem 2.155 (Local convergence). The family of probability measures {ν̂T }T≥0 on the ̂ converges to the probability measure ν̂ as T → ∞ on probability space (X × Y, σ(G )) ̂ ̂ (X × Y, σ(G )) in local sense, i. e., ν̂T (A) → ν(A) as T → ∞, for every A ∈ G .̂ Note that (f Ψg , e−|s−t|(HN (0)−EN (0)) gΨg )L2 (G) = (f , e−|s−t|L(0) g)L2 (M−2 ,Ψ2g dG) , where L(0) =

1 (H (0) − EN (0))Ψg . Ψg N

Hence by the definition of μ̂ we see that for 0 ≤ t0 ≤ t1 ≤ . . . ≤ tn n

𝔼ν̂ [∏ fj (Ytj )] = (f0 , e−(t1 −t0 )L(0) f1 ⋅ ⋅ ⋅ fn−1 e−(tn −tn−1 )L(0) fn )L2 (M−2 ,Ψ2g dG) ,

(2.12.21)

𝔼ν̂ [f (Ys )] = (1, f )L2 (M−2 ,Ψ2g dG)

(2.12.22)

j=0

s ≥ 0.

Theorem 2.156 (P(ϕ)1 -process). The random process (Yt )t≥0 on (X × Y, ℬ(X) × ℬ(Y), ν)̂ is a Markov process under the natural filtration, with invariant measure Ψ2g dG. Proof. The proof is similar to Theorem 2.81, we only give an outline. Defining p(t, Y, A) = (e−tL(0) 1A )(Y) for t ≥ 0, Y ∈ M−2 and A ∈ ℬ(M−2 ), we have immediately that p(t, Y, M−2 ) = 1 and p(t, ⋅, A) Borel measurable. Since ∫M f (Z)p(t, X, dZ) = (e−tL(0) f )(Y) for bounded function f , the semigroup property e−sL e−tL 1A (Y) = e−(s+t)L 1A (Y) yields ∫ e−tL 1A (Y)p(s, Y, dZ) = ∫ P(t, Z, A)P(s, Y, dZ), M

M

checking the Chapman–Kolmogorov identity. Hence p : [0, ∞) × M−2 × ℬ(M−2 ) → ℝ is a probability transition kernel. We also see by (2.12.21) that 𝔼ν̂ [1A0 (Y0 ) ⋅ ⋅ ⋅ 1An (Ytn )] n

n

j=0

j=q

= ∫ ( ∏ 1Aj (Yj ))( ∏ p(tj − tj−1 , Yj−1 dYj ))1A0 (Y0 )Ψ2g (Y0 )dG(Y0 ). M (n+1)

280 | 2 The Nelson model by path measures Thus (Yt )t≥0 is a Markov process with probability transition kernel p(t, Y, A). Furthermore, ∫ p(t, Y, A)Ψ2g (Y)dG = (Ψg , e−t(HN (0)−EN (0)) 1A Ψg )L2 (M−2 ) = ∫ 1A (Y)Ψ2g (Y)dG, M−2

M−2

and thus Ψ2g dG is an invariant measure for (Yt )t≥0 . The process (Yt )t≥0 is called the P(ϕ)1 -process associated with the Nelson Hamiltonian with zero total momentum. 2.12.6 Removal of ultraviolet cutoff In this section we set again d = 3, and similarly to the the discussion in Section 2.11 of the Nelson model, we remove the ultraviolet cutoff of HN (p) for every p ∈ ℝ3 . 2 ̂ We define the regularized HNε (p) by HN (p) with φ(k) replaced by e−ε|k| /2 as in Section 2.11. By the functional integral representation, we have ε

(1, e−2THN (p) 1) = 𝔼 [eip⋅(BT −B−T ) e with SεT

T

g2 2

SεT

],

T

= ∫ ds ∫ Wε (Bs − Bt , s − t)dt −T

−T

given by the pair potential Wε (x, t) = ∫ ℝ3 g2

2 1 e−ε|k| e−ik⋅x e−|t|ω(k) dk, 2ω(k)

Wε : ℝ3 × ℝ → ℝ.

T

Note that 𝔼[eip⋅(BT −B−T ) e 2 Sε ] ∈ ℝ for all p ∈ ℝ3 . From this we can derive the lemma below in a similar way to Lemma 2.114. Denote 2

e−ε|k| e−ik⋅x−|t|ω(k) φε (x, t) = ∫ β(k)dk, 2ω(k)

ε ≥ 0.

ℝ3

Lemma 2.157. It follows that ε

lim(1, e−2T(HN (p)+g ε↓0

where S0ren

T

2

φε (0,0))

1) = 𝔼 [eip⋅(BT −B−T ) e T

g2 2

S0ren

],

(2.12.23)

t

= −2 ∫ φ0 (BT − Bs , T − s)ds + 2 ∫ ( ∫ ∇φ0 (Bt − Bs , t − s)ds) ⋅ dBt . −T

−T

−T

2.12 Translation invariant Nelson model | 281

Furthermore, there exists a constant c independent of ε such that 󵄨󵄨 −2T(HN (p)+g 2 φε (0,0)) 󵄨󵄨 1)󵄨󵄨 ≤ ec(T+1) . 󵄨󵄨(1, e

(2.12.24)

Proof. (2.12.23) is obtained by Lemma 2.114, and (2.12.24) from Lemma 2.111. To remove the ultraviolet cutoff, we need to obtain a uniform lower bound of HNε (p) with respect to ε. This can be done by using the existence of the ground state for p = 0 and the energy comparison inequality EN (p) ≤ EN (0). Lemma 2.158. There exists C ∈ ℝ such that HNε (p) + g 2 φε (0, 0) > −C, uniformly in ε > 0 and p ∈ ℝ3 . ε

Proof. Let Ψεg be the ground state of HNε (0). Since e−2THN (0) is positivity improving, we see that (1, Ψεg ) ≠ 0 and then ENε (0) + g 2 φε (0, 0) = − lim

T→∞

ε 2 1 log (1, e−2T(HN (0)+g φε (0,0)) 1). 2T ε

2

By (2.12.24) we have − limT→∞ 2T1 log (1, e−2T(HN (0)−g φε (0,0)) 1) ≥ −C, where C is independent of ε > 0. Hence we obtain that ENε (0) + g 2 φε (0, 0) ≥ −C. This bound can be extended to p ∈ ℝ3 . By the energy comparison inequality ENε (0) ≤ ENε (p), we then get that ENε (p) − g 2 φε (0, 0) ≥ −C. Theorem 2.159 (Removal of ultraviolet cutoff). Let p ∈ ℝ3 and 2

e−ε|k| β(k)dk. 2ω(k)

Eε = −g 2 φε (0, 0) = −g 2 ∫ ℝ3

Then there exists a self-adjoint operator HNren (p) such that ε

ren

s-lim e−T(HN (p)−Eε ) = e−THN ε↓0

(p)

ε

T ≥ 0.

, 2

Proof. For Φ, Ψ ∈ S (Q ), the limit limε↓0 (Φ, e−2T(HN (p)+g φε (0,0)) Ψ) exists. This is proven in the same way as Lemma 2.116, and the theorem follows similarly to Theorem 2.118. ren

Furthermore, a functional integral representation of (F, e−2THN (p) G) can also be given in terms of exponentials of creation operators and annihilation operators. Theorem 2.160 (Functional integral representation of Nelson Hamiltonian without ultraviolet cutoff with a fixed total momentum). Suppose that Φ, Ψ ∈ L2 (Q ). Let A and Ã

282 | 2 The Nelson model by path measures ̂ ̃ be given by (2.11.44), and uT (p) = ei(p−Pf )⋅BT , i. e., A = eaM (UT ) e−THf and à = e−THf eaM (UT ) with ∗

T

g UT (k) = − ∫ e−|s+T|ω(k) e−ik⋅Bs ds, √2 −T

T

g Ũ T (k) = −Ū −T (k) = − ∫ e−|s−T|ω(k) eik⋅Bs ds. √2 −T

Then ren

(Φ, e−2THN

(p)

Ψ)L2 (Q) = 𝔼0 [e

g2 2

S0ren

̃ T (p)Ψ)] . (u−T (p)Φ, AAu

(2.12.25)

Proof. The proof is similar to Theorem 2.120. By the Baker–Campbell–Hausdorff formula we obtain T

I∗−T e−gϕ(∫−T δs ⊗ϱε (⋅−Bs )ds) IT = e

g2 2

SεT a∗M (UTε ) −2THf aM (Ū Tε )

e

e

e

,

2

where ϱε = (e−ε|k| /2 )∨ , and UTε and Ũ Tε are given by (2.11.45) and (2.11.46), respectively, i. e., UTε (k) = −

T

2 g ∫ e−|s+T|ω(k) e−ε|k| /2 e−ik⋅Bs ds, √2

−T

T

2 g ε Ũ Tε (k) = −Ū −T (k) = − ∫ e−|s−T|ω(k) e−ε|k| /2 eik⋅Bs ds. √2

−T

Thus we have the functional integral representation ε

(F, e−2T(HN (p)−Eε ) G) = 𝔼0 [e

g2 2

Sεren

ε

̃ε

(u−T (p)Φ, eaM (UT ) e−2THf eaM (UT ) uT (p)Ψ)] . ∗

The limiting argument on ε ↓ 0 is similar to the proof of Theorem 2.120, and we see that ε

̃ε

(u−T (p)Φ, eaM (UT ) e−2THf eaM (UT ) uT (p)Ψ) → (u−T (p)Φ, eaM (UT ) e−2THf eaM (UT ) uT (p)Ψ) ∗

∗

and 󵄨󵄨 󵄨 a∗ (U ε ) −2THf aM (Ũ Tε ) e uT (p)Ψ)󵄨󵄨󵄨 < C 󵄨󵄨(u−T (p)Φ, e M T e with a constant C independent of ε ≥ 0. Write ENren (p) = inf Spec(HNren (p)). The following result is immediate by Theorem 2.159.

̃

2.12 Translation invariant Nelson model | 283

Corollary 2.161 (Properties of ENren (p)). ren (1) If p = 0, then e−THN (0) is positivity improving. (2) The ground state of HNren (0) is unique whenever it exists. (3) The energy comparison inequality ENren (0) ≤ ENren (p) holds. (4) The map ℝd ∋ p 󳨃→ ENren (p) ∈ ℝ is continuous. Proof. Let Φ, Ψ ∈ L2 (Q ). We have ren

(Φ, e−2THN

(0)

Ψ) = 𝔼0 [e

g2 2

S0ren

̃ −iBT ⋅Pf Ψ)] . (Φ, eiB−T ⋅Pf AAe ̂

̂

(2.12.26)

̂ ̃ Since e−iBT ⋅Pf is positivity preserving and AÃ = eaM (UT ) e−2THf eaM (UT ) is positivity improving, we see that e−2THN (0) is positivity improving. This implies that the ground state is unique by the Perron–Frobenius theorem. By Theorem 2.160 we have the bound ∗

󵄨󵄨 󵄨 ̃ −iBT ⋅P̂ f |Φ|)] = (|Ψ|, e−2THN (0) |Φ|), 󵄨󵄨(Ψ, e−2THN (p) Φ)󵄨󵄨󵄨 ≤ 𝔼0 [(|Ψ|, eiB−T ⋅P̂ f AAe 󵄨󵄨 󵄨󵄨 and (3) follows. Finally, we prove (4). We have ren

(Ψ, (e−2THN

(p)

ren

− e−2THN

(q)

̂ ̃ −iBT ⋅P̂ f Φ) (eip⋅(BT −B−T ) − eiq⋅(BT −B−T ) )] . )Φ) = 𝔼0 [(I0 Ψ, eiB−T ⋅Pf AAe p⋅(B −B )

Using the identity eip⋅(BT −B−T ) − eiq⋅(BT −B−T ) = i ∫q⋅(B T−B −T) eiθ dθ, we get T

−T

ren 󵄨󵄨 󵄨 −2THNren (p) ̃ − e−2THN (q) )Φ)󵄨󵄨󵄨 ≤ ‖Ψ‖‖Φ‖𝔼0 [‖AA‖|p − q||BT − B−T |] 󵄨󵄨(Ψ, (e 1/2 1/2 ̃ 2 ]) 𝔼0 [|BT − B−T |2 ] . ≤ |p − q|‖Ψ‖‖Φ‖ (𝔼0 [‖AA‖ ren

Hence e−2THN

(p)

is uniformly continuous in p, which implies statement (4).

Lemma 2.162. If V = 0, then ⊕

HNren = T ̂ −1 ( ∫ HNren (p)dp)T ̂ . ℝ3

Here T ̂ : ℋN = L2 (ℝdx ) ⊗ L2 (Q ) → L2 (ℝdp ) ⊗ L2 (Q ) is the unitary map given by ⊕

T ̂ = (F ⊗ 1) ∫ e

ix⋅P̂ f

dx.

ℝd

Proof. It is enough to show that ren

ren

(F, e−THN G)ℋN = ∫ (T ̂ F(p), e−THN ℝ3

(p)

T ̂ G(p))L2 (Q) dp

284 | 2 The Nelson model by path measures for every F, G ∈ ℋN . By Theorem 2.145 we show that ε

ε

(F, e−T(HN −Eε ) G) = ∫ (T ̂ F(p), e−T(HN (p)−Eε ) T ̂ G(p))dp. ℝ3

We take the limit of ε ↓ 0 of both sides and obtain ren

ε

(F, e−THN G) = lim ∫ (T ̂ F(p), e−T(HN (p)−Eε ) T ̂ G(p))dp. ε↓0

ℝ3

Notice that HNε (p) − Eε ≥ −C with a constant C independent of ε > 0 and p ∈ ℝ3 . We have 󵄨󵄨 ̂ 󵄨 tC −T(HNε (p)−Eε ) ̂ T G(p))󵄨󵄨󵄨 ≤ ‖F(p)‖‖G(p)‖e 󵄨󵄨(T F(p), e and ‖F(p)‖‖G(p)‖eCt ∈ L2 (ℝ3 p ). The Lebesgue dominated convergence theorem yields ren

ε

lim ∫ (T ̂ F(p), e−T(HN (p)−Eε ) T ̂ G(p))dp = ∫ (T ̂ F(p), e−THN ε↓0

(p)

T ̂ G(p))dp,

ℝ3

ℝ3

and the lemma follows. Corollary 2.163 (Ground state energy of the renormalized Nelson Hamiltonian with V = 0). Let V = 0. Then the ground state energy of the Nelson Hamiltonian without ultraviolet cutoff ENren coincides with the ground state energy of the Nelson Hamiltonian without ultraviolet cutoff with zero total momentum ENren (0), i. e., ENren (0) = ENren . Proof. The proof is the same as Corollary 2.149 and we skip it. In the d = 2 case under an infrared cutoff, we can make the following interesting observation. In principle, we could to do the same renormalization procedure as for d = 3, however, this is unnecessary since now φε (0, 0) converges to the finite number φ0 (0, 0) as ε ↓ 0. An important consequence of Lemma 2.150 is then the existence of a ground state of HNren (0). Let κ > 0 be the infrared cutoff parameter, and define HNε,κ (p) by HNε (p) with cutoff 2

2

function e−ε|k| /2 replaced by e−ε|k| /2 1{|k|≥κ} . For d = 2 the renormalization term is given by Eε,κ = −g 2 ∫ ℝ2

2

e−ε|k| β(k)1{|k|≥κ} dk. 2ω(k)

However, using the infrared cutoff, we have |Eε,κ | < ∞ for all 0 ≤ ε. It can be also shown that there exists HNren,κ (p) such that ε,κ

s-lim e−THN ε↓0

(p)

ren,κ

= e−THN

(p)

.

2.12 Translation invariant Nelson model | 285

Theorem 2.164 (Existence of ground state). Let d = 2 and assume κ > 0. Then for every value of g the operator HNren,κ (0) has a ground state Ψg (0), such that (1, Ψg (0)) ≠ 0. Proof. By (2.12.15) we have ε,κ

(1, e−THN

(0)

ε,κ

(1, e−2THN

1)2

(0)

1)

≥ exp ( −

g2 2

2

∫ {|k|≥κ}

e−ε|k| dk). ω(k)3

Taking the limit ε ↓ 0 on both sides gives ren

g2 (1, e−THN (0) 1)2 > exp ( − ren 2 (1, e−2THN (0) 1)

1 dk), ω(k)3

(2.12.27)

1 dk) > 0. ω(k)3

(2.12.28)

∫ {|k|≥κ}

and hence ren

(1, e−THN (0) 1)2 g2 lim > exp ( − ren T→∞ (1, e−2THN (0) 1) 2

∫ {|k|≥κ}

On the other hand, we see that ren

(1, e−THN (0) 1)2 = ‖Pg 1‖2 , ren T→∞ (1, e−2THN (0) 1) lim

where Pg denotes the projection to the subspace Ker (HNren (0) − E ren (0)). From (2.12.28) we readily obtain ‖Pg 1‖2 > 0, which implies that HNren (0) has a ground state Ψg (0) such that (1, Ψg (0)) ≠ 0. 2.12.7 Ground state energy and ultraviolet renormalization term Let Eε (g 2 ) = inf Spec(HNε (0)) be the bottom of the spectrum of the Nelson Hamiltonian with zero total momentum. Suppose that formally Eε (g 2 ) can be expanded in g 2 as Eε (g 2 ) = Eε (0) + a2 g 2 + a4 g 4 + ⋅ ⋅ ⋅ , and the ground state as Ψg = 1 + gϕ1 + g 2 ϕ2 + ⋅ ⋅ ⋅ Note that Eε (0) = 0. From the eigenvalue equation HNε (0)Ψg = Eε (g 2 )Ψg we get ϕ1 = −( 21 Pf 2 + Hf )−1 HIε (0)1 and −1 1 a2 = (1, HIε (0)ϕ1 ) = − (HIε (0)1, ( Pf 2 + Hf ) HIε (0)1) . 2

286 | 2 The Nelson model by path measures Recall that 2

φε (x, t) = ∫ ℝ3

e−ε|k| e−ik⋅x−|t|ω(k) β(k)dk, 2ω(k)

ε ≥ 0.

Hence a2 = Eε = −φε (0, 0) is obtained giving Eε (g 2 ) − Eε (0) = Eε . g→0 g2 lim

The ultraviolet renormalization term Eε coincides with the coefficient of g 2 in the expansion of the ground state energy Eε (g 2 ) of the Nelson Hamiltonian with zero total momentum. Furthermore, we expect that lim |Eε (g 2 ) − g 2 Eε | < ∞. ε↓0

For the remainder of this section we discuss rigorously the above informal calculations. Putting p = 0, we have ε

(1, e−2THN (0) 1) = 𝔼[e

g2 2

SεT

].

Lemma 2.165. It follows that Eε (g 2 ) = − lim

T→∞

ε 1 log(1, e−2THN (0) 1). 2T

In particular, g2 T 1 log 𝔼[e 2 Sε ]. T→∞ 2T

Eε (g 2 ) = − lim Proof. This follows from Lemma 1.56.

We derive a precise estimate of the expectation 𝔼[e order to estimate (Eε (g 2 ) − Eε (0))/g 2 .

g2 2

Sεren

] in the lemma below in

Lemma 2.166. Let 21 < θ < 1. Then there exist constants b, c > 0, independent of g, such that for every ε > 0, 𝔼 [e

g2 2

Sεren

4

] ≤ exp [b (c + (g 4 + g 1−θ )(T + 1) + g 2 log T) + c(τ)(g 2 /2)T] ,

(2.12.29)

2

where c(τ) = 8π ∫0 e−εr e−τr dr. In particular, the ground state energy of HNren and HNren (0) are estimated as ∞

4 c(τ) 2 b ENren (0) = ENren ≥ − (g 4 + g 1−θ ) − g . 2 4

(2.12.30)

2.12 Translation invariant Nelson model | 287

Proof. Let Sεren = SεOD + YεT + ZεT . It has been seen before that for every T

2

𝔼[eαYε ] ≤ e(α +α

2 1−θ

1 2

< θ < 1, (2.12.31)

)(T+1)cY

T

with a constant cY . We estimate 𝔼[eαZε ]. There exists a constant M > 0 such that |φε (Bs − BT , s − T)| ≤ |φε (0, s − T)| < M for all T, and φε (0, u) ≤

C u

with a constant C. We have 2T

2T

1

|ZεT | ≤ 2 ∫ φε (0, u)du + 2 ∫ φε (0, u)du ≤ 2M + ∫ 1

1

0

1 du = 2M + C log(2T). u

(2.12.32)

Finally, we compute SεOD . We have T−τ

T

|SεOD | ≤ 2 ∫ ds ∫ dt ∫ −T ∞

s+τ

= 4π ∫ e−εr 0

2

ℝ3

2 1 e−ε|k| e−|s−t|ω(k) dk 2ω(k)

e−τr −(2T−τ)r (e − 1 + (2T − τ)r) dr ≤ c(τ)T. r

(2.12.33)

The bound (2.12.29) follows by the Schwarz inequality 𝔼[e(g

2

/2)(SεOD +YεT +ZεT )

] ≤ (𝔼[eg

2

YεT

])

1/2

(𝔼[eg

2

(SεOD +ZεT )

1/2

])

,

and (2.12.31)–(2.12.33). From g 2 ren 1 log 𝔼[e 2 S0 ], T→∞ 2T

ENren (0) = − lim (2.12.30) also follows.

The constants b and c given in Lemma 2.166 depend on τ. Now we show a key lemma. Lemma 2.167. Let

1 2

< θ < 1. Also, let b > 0 and c(τ) as in Lemma 2.166. Then

󵄨󵄨 Eε (g 2 ) 󵄨󵄨 1 1 2 2 1+θ 󵄨󵄨 󵄨 1−θ ) + c(τ)) . 󵄨󵄨 g 2 + φε (0, 0)󵄨󵄨󵄨 ≤ 2 (b(g + g 2

(2.12.34)

288 | 2 The Nelson model by path measures Proof. We have g2 ren 1 log 𝔼 [e 2 (Sε +4Tφε (0,0)) ] T→∞ 2T

Eε (g 2 ) = − lim and

g 2 ren 1 log 𝔼 [e 2 Sε ] . T→∞ 2T

Eε (g 2 ) = −g 2 φε (0, 0) − lim Hence

g 2 ren 1 1 1 |Eε (g 2 ) + g 2 φε (0, 0)| ≤ 2 lim log 𝔼 [e 2 Sε ] . 2 g g T→∞ 2T

Using Lemma 2.166 we obtain (2.12.34). Theorem 2.168 (Ultraviolet renormalization term). It follows that Eε (g 2 ) = Eε g→0 g2

(2.12.35)

󵄨󵄨 󵄨󵄨 lim 󵄨󵄨󵄨Eε (g 2 ) − g 2 Eε 󵄨󵄨󵄨 < ∞. 󵄨 ε↓0 󵄨

(2.12.36)

lim

and

Proof. By (2.12.34) we see that 󵄨󵄨 1 󵄨󵄨 E (g 2 ) lim 󵄨󵄨󵄨 ε 2 − Eε 󵄨󵄨󵄨 ≤ c(τ) 󵄨 4 g→0 󵄨 g holds for every τ > 0. Also, limτ→∞ c(τ) = 0 implies (2.12.35). Furthermore, (2.12.36) can be derived from (2.12.34) and the fact that limε↓0 c(τ) < ∞. Corollary 2.169. Let HNε be the Nelson Hamiltonian with V ≡ 0. Then inf Spec(HNε ) . g→0 g2

Eε = lim

Proof. This follows from inf Spec(HNε ) = Eε (g 2 ) and Theorem 2.168. 2.12.8 Gibbs measures and applications ren

In Section 2.11.2 we constructed the functional integral representation of (Φ, e−THN (p) Ψ). ren In particular e−THN (0) is positivity improving. Hence in the same way as HNren we can construct the Gibbs measure associated with HNren (0) and we shall show that the famren ily of probability measures {νTren }T>0 defined below has a limit ν∞ as T → ∞ in local sense. Let 1 HNε (0) = Pf2 + Hf + gHIε (0), 2

2.12 Translation invariant Nelson model | 289

and the renormalization term with the infrared cutoff ξλ is given by 2

Eε = −

e−ε|k| |ξλ (k)|2 g2 β(k)dk. ∫ 2 ω(k) ℝ3

In Section 2.12.8 we suppose the assumptions below. Assumption 2.170. The following conditions hold. (1) Dispersion relation: ω(k) = |k|; (2) Infrared cutoff: function ξλ : ℝ3 → ℝ satisfies that (i) ξλ=0 (k) = 1; (ii) ∫{|k|≤1} |ξλ (k)|2 /ω(k)3 dk < ∞ for λ > 0; 1 ∗ ̂ λ ) + aM (φξ ̂̃ λ )); (a (φξ (iii) HIε (0) = √2 M (3) Ground state: For λ > 0, HNren (0) has the normalized ground state Ψren (0). For λ = 0 it was shown that HNε (0) − Eε → HNren (0) as ε → 0 in the sense of strong semigroup in Theorem 2.159. In the same way as this for λ > 0 we can also show HNε (0) − Eε converges as ε → 0, and we denote the limit by the same notation HNren (0). Let ren

MT

= MTren (λ) = e

g2 2

T

S0ren (T) − ∫−T V(Bs )ds

e

,

where (Bt )t∈ℝ is two-sided Brownian motion on the probability space (X, ℬ(X), 𝒲 x ) and S0ren (T) = S0ren (λ, T) is given by (2.11.68). Define the probability measure νTren on (X, ℬ(X)) by ren

ℬ(X) ∋ A 󳨃→ νT (A) =

1 0 𝔼 [1A MTren ], ZT

where ZT is the normalizing constant such that Z1 𝔼0 [MTren ] = 1. We call the measure T νTren the finite volume Gibbs measure associated with the renormalized Nelson model with zero total momentum. Using the functional integral representation we see that ren

(1, e−2THN

(0)

1) = 𝔼0 [MTren ].

ren We shall show that νTren converges to ν∞ in local sense as T → ∞ and Gibbs measure exists. Wr can prove this in a similar way to the case of the renormalized Nelson Hamiltonian discussed in Section 2.11.6 by replacing HNren and f ⊗ 1 with HNren (0) and 1, BM BM respectively. Let F[−T,T] = σ(Br , −t ≤ r ≤ t) and we set G = ⋃s≥0 F[−s,s] . We consider ren the collection of probability spaces (X, σ(G ), νT ), T > 0. Let ren

1T = e−T(HN

(0)−ENren (0))

1,

where ENren (0) = inf Spec(HNren (0)). For λ > 0 it follows that limT→∞ 1T /‖1T ‖ = Ψren . Using this fact we can obtain local convergence of {νTren }T>0 below.

290 | 2 The Nelson model by path measures Theorem 2.171 (Local convergence). The family of probability measures {νTren }T≥0 conren verges to ν∞ on (X, σ(G )) in local sense for arbitrary λ > 0. Notice that ren

ren ν∞ (A) = e2sEN

(0)

𝔼0 [1A (Ψren (0), Iren [−s,s] Ψren (0))] dx

ren for A ∈ Gs . The probability measure ν∞ is the Gibbs measure associated with the renormalized Nelson model with zero total momentum. Now we shall investigate expectation values of self-adjoint operator O with respect to Ψren by the Gibbs measure.

Lemma 2.172. Suppose that ρ is a positive function on ℝ3 such that ∫ ℝ3

1 − e−ρ(k) |ξλ (k)|2 dk < ∞. ω(k)3

(2.12.37)

Then it follows that 0

T

(1T , e−dΓ(ρ) 1T ) = 𝔼νren [exp (−g 2 ∫ ds ∫ Wρ (Bs − Bt , s − t)dt)] , T ‖1T ‖2 0 −T ] [ where Wρ (x, t) = ∫ ℝ3

e−|t|ω(k) −ik⋅x e (1 − e−ρ(k) )|ξλ (k)|2 dk. 2ω(k)

By Lemma 2.172 and a limiting argument same as Theorem 2.134 we have the theorem below. Theorem 2.173. Suppose that ρ is a positive function on ℝ3 and (2.12.37) holds. Let λ > 0. Then 0

∞

2 ren [exp (−g (Ψren (0), e−dΓ(ρ) Ψren (0)) = 𝔼ν∞ ∫ ds ∫ Wρ (Bs − Bt , s − t)dt)] . −∞ 0 [ ] ren

Using Theorem 2.173 and the functional integral representation of (F, e−THN (0) G) due to Theorem 2.160, we can show the super-exponential decay of boson number in the ground state. Recall that the number operator N can be decomposed as N = N0 + N∞ . See (2.11.70). Corollary 2.174 (Super-exponential decay of boson number). Let λ > 0. Then Ψren (0) ∈ D(eβN ) for arbitrary β ∈ ℂ.

2.13 Polaron model |

291

ren

Proof. In a similar way to Lemma 2.135 we can show that eβN∞ e−2THN (0) is bounded for T > β. Furthermore (Ψren , eβN0 Ψren ) < ∞ for any β > 0 can be proven by using the ren Gibbs measure ν∞ in a similar way to Corollary 2.136.

ren We can also see Gaussian domination of Ψren (0) by using the Gibbs measure ν∞ .

Corollary 2.175 (Gaussian domination of the ground state). Let λ > 0. Take h ∈ S 󸀠 (ℝ3 ) 2 ̂ ̂ 2 ∈ L1 (ℝ) and β < 1/‖h/√ω‖ ̂ such that h/√ω ∈ L2 (ℝ3 ), ξλ h/ω . Then Ψren (0) ∈ 2 (β/2)ϕ(h) D(e ) and 1

2

‖e(β/2)ϕ(h) Ψren (0)‖2 =

2 ̂ √1 − β‖h/√ω‖

ren [ exp ( 𝔼ν∞

βg 2 K(h)2 ) ], 2 ̂ 1 − β‖h/√ω‖

where K(h) denotes the random variable defined by ∞

1 K(h) = ∫ (ξλ e−|r|ω e−ik⋅Br /√ω, h/̂ √ω)dr. 2 −∞

In particular, it follows that lim

2

2 )−1 ̂ β↑(2‖h/√ω‖

‖eβϕ(h) Ψren (0)‖ = ∞.

2.13 Polaron model 2.13.1 Definition of the polaron model The polaron model is related to the Nelson model, and describes the interaction of electrons with the quantized vibrations of a crystal lattice. A conduction electron in an ionic crystal or a polar semiconductor are prototype cases of a polaron. In this section we discuss the three-dimensional polaron model and study its ultraviolet renormalization. The main steps of the discussion are similar to the Nelson Hamiltonian. While the state space of bosons of the Nelson Hamiltonian is given by ℱb (ℋ̂ M ), for the polaron we use ℱb (L2 (ℝ3 )). The polaron Hamiltonian ⊕

Hpol = Hp ⊗ 1 + 1 ⊗ N + g ∫ Φ(x)dx ℝ3

is a self-adjoint operator on the Hilbert space 2

3

2

3

ℋpol = L (ℝ ) ⊗ ℱb (L (ℝ )).

Here g ∈ ℝ is a coupling constant, N is the phonon number operator, and Φ(x) =

1 ̃̂ ik⋅x /ω)), ̂ −ik⋅x /ω) + a(φe (a∗ (φe √2

292 | 2 The Nelson model by path measures is the interaction term, with a and a∗ denoting the annihilation operator and the creation operator acting in ℱb (L2 (ℝ3 )), respectively. They satisfy ̂ = (f ̄̂, g)̂ L2 (ℝ3 ) , [a(f ̂), a∗ (g)]

̂ = 0 = [a∗ (f ̂), a∗ (g)]. ̂ [a(f ̂), a(g)]

We use Assumption 2.176 below in Section 2.13 until otherwise stated. Assumption 2.176. The following conditions hold. (1) Dispersion relation: ω(k) = |k|; ̂ ̂ ̂ (2) Charge distribution: φ ∈ S 󸀠 (ℝ3 ), φ(k) = φ(−k) and φ/ω ∈ L2 (ℝ3 ); (3) External potential: V ∈ ℛKato . ̂ Note that the charge distribution of the polaron model is φ/ω and differs from the ̂ in the Nelson model. It can shown that Hpol is self-adjoint on D(Δ⊗1)∩ expression φ/√ω D(1 ⊗ N) and bounded from below by Kato–Rellich theorem, moreover, it is essentially self-adjoint on any core of − 21 Δ ⊗ 1 + 1 ⊗ N. 2.13.2 Functional integral representation To define the polaron Hamiltonian on function space, let (ϕ(f ); f ∈ L2real (ℝ3 )) be a family of Gaussian random variables on (Q , Σ, μ) indexed by L2real (ℝ3 ), and similarly (ϕE (F); F ∈ L2real (ℝ4 )) on (QE , ΣE , μE ) indexed by L2real (ℝ4 ). The polaron Hamiltonian Hpol is then defined on L2 (ℝ3 ) ⊗ L2 (Q ) by ⊕

̃ − x))dx, Ĥ pol = Hp ⊗ 1 + 1 ⊗ N̂ + ∫ ϕ(φ(⋅ ℝ3

where ∨ ̂ φ̃ = (φ/ω) .

We will keep using the same notations Hpol for Ĥ pol , and ℋpol for L2 (ℝ3 ) ⊗ L2 (Q ). As for the Nelson model, the V = 0 case gives a translation invariant model. Thus similarly the polaron Hamiltonian can be decomposed through the fiber Hamiltonians Hpol (p) with total momentum p ∈ ℝ3 , giving ⊕

Hpol ≅ ∫ Hpol (p)dp. ℝ3

Here

1 Hpol (p) = (p − Pf )2 + N + gΦ, 2

p ∈ ℝ3 ,

2.13 Polaron model |

293

and Φ = Φ(0). The state space of phonons is ℱb (L2 (ℝ3 )). Similarly to the Nelson model, using the polaron state space, there exists a family of isometries τ̄s : L2real (ℝ3 ) → L2real (ℝ4 ),

s ∈ ℝ,

such that τ̄s∗ τ̄t = e−|t−s|

s, t ∈ ℝ.

They are given by e−itk0 1 ̄t f (k0 , k) = τ̂ f ̂(k), √π √1 + |k |2 0

(k0 , k) ∈ ℝ × ℝ3 .

A straightforward calculation shows that 1 e−ik0 (t−s) ̄ ̂ ̄s f , τ̂ ̄t g) = ∫ (τ̄s f , τ̄t g) = (τ̂ dk0 ∫ f ̂(k)g(k)dk = e−|t−s| (f , g). 2 π 1 + |k0 | ℝ3

ℝ

We denote the second quantization of τ̄t by It̄ = ΓInt (τ̄t ). Thus It̄ : L2 (Q ) → L2 (QE ) is an isometry for every t ∈ ℝ, and the factorization formula ̂ I∗s̄ It̄ = e−|t−s|N ,

s, t ∈ ℝ,

̂ holds. In particular, we have (F, e−t N G)L2 (Q) = (I0̄ F, It̄ G)L2 (QE ) .

Theorem 2.177 (Functional integral representation of polaron Hamiltonian). (1) If F, G ∈ ℋpol , then T

T

̃ ̄ F(B−T ), e−ϕE (∫−T τ̄s φ(⋅−Bs)ds) (F, e−2THpol G)ℋpol = ∫ 𝔼x [e− ∫−T V(Bs )ds (I−T IT̄ G(BT ))] dx. ℝ3

(2) Let Ψ, Φ ∈ L2 (Q ), and p ∈ ℝ3 . Then T

̃ ̄ u−T (p)Ψ(B−T ), e−ϕE (∫−T τ̄s φ(⋅−Bs)ds) (Ψ, e−2THpol (p) Φ)L2 (Q) = 𝔼0 [(I−T IT̄ uT (p)Φ(BT ))] .

Here uT (p) = ei(p−Pf )⋅BT . ̂

Proof. The proof of (1) is the same as that of (F, e−2THN G) by replacing It and τt with It̄ and τ̄t , respectively. The proof of (2) is also the same as that of (Ψ, e−2THN (p) Φ), and we skip the details.

294 | 2 The Nelson model by path measures 2.13.3 Removal of ultraviolet cutoff In this section we discuss the ultraviolet problem for the polaron Hamiltonian. Due ̂ appearing in the polaron Hamiltonian, the situation to the more regular function φ/ω ̂ simplifies from the Nelson Hamiltonian’s case where we had φ/√ω. Also, since the free field term of the polaron Hamiltonian is given by the number operator N, there is no infrared divergence in this case. 2 ε ̂ Denote the polaron Hamiltonian with φ(k) = e−ε|k| /2 by Hpol . The vacuum expecε

tation of e−THpol is given by ε

(f ⊗ 1, e−THpol h ⊗ 1) = ∫ 𝔼x [f (B0 )h(BT )e

g2 2

t

ε Spol − ∫0 V(Bs )ds

e

]dx,

ℝ3

where ε Spol (Bs

T

T

0

0

ε (Bs − Bt , s − t)dt − Bt , s − t) = ∫ ds ∫ Wpol

is the pair interaction associated with the polaron Hamiltonian with the pair potential ε Wpol (x, t) = e−|t| ∫ ℝ3

2 1 e−ε|k| e−ik⋅x dk. 2 2ω(k)

Also, we have 0 Wpol (x, t)

=e

∞

−|t|

−|t| e−|t| sin u 1 −ik⋅x 2e e dk = 2π du = π , ∫ ∫ |x| u |x| 2ω(k)2 0

ℝ3

ε 0 and clearly Wpol (x, t) → Wpol (x, t) for every (x, t) as ε ↓ 0. Let T

T

0

0

0 Spol (Bt − Bs , t − s) = π 2 ∫ ds ∫

Then

e−|t−s| dt. |Bt − Bs |

g2 0 󵄨 󵄨󵄨 g2 ε 󵄨 lim 𝔼 [󵄨󵄨󵄨e 2 Spol − e 2 Spol 󵄨󵄨󵄨] = 0. 󵄨 󵄨 ε↓0

For f , h ∈ L2 (ℝ3 ) it is then immediate to get ε

lim(f ⊗ 1, e−THpol h ⊗ 1) = ∫ 𝔼x [f (B0 )h(BT )e ε↓0

g2 2

t

0 Spol − ∫0 V(Bs )ds

e

]dx.

(2.13.1)

ℝ3

It is seen that no ultraviolet renormalization is needed in (2.13.1). Thus the theorem below can be proven in the same way as for the Nelson Hamiltonian.

2.13 Polaron model |

295

Theorem 2.178 (Removal of ultraviolet cutoff). ren (1) There exists a self-adjoint operator Hpol such that ε

ren

s-lim e−THpol = e−THpol ,

T ≥ 0.

ε↓0

ren (2) If p ∈ ℝ3 , then there exists a self-adjoint operator Hpol (p) such that ε

ren

s-lim e−THpol (p) = e−THpol (p) , ε↓0

T ≥ 0.

By using the Baker–Campbell–Hausdorff formula we can also derive a functional ren integral representation of (Φ, e−THpol Ψ) in terms of exponentials of creation and annihilation operators. Let T

VT (k) = −

g e−ik⋅Bs ds ∫ e−|s+T| √2 ω(k) −T

T

eik⋅Bs g and Ṽ T (k) = − ds, ∫ e−|s−T| √2 ω(k) −T

and define B = ea

∗

e

(VT ) −TN

̃ B̃ = e−TN ea(VT ) .

and

We present the next result without proof since one can argue similarly to the case of the Nelson Hamiltonian with no cutoff. Theorem 2.179 (Functional integral representation of polaron Hamiltonians without ultraviolet cutoff). Let 0 Spol (Bs

T

T

2

− Bt , s − t) = π ∫ ds ∫ −T

−T

e−|s−t| dt. |Bs − Bt |

(1) We have VT , Ṽ T ∈ L2 (ℝ3k ) a. s. (2) If F, G ∈ ℋpol , then T

ren

(F, e−2THpol G)ℋpol = ∫ 𝔼x [e− ∫−T V(Bs )ds e

g2 2

0 Spol

̃ (F(B−T ), BBG(B T ))] dx.

ℝ3

(3) If Φ, Ψ ∈ L2 (Q ), then ren

(Φ, e−2THpol (p) Ψ)L2 (Q) = 𝔼0 [e

g2 2

0 Spol

̃ T (p)Ψ)] . (u−T (p)Φ, BBu

Here uT (p) = ei(p−Pf )⋅BT . ̂

ren ren With Epol (p) = inf Spec(Hpol (p)) and Epol (p) = inf Spec(Hpol (p)), the functional integral representation gives similar properties as seen before in the Nelson model’s case.

296 | 2 The Nelson model by path measures ren Corollary 2.180 (Properties of Epol (p) and Epol (p)). ren

(1) If p = 0, then e−THpol (0) and e−THpol (0) are positivity improving. ren (2) The ground states of Hpol (0) and Hpol (0) are in each case unique whenever they exist. ren ren (3) The energy comparison inequalities Epol (0) ≤ Epol and Epol (0) ≤ Epol (p) hold. 3 3 ren (4) The maps ℝ ∋ p 󳨃→ Epol (p) ∈ ℝ and ℝ ∋ p 󳨃→ Epol (p) ∈ ℝ are continuous. Proof. The proof is the same as the proofs of Corollaries 2.148 and 2.161.

2.14 Functional central limit theorem 2.14.1 Gibbs measures with no external potential In this section, contrary to Section 2.12.2, beginning with a functional integral representation of the semigroup generated by the Nelson Hamiltonian, we define a Hamiltonian equivalent to a single particle translation invariant Nelson Hamiltonian with total momentum zero, i. e., p = 0, and consider a Gibbs measure associated with this Hamiltonian. This construction of the Gibbs measure can be regraded as a construction of a Gibbs measure with external potential V = 0 in Section 5.2 in Volume 1. We shall assume a general conditions on a pair potential considered in Section 2.14. We assume that W(x, t) =

2 ̂ |φ(k)| 1 e−ik⋅x e−ω(k)|t| dk ∫ 2 ω(k)

(2.14.1)

ℝd

with general ω and φ̂ satisfying Assumption 2.181 below. Assumption 2.181. The following conditions hold. (1) Dispersion relation: ω(k) ≥ 0, ω(k) = ω(−k). ̂ ̂ ̂ ̂ and φ/√ω, φ/ω ∈ L2 (ℝd ). (2) Charge distribution: φ ∈ S 󸀠 (ℝ3 ), φ(k) = φ(−k), 2 3 ̂ (3) Infrared regularity: ∫ℝd |φ(k)| /ω (k)dk < ∞. In this section we use d-dimensional Brownian motion over the whole time. We recall that W(Bs − Bt , s − t) is uniformly bounded with respect to paths as a function of x and t, and ∞

∫ |W(Bs − Bt , s − t)|ds ≤ ∫ ℝd

−∞ 0

∞

2 ̂ |φ(k)| dk, ω(k)2

∫ ds ∫ |W(Bs − Bt , s − t)|dt ≤ −∞

0

2 ̂ |φ(k)| 1 dk. ∫ 2 ω(k)3 ℝd

(2.14.2)

(2.14.3)

2.14 Functional central limit theorem

| 297

We define a measure on (X, ℬ(X)) by d𝒫T0

T

T

−T

−T

1 = exp ( ∫ ds ∫ W(Bs − Bt , s − t)dt) d𝒲 0 , ZT

(2.14.4)

where T

T

ZT = 𝔼𝒲 0 [exp ( ∫ ds ∫ W(Bs − Bt , s − t)dt)] . −T −T [ ] For x ∈ ℝd let τ̂x be the shift by x on M−2 such that (τ̂x ξ )(f ) = ξ (θx f ) for ξ ∈ M−2 , where (θx f )(⋅) = f (⋅ − x). Lemma 2.182. For every x ∈ ℝd the map τ̂x can be extended to a unitary on L2 (G), and x 󳨃→ τ̂x is a strongly continuous group on L2 (G). Proof. First note that for f , g ∈ M , 1 1 ̂ (f , θx g)M = ∫ f ̂(k)θ̂ dk = ∫ f ̂(k)eikx g(k) dk x g(k) ω(k) ω(k) ℝd

ℝd

1 ̂ dk = (θ−x f , g)M . = ∫ θ̂ −x f (k)g(k) ω(k) ℝd

Similarly, (θx f , θx g)M = (f , g)M . We have 1 (eiξ (f ) , τ̂x eiξ (g) )L2 (G) = exp {− ((f , f )M + (f , θx g)M + (θx g, f )M + (θx g, θx g)M )} 4 1 = exp {− ((θ−x f , θ−x f )M + (θ−x f , g)M + (g, θ−x f )M + (g, g)M )} 4 = (θ−x eiξ (f ) , eiξ (g) )L2 (G) . Since the functions of the form eiξs (f ) span L2 (G), the equality of the first two lines above shows strong continuity of the map x 󳨃→ τ̂x , and the semigroup property τ̂x+y = τ̂x τ̂y . Moreover, the equality of the first and last lines shows that the adjoint of τ̂x is given by τ̂−x . Since clearly τ̂x τ̂−x = 1, τ̂x is a unitary map. Let L = {F(ξ (f1 ), . . . , ξ (fn )) | F ∈ S (ℝn ), fj ∈ C0∞ (ℝd ), j = 1, . . . , n, n ≥ 1}. Note that L is dense in L2 (G). We also have τ̂x F(ξ (f1 ), . . . , ξ (fn )) = F(ξ (θx f1 ), . . . , ξ (θx fn )). For T > 0 define measures on X ⊗ Y by

298 | 2 The Nelson model by path measures W

0

𝒫0 = 𝒲 ⊗ 𝒢 ,

d𝒫TW =

(2.14.5) T

1 exp(− ∫ τ̂Bs ξs (φ)ds)d𝒫0W , ZT

(2.14.6)

−T

where ZT is the normalizing constant T

ZT = 𝔼𝒫 W [ exp (− ∫ τ̂Bs ξs (φ)ds)]. 0

−T

Next lemma can be proven in a similar way showing the fact ϱ̄ T = νT in (2.12.16) in BM Section 2.12.4. We recall that GT = ⋃0≤s≤T F[−s,s] . Lemma 2.183. Let A ∈ Gt . Then 𝒫TW (A) = 𝒫T0 (A) for t ≤ T. T

Proof. Since ∫−T τ̂Bs ξs (φ)ds is a Gaussian random variable on Y with mean zero and covariance T

T

T

󵄨󵄨 󵄨󵄨2 1 𝔼𝒢 [󵄨󵄨󵄨 ∫ τ̂Bs ξs (φ)ds󵄨󵄨󵄨 ] = ∫ ds ∫ W(Bs − Bt , s − t)dt, 󵄨 󵄨 2 −T −T [ −T ] the lemma follows. We are interested in the limit of 𝒫T0 as T → ∞. In light of Lemma 2.183, we will study the family of measures (𝒫TW )T≥0 instead of (𝒫T0 )T≥0 . At first sight this seems more difficult as the state space ℝd × M−2 is now infinite-dimensional. However, the great advantage is that 𝒫TW is tightly connected to a Markov process, as we will now see. Definition 2.184. We define the operator Pt on L2 (ℝd ) ⊗ L2 (G) by t

(F, Pt G)L2 (ℝd )⊗L2 (G) = ∫ 𝔼𝒲 x ⊗𝒢 [F(B0 , ξ0 )e− ∫0 τBs ξs (φ)ds G(Bt , ξt )]dx ̂

(2.14.7)

ℝd

for F, G ∈ L2 (ℝd ) ⊗ L2 (G). In a similar way to Corollary 1.88 we have t 󵄨󵄨 󵄨 󵄨󵄨 ∫ 𝔼𝒲 x ⊗𝒢 [F(B0 , ξ0 )e− ∫0 τ̂Bs ξs (φ)ds G(Bt , ξt )]dx 󵄨󵄨󵄨 ≤ ‖F‖‖G‖etI , 󵄨󵄨 󵄨󵄨

(2.14.8)

ℝd

2 ̂ where I = ∫ℝd |φ(k)| /ω(k)3 dk.

Remark 2.185. We have already shown in Theorem 2.12 that the right-hand side of (2.14.7) is a functional integral representation of the semigroup generated by the Nelson Hamiltonian defined on L2 (ℝd ) ⊗ L2 (G). One of the main purpose of this section is, however, to derive the translation invariant Nelson Hamiltonian with total momentum zero from the right-hand side of (2.14.7).

2.14 Functional central limit theorem

| 299

Proposition 2.186. We have Pt = e−tH , where H is given by

t ≥ 0,

(2.14.9)

1 H = − Δx + Ĥ f + HI , 2

(2.14.10)

where Ĥ f = dΓ(ω(−i∇)) is the free field Hamiltonian and (HI f )(x, ξ ) = (τ̂x ξ (φ))f (x, ξ ) = ξ (φ(⋅ − x))f (x, ξ ). Moreover, t

‖Pt f ‖2L2 (G) (x) = 𝔼𝒲 x ⊗𝒢 [f (B−t , ξ−t )e− ∫−t τBs ξs (φ)ds f (Bt , ξt )] ̂

(2.14.11)

for f ∈ L2 (ℝd ) ⊗ L2 (G). Proof. Equation (2.14.9) is due to Theorem 2.12. For (2.14.11), note that for g ∈ C0∞ (ℝd ), the definition of Pt and the semigroup property give ∫ g(x)‖Pt f ‖2L2 (G) (x)dx = (Pt f , gPt f ) = (f , Pt gPt f )

ℝd

t

2t

= ∫ 𝔼𝒲 x ⊗𝒢 [f (B0 , ξ0 )e− ∫0 τBs ξs (φ)ds g(Bt )e− ∫t ℝd

2t

= ∫ 𝔼𝒲 x ⊗𝒢 [f (B0 , ξ0 )e− ∫0

̂

τ̂Bs ξs (φ)ds

τ̂Bs ξs (φ)ds

f (B2t , ξ2t )]dx

g(Bt )f (B2t , ξ2t )]dx.

ℝd

Since both (Bt )t∈ℝ and (ξt )t∈ℝ are invariant under the time shift, we can replace Bs by Bs−t and ξs with ξs−t above, and we find t

∫ g(x)‖Pt f ‖2L2 (G) (x)dx = ∫ g(x)𝔼𝒲 x ⊗𝒢 [f (B−t , ξ−t )e− ∫−t τBs ξs (φ)ds f (Bt , ξt )]dx.

ℝd

̂

ℝd

Since g ∈ C0∞ (ℝd ) is arbitrary, we obtain the claim. We have already seen in Proposition 2.4 that H in (2.14.10) is the Nelson Hamiltonian and is self-adjoint on D(H) = D(−Δ) ∩ D(Ĥ f ), and for each x ∈ ℝd , Ĥ f + HI (x) is a self-adjoint operator in L2 (G) with the domain D(Ĥ f ) and bounded from below uniformly in x. We shall extend Pt on some spaces which are not necessarily Hilbert spaces. Hence we need to work with the following additional function spaces: L∞ (ℝd ; L2 (G)) = {f : ℝd × M−2 → ℂ | ess sup ‖f (x, ⋅)‖L2 (G) < ∞}, d

d

L (ℝ × M−2 ) = {f : ℝ × M−2 → ℂ | ∞

x∈ℝd

ess sup

(x,ξ )∈ℝd ×M−2

|f (x, ξ )| < ∞}.

300 | 2 The Nelson model by path measures Clearly, L∞ (ℝd × M−2 ) ⊂ L∞ (ℝd ; L2 (G)), and unitary equivalence: L2 (ℝd ) ⊗ L2 (G) ≅ L2 (ℝd ; L2 (G)) ≅ L2 (ℝd × G) are used without further notice. Moreover, we write C(ℝd ; L2 (G)) = {f : ℝd × M−2 → ℂ | ℝd ∋ x 󳨃→ f (x, ⋅) ∈ L2 (G) is continuous},

Cb (ℝd ; L2 (G)) = C(ℝd ; L2 (G)) ∩ L∞ (ℝd ; L2 (G)).

Here C(ℝd ; L2 (G)) is a Banach space with the norm ‖f ‖L∞ (ℝd ;L2 (G)) . We need to study Pt on two closed subspaces of C(ℝd ; L2 (G)). The first one is 󵄨󵄨 C0 (ℝd ; L2 (G)) = {f ∈ Cb (ℝd ; L2 (G)) 󵄨󵄨󵄨 lim ‖f (x)‖L2 (G) = 0} . 󵄨 |x|→∞ The second subspace is 𝒯 . To define 𝒯 we need to define U by U : L2 (G) → C(ℝd ; L2 (G)),

(Uf )(x) = (τ̂x f )(⋅).

(2.14.12)

Definition 2.187. 𝒯 is the Hilbert space defined by 2

2

𝒯 = UL (G) = {Uf | f ∈ L (G)}

equipped with the scalar product (f , g)𝒯 = (U −1 f , U −1 g)L2 (G) . Note that for 𝒯 ∋ f = f (x, ξ ) = (Ug)(x, ξ ) = (τ̂x g)(ξ ), ‖f ‖𝒯 = ‖U −1 f ‖L2 (G) = ‖g‖L2 (G) = ‖τ̂x g‖L2 (G) = sup ‖τ̂y g‖L2 (G) = ‖f ‖L∞ (ℝd ;L2 (G)) , y∈ℝd

and U is a unitary operator from L2 (G) onto 𝒯 . We notice that (U −1 f )(x, ξ ) = f (τ̂x ξ ).

(2.14.13)

Note that 𝒯 ⊄ L2 (ℝd ) ⊗ L2 (G) and C(ℝd ; L2 (G)) ⊄ L2 (ℝd ) ⊗ L2 (G). Replacing F and G in Definition 2.188 with Uf and Ug, respectively, we define Pt on 𝒯 : Definition 2.188 (Pt on 𝒯 ). Let g ∈ L2 (G). We define Pt on 𝒯 by t

Pt (Ug)(x, ξ ) = 𝔼𝒲 x ⊗𝒢 ξ [e− ∫0 τBs ξs (φ)ds g(τ̂Bt ξt )]. ̂

(2.14.14)

We denote it by the same notation Pt . In what follows we define Pt f for a map f on ℝ × M−2 by (2.14.14) with Ug replaced by f and we use the same notation Pt unless any confusion may arise. The examples are functions in L∞ (ℝd ; L2 (G)) and C0 (ℝd ; L2 (G)). We need a further auxiliary result. d

2.14 Functional central limit theorem

| 301

Lemma 2.189. (1) {Pt : t ≥ 0} is a C0 -semigroup on L∞ (ℝd ; L2 (G)). (2) If f ∈ L∞ (ℝd × M−2 ) ∩ C(ℝd ; L2 (G)), then Pt f ∈ Cb (ℝd ; L2 (G)). (3) If f ∈ L∞ (ℝd × M−2 ) ∩ C(ℝd ; L2 (G)), then limt→0 supx∈K ‖Pt f (x) − f (x)‖L2 (G) = 0 for any compact set K ⊂ ℝd . Proof. (1) Once we have shown that Pt is bounded on L∞ (ℝd ; L2 (G)), the semigroup property follows from the Markov property of (ξt )t∈ℝ and (Bt )t≥0 . To show boundedness, note that |Pt f | ≤ Pt |f | pointwise. Therefore it suffices to consider nonnegative functions. First let f ∈ L∞ (ℝd × M−2 ). We fix a not necessarily continuous but bounded path X : [−t, t] → ℝd and define the bilinear form t

(f , g)X = 𝔼𝒢 [f (X−t , ξ−t )e− ∫−t τXs ξs (φ)ds g(Xt , ξt )]. ̂

For two paths X and X,̃ t

t

|(f , g)X − (f , g)X̃ | ≤ ‖f ‖∞ ‖g‖∞ 𝔼𝒢 [(e− ∫−t τXs ξs (φ)ds − e− ∫−t τX̃s ξs (φ)ds )2 ]. ̂

̂

(2.14.15)

Gaussian integration with respect to 𝒢 then shows that X 󳨃→ (f , g)X is continuous from L∞ ([−t, t], ℝ) to ℝ. Now let X be a continuous path, and define Xs(n) = ∑n−1 j=−n Xtj/n 1[tj/n,t(j+1)/n) (s). Then |(f , g)X (n) | ≤ eI ‖f (X−t )‖L2 (G) ‖g(Xt )‖L2 (G) for bounded f and g; X (n) → X uniformly on [−t, t] as n → ∞, where I = (1/4) × 2 ̂ /ω(k)3 dk, and thus the above inequality remains valid when we replace X (n) ∫ℝd |φ(k)| by X. For f , g ∈ L2 (ℝd ; L2 (G)) the monotone convergence theorem implies |(f , g)X | ≤ eI ‖f (X−t )‖L2 (G) ‖g(Xt )‖L2 (G)

(2.14.16)

for all X ∈ X. Using this in (2.14.11) shows (1). (2) Choose x, y ∈ ℝd , f , g ∈ L∞ (ℝd × M−2 ) and write fy (x, ξ ) = f (x + y, ξ ). Then ‖Pt f (x) − Pt f (x + y)‖2L2 (G) t

t

= 𝔼G [(𝔼𝒲 x ⊗𝒢 ξ [e− ∫0 τBs ξs (φ)ds f (Bt , ξt ) − e− ∫0 τ(Bs +y) ξs (φ)ds fy (Bt , ξt )])2 ] ̂

̂

= 𝔼𝒲 x [(f , f )B + (fy , fy )B+y − 2(f , fy )B+1{t≥0} y ].

Using (2.14.15), (2.14.16), and the continuity of B 󳨃→ f (B), we see that each term in the last line above converges to (f , f )B as r → 0. It follows that the integrand in the first line converges to zero pathwise, and the second claim follows by the dominated convergence theorem. (3) Define (Qt f )(x, ξ ) = 𝔼𝒲 x ⊗𝒢 ξ [f (Bt , ξt )]. If f is bounded, a calculation similar to (2.14.15) shows ‖Qt f − Pt f ‖L∞ (ℝd ;L2 (G)) → 0 as t → 0. Therefore, we only need to show

302 | 2 The Nelson model by path measures that ‖Qt f − f ‖L2 (G) vanishes uniformly on compact sets as t → 0. Write ft (x, ξ ) = f (x, ξt ). Note that ft is a function on ℝd × Y while f is a function on ℝd × M−2 . By the Schwarz inequality we get ‖Qt f (x) − f (x)‖2L2 (G) ≤ 𝔼𝒲 x [‖ft (Bt ) − f0 (B0 )‖2L2 (𝒢) ]. Moreover, ‖ft (Bt ) − f0 (B0 )‖L2 (𝒢) ≤ ‖ft (Bt ) − ft (B0 )‖L2 (𝒢) + ‖ft (B0 ) − f0 (B0 )‖L2 (𝒢) = ‖f (Bt ) − f (B0 )‖L2 (G) + ‖ft (B0 ) − f0 (B0 )‖L2 (𝒢) .

It thus suffices to show that lim ‖ft (x) − f0 (x)‖L2 (𝒢) = 0

(2.14.17)

t→0

uniformly in x on compact sets and lim 𝔼𝒲 x [‖f (Bt ) − f (B0 )‖m L2 (G) ] = 0

(2.14.18)

t→0

uniformly on compact sets for m = 1, 2. For proving (2.14.17), assume the contrary. Then there exist bounded sequences (xn )n∈ℕ ⊂ ℝd , (tn )n∈ℕ ⊂ [0, ∞) with tn → 0, and ‖ftn (xn ) − f0 (xn )‖L2 (𝒢) > δ for all n. By compactness, we may assume that xn converges to x ∈ ℝd . Then δ < ‖ftn (xn ) − f0 (xn )‖L2 (𝒢)

≤ ‖ftn (xn ) − ftn (x)‖L2 (𝒢) + ‖ftn (x) − f0 (x)‖L2 (𝒢) + ‖f0 (x) − f0 (xn )‖L2 (𝒢)

= ‖ftn (x) − f0 (x)‖L2 (𝒢) + 2‖f (xn ) − f (x)‖L2 (G) .

We assume the continuity of f . Thus we can choose n0 large enough such that ‖f (xn ) − f (x)‖L2 (G) < δ/3, for all n > n0 , and the above calculation shows that ‖ftn (x) − f0 (x)‖L2 (𝒢) > δ/3 for all such n. Since ‖ftn (x) − f0 (x)‖L2 (𝒢) = ‖e−tn Hf f (x) − f (x)‖L2 (𝒢)

must converge to zero by the strong continuity of the semigroup e−t Hf , this is a contradiction and thus (2.14.17) holds. For (2.14.18), note that x 󳨃→ f (x) is uniformly continuous on compact sets. Thus for ε > 0 we can choose δ > 0 such that ̃ L2 (G) < ε/2 whenever |x − x|̃ < δ. By the properties of Brownian motion ‖f (x) − f (x)‖ there exists t0 > 0 such that for 0 ≤ t < t0 , ̂

x

𝒲 (|Bt − x| > δ)
0, Rx (ε) = inf{R ≥ 0 | ‖f (x) − fR (x)‖L2 (G) ≤ ε} < ∞,

(2.14.19)

where Rx (ε) is bounded on compact subsets of ℝd . To see this, assume the contrary. Then there exists (xn )n∈ℕ ⊂ ℝd with xn → x, and Rn = Rxn (ε) > n for all n. Choosing n0 large enough such that ‖f (xn ) − f (x)‖L2 (G) < ε/3 for all n > n0 gives ε = ‖f (xn ) − fRn (xn )‖L2 (G)

≤ ‖f (xn ) − f (x)‖L2 (G) + ‖f (x) − fRn (x)‖L2 (G) + ‖fRn (x) − fRn (xn )‖L2 (G)

≤ ‖f (x) − fRn (x)‖L2 (G) +

2ε 2ε ≤ ‖f (x) − fn (x)‖L2 (G) + 3 3

for each n > n0 . This implies Rx (ε/3) = ∞, in contradiction to (2.14.19). Thus Rx (ε) is bounded on compact sets. Since f ∈ C0 , Rx (ε) = 0 for |x| large enough, and thus Rx (ε) is bounded on all ℝd . Thus bounded functions are dense in C0 . Next we show that Pt leaves C0 invariant. Let f ∈ C0 . From (2.14.11) and (2.14.16) we have ‖Pt f (x)‖L2 (G) ≤ eI 𝔼𝒲 x [‖f (B−t )‖L2 (G) ‖f (Bt )‖L2 (G) ]

= eI 𝔼𝒲 x [1{|Bt |≤|x|/2} ‖f (B−t )‖L2 (G) ‖f (Bt )‖L2 (G) ] + eI 𝔼𝒲 x [1{|Bt |>|x|/2} ‖f (B−t )‖L2 (G) ‖f (Bt )‖L2 (G) ];

𝒲 0 (|Bt | ≥ R) decays exponentially in R for all t, and since f ∈ C0 ⊂ L∞ (ℝd , L2 (G)),

the term in the second line above also decays exponentially. The last line is bounded by a constant times sup|y|>|x|/2 ‖f (x)‖L2 (G) , and this term decays to zero as |x| → ∞, by the assumption f ∈ C0 . In a similar way, we obtain the strong continuity from (3) of Lemma 2.189. Thus Pt is a strongly continuous semigroup of bounded operators on C0 = C0 (ℝ; L2 (G)). Next we investigate {Pt : t ≥ 0} on the Hilbert space 𝒯 . Theorem 2.191. (1) For each t ≥ 0, Pt leaves 𝒯 invariant. (2) {Pt : t ≥ 0} is a C0 -semigroup on 𝒯 . Proof. The semigroup property of {Pt : t ≥ 0} follows from Markov properties of (Bt )t≥0 and (ξt )t∈ℝ . We shall show that Pt is strongly continuous. Choose Uf ∈ 𝒯 . We find t

t

(Pt Uf )(x, ξ ) = 𝔼𝒲 x ⊗𝒢 ξ [e− ∫0 τBs ξs (φ)ds τ̂Bt f (ξt )] = 𝔼𝒲 0 ⊗𝒢 ξ [e− ∫0 τBs +x ξs (φ)ds τ̂Bt +x f (ξt )]. ̂

̂

304 | 2 The Nelson model by path measures The semigroup property τ̂Bs +x = τ̂Bs τ̂x yields that t

t

(Pt Uf )(x, ξ ) = 𝔼𝒲 0 ⊗𝒢 τ̂x ξ [e− ∫0 τBs ξs (φ)ds τ̂Bt f (ξt )] = U(𝔼𝒲 0 ⊗𝒢 ξ [e− ∫0 τBs ξs (φ)ds τ̂Bt f (ξt )]). ̂

̂

(2.14.20) We conclude that Pt leaves 𝒯 invariant. It is seen that L∞ (ℝd ; L2 (G)) ∩ 𝒯 is dense in 𝒯 . Strong continuity then follows directly from (3) of Lemma 2.189. We shall derive the generator of Pt : 𝒯 → 𝒯 . The generator of Pt on 𝒯 is denoted by −H𝒯 . Hence Pt = e−tH𝒯 for t ≥ 0. Corollary 2.192. Let H(0) be a single particle translation invariant Nelson Hamiltonian (2.12.2) on L2 (G) with total momentum p = 0; H(0) = 21 dΓ(−i∇)2 + Ĥ f + HI . Then H𝒯 = UH(0)U −1 . I. e., 𝒯 ≅ L2 (G) and H𝒯 ≅ H(0). In particular D(H𝒯 ) = UD(H(0)) and 1 H𝒯 F = (− Δx + Ĥ f + HI )F 2 for F ∈ U L . Proof. Let f , g ∈ L . Since by (2.14.20) we have t

U −1 Pt Uf (x, ξ ) = 𝔼𝒲 0 ⊗𝒢 ξ [e− ∫0 τBs ξs (φ)ds (Uf )(Bt , ξt )], ̂

(2.14.21)

we can see that t

(Uf , Pt Ug)𝒯 = (f , U −1 Pt Ug)L2 (G) = (f , 𝔼𝒲 0 ⊗𝒢 ξ [e− ∫0 τBs ξs (φ)ds (Ug)(Bt , ξt )])L2 (G) t

̂

= 𝔼𝒲 0 ⊗𝒢 [(Uf )(B0 , ξ0 )e− ∫0 τBs ξs (φ)ds (Ug)(Bt , ξt )]. ̂

In the same way as Remark 2.13 we can see that 1 1 lim (Uf , Pt Ug − Ug)𝒯 = (f , −U −1 (− Δx + Ĥ f + HI )Ug)L2 (G) . t 2

t→0

Then

1 −U −1 (− Δx + Ĥ f + HI )U = −U −1 H𝒯 U 2

on L . Let f ∈ C0∞ (ℝd ) and we have Ueiξ (f ) (x, ξ ) = τ̂x eiξ (f ) = eiξ (f (⋅−x)) = g(x, ξ ).

2.14 Functional central limit theorem

| 305

We set fj = 𝜕xj f and fjj = 𝜕x2j f . Since 𝜕xj g(x, ξ ) = −iξ (fj (⋅ − x))g(x, ξ ), we find that 1 1 d − Δx g(x, ξ ) = − ∑(iξ (fjj (⋅ − x)) − ξ (fj (⋅ − x))2 )g(x, ξ ) 2 2 j=1 and 1 1 d U −1 (− Δx )Ueiξ (f ) = − ∑(iξ (fjj ) − ξ (fj )2 )eiξ (f ) . 2 2 j=1 On the other hand, since dΓ(−i∇j )eiξ (f ) = ξ (fj )eiξ (f ) , we have (dΓ(−i∇j ))2 g(x, ξ ) = (−iξ (fjj ) + ξ (fj )2 )eiξ (f ) . Hence

1 1 U −1 (− Δx )Ueiξ (f ) = (dΓ(−i∇))2 eiξ (f ) 2 2

and we conclude that 1 1 U −1 ( − Δx + Ĥ f + HI )Ueiξ (f ) = ( dΓ(−i∇)2 + Ĥ f + HI )eiξ (f ) = H(0)eiξ (f ) . 2 2 Since L ∋ f is given by n

f (ξ (f1 ), . . . , ξ (fn )) = (2π)−n/2 ∫ f ̂(k)eiξ (∑j=1 kj fj ) dk, ℝn

we have

1 U −1 ( − Δx + Ĥ f + HI )Uf = H(0)f = U −1 H𝒯 Uf . 2

Since L is a core for H(0), H𝒯 = UH(0)U −1 follows. By Corollary 2.192 we can now get some spectral information on H𝒯 . Theorem 2.193 (Ground state of H𝒯 ). H𝒯 has the ground state Ψg ∈ 𝒯 and Ψg can be chosen strictly positive. Proof. Since H𝒯 = UH(0)U −1 and H(0) has the unique and strictly positive ground state Ψg (0) by Theorem 2.151. Hence H𝒯 also has the unique and strictly positive ground state Ψg , i. e., Ψg = UΨg (0). Next we consider the limit of the families (𝒫TW )T>0 on X × Y and (𝒫T0 )T>0 on X. For a bounded interval I ⊂ ℝ, let FI = σ((Bt , ξt ), t ∈ I). Let F∞ = σ( ⋃ F[−s,s] ). s>0

(2.14.22)

306 | 2 The Nelson model by path measures We write E = inf Spec(H𝒯 ). W We define 𝒫∞ to be the unique probability measure on (X×Y, F∞ ), such that for every T > 0 and A ∈ F[−T,T] W

𝒫∞ (A) = e

2TE

T

𝔼𝒫 W [Ψg (B−T , ξ−T )e− ∫−T τBs ξs (φ)ds Ψg (BT , ξT )1A ]. ̂

0

(2.14.23)

We are now ready to state and prove the main result of this section. Theorem 2.194 (Local convergence). W (1) 𝒫TW → 𝒫∞ as T → ∞ in the sense of local convergence, i. e., 𝔼𝒫 W [A] → 𝔼𝒫∞W [A] T for every A ∈ F[−t,t] and every t > 0. 0 (2) The family (𝒫T0 )T>0 converges to a probability measure 𝒫∞ in the sense of local conBM vergence, i. e., 𝔼𝒫 0 [A] → 𝔼𝒫∞0 [A] for every A ∈ F[−t,t] and every t > 0. T

Proof. The proof of (1) is similar to Theorem 2.155, and (2) to Theorem 2.152. In the proofs of Theorems 2.152 and 2.155 replacing ϱ̄ T and νT by 𝒫T0 and 𝒫TW , respectively, we can prove this theorem. 2.14.2 Diffusive behavior 0 W By Theorem 2.194 we know that limit measures 𝒫∞ and 𝒫∞ exist. In this section we discuss the peculiar property of these measures that on a large scale under this measure paths behave like Brownian motion, however with a reduced diffusion constant. The classical central limit theorem yields that the partial sum

Sn =

1 n ∑X √n i=1 i

of independently and identically distributed random variables Xi with mean zero and covariance 𝔼[Xi2 ] = σ 2 weakly converges to the Gaussian random variable with mean zero and covariance σ 2 . We see that for a continuous bounded function g, lim 𝔼[g(Sn )] → 𝔼[g(X)],

n→∞

where X is a Gaussian random variable with covariance σ. We are interested in the asymptotic behaviour of 1 γ ⋅ Btn √n W as t → ∞ for γ ∈ ℝd under the probability measure P∞ .

2.14 Functional central limit theorem

| 307

We define two random processes (Xt )t∈ℝ and (Yt )t∈ℝ in this section. Let F∞ be defined by (2.14.22). Firstly (Xt )t∈ℝ is an ℝd × M−2 -valued random process on the probW ability space (X × Y, F∞ , P∞ ) defined by Xt = (Bt , ξt ) t ∈ ℝ.

(2.14.24)

W Secondly (Yt )t∈ℝ is an M−2 -valued random process on (X × Y, F∞ , P∞ ) defined by

Yt = τ̂Bt ξt ,

t ∈ ℝ.

(2.14.25)

From (2.14.13) it follows that f (Yt ) = (Uf )(Xt ),

(U −1 g)(Yt ) = g(Xt )

(2.14.26)

for f ∈ L2 (G) and g ∈ 𝒯 . We often use this formulae in what follows. Using (Yt )t∈ℝ we can rewrite (Uf , e−tH𝒯 Ug)𝒯 for f , g ∈ L2 (G) in (2.14.14) by t

(Uf , e−tH𝒯 Ug)𝒯 = 𝔼𝒲 0 ⊗𝒢 [f (Y0 )e− ∫0 Ys (φ)ds g(Yt )],

(2.14.27)

where Ys (φ) = (τ̂Bt ξt )(φ). We define L=

1 (H − E)Ψg . Ψg 𝒯

The domain of L is D(L) = {f ∈ 𝒯 | U −1 f Ψg ∈ D(Ĥ f ) ∩ D(dΓ(−i∇))}. We set L2 (M−2 , (U −1 Ψg )2 G) = L2 ((U −1 Ψg )2 G) and L2 (M−2 , Ψ2g G) = L2 (Ψ2g G). Lemma 2.195. Take f , g ∈ L2 (G). It follows (1) and (2) below: (1) 𝔼𝒫∞W [f (Ys )g(Yt )] = (f ̄, e−|s−t|U

−1

LU

g)L2 ((U −1 Ψg )2 G) .

In particular 𝔼𝒫∞W [f (Yt )] = ∫ f (U −1 Ψg )2 dG,

(2.14.28)

𝔼𝒫∞W [f (Yt )g(Yt )] = ∫ fg(U −1 Ψg )2 dG.

(2.14.29)

M−2

M−2

(2) 𝔼𝒫∞W [(Uf )(Xs )(Ug)(Xt )] = (Uf̄ , e−|s−t|L Ug)L2 (Ψ2g G) .

308 | 2 The Nelson model by path measures In particular 𝔼𝒫∞W [(Uf )(Xt )] = ∫ (Uf )Ψ2g dG,

(2.14.30)

𝔼𝒫∞W [(Uf )(Xt )(Ug)(Xt )] = ∫ (Uf )(Ug)Ψ2g dG.

(2.14.31)

M−2

M−2

Proof. Since we have 𝔼𝒫∞W [f (Ys )g(Yt )] = e|s−t|E (Ψg Uf , e−|s−t|H𝒯 (Ψg Ug))𝒯 in the same way as (2.12.21), (1) follows. Since (Uf )(Xt ) = f (Yt ) and (Uf , e−|s−t|L Ug)L2 (Ψ2g G) = ∫ (Uf )(x, ξ )(e−|s−t|L Ug)(x, ξ )Ψ2g (x, ξ )dG(ξ ) M−2

= ∫ f (τ̂x ξ )(U −1 e−|s−t|L Ug)(τ̂x ξ )(U −1 Ψg (τ̂x ξ ))2 dG(ξ ) M−2

= ∫ f (τ̂0 ξ )(U −1 e−|s−t|L Ug)(τ̂0 ξ )(U −1 Ψg (τ̂0 ξ ))2 dG(ξ ) M−2

= ∫ f (ξ )(U −1 e−|s−t|L Ug)(ξ )(U −1 Ψg (ξ ))2 dG(ξ ) = (f , e−|s−t|U

−1

LU

g)L2 ((U −1 Ψg )2 G) ,

M−2

(2) follows. Lemma 2.196 (Markov properties). (1) Y = (Yt )t≥0 is a stationary Markov process with respect to the natural filtration. The generator of Y is −U −1 LU and the invariant measure is (U −1 Ψg )G. (2) X = (Xt )t≥0 is a stationary Markov process with respect to the natural filtration. The generator of X is −L. −1 Proof. By Lemma 2.195, it holds that 𝔼𝒫∞W [f (Ys )g(Yt )] = (f ̄, e−|s−t|U LU g)L2 ((U −1 Ψg )2 G) , and we can also show that

n

𝔼𝒫∞W [ ∏ fj (Ytj )] = (f0̄ , e−|t1 −t0 |K f1 ⋅ ⋅ ⋅ fn−1 e−|tn −tn−1 |K fn )L2 ((U −1 Ψg )2 G) , j=0

where we set K = U −1 LU. Hence it can be proven that Y is a stationary Markov process with respect to the natural filtration in the same way as that of Theorem 2.156. Similar to (1) we see that 𝔼𝒫∞W [(Uf )(Xs )(Ug)(Xt )] = (Uf , e−|s−t|L Ug)L2 (Ψ2g G) by Lemma 2.195. Then (2) follows in the same way as that of Theorem 2.156. Definition 2.197 (Differential operator in 𝒯 ). We define the differential operator in 𝒯 by −i∇̂x = UdΓ(−i∇)U −1 .

2.14 Functional central limit theorem

| 309

Since H𝒯 = UH(0)U −1 , we see that Ψg ∈ D(∇̂xj ). Define 1 1 L̃ = ( − Δx + Ĥ f + HI )Ψg Ψg 2 on C 2 (ℝd ; L2 (G)) by 1 1 Lf̃ (x, ξ ) = ( − Δx Ψg (x, ξ )f (x, ξ ) + Ĥ f Ψg (x, ξ )f (x, ξ ) + HI (x, ξ )Ψg (x, ξ )f (x, ξ )) Ψg 2 It can be seen that L⌈̃ 𝒯 = L. We define the map hγ : ℝd × M−2 → ℝ by γ ∈ ℝd .

hγ (x, ξ ) = γ ⋅ x,

(2.14.32)

Note that hγ is independent of ξ . We have ̃ γ =γ⋅ Lh

∇̂x Ψg Ψg

.

With ρ = U −1 (γ ⋅

∇̂x Ψg Ψg

) ∈ L2 (G)

(2.14.33)

we have ̃ γ (Xt ) = (Uρ)(Xt ) = ρ(Yt ). Lh We are interested in proving a central limit theorem for the process (γ ⋅ Bt )t≥0 under W 0 𝒫∞ , or equivalently the process (γ ⋅ Bt )t≥0 under 𝒫∞ . The fundamental tool will be the martingale central limit theorem. Let t

̃ γ (Xs )ds. Mt = hγ (Xt ) − hγ (X0 ) − ∫ Lh

(2.14.34)

0

̃ γ (Xs ) = ρ(Ys ), (2.14.34) can be also written as Since hγ (Xt ) = γ ⋅ Bt , hγ (X0 ) = 0 and Lh t

γ ⋅ Bt = Mt + ∫ ρ(Ys )ds.

(2.14.35)

0

Lemma 2.198. (Mt )t≥0 is a martingale with respect to the natural filtration of (Xt )t≥0 . Proof. L̃ is the generator of the Markov process (Xt )t≥0 . The lemma follows in a similar way to Theorem 2.93 in Volume 1. Let ℬt = σ(Xr , 0 ≤ r ≤ t). We define P̃ t by (P̃ t F)(x, ξ ) = 𝔼P∞ W [f (Xt )|X0 = (x, ξ )].

(2.14.36)

310 | 2 The Nelson model by path measures Let f = hγ for simplicity. By the Markov property of (Xt )t≥0 we have 𝔼P∞ W [f (Xt )|ℬs ] = P̃ t−s f (Xs ). It is trivial to see that 󵄨󵄨 t 󵄨󵄨 󵄨 ̃ 𝔼P∞ W [Mt |ℬs ] = Ms + 𝔼P W [ f (Xt ) − f (Xs ) − ∫ Lf (Xr )dr 󵄨󵄨 ℬs ] . 󵄨󵄨 ∞ 󵄨󵄨 s 󵄨 ] [ ̃ we have Since f ∈ D(L),

(2.14.37)

t = L̃ P̃ t f = P̃ t Lf̃ and P̃ t f − f = ∫0 L̃ P̃ s fds. It follows that

d ̃ Pf dt t

󵄨󵄨 t t 󵄨󵄨 󵄨 ̃ ̃ ̃ ̃ 𝔼P∞ W [ f (Xt ) − f (Xs ) − ∫ Lf (Xr )dr 󵄨󵄨 ℬs ] = Pt−s f (Xs ) − f (Xs ) − ∫ LPr−s f (Xs )dr 󵄨󵄨 󵄨󵄨 s s 󵄨 ] [ t−s

= P̃ t−s f (Xs ) − f (Xs ) − ∫ L̃ P̃ r f (Xs )dr = 0. 0

Hence the second term on the right-hand side of (2.14.37) is zero and the martingale property is proven. Lemma 2.199. We have 𝔼𝒫∞W [ρ(Yt )2 ] = ‖γ ⋅ ∇̂x Ψg ‖𝒯 , 𝔼𝒫∞W [Mt2 ] = |γ|2 t.

(2.14.38) (2.14.39)

Proof. By (2.14.33) and (2.14.28) we can show that ∇̂x Ψg

𝔼𝒫∞W [ρ(Yt )2 ] = ∫ (U −1 γ ⋅

Ψg

M−2

)2 (U −1 Ψg )2 dG = ‖γ ⋅ ∇̂x Ψg ‖𝒯 .

We have (Lh2γ )(x, ξ ) = |γ|2 + 2hγ (x, ξ )(γ ⋅

∇̂x Ψg (x, ξ ) Ψg (x, ξ )

) = |γ|2 + 2hγ (x, ξ )(Lhγ )(x, ξ ).

(2.14.40)

For simplicity let hγ = f , and 𝔼[. . .] = 𝔼𝒫∞W [. . .]. Write g(Xs ) = (Lf̃ )(Xs ). We have 𝔼[Mt2 ]

2

t

2

t

= 𝔼[f (Xt )] + 𝔼[f (X0 )] + 2 ∫ ds ∫ 𝔼[g(Xs )g(Xr )]dr t

0

s

t

− 2𝔼[f (X0 )f (Xt )] + 2 ∫ 𝔼[f (X0 )g(Xr )]dr − 2 ∫ 𝔼[f (Xt )g(Xr )]dr. 0

Since

0

t

𝔼[f (X0 ) ∫ g(Xr )dr] = 𝔼[f (X0 )(f (Xt ) − f (X0 ))], 0

(2.14.41)

2.14 Functional central limit theorem

| 311

the second, fourth, and fifth terms of (2.14.41) combine to give −𝔼[f 2 (X0 )]. Also, since t

∫ 𝔼[g(Xs )g(Xr )]dr = 𝔼[g(Xs )(f (Xt ) − f (Xs ))], 0

the third term of (2.14.41) decomposes into two expressions, one of which cancels the last term of (2.14.41), while the other gives the last term in the formula. Then it follows that t

𝔼[Mt2 ] = 𝔼[f 2 (Xt )] − 𝔼[f 2 (X0 )] − 2 ∫ 𝔼[f (Xs )(Lf )(Xs )]ds

(2.14.42)

0

and by (2.14.40) we have t

𝔼[Mt2 ] = ∫ 𝔼[(Lh2γ )(Xs ) − 2hγ (Xs )(Lhγ )(Xs )]ds = |γ|2 t. 0

Consider the decomposition (2.14.35). We have 𝔼𝒫∞W [(γ⋅Bt )2 ] < ∞ for all t by Proposition 2.194. Lemma 2.200. We have lim

t→∞

1 𝔼 W [(γ ⋅ Bt )2 ] = |γ|2 − 2(γ ⋅ ∇̂x Ψg , (H𝒯 − E)−1 γ ⋅ ∇̂x Ψg )𝒯 t 𝒫∞

(2.14.43)

Proof. By (2.14.35), t

t

2

𝔼𝒫∞W [(γ ⋅ Bt )2 ] = 𝔼𝒫∞W [Mt2 ] − 𝔼𝒫∞W [( ∫ ρ(Ys )ds) ] + 2𝔼𝒫∞W [γ ⋅ Bt ∫ ρ(Ys )ds]. (2.14.44) 0

0

The third term in (2.14.44) is zero. This can be seen as follows. We have t

𝔼𝒫∞W [γ ⋅ Bt ∫ ρ(Ys )ds] = 𝔼𝒲 0 [γ ⋅ Bt I(B)], 0 [ ] where t

(2.14.45)

t

I(B) = 𝔼𝒢 [Ψg (Y0 )e− ∫0 Ys (φ)ds (∫ ρ(Ys )ds) Ψg (Yt )] . 0 [ ]

(2.14.46)

Put B̃ s = Bt−s − Bt . Then by the reversibility of 𝒢 and the fact that 𝒢 is invariant under the constant shift by τ̂Bt , we have I(X)̃ = I(X). Moreover, X̃ t = −Bt , 𝒲 0 -a. s. and 𝒲 0 is invariant under the transformation B 󳨃→ B.̃ Thus 𝔼𝒲 0 [γ ⋅ Bt I(X)] = −𝔼𝒲 0 [γ ⋅ Bt I(X)] = 0.

(2.14.47)

312 | 2 The Nelson model by path measures 2

t

From (2.14.39) we know that 𝔼𝒫∞W [Mt2 ] = |γ|2 t. We finally estimate 𝔼𝒫∞W [( ∫0 ρ(Ys )ds) ]. In general we can see that limt→∞ (f , e−tK g)/t = 2(f , K −1 g). Hence we have t

2

t

t

0

0

1 1 𝔼 W [( ∫ ρ(Ys )ds) ] = ∫ ds ∫ 𝔼𝒫∞W [ρ(Ys )ρ(Yr )]dr t 𝒫∞ t t

=

0

t

−1 1 ∫ ds ∫(ρ, e−|s−r|U LU ρ)L2 ((U −1 Ψg )2 G) dr t

0

0

t→∞

→ 2(ρ, U −1 L−1 Uρ)L2 ((U −1 Ψg )2 G) = 2(γ ⋅ ∇̂x Ψg , (H𝒯 − E)−1 γ ⋅ ∇̂x Ψg )𝒯 .

Note that the last quantity is automatically finite; this follows from (2.14.44). So we have shown (2.14.43). We will need the following abstract result, where a random process of the form t ∫0 F(Zs )ds can be asymptotically approximated by some martingale. Proposition 2.201 (Functional central limit theorem). Let (S, 𝒮 , π) be a probability space. Let Z = (Zt )t∈ℝ be an S-valued Markov process on a filtered probability space (Ω, F , (Ft )t≥0 , P). Let K be the generator of Z. Assume that (1) (Zt )t∈ℝ is reversible and stational, and e−tK is positivity improving for some t > 0; (2) π is an invariant measure for Z, i. e., 𝔼P [f (Zt )] = ∫S 𝔼Ps [f (Zt )]dπ(s) = ∫S f (s)dπ(s) and F ∈ L2 (S) and 𝔼π [F] = 0; (3) F is in the domain of K −1/2 , where K is the generator of the process (Yt )t∈ℝ . Let

t

Ft = ∫ F(Zs )ds.

(2.14.48)

0

Then there exists a square integrable martingale (Nt )t≥0 with respect to (Ft )t≥0 , with stationary increments, such that 1 sup |F − Ns | = 0 √t 0≤s≤t s 1 lim 𝔼P [|Ft − Nt |2 ] = 0. t→∞ t lim

t→∞

in probability with respect to P,

t

Applying Proposition 2.201 we can approximate ∫0 ρ(Ys )ds by a martingale. Lemma 2.202. There exists a martingale (Nt )t≥0 with respect to the natural filtration of (Yt )t≥0 such that 󵄨󵄨 s 󵄨󵄨 1 󵄨 󵄨 sup 󵄨󵄨󵄨 ∫ ρ(Yr )dr − Ns 󵄨󵄨󵄨 = 0, lim 󵄨󵄨 t→∞ √t 0≤s≤t 󵄨󵄨 0

󵄨󵄨 t 󵄨󵄨2 1 󵄨 󵄨 lim 𝔼P∞ W [󵄨󵄨 ∫ ρ(Ys )ds − Nt 󵄨󵄨 ] = 0. 󵄨 󵄨󵄨 󵄨󵄨 t→∞ t 󵄨 0

(2.14.49)

2.14 Functional central limit theorem

| 313

Proof. We apply Proposition 2.201. By Lemma 2.195, Y = (Yt )t≥0 is a stational and reversible Markov process with respect to the natural filtration and the semigroup as−1 sociated with Y is e−tU LU , and it is positivity improving. Hence assumption (1) is satisfied. By Lemma 2.195 again (U −1 Ψg )2 G is the invariant measure for Y and ∫ ρ(U −1 Ψg )2 dG = (Ψg , (H𝒯 − E)Ψg )𝒯 = 0. M−2

Then assumption (2) is satisfied. Finally ρ ∈ D(L̃ −1/2 ) and assumption (3) is satisfied. Then the proof is completed by Proposition 2.201 under the correspondences: Z and W Y, (Ω, F , (Ft )t≥0 , P) and (X × Y, F∞ , (ℬt )t≥0 , P∞ ) with ℬt = σ(Yr , 0 ≤ r ≤ t), (S, 𝒮 , π) −1 2 and (M−2 , ℬ(M−2 ), (U Ψg ) G), F and ρ, finally K and L.̃ Furthermore we need a version of the martingale central limit theorem. Proposition 2.203 (Central limit theorem for martingale). Let (Nn )n∈ℕ be a martingale on a filtered probability space (Ω, F , (Fn )n∈ℕ , P) with σ 2 = limn→∞ n1 𝔼[Nn2 ], and assume that (Nn )n∈ℕ has stationary increments. Then lim

n→∞

1 N = σ 2 Bt , √n [tn]

(2.14.50)

in the sense of weak convergence, i. e., for any continuous bounded function g on ℝ, lim 𝔼P [g(

n→∞

1 N )] = 𝔼𝒲 0 [g(σ 2 Bt )]. √n [tn]

Theorem 2.204 (Central limit theorem). It follows that lim

n→∞

1 γ ⋅ Btn = (γ, Dγ)B1t √n

in the sense of weak convergence, i. e., for any continuous bounded function g on ℝ, lim 𝔼P∞ W [g(

n→∞

1 γ ⋅ Btn )] = 𝔼𝒲 0 [g((γ, Dγ)B1t )], √n

where (B1t )t≥0 is 1-dimensional Brownian motion and D = (Dij )1≤i,j≤d a diffusion matrix with elements Dij = δij − 2(∇̂i Ψg , (H𝒯 − E)−1 ∇̂j Ψg )𝒯 , t

i, j = 1, . . . , d.

(2.14.51)

Proof. We decompose γ ⋅ Bt as γ ⋅ Bt = Mt + ∫0 ρ(Ys )ds. By Lemma 2.202 there exists a martingale (Nt )t≥0 with respect to the natural filtration (ℬt )t≥0 of (Yt )t≥0 such that (2.14.49). We have γ ⋅ Bt = Mt + Nt + ϵt ,

314 | 2 The Nelson model by path measures t

where ϵt = ∫0 ρ(Ys )ds − Nt . Since σ(Xt ) = σ(Yt ), (Mt )t≥0 is a martingale with respect to (ℬt )t≥0 . Then (Mt + Nt )t≥0 is a martingale with respect to (ℬt )t≥0 , since both (Mt )t≥0 and (Nt )t≥0 are martingales with respect to (ℬt )t≥0 . We have t

t

0

0

1 1 2 2 2 ∫ 𝔼P∞ ∫ 𝔼P∞ W [|Ms + Ns | ]ds = W [|γ ⋅ Bs | − 2ϵs γ ⋅ Bs + |ϵs | ]ds → (γ, Dγ) t t as t → ∞ by (2.14.49) and (2.14.43). By Proposition 2.203 we have lim

n→∞

1 (M + N[tn] ) = (γ, Dγ)B1t √n [tn]

in the sense of weak convergence. Let g ∈ C 2 (ℝ). By the Taylor expansion we have 𝔼P∞ W [g(

1 1 γ ⋅ B[nt] )] = 𝔼P∞ (M + N[nt] + ϵ[nt] ))] W [g( √n √n [nt]

= 𝔼P∞ W [g(

1 1 (M + N[nt] )) + ϵ g 󸀠 (X)] → 𝔼𝒲 [g((γ, Dγ)B1t )]. √n [nt] √n [nt]

(2.14.52)

1 ϵ[nt] with some −1 < θ < 1. Next g be a bounded continuous Here X = M[nt] + N[nt] + θ √n function on ℝ. In this case we can prove (2.14.52) by a simple approximation procedure. There exists a sequence gn ∈ C 2 (ℝ) such that limm→∞ ‖g − gm ‖∞ = 0. For any ϵ > 0 there exists m0 such that for any m > m0 ,

󵄨󵄨 󵄨󵄨 1 󵄨󵄨 󵄨 γ ⋅ B[nt] )] − 𝔼𝒲 [g((γ, Dγ)B1t )]󵄨󵄨󵄨 W [g( 󵄨󵄨𝔼P∞ 󵄨󵄨 󵄨󵄨 √n 󵄨󵄨 1 1 1 󵄨 γ ⋅ B[nt] ) − gm ( γ ⋅ B[nt] ) + gm ( γ ⋅ B[nt] )] = 󵄨󵄨󵄨𝔼P∞ W [g( √n √n √n 󵄨󵄨 󵄨󵄨 󵄨 − 𝔼𝒲 [g((γ, Dγ)B1t ) − gm ((γ, Dγ)B1t ) + gm ((γ, Dγ)B1t )]󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 1 󵄨 󵄨 γ ⋅ B[nt] )] − 𝔼𝒲 [gm ((γ, Dγ)B1t )]󵄨󵄨󵄨. ≤ 2ϵ + 󵄨󵄨󵄨𝔼P∞ W [gm ( 󵄨󵄨 󵄨󵄨 √n

Taking the limit n → ∞ on both sides above, we conclude that lim 𝔼P∞ W [g(

n→∞

1 γ ⋅ B[nt] )] = 𝔼𝒲 [g((γ, Dγ)B1t )]. √n

2.14.3 Diffusion matrix and effective mass The diffusion matrix D derived from a central limit theorem in Section 2.14.2 coincides with the so called effective mass. We shall show this in this section. In terms of translation invariant Nelson Hamiltonian with total momentum zero H(0) the matrix can be represented as j

Dij = δij − 2(Pfi Ψg (0), (H(0) − E(0))−1 Pf Ψg (0))L2 (G) ,

2.14 Functional central limit theorem

| 315

j

where Pf = dΓ(−i∇j ) is the field momentum operator. Let Ψg (p) be the ground state of H(p) with ground state energy E(p): H(p)Ψg (p) = E(p)Ψg (p)

(2.14.53)

for p ∈ ℝd . Formally we expect that E(p) = E(0) +

1 |p|2 + O(|p|3 ). 2meff

Here O(|p|3 )/|p|α → 0 as |p| → 0 for α < 2. meff is called the effective mass. Suppose that E(p) and Ψg (p) are differentiable with respect to p. Then meff is the Hessian of E(p), i. e., 1 = Δp E(p)⌈ . meff p=0 Let

1 mij

𝜕2 E(p) 1 = ⌈ . mij 𝜕pi 𝜕pj p=0 in a heuristic way in what follows, and we shall give the definition of the effective

mass. We notice that

1 H(p) = (p − Pf )2 + Hf + HI (0) 2 has the reflection symmetry: H(p) ≅ H(−p). For simplicity we write Ψg (p) = Ψ(p) and Pfi = Pi . Moreover we write 𝜕pi Ψ(p) = Ψi (p), 𝜕pi 𝜕pj Ψ(p) = Ψij (p), 𝜕pi E(p) = Ei (p), 𝜕pi 𝜕pj E(p) = Eij (p). Take derivative 𝜕pi of both sides of (2.14.53). We have (pi − Pi )Ψ(p) + H(p)Ψi (p) = Ei (p)Ψ(p) + E(p)Ψi (p). Substituting p = 0 we have −Pi Ψ(0) + H(0)Ψi (0) = Ei (0)Ψ(0) + E(0)Ψi (0). From the reflection symmetry it follows that Ei (0) = 0 and −Pi Ψ(0) + H(0)Ψi (0) = E(0)Ψi (0). We can derive that Ψi (0) = (H(0) − E(0))−1 Pi Ψ(0). Take derivative 𝜕pj of both sides of (2.14.54). We have δij Ψ(p) + (pi − Pi )Ψj (p) + (pj − Pj )Ψi (p) + H(p)Ψij (p) = Eij (p)Ψ(p) + Ei (p)Ψj (p) + Ej (p)Ψi (p) + E(p)Ψij (p).

(2.14.54)

316 | 2 The Nelson model by path measures Substituting p = 0 we have δij Ψ(0) − Pi Ψj (p) − Pj Ψi (p) + H(0)Ψij (0) = Eij (0)Ψ(0) + E(0)Ψij (p). Take the scalar product with Ψ(0) on both sides above. We have δij − (Ψ(0), Pj Ψi (0) + Pi Ψj (0)) = Eij (0). Inserting Ψi (0) = (H(0) − E(0))−1 Pi Ψ(0), we have Eij (0) = δij − 2(Pi Ψ(0), (H(0) − E(0))−1 Pj Ψ(0))L2 (G) . We conclude that 1 j = δij − 2(Pfi Ψg (0), (H(0) − E(0))−1 Pf Ψg (0))L2 (G) . mij We now define the effective mass by the inverse of ∑di=1

1 : mii

Definition 2.205 (Effective mass). The effective mass meff of H(p) is defined by d 1 = d − 2 ∑(Pfi Ψg (0), (H(0) − E(0))−1 Pfi Ψg (0))L2 (G) . meff i=1

From the diffusion matrix D = (Dij )1≤i,j≤d we have the theorem below: Theorem 2.206 (Effective mass and diffusion matrix). Let D be the diffusion matrix given by (2.14.51). Then meff =

1 . Tr D

Proof. By Corollary 2.192 and Definition 2.197 we see that (Pfi Ψg (0), (H(0) − E(0))−1 Pfi Ψg (0))L2 (G) = (∇̂i Ψg , (H𝒯 − E)−1 ∇̂j Ψg )𝒯 . Then the theorem is proven.

3 The Pauli–Fierz model by path measures 3.1 Preliminaries 3.1.1 Introduction It has been seen in the previous section that functional integration in the framework of Euclidean quantum field theory is a powerful tool in the spectral analysis of the Nelson scalar quantum field model. In this section, we consider a model of nonrelativistic quantum electrodynamics (QED) by using functional integration. This involves a stochastic integral with a Euclidean quantum field version of a quantized radiation field. A significant point is that the theory is nonrelativistic in the motion of the particle only, while the quantized radiation field is necessarily relativistic. In addition, it also has the feature that describes the particle “dressed” in a cloud of bosons, while its bare charge is conserved. The Hamiltonian of nonrelativistic QED is defined as a self-adjoint operator on a Hilbert space. Its spectrum is of basic interest since it involves perturbations of embedded eigenvalues in the continuous spectrum. Recently, progress in the spectral analysis of nonrelativistic QED has produced an understanding of aspects of binding, spectral scattering theory, resonances, and effective mass to a fine degree. The functional integration introduced in this section offers a new point of view in the spectral analysis of nonrelativistic QED. Importantly, this approach is free of perturbative elements, and it leads to a proof of existence and uniqueness of the ground state of the Hamiltonian in a surprisingly simple manner. In addition to its applications in spectral analysis, we believe that functional integration is an interesting mathematical method in its own right. As we will see, the course of developing this theory gives rise to a new class of objects in probability theory and new problems. In this section, we consider the Pauli–Fierz Hamiltonian HPF in nonrelativistic QED, in which a quantum system of a low energy electron interacting through a minimally coupling with a massless quantized radiation field is described. In this model, the radiation field is quantized in the Coulomb gauge instead of the Lorentz gauge, derivative coupling is involved in its interaction by the minimal coupling, and an ultraviolet cutoff is imposed to define the model as a self-adjoint operator on a Hilbert space. The existence of a ground state Ψg of the Pauli–Fierz Hamiltonian has been established by Bach–Fröhlich–Sigal [30] and Griesemer–Lieb–Loss [128]. For the details of the assumptions on the external potential, we refer to the original paper. In particular, no infrared cutoff is assumed and no restriction on the values of the coupling constant is imposed. Theorem 3.1. Under given conditions on V, there exists a ground state Ψg of HPF , and (Ψg , NΨg ) < ∞. https://doi.org/10.1515/9783110403541-003

318 | 3 The Pauli–Fierz model by path measures We emphasize that in this theorem no infrared regularity condition such as (2.2.6) in the case of the Nelson model is assumed. Conventional quantum electrodynamics uses the Lagrangian density LQED (x). In the language of Feynman diagrams, the leading term of the effective mass is computed from the self-energy of the electron and the g-factor shift by vertex diagrams. Both diagrams include virtual photons, and give corrections to a bare mass and the g-factor. On the other hand, the effective charge eeff is computed from the self-energy of photons. The photon self-energy diagram can be interpreted as the emission of the pairs of virtual electrons and positrons. The assumption that the energy of the electron is low implies that no emission of pairs of virtual electrons and positrons takes place. In the Pauli–Fierz model, thus no diagram of self-energy of photons is included, and the effective charge equals the bare charge and the number of electrons stays constant. Here is an outline of cases discussed in terms of functional integration techniques in this chapter: (1) spinless Pauli–Fierz model (2) relativistic Pauli–Fierz model (3) Pauli–Fierz model with spin 1/2 (4) translation invariant Pauli–Fierz model. In Chapter 2, the path measure was constructed on a Hilbert space-valued path space, and the functional integrals of semigroups are expressed over an infinite dimensional Ornstein–Uhlenbeck process. Intuitively, this is an analogue of the finite dimensional Ornstein–Uhlenbeck process. For the more complicated Pauli–Fierz model, however, we adopt the Euclidean quantum field method described in Chapter 1. In this case, the functional integrals are constructed on the product measure space of a Lévy measure and a Gaussian measure associated with the Euclidean quantum field version of the quantized radiation field. Moreover, in this case we will have to deal with a Hilbert space-valued stochastic integral with a time dependent integrand. 3.1.2 Lagrangian QED As in the case of Nelson’s model discussed in the previous chapter, also here we start by the formulation of the model in Fock space and then proceed to an equivalent formulation in terms of path integrals. In this section, we review the Lagrangian formalism of quantum electrodynamics. We begin by some notation of the relativistically covariant theory used only in this section. Let gμν

+1, { { { = {−1, { { {0,

μ = ν = 0, μ = ν = 1, 2, 3, otherwise

3.1 Preliminaries | 319

and write x = (t, x) = (x0 , x1 , x2 , x3 ) ∈ ℝ4 , xμ = gμν xν . Here and in what follows, in this section we understand that summation is performed over repeated indices. The index μ runs from 0 to 3 and j from 1 to 3. Also, we put 𝜕x μ = 𝜕μ and 𝜕xμ = 𝜕μ , involving 𝜕0 = 𝜕0 and 𝜕j = −𝜕j . The 4 × 4 gamma matrices are defined by σμ ), 0

0 γμ = ( μ σ

where σ 0 = 12×2 and σ μ , μ = 1, 2, 3 are the 2 × 2 Pauli matrices. Let Aμ = Aμ (x) stand for the radiation field and write Aμ = Aμ (x) = gμν Aν (x), i. e., A0 = A0 and Aj = −Aj . The spinor field will be denoted by Ψ = Ψ(x), and we write Ψ = Ψ(x) = Ψ∗ γ 0 . The Lagrangian density in QED terms is given by μ

LQED = Ψ(iγ 𝜕μ − m)Ψ −

1 F F μν − eΨγ μ ΨAμ . 4 μν

Here, e ∈ ℝ and m > 0 denote the charge and mass of the electron, respectively, F μν is the antisymmetric second-rank field tensor (field strength tensor) F μν = 𝜕μ Aν − 𝜕ν Aμ and Fμν = gμα gνβ F αβ = 𝜕μ Aν − 𝜕ν Aμ . The longitudinal component of the radiation field will be denoted by Φ = A0 and the transversal by A = (A1 , A2 , A3 ). Let Ẋ = 𝜕0 X. Then the electrostatic field 𝔼 = (𝔼1 , 𝔼2 , 𝔼3 ) is defined by 𝔼j = F j0 = −𝜕j Φ − Ȧ j , while the magnetic field is the curl of the transversal component, 𝔹 = (𝔹1 , 𝔹2 , 𝔹3 ) = ∇ × A, where ∇ = (𝜕1 , 𝜕2 , 𝜕3 ). In particular, the field tensor is obtained as

F μν

0 𝔼1 =( 2 𝔼 𝔼3

−𝔼1 0 𝔹3 −𝔹2

−𝔼2 −𝔹3 0 𝔹1

−𝔼3 𝔹2 ), −𝔹1 0

Fμν

𝔼1 0 𝔹3 −𝔹2

0 −𝔼1 =( 2 −𝔼 −𝔼3

𝔼2 −𝔹3 0 𝔹1

𝔼3 𝔹2 ). −𝔹1 0

By straightforward computation, using the explicit forms of Fμν and F μν it follows that 1 1 − Fμν F μν = (𝔼2 − 𝔹2 ). 4 2 From this, the Lagrangian density is obtained as μ

1 2

2

2

j

j

LQED = Ψ(iγ 𝜕μ − m)Ψ + (𝔼 − 𝔹 ) + eΨγ ΨA − eΨ ΨΦ ∗

320 | 3 The Pauli–Fierz model by path measures using the fact that Ψγ 0 Ψ = Ψ∗ Ψ. From the Lagrangian, we can derive the Euler– Lagrange equations through the variational principle ̄ ν Ψ, 𝜕μ F μν = eΨγ

(M) (D)

μ

(3.1.1) μ

(iγ 𝜕μ − m)Ψ = eγ Aμ Ψ.

(3.1.2)

Equation (3.1.2) is the Dirac equation and (3.1.1) is written as ̄ ν Ψ, ◻Aν − 𝜕ν 𝜕μ Aμ = eΨγ

(3.1.3)

where ◻ = 𝜕0 𝜕0 − 𝜕j 𝜕j , and is equivalent with (ν = 0)

∇ ⋅ 𝔼 = eΨ∗ Ψ,

(3.1.4)

̄ j Ψ. (ν = j) 𝔼̇ j = (∇ × 𝔹)j − eΨγ

(3.1.5)

Moreover, the definitions of 𝔼 and 𝔹 give 𝔹̇ = −∇ × 𝔼,

(3.1.6)

∇ ⋅ 𝔹 = 0.

(3.1.7)

The system of (3.1.4)–(3.1.7) are the celebrated Maxwell equations. We derive now the QED Hamiltonian by using Legendre transforms. To see this from LQED , we identify the conjugate momenta Π0 = Πj =

𝜕LQED = 0, 𝜕Ȧ 0

𝜕LQED = 𝔼j 𝜕Ȧ j

and 𝜕LQED = iΨ∗ . ̇ 𝜕Ψ The canonical momentum Π0 vanishes. Thus the Legendre transform gives the Hamiltonian density HQED = HQED (x) by j

HQED = Π Ȧ j + iΨ Ψ̇ − LQED ∗

1 = Ψ∗ αj (−i𝜕j + eAj )Ψ + mβΨ∗ Ψ + (𝔼2 + 𝔹2 ) + eΨ∗ ΨΦ + ∇Φ ⋅ 𝔼. 2

Here, αj = γ 0 γ j and β = γ 0 . Next, in order to derive the Pauli–Fierz Hamiltonian as an analogue of full QED, instead of the Lorentz gauge the Coulomb gauge condition ∇⋅A=0

(3.1.8)

3.1 Preliminaries | 321

is used. This gauge is not covariant and (3.1.3) reduces to ̄ ν Ψ. ◻Aν − 𝜕ν Φ̇ = eΨγ Notice that under the Lorentz gauge condition 𝜕μ Aμ = 0, (3.1.3) reduces to the wave equation ̄ ν Ψ. ◻Aν = eΨγ The Coulomb gauge condition (3.1.8) yields ∇ ⋅ 𝔼 = −ΔΦ = eΨ∗ Ψ, where Δ = 𝜕j 𝜕j denotes the 3-dimensional Laplacian and the second equality is derived from the Euler–Lagrange equations (3.1.4)–(3.1.5). This is the Poisson equation and the longitudinal component Φ in the Coulomb gauge becomes Φ(t, x) = e ∫ ℝ3

Ψ∗ (t, y)Ψ(t, y) dy. 4π|x − y|

On decomposing the electric field 𝔼 = −∇Φ − Ȧ into a longitudinal and a transversal component, we furthermore have ∫ 𝔼2 dx = − ∫ Φ ⋅ ΔΦdx + ∫ Ȧ 2 dx ℝ3

ℝ3

ℝ3

by the Coulomb gauge condition. Moreover, − ∫ Φ ⋅ ΔΦdx = +e2 ∫ ℝ3

ℝ3

Ψ∗ (t, y)Ψ(t, y)Ψ∗ (t, x)Ψ(t, x) dxdy. 4π|x − y|

The term eΨ∗ ΨΦ + ∇Φ ⋅ 𝔼 in HQED vanishes since by integration, ∫ ∇Φ ⋅ 𝔼dx = − ∫ Φ∇ ⋅ 𝔼dx = −e ∫ Ψ∗ ΨΦdx. ℝ3

ℝ3

ℝ3

Finally, the Hamiltonian HQED = ∫ HQED dx ℝ3

of full QED with the Coulomb gauge is given by HQED = ∫ Ψ∗ αj (−i𝜕j + eAj )Ψdx + mβ ∫ Ψ∗ Ψdx ℝ3

+

ℝ3

1 ∫ (Ȧ 2 + 𝔹2 )dx + e2 ∫ 2 ℝ3

ℝ3 ×ℝ3

Ψ∗ (t, y)Ψ(t, y)Ψ∗ (t, x)Ψ(t, x) dxdy. 4π|x − y|

322 | 3 The Pauli–Fierz model by path measures 3.1.3 Classical variant of nonrelativistic QED In this section, we derive a classical version of the nonrelativistic QED Hamiltonian using the Lagrange formalism, which is an analogue of the discussion in the preceding section. Here, Ψ(iγ μ 𝜕μ − m)Ψ is replaced by the N-particle classical kinetic energy (1/2) ∑Nj=1 mj q̇ j2 , eΨγ j Ψ by the current density j, and eΨ∗ Ψ by the density ρ. The Pauli– Fierz Hamiltonian in nonrelativistic QED will be given by quantization of the classical version and the spinor field is changed to electrons governed by the Schrödinger operator. Let N electrons interact with a classical electromagnetic field described by the Maxwell equations. Denote by B(t, x) ∈ ℝ3 the classical magnetic field and by E(t, x) ∈ ℝ3 the classical electric field. Consider the Maxwell equations with form factor λ, N

∇ ⋅ E = ∑ ej λ(⋅ − xj ), j=1

N

(3.1.9)

Ė = ∇ × B − ∑ ej λ(⋅ − xj )q̇ j ,

(3.1.10)

Ḃ = −∇ × E,

(3.1.11)

∇ ⋅ B = 0.

(3.1.12)

j=1

Here, xj = xj (t) ∈ ℝ3 denotes the position and ej the charge of the electron labelled by j. The form factor describes the charge distribution satisfying λ ≥ 0 and ∫ℝ3 λ(x)dx = 1. We define the current by N

N

j=1

j=1

(j, ρ) = ( ∑ ej λ(⋅ − xj )q̇ j , ∑ ej λ(⋅ − xj )); which satisfies the continuity equation ρ̇ + ∇ ⋅ j = 0. The equation of motion of the electron is given by mj q̈ j = ej (E + q̇ j × B), where mj is the mass of the jth electron. We next rewrite the Maxwell equations by introducing a vector potential A and a scalar potential ϕ. By (3.1.12), A = (A1 (t, x), A2 (t, x), A3 (t, x)) is required to satisfy B = ∇ × A. Furthermore, from (3.1.11) it follows that ∇×(E + A)̇ = 0. The scalar potential ϕ = ϕ(t, x) is given by E = −Ȧ − ∇ϕ.

(3.1.13)

3.1 Preliminaries | 323

With this, the Lagrangian becomes LPF =

1 N 1 ∑ m q̇ 2 + ∫ (E 2 − B2 )dx + ∫ j ⋅ Adx + ∫ (−ρϕ)dx 2 j=1 j j 2 ℝ3

ℝ3

ℝ3

supplemented with the Coulomb gauge condition ∇ ⋅ A = 0. The system (3.1.9)–(3.1.12) thus turns into ◻A = ∇ϕ̇ − j, Δϕ = −ρ. Inserting the definition of E (3.1.13) and taking into account the Coulomb gauge, we furthermore have 1 1 1 ∫ (E 2 − B2 )dx = ∫ { (Ȧ 2 − B2 ) + ρϕ} dx. 2 2 2

ℝ3

ℝ3

By the Poisson equation Δϕ = −ρ, we have the Coulomb potential regularized by ρ, ei ej λ(xi − y)λ(xj − y󸀠 ) 1 1 N dydy󸀠 . ∫ ρϕdx = ∑ ∫ 2 2 i,j=1 4π|y − y󸀠 | ℝ3

ℝ3

Now we define the Hamiltonian associated with the Lagrangian LPF through Legendre transform. We introduce the canonical momenta pj and Πi by pj =

𝜕LPF = mj q̇ j + ej ∫ A(x)λ(x − xj )dx 𝜕q̇ j ℝ3

and Πi =

δLPF = Ȧ i , δȦ i

i = 1, 2, 3.

In terms of these variables, the Hamiltonian associated with LPF becomes N

̇ − LPF . Hclassical = ∑ pj q̇ j + ∫ Π ⋅ Adx j=1

ℝ3

A straightforward computation yields Hclassical

2

N

1 1 =∑ (pj − ej ∫ A(x)λ(x − xj )dx) + ∫ (Ȧ 2 + B2 )dx 2m 2 j j=1 +

1 N ∑ ∫ 2 i,j=1

ℝ3 ×ℝ3

ℝ3

ei ej λ(xi − y)λ(xj − y ) 󸀠

4π|y − y󸀠 |

ℝ3

dydy󸀠 .

324 | 3 The Pauli–Fierz model by path measures This Hamiltonian contains the convolution ∫ℝ3 A(x)λ(x − xj )dx, the field energy 1 ∫ (Ȧ 2 + B2 )dx, the regularized Coulomb potential, and the minimal interaction. 2 ℝ3 In the next section, we leave the realm of classical electrodynamics and quantize Hclassical . We will define a self-adjoint operator HPF on a suitable Hilbert space ℋPF . In the quantum description, HPF will be defined by Hclassical replaced by pj → −i𝜕j , j = 1, . . . , N, the smeared Coulomb potential goes into a multiplication operator given by a real function V, ̃ − x)), ∫ Aμ (x)λ(x − xj )dx → Â μ (φ(⋅ ℝ3

and

1 ∫ (Ȧ 2 + B2 )dx → Hrad . 2 ℝ3

̃ − x)) and Hrad are the quantized radiation field and the free field HamiltoHere, Â μ (φ(⋅ nian on the boson Fock space, respectively, which are defined and discussed in detail in the next section.

3.2 The Pauli–Fierz model in nonrelativistic QED 3.2.1 The Pauli–Fierz model in Fock space We now turn to defining the Pauli–Fierz Hamiltonian rigorously as a self-adjoint operator bounded from below on a Hilbert space over ℂ. For simplicity, we assume only one electron interacting with the field. Let 2

3

ℋPF = L (ℝ ) ⊗ ℱrad

be the Hilbert space describing the joint electron-photon state vectors. Here, 2

3

ℱrad = ℱrad (L (ℝ × {±}))

is the boson Fock space over L2 (ℝ3 ×{±}). The elements of the set {±} account for the fact that a photon is a transversal wave perpendicular to the direction of its propagation, thus it has two components. The Fock vacuum in ℱrad is denoted by Ωph . We identify ℋPF as the set of ℱrad -valued L2 -functions on ℝ3 , ⊕

ℋPF ≅ ∫ ℱrad dx; ℝ3

this will be used without further notice in what follows. Let a(f ) and a∗ (f ) be the annihilation operator and the creation operator on ℱrad smeared by f ∈ L2 (ℝ3 × {±}),

3.2 The Pauli–Fierz model in nonrelativistic QED

| 325

respectively. Then [a(f ), a∗ (g)] = ∑ (fj̄ , gj ), j=±

[a(f ), a(g)] = 0,

[a∗ (f ), a∗ (g)] = 0.

2

We use the identification L (ℝ3 × {±}) ≅ L2 (ℝ3 ) ⊕ L2 (ℝ3 ) and set a♯ (f , +) = a♯ (f ⊕ 0),

a♯ (f , −) = a♯ (0 ⊕ f ),

where a♯ stands for either operator. Then a♯ (f ) = ∑j=± a♯ (fj , j) for f = f1 ⊕ f2 . In terms of (formal) kernels a♯ (k, j), a♯ (f , j) will be written more conveniently as a♯ (f , j) = ∫ a♯ (k, j)f (k)dk. ℝ3

The finite particle subspace of ℱrad is given by ℱrad,fin = {Ψ = {Ψ

(n) ∞ }n=0

| Ψ(m) = 0 for all m > N with some N}.

Next, we define the quantized radiation field with a given form factor φ.̂ Put ̂ φ(k) ej (k)e−ikx ∈ L2 (ℝ3k ), √ω(k) μ ̂ φ(−k) ej (k)eik⋅x ∈ L2 (ℝ3k ), ℘̃ jμ (x) = √ω(k) μ

℘jμ (x) =

for x ∈ ℝ3 , j = ± and μ = 1, 2, 3, where ω is the dispersion relation for massless photons defined by ω(k) = ων=0 (k) = |k|. Here, φ̂ is the Fourier transform of the charge distribution φ. The vectors e+ (k) and e− (k) are called polarization vectors, that is, e+ (k), e− (k) and k/|k| form a right-hand system at k ∈ ℝ3 ; ei (k) ⋅ ej (k) = δij 1,

ej (k) ⋅ k = 0,

e+ (k) × e− (k) = k/|k|.

In Proposition 3.6, we will see that the spectral analysis is independent of the choice of polarization vectors, i. e., the Hamiltonians defined through different polarizations are unitary equivalent. Thus we may fix the polarization vectors as it is most convenient. Definition 3.2 (Quantized radiation field). The quantized radiation field with form factor φ̂ is defined by Aμ (x) =

1 ∑ (a∗ (℘jμ (x), j) + a(℘̃ jμ (x), j)) √2 j=±

and the conjugate momentum by Πμ (x) =

i ∑ (a∗ (℘jμ (x), j) − a(℘̃ jμ (x), j)) √2 j=±

326 | 3 The Pauli–Fierz model by path measures ̂ ̂ ̂ ∈ L2 (ℝd ) and φ(k) In the case of φ/√ω = φ(−k), Aμ (x) is symmetric, and moreover essentially self-adjoint on the finite particle subspace of ℱrad , ℱrad,fin , by the Nelson ̂ analytic vector theorem (Proposition 1.16), (1.2.14) and (1.2.15). The condition φ(k) = ̂ φ(−k) is necessary and sufficient for φ being real. A physically reasonable choice of φ requires a positive function. We denote the closure of Aμ (x)⌈ℱrad,fin by the same symbol. Write ⊕

Aμ = ∫ Aμ (x)dx,

A = (A1 , A2 , A3 ).

ℝ3

Aμ is a self-adjoint operator on 󵄨󵄨 󵄨 D(Aμ ) = {F ∈ ℋPF 󵄨󵄨󵄨 F(x) ∈ D(Aμ (x)) and ∫ ‖Aμ (x)F(x)‖2ℱrad dx < ∞} 󵄨󵄨 3 ℝ

and acts by (Aμ F)(x) = Aμ (x)F(x),

F ∈ D(Aμ ),

for almost every x ∈ ℝ3 . In terms of the kernel a♯ (k, j), the quantized radiation field Aμ (x) can be written as ̂ ̂ φ(−k) φ(k) 1 e−ik⋅x a∗ (k, j) + eik⋅x a(k, j)) dk ∫ eμj (k) ( √ √ √ 2 ω(k) ω(k) j=± 3

Aμ (x) = ∑

ℝ

and the conjugate momentum by ̂ ̂ φ(−k) φ(k) i e−ik⋅x a∗ (k, j) − eik⋅x a(k, j)) dk. ∫ eμj (k) ( √ √ω(k) √ω(k) j=± 2 3

Πμ (x) = ∑

ℝ

Since k ⋅ej (k) = 0, the polarization vectors introduced above are chosen in the way μ that ∑3μ=1 ∇μ ℘j (x) = 0, implying the Coulomb gauge condition 3

∑ ∇μ ⋅ Aμ = 0.

μ=1

This in turn yields ∑3μ=1 [∇μ , Aμ ] = 0. Next, we introduce the free field Hamiltonian on ℱrad . The free field Hamiltonian Hrad on ℱrad is given in terms of the second quantization Hrad = dΓ(ω ⊕ ω), formally corresponding to ∑ ∫ ω(k)a∗ (k, j)a(k, j)dk.

j=±

ℝ3

3.2 The Pauli–Fierz model in nonrelativistic QED

| 327

(n) It leaves the n-particle subspace ℱrad = L2sym ((ℝ3 ⊕ ℝ3 )n ) invariant and acts as n f f (k , . . . , kn ) (Hrad ( 1 )) (k1 , . . . , kn ) = (∑ ω(kj )) ( 1 1 ). f2 f 2 (k1 , . . . , kn ) j=1

The Pauli–Fierz model describes the minimal interaction between an electron and the quantized radiation field, in which the electron is assumed to carry low energy. The electron, as in the case of the Nelson model, is described by the Schrödinger operator 1 Hp = − Δ + V 2 in L2 (ℝ3 ). The assumptions on V will be specified later on below. The Hamiltonian for the noninteracting quantized radiation field and electron is thus given by 1 HPF,0 = − Δ ⊗ 1 + 1 ⊗ Hrad 2 with domain

1 DPF = D(HPF,0 ) = D (− Δ ⊗ 1) ∩ D(1 ⊗ Hrad ). 2

The interaction is obtained by minimal coupling −i∇μ ⊗ 1 󳨃→ −i∇μ ⊗ 1 − eAμ . Definition 3.3 (Pauli–Fierz Hamiltonian in Fock space). The Pauli–Fierz Hamiltonian in nonrelativistic quantum electrodynamics is defined by the operator 1 HPF = (−i∇ ⊗ 1 − eA)2 + V ⊗ 1 + 1 ⊗ Hrad 2 in ℋPF . We define HPF (A) and HPF,I by 1 HPF (A) = (−i∇ ⊗ 1 − eA)2 , 2 HPF,I = −e(−i∇ ⊗ 1) ⋅ A +

e2 2 A + V ⊗ 1. 2

Clearly, HPF = HPF,0 + HPF,I by the commutation relation ∑3μ=1 [∇μ ⊗ 1, Aμ ] = 0. In what follows, we omit the tensor notation ⊗ for easing the notation. In Sections 3.2–3.3, unless otherwise stated we suppose the following assumptions. Assumption 3.4. The following conditions hold. (1) Charge distribution: φ ∈ S 󸀠 (ℝ3 ) satisfies that (i)

̂ ̂ φ(−k) = φ(k),

̂ √ω, φ/ω ̂ (ii) √ωφ,̂ φ/ ∈ L2 (ℝ3 ),

(3.2.1)

328 | 3 The Pauli–Fierz model by path measures (2) External potential: V ∈ ℛKato , i. e., there exist 0 ≤ a < 1 and 0 ≤ b such that ‖Vf ‖ ≤ a‖ − (1/2)Δf ‖ + b‖f ‖,

f ∈ D(−(1/2)Δ).

(3.2.2)

̄ As was stated before condition (i) of (1) means that φ(x) = φ(x), and hence φ is real and it ensures that HPF is symmetric. By condition (ii) of (1) (−i∇ ⊗ 1) ⋅ A and A2 are relatively bounded with respect to HPF,0 . Finally, by (2) and the Kato–Rellich theorem Hp is self-adjoint on D(−(1/2)Δ). As a consequence, HPF,I is relatively bounded with respect to HPF,0 . As in Section 3.5.1 below, we can prove the self-adjointness of HPF . Theorem 3.5 (Self-adjointness). The Pauli–Fierz Hamiltonian HPF is self-adjoint on DPF and essentially self-adjoint on any core of HPF,0 . In fact, for small values of |e| it is not hard to see self-adjointness of HPF on DPF as we have ‖HPF,I Ψ‖ ≤ (a + a1 |e| + a2 |e|2 )‖HPF,0 Ψ‖ + b(e)‖Ψ‖ with a given in (3.2.2) and constants a1 , a2 and b(e) whose existence is made sure by (ii) of (3.2.1). Thus for e such that (a + a1 |e| + a2 |e|2 ) < 1, HPF is self-adjoint on DPF by the Kato–Rellich theorem. A more challenging problem is to show that HPF is self-adjoint for all e ∈ ℝ. For the moment, we assume that |e| is sufficiently small. Remarkably, Pauli–Fierz Hamiltonians with different polarization vectors are equivalent with each other. We show this next. Let e± and η± be polarization vectors, and HPF (e± ) and HPF (η± ) the corresponding Pauli–Fierz Hamiltonians, respectively. Proposition 3.6. HPF (e± ) and HPF (η± ) are unitary equivalent. Proof. Since for each k ∈ ℝ3 both polarization vectors form orthogonal bases on the plane perpendicular to the vector k, there exists ϑk such that cos ϑk 13 e+ (k) )=( sin ϑk 13 e− (k)

(

cos ϑ − sin ϑk k cos ϑk

where Rk = ( sin ϑ k

− sin ϑk 13 η+ (k) )( − ) cos ϑk 13 η (k)

i. e.

eμ+ (k)

(

eμ− (k)

η+μ (k)

) = Rk (

η−μ (k)

),

). Define R : L2 (ℝ3 ; ℂ2 ) → L2 (ℝ3 ; ℂ2 ) by, for almost every k, f f (k) R( )(k) = Rk ( ) g g(k)

and Γ(R) : ℱrad → ℱrad by the second quantization of R. Then Γ(R) is a unitary map on ℱrad . Note that η+μ f

R(

η−μ f

eμ+ f

)=(

eμ− f

which implies that Γ(R)HPF (η± )Γ(R)−1 = HPF (e± ).

)

3.2 The Pauli–Fierz model in nonrelativistic QED

| 329

3.2.2 The Pauli–Fierz model in function space We introduce a Q -space associated with the quantized radiation field and reformulate the Pauli–Fierz Hamiltonian on L2 (ℝ3 ) ⊗ L2 (Q ) instead of ℋPF . Furthermore, we introduce the Euclidean quantum field associated with the Pauli–Fierz Hamiltonian to derive the functional integral representation of e−tHPF . A slight modification of the setup discussed in Chapters 1 and 2 is needed as the scalar field (ϕ(f ), f ∈ M ) is indexed by f ∈ M while the quantized radiation field is indexed by L2real (ℝ3 ). In particular, the family of isometries τt : M → E will be modified to jt : L2real (ℝ3 ) → L2real (ℝ4 ). For a real-valued f ∈ L2 (ℝ3 ), Aμ (f ) =

1 ∑ ∫ eμj (k)(f ̂(k)a∗ (k, j) + f ̂(−k)a(k, j))dk. √2 j=± 3 ℝ

̃ − x)), where φ̃ = (φ/√ω ̂ With this notation, we write Aμ (x) = Aμ (φ(⋅ ).̌ The relations (mean) (ΩPF , Aμ (f )ΩPF ) = 0, (covariance) (ΩPF , Aμ (f )Aν (g)ΩPF ) =

1 ⊥ ̂ (k)f ̂(k)g(k)dk ∫ δμν 2

(3.2.3) (3.2.4)

ℝ3

⊥ are immediate. Here, δμν (k) is the transversal delta function given by ⊥ δμν (k) = δμν −

kμ kν |k|2

.

The identity ⊥ δμν (k) = ∑ eμj (k)eνj (k) j=±

(3.2.5)

follows from the fact that the 3 × 3 matrix e1+ (k) (e2+ (k) e3+ (k)

e1− (k) e2− (k) e3− (k)

k1 /|k| k2 /|k|) k3 /|k|

is an orthogonal matrix. Note that relation (3.2.5) is independent of the choice of polarization vectors e± (k). We define the 3 × 3 matrix δ⊥ (k) by ⊥ δ⊥ (k) = (δμν (k))1≤μ,ν≤3 .

330 | 3 The Pauli–Fierz model by path measures In order to have a functional integral representation of (F, e−tHPF G), we construct probability spaces (Qβ , Σβ , μβ ), β = 0, 1, and Gaussian random variables  β (f) indexed by f ∈ ⊕3 L2real (ℝ3+β ) of mean zero

𝔼μβ [Â β (f)] = 0

(3.2.6)

𝔼μβ [Â β (f)Â β (g)] = qβ (f, g).

(3.2.7)

and covariance

Here, the bilinear forms qβ on (⊕3 L2real (ℝ3+β )) × (⊕3 L2real (ℝ3+β )) are defined by q0 (f, g) =

1 ̂ ⋅ δ⊥ (k)g(k)dk, ̂ ∫ f(k) 2

(3.2.8)

1 ̂ ⋅ δ⊥ (k)g(k)dk, ̂ ∫ f(k) 2

(3.2.9)

ℝ3

q1 (f, g) =

ℝ4

where in what follows we set k = (k, k0 ) ∈ ℝ3 × ℝ. Note that δ⊥ (k) in (3.2.9) is independent of k0 ∈ ℝ. The definitions of (3.2.8) and (3.2.9) are motivated by (3.2.3) and (3.2.4). It is, however, not straightforward to construct Gaussian random variables such as (3.2.6) and (3.2.7) since, as is seen in the definition, qβ is degenerate; qβ has nonzero null space. In order to construct a Q -space for the Pauli–Fierz Hamiltonian we have to take a quotient space with respect to the null space of qβ . Proposition 3.7 (Gaussian process and quantized radiation field). There exist probability spaces (Qβ , Σβ , μβ ), β = 0, 1, and a family of Gaussian random variables (Â β (f), f ∈ ⊕3 Lreal (ℝ3 )) satisfying (3.2.6) and (3.2.7). Proof. Let Nβ = {f ∈ ⊕3 L2real (ℝ3 ) | qβ (f, f) = 0}. Define the Hilbert space 3 2

3+β

Kβ = (⊕ Lreal (ℝ

))/Nβ

as a quotient space, and take the canonical map πβ : ⊕3 L2real (ℝ3+β ) → Kβ . Then (πβ (f), πβ (g))Kβ = qβ (f, g). By Theorem 1.68, there exists a probability space (Qβ , Σβ , μβ ) and a family of Gaussian random variables {ϕβ (πβ (f)) | πβ (f) ∈ Kβ } such that Σβ is full and 𝔼μβ [ϕβ (πβ (f))] = 0, 𝔼μβ [ϕβ (πβ (f))ϕβ (πβ (g))] = qβ (f, g). Let furthermore  β (f) = ϕβ (πβ (f)), Then  β (f) satisfies (3.2.6) and (3.2.7).

f ∈ ⊕3 L2real (ℝ3+β ).

3.2 The Pauli–Fierz model in nonrelativistic QED

| 331

Define the μth component of  β by 3

f ∈ L2 (ℝ3+β ).

 β,μ (f ) =  β (⨁δμν f ), ν=1

In what follows, we denote  =  0 ,

q = q0 ,

 E =  1 ,

qE = q1 ,

(Minkowskian) (Euclidean)

Q = Q0 , QE = Q1

using the subscript E for Euclidean objects. In the same way as for the scalar field, we define the second quantization Γββ󸀠 (T) for T ∈ C (L2 (ℝ3+β ) → L2 (ℝ3+β )) such that 󸀠

T Nβ ⊂ Nβ󸀠 .

(3.2.10)

Here and in what follows for the operator S : L2 (ℝ3+β ) → L2 (ℝ3+β ), we use the same notation S as for the operator 󸀠

⊕3 S : ⊕3 L2 (ℝ3+β ) → ⊕3 L2 (ℝ3+β ), 󸀠

(f1 , f2 , f3 ) 󳨃→ (Sf1 , Sf2 , Sf3 )

for notational convenience, and we write simply  β (Tf) for  β ((⊕3 T)f), etc. By (3.2.10), πβ (f) 󳨃→ πβ󸀠 (Tf) defines a map from Kβ to Kβ󸀠 . Let thus Γββ󸀠 : C (L2 (ℝ3+β ) → L2 (ℝ3+β )) → C (L2 (Qβ ) → L2 (Qβ󸀠 )) 󸀠

be the functor defined by Γββ󸀠 (T)1 = 1 and n

n

i=1

i=1

Γββ󸀠 (T):∏ Â β (fi ): = :∏ Â β󸀠 (Tfi ):. We write Γ00 = Γ, Γ11 = ΓE and Γ01 = ΓInt . Our goal here is to discuss the Euclidean quantum field associated with the Pauli– Fierz model. We introduce a family of isometries connecting the Minkowski field and the Euclidean quantum field, as well as the scalar field discussed in Section 1.1. In contrast with the Q -space representation of the scalar field indexed by vectors in M , ̂ for the Pauli–Fierz model is indexed by f ∈ ⊕3 L2 (ℝ3 ). the Q -space representation A(f) So instead of the isometry τt : M → E with the property τs∗ τt = e−|s−t|ω(−i∇) , we define the isometry jt : L2real (ℝ3 ) → L2real (ℝ4 ) through τt by pull-back from the operator ℋM → ℋE to the operator L2 (ℝ3 ) → L2 (ℝ4 ). Let i−1/2 : L2 (ℝ3 ) → ℋM and i−1 : L2 (ℝ4 ) → ℋE be given by ̂ √ î −1/2 f (k) = ω(k)f (k), 2 2 ̂ √ î −1 f (k) = ω(k) + |k0 | f (k).

Both i−1/2 and i−1 are unitary operators.

332 | 3 The Pauli–Fierz model by path measures Definition 3.8. Define the family of isometries jt : L2real (ℝ3 ) → L2real (ℝ4 ) by jt = (i−1 )−1 ∘ τt ∘ i−1/2 ,

t ∈ ℝ.

By this definition, jt can be represented as √ω(k) e−itk0 ĵ f ̂(k) t f (k) = √π √ω(k)2 + |k |2 0 or in the position representation, jt f (x, x0 ) =

√ω(k) 1 f ̂(k)dk. ∫ e−it(k0 −x0 )+ik⋅x √π(2π)2 2 + |k |2 √ ω(k) 4 0 ℝ

Note that for f ∈ L2real (ℝ3 ), jt f = jt f , meaning that jt preserves realness, i. e., jt : L2real (ℝ3 ) → L2real (ℝ4 ), and jt N0 ⊂ N1 . In fact, it follows that q1 (jt f, jt f) = q0 (f, f). By using the definition, it can also be seen that (js f , jt g) =

1 e−i(t−s)k0 ̄ ̂ dk0 )dk ∫ ω(k)f ̂(k)g(k)( ∫ π ω(k)2 + k02 ℝ3

=

ℝ

1 πe−|s−t|ω(k) ̄ ̂ ̂ dk = (f ̂, e−|s−t|ω g), ∫ ω(k)f ̂(k)g(k) π ω(k) ℝ3

where we used ∫ ℝ

e−itx πe−|t|a dx = , 2 2 a a +x

a > 0.

Hence j∗s jt = e−|s−t|ω , ̂

s, t ∈ ℝ,

(3.2.11)

where ω̂ = ω(−i∇). Definition 3.9. Define the family of isometries Jt : L2 (Q ) → L2 (QE ) by the second quantization of jt , i. e., Jt = ΓInt (jt ). By (3.2.11), J∗t Js = Γ(e−|s−t|ω ) = e−|s−t|Hrad , ̂

̂

s, t ∈ ℝ,

holds with a self-adjoint operator Ĥ rad . As in the case of the scalar field, Jt plays an important role in the functional integral representation, which connects Minkowskian quantum fields with Euclidean quantum fields.

3.2 The Pauli–Fierz model in nonrelativistic QED

| 333

Note the intertwining property of Jt . Let T be a bounded multiplication operator (Tf )(k) = T(k)f (k) on L2 (ℝ3 ), and T̂ the bounded pseudo-differential operator T(−i∇). It follows that jt Tf̂ = (T̂ ⊗ 1)jt f , where T̂ ⊗ 1 is a bounded operator on L2 (ℝ4 ) under the identification L2 (ℝ4 ) ≅ L2 (ℝ3 ) ⊗ L2 (ℝ). Thus Jt Γ(T)̂ = ΓE (T̂ ⊗ 1)Jt .

(3.2.12)

Let h be a real-valued multiplication operator on L2 (ℝ3 ) and define ĥ = h(−i∇). Hence Jt Γ(e−ih ) = ΓE (e−ih⊗1 )Jt ̂

̂

follows from (3.2.12). Then Jt dΓ(h)̂ = dΓE (ĥ ⊗ 1)Jt also holds. We summarize these results below. Proposition 3.10 (Intertwining property). Let ĥ = h(−i∇) be the self-adjoint operator ̂ ̂ with real-valued symbol h. Then J Γ(e−ih ) = Γ (e−ih⊗1 )J and J dΓ(h)̂ = dΓ (ĥ ⊗ 1)J hold. t

E

t

t

E

t

The Wiener–Itô–Segal isomorphism can be extended to the isomorphism θPF between L2 (Q ) and ℱrad . Let θPF : ℱrad → L2 (Q ) be defined by θPF ΩPF = 1, n

n

i=1

i=1

̂ i ):. θPF :∏ A(fi ):ΩPF = :∏ A(f By the commutation relations, we see that n

n

n

n

i=1

j=1

i=1

j=1

̂ i ):, :∏ A(g ̂ j ):) . (:∏ A(fi ):ΩPF , :∏ A(gj ):ΩPF ) = (:∏ A(f Thus θPF can be extended to the unitary operator from ℱrad to L2 (Q ). We denote 1⊗θPF : ℋPF → L2 (ℝ3 ) ⊗ L2 (Q ) by θPF for simplicity. Note that the inverse Fourier transform of √ω(k) equals g(y, ̌ x) = φ(y ̂ ̃ − x), where g(k, x) = e−ik⋅x φ(k)/ ̂ √ω) ̌ ∈ L2real (ℝ3 ). φ̃ = (φ/

334 | 3 The Pauli–Fierz model by path measures ̃ − x)) for each x by the definition of Aμ (f ). Notice that the Recall that Aμ (x) = Aμ (φ(⋅ ̂ test function of A (f ) is f instead of f . Hence the relations μ

−1 ̃ − x)), θPF Aμ (x)θPF = Â μ (φ(⋅ −1 θPF Hrad θPF = Ĥ rad ,

follow directly. As a result, 1 −1 θPF HPF θPF = (−i∇ − eA)̂ 2 + V + Ĥ rad , 2 with  = ( 1 ,  2 ,  3 ) and ⊕

̃ − x))dx, Â μ = ∫ Â μ (φ(⋅

μ = 1, 2, 3.

ℝ3 −1 In what follows, we use the same notation HPF for θPF HPF θPF and Hrad for Ĥ rad .

Definition 3.11 (Pauli–Fierz Hamiltonian in function space). The Pauli–Fierz Hamiltonian in function space is defined by 1 HPF = (−i∇ − eA)̂ 2 + V + Hrad 2 in L2 (ℝ3 ) ⊗ L2 (Q ). We use the shorthand ℋPF for L2 (ℝ3 ) ⊗ L2 (Q ) and ℋE = L2 (ℝ3 ) ⊗ L2 (QE ) standing for the Euclidean space. Remark 3.12. Note that in the Q -space representation the test function f of  μ (f ) is in the position representation but the test function f of Aμ (f ) is in the momentum representation. For instance, the dispersion relation in Fock representation is ω(k) = |k|, while it becomes ω(−i∇) = |−i∇| in Q -space representation.

3.2.3 Markov property In the scalar quantum field theory discussed in Chapter 1, the family of projections Et = It I∗t , t ∈ ℝ, was defined and it was seen that the Markov property plays an important role in the functional integral representation. Here, we address a similar problem for the Pauli–Fierz model. In the Pauli–Fierz model, we will keep to the same notation et , Et , etc. as the similar objects for the scalar field. Let et = jt j∗t ,

t ∈ ℝ.

3.2 The Pauli–Fierz model in nonrelativistic QED

| 335

Then {et }t∈ℝ is a family of projections from L2real (ℝ4 ) to Ran jt . The latter is a closed subspace of L2real (ℝ4 ). Define U[a,b] = L. H. {f ∈ L2real (ℝ4 ) | f ∈ Ran jt for some t ∈ [a, b]}. Let e[a,b] : L2real (ℝ4 ) → U[a,b] denote the orthogonal projection. In the same way as for the scalar quantum field theory, it follows that for a ≤ b ≤ t ≤ c ≤ d, (1) ea eb ec = ea ec ,

(2) e[a,b] et e[c,d] = e[a,b] e[c,d] .

The projections on L2 (QE ) similarly are Et = Jt J∗t = ΓE (et ),

E[a,b] = ΓE (e[a,b] ).

Let Σ[a,b] be the minimal σ-field generated by {Â Eμ (f ) ∈ L2 (QE ) | f ∈ U[a,b] , μ = 1, 2, 3}. The set of Σ[a,b] -measurable functions in L2 (QE ) will be denoted by E[a,b] . The projection E[a,b] and the set of Σ[a,b] -measurable functions E[a,b] satisfy (1)

Ran E[a,b] = E[a,b] ,

(2) E[a,b] Et E[c,d] = E[a,b] E[c,d]

for a ≤ b ≤ t ≤ c ≤ d. (2) above is the Markov property for the present model. Next, we turn to discussing the family of projections Et , t ∈ ℝ. Let 1/p

‖F‖p = ( ∫ |F(ϕ)|p dμβ ) Qβ

be Lp -norm on (Qβ , μβ ) and (⋅, ⋅)2 the L2 (Qβ )-scalar product. We drop the subscript β unless confusion may arise. As it was seen, Γβ (T) for ‖T‖ ≤ 1 is a contraction on L2 (Qβ ). It also has the hypercontractivity property. The following sharper result is available. See Theorem 1.67. Proposition 3.13 (Lp -Lq boundedness). Let T : L2 (ℝ3+β ) → L2 (ℝ3+β ) be a contraction such that ‖ ⊕3 T‖2 ≤ (p − 1)(q − 1)−1 ≤ 1

(3.2.13)

for some 1 ≤ p ≤ q. Then Γβ (T) is a contraction from Lp (Qβ ) to Lq (Qβ ), i. e., for Φ ∈ Lp (Qβ ), Γβ (T)Φ ∈ Lq (Qβ ) and ‖Γβ (T)Φ‖q ≤ ‖Φ‖p .

336 | 3 The Pauli–Fierz model by path measures Note that Γ(e−t ω̃ ν ), ν > 0, is a contraction from Lp to Lq if t is sufficiently large but it fails to be so for ν = 0. Lemma 3.14. Suppose that ν > 0 and let a ≤ b < t < c ≤ d, F ∈ E[a,b] and G ∈ E[c,d] . Take 1 ≤ r < ∞, r < p and r < q. Suppose moreover that e−2ν(c−b) ≤ (p/r − 1)(q/r − 1) ≤ 1 and F ∈ Lp (QE ) and G ∈ Lq (QE ). Then FG ∈ Lr (QE ) and ‖FG‖r ≤ ‖F‖p ‖G‖q . In particular, for r such that r ∈ [1,

2 ], 1 + e−ν(c−b)

we have ‖FG‖r ≤ ‖F‖2 ‖G‖2 . Proof. Let F,

|F| < N

0,

|F| ≥ N

FN = {

and

G, |G| < N GN = { 0, |G| ≥ N.

Then |FN |r ∈ E[a,b] , |GN |r ∈ E[c,d] and ∫ |FN |r |GN |r dμE = (E[a,b] |FN |r , E[c,d] |GN |r )2 = (|FN |r , ΓE (e[a,b] e[c,d] )|GN |r )2 . QE

Note that e[a,b] e[c,d] satisfies ‖e[a,b] e[c,d] ‖2 = ‖e[a,b] eb ec e[c,d] ‖2 ≤ ‖j∗b jc ‖2 = ‖e−|c−b|ων ‖2 ≤ e−2ν(c−b) ≤ (p/r − 1)(q/r − 1). Thus by the Hölder inequality, ‖FN GN ‖rr ≤ ‖|FN |r ‖q/r ‖ΓE (e[a,b] e[c,d] )|GN |r ‖s ,

(3.2.14)

where 1 = 1/s + r/q. Since ‖e[a,b] e[c,d] ‖2 ≤ (p/r − 1)(q/r − 1) = (p/r − 1)(s − 1)−1 ≤ 1, by Proposition 3.13 it is seen that ‖ΓE (e[a,b] e[c,d] )|GN |r ‖s ≤ ‖|GN |r ‖p/r . Together with (3.2.14), this yields ‖FN GN ‖r ≤ ‖FN ‖q ‖GN ‖p ≤ ‖F‖q ‖G‖p .

(3.2.15)

Taking the limit N → ∞ on both sides of (3.2.15), by monotone convergence the lemma follows. An immediate consequence is as follows.

3.3 Functional integral representation for the Pauli–Fierz Hamiltonian | 337

Corollary 3.15. Let ν > 0 and F, G ∈ L2 (Q ). Suppose that Φ ∈ ⋂p≥1 Lp (QE ). Then (Ja F)(Jb G)Φ ∈ L1 (QE ) and ∫ |(Ja F)(Jb G)Φ|dμE ≤ ‖Φ‖s ‖F‖2 ‖G‖2

(3.2.16)

QE

for arbitrary a ≠ b, where s = s(a, b) =

2 . 1 − e−ν(b−a)

(3.2.17)

If a ≠ b, then J∗a ΦJb is a bounded operator and ‖J∗a ΦJb ‖ ≤ ‖Φ‖s . 2 > 1 and s > 1 be such that 1/r + 1/s = 1, i. e., Proof. Let a < b, and r = 1+e−ν(b−a) s = r/(r − 1). Without loss of generality, we can assume that Φ is real-valued. Then by Lemma 3.14,

|(Ja F, ΦJb G)2 | ≤ ‖Φ‖s ‖(Ja F)(Jb G)‖r ≤ ‖Φ‖s ‖Ja F‖2 ‖Jb G‖2 = ‖Φ‖s ‖F‖2 ‖G‖2 . Then the corollary follows. Example 3.16. Since esAE (f ) ∈ L1 (QE ) for any s ∈ ℝ, we have eAE (f ) ∈ ⋂ Lp (QE ). p≥1

Let ν > 0, and a ≠ b. By Corollary 3.15, we can see that J∗a eAE (f ) Jb is a bounded operator for f ∈ ⊕2 L2 (ℝ3 ) such that ‖J∗a eAE (f ) Jb ‖ ≤ ‖esAE (f ) ‖1/s , L1 (Q ) E

where s is given by (3.2.17). In the case where ν = 0, we can also show the boundedness of J∗a eAE (f ) Jb , but we need an additional condition on f . Theorem 3.17. Suppose that f , f /√ω ∈ ⊕2 L2 (ℝ3 ), and a ≠ b. Then J∗a eAE (f ) Jb is a bounded operator. Proof. This is a minor modification of Theorem 1.87. Then we omit the proof.

3.3 Functional integral representation for the Pauli–Fierz Hamiltonian 3.3.1 Hilbert space-valued stochastic integrals In this section, we define a Hilbert space-valued stochastic integral. It will be first explained in some generality and then applied to the Pauli–Fierz model. Using the

338 | 3 The Pauli–Fierz model by path measures Trotter product formula and the factorization formula e−|s−t|Hrad = J∗t Js , we derive the functional integral representation of e−tHPF . Let K be a Hilbert space and define C n (ℝ3 ; K ) = {f : ℝ3 → K | f is n times continuously strongly differentiable} and 󵄨 Cbn (ℝ3 ; K ) = {f ∈ C n (ℝ3 ; K ) 󵄨󵄨󵄨

sup

|z|≤n,x∈ℝ3

‖𝜕z f (x)‖K < ∞},

z

z

z

where |z| = z1 + z2 + z3 for z = (z1 , z2 , z3 ) and 𝜕z = 𝜕x11 𝜕x22 𝜕x33 denotes strong derivative. We set L2 (X ) = L2 (X , d𝒲 x ),

L2 (X) = L2 (X, d𝒲 x ),

where we recall that X = C([0, ∞); ℝ3 ) and X = C(ℝ; ℝ3 ). The proof of the following lemma is straightforward and similar to the case of real-valued processes. Lemma 3.18. Let f ∈ Cb1 (ℝ × ℝ3 ; K ). The sequence defined by 2n

Jμn (f ) = ∑ f ( j=1

j−1 μ μ t, B j−1 t ) (B j − B j−1 ) t t 2n 2n 2n 2n

is a Cauchy sequence in L2 (X ) ⊗ K . Definition 3.19 (Hilbert space-valued stochastic integral). For f ∈ Cb1 (ℝ × ℝ3 ; K ) the limit t

∫ f (s, Bs )dBμs = s-lim Jμn (f ) n→∞

0

defines a K -valued stochastic integral. By the above definition, t

t

t

𝔼[(∫ f (s, Bs )dBμs , ∫ g(s, Bs )dBνs ) 0

0

K

] = δμν 𝔼[∫(f (s, Bs ), g(s, Bs ))K ds]. 0

In particular, the Itô isometry 󵄩󵄩 t t 󵄩󵄩󵄩2 󵄩 [󵄩󵄩󵄩 󵄩 ] μ󵄩 𝔼 [󵄩󵄩∫ f (s, Bs )dBs 󵄩󵄩󵄩 ] = 𝔼 [∫ ‖f (s, Bs )‖2K ds] 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩K [0 ] [󵄩0 ] holds. Similar to the case of real-valued processes, the result below follows.

3.3 Functional integral representation for the Pauli–Fierz Hamiltonian | 339

Corollary 3.20. Let f ∈ Cb2 (ℝ3 ; K ) and Snμ (f )

2n

= ∑(f (B j=1

j 2n

t

μ

) + f (B j−1 t ))(B 2n

j 2n

t

μ

− B j−1 ). 2n

t

Then s-lim Snμ (f ) n→∞

t

=

∫ f (Bs )dBμs 0

t

1 + ∫ 𝜕μ f (Bs )ds 2 0

holds in L2 (X ) ⊗ K . Example 3.21. We will make the following specific choices for the functional integral representation of the Pauli–Fierz model below. Let λ,̂ ωλ̂ ∈ L2 (ℝ3 ) and define the map ξ : ℝ3 ∋ x 󳨃→ λ(⋅ − x) ∈ L2 (ℝ3 ). Thus ξ ∈ Cb1 (ℝ3 ; L2 (ℝ3 )) by the assumption ωλ̂ ∈ L2 (ℝ3 ), and we can define a L2 (ℝ3 )-valued stochastic integral t

t

0

0

∫ ξ (Bs )dBμs = ∫ λ(⋅ − Bs )dBμs . Later on we will construct the functional integral through the Euclidean quantum field, and will use an L2 (ℝ4 )-valued stochastic integral of the form t

̃ − Bs )dBμs . ∫ js φ(⋅

(3.3.1)

0

However, since −|s−t|ω ‖jt f − js f ‖2 1 ) ̂ ̂, (1 − e f) = 2 ( f 2 |s − t| |s − t| |s − t|

̃ − x) ∈ L2 (ℝ4 ) is not strongly diverges as t → s for f ̂ ∈ D(ω), ℝ × ℝ3 ∋ (s, x) 󳨃→ js φ(⋅ ̃ − x) ∉ Cb1 (ℝs × ℝ3x ; L2 (ℝ4 )). Therefore, we need to give differentiable in s ∈ ℝ. Then js φ(⋅ a proper definition of (3.3.1). Lemma 3.22. If λ,̂ ωλ̂ ∈ L2 (ℝ3 ), then for each μ = 1, 2, 3, Snμ (λ)

2n

μ

μ

= ∑ j(j−1)t/2n λ(⋅ − B(j−1)t/2n )(Bjt/2n − B(j−1)t/2n ), j=1

is a Cauchy sequence in L2 (X ) ⊗ L2 (ℝ4 ).

n = 1, 2, 3, . . . ,

340 | 3 The Pauli–Fierz model by path measures n

μ

Proof. Fix an μ. Write Sn = Sn (λ) and η∗ = j∗ λ(⋅ − B∗ ). Then Sn+1 − Sn = ∑2m=1 am , where μ

μ

am = (η(2m−1)t/2n+1 − η(2m−2)t/2n+1 )(B2mt/2n+1 − B(2m−1)t/2n+1 ). μ

μ

Thus 𝔼x [(ai , aj )L2 (ℝ4 ) ] = 0 for i ≠ j, since B2jt/2n+1 − B(2j−1)t/2n+1 is independent of the rest μ

μ

of ai and aj for j > i, and 𝔼x [B2jt/2n+1 − B(2j−1)t/2n+1 ] = 0. Hence 2n

𝔼x [‖Sn+1 − Sn ‖2 ] = ∑ 𝔼x [‖η(2m−1)t/2n+1 − η(2m−2)t/2n+1 ‖2 ] m=1

t . 2n+1

̂ Since ‖jt f − js g‖2 = ‖f − g‖2 + 2(f ̂, (1 − e−|s−t|ω )g)̂ and ‖λ(⋅ − X) − λ(⋅ − Y)‖ ≤ |X − Y|‖ωλ‖, we have μ

μ

‖η(2m−1)t/2n+1 − η(2m−2)t/2n+1 ‖2 ≤ ‖ωλ‖̂ 2 |B(2m−1)t/2n+1 − B(2m−2)t/2n+1 |2 + 2

t ̂ ‖λ‖‖ωλ‖. 2n+1

Finally, we conclude that 1/2

(𝔼x [‖Sm − Sn ‖2 ])

≤(

1/2 m 2‖λ‖‖ωλ‖̂ + ‖ωλ‖̂ 2 t ) ∑ (j+1)/2 , 2 2 j=n+1

thus (Sn )n∈ℕ is a Cauchy sequence. Definition 3.23 (L2 (ℝ4 )-valued stochastic integral). Let λ,̂ ωλ̂ ∈ L2 (ℝ3 ). Then t

∫ js λ(⋅ − Bs )dBμs = s-lim Snμ , n→∞

0

μ = 1, 2, 3,

defines an L2 (ℝ4 )-valued stochastic integral, where the strong limit is in the topology of L2 (X ) ⊗ L2 (ℝ4 ). By the definition, it is seen that t

t

t

𝔼x [(∫ js λ(⋅ − Bs )dBμs , ∫ js ρ(⋅ − Bs )dBνs )] = δμν 𝔼x [∫(λ(⋅ − Bs ), ρ(⋅ − Bs ))ds] . 0 ] [0 ] [ 0 t

μ

The integral ∫0 js λ(⋅ − Bs )dBs can also be defined as a limit of a sequence of L2 (ℝ4 )-valued stochastic integrals, which will be used in the construction of the functional integral representation for the Pauli–Fierz Hamiltonian. Corollary 3.24. Suppose that λ,̂ ωλ̂ ∈ L2 (ℝ3 ) and let S̃nμ

2n

=∑

kt/2n

∫

k=1 (k−1)t/2n

j(k−1)t/2n λ(⋅ − Bs )dBμs .

3.3 Functional integral representation for the Pauli–Fierz Hamiltonian

Then

| 341

t

s-lim S̃nμ = ∫ js λ(⋅ − Bs )dBμs n→∞

0

in L2 (X ) ⊗ L2 (ℝ4 ). Proof. Using that js is an isometry, we obtain t 2n 󵄩󵄩 󵄩2 󵄩󵄩 ̃ μ 󵄩󵄩 μ󵄩 𝔼 [󵄩󵄩Sn − ∫ js λ(⋅ − Bs )dBs 󵄩󵄩 ] = ∑ 󵄩󵄩 󵄩󵄩 k=1

kt/2n

x

0

∫ (k−1)t/2n

𝔼x [‖λ(⋅ − Bs ) − λ(⋅ − B(k−1)t/2n )‖2 ]ds ≤ C

t . 2n

Then the corollary follows. 3.3.2 Functional integral representation The functional integral representation for the decoupled Hamiltonian comes about surprisingly directly. A combination of the functional integral representation for e−tHp and the equality e−tHrad = J∗0 Jt gives (F, e−tHPF,0 G)ℋPF = (F, e−tHp e−tHrad G)ℋPF = (J0 F, e−tHp Jt G)ℋE . Proposition 3.25. Let F, G ∈ ℋPF . Then t

(F, e−tHPF,0 G)ℋPF = ∫ 𝔼x [e− ∫0 V(Bs )ds (J0 F(B0 ), Jt G(Bt ))L2 (QE ) ]dx. ℝ3

Our next goal is to derive a functional integral representation for e−tHPF . Since 1 HPF (A)̂ = (−i∇ − eA)̂ 2 2 has a similar form as Hp (a) = 21 (−i∇ − a)2 in the Schrödinger Hamiltonian with a vector potential a, the procedure of obtaining this functional integral will narrow down to a combination of the construction for e−tHp (a) . We have seen that t

(f , e−tHp (a) g) = ∫ 𝔼x [f (B0 )e−i ∫0 a(Bs )∘dBs g(Bt )]dx.

(3.3.2)

ℝ3

The proof of formula (3.3.2) is proceeded as follows. Define the family of integral operators ϱs : L2 (ℝd ) → L2 (ℝd ) by ϱs f (x) = ∫ Πs (x − y)eih(x,y) f (y)dy,

s ≥ 0,

ℝd

where ϱ0 f (x) = f (x) and h(x, y) = (1/2)(a(x) + a(y)) ⋅ (x − y). Notice that ϱs is symmetric and ‖ϱs ‖ ≤ 1. For f , g ∈ L2 (ℝd ),

342 | 3 The Pauli–Fierz model by path measures 2n

n

2n

2n

j=1

j=0

(g, ϱ2t/2n f ) = ∫ g(x)ei ∑j=1 h(xj−1 ,xj ) f (xn )( ∏ Πt/2n (xj − xj−1 )) ∏ dxj ℝd

with x0 = x. Therefore, t

n

lim (g, ϱ2t/2n f ) = ∫ 𝔼x [g(B0 )f (Bt ) exp (−i ∫ a(Bs ) ∘ dBs )] dx. n→∞ 0 [ ] ℝd By the Riesz representation theorem, there exists a C0 -semigroup {St : t ≥ 0} such that n

lim (g, ϱ2t/2n f ) = (g, St f ).

n→∞

It is directly seen that lims→0 (d/ds)(g, ϱs f ) = (g, −H(a)f ), and hence lim(g, t −1 (1 − ϱt )f ) = (g, Hp (a)f ).

t→0

Thus we conclude that 1

1 ( (e−tA − 1)g, f ) = ∫(−Hp (a)g, e−tsA f )ds. t

(3.3.3)

0

As t → 0 on both sides above, we get (g, Af ) = (Hp (a)g, f ), implying that Ag = Hp (a)g. Hence A = Hp (a) and (3.3.2) follows for V = 0. Let now V be continuous and bounded. From the Trotter product formula, we have (f , e−tHp (a) g) = lim (f , (e−(t/n)Hp (a) e−(t/n)V )n g) n→∞

t

t

= ∫ 𝔼x [f (B0 )e−i ∫0 a(Bs )∘dBs e− ∫0 V(Bs )ds ]dx. ℝd

On a first view, it can be expected that the functional integral representation for ̂ (F, e−tHPF (A) G) goes similarly with a replaced by  which is L2 (Q )-valued multiplication t operator. There is, however, a difference. Although in the classical case ∫0 a(Bs ) ∘ dBs in (3.3.2), integrand a does not depend on time s explicitly, in the Pauli–Fierz model t ̃ − Bs )dBμs appears instead of the there is dependence on s and the integral ∫0 js φ(⋅ Stratonovich integral. This time dependence originates from the semigroup of the free ̂ field Hamiltonian e−tHrad . An extra difficulty is that js φ(⋅−x) is not differentiable in variable s. Here is an outline of our construction of the functional integral representation of (F, e−tHPF G). Replicating the second proof of functional integral representation for Schrödinger operator with vector potentials we notice that e−tHPF (A) = s-lim(Pt/n )n ̂

n→∞

3.3 Functional integral representation for the Pauli–Fierz Hamiltonian

| 343

with a family of suitable integral operators Ps : L2 (ℝ3 ; L2 (Q )) → L2 (ℝ3 ; L2 (Q )), s ≥ 0. Thus in virtue of J∗s Jt = e−|s−t|Hrad we have n

e−tHPF = s-lim lim . . . lim J∗0 (∏ Jtj/n (Pt/nmj )mj J∗tj/n ) Jt , n→∞ m1 →∞

mn →∞

j=1

and by using the Markov property of Es = Js J∗s we deduce n

e−tHPF = s-lim lim . . . lim J∗0 (∏ Etj/n (Pt/nmj ,j )mj Etj/n ) Jt n→∞ m1 →∞

mn →∞

j=1 n

= s-lim lim . . . lim J∗0 (∏(Pt/nmj ,j )mj ) Jt , n→∞ m1 →∞

mn →∞

j=1

where PS,T denotes an operator acting on the Euclidean space ℋE . Finally, we compute ∏nj=1 (Pt/nmj ,j )mj which gives the functional integral formula of (F, e−tHPF G). In what follows, we will turn this argument rigorous. We first show the functional integral representation for the Pauli–Fierz Hamiltonian with sufficiently smooth and bounded V. After establishing it, we extend it to more general potentials V. Theorem 3.26 (Functional integral representation for Pauli–Fierz Hamiltonian). We suppose that V ∈ C0∞ (ℝ3 ). Then t

(F, e−tHPF G) = ∫ 𝔼x [e− ∫0 V(Bs )ds (J0 F(B0 ), e−ieAE (Kt ) Jt G(Bt ))L2 (QE ) ]dx.

(3.3.4)

̂

ℝ3

Here, Kt denotes the ⊕3 L2 (ℝ3 )-valued stochastic integral given by 3

t

̃ − Bs )dBμs . Kt = ⨁ ∫ js φ(⋅ μ=1 0

Proof. For the main part of the proof, we may and do assume V = 0, and in this proof ̃ we write  Es (x) =  E (⊕3 js φ(⋅−x)). Define the family of symmetric contraction operators Ps : ℋPF → ℋPF by Ps F(x) = ∫ Πs (x − y)eih(x,y) F(y)dy,

s > 0,

ℝ3

with P0 F = F, where Πs (x) = (2πs)−3/2 exp(−|x|2 /2s) is the heat kernel and h(x, y) =

e ̂ ̂ (A(x) + A(y)) (x − y). 2

Here, Ps is the counterpart of the quantum field for ϱs defined above for a Schrödinger operator with a vector potential a. By a direct computation, 2n

2n

2n

2n

j=1

j=1

j=1

(F, (Pt/2n ) G) = ∫ dx ∫ F(x) exp (i ∑ hj ) G(x2n ) (∏ Πt/2n (xj−1 − xj )) ∏ dxj ℝ3

n

(ℝ3 )2

344 | 3 The Pauli–Fierz model by path measures ̂ j−1 ) + A(x ̂ j )) ⋅ (xj−1 − xj ). This can be expressed with x = x0 , hj = hj (xj−1 , xj ) = (e/2)(A(x by using Brownian motion as n

(F, (Pt/2n )2 G) = ∫ 𝔼x [(F(B0 ), e−ieA(Ln ) G(Bt ))]dx, ̂

ℝ3

where 3

2n

μ

μ ). t m−1 2n

̃ − Bt mn ) + φ(⋅ ̃ − Bt m−1 ))(B m − B Ln = ⨁ ∑ (φ(⋅ t n 2

μ=1 m=0

2n

2

It is seen that t

3

̃ − Bs )dBμs Ln → Lt = ⨁ ∫ φ(⋅ μ=1 0

as n → ∞ strongly in ⊕3 (L2 (X ) ⊗ L2 (ℝ3 )) implying that for F, G ∈ ℋPF , n

lim (F, (Pt/2n )2 G) = ∫ 𝔼x [(F(B0 ), e−ieA(Lt ) G(Bt ))]dx.

n→∞

̂

(3.3.5)

ℝ3 n

From (3.3.5), it follows that |limn→∞ (F, (Pt/2n )2 G)| ≤ ‖F‖‖G‖. Hence for each t ≥ 0, there exists a symmetric bounded operator St such that n

lim (F, (Pt/2n )2 G) = (F, St G).

n→∞ n

n

Since (Pt/2n )2 is uniformly bounded as ‖(Pt/2n )2 ‖ ≤ 1, the above weak convergence improves to n

s-lim(Pt/2n )2 = St , t ≥ 0.

(3.3.6)

(F, St G) = ∫ 𝔼x [(F(B0 ), e−ieA(Lt ) G(Bt ))]dx.

(3.3.7)

n→∞

Furthermore, by (3.3.5) ̂

ℝ3

Thus n

m

(F, Ss St G) = lim lim (F, (Ps/2n )2 (Pt/2m )2 G) n→∞ m→∞

= ∫ 𝔼x [(F(B0 ), e−ieA(L(s+t) ) G(Bt ))]dx = (F, Ss+t G), ̂

ℝ3

giving the semigroup property Ss St = Ss+t , s, t ≥ 0. It can also be seen that w − limt→0 St = 1 by (3.3.7), which further implies that s-limt→0 St = 1, since limt→0 ‖St F‖2 =

3.3 Functional integral representation for the Pauli–Fierz Hamiltonian | 345

limt→0 (F, S2t F) = ‖F‖2 . Finally, it is trivial to see that S0 = 1. Putting these together, we conclude that {St : t ≥ 0} is a symmetric C0 -semigroup, thus there exists a unique ̂ ̂ self-adjoint operator Ĥ PF (A)̂ such that St = e−t HPF (A) , t ≥ 0, i. e., (F, e−t HPF (A) G) = ∫ 𝔼x [(F(B0 ), e−ieA(Lt ) G(Bt ))]dx. ̂

̂

(3.3.8)

̂

ℝ3

Furthermore, we have ̂ lim(F, t −1 (1 − Pt )G) = (F, HPF (A)G)

(3.3.9)

t→0

for F, G ∈ C0∞ (ℝ3 )⊗̂ ℱrad,fin in Lemma 3.27 below. This leads to (t (e −1

−t Ĥ PF (A)̂

− 1)F, G) = lim (t ((P n→∞

−1

t/2n

2n

1

̂ e−tsĤ PF (A)̂ G)ds. ) − 1)F, G) = − ∫(HPF (A)F, 0

(3.3.10)

In the second equality above, we used (3.3.9). Since ̂ ‖HPF (A)F‖ ≤ C(‖ − ΔF‖ + ‖Hrad F‖ + ‖F‖)

(3.3.11)

for F ∈ D((−1/2)Δ) ∩ D(Hf ), equality (3.3.10) can be immediately extended to vectors ̂ We made use of the assumptions √ωφ̂ ∈ F ∈ D((−1/2)Δ) ∩ D(Hf ) and G ∈ D(Ĥ PF (A)). ̂ L2 (ℝ3 ) and φ/ω ∈ L2 (ℝ3 ) in (3.3.11). Take now t ↓ 0 on both sides of equality (3.3.10). ̂ ̂ G) for vectors F ∈ D((−1/2)Δ) ∩ D(Hf ) and Then it holds that (F, Ĥ PF (A)G) = (HPF (A)F, ̂ ̂ ̂ ̂ ̂ D as Ĥ PF (A)̂ is self-adjoint. Define G ∈ D(HPF (A)), which implies that HPF (A) ⊃ HPF (A)⌈ PF Ĥ PF = Ĥ PF (A)̂ +̇ Hrad . The Trotter product formula and the factorization e−(t/n)Hrad = J∗kt/n J(k+1)t/n yield that n−1

(F, e−t HPF G) = lim (F, (e−(t/n)HPF (A) e−(t/n)Hrad )n G) = lim (J0 F, (∏ Ri ) Jt G) , ̂

̂

n→∞

̂

n→∞

i=0

where Rj = Jjt/n e−(t/n)HPF (A) J∗jt/n . Using the definition of e−t HPF (A) , we get ̂

̂

̂

̂

n

Js e−t HPF (A) J∗s G(x) = s-lim Js (Pt/2n )2 J∗s G(x) ̂

̂

n→∞

2n

2n

2n

j=1

j=1

j=1

= s-lim ∫ Js exp (i ∑ hj ) J∗s G(x2n ) (∏ Πt/2n (xj−1 − xj )) ∏ dxj n→∞

n

(ℝ3 )2

with x = x0 . We are half way through the trying argument. Write δj = δj (n, t, nj ) = t/(n2nj ) for j = 0, 1, . . . , n − 1 and define Ps,j : ℋE → ℋE by Ps with h(x, y) replaced by the Euclidean version htj/n (x, y) given by hs (x, y) =

e ̂ (A (x) + Â Es (y)) ⋅ (x − y) 2 Es

346 | 3 The Pauli–Fierz model by path measures as tj/n

Ps,j F(x) = ∫ Πs (x − y)eih

(x,y)

F(y)dy.

ℝ3

We have n−1

n−1

ni

(J0 F, (∏ Ri ) Jt G) = lim . . . lim (J0 F, ∏(Ei (Pδi ,i )2 Ei )Jt G) n0 →∞

i=0

nn−1 →∞

i=0

n−1

ni

= lim . . . lim (J0 F, (∏(Pδi ,i )2 ) Jt G) . n0 →∞

nn−1 →∞

i=0

Here, in the first equality, we used the fact that 2n

2n

j=1

j=1

Js exp (i ∑ hj ) J∗s = Es exp (i ∑ hsj ) Es , where hsj is defined by hsj = hs (xj−1 , xj ), and in the second equality we used the Markov property of Es . As a result, we have n−1

(J0 F, (∏ Ri )Jt G) = lim . . . lim ∫ 𝔼x [(J0 F(B0 ), e−ieAE (K) Jt G(Bt ))]dx n0 →∞

i=0

nn−1 →∞

̂

ℝ3

with K = K(n0 , n1 , . . . , nn−1 , n) given by 3

nj

n−1 2

̃ −Bt( K = ⨁ ∑ ∑ j tj (φ(⋅ μ=1 j=0 m=1

m n 2nj

n

+j) )

μ

̃ − B t ( m−1 +j) ))(B t + φ(⋅ n n

( m n 2nj

2 j

μ

+j)

−Bt

n

( m−1 nj +j)

)

2

and 3

n−1

K(n0 , n1 , . . . , nn−1 , n) → ⨁ ∑

μ=1 j=0

t(j+1)/n

∫

̃ − Bs )dBμs jtj/n φ(⋅

tj/n

as n0 , n1 , . . . , nn−1 → ∞ in ⊕3 (L2 (X ) ⊗ L2 (ℝ3 )). Finally, as n → ∞ we have (F, e−t HPF G) = ∫ 𝔼x [(J0 F(B0 ), e−ieAE (Kt ) Jt G(Bt ))]dx. ̂

̂

(3.3.12)

ℝ3

By the construction of Ĥ PF , 1 Ĥ PF ⊃ (−i∇ − eA)̂ 2 + Hrad ⌈DPF . 2 In the next section, we will show that 1 Ĥ PF = (−i∇ − eA)̂ 2 + Hrad 2

(3.3.13)

3.3 Functional integral representation for the Pauli–Fierz Hamiltonian

| 347

and Ĥ PF is self-adjoint on DPF . The functional integral representation of e−tHPF including nonzero V can be obtained by the Trotter product formula (F, e−tHPF G) = lim (F, (e−(t/n)HPF (A) e−(t/n)V e−(t/n)Hrad )n G). ̂

n→∞

̂

This completes the proof. Lemma 3.27. Equation (3.3.9) holds true for F, G ∈ C0∞ (ℝ3 )⊗̂ ℱrad,fin . Proof. It is directly seen that d 1 (F, Ps G) = ∫ dx ∫ Πs (x − y) (F(x), Δy eih(x,y) G(y)) dy. ds 2 ℝ3

ℝ3

Here, we used that (dΠs /ds)(x − y) = (1/2)Δy Πs (x − y). Since 𝜕yμ eih G = (i𝜕yμ h ⋅ G + 𝜕yμ G)eih ,

𝜕y2μ eih G = {𝜕y2μ G + 2i𝜕yμ h ⋅ 𝜕yμ G + (i𝜕y2μ h + (i𝜕yμ h)2 )G}eih ,

1 ̂ 𝜕yμ h(x, y) = {𝜕yμ A(y) ⋅ (x − y) − (Â μ (x) + Â μ (y))}, 2 1 ̂ ⋅ (x − y) − 𝜕yμ Â μ (y), 𝜕y2μ h(x, y) = 𝜕y2μ A(y) 2 we have (F(x), Δy eih(x,y) G(y)) = Iμ+ (x, y) + Iμ− (x, y), where

e e Iμ+ (x, y) = (F(x), (Â μ (y)i𝜕yμ + Â μ (y) + Â μ (y)(Â μ (x) + Â μ (y))) G(y)) (x − y) 2 2 e2 μ 2 − (F(x), (Â (y)(x − y)) G(y)), 4 e2 ̂ 2 ̂ ̂ ̂ Iμ− (x, y) = (F(x), (Δyμ − e(A(x) + A(y))i𝜕 (A(x) + A(y)) ) G(y)) . yμ − 4 ̂ Here, Â μ (y) = 𝜕yμ A(y), and in the Fock representation, ̂ ̂ −ikμ φ(k) ikμ φ(−k) 1 Â μν (y) ≅ e−iky a∗ (k, j) + eiky a(k, j)) dk. ∑ ∫ eνj (k) ( √2 j=± √ω(k) √ω(k) 3 ℝ

We have the bound |Iμ+ (x, y)| ≤ ‖F(x)‖(‖N 1/2 𝜕yμ G(y)‖ + ‖NG(y)‖)|x − y|2 , which implies that lim ∫ dx ∫ Πt (x − y)Iμ+ (x, y)dy = 0,

t→0

ℝ3

ℝ3

348 | 3 The Pauli–Fierz model by path measures since limt→0 ∫ℝ3 Πt (x)|x|n dx = 0 for n > 0. It can be also seen that the map y 󳨃→ Iμ− (x, y) is continuous for each x ∈ ℝ3 and it follows that lim I − (x, y) y→x μ

= −(F(x), (−i∇xμ − e μ (x))2 G(x)).

Hence we have lim ∫ dx ∫ Πt (x − y)Iμ− (x, y)dy = ∫ Iμ− (x, x)dx = − ∫ (F(x), (−i∇xμ − e μ )2 G(x))dx,

t→0

ℝ3

ℝ3

ℝ3

ℝ3

since limt→0 ∫ℝ3 Πt (x)f (x)dx = f (0) for any measurable bounded function f being continuous at x = 0. Together with them, we have lim(F, t −1 (1 − Pt )G) = − lim

t→0

t→0+

d (F, Pt G) dt

3

1 ̂ ∫ dx ∫ Πt (x − y)(Iμ− (x, y) + Iμ+ (x, y))dy = (F, HPF (A)G). 2 μ=1

= − lim ∑ t→0

ℝ3

ℝ3

Thus the lemma follows. A rewarding application of the functional integral representation of (F, e−tHPF G) is a diamagnetic inequality as the exponent −ie E (Kt ) of (F, e−tHPF G) is purely imaginary. Corollary 3.28 (Diamagnetic inequality). Under the conditions of Theorem 3.26, it follows that |(F, e−tHPF G)| ≤ (|F|, e−tHPF,0 |G|). Proof. By the functional integral representation in Theorem 3.26, we plainly have t

|(F, e−tHPF G)| ≤ ∫ 𝔼x [e− ∫0 V(Bs )ds (J0 |F(B0 )|, Jt |G(Bt )|)]dx = (|F|, e−tHPF,0 |G|). ℝ3

Here, we used that |Js F| ≤ Js |F| since Js is positivity preserving. The diamagnetic inequality shows that coupling the particle to the quantized radiation field by minimal interaction increases the ground state energy of the noninteracting system. Corollary 3.29 (Enhanced ground state energy). We have inf Spec(Hp ) ≤ inf Spec(HPF ). Proof. Corollary 3.28 implies that inf Spec(HPF,0 ) ≤ inf Spec(HPF ). Since inf Spec(HPF,0 ) = inf Spec(Hp ), the corollary follows.

3.3 Functional integral representation for the Pauli–Fierz Hamiltonian

| 349

3.3.3 Extension to general external potential In Theorem 3.26, we assumed smoothness of the external potential. In this section, we offer an extension of the functional integral representation to a wider potential class. For convenience, in this section we denote HPF with V ≡ 0 by 1 H0 = (−i∇ − eA)̂ 2 + Hrad . 2 First, consider a form boundedness property of H0 . Let F(x)/‖F(x)‖L2 (Q) sgn F(x) = { 0

if ‖F(x)‖L2 (Q) ≠ 0,

if ‖F(x)‖L2 (Q) = 0.

Lemma 3.30. Suppose that V is −(1/2)Δ-form bounded with a relative bound a. Then |V| is H0 -form bounded with a relative bound smaller than a. Proof. Let ψ ∈ C0∞ (ℝ3 ) and ψ > 0. Substituting F = sgn(e−tH0 G) ⋅ ψ in the diamagnetic inequality |(F, e−tH0 G)| ≤ (|F|, e−t(−(1/2)Δ) |G|), we obtain that (ψ, ‖e−tH0 G(⋅)‖L2 (Q) )L2 (ℝ3 ) ≤ (ψ, e−t(−(1/2)Δ) ‖G‖L2 (Q) )L2 (ℝ3 ) . Hence we have ‖|V|1/2 (H0 + z)−1/2 G‖ℋPF ‖G‖ℋPF

󵄩󵄩 1/2 󵄩 󵄩󵄩|V| (−(1/2)Δ + z)−1/2 ‖G(⋅)‖L2 (Q) 󵄩󵄩󵄩 2 3 󵄩 󵄩L (ℝ ) ≤ . ‖G‖ℋPF

Taking the supremum of both sides above with respect to ‖G‖ℋPF ≠ 0, we further obtain ‖|V|1/2 (H0 + z)−1/2 ‖ℋPF ≤ ‖|V|1/2 (−(1/2)Δ + z)−1/2 ‖L2 (ℝ3 ) . Then the lemma follows. Remark 3.31. In a similar way as in Lemma 3.30, we can show that if V is relatively bounded with respect to −(1/2)Δ with a relative bound a, then V is relatively bounded with respect to H0 with a relative bound a. Theorem 3.32 (Functional integral representation for Pauli–Fierz Hamiltonian with singular external potential). Take H0 +̇ V+ −̇ V− with V+ ∈ L1loc (ℝ3 ) and V− is −(1/2)Δ-relatively form bounded with a relative bound strictly less than 1. The equality (3.3.4) holds for H0 +̇ V+ −̇ V− . Proof. The proof is a minor modification of that of the Feynman–Kac formula for Schrödinger operator with singular external potentials and singular vector potentials in Theorem 2.18. Let V+ (x),

V+n (x) = {

n,

V+ (x) < n,

V+ (x) ≥ n,

V− (x), V− (x) < m, V−m (x) = { m, V− (x) ≥ m.

350 | 3 The Pauli–Fierz model by path measures Set Vn,m = V+n − V−m and h = 21 (−i∇ − eA)̂ 2 + Hrad . Then t

(F, e−t(h+Vn,m ) G) = ∫ 𝔼x [e− ∫0 Vn,m (Bs ) ds (F(B0 ), L G(Bt ))] dx,

(3.3.14)

ℝ3

where L = J∗0 e−ieAE (Kt ) Jt . Define the closed quadratic forms ̂

1/2 1/2 1/2 1/2 qn,m (F, F) = (h1/2 F, h1/2 F) + (V+n F, V+n F) − (V−m F, V−m F),

1/2 1/2 qn,∞ (F, F) = (h1/2 F, h1/2 F) + (V+n F, V+n F) − (V−1/2 F, V−1/2 F),

q∞,∞ (F, F) = (h1/2 F, h1/2 F) + (V+1/2 F, V+1/2 F) − (V−1/2 F, V−1/2 F), whose form domains are respectively Q(qn,m ) = Q(h), Q(qn,∞ ) = Q(h) and Q(q∞,∞ ) = Q(h) ∩ Q(V+ ). In the same arguments on monotone convergence of quadratic forms as the proof of Theorem 2.18, we can conclude that for all t ≥ 0, exp (−t (h + V+n − V−m )) → exp (−t (h + V+n −̇ V− ))

(3.3.15)

exp(−t(h +̇ V+n −̇ V− )) → exp(−t(h +̇ V+ −̇ V− ))

(3.3.16)

as m → ∞, and

as n → ∞ strongly. By taking first n → ∞ and then m → ∞, it can be proven that both sides of (3.3.14) converge. That is, the left-hand side of (3.3.14) converges by (3.3.15) and (3.3.16). We have the inequality t

∫ 𝔼x [e− ∫0 Vn,m (Bs ) ds |(F(B0 ), L G(Bt ))|]dx ℝ3

t

≤ ∫ 𝔼x [e− ∫0 Vn,∞ (Bs ) ds (|F(B0 )|, e−tHrad |G(Bt )|)]dx < ∞. ℝ3

Here, the finiteness of the second term can be derived from the proof of Theorem 2.18. Since t

t

|e− ∫0 Vn,m (Bs ) ds (F(B0 ), L G(Bt ))| ≤ e− ∫0 Vn,∞ (Bs ) ds (|F(B0 )|, e−tHrad |G(Bt )|), the dominated convergence theorem yields that t

t

lim ∫ 𝔼x [e− ∫0 Vn,m (Bs ) ds (F(B0 ), L G(Bt ))]dx = ∫ 𝔼x [e− ∫0 Vn,∞ (Bs ) ds (F(B0 ), L G(Bt ))]dx.

m→∞

ℝ3

ℝ3

Furthermore, since t

∫ 𝔼x [e− ∫0 Vn,∞ (Bs ) ds |(F(B0 ), L G(Bt ))|]dx ℝ3

t

≤ ∫ 𝔼x [e− ∫0 Vn,∞ (Bs ) ds (|F(B0 )|, e−tHrad |G(Bt )|)]dx < ∞ ℝ3

3.4 The Pauli–Fierz model with Kato-class potential | 351

and t 󵄨 󵄨󵄨 − ∫t V (B ) ds 󵄨󵄨e 0 n󸀠 ,∞ s (F(B ), L G(B ))󵄨󵄨󵄨 ≤ e− ∫0 Vn,∞ (Bs ) ds (|F(B )|, e−tHrad |G(B )|) 󵄨󵄨 0 t 0 t 󵄨󵄨 󵄨 󵄨

for n ≤ n󸀠 , the dominated convergence theorem again yields that t

t

lim ∫ 𝔼x [e− ∫0 Vn,∞ (Bs ) ds (F(B0 ), L G(Bt ))]dx = ∫ 𝔼x [e− ∫0 V(Bs ) ds (F(B0 ), L G(Bt ))]dx.

n→∞

ℝ3

ℝ3

Together with them the right-hand side of (3.3.14) converges to t

∫ 𝔼x [e− ∫0 V(Bs ) ds (F(B0 ), L G(Bt ))]dx ℝ3

for any F, G ∈ ℋPF as first m → ∞ and then n → ∞. Then the proof is complete.

3.4 The Pauli–Fierz model with Kato-class potential We consider the Pauli–Fierz model with Kato-class potential V. In Section 3.4 we use Assumption 3.33 below. Assumption 3.33. The following conditions hold. ̂ ̂ ̂ ̂ (1) Charge distribution: φ ∈ S 󸀠 (ℝ3 ), φ(−k) = φ(k) φ/ω ∈ L2 (ℝ3 ); and √ωφ,̂ φ/√ω, (2) External potential: V is Kato-decomposable. First, we are interested in defining HPF with Kato-class potential as a self-adjoint operator. This will be done through the functional integral representation established in the previous section. The expression t

(F, e−tHPF G) = ∫ 𝔼x [e− ∫0 V(Bs )ds (J0 F(x), e−ieAE (Kt ) Jt G(Bt ))]dx ̂

ℝ3

implies t

(e−tHPF G)(x) = 𝔼x [e− ∫0 V(Bs )ds J∗0 e−ieAE (Kt ) Jt G(Bt )]. ̂

(3.4.1)

Conversely, we shall show that a sufficient condition to define the right-hand side of (3.4.1) is that V is of Kato-class. This idea used for Schrödinger operators with Katoclass potentials will be extended to the Pauli–Fierz Hamiltonian in this section. Let V be a Kato-decomposable potential and define the family of operators t

(Tt F) (x) = 𝔼x [e− ∫0 V(Br )dr J∗0 e−ieAE (Kt ) Jt F(Bt )]. ̂

Lemma 3.34. Operator Tt is bounded on ℋPF for each t ≥ 0.

352 | 3 The Pauli–Fierz model by path measures Proof. Let F ∈ ℋPF . We show that t

‖Tt F‖2ℋPF = ∫ ‖𝔼x [e− ∫0 V(Br )dr J∗0 e−ieAE (Kt ) Jt F(Bt )]‖2L2 (Q) dx < ∞. ̂

(3.4.2)

ℝ3

By the Schwarz inequality, we have t

‖Tt F‖2ℋPF ≤ ∫ 𝔼0 [e−2 ∫0 V(Br +x)dr ]𝔼0 [‖F(Bt + x)‖2 ]dx. ℝ3

t

Since V is of Kato-class, we have supx∈ℝ3 𝔼0 [e−2 ∫0 V(Br +x)dr ] = C < ∞, and thus ‖Tt F‖2ℋPF ≤ C‖F‖2ℋPF . From now on, we shall show that {Tt : t ≥ 0} is a symmetric C0 -semigroup. To do that, we introduce the time shift operator ut on L2 (ℝ4 ) by ut f (x) = f (x0 − t, x),

x = (x0 , x) ∈ ℝ × ℝ3 .

It is straightforward that u∗t = u−t and u∗t ut = 1. We denote the second quantization of ut denoted by Ut = ΓE (ut ) which acts on L2 (QE ) and is unitary. The relationship between js and ut is given next. Lemma 3.35. It follows that ut js = js+t for every t, s ∈ ℝ. Proof. In the position representation, js is given by js f (x) =

√ω(k) 1 f ̂(k)dk. ∫ e+i(k0 (x0 −s)+k⋅x) 2 √π(2π) 2 + |k |2 √ ω(k) 4 0 ℝ

Hence ut js f (x) = js+t f (x). Lemma 3.35 implies the formula Ut Js = Js+t . Lemma 3.36. It follows that Ts Tt = Ts+t for s, t ≥ 0. Proof. By the definition of Tt , we have s

t

Ts Tt F = 𝔼x [e− ∫0 V(Br )dr J∗0 e−ieAE (Ks ) Js 𝔼Bs [e− ∫0 V(Br )dr J∗0 e−ieAE (Kt ) Jt F(Bt )]] . ̂

̂

(3.4.3)

∗ ∗ By the formulae, Js J∗0 = Js (U−s Js )∗ = Js J∗s U−s = Es U−s and Jt = U−s Jt+s , we see that (3.4.3) is equal to s

t

∗ −ieAE (Kt ) 𝔼x [e− ∫0 V(Br )dr J∗0 e−ieAE (Ks ) Es 𝔼Bs [e− ∫0 V(Br )dr U−s e U−s Jt+s F(Bt )]] . ̂

̂

(3.4.4)

3.4 The Pauli–Fierz model with Kato-class potential | 353

∗ −ieAE (Kt ) Now we compute U−s e U−s . Since Us is unitary, we have ̂

∗ −ieAE (Kt ) U−s e U−s = e−ieAE (u−s Kt ) ̂

̂

∗

as an operator. The test function of the exponent u∗−s Kt is given by u∗−s Kt

3

t

̃ − Br )dBμr . = ⨁ ∫ jr+s φ(⋅ μ=1 0

Moreover, by the Markov property of Et , t ∈ ℝ, we may neglect Es in (3.4.4), and by the Markov property of (Bt )t≥0 we have s

s+t

(Ts Tt F)(x) = 𝔼x [e− ∫0 V(Br )dr J∗0 e−ieAE (Ks ) 𝔼x [e− ∫s ̂

s+t

= 𝔼x [e− ∫0

V(Br )dr −ie E (Kss+t )

e

V(Br )dr ∗ −ie E (Ks+t ) J0 e Js+t F(Bs+t )]

Js+t F(Bs+t )|FsBM ]]

= Ts+t F,

where (FtBM )t≥0 denotes the natural filtration of (Bt )t≥0 . Strong continuity of Tt on ℋPF can be checked similarly as previously done, while T0 = 1 is trivial. Theorem 3.37. Semigroup {Tt : t ≥ 0} is a symmetric C0 -semigroup. Proof. It was shown that {Tt : t ≥ 0} is a C0 -semigroup. Hence it is sufficient to show that Tt is symmetric for each t ≥ 0. Recall that R = Γ(r) is the second quantization of the reflection r and the second quantization of time-shift ut is denoted by Ut = Γ(ut ). We have t

(F, Tt G) = (Ut RF, Ut RTt G) = ∫ 𝔼x [e− ∫0 V(Bs )ds (Jt F(B0 ), e−ieAE (ut rKt ) J0 G(Bt ))] dx. ̂

ℝ3 t d ̃ − Bs )dBμs . Since Ḃ s = Bt−s − Bt = Bs for any s such that Notice that ut rKt = ∫0 jt−s φ(⋅ 0 ≤ s ≤ t with fixed t, we have t

(F, Tt G) = ∫ 𝔼x [e− ∫0 V(Bs )ds (Jt F(Ḃ 0 ), e−ieAE (ut rKt ) J0 G(Ḃ t ))] dx. ̇

̂

̇

ℝ3 t μ ̃ − Ḃ s )dḂ μs . Thus Here, K̇ t = ⨁3μ=1 K̇ t = ⨁3μ=1 ∫0 jt−s φ(⋅ 2m

μ ̃ − Bt−tj−1 + Bt − x) (Bμt−t − Bμt−t ) K̇ t = lim ∑ jt−tj−1 φ(⋅ j−1 j−1 m→∞

j=1

2m

μ

μ

μ,m

̃ − Btj + Bt − x) (Bt − Bt ) = lim Kt = − lim ∑ jtj φ(⋅ j j−1 m→∞

j=1

m→∞

.

354 | 3 The Pauli–Fierz model by path measures Here, tj = tj/2m . Note that 2m

μ

t

μ

̃ − Btj−1 + y) (Bt − Bt ) = ∫ js φ(⋅ ̃ − Bs )dBμs . lim ∑ jtj−1 φ(⋅ j j−1

m→∞

j=1

0

It is however shown by the Coulomb gauge condition (3.1.8) and the fact that jt is an t μ,m ̃ − Bs )dBμs as m → ∞. Exchanging integrals isometry that Kt also converges to ∫0 js φ(⋅ 0 ∫X d𝒲 and ∫ℝ3 dx and changing the variable x to y + Bt , we can have t

m

(F, Tt G) = lim ∫ 𝔼x [e− ∫0 V(Bs )ds (Jt F(Ḃ 0 ), e−ieAE (Kt ) J0 G(Ḃ t ))] dx m→∞

̇

̂

ℝ3 t

̂ = 𝔼0 [ ∫ e− ∫0 V(Bt−s +y)ds (Jt F(Bt + y), e+ieAE (Kt ) J0 G(y)) dy] . ] [ℝ3

Then t

(F, Tt G) = ∫ 𝔼y [e− ∫0 V(Bs )ds (J∗0 e−ieAE (Kt ) Jt F(y + Bt ), G(y))] dy = (Tt F, G) ̂

ℝ3

and Tt is symmetric. Then the proof is complete. By Theorem 3.37 and the Stone theorem for semigroups, there exists a self-adjoint Kato operator HPF such that Kato

Tt = e−tHPF ,

t ≥ 0.

Definition 3.38 (Pauli–Fierz Hamiltonian for Kato-class potential). We call the selfKato adjoint operator HPF Pauli–Fierz Hamiltonian for Kato-class potential V. In a similar way to HPF with V ∈ ℛKato , the Pauli–Fierz Hamiltonian for Kato-class Kato potential HPF also satisfies the diamagnetic inequality. Corollary 3.39 (Diamagnetic inequality). It follows that Kato

Kato

|(F, e−tHPF G)| ≤ (|F|, e−t(Hp

+Hrad )

|G|),

where HpKato is the Schrödinger operator with Kato-class potential. Proof. Since J∗0 and Jt are positivity preserving, we have |J∗0 F| ≤ J∗0 |F|, |Jt F| ≤ Jt |F| and t

Kato

|e−tHPF F| ≤ 𝔼x [e− ∫0 V(Br )dr J∗0 Jt |F(Bt )|]. Kato

The right-hand side above is just e−t(Hp

+Hrad )

|F|.

Kato For the notational convenience, we use notation HPF for HPF unless confusions arise in what follows.

3.5 Applications of functional integral representations | 355

3.5 Applications of functional integral representations 3.5.1 Self-adjointness of the Pauli–Fierz Hamiltonian One of the basic problems in the spectral analysis of Hamiltonians under consideration is to show their essential self-adjointness or self-adjointness. In Section 3.5.1 we assume Assumption 3.4. As we have pointed out in the previous section, for sufficiently small e the Pauli–Fierz operator HPF is self-adjoint on DPF . We will show in this section that HPF is self-adjoint on DPF for all e ∈ ℝ. Recall that DPF = D(−(1/2)Δ) ∩ D(Hrad ). We have already constructed a functional integral representation for Ĥ PF , however, it remains to be shown that Ĥ PF = HPF . In the proof of Theorem 3.26, we defined the self-adjoint operator Ĥ PF = Ĥ PF (A)̂ +̇ Hrad and have shown that Ĥ PF ⊃ HPF ⌈DPF in (3.3.13), and (F, e−t HPF G) = ∫ 𝔼x [(J0 F(B0 ), e−ieAE (Kt ) Jt G(Bt ))]dx ̂

̂

(3.5.1)

ℝ3

in (3.3.12). Let Rt (F, G) be the quadratic form defined by the right-hand side of (3.5.1) in which there is no restriction on the value of e. The right-hand side of (3.5.1) is welldefined for any e ∈ ℝ. There exists a semigroup St such that Rt (F, G) = (F, St G) and its generator Ĥ PF satisfies that Ĥ PF ⊃ HPF ⌈DPF . Clearly, for weak enough coupling Ĥ PF = HPF ⌈DPF . We will show that St leaves DPF invariant and from that conclude that Ĥ PF is essentially self-adjoint on DPF . This will imply that HPF is essentially self-adjoint on this joint domain. Finally, we will show that HPF is closed on DPF and, therefore, HPF is self-adjoint on DPF . We start by a general result. Lemma 3.40. Let K be a nonnegative self-adjoint operator. Suppose that there exists a dense domain D such that D ⊂ D(K) and e−tK D ⊂ D for all t ≥ 0. Then K⌈D is essentially self-adjoint. Proof. It suffices to show that for some λ > 0, Ran((λ + K)⌈D ) is dense. Suppose the contrary. Then there exists nonzero f such that (f , (λ + K)ψ) = 0 for all ψ ∈ D. We have d (f , e−tK ψ) = (f , −Ke−tK ψ) = λ(f , e−tK ψ). dt Thus (f , e−tK ψ) = eλt (f , ψ). If (f , ψ) ≠ 0, then limt→∞ |(f , e−tK ψ)| = ∞, contradicting the fact that e−tK is a contraction. Hence (f , ψ) = 0 for all ψ ∈ D(K), but (f , ψ) cannot equal zero for all ψ ∈ D(K), since D(K) is dense. Hence we conclude that Ran((λ +K)⌈D ) is dense. The next result is an important application of Lemma 3.40 to the Pauli–Fierz Hamiltonian.

356 | 3 The Pauli–Fierz model by path measures Lemma 3.41. Suppose that |(HPF,0 F, e−t HPF G)| ≤ CG ‖F‖

(3.5.2)

̂

for all F, G ∈ D(HPF,0 ) = DPF with some constant CG which may depend on G. Then HPF is essentially self-adjoint on DPF . Proof. Inequality (3.5.2) yields that e−t HPF G ∈ D(HPF,0 ) by the Riesz representation thê orem, i. e., e−t HPF DPF ⊂ DPF . Thus Lemma 3.40 implies that Ĥ PF ⌈DPF is essentially selfadjoint. The lemma follows from HPF = Ĥ PF on DPF . ̂

t

μ

̃ − Bs )dBs similar to the In order to show (3.5.2), we need an inequality for ∫0 js φ(⋅ μ

BDG inequality. With an ℝd -valued function a = (a1 , . . . , ad ), random process Yt =

t ∫0

μ aμ (s, Bs )dBs

satisfies

μ

μ

μ

d(Yt )2 = 2Yt aμ (t, Bt )dBt + aμ (t, Bt )2 dt and the BDG inequality t 󵄨󵄨 t 󵄨󵄨2m 󵄨󵄨 󵄨󵄨 m m−1 x [ 𝔼 [󵄨󵄨 ∫ a(s, Bs )dBs 󵄨󵄨 ] ≤ (m(2m − 1)) t 𝔼 ∫ |a(s, Bs )|2m ds] 󵄨󵄨 󵄨󵄨 0 ] [0 x

holds. We show its infinite dimensional version below. Let ω̂ E = ω(−i∇) ⊗ 1 under the identification L2 (ℝ4 ) ≅ L2 (ℝ3 ) ⊗ L2 (ℝ). Lemma 3.42 (BDG-type inequality). If ω(k−1)/2 φ̂ ∈ L2 (ℝ3 ), then t 󵄩󵄩 󵄩2m (2m)! m (k−1)/2 2m 󵄩 󵄩󵄩 μ󵄩 ̂ L2 (ℝ3 ) . ̃ φ‖ 𝔼x [󵄩󵄩󵄩ω̂ k/2 j φ(⋅ − B )dB ∫ s s s󵄩 󵄩󵄩 2 3 ] ≤ 2m t ‖ω 󵄩󵄩 E 󵄩L (ℝ ) 0

In particular, 2m sup 𝔼x [‖ω̂ k/2 E Kt ‖⊕3 L2 (ℝ3 ) ] ≤

x∈ℝ3 t

3(2m)! m (k−1)/2 2m ̂ L2 (ℝ3 ) , t ‖ω φ‖ 2m

μ

̃ − Bs )dBs . where Kt = ⨁3μ=1 ∫0 js φ(⋅ Proof. Let (Xt )t≥0 = (Xt1 , Xt2 , Xt3 )t≥0 be a random process defined by μ Xt

󵄩󵄩 t 󵄩󵄩2 󵄩󵄩 󵄩 󵄩󵄩 󵄩 μ󵄩 = 󵄩󵄩∫ js λ(⋅ − Bs )dBs 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩0 󵄩󵄩L2 (ℝ3 )

3.5 Applications of functional integral representations | 357

with λ ∈ L2 (ℝ3 ). By the definition of js , we have μ

󵄨󵄨 n 1 󵄨 μ(k)2 |λ(k)|2 󵄨󵄨󵄨 ∑ 󵄨󵄨 π j=1

Xt = lim ∫ n→∞

ℝ4

tj/n

e−ik⋅Bs e−ik0

∫

t(j−1) n

t(j−1)/n

󵄨󵄨2 󵄨 dBμs 󵄨󵄨󵄨 dk, 󵄨󵄨

1/2 where μ(k) = ( ω(k)ω(k) . Since 2 +|k |2 ) 0

󵄨󵄨 n 󵄨 lim 𝔼󵄨󵄨󵄨 ∑ n→∞ 󵄨󵄨 j=1

tj/n

e−ikBs e−ik0

∫

t(j−1) n

t 󵄨󵄨2 󵄨 dBμs − ∫ e−ikBs e−ik0 s dBμs 󵄨󵄨󵄨 = 0, 󵄨󵄨 0

t(j−1)/n

there exists a subsequence n󸀠 such that tj/n󸀠

n󸀠

∑ j=1

∫

e

t(j−1) −ik⋅Bs −ik0 n󸀠

e

dBμs

t

→ ∫ e−ik⋅Bs e−ik0 s dBμs 0

t(j−1)/n󸀠

almost surely. Hence we obtain that μ

Xt =

1 μ ∫ μ(k)2 |λ(k)|2 |Yt |2 dk, π

(3.5.3)

ℝ3

μ

μ

t

μ

where Yt is the stochastic integral defined by Yt = ∫0 e−isk0 e−ik⋅Bs dBs . The Itô formula then gives μ

μ

μ

d|Yt |2 = 2 Re(Yt e−ik0 t e−ik⋅Bt )dBt + dt.

(3.5.4)

Inserting (3.5.4) into (3.5.3), it is seen that μ

t

Xt = 2 Re

1 ∫ μ(k)2 |λ(k)|2 (∫ Ysμ e−ik0 s e−ik⋅Bs dBμs ) dk + t‖λ‖2 , π 0

ℝ3

where we used that ∫ℝ4 μ(k)2 dk = π. Hence the stochastic differential equation μ

μ

μ

dXt = Zt dBt + ‖λ‖2 dt is obtained, where μ

t

Zt = 2 Re (jt λ(⋅ − Bt ), ∫ js λ(⋅ − Bs )dBμs ) 0

.

L2 (ℝ3 )

An application of the Itô formula gives 1 μ μ μ μ μ μ μ d(Xt )m = m(Xt )m−1 Zt dBt + (m(Xt )m−1 ‖λ‖2 + m(m − 1)(Xt )m−2 (Zt )2 ) dt. 2

358 | 3 The Pauli–Fierz model by path measures Taking expectation on both sides, we have t

μ

𝔼x [(Xt )m ] = m‖λ‖2 ∫ 𝔼x [(Xsμ )m−1 ]ds + 0

By the bound

μ |Zt |

≤

μ 2‖λ‖√Xt , x

𝔼

μ [(Xt )m ]

t

m(m − 1) ∫ 𝔼x [(Xsμ )m−2 (Zsμ )2 ]ds. 2 0

it follows that t

2m(2m − 1) 2 ‖λ‖ ∫ 𝔼x [(Xsμ )m−1 ]ds, ≤ 2 0

and by induction furthermore μ

𝔼x [(Xt )m ] ≤

(2m)! 2m m ‖λ‖ t 2m

is obtained. Notice that the intertwining property (ω(−i∇)⊗1)js = js ω(−i∇) holds. Write ̂ and the proposition follows. λ = ω(−i∇)k/2 φ.̃ Then ‖λ‖ = ‖ω(k−1)/2 φ‖ Before checking (3.5.2), note that Jt intertwines the free field Hamiltonian Hrad and dΓE (ω̂ E ), i. e., Jt Hrad = dΓE (ω̂ E )Jt .

(3.5.5)

Π̂ E (f) = i[N, Â E (f)].

(3.5.6)

Define Then Π̂ E (f) is the conjugate momentum of  E (f), i. e., the canonical commutation relation [ E (f), Π̂ E (g)] = 2iqE (f, g) is satisfied for all f, g ∈ ⊕3 L2 (ℝ4 ). Let G be an analytic vector for  E (f). Then (−i E (f))n G n! n=0 ∞

e−iAE (f) G = ∑ ̂

(3.5.7)

holds. By the commutation relations, [dΓE (ω̂ E ), Â E (f)] = −iΠ̂ E (ω̂ E f), [[dΓE (ω̂ E ), Â E (f)], Â E (f)] = −2qE (ω̂ E f, f), together with (3.5.7), we moreover have the identity dΓE (ω̂ E )e−iAE (f) G = e−iAE (f) (dΓE (ω̂ E ) − Π̂ E (ω̂ E f) + qE (ω̂ E f, f))G ̂

(3.5.8)

̂

for f ∈ D(ω̂ E ). Furthermore, by a limiting argument it can be seen that e−iAE (f) with f ∈ D(ω̂ E ) leaves D(dΓE (ω̂ E )) invariant and the identity (3.5.8) holds for G ∈ D(dΓE (ω̂ E )). Put ̂

Jt = J∗0 e−ieAE (Kt ) Jt . ̂

3.5 Applications of functional integral representations | 359

Lemma 3.43. Let F, G ∈ D(Hrad ). Then we have |(Hrad F, e−t HPF G)| ≤ C((√t + t)‖(Hrad + 1)1/2 G‖ + ‖Hrad G‖)‖F‖. ̂

In particular, e−tHPF DPF ⊂ D(Hrad ). Proof. We see that Jt leaves D(Hrad ) invariant, and for G ∈ D(Hrad ) [Hrad , Jt ]G = J∗0 e−ieAE (Kt ) Ct Jt G ̂

by (3.5.5) and (3.5.8), where Ct = −eΠ̂ E (ω̂ E Kt ) + e2 qE (ω̂ E Kt , Kt ).

(3.5.9)

Let F, G ∈ D(Hrad ). Then the functional integral representation yields (Hrad F, e−t HPF G) = ∫ 𝔼x [(F(B0 ), [Hrad , Jt ]G(Bt ))] dx + (F, e−t HPF Hrad G). ̂

̂

ℝ3

Furthermore, the right-hand side above can be estimated as |(Hrad F, e

−t Ĥ PF

1/2

2

x

G)| ≤ ‖F‖ (∫ (𝔼 [‖[Hrad , Jt ]Gt ‖]) dx)

+ ‖F‖‖Hrad G‖.

ℝ3

Note the bound ‖[Hrad , Jt ]G‖ ≤ ‖Ct Jt G‖. We estimate each term in (3.5.9) separately. We have 1/2 ̂ ‖Π̂ E (ω̂ E Kt )Jt Ψ‖ ≤ C(‖ω̂ 1/2 E Kt ‖ + ‖ωE Kt ‖)‖(Hrad + 1) Ψ‖, 2 |qE (ω̂ E Kt , Kt )| ≤ ‖ω̂ 1/2 E Kt ‖

with a constant C. Applying our BDG-type inequality to ‖ω̂ E Kt ‖2n , we have ∫ (𝔼x [‖[Hrad , Jt ]Gt ‖])2 dx ≤ C 󸀠 (t + t 2 )‖(Hrad + 1)1/2 G‖2 ℝ3

with a constant C 󸀠 . Hence the lemma follows. We next show that e−t HPF DPF ⊂ D((−1/2)Δ). Let ̂

󵄨

n

3 2

3

̂ 1 ) . . . A(f ̂ n )) 󵄨󵄨󵄨 F ∈ S (ℝ ), f1 , . . . , fn ∈ ⊕ L (ℝ ), n ≥ 1} . S (Q ) = { F(A(f real 󵄨

(3.5.10)

Lemma 3.44. Let ω3/2 φ̂ ∈ L2 (ℝ3 ). Then for F ∈ D(Δ) and G ∈ DPF , |(ΔF, e−t HPF G)| ≤ C(‖ − ΔG‖ + (√t + t)‖(−Δ + Hrad + 1)G‖)‖F‖ ̂

with a constant C > 0. In particular, e−t HPF DPF ⊂ D((−1/2)Δ). ̂

(3.5.11)

360 | 3 The Pauli–Fierz model by path measures Proof. Suppose that F = f ⊗ Φ and G = g ⊗ Ψ, where f , g ∈ C0∞ (ℝ3 ) and Φ, Ψ ∈ S (Q ). By a functional integration, we have (−ΔF, e−t HPF G) = − ∫ 𝔼[(J0 F(x), e−ieAE (Kt ) UJt G(Bt + x))]dx, ̂

̂

ℝ3

where 3

U = Δ + ∑ (−ie E (Kμμ ) − e2  E (Kμ )2 − 2ie E (Kμ ) ⋅ ∇μ ) μ=1

and t

3

Kμ = ⨁ ∫ jt φ̃ μ (⋅ − Bs )dBνs , ν=1 0

̂ √ω),̌ φ̃ μ = (−ikμ φ/

t

3

Kμμ = ⨁ ∫ jt φ̃ μμ (⋅ − Bs )dBνs , ν=1 0

̂ √ω).̌ φ̃ μμ = (−kμ2 φ/

̂ ∈ L2 (ℝ3 ), and thus φ̃ μμ Since we assumed that ω3/2 φ̂ ∈ L2 (ℝ3 ), it follows that kμ2 φ/√ω is well-defined. Hence 1/2

|(−ΔF, e

−t Ĥ PF

2

G)| ≤ ‖F‖ (∫ |𝔼 [‖UJt G(Bt + x)‖]| dx)

.

(3.5.12)

ℝ3

We estimate 𝔼 [‖UJt G(Bt + x)‖]. By the BDG-type inequality again we have 2m 󸀠 m (n+3)/2 ̂ 2m sup 𝔼x [‖ω̂ n/2 φ‖ E Kμμ ‖ ] ≤ C t ‖ω

(3.5.13)

2m 󸀠󸀠 m (n+1)/2 ̂ 2m sup 𝔼x [‖ω̂ n/2 φ‖ E Kμ ‖ ] ≤ C t ‖ω

(3.5.14)

x∈ℝ3

and x∈ℝ3

with constants C 󸀠 and C 󸀠󸀠 . Let ‖f ‖α = ‖ωα/2 f ‖ and A = ‖f ‖−1 + ‖f ‖0 ,

B = (‖f ‖−1 + ‖f ‖0 )(‖g‖1 + ‖g‖2 ).

Combining the inequalities, ‖a♯ (f )Ψ‖ ≤ A‖(Hrad + 1)1/2 Ψ‖,

‖a♯ (f )a♯ (g)Ψ‖ ≤ C1 B‖(Hrad + 1)1/2 Ψ‖ + C2 A‖(Hrad + 1)Ψ‖ with (3.5.13) and (3.5.14), we can estimate the right-hand side of (3.5.12) and show (3.5.11). By a limiting argument, the lemma follows for F ∈ D(Δ) and G ∈ DPF .

3.5 Applications of functional integral representations | 361

Lemma 3.45. Let ω3/2 φ̂ ∈ L2 (ℝ3 ). Then HPF is essentially self-adjoint on DPF . Proof. By Lemmas 3.43 and 3.44, we obtain that |(HPF,0 F, e−t HPF G)| ≤ CG ‖F‖ for F, G ∈ ̂

D(HPF,0 ) = DPF with a constant CG depending on G. Then e−t HPF leaves DPF invariant. This implies that Ĥ PF is essentially self-adjoint on DPF and then so is HPF since HPF = Ĥ PF on DPF . ̂

Next, we show self-adjointness of HPF on DPF without assuming ω3/2 φ̂ ∈ L2 (ℝ3 ). Lemma 3.46. Suppose that ω3/2 φ̂ ∈ L2 (ℝ3 ). Then HPF,0 (Ĥ PF + z)−1 is bounded for every z ∈ ℂ with Im z ≠ 0. Proof. We prove that both of Δ(Ĥ PF + z)−1 and Hrad (Ĥ PF + z)−1 are bounded. Note that HPF = Ĥ PF on DPF . Let pμ = −i∇μ . In this proof, all numbered constants are nonnegative. We have 2 ̂ ‖p2 Ψ‖2 ≤ C1 (‖(p − eA)̂ 2 Ψ‖2 + ‖Â ⋅ (p − eA)Ψ‖ + ‖Â 2 Ψ‖2 ). 2 ̂ We estimate ‖Â ⋅ (p − eA)Ψ‖ as 3

2 ̂ ‖Â ⋅ (p − eA)Ψ‖ ≤ C2 ∑ ((p − eA)̂ μ Ψ, (Hrad + 1)(p − eA)̂ μ Ψ) μ=1

3

= C2 ((p − eA)̂ 2 Ψ, (Hrad + 1)Ψ) + C2 ∑ ((p − eA)̂ μ Ψ, −e[Hrad , Â μ ]Ψ). μ=1

Since ‖[Hrad , Â μ ]Ψ‖ ≤ C2󸀠 ‖(Hrad + 1)1/2 Ψ‖, we have ‖p2 Ψ‖2 ≤ C3 (‖(p − eA)̂ 2 Ψ‖2 + ‖Hrad Ψ‖2 + ‖Ψ‖2 ). We note that 󵄩󵄩 1 󵄩󵄩2 󵄩 󵄩 ‖Ĥ PF Ψ‖2 = 󵄩󵄩󵄩 (p − eA)̂ 2 Ψ󵄩󵄩󵄩 + ‖Hrad Ψ‖2 + Re((p − eA)̂ 2 Ψ, Hrad Ψ). 󵄩󵄩 2 󵄩󵄩 Since Re((p − eA)̂ 2 Ψ, Hrad Ψ) 3

= ∑ {((p − eA)̂ μ Ψ, Hrad (p − eA)̂ μ Ψ) + Re((p − eA)̂ μ Ψ, −e[Hrad , Â μ ]Ψ)} μ=1 3

≥ ∑ Re((p − eA)̂ μ Ψ, −e[Hrad , Â μ ]Ψ) μ=1

≥ −ε‖(p − eA)̂ 2 Ψ‖2 − C4

1 1/2 (‖Hrad Ψ‖2 + ‖Ψ‖2 ) 4ε

(3.5.15)

362 | 3 The Pauli–Fierz model by path measures 1/2 for any ε > 0, by the trivial bound ‖Hrad Ψ‖2 ≤ η‖Hrad Ψ‖2 + further have

1 ‖Ψ‖2 4η

for any η > 0, we

‖Ĥ PF Ψ‖2 ≥ C5 ‖(p − eA)̂ 2 Ψ‖2 + C6 ‖Hrad Ψ‖2 − C7 ‖Ψ‖2 . Together with (3.5.15), this gives ‖p2 Ψ‖2 + ‖Hrad Ψ‖2 ≤ C8 (‖Ĥ PF Ψ‖2 + ‖Ψ‖2 ).

(3.5.16)

Thus Δ(Ĥ PF + z)−1 and Hrad (Ĥ PF + z)−1 are bounded. Theorem 3.47 (Self-adjointness). We have (1) and (2). (1) HPF is self-adjoint on DPF and essentially self-adjoint on any core of HPF,0 ; (2) the functional integral representation t

(F, e−tHPF G) = ∫ 𝔼x [e− ∫0 V(Bs )ds (J0 F(B0 ), e−ieAE (Kt ) Jt G(Bt ))]dx ̂

(3.5.17)

ℝ3

holds. Proof. First, we assume that ω3/2 φ̂ ∈ L2 (ℝ3 ). Take V = 0. By Lemma 3.46, ‖HPF,0 Ψ‖ ≤ C 󸀠 ‖(Ĥ PF + z)Ψ‖

(3.5.18)

holds with some C 󸀠 for all Ψ ∈ D(Ĥ PF ). Let (Ψn )∞ n=1 ⊂ D(HPF,0 ) be such that Ψn → Ψ and ̂ HPF Ψn → Φ for some Φ in strong sense as n → ∞. Note that D(Ĥ PF ) ⊃ D(HPF,0 ) = DPF . Due to (3.5.18), (HPF,0 Ψn )n∈ℕ is a Cauchy sequence and then Ψ ∈ D(HPF,0 ) since HPF,0 is closed. In particular, Ĥ PF is thus closed on DPF . This implies that HPF ⌈DPF is closed and HPF is self-adjoint since we already proved in Lemma 3.45 that HPF ⌈DPF is essentially self-adjoint and Ĥ PF = HPF on DPF . Next, let V ≠ 0. As mentioned in Remark 3.31, V is relatively bounded with respect to Ĥ PF with a relative bound strictly smaller than 1. Self-adjointness of HPF on DPF follows then by the Kato–Rellich theorem. Finally, we remove the assumption ω3/2 φ̂ ∈ L2 (ℝ3 ). We divide φ̂ into φ̂ = φ̂ Λ + φ̂ Λ⊥ , ̂ Λ and φ̂ Λ⊥ = φ̂ − φ̂ Λ with the indicator function 1Λ on |k| ≤ Λ. Write where φ̂ Λ = φ1  = Â Λ +  Λ⊥ , where Â Λ has the form factor φ̂ Λ instead of φ,̂ and  Λ⊥ does φ̂ Λ⊥ . Then

HPF can be split off like HPF = HPF (Λ) + HΛ⊥ , where 1 HPF (Λ) = (p − eÂ Λ )2 + V + Hrad , 2

HΛ⊥ = −e Λ⊥ ⋅ (p − eÂ Λ ) +

e2 ̂ 2 A ⊥. 2 Λ

Since ω3/2 φ̂ Λ ∈ L2 (ℝ3 ), HPF (Λ) is self-adjoint on DPF . We shall show below that HΛ⊥ is relatively bounded with respect to HPF (Λ) with a relative bound strictly smaller than

3.5 Applications of functional integral representations | 363

one for the large Λ. The estimate is similar to the proof of Lemma 3.46. Since for any δ > 0, ‖e Λ⊥ ⋅ (p − eÂ Λ )Ψ‖ ≤ δ‖(p − eÂ Λ )2 Ψ‖ + we have ‖HΛ⊥ Ψ‖ ≤ δ‖(p − eÂ Λ )2 Ψ‖ + e2 (

e2 ̂ 2 ‖A ⊥ Ψ‖, 2δ Λ

1 + 1) ‖Â 2Λ⊥ Ψ‖. 2δ

Similar to the estimate in the proof of Lemma 3.46, we have for any ε > 0 and η > 0, 󵄩󵄩 1 󵄩2 C 1 󵄩󵄩 󵄩 2 󵄩 2 2 2 2 ‖Ψ‖2 ) 󵄩󵄩 (p − eÂ Λ ) Ψ󵄩󵄩󵄩 + ‖Hrad Ψ‖ − ε‖(p − eÂ Λ ) Ψ‖ − 1 (η‖Hrad Ψ‖ + 󵄩󵄩 2 󵄩󵄩 4ε 4η ≤ ‖HPF (Λ)Ψ‖2 , where C1 depends on ‖√ωφ̂ Λ ‖ and ‖φ̂ Λ /√ω‖, but is independent of ‖ω3/2 φ̂ Λ ‖. In particular, limΛ→∞ C1 < ∞. From this, we have C η 1 − 4ε ‖(p − eÂ Λ )2 Ψ‖2 + (1 − C1 ) ‖Hrad Ψ‖2 ≤ ‖HPF (Λ)Ψ‖2 + 1 ‖Ψ‖2 . 4 4ε 16εη Let ε be sufficiently small and choose η such that 1 − ‖(p − eÂ Λ )2 Ψ‖2 ≤

C1 η 4ε

(3.5.19)

> 0. Thus we obtain that

4C1 4 ‖HPF (Λ)Ψ‖2 + ‖Ψ‖2 . 1 − 4ε 16ηε(1 − 4ε)

Next, we estimate ‖Â 2Λ⊥ Ψ‖. We have ‖Â 2Λ⊥ Ψ‖2 ≤ C2 (‖Hrad Ψ‖2 + ‖Ψ‖2 ), where C2 depends on ‖ωα/2 φ̂ Λ⊥ ‖, α = −2, −1, 0, 1. Thus limΛ→∞ C2 = 0 and by (3.5.19), ‖Â 2Λ⊥ Ψ‖2 ≤ (

C2 2 2 η ) ‖HPF (Λ)Ψ‖ + C3 ‖Ψ‖ . 1 − C1 4ϵ

Hence we are led to the bounds ‖(p − eÂ Λ )2 Ψ‖2 ≤ C4 (‖HPF (Λ)Ψ‖2 + ‖Ψ‖2 ), ‖ 2Λ⊥ Ψ‖2 ≤ C5 (‖HPF (Λ)Ψ‖2 + ‖Ψ‖2 ),

where limΛ→∞ C4 < ∞ and limΛ→∞ C5 = 0. From the bound, ‖HΛ⊥ Ψ‖ ≤ (δC4 + e2 (

1 + 1) C5 ) ‖HPF (Λ)Ψ‖ + C6 ‖Ψ‖ 2δ

364 | 3 The Pauli–Fierz model by path measures it follows that for sufficiently large Λ and small δ such that δC4 + e2 (

1 + 1) C5 < 1, 2δ

HΛ⊥ is relatively bounded with respect to HPF (Λ) with a relative bound strictly smaller than 1. Thus HPF is self-adjoint on DPF . The inequality ‖HPF Ψ‖ + ‖Ψ‖ ≤ C 󸀠󸀠 (‖HPF,0 Ψ‖ + ‖Ψ‖)

(3.5.20)

can be derived directly without effort with some constant C 󸀠󸀠 . Let D be a core of HPF,0 . For any Ψ ∈ D(HPF ) = D(HPF,0 ) there exists a sequence (Ψn )n∈ℕ ⊂ D such that Ψn → Ψ and HPF,0 Ψ → HPF,0 Ψ. This implies that (HPF Ψn )∞ n=1 is a Cauchy sequence by (3.5.20). Thus HPF Ψn converges to some Φ as n → ∞. Since HPF is closed, Φ = HPF Ψ. Thus D is ̂ a core of HPF . Since (3.5.17) with e−tHPF replaced by e−t HPF is verified, (3.5.17) holds since Ĥ PF = HPF . 3.5.2 Positivity improving and uniqueness of the ground state The main question addressed in this section is whether the ground state of the Pauli– Fierz Hamiltonian is unique. In Section 3.5.2 we assume either Assumption 3.4 or Assumption 3.33. As seen earlier on, a way of proving this is through the Perron– Frobenius theorem, relying on the positivity improving property of the semigroup. ̂ However, in the functional integral representation of e−tHPF the phase factor e−ieAE (Kt ) appears, therefore, intuition would say that the semigroup does not even preserve realness. Nevertheless, we can show that this semigroup is unitary equivalent with a positivity improving operator, and that will allow us to conclude that the ground state of HPF is unique whenever it exists. Consider the multiplication operator Tt = eitx , t ∈ ℝ, in L2 (ℝ) with respect to x. Al̂ t F̂ −1 , obtained under though Tt is not a positivity preserving operator, the operator FT 2 Fourier transform F̂ on L (ℝ), is a shift operator, i. e., ̂ t F̂ −1 g) = (f , g(⋅ + t)) ≥ 0 (f , FT ̂ t F̂ −1 is a positivity preservfor nonnegative functions f and g. This implies that FT ing operator but not a positivity improving operator. We extend this argument to the Pauli–Fierz model by introducing a transformation on L2 (Q ) corresponding to Fourier transform on L2 (ℝ). We return to the Fock space ℱb in Chapter 1, in which we defined the Segal field Φ(f ) and its conjugate Π(f ). Since for α ∈ ℝ, eiαN a∗ (f )e−iαN = a∗ (eiα f ), eiαN a(f )e−iαN = a(e−iα f )

3.5 Applications of functional integral representations | 365

in particular for α = π/2 we have ei(π/2)N a∗ (f )e−i(π/2)N = ia∗ (f ), ei(π/2)N a(f )e−i(π/2)N = −ia(f ). Thus Φ and Π are related by the unitary operator ei(π/2)N as ei(π/2)N Φ(f )e−i(π/2)N = Π(f ). This may suggest that ei(π/2)N can be regarded as a Fourier transform on ℱb . By the argument above, we define the unitary operator Sβ on L2 (Qβ ) by π Sβ = exp (−i Nβ ) , 2

β = 0, 1,

where Nβ denotes the number operator in L2 (Qβ ). We simply write Sβ for the unitary operator 1 ⊗ Sβ on L2 (ℝ3 ) ⊗ L2 (Qβ ) as long as there is no risk of confusion. We have ̂ Π̂ β (f) = S−1 β Aβ (f)Sβ and write Π̂ 0 = Π,̂ Π̂ 1 = Π̂ E , S0 = S and S1 = SE . Clearly, Π̂ E coincides with the expression under (3.5.6). It is plain that [Â β (f), Π̂ β (g)] = 2iqβ (f, g) and, importantly, eiΠβ (g) 1 = e−qβ (g,g) e−Aβ (g) 1. ̂

̂

Moreover, the Weyl relation eiΠβ (g) eiAβ (f) = ei2qβ (f,g) eiAβ (f) eiΠβ (g) ̂

̂

̂

̂

holds. Lemma 3.48. Let f ∈ ⊕3 L2real (ℝ3 ). Then the unitary operator eiΠ(f) is positivity preserving. ̂

Proof. Let F, G ∈ S (Q ) (see (3.5.10)) such that n

̂ j) i ∑j=1 kj A(f ̂ F = (2π)−n/2 ∫ F(k)e dk, ℝn

m

̂ j) i ∑j=1 kj A(g ̂ G = (2π)−m/2 ∫ G(k)e dk ℝm

366 | 3 The Pauli–Fierz model by path measures with positive F ∈ S (ℝn ) and G ∈ S (ℝm ). Using the fact that (eiA(g1 ) , eiΠ(f) eiA(g2 ) ) = (eiA(g1 −g2 ) , eiΠ(f) )e2iq(g2 ,f) , ̂

̂

̂

̂

̂

it is directly seen that (F, eiΠ(f) G) = ̂

n 1 ̂ 󸀠 ) exp (i ∑ q(f , f)k ) ̂ G(k ∫ F(k) j j n+m (2π) j=1 ℝn+m

m 1 × exp (i ∑ q(gj , f)kj󸀠 ) exp (− q(f, f)) 𝒢 (k, k 󸀠 )dkdk 󸀠 , 2 j=1

where 𝒢 (k, k ) = exp (− 󸀠

1 n m ∑ ∑ q(ki fi − ki󸀠󸀠 gi󸀠 , kj fj − kj󸀠󸀠 gj󸀠 )). 2 i,j=1 i󸀠 ,j󸀠 =1

Hence (F, eiΠ(f) G) ≥ 0. Since any nonnegative function can be approximated by functions in S (Q ), the lemma follows. ̂

−ieAE (Kt ) Note the intertwining property Jt S = SE Jt and S−1 SE = e−ieΠE (Kt ) . Using E e the functional integral representation (3.3.4), we have ̂

t

̂

(F, S−1 e−tHPF SG) = ∫ 𝔼x [e− ∫0 V(Bs )ds (J0 F(B0 ), e−ieΠE (Kt ) Jt G(Bs ))]dx. ̂

ℝ3

Although Proposition 1.52 shows that e−tHrad is positivity preserving, we can show a stronger statement below. Proposition 3.49 (Positivity improving for e−tHrad ). The semigroup {e−tHrad : t > 0} is positivity improving. Proof. The semigroup {e−tHrad : t > 0} is positivity preserving, and 1 is a simple eigenvector of Hrad with Hrad 1 = 0, and strictly positive, i. e., 1 > 0. Then the proposition follows from Proposition 1.53. This proposition implies that J∗t J0 is positivity improving. We show furthermore

that J∗t eieΠE (Kt ) J0 is positivity improving. The key factorization identity to achieving that is ̂

J∗t eiΠE (f) J0 Ψ = cABΨ,

(3.5.21)

̂

for Ψ ∈ S (Q ), where c = e−(qE (f,f)+q(j0 f,j0 f)) is a constant, A = J∗t e−AE (f) J0 and B = ∗

e

̂ ∗ f) iΠ(j 0

e

̂ ∗ f) A(j 0

∗

̂

. We will see below that there exists AM and BM such that A ≥ AM and

3.5 Applications of functional integral representations | 367

B ≥ BM , moreover, AM is positivity improving and BM positivity preserving, which shows that the closure of cAB is also positivity improving, since cAB⌈S (Q) ≥ cAM BM and BM G ≠ 0 if G ≠ 0. First, we show (3.5.21). Note that J0 G = G(Â E (j0 g1 ), . . . , Â E (j0 gm ))1. By the commutation relations, ̂ ̂ eiΠE (f) J0 G1 = G(Â E (j0 g1 ) + 2qE (j0 g1 , f), . . . , Â E (j0 gm ) + 2qE (j0 gm , f))eiΠE (f) 1.

From eiΠE (f) 1 = e−qE (f,f) e−AE (f) 1, it then follows that ̂

̂

̂ ̂ eiΠE (f) J0 G1 = e−qE (f,f) e−AE (f) G(Â E (j0 g1 ) + 2qE (j0 g1 , f), . . . , Â E (j0 gm ) + 2qE (j0 gm , f))1

and from qE (j0 g, f) = q(g, j∗0 f), this is further ̂ = e−qE (f,f) e−AE (f) G(Â E (j0 g1 ) + 2q(g1 , j∗0 f), . . . , Â E (j0 gm ) + 2q(gm , j∗0 f))1.

(3.5.22)

Here, ∗ h(k) = ĵ t

2 |k| ̂ eitk0 h(k)dk ∫ 0, √2π |k|2 + |k0 |2

t ∈ ℝ.

ℝ

The right-hand side of (3.5.22) can be written as e−qE (f,f) e−AE (f) J0 eiΠ(j0 f) Ge−iΠ(j0 f) 1. ̂

̂

∗

̂

∗

Using again e−iΠ(f) 1 = e−q(f,f) eA(f) 1 and noting that G commutes with eA(j0 f) , finally we have ̂

̂

̂

∗

eiΠE (f) J0 G1 = e−qE (f,f)−q(j0 f,j0 f) e−AE (f) J0 eiΠ(j0 f) eA(j0 f) G1. ∗

∗

̂

̂

∗

̂

∗

̂

This gives (3.5.21). From this, we have the operator identity J∗t eiΠE (f) J0 = e−(qE (f,f)+q(j0 f,j0 f)) (J∗t e−AE (f) J0 eiΠ(j0 f) eA(j0 f) )⌈S (Q ). ∗

̂

∗

̂

̂

∗

̂

∗

Note that e−AE (f) and eA(j0 f) are unbounded. Define the bounded functions obtained by the truncations ̂

̂

∗

e−AE (f) , ̂

(e−AE (f) )M = { ̂

(e

̂ ∗ f) −A(j 0

M,

̂

e−AE (f) ≥ M, ̂

e−A(j0 f) , ̂

)M = { M,

e−AE (f) < M,

∗

e−A(j0 f) < M, ̂

∗

e−A(j0 f) ≥ M. ̂

∗

368 | 3 The Pauli–Fierz model by path measures Lemma 3.50. For t > 0, J∗t (e−AE (f) )M J0 is positivity improving. ̂

Proof. It suffices to show that (F, J∗t (e−AE (f) )M J0 G) ≠ 0 for any nonnegative F and G, but ̂

−Â E (f)

F ≢ 0 and G ≢ 0 since J∗t (e

)M J0 is positivity preserving. Suppose

(F, J∗t (e−AE (f) )M J0 G) = ((e−AE (f) )M Jt F, J0 G) = 0 ̂

(3.5.23)

̂

for some nonnegative F and G. (3.5.23) implies that μE (supp[(e−AE (f) )M Jt F] ∩ supp[J0 G]) = 0.

(3.5.24)

̂

As  E (f) ∈ L2 (QE ), ̂ μE ({ E ∈ QE | (e−AE (f) )M = 0}) = 0,

(3.5.24) implies that μE (supp[Jt F] ∩ supp[J0 G]) = 0, and thus 0 = (Jt F, J0 G) = (F, e−tHrad G). Since e−tHrad is positivity improving by Proposition 3.49, this is in contradiction with the assumption. Theorem 3.51 (Positivity improving for e−tHPF ). Operator S−1 e−tHPF S is positivity improving for every t > 0. Proof. Let f be real valued. Notice that if f ≥ g, then Tf ≥ Tg for any positivity preserving operator T. For a nonnegative G ∈ S (Q ), J∗t eiΠE (f) J0 G ≥ e−(qE (f,f)+q(j0 f,j0 f)) J∗t (e−AE (f) )M J0 eiΠ(j0 f) (eA(j0 f) )M G = SM G. ∗

̂

∗

̂

̂

∗

̂

∗

Since SM is bounded and any nonnegative function can be approximated by a function in S (Q ), we have for an arbitrary nonnegative G, J∗t eiΠE (f) J0 G ≥ SM G. ̂

Since J∗t (e−AE (f) )M J0 is positivity improving by Lemma 3.51 and eiΠ(j0 f) (eA(j0 f) )M is poŝ

̂

∗

̂

∗

itivity preserving with eiΠ(j0 f) (eA(j0 f) )M G ≢ 0 for G ≥ 0 but G ≢ 0, SM is positivity ̂ improving and so is J∗t eiΠE (f) J0 . In particular, S−1 J∗t S is positivity improving as well. Let F, G ≥ 0, not identically vanishing. Note that ̂

∗

̂

∗

t

(F, S−1 e−tHPF SG) = ∫ 𝔼x [e− ∫0 V(Bs )ds (S−1 J∗t SF(B0 ), G(Bt ))]dx. ℝ3

(3.5.25)

3.5 Applications of functional integral representations | 369

We show that (3.5.25) is strictly positive. Let DG = {x ∈ ℝ3 | G(x, ⋅) ≢ 0},

DF = {x ∈ ℝ3 | F(x, ⋅) ≢ 0}

and DFG = {ω ∈ X | B0 (ω) ∈ DF , Bt (ω) ∈ DG }. In order to complete the proof of the theorem, it suffices to see that the measure of DFG ⊂ X is positive as S−1 Jt S is positivity improving. We have 2

∫ 𝔼x [1DFG ]dx = ∫ 𝔼x [1DG (Bt )]dx = (2πt)−3/2 ∫ dx ∫ e−|x−y| /2t dy > 0. ℝ3

DF

DF

DG

Thus it is obtained that the right-hand side of (3.5.25) is strictly positive and the theorem follows. We proved that Se−tHPF S−1 is positivity improving. A direct consequence of this property is uniqueness of the ground state of HPF . Corollary 3.52 (Uniqueness of ground state). The ground state Ψg of HPF satisfies S−1 Ψg > 0, and it is unique up to multiple constants. Proof. It follows from the Perron–Frobenius theorem and Theorem 3.51. By Theorem 3.51 again, we have (f ⊗ 1, SΨg ) ≠ 0. This allows to derive an expression of the ground state energy inf Spec(HPF ) from which some of its properties follow. Write E(e2 ) = inf Spec(HPF ). Theorem 3.53 (Concavity of ground state energy). The function e2 󳨃→ E(e2 ) is monotonously increasing, continuous and concave. Proof. For strictly positive f > 0, we have 1 E(e2 ) = lim (− log(f ⊗ 1, e−tHPF f ⊗ 1)) t→∞ t t 2 1 = lim (− log ∫ 𝔼x [f (B0 )f (Bt )e− ∫0 V(Bs )ds e−(e /2)qE (Kt ,Kt ) ]dx) . t→∞ t

(3.5.26)

ℝ3

2

As e−(e /2)qE (Kt ,Kt ) is log-convex in e2 , E(⋅) is concave. Thus E(e2 ) is continuous on (0, ∞). Since E(e2 ) is continuous at e = 0, E ∈ C(ℝ+ ) and E(e2 ) can be expressed as E(e2 ) = e2

∫0 ρ(t)dt with a suitable positive function ρ(t). This implies that E(e2 ) is monotonously increasing in e2 . Notice that we know that HPF has a unique ground state Ψg for any value of the coupling constant e ∈ ℝ. Since 0 ≠ (f ⊗ 1, S−1 Ψg ) = (f ⊗ 1, Ψg ) for any e ∈ ℝ and not identically zero f ≥ 0, (3.5.26) follows.

370 | 3 The Pauli–Fierz model by path measures 3.5.3 Spatial decay of bound states Similar to the Nelson model in the previous chapter, we can derive the spatial exponential decay of bound states Φb of the Pauli–Fierz Hamiltonian. Recall the class 𝕍upper defined in Definition 2.88, and let HPF Φb = EΦb . We have the following Carmona-type estimate for the Pauli–Fierz Hamiltonian. Lemma 3.54. Suppose (1) of Assumption 3.4. If V ∈ 𝕍upper , then for any t, a > 0 and every 0 < α < 1/2, there exist constants D1 , D2 , D3 > 0 such that α a2 t

‖Φb (x)‖L2 (Q) ≤ t −d/2 D1 eD2 ‖U‖p t eEt (D3 e− 4

e−tW∞ + e−tWa (x) )‖Φb ‖ℋPF ,

where Wa (x) = inf{W(y) | |x − y| < a}. Proof. Since Φb = etE e−tHPF Φb , the functional integral representation yields t

Φb (x) = 𝔼x [J∗0 e−ieAE (Kt ) Jt e− ∫0 V(Bs )ds Φb (Bt )]. ̂

From this we directly obtain that t

‖Φb (x)‖L2 (Q) ≤ 𝔼x [e− ∫0 V(Bs )ds ‖Φb (Bt )‖L2 (Q) ]. Then the lemma follows in the similar way to Carmona’s estimate of Schrödinger operators in Lemma 4.176 in Volume 1. This lemma implies a similar result of decay of the ground state as for Schrödinger operators. For V = W − U ∈ 𝕍upper , recall that Σ = lim inf|x|→∞ V(x). We only state the results. Corollary 3.55 (Exponential decay of bound states). Suppose (1) of Assumption 3.4. Let V = W − U ∈ 𝕍upper and E = inf Spec(HPF ). (1) Suppose that W(x) ≥ γ|x|2n outside a compact set K, for some n > 0 and γ > 0. Take 0 < α < 1/2. Then there exists a constant C1 > 0 such that ‖Φb (x)‖L2 (Q) ≤ C1 exp (−

αc n+1 |x| ) ‖Φb ‖ℋPF , 16

where c = infx∈ℝ3 \K W|x|/2 (x)/|x|2n . (2) Decaying potential: Suppose that Σ > E, Σ > W∞ , and 0 < β < 1. Then there exists a constant C2 > 0 such that ‖Φb (x)‖L2 (Q) ≤ C2 exp (−

β (Σ − E) |x|) ‖Φb ‖ℋPF . √ 8 2 √Σ − W∞

(3) Confining potential: Suppose that lim|x|→∞ W(x) = ∞. Then there exist constants C, δ > 0 such that ‖Φb (x)‖L2 (Q) ≤ C exp (−δ|x|) ‖Φb ‖ℋPF .

3.6 Path measure associated with the ground state of Pauli–Fierz Hamiltonian

| 371

3.6 Path measure associated with the ground state of Pauli–Fierz Hamiltonian 3.6.1 Path measure with double stochastic integrals In Sections 3.6.1 and 3.6.2, we assume either Assumption 3.4 or Assumption 3.33 unless otherwise stated. That is, external potential V is V ∈ ℛKato or Kato-decomposable. The functional integral method for nonrelativistic QED provides a new class of objects for probability theory. An infinite volume path measure is derived from the Nelson model discussed in Chapter 2. The Pauli–Fierz model also yields a path measure with densities dependent on a double stochastic integral. Its limit can be applied to studying the ground state of the Pauli–Fierz model. As the first step, we show the existence of path measures with densities dependent on a double stochastic integral. Let Ψg be the ground state of HPF , which is unique and S−1 Ψg > 0 by Corollary 3.52. Although the existence of the ground state has been proven, the proof is not constructive and, therefore, it does not make possible a direct study of its properties. As an alternative, the functional integral representation can be used, and ground state expectations (Ψg , 𝒪Ψg ) for various operators 𝒪 can be expressed in terms of averages of the path measure. Let M be a positive Borel measurable function on ℝ3 . Write ℳ = dΓ(M(−i∇))

(3.6.1)

and consider 𝒪 = e−βℳ in the discussion above. The strategy is to construct a sequence {Ψtg }t>0 such that Ψtg → Ψg as t → ∞. One candidate is Ψtg = ‖e−tHPF (f ⊗ 1)‖−1 e−tHPF (f ⊗ 1) with f ≥ 0, since f ⊗ 1 overlaps with Ψg by Corollary 3.52, i. e., (Ψg , f ⊗ 1) = (S−1 Ψg , f ⊗ 1) > 0. This gives lim(Ψtg , e−βℳ Ψtg ) = (Ψg , e−βℳ Ψg ),

t→0

i. e., (f ⊗ 1, e−tHPF e−βℳ e−tHPF (f ⊗ 1)) . t→0 (e−tHPF (f ⊗ 1), e−tHPF (f ⊗ 1))

(Ψg , e−βℳ Ψg ) = lim

(3.6.2)

To have a functional integral representation of the right-hand side of (3.6.2), we need to extend the functional integration (F, e−tHPF e−βℳ e−sHPF G). It was a key step before that e−tHrad could be decomposed as e−|s−t|Hrad = J∗s Jt leading to the functional integral representation of (F, e−tHPF G). Our goal is to extend this idea to ℳ.

372 | 3 The Pauli–Fierz model by path measures Let ιt : L2 (ℝ4 ) → L2 (ℝ5 ) be defined by √M(k) e−ik1 t f ̂(k), ι̂ t f (k, k1 ) = √π √M(k)2 + |k |2 1

(k, k1 ) = (k, k0 , k1 ) ∈ ℝ3 × ℝ × ℝ.

(3.6.3)

Then the factorization formula ι∗s ιt = e−|s−t|(M(−i∇)⊗1) ,

s, t ∈ ℝ,

are satisfied. Here, operator M(−i∇) ⊗ 1 is defined under the identification L2 (ℝ4 ) ≅ L2 (ℝ3 ) ⊗ L2 (ℝ). Let (Q2 , Σ2 , μ2 ) be a probability space and (Â 2 (f), f ∈ ⊕3 L2 (ℝ5 )) denotes a family of Gaussian random variables indexed by f ∈ ⊕3 L2 (ℝ5 ) with mean zero and covariance given by 1 ̂ k ) ⋅ δ⊥ (k)g(k, ̂ k1 )dkdk1 = q2 (f, g). 𝔼μ2 [Â 2 (f)Â 2 (g)] = ∫ f(k, 1 2 ℝ5

Define the contraction operator Ξt : L2 (QE ) → L2 (Q2 ) by the second quantization of ιt , i. e., Ξt = Γint (ιt ). Then the factorization formula e−|s−t|dΓE (M(−i∇)⊗1) = Ξ∗t Ξs ,

s, t ∈ ℝ,

(3.6.4)

follows. In Section 3.2.3, we discuss Markov properties of projection et and Et . Similarly, projections ιt ι∗t and Ξt Ξ∗t also have Markov properties. Theorem 3.56 (Euclidean Green functions). Let 0 = t0 < t1 < . . . < tn = t and 0 = s0 < s1 < . . . < sn = s. Then n

(F, (∏ e−(tj −tj−1 )HPF e−(sj −sj−1 )ℳ ) G) j=1

x

= ∫ 𝔼 [e

ℋPF

t

− ∫0 V(Bs )ds

(Ξ0 J0 F(B0 ), e

−ie E (Kt (s0 ,s1 ,...,sn ))

Ξs Jt G(Bt ))L2 (Q2 ) ] dx,

ℝ3

where 3

n

tj

̃ − Bs )dBμs . Kt (s0 , s1 , . . . , sn ) = ⨁ ∑ ∫ ιsj−1 js φ(⋅ μ=1 j=1 t j−1

Proof. We give an outline of the proof. Let Tj = (tj − tj−1 )/Mj . Substitute e

−(tj −tj−1 )HPF

= s-lim

Mj →∞

lim . . . lim J∗tj−1 m →∞ m →∞ 1

n

Mj

mi

(∏(JiTj (PTj /2mi )2 J∗iTj )) Jtj i=1

(3.6.5)

3.6 Path measure associated with the ground state of Pauli–Fierz Hamiltonian

| 373

to the left-hand side of (3.6.5). The left-hand side of (3.6.5) can be expressed as the limit of n

T(M1 , m11 , . . . , m1M1 , . . . , Mn , mn1 , . . . , mnM1 ) = (Jt0 F, (∏ ηi ζi ) Jtn G) , i=1

(3.6.6)

where j

Mj

j

ηj = ηj (tj−1 , tj , m1 , . . . , mM , Mj ) = ∏ (JiTj (P j

i=1

j m Tj /2 i

j i

m

)2 JiTj ) ,

ζj = Jtj e−(sj −sj−1 )ℳ J∗tj . j

In other words, T converges to the left-hand side of (3.6.5) as mi → ∞ for i = 1, . . . , n, j = 1, . . . , Mj , Mj → ∞ for j = 1, . . . , n and n → ∞. Note that Jtj e−(sj −sj−1 )ℳ J∗tj = Ξ∗sj−1 Ξsj Etj . Inserting this identity on (3.6.6), and using the Markov properties of both Et = Jt J∗t and Ξt Ξ∗t , we obtain the theorem. Next, we construct a functional integral representation for other types of Green functions. Define ̂ 1 ), . . . , A(f ̂ n )) | Φ ∈ L∞ (ℝn ), fj ∈ ⊕3 L2 (ℝ3 ), j = 1, . . . , n, n ≥ 0}. L2∞ (Q ) = {Φ(A(f Theorem 3.57 (Euclidean Green functions). Let 0 = t0 < t1 < . . . < tn = t. Suppose that ̂ j ), . . . , A(f ̂ j )) ∈ L∞ (ℝ3 ) ⊗ L2 (Q ), 1 ≤ j ≤ n − 1 and F0 , Fn ∈ ℋPF . Then Fj = fj ⊗ Φj (A(f nj ∞ 1 n

(F0 , (∏ e−(tj −tj−1 )HPF Fj )) j=1

t

n−1

= ∫ 𝔼x [e− ∫0 V(Bs )ds (J0 F0 (B0 ), e−ieAE (Kt ) (∏ F̃j (Btj ))Jt Fn (Bt )) ̂

j=1

ℝ3

L2 (Q2 )

]dx,

(3.6.7)

j where F̃j (x) = fj (x)Φj (Â E (jtj f1 ), . . . , Â E (jtj fjnj )).

Proof. First, assume that fj and Φj , j = 1, . . . , n, are sufficiently smooth functions with a compact support. Note that j Js Fj J∗s = fj Φj (Â E (js f1 ), . . . , Â E (js fjnj ))

(3.6.8)

as a bounded operator. Thus substituting (3.6.8) on the left-hand side of (3.6.7), the theorem follows. For general Fj , a limiting argument can be used.

374 | 3 The Pauli–Fierz model by path measures As an application, we show the functional integral presentation of (Ψg , e−βN Ψg ), where N is the boson number operator and β > 0. Let (1, T1)L2 (QE ) = ⟨T⟩vac . Set M = 1 in (3.6.1). Then ℳ = N and let ιt : L2 (ℝ4 ) → L2 (ℝ5 ) be the family of isometries defined in (3.6.3). Write 3

0

t

−t

0

̃ − Bs )dBμs + ∫ ιβ js φ(⋅ ̃ − Bs )dBμs ) , Y = Y(t, β) = ⨁ (∫ ι0 js φ(⋅ μ=1 3

t

̃ − Bs )dBμs . Z = Z(t) = ⨁ ∫ js φ(⋅ μ=1 −t

μ

μ

We note that B−s and Bt are independent, for all s, t > 0. Let f ≥ 0 be fixed. Using Theorem 3.57, one can compute t

(Ψtg , e−βℳ Ψtg )

=

∫ℝ3 𝔼x [f (B−t )f (Bt )e− ∫−t V(Bs )ds ⟨e−ieAE (Y) ⟩vac ]dx ̂

t

∫ℝ3 𝔼x [f (B−t )f (Bt )e− ∫−t V(Bs )ds ⟨e−ieAE (Z) ⟩vac ]dx ̂

.

(3.6.9)

Notice that 2

e ⟨exp (−ie E (Y))⟩vac = exp (− q2 (Y, Y)) . 2 As qE is a bilinear form, the identity qE (Y, Y) = qE (Z, Z) + 2(e−β − 1)qE (Z(−t,0) , Z(0,t) ),

(3.6.10)

holds, where 3

T

̃ − Bs )dBμs . Z(S,T) = ⨁ ∫ js φ(⋅ μ=1

S

Define the family of finite Gibbs measures, dμt =

t 2 1 (∫ f (x + B−t )f (x + Bt )e− ∫−t V(x+Bs )ds dx) e−(e /2)qE (Z,Z) d𝒲 0 Nt

(3.6.11)

ℝ3

of probability measures on (X, ℬ(X)). Note that we use the notation 𝒲 0 for two sided Wiener measure unless confusion may arise. In (3.6.11), Nt denotes the denominator of the right-hand side of (3.6.9), regarded as the normalizing constant such that ∫X dμt = 1, and qE (Z, Z) is independent of x.

3.6 Path measure associated with the ground state of Pauli–Fierz Hamiltonian

| 375

Corollary 3.58. It follows that 2

(Ψg , e−βN Ψg ) = lim 𝔼μt [e+e (1−e t→∞

−β

)qE (Z(−t,0) ,Z(0,t) )

].

Proof. By (3.6.10) and (3.6.9), we have 2

(Ψtg , e−βN Ψtg ) = 𝔼μt [ee (1−e

−β

)qE (Z(−t,0) ,Z(0,t) )

].

Since Ψtg → Ψg as t → ∞, the corollary follows. The measure dμt includes pair interaction e2 qE (Z, Z) associated with the Pauli– Fierz Hamiltonian which has the formal expression qE (Z, Z)

formal

=

3

∑

t

∫ dBμs

μ,ν=1 −t

t

∫ Wμν (Bs − Br , s − r)dBνr ,

(3.6.12)

−t

where Wμν is given by ⊥ Wμν (X, T) = ∫ δμν (k) ℝ3

2 ̂ |φ(k)| e−|T|ω(k) e−ik⋅X dk. 2ω(k)

(3.6.13)

Here, Wμν (X, T) is called the pair potential of the Pauli–Fierz Hamiltonian. Moreover, qE (Z(−t,0) , Z(0,t) )

formal

=

3

∑

t

∫ dBμs

μ,ν=1 0

0

∫ Wμν (Bs − Br , s − r)dBνr . −t

Equation (3.6.13) expresses the effect of the quantum field. Recall that in the Nelson model instead of the double stochastic integral a double Riemann integral appears. The formal expression (3.6.12), however, is not well-defined, since the integrand is not admissible. In the next section, we find a suitable expression of the integral. 3.6.2 Expression in terms of iterated stochastic integrals We define the iterated stochastic integral ST by T

ST =

s

2 ̂ 1 |φ(k)| (2T + ∫ eik⋅Bs dBs ⋅ δ⊥ (k) ∫ e−ω(k)(s−r) e−ik⋅Br dBr ) dk ∫ 2 ω(k) ℝ3

−T

(3.6.14)

−T

ST is a well-defined expression and we use it to replace (3.6.12). We will refer to (3.6.14) as the double stochastic integral with the diagonal removed. Theorem 3.59 (Iterated stochastic integral). Suppose that φ̂ is rotation invariant. Then we have 2

⟨e−ieA(Z) ⟩vac = e−e ST . ̂

376 | 3 The Pauli–Fierz model by path measures Proof. We replace the time interval [−T, T] with [0, 2T] by shift invariance, and reset 2T by T for notational convenience. Using the definition of Z and the Lebesgue dominated convergence theorem give n

̂ ̂ ̃ ⟨exp (−ieA(Z))⟩ vac = lim ⟨ exp (−ie ∑ AE (jΔj φ(⋅ − BΔj ))ΔBj ) ⟩ n→∞

j=1

vac

,

where we set ΔBj = BjT/n − B(j−1)T/n and Δj = (j − 1)T/n, j = 1, . . . , n. Since  E (f) is Gaussian, it can be computed as ̂ ⟨exp (−ieA(Z))⟩ vac = lim exp (− n→∞

2 n n ̂ |φ(k)| e2 ik⋅(B −B ) ∑ ∑ e−|Δj −Δl |ω(k) e Δj Δl ΔBj δ⊥ (k)ΔBl dk) . ∫ 2 2ω(k) j=1 l=1 ℝ3

Now the integral ∫ℝ3 . . . dk above can be computed as ∫ ℝ3

2 n n ̂ |φ(k)| ik⋅(B −B ) ∑ ∑ e−|Δj −Δl |ω(k) e Δj Δl ΔBj δ⊥ (k)ΔBl dk 2ω(k) j=1 l=1 n

2 ̂ |φ(k)| δ⊥ (k)dk) ΔBj 2ω(k)

= ∑ ΔBj (∫ j=1

ℝ3

n

+∑∫ j=1

ℝ3

(3.6.15) j−1

2 ̂ |φ(k)| ik⋅B e−Δj ω(k) e Δj ΔBj δ⊥ (k) (∑ eΔl ω(k) e−ik⋅BΔl ΔBl ) dk. ω(k) l=1

(3.6.16)

For the diagonal term (3.6.15), we note that ∫ ℝ3

2 2 ̂ ̂ |φ(k)| 2 |φ(k)| ⊥ δμν (k)dk = δμν ∫ dk 2ω(k) 3 2ω(k) ℝ3

μ

by rotation invariance of φ.̂ As n → ∞, ∑3μ=1 ∑nj=1 |ΔBj |2 → 3T almost surely. Thus we find n

lim ∑ ΔBj (∫

n→∞

j=1

ℝ3

2 2 ̂ ̂ |φ(k)| |φ(k)| δ⊥ (k)dk) ΔBj = T ∫ dk 2ω(k) ω(k) ℝ3

almost surely. For the off-diagonal term (3.6.16), we start by noting that by the definition of the Itô integral for locally bounded functions f , g : ℝ × ℝ3 → ℝ, t s t s 󵄨󵄨 󵄨󵄨2 󵄨 󵄨 𝔼[ ∫ 󵄨󵄨󵄨f (s, Bs ) ∫ g(r, Br )dBμr 󵄨󵄨󵄨 ds] = ∫ ds𝔼[|f (s, Bs )|2 ] ∫ 𝔼[|g(r, Br )|2 ]dr < ∞. 󵄨󵄨 󵄨󵄨 0

0

0

0

3.6 Path measure associated with the ground state of Pauli–Fierz Hamiltonian

s

| 377

μ

Hence the stochastic integral of ρ(s) = f (s, Bs ) ∫0 g(r, Br )dBr exists and for every k ∈ ℝ3 , Δj

n

T

s

lim ∑ f (Δj , BΔj )ΔBj δ (k) ∫ g(r, Br )dBr = ∫ f (s, Bs )dBs δ⊥ (k) ∫ g(r, Br ) dBr

n→∞

⊥

j=1

0

0

(3.6.17)

0

strongly in L2 (X). By the independence of increments of Brownian motion and the fact μ μ that ∑3μ=1 𝔼[(ΔBj )2 ] = 3/n and 𝔼[ΔBj ] = 0, we can estimate the difference of (3.6.17) and the off-diagonal term (3.6.16) to get Δj

n

j

𝔼[ ∑ f (Δj , BΔj )ΔBj δ (k)( ∫ g(r, Br ) dBr − ∑ g(Δl , BΔl )ΔBl )] ⊥

j=1

Δ

2

l=1

0

j 󵄨󵄨 j 󵄨󵄨2 3 3 n 󵄨 󵄨 ≤ ‖f ‖∞ ∑ ∑ 𝔼[󵄨󵄨󵄨 ∫ g(r, Br ) dBνr − ∑ g(Δl , BΔl )ΔBνl 󵄨󵄨󵄨 ]. 󵄨󵄨 n ν=1 j=1 󵄨󵄨 l=1 2

(3.6.18)

0

The right-hand side above converges to zero as n → ∞. Combining (3.6.17) and (3.6.18), we find that for every k ∈ ℝ3 , n

j

t

j=1

l=1

0

s

lim ∑ f (Δj , BΔj )ΔBj δ⊥ (k)( ∑ g(Δl , BΔl )ΔBl ) = ∫ f (s, Bs )δ⊥ (k)( ∫ g(r, Br ) dBr )dBs

n→∞

0

(3.6.19)

strongly in L2 (X , d𝒲 x ). By putting f (t, x) = eik⋅x e−ω(k)t , g(t, x) = e−ik⋅x eω(k)t in (3.6.19), it follows that n

lim ∑ e−Δj ω(k) e

n→∞

ik⋅BΔj

j=1

j−1

ΔBj δ⊥ (k) (∑ eΔl ω(k) e−ik⋅BΔl ΔBl ) l=1

T

s

0

0

= ∫ eik⋅Bs δ⊥ (k)( ∫ e−ω(k)(s−r) e−ik⋅Br dBr )dBs . Hence lim (3.6.16) = ∫ dk

n→∞

ℝ3

T

s

0

0

2 ̂ |φ(k)| ∫ eik⋅Bs dBs ⋅ δ⊥ (k) ∫ e−ω(k)(s−r) e−ik⋅Br dBr . ω(k)

Here is a concluding summary. Proposition 3.60. Let μT be the family of probability measures on X defined in (3.6.11). Then the following identity holds: dμT =

T 2 ̂ 1 (∫ f (B−T + x)f (BT + x)e− ∫−T V(Bs +x) ds dx) e−e ST d𝒲 0 , ̂ ZT

ℝ3

378 | 3 The Pauli–Fierz model by path measures where ŜT is defined by ST with the diagonal part removed T

s

−T

−T

2 ̂ |φ(k)| 1 ŜT = ∫ dk ∫ eik⋅Bs dBs δ⊥ (k) ∫ e−ω(k)(s−r) e−ik⋅Br dBr 2 ω(k) ℝ3

and Ẑ T denotes the normalizing constant such that ∫X dμT = 1. ŜT may be symbolically written as 3

ŜT = ∑

μ,ν=1

∫ [−T,T]2 \{s=r}

(∫ Wμν (Bs − Br , s − r)dsdr) dBμs dBνr . ℝ3

3.6.3 Weak convergence and Gibbs measures In this section, we show tightness of the family of probability measures {μt }t≥0 on X. In Section 3.6.3 we suppose Assumption 3.33. Lemma 3.61. Let f1 , . . . , fn−1 ∈ L∞ (ℝ3 ), 0 ≤ f ∈ L2 (ℝ3 ), and −T = t0 ≤ t1 ≤ . . . ≤ tn = T. Then the Euclidean Green function is expressed as n−1 (f ⊗ 1, e−(t1 −t0 )HPF (f1 ⊗ 1) ⋅ ⋅ ⋅ (fn−1 ⊗ 1)e−(tn −tn−1 )HPF f ⊗ 1) = 𝔼 [ fj (Btj )] . ∏ μ T (f ⊗ 1, e−2TH f ⊗ 1) j=1

Proof. By Theorem 3.57, we can directly see that (f ⊗ 1, e−(t1 −t0 )HPF (f1 ⊗ 1) ⋅ ⋅ ⋅ (fn−1 ⊗ 1)e−(tn −tn−1 )HPF f ⊗ 1) n−1

T

= ∫ 𝔼x [f (B−T )f (BT ) (∏ fj (Btj )) e− ∫−T V(Bs )ds ⟨e−ieAE (Z) ⟩vac ]dx. ℝ3

̂

j=1

By the definition of the measure μT , the lemma follows. An immediate result of s-limt→∞ Ψtg = Ψg and Lemma 3.61 is as follows. Let ρ, ρ1 , ρ2 ∈ L∞ (ℝ3 ). Then for t > s, lim 𝔼μT [ρ(B0 )] = (Ψg , (ρ ⊗ 1)Ψg ),

T→∞

lim 𝔼μT [ρ1 (Bs )ρ2 (Bt )] = (Ψg , (ρ1 ⊗ 1)e−(t−s)HPF (ρ2 ⊗ 1)Ψg )e(t−s)E(HPF ) ,

T→∞

where E(HPF ) = inf Spec(HPF ) denotes the ground state energy of HPF . In the next theorem, we can show the existence of Gibbs measure μ∞ which is the weak limit of s subsequence of {μT }t≥0 .

| 379

3.6 Path measure associated with the ground state of Pauli–Fierz Hamiltonian

Theorem 3.62 (Tightness). Assume that there exists the ground state of HPF . Then there exists a subsequence T 󸀠 such that μT 󸀠 has a weak limit as T 󸀠 → ∞. Proof. The proof is a modification of that of Theorem 2.37. By the Prokhorov theorem, it suffices to show (1) and (2) below: (1) limΛ→∞ supT≥0 μT (|B0 |2 > Λ) = 0; (2) for any ε > 0, limδ↓0 supT≥0 μT (max|s−t| ε) = 0. We have μT (|B0 |2 > Λ) = (ΨTg , 1{|x|2 >Λ} ΨTg ). Note that ΨTg → Ψg strongly as T → ∞. Set α(T) = (ΨTg , 1{|x|2 >Λ} ΨTg ), and without loss of generality we can assume that T ≥ 1. Let ε > 0 be given. Since as a bounded operator ‖1{|x|2 >Λ} ⊗ 1‖ ≤ 1, there exists T ∗ > 0 independent of Λ such that α(T) ≤ (Ψg , 1{|x|2 >Λ} Ψg ) + ε

∀T > T ∗ .

Thus sup α(T) ≤ sup α(T) + (Ψg , 1{|x|2 >Λ} Ψg ) + ε. 1≤T

1≤T≤T ∗

We shall estimate sup1≤T≤T ∗ α(T). Let S = e−i(π/2)N . Since S−1 e−THPF S is positivity improving and S−1 (f ⊗ 1) = f ⊗ 1, we see that ‖e−THPF (f ⊗ 1)‖ = ‖S−1 e−THPF S(f ⊗ 1)‖ > 0 for all T ≥ 0. Then inf1≤T≤T ∗ ‖e−THPF (f ⊗ 1)‖ > 0. Denote the left-hand side by c. Since T ∗ is independent of Λ, c is also independent of Λ. Thus α(T) ≤ c−2 (e−THPF (f ⊗ 1), 1{|x|2 >Λ} e−THPF (f ⊗ 1)). By the functional integral representation and Schwarz inequality, we have 2T

(e−THPF (f ⊗ 1), 1{|x|2 >Λ} e−THPF (f ⊗ 1)) = ∫ 𝔼x [f (B0 )f (B2T )1{|x|2 >Λ} (BT )e− ∫0

V(Bs )ds −S2T

e

]dx

ℝ3

≤ VM ∫ (𝔼x [f (B2T )2 ])1/2 f (x)(𝔼x [1{|x|2 >Λ} (BT )])1/4 dx, ℝ3 2T

where S2T is given by (3.6.14) and VM = supx∈ℝ3 (𝔼x [e−4 ∫0 Schwarz inequality again that

V(Bs )ds

])1/4 . We see by the 1/2

(e

−THPF

(f ⊗ 1), 1{|x|2 >Λ} e

−THPF

x

1/2

2

(f ⊗ 1)) ≤ VM ‖f ‖ (∫ (𝔼 [1{|x|2 >Λ} (BT )]) |f (x)| dx) ℝ3

.

380 | 3 The Pauli–Fierz model by path measures Since 1 ≤ T ≤ T ∗ , the bound x

1/2

2

2

∫ (𝔼 [1{|x|2 >Λ} (BT )]) f (x) dx ≤ (2π)

∫ f (x) (∫ dye

ℝ3

ℝ3

−3/4

−|x−y|2 /(2T ∗ )

1/2

1{|x|2 >Λ} (y))

dx

ℝ3

is obtained. Denote the right-hand side above by Cf (Λ). Thus sup α(T) ≤ ‖f ‖

VM Cf (Λ)1/2 c2

1≤T≤T ∗

.

Since c and VM are independent of Λ and Cf (Λ) → 0 as Λ → ∞, we obtain lim sup α(T) ≤ lim (

Λ→∞ T≥1

Λ→∞

1 ‖f ‖VM Cf (Λ)1/2 + (Ψg , 1{|x|2 >Λ} Ψg )) + ε = ε. c2

Since ε is arbitrary, (1) follows. (2) can be proven in the same way as Theorem 2.37. Denote the weak limit of the measure μT 󸀠 on X by μ∞ . Using the functional integration representation of e−tHPF , we show in Corollary 3.55 that if V(x) = |x|2n , then n+1 ‖Ψg (x)‖ℱrad ≤ C1 e−C2 |x| , while if V(x) = −1/|x|, then ‖Ψg (x)‖ℱrad ≤ C3 e−C4 |x| for appropriate constants Cj . γ

Corollary 3.63. Assume that ‖Ψg (x)‖ℱrad ≤ Ce−c|x| for some positive constants C, c and γ. Then γ

𝔼μ∞ [ec|B0 | ] < ∞. Proof. Let

γ

ec|x| ,

ρm (x) = { m,

(3.6.20)

γ

ec|x| ≤ m, γ

ec|x| > m

γ

be the truncated function of ec|x| . Then (Ψg , (ρm ⊗ 1)Ψg ) = 𝔼μ∞ [ρm (B0 )] follows. By a limiting argument as m → ∞, (3.6.20) follows. Using this measure μ∞ , the expectation ⟨e−βN ⟩vac can be formally represented as 2

⟨e−βN ⟩vac = 𝔼μ∞ [ee (1−e 0

−β

μ

0

μ

∞

) ∫−∞ dBs ∫0 Wμν (Bs −Br ,s−r)dBνr

].

t

It is, however, not easy to control ∫−t dBs ∫0 Wμν (Bs − Br , s − r)dBνr as t → ∞. For the Nelson Hamiltonian, the role of the double stochastic integral is taken by a double Riemann integral and then 󵄨󵄨 0 󵄨󵄨 t 󵄨󵄨 󵄨󵄨 1 ̂ 2 φ(k) 󵄨󵄨 󵄨 dk. 󵄨󵄨∫ ds ∫ W(Bs − Br , s − r)dr 󵄨󵄨󵄨 ≤ ∫ 󵄨󵄨 󵄨󵄨 2 ω(k)3 󵄨󵄨−t 0 󵄨󵄨 3 ℝ Thus if the integral of the right-hand side is bounded, then the left-hand side is uniformly bounded with respect to the path and t.

3.6 Path measure associated with the ground state of Pauli–Fierz Hamiltonian

| 381

3.6.4 Gaussian domination of ground states We can show the Gaussian domination of the ground state of the Pauli–Fierz Hamiltonian in the same way as the Nelson Hamiltonian. In Section 3.6.4 we suppose either Assumption 3.4 or Assumption 3.33, and that H has the ground state Ψg , i. e., HΨg = EΨg ,

E = inf Spec(H).

Hence Ψtg = e−tH f ⊗1/‖e−tH f ⊗1‖ converges to Ψg as t → ∞. We shall show the Gaussian domination of Ψtg for t < ∞. We define the self-adjoint operator  ξ in ℋPF by  ξ = ⊕ ̂ (⋅ − x))dx, where ξ ∈ ⨁3 L2 (ℝ3 ). Then we have ∫ 3 A(ξ μ=1 real

ℝ

(e−tH f ⊗ 1, e−iβAξ e−tH f ⊗ 1) , t→0 (e−tH f ⊗ 1, e−tH f ⊗ 1) ̂

(Ψg , e−iβAξ Ψg ) = lim ̂

β ∈ ℝ.

By using the measure μt , we can see that 1 2 (e−tH f ⊗ 1, e−iβAξ e−tH f ⊗ 1) = 𝔼μt [e− 2 (2ieβqE (Z,j0 ξ )+β qE (j0 ξ ,j0 ξ )) ]. (e−tH f ⊗ 1, e−tH f ⊗ 1)

̂

t

̃ − Bs )dBμs and qE (Z, j0 ξ ) is real. Here, Z = Z(t) = ⨁3μ=1 ∫0 js φ(⋅ ̂2

Lemma 3.64. Suppose that β < (2qE (j0 ξ , j0 ξ ))−1 . Then Ψtg ∈ D(e(β/2)Aξ ) and ̂2

‖e(β/2)Aξ Ψtg ‖2 = (1 − 2βqE (j0 ξ , j0 ξ ))−1/2 𝔼μt [exp (

βe2 qE (Z, j0 ξ )2 )] . (1 − 2βqE (j0 ξ , j0 ξ ))

(3.6.21)

Proof. The proof is a minor modification of Lemma 2.60 and Corollary 2.61. We have 1 2

(Ψtg , e−ikAξ Ψtg ) = 𝔼μt [e−iekqE (Z,j0 ξ ) ] e− 2 k ̂

qE (j0 ξ ,j0 ξ )

,

k ∈ ℝ.

By the Gaussian transformation with respect to k, we see that ̂2

(Ψtg , e−Aξ /2 Ψtg ) =

e2 qE (Z, j0 ξ )2 1 𝔼μt [exp (− )] . 2(1 + qE (j0 ξ , j0 ξ )) √1 + qE (j0 ξ , j0 ξ )

Replacing ξ with √−2βξ for β < 0, we have (3.6.21) with β < 0. We can extend this to β < (2qE (j0 ξ , j0 ξ ))−1 by an analytic continuation in the same way as the proof of Corollary 2.61. For the Nelson Hamiltonian HN , we can construct the equality 2

‖e(β/2)ξ (g) Ψg ‖2 =

1 2 √1 − β‖g/√ω‖ ̂

𝔼𝒩 [exp (

βK(g)2 )] 2 ̂ 1 − β‖g/√ω‖

382 | 3 The Pauli–Fierz model by path measures in terms of the path measure 𝒩 , by which we can show the Gaussian domination of 2 the ground state and, moreover, see the behavior of ‖e(β/2)ξ (g) Ψg ‖ at the critical point β=

1 1 2 , i. e., limβ↑ ̂ ‖g/√ω‖ 2 ̂ 2‖g/√ω‖

2

‖eβξ (g) Ψg ‖ = ∞. Unfortunately, it is not however straight̂2

forward to represent the expectation of eβAξ with respect to the ground state Ψg in terms of the measure μ∞ . The reason is why we do not have an uniform bound of the pair potential Wμν with respect to paths. The Gaussian domination of the ground state however can be shown by using Ψtg , but only for sufficiently small |β|, and unfor̂2

tunately we do not mention anything on the critical behavior of ‖e(β/2)Aξ Ψg ‖ like the ground state of the Nelson Hamiltonian.

Theorem 3.65 (Gaussian domination of the ground state). Let β < (2qE (j0 ξ , j0 ξ ))−1 . ̂2

Then Ψg ∈ D(e(β/4)Aξ ) follows.

Proof. By Lemma 3.64, we have the uniform bound 1

̂2

‖e(β/2)Aξ Ψtg ‖2 ≤

√1 − 2βqE (j0 ξ , j0 ξ )2 ̂2

in t. Thus there exists a subsequence t 󸀠 such that ‖e(β/4)Aξ Ψtg ‖2 converges to some c as 󸀠

̂2

t 󸀠 → ∞. We reset t 󸀠 as t. We claim that {e(β/4)Aξ Ψtg }t≥0 is a Cauchy sequence. Directly we have ̂2

̂2

̂2

̂2

̂2

‖e(β/4)Aξ Ψtg − e(β/4)Aξ Ψsg ‖2 = ‖e(β/4)Aξ Ψtg ‖2 + ‖e(β/4)Aξ Ψsg ‖2 − 2(Ψsg , e(β/2)Aξ Ψtg ). ̂2

Note that Ψtg strongly converges to Ψg as t → ∞. Since the uniform bound of ‖eβAξ Ψtg ‖2 implies that ̂2

̂2

̂2

(Ψsg , e(β/2)Aξ Ψtg ) = (Ψsg − Ψtg , e(β/2)Aξ Ψtg ) + ‖e(β/4)Aξ Ψtg ‖2 → c as t, s → ∞, we obtain that ̂2

̂2

lim ‖e(β/4)Aξ Ψtg − e(β/4)Aξ Ψsg ‖ = 0

t,s→∞ ̂2

̂2

and e(β/4)Aξ Ψtg , t > 0, is a convergent sequence. Hence the closedness of e(β/4)Aξ yields the desired results.

3.7 Translation invariant Pauli–Fierz model In this section, we consider the translation invariant Pauli–Fierz Hamiltonian. This is obtained by setting the external potential V identically zero, resulting in the fact that HPF commutes with the total momentum operator. A general argument allows

3.7 Translation invariant Pauli–Fierz model | 383

then to decompose HPF as HPF = ∫ℝ3 HPF (p)dp. Moreover, for each p ∈ ℝ3 , the fiber Hamiltonian HPF (p) is self-adjoint on ℱrad . We will specify the exact form of HPF (p) and its domain. Although HPF with V ≡ 0 has no ground state, the fiber Hamiltonian HPF (p) may have ground states and a functional integral representation of e−tHPF (p) can be established to study the spectrum of HPF (p). We start by defining the fiber Hamiltonian. As said, the standing assumptions throughout Section 3.7 are as follows. ⊕

Assumption 3.66. The following conditions hold. ̂ ̂ ̂ ̂ √ωφ,̂ φ/√ω, (1) Charge distribution: φ ∈ S 󸀠 (ℝ3 ) and φ(−k) = φ(k) φ/ω ∈ L2 (ℝ3 ); (2) External potential: V = 0. Put Pfμ = dΓ(kμ ⊕ kμ ) = ∑ ∫ kμ a∗ (k, j)a(k, j)dk, j=±

μ = 1, 2, 3,

ℝ3

which describes the field momentum. The total momentum operator Ptot on ℋPF is defined by the sum of the momentum operator for the particle and that of field: Ptot μ = −i∇xμ ⊗ 1 + 1 ⊗ Pfμ ,

μ = 1, 2, 3.

j −ik⋅x √ ̂ Recall that ℘jμ (x) = φ(k)e / ω(k). Since μ (k)e

[−i∇xν , a∗ (℘jμ , j)] = a∗ (−kν ℘jμ , j), [Pfν , a∗ (℘jμ , j)] = a∗ (kν ℘jμ , j), it follows [HPF , Ptot μ ] = 0,

μ = 1, 2, 3.

This leads to a decomposition HPF of the spectrum of the total momentum operator 3 2 2 Spec(Ptot μ ) = ℝ, μ = 1, 2, 3. We denote ∑μ=1 Pf μ by Pf . In the same way as Proposition 2.142, we can also show that HPF with V = 0 has no ground state. We state this as a proposition. Proposition 3.67. HPF has no ground state. Proof. If HPF has the ground state, then it must be unique. Then the proposition follows in the same way as that of Proposition 2.142. Definition 3.68 (Pauli–Fierz Hamiltonian with a fixed total momentum). The Pauli–Fierz Hamiltonian with a fixed total momentum is 1 HPF (p) = (p − Pf − eA(0))2 + Hrad , 2

p ∈ ℝ3 ,

384 | 3 The Pauli–Fierz model by path measures with domain D(HPF (p)) = D(Hrad ) ∩ D(Pf2 ), where Aμ (0) = Aμ (x = 0) is given by Aμ (0) =

̂ ̂ φ(k) φ(−k) 1 + a(k, j) )dk. ∑ ∫ eμj (k)(a∗ (k, j) √2 j=± √ω(k) √ω(k) 3 ℝ

Here, p ∈ ℝ3 is called the total momentum. It is easy to see that HPF (p) is self-adjoint for sufficiently small coupling constants e as a consequence of the Kato–Rellich theorem. In fact, by splitting off HPF (p) as HPF (p) = HPF,0 (p) + HPF,I (p), with

1 HPF,0 (p) = Hrad + (p − Pf )2 , 2 e2 HPF,I (p) = −e(p − Pf )A(0) + A(0)2 , 2

we readily see that HPF,I (p) is relatively bounded with respect to HPF,0 (p). We give the relationship between HPF and HPF (p). Define the unitary operator 2

3

2

3

T : ℋPF = L (ℝx ) ⊗ ℱrad → L (ℝp ) ⊗ ℱrad

by ⊕

T = (F̂ ⊗ 1) ∫ exp(ixPf )dx,

(3.7.1)

ℝ3

with F̂ denoting Fourier transformation from L2 (ℝ3x ) to L2 (ℝ3p ). For Ψ ∈ ℋPF , (T Ψ)(p) =

1

√(2π)3

∫ e−ixp eixPf Ψ(x)dx. ℝ3

Let Mμ f (x) = xμ f (x), μ = 1, 2, 3, be the multiplication by xμ in L2 (ℝ3 ). Write M = (M1 , M2 , M3 ) and n

󵄨󵄨 󵄨

2 ∞

3

(3.7.2)

∞ D∞ = C0∞ (ℝ3 )⊗̂ ℱrad .

(3.7.3)

ℱrad = L. H. {∏ a (fi )ΩPF , ΩPF 󵄨󵄨󵄨 fi ∈ ⊕ C0 (ℝ ), i = 1, . . . , n, n ≥ 1} ∞

i=1

∗

and

3.7 Translation invariant Pauli–Fierz model | 385

We define HPF (p) for arbitrary values of e ∈ ℝ and p ∈ ℝ3 as a self-adjoint operator. As was mentioned above, for sufficiently small e, HPF (p) is defined as a self-adjoint operator, however, for arbitrary values of e, this is not clear. Define 1 L = (M ⊗ 1 − 1 ⊗ Pf − e1 ⊗ A(0))2 + 1 ⊗ Hrad ⌈D∞ . 2 We have T HPF Φ = LT Φ for Φ ∈ D∞ . Since by Theorem 3.47, D∞ is a core of HPF , and L is closed, we see that T maps D(HPF ) onto D(L) with T HPF T −1 = L. Note also that D(L) = T D(HPF ) = D((M ⊗ 1 − 1 ⊗ Pf )2 ) ∩ D(1 ⊗ Hrad ). Theorem 3.69 (Fiber decomposition and self-adjointness). For each p ∈ ℝ3 , HPF (p) is a nonnegative self-adjoint operator, and ⊕

T ( ∫ ℱrad dp) T

⊕

−1

= ℋPF ,

T ( ∫ HPF (p)dp) T

ℝ3

−1

= HPF .

ℝ3

Proof. Define the quadratic form Qp (Ψ, Φ) =

1 3 1/2 1/2 Ψ, Hrad Φ) ∑ ((p − Pf − eA(0))μ Ψ, (p − Pf − eA(0))μ Φ) + (Hrad 2 μ=1

1/2 ). Since Qp is densely on ℱrad × ℱrad with form domain ⋂3μ=1 (D(Pfμ ) ∩ D(Aμ (0))) ∩ D(Hrad defined and nonnegative, there exists a positive self-adjoint operator L(p) such that Qp (Ψ, Φ) = (L(p)1/2 Ψ, L(p)1/2 Φ). Note that

L(p) = HPF (p)

(3.7.4)

⊕ on D(Hrad ) ∩ D(Pf2 ). Define L̃ = ∫ℝ3 L(p)dp. For ϕ, ψ ∈ D∞ , we have

(T ψ, LT ϕ) = (T ψ, L̃ T ϕ). Hence T −1 LT = T −1 L̃ T on D∞ and then HPF = T −1 L̃ T on D∞ . Since D∞ is a core of HPF and L̃ is self-adjoint, it follows that T maps D(HPF ) onto D(L)̃ with T HPF T

−1

= L.̃

(3.7.5)

Next, we prove the self-adjointness of HPF (p). The operator HPF,0 (p) is self-adjoint on D(HPF,0 (p)) = D(Pf2 ) ∩ D(Hrad ) and ⊕

T ( ∫ HPF,0 (p)dp) T ℝ3

−1

= HPF,0 .

386 | 3 The Pauli–Fierz model by path measures ∞ Thus we have for F ∈ ℋPF such that (T F)(p) = f (p)Φ with f ∈ C0∞ (ℝ3 ) and Φ ∈ ℱrad , by the inequality ‖HPF,0 F‖ ≤ C‖(HPF + 1)F‖ derived from Theorem 3.47 and the closed graph theorem, and (3.7.5),

∫ |f (p)|2 ‖HPF,0 (p)Φ‖2 dp ≤ C 2 ∫ |f (p)|2 ‖(L(p) + 1)Φ‖2 dp. ℝ3

ℝ3

Hence ‖HPF,0 (p)Φ‖ ≤ C‖(L(p) + 1)Φ‖

(3.7.6)

follows for almost every p ∈ ℝ3 , since f ∈ C0∞ (ℝd ) is arbitrary. Since both sides of (3.7.6) are continuous in p, this inequality holds for all p ∈ ℝ3 . Thus HPF,0 (p)(L(p) + 1)−1 is bounded, and so is also HPF,0 (p)e−tL(p) , which implies that e−tL(p) leaves D(Hrad ) ∩ D(Pf2 ) invariant. This means that L(p) is essentially self-adjoint on D(Hrad ) ∩ D(Pf2 ) by Lemma 3.40. Moreover, (3.7.6) implies that L(p) is closed on D(Hrad ) ∩ D(Pf2 ), and hence L(p) is self-adjoint on D(Hrad ) ∩ D(Pf2 ). By the basic inequality, ‖L(p)Φ‖ ≤ C 󸀠 (‖HPF,0 (p)Φ‖ + ‖Φ‖),

Φ ∈ D(HPF,0 (p)),

it is clear that L(p) is essentially self-adjoint on any core of HPF,0 (p). By (3.7.4), the theorem follows. Self-adjointness of HPF (p) ensures that {e−tHPF (p) : t ≥ 0} is a well-defined C0 -semigroup. As in the previous section, we move to from Fock representation to Schrödinger representation in order to construct a functional integral representation. Then HPF (p) becomes 1 2 ̂ + Hrad HPF (p) = (p − P̂ f − A(0)) 2 ̂ ̂ = 0) = (Â 1 (φ), ̃ Â 2 (φ), ̃ Â 3 (φ)), ̃ and recall that P̂ f = dΓ(−i∇) on L2 (Q ), where A(0) = A(x and Hrad = dΓ(ω(−i∇)). We remark that we also use notation Hrad for dΓ(ω(−i∇)). The functional integral representation of e−tHPF (p) can be also constructed as an application of that of e−tHPF . Theorem 3.70 (Functional integral representation for Pauli–Fierz Hamiltonian with a fixed total momentum). Let Ψ, Φ ∈ L2 (Q ). Then (Ψ, e−tHPF (p) Φ) = 𝔼0 [eip⋅Bt (J0 Ψ, e−ieAE (Kt ) Jt e−iPf ⋅Bt Φ)L2 (QE ) ]. ̂

̂

(3.7.7)

Proof. Write Fs = Πs ⊗Ψ and Gr = Πr ⊗Φ, where Πs is the three dimensional heat kernel, Πs (x) = (2πs)−3/2 exp(−|x|2 /2s), and Ψ, Φ ∈ L2 (Q ). Due to the fact that HPF = ⊕ T (∫ℝ3 HPF (p)dp)T −1 , we have tot

(Fs , e−tHPF e−iξ P Gr ) = ∫ e−iξp ((T Fs )(p), e−tHPF (p) (T Gr )(p))dp ℝ3

3.7 Translation invariant Pauli–Fierz model | 387

for any ξ ∈ ℝ3 . Note that lims→0 (T Fs )(p) = (2π)−3/2 Ψ strongly in L2 (Q ) for each p ∈ ℝ3 . Hence tot

lim(Fs , e−tHPF e−iξ P Gr ) =

s→0

1 ∫ (Ψ, e−tHPF (p) e−iξp (T Gr )(p))dp. √(2π)3

(3.7.8)

ℝ3

tot

By using the fact eiξ P Gr (x) = Πr (x − ξ )e−iξ Pf Φ, we obtain by the functional integral representation in Theorem 3.26 that ̂

tot

(Fs , e−tHPF e−iξ P Gr ) = ∫ 𝔼x [Πs (x)Πr (Bt − ξ )(J0 Ψ, e−ieAE (Kt ) Jt e−iξ Pf Φ)]dx. ̂

̂

ℝ3

Then it follows that by Πs (x) → δ(x) as s → 0, tot

lim(Fs , e−tHPF e−iξ P Gr ) = 𝔼0 [Πr (Bt − ξ )(J0 Ψ, e−ieAE (Kt ) Jt e−iξ Pf Φ)]. ̂

(3.7.9)

̂

s→0

Combining (3.7.8) and (3.7.9) leads to 1

√(2π)3

∫ e−iξp (Ψ, e−tHPF (p) (T Gr )(p))dp = 𝔼0 [Πr (Bt − ξ )(J0 Ψ, e−ieAE (Kt ) Jt e−iξ Pf Φ)]. ̂

ℝ3

̂

(3.7.10)

Since |(Ψ, e−tHPF (p) (T Gr )(p))| ≤ ‖(T Gr )(p)‖ and ‖(T Gr )(⋅)‖ ∈ L2 (ℝ3 ), we have (Ψ, e−tHPF (p) (T Gr )(p)) ∈ L2 (ℝ3 p ). Taking the inverse Fourier transform on both sides of (3.7.10) with respect to p gives (Ψ, e−tHPF (p) (T Gr )(p)) =

̂ ̂ 1 𝔼0 [ ∫ eiξp Πr (Bt − ξ )(J0 Ψ, e−ieAE (Kt ) Jt e−iξ Pf Φ)dξ ] √(2π)3 3 ℝ

(3.7.11)

for almost every p ∈ ℝ3 . Since both sides of (3.7.11) are continuous in p, the equality stays valid for all p ∈ ℝ3 . After taking r → 0 on both sides, we arrive at (3.7.7). By Theorem 3.70, the functional integral representation of e−tHPF (p) can be formally written as (Ψ, e−tHPF (p) Φ)

formal

=

t

𝔼0 [ ∫ J0 Ψei ∫0 (p−dΓE (−i∇⊗1)−eAE (φ(⋅−Bs )))dBs Jt ΦdμE ]. ̂

̃

QE

̃ − Bs )) appears suggestive of In the exponent, the expression p − dΓE (−i∇ ⊗ 1) − e E (φ(⋅ ̃ the Euclidean version of p − P̂ f − eA(̂ φ). ̂ Since intuitively e−iPf ⋅Bt is a shift operator on L2 (Q ), it can be expected to be a positivity preserving.

388 | 3 The Pauli–Fierz model by path measures Lemma 3.71. Let ξ ∈ ℝ3 . Then e−iPf ξ is positivity preserving. In particular, it follows that ̂ ̂ |e−iPf ξ Ψ| ≤ e−iPf ξ |Ψ|. ̂

Proof. Let n

̂ j) i ∑j=1 kj A(f ̂ 1 ), . . . , A(f ̂ n )) = (2π)−n/2 ∫ F(k)e ̂ F = F(A(f dk, ℝn

m

̂ j) i ∑j=1 kj A(g ̂ ̂ 1 ), . . . , A(g ̂ m )) = (2π)−m/2 ∫ G(k)e G = G(A(g dk ℝm

with 0 ≤ F ∈ S (ℝn ) and 0 ≤ G ∈ S (ℝm ). By (eiA(g1 ) , eiPf ξ eiA(g2 ) ) = (eiA(g1 ) , eiA(g2 (⋅+ξ )) ), ̂

̂

̂

̂

̂

we see that ̂ ̂ 1 ( ⋅ + ξ )), . . . , A(g ̂ m ( ⋅ + ξ )))) ≥ 0. (F, eiPf ξ G) = (F, G(A(g

Hence by a limiting argument, (F, eiPf ξ G) ≥ 0 for arbitrary nonnegative F, G ∈ L2 (Q ). ̂

An analogue of the diamagnetic inequality for e−tHPF can be derived from the functional integral representation for e−tHPF (p) . Corollary 3.72 (Diamagnetic inequality). It follows that ̂2

|(Ψ, e−tHPF (p) Φ)| ≤ (|Ψ|, e−t(Pf +Hrad ) |Φ|). Proof. This follows from Lemma 3.71 and ̂2

|(Ψ, e−tHPF (p) Φ)| ≤ 𝔼[(J0 |Ψ|, Jt e−iBt Pf |Φ|)] = (|Ψ|, e−t(Pf +Hrad ) |Φ|). ̂

From this diamagnetic inequality, however, we can only deduce the trivial energy comparison inequality inf Spec(P̂ f2 +Hrad ) ≤ inf Spec(HPF (p)) = 0. However, combining the unitary transformation e−i(π/2)N with the functional integral representation (3.7.7), we obtain an interesting result. Denote EPF (p) = inf Spec(HPF (p)). Corollary 3.73 (Properties of EPF (p)). It follows that: (1) Let p = 0 and S = e−i(π/2)N . Then S−1 e−tHPF (0) S is positivity improving. (2) The ground state of HPF (0) is unique whenever it exists. (3) EPF (0) ≤ EPF (p) holds. (4) Map p 󳨃→ EPF (p) is continuous and EPF (0) = inf Spec(HPF,V=0 ), where HPF,V=0 is HPF with V = 0.

3.8 The Pauli–Fierz model with spin

| 389

Proof. In the case of p = 0, we remark that eip⋅Bt = 1. Then we have (Ψ, S−1 etHPF (0) SΦ) = 𝔼0 [(J0 Ψ, e−ieΠE (Kt ) Jt e−iPf ⋅Bt Φ)]. ̂

̂

Since J∗0 e−ieΠE (Kt ) Jt is positivity improving and e−iPf Bt is positivity preserving by Lemma 3.71, (1) follows. (2) is implied by (1) and the Perron–Frobenius theorem. We have ̂

̂

|(Ψ, S−1 e−tHPF (p) SΦ)| ≤ 𝔼0 [(J0 |Ψ|, e−ieΠE (Kt ) Jt e−iPf Bt |Φ|)] = (|Ψ|, S−1 e−tHPF (0) S|Φ|). ̂

̂

This yields (3). Finally (4) is derived in the same way as Corollary 2.148. Finally, we show the vacuum expectation (1, e−tHPF (p) 1) as a corollary of Theorem 3.70. Corollary 3.74. It follows that 2

(1, e−tHPF (p) 1) = 𝔼0 [eip⋅Bt e−(e /2)qE (Z,Z) ], where qE (Z, Z) is given in (3.6.11). As was mentioned in the previous section, qE (Z, Z) has the formal expression t μ t qE (Z, Z) = ∑3μ,ν=1 ∫−t dBs ∫−t dBνr Wμν (Bs − Br , s − r), where Wμν is given by ⊥ Wμν (X, T) = ∫ δμν (k) ℝ3

2 ̂ |φ(k)| e−|T|ω(k) e−ik⋅X dk 2ω(k)

and it is called the pair potential of the Pauli–Fierz Hamiltonian with a fixed total momentum.

3.8 The Pauli–Fierz model with spin We assume Assumption 3.4 in Sections 3.8.1–3.8.8, and Assumption 3.33 in Section 3.8.9 unless otherwise stated.

3.8.1 Counting measures In the previous sections, we considered the spinless version of the Pauli–Fierz Hamiltonian ignoring the spin of the electron. In this section, we define the Hamiltonian including spin 1/2 and construct its functional integral representation with a scalar integrand dependent on a jump process which is so-called spin process. To derive a functional integral representation of the Pauli–Fierz Hamiltonian with spin, we review classical cases in which the Feynman–Kac formula for Schrödinger operators

390 | 3 The Pauli–Fierz model by path measures with spin is investigated. As is established in classical cases in addition to Brownian motion, we need a Lévy process for presenting spin. We give this 3-dimensional Lévy process (Zt )t≥0 on a probability space (𝒮 , ℬ𝒮 , P) with characteristics (b, A, ν) such that ν(ℝ3 \ {0}) = 1. For I ∈ ℬ(ℝ3 ), let N(t, I) = |{0 ≤ s ≤ t | ΔZs ∈ I}| be the counting measure associated with (Zt )t≥0 . Define the random process (Nt )t≥0 on (𝒮 , ℬ𝒮 , P) by Nt = N(t, ℝ3 \ {0}). Note that (Nt )t≥0 is a Poisson process with intensity 1. Consider the measure dNt =

∫ N(dtdz) ℝ3 \{0}

on [0, ∞) = ℝ+ . The compensator of Nt is given by t and the expectation of exponent −α e−αNt is given by 𝔼P [e−αNt ] = et(e −1) . We have for f = f (s, τ), (s, τ) ∈ ℝ+ × 𝒮 , independently of z ∈ ℝ3 \ {0}, that t+

t

𝔼P [∫ f (s, τ)dNs ] = 𝔼P [∫ f (s, τ)ds] [0 ] [0 ] and furthermore t+

t ∞

t

n

s 𝔼P [∫ f (s, Ns )dNs ] = 𝔼P [∫ f (s, Ns )ds] = ∫ ∑ f (s, n) e−s ds. n! [0 ] [0 ] 0 n=0 t+

Here, ∫0 = ∫[0,t] . Let N + α = (Nt + α)t≥0 and 𝔼αP [f (N)] = 𝔼P [f (N + α)]. 3.8.2 Review of classical cases Let us review Schrödinger operators with spin 1/2. The spin will be described in terms of a ℤ2 -valued process and the Schrödinger operator is defined on L2 (ℝ3 × ℤ2 ; ℂ) instead of L2 (ℝ3 ; ℂ2 ). Let σ1 , σ2 , σ3 be the 2 × 2 Pauli matrices given by 0 1

σ1 = (

1 ), 0

0 i

σ2 = (

−i ), 0

1 0

σ3 = (

0 ). −1

(3.8.1)

Let ℤ2 be the set of the square roots of identity, i. e., ℤ2 = {θ1 , θ2 }, where θα = (−1)α . The particle and spin yield an (ℝ3 × ℤ2 )-valued random process (qt )t≥0 on the product space (X × 𝒮 , ℬ(X ) × ℬ𝒮 , 𝒲 x × P) by qt = (Bt , θNt ),

t ≥ 0,

3.8 The Pauli–Fierz model with spin

| 391

x,α where θNt = (−1)Nt is called the spin process. For simplicity, we write 𝔼x,α for 𝒲×P = 𝔼 3 (x, α) ∈ ℝ × {1, 2}. The generator of random process (qt )t≥0 is

1 − (− Δ ⊗ 1 + 1 ⊗ σF ) 2 under the identification L2 (ℝ3 × ℤ2 ) ≅ L2 (ℝ3 ) ⊗ ℂ2 given by f (x, +1) ) 󳨃→ f (x, θ) ∈ L2 (ℝ3 × ℤ2 ; ℂ), f (x, −1)

L2 (ℝ3 ; ℂ2 ) ∋ ( where

(3.8.2)

1 σF = (σ3 + iσ2 )(σ3 − iσ2 ). 2

Actually σF = −σ1 + 1, and hence we have 1

(f , e−t(− 2 Δ−σ1 ) g) = et ∑ ∫ 𝔼x,α [f ̄(q0 )g(qt )]dx. α=1,2

ℝ3

Let a ∈ (Cb2 (ℝ3 ))3 and V ∈ L∞ (ℝ3 ). We call the operator HS (a) =

1 (σ ⋅ (−i∇ − a))2 + V 2

(3.8.3)

on L2 (ℝ3 ; ℂ2 ) Schrödinger operator with a vector potential a and spin 1/2. Set D = (D1 , D2 , D3 ) with Dμ = (−i∇ − a)μ , and σ = (σ1 , σ2 , σ3 ). Since σ1 σ2 = iσ3 , σ2 σ3 = iσ1 and σ3 σ1 = iσ2 , we have (σ ⋅ D)2 = D2 + iσ1 (D2 D3 − D3 D2 ) + iσ2 (D3 D1 − D1 D3 ) + iσ3 (D1 D2 − D2 D1 ). Using relations D2 D3 −D3 D2 = i(∇×a)1 , D3 D1 −D1 D3 = i(∇×a)2 and D1 D2 −D2 D1 = i(∇×a)3 , we expand (3.8.3) to obtain 1 HS (a) = Hp (a) − σ ⋅ b, 2

(3.8.4)

where b = (b1 , b2 , b3 ) = ∇ × a is a magnetic field. Let a ∈ (Cb2 (ℝ3 ))3 ,

b ∈ (L∞ (ℝ3 ))3 ,

V ∈ L∞ (ℝ3 ).

(3.8.5)

Then HS (a) is self-adjoint on D(−Δ) and bounded below. Now we transform HS (a) into an operator on the set of ℂ-valued functions on a suitable Hilbert space. By the identification (3.8.2), HS (a) can be reduced to the self-adjoint operator Hℤ2 (a) defined in L2 (ℝ3 × ℤ2 ) to obtain 1 1 (Hℤ2 (a)f )(x, θ) = (Hp (a) − θb3 (x)) f (x, θ) − (b1 (x) − iθb2 (x))f (x, −θ), 2 2

392 | 3 The Pauli–Fierz model by path measures where x ∈ ℝ3 and θ ∈ ℤ2 . Thus HS (a) can be regarded as Hℤ2 (a) on the space of ℂ-valued functions with the configuration space ℝ3 × ℤ2 . Suppose (3.8.5) and that t ∫ ds ∫ 3 | log 21 √b1 (y)2 + b2 (y)2 |Πs (y − x)dy < ∞ for all (x, t) ∈ ℝ3 × ℝ+ . Then 0

ℝ

(f , e−tHℤ2 (a) g) = et ∑ ∫ 𝔼x,α [f (q0 )g(qt )eZt ] dx α=1,2

(3.8.6)

ℝ3

holds. Here, exponent Zt consists of 4 parts; vector potential a, external potential V, the diagonal part of spin interaction b3 and the off-diagonal part of spin interaction b1 − ib2 : t

t

t

0

0

0

1 Zt = −i ∫ a(Bs ) ∘ dBs − ∫ V(Bs )ds + ∫ θNs b3 (Bs )ds 2 t+

1 + ∫ log ( (b1 (Bs ) − iθNs b2 (Bs ))) dNs . 2 0

t+

Here, ∫0 . . . dNs describes the sum of jump points of integrand: t+

1 ∫ log ( (b1 (Bs ) − iθNs b2 (Bs ))) dNs = 2 0

1 ∑ log ( (b1 (Br ) − iθNr− b2 (Br ))) . 2 r∈[0,t]

Nr+ =N ̸ r−

We apply classical cases to construct a functional integral representation of the Pauli–Fierz Hamiltonian with spin in the next section. 3.8.3 Definition of the Pauli–Fierz Hamiltonian with spin In this section, we choose a specific Q space instead of the abstract setup. This will serve the purpose of drawing sufficient regularity of  with respect to ϕ ∈ Q . Let Sreal (ℝ3+β ), β = 0, 1, be the set of real-valued Schwarz test functions on ℝ3+β and put Sβ = ⊕3 Sreal (ℝ3+β ). Put Qβ = Sβ , 󸀠

with Sβ󸀠 denoting the dual space of Sβ . Also, denote the pairing between elements of Qβ and Sβ by ⟨⟨ϕ, f⟩⟩ ∈ ℝ, where ϕ ∈ Qβ and f ∈ Sβ . By the Bochner–Minlos theorem, there exists a complete probability space (Qβ , Σβ , μβ ) such that Σβ is the smallest σ-field generated by {⟨⟨ϕ, f⟩⟩ | f ∈ Sβ } and ⟨⟨ϕ, f⟩⟩ is a Gaussian random variable with mean zero and covariance 𝔼μβ [⟨⟨ϕ, f⟩⟩⟨⟨ϕ, g⟩⟩] = qβ (f, g).

(3.8.7)

3.8 The Pauli–Fierz model with spin

| 393

We can extend ⟨⟨ϕ, f⟩⟩ to more general f. For any f = ℜf+iℑf ∈ ⊕3 S (ℝ3+β ), put ⟨⟨ϕ, f⟩⟩ = ⟨⟨ϕ, ℜf⟩⟩ + i⟨⟨ϕ, ℑf⟩⟩. Since S (ℝ3+β ) is dense in L2 (ℝ3+β ) and the inequality 𝔼μβ [|⟨⟨ϕ, f⟩⟩|2 ] ≤ ‖f‖2⊕3 L2 (ℝ3+β ) holds by (3.8.7), we define ⟨⟨ϕ, f⟩⟩ for f ∈ ⊕3 L2 (ℝ3+β ) by ⟨⟨ϕ, f⟩⟩ = limn→∞ ⟨⟨ϕ, fn ⟩⟩ in L2 (Qβ ), where (fn )n∈ℕ ⊂ ⊕3 S (ℝ3+β ) is any sequence so that s-limn→∞ fn = f. Thus the quantized radiation field  β (f) with f ∈ ⊕3 L2 (ℝ3+β ) is realized as ( β (f)F) (ϕ) = ⟨⟨ϕ, f⟩⟩F(ϕ),

ϕ ∈ Qβ ,

in L2 (Qβ ), with domain 󵄨󵄨 󵄨 D( β (f)) = {F ∈ L2 (Qβ ) 󵄨󵄨󵄨 𝔼μβ [|⟨⟨ϕ, f⟩⟩F(ϕ)|2 ] < ∞}. 󵄨󵄨 We define  β (f, ϕ) for each ϕ ∈ Qβ by f ∈ ⊕3 L2 (ℝ3+β ).

 β (f, ϕ) = ⟨⟨ϕ, f⟩⟩,

In the same way as in the previous section, we put  =  0 ,  E =  1 ,

(Minkowskian) (Euclidean)

μ = μ0 , μE = μ1 ,

Q = Q0 , QE = Q1 .

The Hilbert space consisting of state vectors of the Pauli–Fierz Hamiltonian with spin 1/2 is 2

3

2

2

ℋS = L (ℝ ; ℂ ) ⊗ L (Q ).

The quantized radiation field  = ( 1 ,  2 ,  3 ) in ℋS with cutoff function φ̃ is defined by ⊕

̃ − x))dx, Â μ = ∫ Â μ (φ(⋅

μ = 1, 2, 3,

ℝ3

under the identification ℋS ≅ ∫ℝ3 ℂ2 ⊗L2 (Q )dx. The quantized magnetic field is defined by the curl of  as usual: ⊕

B̂ = (B̂ 1 , B̂ 2 , B̂ 3 ) = rotx A.̂ It is straightforward to see that 3

̃ − x)), B̂ μ (x) = ∑ Â λ (δλν εμαν 𝜕xα φ(⋅ λ,α,ν=1

(3.8.8)

394 | 3 The Pauli–Fierz model by path measures where εαβγ denotes the antisymmetric tensor defined by 1 { { εαβγ = {−1 { {0

(αβγ) is an even permutation of (123), (αβγ) is an odd permutation of (123), otherwise.

For example, ε132 = −1, ε231 = 1 and ε112 = 0. We define B̂ μ (f ) with a test function f ∈ L2 (ℝ3 ) by 3

B̂ μ (f ) = ∑ Â λ (δλν εμαν 𝜕xα f ). λ,α,ν=1

̂ Here, 𝜕xα f = 𝜕xα f (⋅ − x)⌈x=0 , hence 𝜕̂ xα f = −ikα f (k) follows. The Euclidean version of 2 4 ̂ Bμ (g) with test function g ∈ L (ℝ ) is defined by 3

B̂ Eμ (g) = ∑ Â Eλ (δλν εμαν 𝜕xα g). λ,α,ν=1

(3.8.9)

We will use this terminology later. Definition 3.75 (Pauli–Fierz Hamiltonian with spin 1/2). The Pauli–Fierz Hamiltonian with spin 1/2 is defined by 1 e S HPF = (−i∇ − eA)̂ 2 + V + Hrad − σ ⋅ B.̂ 2 2

(3.8.10)

The usual definition by minimal coupling, S HPF =

1 ̂ 2 + V + Hrad (σ ⋅ (−i∇ − eA)) 2

indeed coincides with (3.8.10), which can be seen by expanding the square above and making use of the identity (σ ⋅ a)(σ ⋅ b) = a ⋅ b + iσ ⋅ (a × b). Remark 3.76 (Fock representations). Indeed B̂ μ (f ) corresponds to Bμ (f ) =

1 ̃ ∑ (a∗ (ηjμ f ̂, j) − a(ηjμ f ̂, j)) √2 j=±

in Fock representation on ℱrad , where ηj (k) = −ik × ej (k),

j = ±.

Then in Fock representation the Pauli–Fierz Hamiltonian with spin 1/2 is realized as e 1 (−i∇ − eA)2 + V + Hrad − σ ⋅ B 2 2

(3.8.11)

3.8 The Pauli–Fierz model with spin

| 395

on L2 (ℝ3 ; ℂ2 ) ⊗ ℱrad , where the quantized magnetic field B is given by ⊕

Bμ = ∫ Bμ (x)dx, ℝ3

̂ ̂ φ(−k) φ(k) 1 e−ik⋅x a∗ (k, j) − eik⋅x a(k, j)) dk. ∫ ηjμ (k) ( √ √ √ ω(k) ω(k) j=± 2 3

Bμ (x) = ∑

ℝ

S Proposition 3.77 (Self-adjointness). HPF is self-adjoint on DPF and bounded from below. Moreover, it is essentially self-adjoint on any core of HPF,0 = Hp ⊗ 1 + 1 ⊗ Hrad . S Proof. Let HPF = HPF − e2 σ ⋅ B.̂ It is shown in Theorem 3.47 that HPF is self-adjoint on ̂ DPF . By the bound ‖HPF Ψ‖2 + C‖Ψ‖2 ≥ C 󸀠 ‖Hrad Ψ‖2 given in (3.5.16) and ‖ − e2 σ ⋅ BΨ‖ ≤ ε‖Hrad Ψ‖ + Cε ‖Ψ‖ for arbitrary ε > 0 and a constant Cε , we know that the spin interaction part − e2 σ ⋅ B̂ is infinitesimally small with respect to HPF . Then the proposition follows.

3.8.4 Symmetry and polarization In this section, we take the Fock representation (3.8.11). When the form factor φ̂ and ̂ the external potential V are rotation-invariant, i. e., φ(̂ R k) = φ(k) and V(R x) = V(x) S for any R ∈ O(3), then HPF has the symmetry SU(2) × SOpart (3) × SOfield (3) × helicity, where SU(2) and SOpart (3) come from the spin and the angular momentum of the particle, respectively, while the SOfield (3) and helicity part respectively from the angular momentum and the helicity of photons. ̂k. Thus the orthogonal bases Let R ∈ SO(3) and k̂ = k/|k|. Note that R k̂ = R + − + − 3 ̂ ̂ {e (R k), e (R k), R k} and {R e (k), R e (k), R k} in ℝ at R k satisfy e+ (R k) cos ϑ13 (e− (R k)) = ( sin ϑ13 ̂k 0 R

R e+ (k) 0 0 ) (R e− (k)) , 13 R k̂

− sin ϑ13 cos ϑ13 0

(3.8.12)

where 13 is the 3 × 3 unit matrix and ϑ = ϑ(R , k) satisfies cos ϑ = R e+ (k) ⋅ e+ (R k). Let R = R (n, ϕ) ∈ SO(3) be the rotation around n ∈ S2 = {k ∈ ℝ3 | |k| = 1} by the angle ϕ ∈ ℝ, and det R = 1. Also, let ℓk = k × (−i∇k ) = (ℓk 1 , ℓk 2 , ℓk 3 ) be the triplet of angular momentum operators in L2 (ℝ3k ). Then (3.8.12) can be rewritten as R e+ (k) ) = e−iϑΣ2 ( 0 e− (k)

eiϕn⋅ℓk (

0 R

)(

e+ (k) ), e− (k)

396 | 3 The Pauli–Fierz model by path measures where

0 Σ2 = i ( 13

−13 ). 0

S In order to discuss the symmetry of HPF , we introduce coherent polarization vectors in given directions. We make the following.

Definition 3.78 (Coherent polarization vectors). The polarization vectors e± are coherent polarization vectors in direction n ∈ S2 whenever there exists z ∈ ℤ such that for any ϕ ∈ [0, 2π) and any k with k̂ ≠ n, cos(zϕ)13 e+ (R k) )=( sin(zϕ)13 e− (R k)

− sin(zϕ)13 R )( cos(zϕ)13 0

(

0

e+ (k) )( − ) R e (k)

(3.8.13)

or componentwise eμ+ (R k)

− sin(zϕ) (R e+ (k))μ )( ), cos(zϕ) (R e− (k))μ

cos(zϕ) )=( − sin(zϕ) eμ (R k)

( where R = R (n, ϕ).

If polarization vectors e± are coherent polarization vectors in direction n ∈ S2 , then we have eμ+ (k)

(R e+ (k))μ ) = ( ). (R e− (k))μ eμ− (k)

exp {iϕ (zΣ2 + n ⋅ ℓk )} (

(3.8.14)

Here is an example of coherent polarization vectors in direction n ∈ S2 . Example 3.79. Let n3 = (0, 0, 1) and S‖2 = {(√1 − z 2 , 0, z) ∈ S2 | −1 ≤ z ≤ 1}. Take any polarization vectors on S‖2 : e± (k) for k ∈ S‖2 . For k ∈ S2 \ S‖2 , there exists a unique hk ∈ S‖2 and 0 < ϕ < 2π such that R (n3 , ϕ)hk = k. Then we define e± (k) for k ∈ S2 \ S‖2 by cos(zϕ)13 e+ (k) )=( sin(zϕ)13 e− (k)

(

− sin(zϕ)13 R (n3 , ϕ)e+ (hk ) )( ). cos(zϕ)13 R (n3 , ϕ)e− (hk )

(3.8.15)

It is readily checked that e± satisfies (3.8.13) with (n3 , z) ∈ S2 × ℤ. Example 3.80. Let n ∈ S2 , and e+ (k) = k̂ × n/ sin ϑ,

e− (k) = k̂ × e+ (k),

̂ Then, since R = R (n, ϕ) satisfies that R n = n and R u × R v = where ϑ = cos−1 (kn). ± R (u × v), e (k) obeys (3.8.13) with (n, 0) ∈ S2 × ℤ, i. e., R e± (k) = e± (R k). In particular, when (n, z) = (n3 , 0) ∈ S2 × ℤ, sin ϑ = √k12 + k22 and e+ (k) =

(−k2 , k1 , 0) √k12 + k22

,

e− (k) =

(k3 k1 , −k2 k3 , k12 + k22 ) |k|√k12 + k22

.

3.8 The Pauli–Fierz model with spin

| 397

Let H2 : L2 (ℝ3 ) ⊕ L2 (ℝ3 ) → L2 (ℝ3 ) ⊕ L2 (ℝ3 ) be given by H2 = i (

0 1L2 (ℝ3 )

−1L2 (ℝ3 ) ). 0

(3.8.16)

Suppose that polarization vectors are coherent polarization vectors in direction n ∈ S2 and z ∈ ℤ. We define Sf = dΓ(zH2 ) and Lf = (Lf,1 , Lf,2 , Lf,3 ) = dΓ(ℓk ).

(3.8.17)

Sf is called the helicity and Lf the angular momentum of the field. The helicity Sf can be formally written as Sf = i ∫ z(a∗ (k, −)a(k, +) − a∗ (k, +)a(k, −))dk.

(3.8.18)

ℝ3

For (n, z) ∈ S2 × ℤ define Jf = Jf (n, z) by Jf = n ⋅ Lf + Sf and Jp = Jp (n) by

1 Jp = n ⋅ ℓx + n ⋅ σ. 2

(3.8.19)

Jp is the angular momentum plus spin for the particle. Write J = Jp ⊗ 1 + 1 ⊗ Jf . Clearly, J = J(n, z) is defined for each (n, z) ∈ S2 × ℤ. S Proposition 3.81 (J-invariance of HPF ). If the polarization vectors are coherent in direction n, and φ̂ and V are rotation-invariant, then S −iϕJ S eiϕJ HPF e = HPF

for every ϕ ∈ ℝ. Proof. Write a♯ ((gf )) for a♯ (f ⊕ g). Notice that for a rotation-invariant f , eμ+

eiϕJf a∗ (fe−ik⋅x (

eμ+

)) e−iϕJf = a∗ (feiϕ(zΣ2 +nℓk ) e−ik⋅x ( −

eμ

eμ−

)) .

Since the polarization vectors are coherent, we have by (3.8.14), 3 −1 (R e+ )μ e+ )) = ∑ Rμν a♯ (fe−ikR x ( ν− )) , − eν (R e )μ ν=1

= a∗ (fe−iRk⋅x (

(3.8.20)

398 | 3 The Pauli–Fierz model by path measures where R = R (n, ϕ) = (Rμν )1≤μ,ν≤3 . By (3.8.20), we see that the field part transforms as eiϕJf Hrad e−iϕJf = Hrad ,

eiϕJf Aμ (x)e−iϕJf = (R A)μ (R −1 x), and the particle part as eiϕn⋅ℓx xμ e−iϕn⋅ℓx = (R x)μ ,

eiϕn⋅ℓx (−i∇x )μ e−iϕn⋅ℓx = (R (−i∇x ))μ , eiϕn⋅(1/2)σ σμ e−iϕn⋅(1/2)σ = (R σ)μ .

Together with all the identities above, we have 1 2 S S −iϕJ ̂ + Hrad + V(R x) = HPF . eiϕJ HPF e = (R (−i∇) − eR A(x)) 2 S From this proposition, it is clear that HPF with coherent polarization vectors has the symmetry

SU(2) × SOpart (3) × SOfield (3) × helicity.

(3.8.21)

Denote the set of half integers by ℤ1/2 = {w/2 | w ∈ ℤ}. For each (n, z) ∈ S2 × ℤ, notice that Spec(n ⋅ (ℓx + (1/2)σ)) = ℤ1/2 , Spec(n ⋅ Lf ) = ℤ, ℤ, Spec(Sf ) = { 0,

z ≠ 0, z = 0.

(3.8.22) (3.8.23) (3.8.24)

Thus for each (n, z) ∈ S2 × ℤ, Spec(J) = ℤ1/2

(3.8.25)

and we have the theorem below. Theorem 3.82. If the polarization vectors are coherent in direction n, and φ̂ and V are S rotation-invariant, then ℋS and HPF can be decomposed as ℋS = ⨁ ℋS (w), w∈ℤ1/2

S S HPF = ⨁ HPF (w). w∈ℤ1/2

(3.8.26)

Here ℋS (w) is the subspace spanned by eigenvectors of J associated with eigenvalue S S w ∈ ℤ1/2 and HPF (w) = HPF ⌈ℋS (w) .

3.8 The Pauli–Fierz model with spin

| 399

Proof. This follows from Proposition 3.81 and the fact that Spec(J) = ℤ1/2 . Next, we consider general polarization vectors, i. e., not necessarily coherent sets. The Pauli–Fierz Hamiltonians with different polarization vectors, however, are unitary equivalent. Denote the Pauli–Fierz Hamiltonian with polarization vectors e± by S HPF (e± ). Combining Proposition 3.6 and Theorem 3.82, we have the corollary below. Corollary 3.83 (Fiber decomposition of Pauli–Fierz Hamiltonian with spin; general polarization). Suppose that φ̂ and V are rotation-invariant, and e± are coherent polarizaS tion vectors. Let η± be arbitrary polarization vectors. Then HPF (η± ) is unitary equivalent S ± to ⊕w∈ℤ1/2 HPF (e , w). By using the symmetries of the Pauli–Fierz Hamiltonian, we can show the degenS eracy of ground states of HPF . Assume that V is rotation-invariant and the polarization ± vectors e are given by e+ (k) =

(−k2 , k1 , 0) √k12

+

k22

e− (k) = k̂ × e+ (k).

,

(3.8.27)

These are coherent in direction n3 and their helicity is zero. Let Λ : ℝ3 → ℝ3 be the flip defined by k1 k1 Λ (k2 ) = (−k2 ) . k3 k3 Consider the unitary operators L2 (ℝ3 ; ℂ2 ) → L2 (ℝ3 ; ℂ2 ): f f ∘Λ ũ : ( ) 󳨃→ ( ), g g∘Λ

f −f ∘ Λ u : ( ) 󳨃→ ( ). g g∘Λ

(3.8.28)

A computation gives kμ , μ = 1, 3, −1 u♯ kμ u♯ = { −kμ , μ = 2,

∇μ , −1 u♯ ∇μ u♯ = { −∇μ ,

where u♯ = u or u.̃ From this and e− (k) =

(−k3 k1 , −k2 k3 , k12 + k22 ) |k|√k12 + k22

,

we have for rotation-invariant f and g, eμ f μ = 1, 3, { {(eμ− g ), u ( − ) = { e+ f {−( μ ), μ = 2. eμ g − { eμ g −1

eμ+ f

+

μ = 1, 3, μ = 2,

400 | 3 The Pauli–Fierz model by path measures Then the second quantization Γ(u) of u induces the unitary operator on ℱrad and for rotation-invariant φ̂ we obtain Aμ (Λx),

μ = 1, 3,

Γ(u)−1 Aμ (x)Γ(u) = {

−Aμ (Λx),

μ = 2,

(3.8.29)

and Γ(u)−1 Hrad Γ(u) = Hrad .

(3.8.30)

Next, we consider the transformation on σ = (σ1 , σ2 , σ3 ) given by −σμ , μ = 1, 3,

σμ 󳨃→ σ2 σμ σ2 = {

σμ ,

μ = 2.

(3.8.31)

Under the identification L2 (ℝ3 ; ℂ2 ) ≅ ℂ2 ⊗ L2 (ℝ3 ), we define τ = σ2 ⊗ ũ : L2 (ℝ3 ; ℂ2 ) → L2 (ℝ3 ; ℂ2 ). This satisfies −σμ ⊗ f ∘ Λ−1 ,

τ−1 (σμ ⊗ f )τ = {

σμ ⊗ f ∘ Λ , −1

μ = 1, 3, μ = 2,

(3.8.32)

and τ−1 (σμ ⊗ ∇μ )τ = −σμ ⊗ ∇μ ,

μ = 1, 2, 3.

(3.8.33)

We finally define the unitary operator J : ℋS → ℋS by J = τ ⊗ Γ(u). Combining (3.8.29)–(3.8.33), for rotation-invariant φ̂ and V J−1 σμ (−i∇μ − Aμ )J = −σμ (−i∇μ − Aμ ), J−1 Hrad J = Hrad , J−1 VJ = V are obtained. From these relations, we can show the theorem below. S Theorem 3.84 (Reflection symmetry of HPF ). If φ̂ and V are rotation-invariant, and the ± S S polarization vectors e are given by (3.8.27), then HPF (z) and HPF (−z) are unitary equivalent.

Proof. Since e± is coherent in direction (0, 0, 1) and its helicity is zero, J is of the form J = (ℓx,3 + 21 σ3 ) ⊗ 1 + 1 ⊗ Lf,3 . It follows that J−1 JJ = −J.

3.8 The Pauli–Fierz model with spin

| 401

This implies that J maps ℋS (w) onto ℋS (−w). Furthermore, S J−1 HPF J=

1 S . (−σ ⋅ (−i∇ − eA))2 + V + Hrad = HPF 2

S S Thus J−1 HPF (w)J = HPF (−w) follows.

An interesting application of Theorem 3.84 is to estimate the multiplicity of bound S states of HPF . Corollary 3.85 (Degeneracy of bound states). Suppose that V and φ̂ are rotationS invariant. Let M be the multiplicity of eigenvalues of HPF . Then M is an even number. In particular, whenever a ground states exists, it is degenerate. S Proof. By the unitary equivalence, we may suppose that the polarization vectors of HPF S S S are given by (3.8.27). Thus HPF = ⨁w∈ℤ1/2 HPF (w). Let Ψ be a bound state of HPF with

S S eigenvalue a. Thus Ψ is a bound state of HPF (w) with some w ∈ ℤ1/2 , HPF (w)Ψ = aΨ. S S Theorem 3.84 implies HPF (w) ≅ HPF (−w), and thus there exists Φ ∈ ℋS (−w) such that S HPF (−w)Φ = aΦ. Thus the multiplicity of a is even.

3.8.5 Scalar representations As in the classical case (3.8.4) in order to construct the functional integral represenS tation of (F, e−tHPF G) with a scalar integrand, we introduce a two-valued spin variable S θ ∈ ℤ2 = {−1, +1} = {θ1 , θ2 } and redefine HPF in order to reduce it to a scalar operator. Since B̂ 1 e S HPF = (−i∇ − eA)̂ 2 + V + Hrad − ( ̂ 3 ̂ 2 2 B1 + iB2

B̂ 1 − iB̂ 2 ), −B̂ 3

our Hamiltonian can be regarded as an operator acting on L2 (ℝ3 × ℤ2 ) ⊗ L2 (Q ) and is defined by 1 ℤ (HPF2 F)(θ) = ( (−i∇ − eA)̂ 2 + V + Hrad + Ĥ d (θ)) F(θ) + Ĥ od (−θ)F(−θ) 2

F(+1) with θ ∈ ℤ2 for F = (F(−1) ) ∈ L2 (ℝ3 × ℤ2 ) ⊗ L2 (Q ), where each component is F(θ) ∈ 2 3 2 L (ℝ ) × L (Q ). Here, Ĥ d and Ĥ od denote the diagonal, respectively, off-diagonal parts

of the spin interaction explicitly given by ⊕

Ĥ d (θ) = ∫ Ĥ d (x, θ)dx, ℝ3

where

e Ĥ d (x, θ) = − θB̂ 3 (x), 2

Here, B̂ μ (x) is defined in (3.8.8).

⊕

Ĥ od (−θ) = ∫ Ĥ od (x, −θ)dx, ℝ3

e Ĥ od (x, −θ) = − (B̂ 1 (x) − iθB̂ 2 (x)) . 2

402 | 3 The Pauli–Fierz model by path measures Now we define the Pauli–Fierz Hamiltonian with spin 1/2 on L2 (ℝ3 ×ℤ2 )⊗L2 (Q ) instead of ℋS . Furthermore, we treat  and B̂ or Ĥ d and Ĥ od as not necessarily dependent vectors. This is the same idea as the one applied to Schrödinger operator Hℤ2 (a, b). Write 2

3

2

K = L (ℝ × ℤ2 ) ⊗ L (Q )

and its Euclidean version 2

3

2

KE = L (ℝ × ℤ2 ) ⊗ L (QE ).

Definition 3.86 (Pauli–Fierz Hamiltonian with spin 1/2 on K ). We define the Pauli– ℤ Fierz Hamiltonian with spin 1/2 on K by HPF2 . ℤ2

The key idea of constructing a functional integral representation of e−tHPF is to use the identity ⊕

2

3

K ≅ ∫ L (ℝ × ℤ2 )dμ.

(3.8.34)

Q

In other words, we regard K as the set of L2 (ℝ3 × ℤ2 )-valued L2 -functions on Q . We make the decomposition ⊕

HPF2 = ∫ K(ϕ)dμ(ϕ) + Hrad , ℤ

Q

where K(ϕ) is a self-adjoint operator on L2 (ℝ3 × ℤ2 ) for each ϕ ∈ Q . We construct a ℤ2

functional integral representation of e−tHPF through the functional integrals of e−tK(ϕ) and e−tHrad , and the Trotter product formula. In order to do that, we will use the identity ⊕

(F, e−t ∫Q K(ϕ)dμ G) = ∫ (F(ϕ), e−tK(ϕ) G(ϕ)) dμ(ϕ) Q

while we have already done this of (F(ϕ), e−tK(ϕ) G(ϕ)), ϕ ∈ Q , in the classical case (f , e−tHℤ2 (a) g) in (3.8.6). First, we make the fiber decomposition of the quantized radiation field  and the quantized magnetic field B̂ on Q . For each ϕ ∈ Q , define ⊕

 μ (ϕ) = ∫  μ (x, ϕ)dx, ℝ3

⊕

B̂ μ (ϕ) = ∫ B̂ μ (x, ϕ)dx, ℝ3

where ̃ − x), ϕ), Â μ (x, ϕ) = Â μ (φ(⋅

̃ − x), ϕ))μ . B̂ μ (x, ϕ) = (∇x × A(̂ φ(⋅

3.8 The Pauli–Fierz model with spin

| 403

̃ − x)⟩⟩. For each fiber ϕ ∈ Q , define the ̃ − x), ϕ) = ⟨⟨ϕ, ⨁3ν=1 δμν φ(⋅ Recall that  μ (φ(⋅ Hamiltonian K(ϕ) on L2 (ℝ3 × ℤ2 ) by 1 2 ̂ (K(ϕ)F)(x, θ) = ( (−i∇ − eA(ϕ)) + V + Ĥ d (ϕ)) F(x, θ) + Ĥ od (ϕ)F(x, −θ). 2

(3.8.35)

Here, ⊕

Ĥ d (ϕ) = ∫ Ĥ d (x, θ, ϕ)dx,

⊕

Ĥ od (ϕ) = ∫ Ĥ od (x, −θ, ϕ)dx,

ℝ3

ℝ3

where e Ĥ d (x, θ, ϕ) = − θB̂ 3 (x, ϕ), 2

e Ĥ od (x, −θ, ϕ) = − (B̂ 1 (x, ϕ) − iθB̂ 2 (x, ϕ)). 2

ℤ2 ℤ To prevent the off-diagonal part Ĥ od vanish, we introduce a regularization HPF,ε of HPF2 by

1 ℤ2 HPF,ε F(θ) = ( (−i∇ − eA)̂ 2 + V + Hrad + Ĥ d (θ))F(θ) + Ψε (Ĥ od (−θ))F(−θ), 2 where Ψε (X) = X + εψε (|X|) and ψε ∈ C0∞ (ℝ) is given by 1, |z| < ε/2, { { { ψε (z) = ψε (|z|) = {≤ 1, ε/2 ≤ |z| ≤ ε, { { |z| > ε. {0, Also, let Kε (ϕ) be the counterpart of K(ϕ) with Ĥ od (ϕ) replaced by Ψε (Ĥ od (ϕ)), i. e., 1 2 ̂ + V + Ĥ d (ϕ)) F(x, θ) + Ψε (Ĥ od (ϕ))F(x, −θ). (Kε (ϕ)F)(x, θ) = ( (−i∇ − eA(ϕ)) 2 Recall that θz = (−1)z for z ∈ ℤ and (qt )t≥0 = (Bt , θNt )t≥0 is an (ℝ3 × ℤ2 )-valued random process on X × 𝒮 . Lemma 3.87. Suppose that V ∈ ℛKato . If φ̃ ∈ C0∞ (ℝ3 ), then for every ϕ ∈ Q , Kε (ϕ) is self-adjoint on D(−Δ) ⊗ ℤ2 and for g ∈ L2 (ℝ3 × ℤ2 ), t

(e−tKε (ϕ) g)(x, θα ) = et 𝔼x,α [e− ∫0 V(Bs )ds eZt (ϕ,ε) g(qt )], where Zt (ϕ, ε)

t

t

t+

0

0

0

̃ − Bs ), ϕ)dBs − ∫ Ĥ d (Bs , θNs , ϕ)ds + ∫ log (−Ψε (Ĥ od (Bs , −θNs− , ϕ))) dNs . = −ie ∫ A(̂ φ(⋅

404 | 3 The Pauli–Fierz model by path measures ̃ − x)⟩⟩ ∈ Cb∞ (ℝ3x ), for evProof. As φ̃ ∈ C0∞ (ℝ3 ), we have  μ (ϕ) = ⟨⟨ϕ, ⨁3ν=1 δμν φ(⋅ ery ϕ ∈ Q . Thus Kε (ϕ) is the Pauli operator with spin 1/2 with a smooth bounded ̂ vector potential A(ϕ), and the off-diagonal part is perturbed by the bounded oper̂ ator εψε (Hod (ϕ)). Hence it is self-adjoint on D(−Δ) ⊗ ℤ2 and its functional integral representation follows from the Feynman–Kac formula for the Schrödinger operator with vector potentials and spin: (f , e−tHℤ2 (a) g) = et ∑α=1,2 ∫ℝ3 𝔼x,α [f (q0 )g(qt )eZt ]dx in (3.8.6). Next, we define the operator Kε (A)̂ on K through Kε (ϕ) and the constant fiber direct integral representation (3.8.34) of K . Take φ̃ ∈ C0∞ (ℝ3 ) and define the selfadjoint operator Kε (A)̂ on K by ⊕

Kε (A)̂ = ∫ Kε (ϕ)dμ(ϕ), Q

̂ that is, (Kε (A)F)(ϕ) = Kε (ϕ)F(ϕ) with domain { ̂ = F∈K D(Kε (A)) { {

󵄨󵄨 󵄨󵄨 } 󵄨󵄨 2 󵄨󵄨 ∫ ‖Kε (ϕ)F(ϕ)‖L2 (ℝ3 ×ℤ2 ) dμ(ϕ) < ∞ } . 󵄨󵄨 󵄨󵄨 Q }

Then we are able to define the self-adjoint operator Kε by Kε = Kε (A)̂ +̇ Hrad . In what follows, we construct the functional integral representation of e−tKε and show ℤ2

that e−tKε = e−tHPF,ε . Let us define the finite particle subspace of L2 (Qβ ) by ∞

(k) = 0 for any k ≥ m} , L2fin (Qβ ) = {⨁L2n (Qβ ) ∋ {Φ(n) }∞ n=0 | ∃m such that Φ n=0

where Qβ = Q or QE , and define the dense subspace by K

∞

= C0∞ (ℝ3 × ℤ2 ) ⊗̂ L2fin (Q )

and its Euclidean version by ∞

KE

= C0∞ (ℝ3 × ℤ2 ) ⊗̂ L2fin (QE ).

Lemma 3.88. It follows that ℤ2

ℤ2

s-lim e−tHPF,ε = e−tHPF . ε→0

(3.8.36)

2 Suppose that φ̃ ∈ C0∞ (ℝ3 ). Then HPF,ε = Kε and in particular it follows that

ℤ

ℤ2

(F, e−tHPF G) = lim(F, e−tKε G). ε↓0

(3.8.37)

3.8 The Pauli–Fierz model with spin

| 405

2 2 Proof. It is seen that Kε = HPF,ε on K ∞ , implying that Kε = HPF,ε as a self-adjoint

ℤ

operator since K ∞ is a core of

K

∞

ℤ

ℤ2 HPF,ε .

Moreover,

is a common core of the sequence

ℤ2 HPF,ε

ℤ2 (HPF,ε )ε≥0 .

→

ℤ HPF2

on K ∞ as ε → 0 and

Thus (3.8.36) follows.

ℤ2

By this lemma to have a functional integral representation of (F, e−tHPF G) for any φ̃ defined in Assumption 3.4, it suffices to construct a functional integral representation of the right-hand side of (3.8.37) and to take approximation arguments on φ.̂ To obtain a functional integral representation of e−tKϵ , we apply the Trotter product formula, i. e., e−tKε = s-lim(e−(t/n)Kϵ (A) e−(t/n)Hrad )n . ̂

n→∞

By the property of Js J∗t = e−|s−t|Hrad for s, t ∈ ℝ, it is reduced to n−1

e−tKε = s-lim J∗0 ( ∏ Jti/n e−(t/n)Kε (A) J∗ti/n )Jt . n→∞

̂

i=0

From this, we can construct a functional integral representation of e−tKε . Define the Euclidean version of Ĥ d (x, θ) and Ĥ od (x, −θ) by e ̃ − x)) , Ĥ E,d (x, θ, s) = − θB̂ E3 (js φ(⋅ 2 e ̃ − x)) − iθB̂ E2 (js φ(⋅ ̃ − x))). Ĥ E,od (x, −θ, s) = − (B̂ E1 (js φ(⋅ 2

(3.8.38) (3.8.39)

3.8.6 Fock representations We will use Fock representations to construct functional integral representations. Here, we review them for reader’s convenience. In Fock representation, Ĥ d and Ĥ od can be represented in terms of annihilation operators and creation operators in ℱrad as 1 ̃̂ ik⋅x /√ω, j)} , ̂ −ik⋅x /√ω, j) − a(Sj φe Ĥ d (x, θ) ≅ ∑ {a∗ (Sj φe √2 j=±

1 ̃̂ ik⋅x /√ω, j)} , ̂ −ik⋅x /√ω, j) − a(Tj φe Ĥ od (x, −θ) ≅ ∑ {a∗ (Tj φe √2 j=± where

e j Sj = Sj (θ) = − θη3 , 2

e j j Tj = Tj (θ) = − (η1 − iθη2 ). 2

Since ηj (k) = −ik × ej (k) and |ηjμ (k)| ≤ |k|,

μ = 1, 2, 3,

(3.8.40)

406 | 3 The Pauli–Fierz model by path measures we have bounds which are used frequently in this section: 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 ∑ a∗ (Sj φ̂ e−ik⋅x , j) Φ󵄩󵄩󵄩 ≤ |e|a‖(N + 1)1/2 Φ‖, 󵄩󵄩 󵄩󵄩 √ω 󵄩󵄩j=± 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 ∑ a (Sj φ̂ e−ik⋅x , j) Φ󵄩󵄩󵄩 ≤ |e|a‖N 1/2 Φ‖, 󵄩󵄩 󵄩󵄩 √ω 󵄩󵄩j=± 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 󵄩󵄩 ∑ a∗ (Tj φ̂ e−ik⋅x , j) Φ󵄩󵄩󵄩 ≤ 2|e|a‖(N + 1)1/2 Φ‖, 󵄩󵄩 󵄩󵄩 √ω 󵄩󵄩 󵄩󵄩j=± 󵄩󵄩 󵄩󵄩 φ̂ −ik⋅x 󵄩󵄩 󵄩󵄩󵄩 e , j) Φ󵄩󵄩󵄩 ≤ 2|e|a‖N 1/2 Φ‖. 󵄩󵄩 ∑ a (Tj √ω 󵄩󵄩󵄩 󵄩󵄩󵄩j=± ̂ Here, N denotes the number operator in L2 (Q ), and a = ‖|k|φ/√ω‖/2. Thus we have ‖Ĥ d (x, θ)Φ‖ ≤ √2|e|a‖(N + 1)1/2 Φ‖,

‖Ĥ od (x, −θ)Φ‖ ≤ 2√2|e|a‖(N + 1)1/2 Φ‖. Moreover, we give a Fock representation of Euclidean versions. Let aE and a∗E be the annihilation operator and the creation operator in ℱb (L2 (ℝ4 × {±})), respectively. Then φ̃̂ φ̂ 1 )e−ik⋅x , j) − aE (Sj (jŝ )eik⋅x , j)} , Ĥ E,d (x, θ, s) ≅ ∑ {a∗E (Sj (jŝ √2 j=± √ω √ω φ̂ φ̃̂ 1 Ĥ E,od (x, −θ, s) ≅ )e−ik⋅x , j) − aE (Tj (jŝ )eik⋅x , j)} . ∑ {a∗E (Tj (jŝ √2 j=± √ω √ω Here, jŝ = F4 js F3−1 : L2 (ℝ3 ) → L2 (ℝ4 ), s ∈ ℝ, is the family of isometries such that jŝ jt̂ = e−|s−t|ω , where F3 and F4 are the Fourier transform on L2 (ℝ3 ) and on L2 (ℝ4 ), respectively. Hence it follows that jŝ f ̂ = ĵ sf for f ∈ L2 (ℝ3 ). Then test functions φ̂ −ik⋅x Sj (jŝ )e , √ω

φ̂ −ik⋅x Tj (jŝ )e ∈ L2 (ℝ4(k0 ,k) ), √ω

j = ±,

are explicitly given by j

Sj (jŝ

−isk −ik⋅x ̂ η3 (k)φ(k) φ̂ e e 0 )e−ik⋅x = − θ , √ω 2 √π√ω(k)2 + |k |2

Tj (jŝ

φ̂ e e )e−ik⋅x = − θ √ω 2

0

−isk0 −ik⋅x

j (η1 (k)

j ̂ − iθη2 (k))φ(k)

√π√ω(k)2 + |k0 |2

.

3.8 The Pauli–Fierz model with spin

| 407

From these expressions, we see that 󵄨󵄨 󵄨 ̂ |k||φ(k)| |e| 󵄨󵄨Sj (jŝ φ̂ )e−ik⋅x 󵄨󵄨󵄨 ≤ |e| ̂ |φ(k)|, ≤ 󵄨󵄨 󵄨 󵄨 √ω 2√π √ω(k)2 + |k |2 2√π 0 󵄨󵄨 󵄨 ̂ |k||φ(k)| |e| 󵄨󵄨Tj (jŝ φ̂ )e−ik⋅x 󵄨󵄨󵄨 ≤ |e| ̂ ≤ |φ(k)|. 󵄨󵄨 󵄨󵄨 √π √ω √π 2 2 √ω(k) + |k0 |

We also have bounds: 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 󵄩󵄩 ∑ a∗ (Sj (jŝ φ̂ )e−ik⋅x , j) Φ󵄩󵄩󵄩 ≤ |e|a‖(N + 1)1/2 Φ‖, E 󵄩󵄩 󵄩󵄩 √ω 󵄩󵄩 󵄩󵄩j=± 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 ∑ aE (Sj (jŝ φ̂ )e−ik⋅x , j) Φ󵄩󵄩󵄩 ≤ |e|a‖N 1/2 Φ‖, 󵄩󵄩 󵄩󵄩 √ω 󵄩󵄩j=± 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 󵄩󵄩 ∑ a∗ (Tj (jŝ φ̂ )e−ik⋅x , j) Φ󵄩󵄩󵄩 ≤ 2|e|a‖(N + 1)1/2 Φ‖, E 󵄩󵄩 󵄩󵄩 √ω 󵄩󵄩 󵄩󵄩j=± 󵄩󵄩 󵄩󵄩 φ̂ 󵄩󵄩 󵄩󵄩󵄩 )e−ik⋅x , j) Φ󵄩󵄩󵄩 ≤ 2|e|a‖N 1/2 Φ‖. 󵄩󵄩 ∑ aE (Tj (jŝ 󵄩󵄩 󵄩󵄩 √ω 󵄩 󵄩j=± Hence we have ‖Ĥ E,d (x, θ, s)Φ‖ ≤ √2|e|a‖(N + 1)1/2 Φ‖,

‖Ĥ E,od (x, −θ, s)Φ‖ ≤ 2√2|e|a‖(N + 1)1/2 Φ‖.

3.8.7 Preparation of functional integral representations ℤ2

To construct a functional integral representation of (F, e−tHPF G), we prepare some technical lemmas. We can predict the form of a functional integral representation ℤ2

(F, e−tHPF G) from the classical case (3.8.6), which is unfortunately rather complicated than the spinless case (F, e−tHPF G). To avoid complicated computations, we introduce self-adjoint operators L and LR for R > 0 which satisfy that (|F|, e−tLR |G|)| ↑ (|F|, e−tL |G|)

as R ↑ ∞

and diamagnetic type inequality ℤ2

|(F, e−tHPF G)| ≤ (|F|, e−tL |G|). Applying this inequality, we can avoid technical difficulties to construct a functional ℤ2

integral representation of (F, e−tHPF G).

408 | 3 The Pauli–Fierz model by path measures We introduce two cutoff functions for R > 0. Let χR− ∈ C ∞ (ℝ) be defined by: (1) χR− (x) = x for x > −R + 1; (2) −R ≤ χR− (x) ≤ −R + 1 for −R ≤ x ≤ −R + 1; (3) χR− (x) = −R for x < −R. We also define χR+ ∈ C ∞ (ℝ) by: (1) χR+ (x) = R for x > R; (2) R − 1 ≤ χR+ (x) ≤ R for R − 1 ≤ x ≤ R; (3) χR+ (x) = x for x < R − 1. Furthermore, we suppose a uniform Lipschitz continuity of both χR− and χR+ such that |χR± (x) − χR± (y)| ≤ c|x − y| with some c for any x, y ∈ ℝ. We define Ψε (|Ĥ od (−θ)|)R and Ĥ d (θ)R by Ψε (|Ĥ od (−θ)|)R = χR+ (Ψε (|Ĥ od (−θ)|)) , Ĥ d (θ)R =

χR− (Ĥ d (θ)) .

(3.8.41) (3.8.42)

Hence Ĥ d (θ)R is bounded below and Ψε (|Ĥ od (−θ)|)R is bounded. Through the fiber direct integral, Kε is defined. We also define another useful self-adjoint operator in a similar manner to Kε . Let φ̃ ∈ C0∞ (ℝ3 ). We set for each ϕ ∈ Q , (L󸀠R (ϕ)F)(θ) = (Hp + Ĥ d (ϕ, θ)R )F(θ) − Ψε (|Ĥ od (ϕ, −θ)|)R F(−θ) for each R > 0, and (L󸀠 (ϕ)F)(θ) = (Hp + Ĥ d (ϕ, θ))F(θ) − Ψε (|Ĥ od (ϕ, −θ)|)F(−θ). Then L󸀠R (ϕ) and L󸀠 (ϕ) are self-adjoint and we define ⊕

L󸀠 = ∫ L󸀠 (ϕ)dμ(ϕ), Q

⊕

L󸀠R = ∫ L󸀠R (ϕ)dμ(ϕ). Q

We also define LR = L󸀠R +̇ Hrad for each R > 0, and L = L󸀠 +̇ Hrad . Actually we can show that both LR and L are self-adjoint on DPF , and (L󸀠R F)(θ) = (Hp + Ĥ d (θ)R )F(θ) − Ψε (|Ĥ od (−θ)|)R F(−θ)

3.8 The Pauli–Fierz model with spin

| 409

for each R > 0, and (L󸀠 F)(θ) = (Hp + Ĥ d (θ))F(θ) − Ψε (|Ĥ od (−θ)|)F(−θ). Furthermore, in the Fock representation LR and L are given by LR = Hp + Hrad − (

( e2 B̂ 3 )R

√B̂ 21 + B̂ 22 ) Ψε ( |e| 2

√B̂ 21 + B̂ 22 ) Ψε ( |e| 2

R

L = Hp + Hrad − (

e ̂ B 2 3

√B̂ 21 + B̂ 22 ) Ψε ( |e| 2

R) ,

−( e2 B̂ 3 )R

√B̂ 21 + B̂ 22 ) Ψε ( |e| 2 − e2 B̂ 3

).

Remark 3.89. We give a remark on |Ĥ od (−θ)|. In the Fock representation, we have |Ĥ od (−θ)| = |e| √B̂ 2 + B̂ 2 which leads that |Ĥ od (−θ)| is independent of θ. 2

1

2

The family of self-adjoint operators LR , R > 0, have also a common core K ∞ and limR→∞ LR F = KF for F ∈ K ∞ . Hence e−tLR → e−tL 󸀠

strongly as R → ∞. The functional integral representation of e−tLR (ϕ) can be constructed by (3.8.6) and that of e−tLR can be done by the Trotter product formula. Lemma 3.90. Assume that V ∈ L∞ (ℝ3 ) and φ̃ ∈ C0∞ (ℝ3 ). Let F, G ∈ KE . Then t

∑ ∫ 𝔼x,α [e− ∫0 V(Br )dr (F(q0 ), Es eLt (ε,s) Es G(qt ))] dx

α=1,2

ℝ3

is finite and t

(F, Js e−tLR J∗s G) = et ∑ ∫ 𝔼x,α [e− ∫0 V(Br )dr (F(q0 ), Es eLt (ε,s) Es G(qt ))] dx. 󸀠

α=1,2

(3.8.43)

ℝ3

Here, t

t+

0

0

Lt (ε, s) = − ∫ Ĥ E,d (Br , θNr , s)R dr + ∫ log Ψε (|Ĥ E,od (Br , −θNr , s)|)R dNr . Here, Ĥ E,d (x, θ, s) and Ĥ E,od (x, −θ, s) are given by (3.8.38) and (3.8.39), respectively, and truncated functions Ĥ E,d (Br , θNr , s)R and Ψε (|Ĥ E,od (x, −θ, s)|)R are defined in a similar way to (3.8.41) and (3.8.42), respectively. Proof. Let us set t

VM = sup 𝔼x [e−2 ∫0 V(Bs )ds ] < ∞. x∈ℝ3

410 | 3 The Pauli–Fierz model by path measures Notice that the right-hand side of (3.8.43) is finite, since Ĥ d (x, θ, s)R is bounded below and Ψε (|Ĥ od (x, −θ, s)|)R is bounded; eLt (ε,s) is a bounded function. For each (x, α, w, τ) ∈ ℝ3 × {1, 2} × X × 𝒮 , we have |(F(q0 ), Es eLt (ε,s) Es G(qt ))| ≤ ‖F(q0 )‖‖G(qt )‖‖eLt (ε,s) ‖. Here, ‖eLt (ε,s) ‖ is the operator norm of bounded operator eLt (ε,s) on L2 (QE ). Then it follows that t

|RHS (3.8.43)| ≤ et ∑ ∫ 𝔼x,α [e− ∫0 V(Br )dr ‖F(q0 )‖‖G(qt )‖‖eLt (ε,s) ‖] dx. α=1,2

(3.8.44)

ℝ3

We will also prove in Lemma 3.91 below that there exists a random variable At on (𝒮 , ℬ𝒮 , P) such that: (1) ‖eLt (ε,s) ‖ ≤ At ; (2) At is independent of (x, α, w) ∈ ℝ3 × {1, 2} × X ; (3) 𝔼P [A2t ] < ∞. By (1), (2) and (3) above and (3.8.44), 1/2 |RHS (3.8.43)| ≤ ‖G‖‖F‖VM (𝔼P [A2t ])1/2 < ∞.

(3.8.45)

Next, we prove the equality (3.8.43). Note that L󸀠R is defined by a direct fiber integral representation and by Lemma 3.87, we have t

(J∗s F, e−tLR J∗s G) = et ∑ ∫ 𝔼x,α [e− ∫0 V(Br )dr 𝔼μ [(J∗s F)(ϕ, q0 )eLt (ϕ,ε) (J∗s G)(ϕ, qt )]] dx. 󸀠

α=1,2

ℝ3

Here, we used Fubini’s lemma and t

t+

0

0

Lt (ϕ, ε) = − ∫ Ĥ d (ϕ, Bs , θNs )R ds + ∫ log Ψε (|Ĥ od (ϕ, Bs , −θNs )|)R dNs . Let t

t+

0

0

Lt (ε) = − ∫ Ĥ d (Bs , θNs )R ds + ∫ log Ψε (|Ĥ od (Bs , −θNs )|)R dNs . Given that J∗s F, eLt (ε) J∗s G(qt ) ∈ L2 (Q ), we rewrite as t

(J∗s F, e−tLR J∗s G) = et ∑ ∫ 𝔼x,α [e− ∫0 V(Br )dr (F(q0 ), Js eLt (ε) J∗s G(qt ))] dx. 󸀠

α=1,2

ℝ3

We can see that Js eLt (ε) J∗s = Es eLt (ε,s) Es by a limiting argument, leading to (3.8.43). Then the proof can be completed.

3.8 The Pauli–Fierz model with spin

| 411

Now we have to prove (1), (2) and (3) used in the proof of Lemma 3.90. Lemma 3.91. Operator eLt (ε,s) is bounded, and there exists a random variable At = At (τ) on (𝒮 , ℬ𝒮 , P) satisfying (1)–(3) in the proof of Lemma 3.90. t+

Proof. Since R < ∞, we have |eLt (ε,s) | ≤ etR e∫0 log RdNs . For each τ ∈ S, the number of jumps of map s 󳨃→ Ns (τ) for 0 ≤ s ≤ t is denoted by N = N(τ). Hence t+ ∫0 log RdNs = log RN and then |eLt (ε,s) | ≤ etR RN . Set At = At (τ) = etR RN(τ) . Then 𝔼P [A2t ] = e2tR ∑∞ N=0

t N R2N −t e N!

2

= et(R +2R−1) < ∞. Then At satisfies (1), (2) and (3).

In the next lemma, we construct a functional integral representation of e−tLR for R > 0. Let t

t+

0

0

YR = − ∫ Ĥ E,d (Bs , θNs , s)R ds + ∫ log Ψε (|Ĥ E,od (Bs , −θNs− , s)|)R dNs . Lemma 3.92. Suppose that V ∈ L∞ (ℝ3 ) and let φ̃ ∈ C0∞ (ℝ3 ). Then for every t ≥ 0 and all F, G ∈ K it follows that J∗0 eYR Jt is a bounded operator on L2 (Q ) for every (x, α, w, τ) ∈ ℝ3 × {1, 2} × X × 𝒮 and t

(F, e−tLR G) = et ∑ ∫ 𝔼x,α [e− ∫0 V(Bs )ds (F(q0 ), J∗0 eYR Jt G(qt ))] dx. α=1,2

(3.8.46)

ℝ3

Proof. In a similar manner to the proof of Lemma 3.91 for each τ ∈ S, the number of jumps of map s 󳨃→ Ns (τ) for 0 ≤ s ≤ t is denoted by N = N(τ). Fix τ ∈ 𝒮 . Hence by virtue of cutoff parameter R > 0 it can be seen that ‖J∗0 eYR Jt Φ‖ ≤ etR RN(τ) ‖Φ‖, which implies that J∗0 eYR Jt is bounded for every (x, α, w, τ) ∈ ℝ3 × {1, 2} × X × 𝒮 . By the Trotter product formula, we have t

t

n−1

n

t

(F, e−tLR G) = lim (F, (e− n LR e− n Hrad ) G) = lim (F, J∗0 (∏ Jit/n e− n LR J∗it/n ) Jt G) . 󸀠

n→∞

n→∞

󸀠

i=0

By the Markov property of Js J∗s = Es and Lemma 3.90, we can see that t

(F, e−tLR G) = lim ∑ ∫ 𝔼x,α [e− ∫0 V(Bs )ds (F(q0 ), J∗0 eYR (n) Jt G(qt ))] dx, n→∞

α=1,2

(3.8.47)

ℝ3

where n

YR (n) = − ∑

it/n

∫

i=1 (i−1)t/n

n

Ĥ E,d (Bs , θNs , it/n)R ds + ∑

it/n+

∫

i=1 (i−1)t/n

log Ψε (|Ĥ E,od (Bs , −θNs− , it/n)|)R dNs .

Then |(F(q0 ), J∗0 eYR (n) Jt G(qt ))| ≤ ‖F(q0 )‖‖G(qt )‖etR RN(τ)

(3.8.48)

412 | 3 The Pauli–Fierz model by path measures and t

∑ ∫ 𝔼x,α [e− ∫0 V(Bs )ds ‖F(q0 )‖‖G(qt )‖etR RN(τ) ] dx

α=1,2

ℝ3

t

2

1/2 1/2 2 (R +2R−1) ≤ ‖F‖‖G‖VM (𝔼P [e2tR R2N ])1/2 ≤ ‖F‖‖G‖VM e < ∞.

(3.8.49)

Since the right-hand side above is independent of n, it is sufficient to prove (3.8.46) for F, G ∈ K ∞ , hence we suppose that F, G ∈ K ∞ in what follows in this proof. We can see that 󵄩󵄩󵄩 n it/n 󵄩󵄩󵄩 n it/n 󵄩 󵄩󵄩 󵄩󵄩∑ ∫ Ĥ (B , θ , it/n) dsF 󵄩󵄩󵄩 ≤ ∑ ∫ 󵄩󵄩󵄩Ĥ (B , θ , it/n)F 󵄩󵄩󵄩 ds 󵄩󵄩 󵄩󵄩 E,d s Ns 󵄩󵄩 󵄩󵄩 R E,d s Ns 󵄩󵄩 i=1 󵄩󵄩i=1 󵄩󵄩 󵄩󵄩 (i−1)t/n (i−1)t/n ̂ √ω‖‖(N + 1)1/2 F‖ ≤ tC‖|k|φ/ and similarly 󵄩󵄩 t 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 ̂ 󵄩 ̂ √ω‖‖(N + 1)1/2 F‖ 󵄩󵄩∫ HE,d (Bs , θNs , s)R dsF 󵄩󵄩󵄩 ≤ tC‖|k|φ/ 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩0 with some constant C. By the uniform Lipschitz continuity of function χR− , we see that |χR− (x) − χR− (y)| ≤ c|x − y| with some constant c, we can also see that 󵄩󵄩 󵄩󵄩 t 󵄩󵄩 n it/n 󵄩󵄩 󵄩󵄩 󵄩󵄩 ̂ ̂ 󵄩󵄩󵄩(∑ ∫ HE,d (Bs , θNs , it/n)R ds − ∫ HE,d (Bs , θNs , s)R ds) F 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 i=1 󵄩󵄩 0 (i−1)t/n 󵄩󵄩 n

≤ c∑

it/n

∫

i=1 (i−1)t/n n

󵄩󵄩 ̂ 󵄩 󵄩󵄩|HE,d (Bs , θN , it/n) − Ĥ E,d (Bs , θN , s)|F 󵄩󵄩󵄩 ds s s 󵄩 󵄩

it/n

≤ c∑( ∫ i=1

(i−1)t/n

1/2

󵄩󵄩 ̂ 󵄩2 󵄩󵄩|HE,d (Bs , θN , it/n) − Ĥ E,d (Bs , θN , s)|F 󵄩󵄩󵄩 ds) s s 󵄩 󵄩

√t . √n

It can be seen that t 󵄩󵄩 ̂ 󵄩2 󵄩󵄩|HE,d (Bs , θN , it/n) − Ĥ E,d (Bs , θN , s)|F 󵄩󵄩󵄩 ≤ c󸀠 ‖|k|φ‖ ̂ 2 ‖(N + 1)1/2 F‖2 s s 󵄩 󵄩 n with some constant c󸀠 . Then 󵄩󵄩 󵄩󵄩 t 󵄩󵄩 󵄩󵄩 n it/n 󵄩 󵄩󵄩 󵄩󵄩(∑ ∫ Ĥ E,d (Bs , θN , it/n)R ds − ∫ Ĥ E,d (Bs , θN , s)R ds) F 󵄩󵄩󵄩 s s 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 i=1 (i−1)t/n 0 󵄩 󵄩 n t √t t √t ̂ ̂ ≤ cc󸀠 ∑ ‖|k|φ‖ ‖(N + 1)1/2 F‖ = cc󸀠 ‖|k|φ‖ ‖(N + 1)1/2 F‖ → 0 √n n√n i=1

3.8 The Pauli–Fierz model with spin

| 413

as n → ∞. Thus it can be straightforwardly seen that it/n

n

exp (− ∑

t

∫

i=1 (i−1)t/n

Ĥ E,d (Bs , θNs , it/n)R ds) F → exp (− ∫ Ĥ E,d (Bs , θNs , s)R ds) F (3.8.50) 0

by the expansion eX = ∑∞ j=0

Xj j!

on K ∞ for

t

X = − ∫ Ĥ E,d (Bs , θNs , s)R ds or 0

n

X = −∑

it/n

∫

i=1 (i−1)t/n

Ĥ E,d (Bs , θNs , s)R ds.

it/n+

Next, we consider exp(∑ni=1 ∫(i−1)t/n log Ψε (|Ĥ E,od (Bs , −θNs− , it/n)|)R dNs ). Points of discontinuity of map r 󳨃→ Nr (τ) are denoted by s1 = s1 (τ), . . . , sN = sN (τ) ∈ (0, ∞). For sufficiently large n, the number of discontinuous points located in interval (t(i − 1)/n, ti/n] is at most one. Then by taking n large enough and denoting (n(si ), n(si ) + t/n] for the interval containing si , we get n

it/n+

exp (∑

∫

N

i=1 (i−1)t/n

log Ψε (|Ĥ E,od (Bs , −θNs− , it/n)|)R dNs ) = ∏ χR+ (|ϕi | + εψε (|ϕi |)), i=1

where ϕi = Ĥ E,od (Bsi , −θNs − , n(si )). By the Lipschitz continuity of χR+ and ψε , we have i

|Ψε (x)R − Ψε (y)R | ≤ c|Ψε (x) − Ψε (y)| ≤ c|x − y + εψε (x) − εψ(y)| ≤ c|x − y| + cε|ψε (x) − ψ(y)| ≤ c|x − y| + cc󸀠 ε|x − y|

with some constants c and c󸀠 . Hence 󵄩󵄩 󵄩 󵄩󵄩(Ψε (|Ĥ E,od (Bs , −θN , n(si ))|)R − Ψε (|Ĥ E,od (Bs , −θN , si )|)R ) F 󵄩󵄩󵄩 s− s− 󵄩 󵄩 ≤ C‖ (|Ĥ E,od (Bs , −θNs− , n(si ))| − |Ĥ E,od (Bs , −θNs− , si )|) F‖. Note that ‖ (|Ĥ E,od (Bs , −θNs− , n(si ))| − |Ĥ E,od (Bs , −θNs− , si )|) F‖2

≤ 𝔼μ [|Ĥ E,od (Bs , −θNs− , n(si )) − Ĥ E,od (Bs , −θNs− , si )|2 |F|2 ] ̂ 2 |n(si ) − si |‖(N + 1)1/2 F‖2 ≤ C‖|k|φ‖

with some C > 0. Clearly, n(si ) → si as n → ∞. Then lim Ĥ E,od (Bsi , −θNs − , n(si )) = Ĥ E,od (Bsi , −θNs − , si )

n→∞

i

i

414 | 3 The Pauli–Fierz model by path measures on K ∞ and we have n

lim exp (∑

n→∞

it/n+

∫

i=1 (i−1)t/n

N

log Ψε (|Ĥ E,od (Bs , −θNs− , it/n)|)R dNs )

= ∏ Ψε (|Ĥ E,od (Bsi , −θNs − , si )|)R .

(3.8.51)

i

i=1

Then by (3.8.50) and (3.8.51), we can see that for each (x, α, w, τ) ∈ ℝ3 × {1, 2} × X × 𝒮 , (F(q0 ), J∗0 eYR (n) Jt G(qt )) → (F(q0 ), J∗0 eYR Jt G(qt )) as n → ∞. Together with (3.8.48) and (3.8.49), the Lebesgue dominated convergence theorem yields that t

∑ ∫ 𝔼x,α [e− ∫0 V(Br )dr (F(q0 ), J∗0 eYR (n) Jt G(qt ))] dx

α=1,2

ℝ3

t

→ ∑ ∫ 𝔼x,α [e− ∫0 V(Br )dr (F(q0 ), J∗0 eYR Jt G(qt ))] dx α=1,2

ℝ3

for F, G ∈ K ∞ . Then the proof is complete. Let us define t

t+

Y = − ∫ Ĥ E,d (Bs , θNs , s)ds + ∫ log Ψε (|Ĥ E,od (Bs , −θNs− , s)|)dNs . 0

(3.8.52)

0

We already see that J∗0 eYR Jt is bounded, but it is not trivial to see that J∗0 eY Jt is bounded. Lemma 3.93. For each (x, α, w, τ) ∈ ℝ3 × {1, 2} × X × 𝒮 , J∗0 eY Jt is bounded on L2 (Q ). In particular, J∗0 eY Jt Ψ ∈ L2 (Q ) for Ψ ∈ L2 (Q ). Proof. We introduce proofs for massless cases and massive cases. For massive cases, we can apply hypercontractivity of e−tHrad , but for the massless case we use the Baker– Campbell–Hausdorff formula. (Massive case) Suppose that ω(k) = √|k|2 + ν2 with ν > 0. To see the boundedness ∗ Y of J0 e Jt , by the hypercontractivity of e−tHrad it is sufficient to show that eY ∈ ⋂ Lp (QE ).

(3.8.53)

p>0

It is known that ‖J∗0 eY Jt ‖ ≤ ‖eY ‖Lq (QE ) ,

q=

2 . 1 − e−νt

3.8 The Pauli–Fierz model with spin

| 415

(3.8.53) is equivalent to epY ∈ L1 (QE )

for any p ∈ ℕ.

(3.8.54)

We shall prove (3.8.54). Note that t 󵄩󵄩2 󵄩󵄩2 󵄩󵄩 t 󵄩󵄩 p ̂ ̂ ‖e 2 Y ‖2L1 (QE ) ≤ 󵄩󵄩󵄩󵄩e−p ∫0 HE,d (Br ,θNr ,r)dr 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩ep ∫0 | log Ψε (|HE,od (Br ,−θNr− ,r)|)|dNr 󵄩󵄩󵄩󵄩 . 󵄩 󵄩 󵄩 󵄩

We estimate the right-hand side of this expression. It follows that t

t

0

0

e ̃ − Br )dr) . ∫ Ĥ E,d (Br , θNr , r)dr = B̂ E3 (− ∫ θNr jr φ(⋅ 2 Since B̂ Eμ (f ) is a Gaussian random variable with mean zero and covariance, 𝔼μE [B̂ Eμ (f )B̂ Eν (g)] =

1 2 ̂ δμν (k)dk, ∫ f ̂(k)g(k)|k| 2 ℝ4

we have

t

‖e−p ∫0 HE,d (Br ,θNr ,r)dr ‖2 ≤ c1 , ̂

where c1 = exp (t 2

p2 e 2 ̂ √ω‖2 ) ‖|k|φ/ 4

and c1 is thus independent of (x, θ) ∈ ℝ3 × ℤ2 . Next, consider the off-diagonal spin t ̂ ̃ − Bt )) for notational interaction ‖ep ∫0 | log Ψε (|HE,od (Br ,−θNr− ,r)|)|dNr ‖. Write B̂ E (t) = B̂ E (jt φ(⋅ μ

μ

convenience. For each τ ∈ 𝒮 , there exists N = N(τ) ∈ ℕ and points of discontinuity of r 󳨃→ Nr (τ), s1 = s1 (τ), . . . , sN = sN (τ) ∈ (0, ∞), such that t

N

0

j=1

∫ | log Ψε (|Ĥ E,od (Br , −θNr− , r)|)|dNr = ∑ | log Ψε (|Ĥ E,od (Bsj , −θNs − , sj )|)|. j

Since |Ψε (w)| ≤ |w| + ε, we have 󵄩󵄩 p ∫t | log Ψ (|Ĥ (B ,−θ ,r)|)|dN 󵄩󵄩2 N |e| 2 2 2 ̂ ̂ 󵄩󵄩e 0 ε E,od r Nr− r󵄩 󵄩󵄩 ≤ (1, e2p ∑i=1 log( √2 √BE1 (si ) +BE2 (si ) +ε ) 1) , 󵄩󵄩 󵄩 󵄩 󵄩 where we used that |a + ib| + ε ≤ √2√a2 + b2 + ε2 , a, b, ε ∈ ℝ. Then the right-hand side is computed as pN 󵄩󵄩 p ∫t | log Ψ (|Ĥ (B ,−θ ,r)|)|dN 󵄩󵄩2 |e| 2pN 󵄩 ̂ 󵄩󵄩e 0 ε E,od r Nr− r󵄩 ̂ 󵄩󵄩2 , 󵄩󵄩 ≤ ( ) ε2(pN−m) ∑ 󵄩󵄩󵄩 B ∑ 󵄩󵄩 E# ⋅ ⋅ ⋅ BE# 1󵄩 󵄩 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 󵄩 √ 󵄩 󵄩 2 m=0 combm m-fold

416 | 3 The Pauli–Fierz model by path measures where ∑combm denotes summation over the 2m terms in the expansion of the product ∏m (B̂ E (si )2 + B̂ E (si )2 ), B̂ E denotes one of B̂ E (si ), μ = 1, 2, i = 1, . . . , pN. Thus we i=1

obtain

where

1

2

#

μ

󵄩󵄩 p ∫t | log(Ψ (|Ĥ (B ,−θ ,r)|))|dN 󵄩󵄩2 󵄩󵄩e 0 ε E,od r Nr− r󵄩 󵄩󵄩 ≤ c2 (τ), 󵄩󵄩 󵄩󵄩 󵄩 c2 (τ) = (

(3.8.55)

pN

|e| 2pN ̂ √ω‖2m . ) ∑ ε2(pN−m) 2m (√2)2m m! ‖|k|φ/ √2 m=0

Note that c2 (τ) is independent of (x, α, w) ∈ ℝ3 × {1, 2} × X . Write Ct (τ) = c1 c2 (τ). Then ‖J∗0 eY Jt ‖ ≤ Ct (τ). This completes the proof. (Massless case) Suppose that ω(k) = |k|, i. e., ν = 0. In what follows, we show the lemma for massless cases, but it also covers the massive case. The idea is taken from Section 1.2.8, where we define an operator W(t) and show its boundedness. As will be seen below, the estimate of J∗0 eY Jt can be reduced to the estimate of W(t). Let Y = Y1 + Y2 , where t

Y1 = − ∫ Ĥ E,d (Br , θNr , r)R dr, 0

t+

Y2 = ∫ log(Ψε (|Ĥ E,od (Br , −θNr− , r)|))dNr . 0

Using the Baker–Campbell–Hausdorff formula, we expand J∗0 eY Jt . Thus formally we have to estimate 󵄨 󵄨󵄨 C † N † Cj 󵄨󵄨󵄨 󵄨󵄨 j ∗ Y ∗ B√2 + √B2 󵄨󵄨 + εψε (j)) Jt , 󵄨 J0 e Jt = J0 e + ∏ (󵄨󵄨 󵄨√ √2 󵄨󵄨󵄨󵄨 j=1 󵄨󵄨 2 where in the Fock representation B♯ and Ci are defined by ♯

t

B† = − ∑ a∗E (∫ Sj (θNr )jr̂ ( j=±

0

φ̂ ) e−ik⋅Br dr, j) , √ω

t

φ̂ ) eik⋅Br dr, j) , B = − ∑ aE (∫ Sj (θNr )jr̂ ( √ω j=± 0

Ci† = ∑ a∗E (Tj (θNs )jŝ i ( j=±

i

φ̂ ) e−ik⋅Bsi , j) , √ω

φ̂ ) eik⋅Bsi , j) , Ci = ∑ aE (Tj (θNs )jŝ i ( i √ω j=± ψε (j) = ψε (|Cj† + Cj |/√2). Note that ψε (j) is bounded with bound ‖ψε (j)‖ ≤ 1 for any j.

3.8 The Pauli–Fierz model with spin

| 417

Now fix (w, τ) ∈ X × 𝒮 . In a similar way to massive cases, there exist N = N(τ) ∈ ℕ and points 0 ≤ s1 < . . . < sN ≤ t, sj = sj (τ), j = 1, . . . , N, depending on τ such that s 󳨃→ Ns (τ) is not continuous. Then by taking n large enough and denoting (n(si ), n(si ) + t/n] for the interval containing si , we get N

eY2 = ∏(|ϕi | + εψε (|ϕi |)), i=1

where ϕi = Ĥ E,od (Bsi , −θNs − , n(si )). We have i

N

N

i=1

i=1

∏(|ϕi | + εψε (|ϕi |)) ≤ 1 + ∏(ϕi + εψε (ϕi ))(ϕi + εψε (ϕi )) and (ϕi + εψε (ϕi ))(ϕi + εψε (ϕi )) ≤ (1 + ε)ϕi ϕi + ε2 . Then N

|eY2 | ≤ 1 + ∏((1 + ε)ϕi ϕi + ε2 ) i=1

Hence it follows that N

|(F, J∗0 eY Jt G)| ≤ (|F|, J∗0 eY1 (1 + ∏((1 + ε)ϕi ϕi + ε2 )) Jt |G|) . i=1

(3.8.56)

We shall show that J∗0 eY1 ∏ki=1 ϕi Jt is bounded. Commutation relations are given by [B, B† ] =

t

t

0

0

2 󸀠 ̂ |φ(k)| e2 d(k)e−|r−r |ω(k) e−ik⋅(Br −Br󸀠 ) dk, ∫ dr ∫ dr 󸀠 ∫ 4 |k| ℝ3

2 ̂ |φ(k)| e −ik⋅(Bsj −Bsi ) dk, [Ci , Cj† ] = c(k)e−|si −sj |ω(k) e ∫ 4 |k| 2

ℝ3

t

[B, Ci† ] = −

2 ̂ |φ(k)| e2 b(k)e−|r−si |ω(k) e−ik⋅(Bsi −Br ) dk, ∫ dr ∫ 4 |k| 0

t

[Ci , B† ] = −

ℝ3

2 ̂ |φ(k)| e2 b(k)e−|r−si |ω(k) e−ik⋅(Br −Bsi ) dk. ∫ dr ∫ 4 |k| 0

ℝ3

Here, j

j

c(k) = ∑ (|η1 (k)|2 + |η2 (k)|2 ),

(3.8.57)

j=±

j

j

j

b(k) = ∑ θNr η3 (k)(η1 (k) − iθNs η2 (k)), j=±

i

(3.8.58)

418 | 3 The Pauli–Fierz model by path measures j

d(k) = ∑ |η3 (k)|2 .

(3.8.59)

j=±

Since the Baker–Campbell–Hausdorff formula yields that B†

B

eY1 = eβ e √2 e √2 , where β = [B, B† ]/4. We have k

B†

k

B

J∗0 eY1 ∏ ϕi Jt = J∗0 eβ e √2 e √2 ∏ ( i=1

i=1

Cj† i

√2

+

Cji

√2

) Jt .

(3.8.60) Cj†

We apply Wick’s theorem to compute the commutator between ∏ki=1 ( √2i + B† √2

B √2

Cji ) √2

and

e e . Set ji = i for simplicity. Hence k

∏( i=1

pair p [k/2] Ci† C 1 + i ) = ∑ p ∑ (∏ Ci2l−1 ,i2l ) cik1 ,...,i2k . √2 √2 2 i ...,i p=0 l=1 1 2p

Here, Ci,j = [Ci , Cj† ], we recall that ∑pair denotes summation over all p-pairs chosen i1 ...,i2p from {1, . . . , k}, and k−2p

cik1 ,...,i2p = 2−(k−2p)/2 ∑

∑

l=0 {n1 ,...,nl }⊂{i1 ,...,i2p

}c

∏

i∈{n1 ,...,nl }

Ci†

∏

i∈{n1 ,...,nl }c ∩{i1 ,...,i2p }c

Ci .

Hence B†

B

k

e √2 e √2 ∏ ( i=1

pair p k−2p [k/2] B† B Ci† C 1 Xe √2 e √2 Z, + i ) = ∑ p ∑ (∏ Ci2l−1 ,i2l ) ∑ ∑ √2 √2 2 p=0 i1 ...,i2p l=1 l=0 {n1 ,...,nl }⊂{i1 ,...,i2p }c

(3.8.61)

where X = ∏i∈{n1 ,...,nl } (Ci† + yi ) and Z = ∏i∈{n1 ,...,nl }c ∩{i1 ,...,i2p }c Ci with yj = [B, Cj† ]. Let us define operators in ℱrad by t

B (l) = − ∑ a (∫ e−|r−l|ω Sj (θNr ) †

∗

j=±

t

0

B(l) = − ∑ a (∫ e−|r−l|ω Sj (θNr ) j=±

Ci† (l)

0

= ∑ a (e−|si −l|ω Tj (θNs ) ∗

i

j=±

Ci (l) = ∑ a (e−|si −l|ω Tj (θNs ) j=±

i

φ̂ −ik⋅Br e dr, j) , √ω

φ̂ ik⋅Br e dr, j) , √ω

φ̂ −ik⋅Bs i , j) , e √ω

φ̂ ik⋅Bs e i , j) . √ω

3.8 The Pauli–Fierz model with spin

| 419

Then commutation relations are t

t

0

0

2 󸀠 ̂ |φ(k)| e2 [B(l), B (l )] = d(k)e−ik⋅(Br −Br󸀠 ) dk, ∫ dr ∫ dr 󸀠 ∫ e−(|r−l|+|r−l |)ω(k) 4 |k| † 󸀠

[Ci (l), Cj† (l󸀠 )] =

ℝ3

2 󸀠 ̂ |φ(k)| e −ik⋅(Bsj −Bsi ) c(k)e dk, ∫ e−(|si −l|+|sj −l |)ω(k) 4 |k| 2

ℝ3 t

2 󸀠 ̂ |φ(k)| e2 b(k)e−ik⋅(Bsi −Br ) dk, ∫ dr ∫ e−(|r−l|+|si −l |)ω(k) 4 |k|

[B(l), Ci† (l󸀠 )] =

0

ℝ3

t

[Ci (l), B† (l󸀠 )] =

2 󸀠 ̂ |φ(k)| e2 b(k)e−ik⋅(Br −Bsi ) dk. ∫ dr ∫ e−(|si −l|+|r−l |)ω(k) 4 |k| 0

ℝ3

Here, c(k), b(k) and d(k) are given by (3.8.57), (3.8.58) and (3.8.59), respectively. By intertwining properties J∗0 B† = B† (0)J∗0 ,

J0 B(0) = BJ0 ,

J∗0 Ci† = Ci† (0)J∗0 ,

J0 Ci (0) = Ci J0 ,

and factorization formula J∗0 Jt = e−tHrad , we see by (3.8.61) that B†

k

B

W(k) = J∗0 e √2 e √2 ∏ ( i=1

[k/2]

= ∑

p=0

pair

Ci† C + i )J √2 √2 t p

k−2p

1 ∑ (∏ Ci2l−1 ,i2l ) ∑ 2p i ...,i l=1 l=0 1

2p

∑

{n1 ,...,nl }⊂{i1 ,...,i2p }c

X(0)e

B† (0) √2

B(t)

e−tHrad e √2 Z(t),

where X(0) and Z(t) are defined by X and Z with Cj replaced by Cj (0) and Cj (t), respectively. Let α=

|e| ̂ ̂ √ω‖ + ‖|k|φ/ω‖), (‖|k|φ/ √2

β=

|e| ̂ √ω‖. ‖|k|φ/ √2

Then |[Ci (l), Cj† (l)]| ≤ β2 for any i, j. Hence we have the bound by Theorem 1.43. Let ξ (l, m) and ξr (l, m) be in Lemma 1.41, i. e., (|e|γ)j+j √(j + l)!√(j󸀠 + m)! j!j󸀠 ! j,j󸀠 =0 ∞

󸀠󸀠

ξ (l, m) = ∑ and

(|e|γ)j+j √(j + l)!√(j󸀠 + m)! . 󸀠 r (j+l)/2 r (j +m)/2 j!j󸀠 ! j,j󸀠 =0 ∞

ξr (l, m) = ∑ ̂ ̂ Here, γ = ‖|k|φ/√ω‖ + ‖|k|φ/ω‖.

󸀠󸀠

420 | 3 The Pauli–Fierz model by path measures Case (t ≥ 1): 󵄩󵄩 1 󵄩󵄩2 ‖W(k)Ψ‖ ≤ w(k) 󵄩󵄩󵄩e− 2 (t−1)Hrad 󵄩󵄩󵄩 ‖Ψ‖, 󵄩 󵄩

(3.8.62)

where [k/2]

w(k) = k! ∑

β2

p

( 4 ) (α + y)l αm

∑

p!l!m!

p=0 l+m+2p=k

ξ (l, m)

and y = tβ2 . Case (t < 1): For any 0 < r < t, we have 󵄩󵄩 1 󵄩󵄩2 ‖W(k)Ψ‖ ≤ w(r, k) 󵄩󵄩󵄩e− 2 (t−r)Hrad 󵄩󵄩󵄩 ‖Ψ‖, 󵄩 󵄩

(3.8.63)

where [k/2]

w(r, k) = k! ∑

β2

∑

p

( 4 ) (α + y)l αm

p=0 l+m+2p=k

p!l!m!

ξr (l, m).

Hence W(k) is bounded. Since N

J∗0 eY1 (1 + ∏((1 + ε)ϕi ϕi + ε2 )) Jt i=1

N

= J∗0 eY1 Jt + ∑

∑

k=0 {i1 ,...,ik }⊂[N]

k

ε2(N−k) (1 + ε)k J∗0 eY1 (∏ ϕij ϕ̄ ij ) Jt , j=1

where [N] = {1, . . . , N}, and in a similar manner to (3.8.62) and (3.8.63) we see the bound: 󵄩󵄩 󵄩󵄩 󵄩󵄩 ∗ Y1 k 󵄩 󵄩 1 󵄩2 󵄩󵄩J e (∏ ϕi ϕ̄ i ) Jt 󵄩󵄩󵄩 ≤ w(r, 2k) 󵄩󵄩󵄩e− 2 (t−r)Hrad 󵄩󵄩󵄩 󵄩󵄩 0 󵄩󵄩 󵄩󵄩 󵄩󵄩 j j 󵄩󵄩 󵄩󵄩 j=1 with 0 < r < t for t < 1 and r = 1 for t ≥ 1, we have 󵄩󵄩 󵄩󵄩 N 󵄩󵄩 ∗ Y1 󵄩 󵄩󵄩J0 e (1 + ∏((1 + ε)ϕi ϕi + ε2 )) Jt 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 i=1 N

󵄩󵄩 1 󵄩󵄩2 N! ε2(N−k) (1 + ε)k w(r, 2k)) 󵄩󵄩󵄩e− 2 (t−r)Hrad 󵄩󵄩󵄩 . 󵄩 󵄩 (N − k)!k! k=0

≤ (w(r, 0) + ∑ Then the proof is complete.

3.8 The Pauli–Fierz model with spin

| 421

Lemma 3.94. Let V ∈ L∞ (ℝ3 ) and φ̃ ∈ C0∞ (ℝ3 ). Then for every t ≥ 0 and all F, G ∈ K it follows that t

(F, e−tL G) = et ∑ ∫ 𝔼x,α [e− ∫0 V(Bs )ds (F(q0 ), J∗0 eY Jt G(qt ))]dx. α=1,2

(3.8.64)

ℝ3

Proof. Let F, G ∈ K such that F ≥ 0 and G ≥ 0. By Lemma 3.92, we have t

(F, e−tLR G) = et ∑ ∫ 𝔼x,α [e− ∫0 V(Bs )ds (F(q0 ), J∗0 eYR Jt G(qt ))]dx. α=1,2

(3.8.65)

ℝ3

Here, 0 ≤ g(R) = (F(q0 ), J∗0 eYR Jt G(qt )) and g(R) is a monotonously increasing function on R. Hence (F, e−tLR G) is also increasing on R and (F, e−tLR G) ↑ (F, e−tL G) as R ↑ ∞. For each ϕ ∈ Q , it follows that J0 F(q0 ) ⋅ eYR Jt G(qt ) → J0 F(q0 ) ⋅ eY Jt G(qt ) as R → ∞. Then the monotone convergence theorem yields that the function on the right-hand side above J0 F(q0 ) ⋅ eY Jt G(qt ) is finite for a. e. (ϕ, x, α, w, τ) ∈ Q × ℝ3 × {1, 2} × X × 𝒮 , and t

(F, e−tL G) = lim (F, e−tLR G) = et lim ∑ ∫ 𝔼x,α [e− ∫0 V(Bs )ds (F(q0 ), J∗0 eYR Jt G(qt ))]dx R→∞

R→∞

= et ∑ ∫ 𝔼x,α [ lim e α=1,2

ℝ3

R→∞

t − ∫0

α=1,2

ℝ3

V(Bs )ds

(F(q0 ), J∗0 eYR Jt G(qt ))]dx

t

= et ∑ ∫ 𝔼x,α [e− ∫0 V(Bs )ds 𝔼μ [J0 F(q0 ) ⋅ eY Jt G(qt )]] dx. α=1,2

ℝ3

For general F, G ∈ K , the lemma is proven by decomposing F and G as a linear sum of positive functions: F = ℜF+ − ℜF− + i(ℑF+ − ℑF− ) and G = ℜG+ − ℜG− + i(ℑG+ − ℑG− ). Finally, we show that J∗0 eY Jt G(qt ) ∈ L2 (Q ) for each (x, α, w, τ) ∈ ℝ3 × {1, 2} × X × 𝒮 in Lemma 3.93. Then 𝔼μ [J0 F(q0 ) ⋅ eY Jt G(qt )] = (F(q0 ), J∗0 eY Jt G(qt )) follows and the lemma is proven.

3.8.8 Functional integral representations In this section, we state the main results on functional integral representation of semiℤ2

group (F, e−tHPF G). Let Xt = Yt (1) + Yt (2) + Yt (3), where 3

t

̃ − Bs )dBμs ) , Yt (1) = −ie E (⨁ ∫ js φ(⋅ μ=1 0

422 | 3 The Pauli–Fierz model by path measures t

Yt (2) = − ∫ Ĥ E,d (Bs , θNs , s)ds, 0

t+

Yt (3) = ∫ WE (Bs , −θNs− , s)dNs . 0

Here, WE (Bs , −θNs− , s) = log(−Ĥ E,od (Bs , −θNs− , s)). We shall see that Xt turns to be the ℤ2

exponent of integral kernel of e−tHPF . Here, Yt (1) describes the part of quantized radiation field, Yt (2) the diagonal part of spin interaction, and Yt (3) the off-diagonal part of spin interaction. We furthermore define Xt (ε) by Xt with Yt (3) replaced by Yt (3, ε), i. e., Xt (ε) = Yt (1) + Yt (2) + Yt (3, ε), where t+

Yt (3, ε) = ∫ WE,ε (Bs , −θNs− , s)dNs 0

and WE,ε (Bs , −θNs− , s) = log(−Ψε (Ĥ E,od (Bs , −θNs− , s))). Lemma 3.95. For each (x, α, w, τ) ∈ ℝ3 × {1, 2} × X × 𝒮 , J∗0 eXt (ε) Jt is a bounded operator. t+ Proof. Let Φ, Ψ ∈ L2 (Q ). By Y = Yt (2) + ∫0 log Ψε (|Ĥ E,od (Bs , −θNs− , s)|)dNs defined in (3.8.52), we see that

|(Φ, J∗0 eXt (ε) Jt Ψ)| ≤ (|Φ|, J∗0 eY Jt |Ψ|) ≤ ‖Φ‖‖Ψ‖‖J∗0 eY Jt ‖ follows from Lemma 3.93. Then the lemma is proven. Theorem 3.96 (Functional integral representation for Pauli–Fierz Hamiltonian with spin 1/2). For every t ≥ 0 and all F, G ∈ K it follows that ℤ2

ℤ2

(F, e−tHPF G) = lim(F, e−tHPF,ε G)

(3.8.66)

ε↓0

and t

ℤ2

(F, e−tHPF,ε G) = et ∑ ∫ 𝔼x,α [e− ∫0 V(Bs )ds (F(q0 ), J∗0 eXt (ε) Jt G(qt ))]dx. α=1,2

(3.8.67)

ℝ3

Proof. Since ℤ2

ℤ2

e−tHPF,ε → e−tHPF

strongly as ε → 0, (3.8.66) follows. Now we turn to proving (3.8.67). Suppose that φ̃ ∈ C0∞ (ℝ3 ) and V ∈ L∞ (ℝ3 ). Write T

T

T+

S

S

S

̃ − Br ))dBr − ∫ Ĥ E,d (Br , θNr , s)dr + ∫ WE,ε (Br , −θNr− , s)dNr . XS,T (ε, s) = −ie ∫ Â E (js φ(⋅

3.8 The Pauli–Fierz model with spin

| 423

ε Define St,s : KE∞ → KE∞ by t

ε (St,s G)(x, θα ) = et 𝔼x,α [e− ∫0 V(Br )dr eX0,t (ε,s) G(qt )]. ε It can be seen that St,s has the property: t+t 󸀠

ε ε St,s St 󸀠 ,s󸀠 G(x, θα ) = et+t 𝔼x,α [e− ∫0 󸀠

V(Br )dr X0,t (ε,s)+Xt,t+t 󸀠 (ε,s󸀠 )

e

G(qt+t 󸀠 )].

(3.8.68)

Note that for s1 ≤ . . . ≤ sn , eX0,t1 (ε,s1 )+Xt1 ,t1 +t2 (ε,s2 )+⋅⋅⋅+Xt1 +⋅⋅⋅+tn−1 ,t1 +⋅⋅⋅+tn (ε,sn ) ∈ E[s1 ,sn ] .

(3.8.69)

Since ℤ2

(F, e−tHPF,ε G) = (F, e−tKε G), similar to the proof of Theorem 1.98, by the Trotter product formula, (3.8.68), (3.8.69) and the Markov property of Es , s ∈ ℝ, we obtain that n−1

ℤ2

ε (F, e−tHPF,ε G) = lim (F, J∗0 ( ∏ St/n,it/n )Jt G) n→∞ t

i=0

t

n

= e lim ∑ ∫ 𝔼x,α [e− ∫0 V(Br )dr (F(q0 ), J∗0 eXt (ε) Jt G(qt ))] dx, n→∞

α=1,2

ℝ3

where Xtn (ε) = Ytn (1) + Ytn (2) + Ytn (3, ε) with Ytn (1)

3

tj

n

̃ − Bs )dBμs ) , = −ie E (⨁ ∑ ∫ jtj−1 φ(⋅ μ=1 j=1 t

j−1

tj

n

Ytn (2) = − ∑ ∫ Ĥ E,d (Bs , θNs , tj−1 )ds, j=1 t

j−1

Ytn (3, ε)

tj

n

= ∑ ∫ WE,ε (Bs , −θNs− , tj−1 )dNs j=1 t

j−1

and tj = jt/n. Put t

n

⟨F, Tn G⟩ = et ∑ ∫ 𝔼x,α [e− ∫0 V(Br )dr (F(q0 ), J∗0 eXt (ε) Jt G(qt ))] dx, α=1,2

ℝ3

t

⟨F, TG⟩ = et ∑ ∫ 𝔼x,α [e− ∫0 V(Br )dr (F(q0 ), J∗0 eXt (ε) Jt G(qt ))] dx. α=1,2

ℝ3

(3.8.70)

424 | 3 The Pauli–Fierz model by path measures Notice that by the definition of ⟨F, Tn G⟩ we see that 󵄨󵄨 󵄨󵄨 n−1 t ̂ 󵄨 󵄨 |⟨F, Tn G⟩| = 󵄨󵄨󵄨󵄨(F, J∗0 (∏ Jit/n e− n Kε (A) J∗it/n ) Jt G)󵄨󵄨󵄨󵄨 ≤ ‖F‖‖G‖. 󵄨󵄨 󵄨󵄨 i=0 Suppose that Fm → F and Gm → G as m → ∞. Then by a telescoping, we can see that |⟨F, Tn G⟩ − ⟨Fm , Tn Gm ⟩| ≤ |⟨F, Tn G⟩ − ⟨Fm , Tn G⟩| + |⟨Fm , Tn G⟩ − ⟨Fm , Tn Gm ⟩| ≤ ‖F − Fm ‖‖G‖ + ‖G − Gm ‖‖Fm ‖.

By using facts that |Jt F| ≤ Jt |F|, |e±Yt (1) F| = |F| and |eYt (2)+Yt (3,ε) F| = eY |F|, we can see that t

|⟨F, TG⟩| ≤ et ∑ ∫ 𝔼x,α [e− ∫0 V(Br )dr (|e−Yt (1) J0 F(q0 )|, eYt (2) |eYt (3) |Jt |G(qt )|)] dx α=1,2

ℝ3

t

≤ et ∑ ∫ 𝔼x,α [e− ∫0 V(Br )dr (J0 |F(q0 )|, eY Jt |G(qt )|)] dx α=1,2

= (|F|, e

ℝ3 −tL

|G|) ≤ a‖F‖‖G‖,

where a = e−t inf Spec(L) . Hence we have |⟨Fm , TGm ⟩ − ⟨F, TG⟩| ≤ |⟨Fm , TGm ⟩ − ⟨F, TGm ⟩| + |⟨F, TGm ⟩ − ⟨F, TG⟩| ≤ a‖Fm − F‖‖Gm ‖ + a‖F‖‖G − Gm ‖.

Suppose that ‖F − Fm ‖ < ε and ‖G − Gm ‖ < ε. Together with them we have |⟨F, Tn G⟩ − ⟨F, TG⟩|

≤ |⟨F, Tn G⟩ − ⟨Fm , Tn Gm ⟩| + |⟨Fm , Tn Gm ⟩ − ⟨Fm , TGm ⟩| + |⟨Fm , TGm ⟩ − ⟨F, TG⟩| ≤ ε(‖G‖ + a‖Gm ‖ + ‖Fm ‖ + a‖F‖) + |⟨Fm , Tn Gm ⟩ − ⟨Fm , TGm ⟩|.

(3.8.71)

It can be concluded from (3.8.71) that it is sufficient to show the lemma for arbitrary F, G included in some dense domain. We shall show the lemma for F, G ∈ K ∞ . We claim that: (1) For each (x, α, w, τ) ∈ ℝ3 × {1, 2} × X × 𝒮 , there exists Ct (τ) such that n

|(F, J∗0 eXt (ε) Jt G)L2 (Q) | ≤ Ct (τ), where Ct (τ) is independent of (x, α, w) ∈ ℝ3 × {1, 2} × X , ε > 0 and n, and 𝔼P [Ct2 ] < ∞. (2) For each (x, α, τ) ∈ ℝ3 × {1, 2} × 𝒮 , n

lim 𝔼x𝒲 [(F(q0 ), J∗0 eXt (ε) Jt G(qt ))] = 𝔼x𝒲 [(F(q0 ), J∗0 eXt (ε) Jt G(qt ))].

n→∞

3.8 The Pauli–Fierz model with spin

| 425

The proof of (1) and (2) will be given in Lemma 3.97 and Lemma 3.98 below, respectively. We set RHS (3.8.70) = et ∑ ∫ 𝔼x,α [ξn ]dx, α=1,2

t

(3.8.72)

ℝ3

n

where ξn = e− ∫0 V(Bs )ds (F(q0 ), J∗0 eXt (ε) Jt G(qt )). Thus we have t

𝔼x𝒲 [|ξn |] ≤ 𝔼x𝒲 [e− ∫0 V(Bs )ds Ct (τ)‖F(q0 )‖‖G(qt )‖] and t

1/2 (𝔼P [Ct2 ])1/2 ‖F‖‖G‖ < ∞. ∑ ∫ 𝔼x,α [e− ∫0 V(Bs )ds Ct (τ)‖F(q0 )‖‖G(qt )‖]dx ≤ VM

α=1,2

ℝ3

Since 𝔼x𝒲 [ξn ] → 𝔼x𝒲 [ξ∞ ] as n → ∞ for each (x, α, τ) ∈ ℝ3 × {1, 2} × 𝒮 , by the Lebesgue dominated convergence theorem, lim ∑ ∫ 𝔼αP [𝔼x𝒲 [ξn ]]dx = ∑ ∫ 𝔼αP [𝔼x𝒲 [ξ∞ ]]dx

n→∞

α=1,2

α=1,2

ℝ3

ℝ3

follows. For V ∈ ℛKato by a limiting argument, we can show the theorem. Finally, for φ̂ in Assumption 3.4, we can see (3.8.67) by an approximation, which is also shown in Lemma 3.99 below. Lemma 3.97. (1) in the proof of Theorem 3.96 is true. Proof. For each (x, α, w, τ) ∈ ℝ3 × {1, 2} × X × 𝒮 , it can be directly seen that F, G ∈ n D(eXt (ε) J0 ) ∩ D(eXt (ε) J0 ) for any n ∈ ℕ. We see the inequality: ‖Yt (2)Φ‖ ≤ γ √m + 1‖Φ‖ ̂ Since L2fin (Q ) is the set of analytic vectors for Φ ∈ L2m (Q ), where γ = √2t|e|‖|k|φ/√ω‖. n

n

for eYt (2) , for F ∈ L2fin (Q ) we have eYt (2) F = ∑∞ j=0

Ytn (2)j F. j!

We assume that F ∈ L2m󸀠 (Q )

and G ∈ L2m (Q ), hence NF = m󸀠 F and NG = mG for the number operator N. Then 󵄩󵄩 Y n (2)j 󵄩󵄩 √(m + j − 1) ⋅ ⋅ ⋅ (m + 1)m j 󵄩󵄩 t 󵄩󵄩 󵄩󵄩 󵄩 γ ‖F‖. 󵄩󵄩 j! F 󵄩󵄩󵄩 ≤ j! 󵄩 󵄩 Thus

󵄩󵄩 Ytn (2) 󵄩󵄩 ∞ √(m + j − 1) ⋅ ⋅ ⋅ (m + 1)m j 󵄩󵄩e γ ‖F‖. F 󵄩󵄩󵄩 ≤ ∑ 󵄩 j! j=0

On the other hand by the definition of Ytn (3, ε), we have n

n

ti/n+

eYt (3,ε) = ∏ exp ( ∫ i=1

t(i−1)/n

WE,ε (Bs , −θNs− , t(i − 1)/n)dNs ) .

(3.8.73)

426 | 3 The Pauli–Fierz model by path measures For every τ ∈ 𝒮 , there exists N = N(τ) ∈ ℕ such that map t 󳨃→ Nt (τ) is not continuous at points s1 = s1 (τ), . . . , sN = sN (τ). For sufficiently large n, the number of discontinuous points located in interval (t(i − 1)/n, ti/n] is at most one. Then by taking n large enough and putting (n(si ), n(si ) + t/n] for the interval containing si , we get N

n

eYt (3,ε) = ∏(−ϕi − εψε (ϕi )), i=1

̂

where ϕi

= HE,od (Bsi , −θNs − , n(si )). Fix τ ∈ i ∗ Ytn (1) Ytn (2) |(F, J0 e e ∏Ni=1 (−ϕi − εψε (ϕi ))G)|. We have N

N

i=1

p=0

∏(−ϕi − εψε (ϕi )) = (−1)N ∑ εN−p

𝒮 above and we shall estimate

p

( ∏

∑ {j1 ,...,jp }⊂[N]

i∈{j ̸ 1 ,...,jp }

ψi ) (∏ ϕji ) , i=1

where ψi = ψε (ϕji ) and [N] = {1, . . . , N}, and n

n

N

n

(F, J∗0 eYt (1) eYt (2) eYt (3) Jt G) = ∑ εN−p p=0

∑

n

n

(e−Yt (1) J0 F, eYt (2)

{j1 ,...,jp }⊂[N]

∏

i∈{j ̸ 1 ,...,jp }

p

ψi ∏ ϕji Jt G) . i=1

Notice again that ‖ϕji Φ‖ ≤ 2γ √m + 1‖Φ‖ for Φ ∈ L2m (Q ) and i = 1, . . . , p. Since ψε ≤ 1, each terms on the right-hand side above can be estimated as 󵄨󵄨 󵄨󵄨 p 󵄨󵄨 −Y n (1) 󵄨 󵄨󵄨(e t J F, eYtn (2) ∏ ψ ∏ ϕ J G)󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 0 i ji t 󵄨󵄨 󵄨󵄨 i=1 i∈{j ̸ 1 ,...,jp } 󵄨 󵄨 ∞ √(j + p + m − 1) ⋅ ⋅ ⋅ (m + 1)m (2γ)j+p−m+1 ‖F‖‖G‖. ≤ ∑ εN−p j! j=0 Thus n

n

n

|(F, J∗0 eYt (1) eYt (2) eYt (3,ε) Jt G)| N

εN−p N! ∞ √(j + p + m − 1) ⋅ ⋅ ⋅ (m + 1)m (2γ)j+p−m+1 ‖F‖‖G‖ ∑ p!(N − p)! j! p=0 j=0

≤ ∑

≤ At (N)‖F‖‖G‖, where ∞

At (N) = ∑ (1 + ε)N j=0

√(j + N + m − 1) ⋅ ⋅ ⋅ (m + 1)m (2γ)j+N−m+1 . j!

| 427

3.8 The Pauli–Fierz model with spin

The number N depends on τ ∈ S, then At (N(τ)) turns to be a random process on (𝒮 , ℬ𝒮 , P). We set Ct (τ) = At (N(τ)). For each (x, α, w, τ) ∈ ℝ3 × {1, 2} × X × 𝒮 , n n |(F, J∗0 eXt (ε) Jt G)| is finite with the bound |(F, J∗0 eXt (ε) Jt G)| ≤ Ct (τ) and t N At (N)2 −t e N! N=0 ∞

𝔼P [Ct2 ] = ∑

∞

∞

N=0

j=0

≤ e−t ∑ ( ∑ (1 + ε)N (2γ)j+N−m+1

2

√(j + N + m − 1) ⋅ ⋅ ⋅ (m + 1)m ) < ∞. N!j!

Thus (1) follows. Lemma 3.98. Statement (2) of the proof of Theorem 3.96 is true. n

Proof. We show the convergence of J∗0 eXt (ε) Jt as n → ∞. Let F ∈ L2fin (Q ). We show the n n n convergence of eYt (1) , eYt (2) and eYt (3,ε) , separately. Thus 󵄩󵄩 󵄩󵄩 tj 󵄩󵄩 󵄩󵄩 3 n 󵄩 󵄩 󵄩 ̃ − Bs )dBμs ) F 󵄩󵄩󵄩󵄩 ‖e−Yt (1) J0 F − e J0 F‖ ≤ |e| 󵄩󵄩󵄩Â E (⨁ ∑ ∫ (jtj−1 − js )φ(⋅ 󵄩󵄩 󵄩󵄩 μ=1 j=1 t 󵄩󵄩 󵄩󵄩 j−1 󵄩 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 3 n tj 󵄩󵄩 󵄩󵄩 󵄩 ̃ − Bs )dBμs 󵄩󵄩󵄩󵄩 ‖(N + 1)1/2 F‖. ≤ |e| 󵄩󵄩󵄩⨁ ∑ ∫ (jtj−1 − js )φ(⋅ 󵄩󵄩 μ=1 󵄩󵄩 j=1 t 󵄩󵄩 󵄩󵄩 j−1 󵄩 󵄩 −Ytn (1)

Then it follows that 𝔼x𝒲 [‖e−Yt (1) J0 F

−e

−Ytn (1)

tj

n

2

̃ − Bs )‖2 ds‖(N + 1)1/2 F‖2 J0 F‖ ] ≤ 3|e| ∑ ∫ ‖(jtj−1 − js )φ(⋅ j=1 t

j−1

tj

n

= 3|e| ∑ ∫ ( j=1 t

j−1

φ̂ φ̂ t ̂ 2 ‖(N + 1)1/2 F‖2 . , (1 − e−|s−tj−1 |ω ) ) ds‖(N + 1)1/2 F‖2 ≤ 3|e| ‖√ωφ‖ √ω √ω n

Hence we have n

lim 𝔼x𝒲 [‖e−Yt (1) J0 F − e−Yt (1) J0 F‖] = 0.

n→∞

(3.8.74)

On the other hand in the Fock representation, Yt (2) and Ytn (2) are represented as t

Yt (2) = Ytn (2)

t

{ } −1 ∑ {a∗E (∫ b−j (s)ds, j) − aE (∫ b+j (s)ds, j)} , √2 j=± 0 0 { } ti

ti

n { } −1 = ∑ ∑ {a∗E ( ∫ b−j (ti−1 )ds, j) − aE ( ∫ b+j (ti−1 )ds, j)} . √2 j=± i=1 ti−1 ti−1 { }

428 | 3 The Pauli–Fierz model by path measures φ̂ φ̂ Here, b±j (s) = Sj (θNs )e±ik⋅Bs jŝ √ω and b±j (ti−1 ) = Sj (θNs )e±ik⋅Bs jt̂ i−1 √ω . Then the distance n between test functions of Yt (2) and Yt (2) can be estimated as

󵄩󵄩2 󵄩󵄩 t t t 󵄩󵄩 󵄩󵄩 n i n i 󵄩󵄩 󵄩󵄩 − − 󵄩󵄩∫ bj (s)ds − ∑ ∫ bj (ti−1 )ds󵄩󵄩 ≤ t ∑ ∫ 󵄩󵄩 󵄩󵄩 󵄩󵄩 i=1 t i=1 t 󵄩󵄩󵄩0 󵄩 i−1 i−1

󵄩󵄩 󵄩2 󵄩󵄩S (θ )e−ik⋅Bs (j ̂ φ̂ − j ̂ φ̂ )󵄩󵄩󵄩 ds. 󵄩󵄩 j Ns 󵄩 s √ω ti−1 √ω 󵄩󵄩 󵄩

Since 󵄩2 󵄩󵄩 󵄩󵄩S (θ )e−ik⋅Bs (j ̂ φ̂ − j ̂ φ̂ )󵄩󵄩󵄩 ≤ 2|e|2 |s − t |‖|k|φ‖ ̂ 2, 󵄩 󵄩󵄩 j Ns ti−1 i−1 s √ω √ω 󵄩󵄩 󵄩 we see that 󵄩󵄩 t 󵄩󵄩2 t 󵄩󵄩 󵄩󵄩 n i 2|e|2 2 󵄩󵄩 − 󵄩󵄩 − ̂ 2. t ‖|k|φ‖ 󵄩󵄩󵄩∫ bj (s)ds − ∑ ∫ bj (ti−1 )ds󵄩󵄩󵄩 ≤ n 󵄩󵄩 󵄩󵄩 i=1 󵄩󵄩0 󵄩󵄩 ti−1 It is concluded that 󵄩󵄩 t 󵄩󵄩 t 󵄩󵄩 󵄩󵄩 n i 󵄩󵄩 − 󵄩 − lim 󵄩󵄩󵄩∫ bj (s)ds − ∑ ∫ bj (ti−1 )ds󵄩󵄩󵄩󵄩 = 0. n→∞ 󵄩 󵄩󵄩 󵄩󵄩0 i=1 t 󵄩󵄩 󵄩 i−1

(3.8.75)

Furthermore, by (3.8.75) it can be straightforwardly seen that n

eYt (2) F → eYt (2) F

(3.8.76)

Y n (2)j

n

t by the expansion eYt (2) F = ∑∞ j=0 j! F. In the same argument used in the proof of Lemma 3.97 for every τ ∈ 𝒮 , there exists N = N(τ) ∈ ℕ such that map t 󳨃→ Nt (τ) is not continuous at points s1 = s1 (τ), . . . , sN = sN (τ). We suppose that si is located in (n(si ), n(si ) + t/n] for sufficiently large n. We get n

N

eYt (3,ε) = ∏(−ϕi − εψε (ϕi )), i=1

where ϕi = Ĥ E,od (Bsi , −θNs − , n(si )). In the Fock representation, we can also have i

1 Ĥ E,od (Bsi , −θNs − , si ) ≅ ∑ {a∗ (c− (s ), j) − aE (cj+ (si ), j)} , i √2 j=± E j i

1 Ĥ E,od (Bsi , −θNs − , n(si )) ≅ ∑ {a∗ (c− (n(si )), j) − aE (cj+ (n(si )), j)} , i √2 j=± E j φ̂

where cj± (t) = Tj (−θNs − )(jt̂ √ω )e±ik⋅Bsi . We then estimate the distance between their test i functions as |k|φ̂ |k|φ̂ 󵄩󵄩 − 󵄩2 󵄩󵄩cj (n(si )) − cj− (si )󵄩󵄩󵄩 ≤ 2|e|2 ( , (1 − e−|n(si )−si |ω ) ). 󵄩 󵄩 √ω √ω

3.8 The Pauli–Fierz model with spin

| 429

Clearly, n(si ) → si as n → ∞. Then lim Ĥ E,od (Bsi , −θNs − , n(si )) = Ĥ E,od (Bsi , −θNs − , si ),

n→∞

i

i

lim ψε (Ĥ E,od (Bsi , −θNs − , n(si ))) = ψε (Ĥ E,od (Bsi , −θNs − , si ))

n→∞

i

i

on K ∞ and we have N

n

lim eYt (3,ε) = ∏(−Ψε (Ĥ E,od (Bsi , −θNs − , si ))).

n→∞

i

i=1

(3.8.77)

Then by (3.8.74), (3.8.76) and (3.8.77) we can see that for each (x, α, τ) ∈ ℝ3 × {1, 2} × 𝒮 , n

𝔼x𝒲 [(F(q0 ), J∗0 eXt (ε) Jt G(qt ))] → 𝔼x𝒲 [(F(q0 ), J∗0 eXt (ε) Jt G(qt ))] n

as n → ∞ and |𝔼x𝒲 [(F(q0 ), J∗0 eXt (ε) Jt G(qt ))]| ≤ Ct (τ)𝔼xW [‖F(q0 )‖‖G(qt )‖] and the dominated function Ct (τ)‖F(q0 )‖‖G(qt )‖ is integrable. The Lebesgue dominated convergence theorem yields that t

n

∑ ∫ 𝔼x,α [e− ∫0 V(Br )dr (F(q0 ), J∗0 eXt (ε) Jt G(qt ))] dx

α=1,2

ℝ3

t

→ ∑ ∫ 𝔼x,α [e− ∫0 V(Br )dr (F(q0 ), J∗0 eXt (ε) Jt G(qt ))] dx α=1,2

ℝ3

for F, G ∈ K ∞ as n → ∞. Hence the proof of (2) is complete. Lemma 3.99. (3.8.67) is valid for φ̂ in Assumption 3.4. Proof. It is sufficient to show (3.8.67) for F, G ∈ K ∞ by an approximation argument. ̂ as n → ∞. Then Take a sequence (φ̂ n )n∈ℕ ⊂ C0∞ (ℝ3 ) such that |k|φ̂ n /√ω → |k|φ/√ω (3.8.67) is valid for each φ̂ n : t

ℤ2

(F, e−tHPF (n) G) = ∑ ∫ 𝔼x,α [e− ∫0 V(Bs )ds (F(q0 ), J∗0 eXt (ε)n Jt G(qt ))]dx. α=1,2

ℝ3

ℤ ℤ Here, Xt (ε)n is defined by Xt (ε) with φ̂ replaced by φ̂ n , and HPF2 (n) by HPF2 with φ̂ reℤ2 ℤ2 placed by φ̂ n . It can be seen that HPF (n) → HPF as n → ∞ on a common core K ∞ . ℤ2

ℤ2

Then e−tHPF (n) → e−tHPF strongly as n → ∞. Let Yt (1)n , Yt (2)n and Yt (3, ε)n be Yt (1), Yt (2) and Yt (3, ε) with φ̂ replaced by φ̂ n , respectively. In a similar approximation argument to the proof of Lemma 3.98 above, we can show that: ̂ (1) 𝔼x𝒲 [‖(eYt (1)n − eYt (1) )F] → 0 as ‖(φ̂ n − φ)/√ω‖ → 0; Yt (2)n Yt (2) ̂ (2) ‖(e −e )F‖ → 0 as ‖|k|(φ̂ n − φ)/√ω‖ → 0; ̂ (3) ‖(Yt (3)n − eYt (3,ε) )F‖ → as ‖|k|(φ̂ n − φ)/√ω‖ →0

430 | 3 The Pauli–Fierz model by path measures for F ∈ L2fin (Q ). Hence t

ℤ2

(F, e−tHPF G) = lim ∑ ∫ 𝔼x,α [e− ∫0 V(Bs )ds (F(q0 ), J∗0 eXt (ε)n Jt G(qt ))]dx n→∞

α=1,2

ℝ3

x,α

= ∑ ∫𝔼 α=1,2

t

[e− ∫0 V(Bs )ds (F(q0 ), J∗0 eXt (ε) Jt G(qt ))]dx

ℝ3

for F, G ∈ K ∞ . Then the proof is complete. As was mentioned above, we need the regularization Ψε (Ĥ od ) of Ĥ od to prevent t+ ̂ zeros of Ĥ od . The zeros of Ĥ od produces the zeros of e∫0 log(HE,od (Bs ,−θNs− ,s))dNs . In order

to avoid limε→0 in the functional integral representation, instead of introducing regularization Ψε we introduce a subset W of ℝ3 × X × 𝒮 × QE by t+

} { |e| W = {∫ log ( √|b1 (s, x, Bs )|2 + |b2 (s, x, Bs )|2 ) dNs > −∞} , 2 } {0

(3.8.78)

̃ − Bs − x)). where bα (s, x, Bs ) = B̂ Eα (js φ(⋅ Theorem 3.100 (Functional integral representation for Pauli–Fierz Hamiltonian with spin 1/2). For every t ≥ 0 and all F, G ∈ K it follows that t

ℤ2

(F, e−tHPF G) = et ∑ ∫ 𝔼x,α [e− ∫0 V(Bs )ds (J0 F(q0 ), eXt 1W Jt G(qt ))]dx. α=1,2

(3.8.79)

ℝ3

Here, W is given by (3.8.78) and the exponent Xt is defined by t

t

t+

0

0

0

̃ − Bs ))dBs − ∫ Ĥ E,d (Bs , θNs , s)ds + ∫ log(−Ĥ E,od (Bs , −θNs− , s))dNs Xt = −ie ∫ Â E (js φ(⋅ and t+

∫ log(−Ĥ E,od (Bs , −θNs− , s))dNs = 0

∑ log(−Ĥ E,od (Br , −θNr− , r))

r∈[0,t] Nr+ =N ̸ r−

possibly takes infinity. Proof. By the Lebesgue dominated convergence theorem, we have t

lim ∫ 𝔼x,α [e− ∫0 V(Bs )ds (J0 F(q0 ), eXt (ε) Jt G(qt ))]dx

ε→0

ℝ3

t

= ∫ 𝔼x,α [e− ∫0 V(Bs )ds lim (J0 F(q0 ), eXt (ε) Jt G(qt ))]dx. ℝ3

ε→0

3.8 The Pauli–Fierz model with spin

| 431

We can also show that lim (J0 F(q0 ), 1W c eXt (ε) Jt G(qt )) = 0

ε→0

and lim (J0 F(q0 ), 1W eXt (ε) Jt G(qt )) = (J0 F(q0 ), 1W eXt Jt G(qt )).

ε→0

Then the theorem follows. By using the functional integral representation, we can estimate the ground state ℤ energy of HPF2 . Write ℤ E(A,̂ B̂ 1 , B̂ 2 , B̂ 3 ) = inf Spec(HPF2 ).

For the spinless Pauli–Fierz Hamiltonian HPF , we have inf Spec(HPF ) = E(A,̂ 0, 0, 0) and the diamagnetic inequality E(0, 0, 0, 0) ≤ E(A,̂ 0, 0, 0) was already seen. We extend ℤ this inequality to HPF2 . Define ℤ (HPF2 (0)F)(θ) = (Hp + Hrad + Ĥ d (θ))F(θ) − |Ĥ od (−θ)|F(−θ),

where |Ĥ od (−θ)| =

⊕

|e| √ ̂ 2 ∫ B1 (x) + B̂ 22 (x)dx 2 ℝ3

is independent of θ ∈ ℤ2 . HPF2 (0) corresponds to ℤ

Hp + Hrad − (

e ̂ B 2 3 |e| √ ̂ 2 B1 2

|e| √ ̂ 2 B1 2

+ B̂ 22

+ B̂ 22

− e2 B̂ 3

)

acting in L2 (ℝ3 ; ℂ2 ) ⊗ L2 (Q ). Furthermore, to avoid zeroes of the off-diagonal part to occur we also define 2 (HPF,ε (0)F)(θ) = (Hp + Hrad + Ĥ d (θ))F(θ) − Ψε (|Ĥ od (−θ)|)F(−θ).

ℤ

Since the spin interaction is infinitesimally small with respect to the free Hamiltonian ℤ ℤ2 Hp + Hrad , HPF2 (0) and HPF,ε (0) are self-adjoint on DPF and bounded from below. The ℤ2

functional integral representation of e−tHPF (0) is given by t

ℤ2

(F, e−tHPF (0) G) = lim et ∑ ∫ 𝔼x,α [e− ∫0 V(Bs )ds (J0 F(q0 ), eXt ε↓0

α=1,2

⊥

(ε)

Jt G(qt ))]dx,

ℝ3

where Xt⊥ (ε)

t

t+

0

0

= − ∫ Ĥ E,d (Bs , θNs , s)ds + ∫ log(Ψε (|Ĥ E,od (Bs , s)|))dNs .

(3.8.80)

432 | 3 The Pauli–Fierz model by path measures Corollary 3.101 (Diamagnetic inequality). It follows that ℤ2

ℤ2

|(F, e−tHPF G)| ≤ (|F|, e−tHPF (0) |G|)

(3.8.81)

and max

(αβγ)=(123),(231),(312)

E(0, √B̂ 2α + B̂ 2β , 0, B̂ γ ) ≤ E(A,̂ B̂ 1 , B̂ 2 , B̂ 3 ).

Proof. Since |Ψε (Ĥ E,od )| ≤ Ψε (|Ĥ E,od |), |eXt (ε) | ≤ eXt

⊥

(ε)

(3.8.82)

and |Jt G| ≤ Jt |G| as Jt is positivity ℤ2

preserving, by the functional integral representation of e−tHPF we have ℤ2

ℤ2

ℤ2

|(F, e−tHPF G)| ≤ lim(|F|, e−tHPF,ε (0) |G|) = (|F|, e−tHPF (0) |G|). ε↓0

Thus (3.8.81) follows. From this, E(0, √B̂ 21 + B̂ 22 , 0, B̂ 3 ) ≤ E(A,̂ B̂ 1 , B̂ 2 , B̂ 3 ) is obtained. We will show that E(A,̂ B̂ 1 , B̂ 2 , B̂ 3 ) = E(A,̂ B̂ 2 , B̂ 3 , B̂ 1 ) = E(A,̂ B̂ 3 , B̂ 1 , B̂ 2 )

(3.8.83)

by SU(2)-symmetry. Let R ∈ O(3) be such that x1

x2

x3

x1

R (x2 ) = (x3 ) .

Then there exists (n, ϕ) ∈ S2 × ℝ such that R = R (n, ϕ). Recall that R (n, ϕ) describes the rotation around n by angle ϕ. Hence we see that eiϕn⋅(1/2)σ σμ e−iϕn⋅(1/2)σ = (R σ)μ . ℤ ℤ Now we write HPF2 by HPF2 (A,̂ B̂ 1 , B̂ 2 , B̂ 3 ). Thus we obtain that ℤ ℤ eiϕn⋅(1/2)σ HPF2 (A,̂ B̂ 1 , B̂ 2 , B̂ 3 )e−iϕn⋅(1/2)σ = HPF2 (A,̂ B̂ 2 , B̂ 3 , B̂ 1 )

which implies the first equality in (3.8.83). The second one is proven in the same way.

3.8.9 Translation invariant Pauli–Fierz Hamiltonian with spin Finally, we study the translation invariant Pauli–Fierz Hamiltonian with spin 1/2, that S is, HPF with V ≡ 0. We suppose Assumption 3.66 in Section 3.8.9.

3.8 The Pauli–Fierz model with spin

| 433

Definition 3.102 (Pauli–Fierz Hamiltonian with spin 1/2 and a fixed total momentum). The Pauli–Fierz Hamiltonian with spin and a fixed total momentum p is defined by 1 e S HPF (p) = (p − Pf − eA(0))2 + Hrad − σ ⋅ B(0), 2 2

p ∈ ℝ3 ,

S with domain D(HPF (p)) = D(Hrad ) ∩ D(Pf 2 ). S Theorem 3.103 (Self-adjointness). HPF (p) is self-adjoint and essentially self-adjoint on 2 any core of Hrad + Pf .

Proof. The proof is similar to that of Proposition 3.77. By (3.7.6), we have ‖HPF,0 (p)Φ‖ ≤ C‖(HPF (p) + 1)Φ‖ for Φ ∈ D(Hrad ) ∩ D(Pf2 ), with some constant C. From this inequality, we see that for 1/2 any ε > 0 there exists Cϵ > 0 such that ‖Hrad Φ‖ ≤ ε‖HPF (p)Φ‖ + Cε ‖Φ‖ for Φ ∈ D(Hrad ) ∩ 1/2 2 ̂ ̂ D(Pf ). Since ‖σ⋅ B(0)Φ‖ ≤ ‖Hrad Φ‖+bε ‖Φ‖, it follows that σ⋅ B(0) is infinitesimally small S with respect to HPF (p). Thus HPF (p) is self-adjoint on D(Hrad ) ∩ D(Pf2 ). Essential selfS adjointness follows from the trivial inequality ‖HPF (p)Φ‖ ≤ C‖(Hrad + Pf2 )Φ‖ + ‖Φ‖. As in the case of HPF (p), we have the theorem below. Theorem 3.104 (Fiber decomposition). It follows that 2

3

⊕

2

L (ℝ ; ℂ ) ⊗ ℱrad = T ( ∫ ℂ2 ⊗ ℱrad dp) T −1 ℝ3

and S HPF

⊕

S = T ( ∫ HPF (p)dp) T −1 , ℝ3

where T is defined in (3.7.1). S In Q -representation, HPF (p) is rewritten as

1 e S 2 ̂ ̂ HPF (p) = (p − P̂ f − eA(0)) + Hrad − σ ⋅ B(0), 2 2 ̂ ̂ = 0) defined in (3.8.8). Before going to construct the functional where B(0) = B(x S S integral representation of e−tHPF (p) we look at the symmetry properties of HPF (p). Lemma 3.105. Let φ̂ be rotation-invariant. Then for any choice of the polarization vecS S tors HPF (p) is unitary equivalent with HPF (R −1 p), for all R ∈ SO(3).

434 | 3 The Pauli–Fierz model by path measures Proof. It is sufficient to show the lemma for an arbitrary rotation R = R (m, ϕ) with (m, ϕ) ∈ S2 × [0, 2π). For any polarization vectors e± , define hf = dΓ(θ(R , ⋅)H2 ), where θ(R , k) = cos−1 (e+ (R k) ⋅ R e+ (k)) and H2 in (3.8.16). Formally, hf = ∑ ∫ θ(R , k)(a∗ (k, −)a(k, +) − a∗ (k, +)a(k, −))dk. j=±

ℝ3

Thus we obtain that eihf eiϕm⋅Lf Hrad e−iϕm⋅Lf e−ihf = Hrad ,

eihf eiϕm⋅Lf Pf,μ e−iϕm⋅Lf e−ihf = (R Pf )μ ,

eihf eiϕm⋅Lf Aμ (0)e−iϕm⋅Lf e−ihf = (R A(0))μ , eihf eiϕm⋅Lf Bμ (0)e−iϕm⋅Lf e−ihf = (R B(0))μ , eiϕm⋅(1/2)σ σμ e−iϕm⋅(1/2)σ = (R σ)μ .

From these identities, it follows that S eihf eiϕm⋅((1/2)σ+Lf ) HPF (p)e−iϕm⋅((1/2)σ+Lf ) e−ihf 1 1 = (p − R Pf − eR A(0))2 + Hrad − (R σ) ⋅ (R B(0)) 2 2 S = HPF (R −1 p),

(3.8.84)

proving the lemma. S Let E S (p, e2 ) = inf Spec(HPF (p)). An immediate consequence of Lemma 3.105 is as follows.

Corollary 3.106 (Rotation invariance). Let φ̂ be rotation-invariant. Then for every R ∈ SO(3), it follows that E S (R p, e2 ) = E S (p, e2 ). Theorem 3.107 (Fiber decomposition of the Pauli–Fierz Hamiltonian with spin and a fixed total momentum). Suppose that polarization vectors are coherent in direction n ∈ S2 and φ̂ is rotation invariant. Let p/|p| = n. Then the Hilbert space ℂ2 ⊗ ℱrad and the S S self-adjoint operator HPF (p) = HPF (|p|n) are decomposed as ℂ2 ⊗ ℱrad = ⨁ ℱrad (w), w∈ℤ1/2

S S HPF (p) = ⨁ HPF (p, w). w∈ℤ1/2

Here, ℱrad (w) is the subspace spanned by eigenvectors of 1 J0 = n ⋅ σ + n ⋅ Lf + Sf 2 S S associated with the eigenvalue w ∈ ℤ1/2 and HPF (|p|n, w) = HPF (|p|n)⌈ℱrad (w) .

(3.8.85)

3.8 The Pauli–Fierz model with spin

| 435

Proof. Let R = R (n, ϕ). Since the polarization vectors are coherent in direction n, it follows that θ(R , k) = ϕz with some z ∈ ℤ, and then hf = ϕSf . Since R p = |p|R n = |p|n = p, we have from (3.8.84), S S S eiϕJ0 HPF (p)e−iϕJ0 = HPF (R −1 p) = HPF (p),

ϕ ∈ ℝ.

(3.8.86)

S Hence HPF (p) is reduced by J0 and (3.8.85) follows. S From this theorem, HPF (p) with coherent polarization vectors in direction p/|p| and p ≠ 0 has the symmetry

SU(2) × SOfield (3) × helicity. S Let φ̂ be rotation-invariant and HPF (p, e± ) be the Hamiltonian with coherent polarization vectors given in Example 3.80 with n = n3 = (0, 0, 1), i. e.,

e+ (k) =

(−k2 , k1 , 0) √k12 + k22

,

e− (k) = k̂ × e+ (k).

(3.8.87)

S S HPF (p) with arbitrary polarization vectors is unitary equivalent to HPF (p, e± ), S S HPF (p) ≅ HPF (p, e± ).

Since there exists R such that R p = |p|n3 , we have by Lemma 3.105, S S HPF (p, e± ) ≅ HPF (|p|n3 , e± ).

By (3.8.85), we obtain that S S HPF (|p|n3 , e± ) = ⨁ HPF (|p|n3 , e± , w). w∈ℤ1/2

Then for rotation-invariant φ,̂ we have the unitary equivalence S S HPF (p) ≅ ⨁ HPF (|p|n3 , e± , w). w∈ℤ1/2

We can show the theorem below in a similar way to Theorem 3.84. Theorem 3.108 (Reflection symmetry). Let φ̂ be rotation-invariant and the coherent S S polarization vectors e± be given by (3.8.87). Then HPF (p, w) ≅ HPF (p, −w) for w ∈ ℤ1/2 and p with (0, 0, 1) = p/|p|. Proof. The proof is similar to that of Theorem 3.84. Let J0 = σ2 ⊗Γ(u), where u is defined in (3.8.28). Thus we have Lfμ ,

J−1 0 Lfμ J0 = {

−Lfμ ,

μ = 2, μ = 1, 3,

436 | 3 The Pauli–Fierz model by path measures Pfμ ,

J−1 0 Pfμ J0 = {

μ = 1, 3, μ = 2,

−Pfμ ,

Aμ (0),

J−1 0 Aμ (0)J0 = {

μ = 1, 3,

−Aμ (0),

μ = 2,

−σμ , μ = 1, 3,

J−1 0 σμ J0 = {

σμ ,

μ = 2.

Since n = n3 = (0, 0, 1) and p = |p|n3 , and the helicity of e± is zero, we have J0 = 21 σ3 +Lf,3 and J−1 0 J0 J0 = −J0 .

(3.8.88)

We also see that J−1 0 σμ (p − Pf − eA)μ J0 = −σμ (p − Pf − eA)μ for μ = 1, 2, 3. From here, it follows that S S J−1 0 HPF (p)J0 = HPF (p).

(3.8.89)

Note also that S S HPF (p)ϕ = HPF (p, w)ϕ,

ϕ ∈ ℱrad (w).

(3.8.90)

S Let ϕ ∈ ℱrad (w). We have J0 ϕ ∈ ℱrad (−w) by (3.8.88), and J−1 0 HPF (p, −w)J0 ϕ = −1 S −1 S S J0 HPF (p)J0 ϕ by (3.8.90). Equation (3.8.89) implies that J0 HPF (p)J0 ϕ = HPF (p)ϕ = S HPF (p, w)ϕ. Thus S S J−1 0 HPF (p, −w)J0 = HPF (p, w)

follows. S Since the operators HPF (p) with p ≠ 0 and different polarization vectors from S (3.8.87) are unitary equivalent with HPF (n3 |p|) with e± in (3.8.87), we have

Corollary 3.109 (Degeneracy of bound states). Let φ̂ be rotation-invariant and M be S the multiplicity of bound states of HPF (p). Then M is an even number. In particular, whenever a ground states exists, it is degenerate. The discussion mentioned above can be extended to the Hamiltonian without spin. With a rotation-invariant φ,̂ similar to (3.8.86) also HPF (p) with p/|p| = n and coherent polarization vectors in direction n satisfies eiϕJf HPF (p)e−iϕJf = HPF (p), ϕ ∈ ℝ. Thus HPF (p) with coherent polarization vectors in direction n = p/|p| has the symmetry SOfield (3) × helicity.

3.8 The Pauli–Fierz model with spin

| 437

Since σ(Jf ) = ℤ, this gives rise to the decomposition 0

ℱrad = ⨁ ℱrad (w), HPF (p) = ⨁ HPF (p, w) w∈ℤ

w∈ℤ

0 where ℱrad (w) denotes the subspace spanned by eigenvectors at eigenvalue w ∈ ℤ of Jf . By Corollary 3.73, as soon as p = 0 the ground state of HPF (p = 0) is unique for any e ∈ ℝ. It is also known that HPF (p) has the unique ground state for sufficiently small |p| and |e|. We may suppose that the polarization vectors of HPF (p) is coherent in direction p/|p|, since HPF (p) with different polarization vectors are unitary equivalent. Suppose that the ground state Ψg (p) of HPF (p) is unique. Then Ψg (p) must belong to 0 some ℱrad (w). The only possibility is z = 0, since HPF (|p|n, z) ≅ HPF (|p|n, −z), z ∈ ℤ, by Lemma 3.108. This implies Jf Ψg (p) = 0. We now construct the functional integral representation for the translation invariS ant Pauli–Fierz Hamiltonian with spin 1/2. As before, we transform HPF on ℂ2 ⊗ ℱrad ℤ2 2 2 to HPF on ℓ (ℤ2 ) ⊗ L (Q ) which is defined by

(HPF2 (p)Ψ)(θ) = (HPF (p) + Ĥ d (0, θ))Ψ(θ) + Ĥ od (0, −θ)Ψ(−θ), ℤ

where

e Ĥ d (0, θ) = − θB̂ 3 (0), 2 e Ĥ od (−θ, 0) = − (B̂ 1 (0) − iθB̂ 2 (0)) 2

2 with θ ∈ ℤ2 . Moreover, HPF,ε (p) is defined by HPF2 (p) with off-diagonal Ĥ od replaced ℤ by Ψε (Ĥ od ), which is introduced to avoid the off-diagonal part of H 2 (p) becoming

ℤ

ℤ

PF

S

singular. The strategy of constructing the functional integral representation of e−tHPF (p) is similar to that of the spinless case.

Theorem 3.110 (Functional integral representation for Pauli–Fierz Hamiltonian with spin 1/2 and a fixed total momentum). Let Φ, Ψ ∈ ℓ2 (ℤ2 ) ⊗ L2 (Q ). (1) It follows that ℤ2

ℤ2

lim(Φ, e−tHPF,ε (p) Ψ) = (Φ, e−tHPF (p) Ψ) ε↓0

and ℤ2

(Φ, e−tHPF,ε (p) Ψ) = et ∑ 𝔼0,α [eip⋅Bt (J0 Φ(θα ), eXt (ε) Jt e−iPf ⋅Bt Ψ(θNt ))], ̂

α=1,2

where the exponent Xt (ε) is given by t

t

0

0

̃ − Bs ))dBs − ∫ Ĥ E,d (Bs , θNs , s)ds Xt (ε) = −ie ∫ Â E (js φ(⋅ t+

+ ∫ log(−Ψε (Ĥ E,od (Bs , −θNs− , s)))dNs . 0

438 | 3 The Pauli–Fierz model by path measures (2) Let W0 ⊂ X × 𝒮 × QE be defined by t+

} { |e| W0 = {∫ log ( √|b1 (s, Bs )|2 + |b2 (s, Bs )|2 ) dNs > −∞} , 2 } {0 ̃ − Bs )). Then where bα (s, Bs ) = B̂ Eα (js φ(⋅ S

(Φ, e−tHPF (p) Ψ) = et ∑ 𝔼0,α [eip⋅Bt (J0 Φ(θα ), eXt 1W0 Jt e−iPf ⋅Bt Ψ(θNt ))], ̂

α=1,2

where the exponent Xt is given by t

t

t+

0

0

0

̃ − Bs ))dBs − ∫ Ĥ E,d (Bs , θNs , s)ds + ∫ log(−Ĥ E,od (Bs , −θNs− , s))dNs . Xt = −ie ∫ Â E (js φ(⋅ Proof. The proof of (1) goes along the same lines as in Theorem 3.70, and for (2) in addition to the proof of Theorem 3.70 it can be shown by mimicking the proof of Theorem 3.100. From Theorem 3.110, we can derive energy inequalities in a similar manner to Corollary 3.72 for the spinless case. Write ℤ E(p, A,̂ B̂ 1 , B̂ 2 , B̂ 3 ) = inf Spec(HPF2 (p)),

and define HPF2 (p, 0) by ℤ

(HPF2 (p, 0)Ψ)(θ) = (HPF (p) + Ĥ d (0, θ))Ψ(θ) − |Ĥ od (0, −θ)|Ψ(−θ), ℤ

where |Ĥ od (0, −θ)| =

|e| √ ̂ B1 (0)2 2

+ B̂ 2 (0)2 . This corresponds to

e ̂ B (0) 1 2 3 (p − Pf )2 + Hrad − ( |e| √ ̂ 2 B1 (0)2 + B̂ 2 (0)2 2

|e| √ ̂ B1 (0)2 + B̂ 2 (0)2 2 ) − e2 B̂ 3 (0)

in ℂ2 ⊗ L2 (Q ). Note that |Ĥ od (0, −θ)| is independent of θ. We have the energy comparison inequality Corollary 3.111 (Diamagnetic inequality). It follows that ℤ2

ℤ2

|(Φ, e−tHPF (p) Ψ)| ≤ (|Φ|, e−tHPF (p,0) |Ψ|)

(3.8.91)

and max

(αβγ)=(123),(231),(312)

E(0, 0, √B̂ 2α + B̂ 2β , 0, B̂ γ ) ≤ E(p, A,̂ B̂ 1 , B̂ 2 , B̂ 3 ).

(3.8.92)

3.9 Relativistic Pauli–Fierz model | 439

Proof. We have S

|(Φ, e−tHPF (p) Ψ)| ≤ et lim ∑ 𝔼0,α [(J0 |Φ(θα )|, eXt

⊥

ε↓0

θ∈ℤ2

(ε)

Jt e−iPf ⋅Bt |Φ(θNt )|)]

= RHS (3.8.91), where Xt⊥ (ε) is given by (3.8.80). Equation (3.8.92) is immediate from (3.8.91) and a similar argument to (3.8.82).

3.9 Relativistic Pauli–Fierz model 3.9.1 Definition of relativistic Pauli–Fierz Hamiltonian In quantum mechanics, the relativistic Schrödinger operator with a vector potential a is defined by HR (a) = √(−i∇ − a)2 + m2 − m + V, and the functional integral representation of e−tHR (a) is known. A key element in the construction of the Feynman–Kac formula of e−tHR (a) is to use the Lévy subordinator (Tt )t≥0 on a suitable probability space (𝒯 , ℬ𝒯 , ν) such that 𝔼0ν [exp(−uTt )] = exp(−t(√2u + m2 − m)). 3 3 ∞ 3 2 For simplicity, we write 𝔼x,0 for 𝔼x,0 𝒲×ν . Suppose that V ∈ L (ℝ ), a ∈ (Lloc (ℝ )) and 3 1 ∇ ⋅ a ∈ Lloc (ℝ ). Then we see that Tt

(f , e−tHR (a) g) = ∫ 𝔼x,0 [f (B0 )g(BTt )e−i ∫0

t

a(Bs )∘dBs − ∫0 V(BTs )ds

e

]dx.

ℝ3

In this section, the analogue version of the Pauli–Fierz model is defined and its functional integral representation is given. The relativistic Pauli–Fierz Hamiltonian is defined by HRPF = √(−i∇ − eA)2 + m2 − m + V + Hrad on ℋPF as a self-adjoint operator. A rigorous definition of HRPF has however a difficulty caused by the term containing the square root. In Sections 3.9.1–3.9.4, we use Assumption 3.112 below unless otherwise stated.

440 | 3 The Pauli–Fierz model by path measures Assumption 3.112. The following conditions hold. ̂ ̂ ̂ (1) Charge distribution: φ ∈ S 󸀠 (ℝ3 ), φ(−k) = φ(k) ∈ L2 (ℝ3 ); and ω3/2 φ,̂ √ωφ,̂ φ/ω (2) External potential: V satisfies that there exist 0 ≤ a < 1 and 0 ≤ b such that ‖Vf ‖ ≤ a‖√−Δf ‖ + b‖f ‖.

(3.9.1)

In Assumptions 3.112, we add the extra condition ω3/2 φ̂ ∈ L2 (ℝ3 ) to Assumption 3.4, and instead of V ∈ ℛKato , we suppose that V is relatively bounded with respect to √−Δ. Although a standard way to define (−i∇ − eA)2 + m2 as a self-adjoint operator is to take the self-adjoint operator associated with the quadratic form 3

F, G 󳨃→ ∑ ((−i∇ − eA)μ F, (−i∇ − eA)μ G) + m2 (F, G), μ=1

instead of this we proceed by finding a core of (−i∇−eA)2 +m2 by using functional integration. In Section 3.3.2, we defined the self-adjoint operator Ĥ PF (A)̂ by the generator of a C0 -semigroup, and have seen that 1 Ĥ PF (A)̂ ⊃ (−i∇ − eA)̂ 2 ⌈DPF . 2 Let D = D(Δ) ∩ C ∞ (N). ̂ ̂ ̂ and ω3/2 φ,̂ φ/√ω Lemma 3.113. Suppose φ(−k) = φ(k) ∈ L2 (ℝ3 ). Then 21 (−i∇ − eA)̂ 2 is essentially self-adjoint on D . Proof. By (3.3.8), we have (F, e−t HPF (A) G) = ∫ 𝔼x [(F(B0 ), e−ieA(Lt ) G(Bt ))] dx, ̂

̂

̂

(3.9.2)

ℝ3 t

μ

̃ − Bs )dBs . By using (3.9.2), we shall show that where we recall that Lt = ⨁3μ=1 ∫0 φ(⋅

e

−t Ĥ PF (A)̂

leaves D invariant. Then the lemma follows. ̂ ̂ First, it can be proven that e−t HPF (A) D ⊂ D(Δ) in a similar manner to (3.5.11) of ̂ ̂ Lemma 3.44 with Hrad replaced by N. To see that e−t HPF (A) D ⊂ C ∞ (N), take n ∈ ℕ and F, G ∈ D(N n ). We have (N n F, e−t HPF (A) G) = ∫ 𝔼x [(N n F(B0 ), e−ieA(Lt ) G(Bt ))] dx. ̂

̂

̂

ℝ3

Note that 2 ̂ ̂ ̂ t ) + e ‖Lt ‖2 , eieA(Lt ) Ne−ieA(Lt ) = N − eΠ(L 2

3.9 Relativistic Pauli–Fierz model | 441

̂ ) = i[N, A(f ̂ )] is the conjugate momentum, and thus where Π(f ̂ t) + (N n F, e−t HPF (A) G) = ∫ 𝔼x [(F(B0 ), e−ieA(Lt ) (N − eΠ(L ̂

̂

̂

ℝ3

n

e2 ‖L ‖2 ) G(Bt ))] dx. 2 t (3.9.3)

By the BDG-type inequality, 󵄩󵄩 t 󵄩󵄩2n 󵄩 󵄩 [󵄩󵄩󵄩 ̃ ] (2n)! n ̂ 2n 󵄩 μ󵄩 𝔼 [󵄩󵄩∫ φ(⋅ − Bs )dBs 󵄩󵄩󵄩 ] ≤ n t ‖φ‖L2 (ℝ3 ) , 󵄩󵄩 󵄩󵄩 2 󵄩󵄩0 󵄩󵄩L2 (ℝ3 ) [ ] x

and we obtain n 󵄩󵄩2 󵄩󵄩 2 󵄩 ̂ t ) + e ‖Lt ‖2 ) F(Bt )󵄩󵄩󵄩󵄩 ] dx ≤ C 2 ‖(N + 1)n F‖2 ∫ 𝔼x [󵄩󵄩󵄩󵄩(N − eΠ(L n 󵄩󵄩 2 󵄩󵄩 󵄩 3

(3.9.4)

ℝ

with a constant Cn . A combination of (3.9.3) and (3.9.4) gives |(N n F, e−t HPF (A) G)| ≤ Cn ‖F‖‖(N + 1)n G‖. ̂

̂

̂ ̂ ̂ ̂ This implies e−t HPF (A) C ∞ (N) ⊂ C ∞ (N) and e−t HPF (A) D ⊂ D follows. Hence Ĥ PF (A)̂ is essentially self-adjoint on D .

̂ D by the same symbol We keep denoting the self-adjoint extension of Ĥ PF (A)⌈ Ĥ PF (A)̂ for simplicity, and we define √2Ĥ PF (A)̂ + m2 by the spectral resolution of ̂ Ĥ PF (A). Definition 3.114 (Relativistic Pauli–Fierz Hamiltonian). The relativistic Pauli–Fierz Hamiltonian is defined by HRPF = √2Ĥ PF (A)̂ + m2 − m + Hrad + V. As in the classical case in addition to Brownian motion, we need a subordinator to construct a functional integration for e−tHRPF . Recall that (Tt )t≥0 is the subordinator on a probability space (𝒯 , ℬ𝒯 , ν). Since 2 −m) ̂ √2Ĥ PF (A)+m

(F, e−t(

G) = 𝔼0ν [(F, e−Tt HPF (A) G)], ̂

̂

(3.9.5)

combining the functional integral representation of e−t HPF (A) with (3.9.5) we immediately have that ̂

2 −m) ̂ √2Ĥ PF (A)+m

(F, e−t(

̂

G) = ∫ 𝔼x,0 [(F(B0 ), e−ieA(LTt ) G(BTt ))] dx. ̂

ℝ3

(3.9.6)

442 | 3 The Pauli–Fierz model by path measures 3.9.2 Functional integral representations Now we will construct the functional integral representation of e−tHRPF through the Trotter product formula. We set Tkin = √2Ĥ PF (A)̂ + m2 − m.

(3.9.7)

Lemma 3.115. Suppose V ∈ C0∞ (ℝ3 ). Then t

t

2n

t

(F, (e− 2n Tkin e− 2n Hrad e− 2n V ) G) x,0

= ∫𝔼

[(J0 F(BT0 ), e

−ie E (Ktrel (n))

Jt G(BTt )) e

n

− ∑2j=0

t 2n

V(BTt ) j

] dx,

ℝ3

where Ktrel (n) Ttj

with tj = tj/2n , and ∫T S

tj−1

μ

2n

3

Ttj

̃ − Bs )dBμs = ⨁ ∑ ∫ jtj−1 φ(⋅ μ=1 j=1 Ttj−1

̃ − Bs )dBμs denotes an L2 (ℝ4 )-valued stochastic integral jtj−1 φ(⋅

̃ − Bs )dBs evaluated at T = Ttj−1 and S = Ttj . ∫T jtj−1 φ(⋅ Proof. By the formula J∗t Js = e−|s−t|Hrad and (3.9.6), we have t

t

2n

t

(F, (e− 2n Tkin e− 2n Hrad e− 2n V ) G) = ∫ 𝔼x,0 [Un e

n

− ∑2j=0

t 2n

V(BTt ) j

],

ℝ3

where 2n

Un = (J0 F(BT0 ), ∏ (Jtj−1 e

Tt

j tj−1

3 ̂ −ieA(⨁ μ=1 ∫T

j=1

T 3 ̂ −ieA(⨁ μ=1 ∫S

μ

μ ̃ φ(⋅−B r )dBr ) ∗ Jtj−1 ) Jt G(BTt )) ,

3

T

μ

̃ ̃ φ(⋅−B r )dBr ) ∗ r )dBr ) and we see that Jt e Jt = Et e−ieAE (⨁μ=1 ∫S jt φ(⋅−B Et by the definition of Jt and Et . Then by the Markov property of projection E𝒪 , projection Et󸀠 s can be removed, and thus the lemma follows. ̂

(Ktrel (n))t≥0 can be regarded as a sequence of an ⊕3 L2 (ℝ4 )-valued random process on the product probability space (X × 𝒯 , ℬ(X ) × ℬ𝒯 , 𝒲 x ⊗ ν). By the Itô isometry, we have ̂ √ω‖2L2 (ℝ3 ) 𝔼x [‖Ktrel (n)‖2⊕3 L2 (ℝ4 ) ] = 3Tt ‖φ/

(3.9.8)

for each Tt . We will show that Ktrel (n) has a limit as n → ∞ in some sense. Let Nν ∈ ℬ𝒯 be a null set, i. e., ν(Nν ) = 0, such that for arbitrary w ∈ 𝒯 \ Nν , the path t 󳨃→ Tt (w) is nondecreasing and right-continuous, and has the left-limit.

3.9 Relativistic Pauli–Fierz model | 443

Lemma 3.116. For each w ∈ 𝒯 \ Nν and each t ≥ 0, the sequence (Ktrel (n))n∈ℕ strongly converges in L2 (X ) ⊗ (⊕3 L2 (ℝ4 )) as n → ∞, i. e., there exists Ktrel ∈ L2 (X ) ⊗ (⊕3 L2 (ℝ4 )) such that limn→∞ 𝔼x [‖Ktrel (n) − Ktrel ‖2 ] = 0. Proof. Fix w ∈ 𝒯 \ Nν . Set In = Ktrel (n). It is sufficient to show that (In )n∈ℕ is a Cauchy sequence in L2 (X ) ⊗ (⊕3 L2 (ℝ4 )). We have 2n

3

Tt2m

̃ − Bs )dBμs , In+1 − In = ⨁ ∑ ∫ (jt2m−1 − jt2m−2 )φ(⋅ μ=1 m=1

Tt2m−1

where tj = tj/2n+1 . Thus 2n

Tt2m

[ ̃ − Bs )‖2 ds] 𝔼 [‖In+1 − In ‖ ] = 3 ∑ 𝔼 [ ∫ ‖(jt2m−1 − jt2m−2 )φ(⋅ ] x

2

m=1

x

[Tt2m−1

]

by the Itô isometry (3.9.8). Notice that ‖(jt − js )f ‖2 = 2(f ̂, (1 − e−|s−t|ω )f ̂). Thus 2n

t

̂ √ω, (1 − e− 2n+1 ω )φ/ ̂ √ω)(Tt2m − Tt2m−1 ). 𝔼 [‖In+1 − In ‖ ] ≤ 3 ∑ 2(φ/ x

2

m=1

n

Since Tt = Tt (w) is not decreasing in t, ∑2m=1 (Tt2m − Tt2m−1 ) ≤ Tt follows. Thus 𝔼x [‖In+1 − In ‖2 ] ≤ 3Tt

t ̂ √ω‖2 . ‖φ/ 2n

Hence we have m

̂ √ω‖ ∑ ( 𝔼x [‖Im − In ‖2 ] ≤ (√3tTt ‖φ/ j=n+1

1 j )) √2

2

for m > n. The right-hand side above converges to zero as n, m → ∞. Then the sequence (In )n∈ℕ is a Cauchy sequence for almost surely 𝒲 x ⊗ ν. Then the lemma follows. For each τ ∈ 𝒯 , there exist jump points 0 < s1 < . . . < sn ≤ t. Then for each τ ∈ 𝒯 , Δj = Tsj + − Tsj > 0 for j = 1, . . . , n and integral Ktrel is informally written as 3

n

3

Tt

̃ − Bs )dBμs = ⨁ ∫ jT ∗ φ(⋅ ̃ − Bs )dBμs . Ktrel = ⨁ ∑ ∫ jsj φ(⋅ s μ=1 j=1 Δj

Here Ts∗ = inf{t | Tt = s}.

μ=1 0

444 | 3 The Pauli–Fierz model by path measures Lemma 3.117. Let V = 0. Then rel

(F, e−t(Tkin + Hrad ) G) = ∫ 𝔼x,0 [(J0 F(BT0 ), e−ieAE (Kt ) Jt G(BTt ))] dx. ̂

̇

ℝ3

Proof. By the Trotter product formula and Lemma 3.115, we have n

(F, e−t(Tkin + Hrad ) G) = lim (F, (e−t/2 ̇

n→∞

2n Ĥ PF (A)̂ −t/2n Hrad

e

) G) rel

= lim ∫ 𝔼x,0 [(J0 F(B0 ), e−ieAE (Kt n→∞

̂

(n))

Jt G(BTt ))] dx.

ℝ3

By Lemma 3.116 and a limiting argument, the lemma follows. The immediate consequence of Lemma 3.117 is the diamagnetic inequality. Let F, G ∈ ℋPF . Then it follows that √−Δ+m2 −m+Hrad )

|(F, e−t(Tkin + Hrad ) G)| ≤ (|F|, e−t( ̇

|(F, e−t(Tkin + Hrad ) G)| ≤ (‖F‖L2 (Q) , e ̇

|G|),

−t(√−Δ+m2 −m)

‖G‖L2 (Q) )L2 (ℝ3 ) .

In a similar way to Lemma 3.30, we also have the lemma below. Lemma 3.118. (1) and (2) follow. (1) If V is √−Δ + m2 − m-form bounded with a relative bound a, then |V| is also (Tkin +̇ Hrad )-form bounded with a relative bound smaller than a. (2) If V is relatively bounded with respect to √−Δ + m2 − m with a relative bound a, then V is also relatively bounded with respect to Tkin +̇ Hrad with a relative bound a. Theorem 3.119 (Functional integral representation for relativistic Pauli–Fierz Hamiltonian). It follows that t

rel

(F, e−tHRPF G) = ∫ 𝔼x,0 [e− ∫0 V(BTs )ds (J0 F(B0 ), e−ieAE (Kt ) Jt G(BTt ))] dx. ̂

(3.9.9)

ℝ3

Proof. When V is bounded and continuous, the theorem can be proven by the Trotter formula t

t

t

2n

(F, e−tHRPF G) = lim (F, (e− 2n Tkin e− 2n Hrad e− 2n V ) G) . n→∞

Furthermore, it can be extended to a general V given in this theorem in the same way as Theorem 3.32. We omit the details. We can shift the time in the functional integral representation of (3.9.9). We see it in the corollary below. Now let (Bt )t∈ℝ be 3-dimensional Brownian motion on the

3.9 Relativistic Pauli–Fierz model |

0

445

μ

̃ − Bs )dBs by whole real line. We define ∫−T jTs∗ φ(⋅ t

0

̃ − ∫ jTs∗ φ(⋅

Bs )dBμs

2n

= lim ∑ n→∞

−Tt

−T−(tj −t)

j=1 −T

∫

̃ − Bs )dBμs . j−(tj−1 −t) φ(⋅

−(tj−1 −t)

T

μ

̃ − Bs )dBs , the right-hand side above Here, tj = tj/2n . In the same way as ∫0 t jTs∗ φ(⋅ 2 2 4 strongly converges in L (X) ⊗ L (ℝ ). Corollary 3.120. The functional integral representation of e−2tHRPF is given by t

0

̂ ̂rel

(F, e−2tHRPF G) = ∫ 𝔼x,0 [e− ∫0 V(BTs )ds−∫−t V(B−T−s )ds (J−t F(B−Tt ), e−ieAE (Kt ) Jt G(BTt ))] dx. ℝ3

Here, 3

Tt

μ=1

0

0

̂rel = ⨁ (∫ jT ∗ φ(⋅ ̃ − Bs )dBμs ) . ̃ − Bs )dBμs + ∫ jT ∗ φ(⋅ K t s s −Tt

Proof. This is proven by means of the shift Ut in the field and the fact that Ts −Tt = Ts−t in law. That is, for t ≤ t1 ≤ . . . ≤ tn , 𝔼0 [f (Tt1 − Tt , . . . , Ttn − Tt )]

= 𝔼0 [f (Tt1 − Tt , Tt2 − Tt1 + Tt1 − Tt , . . . , Ttn − Ttn−1 + ⋅ ⋅ ⋅ Tt1 − Tt )]

= ∫ f (x1 , x2 + x1 , . . . , xn + ⋅ ⋅ ⋅ + x1 )μt1 ,t (dx1 ) ⋅ ⋅ ⋅ μtn ,tn−1 (dxn ),

(3.9.10)

ℝn

where μti ,tj is the distribution of Tti − Ttj , and μti ,tj = μti −tj is satisfied. Here, μr is the distribution of Tr . On the other hand, 𝔼0 [f (Tt1 −t , . . . , Ttn −t )]

= 𝔼0 [f (Tt1 −t , Tt2 −t − Tt1 −t + Tt1 −t , . . . , Ttn −t − Ttn

1

−t

+ ⋅ ⋅ ⋅ + Tt1 −t )]

= ∫ f (x1 , x2 + x1 , . . . , xn + ⋅ ⋅ ⋅ + x1 )μt1 ,t (dx1 ) ⋅ ⋅ ⋅ μtn ,tn−1 (dxn ).

(3.9.11)

ℝn

Comparing (3.9.10) with (3.9.11) above, we can see that 𝔼0 [f (Tt1 − Tt , . . . , Ttn − Tt )] = 𝔼0 [f (Tt1 −t , . . . , Ttn −t )]

(3.9.12)

for each t. By Theorem 3.119, we have rel

2t

(F, e−2tHRPF G) = ∫ 𝔼x,0 [(J0 F(BT0 ), e−ieAE (K2t ) J2t G(BT2t )) e− ∫0 ̂

ℝ3

V(BTs )ds

] dx

446 | 3 The Pauli–Fierz model by path measures and 2t

rel

= ∫ 𝔼x,0 [(J−t F(BT0 ), Ut e−ieAE (K2t ) U−t Jt G(BT2t )) e− ∫0 ̂

V(BTs )ds

] dx.

ℝ3

By the shift of Brownian motion, Bt → Bt−Tt , we have 2t

= ∫ 𝔼x,0 [(J−t F(B−Tt ), e−iαAE (S) Jt G(BT2t −Tt )) e− ∫0 ̂

V(BTs −Tt )ds

] dx,

ℝ3 Ttj −Tt

n

μ

where S = limn→∞ ⨁3μ=1 ∑2⋅2 j=1 ∫T

tj−1 −Tt

̃ − Bs )dBs and, by (3.9.12), we can check jtj−1 −t φ(⋅

that the following identities in law follow: 2t

t

2t

0

t

∫ V(BTs −Tt )ds = ∫ V(B−(Tt−s ) )ds + ∫ V(BTs−t )ds = ∫ V(B−T−s )ds + ∫ V(BTs )ds. 0

t

0

0

−t

̃ − Bs ) we have Furthermore, with F(s) = φ(⋅ Ttj −Tt

2⋅2n

∑

j=1 T

∫

jtj−1 −t F(s)dBμs

tj−1 −Tt

2n

=∑

−T−(tj −t)

j=1 −T

∫

j−(tj−1 −t) F(s)dBμs

Ttj −t

2⋅2n

+ ∑

−(tj−1 −t)

j=2n +1

∫ jtj−1 −t F(s)dBμs .

Ttj−1 −t

Then the corollary follows. In what follows, we set t

0

t

∫ V(BTs )ds + ∫ V(B−T−s )ds = ∫ V(BTs )ds. 0

−t

−t

By Lemma 3.118, we can also extend the functional integral representation to a wider class of external potentials. Let V = V+ − V− be such that V− is relatively form bounded with respect to √−Δ + m2 − m with a relative bound strictly smaller than one and V+ ∈ L1loc (ℝ3 ). Then we define the relativistic Pauli–Fierz Hamiltonian with singular external potential by HRPF = Tkin +̇ V+ −̇ V− +̇ Hrad . 1/2 1/2 ) ∩ D(Hrad ) ∩ D(V+1/2 ) ∩ D(V−1/2 ) is dense. Hence we can also have the Note that D(Tkin following functional integral representation for such singular external potentials. The proof is similar to that of Theorem 3.32.

Theorem 3.121 (Functional integral representation for relativistic Pauli–Fierz Hamiltonian with singular external potential). Suppose Assumption 3.112 but instead of relative boundedness of V = V+ −V− with respect to √−Δ, it is supposed that V− is relatively form

3.9 Relativistic Pauli–Fierz model | 447

bounded with respect to √−Δ + m2 − m with a relative bound strictly smaller than one and V+ ∈ L1loc (ℝ3 ). Then t

rel

(F, e−tHRPF G) = ∫ 𝔼x,0 [e− ∫0 V(BTs )ds (J0 F(B0 ), e−ieAE (Kt ) Jt G(BTt ))] dx. ̂

(3.9.13)

ℝ3

By using this functional integral representation, we can obtain similar results to those of HPF . Corollary 3.122 (Diamagnetic inequality). Let E(e) = inf Spec(HRPF ). Under Assumption 3.112 or the assumptions of Theorem 3.121, it follows that √−Δ+m2 −m +̇ V+ −̇ V− +Hrad )

|(F, e−tHRPF G)| ≤ (|F|, e−t(

|G|)

and E(0) ≤ E(e). ̂ ̂rel

Proof. Since |e−ieAE (Kt ) | = 1, the corollary follows from the functional integral representation of (F, e−tHRPF G). Corollary 3.123 (Positivity improving). Under Assumption 3.112 or the assumptions of Theorem 3.121, e−i(π/2)N e−tHRPF ei(π/2)N is positivity improving. In particular, the ground state of HRPF is unique, if it exists. ̂ ̂rel

Proof. Since J∗0 e−ieAE (Kt ) Jt is positivity improving, the corollary follows from the functional integral representation of (F, e−tHRPF G).

3.9.3 Self-adjointness In this section by using the functional integral representation derived in Theorem 3.119, we show the self-adjointness of HRPF for arbitrary values of coupling constants. To prove this, we find an invariant domain D so that D ⊂ D(HRPF ) and e−tHRPF D ⊂ D in a similar way to that of HPF . Let ω̂ E = ω(−i∇) ⊗ 1 under L2 (ℝ4 ) ≅ L2 (ℝ3 ) ⊗ L2 (ℝ). By the Itô isometry, we have x,0

𝔼

rel 2 [‖ω̂ α/2 E Kt ‖⊕3 L2 (ℝ4 ) ]

= 3𝔼

x,0 [

Tt

̃ − Br )‖2L2 (ℝ3 ) dr ] . ∫ ‖ω̂ α/2 φ(⋅ [0 ]

In particular, rel 2 0 (α−1)/2 ̂ 2L2 (ℝ3 ) 𝔼x,0 [‖ω̂ α/2 φ‖ E Kt ‖⊕3 L2 (ℝ4 ) ] ≤ 3𝔼ν [Tt ]‖ω

and the right-hand side above is finite in the case of m > 0, since 𝔼0ν [Tt ] < ∞. We can rel 4 also estimate forth moment 𝔼x,0 [‖ω̂ α/2 E Kt ‖⊕3 L2 (ℝ4 ) ] in the next lemma.

448 | 3 The Pauli–Fierz model by path measures Lemma 3.124 (BDG-type inequality). Let α > 0 and m > 0, i. e., 𝔼0ν [e−uTt ] = ̂ ω(α−1)/2 φ̂ ∈ L2 (ℝ3 ). Then the BDG-type exp(−t(√2u + m2 − m)). Suppose that φ/√ω, inequality holds: rel 4 (α−1)/2 ̂ 4L2 (ℝ3 ) , 𝔼x,0 [‖ω̂ α/2 φ‖ E Kt ‖⊕3 L2 (ℝ4 ) ] ≤ C‖ω

where C is a constant. Proof. Recall the BDG-type inequality 󵄩󵄩 t 󵄩󵄩󵄩2n 󵄩 [󵄩󵄩󵄩 ̂ α/2 ] (2n)! n (α−1)/2 ̂ 2n μ󵄩 ̃ − Bs )dBs 󵄩󵄩󵄩󵄩 𝔼 [󵄩󵄩ωE ∫ js φ(⋅ φ‖L2 (ℝ3 ) . ] ≤ n t ‖ω 󵄩󵄩 󵄩󵄩 2 󵄩󵄩 󵄩󵄩L2 (ℝ4 ) 0 [ ] x

n

(3.9.14)

μ

3 2 rel Notice that ω̂ α/2 E Kt = s-limn→∞ ⨁μ=1 ∑j=1 aj with

μ aj

Ttj

= ∫ jtj−1 λα (⋅ − Bs )dBμs , Ttj−1

μ and λα = ω̂ α/2 φ̃ and λ̂α = ω(α−1)/2 φ.̂ We fix a μ and set aj = aj for simplicity. aj and ai are independent for i ≠ j and then we have

󵄩󵄩 2n 󵄩󵄩2 2n 2n 󵄩󵄩 ] = ∑ ∑ 𝔼x,0 [( ∫ aj (z)aj󸀠 (z)dz) ( ∫ ai (y)ai󸀠 (y)dy)] 󵄩󵄩∑ aj 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󸀠 󸀠 [󵄩j=1 󵄩L2 (ℝ4 ) ] j,j =1 i,i =1 [ ℝ4 ] ℝ4

󵄩󵄩 x,0 [󵄩

𝔼

We divide the sum above into the diagonal part and the off diagonal part. Then we have 2n

2 x,0 [

= ∑𝔼 j=1

2n

2

2n

( ∫ aj (z) dz) ] + ∑ 𝔼x,0 [ ∫ aj (z)2 dz ] ∑ 𝔼x,0 [ ∫ ai (z)2 dz ] [ ℝ4 ] j=1 [ℝ4 ] i=j̸ [ℝ4 ]

2n 2n

+ ∑ ∑ 𝔼x,0 [ ∫ aj (z)ai (z)dz ∫ aj (y)ai (y)dy] . j=1 i=j̸ ] [ℝ4 ℝ4 We estimate the first term of the right-hand side above. We have by (3.9.14) 2n

2

∑𝔼 j=1

n

n

2 2 󵄨 󵄨2 ̂ 4 ∑ 𝔼0ν [󵄨󵄨󵄨󵄨T t 󵄨󵄨󵄨󵄨 ] . ( ∫ aj (z) dz) ] = ∑ 𝔼x,0 [‖aj ‖4 ] ≤ 6‖ω(α−1)/2 φ‖ n+1 󵄨 2 󵄨 j=1 [ ℝ4 ] j=1

x,0 [

2

By using 𝔼0ν [Ttn ]

sn 1 t2 tetm = ∫ 3/2 exp (− ( + m2 s)) ds < ∞ √2π s 2 s ∞

0

3.9 Relativistic Pauli–Fierz model | 449

for n ≥ 0, and the assumption m > 0, we have 2n

2

∑𝔼 j=1

t

2

mt ̂ 4 n+1 6t‖ω(α−1)/2 φ‖ 1 ( n+1 ) e 2 ∫ √s exp (− ( 2 ( ∫ aj (z) dz) ] ≤ + m2 s)) ds. √2π 2 s 2 0 ] [ ℝ4

∞

2

x,0 [

The right-hand side converges to ∞

1 3t ̂ 4 ∫ √s exp (− m2 s) ds ‖ω(α−1)/2 φ‖ √2π 2 0

as n → ∞. The second term is estimated as 2n

x,0 [

2n

2

x,0 [

∫ aj (z) dz ] ∑ 𝔼 [ℝ4 ] i=j̸

∑𝔼 j=1

2n

2

2 x,0 [

∫ aj (z) dz ] ≤ (∑ 𝔼 j=1 [ℝ4 ]

2

∫ aj (z) dz ]) . [ℝ4 ]

By the Itô isometry, we have 2n

x,0 [

Tt

̂ 2. ∫ aj (z) dz ] = 𝔼x,0 [∫ ‖js λα (⋅ − Bs )‖2 ds] ≤ 𝔼0ν [Tt ]‖ω(α−1)/2 φ‖ [ℝ4 ] [0 ]

∑𝔼 j=1

2

Hence 2n

x,0 [

2n

̂ 4. ∫ aj (z) dz ] ∑ 𝔼x,0 [ ∫ aj (z)2 dz ] ≤ (𝔼0ν [Tt ])2 ‖ω(α−1)/2 φ‖ [ℝ4 ] i=j̸ [ℝ4 ]

∑𝔼 j=1

2

Finally, we estimate the third term. We see that 2n 2n

∑∑𝔼 j=1 i=j̸

2 󵄨󵄨 2n 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 x,0 ] 󵄨 ∫ aj (z)ai (z)dz ∫ aj (y)ai (y)dy ≤ ∫ dz ∫ 󵄨󵄨∑ 𝔼 [aj (z)aj (y)]󵄨󵄨 dy. 󵄨󵄨 󵄨󵄨 󵄨 [ℝ4 ] ℝ4 ℝ4 󵄨j=1 ℝ4

x,0 [

Ttj

Note that 𝔼x,0 [aj (z)aj (y)] = 𝔼x,0 [∫T

tj−1

As (z, j)As (y, j)ds], where we set

As (z, j) = (jtj−1 λα (⋅ − Bs ))(z). By the Schwarz inequality, we have 2

Ttj

2n

[ ] ] ≤ ∫ dz ∫ 𝔼x,0 [ [(∑ ∫ As (z, j)As (y, j)ds) ] dy ℝ4

ℝ4

[

j=1 T

tj−1

Ttj

n

]

Ttj

n

2 [ 2 ] 2 2 ] ≤ ∫ dz ∫ 𝔼x,0 [ [(∑ ∫ As (z, j) ds) (∑ ∫ As (y, j) ds)] dy ℝ4

ℝ4

[

j=1 T

tj−1

j=1 T

tj−1

]

450 | 3 The Pauli–Fierz model by path measures and Fubini’s lemma yields that Ttj

n

n

Ttj

2 2 [ ] 2 ( ( = 𝔼x,0 [ A (z, j) ds) dz ∑ ∑ ∫ ∫ ∫ ∫ As (y, j)2 ds) dy] s [ ] 4

j=1 T

j=1 T

4

ℝ tj−1 tj−1 [ℝ x,0 2 (α−1)/2 4 0 2 (α−1)/2 4 ̂ ] = 𝔼ν [Tt ]‖ω ̂ . = 𝔼 [Tt ‖ω φ‖ φ‖

]

Then the lemma follows. Let Ptot μ = −i∇μ ⊗ 1 + 1 ⊗ Pfμ be the total momentum operator in ℋPF . tot

tot

Lemma 3.125. Let V = 0. Then e−itPμ e−sHRPF eitPμ = e−sHRPF . Proof. We set Pfμ = dΓE (−i∇μ ⊗ 1). We have tot

tot

(F, e−itPμ e−sHRPF eitPμ G) rel

= ∫ 𝔼x,0 [(J0 F(BT0 ), e−it(−i∇μ ) e−itPfμ e−iAE (Kt ) eitPfμ eit(−i∇μ ) Jt G(BTt ))] dx. ̂

ℝ3

Since

rel

rel

e−it(−i∇μ ) e−itPfμ e−iAE (Kt ) eitPfμ eit(−i∇μ ) = e−iAE (Kt ) , ̂

̂

the lemma follows. 1/2 Lemma 3.126. Let V = 0. For F ∈ D(−i∇μ ) and G ∈ D(−i∇μ ) ∩ D(Hrad ), it follows that 1/2 ̂ + ‖φ‖)‖(H ̂ (−i∇μ F, e−tHRPF G) ≤ C ((‖√ωφ‖ rad + 1) G‖ + ‖−i∇μ G‖) ‖F‖.

In particular, e−tHRPF G ∈ D(√−Δ) for t ≥ 0. tot

Proof. Notice that (eis(−i∇μ ) F, e−tHRPF G) = (e−isPf,μ F, e−tHRPF e−isPμ G). We have ̂

(eis(−i∇μ ) F, e−tHRPF G) = ∫ 𝔼x,0 [(J0 F(BT0 ), e+isPfμ e−ieAE (K) e−isPfμ Jt e−is(−i∇μ ) G(BTt ))] dx. ̂

ℝ3

(3.9.15)

Here and in what follows in this proof, we set K = Ktrel . We see that e+isPfμ e−ieAE (K) e−isPfμ = e−ieAE (e ̂

̂

is(−i∇μ ⊗1) K)

.

Take the derivative at s = 0 on both sides of (3.9.15). We have (∇μ F, e−tHRPF G) = ∫ 𝔼x,0 [(J0 F(BT0 ), −ie Eμ (∇μ K)e−ieAE (K) Jt G(BTt ))] dx ̂

ℝ3

+ ∫ 𝔼x,0 [(J0 F(BT0 ), e−ieAE (K) Jt (−∇μ G)(BTt ))] dx. ̂

ℝ3

(3.9.16)

3.9 Relativistic Pauli–Fierz model | 451

It is trivial to see that 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 x,0 −ie E (K) 󵄨󵄨 ≤ ‖F‖‖∇μ G‖. 󵄨󵄨 ∫ 𝔼 [(J0 F(BT ), e J (−∇ G)(B ))] dx t μ Tt 0 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 3 󵄨 󵄨ℝ We can estimate the first term on the right-hand side of (3.9.16) as 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 −ie Eμ (K) x,0 ̂ Jt G(BTt ))] dx 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨 ∫ 𝔼 [(J0 F(BT0 ), AEμ (∇μ K)e 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨ℝ3 ≤ ∫ 𝔼x,0 [‖ Eμ (∇μ K)J0 F(BT0 )‖‖Jt G(BTt )‖] dx ℝ3 1/2 ̂−1/2 By the inequality ‖ Eμ (f )Φ‖ ≤ C(‖f ‖ + ‖ω E f ‖)‖(Hrad + 1) Φ‖ with some constant C > 0, we have 1/2 ≤ C ∫ 𝔼x,0 [(‖∇μ K‖ + ‖ω̂ −1/2 E ∇μ K‖)‖(Hrad + 1) F(BT0 )‖‖G(BTt )‖] dx ℝ3

1/2 ̂ + ‖φ‖)‖(H ̂ ≤ C(‖ω1/2 φ‖ rad + 1) F‖‖G‖.

Then the lemma follows. Lemma 3.127. Let V = 0. Then for F, G ∈ D(Hrad ) it follows that 1/2 2 ̂ + ‖φ‖)‖(H ̂ ̂ √ω‖2 ‖G‖) ‖F‖. (Hrad F, e−tHRPF G) ≤ (‖Hrad G‖ + |e|(‖√ωφ‖ rad + 1) G‖ + |e| ‖φ/

In particular, e−tHRPF G ∈ D(Hrad ) for t ≥ 0. Proof. Let Hrad = dΓE (ω̂ E ). Note that ΠE (ωf̂ ) = i[Hrad , Â E (f )] in the function space. We have rel

(Hrad F, e−tHRPF G) = ∫ 𝔼x,0 [(J0 F(BT0 ), e−ieAE (Kt ) SJt G(BTt ))] dx, ̂

ℝ3

where rel

rel

S = eieAE (Kt ) Hrad e−ieAE (Kt ̂

̂

)

̂ trel ) + e2 g = Hrad − eΠE (ωK

with g = qE (Ktrel , Ktrel ). It is trivial to see that 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 x,0 −ie E (Ktrel ) 󵄨󵄨 ∫ 𝔼 [(J0 F(BT ), e 󵄨󵄨 ≤ ‖F‖‖Hrad G‖. H J G(B ))] dx rad t Tt 0 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨ℝ3 󵄨

452 | 3 The Pauli–Fierz model by path measures In the same way as the estimate of the first term of the right-hand side of (3.9.16), we can see that 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 x,0 −ie E (Ktrel ) rel ̂ t )Jt G(BTt ))] dx󵄨󵄨󵄨 ΠE (ωK 󵄨󵄨󵄨 ∫ 𝔼 [(J0 F(BT0 ), e 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨ℝ3 󵄨 1/2 ̂ + ‖φ‖)‖F‖‖(H ̂ ≤ C(‖√ωφ‖ + 1) G‖ rad with some constant C > 0. Here, we used Lemma 3.124. Finally, we see that g ≤ C‖Ktrel ‖2 and by Lemma 3.124 again, 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 x,0 −ie E (Ktrel ) 󵄨󵄨 ∫ 𝔼 [(J0 F(BT ), e gJt G(BTt ))] dx󵄨󵄨󵄨󵄨 0 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨ℝ3 󵄨󵄨 1/2

x,0

≤ C (∫ 𝔼

[‖Ktrel ‖4 ] ‖F(x)‖2 dx)

̂ √ω‖2 ‖F‖‖G‖. ‖G‖ ≤ C‖φ/

ℝ3

Then the lemma follows. ∞ We recall D∞ = C0∞ (ℝ3 ) ⊗ ℱrad . See (3.7.2) and (3.7.3).

Lemma 3.128. Let m > 0. Then HRPF is essentially self-adjoint on D(√−Δ) ∩ D(Hrad ). ̂ Proof. Suppose V = 0. Let F ∈ D∞ . Since φ/ω ∈ L2 (ℝ3 ), we see that ‖(Tkin +̇ Hrad )F‖2 ≤ C1 ‖√−ΔF‖2 + C2 ‖Hrad F‖2 + C3 ‖F‖2

(3.9.17)

with some constants C1 , C2 and C3 . Since D∞ is a core of √−Δ + Hrad , D(√−Δ) ∩ D(Hrad ) ⊃ D(Tkin +̇ Hrad )

(3.9.18)

follows from a limiting argument. By Lemmas 3.126 and 3.127, we also see that e−t(Tkin + Hrad ) (D(√−Δ) ∩ D(Hrad )) ⊂ (D(√−Δ) ∩ D(Hrad )) . ̇

(3.9.19)

(3.9.18) and (3.9.19) imply that Tkin +̇ Hrad is essentially self-adjoint on the domain D(√−Δ) ∩ D(Hrad ) by Lemma 3.40. Next, we suppose that V satisfies assumptions in the lemma. Then V is also relatively bounded with respect to Tkin +̇ Hrad with a relative bound strictly smaller than one. Then the theorem follows by the Kato–Rellich theorem. We will be able to show that HRPF is essentially self-adjoint on D∞ by limiting arguments. We define additional two dense subspaces E and D0 such that D∞ ⊂ E ⊂ D0 ⊂ D(√−Δ) ∩ D(Hrad ).

3.9 Relativistic Pauli–Fierz model | 453

We shall show below that HRPF ⌈D0 and HRPF ⌈E are essentially self-adjoint, and finally we can conclude that D∞ is a core of HRPF . Let D = {{Ψ

(n) ∞ }n=0

∈ H | Ψ(n) = 0 for all n ≥ n0 with some n0 ≥ 1}

and we set D0 = D(√−Δ) ∩ D(Hrad ) ∩ D . We define H0 = Tkin +̇ Hrad . Lemma 3.129. Let m > 0. Then D is a core of H0 . Proof. Let Pn = 1[0,n] (N) for n ∈ ℕ. Take an arbitrary Ψ ∈ D(√−Δ) ∩ D(Hrad ). Hence Pn Ψ ∈ D . We see that Pn Ψ → Ψ as n → ∞. Since ‖H0 (Pn − Pm )Ψ‖ ≤ C(‖(Pn − Pm )√−ΔΨ‖ + ‖(Pn − Pm )Hrad Ψ‖), we also see that (H0 Pn Ψ)n∈ℕ is a Cauchy sequence in ℋPF . From the closedness of H0 , it follows that Ψ ∈ D(H0 ) and H0 Pn Ψ → H0 Ψ. Thus D is a core of H0 by Lemma 3.128. Let ℰ = {{Ψ

(n) ∞ }n=0

∈ D | Ψ(n) (⋅, k) ∈ C0∞ (ℝ3 ) a. e. k ∈ ℝ3n , n ≥ 1}.

Lemma 3.130. Let m > 0. Then ℰ is a core of H0 . Proof. Take an arbitrary Φ ∈ D . Let 0 ≤ j ∈ C0∞ (ℝ3 ) and g ∈ C0∞ (ℝ3 ) such that ∫ℝ3 j(x)dx = 1, 0 ≤ g(x) ≤ 1 and g(x) = 1 for |x| ≤ 1. For each ε > 0, we set jε (x) =

(n) ∞ (n) ε−3 j(x/ε) and Φ(n) ε,L (x, k) = g(x/L) ∫ℝ3 jε (x − y)Φ (y, k)dy. Define Φε,L = {Φε,L }n=0 . We see that Φε,L → Φ, −i∇μ Φε,L → −i∇μ Φ and Hrad Φε,L → Hrad Φ strongly as ε ↓ 0 and L → ∞. Then by (3.9.17) and the closedness of H0 , we see that Φ ∈ D(H0 ) and limL→∞ limε↓0 H0 Φε,L = H0 Φ in ℋPF . Thus the lemma follows from Lemma 3.129.

Theorem 3.131. Let m > 0. Then H0 is essentially self-adjoint on D∞ . Proof. Suppose that m > 0. Let F ∈ ℰ . Then it follows that ‖ − ΔF‖ + ‖Hrad F‖ ≤ C‖(−Δ + Hrad + 1)F‖

(3.9.20)

with some constant C > 0. Note that −Δ + Hrad is essentially self-adjoint on D∞ . There exists a sequence (Fn )n∈ℕ ⊂ D∞ such that Fn → F, and (−Δ + Hrad )Fn → (−Δ + Hrad )F as n → ∞. From (3.9.20), it follows that −ΔFn → −ΔF and Hrad Fn → Hrad F as n → ∞. Then we can also see that (H0 Fn )n∈ℕ is a Cauchy sequence by (3.9.17) and H0 Fn → H0 F as n → ∞ follows. Thus D∞ is a core of H0 by Lemma 3.130. Furthermore, we can establish the self-adjointness of HRPF . The key inequality to show the self-adjointness of HRPF on D(√−Δ) ∩ D(Hrad ) is the following inequality.

454 | 3 The Pauli–Fierz model by path measures Lemma 3.132. Let m > 0 and V = 0. Then there exists a constant C such that ‖√−ΔF‖2 + ‖Hrad F‖2 ≤ C‖(Tkin +̇ Hrad + 1)F‖2

(3.9.21)

for all F ∈ D(√−Δ) ∩ D(Hrad ). Proof. Suppose that m = 0. In the case of m > 0, the proof is parallel with that of m = 0, but rather easier. Let F ∈ D∞ . We have ‖H0 F‖2 = ‖|−i∇ − eA|F‖2 + ‖Hrad F‖2 + 2ℜ(|−i∇ − eA|F, Hrad F).

(3.9.22)

We estimate ‖|−i∇ − eA|F‖2 and ℜ(|−i∇ − eA|F, Hrad F) on the right-hand side of (3.9.22) from below. Since the operator |−i∇ − eA| is singular, we introduce an artificial positive mass M > 0 and TM = √(−i∇ − eA)2 + M 2 . Note that |−i∇ − eA| = TM + (|−i∇ − eA| − TM ). Let Eλ be the spectral resolution of nonnegative self-adjoint operator (−i∇ − eA)2 . Then ∞ 󵄨󵄨 󵄨󵄨 M2 󵄨 󵄨󵄨 2 2 2 ‖(|−i∇ − eA| − TM )F‖2 = ∫ 󵄨󵄨󵄨 󵄨 d‖Eλ F‖ + M ‖F‖ 󵄨󵄨 √λ + M 2 + √λ 󵄨󵄨󵄨 0

≤ (1 + M 2 )‖F‖2 . Then B = |−i∇ − eA| − TM is bounded, and |−i∇ − eA| = TM + B. We have ℜ(|−i∇ − eA|F, Hrad F) = ℜ(TM F, Hrad F) + ℜ(BF, Hrad F). Since F ∈ D∞ , Hrad F ∈ D(−Δ) ∩ D(Hrad ). In particular, Hrad F ∈ D(TM ) and then Hrad F ∈ 1/2 D(TM ). Furthermore, we show that 1/2 TM F ∈ D(Hrad )

(3.9.23)

in Lemma 3.133 below. So we can see that 1/2 1/2 1/2 1/2 ℜ(|−i∇ − eA|F, Hrad F) = (TM F, Hrad TM F) + ℜ(TM F, [TM , Hrad ]F) + ℜ(BF, Hrad F) 1/2 1/2 ≥ ℜ(TM F, [TM , Hrad ]F) + ℜ(BF, Hrad F).

We estimate (BF, Hrad F). Since ‖BF‖ ≤ M‖F‖, we see that for each ε > 0 there exists C1 > 0 such that ℜ(BF, Hrad F) ≥ −ε‖Hrad F‖2 − C1 ‖F‖2 .

3.9 Relativistic Pauli–Fierz model | 455

1/2 1/2 On the other hand, we estimate ℜ(TM F, [TM , Hrad ]F). Let ε > 0 be given. Then there exists C2 > 0 such that 1/2 1/2 1/2 ℜ(TM F, [TM , Hrad ]F) ≥ −c‖TM F‖ ‖(Hrad + 1)F‖

≥ −ε‖|−i∇ − eA|F‖2 − ε‖Hrad F‖2 − C2 ‖F‖2 .

(3.9.24)

The first inequality of (3.9.24) is derived from 1/2 ‖[TM , Hrad ]F‖ ≤ c‖(Hrad + 1)1/2 F‖

(3.9.25)

with some constant c > 0. This is shown in Lemma 3.133 below. Hence we have ℜ(|−i∇ − eA|F, Hrad F) ≥ −ε‖|−i∇ − eA|F‖2 − 2ε‖Hrad F‖2 − (C1 + C2 )‖F‖.

(3.9.26)

We next estimate ‖|−i∇ − eA|F‖. Note that ‖−i∇μ F‖2 = ‖(−i∇μ − eAμ )F‖2 + 2ℜ(eAμ F, (−i∇μ − eAμ )F) + ‖eAμ F‖2 . For each ε > 0, there exist C5 > 0 and C6 > 0 such that |ℜ(AF, (−i∇ − eA)F)| ≤ ε(‖|−i∇ − eA|F‖2 + ‖Hrad F‖2 ) + C5 ‖F‖2 ,

‖√−ΔF‖2 ≤ (1 + ε)‖|−i∇ − eA|F‖2 + ε‖Hrad F‖2 + C6 ‖F‖2 .

Hence we have ‖|−i∇ − eA|F‖2 ≥

C ε 1 √ ‖ −ΔF‖2 − ‖H F‖2 − 6 ‖F‖2 . 1+ε 1 + ε rad 1+ε

(3.9.27)

By (3.9.26), (3.9.27) and (3.9.22), we can see (3.9.21) for F ∈ D∞ . Let F ∈ D(HRPF ). Since HRPF is essentially self-adjoint on D∞ , by a limiting argument we can see (3.9.21) for F ∈ D(HRPF ). Lemma 3.133. Equations (3.9.23) and (3.9.25) hold true. 1/2 Proof. Note that TM = (2K + M 2 )1/4 , where K = 21 (−i∇ − eA)2 . We have ∞

2

(2K + M 2 )α/2 = Cα ∫ (1 − e−λ(2K+M ) ) 0

dλ λ1+α/2

for 0 ≤ α < 2 with some constant Cα . From this formula, we have ∞

{ } dλ 2 ̂ 1/2 (F, TM G) = C1/2 ∫ {(F, G) − e−λM ∫ 𝔼x [(F(B0 ), e−iA(L2λ ) G(B2λ ))]dx } 5/4 . λ 0 { } ℝ3

456 | 3 The Pauli–Fierz model by path measures 2λ

̃ − Bs )dBμs . Let F ∈ D(Hrad ) and G ∈ D∞ . Thus Hrad G ∈ D∞ . We Here, L2λ = ⨁3μ=1 ∫0 φ(⋅ have 1/2 (Hrad F, TM G)

∞

{ } dλ 2 ̂ = C1/2 ∫ {(Hrad F, G) − e−λM ∫ 𝔼x [(Hrad F(B0 ), e−iA(L2λ ) G(B2λ ))]dx } 5/4 . λ 0 { } ℝ3

Then 1/2 1/2 (Hrad F, TM G) − (F, TM Hrad G) ∞

= −C1/2 ∫ 0

2

̂ e−λM dλ ∫ 𝔼x [(F(B0 ), [Hrad , e−iA(L2λ ) ]G(B2λ ))]dx. λ5/4

(3.9.28)

ℝ3

̂ ̂ ̂ 2λ ) + ξ ), where ξ = q0 (L2λ , L2λ ) and Π(L ̂ 2λ ) = Note that [Hrad , e−iA(L2λ ) ] = e−iA(L2λ ) (Π(L ̂ 2λ )]. Thus we see that [Hrad , A(L

󵄨󵄨 󵄨󵄨 ∞ 2 󵄨󵄨 { } e−λM dλ 󵄨󵄨󵄨󵄨 ̂ 2λ ) 󵄨󵄨 x −iA(L 󵄨󵄨 ]G(B2λ ))]dx} 󵄨󵄨󵄨 ∫ { ∫ 𝔼 [(F(B0 ), [Hrad , e 󵄨󵄨 λ5/4 󵄨󵄨󵄨 󵄨󵄨 } 󵄨󵄨 0 {ℝ3 ∞

≤C∫ 0

√λ + λ −λM 2 e dλ‖F‖‖(Hrad + 1)1/2 G‖ λ5/4

1/2 G)| ≤ C‖F‖‖(Hrad + 1)G‖ with some constant with some constant C. Then |(Hrad F, TM 1/2 C > 0. Hence TM G ∈ D(Hrad ) follows. Next, we shall prove (3.9.25). The proof of (3.9.25) is similar to that of (3.9.23). Let 1/2 G ∈ D∞ . From (3.9.28), it follows that |(F, [Hrad , TM ]G)| ≤ C‖F‖‖(1 + Hrad )1/2 G‖. This implies (3.9.25).

Theorem 3.134 (Self-adjointness). Let m ≥ 0. Then D(√−Δ) ∩ D(Hrad ), and essentially self-adjoint on D∞ .

HRPF

is

self-adjoint

on

Proof. Suppose that V = 0. We write H (m) for HRPF to emphasize m-dependence. Let m > 0. Then H (m) is essentially self-adjoint on D(√−Δ) ∩ D(Hrad ). On the other hand by (3.9.21), H (m) ⌈D(√−Δ)∩D(Hrad ) is closed. Then H (m) is self-adjoint on D(√−Δ)∩D(Hrad ). Note that H (0) = H (m) +(H (0) −H (m) ) and H (0) −H (m) is bounded. Then H (0) is also self-adjoint on D(√−Δ) ∩ D(Hrad ) for V = 0. Finally, let V be potential satisfying assumptions given

in this theorem. Then V is also relatively bounded with respect to H (m) with a relative bound strictly smaller than one. Then the theorem follows from Kato–Rellich theorem and Theorem 3.131.

Example 3.135 (Hydrogen-like atom). Let d = 3. A spinless hydrogen-like atom is defined by introducing the Coulomb potential VCoulomb (x) = −g/|x|, g > 0, which is relatively form bounded with respect to √−Δ + m2 with a relative bound strictly smaller

3.9 Relativistic Pauli–Fierz model | 457

than one if g ≤ 2/π. Furthermore, if g < 1/2, VCoulomb is relatively bounded with respect to √−Δ + m2 with a relative bound strictly smaller than one. Let Â Λ be the quantized ̂ radiation field with the cutoff function φ(k) = 1|k|≤Λ (k)/√(2π)3 , where Λ > 0 describes a ultraviolet cutoff parameter. By Lemma 3.118, when g < 2/π, V is relatively form bounded with respect to Tkin +̇ Hrad and HRPF is well-defined as a self-adjoint operator. Furthermore, by Theorem 3.134 when g < 1/2, HRPF is self-adjoint on D(√−Δ) ∩ D(Hrad ). All the statements mentioned above are true for arbitrary values of coupling constant e ∈ ℝ and Λ > 0. 3.9.4 Nonrelativistic limit of relativistic Pauli–Fierz Hamiltonian In the quantum mechanics we can see that √−c2 Δ + m2 c4 − mc2 + V → − 1 Δ + V 2m as c → ∞ strongly in the sense of semigroup. We introduce the speed of light c in HRPF , and it turns to be HRPF = √c2 (−i∇ − eA)2 + m2 c4 − mc2 + V + Hrad . It is interesting to investigating the nonrelativistic limit of the relativistic Pauli–Fierz Hamiltonian. Here the nonrelativistic limit describes the nonrelativistic limit of the charged particle, hence we set c = 1 in A and Hrad . The subordinator (Tt (c))t≥0 with parameter c in (𝒯 , ℬ𝒯 , ν) satisfies that 𝔼0ν [exp(−uTt (c))] = exp(−t(√2c2 u + m2 c4 − mc2 )).

(3.9.29)

In Lemma 4.228 in Volume 1 it is shown that lim 𝔼0ν [f (Tt (c))] = f (t/m)

c→∞

for every bounded continuous function f . From this fact we can show that √−c2 Δ+m2 c4 −mc2 +V)

(f , e−t(

→ ∫ 𝔼x [f ̄(x)g(Bt/m )e

t

g) = ∫ 𝔼x,0 [f ̄(x)g(BTt (c) )e− ∫0 V(BTs (c) )ds ]dx

t − ∫0

ℝ3 V(Bs/m )ds

1

]dx = (f , e−t(− 2m Δ+V) g).

ℝ3

In a similar way to this we shall show the nonrelativistic limit of the relativistic Pauli– Fierz Hamiltonian. The Pauli–Fierz Hamiltonian with particle mass m > 0 is defined by HPF =

1 (−i∇ − eA)2 + V + Hrad . 2m

458 | 3 The Pauli–Fierz model by path measures Using (Tt (c))t≥0 we can see that t

rel

(F, e−tHRPF G) = ∫ 𝔼x,0 [e− ∫0 V(BTs (c) )ds (J0 F(x), e−ieAE (Kt ̂

(c))

Jt G(BTt (c) ))]dx,

(3.9.30)

ℝ3

where Ktrel (c) is defined by Ktrel with Tt replaced by Tt (c), and the functional integral representation of e−tHPF with mass m is given by t

(F, e−tHPF G) = ∫ 𝔼x [e− ∫0 V(Bs/m )ds (J0 F(x), e−ieAE (Kt ) Jt G(Bt/m ))]dx, ̂

(3.9.31)

ℝ3 t/m

where Kt = ⨁3μ=1 ∫0

μ

̃ − Bs ) ∘ dBs . The purpose of this section is to show that js φ(⋅

e−tHRPF → e−tHPF as c → ∞ strongly by using (3.9.30) and (3.9.31).

Lemma 3.136. It follows that limc→∞ Ktrel (c) = Kt strongly in L2 (X × 𝒯 ) ⊗ (⊕3 L2 (ℝ4 )). Proof. Let 3

2n

Ttj (c)

̃ − Bs ) ∘ Im (c) = ⨁ ∑ ∫ jtj−1 φ(⋅ μ=1 j=1 Ttj−1 (c)

dBμs ,

3

tj /m

2n

̃ − Bs ) ∘ dBμs . Im = ⨁ ∑ ∫ jtj−1 φ(⋅ μ=1 j=1 tj−1 /m

We see that Im (c) → Ktrel (c) and Im → Kt as m → ∞ strongly in L2 (X × 𝒯 ) ⊗ (⊕3 L2 (ℝ4 )) and L2 (X ) ⊗ (⊕3 L2 (ℝ4 )), respectively. We have ‖Ktrel (c) − Kt ‖ ≤ ‖Ktrel (c) − In (c)‖ + ‖In (c) − In ‖ + ‖In − Kt ‖. Here ‖ ⋅ ‖ denotes the norm on ⊕3 L2 (ℝ4 ). We have 2

k

̂ √ω‖2 ( ∑ 2−j/2 ) . 𝔼x [‖In (c) − Ik (c)‖2 ] ≤ 3Tt (c)‖φ/ j=n+1

From this we have 2

∞

̂ √ω‖2 ( ∑ 2−j/2 ) . 𝔼x,0 [‖In (c) − Ktrel (c)‖2 ] ≤ 3𝔼0 [Tt (c)]‖φ/ j=n+1

Since 𝔼0 [Tt (c)] = t/m which is independent of c > 0, we obtain that 𝔼x,0 [‖In (c) − K rel (c)‖2 ] ≤ 3

2

∞ t ̂ √ω‖2 ( ∑ 2−j/2 ) ‖φ/ m j=n+1

and we conclude that lim sup 𝔼x,0 [‖In (c) − K rel (c)‖2 ] = 0.

n→∞ c>0

(3.9.32)

3.9 Relativistic Pauli–Fierz model | 459

Now we estimate ‖In (c) − In ‖. We have 3

Ttj (c)

2n

̃ − In (c) − In = ⨁ ∑ ( ∫ jtj−1 φ(⋅ μ=1 j=1

Bs )dBμs

Ttj−1 (c)

s

tj /m

̃ − Bs )dBμs ). − ∫ jtj−1 φ(⋅ tj−1 /m

b

μ

μ

̃ − Bs )dBs are almost surely ̃ − Bs )dBs and s → ∫s jtj−1 φ(⋅ We note that s → ∫a jtj−1 φ(⋅ continuous. Hence x

T

Bs )dBμs ,

̃ − (S, T) → 𝔼 [( ∫ jtj−1 φ(⋅ S

tj /m

̃ − Bs )dBμs )] ∫ jtj−1 φ(⋅

tj−1 /m

is continuous, where (⋅, ⋅) deotes the scalar product on L2 (ℝ3 ). This implies that for every j, x,0

𝔼

Ttj (c)

̃ − [( ∫ jtj−1 φ(⋅ Ttj−1 (c)

tj /m

x

tj /m

Bs )dBμs ,

̃ − → 𝔼 [( ∫ jtj−1 φ(⋅

̃ − Bs )dBμs )] ∫ jtj−1 φ(⋅

tj−1 /m

Bs )dBμs ,

tj−1 /m

tj /m

̃ − Bs )dBμs )] = ∫ jtj−1 φ(⋅

tj−1 /m

(tj − tj−1 ) m

̂ √ω‖2 ‖φ/

(3.9.33)

as c → ∞ by Lemma 4.228 in Volume 1. We see that Tt (c)

t /m

Ttj−1 (c)

tj−1 /m

j 󵄩 j 󵄩󵄩2 󵄩󵄩 x,0 2 x,0 󵄩 μ ̃ − Bs )dBs − ∫ jtj−1 φ(⋅ ̃ − Bs )dBμs 󵄩󵄩󵄩󵄩 ] 𝔼 [‖In (c) − In ‖ ] = 3 ∑ 𝔼 [󵄩󵄩 ∫ jtj−1 φ(⋅ 󵄩󵄩 󵄩󵄩 j=1

2n

and x,0

𝔼

Tt (c)

t /m

Ttj−1 (c)

tj−1 /m

j 󵄩󵄩 j 󵄩󵄩2 󵄩󵄩 μ ̃ − Bs )dBs − ∫ jtj−1 φ(⋅ ̃ − Bs )dBμs 󵄩󵄩󵄩󵄩 ] [󵄩󵄩 ∫ jtj−1 φ(⋅ 󵄩󵄩 󵄩󵄩

=𝔼

x,0

Ttj (c)

t /m

Ttj−1 (c)

tj−1 /m

󵄩󵄩 󵄩2 󵄩 j 󵄩󵄩2 󵄩󵄩 󵄩 μ󵄩 x,0 󵄩 󵄩 ̃ − Bs )dBs 󵄩󵄩 ] + 𝔼 [󵄩󵄩󵄩󵄩 ∫ jtj−1 φ(⋅ ̃ − Bs )dBμs 󵄩󵄩󵄩󵄩 ] [󵄩󵄩 ∫ jtj−1 φ(⋅ 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 x,0

− 2𝔼

Ttj (c)

tj /m

Ttj−1 (c)

tj−1 /m

̃ − Bs )dBμs , ∫ jtj−1 φ(⋅ ̃ − Bs )dBμs )] [( ∫ jtj−1 φ(⋅

460 | 3 The Pauli–Fierz model by path measures

=

1 ̂ √ω‖2 (𝔼0 [Ttj (c) − Ttj−1 (c)] + tj − tj−1 ) ‖φ/ m x,0

− 2𝔼

Ttj (c)

tj /m

Ttj−1 (c)

tj−1 /m

̃ − Bs )dBμs , ∫ jtj−1 φ(⋅ ̃ − Bs )dBμs )]. [( ∫ jtj−1 φ(⋅

Note that 𝔼0 [Ttj (c) − Ttj−1 (c)] = tj − tj−1 and (3.9.33). We can see that 𝔼x,0 [‖In (c) − In ‖2 ] → 0 as c → ∞ for each n. Let ε > 0 be arbitrary. There exists n0 ∈ ℕ such that for all n > n0 𝔼x,0 [‖Ktrel (c) − In (c)‖2 ] < ε2 and 𝔼x,0 [‖In − Kt ‖2 ] < ε2 uniformly in c. We have for n > n0 , lim (𝔼x,0 [‖Ktrel (c) − Kt ‖]2 )1/2 ≤ 2ε + lim (𝔼x,0 [‖In (c) − In ‖])1/2 = 2ε.

c→∞

c→∞

Thus the lemma is proven. Theorem 3.137 (Nonrelativistic limit). Suppose that V is bounded and continuous. Then for every t ≥ 0 it follows that s-lim e−tHRPF = e−tHPF . c→∞

Proof. Suppose that F, G ∈ C0∞ (ℝ3 ) ⊗ ℱrad . From Lemma 3.136 and t

rel

(F, e−tHRPF G) = ∫ 𝔼x,0 [e− ∫0 V(BTs (c) )ds (J0 F(x), e−ieAE (Kt ̂

(c))

Jt G(BTt (c) ))]dx

ℝ3

it follows that t

lim (F, e−tHRPF G) = ∫ 𝔼x [e− ∫0 V(Bs/m )ds (J0 F(x), e−ieAE (Kt ) Jt G(Bt/m ))]dx

c→∞

̂

ℝ3

= (F, e−tHPF G). Since HRPF ≥ infx∈ℝ3 V(x) = g ≥ −∞, e−HRPF ≤ e−tg . Let F, G ∈ ℋPF . There exists Fn , Gn ∈ C0∞ (ℝ3 ) ⊗ ℱrad such that Fn → F and Gn → G strongly as n → ∞. By the uniform bound e−tHRPF ≤ e−tg , we can show limc→∞ (F, e−tHRPF G) = (F, e−tHPF G). Finally since the weak convergence of e−tHRPF implies the strong convergence, the theorem follows. 3.9.5 Relativistic Pauli–Fierz model with relativistic Kato-class potential Similar to Section 3.4, we can consider the relativistic Pauli–Fierz model with Katodecomposable potential V. Arguments are parallel to that of the Pauli–Fierz model

3.9 Relativistic Pauli–Fierz model | 461

Kato HPF . Let us recall the d-dimensional relativistic Kato-class. Let

pRt (x) =

1 √ 2 2 ∫ e−ix⋅ξ e−t( |ξ | +m −m) dξ . d (2π) ℝd

We define ∞

gR (x) = ∫ e−t pRt (x)dt. 0

Then gR (x) gives the integral kernel of (√−Δ + m2 −m+1)−1 . V is a relativistic Kato-class potential whenever it satisfies lim sup

δ↓0 x∈ℝd

gR (x − y)V(y)dy = 0.

∫

(3.9.34)

|x−y| R}. Lemma 3.148. Both τR and τR󸀠 are stopping times with respect to (Mt )t≥0 . Proof. It is sufficient to show that Et = {τR ≤ t} ∈ F[0,t] for arbitrary t ≥ 0. The sector of Et at w󸀠 ∈ 𝒯 is given by {w ∈ X | sup |BTs (w󸀠 ) (w) + x| > R} ∈ σ(Br , 0 ≤ r ≤ Tt (w󸀠 )). 0≤s≤t

(1) Then {τR ≤ t} ∈ F[0,t] . The sector of Et at w ∈ X is also given by

{w󸀠 ∈ 𝒯 | sup |BTs (w󸀠 ) (w) + x| > R} ∈ σ(Tr , 0 ≤ r ≤ t). 0≤s≤t

(2) Then {τR ≤ t} ∈ F[0,t] . Hence {τR ≤ t} ∈ F[0,t] . In a similar manner to τR , we can show 󸀠 that τR is also a stopping time.

Theorem 3.149 (Spatial decay of bound states). Statements (1) and (2) follow. (1) Suppose that lim|x|→∞ V− (x) + E = a < 0. Then

3.9 Relativistic Pauli–Fierz model |

471

(m = 0) there exists a constant C > 0 such that ‖Φb (x)‖L2 (Q) ≤

C , 1 + |x|4

(m > 0) there exist constants C > 0 and c > 0 such that ‖Φb (x)‖L2 (Q) ≤ Ce−c|x| . (2) Suppose that lim|x|→∞ V(x) = ∞. Then there exist constants C > 0 and c > 0 such that ‖Φb (x)‖L2 (Q) ≤ Ce−c|x| . Proof. Let τR = τR (x) = inf{s | |zs + x| < R} and τR󸀠 = inf{s | |zs | > R}. (1) Suppose that V− (x) + E < a + ϵ < 0 for all x such that |x| > R. By (3.9.39), we have ‖Φb (x)‖ ≤ C‖Φb ‖ℋPF 𝔼0Z [e+(ϵ+a)(t∧τR (x)) ] for |x| > R with some constant C > 0. It can be seen that 𝔼0Z [e+(ϵ+a)(t∧τR (x)) ] ≤

C1 , 1 + |x|4

𝔼0Z [e+(ϵ+a)(t∧τR (x)) ] ≤ C2 e−C3 |x| ,

m = 0, m > 0.

Here, C1 , C2 and C3 are positive constants. Thus (1) follows. (2) Let WR (x) = inf{V(y) | |x − y| < R}. Then it can be shown that 󸀠 t∧τR

𝔼0Z [e(t∧τR )E e− ∫0 󸀠

V(zr +x)dr

] ≤ e−t(WR (x)−E) + Ce−αR ect

with some constants α, c and C. Inserting R = p|x| with any 0 < p < 1, we see that WR (x) → ∞ as |x| → ∞. Thus substituting t = δ|x| for sufficiently small δ > 0 and R = p|x| with some 0 < p < 1, we can derive (2).

3.9.8 Gaussian domination of ground states π

π

By Corollaries 3.123 and 3.144, it is shown that ei 2 N e−tHRPF e−i 2 N is positivity improving π for all t > 0 under Assumption 3.112 and Assumption 3.138. In particular, ei 2 N Ψg is strictly positive and then the ground state Ψg of HRPF is unique up to multiplication constants if it exists. In Section 3.9.8 we suppose either Assumption 3.112 or Assumption 3.138. For an arbitrary fixed 0 ≤ f ∈ L2 (ℝ3 ) but f ≢ 0, we define ft = e−t(HRPF −E) (f ⊗ 1),

Ψtg = ft /‖ft ‖.

Then it follows that Ψtg → Ψg strongly as t → ∞, since (f ⊗ 1, Ψg ) ≠ 0.

472 | 3 The Pauli–Fierz model by path measures Let Lt = f (B−Tt )f (BTt )e

2

t

− e2 qE (Ktrel ,Ktrel ) − ∫−t V(BTs )ds

e

,

t ≥ 0.

We can formally write the pair interaction W RPF = e2 qE (Ktrel , Ktrel ) by qE (Ktrel , Ktrel )

3

Tt

= ∑ ∫ μ,ν=1

dBμs

−Tt

Tt

∫ Wμν (Ts∗ − Tr∗ , Bs − Br )dBνr , −Tt

where the pair potential, Wμν (t, X), is given by Wμν (t, X) =

2 kμ kν −ik⋅X −ω(k)|t| ̂ 1 |φ(k)| )e e dk. (δμν − ∫ 2 ω(k) |k|2 ℝ3

For each t ≥ 0, we define a family of probability measures μt , t > 0, on the measurable space (X × 𝒯 , ℬ(X) × ℬ𝒯 ) by ℬ(X) × ℬ𝒯 ∋ A 󳨃→ μt (A) =

1 ∫ 𝔼x,0 [1A Lt ] dx, Zt

t ≥ 0.

ℝ3

Here, Zt is the normalizing constant such that μt (X × 𝒯 ) = 1. This is the finite Gibbs measure associated with the relativistic Pauli–Fierz model. We shall discuss it in the ⊕ ̂ (⋅ − x))dx, next section. We define the self-adjoint operator  ξ in ℋPF by  ξ = ∫ℝ3 A(ξ 3 2 3 where ξ ∈ ⊕ Lreal (ℝ ). Then we have (e−tHRPF f ⊗ 1, e−iβAξ e−tHRPF f ⊗ 1) , t→0 (e−tHRPF f ⊗ 1, e−tHRPF f ⊗ 1)

(Ψg , e−iβAξ Ψg ) = lim ̂

̂

β ∈ ℝ.

Let β ∈ ℝ. Then it follows that 1 rel 2 (e−tHRPF f ⊗ 1, e−iβAξ e−tHRPF f ⊗ 1) = 𝔼μt [e− 2 (2eβiℜqE (Kt ,j0 ξ )+β qE (j0 ξ ,j0 ξ )) ] (e−tHRPF f ⊗ 1, e−tHRPF f ⊗ 1)

̂

Note that both qE (Ktrel , j0 ξ ) and qE (j0 ξ , j0 ξ ) do not depend on x. ̂2

Lemma 3.150. Suppose that β < (2qE (j0 ξ , j0 ξ ))−1 . Then Ψtg ∈ D(e(β/2)Aξ ) and ̂2

‖e(β/2)Aξ Ψtg ‖2 = (1 − 2βqE (j0 ξ , j0 ξ ))−1/2 𝔼μt [exp (

−βe2 qE (Ktrel , j0 ξ )2 )] . (1 − 2βqE (j0 ξ , j0 ξ ))

Proof. The proof is a minor modification of Lemma 3.64. Thus in the same way as Theorem 3.65 we have the theorem below. Theorem 3.151 (Gaussian domination of the ground state). Suppose that the ground ̂2 state Ψg of HRPF exists. Let β < (2qE (j0 ξ , j0 ξ ))−1 . Then Ψg ∈ D(e(β/4)Aξ ) follows.

3.9 Relativistic Pauli–Fierz model |

473

3.9.9 Path measure associated with the ground state of relativistic Pauli–Fierz Hamiltonian In Section 3.9.9 we suppose either Assumption 3.112 or Assumption 3.138, and that HRPF has a ground state Ψg . We set Z =X×𝒯

and Wx = 𝒲 x ⊗ ν in what follows. Let (Bt )t∈ℝ be 3-dimensional two-sided Brownian motion and we set zt = {

BTt , B−T−t ,

t ≥ 0, t < 0.

Thus t 󳨃→ zt (ω1 , ω2 ) = BTt (ω2 ) (ω1 ),

(ω1 , ω2 ) ∈ Z

is a cádlág path. Let F[−s,s] = σ(zr ; r ∈ [−s, s]). Then Gt = ⋃0≤s≤t F[−s,s] and G = ⋃0≤s F[−s,s] are finitely additive families of sets. We define the correction of probability spaces by (Z , σ(G ), μT ), T > 0, where μT is given by μT (A) =

1 ∫ 𝔼xW [1A LT ] dx. ZT ℝ3

Here LT = f (z−T )f (zT )e

2

T

− e2 qE (KTrel ,KTrel ) − ∫−T V(zs )ds

e

.

This is the finite Gibbs measure for relativistic Pauli–Fierz model. In a similar way to the Nelson model, we can show that there exists a probability measure μ∞ on (Z , σ(G )) such that μT → μ∞ as T → ∞ in the local sense. The idea to show the convergence is the same as the Nelson model. We show only the outline. First, we define the family of finitely additive set functions ρT on (Z , GT ), T > 0, by ρT (A) = e2Es ∫ 𝔼x [1A ( ℝ3

fT−s (z−s ) f (z ) , J[−s,s] T−s s )] dx, ‖fT ‖ ‖fT ‖

where A ∈ F[−s,s] and fT = e−T(H−E) (f ⊗ 1) and s

rel

J[−s,s] = J∗−s e− ∫−s V(zr )dr e−ieAE (Ks ) Js . ̂

We denote the extension to the probability measure on (Z , σ(GT )) by ρ̄ T . Thus we define the probability space (Z , σ(GT ), ρ̄ T ). By using functional integrations for HRPF , we

474 | 3 The Pauli–Fierz model by path measures can see that ρ̄ T (A) = ρT (A) = μT (A) for A ∈ Gs for all s ≤ T. Next, in terms of the ground state Ψg , we define a finitely additive set function μ on (Z , G ) by μ(A) = e2ET ∫ 𝔼xW [1A (Ψg (z−T ), J[−T,T] Ψg (zT ))] dx ℝ3

for A ∈ F[−T,T] . We denote the extension to the probability measure on (Z , σ(G )) by μ∞ . Thus we define the probability space (Z , σ(G ), μ∞ ). By applying the fact that fT /‖fT ‖ strongly converges to Ψg as T → ∞, we prove that ρT (A) → μ(A) as T → ∞ for A ∈ G , which implies that μT (A) → μ∞ (A) for A ∈ G , and μT converges to the measure μ∞ in the local sense. We state it as a theorem. Theorem 3.152 (Local convergence). Suppose that there exists a ground state of HRPF . Then the probability measures μT converge to μ∞ on (Z , σ(G )) as T → ∞ in the local sense. The probability measure μ∞ is called the Gibbs measure for relativistic Pauli–Fierz model. 3.9.10 Translation invariant relativistic Pauli–Fierz model In the case of the relativistic Pauli–Fierz Hamiltonian with V = 0, it can be seen that tot [HRPF , Ptot μ ] = 0, where Pμ = −i∇xμ ⊗ 1 + 1 ⊗ Pfμ is the total momentum operator. This allows that HRPF with V = 0 has no ground state, which is shown in the same way as the absence of round state of HPF with V = 0, and there exists a self-adjoint operator HRPF (p) in ℱrad such that ⊕

HRPF = ∫ HRPF (p)dp. ℝ3

The self-adjoint operator HRPF (p), p ∈ ℝ3 , is called relativistic Pauli–Fierz Hamiltonian with a total momentum p. We can show similar results to those of HPF (p) by using the functional integral representation of e−tHRPF . We use Assumption 3.153 below in Section 3.9.10. Assumption 3.153. The following conditions hold. ̂ ̂ ̂ (1) Charge distribution: φ ∈ S 󸀠 (ℝ3 ), φ(−k) = φ(k) and ω3/2 φ,̂ √ωφ,̂ φ/ω ∈ L2 (ℝ3 ); (2) External potential: V = 0. Theorem 3.154 (Functional integral representation for relativistic Pauli–Fierz model with a fixed total momentum). Suppose Assumption 3.153. Let Ψ, Φ ∈ L2 (Q ) and ERPF (p) = inf Spec(HRPF (p)). Then rel

(Ψ, e−tHRPF (p) Φ) = 𝔼0,0 [eip⋅BTt (J0 Ψ, e−ieAE (Kt ) Jt e−iPf BTt Φ)] ̂

̂

3.9 Relativistic Pauli–Fierz model |

475

and (1)–(4) below are satisfied. (1) Let p = 0. Then ei(π/2)N e−tHRPF (0) e−i(π/2)N is positivity improving. (2) The ground state of HRPF (0) is unique whenever it exists. (3) Energy comparison inequality ERPF (0) ≤ ERPF (p) holds. (4) Map p 󳨃→ ERPF (p) is continuous and E(0) = inf Spec(HRPFV=0 ), where HRPFV=0 is HRPF with V = 0. Proof. The proof is similar to those of Theorem 3.70 and Corollary 3.73. We can see the explicit form of the fiber Hamiltonian HRPF (p). Let 2 ̂ K(p) = (p − P̂ f − A(0)) + m2 .

Then we have 2

(Ψ, e−tK(p) Φ) = e−tm 𝔼0,0 [eip⋅Bt (Ψ, e−ieA(Lt ) e−iPf ⋅Bt Φ)] . ̂

̂

∞ ∞ Let D = D(P̂ f2 ) ∩ D(Hrad ) and D∞ = C0∞ (ℝ3 ) ⊗ ℱrad . See (3.7.2) for the definition of ℱrad . Applying the functional integral representation we show that K(p) is essentially self̄ adjoint on D, i. e., it can be shown that e−tK(p) D ⊂ D. Set K(p) = K(p)⌈D . We define LRPF (p) by

̄ LRPF (p) = √K(p) +̇ Hrad − m for each p ∈ ℝ3 . Theorem 3.155 (Fiber decomposition). Suppose Assumption 3.153. Then ⊕

HRPF ≅ ∫ LRPF (p)dp. ℝ3

In particular, HRPF (p) = LRPF (p). Proof. We define the unitary operator U on ℋPF by (UF)(p) = (2π)−3/2 ∫ eipx e−iPf x F(x)dx. ̂

ℝ3 ⊕ 2 ̄ , where Tkin = √2Ĥ PF (A)̂ + m2 . Actually it is shown that Then U −1 (∫ℝ3 K(p)dp)U = Tkin 2 ̄ (F, Tkin G) = ∫ ((UF)(p), K(p)(UG)(p)) dp ℝ3 2

for F, G ∈ D∞ . We see that U −1 (∫ℝ3 e−t K(p) dp)U = e−tTkin for all t ≥ 0. By the formula, ⊕

α Tkin

̄

∞

2

= Cα ∫ (1 − e−λTkin ) 0

dλ

λ1+α/2

,

476 | 3 The Pauli–Fierz model by path measures we can see that for F ∈ D∞ , ∞

(F, Tkin F) = C1 ∫ 0

̄ dλ ∫ ((UF)(p), (1 − e−λK(p) )(UF)(p))dp. λ3/2 ℝ3

By Fubini’s lemma, we have (F, Tkin F) = C1 ∫ dp ∫((UF)(p), (1 − e−λK(p) )(UF)(p)) ̄

ℝ3

dλ . λ3/2

∞ ̄ ̄ which Note that (UF)(p) ∈ ℱrad for each p ∈ ℝ3 . Hence (UF)(p) ∈ D(K(p)) ⊂ D(√K(p)), implies that

̄ (F, Tkin F) = ∫ ((UF)(p), √K(p)(UF)(p)) dp.

(3.9.41)

ℝ3

By the polarization identity and (3.9.41), we have ̄ (F, Tkin G) = ∫ ((UF)(p), √K(p)(UG)(p)) dp. ℝ3

Furthermore, we see that (F, (Tkin + Hrad ) G) = ∫ ((UF)(p), LRPF (p)(UG)(p)) dp, ℝ3

which implies that ⊕

Tkin +̇ Hrad = U −1 ( ∫ LRPF (p)dp) U

(3.9.42)

ℝ3

on D∞ . Since D∞ is a core of the left-hand side of (3.9.42), the lemma holds true as self-adjoint operators. Note that Tkin +̇ Hrad = Tkin + Hrad on D(√−Δ) ∩ D(Hrad ) and Tkin + Hrad is self-adjoint on D(√−Δ) ∩ D(Hrad ). Then the lemma follows. Note that D(|p − P̂ f |) = D(|P̂ f |) for all p ∈ ℝ3 . The essential self-adjointness of HRPF (p) is established. Lemma 3.156. Let m > 0. Then HRPF (p) is closable on D(|P̂ f |) ∩ D(Hrad ). Proof. The proof is similar to show the essential self-adjointness of HRPF . By using the functional integral representation, e−tHRPF (p) D(|P̂ f |) ∩ D(Hrad ) ⊂ D(|P̂ f |) ∩ D(Hrad ), can be shown. Then the lemma follows.

3.9 Relativistic Pauli–Fierz model |

477

We can furthermore specify the domain of HRPF (p). Theorem 3.157 (Self-adjointness). Let m ≥ 0. Then HRPF (p) is self-adjoint on ∞ D(|P̂ f |) ∩ D(Hrad ) and essentially self-adjoint on ℱrad . Proof. We show the outline of the proof. Note that HRPF (p) = LRPF (p). It can be seen ∞ that there exists a constant C > 0 such that for arbitrary Ψ ∈ ℱrad , 1/2 ̄ ‖√K(p)Ψ‖ ≤ C(‖|p − P̂ f |Ψ‖ + ‖Hrad Ψ‖ + ‖Ψ‖).

Then we can derive that ‖LRPF (p)Ψ‖ ≤ C(‖|p − P̂ f |Ψ‖ + ‖Hrad Ψ‖ + ‖Ψ‖)

(3.9.43)

∞ ∞ for Ψ ∈ ℱrad . From (3.9.43), we can see that ℱrad is a core of LRPF (p) for m > 0 by a limiting argument. In the case of m = 0,

̄ ̂ ̄ ̂ + (|p − P̂ f − A(0)| − √K(p)) |p − P̂ f − A(0)| = √K(p) ̂ ̄ ̂ √K(p) and |p− P̂ f − A(0)|− is bounded. Then |p− P̂ f − A(0)|+H rad is essentially self-adjoint ∞ on ℱrad . The key inequality to show the self-adjointness of LRPF (p) is ‖|p − P̂ f |Ψ‖2 + ‖Hrad Ψ‖2 ≤ C‖(LRPF (p) + 1)Ψ‖2

(3.9.44)

∞ with some C > 0 for Ψ ∈ ℱrad . This is shown by using the inequality

̄ 1/4 , Hrad ]Ψ‖ ≤ c‖(Hrad + 1)1/2 Ψ‖. ‖[K(p)

(3.9.45)

Inequality (3.9.45) will be proven in Lemma 3.158 below. Thus we can see that LRPF (p)⌈D(|P̂ |)∩D(H ) is closed by (3.9.44). Then LRPF (p) is self-adjoint on the domain f rad D(|P̂ f |) ∩ D(Hrad ) for m ≥ 0. Lemma 3.158. Equation (3.9.45) holds true. Proof. The idea of the proof of (3.9.45) is similar to (3.9.23) and (3.9.25). We have ̄ 1/4 ]Ψ) = (Hrad Φ, K(p) ̄ 1/4 Ψ) − (Φ, K(p) ̄ 1/4 Hrad Ψ) (Φ, [Hrad , K(p) The functional integral representation yields that 2

−m λ ̂ ̄ 1/4 ]Ψ) = ∫ e (Φ, [Hrad , K(p) 𝔼0 [eip⋅B2λ (Φ(B0 ), [Hrad , e−iA(L2λ ) ]e−iPf ⋅B2λ Ψ(B2λ ))]dλ. λ5/4 ∞

0

̂ 2λ ) + ξ ), then we can derive the desired results. Since [Hrad , e−iA(L2λ ) ] = e−iA(L2λ ) (Π(L ̂

̂

478 | 3 The Pauli–Fierz model by path measures 3.9.11 Nonrelativistic limit of translation invariant relativistic Pauli–Fierz Hamiltonian In a similar manner to the relativistic Pauli–Fierz Hamiltonian we can show the nonrelativistic limit of translation invariant relativistic Pauli–Fierz Hamiltonian. We introduce the speed of light c in HRPF (p) by 2 + m2 c4 − mc2 + H ̂ HRPF (p) = √c2 (p − P̂ f − A(0)) rad .

The subordinator (Tt (c))t≥0 with parameter c in (𝒯 , ℬ𝒯 , ν) satisfies (3.9.29). The translation invariant Pauli–Fierz Hamiltonian with particle mass m > 0 is defined by HPF (p) =

1 2 ̂ (p − P̂ f − A(0)) + Hrad . 2m

Using (Tt (c))t≥0 we can see that rel

(Ψ, e−tHRPF (p) Φ) = 𝔼0,0 [eip⋅BTt (c) (J0 Ψ, e−ieAE (Kt ̂

(c))

Jt e−iPf ⋅BTt (c) Φ)] ̂

where Ktrel (c) is defined by Ktrel with Tt replaced by Tt (c), and the functional integral representation of e−tHPF (p) with mass m is given by (Ψ, e−tHPF (p) Φ) = 𝔼0 [eip⋅Bt/m (J0 Ψ, e−ieAE (Kt ) Jt e−iPf ⋅Bt/m Φ)]dx, ̂

t/m

where Kt = ⨁3μ=1 ∫0

̂

μ

̃ − Bs ) ∘ dBs . js φ(⋅

Theorem 3.159 (Nonrelativistic limit). For every t ≥ 0 and p ∈ ℝ3 it follows that s-lim e−tHRPF (p) = e−tHPF (p) . c→∞

Proof. Let Ψ, Φ ∈ ℱrad . By Lemma 3.136 it is shown that limc→∞ Ktrel (c) = Kt strongly in L2 (X × 𝒯 ) ⊗ (⊕3 L2 (ℝ4 )). It follows that rel

lim (Ψ, e−tHRPF (p) Φ) = lim 𝔼0,0 [eip⋅BTt (c) (J0 Ψ, e−ieAE (Kt

c→∞

̂

c→∞

(c))

Jt e−iPf BTt (c) Φ)]

= 𝔼0 [eip⋅Bt/m (J0 Ψ, e−ieAE (Kt ) Jt e−iPf ⋅Bt/m Φ)]dx = (Ψ, e−tHPF (p) Φ). ̂

Then the proof is complete.

̂

̂

4 Spin-boson model by path measures 4.1 Definitions As discussed in Section 3.8, for the Pauli–Fierz Hamiltonian with spin 21 a functional integral representation can be derived by using suitable Lévy processes. In this chapter we derive further functional integral representations for semigroups generated by the so-called spin-boson and Rabi models, which will be less technically demanding than for the Pauli–Fierz case. The spin-boson model describes a two-state quantum system linearly coupled to a scalar quantum field. To study this model in a stochastic representation, we describe the spin states by the set ℤ2 = {−1, +1} as in the Pauli–Fierz model with spin, and derive a Poisson-driven random process with càdlàg path space Z = D(ℝ, ℤ2 ) indexed by ℝ and taking values in ℤ2 . This will give the spin-process. On the path space Z we are then able to construct a probability measure μ∞ associated with the unique ground state Ψg of the spin-boson Hamiltonian. Using this probability measure, we can represent ground state expectations for operators of interest. The strategy is similar to the Nelson model and the Pauli–Fierz model discussed in the previous sections. Consider the boson Fock space ℱSB = ℱb (L2 (ℝd )) over L2 (ℝd ), d ≥ 1, and the Hilbert space 2

ℋSB = ℂ ⊗ ℱSB .

We begin by defining the spin-boson Hamiltonian as a self-adjoint operator acting in ℋSB . The free field Hamiltonian for the spin-boson model is given by Hf = dΓ(ω), where ω(k) = |k|. The operator ϕφ =

1 ̃̂ √ω)) ̂ √ω) + a(φ/ (a∗ (φ/ √2

acting on the boson Fock space is the scalar field operator, where the form factor φ is required to satisfy the following conditions. ̂ ̂ ̂ ̂ Assumption 4.1. φ ∈ S 󸀠 (ℝd ) satisfies φ(k) = φ(−k) and φ/ω, φ/√ω ∈ L2 (ℝd ). Let σx , σy , σz be the 2 × 2 Pauli matrices given by 0 1

σx = (

1 ), 0

https://doi.org/10.1515/9783110403541-004

0 i

σy = (

−i ), 0

1 0

σz = (

0 ). −1

480 | 4 Spin-boson model by path measures Definition 4.2 (Spin-boson Hamiltonian). The spin-boson Hamiltonian is defined by the linear operator HSB = κσz ⊗ 1 + 1 ⊗ Hf + σx ⊗ ϕφ on ℋSB , where κ ≥ 0 is a parameter. We note that the physical meaning of the parameter is that 2κ is the gap between the ground state energy and the energy of the first excited state of the two-level atom. Proposition 4.3 (Self-adjointness). Let Assumption 4.1 hold. Then HSB is a self-adjoint operator on D(Hf ) and bounded from below. Furthermore, HSB is essentially self-adjoint on any core of Hf . Proof. By the assumption, κσz ⊗ 1 + σx ⊗ ϕφ is symmetric and infinitesimally small with respect to 1 ⊗ Hf , hence the proposition follows by the Kato–Rellich theorem. Recall that the rotation group in ℝ3 has an adjoint representation on SU(2). Let n ∈ ℝ3 be a unit vector and θ ∈ [0, 2π). Thus e(i/2)θn⋅σ satisfies e(i/2)θn⋅σ σμ e−(i/2)θn⋅σ = (Rθ σ)μ ,

(4.1.1)

where Rθ denotes the 3 × 3 matrix representing the rotation around n with angle θ, and σ = (σx , σy , σz ). Note that n ⋅ σ = n1 σx + n2 σy + σ3 σz for n = (n1 , n2 , σ3 ) ∈ ℝ3 . In particular, for n = (0, 1, 0) and θ = π/2, we have e(i/2)θn⋅σ σx e−(i/2)θn⋅σ = σz

and e(i/2)θn⋅σ σz e−(i/2)θn⋅σ = −σx .

(4.1.2)

Let 1 π 1 ( U = exp (i σy ) ⊗ 1 = √2 −1 4

1 )⊗1 1

(4.1.3)

be a unitary operator on ℋSB . By (4.1.2), the operator HSB transforms as H̃ SB = UHSB U ∗ = −κσx ⊗ 1 + 1 ⊗ Hf + σz ⊗ ϕφ . Then H̃ SB is realized as H + ϕφ H̃ SB = ( f −κ

−κ ). Hf − ϕφ

Each diagonal part is the so-called van Hove Hamiltonian which will be discussed later. In particular, κ = 0 makes HSB diagonal, obtained as a direct sum of van Hove Hamiltonians. It is also a known fact that HSB has a parity symmetry. Let P = σz ⊗ (−1)N ,

4.1 Definitions | 481

where N denotes the number operator in ℱSB . From Spec(σz ) = {−1, 1} and Spec(N) = ℕ ∪ {0}, it follows that Spec(P) = {−1, 1}. In fact, P is a projection on ℋSB . We identify ℋSB as +

−

ℋSB = ℱSB ⊕ ℱSB , + − where ℱSB and ℱSB are identical copies of ℱSB . Then each Pauli matrix θX = ( ac db ) acts as

Ψ(+) aΨ(+) + bΨ(−) )=( ) Ψ(−) cΨ(+) + dΨ(−)

θX (

+ − for ( Ψ(+) Ψ(−) ) ∈ ℱSB ⊕ ℱSB . Furthermore, ℱSB can be decomposed as e

o

ℱSB = ℱSB ⊕ ℱSB , e o where ℱSB and ℱSB denote respectively the subspaces of ℱSB consisting of even and ∞ (2m) (2m+1) e o odd numbers of bosons, i. e., ℱSB = ⨁∞ . The projecm=0 ℱSB and ℱSB = ⨁m=0 ℱSB e o tions from ℱSB to ℱSB and ℱSB will be denoted by Pe and Po , respectively. We will make use of the subspaces +

+

−

ℋSB = Pe ℱSB ⊕ Po ℱSB

− − + = Po ℱSB ⊕ Pe ℱSB and ℋSB

+ − of ℱSB ⊕ ℱSB .

Lemma 4.4. The following properties hold: + − + − (1) ℋSB can be identified with ℋSB ⊕ ℋSB , ℋSB ≅ ℋSB ⊕ ℋSB , by the correspondence +

−

Ψ(+) Ψ (+) Ψ (+) + − ) 󳨃→ ( e ) ⊕ ( o ) ∈ ℋSB ⊕ ℋSB , Ψ(−) Ψo (−) Ψe (−)

ℱSB ⊕ ℱSB ∋ (

where Ψe (±) = Pe Ψ(±) and Ψo (±) = Po Ψ(±). (2) The commutator [HSB , P] = 0. ± + (3) ℋSB is the eigenspace associated with the eigenvalue ±1 of P, i. e., P ℋSB = ℋSB and − (1 − P)ℋSB = ℋSB . (4) HSB can be decomposed as HSB = HSB ⌈ℋ+ ⊕HSB ⌈ℋ− . SB

SB

Proof. (1) is straightforward. We see that [HSB , P] = [σx ⊗ ϕφ , P] = σx σz ⊗ ϕφ (−1)N − σz σx ⊗ (−1)N ϕφ = −iσy ⊗ (ϕφ (−1)N + (−1)N ϕφ ).

+ − Let Ψ ∈ ℱSB . Since ϕφ Ψ ∈ ℱSB , we have

(ϕφ (−1)N + (−1)N ϕφ )Ψ = (ϕφ − ϕφ )Ψ = 0.

482 | 4 Spin-boson model by path measures − + Similarly, let Ψ ∈ ℱSB . Since ϕφ Ψ ∈ ℱSB , we have

(ϕφ (−1)N + (−1)N ϕφ )Ψ = (−ϕφ + ϕφ )Ψ = 0. Ψ (+)

+ Then [HSB , P] = 0 follows. Let Ψ = ( Ψe (−) ) ∈ ℋSB . Then PΨ = Ψ follows by a direct o calculation, thus Ψ is an eigenvector of P with eigenvalue +1. Similarly, it follows that − ℋSB is the eigenspace associated with eigenvalue −1 of P, and (3) also follows; (4) is obtained by a combination of (1), (2) and (3).

Statement (2) of Lemma 4.4 is called the parity symmetry. We write ± HSB = HSB ⌈ℋ± . SB

+ − Thus HSB is of parity plus and HSB of parity minus. We also can derive the explicit ± forms of reduced Hamiltonian HSB .

Corollary 4.5. It follows that + HSB = κ ⊗ (−1)N + 1 ⊗ Hf + σx ⊗ ϕφ ,

− HSB = −κ ⊗ (−1)N + 1 ⊗ Hf + σx ⊗ ϕφ . + − Proof. Since HSB = PHSB P and HSB = (1 − P)HSB (1 − P), the corollary is immediate.

In Theorem 4.12 and Corollary 4.13 below we will see that for κ ≠ 0 the boson Hamiltonian HSB has a unique ground state of parity minus. This results in an inter+ − + esting fact. If inf Spec(HSB ) = inf Spec(HSB ), then HSB has no ground state, which can ± be seen directly in a simple case. Let φ = 0 and κ ≠ 0. Then HSB can be regarded as operators in ℱSB with the expressions + HSB = κ(−1)N + Hf

and

− HSB = −κ(−1)N + Hf .

− − + It can be seen that inf Spec(HSB ) = −κ and HSB Ωb = −κΩb . On the other hand, HSB can (n) (n) ∞ be reduced to the n-particle subspace ℱSB for each n ≥ 0, and for Φ = {Φ }n=0 we have ∞

n

n=1

j=1

+ HSB Φ = κΦ(0) + ⨁((−1)n κ + ∑ ω(kj ))Φ(n) .

Since ω(k) = √|k|2 + ν2 , we have + inf Spec(HSB ) = inf {κ, (−1)n κ + nν, n ∈ ℕ} = −κ + ν.

In particular, when ν = 0, we have − + inf Spec(HSB ) = inf Spec(HSB ) = −κ.

4.2 Functional integral representation for the spin-boson Hamiltonian

| 483

+ Thus HSB has no ground state, since the multiplication operator + HSB ⌈ℱ (1) = −κ + |k| SB

(1) in ℱSB = L2 (ℝd ) has no ground state. These statements are also valid for φ ≠ 0 under some further conditions. Finally, for later use we show some related spin-flip properties. ± Lemma 4.6. For every Ψ, Φ ∈ ℋSB we have ∗ (1) (Ψ, σx Φ) = 0 and (U Ψ, σz U ∗ Φ) = 0; (2) (Ψ, Φ(f ̂)Φ) = 0 and (U ∗ Ψ, Φ(f ̂)U ∗ Φ) = 0, where Φ(f ̂) =

1 (a∗ (f ̂) √2

̃ + a(f ̂)).

± ∓ Proof. (1) We see that σx is a spin-flip transform, i. e., σx ℋSB ⊂ ℋSB , which gives the first statement. The second statement follows by observing that Uσz U ∗ = σx . To obtain ± ∓ (2), we show again that Φ(f ̂) (ℋSB ∩ D(Φ(f ̂))) ⊂ ℋSB . The second part follows from ∗ ̂ ̂ UΦ(f )U = Φ(f ).

4.2 Functional integral representation for the spin-boson Hamiltonian 4.2.1 Preliminaries ℤ2

As we have seen in Section 3.8, for the functional integral representation of e−tHPF generated by the Pauli–Fierz Hamiltonian with spin it was convenient to use a spin variable σ ∈ ℤ2 in a related space. This will prove to be useful also for e−tHSB . Let (ϕ(f ), f ∈ L2real (ℝd )) be a family of Gaussian random variables indexed by 2 Lreal (ℝd ) on a probability space (Q, Σ, μ). As before, we identify ℱSB with L2 (Q ). Under this equivalence, ϕφ can be identified with a Gaussian random variable ϕ(φ)̃ indexed ̂ by φ̃ = (φ/√ω ).̌ Thus the spin-boson Hamiltonian H̃ SB can be equivalently defined on the Hilbert space ℂ2 ⊗ L2 (Q ) by

̃ −κσx ⊗ 1 + 1 ⊗ Ĥ f + σz ⊗ ϕ(φ),

(4.2.1)

which we will also denote by H̃ SB , and ℂ2 ⊗ L2 (Q ) ≅ L2 (Q ) ⊕ L2 (Q ) also by ℋSB . Recall ̂ For Ψ = ( Ψ(+) that Ĥ f = dΓ(ω). Ψ(−) ) ∈ ℋSB we have ̃ (Ĥ + ϕ(φ))Ψ(+) − κΨ(−) ). H̃ SB Ψ = ( ̂ f ̃ (Hf − ϕ(φ))Ψ(−) − κΨ(+) ℤ Thus we can transform H̃ SB on ℋSB to the operator HSB2 on

󵄨󵄨 L2 (ℤ2 ; L2 (Q )) = {f : ℤ2 → L2 (Q ) 󵄨󵄨󵄨 ∑ ‖f (σ)‖2L2 (Q) < ∞} 󵄨 σ∈ℤ 2

484 | 4 Spin-boson model by path measures by ℤ

̃ Ψ(σ) − κΨ(−σ), (HSB2 Ψ)(σ) = (Ĥ f + σϕ(φ))

σ ∈ ℤ2 .

(4.2.2)

In what follows, we write Hf for Ĥ f and also identify ℋSB with L2 (ℤ2 ; L2 (Q )) through Ψ(+) Ψ(+), ) 󳨃→ Ψ(σ) = { Ψ(−) Ψ(−),

ℋSB ∋ (

σ = +1 ∈ L2 (ℤ2 ; L2 (Q )), σ = −1

ℤ

and consider HSB2 instead of H̃ SB . 4.2.2 Spin process Let (Nt )t∈ℝ be a two-sided Poisson process with unit intensity on a probability space (𝒮 , ℬ𝒮 , P), defined by N̄ t Nt = { ̄ N̄ −t

t ≥ 0, t < 0,

where (N̄ t )t≥0 is a càdlàg version of the Poisson process, and (N̄̄ t )t≥0 a càglàd version such that P(N̄̄ 0 = 0) = 1 = P(N̄ t = 0). Here ℝ ∋ t 󳨃→ Nt ∈ ℤ is a càdlàg function. We define the spin process by

(σt )t∈ℝ = (σ(−1)Nt )t∈ℝ ,

σ ∈ ℤ2 .

We write 𝔼σ [f (σt )] = 𝔼[f (σ(−1)Nt )]. Since the spin process is indexed by the real line, we summarize some basic properties of (Nt )t∈ℝ below. (1) (Independence) The random variables Nt and Ns are independent for all s ≤ 0 ≤ t, s ≠ t. (2) (Markov property) The random processes (Nt )t≥0 and (Nt )t≤0 have the Markov property with respect to the natural filtrations Nt + = σ(Ns , 0 ≤ s ≤ t) and Nt − = σ(Ns , t ≤ s ≤ 0), respectively, i. e., for s, t ≥ 0, N

𝔼0P [Nt+s |Ns+ ] = 𝔼P s [Nt ]

and

N

𝔼0P [N−t−s |N−s− ] = 𝔼P −s [N−t ] .

(3) (Reflection symmetry) The random variables Nt and N−t are identically distributed for all t ∈ ℝ, i. e., ∞

𝔼0P [f (N−t )] = 𝔼0P [f (Nt )] = ∑ f (n) n=0

t n −t e , n!

(1)–(3) can be shown directly by the definition of (Nt )t∈ℝ .

t ≥ 0.

4.2 Functional integral representation for the spin-boson Hamiltonian | 485

Now we discuss some technical facts on the spin process. Consider L2 (ℤ2 ), which 2 can be identified with ℂ2 under the map L2 (ℤ2 ) ∋ f 󳨃→ ( ff (+1) (−1) ) ∈ ℂ . Define the C0 -semigroup Pt : ℂ2 → ℂ2 , t ≥ 0, under this identification by (Pt f )(σ) = 𝔼σ [f (σt )],

σ ∈ ℤ2 .

The generator of Pt is σx − 1 and thus Pt = et(σx −1) ,

t ≥ 0.

Write S = −σx + 1. We have for 0 ≤ t1 ≤ . . . ≤ tn and fj ∈ ℂ2 , j = 1, . . . , n, n

∑ 𝔼σ [ ∏ fj (σtj )] = (1, e−t1 S f1 e−(t2 −t1 )S f2 ⋅ ⋅ ⋅ e−(tn −tn−1 )S fn )L2 (ℤ2 ) .

σ∈ℤ2

j=1

Proposition 4.7 (Shift invariance). Let fj ∈ L2 (ℤ2 ), j = 1, . . . , n + m, m, n ∈ ℕ, and consider the arbitrary time-points −∞ < t1 ≤ . . . ≤ tn ≤ 0 ≤ tn+1 ≤ . . . ≤ tn+m < ∞. Then n+m

n+m

∑ 𝔼σ [ ∏ fj (σtj )] = ∑ 𝔼σ [ ∏ fj (σtj +s )];

σ∈ℤ2

σ∈ℤ2

j=0

j=0

for −tn+m ≤ s ≤ −t1 . Proof. Let −t1 ≤ . . . ≤ −tn ≤ 0 ≤ tn+1 ≤ . . . ≤ tn+m . We show that n

n+m

i=1

j=n+1

n

n+m

i=1

j=n+1

∑ 𝔼σ [ ∏ fi (σ−ti ) ∏ fj (σtj )] = ∑ 𝔼σ [ ∏ fi (σt1 −ti ) ∏ fj (σt1 +tj )],

σ∈ℤ2

σ∈ℤ2

(4.2.3)

from which the proposition follows. First we compute the left-hand side of (4.2.3). By d

the independence of N−t and Ns for −t ≤ 0 ≤ s, and the reflection symmetry N−t = Nt we have n

n+m

n

n+m

i=1

j=n+1

∑ 𝔼σ [ ∏ fi (σ−ti ) ∏ fj (σtj )] = ∑ 𝔼σ [ ∏ fi (σ−ti )]𝔼σ [ ∏ fj (σtj )]

σ∈ℤ2

i=1

σ∈ℤ2

j=n+1

n

n+m

= ∑ 𝔼σ [ ∏ fi (σti )]𝔼σ [ ∏ fj (σtj )] σ∈ℤ2

= (e

−tn S

i=1

fn e

−(tn−1 −tn )S

j=n+1

⋅⋅⋅e

−(t1 −t2 )S

f1 , e−tn+1 S fn+1 e−(tn+2 −tn+1 )S ⋅ ⋅ ⋅ e−(tn+m −tn+m−1 )S fn+m ). (4.2.4)

Hence we can compute RHS (4.2.4) = (1, f1 e−(t1 −t2 )S ⋅ ⋅ ⋅ fn e−(tn +tn+1 )S fn+1 e−(tn+2 −tn+1 )S ⋅ ⋅ ⋅ e−(tn+m −tn+m−1 )S fn+m ) .

(4.2.5)

On the other hand, we have n

n+m

i=1

i=n+1

∑ 𝔼σ [ ∏ fi (σt1 −ti ) ∏ fi (σt1 +tj )] = (4.2.5).

σ∈ℤ2

486 | 4 Spin-boson model by path measures Let κ > 0 and consider the random process (σκt )t∈ℝ . Then (Pt f )(σ) = 𝔼σ [f (σκt )] also defines the C0 -semigroup on ℂ2 with generator κσx . Hence Pt = etκ(σx −1) . We use this random process to construct a functional integral representation of the spin-boson Hamiltonian, and will also refer to (σκt )t∈ℝ as the spin process. 4.2.3 Functional integral representation ℤ2

We can construct a functional integral representation of e−tHSB by a modification of ℤ2

the similar construction for e−tHPF , using that simplifications occur for the spin-boson Hamiltonian. We identify L2 (ℤ2 ; L2 (Q )) with L2 (ℤ2 × Q ) and denote ℋSB = L2 (ℤ2 × Q ). Theorem 4.8 (Functional integral representation for spin-boson Hamiltonian). For every Φ, Ψ ∈ ℋSB , the expression ℤ2

(Φ, e−THSB Ψ) = eκT ∑ 𝔼σ [(J0 Φ(σ0 ), e

T

̃ −ϕE (∫0 σκs js φds)

σ∈ℤ2

JT Ψ(σκT ))]

(4.2.6)

holds. Proof. We see by (4.2.2) that ℤ

̃ (HSB2 Ψ)(σ) = (Hf + ϕ(φ))Ψ(σ) − κΨ(−σ),

σ ∈ ℤ2 .

Then by Lemma 3.96 with σt replaced by σκt we have T

ℤ2

(Φ, e−THSB Ψ) = eκT ∑ 𝔼σ [(J0 Φ(σ0 ), e− ∫0

T+

̃ σκs ϕE (js φ)ds+∫ log 1dNs 0

σ∈ℤ2

JT Ψ(σκT ))] .

T+

Since ∫0 log 1dNs = 0, the theorem follows. Remark 4.9. By the shift invariance of (σt )t∈ℝ we get that ℤ2

(Φ, e−THSB Ψ) = eκT ∑ 𝔼σ [(J−r Φ(σ−κr ), e

T−r

̃ −ϕE (∫−r σκs js φds)

σ∈ℤ2

JT−r Ψ(σκ(T−r) ))]

for 0 ≤ r ≤ T. In particular, ℤ2

(Φ, e−2THSB Ψ) = e2κT ∑ 𝔼σ [(J−T Φ(σ−κT ), e σ∈ℤ2

T

̃ −ϕE (∫−T σκs js φds)

JT Ψ(σκT ))] .

This will be used to construct a Gibbs measure for the spin-boson model.

4.2 Functional integral representation for the spin-boson Hamiltonian

| 487

ℤ2

Remark 4.10. We can give an alternative functional integral representation of e−THSB in terms of the Poisson process (Nt )t≥0 . This is (Φ, e

ℤ

−THSB2

T

̃ {eT ∑σ∈ℤ 𝔼σ [(J0 Φ(σ0 ), e−ϕE (∫0 σs js φds) κ NT JT Ψ(σT ))], 2 Ψ) = { T ̃ T −ϕE (σ ∫0 js φds) JT Ψ(σ)), {e ∑σ∈ℤ2 (J0 Φ(σ), e

κ ≠ 0,

κ = 0.

(4.2.7)

The case κ = 0 follows by Theorem 4.8. If κ ≠ 0, we have T

ℤ2

T+

(Φ, e−THSB Ψ) = eT ∑ 𝔼σ [(J0 Φ(σ0 ), e− ∫0

̃ σs ϕE (js φ)ds+∫ log κdNs 0

σ∈ℤ2

JT Ψ(σT ))]

T+

by Lemma 3.87. Since ∫0 log κdNs = NT log κ, the result follows. We emphasize that (4.2.6) is shift invariant but the representation (4.2.7) is not. By the functional integral representation we see that T

ℤ2

e−THSB Φ(σ) = eκT 𝔼σ [J∗0 e−ϕE (∫0

̃ σκr jr φdr)

JT Φ(σκT )] .

(4.2.8)

Denote 1SB = 1L2 (ℤ2 ) ⊗ 1L2 (Q) for simplicity. Using Theorem 4.8, we can compute the vacuum expectation of the ℤ2

semigroup e−THSB .

Corollary 4.11 (Pair interaction and pair potential). For every T ≥ 0 we have T

1

ℤ2

(1SB , e−THSB 1SB ) = eκT ∑ 𝔼σ [e 2 ∫0

T

ds ∫0 W(σκs σκt ,s−t)dt

σ∈ℤ2

T

],

T

where ∫0 ds ∫0 W(σκs σκt , s − t)dt is the pair interaction associated with spin-boson Hamiltonian with the pair potential W(η, t) =

2 ̂ η |φ(k)| dk. ∫ e−|t|ω(k) 2 ω(k) ℝd

Proof. By Theorem 4.8 it is direct to see that ℤ2

(1SB , e−THSB 1SB ) = eκT ∑ 𝔼σ [(1, e

T

̃ −ϕE (∫0 σκs js φds)

σ∈ℤ2

1

T

= eκT ∑ 𝔼σ [e 2 ∫0 σ∈ℤ2

1)L2 (QE ) ]

T

ds ∫0 W(σκs σκt ,s−t)dt

].

488 | 4 Spin-boson model by path measures

4.3 Existence and uniqueness of ground state Now we make use of the functional integral representation established in the previous section to show existence and uniqueness of the ground state of the spin-boson Hamiltonian. Denote ISB = ∫ ℝd

2 ̂ |φ(k)| dk. ω(k)3

ℤ

Whenever κ = 0, the Hamiltonian HSB2 is diagonal and we have ℤ

Hf + ϕφ 0

0 ). Hf − ϕφ

HSB2 = (

The operator Hf + ϕφ is a van Hove Hamiltonian and has a unique ground state if and ℤ

only if ISB < ∞, which implies that HSB2 with κ = 0 has a two-fold degenerate ground state if and only if ISB < ∞. ℤ Next we consider the case κ ≠ 0. Denote E = inf σ(HSB2 ), and write ℤ2

ΨTg = e−T(HSB −E) 1SB ,

T ≥ 0,

and γ(T) =

(1SB , ΨTg )2 ‖ΨTg ‖2

ℤ2

=

(1SB , e−THSB 1SB )2 ℤ2

(1SB , e−2THSB 1SB )

.

ℤ

By Propositions 2.30–2.31 a ground state of HSB2 exists if and only if limT→∞ γ(T) > 0. By Corollary 4.11 we have T

1

T

‖ΨTg ‖2 = e2TE e2κT ∑ 𝔼σ [e 2 ∫−T ds ∫−T W(σκs σκt ,s−t)dt ] , σ∈ℤ2

1

T

(1SB , ΨTg ) = eTE eκT ∑ 𝔼σ [e 2 ∫0

T

ds ∫0 W(σκs σκt ,s−t)dt

σ∈ℤ2

].

Note that 0

T

−T

0

2 󵄨󵄨 󵄨 ̂ 󵄨󵄨 ∫ ds ∫ W(σκs σκt , s − t)dt 󵄨󵄨󵄨 ≤ 1 ∫ |φ(k)| dk, 󵄨󵄨 󵄨󵄨 2 ω(k)3

(4.3.1)

ℝd

uniformly in T and in the paths. ℤ

Theorem 4.12 (Existence of ground state). If ISB < ∞, then HSB2 has a ground state and it is unique.

| 489

4.3 Existence and uniqueness of ground state

Proof. For a shorthand, we write W = W(κs κt , s − t), and T

T

0

0

T

T

0

T

∫ ds ∫ Wdt = ∫ ds ∫ Wdt + ∫ ds ∫ Wdt + 2 ∫ ds ∫ Wdt. −T

−T

−T

−T

0

0

0

0

0

−T

Using (4.3.1) we obtain 1

T

T

(∫−T ds ∫−T Wdt+∫0 ds ∫0 Wdt+ISB )

‖ΨTg ‖2 ≤ e2TE e2κT ∑ 𝔼σ [e 2 σ∈ℤ2

].

By the independence of Nt and N−s for −s < 0 < t, and reflection symmetry of the paths, we furthermore obtain that T

1

‖ΨTg ‖2 ≤ e2TE e2κT ∑ (𝔼σ [e 2 ∫0

T

ds ∫0 W(σκs σκt ,s−t)dt

σ∈ℤ2

TE κT

≤ (e e

σ

∑ 𝔼 [e

1 2

T

T

∫0 ds ∫0 W(σκs σκt ,s−t)dt

σ∈ℤ2

1

2

1

]) e 2 ISB 2

1

1

]) e 2 ISB = (1, ΨTg )2 e 2 ISB .

Hence γ(T) ≥ e− 2 ISB and a ground state Ψg of HSB2 exists. ℤ

ℤ2

Corollary 4.13 (Positivity improving). Let κ ≠ 0. Then e−tHSB , t > 0, is positivity improvℤ2

ing on L2 (ℤ2 × Q), i. e., (Ψ, e−tHSB Φ) > 0 for Ψ, Φ ≥ 0, with Ψ ≢ 0 ≢ Φ. In particular, if ℤ HSB2 has a ground state for κ ≠ 0, then it is unique. Proof. Positivity improving follows by the functional integral representation in Theorem 4.8. The uniqueness follows by the Perron–Frobenius theorem. ℤ

+ − As was seen above, ℋSB can be decomposed as ℋSB = ℋSB ⊕ ℋSB , and HSB2 can be ± reduced by ℋSB . ℤ

Corollary 4.14 (Parity of ground state). Let Ψg be the ground state of HSB2 . Then Ψg ∈ − ℋSB , i. e., the parity of the ground state of the spin-boson Hamiltonian is −1. Proof. Let U be the unitary operator as in (4.1.3). Notice that Ψg = U ∗ Ψg , and thus Ψg = s-lim T→∞

ℤ2

U ∗ e−THSB 1SB ℤ2

‖U ∗ e−THSB 1SB ‖

= s-lim T→∞

ℤ2

e−THSB U ∗ 1SB ℤ2

‖e−THSB U ∗ 1SB ‖

Ω

.

+ − The function 1SB ∈ L2 (ℤ2 ; L2 (Q )) corresponds to ( Ωb ) ∈ ℱSB ⊕ ℱSB and b

U ∗ 1SB =

1 1 ( 2 1

−1 Ωb 0 − ) ( ) = ( ) ∈ ℋSB . 1 Ωb Ωb

ℤ

Hence by the parity symmetry of HSB2 , (2) of Lemma 4.4, we have ℤ2

ℤ2

ℤ2

Pe−THSB U ∗ 1SB = e−THSB PU ∗ 1SB = −e−THSB U ∗ 1SB , ℤ2

− − and thus e−THSB U ∗ 1SB ∈ ℋSB . This implies Ψg ∈ ℋSB .

490 | 4 Spin-boson model by path measures

4.4 Gibbs measure associated with the ground state 4.4.1 Local convergence and Gibbs measures Let Z = D(ℝ; ℤ2 ) be the set of càdlàg paths with values in ℤ2 , and ℬZ the σ-field generated by its cylinder sets. Thus σκ⋅ : (𝒮 , ℬ𝒮 , P) → (Z , ℬZ ) is an Z -valued random variable. We denote its image measure on (Z , ℬZ ) by 𝒫 σ , i. e., σ

−1

𝒫 (A) = σκ⋅ (A),

A ∈ ℬZ .

We denote the coordinate process by the same symbol (σκt )t∈ℝ , i. e., σκt (ω) = ω(κt) for ω ∈ Z . Then 𝒫 σ (σ0 = σ) = 1. Hence Theorem 4.8 can be reformulated in terms of (σκt )t≥0 as t

ℤ2

(Φ, e−tHSB Ψ) = eκt ∑ 𝔼σ𝒫 [(J0 Φ(σ0 ), e−ϕE (∫0 σκs js φds) Jt Ψ(σκt ))] . ̃

σ∈ℤ2

(4.4.1)

Then (4.2.8) can be cast in the form t

ℤ2

e−tHSB Ψ(σ) = eκt 𝔼σ𝒫 [J∗0 e−ϕE (∫0 σκr jr φdr) Jt Ψ(σκt )] . ̃

(4.4.2)

Let G[−T,T] = σ (σκt ; t ∈ [−T, T]) be the family of sub-σ-fields of ℬZ , and define GT = ⋃0≤s≤T G[−s,s] and G = ⋃s≥0 G[−s,s] . Also, define the probability measure μT (A) =

T 1 T e2κT ∑ 𝔼σ𝒫 [1A e 2 ∫−T ds ∫−T W(σκs σκt ,s−t)dt ] , ZT σ∈ℤ

A ∈ σ(G ),

2

on the measurable space (Z , σ(G )), where ZT is the normalizing constant such that μT (Z ) = 1. The probability measure μT is a finite volume Gibbs measure for the spinboson model. We can show the convergence of μT to a probability measure μ∞ in local sense as T → ∞. Let μ be the additive set function on G defined by μ(A) = e2κT e2ET ∑ 𝔼σ𝒫 [(Ψg (σ−κT ), J[−T,T] Ψg (σκT )) 1A ] , σ∈ℤ2

A ∈ G[−T,T] .

Here E = inf Spec(HSB ) and T

J[−T,T] = J∗−T e−ϕE (∫−T σκr jr φdr) JT . ̃

The set function μ can be uniquely extended to a probability measure on (Z , σ(G )), which we denote by μ∞ . This is the infinite volume Gibbs measure associated with the spin-boson Hamiltonian.

4.4 Gibbs measure associated with the ground state

| 491

Theorem 4.15 (Local convergence). The family of probability measures {μT }T>0 on (Z , σ(G )) converges to μ∞ as T → ∞ in local sense. Proof. We show an outline of the proof. The set function μ is completely additive on (Z , G ), thus by Hopf’s extension theorem there exists a unique probability measure μ∞ on (Z , σ(G )) such that μ∞ (A) = μ(A) for A ∈ G . Define an additive set function ρT : GT → ℝ by ρT (A) = e2Es e2κs ∑ 𝔼σ𝒫 [1A ( σ∈ℤ2

1T−s (σ−κs ) 1 (σ ) , J[−s,s] T−s κs )] ‖1T ‖ ‖1T ‖

for A ∈ G[−s,s] with s ≤ T. Here ℤ2

1T = e−T(HSB −E) 1SB . The set function ρT is well-defined, i. e., ρT (A) = e2Er e2κr ∑ 𝔼σ𝒫 [1A (

1 (σ ) 1T−r (σ−κr ) , J[−r,r] T−r κr )] ‖1T ‖ ‖1T ‖

= e2Es e2κs ∑ 𝔼σ𝒫 [1A (

1T−s (σ−κs ) 1 (σ ) , J[−s,s] T−s κs )] ‖1T ‖ ‖1T ‖

σ∈ℤ2

σ∈ℤ2

for all r ≤ s ≤ T and A ∈ G[−r,r] . By Hopf’s extension theorem again, there exists a probability measure ρ̄ T on (Z , σ(GT )) such that ρT = ρ̄ T ⌈GT . Let s ≤ T and A ∈ Gs . Then ρ̄ T (A) = μT (A) and lim μT (A) = lim e2Es e2κs ∑ 𝔼σ𝒫 [1A (

T→∞

T→∞

σ∈ℤ2

1 (σ ) 1T−s (σ−κs ) , J[−s,s] T−s κs )] . ‖1T ‖ ‖1T ‖

Since 1T /‖1T ‖ → Ψg strongly as T → ∞, we have lim μT (A) = e2Es e2κs ∑ 𝔼σ𝒫 [1A (Ψg (σ−κs ), J[−s,s] Ψg (σκs ))] = μ∞ (A).

T→∞

σ∈ℤ2

4.4.2 Ground state properties Like in the sections above, the Gibbs measure μ∞ can be used to derive the ground state properties of the Hamiltonian. We start by considering ground state expectations of the form (Ψg , hF(ϕ(f ))Ψg ) with suitable functions F and h. By the parity symmetry, (Ψg , σΨg )L2 (ℤ2 ;L2 (Q)) = (Ψg , σx Ψg )ℂ2 ⊗ℱSB = 0. Theorem 4.16. Let f be a bounded G[−t,t] -measurable function on Z . Then 𝔼μ∞ [f ] = e2κt e2Et ∑ 𝔼σ𝒫 [(Ψg (σ−κt ), J[−t,t] Ψg (σκt )) f ] . σ∈ℤ2

(4.4.3)

492 | 4 Spin-boson model by path measures Proof. For A ∈ G[−t,t] we have 𝔼μ∞ [1A ] = μ∞ (A) = e2κt e2Et ∑ 𝔼σ𝒫 [(Ψg (σ−κt ), J[−t,t] Ψg (σκt )) 1A ] σ∈ℤ2

and n

n

𝔼μ∞ [ ∑ aj 1Aj ] = e2κt e2Et ∑ 𝔼σ𝒫 [ (Ψg (σ−κt ), J[−t,t] Ψg (σκt )) ∑ aj 1Aj ] σ∈ℤ2

j=1

j=1

for every Aj ∈ G[−t,t] and aj ≥ 0, j = 1, . . . , n. Then (4.4.3) follows by a limiting argument. An immediate consequence of Theorem 4.16 is the following. Corollary 4.17 (Euclidean Green functions). Let fj : ℤ2 → ℂ, j = 0, . . . , n, n ∈ ℕ, be bounded functions. Then for t0 ≤ t1 ≤ . . . ≤ tn , n

ℤ2

ℤ2

𝔼μ∞ [ ∏ fj (σκtj )] = (Ψg , f0 e−(t1 −t0 )(HSB −E) f1 ⋅ ⋅ ⋅ fn−1 e−(tn −tn−1 )(HSB −E) fn Ψg ) . j=0

In particular, for all bounded functions h, f and g : ℤ2 → ℂ we have 𝔼μ∞ [h(σ0 )] = (Ψg , hΨg ), ℤ2

𝔼μ∞ [f (σκt )g(σκs )] = (f Ψg , e−|t−s|(HSB −E) gΨg ). Proof. For Aj ∈ σ(G ), j = 0, 1, . . . , n, it follows that n

n

𝔼μ∞ [ ∏ 1Aj (σκtj )] = e2κt e2Et ∑ 𝔼σ𝒫 [ (Ψg (σκt0 ), J[−t,t] Ψg (σκtn )) ∏ 1Aj (σκtj )] σ∈ℤ2

j=0

= (Ψg , 1A0 e

j=0

ℤ −(t1 −t0 )(HSB2 −E)

1A1 ⋅ ⋅ ⋅ e

ℤ −(tn −tn−1 )(HSB2 −E)

1An Ψg ) .

The corollary is obtained by a limiting argument. Lemma 4.18. Let F : ℝ → ℝ be a bounded function, f ∈ L2real (ℝd ), and h : ℤ2 → ℂ be another bounded function. Then ℤ2

T

ℤ2

(e−THSB 1SB , hF(ϕ(f ))e−THSB 1SB ) = e2κT ∑ 𝔼σ𝒫 𝔼μE [h(σ0 )e−ϕE (∫−T σκs js φds) F(ϕE (j0 f ))] . ̃

σ∈ℤ2

t

ℤ2

̃ Proof. By e−tHSB Ψ(σ) = eκt 𝔼σ𝒫 [J∗0 e−ϕE (∫0 σκr jr φdr) Jt Ψ(σκt )], we have ℤ2

ℤ2

σ

(e−THSB hF(ϕ(f ))e−THSB 1SB )(σ) = e2κT 𝔼σ𝒫 [J[0,T] h(σκT )F(ϕ(f ))𝔼𝒫κT [J[0,T] 1(σκT )]] .

4.4 Gibbs measure associated with the ground state

| 493

Note that 1SB (σ) = 1L2 (Q) for every σ ∈ ℤ2 . Thus we see that ℤ2

ℤ2

∑ (1, (e−THSB hF(ϕ(f ))e−THSB 1SB )(σ))L2 (Q)

σ∈ℤ2

σ

= e2κT ∑ 𝔼σ𝒫 [(1, J[0,T] h(σκT )F(ϕ(f ))𝔼𝒫κT [J[0,T] 1SB ])] σ∈ℤ2

T

σ

= e2κT ∑ 𝔼σ𝒫 [(1, X𝔼𝒫κT [JT F(ϕ(f ))J∗0 e−ϕE (∫0

̃ σκs js φds)

σ∈ℤ2

T

Here X = e−ϕE (∫0

̃ σκs js φds)

1SB ])] .

h(σκT ). By the Markov property of Et = Jt J∗t , we also have T

σ

= e2κT ∑ 𝔼σ𝒫 [(1, XF(ϕE (jT f ))𝔼𝒫κT [e−ϕE (∫0

̃ σκs js+T φds)

σ∈ℤ2

1SB ])] ,

and by the Markov property of (Nt )t∈ℝ we furthermore have T

= e2κT ∑ 𝔼σ𝒫 [(1, XF(ϕE (jt f ))e−ϕE (∫0

̃ σκ(s+T) js+T φds)

σ∈ℤ2

2T

= e2κT ∑ 𝔼σ𝒫 [(1, e−ϕE (∫0

̃ σκs js φds)

σ∈ℤ2

1)]

h(σκT )F(ϕE (jT f ))1)] .

Using shift invariance, the lemma follows. Theorem 4.19. Let ISB < ∞, f ∈ L2real (ℝd ), h : ℤ2 → ℂ be a bounded function, and β ∈ ℝ. Then β2

2

(Ψg , heiβϕ(f ) Ψg ) = e− 4 ‖f ‖ 𝔼μ∞ [h(σ0 )eiβK(f ) ] ,

(4.4.4)

where K(f ) is a random variable on (Z , σ(G )) given by ∞

1 ̂ √ω, f ̂)σκr dr. K(f ) = − ∫ (e−|r|ω φ/ 2 −∞

Proof. Note that (Ψg , heiβϕ(f ) Ψg ) = lim (ΨTg , heiβϕ(f ) ΨTg ), T→∞

where ΨTg = 1T /‖1T ‖, and we see that by Lemma 4.18, (ΨTg , heiβϕ(f ) ΨTg ) =

T e2κT ̃ ∑ 𝔼σ𝒫 𝔼μE [h(σ0 )e−ϕE (∫−T σκs js φds) eiβϕE (j0 f ) ] . ZT σ∈ℤ 2

The expectation with respect to μE can be computed explicitly, and thus β2

2

β

T

(Ψg , heiβϕ(f ) Ψg ) = lim e− 4 ‖f ‖ 𝔼μT [h(σ0 )e−i 2 ∫−T (e T→∞

−|s|ω

̂ φ/√ω, f ̂)σκs ds

].

494 | 4 Spin-boson model by path measures Notice that ∞

󵄨 󵄨󵄨 ̂ 󵄨󵄨 ∫ (e−|s|ω φ/ ̂ √ω, f ̂)σκs ds󵄨󵄨󵄨󵄨 ≤ 2I1/2 󵄨󵄨 SB ‖f ‖ < ∞. 󵄨 −∞

By the local convergence of μT as T → ∞ and a similar telescoping as (2.8.22)–(2.8.24) in the proof of Corollary 2.86, we obtain the desired result. By using Theorem 4.19, the functionals (Ψg , hF(ϕ(f ))Ψg ) can be represented in terms of averages with respect to the Gibbs measure μ∞ . We consider the cases when F is a polynomial or a Schwartz test function. We will show in Corollary 4.27 below that Ψg ∈ D(e+βN ) for all β > 0, thus Ψg ∈ D(ϕ(f )n ) for every n ∈ ℕ. Corollary 4.20 (Moments of field operators). Assume ISB < ∞, and let f ∈ L2real (ℝd ), and h : ℤ2 → ℂ be a bounded function. Also, let Hn (x) = (−1)n ex Hermite polynomials of degree n. Then (Ψg , hϕ(f )n Ψg ) = in 𝔼μ∞ [h(σ0 )Hn (

−iK(f ) )](‖f ‖/√2)n , ‖f ‖/√2

2

/2 dn −x2 /2 e dxn

n ∈ ℕ.

be the

(4.4.5)

Proof. We have −iK(f ) 1 dn −β2 ‖f ‖2 /4 iβK(f ) ) (‖f ‖/√2)n . e e ⌈ = in Hn ( in dβn ‖f ‖/√2 β=0 Using this and the computation, (Ψg , hϕ(f )n Ψg ) =

1 dn − β42 ‖f ‖2 𝔼μ∞ [h(σ0 )eiβK(f ) ]⌈ , e in dβn β=0

we obtain (4.4.5). In the next corollary we express (Ψg , hF(ϕ(f ))Ψg ), F ∈ S (ℝ) in terms of the Gibbs measure. Corollary 4.21. Let ISB < ∞, f ∈ L2real (ℝd ), F ∈ S (ℝ), and h : ℤ2 → ℂ be a bounded function. Then (Ψg , hF(ϕ(f ))Ψg ) = 𝔼μ∞ [h(σ0 )G (K(f ))] , 2

2

where G = F̌ ∗ ǧ and g(β) = e−β ‖f ‖ /4 . Proof. Since F(ϕ(f )) =

1 √2π

∞

iβϕ(f ) ̌ dβ, we have ∫−∞ F(β)e ∞

β2 1 − 4 ‖f ‖2 ̌ 𝔼μ∞ [h(σ0 )eiβK(f ) ] dβ. (Ψg , hF(ϕ(f ))Ψg ) = ∫ F(β)e √2π

−∞

(4.4.6)

4.4 Gibbs measure associated with the ground state

| 495

The field fluctuations in the ground state can be defined for every real-valued function f ∈ L2real (ℝd ) by F(f ) = (Ψg , ϕ(f )2 Ψg ) − (Ψg , ϕ(f )Ψg )2 .

(4.4.7)

More generally, we also consider fluctuations of the form G(f ) = (Ψg , (σϕ(f ))2 Ψg ) − (Ψg , σϕ(f )Ψg )2 . Corollary 4.22 (Average field strength and field fluctuations). Assume ISB < ∞ and let f ∈ L2real (ℝd ). The following hold. (1) Average field strength: (Ψg , σϕ(f )Ψg ) = 𝔼μ∞ [σ0 K(f )]. (2) Field fluctuations: (Ψg , (σϕ(f ))2 Ψg ) = 𝔼μ∞ [(σ0 K(f ))2 ] + 21 ‖f ‖2 .

In particular, we have 2 (3) G(f ) = 𝔼μ∞ [(σ0 K(f ))2 ] − (𝔼μ∞ [σ0 K(f )]) + 21 ‖f ‖2 . 2 (4) (Ψg , ϕ(f )Ψg ) = 0 and F(f ) = (Ψg , ϕ(f ) Ψg ) = 𝔼μ∞ [K(f )2 ] + 21 ‖f ‖2 . (5) Let f ≢ 0, F0 and G0 be given by F and G with ϕφ replaced by 0, then (i) F(f ) > 0 and F(f ) ≥ F0 (f ), (ii) G(f ) > 0 and G(f ) ≥ G0 (f ).

Proof. Statements (1)–(3) easily follow from Corollary 4.20, which imply (4) for σ = 1. Using Schwarz inequality, we obtain (5) (ii), while (5) (i) is clear by (4). We note that to prove (1)–(2) in Corollary 4.22 we can proceed, alternatively, to first derive the equality (Ψg , ϕ(f )Ψg ) = 𝔼μ∞ [K(f )] by using Corollary 4.20, and from 𝔼μ∞ [σ0 ] = (Ψg , σΨg ) to further obtain that 1 ̂ (Ψg , ϕ(f )Ψg ) = − (φ/ω, f ̂)(Ψg , σΨg ) = 0. 2 (Notice that obviously σ02 = 1.) Thus in Corollary 4.22 we have equivalently 𝔼μ∞ [(σ0 K(f ))2 ] = 𝔼μ∞ [K(f )2 ]. 2

Next we show that (Ψg , eβϕ(f ) Ψg ) < ∞ for some β > 0. Theorem 4.23 (Gaussian domination of the ground state). Assume ISB < ∞ and let f ∈ 2 L2real (ℝd ). If β ∈ (−∞, 1/‖f ‖2 ), then Ψg ∈ D(e(β/2)ϕ(f ) ) and 2

‖e(β/2)ϕ(f ) Ψg ‖2 =

1 √1 − β‖f ‖2

In particular, 2

lim ‖eβϕ(f ) Ψg ‖ = ∞.

β↑1/2‖f ‖2

βK(f )2

𝔼μ∞ [e 1−β‖f ‖2 ].

496 | 4 Spin-boson model by path measures Proof. Let B = {z ∈ ℂ | |z| < 1/‖f ‖2 } and consider ℂ+ = {z ∈ ℂ | ℜz > 0}. Define ρ(z) =

1 √1 + z‖f ‖2

𝔼μ∞ [e

−

zK(f )2 1+z‖f ‖2

],

z > 0.

̂ Since |K(f )| ≤ ‖f ‖‖φ/√ω‖ uniformly in the paths, the function ρ can be analytically continued to a function ρ̃ on ℂ+ ∪B. In the same way as in the proof of Corollary 2.61, we 2 2 ̃ follows for z ∈ ℂ+ ∪B. can show that Ψg ∈ D(e−(z/2)ϕ(f ) ) and ‖e−(z/2)ϕ(f ) Ψg ‖2 = ρ(z) We can also derive the moments of fractional order. Define |ϕ(f )|s = (ϕ(f )2 )s/2 for 0 ≤ s ≤ 2, and let λs be a Lévy measure on (ℝ \ {0}, ℬ(ℝ \ {0})) such that ∞

∫ (1 − e−yu )λs (dy) = us/2 ,

u > 0,

0

i. e., λs (dy) =

s s y−1− 2 1(0,∞) (y)dy, 2Γ(1 − s/2)

corresponding to the s/2-stable process. Let Λφ̂ = ‖|ϕ(f )|s/2 Ψg ‖2 . Corollary 4.24 (Fractional moments of field operators). Assume ISB < ∞, and take f ∈ L2real (ℝd ). Then for 0 < s < 2, ∞

Λφ̂ = 𝔼μ∞ [ ∫ (1 −

e

−

βK(f )2 1+β‖f ‖2

0

√1 + β‖f ‖2

−1− 2s

e

(4.4.8)

)λs (dβ)].

In particular, Λ0 ≤ Λφ̂ follows. Proof. Notice that

(1 −

e

−

βK(f )2 1+β‖f ‖2

√1 +

β‖f ‖2

)β

≤ (1 −

−

β‖f ‖2 ISB /4 1+β‖f ‖2

√1 +

β‖f ‖2

s

)β−1− 2 = η(β).

In a neighborhood of β = 0 we have that η(β) = β−s/2 +o(β), locally uniformly. Then η(β) is integrable in this region, and since η(β) ≤ const β1+s/2 , it follows that η ∈ L1 ([0, ∞)). Then (4.4.8) is immediate by using Fubini’s theorem. Since ∞

Λ0 = ∫ (1 − 0

Λ0 ≤ Λφ̂ also follows.

1 √1 + β‖f ‖2

)λs (dβ),

4.4 Gibbs measure associated with the ground state

| 497

2

Theorem 4.23 says that ‖e(β/2)ϕ(f ) Ψg ‖ < ∞ for β < 1/‖f ‖2 . Using this fact, we can obtain explicit formulae for the exponential moments (Ψg , eβϕ(f ) Ψg ) of the field. Corollary 4.25 (Exponential moments). Assume ISB < ∞ and let f ∈ L2real (ℝd ). Then Ψg ∈ D(eβϕ(f ) ) and β2

2

(Ψg , eβϕ(f ) Ψg ) = (Ψg , cosh(βϕ(f ))Ψg ) = e 4 ‖f ‖ 𝔼μ∞ [eβK(f ) ] , β2

(4.4.9)

2

(Ψg , σeβϕ(f ) Ψg ) = (Ψg , σ sinh(βϕ(f ))Ψg ) = e 4 ‖f ‖ 𝔼μ∞ [σ0 eβK(f ) ] .

(4.4.10)

Proof. For simplicity, we reset βf to f . By using the generating function 1 2

∞

exy− 2 y = ∑ Hn (x) n=0

yn n!

of Hermite polynomials, summation in (4.4.5) gives M

1 2 1 ϕ(f )n Ψg ) = e 4 ‖f ‖ 𝔼μ∞ [eK(f ) ] . n! n=0

(4.4.11)

lim (Ψg , ∑

M→∞

We need to check that the left-hand side converges to (Ψg , eϕ(f ) Ψg ). Since by the spin flip property (2) in Lemma 4.6 for for odd n we get (Ψg , ϕ(f )n Ψg ) = 0, it suffices to show convergence of the terms with even exponents. By Theorem 4.23 we have that 2 2 ‖eϕ(f ) /(4‖f ‖ ) Ψg ‖ < ∞. Let Eλ be the spectral measure of ϕ(f ) with respect to Ψg . Then M 1 2n −λ2 /(4‖f ‖2 ) λ2 /(4‖f ‖2 ) 1 (Ψg , ϕ(f )2n Ψg ) = ∫ ∑ λ e e dEλ . (2n)! (2n)! n=0 n=0 M

∑

ℝ 2

2

Since by Theorem 4.23 the function eλ /(4‖f ‖ ) is integrable with respect to dEλ , we see 2 2 1 2n −λ2 /(4‖f ‖2 ) are monotonously increasing to cosh(λ)e−λ /(4‖f ‖ ) that the sums ∑M n=0 (2n)! λ e as M ↑ ∞, which is a bounded function, hence the monotone convergence theorem yields M

1 n λ dEλ = ∫ eλ dEλ < ∞, n! n=0

lim ∫ ∑

M→∞

ℝ

ℝ

implying Ψg ∈ D(eϕ(f ) ) and (4.4.9). Equality (4.4.10) is obtained similarly. Now we restrict attention to expectations of the form (Ψg , e−βdΓ(ρ)̂ Ψg ), where ρ is a real-valued function and ρ̂ = ρ(−i∇). A fundamental example is ρ = 1 giving the boson number operator N. As for the Nelson model in Theorem 3.56, we obtain the expression (ΨTg , he−βdΓ(ρ)̂ ΨTg ) ‖ΨTg ‖2

0

T

= 𝔼μT [h(σ0 )e− ∫−T ds ∫0

W ρ,β (σκs σκt ,s−t)dt

],

498 | 4 Spin-boson model by path measures with

η 2 −|T|ω(k) ̂ e (1 − e−βρ(k) )dk. ∫ |φ(k)| 2

W ρ,β (η, T) =

ℝd

Denote 0

∞

−∞

0

ρ,β W∞ = ∫ ds ∫ W ρ,β (σκs σκt , s − t)dt,

ρ,β and notice that |W∞ | ≤ ISB /2 < ∞, uniformly in the paths in Z . Thus we have the theorem below.

Theorem 4.26. Assume ISB < ∞ and let h : ℤ2 → ℂ be a bounded function. Then ρ,β

(Ψg , he−βdΓ(ρ) Ψg ) = 𝔼μ∞ [h(σ0 )e−W∞ ] , ̂

β > 0.

In particular, (Ψg , he−βN Ψg ) = 𝔼μ∞ [h(σ0 )e−(1−e

−β

)W∞

],

(4.4.12)

where 0

∞

W∞ = ∫ ds ∫ W(σκs σκt , s − t)dt. −∞

0

The following result says that the distribution of the number of bosons in the ground state has a super-exponentially short tail. Corollary 4.27 (Super-exponential decay of boson number). If ISB < ∞, then Ψg ∈ D(eβN ) for every β ∈ ℂ, and (Ψg , eβN Ψg ) = 𝔼μ∞ [e−(1−e

β

)W∞

]

holds. In particular, Ψg ∈ D(e+βN ), for all β > 0. Proof. The proof is similar to that of Corollary 2.86, and is left to the reader. By using Corollary 4.27, we can also derive the expectations of N m , m ∈ ℕ, with respect to the ground state Ψg . Corollary 4.28. If ISB < ∞, then m

r (Ψg , N m Ψg ) = ∑ S(m, r)𝔼μ∞ [W∞ ], r=1

where S(m, r) are the Stirling numbers of the second kind.

m ∈ ℕ,

4.4 Gibbs measure associated with the ground state

| 499

Proof. It can be checked directly that m −β dm −C(1−e−β ) m = (−1) S(r, m)e−rβ (−C)r e−a(1−e ) . e ∑ m dβ r=1 m

d −βN Ψg )⌈β=0 and Then the corollary follows by the equality (Ψg , N m Ψg ) = (−1)m dβ m (Ψg , e Corollary 4.27.

Corollary 4.29. If ISB < ∞, then (Ψg , (−1)N Ψg ) = 𝔼μ∞ [e−2W∞ ] ,

(Ψg , h(−1)N Ψg ) = 𝔼μ∞ [h(σ0 )e−2W∞ ] .

(4.4.13) (4.4.14)

In particular, (Ψg , (−1)N Ψg ) ≥ e−ISB > 0,

(4.4.15)

(Ψg , σ(−1)N Ψg ) = 𝔼μ∞ [σ0 e−2W∞ ] = −1 < 0. Proof. Equality (4.4.13) is derived from (4.4.12) with h(σ) = 1 and β = −iπ, and (4.4.14) with h(σ) = σ. Equality (4.4.15) follows from an estimate of the right-hand side of − (4.4.14). Noticing that Ψg ∈ ℋSB , we obtain PΨg = −Ψg . In particular, this gives 𝔼μ∞ [σ0 e−2W∞ ] = (Ψg , σ(−1)N Ψg ) = (Ψg , PΨg ) = −1.

4.4.3 van Hove representation The spin-boson model with κ = 0 can be realized as the direct sum of the so-called van Hove Hamiltonian. Then all the spectral properties of the spin-boson Hamiltonian go over from the van Hove Hamiltonian. The van Hove Hamiltonian is defined by the selfadjoint operator ̂ HvH (g)̂ = Hf + Φ(g), in Fock space ℱSB . Here Φ(g)̂ =

1 ∗ ̃̂ (a (g)̂ + a(g)) √2

̂ is the scalar field with cutoff g.̂ Suppose that g/ω ∈ L2 (ℝd ) and define the conjugate momentum by Π(g)̂ =

i ̃̂ ̂ (a∗ (g/ω) − a(g/ω)). √2

500 | 4 Spin-boson model by path measures Then we have 1 ̂ ̂ −iΠ(g)̂ = Hf − ‖g/̂ √ω‖2 . eiΠ(g) HvH (g)e 2 Hence we conclude that HvH (g)̂ has the unique ground state ̂ = e−iΠ(g)̂ Ωb ΨvH g (g) ̂ ∈ L2 (ℝd ). It is also known that HvH (g)̂ has no ground state when g/ω ̂ ∉ whenever g/ω 2 d L (ℝ ). We summarize this in the proposition below. ̂ ∈ L2 (ℝd ). Proposition 4.30. HvH (g)̂ has a ground state if and only if g/ω On the other hand, clearly the spin-boson Hamiltonian H with κ = 0 is the direct sum of van Hove Hamiltonians since ̂ Hf + Φ(φ/√ω) 0

H=(

0 ) ̂ Hf − Φ(φ/√ω)

(4.4.16)

̂ and Hf ± Φ(φ/√ω) are equivalent. Therefore, whenever ISB < ∞, the ground state of H with κ = 0 can be obtained as ̂ ΨvH g (φ/√ω) ). Ψg = ( vH ̂ Ψg (−φ/√ω) The operator (4.4.16) has however no ground state when ISB = ∞. Thus in the case that ISB < ∞ and κ = 0, we have ̂ 1 ̂ √ω), eiβϕ(f ) ΨvH ̂ √ω))ℱ , ∑ (ΨvH g (σ φ/ g (σ φ/ SB 2 σ∈ℤ

(Ψg , eiβϕ(f ) Ψg )ℋSB =

2

and the right-hand side above equals ̂

̂

2

̂

2

̂

(Ωb , eiβ(ϕ(f )+(φ/ω,f )) Ωb )ℱSB = e−β ‖f ‖ /4+iβ(φ/ω,f ) . ̂

̂

When κ ≠ 0, we can derive similar but nontrivial representations. Define the random boson field operator Ψ(f ̂) = ϕ(f ̂) + K(f ) ∞ ̂ f ̂)σκr dr is the random variable on (Z , ℬZ ). on ℱSB , where K(f ) = − 21 ∫−∞ (e−|r|ω φ/√ω, Then we see that: (1) (Ωb , Ψ(f ̂)Ωb ) = K(f ); (2) (Ωb , Ψ(f ̂)2 Ωb ) − (Ωb , Ψ(f ̂)Ωb )2 = ‖f ‖2 /2; ̂

2

2

(3) (Ωb , eiβΨ(f ) Ωb ) = e−β ‖f ‖ /4+iβK(f ) .

4.5 Rabi Hamiltonian

| 501

Let ∞

1 ̂ √ω(k) ∫ e−|s|ω(k) σκs ds. χ = − φ(k) 2 −∞

Note that χ ∈ L2 (ℝd ) and K(f ) = (χ, f ̂); ̂ when κ = 0. We define the moreover, χ/ω ∈ L2 (ℝd ) whenever ISB < ∞, and χ = σ φ/√ω random van Hove Hamiltonian by HvH (χ). Theorem 4.31 (van Hove representation). If ISB < ∞, then ̂

̂

iβϕ(f ) vH (Ψg , eiβϕ(f ) Ψg ) = 𝔼μ∞ [(Ωb , eiβΨ(f ) Ωb )] = 𝔼μ∞ [(ΨvH Ψg (χ))] . g (χ), e

Proof. The first equality can be directly obtained by Theorem 4.19. The second equality follows from eiΠ(χ) Ψ(f ̂)e−iΠ(χ) = ϕ(f ̂). Corollary 4.32. Suppose that F ∈ S (ℝ). Then we have ̂ vH (Ψg , F(ϕ(f ))Ψg ) = 𝔼μ∞ [(Ωb , F(Ψ(f ̂))Ωb )] = 𝔼μ∞ [(ΨvH g (χ), F(ϕ(f ))Ψg (χ))] , 2

̂

2

̂

2

2 ‖eβϕ(f ) /2 Ψg ‖2 = 𝔼μ∞ [‖eβΨ(f ) /2 Ωb ‖2 ] = 𝔼μ∞ [‖eβϕ(f ) /2 ΨvH g (χ)‖ ] .

Proof. This follows by Corollary 4.21 and Theorem 4.23.

4.5 Rabi Hamiltonian Cavity quantum electrodynamics has supplied evidence of stronger interactions than encountered in standard quantum electrodynamics. Experimental physicists study this interaction by coupling a two-level atom with a one-mode light in a mirror cavity. This can be realized as a one-photon mode spin-boson model. The Hamiltonian of the Rabi model is defined as a self-adjoint operator in ℂ2 ⊗ L2 (ℝ). Let a and a∗ denote the single mode annihilation and creation operators given by a=

1 d 1 ( + √ωx) √2 √ω dx

and a∗ =

1 1 d (− + √ωx). √2 √ω dx

Definition 4.33 (Rabi Hamiltonian). The Rabi Hamiltonian is defined by HRabi = κσz ⊗ 1 + 1 ⊗ ωa∗ a + gσx ⊗ (a + a∗ ) in the Hilbert space ℂ2 ⊗ L2 (ℝ). Here κ > 0 and ω > 0 are respectively the atom transition frequency and the cavity resonance frequency, and g ∈ ℝ is a coupling constant.

502 | 4 Spin-boson model by path measures The Hamiltonian HRabi has the purely discrete spectrum σ(HRabi ) = {Ej (g)}∞ j=0 , and the map g 󳨃→ Ej (g) gives rise to the eigenvalue curves. We are interested in studying spectral properties of HRabi , in particular, the crossing of eigenvalue curves. The absence of crossing of ground state energy g 󳨃→ E0 (g) can be derived from the non-degeneracy of the ground state energy of HRabi . We will construct a functional integral representation of e−tHRabi to show that the ground state energy has multiplicity one. This requires a minor modification of the spin-boson model discussed so far. Denote U = e(iπ/4)σy , as for spin-boson model. Then UHRabi U −1 = −κσx ⊗ 1 + 1 ⊗ ωa∗ a + gσz ⊗ (a + a∗ ). 2

Recall the ground state Ψosc (x) = (ω/π)1/4 e−ωx /2 of the classical harmonic oscillator Hosc . Since Ψosc is strictly positive, we can define the ground state transform Ug : L2 (ℝ) → L2 (ℝ, Ψ2osc dx) by Ug f = f /Ψosc , and we will denote the probability measure Ψ2osc dx on ℝ by dμ. Thus UHRabi U −1 is mapped into the operator Ug UHRabi U −1 Ug−1 = −κσx ⊗ 1 + 1 ⊗

1 d2 d (− 2 + ωx ) + gσz ⊗ √2ωx 2 dx dx

(4.5.1)

in ℂ2 ⊗ L2 (ℝ, dμ). Formally, Ug UHRabi U −1 Ug−1 is the one-dimensional version of H̃ SB 2

d d ̃ ∼ √2ωx. We make the idenunder the correspondence Hf ∼ 21 (− dx 2 + ωx dx ) and ϕ(φ) tification

󵄨󵄨 ℂ2 ⊗ L2 (ℝ) ≅ L2 (ℝ × ℤ2 , dμ) = {f = f (x, σ)󵄨󵄨󵄨 ∑ ∫ | f (x, σ)|2 dμ(x) < ∞} 󵄨 σ∈ℤ 2

ℝ

through f+ (x) ) 󳨃→ f (x, σ) ∈ L2 (ℝ × ℤ2 , dμ). f− (x)

ℂ2 ⊗ L2 (ℝ) ∋ (

Under this equivalence, (4.5.1) turns into the operator 1 d2 d ℤ2 (HRabi f )(x, σ) = [ (− 2 + ωx ) + g √2ωσx] f (x, σ) − κf (x, −σ). 2 dx dx ℤ

2 In what follows we use HRabi instead of HRabi . Let

Losc =

1 d2 d (− 2 + ωx ) 2 dx dx

4.5 Rabi Hamiltonian

| 503

and (Xt )t≥0 be an Ornstein–Uhlenbeck process on Wiener space (X , ℬ(X ), 𝒲 x ). We have ∫ 𝔼x [Xt ]dμ(x) = 0

∫ 𝔼x [Xs Xt ]dμ(x) =

and

ℝ

ℝ

e−|t−s|ω . 2ω

The generator of Xt is −Losc and (f , e−tLosc g)L2 (ℝ,dμ) = ∫ 𝔼x [f (X0 )g(Xt )] dμ(x) ℝ

holds. The distribution ρt (x, y) of Xt is given by ρt (x, y) =

1

Ψosc (x)

Kt (x, y)Ψosc (y),

where Kt (x, y) denotes the Mehler kernel Kt (x, y) =

1

√π(1 − e−2t )

exp (

4xye−t − (x 2 + y2 )(1 + e−2t ) ). 2(1 − e−2t )

Theorem 4.34 (Functional integral representation for Rabi Hamiltonian). We have ℤ2

(f , e−tHRabi g) = eκt ∑ ∫ 𝔼x,α [f (X0 , σ0 )g(Xt , σκt )e−g

t

√2ω ∫ σκs Xs ds 0

σ∈ℤ2 ℝ

] dμ(x).

(4.5.2)

ℤ

2 Proof. The Rabi Hamiltonian HRabi is the one-dimensional version of the spin-boson

Hamiltonian rem 4.8.

ℤ HSB2 ,

obtaining the functional integral representation is similar to Theo-

Corollary 4.35. The ground state of HRabi is unique. In particular, the ground state energy of HRabi has no crossing for all values of g and κ. Proof. Let f , g ≥ 0, not identically zero. Then for sufficiently small ε > 0 the sets Cf = {(x, σ) ∈ ℝ × ℤ2 | f (x, σ) > ε}, Cg = {(x, σ) ∈ ℝ × ℤ2 | g(x, σ) > ε} have positive measure. We have by (4.5.2), ℤ2

(f , e−tHRabi g) ≥ ε2 eκt ∑ ∫ 𝔼x,α [1Cf (X0 , σ0 )1Cg (Xt , θNt )e−g σ∈ℤ2 ℝ

t

√2ω ∫ σκs Xs ds 0

] dμ(x).

Since Cf is a subset of ℝ × ℤ2 , we have Cf = ⋃σ∈ℤ2 (Cfσ , σ). Thus either Cf+1 or Cf−1 have at least a positive measure. Suppose that Cf+1 has a positive measure. Similarly, we see

504 | 4 Spin-boson model by path measures that Cg = ⋃σ∈ℤ2 (Cgσ , σ) and suppose that Cg+1 has a positive measure. Let 𝒞 be the set of paths starting from the inside of (Cf+1 , +1) and arriving at the inside of (Cg+1 , +1). We see that ∑ ∫ 𝔼x,α [1𝒞 ] dμ(x) = ∑ ∫ 𝔼x,α [1C+1 (X0 )1Cg+1 (Xt )1Nt =even ] dμ(x). f

σ∈ℤ2 ℝ

σ∈ℤ2 ℝ

By using the distribution ρt of Xt we get t 2n e−t ∫ dx ∫ Ψosc (x)Kt (x, y)Ψosc (y)dy > 0. (2n)! n=0 ∞

∑ ∫ 𝔼x,α [1𝒞 ] dμ(x) = ∑

σ∈ℤ2 ℝ

Cf+1

Cg+1

Hence we conclude that 𝒞 has a positive measure and ℤ2

(f , e−tHRabi g) ≥ ε2 eκt ∑ ∫ 𝔼x,α [1𝒞 e−g σ∈ℤ2 ℝ

ℤ2

t

√2ω ∫ σκs Xs ds 0

] dμ(x) > 0.

Thus e−tHRabi is a positivity improving operator and uniqueness of the ground state follows by the Perron–Frobenius theorem.

5 Notes and references Notes to Chapter 1 1.1 A mathematical introduction to quantum field theory is given in [16, 24, 126, 142, 277, 292]. References [199, 260, 299–301] treat quantum field theory from the point of view of particle physics. Useful texts on quantum field theory include [19, 46, 53, 54, 97, 152, 245, 272, 267]. [95] is a monumental book on quantum field theory using path integrals. We also refer to [3, 62, 73, 144, 265, 270, 293] for texts in this spirit. [91] offers a concise summary of fermion functional integrals. Geometric and topological discussions on quantum field theory are [92, 108, 271, 291, 297]. Standard texts on QED are [150, 201, 203, 236, 269, 308]. Quantum field theory in curved space-time is discussed in [32, 45, 106]. [37, 232] covers spectral methods in infinite dimensional analysis. 1.2 The Fock space was introduced in [96]. The second quantization dΓ(h) was first treated by Cook [69], where self-adjointness of the field operator Φ(f ) and its conjugate momentum Π(f ) was shown. Nelson’s analytic vector theorem (Proposition 1.16) appeared in [246]. The Segal–Bargmann space was introduced in [35, 273]. Wick product was introduced in [303]. The number operator bounds (1.2.14)–(1.2.15) can be generalized to ‖(N + 1)−m/2 T(N + 1)−n/2 ‖ ≤ ‖W‖L2 (ℝd(m+n) ) , n ∗ where T = ∫ℝd(m+n) W(k1 , . . . , km , p1 , . . . , pn ) ∏m j=1 a (kj ) ∏i=1 a(pi )dkdp; see [119] for details. Operators of the form

ea

♯

(f )

1 ♯ n a (f ) , n! n=0 ∞

= ∑

a♯ = a, a∗ ,

are useful to study functional integral representations as is discussed in this chapter, ♯ but they are unbounded. There is much literature in which ea (f ) is treated, however, ♯ few rigorous discussions including domain issues of ea (f ) . In [141] several useful and ♯ interesting properties of ea (f ) are discussed, where Lemma 1.32 is proven and it is rig∗ orously shown that ea (f ) e−tHf is bounded for t > 0 (Proposition 1.33). 1.3 Segal [274, 275] introduced Q space, which linked the theory of free quantum fields with probability. Using this expression, the spectrum of the P(ϕ)2 -Hamiltonian has been studied, for instance, in the series of papers [120–125] by Glimm and Jaffe. Let A be a bounded operator on L2 (M, dμ), where (M, μ) is a σ-finite measure space. A is called ergodic if and only if it is positivity preserving and for every nonnegative https://doi.org/10.1515/9783110403541-005

506 | 5 Notes and references f , g ∈ L2 (M, dμ) that are there is n > 0 for which (f , An g) > 0. A positivity improving operator A is clearly ergodic. ‖A‖ is a simple eigenvalue and the associated eigenvector is strictly positive if and only if A is ergodic. Proposition 1.53 is a special version of this fact; we refer to [264, XIII.12]. For abstract discussions of positivity improving operators see [87, 240]. The doubly Markovian and the hypercontractivity discussed in Section 1.3.3 is taken from [277, Sections I.4 and I.5]. The hypercontractivity was introduced by Nelson in [250, 251], where also Proposition 3.13 was proved in [251, Theorem 2]. The example is furthermore given in [251, Theorem 4] such that c > √(p − 1)/(q − 1) with 1 ≤ p ≤ q ≤ ∞ and Γ(c) is not bounded from Lp (ℝ) to Lq (ℝ). Hence it implies that the condition ‖T‖ ≤ √(p − 1)/(q − 1) ≤ 1 in Theorem 1.67 is best possible. We also refer to [278]. 1.4 Theorem 1.68 is taken from [277, Section I.2]. Minlos’ theorem, Theorem 1.72, is an infinite dimensional extension of Bochner theorem. Minlos theorem first appeared in [237], and the proof of Theorem 1.72 is taken from [153]. By virtue of the Minlos theorem, quantum field theory can be investigated from a measure theoretical point of view. [126, 277] are references in this direction. 1.5 We refer to [31, Section 1.3] for homogeneous Sobolev spaces, where dualities, critical dimension s = d/2, and necessary and sufficient conditions for Ḣ s (ℝd ) to be a Hilbert space are discussed. Nelson emphasized the functional integral approach to quantum field theory in [250], and the fundamental results concerning the Markov property in quantum field theory were established in [251, 249]. We also refer to [277] for further aspects of Euclidean quantum field theory. Nelson’s Euclidean quantum field and its lattice approximation due to [140] changed the main object studied in quantum field theory before. 1.6 As explained in this section, there exists no Gaussian measure on a real Hilbert space ℋ with a given covariance. Instead of this, we consider the triplet M+2 ⊂ M ⊂ M−2 under the identification M ∗ = M , and construct the process t 󳨃→ ξt ∈ M−2 such that ξt (f ) = ⟨⟨ξt , f ⟩⟩ is a Gaussian process for f ∈ M+2 . Similar results appear in [59], where path continuity is considered. Gaussian measures and Borel measures on an infinite product probability space are studied in [71, 154, 307]. [261] treats the Ornstein– Uhlenbeck semigroup in an infinite dimensional L2 setting. [200] is a comprehensive study of Gaussian measures on a Hilbert space; we also refer to [80] for nonGaussian measures. [71, 204, 262] treat infinite dimensional stochastic equations. For the Minlos–Sazonov theorem (Theorem 1.117), we refer to [204, Theorem 2.3.4].

Notes to Chapter 2

| 507

Due to Bochner’s theorem it is an established fact positive definite functions are characteristic functions of finite measures. Note that the product and sum of positive definite functions are also positive definite. A large class of positive definite functions that was discovered by Pólya [230]. Given a function ϕ : [0, ∞) → [0, 1], satisfying (1) ϕ is continuous with ϕ(0) = 1, (2) limx→∞ ϕ(x) = 0, (3) ϕ is convex, an even extension of ϕ to the real line is positive definite. In other words, the function f : ℝ → [0, 1] given by f (x) = ϕ(|x|) is positive definite. Using this, we see that the functions f (x) = max(0, 1 − |x|) and g(x) = e−|x| are positive definite. 1.7 In our presentations we have followed [231, 252, 253, 79], we refer the reader to these sources for further detail and proofs, see also [8, 9]. For white noise theory we refer to [153, 154, 254]. For quantum stochastic calculus see the seminal [197, 258], and for a quantum Malliavin calculus [98].

Notes to Chapter 2 2.1 In 1964, Nelson demonstrated the mathematical existence of a meson theory with nonrelativistic nucleons in [247]. He introduced a system of Schrödinger particles coupled to a quantized relativistic field, which is nowadays called the Nelson model. He also introduced a quantum field theory model defined by H=

1 ∫ ψ∗ (x)(−Δ)ψ(x)dx + Hf + g ∫ ψ(x)HI (x)ψ(x)dx, (2m) ℝd

ℝd

where ψ is a nonrelativistic complex scalar field on ℋ = ⨁∞ n=0 ℋn . The restriction H⌈ℋn is the n-nucleon Nelson model. Since this model can be reduced to each ℋn , this is essentially the same as the Nelson model considered in this chapter. 2.2 As explained in this section, infrared divergence and ultraviolet divergence are a ubiquitous difficulty not only in the Nelson model but in the general theory of quantum fields. In 1952, Miyatake [242] and van Hove [296] found that the ground state of a Hamiltonian in Fock space weakly converges to zero when the ultraviolet cutoff is removed. Shale [276] developed mathematical methods to study both infrared and ultraviolet divergences. In [247], Nelson defined a self-adjoint operator with ultraviolet cutoff on the tensor product L2 (ℝd ) ⊗ ℱb of Hilbert spaces. The solution of the Schrödinger equation diverges as the cutoff tends to infinity, but the divergence amounts merely to a complex

508 | 5 Notes and references infinite phase shift due to the self-energy of the particles. A canonical transformation, which is implemented by a unitary operator when the infrared cutoff is introduced, separates the divergent self-energy term. It is then shown that, after removing the ultraviolet cutoff and subtracting an infinite constant, a limit Hamiltonian exists. The infrared problem in the Nelson model was first studied by Fröhlich [102, 103]. We also refer to, e. g., [57, 132, 147, 196, 207, 280–282] for earlier results on the Nelson model. 2.3 The Schrödinger operator Hp = − 21 Δ + V with Kato-decomposable potential V can be defined through Feynman–Kac formula of e−tHp . Let Ψp be the positive ground state of Hp and define Lp = Ψ1 (Hp − Ep )Ψp with Ep = inf Spec(Hp ) as a self-adjoint operator in p

L2 (ℝd , Ψ2p dx). Then there exists a diffusion process (Xt )t≥0 such that (f , e−tLp g)L2 (ℝd ,Ψ2 dx) = ∫ 𝔼x [f (X̄ 0 )g(Xt )]Ψp (x)2 dx. p

ℝd

The random process (Xt )t≥0 is called the P(ϕ)1 -process associated with Hp . This is discussed in detail in [222]. The functional integral representation of the semigroup generated by the Nelson Hamiltonian in terms of a combination of (Xt )t≥0 and an M−2 -valued random process (ξt )t≥0 is due to [41]. Here the ground state transformation of the Nelson Hamiltonian is applied. A similar functional integral representation, with no quantized field, is discussed in [305]. To construct (Xt )t≥0 , the existence of a positive ground state of Hp is assumed. By using Brownian motion (Bt )t≥0 instead of (Xt )t≥0 , we can construct a functional integral representation of e−tHN without any assumption on the existence of ground state of Hp . Then the integral kernel of (F, e−tHN G) is given by t

t

e− ∫0 V(Bs )ds I∗0 e−ϕE (∫0 δs ⊗φ(⋅−Bs )ds) It . In the massive case, the bound t

t

‖I∗0 e−ϕE (∫0 δs ⊗φ(⋅−Bs )ds) It ‖ ≤ ‖e−ϕE (∫0 δs ⊗φ(⋅−Bs )ds) ‖Lp can be derived from hypercontractivity with p = 1−e2−tν , where ν > 0 is the mass of a single boson. On the other hand, in the massless cases the bound (2.3.18), t

‖I∗0 e−ϕE (∫0 δs ⊗φ(⋅−Bs )ds) It ‖ ≤

q tEq (φ)̂ e q−1

is derived from the Baker–Campbell–Hausdorff formula and estimates of exponentials of creation and annihilation operators. This is discussed in [234]. A diamagnetic type inequality in Corollary 2.14 is also obtained in [169].

Notes to Chapter 2

| 509

In [113, 115] the functional integral representation of the semigroup generated by the Nelson Hamiltonian defined on a static Lorentz manifold is constructed. Here the static Lorentz manifold is a 4-dimensional pseudo-Riemannian manifold with line element ds2 = g00 (x)dt ⊗ dt − ∑ γij (x)dx i ⊗ dxj , i,j

where g00 > 0 is time-independent and γ = (γij ) denotes a 3-dimensional Riemannian metric. There the diffusion process (Xt )t≥0 associated with Hp = − 21 Δ + V is replaced by a random process associated with the divergence form 3

1 1 ∇μ Aμν (x)∇ν + V, c(x) c(x) μ,ν=1

− ∑

and the dispersion relation ω = √−Δ + m2 expressed in the position representation is replaced by the pseudo-differential operator 3

1/2

ωa,m = (− ∑ ∇μ aμν (x)∇ν + m2 (x)) . μ,ν=1

Note that, importantly, the mass becomes a position-dependent function m(x). 2.4 Kato-decomposable potentials provide a class of semigroups generated by Hp with singular potentials. In order to construct a path measure associated with the ground state of the Nelson Hamiltonian, the choice of Kato-decomposable potentials is also suitable. This is discussed in [41]. The absence of the ground state of the Nelson Hamiltonian is discussed under taking Kato-class potential; we refer to [227]. 2.5 For the massive case, the existence of the ground state of the translation invariant Nelson Hamiltonian is studied in [103, 131]. In the massless case, Bach, Fröhlich, and Sigal [28, 29] proved the existence and uniqueness of the ground state for sufficiently weak couplings under the infrared regular condition; they treated more general models, including the Nelson model, and Arai and Hirokawa [20] for the spin-boson model. Spohn [288] and Gérard [110, 112] proved the existence of a ground state for arbitrary values of the coupling constant. Functional integral methods are used in [288], where Theorem 2.28 is proven, and operator theory in [110]. External potentials V discussed in [110, 288] are confining. Let 1|x|≥R Ψ = ΨR . We define the ionization energy N ΨR ) . Let inf Spec(HN ) = EN . Then by Σ = limR→∞ ΣR , where ΣR = infΨ∈D(HN ),Ψ=0̸ (Ψ(ΨR ,H,Ψ R R) the binding condition is given by Σ − EN > 0.

510 | 5 Notes and references Under the binding condition, for general potentials including Coulomb potential, Griesemer, Lieb, and Loss [128], and Lieb and Loss [218] proved existence of a ground state by a compactness argument. In [128] the Pauli–Fierz model in nonrelativistic quantum electrodynamics is treated, but the method can be applied to other models including the Nelson model; see, e. g., [268] for the Nelson model. A comprehensive reference for the existence of ground states in quantum field theory is Hiroshima [180]. In the infrared singular case, the Nelson Hamiltonian has no ground state as is proven in this chapter. However, it has a ground state when a non-Fock representation is used; see [18, 226]. In a series of papers [113–115], the Nelson model on a static Lorenz manifold is considered. In particular, the infrared problem is studied, and criteria for the existence and the absence of ground states are shown. We also refer to [76–78, 107, 109, 111, 116, 155]. 2.6 This material is taken from [41]. Physically, it is expected that the number of soft (i. e., low-energy) bosons in the ground state diverges when the infrared cutoff is removed, and thus a ground state cannot exist in Fock space. (2.6.18) mathematically validates the first half of this intuition. The pull-through formula is quite useful to estimate (Ψg , NΨg ), which is used in, e. g., [4, 28, 30, 102, 110, 168]. Super-exponential decay of the number of bosons in the ground state of models in quantum field theory is also studied in [132, 285]. Ground state expectation for the fractional power of the number operator in Section 2.6.3 is studied in [185], and the Gaussian domination stated in Corollary 2.87 in [167, 177]. 2.7 2 ̂ In the infrared singular case, i. e., when ∫ℝd |φ(k)| /ω(k)3 dk = ∞, the absence of the ground state of the Nelson Hamiltonian is proven in [22, 77, 163, 226, 227]. Theorem 2.63 is due to [226, 227], where |V(x) ≥ |x|2β for some β > 0 is needed. In [22, 77, 163], it is shown that ‖N 1/2 Ψg ‖ = ∞ follows from the pull-through formula, if HN has a 2 ̂ ground state under ∫ℝd |φ(k)| /ω(k)3 dk = ∞. This contradicts Ψg ∈ D(Hf ) in the massive case and a ground state does not exist. In the massless case further investigation is needed. It is not straightforward to show the absence of the ground state of the Nelson model on a static Lorentz manifold by operator theory. Since the dispersion relation is the pseudo-differential operator ωa,m = (− ∑3μ,ν=1 ∇μ aμν (x)∇ν + m2 (x))1/2 , it α φ(k) as |k| → ∞. Hence the ̂ is then not clear how to estimate the decay rate of ω a,m

pull-through formula is not applicable. Instead of operator theory one can apply the functional integral method introduced in this chapter to show the absence of ground state.

Notes to Chapter 2

| 511

2.8 For the Nelson Hamiltonian, we define the family of probability measures μT , T ≥ 0, by (2.8.1) with the pair potential W, and a limit measure μ∞ exists (Theorem 2.77) Due to the infinite range pair potential W, μT does not define a Markov process. The pair potential W of the Nelson model is given by an iterated Lebesgue integral and the limit measure μ∞ can be applied to derive ground state expectations of observables. This discussion was initiated by [42, 283, 285] and extended to other models by [167, 177]. The key ingredient to prove the existence of the limit measure is the existence of the ground state Ψg of the Nelson Hamiltonian. In [255], however, the existence of a limit measure is shown without assuming the existence of the ground state but W is modified. For the proper pair potential of the Nelson model the existence, uniqueness and properties of the Gibbs measure have been obtained under the assumption of a small coupling constant in [225] by using cluster expansion. This has been further developed to cover Gibbs measures on more singular processes involving double Itôintegrals related to the ultraviolet renormalized Nelson model in [138, 221], by using techniques of rough paths. An alternative proof of the existence of the limit measure is to show the tightness of the family of measures {μT }T≥0 . This is done in [40] for the Pauli–Fierz Hamiltonian. It is however not straightforward to show the tightness of the family of measures with a pair potential including discontinuous paths. Sufficient conditions for the tightness of the family of measures on a discontinuous path space are comprehensively discussed in [44, 84]. 2.9 Carmona-type estimate plays an important role to investigate decay properties of eigenvectors of Schrödinger operators. This is shown in Carmona [60]. It can be applied to estimate decay properties of bound states of the Nelson Hamiltonian. This type of applications can be done not only for the Nelson Hamiltonian but also the Pauli–Fierz Hamiltonian. We refer to [181]. 2.10 An eigenfunction φ of the Schrödinger operator Hp = − 21 Δ + V defines a martingale with respect to the σ-field (FtBM )t≥0 by t

ht (x) = etE e− ∫0 V(Bs +x)ds φ(Bt + x),

t ≥ 0,

where (− 21 Δ+V)φ = Eφ. It is seen that 𝔼[ht (x)|FsBM ] = 𝔼[hs (x)], thus φ(x) = 𝔼[h0 (x)] = 𝔼[ht∧τ (x)] follows for every stopping time τ and |φ(x)| can be estimated by choosing a suitable stopping time according to V. This is applied to the estimate of the spatial decay of eigenfunctions of the semi-relativistic Schrödinger operators by [61, 182]. We also refer to [51, 52] for discussions on capacitary measures. This method is also applicable to models in quantum field theory with nonlocal kinetic terms; see [177].

512 | 5 Notes and references 2.11 Removal of the ultraviolet cutoff of the Nelson Hamiltonian (Theorem 2.100) has been proven first by Nelson [247] by using the so-called Gross transformation [129, 216]. Nelson [248] also tried to prove this by functional integration, and finally Gubinelli, Hiroshima and Lőrinczi [137] proved this by path integral methods, where a stochastic Fubini theorem [298] was used and a uniform lower bound of HNε with respect to ε was proven. Furthermore, Matte and Møller [234] gave an alternative proof by using path measures. On functional integral representations of semigroups generated by renormalized Nelson Hamiltonians, Lemma 2.114 is due to [137, 139] and Theorem 2.120 is ren [234]. In Theorem 2.120, a functional integral representation of (F, e−THN G) for arbitrary F, G is given, where the uniqueness of the ground state can be also derived by the Perron–Frobenius theorem. Proposition 2.108 is due to Bley and Thomas [49, 48] where the case of 0 < α < 2 are estimated. Using this formula, the ground state energies of related models including polaron model can be estimated. In order to derive Proposition 2.108, the Clark– Ocone formula is applied, see e. g., [205, 206, 252]. Let f : ℝ → ℝ be absolutely conx tinuous. Then the fundamental theorem of calculus is given by f (x) = f (0) + ∫0 f 󸀠 (t)dt a. e. A counterpart of this in stochastic analysis is the Clark–Ocone formula (see Theorem 1.166). Let F be a Brownian functional which is ℱTBM -measurable. Then under some conditions there exists an (ℱtBM )-adapted process (ρt )t≥0 such that T

F = 𝔼[F] + ∫ ρt dBt . 0

Thus T

𝔼[eF ] = e𝔼[F] 𝔼[e∫0

ρt dBt

T

] = e𝔼[F] 𝔼[e∫0

T

T

ρt dBt + p2 ∫0 |ρt |2 dt− p2 ∫0 |ρt |2 dt

].

Then by Schwarz inequality and the same trick used in (2.11.18) with the Girsanov theorem it follows that T

e𝔼[F] ≤ e𝔼[F] 𝔼[ep ∫0 = e𝔼[F] 𝔼[e

− qp 2

2

T

ρt dBt + p2 ∫0 |ρt |2 dt 1/p T ∫0

2

]

qp

𝔼[e− 2

T

∫0 |ρt |2 dt 1/q

]

|ρt | dt 1/q

]

for 1/p + 1/q = 1. Using this bound and some estimates on the heat kernel, Proposition 2.108 can be obtained. The bounds (2.11.7)–(2.11.8) are presented in [4, 168]. The existence of a ground state of the renormalized Nelson Hamiltonian HNren , is proven in [186] for arbitrary values of coupling constants, in [168] for sufficiently small coupling constants, and in [4] for the massive case. A path measure associated with the ground state of the renormalized Nelson Hamiltonian discussed in Section 2.11.6 is taken from [186] where Gaussian domination and super-exponential decay of the

Notes to Chapter 2

| 513

number of boson of the ground state are also shown. If the infrared singular condition ∫ ℝ3

|ξλ (k)|2 dk = ∞ ω(k)3

is satisfied, then Hren has no ground state. This is also proven in [186]. In [74, 75], a weak coupling limit of the form Hp +Λϕ(f )+Λ2 Hf is discussed. In [171] the weak coupling limit with a simultaneous removal of ultraviolet cutoff is studied by using functional integrals. Related works on the scaling limit in quantum field theory are [15, 17, 82, 257, 295]. In particular, in [15] a general criterion of convergence in the scaling limit is found. 2.12 The translation invariant Nelson model and related models have been studied in, e. g., [7, 57, 59, 102, 103, 131, 132, 243]. The functional integral approach to this model has been developed and Theorem 2.147 is due to [176, 284]. The identity (2.12.10) is due to [103, 284], and the pull-through formula (2.12.12) is studied in [103]. The removal of the ultraviolet cutoff of translation invariant Nelson Hamiltonian is parallel with that of the Nelson Hamiltonian, see, e. g., [179]. The renormalization term −

2 ̂ |φ(k)| 1 g2 dk ∫ 2 ω(k) ω(k) + |k|2 /2 ℝ3

introduced by Nelson [247] coincides with the coefficient of g 2 in the Taylor expansion of the ground state energy EN (g 2 ) of the translation invariant Nelson Hamiltonian with zero total momentum HN (0) = 21 Pf 2 + Hf + gHI (0) by a formal perturbation theory. On the other hand, the ground state energy is represented as EN (g 2 ) = − limt→0 1t log(f ⊗ 1, e−tHN (0) f ⊗ 1), and the renormalization term coincides with the diagonal part of the pair interaction. Hence the formal derivation of the renormalization term from the Taylor expansion of EN (g 2 ) is valid by functional integral representation in Theorem 2.168 involving limg→0

EN (g 2 )−EN (0) , g2

which is due to [178].

2.13 The study of the polaron model was initiated by Landau in 1933 to describe an electron moving in a crystal, where atoms move from their equilibrium positions to effectively screen the charge of an electron, known as a phonon cloud. This lowers the electron mobility and increases the so-called effective mass [215] of the electron. Fröhlich [101, 100] proposed a model for this polaron through which its dynamics are treated quantum mechanically. This model assumes that the wave-function describing the electron is spread out over many ions, which are all somewhat displaced from their equilibrium positions. Much of the theory focussed on solving this so-called Fröhlich

514 | 5 Notes and references polaron. [118] is a useful summary of various versions of polaron models. Feynman introduced a variational principle for path integrals to study the polaron in [94], and a Feynman path integral approach to the ground state energy of polaron Hamiltonian is a remarkable result. In [283] the effective mass is studied by functional integration, where a double stochastic integral appears. Suppose that the strength of interaction between electron and phonon is expressed by a coupling constant α. In physics, despite the lack of an exact solution, some approximations of the polaron properties are known. When |α| is sufficiently small, the self-energy of the polaron can be approximated linearly in α. On the other hand, when |α| is large, a variational approach indicates that the self-energy of polaron is proportional to α2 . It was an open question for many years whether this expression was asymptotically exact as α tends to infinity. Applying large deviation theory to functional integration for the self-energy, [81] contributed to solving this problem. Later [219] gave a shorter proof using more conventional methods. We also refer to [99, 224, 238, 244] for the spectral analysis of polaron-type models. 2.14 The presentation here follows from Betz and Spohn [42]. The functional central limit theorem in Theorem 2.203 follows from [43, 151, 195]. The above theorem still contains the possibility that the diffusion matrix is zero, which would imply subdiffusive behavior of Brownian paths. There are two independent proofs of ruling this out, one in [42, 283] using ideas of [58], and another in [136]. Both require substantial additional techniques to the material of this book, and therefore they are omitted in our presentation.

Notes to Chapter 3 3.1 In 1938, the Pauli–Fierz model was introduced in [259], where the generation of infrared photons was studied. In particular, they considered the cross- section under assumption that a finite extent was ascribed to the electron, and thus the wavelength of photon shorter than the diameter of electron was cut, equivalently ultraviolet cutoff was introduced. For the model, they considered they took the dipole approximation and neglected  2 term for the sake of simplicity. Then it is of the form: −

1 1 ̂ Δ ⊗ 1 + i∇ ⊗ A(0) + V ⊗ 1 + 1 ⊗ Hrad 2m m

which can be diagonalized as −

1 Δ ⊗ 1 + V(⋅ − ϕ) + 1 ⊗ Hrad + c. 2(m + δm)

Notes to Chapter 3

| 515

Here, ϕ is some field operator and c and δm are constants. See also [50] for the infrared divergence. Bethe [39] and Welton [302] proposed a theoretical interpretation of the Lamb shift by a variant of the Pauli–Fierz model. We refer to [68, 236] for details, and to [289] for a summary of recent progress of the spectral analysis of the Pauli–Fierz and related models. 3.2 ̂ ̂ In the dipole approximation, the quantized radiation field A(x) is replaced by A(0). In this case, collisions between electrons and photons are ignored, and there is no derivative coupling. Under specific conditions, the Pauli–Fierz model with dipole apdip proximation and V = 0 can be diagonalized, i. e., HPF can be approximated by the 3 + 3L dimensional harmonic oscillator L 2 1 1 L ̂ (−i∇ + ∑ φ(l)Q ) + ∑(Pl Pl + Ql ω2 (l)Ql − ω(l)). l 2m 2 l=1 l=1

Here, [Pl , Ql󸀠 ] = −iδll󸀠 is satisfied. Then it can be shown that dip HPF ≅−

2(m +

1

2) ̂ ‖φ/ω‖ e2 d−1 d

Δ ⊗ 1 + 1 ⊗ Hf + g,

where g is a constant. In the series of papers [11–14], Arai investigated the spectrum of the Pauli–Fierz Hamiltonian in the dipole approximation. In [190], the diagonalization of the Pauli–Fierz Hamiltonian with the dipole approximation by Bogoliubov transformation is discussed in detail. We also refer to [47, 286] for the scattering theory. The spectral analysis of the Pauli–Fierz Hamiltonian without dipole approximation has seen much progress since the 1990s. The existence of a ground state has been proven by Bach, Fröhlich, and Sigal [30] for weak couplings and no infrared cutoff. Griesemer, Lieb, and Loss [128], Lieb and Loss [218] successively removed the restriction on the coupling constant, and have also introduced a binding condition giving a criterion for the existence of a ground state; see also [33]. We emphasize that no infrared regularity is needed to show the existence of a ground state in Fock space, in contrast to the Nelson model. More precisely, to show the existence of the ground state of the Pauli–Fierz Hamiltonian the binding condition is a sufficient condition for massive cases. In massless cases, (1) and (2) can be checked: (1) localizations of the number of bosons by pull-through formula and the exponential decay [127] of massive ground state Ψg , (2) regularities with respect to variables x ∈ ℝ3 in position of nonrelativistic particle and k = (k1 , . . . , kn ) ∈ ℝ3n of the quantized field of the nth sector of the massive ground state Ψ(n) g (x, k1 , . . . , kn ). Then it can be shown in [128] massive ground states strongly converges to a nonzero vector as the mass reduces to zero, which is the ground state of the massless Pauli–Fierz Hamiltonian. See [180] for a brief review.

516 | 5 Notes and references A basic assumption to show the existence of the ground state is the existence of a ground state for the zero coupling Hamiltonian, i. e., that Hp has a ground state with positive spectral gap. [191] shows that for sufficiently large coupling constants the Pauli–Fierz Hamiltonian in the dipole approximation has a ground state even when Hp has no ground state. This phenomenon is called enhanced binding. Enhanced binding for the Pauli–Fierz Hamiltonian is also studied in [23, 63, 67, 146, 194]; See [210] for the relativistic Pauli–Fierz Hamiltonian. The enhanced binding for the N-body Nelson Hamiltonian is due to [187] and in [189] a relativistic version is studied. 3.3 Functional integral representations for the Pauli–Fierz Hamiltonian is discussed in [90, 105, 141, 143, 170, 233, 289]. We emphasize that the exponent in the functional integral representation t

̃ − Bs ))dBs −ie E (Kt ) = −ie ∫  Es (φ(⋅ 0

depends on time s explicitly, in contrast to the classical case. Gross [133] discusses a Q -space representation of a Proca field, which is a model of massive quantum electrodynamics. The diamagnetic inequality (Corollary 3.28) with a quantized radiation field is obtained in [25, 170]. 3.4 This material is taken from [156]. In a similar manner to the Nelson Hamiltonian, the choice of Kato-class potentials is suitable to construct a path measure associated with the ground state of the Pauli–Fierz Hamiltonian. 3.5 Essential self-adjointness of H + V such that e−tH is positivity preserving is studied in [88, 208]. Self-adjointness of the Pauli–Fierz Hamiltonian is established by [30] for weak couplings, and by [172, 174] for arbitrary values of coupling constants. Theorem 3.47 is taken from [174]. Alternative proofs are given in [85, 148]. An infinite dimensional version of the Perron–Frobenius theorem is due to [123, 130, 131]. Propositions 2.25 and 3.49 are taken from [87, 89]. The notion of a positivity preserving operator can be extended to invariant cone property. Let ℋ be a real vector space. 𝒦 ⊂ ℋ is a cone if and only if u, v ∈ 𝒦 → u + v ∈ 𝒦, v ∈ 𝒦 and a ≥ 0 → av ∈ 𝒦 and 𝒦 ∩ −𝒦 = 0. The Perron–Frobenius theorem can then be extended to the invariant cone semigroup. One application is to show the uniqueness of the ground state of a fermion system, which was done by [89, 131]. Theorem 3.51 is due to [173]. We also refer to [27, 123, 175, 229, 279] for the uniqueness of the ground state of the Pauli– Fierz Hamiltonian, to [239, 240] for an abstract version. The spatial exponential decay

Notes to Chapter 3

| 517

of eigenvectors of the Pauli–Fierz model is discussed in [30, 128, 156]. In [156], it is Kato Kato shown that e−tHPF is also positivity improving, and the ground state of HPF decays exponentially. These facts are proven in the same way as for HPF . 3.6 This material is taken from [40]. The formal double stochastic integral of the type ∞ ∞ ∫−∞ ds ∫−∞ e−iω|y−s| γ(t)γ(s)dt appears in Feynman [93, p. 443]. The mathematical derivation is done in [173, 248, 289]; see also [70, 241]. 3.7 This material is taken from [176]. The idea to construct the functional integral representation comes from [284]; see also [5, 228]. Chen [64] shows the existence of the ground state of HPF (p) when p = 0. The effective mass meff of the Pauli–Fierz Hamiltonian is defined by 1 1 = Δ E(p)⌈p=0 , meff 3 p where E(p) denotes the ground state energy of HPF (p) [134, 135]. When a sharp infrared cutoff is introduced in HPF , E(p) is differentiable with respect to pμ for sufficiently small coupling constants. It is, however, not clear that E(⋅) is twice differentiable in a neighborhood of p = 0, when no infrared cutoff is introduced. The infrared problem of the effective mass is studied in [26, 65, 66, 145]. The functional integral approach of the effective mass is [42, 283]. The asymptotic behavior of effective mass with respect to the ultraviolet cutoff is studied in [183, 193]. 3.8 A basic reference for the material of this section is [184]. The degeneracy of ground states for weak enough coupling is proven in [175, 192]. The uniqueness of the ground state of the translation invariant spinless Pauli–Fierz Hamiltonian is also proven in [192]. Spohn [285] treats the model by functional integration. The electronic part of the Pauli–Fierz Hamiltonian is nonrelativistic, given by a Schrödinger operator. Nelson’s work on Euclidean field theory for spinless boson proved to be a very significant conceptual and technical stimulus in the program of constructive quantum field theory. A version for fermions has been also discussed. The Euclidean Fermi field was introduced in [256]. [304] is an axiomatic approach. See, furthermore, [36] for the Itô–Clifford integral, [10] for the fermionic Itô formula, [220] for fermion martingales, [266] for fermion path integrals, and [213] for functional integration of the Euclidean Dirac field.

518 | 5 Notes and references 3.9 Relativistic Pauli–Fierz Hamiltonian HRPF is studied in many articles. In the functional integral representation, the subordinated Brownian motion (BTt )t≥0 is crucial. General change of time of processes is studied in [34]. In [104], the asymptotic field of the Pauli–Fierz Hamiltonian as well as relativistic one is investigated. In [177, 157], the selfadjointness and essential self-adjointness of the relativistic Pauli–Fierz Hamiltonian is shown by using a path measure, where an invariant domain of the semigroup e−tHRPF is found, and the uniqueness of the ground state is also derived by showing e−tHRPF is unitary equivalent to a positivity improving semigroup. The existence of the ground state of the relativistic Pauli–Fierz Hamiltonian is studied in, e. g., [209–212, 235] for the case of HRPF = √(−i∇ − eA)2 + m2 + V + Hrad with m ≠ 0. Relativistic Pauli–Fierz Hamiltonian with zero particle mass, i. e., HRPF = |−i∇ − eA| + V + Hrad , is the most singular case, and the existence of the ground state of this is shown in [158] for massive case and, in [159] for massless case, where the functional integral representation and asymptotic field are applied. In [177], the Gibbs measure associated with the ground state of HRPF is shown by using the existence of ground state, and the Gaussian domination of the ground state is proven. In [117, 188], binding conditions are discussed. We introduce three classes V1 , V2 , and V3 of external potentials V. We say that V is a relativistic Kato-class potential whenever V ∈ V3 , but V1 = V2 = V3 if V is uniformly locally integrable. This is studied in [2]. In [223], not only relativistic Kato-class but also Kato-class for Bernstein function of Laplacian is investigated. Let σ = (σ 1 , σ 2 , σ 3 ) be the 2 × 2 Pauli matrices. We define the relativistic Pauli–Fierz Hamiltonian with spin 1/2 by S HRPF = √(σ ⋅ (−i∇ − eA))2 + m2 − m + V + Hrad . S

The functional integral representation of e−tHRPF is given by an application of both of those of the Pauli–Fierz Hamiltonian with spin and the relativistic Pauli–Fierz Hamiltonian: t

S

(F, e−tHRPF G) = eTt ∑ ∫ 𝔼x,0,α [e− ∫0 V(BTs ) (J0 F(qT0 ), eSt Jt G(qTt ))] dx, α=1,2

ℝ3

where the exponent is given by Tt

Tt +

0

0

St = −ie E (KTrelt ) − ∫ Ĥ E,d (Bs , θNs , s)ds + ∫ log(−Ĥ E,od (Bs , −θNs− , s))dNs .

Notes to Chapter 4

| 519

Notes to Chapter 4 4.1 The spin-boson model is a much studied variant of the Caldeira–Leggett model describing a two-state quantum system linearly coupled to a scalar quantum field. A general reference for the spin-boson model is [217]. Work on the spectral properties of the spin-boson and related models includes [1, 20, 21, 161, 167] for the ground state energy, [6, 149, 285, 287, 294] for the existence of ground state, and [15, 55, 86, 162, 198, 290] for the spectral analysis. 4.2 This is taken from [167]. In [38, 83, 167, 198, 285, 290] stochastic methods for spinboson Hamiltonian are studied and applied. The functional integral representation of the spin-boson model follows directly from results in [184]; see also [38]. The shift invariance of spin processes is discussed in [223]. 4.3 The existence and absence of the ground state is investigated in [20, 21, 72, 109, 110, 112, 167]. Arai and Hirokawa [20, 21] defined a generalized spin-boson model and proved the existence and uniqueness of a ground state for sufficiently small coupling constants. In [110, 167], the existence and uniqueness of the ground state is shown for arbitrary values of coupling constants. Theorem 4.12 is taken from [167]. In particular, in [149] it is shown that the spin-boson Hamiltonian has a ground state even when no infrared regularity condition is imposed. In [72] a general version of the spin-boson − model is proposed, and existence of a ground state of HSB and the absence of a ground + state of HSB are shown by using the symmetries of the spin-boson model. 4.4 The path measure associated with the ground state and its applications have been done in [167]. Here the van Hove representation is also discussed and the ground state expectation (σα ⊗ Ω, e−tHSB σβ ⊗ Ω) is given in terms of a Poisson-driven process. This was also studied operator-theoretically in [160]. 4.5 In 1936, Rabi introduced a model in [263] to discuss the effect of a rapidly varying weak magnetic field on an oriented atom possessing nuclear spin. The atom is treated as a two-level system and the field as a classically rotating field. Experimental physicists evidence the interaction by coupling a two-level atom with a one-mode light in a mirror cavity, which can be realized as the Rabi model. Jaynes and Cummings [202] introduced a similar quantum model in 1963 describing a two-level atom interacting with a

520 | 5 Notes and references quantized mode of an optical cavity. Traditionally, to solve the model, the so-called rotating wave approximation is taken. In this approximation, the counter-rotating term of the model introduced by Jaynes and Cummings is neglected. Then it turned out to be a valid approximation for the near resonance and weak coupling regions; see [306] for the details and references therein. The Rabi model is studied from a mathematical point of view in, e. g., [56, 165, 164, 214]. In [166], a stochastic method is applied to estimate the ground state energy, and Theorem 4.34 is taken from here. Instead of L2 (ℝ), taking Segal–Bargmann space 𝒦 is useful to study HRabi , see [273]. Let 󵄨󵄨 󵄨

2 −|z|2

ℬ = {f (z) is holomorphic on ℂ 󵄨󵄨󵄨 ∫ |f (z)| e

dxdy < ∞}.

ℂ

Then L2 (ℝ) is unitarily equivalent to ℬ, and under this identification it follows that a ≅ d/dz and a∗ ≅ z. Then the eigenvalue equation ψ ψ HRabi ( 1 ) = E ( 1 ) ψ2 ψ2 can be reduced to find holomorphic solutions of the system dψ1 = (E − gz)ψ1 − Δψ2 , dz dψ (z − g) 2 = (E + gz)ψ2 − Δψ1 . dz

(z + g)

Let κ = 0. Then the eigenvalue equation turns into (

d E + g2 − + g)ϕ = 0. dz z+g

2

Thus ϕ(z) = Ce−gz (z + g)E+g , and ϕ must be holomorphic, implying that E + g 2 is a nonnegative integer. Thus eigenvalues are of the form E = n − g2,

n = 0, 1, 2, . . . .

Similar arguments for HRabi with κ ≠ 0 have been presented in, e. g., [214]. A two-photon Rabi Hamiltonian is defined by κσz + ωa∗ a + gσx (a2 + (a∗ )2 ) which is also a subject of interest in cavity quantum electrodynamics.

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Index absence of ground state 185 – criterion 160 absolutely continuous measure 85 analytic vector 16 angular momentum 395 – field 397 annihilation operator 4, 9, 324 average field strength – Nelson Hamiltonian 181 – spin-boson Hamiltonian 495 Baker–Campbell–Hausdorff formula 26, 34, 81 Bernstein function 176 binding condition 509 Bochner’s theorem 59 Borel measure on Hilbert space 95 boson Fock space 3 boson mass 9 boson number distribution 174 Burkholder–Davis–Gundy inequality 356, 441, 448 Cameron–Martin space 85, 117 canonical commutation relation 4, 16 Carmona type estimate – Nelson model 207 – Pauli–Fierz model 370 central limit theorem 306 – functional 312 – martingale 313 change of time 518 chaos expansion 123 charge distribution 133, 138, 148, 151, 164, 184, 189, 268, 292, 296 Clark–Ocone formula 126, 512 coherent polarization vectors 396 coherent state 14 coherent vector 22 complexification 46 cone 516 conjugate momentum 16, 325, 364 continuity equation 322 Coulomb gauge 320, 326 creation operator 4, 9, 324 C0 -semigroup 153, 345, 353, 386

diamagnetic inequality – Nelson Hamiltonian 147 – Pauli–Fierz Hamiltonian 348 – fixed total momentum 388 – Kato-decomposable potential 354 – Pauli–Fierz Hamiltonian with spin 432 – fixed total momentum 438 – relativistic Pauli–Fierz Hamiltonian 447 – fixed total momentum 475 – relativistic Kato-decomposable potential 465 differential second quantization 6, 47 diffusion process 139 dipole approximation 514 Dirac equation 320 Dirichlet principle 188 dispersion relation 9, 133, 138, 148, 151, 164, 184, 189, 215, 268, 292, 296, 325 – Lorentz manifold 509 divergence operator 119 – on Wiener chaos 125 double stochastic integral 371, 517 doubly Markovian 52 effective mass 316, 517 eigenvalue curve 502 embedded eigenvalue 136 energy comparison inequality – Nelson Hamiltonian 272 – Nelson Hamiltonian without ultraviolet cutoff 283 – Pauli–Fierz Hamiltonian 388 – Pauli–Fierz Hamiltonian with spin 438 – polaron Hamiltonian 296 – polaron Hamiltonian without ultraviolet cutoff 296 – relativistic Pauli–Fierz Hamiltonian 475 enhanced binding 516 equation of motion 322 equivalent measure 85 ergodic 505 Euclidean field 68, 106, 141 Euclidean Green function – Nelson Hamiltonian 141, 147 – Pauli–Fierz Hamiltonian 372, 373 – spin-boson Hamiltonian 492 Euler–Lagrange equation 131

534 | Index

existence of ground state – Nelson Hamiltonian 157 – Nelson Hamiltonian with zero total momentum 274, 285 – Pauli–Fierz Hamiltonian 317, 515 – Pauli–Fierz Hamiltonian with zero total momentum 517 – relativistic Pauli–Fierz Hamiltonian 518 – spin-boson Hamiltonian 488 exponential decay of bound state – Nelson Hamiltonian 208, 213 – Pauli–Fierz Hamiltonian 370 – relativistic Pauli–Fierz Hamiltonian 470 exponential moments of ground state – Nelson Hamiltonian 180 – spin-boson Hamiltonian 497 exponential of annihilation operators 21 exponential of creation operators 21 Feynman propagator 2 Feynman–Kac–Nelson formula – Euclidean field 79 – path measure 110 fiber decomposition – Nelson Hamiltonian 269 – Pauli–Fierz Hamiltonian 385 – Pauli–Fierz Hamiltonian with spin 433 – Pauli–Fierz Hamiltonian with spin and a fixed total momentum 434 – Pauli–Fierz Hamiltonian with spin; general polarization 399 – relativistic Pauli–Fierz Hamiltonian 475 field at time zero 1 field fluctuation – Nelson Hamiltonian 181 – spin-boson Hamiltonian 495 field momentum operator 268, 383 field strength tensor 319 finite Gibbs measure – Pauli–Fierz model 374 – relativistic Pauli–Fierz model 472 – semi-relativistic Pauli–Fierz model 473 – spin-boson model 490 finite particle subspace 4, 325 finite volume Gibbs measure – Nelson model 190 – Nelson model with zero total momentum 275 – renormalized Nelson model 254

– renormalized Nelson model with zero total momentum 289 Fock vacuum 4 Fréchet differentiability 115 free field Hamiltonian 299 – in Fock space 10 – in function space 69, 100 – Nelson Hamiltonian 133 – Pauli–Fierz Hamiltonian 326 – spin-boson Hamiltonian 479 free particle Hamiltonian 133 full 44, 330 functional central limit theorem 312 functional integral representation – decoupled Pauli–Fierz Hamiltonian 341 – free field Hamiltonian 70 – Nelson Hamiltonian 140, 144 – fixed total momentum 271 – singular external potential 149 – Nelson Hamiltonian without ultraviolet cutoff 240 – fixed total momentum 281 – Pauli–Fierz Hamiltonian 343 – fixed total momentum 386 – singular external potential 349 – Pauli–Fierz Hamiltonian with spin 422, 430 – fixed total momentum 437 – polaron Hamiltonian 293 – polaron Hamiltonian without ultraviolet cutoff 295 – fixed total momentum 295 – Rabi Hamiltonian 503 – relativistic Pauli–Fierz Hamiltonian 444, 445 – fixed total momentum 474 – singular external potential 446 – relativistic Pauli–Fierz Hamiltonian with spin 518 – spin-boson Hamiltonian 486 – van Hove Hamiltonian 80, 81 gamma matrix 319 Gâteaux derivative 115 Gaussian domination of the ground state – Nelson Hamiltonian 183, 206 – Nelson Hamiltonian with zero total momentum 277 – Pauli–Fierz Hamiltonian 382 – relativistic Pauli–Fierz Hamiltonian 472 – renormalized Nelson Hamiltonian 261, 291

Index | 535

– spin-boson Hamiltonian 495 Gaussian measure on Hilbert space 84, 96, 98 Gaussian random variable indexed by real Hilbert space 43, 65, 101, 330 generating function 497 generator – infinite dimensional Ornstein–Uhlenbeck process 110 Gibbs measure – Nelson model 195 – Nelson model with zero total momentum 276 – Pauli–Fierz model 378 – relativistic Pauli–Fierz model 474 – renormalized Nelson model 256 – renormalized Nelson model with zero total momentum 290 – spin-boson model 491 Gross transform 216 ground state 305 – concavity 369 – existence 157, 274, 285, 317, 488, 515, 517, 518 ground state expectation – boson number 171, 204, 259, 277, 290, 498 – distribution 174 – fractional power 178 – bounded operators 167 – exponential moment of field 180, 497 – field operator 181, 495 – parity 499 – second quantized operators 170 – unbounded operators 169 ground state transform 138, 502 harmonic oscillator 60 helicity 397 Hellinger integral 85 Hermite function 13, 128 Hermite polynomial 112, 180 – physicists 113 – probabilist 113 Hida distribution Hilbert space 129 Hida distribution space 129 Hida space 129 Hida test function Hilbert space 129 Hida test function space 129 Hilbert space-valued – Gaussian random process 101 – stochastic integral 338 homogeneous Sobolev space 66

hydrogen like atom 456 hypercontractivity 56, 335, 508 infinite dimensional Ornstein–Uhlenbeck process 101 – continuous version 105 – generator 110 – invariant measure 110 – Markov property 107 infinite dimensional Ornstein–Uhlenbeck semigroup 126 infrared divergence 135, 173 – renormalized Nelson Hamiltonian 260 infrared regularity 136, 184 infrared singularity 136 inhomogeneous Sobolev space 66 initial distribution 139 intertwining property 71, 81, 200, 332, 333, 419 invariant cone 516 invariant measure – infinite dimensional Ornstein–Uhlenbeck process 110 – Nelson Hamiltonian 199 – renormalized Nelson Hamiltonian 280 iterated stochastic integral 123, 375 Itô representation theorem 122 Kakutani’s theorem 86 Kato–Rellich theorem 135, 452 Kato-class 151 – local 138 Kato-class potential 137, 351 – relativistic 461 Kato-decomposable 138 – relativistic 461 Kolmogorov extension theorem 192 Kolmogorov–Čentsov theorem 105 Lagrangian – Nelson model 131 – Pauli–Fierz model 323 – QED 319 Lagrangian density – Nelson model 131 – QED 319 Legendre transform – Nelson model 131 – QED 320

536 | Index

local convergence – Nelson Hamiltonian 190, 195, 198 – Nelson Hamiltonian with zero total momentum 276, 279, 306 – relativistic Pauli–Fierz Hamiltonian 474 – renormalized Nelson Hamiltonian 256 – renormalized Nelson Hamiltonian with zero total momentum 290 – spin-boson Hamiltonian 491 local Kato-class 138 longitudinal component 319 Lorentz covariant quantum field 56 Lorentz gauge 321 Lp -Lq boundedness for second quantization 335 magnetic field 391 Malliavin derivative 118 – chain rule 121 – on white noise 130 – on Wiener chaos in white noise 130 Malliavin differentiable 120 – on white noise 130 Markov property – Ornstein–Uhlenbeck process 107 – projection 78, 212, 335 martingale – Nelson Hamiltonian 212 – relativistic Pauli–Fierz Hamiltonian 466 – Schrödinger operator 211 Martingale representation theorem 121 Maxwell equation 320, 322 measure – absolutely continuous 85 – equivalent 85 – singular 85 measure on Hilbert space – Borel 95 – Gaussian 84, 96, 98 Mehler kernel 503 Mehler’s formula 14 Minlos’ theorem 60 Minlos–Sazonov theorem 95 moment generating function 180 Nelson Hamiltonian 134 – fixed total momentum 268, 270 – in Fock space 134 – in function space 143 – in L2 (P0 ) 140

– Kato-class potential 154 – massive 136 – massless 136 – N-particle 262 – scaled 263 – singular external potential 149 – static Lorentz manifold 509 – without ultraviolet cutoff 217 Nelson’s analytic vector theorem 17, 134 nonrelativistic limit – relativistic Pauli–Fierz Hamiltonian 457 – translation invariant relativistic Pauli–Fierz Hamiltonian 478 nuclear space 60 number operator 6 Ornstein–Uhlenbeck process – infinite dimensional 101 Ornstein–Uhlenbeck semigroup – infinite dimensional 126 P(ϕ)1 -process 139 – Nelson Hamiltonian 199 – Nelson Hamiltonian with zero total momentum 280 pair interaction 142 – Nelson Hamiltonian 142 – Pauli–Fierz Hamiltonian 375 – polaron Hamiltonian 294 – regularized 220 – relativistic Pauli–Fierz Hamiltonian 472 – renormalized Nelson Hamiltonian 239 – spin-boson Hamiltonian 487 pair potential – Nelson Hamiltonian 142 – Nelson Hamiltonian with a fixed total momentum 275 – Pauli–Fierz Hamiltonian 375 – Pauli–Fierz Hamiltonian with a fixed total momentum 389 – polaron Hamiltonian 294 – regularized 220 – relativistic Pauli–Fierz Hamiltonian 472 – renormalized Nelson Hamiltonian 239 – spin-boson Hamiltonian 487 parity symmetry 480, 482 particle-field interaction Hamiltonian 133 path integral 2

Index | 537

path measure associated with the ground state – Nelson Hamiltonian 195 – Nelson Hamiltonian with zero total momentum 276 – Pauli–Fierz Hamiltonian 379 – relativistic Pauli–Fierz Hamiltonian 474 – renormalized Nelson Hamiltonian 256 – renormalized Nelson Hamiltonian with zero total momentum 289 – spin-boson Hamiltonian 490 Pauli matrix 390 Pauli–Fierz Hamiltonian – dipole approximation 515 – fixed total momentum 383 – in Fock space 327 – in function space 334 – Kato-class potential 354 – relativistic 441 – fixed total momentum 474 – spin 518 – singular external potential 349 – spin 394, 402 – fixed total momentum 433 Perron–Frobenius theorem 272, 283, 369, 389, 516 Poisson equation 321 Poisson process 390 polarization vectors 325 – coherent 396 polaron Hamiltonian – fixed total momentum 293 – in Fock space 291 – in function space 292 polaron model 291 polynomial decay of bound state 470 positive definite 90, 507 positive definite function 60 positivity improving 47, 50 – e−sω(−i∇) 176 – free field Hamiltonian 155, 366 – Nelson Hamiltonian 155 – zero total momentum 272 – Nelson Hamiltonian without ultraviolet cutoff – zero total momentum 283 – Pauli–Fierz Hamiltonian 368 – Kato-decomposable potential 368 – zero total momentum 388 – polaron Hamiltonian – zero total momentum 296

– polaron Hamiltonian without ultraviolet cutoff – zero total momentum 296 – relativistic Pauli–Fierz Hamiltonian 447 – relativistic Kato-decomposable potential 465 – zero total momentum 475 – spin-boson Hamiltonian 489 positivity preserving 47, 48, 174, 272, 516 Proca field 516 Prokhorov theorem 165, 379 pull-through formula – dΓ(g) 172 – N 171, 274 quantized magnetic filed 393 quantized radiation field 325, 393 Rabi Hamiltonian 501 Radon–Nikodým derivative 85 random van Hove Hamiltonian 501 reflection symmetry 139 – Pauli–Fierz Hamiltonian with spin 400 – Pauli–Fierz Hamiltonian with spin and a fixed total momentum 435 regular conditional probability measure 108 relative bound for annihilation and creation operators 7, 10 relativistic Kato-class 461 relativistic Kato-decomposable 461 relativistic Pauli–Fierz Hamiltonian 441 – fixed total momentum 474 – relativistic Kato-class potential 465 – singular external potential 446 – spin 518 removal of ultraviolet cutoff – Nelson Hamiltonian 238 – Nelson Hamiltonian with a fixed total momentum 281 – polaron Hamiltonian 295 – polaron Hamiltonian with a fixed total momentum 295 reproducing kernel 14 Riesz–Markov theorem 58 Sazonov topology 94 scalar boson field 132 scalar field 56 – time-zero 56 Schwinger function 2

538 | Index

second quantization 5, 6, 9, 47, 170, 331 – contraction operator 69, 100 – spectrum 6 Segal field 16 Segal–Bargmann space 14, 520 self-adjointness – Nelson Hamiltonian 134 – Nelson Hamiltonian with a fixed total momentum 269 – Pauli–Fierz Hamiltonian 328, 362 – Pauli–Fierz Hamiltonian with a fixed total momentum 385 – Pauli–Fierz Hamiltonian with spin 395 – Pauli–Fierz Hamiltonian with spin and a fixed total momentum 433 – relativistic Pauli–Fierz Hamiltonian 456 – relativistic Pauli–Fierz Hamiltonian with a fixed total momentum 477 – spin-boson Hamiltonian 480 semigroup – positivity improving 48 shift invariance 139, 485 singular measure 85 Skorokhod integral 121 Sobolev space – homogeneous 66 – inhomogeneous 66 spatial decay of bound state – Nelson Hamiltonian 208, 213 – Pauli–Fierz Hamiltonian 370 – relativistic Pauli–Fierz Hamiltonian 470 spin process 391, 484 spin-boson Hamiltonian – in Fock space 479 – in function space 483 spin-flip 483 Stirling number 177, 498 stochastic differentiability 117 stochastic integral – Hilbert space-valued 338 Stone’s theorem – semigroup version 239 stopping time 213 subordinator 439, 457, 478 super-exponential decay of boson number – Nelson Hamiltonian 171, 204 – Nelson Hamiltonian with zero total momentum 277 – renormalized Nelson Hamiltonian 259

– renormalized Nelson Hamiltonian with zero total momentum 290 – spin-boson Hamiltonian 498 tempered distributions 66 tightness 165, 379 time-zero scalar field 56 total momentum 384 total momentum operator 268, 383, 474 translation invariant Hamiltonian – Nelson Hamiltonian 268, 270 – Pauli–Fierz Hamiltonian 383 – Pauli–Fierz Hamiltonian with spin 433 – relativistic Pauli–Fierz Hamiltonian 475 transversal delta function 329 transversal wave 324 Trotter product formula 70 two point function 1 Tychonoff theorem 58 ultraviolet divergence 135 ultraviolet renormalization term 288 uniformly locally integrable 461 uniqueness of ground state – Nelson Hamiltonian 155 – Nelson Hamiltonian with zero total momentum 272, 283 – Pauli–Fierz Hamiltonian 369 – Pauli–Fierz Hamiltonian with zero total momentum 388 – polaron Hamiltonian with zero total momentum 296 – polaron Hamiltonian without ultraviolet cutoff – zero total momentum 296 – Rabi Hamiltonian 503 – relativistic Pauli–Fierz Hamiltonian 447 – relativistic Kato-decomposable potential 465 – relativistic Pauli–Fierz Hamiltonian with zero total momentum 475 – spin-boson Hamiltonian 489 van Hove Hamiltonian 80, 480, 488, 499 – random 501 van Hove representation 501 weak coupling limit 263 white noise process 127 Wick product 20, 45 Wick’s theorem 33

Index | 539

Wiener chaos 123 Wiener–Itô chaos expansion 123 – on white noise 129 Wiener–Itô decomposition 46

Wiener–Itô–Segal isomorphism 47 Wightman distribution 1 Yukawa potential 267

De Gruyter Studies in Mathematics Volume 34/1 József Lőrinczi, Fumio Hiroshima, Volker Betz Feynman–Kac-Type Theorems and Gibbs Measures on Path Space. Volume 1: Feynman–Kac-Type Formulae and Gibbs Measures ISBN 978-3-11-033004-5, e-ISBN 978-3-11-033039-7, e-ISBN (ePUB) 978-3-11-038993-7 Volume 71 Christian Remling Periodic Locally Compact Groups: A Study of a Class of Totally Disconnected Topological Groups ISBN 978-3-11-059847-6, e-ISBN 978-3-11-059919-0, e-ISBN (ePUB) 978-3-11-059908-4 Volume 70 Christian Remling Spectral Theory of Canonical Systems ISBN 978-3-11-056202-6, e-ISBN 978-3-11-056323-8, e-ISBN (ePUB) 978-3-11-056228-6 Volume 69 Derek K. Thomas, Nikola Tuneski, Allu Vasudevarao Univalent Functions. A Primer ISBN 978-3-11-056009-1, e-ISBN 978-3-11-056096-1, e-ISBN (ePUB) 978-3-11-056012-1 Volume 68/1 Timofey V. Rodionov, Valeriy K. Zakharov Set, Functions, Measures. Volume 1: Fundamentals of Set and Number Theory ISBN 978-3-11-055008-5, e-ISBN 978-3-11-055094-8 Volume 68/2 Alexander V. Mikhalev, Timofey V. Rodionov, Valeriy K. Zakharov Set, Functions, Measures. Volume 2: Fundamentals of Functions and Measure Theory ISBN 978-3-11-055009-2, e-ISBN 978-3-11-055096-2 Volume 67 Alexei Kulik Ergodic Behavior of Markov Processes. With Applications to Limit Theorems ISBN 978-3-11-045870-1, e-ISBN 978-3-11-045893-0 Volume 66 Igor V. Nikolaev Noncommutative Geometry. A Functorial Approach, 2017 ISBN 978-3-11-054317-9, e-ISBN 978-3-11-054525-8 www.degruyter.com