Feynman-Kac-Type Theorems and Gibbs Measures on Path Space: With Applications to Rigorous Quantum Field Theory

De Gruyter Studies in Mathematics 34 Editors Carsten Carstensen, Berlin, Germany Nicola Fusco, Napoli, Italy Fritz Gesz...

Author: Jozsef Lorinczi | Fumio Hiroshima | Volker Betz

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De Gruyter Studies in Mathematics 34 Editors Carsten Carstensen, Berlin, Germany Nicola Fusco, Napoli, Italy Fritz Gesztesy, Columbia, USA Niels Jacob, Swansea, United Kingdom Karl-Hermann Neeb, Erlangen, Germany

Jo´zsef Lo˝rinczi Fumio Hiroshima Volker Betz

Feynman-Kac-Type Theorems and Gibbs Measures on Path Space With Applications to Rigorous Quantum Field Theory

De Gruyter

Mathematics Subject Classification 2010: 81Q10, 81S40, 47D08, 60J60, 82B05, 35Q40, 46N50, 81V10.

ISBN 978-3-11-020148-2 e-ISBN 978-3-11-020373-8 ISSN 0179-0986 Library of Congress Cataloging-in-Publication Data Lo˝rinczi, Jo´zsef. Feynman-Kac-type theorems and Gibbs measures on path space : with applications to rigorous quantum field theory / by Jo´zsef Lo˝rinczi, Fumio Hiroshima, Volker Betz. p. cm. ⫺ (De Gruyter studies in mathematics ; 34) Includes bibliographical references and index. ISBN 978-3-11-020148-2 1. Integration, Functional. 2. Stochastic analysis. 3. Quantum field theory ⫺ Mathematics. I. Hiroshima, Fumio. II. Betz, Volker. III. Title. QC20.7.F85L67 2011 5151.724⫺dc22 2011005708

Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.d-nb.de. ” 2011 Walter de Gruyter GmbH & Co. KG, Berlin/Boston Typesetting: Da-TeX Gerd Blumenstein, Leipzig, www.da-tex.de Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen ⬁ Printed on acid-free paper Printed in Germany www.degruyter.com

Preface

For a long time disciplines such as astronomy, celestial mechanics, the theory of heat and the theory of electromagnetism counted among the most important sources in the development of mathematics, continually providing new problems and challenges. The main mathematical facts resulting from these fields typically crystallized in linear ordinary or partial differential equations. In the early 20th century Heisenberg’s discovery of a rule showing that the measurement of the position and momentum of a quantum particle gives different answers dependent on the order of their measurement led to the concept of commutation relations, and it made matrix theory find an unexpected application in the natural sciences. This resulted in an explosion of interest in linear structures developed by what we call today linear algebra and linear analysis, among them functional analysis, operator theory, C -algebras, and distribution theory. Similarly, quantum mechanics inspired much of group theory, lattice theory and quantum logic. Following this initial boom Wigner’s beautiful phrase talking of the “unreasonable effectiveness of mathematics” marks a point of reflection on the fact that it is by no means obvious why this kind of abstract and sophisticated mathematics can be expected at all to make faithful descriptions and reliable predictions of natural phenomena. If indeed this can be appreciated as a miracle, as he said, it is perhaps at least a remarkable fact the degree of flexibility that mathematics allows in its uses. Feynman has pointed to the different cultures of using mathematics when he remarked that “if all of mathematics disappeared, physics would be set back by exactly one week.” While mathematics cannot compete with physics in discovering new phenomena and offering explanations of them, physics continues to depend on the terminology, arsenal and discipline of mathematics. Mathematical physics, which is part of mathematics and therefore operates by its rules and standards, has set the goal to understand the models of physics in a rigorous way. Mathematical physicists thus find themselves at the borderline, listening to physics and speaking mathematics, at best able to use these functionalities interchangeably. If one of the imports of early quantum mechanics has been the realization that the basic laws on the atomic scale can be formulated by linear superposition rules, another was a probabilistic interpretation of the wave function. This connection with chance, amplified and strongly advocated by the Copenhagen school, was no less revolutionary than the commutation relations. Richard Feynman has discovered a second connection with probability when he offered a representation of the state of a particle in terms of averages over all of its possible histories from one point in spacetime to another. While this was an instance seriously questioning his dictum quoted

vi

Preface

above and his use of mathematics was in this case very problematic, the potential lying in the description advanced in his work turned out to be far reaching. It took the efforts of the mathematician Mark Kac to be the first to show that this method could be made sense of and was eminently viable. Our project to write the present monograph has grown out of the thematic programme At the Interface of PDE, Self-Adjoint Operators and Stochastics: Models with Exclusion organized by the first named author at the Wolfgang Pauli Institute, Vienna, in 2006. The initial concept was a smaller scale but up-to-date account of Feynman–Kac-type formulae and their uses in quantum field theory, which we wanted to dedicate to the person all three of us have much to thank both scientifically and personally, Herbert Spohn. At that time we secretly meant this as a present for his 60th birthday, and we are glad that we are now able to make this hopefully more mature tribute on the occasion of his 65th birthday! Apart from the inspiring and pleasant environment at Zentrum Mathematik, Munich University of Technology, where our collaboration has started, and Wolfgang Pauli Institut, University of Vienna, hosting the thematic programme, we are indebted to a number of other individuals and institutions. We are thankful for the joint work and friendship of our closest collaborators in this direction, Asao Arai, Massimiliano Gubinelli, Masao Hirokawa, and Robert Adol’fovich Minlos. We also thank Cristian Gérard, Takashi Ichinose, Kamil Kaleta, Jacek Małecki, Annalisa Panati, Akito Suzuki for ongoing related joint work. It is furthermore a pleasure to thank Norbert Mauser, Director of the Wolfgang Pauli Institute, for supporting the programme, and Sergio Albeverio, Rodrigo Bañuelos, Krzysztof Bogdan, Carlo Boldrighini, Thierry Coulhon, Laure Coutin, Cécile DeWitt-Morette, Aernout van Enter, Bill Faris, Gero Friesecke, Jürg Fröhlich, Mohammud Foondun, Hans-Otto Georgii, Alexander Grigor’yan, Martin Hairer, Takeru Hidaka, Wataru Ichinose, Keiichi Itô, Niels Jacob, Hiroshi Kawabi, Tadeusz Kulczycki, Kazuhiro Kuwae, Terry Lyons, Tadahiro Miyao, Hirofumi Osada, Habib Ouerdiane, Sylvie Roelly, Itaru Sasaki, Tomoyuki Shirai, Toshimitsu Takaesu, Setsuo Taniguchi, Josef Teichmann, Daniel Ueltschi, Jakob Yngvason for many useful discussions over the years. We thank especially Erwin Schrödinger Institut, Vienna, Institut des Hautes Études Scientifiques, Buressur-Yvette, and Università La Sapienza, Rome, for repeatedly providing excellent environments to further the project and where parts of the manuscript have been written. Also, we thank the hospitality of Kyushu University, Loughborough University and Warwick University, for accommodating mutual visits and extended stays. We are particularly pleased to thank the professionalism, assistance and endless patience of Robert Plato, Simon Albroscheit and Friederike Dittberner of Walter de Gruyter. Also, we are grateful to Max Lein for technical support in the initial phase of this project. József L˝orinczi, Fumio Hiroshima, Volker Betz

February 2011

Contents

Preface

v

I Feynman–Kac-type theorems and Gibbs measures

1

1

Heuristics and history 1.1 Feynman path integrals and Feynman–Kac formulae . . . . . . . . . 1.2 Plan and scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 7

2

Probabilistic preliminaries 2.1 An invitation to Brownian motion . . . . . . . . . . . . 2.2 Martingale and Markov properties . . . . . . . . . . . . 2.2.1 Martingale property . . . . . . . . . . . . . . . 2.2.2 Markov property . . . . . . . . . . . . . . . . . 2.2.3 Feller transition kernels and generators . . . . . 2.2.4 Conditional Wiener measure . . . . . . . . . . . 2.3 Basics of stochastic calculus . . . . . . . . . . . . . . . 2.3.1 The classical integral and its extensions . . . . . 2.3.2 Stochastic integrals . . . . . . . . . . . . . . . . 2.3.3 Itô formula . . . . . . . . . . . . . . . . . . . . 2.3.4 Stochastic differential equations and diffusions . 2.3.5 Girsanov theorem and Cameron–Martin formula 2.4 Lévy processes . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Lévy process and Lévy–Khintchine formula . . . 2.4.2 Markov property of Lévy processes . . . . . . . 2.4.3 Random measures and Lévy–Itô decomposition . 2.4.4 Itô formula for semimartingales . . . . . . . . . 2.4.5 Subordinators . . . . . . . . . . . . . . . . . . . 2.4.6 Bernstein functions . . . . . . . . . . . . . . . .

3

Feynman–Kac formulae 3.1 Schrödinger semigroups . . . . . . . . . . . . . . . . . 3.1.1 Schrödinger equation and path integral solutions 3.1.2 Linear operators and their spectra . . . . . . . . 3.1.3 Spectral resolution . . . . . . . . . . . . . . . . 3.1.4 Compact operators . . . . . . . . . . . . . . . .

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11 11 21 21 25 29 32 33 33 34 42 46 50 53 53 57 61 64 67 69

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71 71 71 72 78 80

viii

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3.1.5 Schrödinger operators . . . . . . . . . . . . . . . . . . . . . 3.1.6 Schrödinger operators by quadratic forms . . . . . . . . . . . 3.1.7 Confining potential and decaying potential . . . . . . . . . . 3.1.8 Strongly continuous operator semigroups . . . . . . . . . . . 3.2 Feynman–Kac formula for external potentials . . . . . . . . . . . . . 3.2.1 Bounded smooth external potentials . . . . . . . . . . . . . . 3.2.2 Derivation through the Trotter product formula . . . . . . . . 3.3 Feynman–Kac formula for Kato-class potentials . . . . . . . . . . . . 3.3.1 Kato-class potentials . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Feynman–Kac formula for Kato-decomposable potentials . . 3.4 Properties of Schrödinger operators and semigroups . . . . . . . . . . 3.4.1 Kernel of the Schrödinger semigroup . . . . . . . . . . . . . 3.4.2 Number of eigenfunctions with negative eigenvalues . . . . . 3.4.3 Positivity improving and uniqueness of ground state . . . . . 3.4.4 Degenerate ground state and Klauder phenomenon . . . . . . 3.4.5 Exponential decay of the eigenfunctions . . . . . . . . . . . . 3.5 Feynman–Kac–Itô formula for magnetic field . . . . . . . . . . . . . 3.5.1 Feynman–Kac–Itô formula . . . . . . . . . . . . . . . . . . . 3.5.2 Alternate proof of the Feynman–Kac–Itô formula . . . . . . . 3.5.3 Extension to singular external potentials and vector potentials 3.5.4 Kato-class potentials and Lp –Lq boundedness . . . . . . . . 3.6 Feynman–Kac formula for relativistic Schrödinger operators . . . . . 3.6.1 Relativistic Schrödinger operator . . . . . . . . . . . . . . . 3.6.2 Relativistic Kato-class potentials and Lp –Lq boundedness . . 3.7 Feynman–Kac formula for Schrödinger operator with spin . . . . . . 3.7.1 Schrödinger operator with spin . . . . . . . . . . . . . . . . . 3.7.2 A jump process . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.3 Feynman–Kac formula for the jump process . . . . . . . . . . 3.7.4 Extension to singular potentials and vector potentials . . . . . 3.8 Feynman–Kac formula for relativistic Schrödinger operator with spin 3.9 Feynman–Kac formula for unbounded semigroups and Stark effect . . 3.10 Ground state transform and related diffusions . . . . . . . . . . . . . 3.10.1 Ground state transform and the intrinsic semigroup . . . . . . 3.10.2 Feynman–Kac formula for P ./1 -processes . . . . . . . . . . 3.10.3 Dirichlet principle . . . . . . . . . . . . . . . . . . . . . . . 3.10.4 Mehler’s formula . . . . . . . . . . . . . . . . . . . . . . . . 4

81 85 87 89 93 93 95 97 97 108 112 112 113 120 124 126 131 131 135 138 142 143 143 149 150 150 152 154 157 162 166 170 170 174 181 184

Gibbs measures associated with Feynman–Kac semigroups 190 4.1 Gibbs measures on path space . . . . . . . . . . . . . . . . . . . . . 190 4.1.1 From Feynman–Kac formulae to Gibbs measures . . . . . . . 190 4.1.2 Definitions and basic facts . . . . . . . . . . . . . . . . . . . 194

ix

Contents

4.2

4.3

4.4

Existence and uniqueness by direct methods . . . . . . . . . 4.2.1 External potentials: existence . . . . . . . . . . . . 4.2.2 Uniqueness . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Gibbs measure for pair interaction potentials . . . . Existence and properties by cluster expansion . . . . . . . . 4.3.1 Cluster representation . . . . . . . . . . . . . . . . 4.3.2 Basic estimates and convergence of cluster expansion 4.3.3 Further properties of the Gibbs measure . . . . . . . Gibbs measures with no external potential . . . . . . . . . . 4.4.1 Gibbs measure . . . . . . . . . . . . . . . . . . . . 4.4.2 Diffusive behaviour . . . . . . . . . . . . . . . . . .

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II Rigorous quantum field theory 5

6

Free Euclidean quantum field and Ornstein–Uhlenbeck processes 5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Boson Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Second quantization . . . . . . . . . . . . . . . . . . . . 5.2.2 Segal fields . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Wick product . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Q-spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Gaussian random processes . . . . . . . . . . . . . . . . 5.3.2 Wiener–Itô–Segal isomorphism . . . . . . . . . . . . . . 5.3.3 Lorentz covariant quantum fields . . . . . . . . . . . . . . 5.4 Existence of Q-spaces . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Countable product spaces . . . . . . . . . . . . . . . . . 5.4.2 Bochner theorem and Minlos theorem . . . . . . . . . . . 5.5 Functional integration representation of Euclidean quantum fields 5.5.1 Basic results in Euclidean quantum field theory . . . . . . 5.5.2 Markov property of projections . . . . . . . . . . . . . . 5.5.3 Feynman–Kac–Nelson formula . . . . . . . . . . . . . . 5.6 Infinite dimensional Ornstein–Uhlenbeck process . . . . . . . . . 5.6.1 Abstract theory of measures on Hilbert spaces . . . . . . . 5.6.2 Fock space as a function space . . . . . . . . . . . . . . . 5.6.3 Infinite dimensional Ornstein–Uhlenbeck-process . . . . . 5.6.4 Markov property . . . . . . . . . . . . . . . . . . . . . . 5.6.5 Regular conditional Gaussian probability measures . . . . 5.6.6 Feynman–Kac–Nelson formula by path measures . . . . .

201 201 204 208 217 217 223 224 226 226 238

245 . . . . . . . . . . . . . . . . . . . . . . .

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247 247 249 249 255 257 258 258 260 262 263 263 264 268 268 271 274 276 276 279 282 288 290 292

The Nelson model by path measures 293 6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

x

Contents

6.2

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294 294 296 298 298 303 309 309 315 316 322 324 328 333 333 335 339 344

The Pauli–Fierz model by path measures 7.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Lagrangian QED . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Classical variant of non-relativistic QED . . . . . . . . . 7.2 The Pauli–Fierz model in non-relativistic QED . . . . . . . . . . 7.2.1 The Pauli–Fierz model in Fock space . . . . . . . . . . . 7.2.2 The Pauli–Fierz model in function space . . . . . . . . . . 7.2.3 Markov property . . . . . . . . . . . . . . . . . . . . . . 7.3 Functional integral representation for the Pauli–Fierz Hamiltonian 7.3.1 Hilbert space-valued stochastic integrals . . . . . . . . . . 7.3.2 Functional integral representation . . . . . . . . . . . . . 7.3.3 Extension to general external potential . . . . . . . . . . . 7.4 Applications of functional integral representations . . . . . . . . . 7.4.1 Self-adjointness of the Pauli–Fierz Hamiltonian . . . . . . 7.4.2 Positivity improving and uniqueness of the ground state . 7.4.3 Spatial decay of the ground state . . . . . . . . . . . . . . 7.5 The Pauli–Fierz model with Kato class potential . . . . . . . . . . 7.6 Translation invariant Pauli–Fierz model . . . . . . . . . . . . . . 7.7 Path measure associated with the ground state . . . . . . . . . . . 7.7.1 Path measures with double stochastic integrals . . . . . . 7.7.2 Expression in terms of iterated stochastic integrals . . . . 7.7.3 Weak convergence of path measures . . . . . . . . . . . .

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351 351 351 352 356 359 359 363 369 372 372 375 381 382 382 392 398 399 401 408 408 412 415

6.3 6.4 6.5

6.6 6.7 6.8

7

The Nelson model in Fock space . . . . . . . . . . . . . . . . . 6.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Infrared and ultraviolet divergences . . . . . . . . . . . 6.2.3 Embedded eigenvalues . . . . . . . . . . . . . . . . . . The Nelson model in function space . . . . . . . . . . . . . . . Existence and uniqueness of the ground state . . . . . . . . . . Ground state expectations . . . . . . . . . . . . . . . . . . . . . 6.5.1 General theorems . . . . . . . . . . . . . . . . . . . . . 6.5.2 Spatial decay of the ground state . . . . . . . . . . . . . 6.5.3 Ground state expectation for second quantized operators 6.5.4 Ground state expectation for field operators . . . . . . . The translation invariant Nelson model . . . . . . . . . . . . . . Infrared divergence . . . . . . . . . . . . . . . . . . . . . . . . Ultraviolet divergence . . . . . . . . . . . . . . . . . . . . . . . 6.8.1 Energy renormalization . . . . . . . . . . . . . . . . . . 6.8.2 Regularized interaction . . . . . . . . . . . . . . . . . . 6.8.3 Removal of the ultraviolet cutoff . . . . . . . . . . . . . 6.8.4 Weak coupling limit and removal of ultraviolet cutoff .

xi

Contents

7.8

7.9

8

Relativistic Pauli–Fierz model . . . . . . 7.8.1 Definition . . . . . . . . . . . . . 7.8.2 Functional integral representation 7.8.3 Translation invariant case . . . . . The Pauli–Fierz model with spin . . . . . 7.9.1 Definition . . . . . . . . . . . . . 7.9.2 Symmetry and polarization . . . . 7.9.3 Functional integral representation 7.9.4 Spin-boson model . . . . . . . . 7.9.5 Translation invariant case . . . . .

Notes and References

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418 418 420 423 424 424 427 434 447 448 455

Bibliography

473

Index

499

Part I

Feynman–Kac-type theorems and Gibbs measures

Chapter 1

Heuristics and history

1.1

Feynman path integrals and Feynman–Kac formulae

In Feynman’s work concepts such as paths and actions, just discarded by the new-born quantum mechanics, witnessed a comeback in the description of the time-evolution of quantum particles, proposing an alternative to Heisenberg’s matrix mechanics and Schrödinger’s wave mechanics. It is worthwhile briefly to review how this turned out to be a viable project and led to a mathematically meaningful approach to quantum mechanics and quantum field theory. A basic tool in the hands of the quantum physicist is the Schrödinger equation i„

@ @t

„2 .t; x/ C V .x/ .t; x/; 2m .0; x/ D .x/; .t; x/ 2 R R3 .t; x/ D

(1.1.1)

with Hamilton operator H D

„2 C V; 2m

(1.1.2)

and a complex valued wave function describing a quantum particle of mass m at position x in space at time t . Here „ is Planck’s constant, V is a potential in whose force field the particle is moving, and describes the initial state of the quantum particle. The Schrödinger equation can be solved only under exceptional choices of V . However, formally the solution of the equation can be written as .t; x/ D .e .it=„/H /.x/ for any V . The difficulty is that for a general choice of V we do not know how to calculate the right-hand side of the equality. The action of the complex exponential operator on can be evaluated by using an integral kernel G allowing to write the wave function of the particle at position y at time t based on its knowledge in x at s < t , i.e., Z .t; y/ D G.x; yI s; t / .s; x/dx: R3

This representation can then in principle be iterated from one interval Œs; t to another, and thus our knowledge of the wave function “propagates” from the initial time to any

4

Chapter 1 Heuristics and history

time point in the future. This observation suggests that at least a qualitative analysis of the solution might be possible for a wider choice of V by using this integral kernel, and the key to this approach is the information content of the kernel. Feynman proposed a method of constructing this integral kernel in the following way. Consider first a classical Hamiltonian function 1 2 p C V .q/; .p; q/ 2 R3 R3 ; (1.1.3) 2m describing a particle at position q moving with momentum p under the same potential V as above. The corresponding Lagrangian is obtained through Legendre transform, m L .q; q/ P D qp P H .p; q/ D qP 2 V .q/; 2 where @H p qP D D : @p m Classical mechanics says that the particle is equivalently described by the action functional Z H .p; q/ D

t

A .q; qI P 0; t / D

0

L .q.s/; q.s//ds; P

where q.s/ is the path of the particle as a function of the time s. In particular, the equation of motion can be derived by the principle of least action by using variational calculus, and it is found that the particle follows a route from a fixed initial to a fixed endpoint which minimizes the action. In the late 1920s Dirac worked on a Lagrangian formulation of quantum mechanics and arrived at the conclusion that the ideas of the classical Lagrangian theory can be taken over, though not its equations. Specifically, he arrived at a rule he called “transformation theory” allowing to connect the positions at s and t of the quantum P particle through the phase factors e .i=„/A .q;qIs;t/ . By dividing up the interval Œs; t into small pieces, Dirac noticed [128, 129, 130] that due to the fact that „ is of the order of 1034 in standard units, this is so small that the integrands in the exponent are rapidly oscillating functions around zero. Therefore, as a result of the many cancellations the only significant contribution into the integral is given by those q whose large variations over the subintervals involve small differences in the Lagrangian. Feynman, while a Princeton research assistant during 1940–41, turned Dirac’s somewhat vague observations into a more definite formulation. He argued that the propagators G.x; yI s; t /, which were Dirac’s “contact transformations,” can be obtained as limits over products of propagators at intermediary time points. This means that with a partition of Œs; t to subintervals Œtj ; tj C1 with tj D s C jb, j D 0; : : : ; n 1, b D .t s/=n, the expression n1 n1 YZ Y G.x; yI s; t / D lim G.xj ; xj C1 I tj ; tj C1 / dxj n!1

3 j D0 R

j D1

5

Section 1.1 Feynman path integrals and Feynman–Kac formulae

holds, where r G.xj ; xj C1 I tj ; tj C1 / D

m ib xj C1 xj exp L xj C1 ; : 2b i „ „ b

From here it was possible to conclude that

.e

.it=„/H

Y i A .q; qI P 0; t / /.x; y/ D G.x; yI 0; t / D const exp dq.s/ „ xy Z

0st

(1.1.4) giving a formal expression of the integral kernel of the operator e .it=„/H for all t . Here dq.s/ is Lebesgue measure, and xy is the set of continuous paths q.s/ such that q.0/ D x and q.t / D y. The right-hand side of the expression is Feynman’s integral. These results have been obtained first in his 1942 thesis, which remained unpublished; the paper [161] published in 1948, however, reproduces the crucial statements (see also [164]). Feynman was able to apply his approach also to quantum field theory [162, 163]. For all its intuitive appeal and potential, Feynman’s method did not win immediate acclaims, not even in the ranks of theoretical physicists. Mathematically viewed, the object Feynman proposed was by no means easy to define. Mathematicians hitherto have not found a satisfactory meaning of Feynman integrals covering a sufficiently wide class of choices of V . For instance, there is no countably additive measure on xy allocating equal weight to every path, and A .q; qI P 0; t / fails to be defined on much of xy since it involves q.s/. P Various ways of trying to cope with these difficulties have been explored. Several authors considered purely imaginary potentials V and masses m leading to a complex measure to define Feynman’s integral, but these attempts had to face the problem of infinite total variation [79, 102, 372]. Another suggestion was to define the Feynman integral in terms of Fresnel distributions [126, 3, 127, 80, 85, 474]. Other attempts constructing solutions of Schrödinger’s equation from Feynman’s integral through piecewise classical paths or broken lines or polygonal paths [177, 279, 283, 273, 476, 292]. In Feynman’s days integration theory on the space of continuous functions has already been in existence due to Wiener’s work on Brownian motion initiated in 1923. It was Kac in 1949 who first realized that this was a suitable framework, however, not for the path integral (1.1.4) directly, see also [294, 295]. In contrast to Feynman’s expression, the Feynman–Kac formula offers an integral representation on path space for the semigroup e .1=„/tH instead of e .i=„/tH , obtained on replacing t by i t . Why should this actually improve on the situation can be seen by the following argument. Analytic continuation s ! i s, ds ! ids and the replacement q.s/ P 2 ! q.s/ P 2

6


in (1.1.4) leads to the kernel Z tH .e /.x; y/ D const

e .1=„/

Rt 0

V .q.s//ds .m=2„/

e

Rt 0

xy

Y

q.s/ P 2 ds

dq.s/:

0st

(1.1.5) Since defining the latter factor as a measure remains to be a problem as Z Y dq.s/ D 1

(1.1.6)

xy 0st

for every t , one might boldly hope that its support Cxy is a set on which the exponential weights typically vanish. If qP were almost surely plus or minus infinite on this set, then the situation Z 1 tm 2 exp q.s/ P ds D0 (1.1.7) „ 0 2 Cxy would occur. Cxy would thus be required to consist of continuous but nowhere differentiable paths. Once this can be allowed, zero in (1.1.7) can possibly cancel infinity in (1.1.6), and it would in principle be possible that Y Z 1 tm 2 q.s/ P ds dq.s/ D 0 1 exp „ 0 2 0st

is a well-defined measure. As we will explain below, the miracle happens: The paths of a random process .B t / t0 called Brownian motion just have the required properties and the Feynman–Kac formula Z Rt x;y tH .e /.x; y/ D e 0 V .Bs .!//ds d WŒ0;t .!/; 8t 0; x; y 2 R3 C.R;R3 /

(1.1.8) rigorously holds provided some technical but highly satisfying conditions hold on V . In particular, there is a measure supported on the space Cxy identified with the space C.R; R3 / of continuous functions R ! R3 , and it can be identified as Wiener measure W conditional on paths leaving from x at time 0 and ending in y at time t . In other words, the kernel of the semigroup at the left-hand side (the object corresponding to G.x; yI 0; t / above) can be studied by running a Brownian motion driven by the potential V . Kac has actually proved [293] that the heat equation with dissipation @f 1 D f Vf; @t 2

.x; 0/ D .x/;

(1.1.9)

7

Section 1.2 Plan and scope

is solved by the function Z f .x; t / D

e

Rt 0

V .Bs .!// ds

C.R;R3 /

.B t .!// d W x .!/:

(1.1.10)

This equation is actually the same as the Schrödinger equation (1.1.1) when analytic continuation t 7! i t is made, a fact that has been noticed by Ehrenfest as early as 1927. In particular, on setting V D 0 and formally replacing with a ı-distribution, the fundamental solution ….x; t / of the standard heat equation is obtained. The relation of this so called heat kernel 1 jxj2 ….x; t / D exp ; t > 0; (1.1.11) 2t .2 t /3=2 is once again strong with random processes as it gives the transition probabilities of Brownian motion. The Feynman–Kac formula makes thus a link between an operator semigroup, the heat equation, and Brownian motion. This observation has opened up a whole new conceptual and technical framework pointing to and exploring the rich interplay between self-adjoint operators, probability and partial differential equations.

1.2

Plan and scope

The Feynman–Kac formula for the operator H D 12 C V derived from the heat equation with dissipation can be written in any dimension d as Z Rt tH .e f /.x/ D e 0 V .Bs .!// ds f .B t .!// d W x .!/ C.R;Rd /

where .B t / t0 is accordingly an Rd -valued Brownian motion. Due to the connections outlined above, the operator H is called a Schrödinger operator with potential V . If the potential is such that H admits an eigenfunction at eigenvalue E, then e tH D e E t and thus this eigenfunction can be represented in terms of an average over the paths of Brownian motion given by the right-hand side. This makes possible to obtain information on the spectral properties of H by probabilistic means. In particular, the Feynman–Kac formula can be used to study the ground states (i.e., eigenfunctions at the bottom of the spectrum) of Schrödinger operators. It is natural to ask the question whether it is possible to derive representations similar to the Feynman–Kac formula for any other operator. In the pages below we will show that also other operators, derived from quantum field theory, allow to derive Feynman–Kac-type formulae with appropriately modified random processes at the right-hand side. On the one hand, this relation will allow us to ask and answer questions relevant to quantum field theory in a rigorous way. On the other hand, we will face situations in which new concepts and methods need to be developed, which thus generates new mathematics.

8


We split the material of this monograph in two parts of comparable size, each focussing on one aspect explained above. In Part I we discuss the mathematical framework and in Part II we present the applications in rigorous quantum field theory. Part I is called Feynman–Kac-type theorems and Gibbs measures and it consists of four chapters. In Chapter 1 we outline the origins of the idea of a path integral applied in quantum mechanics and quantum field theory, due to the physicist Richard P Feynman, and formulated in a mathematically rigorous way by the mathematician Mark Kac. In Chapter 2 we offer background material from stochastic analysis starting from a review of Brownian motion, Markov processes and martingales. We devote a section to basic concepts of Itô integration theory including some examples, and we conclude by a review of Lévy processes and subordinate Brownian motion. In Chapter 3 we start by a review of background material of the theory of selfadjoint operators and their semigroups, with a special view on Schrödinger operators. As in the previous chapter we present established material in the light of our goals. The remainder of the chapter deals with a systematic presentation of versions of the Feynman–Kac formula. We cover cases of increasing complexity starting from an external potential V , through including vector potentials describing interactions with magnetic fields, spin, and relativistic effects. Purely mathematically this amounts to saying that we proceed from operators consisting of the sum of the negative Laplacian and a multiplication operator of varying regularity, through including gradient operator terms, and pseudo-differential operators such as the square root of the Laplacian and related operators. From the perspective of random processes involved, we proceed from Brownian motion to diffusions (P ./1 -processes and Itô diffusions), and subordinate Brownian motion (relativistic stable processes). Apart from establishing the variants of the Feynman–Kac formula for these choices of operators and processes, we also address their applications in the study of analytic and spectral properties of these operators (properties of integral kernels, positivity improving properties, Lp -smoothing properties, intrinsic utracontractivity, uniqueness and multiplicity of eigenfunctions, decay of eigenfunctions, number of eigenfunctions with negative eigenvalues, diamagnetic inequalities, spectral comparison inequalities etc). From the perspective of quantum mechanics, these results translate into the ground state properties of non-relativistic or relativistic quantum particles with or without spin. In Chapter 4 we focus on the “right-hand side” of the Feynman–Kac formula and its variants. In all cases that we consider the kernel of the operator semigroup can be expressed in terms of an expectation over the paths of a random process with respect to a measure weighted by exponential densities dependent on functionals additive in the time intervals. This structure suggests to view this as a new probability measure which can be interpreted as a Gibbs measure on path space for the potential(s) appearing directly in or derived from the generator of the process. Gibbs measures are well understood in discrete stochastic models, and originate from statistical mechan-

Section 1.2 Plan and scope

9

ics where they have been used to model thermodynamic equilibrium states. In our context this interpretation is not relevant, however, the so obtained Gibbs measures provide a tool for studying analytic and spectral properties of the operators standing at the left-hand side of the Feynman–Kac formula. In this chapter we define and prove the existence and properties of Gibbs measures on path space. First we present this for the case of external potentials and define the framework of P ./1 -processes. Then we establish existence and other results (uniqueness, almost sure path behaviour, mixing properties etc) for the case when the densities of the Gibbs measure also contain a pair interaction potential. Finally, we present the special case when the external potential is zero, and discuss Gibbs measures on the Brownian increment process, showing an emerging diffusive behaviour under proper scaling. Part II of the monograph is called Rigorous quantum field theory and consists of three chapters. In Chapter 5 we discuss the representations of the free boson field Hamilton operator. First we discuss it in Fock space, which is the natural setting for the description of quantum Hamiltonians in terms of self-adjoint operators. Then we introduce suitable function spaces in which the operator semigroups will be mapped by using the Wiener–Itô–Segal isomorphism, and discuss the basic theorems of Bochner and of Minlos on the existence of an infinite Gaussian measure on these spaces. Then we establish the functional integral representation of the free field Hamiltonian on this space. Also, we establish the Feynman–Kac–Nelson formula, which is a representation of the free quantum field in terms of the path measure of an infinite dimensional Ornstein–Uhlenbeck process, and we discuss the Markov property which is crucial in this context. In Chapter 6 we make a detailed study in terms of path measures of Nelson’s model of an electrically charged spinless quantum particle coupled to a scalar boson field. First we formulate the Nelson model in Fock space and discuss the infrared and ultraviolet cutoffs which we need to impose to define the model rigorously. Then we formulate the Nelson model in function space by using the same Wiener–Itô–Segal isomorphism as for the free quantum field previously. We present a proof of existence and uniqueness of a ground state. Next we establish a path measure representation of the Nelson Hamiltonian and identify a Gibbs measure on path space which can be associated to it. This Gibbs measure is defined for a given external potential, and a pair interaction potential arising as an effective potential obtained after integrating over the quantum field. Using the Gibbs measure we derive and prove detailed properties of the ground state, in particular, we discuss ground state expectations for the field operators, establish exponential spatial localization of the particle etc. We furthermore discuss the case of the translation invariant Nelson model obtained when the external potential is zero. We conclude the chapter by discussing infrared divergence manifested in the absence of a ground state, and ultraviolet divergence appearing in the point charge limit.

10


In Chapter 7 we carry out a similar study in terms of path measures of the Pauli– Fierz model. This model is obtained from the quantization of the Maxwell (vector) field. As before, we start by a formulation of the model in terms of Fock space operators. Next we derive and prove a functional integral representation of the operator semigroup generated by the Pauli–Fierz Hamiltonian. As in the Nelson model, also in this case we obtain a pair interaction potential through the functional integral, however, the double Riemann integrals seen previously are replaced by a double stochastic integral. We define this integral and will be able to extend the functional integral representation to a large class of external potentials. As applications of functional integration first of all we establish self-adjointness of the Pauli–Fierz operator. Also, we prove uniqueness of its ground state by exploring the positivity properties that the integral yields, and discuss its spatial decay. We will use the functional integral formula in order to define the Pauli–Fierz operator for Kato-decomposable potentials as a self-adjoint operator. Next we discuss the translation invariant case when the external potential is zero. In a next step by using the appropriate function spaces we establish a path integral representation of the Pauli–Fierz operator, in which the pair interaction potential becomes a twice iterated stochastic integral. Then we turn to the case of the relativistic Pauli–Fierz model, whose functional integral representation involves subordinate Brownian motion. Finally we discuss the Pauli–Fierz model with spin, and the spin-boson model as an application. In a concluding section we offer a commented guide to the selected bibliography. Undoubtedly, our list of references can be further increased, however, in lack of space we need to defer this to the reader.

Chapter 2

Probabilistic preliminaries

2.1

An invitation to Brownian motion

A random process is a mathematical model of a phenomenon evolving in time in a way which cannot be predicted with certainty. Brownian motion is a basic random process playing a central role throughout below. We offer here a brief summary of the mathematical theory of Brownian motion as background material for a reasonably self-contained presentation of what follows. Let .; F / be a measurable space, where is a set and F a -field. When is equipped with a topology, its Borel -field will be denoted by B./ and then .; B.// is called a Borel measurable space. A measurable space .; F / is separable if and only if F is generated by a countable collection of subsets in . In particular, if is a locally compact space with a countable basis, then .; B.// is separable. Throughout we will assume that Borel measurable spaces are separable. With measurable spaces .; F / and .S; S / we write f 2 F =S to indicate that the function f W ! S is measurable with respect to F and S , i.e., f 1 .E/ 2 F for all E 2 S . When f 2 F =B.Rd /, f is called a Borel measurable function. A measurable space equipped with a measure is a measure space denoted by .; F ; /. One can define the Rintegral of a Borel measurable function f on .; F ; / which will be denoted by f d. A probability space is a measurable space .; F ; P / in which is called sample space, F is called algebra of events and P is a probability measure on .; F /. In this case the integral with respect to P of a Borel measurable function f is called expectation of f and will be denoted by any of the symbols Z Z f .!/dP .!/ D f dP D EP Œf :

The probability of event ¹! 2 j conditionsº is simply denoted by P .conditions/: A measurable function X from a probability space .; F ; P / to a measurable space .S; S / is called an S-valued random variable. The inverse map identifies the sample points in on which the observation of event A depends. Let X be an Svalued random variable. The measure PX on .S; S / defined by PX .A/ D P .X 1 .A//;

A 2 S;

12

Chapter 2 Probabilistic preliminaries

is called the image measure or the distribution of P under X . By the definition of the distribution of X W ! S we have Z EP Œf .X / D f .x/dPX .x/ (2.1.1) S

for a bounded Borel measurable function f W S ! Rd . The covariance of two Rd -valued random variables X and Y on the same probability space is defined by cov.X; Y / D EP Œ.X mX /.Y mY /;

(2.1.2)

where mX D EP ŒX and mY D EP ŒY are the expectations of X and Y , respectively. For two S-valued random variables X and Y , not necessarily defined on the d same probability space, we use the notation X D Y whenever they are identically distributed, i.e., PX D PY . A random variable X W ! R with distribution 2 1 .xm/ dPX .x/ D p e 2 2 dx 2 2

d

is called a Gaussian random variable and the notation X D N.m; 2 / is used. An N.0; 1/-distributed random variable is a standard Gaussian random variable. A random vector X D .X1 ; : : : ; Xd / W ! Rd is a multivariate Gaussian random variable if its distribution is given by dPX .x/ D

1 .2/d=2 .detC /1=2

1

e 2 .xm/C

1 .xm/

dx;

where x D .x1 ; : : : ; xd / 2 Rd , C D .Cij / is a d d symmetric, strictly positive definite matrix and m D .m1 ; : : : ; md / 2 Rd is a vector. The matrix element Cij gives the covariance cov.Xi ; Xj / of the multivariate Gaussian random variable, and mj D EP ŒXj . We summarize the notions of convergence of random variables next. Definition 2.1 (Convergence of random variables). Let .Xn /n2N be a sequence of real-valued random variables and X another real-valued random variable, all on a given probability space .; F ; P /. a:s:

.1/ Xn ! X , i.e., Xn is convergent almost surely to X if P . lim Xn .!/ D X.!// D 1: n!1

Lp

.2/ Xn ! X , i.e., Xn is convergent in Lp -sense to X for 1 p < 1 if lim EP ŒjXn X jp D 0:

n!1

13

Section 2.1 An invitation to Brownian motion P

.3/ Xn ! X , i.e., Xn is convergent in probability to X if for every " > 0, lim P .jXn .!/ X.!/j "/ D 0:

n!1 d

.4/ Xn ! X , i.e., Xn is convergent in distribution or weakly to X if lim EP Œf .Xn / D EP Œf .X /

n!1

for any bounded continuous function f . The relationships of the various types of convergence are given by Theorem 2.1. Let .Xn /n2N be a sequence of real-valued random variables and X another real-valued random variable, all on .; F ; P /. a:s:

Lp

P

.1/ If Xn ! X or Xn ! X for some p 1, then Xn ! X as n ! 1. P

d

.2/ If Xn ! X , then Xn ! X as n ! 1. P

a:s:

.3/ If Xn ! X , then there exists a subsequence Xnk ; k 1; such that Xnk ! X as k ! 1. P

.4/ .Dominated convergence theorem/ Let Xn ! X as n ! 1 and suppose there is a random variable Y such that jXn j Y , n D 1; 2; : : : , and EŒY < 1. L1

Then X 2 L1 .; dP / and Xn ! X . a:s:

.5/ .Monotone convergence theorem/ Let Xn ! X as n ! 1. Suppose the sequence .Xn /n1 is monotone and there exists M > 0 such that EP ŒXn < M , for all n. Then X 2 L1 .; dP / and EP ŒXn ! EP ŒX as n ! 1. Next we review the convergence properties of probability measures. Definition 2.2. Let S be a metric space and P a probability measure on .S; B.S //. .1/ A sequence of probability measures .Pn /n2N on .S; B.S // is weakly convergent to P whenever lim EPn Œf D EP Œf

n!1

(2.1.3)

for every bounded and continuous real-valued function f on S . .2/ A family P of probability measures on .S; B.S // is relatively compact whenever every sequence of elements of P contains a weakly convergent subsequence.

14


.3/ A family P of probability measures on .S; B.S // is tight whenever for every " > 0 there exists a compact set K" S such that P .K" / > 1 ", for every P 2 P. Note that for any given sequence .Xn /n2N of random variables on .; F ; P / with d

values in a metric space .S; B.S //, the convergence Xn ! X is equivalent to weak convergence of P ı Xn1 to P ı X 1 . We will be mainly interested in the cases S D C.Œ0; 1// and S D C.R/. Since both are Polish (i.e., complete separable metric) spaces, a family of probability measures .S; B.S // has a weakly convergent sequence if it is tight. Take S D C.R/. Recall that for T; ı > 0 the modulus of continuity of ! 2 S is defined by mT .!; ı/ D

max

jst jı T s;t T

j!.s/ !.t /j:

(2.1.4)

For S D C.Œ0; 1// the modulus of continuity only modifies by T being replaced by 0. Theorem 2.2. Let S be a Polish space. .1/ Prokhorov’s theorem: A family P of probability measures on .S; B.S // is relatively compact if and only if it is tight. .2/ A sequence .Pn /n2N of probability measures on .C.R/; B.C.R/// is tight if and only if lim sup Pn .!.0/ > / D 0;

(2.1.5)

"1 n1

lim sup Pn .mT .!; ı/ > "/ D 0;

ı#0 n1

8T > 0; 8" > 0:

(2.1.6)

Two events A; B 2 F are independent if and only if P .A \ B/ D P .A/P .B/. Two sub--fields F1 ; F2 of F are independent if every A 2 F1 and B 2 F2 are pairwise independent. Let X be a random variable on .; F ; P /. .X / denotes the minimal -field such that X is measurable, i.e., .X / D ¹X 1 .U /jU 2 S º:

(2.1.7)

Let Y be also a random variable on .; F ; P /. If the sub--fields .X / and .Y / are independent, then X; Y are said to be independent. If X; Y are independent, then cov.f .X /; g.Y // D 0 for all bounded Borel measurable functions f; g. For two Gaussian random variables the converse is also true, i.e., if their covariance is zero, then they are independent. Let .; F ; P / be a probability space and J be a given set. A family of S-valued random variables .X t / t2J is called a random process with index set J . When S D

15

Section 2.1 An invitation to Brownian motion

Rd we refer to it as a real-valued random process no matter the dimension. If J is uncountable, then .X t / t0 is called a continuous time random process. For any fixed t 2 J the map ! 7! X t .!/ is an S-valued random variable on , while for any fixed ! 2 the map t 7! X t .!/ is a function called path. This distinction justifies for any random process to address distributional properties on the one hand, and sample path properties on the other. Definition 2.3 (Brownian motion). A real-valued random process .B t / t0 on a probability space .; F ; P / is called Brownian motion or Wiener process starting at x 2 Rd , whenever the following conditions are satisfied: .1/ P .B0 D x/ D 1 .2/ the increments .B ti Bti1 /1in are independent Gaussian random variables d

for any 0 D t0 < t1 < < tn , n 2 N, with B ti Bti1 D N.0; ti ti1 / .3/ the function t 7! B t .!/ is continuous for almost every ! 2 . In particular .B t / t0 is called standard Brownian motion when x D 0. A d -component vector of independent Brownian motions .B t1 ; : : : ; B td / t0 is called d -dimensional Brownian motion. Brownian motion is thus a real-valued continuous time random process with index set RC D Œ0; 1/. Consider X D C.RC I Rd /;

(2.1.8)

the space of Rd -valued continuous functions on RC , equipped with the metric X 1 sup jf .x/ g.x/j ^ 1: dist.f; g/ D k 2 0jxjk k0

This metric induces the locally uniform topology in which fn ! f if and only if fn ! f uniformly on every compact set in RC . The so obtained topology gives rise to the Borel -field B.X /. Proposition 2.3. The Borel -field B.X / coincides with the -field generated by the cylinder sets of the form 1 ;:::;En AE t1 ;:::;tn D ¹! 2 X j.!.t1 /; : : : ; !.tn // 2 E1 En º

with E1 ; : : : ; En 2 B.Rd /, 0 t1 < t2 < < tn and n 2 N. Next we consider the canonical representation of Brownian motion on the measure space .X ; B.X /; W x /, where W x is the Wiener measure defined below. In this representation a realization of the process .B t / t0 is given by B t .!/ D !.t / 2 Rd ;

!2X

(2.1.9)

16


called coordinate process. For each fixed ! 2 X we think of RC 3 t 7! B t 2 Rd as a Brownian path. Write 1 jxj2 … t .x/ D exp (2.1.10) ; t > 0; x 2 Rd 2t .2 t /d=2 called d -dimensional heat kernel. Definition 2.4 (Wiener measure). Let 0 D t0 < t1 < < tn , Ej 2 B.Rd /, j D 1; : : : ; n, n 2 N, and 1E be the indicator function of E 2 B.Rd /. The unique measure W x on .X ; B.X // whose finite dimensional distributions are given by W x .B0 D x/ D 1

(2.1.11)

and W x .B t1 2 E1 ; : : : ; B tn 2 En / Z n n n Y Y Y D 1Ej .xj / … tj tj 1 .xj 1 xj / dxj Rd n

j D1

j D1

(2.1.12)

j D1

is called Wiener measure starting at x. For expectation with respect to Wiener measure we will use the notations Ex ŒF D EW x ŒF ;

EŒF D E0 ŒF :

(2.1.13)

By the definition of Wiener measure we have Ex

n hY j D1

i Z fj .B tj / D

Rnd

n Y

n n Y Y fj .xj / … tj tj 1 .xj 1 xj / dxj

j D1

j D1

j D1

(2.1.14) for all bounded Borel measurable functions f1 ; : : : ; fn with x0 D x. In particular, Z x f .y/… t .x y/dy: (2.1.15) E Œf .B t / D Rd

Recall that the real-valued random processes .X t / t2J and .Y t / t2J on a probability space .; F ; P / are versions of each other if for each t 0 there is a null set N t such that X t .!/ D Y t .!/ for all ! … N t . When t 7! Y t .!/ is continuous in t for almost every ! 2 , .Y t / t2J is called a continuous version of .X t / t2J . Theorem 2.4. The Wiener measure W x exists on .X ; B.X // and the coordinate process B t .!/ D !.t /, t 0, on .X ; B.X /; W x / is Brownian motion starting at x 2 Rd .


17

Proof. The first part of the argument is to prove that there is a unique measure on .RŒ0;1/ ; G / satisfying (2.1.11) and (2.1.12), where G is the -field generated by the cylinder sets in RŒ0;1/ . This can be done by using the Kolmogorov extension theorem quoted in Proposition 2.5 below. The second part is to prove that there exists a continuous version .BQ t / t0 of the coordinate process .B t / t0 on RŒ0;1/ by using ˇ the simple criterion given by the Kolmogorov–Centsov theorem, see Proposition 2.6 below. Note that .BQ t / t0 is not the coordinate process, and while BQ t .!/ 6D !.t / for every ! 2 RŒ0;1/ , it is true that BQ t .!/ D !.t /, for ! … N t , with a null set N t depending on t . With the so constructed Brownian motion .BQ t / t0 on .RŒ0;1/ ; G ; /, BQ 0 D 0, let X W .RŒ0;1/ ; G ; / ! .X ; B.X // be defined by X t .!/ D BQ t .!/. Since BQ t .!/ is continuous in t , it can be checked that X 1 .E/ 2 B.RŒ0;1/ / for any cylinder set E. Thus X 2 G =B.X /, since B.X / coincides with the -field generated by the cylinder sets. The image measure X of under X on .X ; B.X // is denoted by W 0 . It is seen that W 0 is Wiener measure with x D 0, and the coordinate process B t W ! 2 X 7! !.t / is Brownian motion on .X ; B.X /; W 0 / starting at 0. Define Y W .X ; B.X /; W 0 / ! .X ; B.X // by Y t .!/ D B t .!/ C x. It can be seen that Y 2 B.X /=B.X /. Then the image measure of W 0 under Y on .X ; B.X // is denoted by W x . This is Wiener measure starting at x, and the coordinate process B t .!/ D !.t /, t 0, under W x is Brownian motion starting at x. Let be a Polish space and J be an index set, ƒ J , jƒj < 1, with j j denoting the cardinality of a finite set. Consider the family of probability spaces jƒj .ƒ ; B.ƒ /; ƒ / indexed by the finite subsets of J , where ƒ D kD1 . An element ! 2 ƒ is regarded as a map ƒ ! . Take now ƒ1 ƒ2 ƒ3 J and let ƒ1 ƒ2 W ƒ2 ! ƒ1 denote the projection defined by ƒ1 ƒ2 .!/ D !dƒ1 . The relation 1 ƒ1 .E/ D ƒ2 .ƒ .E// 1 ƒ2

(2.1.16)

for E 2 B.ƒ1 / and ƒ1 ƒ2 with jƒ1 j; jƒ2 j < 1, is called Kolmogorov consistency relation. Proposition 2.5 (Kolmogorov extension theorem). Let .ƒ ; B.ƒ /; ƒ /, ƒ J with jƒj < 1, be a family of probability spaces with underlying Polish space . Suppose that ƒ satisfies the consistency relation (2.1.16), and define 1 A D ¹ƒJ .E/ 2 J jƒ J; jƒj < 1; E 2 B.jƒj /º;

18


1 where ƒJ .E/ D ¹! 2 J j!dƒ 2 Eº. Then there exists a unique probability mea1 .E//, for E 2 B.ƒ / with ƒ J sure Q on .J ; .A// such that ƒ .E/ D . Q ƒ and jƒj < 1, where .A/ denotes the minimal -field containing A.

To address the second part of the existence theorem first we need the following concept. Definition 2.5 (Hölder continuity). A function f W RC ! R is called locally Hölder continuous at t 0 if there exist "; > 0 and Cs;t > 0 such that jf .t / f .s/j Cs;t jt sj ; for every s 0 such that jt sj < ". The number is called Hölder exponent and Cs;t is the Hölder constant. ˇ Proposition 2.6 (Kolmogorov–Centsov theorem). Let .; F ; P / be a probability space and .X t / t0 a real-valued random process on it. Suppose there exist ˛ 1, ˇ; C > 0 such that EP ŒjX t Xs j˛ C jt sjˇ C1 :

(2.1.17)

Then there exists a version .Y t /0tT of .X t /0tT for every T 0 with -Hölder continuous paths, with 0 ˇ=˛. Using a version copes with the fact that for a continuous time process path continuity is not a measurable event since, intuitively, to know about a path that it is continuous would need uncountably many observations and the countably generated field of cylinder sets cannot accommodate them. By the definition of Brownian motion it is straightforward to see that standard Brownian motion is a multivariate Gaussian process with EŒB t D 0; EŒBs B t D s ^ t; and an easy computation gives EŒ.B t Bs /2n D

.2n/Š jt sjn ; nŠ2n

n 2 N;

(2.1.18)

while all odd order moments of the increments of Brownian motion vanish. When applying Proposition 2.6 to Brownian motion one can make use of (2.1.18) and take ˛ D 2k, ˇ D k 1, k D 1; 2; : : : , to construct a -Hölder continuous version for every 0 < < .k 1/=.2k/. Optimizing over k implies Hölder continuity for any

D .0; 1=2/. In the following we briefly review some basic properties of Brownian motion starting with the distributional properties.

19


Proposition 2.7 (Symmetry properties). Let .B t / t0 be standard Brownian motion. The following symmetry properties hold: .1/ .spatial homogeneity/ if .B tx / t0 is Brownian motion starting at x 2 R, then d

B t C x D B tx d

.2/ .reflection symmetry/ B t D B t p d .3/ .self-similarity/ cB t=c D B t , for all c > 0 ² 0 if t D 0; d then Z t D B t .4/ .time inversion/ let Z t .!/ D tB1=t .!/ if t > 0; d

.5/ (time reversibility) B ts B t D Bs , for all 0 s t . Proof. These equalities easily follow from (2.1.14). The fact that Z0 D 0 a.s. is an application of the strong law of large numbers quoted below. Next we discuss sample path properties of Brownian motion without proof. Proposition 2.8 (Global path properties). Let .B t / t0 be standard Brownian motion. Then the following hold almost surely: .1/ lim inf t!1 B t D 1 and lim sup t!1 B t D 1 p p .2/ lim inf t!1 B t = t D 1 and lim sup t!1 B t = t D 1 a:s:

.3/ .strong law of large numbers/ B t =t ! 0 as t ! 1 .4/ .law of the iterated logarithm/ p lim sup B t = 2t log log t D 1 and t!1

p lim sup B t = 2t log log.1=t / D 1 and t#0

p lim inf B t = 2t log log t D 1; t!1

p lim inf B t = 2t log log.1=t / D 1: t#0

Proposition 2.9 (Local path properties). Let .B t / t0 be standard Brownian motion. Then the following hold: .1/ .Hölder continuity/ For every T > 0, 2 .0; 1=2/ and almost every ! 2 X there is .T; ; !/ > 0 such that jB t .!/ Bs .!/j .T; ; !/jt sj ;

0 s; t T:

.2/ .Nowhere differentiability/ t 7! B t is almost surely at no point Hölder continuous for any > 1=2, hence nowhere differentiable.

20


.3/ .p-variation/ Let p > 0 and n D ¹t0 < t1 < < tn º be a partition of a bounded interval I R such that t0 and tn coincide with the two endpoints of I , with mesh m.n / D maxkD0;:::;n1 jtkC1 tk j. Define p

VI;n .!/ D

n1 X

jB tkC1 .!/ B tk .!/jp :

kD0

Then for every bounded interval I and d -dimensional Brownian motion ² d jI j; p D 2; p in LP .X ; d W x /-sense (a) lim supm.n /!0 VI;n .!/ D 0; p > 2; p

(b) lim supm.n /!0 VI;n is almost surely bounded if p > 2, and almost surely unbounded if p 2. By a similar construction as above Brownian motion can be extended over the whole time-line R instead of defining it only on RC . This can be done on the measurable space .Y ; B.Y //, with Y D C.RI Rd /. Consider XQ D X X and x D W x W x . Let ! D .!1 ; !2 / 2 XQ and define ´ if t 0 !1 .t / BQ t .!/ D !2 .t / if t < 0: Since BQ t .!/ is continuous in t 2 R under x , X W .XQ ; B.XQ // ! .Y ; B.Y // can be defined by X t .!/ D BQ t .!/. It can be seen that X 2 B.XQ /=B.Y / by showing that X 1 .E/ 2 B.XQ /, for any cylinder set E 2 B.Y /. Thus X is a Y -valued random variable on XQ . Denote the image measure of x under X on .Y ; B.Y // by WQ x . The coordinate process denoted by the same symbol BQ t W ! 2 Y 7! !.t / 2 Rd

(2.1.19)

is Brownian motion over R on .Y ; B.Y /; WQ x / with BQ 0 D x almost surely. We will denote the path space of two sided Brownian motion by X D C.R; Rd /. The properties of Brownian motion on the whole real line can be summarized as follows. Proposition 2.10 (Brownian motion on R). .1/ W x .BQ 0 D x/ D 1 .2/ the increments .BQ ti BQ ti1 /1in are independent Gaussian random variables d for any 0 D t0 < t1 < < tn , n 2 N, with BQ t BQ s D N.0; t s/, for t > s .3/ the increments .BQ ti1 BQ ti /1in are independent Gaussian random varid ables for any 0 D t0 > t1 > > tn , n 2 N, with BQ t BQ s D N.0; s t /, for t > s .4/ the function R 3 t 7! BQ t .!/ 2 R is continuous for almost every !

.5/ B t and Bs for t > 0 and s < 0 are independent.

21

Section 2.2 Martingale and Markov properties

It can be checked directly through the finite dimensional distributions (2.1.14) that the joint distribution of BQ t0 ; : : : ; BQ tn , 1 < t0 < t1 < < tn < 1, with respect to dx ˝ d WQ x is invariant under time shift, i.e., Z Rd

dxEWQ x

n hY

i Z fi .BQ ti / D

iD0

Rd

dxEWQ x

n hY

i fi .BQ ti Cs /

(2.1.20)

iD0

for all s 2 R. Moreover, the left-hand side of (2.1.20) can be expressed in terms of Wiener measure on X as " n " n # Z # Z Y Y dxEWQ x fi .BQ ti / D dxEW x fi .B ti t0 / : (2.1.21) Rd

2.2

iD0

Rd

iD0

Martingale and Markov properties

2.2.1 Martingale property Let G F be a sub--field and X a random variable on .; F ; P /. Recall that the conditional expectation EP ŒX jG is defined as the unique G -measurable random variable such that EP Œ1A X D EP Œ1A EP ŒX jG ;

A 2 G:

(2.2.1)

The left-hand side of (2.2.1) defines a measure PQ .A/ D EP Œ1A X on G and EP ŒX jG is in fact the Radon–Nikodým derivative d PQ =dP . The next lemma is of key importance. Lemma 2.11. Let .; F ; P / be a probability space and .S; S / a measurable space. Let, moreover, X be an S-valued random variable on and Y a real-valued and .X /-measurable random variable on . Then there exists an S -measurable function f on S such that Y D f ı X a.e. Proof. It suffices to show the lemma for non-negative Y . Let L be the set of realvalued .X /-measurable random variables on satisfying the assumption of the lemma. If A 2 .X /, then 1A 2 L and all .X /-measurable step functions belong to L . Since Y is .X /-measurable, there exists a sequence of increasing step functions n such that n ! Y almost surely. Since n 2 L , there exists a sequence of S -measurable functions fn such that n D fn ı X . Define the function f on S by f .x/ D lim supn!1 fn .x/ for x 2 S. Then f is S -measurable and f .X.!// D lim supn!1 fn ı X.!/ D limn!1 fn ı X.!/ D Y .!/, and hence Y 2 L.

22


Let .X t / t0 be a random process on .; F ; P /. Then EP Œf j.X t / is a .X t /measurable function and by Lemma 2.11 it can be expressed as EP Œf j.X t / D h.X t / with a Borel measurable function h. This function is usually denoted by h.x/ D EP Œf jX t D x:

(2.2.2)

Note that if h.X t / D f .X t / D EP Œf j.X t / a.s., then h.x/ D f .x/ for P ıX t1 -a.e. x 2 R, so the ambiguity in the expression (2.2.2) has measure zero under P ı X t1 . Since EP Œf .X0 /g.X t / D EP Œf .X0 /EP Œg.X t /j.X0 /, using the notation (2.2.2) we have Z 0 .dx/f .x/EP Œg.X t /jX0 D x; (2.2.3) EP Œf .X0 /g.X t / D Rd

where 0 .dx/ denotes the distribution of X0 on R. The conditional expectation EP Œ1A jG is in particular denoted by P .AjG /. It is not obvious that P .jG /.!/ defines a probability measure for almost every !, since P .AjG /./ is defined almost surely for each A. Let P .AjG /.!/ be defined on ! 2 nS NA with P .NA / D 0. Then P .jG /.!/ W F ! Œ0;S 1 is defined only for ! 2 n A2F NA . It is, however, in general not clear that P . A2F NA / D 0. This justifies the following definition. Definition 2.6 (Regular conditional probability measure). Let .; F ; P / be a probability space, and G a sub--field of F . A function Q W F ! Œ0; 1 is called a regular conditional probability measure for F given G if .1/ Q.!; / is a probability measure on .; F / for every ! 2 .2/ Q.; A/ is G -measurable for every A 2 F .3/ Q.!; A/ D P .AjG /.!/, P -a.s. ! 2 , for every A 2 F . We will be interested in the case when the sub--field is generated by a random variable X , which makes worthwhile to rephrase this definition. We use the notation EŒ1A jX D x D P .AjX D x/. Definition 2.7 (Regular conditional probability measure). Let .; F ; P / be a probability space and X be a random variable on it with values in a measure space .S; S /. A function Q W S F ! Œ0; 1 is called a regular conditional probability measure for F given X if .1/ Q.x; / is a probability measure on .; F / for every x 2 S .2/ Q.; A/ is S -measurable for every A 2 F .3/ Q.x; A/ D P .AjX D x/, P ı X 1 -a.s., for every A 2 F .


23

Theorem 2.12. Suppose that is a Polish space and F D B./. .1/ Let P be a probability measure on .; F / and G a sub--field of F . Then there exists a regular conditional probability measure for F given G , and it is unique. .2/ Let X be random variable on .; F / with values in a measurable space .S; S /. Then there exists a regular conditional probability measure for F given X . Moreover, if Q0 is another regular conditional probability measure, then there exists a set N 2 S with P ı X 1 .N / D 0, such that Q.x; A/ D Q0 .x; A/, for all A 2 F and every x 2 S n N . .3/ If S is a Polish space and S D B.S /, then the null set N 2 S above can be chosen such that Q.x; X 1 .B// D 1B .x/, for B 2 S , x 2 S n N . In particular, Q.x; X 1 .x// D 1, P ı X 1 -a.e. x 2 S , holds. In what follows the regular conditional probability measures Q.!; A/ and Q.x; A/ will be denoted by P .AjG / and P .AjX D x/, respectively, unless confusion may arise. Definition 2.8 (Filtration). Let .; F ; P / be a probability space and .F t / t0 a family of sub--fields of F . The collection .F t / t0 is called a filtration whenever Fs F t for s t . The given measurable space endowed with a filtration is called a filtered space. Intuitively, Ft contains the information known to an observer at time t . Let .Xs /s0 be an S -valued random process. A convenient choice of filtration is the natural filtration given by F tX D .Xs ; 0 s t /, containing the information accumulated by observing X up to time t . Definition 2.9 (Adapted process). Let .; F / be a filtered space, .F t / t0 a given filtration and .X t / t0 a random process. The process is called .F t / t0 -adapted whenever X t is F t -measurable for every t 0. If a process .X t / t0 is .F t / t0 -adapted it means that the process does not carry more information at time t than F t . Obviously, .X t / t0 is adapted to its natural filtration .F tX / t0 , or equivalently, the natural filtration .F tX / t0 is the smallest filtration making .X t / t0 adapted. Definition 2.10 (Martingale). Let .; F ; P / be a filtered space with a given filtration .F t / t0 . The random process .X t / t0 is an .F t / t0 -martingale whenever .1/ EP ŒjX t j < 1, for every t 0 .2/ .X t / t0 is .F t / t0 -adapted .3/ EP ŒX t jFs D Xs , for every s t .

24


We will make use of the next result on martingales which will be quoted here with no proof. Theorem 2.13 (Martingale inequality). Let .X t / t0 be an .F t / t0 -martingale on .; F ; P / and suppose that X t .!/ is continuous in t , for almost every ! 2 . Then for all p 1, T 0 and all > 0, P

1 sup jX t j p EP ŒjXT jp : 0tT

(2.2.4)

Proposition 2.14. Let .B t / t0 be Brownian motion on .; F ; P / and consider its natural filtration .F tB / t0 . Then both .B t / t0 and .B t2 t / t0 are .F tB / t0 -martingales. Proof. We have that EP ŒB t jFsB D EP ŒB t Bs jFsB C EP ŒBs jFsB D EP ŒB t Bs C Bs D Bs (2.2.5) by independence of B t Bs and Br for r s. By a similar calculation it is easy to show also the second claim. Remarkably, there is a strong converse of this proposition. Proposition 2.15 (Lévy martingale characterization theorem). Let .X t / t0 be a random process and .F tX / t0 be its natural filtration. Suppose that .1/ X0 D 0 almost surely, .2/ the paths t 7! X t are almost surely continuous, .3/ both .X t / t0 and .X t2 t / t0 are .F tX / t0 -martingales. Then .X t / t0 is a standard Brownian motion. There is a useful weakening of the martingale property which removes the condition of integrability in Definition 2.10. To introduce and apply it below, we first need a concept of random times. Definition 2.11 (Stopping time). Let .; F ; P / be a filtered space with filtration .F t / t0 . A random variable W ! Œ0; 1 such that ¹ t º 2 F t , for all t 0, is called a stopping time. Whenever the filtration is right-continuous, is a stopping time if and only if ¹ < t º 2 F t , for every t 0. An example of a stopping time is the first hitting time of Brownian motion .B t / t0 of an open (or a closed) set E, i.e., D inf¹t 0 W B t 2 Eº. If . n /n2N is a sequence of stopping times and n # ,

25


S then since ¹ < t º D n1 ¹ n < t º, we have that also is a stopping time. Similarly, if a sequence of stopping times . n /n2N is such that n " , then since ¹ t º D \n1 ¹ n < t º, again is a stopping time. Notice that t ^ is a stopping time whenever is. We make use of this in the following property. Definition 2.12 (Local martingale). A random process .X t / t0 on a filtered probability space .; F ; .F t / t0 ; P / is said to be a local martingale if there exists a sequence of stopping times . n /n2N such that .X t^n / t0 is a martingale with respect to the filtration .F t^n / t0 . Whenever .X t / t0 has a continuous version, one way of choosing the stopping times is n D inf¹t 0 j jXn j > nº.

2.2.2 Markov property In addition to the martingale property Brownian motion possesses the equally important Markov property. Intuitively, this means that the future values of the random process only depend on its present value and do not on its past history. Definition 2.13 (Markov process). Let .X t / t0 be an adapted process on a filtered space .; F ; P / with a filtration .F t / t0 . .X t / t0 is a Markov process with respect to .F t / t0 whenever EP Œf .X t /jFs D EP Œf .X t /j.Xs / ;

0 s t;

(2.2.6)

for all bounded Borel measurable functions f . Markov processes can be characterized by their probability transition kernels. Definition 2.14 (Probability transition kernel). A map p W RC RC Rd B.Rd / ! R, .s; t; x; A/ 7! p.s; t; x; A/, is called a probability transition kernel if .1/ p.s; t; x; / is a probability measure on B.Rd / .2/ p.s; t; ; A/ is Borel measurable .3/ for every 0 r s t , the Chapman–Kolmogorov identity Z p.s; t; y; A/p.r; s; x; dy/ D p.r; t; x; A/ Rd

(2.2.7)

holds. Let .X t / t0 be a Markov process. We define pX .s; t; x; A/ D EP Œ1A .X t /jXs D x for A 2 B.Rd /, 0 s t < 1 and x 2 Rd .

(2.2.8)

26


Lemma 2.16. Let .X t / t0 be a Markov process with a filtration .F t / t0 . Then pX is a transition probability kernel. Proof. Properties (1) and R(2) of Definition 2.13 are straightforward to check, therefore we R only show (3). Since Rd 1A .y/pX .r; s; x; dy/ D EP Œ1A .Xs /jXr D x, we have Rd f .y/pX .r; s; x; dy/ D EP Œf .Xs /jXr D x, for bounded Borel functions f . Hence Z pX .s; t; y; A/pX .r; s; x; dy/ D EP ŒpX .s; t; Xs ; A/jXr D x: (2.2.9) Rd

Here pX .s; t; Xs ; A/ D pX .s; t; y; A/ evaluated at y D Xs . Thus the identity pX .s; t; Xs ; A/ D EP Œ1A .X t /j.Xs / follows, and furthermore pX .s; t; Xs ; A/ D EP Œ1A .X t /jFs by the Markov property. Inserting this into (2.2.9) we obtain Z pX .s; t; y; A/pX .r; s; x; dy/ D EP ŒEP Œ1A .X t /jFs jXr D x: (2.2.10) Rd

Since .Xr / Fs , we have that EP ŒEP Œ1A .X t /jFs j.Xr / D EP Œ1A .X t /j.Xr /, and therefore EP ŒEP Œ1A .X t /jFs jXr D x D pX .r; t; x; A/: R Combining this with (2.2.10), Rd pX .s; t; y; A/pX .r; s; x; dy/ D pX .r; t; x; A/ is obtained. The measure P 0 .A/ D P .X0 2 A/

(2.2.11)

is the initial distribution of the Markov process describing the random variable at t D 0. Then the finite dimensional distributions of .X t / t0 are determined by its probability transition kernel and initial distribution through the formula P .X t0 2 A0 ; : : : ; X tn 2 An / Z Y n n Y 1Ai .xi / p.ti1 ; ti ; xi1 ; dxi / 1A0 .x0 /P 0 .dx0 /; D Rd

iD1

(2.2.12)

iD1

for n D 1; 2; : : : , and where t0 D 0. Proposition 2.17. Let .X t / t0 be an Rd -valued random process on a probability space .; F ; P /, P 0 a probability measure on Rd and p.s; t; x; A/ a probability transition kernel. Suppose that the finite dimensional distributions of the process are

27


given by (2.2.12). Then .X t / t0 is a Markov process under the natural filtration with probability transition kernel p.s; t; x; A/. Proof. It suffices to show that n n i i h h Y Y EP 1A .X tCs / 1Aj .X tj / D EP EP Œ1A .X t /j.Xs / 1Aj .X tj / (2.2.13) j D0

j D0

for every 0 D t0 < t1 tn t r and every Aj 2 B.Rd /, j D 0; 1; : : : ; n. From the identity EP Œ1A .X t /EP Œf .Xr /j.X t / D EP Œ1A .X t /f .Xr /, it follows that Z Z 1A .y/EP Œf .Xr /jX t D yP t .dy/ D 1A .y/f .y 0 /p.t; r; y; dy 0 /P t .dy/; Rd

Rd

R where we denote P t .dy/ D Rd P 0 .dy0 /p.0; t; y0 ; dy/ and note that P t .dy/ is the distribution of X t on Rd . Thus it is seen that Z EP Œf .Xr /jX t D y D f .y 0 /p.t; r; y; dy 0 /: (2.2.14) Rd

By the definition of EP Œf .Xr /jX t D y we have Z EP Œf .Xr /j.X t / D

Rd

f .y/p.t; r; X t ; dy/

(2.2.15)

and thus EP Œ1A .Xr /j.X t / D p.t; r; X t ; A/:

(2.2.16)

Using (2.2.16) and the Chapman–Kolmogorov identity, (2.2.13) follows. Furthermore we have the following strong description of the converse situation. Proposition 2.18. Let p.s; t; x; A/ be a probability transition kernel and P 0 a probability distribution on Rd . Then there exists a filtered probability space .; F ; P / and a Markov process .X t / t0 on it such that EP Œ1A .X t /jXs D x D p.s; t; x; A/ and P ı X01 D P 0 . When the transition probability kernel p.s; t; X; A/ of a Markov process satisfies p.s; t; x; A/ D p.0; t s; x; A/, the process is said to be stationary Markov process. In this case by a minor abuse of notation we write p.t; x; A/ for p.0; t; x; A/, and the

28


Chapman–Kolmogorov identity reduces to Z Rd

p.s; y; A/p.t; x; dy/ D p.s C t; x; A/:

(2.2.17)

Theorem 2.19. Let .B t / t0 be Brownian motion adapted to a filtration .F t / t0 . Then .B t / t0 is a stationary Markov process and Ex Œf .BsCt0 ; : : : ; BsCtn /jFs D EBs Œf .B t0 ; : : : ; B tn / ;

8x 2 Rd ;

(2.2.18)

holds for every bounded Borel function f . Moreover, its initial distribution P 0 D ıx is Dirac measure with mass at x, while its probability transition kernel is given by Z PBM .t; x; A/ D

Rd

1A .z/… t .x z/dz

(2.2.19)

or equivalently, PBM .t; x; dy/ D … t .x y/dy:

(2.2.20)

Proof. The Markov property with transition probability kernel (2.2.20) follows by the definition of the finite dimensionalRdistributions of Brownian motion and Proposition 2.17. Note that Q PBM .s; t; x; A/ D Rd 1A .z/… ts .x z/dz. We show (2.2.18). Let h.y/ D Ex Œ jnD1 f .BsCtj /jBs D y. By the identity n n h i h hY ii Y x x fj .BsCtj / D E 1A .Bs /E fj .BsCtj /j.Bs / E 1A .Bs / x

j D1

j D1

we have Z

n n n Y Y Y 1A .y/…s .y x/ fj .xj / …t1.x1 y/ … tj tj 1 .xj xj 1 / dy dxj

R.nC1/d

j D1

Z

D

Rd

j D2

j D1

1A .y/h.y/…s .y x/dy:

Comparing the two sides above Z h.y/ D

n Y Rnd

j D1

n n Y Y fj .xj / … t1 .x1 y/ … tj tj 1 .xj xj 1 / dxj j D2

j D1

29


Q is obtained, while the right-hand side equals Ey Œ jnD1 fj .B tj /. Hence E

x

n hY

n i i hY Bs f .BsCtj /j.Bs / D E fj .B tj /

j D1

j D1

and (2.2.18) follows. From the proof of Theorem 2.19 it follows that the Markov property (2.2.6) can be reformulated as Ex Œf .B t /jFs D EBs Œf .B ts /;

t s:

(2.2.21)

In what follows, the Markov property of a random process will be considered with respect to its natural filtration, unless otherwise stated. 1

Corollary 2.20. Let ˛ 2 R. Then the random process .e ˛B t 2 ˛ t / t0 is a martingale with respect to the natural filtration of Brownian motion. 2

Proof. By (2.2.21) we have 1

1

Ex Œe ˛B t 2 ˛ t jFsB D EBs Œe ˛B t s 2 ˛ t : 2

2

The right-hand side above can be computed as Z 1 2 1 2 1 2 EBs Œe ˛B t s 2 ˛ t D e 2 ˛ t e ˛y … ts .Bs y/dy D e ˛Bs 2 ˛ s Rd

and the corollary follows. Brownian motion has a yet stronger Markov property than (2.2.18) since the same equality holds even for random times. Given a stopping time , we define the -field F by saying that A 2 F if and only if A 2 F and A \ ¹ t º 2 F t , for all t 0. Intuitively, F contains the events occurring before or at the given stopping time. Theorem 2.21 (Strong Markov property). Let .B t / t0 be a Brownian motion and a bounded stopping time with respect to the filtration .F t / t0 . Then Ex ŒB tC jF D EB ŒB t ;

t 0; x 2 Rd :

(2.2.22)

The theorem is valid also for unbounded stopping times but we have stated it in this form for simplicity.

2.2.3 Feller transition kernels and generators Markov processes are intimately related to semigroups, which intuitively describe the time evolution of the probability distributions. Here we need to use operator semigroup theory; for essential background material see Section 3.1.8 below.

30


Definition 2.15 (Feller transition kernel). A probability transition kernel p.t; x; A/ is said to be a Feller transition kernel whenever the following conditions hold: .1/ p.t; x; Rd / D 1 R .2/ Rd p.t; ; dy/f .y/ 2 C1 .Rd / for f 2 C1 .Rd / R .3/ lim t#0 Rd p.t; x; dy/f .y/ D f .x/, for every f 2 C1 .Rd / and x 2 Rd . Consider Z .P t f /.x/ D

p.t; x; dy/f .y/: Rd

(2.2.23)

When P t f 0 for f 0, for all t 0, the semigroup ¹P t ; t 0º is called nonnegative. The next proposition gives the relationship between Markov processes and C0 -semigroups. Proposition 2.22. Let p.t; x; A/ be a Feller transition kernel. Then ¹P t W t 0º is a non-negative C0 -semigroup on C1 .Rd /. Proof. It is easy to see that P0 D 1 and P t is non-negative. P t Ps D P tCs follows by the Chapman–Kolmogorov identity. Moreover, kP t f k1 kf k1 can also easily be seen. The difficult part is to show the strong continuity, i.e., lim P t f D f

t!0

(2.2.24)

R1 in the sup-norm. Let ˛ > 0 and define f˛ D 0 ˛e ˛t P t f .x/dt . Then f˛ 2 C1 .Rd /, since f 2 C1 .Rd /. Let M be the closed subspace ofR C1 .Rd / spanned by 1 f such that (2.2.24) holds. Take f 0 and define f˛;h .x/ D h ˛e ˛t P t f .x/dx; then f˛;h .x/ " f˛ .x/ as h # 0. This convergence is uniform as the sequence of continuous monotone functions .f˛;h /h converges to f˛ .x/ pointwise on the compact set Rd [ ¹1º. Since M is closed, f˛ 2 M follows for any non-negative f 2 C1 .Rd / and therefore for all f 2 C1 .Rd /. Let l 2 C1 .Rd / be a linear functional on C1 .Rd /. Since kf˛ k1 kf k1 and lim˛!1 f˛ .x/ D f .x/ pointwise, it follows that f˛ ! f weakly as ˛ ! 1 and hence l.f˛ / ! l.f / as ˛ ! 1. Suppose l.g/ D 0 for all g 2 M . Then for any f 2 C1 .Rd /, l.f / D lim˛!1 l.f˛ / D 0. Thus the Hahn–Banach theorem yields that C1 .Rd / D M . The converse statement of Proposition 2.22 is also true. Proposition 2.23. For any non-negative C0 -semigroup there exists a unique Feller transition kernel such that (2.2.23) holds.

31


Let .X t / t0 be a Markov process and consider its Feller transition kernel. P t in (2.2.23) is by Proposition 2.22 a C0 -semigroup and there exists a closed operator L on C1 .Rd / such that 1 lim .P t f f / D Lf: t!0 t L is called the generator of the Markov process and is formally written as P t D e tL . Proposition 2.24. The probability transition kernel P BM of Brownian motion is a Feller transition kernel and its generator is .1=2/ on C01 .Rd /. R Proof. We have already shown that p BM .t; x; A/ D Rd 1A .z/… t .xz/dz in (2.2.19). R 2 Thus p BM .t; x; Rd / D 1 holds. Since P tBM f .x/ D .2 t /d=2 Rd e jzj =2t f .x C z/dz for f 2 C1 .Rd /, by dominated convergence it is clear that P tBM f .x/ is continuous in x and limjxj!1 P tBM f .x/ D 0. Thus P tBM f 2 C1 .Rd /. Next we check that lim P tBM f .x/ D f .x/:

t!0

(2.2.25)

Notice that f is uniformly continuous on the compact set Rd [ ¹1º. Thus for any " > 0 there exists ı > 0 such that jf .x C y/ f .x/j " for all x 2 Rd and jyj < ı. Hence jP tBM f .x/ f .x/j Z Z jf .x C y/ f .x/j… t .y/dy C jyj 1, then a f .x/dg.x/ exists. A nearly optimal (sufficient and “very nearly” also necessary) condition is that .1/ f and g are not both discontinuous at the same point x, .2/ f has bounded p-variation and g has bounded q-variation for some p; q > 0 such that 1=p C 1=q D 1. Rb Our interest is to define integrals of the type a X t dB t where we would use Brownian motion as integrator and a random process X t as integrand. Clearly, due to nowhere differentiability of Brownian motion, the usual Riemann–Stieltjes integral will not work. However, since Brownian motion is of bounded p-variation for p > 2, see Proposition 2.9, provided the integrand is of bounded q-variation with suitable q < 2, Young’s criteria can be applied. This is possible, in particular, when .X t / t0 has bounded variation, i.e., q D 1. A sufficient condition for this is that t 7! X t is differentiable and has bounded derivative. However, the approach offered by Young or the conditions above quickly show their limitations as they would fail to cover Rb the simple integral a B t dB t . In order to include this and more interesting cases a genuinely new type of integral is needed.

2.3.2 Stochastic integrals Recall the basic theorem of calculus: If f 2 C 1 .Rd I R/ and g 2 C 1 .RI Rd /, then Z f .g.t // f .g.0// D

t 0

rf .g.s// .dg.s/=ds/ ds:

Since Brownian paths are nowhere differentiable, we cannot substitute Bs for g.s/ in the formula above. In spite of that, a similar formula still exists with an extra term appearing as a correction. Indeed, we will see that almost surely we have Z f .B t / f .B0 / D

0

t

1 rf .Bs / dBs C 2

Z

t

f .Bs / ds; 0

with an expression formally written as dBs “D” .dBs =ds/ds which will be explained below. The first term on the right-hand side above leads to a new concept of integral. Following the conventional concept of Riemann–Stieltjes integrals, in order to construct the integral with respect to Brownian motion we divide the interval into n subin-

35

Section 2.3 Basics of stochastic calculus

tervals and look at approximating sums Sn D

n X

f .Bsj /.B tj C1 B tj /;

sj 2 Œtj ; tj C1 /:

j D1

The pathwise (that is, in X pointwise) limit of the random variable Sn does not exist due to the unbounded variation of Brownian paths, thus it cannot be defined as a Stieltjes integral. However, as it will be seen below, when Sn is regarded as an element of L2 .X ; d W /, it does have a limit. But even in this situation we face the peculiar feature that, unlike in the case of Riemann–Stieltjes integral, the limit depends on how the point sj is chosen and a different object is obtained when sj D tj or when sj D tj C1 . First we consider one dimensional standard Brownian motion. Example 2.1. Take a division of the interval Œ0; T into P subintervals of equal length, i.e., choose tj D j T =n. Then in L2 -sense limn!1 jn1 D0 B tj .B tj C1 B tj / D Pn1 1 1 2 2 j D0 B tj C1 .B tj C1 B tj / D 2 .BT C T /. Indeed, a rear2 .BT T / and limn!1 rangement of the terms gives for the first sum n1 X j D0

n1 1 1X B tj .B tj C1 B tj / D BT2 .B tj C1 B tj /2 : 2 2 j D0

By Proposition 2.9 the latter sum converges in L2 sense to the quadratic variation VT2 D T , thus the first claim follows. The other limit can be computed similarly. To remove the problem of dependence on the intermediary point, we restrict here to the choice sj D tj , i.e., choose the integrand to be adapted to the natural filtration .F tB / t0 of standard Brownian motion. We call ..t // t0 a random step process if there is a sequence 0 D t0 < < tn and random variables f0 ; : : : ; fn 2 L2 .X ; d W / such that fj are F tjB -adapted and .t; !/ D

n1 X

fj .!/1Œtj ;tj C1 / .t /:

(2.3.1)

j D0 2 will be used for the space of random step processes. The notation Mstep

Definition 2.18. We denote by M 2 .S; T / the class of functions f W RC X ! R such that .1/ f 2 B.RC / B.X /=B.R/ .2/ f .t; !/ is F tB -adapted RT .3/ EŒ S jf .t; !/j2 dt < 1.

36


Let .t; !/ be (2.3.1) with S D t0 < t1 < < tn D T . Define Z

T

S

.t; !/dB t D

n X

fj .!/ .Btj C1 .!/ Btj .!//:

(2.3.2)

j D0

Lemma 2.25. Let .t; !/ be as in (2.3.2). Then

ˇ Z ˇ Ex ˇˇ

T S

ˇ2

Z ˇ .t; !/dB t ˇˇ D Ex

T S

j.t; !/j2 dt :

(2.3.3)

RT Proof. Denote for a shorthand I./ D S .t; !/dB t , Bj D B tj C1 B tj and tj D tj C1 tj . Then clearly Ex ŒBj D 0, Ex Œ.Bj /2 D tj . We have n1 X

jI./j2 D

fj2 .Bj /2 C 2

j D0

X

fj fk Bj Bk :

j 0 lim P .jXs X t j > "/ D 0:

s!t

Remark 2.3 (Càdlàg paths). It can be shown that every Lévy process has a version with paths t 7! X t which are right continuous and have left limits; functions, and thereby random processes, of this type are called càdlàg retaining the original French terminology “continue à droite, limitée à gauche.” The space of Rd -valued càdlàg functions on an interval I R is denoted by D.I I Rd /; it is a linear space with respect to pointwise addition and multiplication by real numbers. There is a close relationship between infinitely divisible distributions and Lévy processes. Theorem 2.40. .1/ Let .X t / t0 be a Lévy process. Then for every t the distribution t of X t is infinitely divisible. .2/ Conversely, for every infinitely divisible distribution on Rd with a triplet .b; A; /, where b 2 Rd , A is a d d non-negative definite symmetric matrix and a Lévy measure, there is a Lévy process .X t / t0 on .; F ; P / such that the distribution of X1 is and the characteristic triplet of X t is .t b; tA; t /, i.e., EP Œe iuX t D e t .u/ ;

(2.4.7)

where is given by (2.4.6). The vector b is called drift term, the matrix A diffusion matrix and the triplet .b; A; / the characteristics of the Lévy process .X t / t0 , while given by (2.4.6) is called Lévy symbol. Example 2.13. By Theorem 2.40 the processes in Example 2.12 are Lévy processes. The Lévy triplet of Brownian motion with drift (Itô process) is .b; A; 0/ and its Lévy symbol is .u/ D i b u 12 u Au. For a Poisson process with intensity > 0 the Lévy triplet is .0; 0; ı1 /, where ı1 is one dimensional Dirac measure, and the Lévy symbol is .u/ D .e iu 1/. The Cauchy process is a specific case of the class in the next example.

57

Section 2.4 Lévy processes

Example 2.14 (Stable processes). A Lévy process .X t / t0 is called a stable process d

if for any a > 0 there exist b > 0 and c 2 R such that Xat D bX t C ct , and a strictly stable process if c D 0. This definition originates from an extension of the scaling property of Brownian motion, see Proposition 2.7 (3). From the definition it can be seen that for a stable process there exist 0 < ˛ 2 and b t 2 Rd such that d X t D t 1=˛ X1 C b t ; for a strictly stable process one can take b t D 0. ˛ is called the index of the stable process. For a stable process either ˛ D 2 and its Lévy triplet is .b; A; 0/, i.e., it is a Gaussian process with mean b and covariance matrix A, or 0 < ˛ < 2 and its Lévy triplet is .b; 0; / with Lévy measure Z Z 1 dr .E/ D .d / 1E .r / 1C˛ ; (2.4.8) r S d 1 0 where S d 1 is the d 1-dimensional unit sphere centered in the origin. The above integral can be computed; for d D 1 the calculation gives c1 c2 .dy/ D 1.0;1/ C 1.1;0/ dy; c1 ; c2 0: (2.4.9) y 1C˛ .y/1C˛ d

A stable process with index 0 < ˛ 2 is rotation invariant if RX t D X t , for all R 2 O.Rd /, the orthogonal group. In this case the Lévy symbol of the process becomes .u/ D cjuj˛ , with some c > 0, and for 0 < ˛ < 2 the Lévy measure has uniform distribution on Sd 1 . In particular, the rotation invariant 1-stable process is the Cauchy process.

2.4.2 Markov property of Lévy processes In this section we show that Lévy processes are Markov processes possessing Feller transition kernels, and derive their generators. Let .X t / t0 be a Lévy process on .; F ; P /. Let t be the infinitely divisible distribution of X t and define p.s; t; x; A/ by p.s; t; x; A/ D ts .A x/

(2.4.10)

for t s 0, x 2 Rd and A 2 B.Rd /, where A x D ¹y xjy 2 Aº. Note that t .A x/ D P .X t C x 2 A/. By the definition of a Lévy process we have p.s; t; x; A/ D P .X t Xs 2 A x/ and p.s; t; x; A/ is a probability transition kernel. The Chapman–Kolmogorov identity results from the semigroup property ts ut D us ;

s < t < u:

(2.4.11)

Set p.s; t; x; A/ D p.t s; x; A/:

(2.4.12)

58


Proposition 2.41. A Lévy process .X t / t0 is a stationary Markov process and its probability transition kernel is given by p.t; x; A/ under (2.4.12). Proof. By the independent increment property of the process its finite dimensional distributions can be written as P .X0 2 A0 ; : : : ; Xn 2 An / Z n n Y Y D 1Ai .xi / p.ti ti1 ; xi1 ; dxi /P 0 .dx0 /; R.nC1/d iD0

(2.4.13)

iD1

where P 0 .dx/ D ı.x/ is Dirac measure with mass at x D 0. Thus by Proposition 2.17, .X t / t0 is a Markov process with probability transition kernel p.t; x; A/. Theorem 2.42 (C0 -semigroup on C1 .Rd /). The probability transition kernel p.t; In particular, P t f .x/ D R x; A/ of a Lévy process is a Feller transition kernel. d Rd p.t; x; dy/f .y/ defines a C0 -semigroup on C1 .R /. Proof. The fact p.t; x; A/ 1 is trivial. We show that P t f 2 C1 .Rd / for f 2 C1 .Rd /. Let xn ! x as n ! 1. By the dominated convergence theorem we have Z lim p.t; 0; dy/f .xn C y/ D P t f .x/ lim P t f .xn / D n!1

Rd n!1

Z

and lim P t f .x/ D

jxj!1

lim p.t; 0; dy/f .x C y/ D 0:

Rd jxj!1

Thus P t f 2 C1 .Rd /. To conclude, we show that lim t!0 P t f D f for f 2 C1 .Rd / in the sup-norm. Since f is uniformly continuous, for any " > 0 there exists an r > 0 such that supx2Rd jf .x C y/R f .x/j < " for all jyj < r. Moreover, by stochastic continuity of the Lévy process jyj>r p.t; 0; dy/ D P .jX t j > r/ ! 0 R as t ! 0. Thus there exists t0 such that jyj>r p.t; 0; dy/ < " for 0 < t < t0 . Hence ˇ ˇZ ˇ ˇ .f .x C y/ f .x//p.t; 0; dy/ˇˇ sup jP t f .x/ f .x/j D sup ˇˇ Rd x2Rd x2Rd ˇ ˇZ Z ˇ ˇ D sup ˇˇ .f .x C y/ f .x//p.t; 0; dy/ C .f .x C y/ f .x//p.t; 0; dy/ˇˇ x2Rd

Z

"

jyjr

p.t; 0; dy/ C "kf k1

for 0 < t < t0 . Thereby lim t!0 P t f D f follows.

59

Section 2.4 Lévy processes

This amply illustrates the fact that the class of Feller transition kernels is far from empty. When the infinitely divisible distribution t of .X t / t0 satisfies t .A/ D t .A/;

(2.4.14)

P t can be extended to a C0 -semigroup on L2 .Rd /. Theorem 2.43 (C0 -semigroup on L2 .Rd /). Suppose that (2.4.14) holds. Then P t can be extended to a symmetric C0 -semigroup on L2 .Rd /. R Proof. Suppose that f; g 0 and Rd g.x/.P t f /.x/dx < 1. Since Z Z Z dxg.x/.P t f /.x/ D dxg.x/ f .x C y/p.t; 0; dy/ Rd Rd Rd Z Z dxg.x y/ f .x/p.t; 0; dy/; D Rd

Rd

(2.4.14) implies that Z Z Z dxg.x/.P t f /.x/ D dxg.x C y/ f .x/p.t; 0; dy/ Rd Rd Rd Z D dx.P t g/.x/f .x/: Rd

(2.4.15)

Let f 2 L2 .Rd / \ C1 .Rd /. Thus by Schwarz inequality and the symmetry of equality (2.4.15) we have Z Z Z j.P t f /.x/j2 dx P t .jf j2 /.x/dx D 1 P t .jf j2 /.x/dx d d d R R R Z D P t 1 jf j2 .x/dx: Rd

P t 1 D 1 implies that kP t f k kf k. Since L2 .Rd / \ C1 .Rd / is dense, the inequality kP t f k kf k can be extended to f 2 L2 .Rd /, and the theorem follows. By the Hille–Yoshida theorem (see Proposition 3.25) the semigroup ¹P t W t 0º associated with a Lévy process is of the form P t D e tL with a generator L on C1 .Rd /. The operator L is closed on C1 .Rd / endowed with the sup-norm. The semigroup ¹P t W t 0º can be also viewed as a pseudo-differential operator with symbol e t .k/ (see below). Denote by S .Rd / Schwartz space over Rd , i.e., the space of infinitely differentiable rapidly decreasing functions on Rd . Proposition 2.44 (Generator of a Lévy process). Let .X t / t0 be a Lévy process with characteristics .b; A; / and symbol . Let ¹P t W t 0º be the associated C0 semigroup on C1 .Rd / and L its generator, i.e., P t D e tL . Then the following hold:

60


.1/ For t 0, f 2 S .Rd / and x 2 Rd , Z .P t f /.x/ D .2/d=2

e ikx e t .k/ fO.k/d k:

Rd

.2/ For f 2 S .Rd / and x 2 Rd , Lf .x/ D .2/

d=2

(2.4.16)

Z e ikx .k/fO.k/: Rd

(2.4.17)

In particular 1 L D r Ar C b r C L ; 2 where

(2.4.18)

Z

L f .x/ D

Rd n¹0º

.f .x C y/ f .x/ y rf .x/1¹jyj 0 for which dim Ran EA . "; C"/ < 1, then is said to be an element of the discrete spectrum, denoted Specd .A/. If 2 Spec.A/ is such that dim Ran EA . "; C "/ D 1 for all " > 0, then is in the essential spectrum, denoted Specess .A/. Note that the discrete and the essential spectra form a partition of the spectrum. Moreover, Specess .A/ is a closed set, while Specd .A/ need not be so. Furthermore, the following facts hold. Proposition 3.7. 2 Specd .A/ if and only if both of the following properties hold: .1/ is an isolated point of Spec.A/, i.e., there exists " > 0 such that Spec.A/ \ . "; C "/ D ¹ º .2/ is an eigenvalue of finite multiplicity, i.e., dim¹' 2 KjA' D 'º < 1. An eigenvalue of arbitrary multiplicity which is not an isolated point in the spectrum is called an embedded eigenvalue. Example 3.1 (Multiplication operator). Let V be a Lebesgue measurable real function and define a multiplication operator by .V '/.x/ D V .x/'.x/ and domain D.V / D ¹' 2 L2 .Rd /j Vf 2 L2 .Rd /º. Note that it is densely defined. Since V is real-valued, it is symmetric, and in fact it is a self-adjoint operator. Example 3.2 (Laplace operator). Recall that the weak partial derivative @j ' of ' 2 L1loc .Rd / is a function h 2 L1loc .Rd / such that Z

Z Rd

h.x/f .x/dx D

Rd

'.x/@j f .x/dx;

f 2 C01 .Rd /;

80

Chapter 3 Feynman–Kac formulae

and is the j th component of the gradient operator in weak sense. This notion reduces to the partial derivative (gradient) known from elementary calculus whenever ' 2 C 1 .Rd /. The Laplacian in L2 .Rd / is the symmetric operator defined by W ' 7!

d X

@j2 '

(3.1.20)

j D1

for ' 2 C01 .Rd /. It is seen that T D dC 1 .Rd / is essentially self-adjoint. We 0

denote the closure of T by the same symbol, i.e., D T . The domain of is the Sobolev space D..1=2// D H 2 .Rd / D ¹f 2 L2 .Rd /jjkj2 fO 2 L2 .Rd /º; where fO and fL denote the Fourier transform and inverse Fourier transform of f , respectively. Moreover, f .k/ D jkj2 fO.k/ holds for every f 2 H 2 .Rd /. Since the range of the multiplication operator jkj2 is the positive semi-axis, we have that has continuous spectrum and Spec./ D Specess ./ D Œ0; 1/, in particular, is a positive operator.

1

Example 3.3 (Fractional Laplacian). The operator with domain H ˛ .Rd / D ¹f 2 ˇ L2 .Rd /ˇjkj˛ fO 2 L2 .Rd /º, 0 < ˛ < 2, defined by ./˛=2 f .k/ D jkj˛ fO.k/, is called fractional Laplacian with exponent ˛=2. It is essentially self-adjoint on C01 .Rd /, and its spectrum is Spec../˛=2 / D Specess ../˛=2 / D Œ0; 1/. This is a case of a pseudo-differential operator. A related operator is the relativistic fractional Laplacian . C m2=˛ /˛=2 m, with m > 0.

4

3.1.4 Compact operators Finally we discuss compact operators. Definition 3.12 (Compact operator). A bounded operator A W K ! H is a compact operator whenever A takes bounded sets in K into pre-compact sets in H . Equivalently, A is compact if and only if for every bounded sequence .'n /n2N , .A'n /n2N has a subsequence convergent in H . R Example 3.4. Let k 2 L2 .Rd Rd /. Then A W f 7! Rd k.x; y/f .y/dy is a compact operator on L2 .Rd /. 1=2 .y/ for d 3 with 0 V 2 Example 3.5. Let k.x; y/ D V 1=2 .x/jx yj2d RV d=2 d L .R /. Then the integral operator A W f 7! k.x; y/f .y/dy is compact. In the case of d D 3 this is proven by the Hardy–Littlewood–Sobolev inequality ˇZ Z ˇ ˇ ˇ ˇ ˇ C kf kp kgkq f .x/jx yj g.y/dxdy (3.1.21) ˇ ˇ Rd

Rd

Section 3.1 Schrödinger semigroups

for < d and

1 p

C

d

C

1 q

81

D 2, with some constant C . See Lemma 3.47.

Some important properties of compact operators are summarized below. Proposition 3.8. Let A be a compact operator. .1/ Spec.A/ n ¹0º Specd .A/. .2/ Spec.A/ is a finite set or a countable set without accumulation points. .3/ Let C.K/ be the set of compact operators from K to itself. Then C.K/ is an ideal of B.K; K/, i.e., AB; BA 2 C.K/ for A 2 C.K/ and B 2 B.K; K/. Furthermore, K.K/ D B.K; K/=C.K/ is simple, i.e., K.K/ has no nontrivial ideal. .4/ C.K/ is closed in the uniform operator topology. The space K.K/ is called Calkin algebra. It is useful to have criteria guaranteeing that the spectrum of a self-adjoint operator is purely discrete. Proposition 3.9. Let A be a self-adjoint operator and bounded from below. Then properties .1/–.4/ below are equivalent: .1/ . A/1 is a compact operator for some 2 .A/. .2/ . A/1 is a compact operator for all 2 .A/. .3/ The set ¹' 2 D.A/jk'k < 1; kA'k bº is compact for all b. .4/ The set ¹' 2 D.jAj1=2 /jk'k < 1; kjAj1=2 'k bº is compact for all b. We will consider perturbations of given operators, and address the question of invariance of the essential spectrum of a self-adjoint operator under perturbations. Definition 3.13 (Relative compactness). Let A be self-adjoint. An operator B such that D.A/ D.B/ is called relatively compact with respect to A if and only if C.i C A/1 is compact. Proposition 3.10. Let A be self-adjoint and B be relatively compact with respect to A. Then A C B with domain D.A/ is closed and Specess .A C B/ D Specess .A/.

3.1.5 Schrödinger operators Definition 3.14 (Schrödinger operator). Let V be a real-valued multiplication operator in L2 .Rd /. The linear operator 1 H D CV 2

(3.1.22)

D.H / D D..1=2// \ D.V /

(3.1.23)

with the dense domain

is called a Schrödinger operator.

82


By the definition of it is readily seen that every ' 2 D..1=2// is a bounded continuous function whenever d 3. Indeed, by the estimate k'k O 1 D k.1 C jkj2 /1 .1 C jkj2 /'k O 1 k.1 C jkj2 /1 k2 k.1 C jkj2 /'k O 2 < 1; the Fourier transform of ' lies in L1 .Rd /. Thus ' is a continuous bounded function by the Riemann-Lebesgue lemma. Moreover, by the scaling 'Or .k/ D r d '.rk/ O we d 4 have k'k O 1 cr 2 kjkj2 'k O 2 C cr d=2 k'k O 2 , with a constant c independent of r. Since d 3, for arbitrary a > 0 there exists b such that k'k1 k'k O 1 ak'k2 C bk'k2 :

(3.1.24)

This bound is useful in estimating the relative bound of a given potential V with respect to the Laplacian. We are interested in defining Schrödinger operators as self-adjoint operators. One reason is that then the spectrum is guaranteed to be real, in match with the uses of quantum mechanics. Secondly, since the solution of the Schrödinger equation (3.1.1) is given by e itH '0 for '.x; 0/ D '0 .x/, to make sure that e itH is a unitary map the question has to be addressed whether the symmetric operator H is self-adjoint or essentially self-adjoint. From the point of view of quantum mechanics, distinct self-adjoint extensions lead to different time-evolutions, however, since several selfadjoint extensions of H may exist, a particular one must be chosen to describe the evolution of the system. Since for any self-adjoint extension K of H it is true that K H , an important step is to find a core of H as then T D H dD is the unique self-adjoint extension of H dD . In general it is difficult to show self-adjointness or essential self-adjointness of a Schrödinger operator. Here we review some basic criteria, however, as we are interested in defining and studying the properties of perturbations of the Laplacian, first we give a notion of perturbation. Definition 3.15 (Relative boundedness). Given an operator A, the operator B is called A-bounded whenever D.A/ D.B/ and kBf k akAf k C bkf k

(3.1.25)

with some numbers a; b 0, for all f 2 D.A/. In this case a is called a relative bound. Moreover, if the relative bound a can be chosen arbitrarily small, B is said to be infinitesimally small with respect to A. A classic result of self-adjoint perturbations of a given self-adjoint operator is offered by the following theorem. Theorem 3.11 (Kato–Rellich). Suppose that K0 is a self-adjoint operator, and KI is a symmetric K0 -bounded operator with relative bound strictly less than 1. Then K D K0 C KI is self-adjoint on D.K0 / and bounded from below. Moreover, K is essentially self-adjoint on any core of K0 .

83


Applied to Schrödinger operators, the natural class of V for the Kato–Rellich theorem is Lp .Rd / C L1 .Rd /. Example 3.6. Let d D 3 and V D V1 C V2 2 L2 .R3 / C L1 .R3 /. Since kV 'k2 kV1 k2 k'k1 C kV2 k1 k'k2 and k'k1 ak'k2 C bk'k2 for arbitrary a > 0 and some b > 0 by (3.1.24), we have kV 'k2 akV1 k2 k'k2 C .b C kV2 k1 /k'k2 : Thus by Theorem 3.11 the Schrödinger operator .1=2/ C V is self-adjoint on D..1=2// and bounded from below. Moreover, it is essentially self-adjoint on any core of . In particular, .1=2/jxj1 is self-adjoint on D..1=2// for d D 3. Example 3.7. The previous example can be extended to arbitrary d -dimension. Let ´ p D 2; d 3 V 2 Lp .Rd / C L1 .Rd / (3.1.26) p > d=2; d 4: In the case d 4 consider V 2 Lp .Rd /. Set q D 2p=.p 2/ and r D 2p=.p C2/. By the Hölder and the Hausdorff–Young inequalities it follows that kVf k2 kV kp kf kq C kV kp kfOkr : Thus kfOkr D k.1 C "jkj2 /1 .1 C "jkj2 /fOkr k.1 C "jkj2 /1 kp k.1 C "jkj2 /fOk2 C 0 "d=.2p/ ."kf k2 C kf k2 /: For a given ı > 0 by taking a sufficiently small " we see that kVf k2 ık.1=2/f k C Cı kf k2 : Thus V is infinitesimally small with respect to . In particular, H is self-adjoint on D..1=2// and essentially self-adjoint on any core of . From the discussion in Example 3.7 we conclude the following result. Proposition 3.12. Let V 2 Lp .Rd /, with p D 2 for d 3, and p > d=2 for d 4. Then there exists a constant C such that kVf k2 C kV kp k..1=2/ C 1/f k for f 2 D..1=2//.

(3.1.27)

84


The Kato–Rellich theorem fails to be useful in investigating the self-adjointness of Schrödinger operators with a polynomial potential such at V .x/ D jxj2 . In this case the following can be applied. Theorem 3.13 (Kato’s inequality). Let u 2 L1loc .Rd / and suppose the distributional Laplacian u 2 L1loc .Rd /. Then the inequality juj 0 such that kV 1=2 f k "k./1=2 f k C b" kf k for f 2 Q./. Therefore there exists a self-adjoint operator K such that .f; Kg/ D .f; . C V /g/ for f; g 2 C01 .R3 /. R Example 3.11. Let q.f; g/ D .rf; rg/ C Rd V .x/fN.x/g.x/dx. The natural class of V such that inf¹q.f; f /j kf k2 D 1; f 2 H 1 .Rd /º > 1 is 8 d=2 d 1 d ˆ d 3 0; d D 2 ˆ : 1 1 1 1 L .R / C L .R /; d D 1: Note that in theR case of (3.1.29) the fact that f 2 H 1 .Rd / implies by the Sobolev inequality that Rd jV .x/jf .x/j2 dx kV kd=2 kf k22d=.d 2/ C kV kd=2 krf k22 < 1. Since q is semibounded on the form domain H 1 .Rd /, there exists a self-adjoint operator associated with q.

The relationship between the convergence of sequences of quadratic forms and associated self-adjoint operators is the following. Let q1 and q2 be symmetric quadratic forms bounded from below. q1 q2 means that Q.q1 / Q.q2 / and q1 .f; f / q2 .f; f /, for all f 2 Q.q1 /. A sequence .qn /n2N of symmetric quadratic forms bounded from below is non-increasing (resp. non-decreasing) if qn qnC1 (resp. qn qnC1 ), for all n. Theorem 3.18 (Monotone convergence theorem for forms). .1/ Case of non-increasing sequence: Let .qn /n2N be a non-increasing sequence of densely defined, closed symmetric quadratic forms uniformly bounded from below; qn

with some constant , and let Hn be the self-adjoint operator associated with qn . Let q1 be the symmetric quadratic form defined by [ q1 .f; f / D lim qn .f; f /; Q.q1 / D Q.qn /: (3.1.30) n!1

n

Let H be the self-adjoint operator associated with qN 1 , the closure of q1 . Then it follows that s-limn!1 .Hn /1 D .H /1 , for ƒ C 1: Then the product 1ƒ .x/1ƒ .i r/ is an integral operator with kernel k.x; y/ D .2/d=2 1ƒ .x/1O ƒ .y x/ 2 L2 .Rd Rd /. Thus 1ƒ .x/1ƒ .i r/ is compact by Example 3.4. Since Q.x/1ƒ .x/ ! Q.x/ and 1ƒ .i r/P .i r/ ! P .i r/ as ƒ ! 1 uniformly, it is seen that Q.x/P .i r/ D lim Q.x/1ƒ .x/1ƒ .i r/P .i r/: ƒ!1

Since Q.x/P .i r/ is a uniform limit of a sequence of compact operators, it is compact by Proposition 3.8. Theorem 3.22 (Decaying potentials). Let V be such that ´ p D 2; V 2 Lp .Rd / C L1;0 .Rd / p > d=2;

d 3; d 4:

Then H D .1=2/ C V is essentially self-adjoint on C01 .Rd / and Specess .H / D Specess ./ D Œ0; 1/. Proof. Set H0 D .1=2/. Essential self-adjointness follows by the Kato–Rellich theorem. By Proposition 3.21, V .H0 C 1/1 is compact for V 2 L1;0 .Rd /. Thus it suffices to show that V 2 Lp .Rd / is relatively compact with respect to H0 . Let V 2 Lp .Rd /. For each " > 0 we decompose V D V" C W" , where kV" kp < " and W" 2 L1;0 .Rd /. Indeed let 8 ´ ˆ jgj < N ƒ : N; g N:

89


Then V is decomposed as V D .V 1ƒ /.N / C V .V 1ƒ /.N / : „ ƒ‚ … „ ƒ‚ … N DVƒ

(3.1.32)

N DWƒ

Thus VƒN 2 L1;0 .Rd /, WƒN 2 Lp .Rd / and kWƒN kp < " for sufficiently large ƒ and N . Furthermore, VƒN ! V in Lp as ƒ; N ! 1. We have by (3.1.27) k.V .1 C H0 /1 VƒN .1 C H0 /1 /f k C kV VƒN kp kf k2 : Thus VƒN .1 C H0 /1 ! V .1 C H0 /1 in the uniform topology and VƒN .1 C H0 /1 is compact since VƒN 2 L1;0 .Rd /. Hence V .1 C H0 /1 is compact, and the proof is complete. Example 3.12. Two notable examples of spectra of Schrödinger operators are: .1/ Harmonic oscillator: Choose, for simplicity, d D 1, and let ! > 0. The 2 Schrödinger operator H with V .x/ D !2 x 2 !2 describes the harmonic oscillator. In this case the spectrum is purely discrete Spec.H / D Specd .H / D ¹!njn 2 N [ ¹0ºº. .2/ Hydrogen atom: Choose d D 3, and let > 0. The Schrödinger operator H with V .x/ D jxj describes the hydrogen atom. In this case Specess .H / D 2

Œ0; 1/, and Specp .H / D Specd .H / D ¹ 2n 2 j n 2 Nº.

3.1.8 Strongly continuous operator semigroups We conclude this section by a summary of basic facts on operator semigroups. Definition 3.17 (C0 -semigroup). The one-parameter family of bounded operators ¹S t W t 0º on a Banach space B is said to be a strongly continuous one-parameter semigroup or C0 -semigroup whenever .1/ S0 D 1 .2/ Ss S t D SsCt .3/ S t is strongly continuous in t , i.e., s-lim t!0 S t D S0 D 1. Every C0 -semigroup is generated by a uniquely associated operator. Definition 3.18 (Generator). Let ¹S t W t 0º be a C0 -semigroup on a Banach space B. Its generator is defined by 1 Tf D s-lim .S t f f / t!0 t ˇ ® ¯ with domain D.T / D f 2 B ˇs-lim t!0 1t .S t f f / exists .

90


Note that T is densely defined, closed, and determines S t uniquely. Proposition 3.23. Let ¹S t W t 0º be a C0 -semigroup with generator T . The following properties hold for all t 0: .1/ S t D.T / D.T / and

d S ' dt t

D T St '

.2/ there exist M 1 and a > 0 such that kS t k M e at Rt .3/ 0 Sr 'dr 2 D.T / for all ' 2 B Rt Rt .4/ S t ' ' D T 0 Sr 'dr, moreover S t ' ' D 0 Sr T 'dr if ' 2 D.T / .5/ if T is a bounded operator, then D.T / D B and S t is uniformly continuous. With a given self-adjoint operator A, the semigroups e itA and e tA can be defined through the spectral resolution (3.1.19). In particular, ¹e itA W t 2 Rº is a strongly continuous one-parameter unitary group, and ¹e tA W t 0º is a strongly continuous one-parameter semigroup whenever A is bounded from below. Example 3.13 (Heat kernels). Some specific choices of interest of generators giving rise to strongly continuous semigroups are as follows: .1/ T D 12 gives e .1=2/t .x; y/ D .2 t /d=2 exp.jx yj2 =2t /: .2/ T D ./˛=2 , 0 < ˛ < 2, has the integral kernel Z 1 ˛=2 ˛ e t./ .x; y/ D e tjkj Cik.xy/ d k; d .2/ Rd moreover for ˛ D 1 the explicit formula d C1 t 1 t./1=2 e .x; y/ D .d C1/=2 2 2 .t C jx yj2 /.d C1/=2 holds. .3/ T D . C m2 /1=2 m, m > 0, has the integral kernel e t..Cm

2 /1=2 m/

.x; y/

1 t D p d 2 .2/ t C jx yj2

Z

p 2 2 2 2 e mt .t Cjxyj /.jkj Cm / d k:

Rd

The cases above are examples of Feller semigroups associated with Brownian motion, ˛-stable processes and relativistic stable processes, respectively. Conversely, given a semigroup here are some basic results on their generators, of which we will often make use.

91


Proposition 3.24 (Stone’s Theorem). Let ¹U t W t 2 Rº be a strongly continuous oneparameter unitary group on a Hilbert space. Then there exists a unique self-adjoint operator A such that U t D e itA , t 2 R. Proposition 3.25 (Hille–Yoshida theorem). A linear operator T is the generator of a C0 -semigroup ¹S t W t 0º on a Banach space if and only if .1/ T is a densely defined closed operator, .2/ there exists a 2 R such that .a; 1/ .T / and k.a T /m k

M ; . a/m

> a;

m D 1; 2; : : : ;

where .T / denotes the resolvent set of T . Proposition 3.26 (Semigroup version of Stone’s theorem). Let S t be a symmetric C0 semigroup on a Hilbert space. Then there exists a self-adjoint operator A bounded from below such that S t D e tA , t 0. Proof. Define the operator A by 1 Af D s-lim .S t 1/f; t!0 t

² ˇ ³ ˇ 1 D.A/ D f ˇˇ s-lim .S t 1/f exists : t!0 t

By the Hille–Yoshida theorem and the fact that S t is symmetric, A is a closed symmetric operator and for some a 2 R the resolvent .A/ contains .1; a/. It is a standard fact that the spectrum of a closed symmetric operator is one of the following cases: (a) the closed upper half-plane, (b) the closed lower-half plane, (c) the entire plane, (d) a subset of R. Hence A must be (d), which implies that A is self-adjoint and bounded from below. Since a given generator gives rise to a unique C0 -semigroup, we have that S t D e tA . In case when S t is not a semigroup yet s-lim t!0 t 1 .S t 1/ does exist, one can use the following result. Proposition 3.27. Let A be a positive self-adjoint operator and %.t / a family of selfadjoint operators with 0 %.t / 1. Suppose that S t D t 1 .1 %.t // converges to A as t # 0 in strong resolvent sense. Then s-lim %.t =n/n D e tA :

n!1

Proof. s-limn!1 exp tS t=n D e tA follows by strong resolvent convergence of S t=n to A. Then it suffices to show that s-lim .exp.tS t=n / %.t =n/n / D s-lim Gn .tS t=n / D 0;

n!1

n!1

92


where ´ e x .1 x=n/n ; 0 x n; Gn .x/ D x x n: e ; It is easily seen that limn!1 kGn k1 D 0 and the proposition follows. We now define semigroups generated by Schrödinger operators. Definition 3.19 (Schrödinger semigroup). The one-parameter operator semigroup ¹e tH W t 0º defined by a Schrödinger operator H is a Schrödinger semigroup. Due to the additive structure of Schrödinger operators the following result will also be useful. Proposition 3.28 (Trotter product formula). Let A and B be positive self-adjoint operators such that A C B is essentially self-adjoint on D.A/ \ D.B/. Then s-lim .e .t=n/A e .t=n/B /n D e t.ACB/ ;

n!1

(3.1.33)

where A C B denotes the closure of .A C B/dD.A/\D.B/ . Proof. Let %.t / D e .t=2/A e tB e .t=2/A . Then .e .t=n/A e .t=n/B /n D e .t=2n/A %.t =n/n1 e .t=2n/A e .t=n/B holds. Since e tA and e tB are uniformly bounded in t 0 and strongly converge to identity as t ! 0, it suffices to show that s limn!1 %.t =n/n1 D e tC , where C D A C B. We have that t 1 .1 %.t // is a bounded self-adjoint operator and t 1 .1 %.t // D e .t=2/A e tB t 1 .1 e .t=2/A / C e .t=2/A t 1 .1 e tB / C t 1 .1 e .t=2/A / ! C as t ! 1, for 2 D.A/ \ D.B/, which implies that t 1 .1 %.t // ! C in strong resolvent sense. Thus s-limn!1 %.t =n/n D e tC follows by Proposition 3.27. Furthermore, s-lim .%.t =n/n1 e tC /

n!1

D s-lim %.t =n/n1 .1 %.t =n// C s-lim .%.t =n/n e tC / D 0: n!1

Hence the proposition follows.

n!1

93

Section 3.2 Feynman–Kac formula for external potentials

The Trotter product formula can be extended to more general cases. Proposition 3.29. Let A and B be positive self-adjoint operators in a Hilbert space H with form domains Q.qA / and Q.qB /. Denote by PAB the projection to Q.qA / \ Q.qB /. Then P

s-lim .e .t=n/A e .t=n/B /n D e t.A C B/ PAB ;

n!1

(3.1.34)

P B is the quadratic form sum of A and B, i.e., the self-adjoint operator in where A C PAB H associated with the densely defined closed quadratic form f 7! kA1=2 f k2 C kB 1=2 f k2 .

3.2

Feynman–Kac formula for external potentials

3.2.1 Bounded smooth external potentials We begin discussing Feynman–Kac formulae by assuming that V 2 C01 .Rd /, which is the simplest case. For such a choice of potential the Schödinger operator H is self-adjoint on D.H / D H 2 .Rd / and bounded from below by the Kato–Rellich theorem. Therefore the semigroup e tH can be defined by the spectral resolution and the solution of (3.1.7) with initial condition g is given by f .x; t / D .e tH g/.x/. We will denote Brownian motion throughout by .B t / t0 and its natural filtration .F t / t0 as discussed in Chapter 2. Theorem 3.30 (Feynman–Kac formula for Schrödinger operator with smooth external potentials). Let V 2 C01 .Rd /. Then for f; g 2 L2 .Rd /, Z Rt Ex Œf .B0 /e 0 V .Bs / ds g.B t /dx: (3.2.1) .f; e tH g/ D Rd

In particular, .e tH g/.x/ D Ex Œe

Rt 0

V .Bs / ds

g.B t /:

(3.2.2)

Proof. Define the map K t on L2 .Rd / by .K t f /.x/ D Ex Œe

Rt 0

V .Bs / ds

f .B t /:

(3.2.3)

K t is bounded on L2 .Rd / for each t as kK t f k2 e 2 inf V kf k2 . First we show that ¹K t W t 0º is a symmetric C0 -semigroup on L2 .Rd /. Define BQ s D B ts B t , d s < t , for a fixed t > 0. By Proposition 2.7 we have BQ s D Bs . Thus Z Rt .f; K t g/ D EŒf .x/e 0 V .Bs Cx/ds g.B t C x/dx Rd

Z Rt Q DE f .x/e 0 V .Bs Cx/ds g.BQ t C x/dx : Rd

94


Changing the variable x to y D BQ t C x, we obtain

Z Rt Q Q f .y BQ t /e 0 V .Bs B t Cy/ds g.y/dy .f; K t g/ D E Rd Z Rt D EŒf .y C B t /e 0 V .B t s Cy/ds g.y/dy Rd

D .K t f; g/;

(3.2.4) Rt

i.e., K t is symmetric. Write now Z t D e 0 V .Bs /ds . The semigroup property follows directly from the Markov property of Brownian motion: .Ks K t f /.x/ D Ex ŒZs EBs .Z t f .B t // D Ex ŒEx ŒZs e

Rt 0

V .BsCu /du

f .BsCt /jFs

D Ex ŒZsCt f .BsCt / D KsCt f .x/: Strong continuity is implied by kK t f f k E0 Œke

Rt 0

V .CBs /ds

f . C B t / f k ! 0

as t ! 0, by using the dominated convergence theorem. We now know that on both sides of (3.2.1) there is a C0 -semigroup. What remains to show is that their generators are equal, i.e., 1t .K t 1/ converges to H strongly in L2 on D.H /. We will use Itô calculus for this. Put Y t D f .B t /; then the Itô formula gives dZ t D V Z t dt and d Y t D rf dB t C .1=2/f dt . By the product rule (2.3.25), 1 d.Z t Y t / D Z t Vf .B t / C f .B t / dt C Z t rf .B t / dB t 2 D Z t .Hf /.B t /dt C Z t rf .B t / dB t :

(3.2.5)

Taking expectation with respect to R tWiener measure on both sides of the integrated form of (3.2.5) and using that Ex Œ 0 Zs rf dBs D 0, we obtain Z t Z t Ex ŒZs .Hf /.Bs /ds D f .x/ Ks Hf .x/ds: (3.2.6) .K t f /.x/ D f .x/ 0

0

For f 2 D.H / and " > 0 there exists s0 > 0 with kKs Hf Hf k ", for all s < s0 . Thus by (3.2.6), with t < s0 Z t 1 .K t f f / C Hf 1 k Ks Hf C Hf kds " t t 0 is obtained. This shows strong convergence of 1t .K t 1/f to Hf for f 2 D.H / as t ! 0, completing the proof.

95

Section 3.2 Feynman–Kac formula for external potentials

Definition 3.20 (Feynman–Kac semigroup). The symmetric C0 -semigroup ¹K t W t 0º defined by (3.2.3) is called Feynman–Kac semigroup for the given V . It is seen that the solution of (3.1.7), like the solution of the heat equation, can be obtained by running a Brownian motion with an additional exponential weight applied. This gives a probabilistic representation of the solution or, equivalently, of the kernel of the semigroup e tH . Using this picture we see that Brownian paths spending a long time in a region where V is large are exponentially penalized and thus their contribution to the expected value will be accordingly small. The dominant contribution will therefore come from paths that spend most of their time near the lowest values of V . This is a property which will be explored in more detail in Part II of the book.

3.2.2 Derivation through the Trotter product formula By using the Trotter product formula we can give another proof of Theorem 3.30. Alternate proof of Feynman–Kac formula for Schrödinger operator with smooth potentials. Write H0 D .1=2/, let f1 ; : : : ; fn1 2 L1 .Rd / and f0 ; fn 2 L2 .Rd /. By the finite dimensional distributions of Brownian motion, for 0 D t0 t1 tn we have n i hY e .tj tj 1 /H0 fj .x/ D Ex fj .B tj /

n Y j D1

(3.2.7)

j D1

and n Z Y .tj tj 1 /H0 N f0 ; e fj D j D1

Rd

E

x

n hY

i fj .B tj / dx:

j D0

The Trotter product formula combined with (3.2.8) gives .f; e tH g/ D lim .f; .e .t=n/H0 e .t=n/V /n g/ n!1 Z Pn D lim Ex Œf .B0 /g.B t /e .t=n/ j D1 V .B tj=n / dx: n!1 Rd

Since V is continuous and Bs is almost surely continuous in s, Z t n t X V .B tj=n / ! V .Bs /ds n 0 j D1

almost surely as n ! 1. Thus we have Z Rt .f; e tH g/ D Ex Œf .B0 /g.B t /e 0 V .Bs /ds dx: Rd

(3.2.8)

96


Having the Feynman–Kac formula for smooth bounded potentials at hand, we are now interested in extending it to wider classes of V . We present here one immediate extension covering potentials as singular as V .x/ D 1=jxj3 (see Section 3.4.4). Theorem 3.31 (Feynman–Kac formula for Schrödinger operator with singular external potentials). Assume that for V D VC V we have VC 2 L1loc .Rd n S /, where S is a closed set of measure zero, and V is relatively form bounded with respect to P VC .1=2/ with a relative bound strictly less than 1. Define H D .1=2/ C P V . Then (3.2.1) holds. Proof. Suppose that V 2 L1 and Vn D .x=n/.V jn /, where jn D nd .x n/ R 1 d with 2 C0 .R / such that 0 1, Rd .x/dx D 1 and .0/ D 1. Then Vn .x/ ! V .x/ almost everywhere. Thus there is a set N Rd of Lebesgue measure zero such that Vn .x/ ! V .x/ for x … N , and Z Ex Œ1¹Bs 2N º D

Rd

1N .y/… t .y x/dy D 0

for every x 2 N . Then for every x 2 N , Z 0D

Z

t 0

x

dsE Œ1¹Bs 2N º D E

t

x 0

ds1¹Bs 2N º

by Fubini’s theorem. Thus for every x 2 RN , the measure Rof ¹t 2 RC jB t .!/ 2 N º t t is zero, for almost every ! 2 X . Hence 0 Vn .Bs /ds ! 0 V .Bs /ds almost surely x under W , for x … N . Therefore by the dominated convergence theorem, Z x

Rd

dxE Œf .B0 /g.B t /e

Rt 0

Z Vn .Bs /ds

!

Rd

dxEx Œf .B0 /g.B t /e

Rt 0

V .Bs /ds

:

On the other hand, e t..1=2/CVn / ! e t..1=2/CV / strongly, since .1=2/CVn converges to .1=2/ C V on a common core. Thus (3.2.1) holds for V 2 L1 .Rd /. Now assume that VC 2 L1loc .Rd n S/ and V is relatively form bounded with respect to .1=2/. Let ´

VC .x/; VC .x/ < n; VCn .x/ D n; VC .x/ n;

´ Vm .x/ D

V .x/; V .x/ < m; m; V .x/ m:

Then .f; e t..1=2/CVn;m / g/ D

Z Rd

dxEx Œf .B0 /e

Rt 0

Vn;m .Bs / ds

g.B t /:

(3.2.9)

97

Section 3.3 Feynman–Kac formula for Kato-class potentials

Set h D .1=2/. Define the closed quadratic forms 1=2

1=2

1=2

1=2

1=2

1=2

1=2 1=2 qn;m .f; f / D .h1=2 f; h1=2 f / C .VCn f; VCn f / .Vm f; Vm f /;

qn;1 .f; f / D .h1=2 f; h1=2 f / C .VCn f; VCn f / .V1=2 f; V1=2 f /; q1;1 .f; f / D .h1=2 f; h1=2 f / C .VC f; VC f / .V1=2 f; V1=2 f /; whose form domains are respectively Q.qn;m / D Q.h/, Q.qn;1 / D Q.h/ and Q.q1;1 / D Q.h/ \ Q.VC /. Clearly, qn;m qn;mC1 qn;mC2 qn;1 and qn;m ! qn;1 in the sense of quadratic forms on [m Q.qn;m / D Q.h/. Since qn;1 is closed on Q.h/, by Theorem 3.18 the associated positive self-adjoint operators satisfy hCVCn Vm ! hCVCn P V in strong resolvent sense, which implies that for all t 0, exp .t .h C VCn Vm // ! exp .t .h C VCn P V //

(3.2.10)

strongly as m ! 1. Similarly, we have qn;1 qnC1;1 qnC2;1 q1 and qn;1 ! q1;1 in form sense on ¹f 2 \n Q.qn;1 / j supn qn;1 .f; f / < 1º D Q.h/ \ Q.VC /. Hence by Theorem 3.18 again we obtain P VC P VCn P V // ! exp.t .h C P V //; t 0; exp.t .h C

(3.2.11)

in strong sense as n ! 1. By taking first n ! 1 and then m ! 1 it can be proven that both sides of (3.2.9) converge; the left-hand side of (3.2.9) converges by monotone convergence for forms, (3.2.10) and (3.2.11), and the right-hand side by monotone convergence for integrals.

3.3

Feynman–Kac formula for Kato-class potentials

3.3.1 Kato-class potentials The Feynman–Kac formula we have derived so far covers only a limited range of interesting models of mathematical physics. First, the potential was not allowed to grow at infinity. This excludes cases when the particle is trapped in some region of Rd , like under the harmonic potential V .x/ D jxj2 . Secondly, it is desirable to allow for local singularities, especially the case of the Coulomb potential V .x/ D 1=jxj in dimension 3. To include these and further cases of interest, we introduce here a large class of potentials and use it as a space of reference as far as we can. This space is Kato-class which on several counts is a natural function space as it will be made clear below. However, occasionally we will have to leave this space in order to draw stronger conclusions or because the conditions of Kato-class are too strong themselves for a particular statement we want to prove.

98


Our objective in this section is to introduce Kato-class and discuss its basic properties, then define Schrödinger operators with Kato-class potentials as self-adjoint operators, and finally derive a corresponding Feynman–Kac formula. .1/ V W Rd ! R is called a Kato-class potential

Definition 3.21 (Kato-class). whenever

Z lim sup

r!0 x2Rd

Br .x/

jg.x y/V .y/j dy D 0

(3.3.1)

holds, where Br .x/ is the closed ball of radius r centered at x, and 8 ˆ d D 1; <jxj; g.x/ D log jxj; d D 2; ˆ : 2d jxj ; d 3:

(3.3.2)

We denote this linear space by K.Rd /. .2/ V is Kato-decomposable whenever V D VC V with VC 2 L1loc .Rd / and V 2 K.Rd /. In the case of d D 1 we have an equivalent definition of Kato-class which we will use below. V is of Kato-class if and only if Z sup jV .y/jdy < 1: (3.3.3) x2R B1 .x/

R R Suppose (3.3.3) holds. Then supx Br .x/ jx yjjV .y/jdy r supx B1 .x/ jV .y/jdy, for 0 < r < 1, thus V is of Kato-class. Next R suppose that V is of Kato-class. For any " > 0 there exists r > 0 such that supx Br .x/ jx yjjV .y/jdy < ". Then it follows that Z Z r sup jV .y/jdy sup jx yjjV .y/jdy " 2 x r=2jxyjr x r=2jxyjr Z

and therefore sup x

r=2jxyjr

jV .y/jdy

2" : r

Since B1 .0/ can be covered by a finite number of annuli, sayR N , we have that Aj D ¹yjr=2 jxj yj rº with some point xj . Then supx B1 .x/ jV .y/jdy R PN 2N " j supx Aj Cx jV .y/jdy r < 1 holds. Before deriving a Feynman–Kac formula for such potentials it is useful to discuss some basic features of Kato-class. We begin by explaining the definition. Let … t


99

be the transition kernel of Brownian motion given by (2.1.10), and f a non-negative function. By Fubini’s theorem we have

Z 1 Z Ex f .B t /dt D H.x; y/f .y/dy; (3.3.4) 0

R1

Rd

with integral kernel H.x; y/ D 0 … t .x; y/dt . In the limit t ! 1, … t .x/ behaves like .2 t /d=2 , thus H diverges for d D 1 and 2. For d 3, on making the change of variable jx yj2 =.2t / 7! t , H.x; y/ D

1 .d=2 1/ d=2 2 jx yjd 2

(3.3.5)

is obtained, where is the Gamma-function. For lower dimensions we proceed by R1 compensation with a suitable additive term. Let HQ .x; y/ D 0 .… t .x; y/ h.t // dt . For d D 1 we choose h.t / D … t .0; 0/ D .2 t /1=2 yielding Z 1 jxyj2 =.2t/ 1 e 1 Q dt D jx yj H .x; y/ D p p t 2 0 by making the same change of variable as above. For d D 2 write h.t / D … t .0; v/, with unit vector v D 10 . Then Z 1 jxyj2 =.2t/ 1 1 e e 1=.2t/ Q H .x; y/ D dt D log jx yj: 2 0 t This shows that g in Definition 3.21 is basically given by the potential kernels of the Laplacian. The conditions in (3.3.2) limit the growth at infinity (K.Rd /) and the severity of local singularities (L1loc .Rd /). These are illustrated by the following examples. Example 3.14. Let d D 3. Kato-class allows singularities of the type jxj.2"/ with any " > 0, including in particular Coulomb potential V .x/ D 1=jxj, as well as infinitely many isolated singularities of the same type are allowed. In contrast, the function V .x/ D jxj is not contained in K.R3 /. Example 3.15. An application of the Hölder inequality gives an idea about the growth at infinity allowed. Let d D 3; then ˇ Z ˇZ 1=p Z 1=q ˇ ˇ 1 1 q ˇ ˇ jV .y/j dy ˇ dy jV .y/j dy ; ˇ p Br .x/ jx yj Br .x/ jx yj Br .x/ with 1=p C 1=q D 1. The first factor is independent of x and when p < 3, it goes to R zero as r ! 0. Thus V 2 K.R3 / if supx Br .x/ jV .y/jq dy < 1 for some r > 0 and some q > 3=2. A direct calculation shows, for instance, that V .x; y; z/ D jxj˛ e jxj

ˇ .jyj2 Cjzj2 /

100


is in K.R3 / if ˇ > 3˛=2. This shows that Kato-class is essentially an integrability condition; pointwise divergence is allowed as long as the given integrals can be controlled. Example 3.16. Let V 2 Lp .Rd / C L1 .Rd / with p D 1 for d D 1, and p > d=2 for d 2. Then V 2 K.Rd /. If V is Kato-decomposable, the positive part of V is in L1loc .Rd /. We have the following property. Rt Lemma 3.32. Suppose that 0 V 2 L1loc .Rd /. Then W x . 0 V .Bs /ds D 1/ D 0 for almost every x 2 Rd . In particular, Ex Œe x 2 Rd .

Rt 0

V .Bs /ds

> 0, for almost every

Proof. Let N 2 N and ´ 1N .x/ D

1; jxj < N; 0; jxj N:

Since 1N V 2 L1 .Rd /, we have

Z t Z x dxE 1N V .Bs /ds t k1N V kL1 : R3

0

Rt Hence for almost every fixed x 2 Rd , W x . 0 1N V .Bs /ds < 1/ D 1, and then there Rt exits NN , dependent on x, such that W x .NN / D 0 and 0 1N V .Bs .!//ds < 1, for R t every ! 2 X n NN . Let N D [N NN . Note that N depends on x. Thus 0 1N V .Bs .!//ds < 1, for every N 2 N and ! 2 X n N . Since Bs .!/ is continuous in s almost surely, for every ! 2 X nN there exists N D N.!/ 2 N such that V .Bs .!// D 1N V .Bs .!//, for all 0 s t . Thus V .Bs .!// D 1N V .Bs .!// and hence Z t Z t V .Bs .!//ds D 1N V .Bs .!//ds < 1; ! 2 X n N : 0

0

Next we present a first equivalent characterization of Kato-class. Proposition 3.33. A non-negative function V is in K.Rd / if and only if

Z t x lim sup E V .Bs / ds D 0: t#0 x2Rd

0

(3.3.6)


101

Proof. As in (3.3.4), write for t 2 R

Z E

t

x 0

Z

V .Bs / ds D

Z

t

ds Rd

0

dy …s .x y/V .y/:

By the same change of variable as above, Z 0

t

…s .x y/ds D

1 1 d=2 .2/ jx yjd 2

Z

1

jxyj2 =.2t/

e u ud=22 du D H.x; yI t /

readily follows. The key observation now is that H.x; yI t / behaves like g.x y/ in Definition 3.21 whenever jx yj2 =2t is small, and vanishes rapidly when jx yj2 =2t goes to infinity. This is obvious in the cases d 3 and is seen through integration by parts in d D 1 and 2. We give the details only in the simplest case d D 4; the others go by the same arguments, though require further estimates. The convenient feature of the case d D 4 is that then we have an explicit formula for H.x; yI t / and

Z sup Ex x2R4

t 0

Z V .Bs / ds D sup x2R4

R4

jxyj2 1 e 2t V .y/dy 2 .2jx yj/

(3.3.7)

follows. We split up the domain of integration on the right-hand side above into Aq D ¹y j jx yj < t 1=q º and Bq D R4 n Aq for q > 0. Notice that Aq shrinks to ¹xº when t ! 0. If (3.3.6) holds, then the integral over A2 on the right-hand side of (3.3.7) vanishes with t ! 0, uniformly in x. On A2 , exp.jx yj2 =2t / is bounded away from zero, hence V 2 K.R4 / follows. Conversely, assume V 2 K.R4 /. Then by definition and the fact that the negative exponential is bounded from above, we 2 R 1 jxyj 2t obtain lim t#0 supx2R4 A3 .2 jxyj/ V .y/dy D 0: On the other hand, on 2e B3 we have exp.jx yj2 =2t / exp.jx yj=t 1=6 /, thus the right-hand side of (3.3.7) goes to zero on this set by dominated convergence. We present a second equivalent characterization of Kato-class recalling the asymptotic properties of the resolvent of the Laplacian. The kernel of the resolvent operator . C /1 is explicitly given by . C /1 .x; y/ D C .x y/;

(3.3.8)

where d=2

C .x/ D .2/

p !.d 1/=2 p K.d 2/=2 . jxj/; jxj

(3.3.9)

102


with Kd .z/ being the modified Bessel function of the third kind. As jx yj ! 0, 8 2d ; ˆ d 3; <jx yj C .x y/ log jx yj; d D 2; ˆ : C; d D 1; moreover for any ı > 0, lim

sup e jxyj C .x y/ D 0:

!1 jxyj>ı

R Then we readily have supx Rd CE .x y/jV .y/jdy ! 0 as E ! 1 if and only if V 2 K.Rd /, and by comparison with R Definition 3.21 the proposition below follows. We see that . C E/1 jV j.x/ D Rd CE .x y/jV .y/jdy, and we use that ˇ ˇ k. C E/1 jV jk1 D sup ˇ. C E/1 jV j.x/ˇ :

(3.3.10)

x2Rd

Proposition 3.34. V 2 K.Rd / if and only if lim k. C E/1 jV jk1 D 0:

E !1

(3.3.11)

For the sake of completeness, we present a proof of equivalence between (3.3.11) and (3.3.6). Proof. Take in a suitable neighbourhood of 0 and denote H0 D .1=2/. Clearly, Ex ŒjV .Bs /j D .e sH0 jV j/.x/. Then by Laplace transform, 1

..H0 C E/

jV j/.x/ D

XZ

.nC1/T

n0 nT

e sE .e sH0 jV j/.x/ds:

By a change of variable we furthermore have ..H0 C E/1 jV j/.x/ D

X n0

e nTE

Z

T 0

e sE .e .sCnT /H0 jV j/.x/ds:

By the semigroup property and using the integral kernel e tH0 .x; y/ we derive that ..H0 C E/1 jV j/.x/ Z Z T X nTE nTH0 e dy.e /.x; y/ e sE .e sH0 jV j/.y/ds: D n0

Rd

0

103


Take the supremum with respect y. Then ..H0 C E/1 jV j/.x/ X Z T Z sup e sE .e sH0 jV j/.y/ds e nTE 0

y2Rd

Since

R

Rd .e

n0

nTH0 /.x; y/dy

Z .e

nTH0

Rd

.e nTH0 /.x; y/dy: Rd

D 1, we have

1 /.x; y/dy sup 1 e TE y2Rd

Z

T 0

e sE .e sH0 jV j/.y/ds

1 k.H0 C E/1 jV jk1 : 1 e TE

Taking the supremum, this yields .1 e TE /k.H0 C E/1 jV jk1 Z T sup e sE .e sH0 jV j/.y/ds k.H0 C E/1 jV jk1 : y2Rd

0

Furthermore, trivially we have e

TE

Z

T

.e

sH0

0

Then supy2Rd T 0, hence

Z jV j/.y/ds

RT 0

T

e

sE

0

.e

sH0

Z jV j/.y/ds

T 0

.e sH0 jV j/.y/ds:

.e sH0 jV j/.y/ds is bounded by e TE k.H0 C E/1 jV jk1 , for all Z

T

lim sup

T #0 x2Rd

0

.e sH0 jV j/.x/ds k.H0 C E/1 jV jk1 ;

for all E. Therefore, if limE !1 k.H0 C E/1 jV jk1 D 0, then the right-hand side decreases with E, and optimizing over E implies (3.3.6). Conversely, if we assume (3.3.6), then by the above 1 sup T #0 E !1 1 e TE x2Rd

lim k.H0 C E/1 jV jk1 lim lim

E !1

Z

T 0

.e sH0 jV j/.x/ds D 0;

i.e., (3.3.11) holds. The latter characterization is particularly interesting in view of the following property, which through the KLMN theorem implies that if V is Kato-class, then H is self-adjoint at least in form sense.

104


Proposition 3.35. If there exist a; b > 0 and 0 < ı < 1 such that for all 0 < " < 1 p p .f; jV jf / "k f k2 C a exp.b ı /kf k2 ; 8f 2 D. /; (3.3.12) p then V 2 K.Rd /. Conversely, if V is Kato-class, then D. / D.V 1=2 / and V is -form bounded with infinitesimally small relative bound. Proof. Suppose (3.3.12). Since . C E/1 jV j.x/ D

Z

1

dt e tE

Z Rd

0

… t .x y/jV j.y/dy;

p we estimate … t .x y/jV .y/j for a fixed x. Let ./ D … t .x /. By the assumption p .; jV j/ "k k2 C a exp.b"ı /kk2 D .; / C a exp.b"ı / ct 2 C a exp.b"ı / with a constant c independent of x. Hence Z 1 Z 1 ˇ ˇ ˇ. C E/1 jV j.x/ˇ C e tE .ct 2 C a exp.b"ı //dt: 0

(3.3.13)

1

Taking " D .1Clog jt j/2=.1Cı/ , we have a exp.b"ı / t b ae b , thus (3.3.11) follows. Conversely, suppose that V is of Kato-class. By duality it is seen that k. C E/1 jV jk1;1 D kjV j. C E/1 k1;1 , where k kp;p denotes bounded operator norm on Lp . Notice that k. C E/1 jV jk1;1 D k. C E/1 jV jk1 . For the operator-valued function F .z/ D jV jz . C E/1 jV j1z on the strip ¹z 2 Cj 0 and a ı with 0 < ı < 1 such that for all " < 1 and all f 2 D./, kVf k2 "k f k2 C a exp.b"ı /kf k2 : Then V 2 S.Rd /. Remark 3.2. From the perspective of self-adjointness Kato-class is not a natural space, certainly not the largest possible. A well-known example is V .x/ D jxj2 j log jxjj" which is -form bounded with relative bound 0 exactly when " D 0, while it is of Kato-class if and only if " > 1. Rt Proposition 3.33 says that given V 2 K.Rd / one can make Ex Œ 0 V .Bs /ds arbitrarily small by taking t small uniformly in x. R t We will now see that this implies exponential integrability of the random variable 0 V .Bs / ds. The proof of this fact is based on Khasminskii’s lemma which follows directly from the Markov property of Brownian motion but is so useful that we state it as a separate result. First we show a preliminary result. Lemma 3.36. Let f; g W X ! R be functions such that f is F tB -measurable, Ex Œjf j < 1 and supy Ey Œjgj < 1. Let t W X ! X be time shift given by . t !/./ D !.t C / and define t g.!/ D g ı t .!/ D g.!. C t //. Then ˇ x ˇ ˇE Œf gˇ Ex Œjf j sup Ey Œjgj: t y

Proof. Notice that jEx g.BCt /jF tB j D jEB t Œg.B /j supy Ey Œjgj. Since f is F tB -measurable, jEx Œf t gj D jEx Œf Ex Œ t gjF tB j Ex Œjf j sup Ey Œjgj D sup Ey ŒjgjEx Œjf j; y

where we used the Markov property of Brownian motion.

y

106


Lemma 3.37 (Khasminskii’s lemma). Let V 0 be a measurable function on Rd with the property that for some t > 0 and some ˛ < 1

Z t x sup E V .Bs / ds D ˛: (3.3.16) 0

x2Rd

Then sup Ex Œe

Rt 0

V .Bs / ds

x2Rd

1 : 1˛

(3.3.17)

Proof. By expanding the exponential in (3.3.17), it suffices to show that n

Z t 1 ˛ n sup Ex V .Bs / ds nŠ 0 x

Z t Z t 1 D sup Ex ds1 dsn V .Bs1 / V .Bsn / : nŠ x 0 0 Since there are nŠ ways of ordering ¹s1 ; : : : ; sn º it suffices to prove

Z V .Bs1 / V .Bsn /ds1 dsn ; ˛ n sup Ex x

(3.3.18)

n

where n D ¹.s1 ; : : : ; sn / 2 Rn j0 < s1 < < sn º. For fixed s1 < < sn1 , Lemma 3.36 gives

Z t x E V .Bs1 / V .Bsn1 / V .Bsn / dsn sn1

x

Z

E ŒV .Bs1 / V .Bsn1 / sup E x

x

t

V .Bsn / dsn sn1

˛Ex ŒV .Bs1 / V .Bsn1 / : The result follows by induction and an application of Fubini’s theorem. Now we are in the position to prove exponential integrability of the integral over the potential. As it will be seen below this is essentially all we need to prove the Feynman–Kac formula for Kato-decomposable potentials. When V 2 K .Rd /, it Rt V .B /ds s can be seen that the exponent e 0 is integrable with respect to Wiener measure Rt x V .B /ds s is bounded for all x. so that E Œe 0 Lemma 3.38. Let 0 V 2 K.Rd /. Then there exist ˇ; > 0 such that sup Ex Œe x

Rt 0

V .Bs /

< e ˇ t :

(3.3.19)

107


Furthermore, if V 2 Lp .Rd / with p D 1 for d D 1, and p > d=2 for d 2, then there exists C such that ˇ C kV kp :

(3.3.20)

Rt Proof. There exists t > 0 such that ˛ t D supx Ex Œ 0 V .Bs / < 1, for all t t , and ˛ t ! 0 as t ! 0. By Khasminskii’s lemma we have sup Ex Œe

Rt 0

V .Bs /

1 1 ˛t

0, where Œz D max¹w 2 Zjw zº. Set D . 1˛ / and ˇ D t

1 log. 1˛ /1=t . This proves (3.3.19). Next we prove (3.3.20). Suppose V 2 Lp .Rd /. t When d D 1 it follows directly that Z t Z t Ex ŒV .Bs / ds .2s/1=2 dskV k1 : (3.3.23) ˛t D 0

0

1 p

Next we let d 2 and q be such that Z

Z

t

E ŒV .Bs /ds D

0

Z

t

t

x

jyj"

0

C

1 q

D 1. Fix an arbitrary " > 0. We have

Ex ŒV .Bs /1¹jBs xj"º C 1¹jBs xj d=2. Since H.x/ x mC1 kkk1 0 e y y m dy, Z Tr H.K / kkk1

0

1

e y y m dy Tr Km < 1

follows. Then (3.4.17) follows by the dominated convergence theorem, and (3.4.16) follows in the same way as (3.4.17). Step 4: Suppose that W is as in Step 3 and f is continuous. Let fn D f 1Œn;n . By the monotone convergence theorem again it follows that limn Tr Fn .K / D Tr F .K / and Z t Z 1 Z Z dt t x;x lim e dx d WŒ0;t fn W .Bs /ds n t Rd 0 0 Z t Z 1 Z Z dt t x;x D e dx d WŒ0;t f W .Bs /ds : t Rd 0 0 Then (3.4.16) follows and (3.4.17) is similarly proven. Step 5: Suppose that W is as in Step 3 and f is a non-negative lower semicontinuous function on RC with f .0/ D 0, and f , g and F are related by (3.4.9). Since any lower semicontinuous function is a monotone limit continuous function, Fn D R 1 y R 1 of y f .xy/dy=y monotonously. e f .xy/dy=y also converges to F D e n 0 0 Hence limn Tr Fn .K / D Tr F .K / by the monotone convergence theorem, and also R Rt R1 R x;x t limn 0 dt Rd dx d WŒ0;t fn . 0 W .Bs /ds/ converges to the right-hand side t e of (3.4.16) by the monotone convergence theorem. (3.4.17) can be also proven in the same way as (3.4.16). Step 6: Make the same assumptions as in Step 5, moreover suppose that 0. It suffices to check the case of D 0, which is readily given by the monotone convergence theorem. This completes the proof. Proof of Theorem 3.45. Notice that F is monotone increasing, thus F .x/=F .1/ 1, for x 1. Let f .y/ D yg.y/ and F be related by (3.4.9), and f be convex. Then by

120


Lemma 3.51 we have N0 .V / Tr F .K0 /=F .1/ Z t Z Z Z 1 ds dt x;x D F .1/1 … t .0/ dx d WŒ0;t f tW .Bs / t Rd t 0 0 Z Z t Z Z 1 dt ds x;x F .1/1 dx d WŒ0;t f .tW .Bs // … t .0/; d t t R 0 0 where we used Jensen’s inequality. By using the definition of Brownian bridge Z Z x;x d WŒ0;t h.Bs / D dyh.y/…s .x y/… ts .y x/=… t .0/ Rd

the right-hand side above can be computed as Z 1 Z Z t Z dt ds 1 dx dyf .tW .y//…s .x y/… ts .y x/ F .1/ t Rd Rd 0 0 t Z Z Z 1 dt t ds 1 D F .1/ dyf .tW .y//… t .0/ t 0 t Rd 0 Z 1 Z 1 d=2 1 d=2 D F .1/ .2/ t t dt dyf .tW .y//: Rd

0

Changing the variable t to s=W .y/, we furthermore have Z Z 1 Z F .1/1 .2/d=2 f .s/s 1 s d=2 ds W .y/d=2 dy D ad 0

where ad D F .1/1 .2/d=2

R1 0

Rd

W .y/d=2 dy; Rd

f .s/s 1 s d=2 ds.

Remark 3.4. In the case of d D 1 or 2 the operator H D .1=2/ C V with V 0 (not identically zero) has at least one negative eigenvalue. The operator norm of the compact operator K goes to infinity as " 0. This may be compared with the case of d 3 where by the Lieb–Thirring inequality there are no negative eigenvalues if kV kd=2 is sufficiently small.

3.4.3 Positivity improving and uniqueness of ground state We have seen in the preceding section how to use the Feynman–Kac formula to derive the properties of the semigroup e tH . Here we give some examples of another application. We explain below how to use it to study eigenfunctions of e tH which are obviously also eigenfunctions of H . This last fact makes them important for quantum physics through the Schrödinger eigenvalue equation H D E . A detailed study of ground states in models of quantum field theory through Feynman–Kac formulae will be further done in Part II of the book.

121

Section 3.4 Properties of Schrödinger operators and semigroups

Definition 3.24 (Ground state). Let E.H / D inf Spec.H /. An eigenfunction satisfying H ‰p D E.H /‰p is called a ground state of H , and the bottom of the spectrum E.H / is called ground state energy. The non-negative integer m.H / D dim Ker.H E.H //

(3.4.18)

is called the multiplicity of the ground state. If m.H / D 1, the ground state is said to be unique. The simplest but very useful fact is Rthat e tH .x; y/ is strictly positive. This is t readily implied by (3.4.1) whenever e 0 V .Bs /ds is strictly positive for almost every ! 2 X . In particular, e tH is positivity improving in the sense of the following definition. Definition 3.25 (Positivity preserving/improving operator). Let .M; / be a -finite measure space. .1/ A non-zero function f 2 L2 .M; d/ is called positive if f 0 -almost everywhere. Moreover, f is called strictly positive if f > 0 -almost everywhere. .2/ A bounded operator A on L2 .M; d/ is called a positivity preserving operator if .f; Ag/ 0, for all positive f; g 2 L2 .M; d/. A is called a positivity improving operator if .f; Ag/ > 0, for all positive f; g 2 L2 .M; d/. R Example 3.17. Since e t f .x/ D Rd … t .x/f .x/dx, the operator e t is positivity improving. Moreover, e ia.ir/ is a positivity preserving operator, as it produces the shift e ia.ir/ f .x/ D f .x a/. Positivity preserving operators of the form e tE.ir/ have a deep connection with Lévy processes. As it was seen in Example 2.15, the square root of the Laplacian generates a stable Lévy process. The Lévy–Khintchine formula (2.4.6) has the following important relationship with positivity preserving operators. Proposition 3.52. Let E W Rd ! C. .1/ Suppose that the real part of E is bounded from below. Then (a)–(d) below are equivalent: (a) e tE.ir/ is a positivity preserving semigroup. (b) e tE.u/ is a positive definite distribution, i.e., for all 2 C01 .Rd / we have R R tE.u/ .x u/.x/dudx 0. Rd Rd e (c) E.u/ D E.u/, and it is conditionally negative definite, i.e., n X

E.pi pj /Nzi zj 0;

i;j D1

for any n 2 N, z D .z1 ; : : : ; zn / 2 C n with

Pn

iD1 zi

D 0.

122


(d) There exists a triplet .b; A; /, where b 2 Rd , A is a positive definite symmetric matrix, and is a Lévy measure such that Z 1 E.u/ D a C i u b u Au C .e iuy 1 i u y1¹jyj1 º/.dy/ 2 Rd n¹0º (3.4.19) with some a 2 R. .2/ Suppose that E is a spherically symmetric and polynomially bounded continuous function with E.0/ D 0 and real part bounded from below. Suppose, moreover, that E in distributional sense is positive definite. Then E is conditionally negative definite. Write E.u/ D

p u2 C m2 m;

m 0I

then by a straightforward calculation u E.u/ D .d 1/.juj2 C m2 /1=2 C m2 .juj2 C m2 /3=2 : R1 2 2 2 Since .juj2 C m2 /ˇ D dˇ 0 t ˇ 1 e t.juj Cm / dt and e tjuj is positive definite, we conclude by Proposition 3.52 that E.u/ is conditionally negative definite, which implies that E.u/ can be represented as in (3.4.19). Since the left-hand side is real and E.u/=juj2 ! 0 as juj ! 0, its Lévy triplet is .b; A; / D .0; 0; /. Thus we have Z q 2 2 juj C m C m D .e iuy 1 i u y1¹jyj0

The following result gives the Feynman–Kac formula for e E.ir/ . Proposition 3.53. Let .X t / t0 be a Lévy process with characteristic triplet .0; 0; / in (3.4.20). Then Z tE.ir/ .g; e f / D E Œg.x/f .x C X t /dx: The Lévy measure in (3.4.20) can be determined exactly. The integral kernel of the operator e tE.ir/ is given by the Fourier transform F e tE.u/ F 1 and thus p m .d C1/=2 tK .m jx yj2 C t 2 / .d C1/=2 ; (3.4.21) e tE.ir/ .x; y/ D 2 2 .t 2 C jx yj2 /.d C1/=4 where Kd is the modified Bessel function of the third kind. Note that Kd .z/ 1 1 d as jzj # 0. Hence 2 .d /. 2 z/ 1 Œ.g; f / .g; e tE.ir/ f / t!0 t .d C1/=2Z Z q m .g.x/ g.y//.f .x/ f .y// 2 D K .d C1/=2 m jx yj dxdy .d C1/=2 2 jx yj Rd Rd

.g; E.i r/f / D lim

123


In the case of m D 0, the integral form of .g; E.i r/f / is given by .g; E.i r/f / D

..d C 1/=2/ 2 .d C1/=2

Z Rd

Z Rd

.g.x/ g.y//.f .x/ f .y// dxdy: jx yjd C1

Hence it is seen that the Lévy measure associated with E.u/ for m D 0 is 1 1 d C1 1 .dy/ D .d C1/=2 dy 2 jyjd C1 and for m > 0 m 1 .dy/ D 2

m .d C1/=2 1 K.d C1/=2 .mjyj/dy: .d 2 jyj C1/=2

(3.4.22)

Remark 3.5. Let ˛ .dy/ D

1 2˛ . d C˛ 2 / dy; ˛ d=2 d C˛ j. /j 2 jyj

0 < ˛ < 2;

where .x/ D x1 .1 x/ for x > 0. Then the Lévy process .X t / t0 with triplet .0; 0; ˛ / is a stable process with index ˛ and its generator is ./˛=2 . A useful application of the operator positivity properties is to show uniqueness of the ground state of self-adjoint operators. Theorem 3.54 (Perron–Frobenius). Let .M; / be a -finite measure space and K be a self-adjoint operator bounded from below in L2 .M; d/. Suppose that inf Spec.K/ is an eigenvalue and e K is positivity improving. Then inf Spec.K/ is non-degenerate and the eigenfunction corresponding to the eigenvalue inf Spec.K/ is strictly positive. Proof. Put D inf Spec.K/. Then ke K k D e . Let f be an eigenvector corresponding to the eigenvalue . Since e K maps real-valued functions to real-valued functions, 0 and f > 0. Thus e kf k2 D .e K f; f / .e K jf j; jf j/ ke K kkf k2 D e kf k2 ; implying .e K f; f / D .e K jf j; jf j/. This further implies .e K f C ; f / D .e K f ; f C /:

(3.4.23)

124


Since f C , f , e K f C and e K f are all nonnegative, both sides of (3.4.23) must vanish, and this can only happen if either f C or f vanish almost everywhere as e K improves positivity. Hence f has a definite sign and, for instance, f 0 can be assumed. However, since f D e K e C f , f is even strictly positive. The above reasoning applies to any eigenvector corresponding to the eigenvalue . This shows that there cannot be two linearly independent such eigenvectors since then they could be chosen orthogonal which is impossible as they both would have to be strictly positive. Hence f is unique. Theorem 3.55 (Uniqueness of ground state). Suppose that there exists t > 0 such that Z Rt W x .e 0 V .Bs /ds D 0/dx D 0: (3.4.24) Rd

Then the ground state of H is unique. In particular, for VC 2 L1loc .Rd /, H has the unique ground state whenever it exists. R Proof. Since e .t=2/ is positivity improving, Rd dxEx Œf .B0 /g.B t / > 0 for any positive f and g. Then the measure ofR M D ¹.x; !/ 2 Rd X j f .x/g.B t .!/ C t x/ > 0º is positive. By (3.4.24), e 0 V .Bs Cx/ds > 0 for almost every .x; !/ 2 Rd X . Then Z Rt tH .f; e g/ D dxEŒf .x/g.B t C x/e 0 V .Bs Cx/ds Rd Z Rt dxd W f .x/g.B t C x/e 0 V .Bs Cx/ds > 0; M

and the theorem follows from the Perron–Frobenius theorem, R R t Theorem 3.54. For VC 2 L1loc .Rd /, it is seen in Lemma 3.32 that Rd dxW x . 0 VC .Bs /ds D 1/ D 0, which implies (3.4.24).

3.4.4 Degenerate ground state and Klauder phenomenon Now we turn to presenting an example where e tH is not positivity improving. To break positivity down we construct a potential V such that Z Rt dxW x .e 0 V .Bs /ds D 0/ > 0: Rd

This requires of the potential V to be singular in a suitable form. Let M1 D ¹x D .x1 ; x2 ; x3 / 2 R3 jx3 > 0º; M2 D ¹x D .x1 ; x2 ; x3 / 2 R3 jx3 < 0º; M D M1 [ M 2 :

125


Define 1 1 H./ D C jxj2 C : 2 2 jx @M j3

(3.4.25)

Here @M is the boundary of M and jx @M j denotes the distance between x and @M . Since the potential in (3.4.25) is singular, self-adjointness of H./, 6D 0, is unclear. To see this we use the following general result. Proposition 3.56. Let D be an open set in Rd and V 0 on D. Suppose that there is a P uniform Lipshitz continuous function h on every compact subset of D such that .1/ diD1 .@i h/2 e 2h on D, .2/ limx!@D h.x/ D 1 (if D is not bounded, 1 is regarded as a point of @D), .3/ there exists ı > 0 such that .f; . C V /f / .1 C ı/.f; e 2h f / for f 2 C01 .D/. Then C V is essentially self-adjoint on C01 .D/. Lemma 3.57. H is essentially self-adjoint on C01 .M /, for all 2 R. Proof. Let

´ V" .x/ D

=jx @M j3 C 12 jxj2 ; jx @M j < " otherwise: ="3 C 12 jxj2 ;

Since V D V" C T , where T is a bounded operator, it suffices to show that C 2V" 1=2 is essentially self-adjoint on C01 .M /. Define h D log V" . h can be directly shown to satisfy (2)P and (3) in Proposition 3.56. We shall check (1). (1) is equivalent to 3 0 4V" 3iD1 .@i V" /2 . For jx @M j ", it is immediate to see that 4V"3 P3 2 iD1 .@i V" / > 0 for sufficiently small ". We have for jx @M j < ", 3 X 9 2 1 2 3 1 3 2 2 4V" .@i V" / 4 C jxj C jxj : jx @M j3 2 2 jx @M j8 iD1

Take sufficiently small ". Then the right hand side is positive, and the lemma follows. Since H.0/ H./, we have n .H.0// n .H.//, where n .K/ D

sup

inf

1 ;:::;n1

2D.K/;k kD1 2Œ1 ;:::;n1 ?

. ; K /:

By the fact that n .H.0// D n C 12 it is seen that n .H.// ! 1 as n ! 1. In particular, by the min-max principle H./ has a purely discrete spectrum. Further by Theorem 3.31 we have .f; e tH. / g/ Z

Z D Ex f .x/g.B t / exp R3

t 0

C jBs j2 ds jBs @M j3

dx:

(3.4.26)

126


Lemma 3.57 gives that H./ D HD ./, where HD ./ denotes H./ with Dirichlet boundary condition. We see that .f; e tHD . / g/ Z Z D f .x/g.B t / exp M1 [M2

t 0

2 C jB j ds d W x dx; s jBs @M j3

(3.4.27)

with Mj D ¹.x; !/ 2 R3 X jx C Bs 2 Mj ; s 2 Œ0; t º denoting the set of paths starting at x and confined to Mj . Comparing (3.4.26) with (3.4.27), we can see that Z 0

t

1 ds D 1 jBs .!/ @M j3

for ! 2 R3 X n .M1 [ M2 /. Thus such paths do not contribute to (3.4.26) at all. Hence .f; e tH. / g/ D 0 for f and g for which supp f M1 and supp g M2 . Thus e tH. / is not positivity improving. Moreover we have Z f .x/g.B t /d W x dx 6D .f; e tH.0/ g/: lim .f; e tH. / g/ D !0

M1 [M2

This vestigial effect of singular potentials is called Klauder phenomenon and shows that once a singular potential is turned on, its effect is felt after the potential is turned off. We have seen that for e tH. / positivity improving breaks down. This offers the possibility of constructing a two-fold degenerate ground state. Corollary 3.58. The ground state of H./ is two-fold degenerate. Proof. It can be seen that H./ can be reduced by L2 .Mj /. With Hj ./ D H./dL2 .Mj / we have that H./ D H1 ./ ˚ H2 ./. Let Ej D inf Spec.Hj .//; then E1 D E2 by symmetry. Thus the corollary follows. By replacing the potential jx @M j3 with jx @M j2" , " > 0, we can show the same results as in Lemma 3.57 and Corollary 3.58.

3.4.5 Exponential decay of the eigenfunctions A sufficient condition for the existence of a ground state of a Schrödinger operator is that lim infjxj!1 V .x/ D 1. However, ground states exist also for many other choices of V such as Coulomb potential in three dimensions. In many cases these ground states decay exponentially at infinity, while for potentials growing at infinity they decay even faster. An intuitive derivation of the order of decay jxjmC1 of the eigenvectors is obtained as follows.

127


Suppose the potential is of the form V .x/ D jxj2m and the ground state of the form n .x/ D e jxj . By the eigenvalue equation 12 C jxj2m D E we roughly have 1 .n.n 1/jxjn2 C n2 jxj2n2 / C jxj2m D E : 2 Comparing the two sides in the leading order for large jxj we expect that n D m C 1. This can in fact be proved by using the Feynman–Kac formula. Suppose 2 L2 .Rd / with H D for some 2 R. By adding a constant to V we may assume D 0. Thus Rt .x/ D e tH .x/ D Ex Œe 0 V .Bs / ds .B t /: Therefore the value of .x/ is obtained by running a Brownian motion from x under a potential V . This suggests that starting in regions where V is large, the effect will be quite small as most of the Brownian paths take a long time to leave a finite region. Formally, j .x/j2 k k21 Ex Œe 2

Rt 0

VC .Bs / ds

Ex Œe 2

Rt 0

V .Bs / ds

by Schwarz inequality. By Khasminskii’s lemma the expectation involving V grows at most exponentially in t with a constant that can be estimated conveniently. We now present Carmona’s argument in detail. This will work for any eigenfunction of H and not just the ground state; therefore we assume ‰p 2 L2 .Rd / with H ‰p D Ep ‰p : For the present purposes we introduce two further classes of potentials V . Definition 3.26. V upper and V lower are defined as follows. (V upper ) V 2 V upper if and only if V D W U such that .1/ U 0 and U 2 Lp .Rd / for p D 1 if d D 1, and p > d=2 if d 2 .2/ W 2 L1loc .Rd / and W1 D infx2Rd W .x/ > 1. (V lower ) V 2 V lower if and only if V D W U such that .1/ U 0 and U 2 Lp .Rd / for p D 1 if d D 1, and p > d=2 if d 2 .2/ W 2 L1loc .Rd /. Let V D W U 2 V upper . Then W 2 L1loc .Rd / and 0 U 2 Lp .Rd / implies that U 2 K.Rd /. Thus V is Kato decomposable. Hence e tH W L2 .Rd / ! L1 .Rd / by Theorem 3.39, in particular, ‰p 2 L1 .Rd /. A fundamental estimate which we will make use of in showing the spatial exponential decay of eigenfunctions is given in the lemma below.

128


Lemma 3.59 (Carmona’s estimate). Let V 2 V upper . Then for any t; a > 0 and every 0 < ˛ < 1=2, there exist constants D1 ; D2 ; D3 > 0 such that ˛ a2 t

j‰p .x/j D1 e D2 kU kp t e Ep t .D3 e 4

e tW1 C e tWa .x/ /k‰p k;

(3.4.28)

where Wa .x/ D inf¹W .y/jjx yj < aº. Proof. By Schwarz inequality j‰p .x/j e tEp .Ex Œe 4

Rt 0

W .Bs /ds

Note that

/1=4 .Ex Œe C4

Rt 0

U.Bs /ds

/1=4 EŒj‰p .x C B t /j2 1=2 : (3.4.29)

Z 2

EŒj‰p .x C B t /j D

Rd

Z D

… t .y/j‰p .x C y/j2 dy e jzj j‰p .x C 2

Rd

p

2 tz/j2 dz k‰p k2 :

Let A D ¹! 2 X j sup0st jBs .!/j > aº. Then it follows by Lévy’s maximal inequality that Z 1 2 r 2 =2 d 1 EŒ1A 2EŒ1¹jB t jaº D 2.2/d=2 Sd 1 r dx ˛ e ˛a =t p e a= t

with some ˛ , for every 0 < ˛ < 1=2, and where Sd 1 is the area of the d 1 dimensional unit sphere. The first factor in (3.4.29) is estimated as Ex Œe 4

Rt 0

W .Bs /ds

D E0 Œ1A e 4

Rt 0

W .Bs Cx/ds

C Ex Œ1Ac e 4

Rt 0

W .Bs /ds

e 4tW1 E0 Œ1A C e 4tWa .x/ ˛ e ˛a

2 =t

e 4tW1 C e 4tWa .x/ :

(3.4.30)

Next we estimate the second factor. Since U is in Kato-class, there exist constants D1 ; D2 > 0 such that Ex Œe 4

Rt 0

U.Bs /ds

D1 e D2 kU kp t

(3.4.31)

by Lemma 3.38. Setting D3 D ˛ 1=4 , we obtain the lemma by using the inequality .a C b/1=4 a1=4 C b 1=4 for a; b 0. Corollary 3.60 (Confining potential). Let V D W U 2 V upper . Suppose that W .x/ jxj2n outside a compact set K, for some n > 0 and > 0. Take 0 < ˛ < 1=2. Then there exists a constant C1 > 0 such that ˛c (3.4.32) j‰p .x/j C1 exp jxjnC1 k‰p k; 16 where c D infx2Rd nK Wjxj=2 .x/=jxj2n .

129


Proof. Since supx j‰p .x/j < 1, it suffices to show all the statements for sufficiently large jxj. Note that Wjxj=2 .x/ cjxj2n for x 2 Rd n K. Then we have the following bounds for x 2 Rd n K: jxjWjxj=2 .x/1=2 cjxjnC1 ; jxjWjxj=2 .x/1=2 cjxj1n : Inserting t D t .x/ D jxjWjxj=2 .x/1=2 and a D a.x/ D j‰p .x/j e

˛ 16 cjxjnC1

D1 e .D2 kU kp CEp

1n jW j 1

.D3 e cjxj

jxj 2

in (3.4.28), we have

/cjxj1n ˛

C e .1 16 /cjxj

nC1

/k‰p k

for x 2 Rd n K. Hence (3.4.32) follows. For V D W U 2 V upper define † D lim infjxj!1 V .x/: Since U 2 Lp .Rd /, lim infjxj!1 .U.x// D 0 and hence † D lim infjxj!1 W .x/: Moreover † W1 holds. Corollary 3.61. Let V D W U 2 V upper . .1/ Decaying potential: Suppose Suppose that † > Ep , † > W1 , and Let 0 < ˇ < 1. Then there exists a constant C2 > 0 such that ˇ .† Ep / jxj k‰p k: (3.4.33) j‰p .x/j C2 exp p p 8 2 † W1 .2/ Confining potential: Suppose that limjxj!1 W .x/ D 1. Then there exist constants C; ı > 0 such that j‰p .x/j C exp .ıjxj/ k‰p k:

(3.4.34)

Proof. Since supx j‰p .x/j < 1, it is again sufficient to show the statements for large enough jxj. Decaying case: Rewrite formula (3.4.28) as ˛ a2 t

j‰p .x/j D1 e D2 kU kp t .D3 e 4

e t.W1 Ep / C e t.Wa .x/Ep / /k‰p k: (3.4.35)

Then on flipping signs, with † D lim infjxj!1 .W .x// and † > W1 it is possible to choose a decomposition V D W U 2 V upper such that kU kp .† Ep /=2 since lim infjxj!1 .U.x// D 0. Inserting t D t .x/ D "jxj and a D a.x/ D jxj 2 in (3.4.35), we have ˛

j‰p .x/j D1 e kU kp "jxj .D3 e 16" jxj e "jxj.W1 Ep / C e "jxj.Wjxj=2 .x/Ep / /k‰p k ˛

1

D1 .D3 e . 16" C".W1 Ep / 2 ".†Ep //jxj 1

C e "..Wjxj=2 .x/Ep / 2 .†Ep //jxj /k‰p k:

130


Choosing " D

p p ˛=16= † W1 , the exponent in the first term above becomes ˛ 1 1 C ".W1 Ep / ".† Ep / D ".† Ep /: 16" 2 2

Moreover, we see that lim infjxj!1 Wjxj=2 .x/ D †, and obtain "

j‰p .x/j C2 e 2 .†Ep /jxj k‰p k for sufficiently large jxj. Thus (3.4.33) follows. Confining case: In this case for any c > 0 there exists N > 0 such that Wjxj=2 .x/ c, for all jxj > N . Inserting t D t .x/ D "jxj and a D a.x/ D jxj 2 in (3.4.28), we obtain that ˛

j‰p .x/j D1 e kU kp "jxj .D3 e 16" jxj e "jxj.W1 Ep / C e "jxj.Wjxj=2 .x/Ep / /k‰p k ˛

D1 .D3 e . 16" "kU kp C".W1 Ep //jxj C e "jxj.cEp kU kp / /k‰p k for jxj > N . Choosing sufficiently large c and sufficiently small " such that ˛ "kU kp C ".W1 Ep / > 0 and 16"

c Ep kU kp > 0;

0

we obtain j‰p .x/j C 0 e ı jxj for large enough jxj. Thus (3.4.34) follows. By using the Feynman–Kac formula we can also establish a lower bound on positive eigenfunctions. Recall that eigenfunctions corresponding to E.H / are strictly positive. Lemma 3.62. Let V D W U 2 V lower and suppose ‰p .x/ 0. Also, let x 2 Rd n ¹0º, and ˛1 ; : : : ; ˛d , a1 ; : : : ; an , b1 ; : : : ; bd and t > 0 satisfy for j D 1; : : : ; d , ˛j2 > t;

˛j < aj =2;

jŒaj ; aj \ Œxj bj ; xj C bj j > ˛j ;

(3.4.36)

where j j denotes Lebesgue measure. Then log ‰p .x/ Ep t log ".b/ C t WQ a .x/ p

C d log.2t 2 t /

d X j D1

log.˛j2 aj /

d 9 X 2 C aj ; 8t j D1

where WQ a .x/ D sup¹W .y/j jyj xj j < aj ; j D 1; : : : ; d º and ".b/ D inf¹‰p .y/j jyj j bj ; j D 1; : : : ; d º < 1:

(3.4.37)

131

Section 3.5 Feynman–Kac–Itô formula for magnetic field

Proof. Let ˇ j ® ¯ A D ! 2 X ˇ jB t .!/j bj ; sup jBsj .!/ xj j aj ; j D 1; : : : ; d : 0st

(3.4.36) yields that d Y p 2 Ex Œ1A .2t 2 t/d ˛j2 aj e 9jaj =.8t/ :

(3.4.38)

j D1

Hence ‰p .x/ D e tEp Ex Œ‰p .B t /e e tEp Ex Œ‰p .B t /e ".b/e tEp e

Q a .x/ t W

Rt 0

Rt 0

W .Bs /ds W .Bs /ds

Rt

e

0

U.Bs /ds

1A

Ex Œ1A :

Combining this with (3.4.38) we obtain (3.4.37). Corollary 3.63 (Lower bound of ground state). Let V D W U 2 V lower . Suppose that W .x/ jxj2m outside a compact set with > 0 and m > 1, and ‰p .x/ 0. mC1 Then there exist constants ı; D > 0 such that ‰p .x/ De ıjxj . Proof. Since k‰p k1 < 1 it suffices to prove the corollary for large enough jxj. Set t D jxj.m1/ , aj D 1 C jxj j, ˛j D 1=2 and bj D 1 for j D 1; : : : ; d . It can be directly checked that with these choices (3.4.36) is satisfied. Notice that WQ a .x/ 24m jxj2m . Upon insertion to (3.4.37) we obtain p log ‰p .x/ Ep jxj.m1/ log ".b/ C jxjmC1 24m C d log.2 2/ C d log.jxj.m1/3=2 / C 2d log 2 X 9 C jxjm1 .1 C 2jxj j C jxj j2 /; 8 d

d X

log.1 C jxj j/

j D1

j D1

from which 2

log ‰p .x/ log.2.2

p

9 4m 9 2 9 1 2/ =".b//C 2 C d C d jxj C d jxj jxjmC1 : 8 8 4 d

Then for sufficiently large jxj and ı > 24m C 9d=8 the result follows.

3.5

Feynman–Kac–Itô formula for magnetic field

3.5.1 Feynman–Kac–Itô formula The Schrödinger operator (3.1.22) gives the energy of a quantum particle in a force field rV . In some cases of electrodynamics there is an interaction with a magnetic

132


field that has to be taken into account. This is done by adding a vector potential a D .a1 ; : : : ; ad / to the Schrödinger operator. A formal definition of the Schrödinger operator with vector potentials is then given by 1 H.a/ D .i r a/2 C V: 2

(3.5.1)

Lemma 3.64. Suppose that a 2 .L2p .Rd / C L1 .Rd //d , V 2 Lp .Rd / C L1 .Rd / and .r a/ 2 Lp .Rd / C L1 .Rd /, where p D 2 for d 3 and p > d=2 for d 4. Then H.a/ is self-adjoint on D..1=2// and essentially self-adjoint on any core of .1=2/. Proof. Under the assumption H.a/ D .1=2/ C .1=2/.i r a/ C a .i r/ C .1=2/a a C V

(3.5.2)

follows. It is easy to see that a a, r a and V are infinitesimally small multiplication operators with respect to .1=2/. Furthermore, a r is also infinitesimally small. Indeed, we have ka @ f k ka k2p k@ f kq ka k2p ."kf k2 C b" kf k/ for any " > 0, where q D 2p=.p 1/. The second inequality follows from the Hausdorff–Young inequality as k@ f kq kk fOkq 0 k.1 C jkj/˛ k2p k.1 C jkj/˛ k fOk2 ; d ; 1/. Then by the Kato–Rellich theorem H.a/ is where q 0 D 2p=.p C 1/ and ˛ 2 . 2p self-adjoint on D..1=2//, and essentially self-adjoint on any core of .1=2/.

In this section our goal is to construct a Feynman–Kac formula of e tH.a/ . A drift transformation applied to Brownian motion gives Ex Œf .B t /e

Rt 0

a.Bs /dBs 12

Rt

On replacing a by i a, formally Ex Œf .B t /e i

Rt 0

a.Bs /dBs C 12

0

a.Bs /2 ds

Rt 0

D e t..1=2/ar/ f .x/:

a.Bs /2 ds

D e t..1=2/Ciar/ f .x/

is obtained. Let W D V C .1=2/a a C .1=2/.i r a/. By (3.5.2) the “imaginary” drift transformation yields Z t Z t

Z 1 t x a.Bs / dBs C r a.Bs /ds V .Bs /ds E f .B t / exp i 2 0 0 0

Z t Z Z t 1 t x 2 D E f .B t / exp i a.Bs / dBs C a.Bs / ds W .Bs /ds 2 0 0 0 D .e tH.a/ /f .x/: We turn this rigorous in the next theorem.

133


Theorem 3.65 (Feynman–Kac–Itô formula). Suppose a 2 .Cb2 .Rd //d and V 2 L1 .Rd /. Then for f; g 2 L2 .Rd /, Z Rt Rt .f; e tH.a/ g/ D dxEx Œf .B0 /e i 0 a.Bs /ıdBs e 0 V .Bs /ds g.B t /: (3.5.3) Rd

In particular, .e tH.a/ f /.x/ D Ex Œe i

Rt 0

a.Bs /ıdBs

e

Rt 0

V .Bs /ds

f .B t /:

(3.5.4)

Recall that the notation ı in the stochastic integral above stands for Stratonovich integral. Rt Rt Proof. Write Z t D e X t , X t D i 0 a.Bs / ı dBs 0 V .Bs /ds and Y t D f .B t /. Set .Q t f /.x/ D Ex ŒZ t f .B t /: Note that V .B t / and .ra/.B t / are L1loc .R; dt / almost surely, and aj .B t / 2 M 2 .0; t /. Thus, since i dX t D i a dB t .r a/dt Vdt; 2 we arrive at 1 1 i 2 dZ t D Z t dX t C Z t .dX t / D Z t .r a/ a a V dt C Z t .i a/ dB t 2 2 2 by the Itô formula. Then by d Y t D .1=2/f dt C rf dB t and the product formula 1 i d.Z t Y t / D Z t .r a/dt .a a/dt C .i a/ dB t Vdt Y t 2 2 1 f dt C rf dB t C Z t .i a/ rf dt C Zt 2 H.a/f Z t dt C Z t .rf C .i a/f / dB t is obtained. It gives Z .Q t f /.x/ f .x/ D

t

.Qs H.a/f /.x/ds: 0

Then in a similar manner as in Theorem 3.30 we see that 1 s-lim .Q t f f / D H.a/f: t!0 t

134


Then it is enough to show that Q t is a symmetric C0 semigroup. We directly see that Qs Q t f .x/ D Ex Œe i D Ex Œe i

Rs 0

Rs 0

a.Br /ıdBr a.Br /ıdBr

EBs Œf .B t /e i

Rt

Ex Œf .BsCt /e i

0

a.Bs /ıdBs

R sCt s

a.Br /ıdBr

jFs

D QsCt f .x/: Recall that .F t / t0 is the natural filtration of Brownian motion. Hence the semigroup property follows. Let BQ s D B ts B t , 0 s t . We have Z Rt Rt Q Q Q .f; Q t g/ D dxf .x/Ex Œe i 0 a.Bs /ıd Bs e 0 V .Bs / g.BQ t / d R

Z Rt Rt 0 i 0 a.xCBQ s /ıd BQ s 0 V .xCBQ s / DE dxf .x/e e g.x C BQ t / : Rd

Let

1 Ij D .a.x C BQ tj=n / C a.x C BQ t.j 1/=n //.BQ tj=n BQ t.j 1/=n /: 2 Rt Pn es in L2 .X / as n ! 1. Thus a subsequence Then j D1 Ij ! 0 a.x C BQ s / ı d B Rt P n0 es almost surely. We reset n0 as n. Chang. j D1 Ij /n0 converges to 0 a.x C BQ s /ıd B ing the variable x to y BQ t , we have

Z Rt P 0 i jnD1 IQj 0 V .yBQ t BQ s / Q .f; Q t g/ D lim E dyf .y B t /e e g.y/ ; n!1

Rd

where 1 IQj D .a.y BQ t C BQjt=n / C a.y BQ t C BQ t.j 1/=n //.BQ tj=n BQ t.j 1/=n /: 2 Recall that a 2 .Cb2 .Rd //d . Then in L2 .X / sense it follows that lim

n!1

n X

Z IQj D

j D1

0

t

a.y C Bs / ı dBs :

This gives Z .f; Q t g/ D

Rd

dxEx Œf .B t /e i

Rt 0

a.Bs /ıdBs e

Rt 0

V .Bs /ds g.x/

D .Q t f; g/: (3.5.5)

Hence Q t is symmetric. The strong continuity of Q t with respect to t is derived in the same way as in the proof of Theorem 3.30.

135


3.5.2 Alternate proof of the Feynman–Kac–Itô formula As before, the Trotter product formula can once again be employed to prove the Feynman–Kac–Itô formula. Here we give this proof, which will be useful in constructing the functional integral representation of the Pauli–Fierz model in Chapter 7. Suppose a 2 .Cb2 .Rd //d and V D 0. Write Ks .x; y/ D …s .x y/e ih ;

(3.5.6)

where h D h.x; y/ D .1=2/.a.x/ C a.y// .x y/: Define the family of integral operators %s W L2 .Rd / ! L2 .Rd / by Z %s f .x/ D

Rd

Ks .x; y/f .y/dy;

s 0;

(3.5.7)

where %0 f .x/ D f .x/. Notice that %s is symmetric and k%s k 1. Let f; g 2 C01 .Rd /. It is directly seen that d .g; %s f / D ds

Z

Z g.x/dx Rd

Rd

1 ih y e f .y/ dy: …s .x y/ 2

Here we used that .d…s =ds/.x y/ D .1=2/y …s .x y/. Since @j e ih f D .i @j h f C @j f /e ih ; @j2 e ih f D ¹@j2 f C 2i @j h @j f C .i @j2 h C .i @j h/2 /f ºe ih ; 1 @j h D ¹@j a.y/ .x y/ .aj .x/ C aj .y//º; 2 ² ³ 1 @j2 h D @j2 a.y/ .x y/ @j aj .y/ ; 2 where @j D @yj , j D 1; : : : ; d , we have Z lim

s!0 Rd

…s .x y/.y e ihf .y//dy D .f 2a .i rf / .i ra a a/f / D 2H.a/f:

Thus lims!0 .d=ds/.g; %s f / D .g; H.a/f / and hence lim .g; t 1 .1 % t /f / D .g; H.a/f /:

t!0

(3.5.8)

136

Chapter 3 Feynman–Kac formulae n

We show next that %2t=2n converges to a symmetric semigroup. For f; g 2 L2 .Rd /, n .g; %2t=2n f

Z /D

g.x/e

i

P2 n

j D1

h.xj ;xj 1 /

Rd

j D1

with x0 D x. Therefore n lim .g; %2t=2n f n!1

2n n Y Y f .xn / … t=2n .xj xj 1 / dxj

Z

j D0

dxE g.B0 /f .B t / x

/ D lim

n!1 Rd

X 1 exp i .a.B tj=2n / C a.B t.j 1/=2n //.B tj=2n B t.j 1/=2n / 2 j D1 Z t

Z x D dxE g.B0 /f .B t / exp i a.Bs / ı dBs (3.5.9) 2n

Rd

0

by Corollary 2.30. Hence

n j limn!1 .g; %2t=2n f

/j kgk kf k. By the Riesz theon

rem there is a bounded operator S t such that limn!1 .g; %2t=2n f / D .g; S t f /. It is symmetric since %s is symmetric. (3.5.9) implies that S t f .x/ D Ex Œf .Bs /e i

Rt 0

a.Bs /ıdBs

:

Hence S0 D 1, s-lim t!0 S t D 1 and Ss S t D SsCt follow. Thus S t is a symmetric C0 -semigroup, so there exists a self-adjoint operator A such that e tA D S t by Proposition 3.26. It suffices to show that A D H.a/. This can be checked without using the Itô formula. We have 1 tA 1 2n .e .% t=2n 1/g; f 1/g; f D lim n!1 t t n 1 2X 1 2n 2n .j=2n / D lim .% t=2n 1/g; % t=2n f n!1 2n t j D0

Z D lim

n!1 0

1 n 2

t

Œ2n s

.% t=2n 1/g; % t=2n f

ds;

where the brackets denote integer part in this formula. Since for g 2 D..1=2//, 2n .% t=2n 1/g D H.a/g; n!1 t the norm k.2n =t /.% t=2n 1/gk is uniformly bounded in n. Moreover, we see that Œ2n s s-limn!1 % t=2n D e stA , s 2 Œ0; 1, by Corollary 3.67 below. Thus we conclude that Z 1 1 tA .e 1/g; f D .H.a/g; e tsA f /ds: (3.5.10) t 0 w lim

137


As t ! 0 on both sides above, we get .g; Af / D .H.a/g; f /, implying that Ag D H.a/g. Hence A D H.a/ and the proposition follows for V D 0. Let now V be continuous and bounded. Note that for f0 ; fn 2 L2 .Rd / and fj 2 1 L .Rd / for j D 1; : : : ; n 1, it follows that for 0 D t0 < t1 < < tn , n n h Y Z i R tn Y .tj tj 1 /H.a/ e fj D dx f0 .B0 / fj .B tj / e i 0 a.Bs /ıdBs : f0 ; Rd

j D1

j D1

(3.5.11) This can be proven by using the Markov property of Brownian motion as

n Y

f0 ; Z D

j D1

Rd

Z D

e .tj tj 1 /H.a/ fj

Rd

dxEx Œf0 .B0 /f1 .B t1 /e i EB t1 Œf1 .B t1 /e i dxEx Œf0 .B0 /f1 .B t1 /e i

R t1 0

a.Bs /ıdBs

R t2 t1 0

R t2 0

a.Bs /ıdBs

a.Bs /ıdBs

G2 .B t2 t1 /

f1 .B t1 /G2 .B t2 t1 /;

Q where Gj .x/ D fj .x/. niDj C1 e .ti ti1 /H.a/ fi /.x/. Repeating this procedure we arrive at (3.5.11). From this and the Trotter product formula we have .f; e tH.a/ g/ D lim .f; .e .t=n/H.a/ e .t=n/V /n g/ n!1 Z Rt Pn D lim dxEx Œf .B0 /e i 0 a.Bs /ıdBs e j D1 V .Bjt =n / n!1 Rd Z Rt Rt D dxEx Œf .B0 /e i 0 a.Bs /ıdBs e 0 V .Bs /ds : Rd

Finally, we can extend the result for smooth V in a similar way as done in Theorem 3.31. Now we prove the statements used in the proof above. Lemma 3.66. Let An be a positive and contracting self-adjoint operator such that s-limn!1 An D A. Then for all s 0, s-limn!1 Asn D As . Proof. Suppose that 0 < s 1=2. Since An is positive, we have Z sin.2s/ 1 2s1 f D .An C /1 An f d : A2s n 0 2s weakly as n ! 1, thus As ! As strongly. For From this follows that A2s n ! A n s=2 0 s 1, s-limn!1 Asn D s-limn!1 .An /2 D As , since An is a contraction. Repeating this procedure the result follows.

138

Chapter 3 Feynman–Kac formulae Œns

Corollary 3.67. s-limn!1 % t=n D e stA for s 0. Proof. Set An D %2n . Then An is a positive and a contracting self-adjoint operator t=n .nsC1/=n

such that s-limn!1 An D e 2tA . We have An

2Œns

% t=n Asn in form sense. 1=n

From this inequality, Lemma 3.66 and the fact that s-limn!1 An D 1, we obtain 2Œns Œns that .f; % t=n f / ! .f; e 2tsA f / as n ! 1. Hence s-limn!1 % t=n D e tsA follows.

3.5.3 Extension to singular external potentials and vector potentials The Feynman–Kac–Itô formula derived in the previous subsection can further be extended to singular vector potentials a D .a1 ; : : : ; ad /. By using form technique, we are able to define H.a/ as a self-adjoint operator. Let D D i @ a W L2 .Rd / ! D 0 .Rd /;

(3.5.12)

with Schwartz distribution space D 0 .Rd / over Rd . Define the quadratic form qa .f; g/ D

d X

.D f; D g/ C .V 1=2 f; V 1=2 g/

(3.5.13)

D1

with domain Q.qa / D

Td

D1 ¹f

2 L2 .Rd /jD f 2 L2 .Rd /º \ D.V 1=2 /.

Lemma 3.68. Suppose that a 2 .L2loc .Rd //d and V 2 L1loc .Rd / with V 0. Then qa is a closed symmetric form. Proof. Let .fn /n Q.qa / be a Cauchy sequence with respect to the norm k k0 D .qa .; /Ckk2 /1=2 ; then V 1=2 fn , D fn are Cauchy sequences in L2 .Rd /. Thus there exist g; f 2 L2 .Rd / such that D fn ! g and V 1=2 fn ! V 1=2 f as n ! 1 in L2 .Rd /. This gives .f; D / D .g; /, for all 2 C01 .Rd /. Therefore g D D f in D 0 .Rd / distribution sense, hence Q.qa / is complete with respect to k k0 . By Lemma 3.68 there exists a self-adjoint operator h such that .f; hg/ D qa .f; g/;

f 2 Q.qa /; g 2 D.h/:

(3.5.14)

Definition 3.27 (Schrödinger operator with a singular external potential and a vector field). Assume that a 2 .L2loc .Rd //d and V 2 L1loc .Rd /. We denote the self-adjoint operator h in (3.5.14) by H.a/ and call it Schrödinger operator with singular external potentials and vector potentials. A sufficient condition for C01 .Rd / to be a core of H.a/ is the proposition below.

139


Proposition 3.69. .1/ Let V 2 L1loc .Rd / with V 0. If a 2 .L2loc .Rd //d , then C01 .Rd / is a form core of H.a/. .2/ Let V 2 L2loc .Rd / with V 0. If a 2 .L4loc .Rd //d and r a 2 L2loc .Rd /, then C01 .Rd / is an operator core. We can also construct a Feynman–Kac–Itô formula for the form-defined H.a/. Lemma 3.70. Let 0 V and V 2 L1loc .Rd /. Suppose that a.n/ ; a 2 .L2loc .Rd //d .n/ and a ! a in L2loc .Rd / as n ! 1. Then H.a.n/ / converges to H.a/ in strong resolvent sense. .n/

.n/

Proof. Put Hn D H.a.n/ /, H D H.a/ and D D i @ a . Notice that C01 .Rd / is a form core of H.a/ by Proposition 3.69. Set un D .Hn C i /1 f for f 2 L2 .Rd /. Thus kun k kf k and d X

.n/ kD un k2 C kV 1=2 un k2 kf k2 :

D1 .n/

Since un , D un and V 1=2 un are bounded, there exist a subsequence un0 and some .n0 / vectors u, v, w 2 L2 .Rd / such that un0 ! u, V 1=2 un0 ! v and D un0 ! w .n0 / weakly. Since, furthermore, V 1=2 un0 ! V 1=2 u and D un0 ! D u in D 0 .Rd /, we have v D V 1=2 u and D u D w , and hence u 2 Q.qa /. For 2 C01 .Rd / qa .u; / D lim .un ; Hn / D .f i u; /: n!1

(3.5.15)

As C01 .Rd / is a form core of H , (3.5.15) implies that u 2 D.H / and f D .H Ci /u. Hence u D .H C i /1 f follows. Therefore, .Hn C i /1 ! .H C i /1 in weak sense, and similarly, .Hn i /1 ! .H i /1 weakly. Since k.Hn C i /1 f k2 D .i=2/.f; .Hn C i /1 f .Hn i /1 f / ! k.H C i /1 f k2 ; strong resolvent convergence of Hn follows. Lemma 3.71. Let V 0 be in L1loc .Rd /, a 2 .L2loc .Rd //d and r a 2 L1loc .Rd /. Then .f; e tH.a/ g/ has the same functional integral representation as in (3.5.3). S Proof. First suppose that V 2 L1 .Rd /. Notice that since a .Bs / 2 T >0 M2T by Rt the assumption that a 2 .L2loc .Rd //d we have W x .j 0 a.Bs / dBs j < 1/ D 1, R t moreover by Lemma 3.32, W x .j 0 r a.Bs /dsj < 1/ D 1 also holds. By using a mollifier we can choose a sequence a.n/ 2 C01 .Rd /, n D 1; 2; : : : , such that .n/ a ! a in L2loc .Rd / and r a.n/ ! r a in L1loc .Rd / as n ! 1. Let 1R D .x1 =R/ .xd =R/, R 2 N, where 2 C01 .R/ such that 0 1, .x/ D 1

140


for jxj < 1 and .x/ D 0 for jxj 2. Since 1R a.n/ ! a.n/ as R ! 1 in .L2loc .Rd //d and a.n/ ! a as n ! 1 in .L2loc .Rd //d , it follows from Lemma 3.70 .n/ that e tH.1R a / ! e tH.1R a/ , as n ! 1 and e tH.1R a/ ! e tH.a/ as R ! 1 in strong sense. Furthermore, (3.5.3) remains true for a replaced by 1R a.n/ 2 C01 .Rd /. Since 1R a.n/ 2 .C01 .Rd //d and 1R a.n/ ! 1R a in .L2loc .Rd //d as n ! 1, it follows that Z t Z t 1R .Bs /a.n/ .Bs / dBs ! 1R .Bs /a.Bs / dBs (3.5.16) 0

0

and since r .1R a.n/ / D .r1R / a.n/ C 1R .r a.n/ / ! .r1R / a C 1R .r a/ in L1 .Rd / it follows that Z

Z

t 0

r .1R .Bs /a

.n/

.Bs //ds !

0

t

¹.r1R .Bs // a.Bs / C 1R .Bs /.r a.Bs //º ds (3.5.17)

strongly in L1 .X ; d W x /. Thus there is a subsequence n0 such that (3.5.16) and (3.5.17) hold almost surely for n replaced by n0 , therefore (3.5.3) follows by a limiting argument for a replaced by 1R a. Let ® C .R/ D ! 2 X j ®

.R/ D ! 2 X j

max

¯ Bs .!/ R ;

min

Bs .!/ R

0st;1d 0st;1d

¯

ˇZ t ˇ Z t ˇ ˇ I.R/ D ˇˇ 1R .Bs /a.Bs / dBs a.Bs / dBs ˇˇ :

and

0

0

It is a well-known fact that x

x

W . .R// D W .C .R// D

d Y

W x .jB t j R/

D1

D

p

2 2 t

Z

!d

R

e

y 2 =.2t/

dy

0

Since I.R/ D 0 on C \ , we have W x .I.R/ "/ D W x I.R/ "; C .R/c [ .R/c d Z 1 2 2 e y =.2t/ dy : 2 p 2 t R

:

141


Hence limR!1 W x .I.R/ R "/ D 0. Thus there exists a subsequence R0 such that Rt t 0 0 0 1R .Bs /a.Bs / dBs ! 0 a.Bs / dBs as R ! 1, almostR surely. It can be seen t in a similar way that with a subsequence (3.5.17) converges to 0 r a.Bs /ds almost Rt Rt surely. Thus 0 1R00 .Bs /a.Bs / ı dBs ! 0 a.Bs / ı dBs , almost surely. Since jf .B0 /g.B t /e

Rt 0

V .Bs /ds i

Rt

e

0

1R00 .Bs /a.Bs /ıdBs

j D jf .B0 /g.B t /je

Rt 0

V .Bs /ds

and the right-hand side is integrable, (3.5.3) holds for a satisfying a 2 .L2loc .Rd //d and r a 2 L1loc .Rd / by the dominated convergence theorem. Finally, by a similar argument as in the proof of Theorem 3.31, we can extend (3.5.3) to V 2 L1loc .Rd /. Define H 0 .a/ by H.a/ with V D 0. Lemma 3.72. Let a 2 .L2loc .Rd //d , r a 2 L1loc .Rd / and V be a real multiplication operator. .1/ Suppose jV j is relatively form bounded with respect to .1=2/ with relative bound b. Then jV j is also relatively form bounded with respect to H 0 .a/ with a relative bound not larger than b. .2/ Suppose jV j is relatively bounded with respect to .1=2/ with relative bound b. Then jV j is also relatively bounded with respect to H 0 .a/ with a relative bound not larger than b. Proof. By virtue of Lemma 3.71 we have j.f; e tH Since .H 0 .a/ C E/1=2 D that

p1

0 .a/

R1 0

g/j .jf j; e t.1=2/ jgj/: t 1=2 e t.H

0 .a/CE /

(3.5.18)

dt , E > 0, (3.5.18) implies

j.H 0 .a/ C E/1=2 f j.x/ ..1=2/ C E/1=2 jf j.x/

(3.5.19)

for almost every x 2 Rd . Hence we have jV .x/j1=2 j.H 0 .a/ C E/1=2 f j.x/ jV .x/j1=2 ..1=2/ C E/1=2 jf j.x/ and k jV j1=2 .H 0 .a/ C E/1=2 f k k jV j1=2 ..1=2/ C E/1=2 jf jk : (3.5.20) kf k kf k R1 0 Similarly, by using .H 0 .a/ C E/1 D 0 e t.H .a/CE / dt , E > 0, we have k jV j.H 0 .a/ C E/1 f k k jV j..1=2/ C E/1 jf jk : kf k kf k

(3.5.21)

On taking the limit E ! 1, the right-hand sides of (3.5.20) and (3.5.21) converge to b. Hence .1/ follows by (3.5.20) and .2/ by (3.5.21).

142


Let V be such that VC 2 L1loc .Rd / and V is relatively form bounded with respect to .1=2/ with a relative bound strictly smaller than one. Then the KLMN theorem P VC and Lemma 3.72 allow to define H 0 .a/ C P V as a self-adjoint operator. Definition 3.28 (Schrödinger operator with singular external potentials and vector potentials). Let a 2 .L2loc .Rd //d and r a 2 L1loc .Rd /. Let V be such that VC 2 L1loc .Rd / and V is relatively form bounded with respect to .1=2/ with a relative bound strictly smaller than one. Then the Schrödinger operator with V and a is defined by P VC H 0 .a/ C P V :

(3.5.22)

This is also denoted by H.a/. Theorem 3.73 (Feynman–Kac formula for Schrödinger operator with singular external potentials and vector potentials). Let a 2 .L2loc .Rd //d and r a 2 L1loc .Rd /. Suppose that VC 2 L1loc .Rd / and V is relatively form bounded with respect to .1=2/ with a relative bound strictly smaller than one. Then the Feynman–Kac formula of .f; e tH.a/ g/ is given by (3.5.3). Proof. The proof is similar to the limiting argument in Theorem 3.31. We used inequality (3.5.18) in the proof of Lemma 3.72. This can be extended to operators with singular external potentials. Corollary 3.74 (Diamagnetic inequality). Under the assumptions of Theorem 3.73 we have j.f; e tH.a/ g/j .jf j; e tH.0/ jgj/:

(3.5.23)

3.5.4 Kato-class potentials and Lp –Lq boundedness We can also add Kato-class potentials for Schrödinger operators with vector potentials a. Let V be Kato decomposable, a 2 .L2loc .Rd //d and r a 2 L1loc .Rd /. Define K t f .x/ D Ex Œe i

Rt 0

a.Bs /ıdBs

e

Rt 0

V .Bs /ds

f .B t /:

(3.5.24)

Since the absolute value of the first exponential is 1, K t is well defined as a bounded operator on L2 .Rd /. Theorem 3.75. Let V be Kato decomposable, and a 2 .L2loc .Rd //d , r a 2 L1loc .Rd /. Then ¹K t W t 0º is a symmetric C0 semigroup and there exists the unique selfadjoint operator K.a/ bounded from below such that K t D e tK.a/ , t 0.

143

Section 3.6 Feynman–Kac formula for relativistic Schrödinger operators

Proof. In a similar way to Theorem 3.65 it can be proven that ¹K t W t 0º is a symmetric C0 semigroup. Definition 3.29 (Schrödinger operator with vector potentials and Kato-decomposable potentials). Let a 2 .L2loc .Rd //d , r a 2 L1loc .Rd / and V be Kato decomposable. Then we call the self-adjoint operator K.a/ defined by Theorem 3.75 Schrödinger operator with vector potentials and Kato-decomposable potentials. Corollary 3.76 (Lp –Lq boundedness). Let V be Kato decomposable, a 2 .L2loc .Rd //d and r a 2 L1loc .Rd /. Then e tK.a/ is Lp –Lq bounded, for every t 0. Proof. By the diamagnetic inequality je tK.a/ f .x/j e tK.0/ jf j.x/:

(3.5.25)

In Theorem 3.39 we have shown that e tK.0/ is hypercontractive. By (3.5.25) it follows that e tK.a/ is bounded from L1 .Rd / to L1 .Rd /, from L1 .Rd / to L1 .Rd /, and from L1 .Rd / to L1 .Rd /.

3.6

Feynman–Kac formula for relativistic Schrödinger operators

3.6.1 Relativistic Schrödinger operator Relativity theory says that in the high energy regime the kinetic energy of a body in motion should be proportional to the absolute value of its momentum p rather than p its square. In fact, the classical energy of a relativistic particle of mass m is jpj2 C m2 c 2 , where c is the speed of light, and this justifies the definition of the relativistic Schrödinger operator p C m2 m (in units for which c D 1). The constant m is subtracted to set the bottom of its spectrum to zero. In this section we consider Feynman–Kac formulae for relativistic Schrödinger operators of a more general form. Definition 3.30 (Relativistic Schrödinger operator with vector potentials). Suppose that a 2 .L2loc .Rd //d and V 2 L1 .Rd /. The operator HR .a/ D .2H 0 .a/ C m2 /1=2 m C V;

(3.6.1)

is called relativistic Schrödinger operator, where H 0 .a/ is the self-adjoint operator given in Definition 3.27 with V D 0.

144


Note that the square-root term is defined through the spectral projection of the selfadjoint operator H 0 .a/. Denote by HR0 .a/ the operator HR .a/ with V D 0. Proposition 3.77. .1/ Suppose that a 2 .L4loc .Rd //d and r a 2 L2loc .Rd /. Then C01 .Rd / is an operator core of HR0 .a/. .2/ Suppose that a 2 .L2loc .Rd //d . Then C01 .Rd / is a form core of HR0 .a/. Proof. For simplicity, in this proof we set HR0 .a/ D HR and H 0 .a/ D H . .1/ There exist non-negative constants c1 and c2 such that kHR f k c1 kHf k C c2 kf k

(3.6.2)

for all f 2 D.H /. Hence C01 .Rd / is contained in D.HR /. Since HR is a nonnegative self-adjoint operator, HR C1 has a bounded inverse, and we use that C01 .Rd / is a core of HR if and only if HR C01 .Rd / is dense in L2 .Rd /. Let g 2 L2 .Rd / and suppose that .g; .HR C 1/f / D 0, for all f 2 C01 .Rd /. Then C01 .Rd / 3 f 7! .g; HR f / D .g; f / defines a continuous functional which can be extended to L2 .Rd /. Thus g 2 D.HR / and 0 D ..HR C 1/g; f /. Since C01 .Rd / is dense, we have .HR C 1/g D 0, and hence g D 0 since HR C 1 is one-to-one, proving the assertion. 1=2 .2/ Note that kHR f k2 c1 kH 1=2 f k2 C c2 kf k2 for f 2 Q.h/ D D.H 1=2 /, 1=2 1=2 and C01 .Rd / is contained in Q.HR / D D.HR /. Since HR C 1 has also bounded 1=2 inverse, it is seen by the same argument as above that C01 .Rd / is a core of HR or a form core of HR . A key element in the construction of the Feynman–Kac formula is to use the Lévy subordinator studied in Example 2.18. In that notation we write ı D 1, D m so that we have the subordinator T t D T t .1; m/ on a suitable probability space .T ; BT ; / such that p E0 Œe uT t D exp.t . 2u C m2 m//: (3.6.3) For simplicity, we write Ex;0 for Ex;0 . W Theorem 3.78 (Feynman–Kac formula for relativistic Schrödinger operator with vector potentials). Suppose that V 2 L1 .Rd /, a 2 .L2loc .Rd //d and r a 2 L1loc .Rd /. Then Z R Tt Rt tHR .a/ .f; e g/ D dxEx;0 Œf .B0 /g.BT t /e i 0 a.Bs /ıdBs e 0 V .Bs /ds : Rd

(3.6.4)

145


Proof. We divide the proof into four steps. Step 1: Suppose V D 0. Then Z R Tt 0 dxEx;0 Œf .B0 /g.BT t /e i 0 a.Bs /ıdBs : .f; e tHR .a/ g/ D

(3.6.5)

Rd

Denote by E the spectral projection of the self-adjoint operator H 0 .a/. Then Z p 2 tHR0 .a/ g/ D e t. 2Cm m/ d.f; E g/: (3.6.6) .f; e Œ0;1/

By (3.6.3) .f; e

tHR0 .a/

Z g/ D

1

E0 Œe T t d.f; E g/ D E0 Œ.f; e T t H

0

0 .a/

g/:

Then (3.6.5) follows from inserting the Feynman–Kac–Itô formula of e T t H .a/ : Z R Tt T t HR0 .a/ .f; e g/ D dxEx;0 Œf .B0 /g.BT t /e i 0 a.Bs /ıdBs (3.6.7) 0

Rd

developed in Section 3.5.3. Step 2: Let 0 D t0 < t1 < < tn and f0 ; fn 2 L2 .Rd / and fj 2 L1 .Rd / for j D 1; : : : ; n 1. We prove next that n Y 0 e .tj tj 1 /HR .a/ fj f0 ; Z D

j D1

Rd

n Y h i R Tt dxEx;0 f .B0 / fj .BT tj / e i 0 a.Bs /ıdBs :

(3.6.8)

j D1

Q 0 We use the shorthand Gj .x/ D fj .x/. niDj C1 e .ti ti1 /HR .a/ fi /.x/. The lefthand side of (3.6.8) can be written as Z R T t1 t0 a.Bs /ıdBs dxEx;0 Œf .B0 /e i 0 G1 .BT t1 t0 /: Rd

By the Markov property of .B t / t0 we have n Y 0 e .tj tj 1 /HR .a/ fj f0 ; Z D

j D1

Rd

Z D

Rd

dxEx;0 Œf .B0 /e i

R T t1 0

BT t

E0 EW

dxEx;0 Œf .B0 /e i

1

a.Bs /ıdBs

Œf1 .B0 /e i

R T t1 0

R T t2 t1 0

a.Bs /ıdBs

G2 .BT t2 t1 /

a.Bs /ıdBs

i E0 Œf1 .BT t1 /e

R T t2 t1 CT t1 Tt

1

a.Bs /ıdBs

G2 .BT t1 CT t2 t1 /:

146


The right-hand side above can be rewritten as Z R T t1 dxEx;0 Œf .B0 /e i 0 a.Bs /ıdBs f1 .BT t1 / Rd

Tt

E 1 Œe i

R T t2 t1 0

a.Bs /ıdBs

G2 .BT t2 t1 /:

By the Markov property of .T t / t0 again we can see that Z R Tt R Tt i T 2 a.Bs /ıdBs x;0 i 0 1 a.Bs /ıdBs t1 D dxE Œf .B0 /e f1 .BT t1 /e G2 .BT t2 / Rd Z R T t2 D dxEx;0 Œf .B0 /e i 0 a.Bs /ıdBs f1 .BT t1 /G2 .BT t2 /: Rd

Continuing by the above procedure (3.6.8) is obtained. Step 3: Suppose now that V 2 L1 and it is continuous. We show now (3.6.4) under this condition. By the Trotter product formula, .f; e tHR .a/ g/ 0

D lim .f; .e .t=n/HR .a/ e .t=n/V /n g/ n!1 Z Pn R Tt D lim dxEx;0 Œf .B0 /g.BT t /e i 0 a.Bs /ıdBs e j D1 .t=n/V .BT tj=n / n!1 Rd

D r.h.s. (3.6.8): Here we used that for each path .!; /, V .BTs ./ .!// is continuous in s except for at most finitely many points. Note that Ts almost surely has no accumulation points in any compact interval. Thus Z t n X t V .BT tj=n / ! V .BTs /ds n 0 j D1

as n ! 1, for each path in the sense of Riemann integral. This completes the proof of the claim. Step 4: Finally, to complete the proof of the theorem define Vn in the same manner as in the proof of Theorem 3.86. Vn is bounded and continuous, moreover Vn .x/ ! V .x/ as n ! 1 for x … N , where N is a set of Lebesgue measure zero. Notice that

Z x;0 0 E Œ1N .BTs / D E 1¹Ts >0º 1N .x C y/ks .y/dy C 1N .x/E0 Œ1¹Ts D0º D 0 Rd

for x 2 N , where ks .x/ is the distribution of the random variable BTs given by (3.4.21). Hence

Z t Z t 0D Ex;0 Œ1N .BTs /ds D Ex;0 1N .BTs /ds : 0

0


147

Then for almost surely .!; / the Lebesgue measure of ¹t 2 Œ0; 1/ j BT t ./ .!/ 2 N º Rt Rt is zero. Therefore 0 Vn .BTs /ds ! 0 V .BTs /ds as n ! 1 for almost every .!; /. Moreover, Z R Tt Rt dxEx;0 Œf .B0 /g.BT t /e i 0 a.Bs /ıdBs e 0 Vn .Bs /ds Rd Z R Tt Rt ! dxEx;0 Œf .B0 /g.BT t /e i 0 a.Bs /ıdBs e 0 V .Bs /ds Rd

0

0

as n ! 1. On the other hand, e t.HR .a/CVn / ! e t.HR .a/CV / strongly as n ! 1, since HR0 .a/ C Vn converges to HR0 .a/ C V on the common domain D.HR0 .a//. This proves the theorem. The Feynman–Kac formula of e tHR .a/ can be extended to more singular external potentials. The key idea is similar to that of applied in the case of e tH.a/ discussed in the previous section. Lemma 3.79. Let a 2 .L2loc .Rd //d , r a 2 L1loc .Rd / and V be a real multiplication operator. p .1/ Suppose jV j is relatively form bounded with respect to C m2 m with relative bound b. Then jV j is also relatively form bounded with respect to HR0 .a/ with a relative bound not larger than b. p .2/ Suppose jV j is relatively bounded with respect to C m2 m with relative bound b. Then jV j is also relatively bounded with respect to HR0 .a/ with a relative bound not larger than b. Proof. By virtue of Theorem 3.78 we have 0

p

j.f; e tHR .a/ g/j .jf j; e t.

Cm2 m/

jgj/:

(3.6.9)

The lemma can be proven in the parallel way to that of Lemma 3.72. Let V be such that VC 2 L1loc .Rd / and V is relatively form bounded with respect p to C m2 m with a relative bound strictly smaller than one. Then the KLMN P VC theorem and Lemma 3.79 can be used to define HR0 .a/ C P V as a self-adjoint operator. Definition 3.31 (Relativistic Schrödinger operator with singular external potentials and vector potentials). Let a 2 .L2loc .Rd //d and r a 2 L1loc .Rd /. Let V be such p that VC 2 L1loc .Rd / and V is relatively form bounded with respect to C m2 m with a relative bound strictly smaller than one. Then the relativistic Schrödinger operator with a and V is defined by P VC HR .a/ D HR0 .a/ C P V :

(3.6.10)

148


Theorem 3.80 (Feynman–Kac formula for relativistic Schrödinger operator with singular external potentials and vector potentials). Suppose that a 2 .L2loc .Rd //d and r a 2 L1loc .Rd /. Let V be such that VC 2 L1loc .Rd / and V is relatively form p bounded with respect to C m2 m with a relative bound strictly smaller than one. Then the Feynman–Kac formula of .f; e tHR .a/ g/ is given by (3.5.3). Proof. The proof is similar to the limiting argument in that of Theorem 3.31. Corollary 3.81 (Diamagnetic inequality). With the same assumptions as in Theorem 3.80 we have j.f; e tHR .a/ g/j .jf j; e tHR .0/ jgj/

(3.6.11)

Remark 3.6 (Bernstein functions of the Laplacian). Theorems 3.78 and 3.80 can be generalized to fractional Schrödinger operators as in Definition 3.3. .T t / t0 is then an ˛=2-stable subordinator, 0 < ˛ < 2, and thus .BT t / t0 is a symmetric ˛-stable process. In fact, the relativistic and fractional Laplacians are cases of a larger class of operators obtained as Bernstein functions of the Laplacian. Let ˆ 2 B0 be a Bernstein function with vanishing right limit as in (2.4.44), and consider its uniquely associated subordinator .T tˆ / t0 . Then we define a generalized Schrödinger operator 1 2 .i r a/ C V: Hˆ .a/ D ˆ 2

(3.6.12)

Using similar arguments than before we can show the Feynman–Kac formula .f; e tHˆ .a/ g/ D

Z Rd

dxEx;0 Œf .B0 /g.BT ˆ /e i

R T tˆ 0

a.Bs /ıdBs

e

t

Rt 0

V .BT ˆ /ds s

:

(3.6.13) Setting a D 0, we obtain .f; e tHˆ .0/ g/ D

Z Rd

dxEx;0 Œf .X0 /g.X t /e

Rt 0

V .Xs /ds

(3.6.14)

where .X t / t0 D .BT ˆ / t0 is subordinate Brownian motion with respect to the t given subordinator. This is a Lévy process with the property E0;0 Œe iuXs D E0 Œe uuTs

ˆ =2

D e sˆ.uu=2/ :

(3.6.15)


149

3.6.2 Relativistic Kato-class potentials and Lp –Lq boundedness In the previous section we constructed the Feynman–Kac formula of e tHR .a/ . Similarly to K.a/, we can define the relativistic Schrödinger operator KR .a/ as the generator of a C0 -semigroup defined through this Feynman–Kac formula with general external potentials. Definition 3.32 (Relativistic Kato-class). .1/ V is a relativistic Kato-class potential whenever it satisfies Z t lim sup Ex;0 ŒV .BTs /ds D 0: (3.6.16) t#0 x2Rd

0

.2/ V D VC V is relativistic Kato-decomposable whenever VC 2 L1loc .Rd / and V is a relativistic Kato-class. Similarly to the non-relativistic case we can show an exponential bound of relativistic Kato-decomposable potentials by using Khasminskii’s lemma and (3.6.16). Lemma 3.82. Suppose that V is of relativistic Kato-decomposable and write .X t / t0 D .BT t / t0 . Then for any t 0, sup Ex;0 Œe

Rt 0

V .Xs / ds

< 1:

(3.6.17)

x2Rd

Define the semigroup K tR W L2 .Rd / ! L2 .Rd / by .K tR f /.x/ D Ex;0 Œe i

R Tt 0

a.Bs /ıdBs

e

Rt 0

V .BTs /ds

f .BT t /:

(3.6.18)

Theorem 3.83. Let a 2 .L2loc .Rd //d and r a 2 L1loc .Rd /. Suppose that V is relativistic Kato-decomposable. Then the family of operators ¹K tR W t 0º is a symmetric C0 -semigroup. Since subordinate Brownian motion is a Markov process, the proof of this result is similar to that of Theorem 3.65. By the above and the Hille–Yoshida theorem we can define the corresponding operator. Definition 3.33 (Relativistic Schrödinger operator with vector potential and Kato-decomposable potential). Let a 2 .L2loc .Rd //d and r a 2 L1loc .Rd /. Suppose that V is relativistic Kato-decomposable. We call the self-adjoint operator K R .a/ defined by K tR D e tKR .a/ ;

t 0

(3.6.19)

relativistic Schrödinger operator with vector potentials and relativistic Kato-class potentials.

150


Some immediate consequences of the definition are stated below. Corollary 3.84. Suppose the same assumptions as in Theorem 3.83. Then we have: .1/ Diamagnetic inequality: j.f; e tKR .a/ g/j .jf j; e KR .0/ jgj/:

(3.6.20)

.2/ Lp –Lq boundedness: e tKR .a/ is Lp –Lq bounded. These statements can be proven in the same manner as Theorem 3.39 with Brownian motion .B t / t0 replaced by subordinate Brownian motion .X t / t0 D .BT t / t0 .

3.7

Feynman–Kac formula for Schrödinger operator with spin

3.7.1 Schrödinger operator with spin In this section we set d D 3 and we consider the path integral representation of the Pauli operator, which includes spin 1=2. The spin will be described in terms of a Z2 -valued jump process and the Schrödinger operator is defined on L2 .R3 Z2 I C/ instead of L2 .R3 I C 2 /. This construction has the advantage to use scalar valued functions rather than C 2 -valued ones, so in the Feynman–Kac formula of Schrödinger operators with spin a scalar valued integral kernel appears. Let 1 ; 2 ; 3 be the 2 2 Pauli matrices given by 0 1 0 i 1 0 ; 2 D ; 3 D : (3.7.1) 1 D 1 0 i 0 0 1 Each is symmetric and traceless. They satisfy the anticommutation relations ¹ ; º D 2ı

(3.7.2)

and thereby D i

3 X

" ;

(3.7.3)

D1

where "˛ˇ denotes the antisymmetric tensor given by 8 ˆ ˛ˇ D even permutation of 123; 0, and 1 1 1 2 i a.x/ r i r a.x/ a.x/ V .x/ U.x; ˛ / 1 g.x; ˛ / G.0/ D 2 2 2 C e W .x;˛ / g.x; ˛ / D .HS .a/ C 1/g.x; ˛ /: G.s/ is continuous at s D 0 for each .!; / 2 X S, whence Z X Z 1 1 t ds dxf .x; ˛ /Ex;˛ ŒG.s/ lim .f; .K tS 1/g/ D lim t!0 t t!0 t 0 R3 ˛D1;2

D .f; .HZ2 .a/ C 1/g/: Since C01 .R3 Z2 / is a core of HZ2 .a/, (3.7.20) follows.

3.7.4 Extension to singular potentials and vector potentials Next we extend Theorem 3.86 to singular vector potentials a and external potentials V as in the spinless case, following a similar strategy as in Section 3.5.3. Define the quadratic form qaS .f; g/ D . D f; D g/ C .V 1=2 f; V 1=2 g/ with domain Q.qaS / D

T3

D1 ¹f

(3.7.24)

2 L2 .R3 /jD f 2 L2 .R3 /º \ D.V 1=2 /.

Lemma 3.87. Let V 0 be L1loc .R3 / and a 2 .L2loc .R3 //3 . Then qaS is a closed symmetric form. The proof of this is very similar to Lemma 3.68. From here it follows that there exists a self-adjoint operator hS such that .f; hS g/ D qaS .f; g/;

f 2 Q.qaS /; g 2 D.hS /:

(3.7.25)

A sufficient condition for C01 .R3 / to be a core of hS is also derived in a similar way to the spinless case, see Proposition 3.69.

158


Proposition 3.88. .1/ Let V 2 L1loc .R3 / with V 0. If a 2 .L2loc .R3 //3 , then C01 .R3 / is a form core of hS . .2/ Let V 2 L2loc .R3 / with V 0. If a 2 .L4loc .R3 //3 , r a 2 L2loc .R3 / and b 2 .L2loc .R3 //3 , then C01 .R3 / is an operator core of hS . In case (2) above the operator hS can be realized as 1 1 1 hS f D f a .i r/f C a a .i r a/ b f C Vf: 2 2 2 (3.7.26) Instead of hS we define the Schrödinger operator with spin by the sum of the kinetic term and spin. Assumption 3.1. The magnetic field 12 bj , j D 1; 2; 3, is bounded. Let H 0 .a/ be given by Definition 3.27 with V D 0. Under Assumption 3.1 the operator H 0 .a/ 12 b is self-adjoint on D.H 0 .a//. We give the definition of HS .a; b/ below. Definition 3.36 (Schrödinger operator with spin, singular potential and vector potential). Assume that a 2 .L2loc .R3 //3 , 0 V 2 L1loc .R3 /, and that the magnetic field b satisfies Assumption 3.1. We regard b to be independent of a, and denote the selfP V 12 b by HS .a; b/. We call HS .a; b/ Schrödinger adjoint operator H 0 .a/ C operator with spin, singular potential and vector potential. Let HZ2 .a; b/ be the unitary transformed operator to L2 .R3 Z2 / of HS .a; b/, and let HZ0 2 .a; b/ denote HZ2 .a; b/ with V D 0. Next consider general vector potentials having possible zeroes. Note that (3.7.19) is a sufficient condition making sure that Z

tC 0

jW .Bs ; Ns /jdNs < 1;

a:e: .!; / 2 X S:

(3.7.27)

When, however, b1 .x/ ib2 .x/ vanishes for some .x; /, (3.7.27) is not clear. This case is relevant and Theorem 3.86 must be improved since we have to construct the path integral representation of e tHZ2 .a;b/ in which the off-diagonal part b1 ib2 of HZ2 .a; b/ has zeroes or a compact support. The off-diagonal part of HS .a; b/, however, in general may have zeroes. For instance, a for all D 1; 2; 3 have a compact support, and so does the off-diagonal part of b D r a. Therefore, in

159

Section 3.7 Feynman–Kac formula for Schrödinger operator with spin

order to avoid that the off-diagonal part vanishes, we introduce 1 " HZ2 .a; b/f .x; / D H.a; b/ b3 .x/ f .x; / 2 1 .b1 .x/ ib2 .x// f .x; /; ‰" 2

(3.7.28)

where ‰" .z/ D z C "

" .z/

(3.7.29)

for z 2 C and " > 0, with the indicator function ´ 1; jzj < "=2; " .z/ D 0; jzj "=2:

(3.7.30)

ˇ ˇ ˇ ˇ ˇ‰" 1 .b1 ib2 / ˇ > "=2 > 0: ˇ ˇ 2

Thus

Theorem 3.89 (Feynman–Kac formula for Schrödinger operator with spin, singular external potential and vector potential). Suppose that V 2 L1 .R3 /, a 2 .L2loc .R3 //3 , r a 2 L1loc .R3 /, Assumption 3.1 and Z

Z

t

ds

ˇ q ˇ ˇ ˇ ˇlog 1 b1 .y/2 C b2 .y/2 C " ˇ …s .x y/dy < 1 ˇ ˇ 3 2

R

0

for all .x; t / 2 R3 RC and " > 0. Then X Z " " tHZ .a;b/ t 2 g/ D e Ex;˛ Œf .q0 /g.q t /e Zt dx; .f; e 3 ˛D1;2 R

(3.7.31)

(3.7.32)

and .f; e

tHZ2 .a;b/

g/ D lim e "!0

t

X Z 3 ˛D1;2 R

"

Ex;˛ Œf .q0 /g.q t /e Zt dx;

(3.7.33)

where Z Z t" D i Z

Z

t

a.Bs / ı dBs

0 t 0

t

V .Bs /ds 0

Z tC 1 1 log ‰" .b1 .Bs / iNs b2 .Bs // dNs : Ns b3 .Bs /ds C 2 2 0

160


Proof. We show this theorem through a similar limiting argument to Lemma 3.71. .n/ Suppose first that a.n/ 2 .C01 .R3 //3 , n D 1; 2; : : : , such that a ! a in L2loc .R3 / as n ! 1. Take 1R .x/ D .x1 =R/.x2 =R/.x3 =R/, R > 0, where 2 C01 .R/ such that 0 1, .x/ D 1 for jxj < 1 and .x/ D 0 for jxj 2. In the same way as in Lemma 3.70 we can show that e

" tHZ .1R a.n/ ;b/ 2

e

" tHZ .1R a;b/ 2

!e !e

" tHZ .1R a;b/

as n ! 1;

2

" tHZ .a;b/ 2

(3.7.34)

as R ! 1

(3.7.35)

in the strong sense. In the proofRof Lemma 3.71 we have Ralready shown that there 0 t t exists a subsequence n0 such that 0 1R a.n / .Bs / ı dBs ! 0 1R a.Bs / ı dBs almost R R t t surely as n0 ! 1, and R0 such that 0 1R0 a.Bs / ı dBs ! 0 a.Bs / ı dBs . We reset n0 and R0 as n and R, respectively. Let Z t" .n; R/ be Z t with a replaced by 1R a.n/ . Hence we conclude that Z Z " " lim lim Ex;˛ Œf .q0 /g.q t /e Zt .n;R/ dx D Ex;˛ Œf .q0 /g.q t /e Zt dx: R!1 n!1 R3

R3

Combining this with (3.7.34) and (3.7.35), we obtain (3.7.32). Since e strongly convergent to e tHZ2 .a;b/ as " ! 0, (3.7.33) also follows.

" tHZ .a;b/ 2

is

In case when the off-diagonal part identically vanishes, i.e. b D .0; 0; b3 /, we have 1 ‰" .b1 .x/ iNs b2 .x// D " 2 and

1 log ‰" .b1 .Bs / iNs b2 .Bs // dNs D log "N t : 2 0 Then it can be seen that Z

tC

"

lim Ex Œe Zt g.q t / D lim Ex Œe i "#0

Rt 0

Rt Rt a.Bs /ıdBs 0 V .Bs /ds 0 . 12 /Ns b3 .Bs /ds Nt

"#0

" g.qt /:

Notice that as " ! 0 the functions on K t D ¹ 2 S j N t . / 1º vanish and those on K tc D ¹ 2 S j N t . / D 0º stay different from zero. Note also that for 2 K tc , Ns . / D 0 whenever 0 s t , as N t is counting measure. Since ´ 0; 2 K t ; lim "N t ./ D "#0 1; 2 K tc ; we obtain that "

lim e t Ex;˛ Œe Zt g.q t / "#0

D e t Ex Œe i

Rt 0

Rt Rt a.Bs /ıdBs 0 V .Bs /ds 0 . 12 /˛ b3 .Bs /ds

g.B t ; ˛ /:

Clearly, the right-hand side in the expression above describes the diagonal operator.

161

Section 3.7 Feynman–Kac formula for Schrödinger operator with spin

By Lemma 3.71 we have a diamagnetic inequality for HZ2 .a; b/. Corollary 3.90 (Diamagnetic inequality). Under the assumptions of Lemma 3.88, j.f; e tHZ2 .a;b/ g/j .jf j; e tHZ2 .0;b0 / jgj/;

(3.7.36)

q where b0 D . b12 C b22 ; 0; b3 /.

Proof. This follows from the fact that j‰" .z/j jzj C " for all z 2 C, je

R tC 0

log ‰" . 12 .b1 .Bs /iNs b2 .Bs ///dNs

j.f; e

g/j lim "#0

je

X Z

and tHZ2 .a;b/

R tC

˛D1;2

R3

0

log. 12

q b12 .Bs /Cb2 .Bs /2 C"/dNs

O"

Ex;˛ Œjf .q0 /jg.q t /je Zt dx;

ZO t"

is defined by Z t t 1 V .Bs /ds Ns b3 .Bs /ds 2 0 0 q Z tC 1 2 2 C log b1 .Bs / C b2 .Bs / C " dNs : 2 0

where the exponent

Z

Theorem 3.91. Suppose that a 2 .L2loc .R3 //3 , r a 2 L1loc .R3 /, and b satisfies Assumption 3.1 and (3.7.31). .1/ Let V be a real-valued multiplication operator relatively bounded (resp. form bounded) with respect to .1=2/ with a relative bound c. Then V is also relatively bounded (resp. form bounded) with respect to HZ0 2 .a; b/ with a relative bound not exceeding c. .2/ Let V be such that VC 2 L1loc .R3 / and V is relatively form bounded with respect to .1=2/ with a relative bound less than 1. Then P VC HZ0 2 .a; b/ C P V

(3.7.37)

can be defined as a self-adjoint operator. This also will be denoted by HZ2 .a; b/. .3/ Let V satisfy the same assumptions as in .2/. Then the Feynman–Kac formula of .f; e tHZ2 .a;b/ g/ is given by (3.7.32) and (3.7.33). Proof. By the diamagnetic inequality kjW j˛ .HZ0 2 .a; b/ C E/˛ f k kf k

kjW j˛ .HZ0 2 .0; b0 / C E/˛ f k kf k

(3.7.38)

for ˛ D 1 and 1=2. .1/ follows by the same argument as Lemma 3.72 and the boundedness of the magnetic field bj . .2/ is implied by the KLMN theorem. Thus by Theorem 3.86 and the same limiting argument as in Theorem 3.31 also .3/ follows.

162

3.8


Feynman–Kac formula for relativistic Schrödinger operator with spin

By the methods developed above it is possible to derive a Feynman–Kac formula also for relativistic Schrödinger operators with spin 1=2. This operator is formally given by q .i r a/2 b C m2 m C V: (3.8.1) We have seen that C01 .R3 / is a form core of . .i r a//2 under suitable conditions on a in the previous section. Here we have the following counterpart to this, similarly proven as Proposition 3.77. Proposition 3.92. Let a 2 .L2loc .R3 //3 and suppose that b satisfies Assumption 3.1. Let 0 HRS .a/ D .2HS0 .a; b/ C m2 /1=2 m:

(3.8.2)

Then (1) und (2) below follow. 0 .1/ C01 .R3 / is a form core of HRS .a/.

.2/ Suppose that a 2 .L4loc .R3 //3 and r a 2 L2loc .R3 /. Then C01 .R3 / is an 0 .a/. operator core of HRS 0 In particular, from Proposition 3.92 it follows that Q.HRS .a// C01 .R3 / if 2 0 3 3 a 2 .Lloc .R // . Thus Q.V / \ Q.HRS .a// is dense whenever V 2 L1loc .R3 /, and 0 .a/ and V can be densely defined. then the form sum of HRS

Definition 3.37 (Relativistic Schrödinger operator with spin 1=2). Let V 0 be in L1loc .R3 /, a 2 .L2loc .R3 //3 , and Assumption 3.1 hold. We call P V HRS .a; b/ D .2HS0 .a; b/ C m2 /1=2 m C

(3.8.3)

relativistic Schrödinger operator with spin 1=2, where m is assumed to be sufficiently large such that 2HS0 .a; b/ C m2 0. As it was done for HS .a; b/, we transform HRS .a; b/ by a unitary map to the selfadjoint operator P V HRZ2 .a; b/ D .2HZ0 2 .a; b/ C m2 /1=2 m C

(3.8.4)

on L2 .R3 Z2 /. In order to derive the Feynman–Kac formula for e tHRZ2 .a;b/ we need the random process .qQ T t / t0 D .BT t ; T t / t0

(3.8.5)

163

Section 3.8 Feynman–Kac formula for relativistic Schrödinger operator with spin

on X S T under the probability measure W x P , where .T t / t0 is the p 2 subordinator given by E Œe uT t D e t. 2uCm m/ . Write Ex;˛;0 D Ex;˛;0 , and W P let .; t / be the distribution of T t on RC , one of the rare cases which can be described explicitly by a formula: t 1 t2 2 C m exp r C mt ; 2 r .2 r 3 /1=2

.r; t / D

r 2 RC :

(3.8.6)

To avoid zeroes of the off-diagonal part of the spin we introduce the cutoff function ‰" as defined in (3.7.29). We write " P V; HRZ .a; b/ D .2HZ0;" .a; b/ C m2 /1=2 m C 2 2

.a; b/ is defined by HZ0 2 .a; b/ with the off-diagonal part .1=2/.b1 ib2 / where HZ0;" 2 of the spin replaced by ‰" ..1=2/.b1 ib2 //. Theorem 3.93 (Feynman–Kac formula for relativistic Schrödinger operator with spin 1=2). Let V 0 be in L1loc .R3 /, a 2 .L2loc .R3 //3 and r a 2 L1loc .R3 /. Suppose that b satisfies Assumption 3.1 and Z

Z

1

Z

r

dr.r; t /

ds

0

0

ˇ q ˇ ˇ ˇ 1 dy ˇˇlog b1 .y/2 C b2 .y/2 C " ˇˇ …s .y x/ < 1 2 R3 (3.8.7)

for all " > 0 and all .x; t / 2 R3 RC . Then .f; e

" tHRZ .a;b/ 2

g/ D

X Z

Q"

3 ˛D1;2 R

Ex;˛;0 Œe T t f .qQ 0 /g.qQ T t /e Zt dx

(3.8.8)

and .f; e tHRZ2 .a;b/ g/ D lim "#0

X Z

Q"

3 ˛D1;2 R

Ex;˛;0 Œe T t f .qQ 0 /g.qQ T t /e Zt dx;

(3.8.9)

where Z ZQ t" D i Z C

0

0

Z

Tt

a.Bs / ı dBs

Z

t 0

V .BTs /ds

Tt 0

1 Ns b3 .Bs /ds 2

1 .b1 .Bs / iNs b2 .Bs // log ‰" dNs : 2

Tt C

164


Proof. The proof is a combination of the formulae already established for HR .a/ and HZ2 .a; b/. We only give an outline. Note that ˇ

ˇ Z T t C ˇ ˇ 1 x;˛;0 ˇ E .b1 .Bs / iNs b2 .Bs // log ‰" dNs ˇˇ ˇ 2 0 ˇ

Z Tt ˇ q Z ˇ ˇ 1 2 2 ˇ ds dy…s .y x/ˇ log b1 .y/ C b2 .y/ C " ˇˇ E 2 0 R3 ˇ q ˇ Z r Z Z 1 ˇ ˇ 1 dr.r; t / ds dy…s .y x/ ˇˇlog b1 .y/2 C b2 .y/2 C " ˇˇ D 2 R3 0 0 RT C which is bounded by (3.8.7). Thus 0 t log.‰" . 12 .b1 .Bs / iNs b2 .Bs ////dNs is almost surely finite for every .x; ˛/. First consider V D 0. The Feynman–Kac formula for .f; e

0;" tHRZ .a;b/ 2

g/ D E Œ.f; e

0;" T t HZ .a;b/ 2

g/

is given by Theorem 3.89, namely the time t in the Feynman–Kac formula of tH 0;" .a;b/

Z2 g/ is replaced by a random time given by the subordinator T t , and .f; e it is integrated against , the measure associated with .T t / t0 . Next, for a continuous bounded V , using the Trotter product formula

lim .e

n!1

0;" .t=n/HRZ .a;b/ .t=n/V n 2

e

/ De

" tHRZ .a;b/ 2

;

we arrive at (3.8.8). A limiting arguments on V and for " ! 0 give then (3.8.9). Remark 3.8. Recall that .r; t / is the distribution of T t . Then clearly ˇ q ˇ Z 1 Z r Z ˇ ˇ 1 2 2 ˇ dr.r; t / ds dy ˇlog b1 .y/ C b2 .y/ C " ˇˇ …s .y x/ 2 R3 0 0 ˇ q ˇ Z 1 Z ˇ ˇ 1 2 2 ˇ ds dy ˇlog b1 .y/ C b2 .y/ C " ˇˇ …s .y x/: (3.8.10) 2 R3 0 Since the right-hand side of (3.8.10) bounded, this is sufficient for condition (3.8.7) to hold. From the above Feynman–Kac formula we can derive a diamagnetic inequality for HRZ2 .a; b/, however, this expression is not as simple as in the previous cases. In tH 0 .a;b/

Z2 the previous section we obtained the g/j qdiamagnetic inequality j.f; e 0 tHZ .0;b0 / 2 .jf j; e jgj/, with b0 D . b12 C b22 ; 0; b3 /. Note that HZ0 2 .0; b0 / can be

negative, however, in virtue of the diamagnetic inequality inf Spec.HZ0 2 .0; b0 //

inf Spec.HZ0 2 .a; b//. The difficulty here is that the formula .f; e t

p

2H Cm2 m g/

D

165

Section 3.8 Feynman–Kac formula for relativistic Schrödinger operator with spin

E Œ.f; e T t H g/, m 0, only holds for positive self-adjoint operators H . To derive the diamagnetic inequality for HRZ2 .a; b/ we define HQ RZ2 .a; b/ D

q

P V; 2.HZ0 2 .a; b/ E/ C m2 m C

(3.8.11)

where E D inf Spec HZ0 2 .0; b0 / and HZ0 2 .a; b/ E 0 follows. Thus HQ RZ2 .0; b0 / D

q P V: 2.HZ2 .0; b0 / E/ C m2 m C

(3.8.12)

From the Feynman–Kac formula in Theorem 3.93 we immediately derive Corollary 3.94 (Diamagnetic inequality). Under the assumptions of Theorem 3.93 Q

Q

j.f; e t HRZ2 .a;b/ g/j .jf j; e t HRZ2 .0;b0 / jgj/

(3.8.13)

holds. Using this inequality we can give a similar results to Theorem 3.91. Theorem 3.95. Suppose that a 2 .L2loc .R3 //3 , r a 2 L1loc .R3 /. Let b satisfy Assumption 3.1 and (3.8.7). .1/ Let V be a real-valued multiplication operator relatively bounded (resp. form p bounded) with respect to C m2 with a relative bound c. Then V is also 0 relatively bounded (resp. form bounded) with respect to HRS .a; b/ with a relative bound not exceeding c. .2/ Let V be such that VC 2 L1loc .R3 / and V is relatively form bounded with p respect to C m2 with a relative bound less than 1. Then 0 P VC HRS .a; b/ C P V

(3.8.14)

can be defined as a self-adjoint operator. This also will be denoted by HRS .a; b/. .3/ Let V satisfy the same assumptions as in .2/. Then the Feynman–Kac formula of .f; e tHRS .a;b/ g/ is given by (3.8.8) and (3.8.9). Proof. Set S D .1=2/ b0 and E D inf Spec.HS0 .0; b0 //. Notice that q p k C m2 f k2 D k 2HS0 .0; b0 / 2E C m2 f k2 C .f; 2.S E/f /: Since j.f; 2Sf /j 0 kf k2 with a constant 0 , we have q p k C m2 f k2 k 2HS0 .0; b0 / 2E C m2 f k2 C .j2Ej C 0 /kf k2 :

166


p Together with kVf k ck C m2 f k C "0 kf k2 with some constant "0 , we have q kVf k ck 2HS0 .0; b0 / 2E C m2 f k C C kf k (3.8.15) with some constant C . By (3.8.15) and the diamagnetic inequality in Corollary 3.94, q

V is also relatively bounded with respect to 2HS0 .a; b/ C m2 2E with the same relative bound c. Moreover, since the difference q q 0 2 T D 2HS .a; b/ C m 2E 2HS0 .a; b/ C m2 is aqbounded operator with kT k

p j2Ej, V is also relatively bounded with respect

to 2HS0 .a; b/ C m2 with relative bound c. Thus .1/ follows. The form version is similarly proven. The proofs of .2/ and .3/ are the same as Theorem 3.91. p The essential self-adjointness of . .i r a//2 C m2 m C V on C 2 ˝ C01 .R3 / can also be obtained from Proposition 3.88 and .1/ of Theorem 3.95. Theorem 3.96 (Essential self-adjointness of relativistic Schrödinger operator with p spin). Let V be relatively bounded with respect to C m2 with a relative bound < 1. Suppose that a 2 .L4loc .R3 //3 , r a 2 L2loc .R3 /, and r a D b satisfies p Assumption 3.1 and (3.8.7). Then . .i r a//2 C m2 m C V is essentially self-adjoint on C 2 ˝ C01 .R3 /. p Proof. By Proposition 3.92,p . .i r a//2 C m2 is essentially self-adjoint on 0 C 2 ˝ C01 .R3 /. Note that . .i r a//2 C m2 m D HRS .a; b/, since . 0 2 .i r a// can be expanded and equals HS .a; b/. Then .1/ of Theorem 3.95 yields p that V is relatively bounded with respect to . .i r a//2 C m2 with a relative bound < 1. Hence the theorem follows by the Kato–Rellich theorem.

3.9

Feynman–Kac formula for unbounded semigroups and Stark effect

Throughout the previous sections the Schrödinger operator H was required to be bounded from below, so that e tH is bounded for all t 0. In what follows we derive a Feynman–Kac formula for an unbounded semigroup e tH with interesting applications. Definition 3.38 (Stark Hamiltonian). Let E 2 R3 . The operator H.E/ D .1=2/ C E x C V .x/ on L2 .R3 / is called Stark Hamiltonian.

Section 3.9 Feynman–Kac formula for unbounded semigroups and Stark effect

167

The Stark Hamiltonian describes the interaction between a charged quantum particle and an external electrostatic field E, leading to a shift and split-up of the particle’s spectral lines. In case when V .x/ D 1=jxj, H.0/ has a point spectrum. In contrast, the spectrum of H.E/ is the whole of R when E 6D 0, in particular, it has no point spectrum at all. Lemma 3.97 (Faris–Levine). Let V and W be real-valued measurable functions on Rd such that V .x/ cjxj2 d , with c; d 0, and V 2 L2loc .Rd /. Suppose that .1/ there exists a dense subset D such that D D..1=2// \ D.V / \ D.W / and xj D D and @j D D so that .1=2/ C V C W C 2cjxj2 is essentially self-adjoint on D, .2/ .a=2/ C W is bounded from below on D for some a < 1. Then .1=2/ C V C W is essentially self-adjoint on D. Two immediate corollaries of Lemma 3.97 follow. Corollary 3.98. Let V1 and V2 be such that .1/ V2 < .1=2/ with a relative bound less than 1, .2/ V1 cjxj2 d for some c and d and V1 2 L2loc .Rd /. Then .1=2/ C V1 C V2 is essentially self-adjoint on C01 .Rd /. Proof. We apply Lemma 3.97. Let V D cjxj2 d and W D V1 C V2 C cjxj2 C d . Then V cjxj2 d and C01 D..1=2// \ D.V / \ D.W / hold trivially. .1=2/ C V C W C 2cjxj2 D .1=2/ C .V1 C 2cjxj2 / C V2 is essentially self-adjoint on C01 .Rd /, since by assumption .1/ above V1 C 2cjxj2 C d > 0 and V1 C 2cjxj2 2 L2loc .Rd /; see Corollary 3.15. Moreover, .a=2/ C W is bounded from below. Hence by Lemma 3.97 the statement follows. The following corollary covers the Stark Hamiltonian. Corollary 3.99. Let V < .1=2/ with a relative bound less than 1. Then the Stark Hamiltonian H.E/ is essentially self-adjoint on C01 .R3 /. Lemma 3.100. Choose Coulomb potential V .x/ D 1=jxj. Then for all E 2 R3 , inf Spec.H.E// D 1. In particular, e tH.E / is unbounded. Proof. Let H0 .E/ D .1=2/ C E x. Since V < 0, it suffices to show that Spec.H0 .E// D R. By rotation symmetry this can be reduced to addressing the case E D ."; 0; 0/. Let x D .x0 ; x? / 2 R R2 and p D .p0 ; p? / be its conjugate 2 / C "x0 . Let Uk D momentum. The Hamiltonian becomes H0 .E/ D .1=2/.p02 C p?

168

Chapter 3 Feynman–Kac formulae 3

e i.1=6"/p0 and F? denote Fourier transform with respect to x? . Then U D F? Uk W L2 .Rx R2k? / ! L2 .R3 / maps S .R3 / onto itself. Furthermore, we have 2 U 1 H0 .E/U D .1=2/k? C "x0

(3.9.1)

on S .R3 /. Since S .R3 / is a core of both H0 .E/ and the multiplication operator on 2 the right-hand side of (3.9.1), H0 .E/ and T D .1=2/k? C x0 are unitary equivalent as self-adjoint operators. Since Spec.T / D R, the lemma follows. Next we establish a Feynman–Kac formula for a class of unbounded semigroups including in particular the Stark Hamiltonian. Recall that H D .1=2/ C V . Our assumption on V is that for any " > 0 there exists C" such that V .x/ "jxj2 C" . It is useful to rewrite the Feynman–Kac formula by using Brownian bridge starting from a at t D 0 and ending in b at t D T , i.e., the solution of the stochastic differential equation dX t D

b Xt dt C dB t ; T t

0 t < T;

X0 D a:

This equation can be solved exactly to obtain Z t 1 t t dBs ; Xt D a 1 C b C .T t / T T 0 T s

t < T;

(3.9.2)

(3.9.3)

see Example 2.9 and (2.3.49). Note that X t ! b almost surely as t ! T . Hence, for continuous V bounded from below, Z tH g/ D dxdyf .x/g.y/Q.x; yI V; t /… t .x y/; (3.9.4) .f; e Rd Rd

where by (3.9.3)

Z t s s Q.x; yI V; t / D E exp V 1 x C y C ˛s ds ; t t 0

(3.9.5)

and ˛s is the Gaussian random variable defined by ´ Rs 1 dBr ; s < t .t s/ 0 tr ˛s D 0; s D t: Note that EŒ˛s D 0 and EŒ˛s ˛s 0 D s.1 this section we prove the theorem below.

s0 t /

for s s 0 t . In the remainder of

Theorem 3.101 (Feynman–Kac formula for unbounded semigroup). Assume that V is continuous and for any " > 0 there exists C" such that V .x/ "jxj2 C . Let f; g 2 L2 .Rd / have compact support. Then f; g 2 D.e tH / and (3.9.4) holds.

Section 3.9 Feynman–Kac formula for unbounded semigroups and Stark effect

169

In order to prove this theorem we first need the following lemma. Lemma 3.102. Let An and A be self-adjoint operators such that An ! A in strong resolvent sense. Also, let f be a continuous function and 2 D.f .An // for all n. Then .1/ If supn kf .An / k < 1, then .2/ If supn kf .An

/2

2 D.f .A//.

k < 1, then f .An /

! f .A/

in strong sense as n ! 1.

Proof. .1/ Let 8 ˆ f .x/ m; <m; fm .x/ D f .x/; jf .x/j m; ˆ : m; f .x/ m: As fm is bounded, fm .An / ! fm .A/ strongly as n ! 1. Since kfm .A/ k D lim kfm .An / k sup kfm .An /k sup kf .An /k; n!1

n

n

we have supm kfm .A/ k < 1. This implies 2 D.f .A//. .2/ Since k.f .An / fm .An // k m1 kf .An /2 k, it follows that fm .An / ! f .An / , uniformly in n. Then f .An / ! f .A/ since fm .An / ! fm .A/ . Proof of Theorem 3.101. Let Vn D max¹V .x/; nº and Hn D .1=2/ C Vn . Since C01 .Rd / is a common core of Hn , Hn ! H as n ! 1 in strong resolvent sense. For f 2 L2 .Rd / with compact support we prove sup ke tHn f k < 1:

(3.9.6)

n

Note that

Z t s s exp Vn 1 x C y C ˛s t t 0 Z t 2 s s exp " 1 x C y C ˛s C C" ds t t 0 Z t ds exp C" t C 2"t .x 2 C y 2 / C 2"t ˛s2 : t 0

By Jensen’s inequality e

2"t

and therefore EŒe

2"t

Rt 0

Rt 0

˛s2 ds t

˛s2 ds t

Z

t

0

Z

0

t

2

e 2"t˛s

ds ; t 2

EŒe 2"t˛s

ds : t

170


A direct computation gives EŒe

2"t˛s2

1 D p 2s

Z

e 2"ty e y 2

2 =2

s

1 dy D p ; .1 4"t s /

where s D s.1 st / is the covariance of ˛s and we suppose that " < 1. Set " D ı=t 2 with some ı < 1. Since s t=2 D t =4, we have e C t 2 2 Q.x; yI V; t / p e 2ı.x Cy /=t .1 ı/ Hence e 2C t ke tHn f k2 p .1 ı/

Z Rd Rd

dxdyjf .x/f .y/j…2t .xy/e .ı=t/.x

(3.9.7)

2 Cy 2 /

< 1;

uniformly in n. Thus (3.9.6) follows. By Lemma 3.101 .1/, f 2 D.e tH / and e tHn f ! e tH f strongly as n ! 1. Thus the left-hand side of (3.9.4) converges and so does the right-hand side by (3.9.7) and the dominated convergence theorem.

3.10

Ground state transform and related diffusions

3.10.1 Ground state transform and the intrinsic semigroup The Feynman–Kac semigroup ¹K t W t 0º has the particularity that in general K t 1Rd ¤ 1Rd . In this section we introduce a semigroup related with the Feynman– Kac semigroup given by Definition 3.20 which, however, does keep unit mass. This is called the intrinsic Feynman–Kac semigroup. We assume that V is Kato-decomposable and the corresponding Schrödinger operator is defined through the Feynman–Kac formula, as in Definition 3.23. We suppose, moreover, that H has a normalized ground state ‰p > 0, i.e., H ‰p D E.H /‰p ;

E.H / D inf Spec.H /;

k‰p k2 D 1:

(3.10.1)

We set HN D H E.H / so that HN ‰p D 0. Definition 3.39 (Ground state transform). Let H be a Schrödinger operator with Kato-decomposable potential V , and ‰p > 0 be the normalized unique ground state of H . Let, moreover, d N0 D ‰p2 .x/dx

(3.10.2)

be a probability measure on Rd and consider L2 .Rd ; d N0 /. The ground state transform is a unitary map U W L2 .Rd ; d N0 / ! L2 .Rd ; dx/;

f 7! ‰p f:

Section 3.10 Ground state transform and related diffusions

171

In fact, this is a case of Doob’s h-transform familiar from the theory of random processes. Define the operator L D U 1 HN U

(3.10.3)

with domain D.L/ D ¹f 2 L2 .Rd ; d N0 / j U f 2 D.H /º: Also, let E.H /t K .x; y/ t e t .x; y/ D e K ; ‰p .x/‰p .y/

(3.10.4)

where K t .x; y/ denotes the integral kernel of K t . Definition 3.40 (Intrinsic Feynman–Kac semigroup). The one-parameter semigroup e t W t 0º acting on L2 .Rd ; d N0 / with integral kernel K e t .x; y/, i.e., ¹K Z e t f .x/ D e t .x; y/‰p2 .y/dy K f .y/K (3.10.5) Rd

is called intrinsic Feynman–Kac semigroup for the potential V . The generator of the intrinsic semigroup is the operator L given by (3.10.3). Notice that even though the Lp -norms of the operators K t can be larger than 1, the operators e t are always contractions. Moreover, the intrinsic Feynman–Kac semigroup is more K e t 1Rd D 1Rd , for every t > 0. natural than ¹K t W t 0º since K In view of Examples 3.20 and 3.22 it can be expected that there are important differences between Schrödinger (and therefore Feynman–Kac) semigroups depending on the behaviour of V at infinity. Here we only discuss one particular regularizing property called ultracontractivity; for further properties such as hypercontractivity and supercontractivity we refer to the literature. Definition 3.41 (Ultracontractivity properties). Let ¹e tH W t 0º be a Schrödinger semigroup on L2 .Rd /. .1/ ¹e tH W t 0º is said to be ultracontractive if e tH is contractive from L2 .Rd / to L1 .Rd /, for all t > 0. .2/ Suppose that H has a strictly positive ground state. Then ¹e tH W t 0º is called intrinsically ultracontractive (IUC) if e tL is ultracontractive on L2 .Rd ; d N0 /, for all t > 0. .3/ We call ¹e tH W t 0º asymptotically intrinsically ultracontractive (AIUC) if there exists t0 > 0 such that e tL is ultracontractive on L2 .Rd ; d N0 /, for all t > t0 .

172


Theorem 3.103 (Intrinsic ultracontractivity). Let H be a Schrödinger operator with a Kato-decomposable potential V and ground state ‰p . Consider its Feynman–Kac semigroup ¹K t W t 0º. Then ¹e tH W t 0º is IUC (resp. AIUC) if and only if for every t > 0 (resp. if for every t > t0 , for some t0 > 0) there is a constant CV;t > 0 such that K t .x; y/ CV;t ‰p .x/‰p .y/;

x; y 2 Rd ;

(3.10.6)

or equivalently, e t .x; y/ CV;t ; K

x; y 2 Rd :

(3.10.7)

A consequence of intrinsic ultracontractivity is that a similar lower bound on the .1/ kernel also holds, i.e., for every t > 0 there is a constant CV;t > 0 such that .1/

K t .x; y/ CV;t ‰p .x/‰p .y/;

x; y 2 Rd :

(3.10.8)

An immediate consequence of this is that if the semigroup is intrinsically ultracontractive, then ‰p 2 L1 .Rd /. The classic result for the Feynman–Kac semigroup generated by Schrödinger operators H D .1=2/ C V identifying the borderline case for IUC is the following fact. Theorem 3.104. If V .x/ D jxjˇ , then the Schrödinger semigroup ¹e tH W t 0º is intrinsically ultracontractive if and only if ˇ > 2. Moreover, if ˇ > 2, then c'.x/ ‰p .x/ C '.x/, jxj > 1, holds with some C; c > 0 and where '.x/ D jxjˇ=4C.d 1/=2 exp.2jxj1Cˇ=2 =.2 C ˇ//:

For our purposes below an important consequence of AIUC is the following property. Lemma 3.105. The following two conditions are equivalent. .1/ The semigroup ¹K t W t 0º is AIUC. .2/ The property e t .x; y/ t!1 K ! 1; holds, uniformly in .x; y/ 2 Rd Rd .

(3.10.9)

173


Proof. The implication .2/ ) .1/ is immediate, we only show the converse statement. We have for every x; y 2 Rd and t > t0 ˇ ˇ ˇe K t .x; y/ 1ˇ ˇ ˇZ ˇ e E.H /t ‰p .x/‰p .y/ ˇˇ K t0 .x; z/K t2t0 .z; w/K t0 .w; y/ ˇ dzdw Dˇ ˇ ˇ RdRd e E.H /t ‰p .x/‰p .y/ e E.H /t ‰p .x/‰p .y/ ˇ ˇ ˇZ ˇ ˇ K t0 .x; z/‰p .z/.K t2t0 .z; w/ e E.H /.t2t0 / ‰p .z/‰p .w//K t0 .w; y/‰p .w/ ˇ ˇ dzdw Dˇ ˇ E.H /t ˇ ˇ RdRd e ‰p .x/‰p .z/‰p .w/‰p .y/ 2 K t0 .x; y/ e E.H /t ‰ .x/‰ .y/ p p 1 Z ˇ ˇ ˇ ˇ ˇK t2t0 .z; w/ e E.H /.t2t0 / ‰p .z/‰p .w/ˇ ‰p .z/‰p .w/dzdw RdRd

Z

C e E.H /t

RdRd

jK t2t0 .z; w/ e E.H /.t2t0 / ‰p .z/‰p .w/j2 dzdw

1=2 :

The last factor on the right-hand side is the Hilbert–Schmidt norm of the operator K t2t0 e E.H /.t2t0 / 1p , where 1p W L2 .Rd / ! L2 .Rd / is the projection onto the one dimensional subspace of L2 .Rd / spanned by ‰p . This gives e t .x; y/ 1j C e tE.H / jK

1 X

e 2Ek .H /.t2t0 /

1=2

kD1

D C e 2t0 E1 .H / e .E1 .H /E.H //t

1 X

e 2.Ek .H /E1 .H //.t2t0 /

1=2 ;

kD1

where E.H / < E1 .H / are the eigenvalues of H . By the Lebesgue dominated convergence theorem the last sum converges to the multiplicity of E1 .H / as t ! 1. Since E1 .H / > E.H /, (3.10.9) follows. Remark 3.9. Ultracontractivity properties can be investigated also for fractional Schrödinger operators as in Definition 3.3. In that case the borderline case of IUC is V .x/ D log jxj, indicating that it is “easier” for a càdlàg process to be intrinsically ultracontractive. This can be explained by an efficient strong mixing mechanism due to the possibility of jumps in contrast with the continuous paths of Brownian motion. Also, with 1 V .x/ D log jxj1¹jxj>1º .x/ 1¹jxj1º .x/; jxj˛=2 the fractional Feynman–Kac semigroup ¹K t W t 0º corresponding to ./˛=2 C V is AIUC but it is not IUC. This contrasts the case of Schrödinger operators and Brownian motion where the semigroup for the borderline potential is neither IUC nor AIUC.

174


3.10.2 Feynman–Kac formula for P./1 -processes Whenever a Schrödinger operator has a ground state, a Feynman–Kac formula of the related Schrödinger semigroup can be constructed on a path space in terms of a diffusion process. The so obtained process is a P ./1 -process for the given V . We will construct a diffusion process associated with the operator L. In general, the differentiability of the ground state ‰p of H is not clear. However, whenever r log ‰p exists, we have 1 Lf D f r log ‰p rf: 2

(3.10.10)

If ‰p is such that r log ‰p is not well-defined, the above SDE is a formal description only with no rigorous meaning in this form. The three-dimensional Coulomb potential 1 V D jxj is a case in point: then ‰p .x/ D C exp.jxj/, which is not C 1 .R3 /. A solution .X tx / t0 of the SDE dX t D r log ‰p dt C dB t ;

X0 D x

(3.10.11)

satisfies that EŒf .X tx / D .e tL f /.x/:

(3.10.12)

It is also not clear that (3.10.11) is a well defined stochastic differential equation and when does it have a solution. A sufficient condition for the existence of a solution of (3.10.11) is the Lipshitz condition (2.3.34)–(2.3.35) in Theorem 2.35 on log ‰p , however, in general this is difficult to check. Therefore in a next section we adopt a different way of constructing the process for which (3.10.12) is satisfied. Consider two-sided Brownian motion as defined at the end of Section 2.1, and denote by X D C.R; Rd / the space of continuous paths on the real line R as before. In this section we construct a probability measure N0x , x 2 Rd , on .X; B.X// such that the coordinate process .X t / t2R is a diffusion process generated by L. Theorem 3.106. Suppose that V is Kato-decomposable and H has a ground state ‰p . Let L D U 1 HN U be the ground state transformation of H , and X t .!/ D !.t / be the coordinate process on .X; B.X//. Then there exists a probability measure N0x on .X; B.X// satisfying the following properties: .1/ (Initial distribution) N0x .X0 D x/ D 1. .2/ (Reflection symmetry) .X t / t0 and .Xs /s0 are independent, and d

Xt D X t ;

t 2 R:

175


.3/ (Diffusion property) Let F tC D .Xs ; 0 s t /;

F t D .Xs ; t s 0/

be given filtrations. Then .X t / t0 and .Xs /s0 are diffusion processes with respect to .F tC / t0 and .F t / t0 , respectively, i.e., EN0x ŒX tCs jFsC D EN0x ŒX tCs j.Xs / D EN Xs ŒX t ; 0

E

N0x

ŒXts jFs

DE

N0x

ŒXts j.Xs / D EN Xs ŒXt 0

for s; t 0, and R 3 t 7! X t 2 Rd is a.s. continuous, where EN Xs means 0 EN y evaluated at y D Xs . 0

.4/ (Shift invariance) Let 1 < t0 t1 tn < 1. Then Z Rd

EN0x

n hY

i fj .X tj / d N0 D .f0 ; e .t1 t0 /L f1 e .tn tn1 /L fn /L2 .Rd ;d N0 /

j D0

(3.10.13) for fj 2 L1 .Rd /, j D 1; : : : ; n, and the finite dimensional distributions of the process are shift invariant, i.e., Z Rd

E

n hY N0x

Z i fj .X tj / d N0 D

j D1

Rd

E

n hY N0x

i fj .X tj Cs / d N0 ;

s 2 R:

j D1

Definition 3.42 (P ./1 -process). Suppose H has a ground state ‰p and consider the probability space .X; B.X/; N0x /. The coordinate process X t .!/ D !.t / on this space is called a P ./1 -process for H . We proceed now to prove existence and some properties of P ./1 -processes in several steps. Define the set function t0 ;:::;tn W jnD0 B.Rd / ! R for 0 t0 t1 tn by t0 ;:::;tn niD0 Ai D .1A0 ; e .t1 t0 /L 1A1 e .tn tn1 /L 1An /L2 .Rd ;d N0 / (3.10.14) and for t 0, t W B.Rd / ! R by t .A/ D .1; e tL 1A /L2 .Rd ;d N0 / D .1; 1A /L2 .Rd ;d N0 / :

(3.10.15)

Step 1: The family of set functions ¹ƒ ºƒR;jƒj 0 and suppose that f 2 C.Rd / \ D.L1=2 /. Then it follows that e sup jf .Xs /j ƒ ..f; f /L2 .Rd ;d N0 / CT .L1=2 f; L1=2 f /L2 .Rd ;d N0 / /1=2: ƒ 0sT (3.10.36) (Here e is the base of natural logarithm.) N0

Proof. Set Tj D Tj=2n , j D 0; 1; : : : ; 2n , and fix T and n. Let G D ¹x 2 Rd jjf .x/j ƒº;

(3.10.37)

and define the stopping time D inf¹Tj 0j XTj 2 Gº:

(3.10.38)

Then the identity N0

sup

j D0;:::;2n

jf .XTj /j ƒ D N0 . T /

follows. We estimate the right-hand side above. Let 0 < % < 1 be fixed and choose a suitable % later. We see that by Schwarz inequality with respect to d N0 , N0 . T / D EN0 Œ1¹T º EN0 Œ%T Z 1=2 T T x 2 % EN0 Œ% % .EN0 Œ% / d N0 ; Rd

(3.10.39)

where ExN0 D EN0x . Let 0 be any function such that .x/ 1 on G. Then the Dirichlet principle (shown in the lemma below) implies that Z .ExN0 Œ% /2 d N0 Rd

n

. ; /L2 .Rd ;d N0 / C Inserting

%T =2 .T =2n /L / /L2 .Rd ;d N0 / : n . ; .1 e T =2 1%

´ 1; jf .x/j=ƒ D jf .x/j=ƒ;

x 2 G; x 2 Rd n G;

(3.10.40)

182 into Z


in (3.10.40), we have n

.ExN0 Œ% /2 d N0 Rd Since e .T =2

n /L

1 1 %T =2 n 2 .f; f / C .jf j; .1 e .T =2 /L /jf j/: n 2 T =2 ƒ ƒ 1% (3.10.41)

is positivity improving and thus

.jf j; .1 e .T =2 we have by (3.10.39), sup N0

j D0;:::;2n

/jf j/ .f; .1 e .T =2

n /L

/f /;

jf .XTj /j ƒ

1=2 n %T %T =2 .T =2n /L .f; f / C .f; .1 e /f / : ƒ 1 %T =2n

Set % D e 1=T . Then by N0

n /L

n

%T =2 n 1%T =2

2n , we have

e n jf .XTj /j ƒ ..f; f / C 2n .f; .1 e .T =2 /L /f //1=2 : ƒ j D0;:::;2n (3.10.42) sup

Since .f; .1 e .T =2 /L /f / .T =2n /.L1=2 f; L1=2 f /, we obtain that e sup jf .XTj /j ƒ ..f; f / C T .L1=2 f; L1=2 f //1=2 : N0 ƒ j D0;:::;2n n

(3.10.43)

Take n ! 1 on both sides of (3.10.43). By the Lebesgue dominated convergence theorem lim N0 sup jf .XTj /j ƒ D N0 lim sup jf .XTj /j ƒ : n!1

n!1 j D0;:::;2n

j D0;:::;2n

Since f .X t / is continuous in t , limn!1 supj D0;:::;2n jf .XTj /j D sup0sT jf .Xs /j follows. It remains to show the Dirichlet principle (3.10.40). Lemma 3.112 (Dirichlet principle). Let 0 < % < 1. Fix n and set Tj D Tj=2n , j D 0; 1; : : : ; 2n . Let G Rd be measurable and D inf¹Tj 0jXTj 2 Gº. Then for any function 0 such that .x/ 1 on G, it follows that Z .ExN0 Œ% /2 d N0 Rd

. ; /L2 .Rd ;d N0 / C

n

%T =2 .T =2n /L / /L2 .Rd ;d N0 / : n . ; .1 e T =2 1%

183


Proof. Define the function see that

%

D ExN0 Œ% . By the definition of we can

% .x/

by

% .x/

D 1;

x 2 G;

(3.10.44)

since D 0 when Xs starts from the inside of G. Let F t D .Xs ; 0 s t / be the natural filtration of .X t / t0 . By the Markov property of X it is directly seen that e .T =2

n /L

% .x/

X

n

D ExN0 ŒENT0 =2 Œ% D ExN0 ŒExN0 Œ%ıT =2n jFT =2n D ExN0 Œ%ıT =2n ;

(3.10.45)

where t is the shift on X defined by . t !/.s/ D !.s C t / for ! 2 X. Note that . ı T =2n /.!/ D .!/ T =2n 0;

(3.10.46)

when x D X0 .!/ 2 Rd n G. Hence by (3.10.45) and (3.10.46) we have the identity %T =2 e .T =2 n

n /L

% .x/

D

x 2 Rd n G:

% .x/;

It is trivial to see that Z .ExN0 Œ% /2 d N0 D .

(3.10.47)

n

%T =2 .T =2n /L / %; %/ . %; %/ C n . % ; .1 e T =2 1% Rd By using relation (3.10.47) we can compute the right-hand side above as

% /:

n

%T =2 . % 1G ; % 1G / C n. 1 %T =2

% 1G ; .1

e .T =2

n /L

/

% /:

(3.10.48)

Since .

% 1G ; .1

D. D. .

e .T =2

% 1G ; .1 % 1G ; .1 % 1G ; .1

e e e

n /L

/

%/

.T =2n /L .T =2n /L

.T =2n /L

we have Z .ExN0 Œ% /2 d N0 . Rd

/

% 1G /

/

% 1G /

/

% 1G /;

C. .

% 1G ; .1 % 1G ; e

e .T =2

.T =2n /L

n /L

/

% 1Rd nG /

% 1Rd nG /

n

% 1G ;

% 1G /

C

%T =2 . 1 %T =2n

% 1G ; .1

e .T =2

n /L

/

% 1G /:

(3.10.49) Note that

% 1G .x/

.x/ for all x 2 Rd . Then n

.

% 1G ;

% 1G / C

%T =2 n. 1 %T =2

% 1G ; .1

e .T =2

n /L

/

% 1G /

n

%T =2 n . ; .1 e .T =2 /L / /: 1 %T =2n A combination of (3.10.49) and (3.10.50) then proves the lemma. . ; /C

(3.10.50)

184


3.10.4 Mehler’s formula In this section we consider the special case V .x/ D .! 2 =2/jxj2 !=2 describing the harmonic oscillator, which is one of the few models for which an explicit formula is available for the kernel of the Schrödinger semigroup. This kernel is given by Mehler’s formula. Let d D 1 and consider the Schrödinger operator Hosc D

1 d2 m! 2 2 ! C x 2m dx 2 2 2

(3.10.51)

on L2 .R/. Here m is the mass of the oscillator, and ! is the traditional notation for its frequency and should not be confused with its meaning before. Hosc is essentially self-adjoint on C01 .R/ by Corollary 3.14, and from Example 3.12 we know that Spec.Hosc / D ¹m!njn 2 N [ ¹0ºº. The ground state is explicitly given by ‰osc .x/ D .m!=/1=4 e m!x

2 =2

(3.10.52)

Furthermore, the complete set of eigenfunctions is given by ‰n .x/ D .m!=.4n .nŠ/2 //1=4 e

x2 2

Hn .x/;

(3.10.53)

where Hn , n D 1; 2; : : : , denote the Hermite polynomials. We present two derivations of Mehler’s formula, first by using diffusion theory and next by a direct analytic calculation. Proposition 3.113 (Mehler’s formula). Let t D

1 e 2m!t ; 2m!

t 0:

Then e tHosc .x; y/

2xye m!t .! t C e 2m!t /x 2 .e 2m!t ! t /y 2 Dp : exp 2 t 2 t =m (3.10.54) 1

Proof. Let m; !; > 0 and consider the stochastic differential equation dX t D m!X t dt C dB t ;

B0 D x:

(3.10.55)

As seen in Example 2.3.45, its solution is the Ornstein–Uhlenbeck process Z t e m!.ts/ dBs : (3.10.56) X tx D e m!t x C 0

185


By a direct calculation EŒX tx D e m!t x;

EŒX tx Xsx D e m!.tCs/ x 2 C

2 .1 e 2m!.s^t/ / 2m! (3.10.57)

follow from (3.10.56). The generator of X tx is 2 d2 d m!x : 2 dx 2 dx Now consider (3.10.51) above. Write X t D X t and E Œ D Note that E ŒX t D 0 and

E ŒXs X t D

(3.10.58) R1

2 1 EŒ ‰osc .x/ dx.

2 jtsjm! e : 2m!

(3.10.59)

2 dx/ ! L2 .R/, Clearly, ‰osc 2 C 2 .R/. The ground state transform Ug W L2 .R; ‰osc Ug f D ‰osc f , gives

Losc D Ug1 Hosc Ug D

1 d2 d C m!x ; 2 2m dx dx

which coincides with (3.10.58) up to sign on putting 1=m D 2 . We have E

n hY

n i Y fj .X tj / D 1; f0 e .tj tj 1 /Losc fj 1

j D0

2 dx/ L2 .R;‰osc

j D1

for f0 ; : : : ; fn 2 L1 .R/. Noting that Hosc x‰osc D m!‰osc , we can also check (3.10.59) by the formal calculation E ŒX t Xs D .‰osc ; xe .ts/Hosc x‰osc / Z 1 2 m!jtsj m!jtsj 2 De e jxj2 ‰osc .x/dx D : 2m! 1 Next we view e tHosc as an operator in L2 .R/ and derive its kernel. By (3.10.56), EŒf .X tx /

Z t m!t m!.ts/ DE f e xC e dBs : 0

For any fixed t , Z rt D

t

e 0

m!.ts/

dBs D e

m!t

Z t m!t 1 m!s B t .m!/ Bs e ds e 0

186


is a Gaussian random variable with zero mean and covariance EŒr t r t D . 2 =2m!/.1 e 2m!t / D

2 t : 2

Thus 1 e tHosc f .x/ D Ug e tLosc Ug1 f .x/ D ‰osc .x/EŒ.‰osc f /.X tx /:

Since X tx D e m!t x C r t with the Gaussian random variable r t , the right-hand side above can be computed as

2 2 t 2

1=2

Z ‰osc .x/

1

f .e m!t C y/ jyj2 exp m!t 2 ‰osc .e C y/ 2 t

1

! dy:

(3.10.60)

Changing the variable e m!t C y to z and setting 2 D 1=m, we further obtain that Z D

1

1

1=2

f .z/ . t =m/

jz e m!t j2 ‰osc .x/ exp dz: t ‰osc .z/ 2m

From this (3.10.54) follows directly.

Alternative proof: Mehler’s formula can also be obtained through analytic continuation t 7! i t as a corollary of the theorem below. From now on we simply set m D ! D 1. Then 1 4xye t .x 2 C y 2 /.1 C e 2t / e tHosc .x; y/ D p exp : 2.1 e 2t / .1 e 2t / Let K t be the integral kernel of e itHosc , and denote T D ¹kjk 2 Zº. Theorem 3.114. The kernel K t W R R ! C has the following expressions: .1/ for every t 2 R n T , K t .x; y/ D p

1 2 i sin t

e i.

x 2 Cy 2 2

xy cot t sin t/

(3.10.61)

.2/ for every t 2 T , a.e. x 2 R and f 2 D.Hosc / Z lim

s!t

1

1

Ks .x; y/f .y/ D f .x/:

(3.10.62)

187


Proof. By eigenfunction expansion the solution of the initial value problem i @ t ‰ D Hosc ‰;

‰.x; 0/ D .x/

can be written as Z ‰.x; t / D

1

K t .x; y/.y/dy;

(3.10.63)

‰n .x/‰n .y/e iEn t :

(3.10.64)

1

with integral kernel K t .x; y/ D

1 X nD0

A combination of (3.10.53) and (3.10.64) leads to 1 x 2 Cy 2 it X 1 1 K t .x; y/ D p e 2 2 Hn .x/Hn .y/e i nt : 2n nŠ nD0 n

(3.10.65)

d y known for Hermite polynomials alNext, the formula Hn .y/ D .1/n e y dy ne lows to replace the sum appearing at the right-hand side of (3.10.65) by 2

e

y2

2

1 X 1 1 it d n 2 Hn .x/ e y : e nŠ 2 dy

nD0

P 2 The generating function e 2szz D 1 nD0 then for the above the formal expression

1 n nŠ Hn .s/z

of Hermite polynomials gives

1 d d2 2 2 e y exp xe it e 2it 2 e y : dy 4 dy

(3.10.66)

In the following we will argue that this expression actually makes sense. 2 Clearly, the exponential operator acting on e y factorizes. Both operators are unbounded, however, if the second can be defined, the first is well defined on the d same domain. The first operator is a shift by the prefactor of dy . We consider

d2 Kz D exp z 2 ; dy

z 2 C;

and make sense of it on a set including z D D .1=4/e 2it . Take the open disc D D ¹z 2 Cjjzj < 14 º. Note that for every z 2 D the function 2 2 f .y/ D e y is an analytic vector of Kz , i.e., Kz e y 2 L2 .R/ and is analytic with

188


respect to z: 1 n 1 X z d n y 2 X jzjn e nŠ dy n nŠ

nD0

nD0

n d y 2 dy n e

1 1 X 1 X jzjn 2n k 2 =4 kk e k2 D cn jzjn ; Dp nŠ 2 nD0 nD0

p where cn D .=2/1=4 .4n 1/ŠŠ=nŠ. It is seen that cn =cnC1 ! 1=4 as n ! 1 and this explains the way D was defined. For every z 2 D define Az D ¹f 2 D.Kz /jf is analytic vector of Kz º: This set 2 contains, for instance, e y . For every z 2 D and f 2 Az the derivative with respect to z of Kz exists. Denote g.y; z/ D .Kz f /.y/; then we have @z g D @y2 g;

g.y; 0/ D f .y/

since z D 0 2 D, i.e., g satisfies the heat equation with the initial condition above, for every z 2 D. This observation allows to express g.y; z/. Define ˇˇ ˇ ² ³ Z 1 ˇˇ 1 ˇ .yu/2 f .u/e 4z du ˇˇ < 1; 8y 2 R Df D z 2 DN ˇˇ ˇˇ p z 1 as a subset of the closure of D; in particular, we write Dexp for f .y/ D e y . An easy calculation shows that Dexp lies in D and coincides with the complement in D of the closed disk of radius 1=8 centered in .1=8; 0/. It is checked directly that for every z 2 Df Z 1 .yu/2 1 Ff .y; z/ D p f .u/e 4z du; (3.10.67) 2 z 1 2

where the square root is the one with the smallest argument, is a solution of the heat equation above. Moreover, Ff is analytic since j@Ff =@zj < 1. Take now z D 2 .0; 14 /. We have in L2 sense that Ff .y; / D g.y; /, hence by uniqueness of the analytic continuation g.y; z/ D .Kz f /.y/;

z 2 Df :

(3.10.68)

This representation can be extended to the circle @D. Pick a number " such that j< j " < 1=4 and define A"z D K" L2 .R/ AT z . Take a sequence .zn /n2N D such that zn ! for some 2 Dh and h 2 n A"zn . Then it is easily seen R1 p 2 that pointwise Kzn h ! 1=.2 / 1 h.u/ exp. .u/ 4 /du as n ! 1. By the Lebesgue dominated convergence theorem it follows then for every ' 2 C01 that Z 1 1 . u/2 h.u/ exp ; as n ! 1: .'; Kzn h/ ! '; p du 4 2 1 L2


189

T On the other hand,TK h is well defined for every h 2 n A"zn since K h = KC" fh , with some fh 2 n Azn , and " C < 0. Moreover, Kzn h ! K h in strong sense. This follows again through dominated convergence by using the continuity of the exponential function, the equalities kK h Kzn hk22 D kKC 1 fh Kzn C 1 fh k22 4 4 Z 1 1 1 D je .C 4 / e .zn C 4 / j2 dEfh . / 0

1

1

and the estimate je .C 4 / e .zn C 4 / j2 4, uniform in , and making use of the spectral theorem. Hence also .'; Kzn h/ ! .'; K h/ as n ! 1; 8' 2 C01 , and by comparison Z 1 .yu/2 1 .K h/.y/ D p h.u/e 4 du (3.10.69) 2 1 T follows, for everyT h 2 n A"zn , 2 Dh and almost every y 2 R. Take an fh 2 n Azn and " j< j, and look at the solution of exp."@y2 /fh D 2 2 2 e y . On taking Fourier transforms, e "k fOh .k/ D e k =4 follows. fOh , and thus also

fh , will be L2 whenever " < 1=4, and then e y 2 \n A"zn . This also explains the 2 way A"z was defined. By putting now h.y/ D e y in (3.10.69) and using (3.10.66), (3.10.61) comes about by easy manipulations for every with j< j " < 1=4. This argument excludes the values of time corresponding to D .1=4; 0/, i.e., the set T The reason is that this is the only value of that falls outside of Dexp . Informally, for these values the kernel degenerates into a ı-distribution. Take t 2 T and a sequence .tn /n2N R n T such that tn ! t . Since e itHosc is a strongly continuous unitary group, we have with some subsequence .tm /m2N for any L2 function ' that itHosc '.x/, for almost every x 2 R. Moreover, for almost every e itm Hosc '.x/ ! R 1e x 2 R we have 1 K tm .x; y/'.y/dy ! '.x/, which is seen directly. 2

Chapter 4

Gibbs measures associated with Feynman–Kac semigroups

4.1

Gibbs measures on path space

4.1.1 From Feynman–Kac formulae to Gibbs measures In this chapter we take a different view on the Feynman–Kac formula. The point of departure is the proof of the Feynman–Kac formula using the Trotter product formula, given in Section 3.2.2. There we have seen that for a Schrödinger operator H D 12 C V with smooth and bounded potential V , and for f; g 2 L2 .Rd /, we have .f; e tH g/ D lim .f; .e .t=n/H0 e .t=n/V /n g/; n!1

where H0 D .1=2/. By Theorem 3.44 the integral kernel K t .x; y/ of e tH is continuous in x and y. Writing out the integral kernels of e .t=n/H0 and letting f and g approach ı distributions yields Z Pn Pn 1 2 1 K t .x; y/ D lim e j D0 2t =n jxj C1 xj j j D1 V .xj / dx1 dxn ; d=2 n!1 Rd n .2 t =n/ (4.1.1) where x0 D x and xnC1 D y. For a finite n, the right-hand side above has an interpretation in the context of statistical mechanics. Consider a one-dimensional chain of length n C 2, consisting of particles placed in Rd . Suppose that the chain is pinned at both ends, at x and y, respectively. Furthermore, let neighbouring particles be coupled by a quadratic potential of strength n=2t , while each particle is subject to an external potential V . The total potential energy of the chain is then just the exponent in (4.1.1). The energetically most optimal position of the chain is a configuration that minimizes the potential energy. In this case it comes about as a compromise between a straight line through x and y, and a configuration in which all xj are in the global minimum of V (assuming there is one). On the other hand, entropy makes the position of the chain fluctuate around this energetically optimal state. The expression in (4.1.1) defines a finite measure on .Rd /n , which can be normalized to a probability measure. The latter gives the probability distribution of finding the chain in a subset of positions, and it describes its equilibrium state in which energy and entropy balance each other in an optimal way. Measures such as this are known as Gibbs measures, and have been studied intensely over the last decades, in various contexts.

191

Section 4.1 Gibbs measures on path space

A peculiarity of the model (4.1.1) is that the limit n ! 1 is taken. Different from the infinite volume limit, which is standard in the classical theory of Gibbs measures and will be discussed below, this is a type of continuum limit for the particle system: n while the number of particles grows like n, the pair interaction energy 2t jxj C1 xj j2 increases at the same rate. This couples nearby particles increasingly strongly, forcing them to occupy the same position with large probability, and eventually leads to a Rt Brownian path perturbed by the density exp. 0 V .Xs /ds/. Thus one can view the right-hand side of the Feynman–Kac formula Z Rt x;y K t .x; y/ D … t .x y/ e 0 V .Bs /ds d WŒ0;t (4.1.2) X

appearing in Theorem 3.44 as a model of an elastic string fixed at both ends, under the effect of an external potential V . This view leads to a new intuition of the right-hand side of (4.1.2). Instead of viewing the parameter t of the process B t as time and thinking of the process as a time evolution, one can think of t as a spatial variable. The following questions then make sense: How does the elastic string B t behave when we make it longer without stretching it, i.e., increase t ? Is there a sensible limit of a string of infinite length? Is that limit unique and which properties does the infinite string have? All of these questions have been asked, and some answered in a variety of models. Before we present some examples, we informally discuss a few general concepts related to Gibbs measures. From here on we use the notation .X t / t0 for a random process under investigation extended to the full time line R, which occasionally can be Brownian motion, Brownian motion with a drift, an Itô process or a Lévy process with càdlàg paths. The basic object of interest is dŒt;t .!jx; y/ D e U t .!/ dˇŒt;t .!/; x;y

x;y

t > 0;

(4.1.3)

where ˇŒt;t is the appropriate bridge measure for .X t / t2R . Note that in our case it is more natural to consider a two-sided process running from t to t , instead of starting at t D 0. U t .!/ is the “energy” associated with path .Xs .!//s2Œt;t , which in the Rt case of (4.1.2) is U t .!/ D 0 V .Xs .!//ds. In the classic theory of Gibbs measures the thermodynamic limit is the passage from microscopic to macroscopic level. In the context of statistical mechanics and thermodynamics the basic problem is to derive the large-scale observable behaviour in equilibrium of a large collection of components from their small-scale interactions. Mathematically this translates in our context into appropriately extending Œt;t .jx; y/ from a path space over bounded intervals to a path space over the full line R by taking t ! 1. A basic difficulty is that for the non-trivial types of potentials V encountered U t almost surely diverges in this limit. Therefore we follow the Dobrushin–Lanford– Ruelle (DLR) construction originally applied in statistical mechanics, and look for probability measures on a path space over R whose family of probability measures

192

Chapter 4 Gibbs measures associated with Feynman–Kac semigroups

conditional on paths outside Œt; t match with Œt;t , for every t > 0. This problem differs essentially from the Kolmogorov extension problem in that here the extension is made by starting from prescribed conditional measures rather than marginals. In practice, solutions to the DLR problem are constructed by taking suitable t ! 1 limits over x D Xt .!/ and y D X t .!/ in Œt;t .jx; y/, which can be regarded as boundary conditions. We emphasize that although it is possible to find microscopic models as seen above whose continuum limit may be a probability measure such as appearing at the righthand side of (4.1.2), our present focus is different. Here we are not interested in such models or interpretations. We retain the language of Gibbs measures because the measures we automatically gain from Feynman–Kac semigroups have the structure of a Gibbs measure, i.e., they come about as modifications of an underlying measure by densities dependent on additive functionals indexed by a family of bounded sets. Indeed, this is roughly the extent to which the measures we study and the measures appearing in statistical mechanics can be parallelled. For a start, the class of sample space on which we operate is that of Brownian paths, which differs essentially from the classic cases where the sample space is often a countable product of a finite or countable state space. This rules out the handy tools using compactness of the spaces. Also, our index set is the family of bounded intervals instead of finite subsets of a countable set. The reference measure in our case is Brownian bridge and not a product measure. These differences necessitate to develop specific methods for constructing limit measures on path space since the classic tools are not applicable. A second point we need to make is that even though the parallels between our and the classic cases run mostly on a general level, the concepts and intuitions of the original Gibbsian theory are relevant and useful in our context. While a thermodynamic limit does not interest us for the reason it does practitioners of statistical mechanics and related fields, we do have an interest in limit Gibbs measures. As it will turn out and will be explained below, Gibbs measures on path space can be used to study analytic and spectral properties of the generators of the random processes these measures are associated with. Some of these relationships can only be seen under Gibbs measures on path space extended over the full index set. Although it is possible to consider Gibbs measures for continuous time random processes in greater generality, our primary interest here is to use them in studying ground state properties of specific quantum field models and therefore we restrict attention to certain classes of functionals U t . Example 4.1. Various choices of the energy functional considered below are as follows, some of which will be considered in more detail in the following sections. (For simplicity of notation we do not explicitly feature !.) (1) The case

Z Ut D

t

V .Xs /ds 0

(4.1.4)

193


relates with the path measure of a Schrödinger operator with Kato decomposable potential V , as seen in Chapter 3. Moreover, for sufficiently regular potentials describes in this case the path measure of the P ./1 -process dX t D dB t C .r log ‰/.X t /dt;

(4.1.5)

where ‰ is the ground state of the Schrödinger operator. (2) Let Z Ut D

Z tZ

t

0

V .Xs /ds C

0

0

t

W ' .Xs Xr ; s r/dsdr;

(4.1.6)

where

Z 1 jb ' .k/j2 ikxjkjjsj e d k; W .x; s/ D 4 Rd jkj with a sufficiently fast decaying, spherical symmetric real-valued function '. Gibbs measures for this U will be discussed in the present chapter, and in Part II of the book we will see how by using them it is possible to derive and prove the ground state properties of the Nelson model of an electrically charged spinless quantum particle linearly coupled to a scalar boson field. '

(3) Changing W ' .x; s/ above for Wpol .x; s/ D

1 jsj e 4jxj

(4.1.7)

describes the polaron model. The bipolaron model differs by the fact that it consists of two dressed electrons coupled to the same phonon field, which repel each other by Coulomb interaction. In this case Z tZ t Z t 1 Ut D ˛2 ds; E.Xs ; Xr ; Ys ; Yr ; s r/dsdr g jX Ys j s 0 0 0 where E.Xs ; Xr ; Ys ; Yr ; u/ D Wpol .Xs Xr ; u/ C 2Wpol .Xs Yr ; u/ C Wpol .Ys Yr ; u/ with ˛ < 0 being the polaron-phonon coupling parameter, and g > 0 is the strength of the Coulomb repulsion between the two polarons. In this case the reference measure is a product of two Wiener measures of the independent Brownian motions .X t / t2R and .Y t / t2R . (4) Let Z tZ Ut D

0

t 0

ı.Xs Xr /dsdr:

(4.1.8)

This gives the intersection local time of Brownian motion describing a polymer model with short-range soft-core interaction encouraging to avoid self-intersections.

194


(5) Let Z Ut D

0

Z tZ

t

V .Xs /ds C

0

t

W .Xs Xr ; s r/dXs dXr

0

(4.1.9)

with some choices of W which we do not write down explicitly here. This describes the Pauli–Fierz model of a charged particle interacting with a vector boson field. The difference from the cases above consists in having double stochastic integrals here instead of double Riemann integrals. Gibbs measures of this type will be discussed when we study the Pauli–Fierz model in Part II. (6) Let Z tZ Ut D

0

t 0

1 dX t dXs : jX t Xs j

(4.1.10)

This is used in describing turbulent fluids. The above formal expression gives the total energy of a vorticity field concentrated along Brownian curves X t 2 R3 obtained from a given divergence-free velocity field. (7) Let Z Ut D

t

V .Xs /ds

(4.1.11)

0

where .Xt / t0 is a symmetric ˛-stable process is obtained for a fractional Schrödinger operator ./˛=2 C V , ˛ 2 .0; 2/. The case ˛ D 1 describes a massless relativistic quantum particle. A similar expression can be obtained when .X t / t0 is a relativistic ˛-stable process generated by the operator . m2=˛ /˛=2 C V , m > 0, and ˛ D 1 describes a massive relativistic quantum particle. Our main concern here will be to construct infinite volume Gibbs measures covering Examples (1) and (2). A main benefit will be a study, through the appropriate Feynman–Kac-type formulae, of ground state properties of quantum models in terms of averages of suitable functions on path space with respect to these Gibbs measures, which will be carried out in Part II below.

4.1.2 Definitions and basic facts Let .; F ; P / be a given probability space, and .X t / t2R be an Rd -valued reversible two-sided Markov process on .; F ; P / with almost surely continuous paths. For I R we use the notation FI for the sub--field generated by .X t / t2I , and TI D FI c , where I c D R n I , as well T as FT and TT for I D ŒT; T , T > 0. The tailfield will be denoted by T D N 2N TN . Also, jI j stands for the Lebesgue measure of I .


195

Definition 4.1 (Index set and state space). The collection ¹I Rj jI j < 1º of bounded intervals of the real line is the index set, and the state space is Rd . Let be the path measure of the given Markov process .X t / t2R , which we use as reference measure. Recall that X D C.R; Rd /. Definition 4.2 (Potentials). (1) A Borel measurable function V W Rd ! R is called an external potential. We call V a -admissible external potential whenever Z R e I V .Xs .!// ds .d!/ < 1 (4.1.12) 0< X

for every bounded interval I R. (2) A function W W Rd Rd R R ! R is called a pair interaction potential. We say that W is admissible whenever Z 1 sup sup jW .x; y; s; t /j ds < 1: (4.1.13) t2R 1 x;y2Rd

If W .x; y; s; t / D W .x; y; s t; 0/, we use the simpler notation W .x; y; t /; similarly, we write W .x; t / in case W .x; y; s; t / D W .x y; 0; s t; 0/. For brevity we will henceforth refer to the potentials jointly satisfying the conditions above as admissible. The condition on the pair potential is rather strong, however, as we move on we will introduce weaker regularity conditions. In the Feynman– Kac formula considered at the beginning of this chapter, (4.1.12) holds for Katodecomposable potentials V by Lemma 3.38. While n-body potentials can be defined in a straightforward way, in what follows we will not need more general potentials than the single and two-body potentials V and W . Definition 4.3. For fixed !N 2 X denote by T!N the unique measure on .X; F / such that for all f bounded and measurable with respect to FT , and all g bounded and measurable with respect to TT E !N Œfg D E Œf jTT .!/g. N !/ N T

holds.

(4.1.14)

R For general measurable f , f dT!N is obtained from (4.1.14) by approximation of f . Put differently, T!N is the product measure on C.ŒT; T I Rd / C.ŒT; T c I Rd / of the regular conditional probability of given !.t / D !.t N /, for all jt j > T , and the Dirac measure concentrated at !N on C.ŒT; T c I Rd /. Thus, intuitively, T!N is the measure obtained from by fixing the path to !N outside ŒT; T and allowing it to fluctuate inside ŒT; T .

196


It is clear that T!N is indeed a version of .jTT /, and it is probably the most natural one. We emphasize that we had many choices of possible versions here given that conditional expectations are only defined up to sets of measure zero. By a law of the iterated logarithm-type argument, measures of Markov processes are usually concentrated on a set of paths that is characterized by the asymptotic behaviour of X t .!/ as jt j ! 1. Thus had we defined .jTT / arbitrarily on the set N X of measure zero consisting of all paths not having this precise asymptotic behavior, we still would have obtained a version of the conditional expectation. Our definition of T!N reduces this element of arbitrariness considerably, and we will need this when defining Gibbs measures. Without restricting generality, we choose symmetric bounded intervals from the index set. Write ƒ.S; T / D .ŒS; S ŒT; T / [ .ŒT; T ŒS; S /; ƒ.T / D .R ŒT; T / [ .ŒT; T R/; where S; T > 0. For S T and admissible potentials V; W we use the notation Z T ZZ Hƒ.S;T / .X / D V .Xs / ds C W .X t ; Xs ; jt sj/ ds dt; (4.1.15) T

ƒ.S;T /

and define Hƒ.T / by replacing ƒ.S; T / with ƒ.T / in (4.1.15). We will also write HT instead of Hƒ.T;T / . To ease the notation we do not explicitly indicate the !dependence of these functionals. Definition 4.4 (Gibbs measure). A probability measure on .X; F / is called a Gibbs measure with respect to reference measure and admissible potentials V; W if for every T > 0, (1) jFT jFT for all T > 0, (2) for every bounded F -measurable function f , E Œf jTT .!/ N D

E !N Œf e Hƒ.T / T

E !N Œe Hƒ.T /

;

-a.s.

(4.1.16)

T

A probability measure T on .X; F / is called finite volume Gibbs measure for the interval ŒT; T if .10 / jFS jFS for all S < T .20 / for all 0 < S < T and every bounded F -measurable function f , N D ET Œf jTS .!/

E !N Œf e Hƒ.S;T / S

E !N Œe Hƒ.S;T / S

;

T -a.s.

(4.1.17)

197


In our terminology “Gibbs measure” corresponds to “infinite volume Gibbs measure” in the classic theory. We refer to the path !N as a boundary condition or boundary path. The normalization factor ZT .!/ N D E !N Œe Hƒ.T / T

is traditionally called partition function. Remark 4.1 (DLR equations). Conditions (2) and .20 / in the above definition are traditionally called Dobrushin–Lanford–Ruelle (DLR) equations. These are in fact equalities and can be viewed as counterparts for conditional measures of the Kolmogorov consistency relations for marginal measures. Remark 4.2 (Gibbs specification). To make it more visible that a Gibbs measure is a probability measure which has prescribed conditional probability kernels we give the following equivalent definition of a Gibbs measure: A probability measure on .X; F / is a Gibbs measure for the admissible potentials V; W if for every A 2 F and T > 0 the function !N 7! T .A; !/ N is a version of the conditional probability .AjTT /, i.e., .AjTT /.!/ N D T .A; !/; N

A 2 F ; T > 0; a.e. !N 2 X

(4.1.18)

where T .A; !/ N D

1 ZT .!/ N

Z X

1A e Hƒ.T / dT!N ;

A 2 F ; !N 2 X:

(4.1.19)

Remark 4.3. To understand why (1) in Definition 4.4 is necessary consider the special version T!N of E ŒjTT .!/. N Writing the (on a first sight more natural) expression E Œf jTT D

E Œf e Hƒ.T / jTT E Œe Hƒ.T / jTT

(4.1.20)

instead of (4.1.16) is in most cases meaningless. The problem is that the left-hand side of the above equality is only defined uniquely outside a set N of -measure zero, while the right-hand side is only defined uniquely outside a set M of -measure zero. In many cases of interest, and are mutually singular on X, and then (4.1.20) is no condition at all. Thus in our case sets of measure zero do matter and extra care is needed. The reason why we require (1) explicitly is the same. Without it the two sides of (2) may refer to functions that are uniquely defined only on disjoint subsets of the probability space. In case of a finite dimensional state space and the measure of a Feller Markov process as reference measure, this inconvenience can be avoided and then (1) is unnecessary.

198


Constructing a Gibbs measure for given potentials V and W can be quite difficult. However, constructing a finite volume Gibbs measure under the given regularity conditions is straightforward. Proposition 4.1. Let be a given reference measure, T > 0, and V; W be admissible potentials. If 0 < E Œe HT < 1, then dT D

e HT d E Œe HT

is a finite volume Gibbs measure for ŒT; T . Proof. By assumption e HT is -integrable. Thus T , and in particular T jFT jFT . To check DLR consistency let f and g be bounded, and suppose g is TS -measurable. Then for T > S ˇ

E S Œf e Hƒ.S;T / E S Œf e Hƒ.S;T / H ˇ HT Tˇ D E E g E Œe ET g e ˇ TS E S Œe Hƒ.S;T / E S Œe Hƒ.S;T /

E S Œf e Hƒ.S;T / H Hƒ.S;T / 2 nƒ.S;T / ŒT;T D E g E Œe jTS e E S Œe Hƒ.S;T / D E Œe HT ET Œfg: The last equality is due to the fact that E S Œe Hƒ.S;T / is a version of the conditional expectation E Œe Hƒ.S;T / jTS . Dividing by E Œe HT shows that condition (2) in Definition 4.4 is verified. Remark 4.4 (Stochastic and sharp boundary conditions). The finite volume Gibbs measure that we constructed above corresponds to free boundary conditions. Virtually the same proof as above shows that for every !N 2 X, the measure N d! T D

e Hƒ.T / d !N E !N Œe Hƒ.T / T

(4.1.21)

T

is a finite volume Gibbs measure. We call it a finite volume Gibbs measure with sharp boundary condition !. N Occasionally, instead of pinning down path positions in a sharp boundary condition, it is more convenient to give only their probability distributions. Let h W X ! R be a TT -measurable non-negative function such that 0 < E Œhe HT < 1: Then as in Proposition 4.1 we can show that dhT D

he Hƒ.T / d E Œhe Hƒ.T /

199


is a finite volume Gibbs measure for ŒT; T . The density h can then be regarded as a stochastic boundary condition. Next we show that limits of finite volume Gibbs measures yield Gibbs measures in the sense of Definition 4.4. The following notion of convergence will be used. Definition 4.5 (Local convergence). A sequence .mn /n2N of probability measures on X is said to be locally convergent to a probability measure m if for every 0 < T < 1 and every A 2 FT , mn .A/ ! m.A/ as n ! 1. Remark 4.5. (1) In many cases Gibbs measures are mutually singular with respect to their reference measures, while finite volume Gibbs measures are absolutely continuous with respect to them. Thus the topology of local convergence is the strongest reasonable for such Gibbs measures. In particular, in many cases it is R possible R to find a tail measurable function f such that f dN D 0 for all N , while f d D 1. (2) Convergence on sets is usually a very strong requirement for the convergence of measures. Indeed, the situation discussed in the previous remark indicates that convergence practically never holds for all sets of F . Since, however, we only need to check convergence on elements of FT for finite T , the special structure of Gibbs measures allows to show local convergence of finite volume Gibbs measures in all cases we consider below. Proposition 4.2. Let be a reference measure and V; W be admissible potentials. Let .n /n2N be a sequence of finite volume Gibbs measures for V; W and ŒTn ; Tn , with Tn ! 1 as n ! 1. Suppose that there exists a probability measure on X such that n ! in the topology of local convergence. Suppose, moreover, that satisfies .1/ in Definition 4.4 with respect to . Then is a Gibbs measure for V; W . Proof. As (1) in Definition 4.4 holds by assumption, only DLR consistency remains to be shown. Each n is a finite volume Gibbs measure, thus for bounded f; g with TS -measurable g we have " # E S Œf e Hƒ.S;Tn / (4.1.22) En g D En Œfg E S Œe Hƒ.S;Tn / for every S < Tn . We show now that (4.1.22) remains valid in the n ! 1 limit. By a monotone class argument we may assume that both f and g are FR -measurable for some R > 0. In this case, the right-hand side of (4.1.22) converges by definition of local convergence. At the left-hand side (4.1.13) implies that ZZ ZZ n!1 W .Xs ; X t ; jt sj/ ds dt ! W .Xs ; X t ; jt sj/ ds dt ƒ.S;Tn /

ƒ.S/

200


uniformly in path, and thus Fn .X.!// D

E S! Œf e Hƒ.S;Tn / E S! Œe Hƒ.S;Tn /

n!1

!

E S! Œf e Hƒ.S/ E S! Œe Hƒ.S/

D F .X.!//

uniformly in ! 2 X. Thus for every " > 0 there is an N 2 N such that kFm F k1 < ", whenever m N . By the triangle inequality, n!1

jEn ŒgFn E ŒgF j jEn ŒgFN E ŒgFN j C 3kgk1 " ! 3kgk1 ": Since " is arbitrary, convergence of the left-hand side in (4.1.22) follows. The previous proposition shows that suitable limits of finite volume Gibbs measures over boundary conditions result in Gibbs measures. The next statement shows that this procedure actually yields all Gibbs measures supported on a specific set of paths. Proposition 4.3. Let X X be measurable and be a Gibbs measure supported on X , with reference measure . For N 2 N, ! 2 X, define E! Œf D N

Hƒ.N / ! Œf e E N ! Œe Hƒ.N / E N

as in (4.1.16). Suppose that for every T > 0, A 2 FT and ! 2 X we have ! N .A/ ! .A/ as N ! 1. Then is the unique Gibbs measure associated with H , supported on X . Proof. Let Q be a Gibbs measure supported by X . For every T < N and A 2 FT , !N 7! .AjT Q N is a backward martingale in N , thus convergent almost N /.!/ N everywhere to .AjT Q /.!/. N By the Gibbs property .AjT Q N D ! Q N /.!/ N .A/ -a.s., and thus for -a.e. Q !N 2 X there exists .AjT Q /.!/ N D limN !1 .AjT Q /. N D N !/ N .A/ D .A/. Taking -expectations limN !1 ! Q on both sides of the above equality N shows .A/ Q D .A/, and since this holds for every A 2 FT and T > 0, we obtain Q D . Finally, we give a useful criterion for a sequence of finite volume Gibbs measures to converge along a sequence of boundary conditions. Definition 4.6. We say that a family of probability measures .T /T >0 is locally uniformly dominated by a family of probability measures .Q S /S>0 whenever (1) Q S is a probability measure on FŒS;S , (2) for every " > 0 and S > 0 there exists ıS > 0 such that lim supT !1 T .A/ < " when A 2 FŒS;S with Q S .A/ < ıS .

201

Section 4.2 Existence and uniqueness by direct methods

The significance of uniform local domination is made clear by the following result. Proposition 4.4. Let .Tn /n2N be an increasing and divergent sequence of positive numbers and assume that .Tn /n2N is a locally uniformly dominated family of finite volume Gibbs measures. Then .Tn /n2N has a convergent subsequence in the local strong topology and the limit is a Gibbs measure. Proof. Let S > 0 and write Q S for the measure dominating n , i.e., suppose that for " > 0 there exists ıS > 0 such that lim supn!1 n .A/ < ", for all A satisfying Q S .A/ < ıS . Since Q S is a measure we have limm!1 S .Am / D 0. In particular, Q S .Am / < ıS , for all m > M with some M . Thus lim supn!1 n .Am / < " for all m > M by the local uniform domination property. This means that 1 .Am / < " for all m > M and hence limm!1 1 .Am / D 0. Thus a limit measure on FS exists. For FSC1 we take a further subsequence of the subsequence n and find a measure 01 on FSC1 whose restriction to FS is 1 . Continuing inductively, we find a sequence of measures S for each S 2 N; they form a consistent family of probability measures and thus by the Kolmogorov extension theorem there is a unique probability measure on F such that all jFS D S . This proves existence of a convergent subsequence. Since the DLR property holds in finite volume, due to convergence along a subsequence it carries over in the limit.

4.2

Existence and uniqueness by direct methods

4.2.1 External potentials: existence In this section we present results on the existence and uniqueness of Gibbs measures on Brownian paths, using only probabilistic techniques. A powerful alternative, yielding additional insight, is the method of cluster expansion, which will be discussed in Section 4.3. We consider admissible potentials in the sense of Definition 4.2, and start with the case W D 0. Moreover, we restrict attention to the case where the reference measure is Brownian bridge measure. Recall the notation 1 jxj2 … t .x/ D exp : 2t .2 t /d=2 For any interval I D Œs; t R define x;y

WI x;y

x;y

D … ts .y x/WI

;

(4.2.1) x;y

where WI denotes the Brownian bridge measure. Although WI is a finite measure on C.I; Rd /, it is not a probability measure. However, it makes no difference in

202

Chapter 4 Gibbs measures associated with Feynman–Kac semigroups x;y

the definition of finite volume Gibbs measures below whether we use WI and using the former has some advantages in calculations.

x;y

or WI

,

Definition 4.7 (Itô bridge). Let V be an admissible potential. For any bounded I R and x; y 2 Rd define Z R 1 x;y x;y I .A/ D 1A e I V .X t /dt d WI ; (4.2.2) ZI .x; y/ X for every A 2 FI , where Z ZI .x; y/ D x;y

We call I

e

R I

V .X t /dt

X

x;y

d WI

:

(4.2.3)

Itô bridge measure for potential V .

The following result identifies a large class of admissible potentials V . Lemma 4.5. Let V 2 K.Rd /. Then for any bounded interval I R Z R x;y e I V .X t /dt d WI < 1: sup

(4.2.4)

x;y2Rd

Proof. This follows from Khasminskii’s Lemma 3.37. Let V be a potential such that H is a Schrödinger operator with a unique strictly positive ground state ‰p . By adding a constant, if necessary, we choose V such that inf Spec.H / D 0. Take any I R, any division t1 < < tn 2 I , and functions f1 ; : : : ; fn 2 L2 .Rd / \ L1 .Rd /. Recall that the P ./1 -process .X t / t2R associated with H satisfies EN0

n hY

i fj .X tj / D .f1 ‰p ; e .t2 t1 /L f2 e .tn tn1 /L fn ‰p /L2 .Rd /

(4.2.5)

j D1

where L is the ground state transform defined in Section 3.10. From now on we rewrite D N0

(4.2.6)

and refer to as Itô measure for potential V . The Feynman–Kac formula then gives Z Z Z R x;y dx‰p .x/ dy‰p .y/ 1A e I V .X t /dt d WI (4.2.7) .A/ D Rd

for every A 2 FI .

Rd

203


Lemma 4.6. For V 2 K.Rd / and any I D Œa; b R the restriction jFI is absolutely continuous with respect to W jFI , and the density is Z hI .!/ D ‰p .Xa / exp V .X t /dt ‰p .Xb /: (4.2.8) I

Proof. This follows directly from (4.2.7) since by the definition of Wiener measure Z Z Z x;y EW Œf .Xa /g.Xb /h D dx dyf .x/g.y/ hd WI Rd

Rd

holds for bounded measurable f; g W Rd ! R and bounded, FI -measurable h. Write

Z K t .x; y/ D

e

Rt 0

V .Xs /ds

x;y

d WŒ0;t

(4.2.9)

for the integral kernel of e tH , and t .x; y/ D

K t .x; y/ ‰p .x/‰p .y/

(4.2.10)

for the transition density of the P ./1 -process. Since Z .e tL f /.x/ D t .x; y/f .y/‰p2 .y/dy Rd

holds, t is the probability transition kernel of e tL in L2 .Rd ; ‰p2 dx/. Recall that L is given by (3.10.3) and (3.10.10). Then x;y

I .A/ D

R …T2 T1 .x y/ EW x;y Œ1A e I V .X t /dt : ‰p .x/‰p .y/T2 T1 .x; y/ I

(4.2.11)

We prove now that an Itô measure indeed is a Gibbs measure in the sense of Definition 4.4. Theorem 4.7 (Existence of Gibbs measure). Let V be Kato-decomposable and be the Itô-measure for potential V . Then is a Gibbs measure for V . Proof. We need to check is that (4.2.2) is a conditional probability .AjTI /. However, this follows immediately from (4.2.11) and the representation (4.2.7) of . We end this section by pointing out that any two Itô measures are distinguished by their potentials. Proposition 4.8. Take V1 ; V2 2 K.Rd / and their Itô measures 1 , 2 , respectively. If V1 V2 is different from a constant outside of a set of Lebesgue measure zero, then 1 and 2 are mutually singular measures.

204


Proof. If V1 and V2 differ by a non-constant term, then the ground state 1 of H1 D 12 C V1 is different from the ground state 2 of H2 D 12 C V2 . As both are continuous functions, there is A Rd such that Z Z 2 2 a1 D 1 .X0 2 A/ D .x/dx ¤ 1 2 .x/dx D 2 .X0 2 A/ D a2 : A

A

Both 1 and 2 are stationary, and thus the ergodic theorem below implies N 1X 1A .Xn / D ai N !1 n

lim

for i -almost all paths X;

nD0

P for i D 1; 2. In other words, the set ¹! 2 Xj limN !1 n1 N nD0 1A .Xn .!// D a1 º is of 1 -measure 1 and of 2 -measure 0, i.e., 1 and 2 are mutually singular. Definition 4.8 (Ergodic map). A map T on a probability space .; F ; P / is called ergodic whenever T is measure preserving and TA D A for A 2 F implies P .A/ D 1 or 0. Proposition 4.9 (Ergodic theorem). Let T be a measure preserving map on a probability space .; F ; P /. Take f 2 L1 ./. Then the limit n1 1X f .T j !/ n!1 n

g.!/ D lim

j D0

exists for almost every ! 2 , and EP Œg D EP Œf . If, moreover, T is ergodic, then g D EP Œf is constant.

4.2.2 Uniqueness Having settled the existence problem, the next natural question is whether only a single Gibbs measure exists for given V . We start by an illuminating example. Example 4.2 (Non-uniqueness). Take V .x/ D 12 .x 2 1/, x 2 R, and consider the corresponding Schrödinger operator Hosc D 12 C V . Its ground state is ‰p .x/ D 2 1=4 e x =2 , and the process generated by Hosc is the one-dimensional Ornstein– Uhlenbeck process. Fix ˛; ˇ 2 R and define for s; x 2 R l s .x/

D

r s .x/

D

1 1=4 1 1=4

s 2 ˛e 1 exp .x C ˛e s /2 exp ; 2 2 s 2 ˇe 1 exp .x C ˇe s /2 exp : 2 2

205


A simple calculation yields e tHosc Hence Z Y n

l s

D

l sCt ;

e tHosc

fj .X tj /d˛;ˇ D e ˛ˇ=2 .f1

r s

D

r st ;

.

l s;

r s/

D e ˛ˇ=2 :

l .t2 t1 /Hosc f2 e .tn tn1 /Hosc fn t1 ; e

r tn /

j D1

gives rise to the finite dimensional distributions of the probability measure ˛;ˇ on C.R; R/ of a Gaussian Markov process. This is stationary if and only if ˛ D ˇ D 0. That ˛;ˇ is a Gibbs measure for every ˛; ˇ 2 R can be checked by straightforward computation. In this case thus uncountably many Gibbs measures exist for the same potential. The above example shows that a Gibbs measure for a Kato-decomposable potential may not be unique. However, Lemma 4.3 gives a simple criterion allowing to check if a Gibbs measure is the only one supported on a given subset. In order to apply x;y Lemma 4.3, we need to know for which !N 2 X is it true that N .A/ ! .A/ (with !.N N / D x; !.N N / D y). The next lemma gives a sufficient condition for this in terms of the transition densities T of . Lemma 4.10. Let V be Kato-decomposable, and assume that the Schrödinger operator 1=2 C V has a ground state ‰p . Suppose that for some !N 2 X ˇ ˇ ˇ ˇ N T .!.N N /; x/ .y; !.N N // N T lim sup ˇˇ 1ˇˇ‰p .x/‰p .y/ D 0 (4.2.12) N !1 x;y2Rd N .!.N N /; !.N N // x;y

for all T > 0. Then for every T > 0 and A 2 FT , N .A/ ! .A/ as N ! 1. Proof. Let A 2 FT , and take A ¹! 2 Xjj!.˙T /j < M º for some M > 1. Fix N > T and !N 2 X, and put !.N N / D and !.N N / D . The Markov property of Brownian motion and the Feynman–Kac formula give Z Z R T 1

;

;x N .A/ D dx dy EŒN;T Œe N V .Xs /ds ZN .!/ N Rd Rd RT

RN

EŒT;T Œe T V .Xs /ds 1A EŒT;N Œe T V .Xs /ds Z Z RT KN T . ; x/KN T .y; / x;y D EŒT;T Œe T V .Xs /ds 1A : dx dy K2N . ; / Rd Rd x;y

y;

By having restricted to A and by boundedness of K2T .x; y/, the last factor in the above formula is a bounded function of x and y with compact support, thus integrable over Rd Rd . The claim follows for A once we show that N !1

N /; x/KN T .y; !.N N //=K2N .!.N N /; !.N N // ! ‰p .x/‰p .y/; KN T .!.N

206


uniformly in x; y 2 Rd . This is equivalent to (4.2.12). For general A 2 FT , consider BM D ¹! 2 Xj max¹j!.T /j; j!.T /jº < M º with M 2 N, and AM D A \ BM . c Since BM % X as M ! 1, for given " > 0 we may take M 2 N with .BM / < ". Moreover, for AM and BM we find N0 2 N such that for all N > N0 we have c jN .AM / .AM /j < " resp. jN .BM / .BM /j < ". Hence N .BM / < 2" for all N > N0 , and thus jN .A/ .A/j D jN .AM / C N .A n BM / .AM / .A n BM /j c c jN .AM / .AM /j C N .BM / C .BM / 4":

This shows that N .A/ ! .A/. Having these results at hand, we can now state and prove our results on uniqueness. Theorem 4.11 (Case of intrinsic ultracontractivity). Suppose V is such that the Schrödinger semigroup is intrinsically ultracontractive in the sense of Definition 3.41. Then the Itô measure corresponding to it is the unique Gibbs measure for V supported on X. Proof. By Theorem 3.105, limN !1 jN .x; y/ 1j D 0 uniformly in x and y. Thus (4.2.12) holds. The proof of Theorem 3.105 has the following consequence which we state separately for later use. Corollary 4.12. Let t be the probability transition kernel of a P ./1 -process generated by an intrinsically ultracontractive Schrödinger semigroup. Then there exist a1 ; a2 > 0 such that for large enough t > 0, a1 t .x; y/ a2 uniformly in x; y 2 Rd . Recall that for an operator H the non-negative number ƒ D inf¹Spec.H / n inf Spec.H /º inf Spec.H / is called spectral gap of H . Theorem 4.13 (Case of Kato-class). Suppose V is Kato-decomposable and ‰p 2 L2 .Rd / \ L1 .Rd /. Put ˇ μ ´ ˇ e ƒjN j ˇ D0 : (4.2.13) X0 D ! 2 X ˇ lim ˇN !˙1 ‰p .!.N // Then is the unique Gibbs measure for V supported on X0 .

207


Proof. With P‰p the projection used in the proof of Theorem 4.11 write L t D e tH P‰p : This is an integral operator with kernel KQ t .x; y/ D K t .x; y/ ‰p .x/‰p .y/. As ƒ > 0 by assumption, we have kL t k 2;2 D e ƒt : In order to estimate KQ t note that Z Q Q sup jK t .x; y/j D K t .x; y/f .y/dy sup D kL t k 1;1 ; x;y2Rd

f 2L1 ;kf k 1 D1

Rd

1

and since e tH P‰p D P‰p e tH D P‰p for all t > 0, we get kL t k 1;1 D ke H .e .t2/H P‰p /e H k1;1 ke H k2;1 kL t2 k2;2 ke H k1;2 : By Theorem 3.39, both ke H k2;1 and ke H k1;2 are finite, thus it follows that for every t N , jKN t .x; y/ ‰p .x/‰p .y/j C t e ƒN ; where C t D ke H k2;1 ke H k1;2 e ƒ.2Ct/ is independent of x; y and N . In terms of N this implies that for all !N 2 X0 , jN T .!.N N /; x/ 1j‰p .x/ CT e ƒN =‰p .!.N N // ! 0; jN T .y; !.N N // 1j‰p .y/ CT e ƒN =‰p .!.N N // ! 0; j2N .!.N N /; !.N N // 1j C0 e 2ƒN =.‰p .!.N N //‰p .!.N N /// ! 0 as N ! 1. From this (4.2.12) can be deduced. It remains to show that is actually supported on X0 . By time reversibility it suffices to show that ! e ƒN lim sup > n D 0; 8n > 0: N !1 ‰p .!.N // To prove this, note that by stationarity of , ! Z exp.ƒN / e ƒN >n D k‰p kL1 : 1¹‰p <exp.ƒN /=nº ‰p2 dx d ‰p .!.N // n R The right-hand side of the last expression is summable in N for each n, thus an application of the Borel–Cantelli lemma completes the proof. Remark 4.6. The additional assumption that ‰p 2 L1 is very weak. In fact, for many potentials V it is known that ‰p decays exponentially at infinity. A large class with ‰p 2 L1 is dealt with in Lemma 3.59. In Theorem 4.13 X0 is seen to depend directly on the decay of ‰p at infinity. Thus for given conditions on V it is interesting to know the asymptotic behaviour of ‰p . Suppose that C2 jxj2˛ V .x/ C1 jxj2˛ ;

˛>1

(4.2.14)

208


outside of a compact set of Rd ; we say in this case that V is a super-quadratic potential. Then there exist constants b1 ; b2 ; D1 ; D2 such that D2 exp.b2 jxj˛C1 / ‰p .x/ D1 exp.b1 jxj˛C1 /;

(4.2.15)

see Corollaries 3.60 and 3.63. Also, for super-quadratic potentials V the corresponding Schrödinger semigroup is intrinsically ultracontractive (see Theorem 3.104). It can be checked directly that all potentials considered in Lemma 3.59 are Katodecomposable. Example 4.3. Let V be a non-constant polynomial bounded from below. This in particular implies that the degree of V is even, so H has a unique ground state and a positive spectral gap. From Theorem 4.11 and Example 4.2 it follows that a Gibbs measure for V is unique if and only if the degree of V is greater than 2. Example 4.4. Let V .x/ D jxj2˛ with ˛ > 0. Again, H has a unique ground state ‰p and a spectral gap ƒ. In case ˛ 1, Theorem 4.11 and Example 4.2 completely solve the question of uniqueness. In case 0 < ˛ < 1, (3.4.32) implies that Theorem 4.13 applies, and thus uniqueness holds on a set X0 D X0 .˛/ given by (4.2.13). By Corollaries 3.61 and 3.63 sharp estimates can be obtained on this supporting set. Indeed, ˇ ³ ² ˇ j!.˙T /j˛C1 ƒ ˇ < (4.2.16) X0 .˛/ ! 2 X ˇlim sup T b1 T !˙1 ˇ ² ³ ˇ j!.˙T /j˛C1 ƒ < X 2 X ˇˇlim sup T b2 T !˙1 with constants b1 ; b2 . Note that, counterintuitively, X0 .˛/ becomes smaller as ˛ increases to 1. Indeed, a fast growing potential should bring the path back to stationarity more quickly, as it is seen in the case ˛ > 1. What happens for ˛ < 1 results from (4.2.13), which turns out to be too crude for this range of exponents.

4.2.3 Gibbs measure for pair interaction potentials In this section we discuss the more difficult situation when the pair interaction potential W is non-zero. Let X0 be the space of functions ! W R ! Rd , t 7! !.t / that are continuous in every t ¤ 0, and for which a left and a right limit exist in t D 0. For > 0 define the shrinking operator W X ! X0 by ´ !.t / if t 0 (4.2.17) . .!//.t / D !.t C / if t < 0:

209


Let V W Rd ! R be an external potential, and W W Rd Rd R ! R be a pair interaction potential in the sense of Definition 4.2. Write Z WT .X.!// D

Z

T

T

ds T

T

W .X t .!/; Xs .!/; t s/dt

(4.2.18)

whenever the right-hand side is well defined. We consider the following regularity conditions on the potentials V and W . Assumption 4.1. (1) V is Kato-decomposable and is such that the Schrödinger operator H D 12 C V has a unique ground state ‰p 2 L2 .Rd / \ L1 .Rd / with the property H ‰p D 0:

(4.2.19)

(2) There exists C1 2 RC such that Z

C1

jW .X t .!/; Xs .!/; t /jdt C1 :

sup

!2X 1

(4.2.20)

(3) There exist C; D 0 such that with C < lim inf V .x/;

(4.2.21)

WT .X.!// WT ..X. !/// C C C D

(4.2.22)

jxj!1

the bound

holds for all T; > 0 and all ! 2 X. (4.2.19) can be achieved by adding a constant to V , if necessary. Recall that N0 denotes the path measure of the P ./1 -process with potential V and stationary measure ‰p2 .x/dx. We then have the following result. Proposition 4.14. With Assumption 4.1, let dhT be the finite volume Gibbs measure for the potentials V and W , with stochastic boundary condition h D hT D ‰p .XT /‰p .XT /. Then on FT we have 1 dhT D exp ZT with ZT D EN0 Œexp.

RT T

ds

Z

Z

T

ds T

RT T

!

T

T

W .X t ; Xs ; t s/dt d N0 ;

W .X t ; Xs ; t s/dt /.

210


Proof. By the Feynman–Kac formula in its version (3.10.35) we find for all T < s1 < < sn < T that EN0 Œf1 .Xs1 / fn .Xsn / Z RT D dxEx Œ‰p .XT /f1 .Xs1 / fn .Xsn /‰p .XT /e T V .Xs /ds : Rd

By an approximation argument the above equality can be extended to all bounded RT RT FT -measurable functions. Since exp. T ds T W .X t ; Xs ; t s/dt / is such a function, we conclude that Eh Œg1 .Xs1 / gn .Xsn / T

Z T Z ds D EN0 exp T

T T

W .X t ; Xs ; t s/dt g1 .Xs1 / gn .Xsn / ;

for all bounded functions g1 ; : : : ; gn , and all T < s1 < < sn < T , and the claim follows. In order to prove the existence of a Gibbs measure for the potentials V and W , we apply Proposition 4.2. To be able to use this result we need to prove that the family Z T Z T 1 dT D exp ds W .X t ; Xs ; t s/dt d N0 (4.2.23) ZT T T from Proposition 4.14 is relatively compact. By Proposition 4.4 it will suffice to show local uniform domination. We start by showing that the one-point distributions T are tight as T ! 1. Proposition 4.15. Define T as in (4.2.23), and suppose Assumption 4.1 holds. Then for every " > 0 and S > 0 there exists R > 0 such that sup T .jXu j > R/ " T >0

for all juj S. Proof. First we relate T .jXu j > R/ to T .jX0 j > R/. Assume that u > 0, the case u < 0 can be treated similarly. Using a formal, though suggestive, notation we obtain !2 Z Z Z Z Z T

T

T Cu

T

T

D Z D

T

Z

T Cu T Cu

Z

C2

T Cu

C

T Cu

Z

T Cu

T Cu

C

T Cu

T

T Cu Z T Cu

T

T

T Cu Z T Cu

T Cu

2

Z

T

T Cu Z T Cu

Z C

T Cu Z T Cu

T

: T

T Cu

T

Z 2

T Cu Z T Cu T

T


211

Inserting the pair interaction potentials in the double integrals, we see that the last integral in the first line is bounded in absolute value by kW k1 u2 , and the same is true for the first two terms in the second line, with a factor 2 in front of the second term. The last two terms in the second line are bounded by 2C1 u, where we used (4.2.20). Thus, sup jWT .X.!// WT Cu .X.!//j D 4kW k1 u2 C 4C1 u D C1 .u/

(4.2.24)

!2X

and we find ZT T .jXu j > R/ D EN0 Œ1¹jXu j>Rº e WT .X/ e C1 .u/ EN0 Œ1¹jXu j>Rº e WT Cu .X/ D e C1 .u/ EN0 Œ1¹jX0 j>Rº e WT .X/ D e C1 .u/ ZT T .jX0 j > R/: The equality of the first and second lines above follows from the stationarity of the measure N0 . Thus the claim will be shown once we show it for u D 0. We proceed in several steps. Step 1: By Theorem 3.106, we have Z 1 T .jX0 j > R/ D ‰ 2 .y/EN y Œe WT .X/ dy: 0 ZT jyj>R p

(4.2.25)

In the next steps we will show that there exist K; r > 0 such that for every T > 0 and y 2 Rd , EN y Œe WT .X/ 0

K inf EN0z Œe WT .X/ : ‰p .y/ jzjr

(4.2.26)

Assume for the moment that (4.2.26) holds. We can estimate Z 1 1 T .jX0 j r/ D E y Œe WT ‰p2 .y/dy ZT jyjr N0

1 Cr inf EN0z Œe WT inf ‰p2 .z/; ZT jzjr jzjr

where Cr is the volume of a ball with radius r. Thus infjzjr EN0z Œe WT CQ r ZT , where the constant only depends on r but not on T . Using (4.2.25) and (4.2.26) gives Z T .jX0 j > R/ K CQ r ‰p .y/dy: (4.2.27) jyj>R

Since ‰p 2 L1 , the right-hand side above vanishes when R ! 1, and the claim follows.

212


Step 2: In order to prove (4.2.26) we change the probability space. Recall the definition of X0 at the beginning of the section, and consider J W X0 ! X2C D C.Œ0; 1/; Rd Rd /; ´ .!.t /; !.t // .!.t // t2R 7! .lim t!0C !.t /; lim t!0 !.t //

t > 0; t D 0:

(4.2.28)

On fixing the value of ! 2 X0 at t D 0, J becomes a bijection; this value can be chosen, for instance, equal to the left limit. Write ! D .!0 ; !00 / for the elements of X2C . When we omit the variable !, we write X D .X 0 ; X 00 /, where X 0t .!/ D X t .!0 /, and X 00t .!/ D X t .!00 /. The image measure of N0z under J can be described explicitly. For z D .z 0 ; z 00 / 2 d R Rd denote by N0z the measure of the Rd Rd -valued P ./1 -process with potential VQ .x; y/ D V .x/ C V .y/, conditional on !0 D z. Write FQT for the field over X2C generated by point evaluations at points inside Œ0; T . Then for every FQT -measurable bounded function f Z R T 1 0 .V .X 0s /CV .X 00 s //dsf .!/‰ .X 0 /‰ .X 00 / dW z .!/: EN0z Œf D e p p T T ‰p .z 0 /‰p .z 00 / (4.2.29) Here, W z denotes 2d -dimensional Wiener measure starting at z. (4.2.29) follows from the fact that the ground state of the Schrödinger operator 1 1 H ˝ 1 C 1 ˝ H D z 0 z 00 C V .z 0 / C V .z 00 / (4.2.30) 2 2 is given by ‰p .z 0 /‰p .z 00 /, combined with (3.10.33). Note that the bottom of the spectrum of (4.2.30) is zero by shifting by a constant. The Markov property and time reversibility of X imply that for every z 2 Rd , .z;z/ N0 is the image of N0z under J , i.e. EN0z Œf D EN .z;z/ Œf ı J 1 ; 0

for all bounded functions f . Write WQ T .X / D WT .J 1 .X //. Then, in particular, Q

EN0z Œe WT .X/ D EN .z;z/ Œe WT .X / :

(4.2.31)

0

From (4.2.26) it can be seen that the quantity we want to estimate is the expectation Q .z;z/ . It is easy to check that of e WT .X / with respect to N0 Z T Z T WQ T .X / D ds dt .W .X 0t X 0s ; js t j/ C W .X 00t X 00s ; js t j/ 0

0

C W .X 0t X 00s ; js C t j/ C W .X 00t X 0s ; js C t j//: (4.2.32)

213


Step 3: Next notice that the shrinking operator is mapped to the time shift on X2C under J . In other words, Q D J J 1 is the usual time shift that maps .!.t // t0 to .!.t C // t0 . Thus in the new representation (4.2.22) becomes WQ T .X .!// WQ T .X .Q !// C C C D

(4.2.33)

for all X 2 X2C and all T; > 0. Our strategy is to use (4.2.33) together with the strong Markov property of N0z . For r > 0 let r D inf¹t 0j jX t j rº be the hitting time of the centered ball with radius r. This is a stopping time and we write Fr for the associated -field. Then for every z 2 Rd Rd , Q

EN0z Œe WT .X / Q

Q

Q

D EN0z ŒEN0z Œe WT .X / jFr EN0z ŒEN0z Œe WT .X ır / e C r CD jFr Q

Q

D EN0z Œe C r CD EN0z Œe WT .X ır / jFr D EN0z Œe C r CD E

X r

N0

Q

Œe WT

Q

sup EN y Œe WT EN0z Œe C r CD : jyjr

(4.2.34)

0

In order to obtain (4.2.26) we need a good estimate of the second factor at the righthand side of (4.2.34), and estimate the supremum in the first factor against an infimum. This will be done in Steps 4 and 5 below. Step 4: We show that there exist r; > 0, such that for all z 2 Rd Rd EN0z Œe

C r

C k‰p k1 1C

1 1 C : ‰p .z 0 / ‰p .z 00 /

(4.2.35)

Above C is the constant from (4.2.22). To show (4.2.35), we choose 0 < < lim infjxj!1 V .x/ C , and take r so large that V .x/ > C C for all x 2 Rd p with jxj > r= 2. This is possible by the definition of C . Clearly, ¹z 2 Rd Rd j jzj > rº p p ¹z 2 Rd Rd j jz 0 j > r= 2º [ ¹z 2 Rd Rd j jz 00 j > r= 2º;

214


and with (4.2.29) it follows that ‰p .z 0 /‰p .z 00 /EN0z Œ r > t Z Rt 0 00 D e 0 .V .X s /CV .X s //ds 1¹jX s j>r Z

X2C

X2C

e

Rt 0

V .X 0s /ds

Rt

e

0

V .X 00 s /ds

0 00 z 8stº ‰p .X t /‰p .X t / d W .!/

.1¹j.X 0 /j>r=p2 s

0

8stº

C 1¹j.X 00 /j>r=p2 s

8stº /

00

‰p .X 0t /‰p .X 00t / d W z .!0 / d W z .!00 / Z Rt 0 0 00 D ‰p .z / e 0 V .X s /ds 1¹j.X 0 /j>r=p2 8stº ‰p .X 0t / d W z .!0 / s Z Rt 00 00 0 C ‰p .z / e 0 V .X s /ds 1¹j.X 00 /j>r=p2 8stº ‰p .X 00t / d W z .!00 / s

.‰p .z 0 / C ‰p .z 00 //k‰p k1 e .C C/t : The second equality above is due the eigenvalue equation e tH ‰p D ‰p and the Feynman–Kac formula. It follows that 1 1 EN0z Œ r > t C k‰p k1 e .C C/t ; ‰p .z 0 / ‰p .z 00 / and using the equality Z E

N0z

Œe

C r

D1C

1 0

C e C t EN0z Œ r > t dt;

(4.2.36)

we arrive at (4.2.35). Step 5: Let r > 0 be as in Step 4. We show that there exists M > 0 such that Q

Q

sup EN0z Œe WT M inf EN0z Œe WT jzjr

jzjr

(4.2.37)

uniformly in T > 0. Denote by p t .z; y/ the transition density from y to z in time t under N0 , i.e., define p t through the relation Z z N0 .X t 2 A/ D p t .z; y/1A .y/d y Rd Rd

for all z 2 Rd Rd and measurable A Rd Rd . By (4.2.29) and the Feynman–Kac formula we have p t .z; y/ D

‰p .z 0 /‰p .z 00 / K t .y 0 ; z 0 /K t .y 00 ; z 00 /; ‰p .y 0 /‰p .y 00 /

(4.2.38)

215


where K t is the integral kernel of e tH . ‰p and K t are both strictly positive and uniformly bounded; the latter statement follows from the boundedness from L1 to L1 of the operator e tH , see Theorem 3.39. Thus they are bounded and non-zero on compact sets, and for every R > 0 ˇ ² ³ p t .x; z/ ˇˇ S t .R; r/ D sup jxj r; jyj r; jzj R p t .y; z/ ˇ is finite. Defining WQ T1 like in (4.2.32) with the integrals starting at 1 rather than at 0, we see that there is a constant C1 independent of T such that WQ T .X .!// C1 WQ T1 .X .!// WQ T .X .!// C C1 for all ! 2 X2C and T > 0. Define B D ¹jX 1 j < Rº. Then for every x with jxj < r we have Q

Q

Q

1

Q

EN0x Œe WT e C1 EN0x Œ1B e WT C e C CD EN0x Œ1B c e WT ı1 :

(4.2.39)

Defining WN T as in (4.2.32) with js C t C 2j appearing instead of js C t j everywhere, in the first term at the right-hand side of (4.2.39) we find Z Q1 N EN0x Œ1B e WT D p1 .x; z/EN0z Œe WT 1 d z jzjR

Z

e

0

jyj>R

D

p1 .z; y/EN y Œe C r CD d y 0

C k‰p k1 p1 .z; y/ 1 C

jyj>R

1 1 C ‰p .y 0 / ‰p .y 00 /

By (4.2.38) and the eigenvalue equation we have Z 1 1 C p1 .z; y/ dy ‰p .y 0 / ‰p .y 00 / Rd Rd Z Z 1 1 00 00 00 K .z ; y /dy C K1 .z 0 ; y 0 /dy 0 : D 1 ‰p .z 0 / Rd ‰p .z 00 / Rd

d y:

(4.2.41)

216


We know by Theorem 3.39 that e tH is bounded from L1 to L1 . In particular, the image of R the constant function f .x/ D 1 is bounded, and we conclude that supz 00 2Rd j Rd K1 .z 00 ; y 00 /dy 00 j < 1. Thus the right-hand side of (4.2.41) is uniformly bounded on ¹z 2 Rd Rd j jzj < rº. This implies that for any ı < 1 there is RN > 0 large enough such that Z C k‰p k1 1 1 sup C d y e .C C2D/ ı: p1 .z; y/ 1 C 0/ 00 /

‰ .y ‰ .y N p p jzjR Inserting this and (4.2.40) into (4.2.39) we arrive at Q

Q

Q

N r/e C1 EN x Œe WT C ı sup E y Œe WT ; EN0z Œe WT S1 .R; N 0 jyjr

0

(4.2.42)

which holds for all x; z with jxj; jzj r. By taking the supremum over z and the infimum over x in (4.2.42), after rearranging we find Q

sup EN0x Œe WT

jxjr

N r/e C1 S1 .R; Q inf EN0z Œe WT ; 1ı jzjr

which concludes Step 5 and completes the proof of the theorem. The next step is to show local uniform domination. Proposition 4.16. Let Assumption 4.1 hold, and define T as in (4.2.23). Then for every S > 0 the restrictions of the family .T /T 0 to FS are uniformly dominated by the restrictions of N0 to FS . Proof. Fix S > 0 and " > 0. Using Proposition 4.15, choose R > 0 so large that T .jXs j > R/ < "=8 for all jsj < S C 1 and T > 0. Put B D ¹jXS1 j < R; jXS j < R; jXS j < R; jXSC1 j < Rº: Then we have T .B c / < "=2 for all T > 0. For any A 2 FS T .A/ D T .A \ B c / C T .A \ B/ "=2 C ET ŒT .A \ BjTSC1 /: (4.2.43) We are only interested in the lim sup of this as T ! 1, thus we choose T > S C 1. Since T is a finite volume Gibbs measure, (4.1.17) gives N D T .A \ BjTSC1 /.!/

E.N0 /!N

SC1

e

1 Œe Hƒ.SC1;T / 1A\B E ! N Œe Hƒ.SC1;T / .N0 /SC1

8C.SC1/2

E.N0 /!N

SC1

Œ1A\B :

217

Section 4.3 Existence and properties by cluster expansion

The inequality above follows from (4.2.20) by a similar reasoning as the one leading to (4.2.24), and the constant C above does not depend on T . Moreover, by the choice of B we have Z Z K1 .!.S N 1/; y/ E.N0 /!N Œ1A\B D 1¹j!.S1/jRº 1¹j!.SC1/jRº N N SC1 ‰p .y/ jzjR jyjR EN0 Œ1A j!.S/ N D y; !.S N / D z

N C 1// K1 .z; !.S dy dz; ‰p .z/

where, as in (4.2.38), K t is the kernel of e tH . As in the argument following (4.2.38), N it can be seen that there exists D > 0 such that K1 .!.S1/;y/ D‰p .y/ for all ‰p .y/ jyj R and j!.S N 1/j R. Thus, E.N0 /!N Œ1A\B SC1 Z Z D2 jzjR

Z D

Z

jyjR

2 Rd

Rd

‰p .y/EN0 Œ1A j!.S/ N D y; !.S N / D z‰p .z/dy dz

‰p .y/EN0 Œ1A j!.S/ N D y; !.S N / D z‰p .z/dy dz D D 2 EN0 Œ1A :

We have obtained that 2

T .A/ "=2 C D 2 e C.SC1/ N0 .A/ for every A 2 FS . Thus for given " > 0, we can choose ı < D 2 e C.SC1/ "=2, and then for every A 2 FS with N0 .A/ < ı we have T .A/ < " uniformly in T . This shows the claim. 2

We can now formulate the main theorem of this section. Theorem 4.17 (Existence of Gibbs measure). Under Assumption 4.1 there exists a Gibbs measure for the potentials V and W . Proof. By Propositions 4.16 and 4.14 the family hT of finite volume Gibbs measures is locally uniformly dominated. By Proposition 4.4, the sequence .hn /n2N then has a convergent subsequence, and the limit point is a Gibbs measure.

4.3

Existence and properties by cluster expansion

4.3.1 Cluster representation Beside existence a number of properties of Gibbs measures for densities dependent on a pair interaction W can be derived by using the special technique called cluster expansion. Some of these properties have an immediate application to the understanding of cut-off behaviours as discussed above. In lack of space we can only summarize the

218


key elements of this method and the main results. We refer the reader to the literature for a more detailed discussion. Cluster expansion is a technique developed originally in classical statistical mechanics. It relies on the existence of a small parameter so that a measure with the interaction switched on can be constructed in terms of a convergent perturbation series around the interaction free case. In our case the Gibbs measure for the W 0 case is a path measure of a Markov process, while the Gibbs measure we want to construct for W 6 0 is a path measure of a random process which is not Markovian. The cluster expansion method has two distinct aspects. One is a combinatorial framework allowing to make series indexed by graphs and derive bounds on them by series indexed by trees. The other part is analytic which is more dependent on the details of the model, and consists of basic estimates on the entries of these sums leading to the convergence of the expansion. In this section we make the following assumptions on the potentials. Assumption 4.2. (1) The external potential V W Rd ! R is (a) Kato-decomposable and the Schrödinger operator H D 12 C V has a unique ground state ‰p 2 L2 .Rd / with the property H ‰p D 0 (b) the Schrödinger semigroup generated by H D 12 C V is intrinsically ultracontractive (see Definition 3.41). (2) The pair interaction potential W W Rd Rd R ! R has the symmetry properties W .; ; t s/ D W .; ; jt sj/, W .x; y; / D W .y; x; /, and satisfies either of the following regularity conditions: (W1) There exist R > 0 and ˇ > 2 such that jW .x; y; t s/j R

jxj2 C jyj2 1 C jt sjˇ

(4.3.1)

for every x; y 2 Rd and t; s 2 R. (W2) There exist R > 0 and ˇ > 1 such that jW .x; y; t s/j

R 1 C jt sjˇ

(4.3.2)

for every x; y 2 Rd and t; s 2 R. In particular, if V is a super-quadratic potential with exponent ˛ > 1 as defined by (4.2.14), then condition (1) above holds. Let be the Itô measure for V , which we will use as reference measure. Similarly to (4.2.23) we define ! Z T Z T 1 dT D exp ds W .X t ; Xs ; t s/dt d (4.3.3) ZT T T where > 0 is a parameter.

219


Theorem 4.18 (Existence of Gibbs measure). Take any increasing sequence .Tn/n2N R such that Tn ! 1, and suppose 0 < j j with small enough. Then under Assumption 4.2 the local limit limn!1 ŒTn ;Tn D exists and is a Gibbs measure on .X; F /. Moreover, does not depend on the sequence .Tn /n2N . In order to prove existence of the Gibbs measure we use a cluster expansion controlled by the parameter which we choose to be suitably small. In what follows we outline the main steps of the proof of this theorem. For simplicity and without restricting generality we consider the family of symmetric bounded intervals of R for the index set of measures. We partition ŒT; T into disjoint intervals k D .tk ; tkC1 /, k D 0; : : : ; N 1, with t0 D T and tN D T , each of length b, i.e. fix b D 2T =N ; for convenience we choose N to be an even number so that the origin is endpoint to some intervals. We break up a path X into pieces Xk by restricting it to k . The total energy contribution of the pair interaction can be written in terms of the sum Z T Z T X WT D ds W .X t ; Xs ; s t /dt D Wi ;j (4.3.4) T

T

where with the notation Jij D

Wi ;j

0i<j N 1

R i

ds

R j

W .Xs ; X t ; s t /dt we have

8 ˆ Jij C Jj i ˆ ˆ ˆ < 1 .J C J / C J C J ii jj ij ji D 2 1 ˆ J C J C J ij ji ˆ 2 00 ˆ ˆ :J C J C 1 J ij ji 2 N 1 N 1

if ji j j 2 if ji j j D 1, and i ¤ 0, j ¤ N 1 if i D 0 and j D 1 (4.3.5) if i D N 1 and j D N 2

In order to keep the notation simple we do not make explicit the X t dependence of these objects. A simple calculation gives e

RT T

ds

RT T

W .X t ;Xs ;ts/dt

Y

D

0i<j N 1

D1C

X

.e Wi ;j C 1 1/ Y

(4.3.6)

.e Wi ;j 1/:

RT ¤; .i ;j /2RT

Here the summation is performed over all non-empty sets of different pairs of bounded intervals RT D ¹. i ; j /j. i ; j / ¤ . i 0 ; j 0 / whenever .i; j / ¤ .i 0 ; j 0 /º. Since the reference measure is the measure of a Markov process with transition density t .x; y/, a break-up of the paths involves a corresponding factorization of the reference measure into piecewise pasted conditional measures. Put X tk D xk for x ;x the positions at the time-points of the division, 8k D 0; : : : ; N , and kk kC1 for the

220


corresponding bridge measures. We have thus T . jX t0 D x0 ; : : : ; X tN D xN / D

N 1 O

x ;xkC1

kk

./:

(4.3.7)

kD0

Denote by d m D ‰p2 dx be the stationary distribution of the Itô measure, and let N p t0 ;:::;tN .x0 ; : : : ; xN / be the density with respect to N kD0 d m.xk / of the joint distribution of positions of the path X t recorded at the time-points of the division. By the Markov property it follows that p t0 ;:::;tN .x0 ; : : : ; xN / D

N 1 Y

b .xkC1 ; xk / D

kD0

D1C

N 1 Y

.b .xkC1 ; xk / 1 C 1/

kD0

X

Y

.b .xkC1 ; xk / 1/;

ST ¤; kWk 2ST

where b is given by (4.2.10). The summation runs over all non-empty sets ST D ¹ k j k D .tk ; tkC1 /º of different pairs of consecutive time-points. We have thus the product of two sums over not unrelated collections of subsets as the net result. To keep a systematic control over these sums we introduce the following objects. (1) Contours. Two distinct pairs of intervals . i ; j / and . i 0 ; j 0 / will be called joint and denoted . i ; j / . i 0 ; j 0 / if one interval of the pair . i ; j / coincides with one interval of the pair . i 0 ; j 0 /. A set of connected pairs of intervals is a collection ¹. i1 ; j1 /; : : : ; . in ; jn /º in which each pair of intervals is connected to another through a sequence of joint pairs, i.e., for any . i ; j / ¤ . i 0 ; j 0 / there exists ¹. k1 ; l1 /; : : : ; . km ; lm /º such that . i ; j / . k1 ; l1 / . km ; lm / . i 0 ; j 0 /. A maximal set of connected pairs of intervals is called a contour, denoted by . We denote by N the set of all intervals that are elements of the pairs of intervals belonging to contour , and by the set of time-points of intervals appearing in . N Two contours 1 ; 2 are disjoint if they have no intervals in common, i.e. N1 \ N2 D ;. S Clearly, RT can be decomposed into sets of pairwise disjoint contours: RT D r1 ¹ 1 ; : : : ; r j Ni \ Nj D ;, i ¤ j ; i; j D 1; : : : ; rº. (2) Chains. A collection of consecutive intervals ¹ j ; j C1 : : : ; j Ck º, j 0, j Ck N 1 is called a chain. As in the case of contours, %N and % mean the set of intervals belonging to the chain % and the set of time-points in %, respectively. Two chains %1 ; %2 are called disjoint if they have no common time-points, i.e. %1 \%2 D ;. Denote by @ % resp. @C % the leftmost resp. rightmost time-points belonging to %.

221


(3) Clusters. Take a (non-ordered) set of disjoint contours and disjoint chains, D ¹ 1 ; : : : ; r I %1 ; : : : ; %s º, with some r 1 and s 0. Note that suchScontours and may have common time-points. The notation D . i i / [ S chains . j %j / means the set of all time-points appearing as beginnings or ends of S intervals belonging to some contour or chain in . Also, we put N D . i Ni / [ S . j %Nj / for the set of intervals appearing in through entering some contours or chains. is called a cluster if is a connected collectionSof sets (in the usual sense), and for every % 2 we have that @ %; @C % 2 jr D1 j . This means that in a cluster chains have no loose ends. We denote by KN the set of all clusters for a given N . With these notations the sum in (4.3.6) is then further written as X

Y

X

.e Wi ;j 1/ D

RT ¤; .i ;j /2RT

X

r Y

Y

.e Wi ;j 1/;

r1 ¹1 ;:::;r º kD1 .i ;j /2k

(4.3.8) where now summation goes over collections ¹ 1 ; : : : ; r º of disjoint contours. In a similar way (4.3.7) appears in the form Y

X

.b .xkC1 ; xk / 1/ D

X

X

s Y

Y

.b .xkC1 ; xk / 1/ :

s1 ¹%1 ;:::;%s º j D1 kWk 2%j

ST ¤; kWk 2ST

(4.3.9) Here ¹%1 ; : : : ; %s º is a collection of disjoint chains, and this formula justifies how we defined them. For every cluster D ¹ 1 ; : : : ; r I %1 ; : : : ; %s º 2 KN define the function D

r Y

Y

.e Wi ;j 1/

s Y

Y

.b .xkC1 ; xk / 1/ :

(4.3.10)

mD1 kWk 2%m

lD1 .i ;j /2l

Also, we define the auxiliary probability measure dN .X / D

N 1 O

x ;xkC1

dkk

.Xk /

kD0

N O

d m.xk /:

(4.3.11)

kD0

It is straightforward to check that ¹N ºN 2N is a family of consistent probability measures, thus by the Kolmogorov extension theorem it has a unique extension to R which we denote by . Note that Z Z .b .xkC1 ; xk / 1/d m.xkC1 / D .b .xkC1 ; xk / 1/d m.xk / D 0: Rd

Rd

This is why in our definition a cluster does not contain chains having loose ends as for any such chain EN Œ D 0.

222


Finally, we define the function K D E Œ ;

(4.3.12)

A combination of (4.3.8), (4.3.7), (4.3.9), (4.3.10) and (4.3.12) then gives the cluster representation of the partition function ZT D 1 C

X

X

n Y

n1

¹1 ;:::;n º2KN \ D;;i¤j i j

lD1

Kl

(4.3.13)

for every T D N b=2 > 0. Let G be a graph with vertex set ¹1; : : : ; nº, in which an edge ij joins vertex i with vertex j whenever i [j ¤ ;. Denote by Gn the set of connected graphs G defined in this way, and define ´ 1 if n D 1 T Q .1 ; : : : ; n / D P G2Gn ij 2G .1¹i \j ¤;º / if n > 1: By using a combinatorial function calculus it is possible to show that log ZT D

X

X

n1

¹1 ;:::;n º2KN 021

n Y

T .1 ; : : : ; n /

Kl :

(4.3.14)

lD1

Also, consider for any subset A R the set function ZTA D 1 C

X

X

n Y

n1

1 ;:::;n 2KN \ D;; i¤j i j A\.[i /D; i

iD1

Ki

and write ZT D ZT[ . Then by combinatorial function calculus it is furthermore possible to derive the expressions log ZT D 1 C

X

X

n1

1 ;:::;n 2KN \.[i /D; i

T .1 ; : : : ; n /

n Y

Ki

iD1

and fT D

X ZT D exp ZT

n1

X 1 ;:::;n 2CN \.[i /¤; i

T .1 ; : : : ; n /

n Y

Ki :

iD1

The function fT is a correlation function associated with cluster .

(4.3.15)

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4.3.2 Basic estimates and convergence of cluster expansion There are two crucial estimates on which the cluster expansion depends. The first estimate is on the energy terms coming from the contributions of the intervals in the partition. Recall the notation Jij in (4.3.5). Lemma 4.19 (Energy estimates). We have R R 8 2 2 i Xi .t/dtC j Xj .s/ds < C1 b ˇ .jj i 1jb/ C1 jJij j C2 b 2 : .jj i 1jb/ˇ C1

(W1) case

(4.3.16)

(W2) case

with some C1 ; C2 > 0 in each case respectively. As a consequence, Wi ;j is bounded from below for all i; j , which is a stability condition ensuring that the variables indexed by the subintervals are well defined and there are no short range (i D j ) singularities. The second estimate for the cluster expansion is given by Lemma 4.20 (Cluster estimates). There exist some constants c1 ; c2 > 0 and ı > 1 such that for every cluster 2 KN jK j

Y %2

N .c1 j j1=3 /j%j

Y

Y

2 .i ;j /2

c2 j j1=3 .ji j 1jb/ı C 1

(4.3.17)

where j%j N denotes the number of intervals contained in %. In estimate (4.3.17) the factor accounting for the contribution of chains comes from the upper bound C e ƒb on jb .x; y/ 1j uniform in x; y (see second factor in (4.3.10)), where ƒ is the spectral gap of the Schrödinger operator H of the underlying P ./1 -process, and C > 0. This bound results from intrinsic ultracontractivity of e tH , see Lemma 3.105. The factor accounting for the contribution of contours comes from a suitable use of the Hölder inequality and optimization applied to the products over e Wi j 1 (see first factor in (4.3.10)), taken together with Lemma 4.19. b is then chosen in such a combination with and ƒ that the expression (4.3.17) results. Proposition 4.21 (Convergence of cluster expansion). If there exists . / > 0 such that lim!0 . / D 0 and for every n 2 N X jK j c n ; (4.3.18) 2KN 30;jjDn

then the series at the right-hand sides of (4.3.13) and (4.3.14), respectively, are absolutely convergent as N ! 1.

224


In order to prove this statement we consider the generating function X H.z; / D lim K z jj N !1

2KN 30 1

in which is considered a parameter and jj is the number of intervals contained by . We show that H.z; / is analytic in z in a circle of radius r. /, and that inside this circle H.z; / is uniformly bounded in . By choosing . / D 1=r. / gives then the bound in (4.3.18). This estimate follows through a procedure of translating the sums over KN at the left-hand side of (4.3.18) to sums indexed by connected graphs. For every 2 KN we define a graph G whose vertex set are the contours ¹ 1 ; : : : ; r º in , and whose edge set are those pairs of contours which are appropriately matched with the chains 1 ; : : : ; s of . For r D 1 we have s D 0 and therefore the empty graph G1 . For r 2 we have s 1 and the sums can be rewritten by sums over connected graphs following a specific enumeration. By using a graph-tree bound, the graph-indexed sums can be bounded by tree-indexed sums, which can be better further estimated. An inductive procedure and some non-trivial combinatorics yields then (4.3.18). By an application of Proposition 4.21 to (4.3.15) it can be shown that f D limT !1 fT exists, and jfT j 2jj holds uniformly in T for small enough. This can be used to prove the following statement. Proposition 4.22. The local limit D limT !1 T exists and satisfies the equality n h i X Y X E FS i f [ (4.3.19) E ŒFS D E ŒFS f S C n1

1 ;:::;n 2C \ D;; i¤j i j iWS\ ¤; i

iD1

for any bounded FS -measurable function FS , where S is a finite union of intervals of the partition considered in the cluster expansion. Moreover, the measure is invariant with respect to time shift. Theorem 4.18 follows directly from this proposition. Equality (4.3.19) implies that satisfies the DLR-equations, in particular, it is a Gibbs measure.

4.3.3 Further properties of the Gibbs measure An important aspect in understanding the Gibbs measure is to characterize almost sure properties of paths under this measure. This is answered by the following theorem. Theorem 4.23 (Typical path behaviour). If V is a super-quadratic potential with exponent ˛ > 1, then jX t j C .log.jt j C 1//1=.˛C1/ ; -a.s. (4.3.20) with a suitable number C > 0.

225


The strategy of proving Theorem 4.23 goes by comparing the typical behaviours of the reference process and of the process with the pair interaction potential (which is not a Markov process). Lemma 4.24. Suppose there exist some numbers C; > 0 such that for any x > 0 . max jX t j x/ C e x 0t1

˛C1

:

(4.3.21)

Then there exist C 0 > 0 and 0 > 0 such that for any x > 0 . max jX t j x/ C 0 e 0t1

0 x ˛C1

:

(4.3.22)

The proof of this lemma requires once again the use of cluster expansion. The assumption of the lemma can be verified by using (3.111) for the underlying P ./1 process. Another important property of is its uniqueness in DLR sense. This means that for any increasing sequence of real numbers .Tn /n2N , Tn ! 1, and any corresponding sequence of boundary conditions .Yn /n2N X0 , limn!1 ETn ŒFB jYn D E ŒFB , for every bounded B R, and every bounded FB -measurable function FB . Here X0 is the subspace given by (4.2.13). In other words, DLR uniqueness means that the limit measure is independent of the boundary paths. Theorem 4.25 (Uniqueness of Gibbs measure). Suppose that V satisfies the conditions in Assumption 4.2 and W satisfies (W2). Then the following cases occur: (a) If ˇ > 2, then whenever the Gibbs measure exists, it is unique in DLR sense. (b) If ˇ > 1, then for sufficiently small j j the limiting Gibbs measure is unique in DLR sense whenever the reference measure is unique. If ˇ > 2 the energy functional given by (4.1.15) is uniformly bounded in T , and in the restrictions of paths over ŒT; T and R n ŒT; T , respectively. This then implies that only one Gibbs measure can exist as the limit does not depend on the choice of the boundary conditions. This argument requires no restriction on the values of . For 1 < ˇ 2 this uniform boundedness does not hold any longer and cluster expansion is to date the only method allowing to derive these results. To conclude, we list some additional properties of Gibbs measures for (W2)-type pair interaction potentials, useful in various contexts. This case in particular covers the Nelson model. Theorem 4.26. Let be a Gibbs measure for W satisfying (W2), and suppose V satisfies Assumption 4.2. Then the following hold:

226


(a) Invariance properties: is invariant with respect to time shift and time reflection. (b) Univariate distributions: The distributions $T under T of positions x at time t D 0 are equivalent to , i.e. there exist C1 ; C2 2 R, independent of T and x such that C1

d$T .x/ C2 d

for every x 2 Rd and T > 0. Moreover limT !1 pointwise.

(4.3.23) d$T d

.x/ D

d$ .x/ d

exists

(c) Univariate conditional probability distributions: The conditional distributions T . jX0 D x/ converge locally weakly to . jX0 D x/, for all x 2 Rd . (d) Mixing properties: For any bounded functions F; G on Rd j cov .Fs I G t /j const

sup kF k1 sup kGk1 1 C jt sj

(4.3.24)

where > 0 and the constant prefactor is independent of s; t and F; G. The invariance properties are inherited trivially from the underlying P ./1 -process. The proof of the remaining properties makes extensive use of cluster expansion.

4.4

Gibbs measures with no external potential

4.4.1 Gibbs measure Here we consider the case when the external potential V D 0. We further assume that the pair ’ interaction potential has the form W .X t Xs ; t s/. Then the energy Hƒ.S;T / D ƒ.S;T / W .X t ; Xs ; jt sj/dsdt; is invariant under path shifts !.t / 7! !.t /Cc, which makes it reasonable to expect that no Gibbs measure exists. However, when the path is fixed at t D 0 we can in certain cases construct a Gibbs measure with interesting properties. In contrast to the previous sections, in this section we use Wiener measure W 0 on path space over the whole time line as reference measure. We assume that Z jb .k/j2 ikx !.k/jtj 1 W .x; t / D e e dk (4.4.1) 2 Rd 2!.k/ with !.k/ 0;

!.k/ D !.k/;

b =! ˛ 2 L2 .Rd /;

b .k/ D b .k/

˛ D 1=2; 1; 3=2:

(4.4.2) (4.4.3)

227

Section 4.4 Gibbs measures with no external potential

The reason for this special choice will become clear later. For the moment, we note that by (4.4.3) W is bounded as a function of x and t , Z

1

1 sup jW .B t .! / Bs .! /; t s/jds D 2 0 1 ! 2X

and Z

0

Z

0

1

ds 1

0

0

Z Rd

jb .k/j2 d k < 1; !.k/2

1 dt sup jW .B t .! / Bs .! /; t s/j D 4 ! 0 2X 0

0

Z Rd

(4.4.4)

jb .k/j2 d k < 1: !.k/3 (4.4.5)

We define d PT0

1 D exp ZT

Z

T

Z

T

"Z

where ZT D E

0

T

Z exp

!

T

W .B t Bs ; t s/dt ds d W 0 ;

T T

Z

!#

T T

(4.4.6)

W .B t Bs ; t s/dt ds

:

Proposition 4.27 (Finite volume Gibbs measure). The measure PT0 is a finite volume Gibbs measure for reference measure W 0 and pair interaction potential W . Proof. This follows directly from Proposition 4.1. For the existence of a Gibbs measure we need to refer to the infinite dimensional Ornstein–Uhlenbeck process which will be introduced in Section 5.6; see Theorem 5.26 for the details of definitions and notation. Here we quickly give a brief overview of what we need in this subsection. Let M be a Hilbert space with scalar product Z 1 .f; g/M D fO.k/g.k/ O dk (4.4.7) d 2!.k/ R D M . Let Y D C.R; M /. and MC2 M M2 be a triplet, where MC2 2 2 The process R 3 t 7! t 2 M2 is the M2 -valued Ornstein–Uhlenbeck process on .Y; G / with path measure G . Let t .f / D hh t ; f ii for f 2 MC2 , where hh; ii denotes the pairing between M2 and MC2 . Note that t .f / can be uniquely extended for f 2 M , which for simplicity we keep denoting by t .f /. Then t .f / has mean zero and covariance Z 1 jtsj!.k/ EG Œ s .f / t .g/ D fO.k/g.k/e O d k: (4.4.8) d 2!.k/ R

228


Furthermore, G is a stationary measure on M2 such that .f /, 2 M2 and f 2 M , with mean zero and the covariance given byR (4.4.7). Let G ./ D G .j 0 D / be a regular conditional probability measure; then G.d /EG Œ D EG Œ . We write L2 .G / and L2 .G/ for L2 .Y; d G / and L2 .M2 ; d G/, respectively. For q 2 Rd let q be the shift by q on M2 such that . q /.f / D .q f / for 2 M2 , where .q f /./ D f . q/. It can be seen that the linear hull of ¹e i .f / jf 2 M º is dense in L2 .G/. By . q e i .f / / D e i .q f / , q can be extended to whole L2 .G/. Lemma 4.28. For every q the map q is unitary on L2 .G/, and q 7! q is a strongly continuous group on L2 .G/. Proof. First note that for f; g 2 M , Z Z 1 1 O .f; q g/M D dk D dk O f .k/q g.k/ fO.k/e ikq g.k/ d d 2!.k/ 2!.k/ R R Z 1 D d k D .q f; g/M : q f .k/g.k/ O 2!.k/ Rd

b 1

Similarly, .q f; q g/M D .f; g/M . By Theorem 5.26 (d), .e i .f / ; q e i .g/ /L2 .G/ 1 D exp ..f; f /M C .f; q g/M C .q g; f /M C .q g; q g/M / 2 1 D exp ..q f; q f /M C .q f; g/M C .g; q f /M C .g; g/M / 2 D .q e i .f / ; e i .g/ /L2 .G/ : Since the functions of the form e i s .f / span L2 .G/, the equality of the first two lines above shows strong continuity of the map q 7! q , and the group property qCr D q r . Moreover, the equality of the first and last lines shows that the adjoint of q is given by q . Since clearly q q D 1, q is a unitary map. For T > 0 define P0W D W 0 ˝ G and d PTW

(4.4.9)

Z T 1 D exp Bs s .%/ds d P0W ; ZT T

where ZT is the normalizing constant

Z ZT D EP W exp 0

T T

Bs s .%/ds

:

(4.4.10)

229


Proposition 4.29. For bounded FT -measurable functions f W X ! R we have EP W Œf D EP 0 Œf . T

T

RT

Proof. Since T Bs s .%/ds is a Gaussian process with mean zero and covariance RT RT RT EG Œj T Bs s .%/dsj2 D T ds T dtW .B t Bs ; t s/, the proposition follows. In the light of Proposition 4.29, we will now study the family of measures PTW instead of PT0 . On a first sight this seems more difficult as the state space is now infinite dimensional. However, the great advantage is that PTW is tightly connected to a Markov process, as we will now see. Define the operator P t on L2 .Rd / ˝ L2 .G/ by Z Rt .F; P t G/L2 .Rd /˝L2 .G/ D dxEW x ˝G ŒF .B0 ; 0 /e 0 Bs s .%/ds G.B t ; t /: Rd

(4.4.11) It is not trivial to see Rthat (4.4.11) is well defined for all F; G 2 L2 .Rd / ˝ L2 .G/, t since the exponent 0 Bs s .%/ds is not necessarily negative. However, ˇZ ˇ Rt ˇ ˇ 0 Bs s .%/ds c ˇ ˇ x dxE ŒF .B ;

/e G.B ;

/ (4.4.12) 0 0 t t ˇ kF kkGke ; W ˝G ˇ Rd

R 2 =!.k/2 d k < 1. This will be shown in Corollary 7.8 later where c D t Rd j.k/j O on; see also (6.5.8). We will need to work with the following additional function spaces: L1 .Rd I L2 .G// D ¹f W Rd M2 ! Cj ess sup kf .q; /kL2 .G/ < 1º; q2Rd

L1 .Rd M2 / D ¹f W Rd M2 ! Cj

ess sup

jf .q; /j < 1º;

q2Rd ; 2M2

ˇZ ² ³ ˇ L2 .Rd I L2 .G// D f W Rd M2 ! C ˇˇ jf .q; /j2 dqd G < 1 : Clearly, L1 .Rd M2 / L1 .Rd I L2 .G//, and L2 .Rd I L2 .G// is isomorphic to L2 .Rd / ˝ L2 .G/. We use this identification without further notice. Moreover, we write C.Rd I L2 .G// D ¹f W Rd M2 ! C j Rd 3 q 7! f .q; / 2 L2 .G/ is continuousº; Cb .Rd I L2 .G// D C.Rd I L2 .G// \ L1 .Rd I L2 .G//: We need to study P t on two closed subspaces of Cb .Rd ; L2 .G//. The first one is ˇ ² ³ ˇ d 2 d 2 ˇ C0 .R I L .G// D f 2 Cb .R I L .G// ˇ lim kf .q; /kL2 .G/ D 0 : jqj!1

230


The second subspace T is the image of L2 .G/ under the operator U W L2 .G/ ! C.Rd I L2 .G//;

.Uf /.q; / D . q f /./:

q is an isometry on L2 .G/ for every q. Thus, T equipped with the scalar product hf; giT D hU 1 f; U 1 giL2 .G/

(4.4.13)

is a Hilbert space. Notice that kf kT D kf kL1 .Rd IL2 .G// , and U is an isometry from L2 .G/ onto T . Theorem 4.30. P t is a strongly continuous semigroup on T and on C0 .Rd I L2 .G//. The generator of P t on both spaces is given by H , with 1 .Hf /.q; / D q f .q; / C Hf f .q; / C V% .q; /f .q; /; .q; / 2 Rd M2 ; 2 (4.4.14) where V% .q; / D q .%/ and Hf is the infinitesimal generator of the infinite dimensional Ornstein–Uhlenbeck process . t / t2R . On T , H is a self-adjoint operator. The proof of this theorem depends on a number of auxiliary results. Proposition 4.31. For every " > 0 there exists a constant C independent of q such that 2 2 2 kV% .q; /f kL 2 .G/ "kHf f kL2 .G/ C C kf kL2 .G/

(4.4.15)

for all f 2 D.Hf /. Consequently, Hf C V% .q/ is self-adjoint on D.Hf / and bounded from below uniformly in q. Proof. This is a reformulation of Proposition 6.1, and the proof is given there. The final statement follows from the Kato–Rellich theorem. Proposition 4.32. We have on L2 .Rd I L2 .G// P t D e tH ;

t 0;

(4.4.16)

where H is given by (4.4.14). Moreover, 2 kP t f kL 2 .G/ .x/ D EW x ˝G Œf .Bt ; t /e

for f 2 L1 .Rd M2 / \ L2 .Rd I L2 .G//.

Rt

t Bs s .%/ds

f .B t ; t /

(4.4.17)

231


Proof. (4.4.16) is the Feynman–Kac formula, see Theorems 6.2 and 6.3. For (4.4.17), note that for g 2 L2 .Rd / with compact support, the definition of P t , self-adjointness and the semigroup property give .P t f; gP t f / D .f; P t gP t f / Z Rt R 2t D dxEW x ˝G Œf .B0 ; 0 /e 0 Bs s .%/ds g.B t /e t Bs s .%/ds f .B2t ; 2t / Rd

D EW ˝G Œf .B0 ; 0 /e

R 2t 0

Bs s .%/ds

g.B t /f .B2t ; 2t /;

where W denotes the full Wiener measure. Since both W and G are invariant under time shift, we can replace Bs by Bst and s with st above, and find .P t f; gP t f /L2 .Rd IL2 .G// D EW ˝G Œf .Bt ; t /e

Rt

t Bs s .%/ds

g.B0 /f .B t ; t /:

Now taking for g a sequence of functions approximating a Dirac measure at q, we obtain the claim. Note that the content of Theorem 4.30 is about extending (4.4.16) to the spaces C0 .Rd I L2 .G// and T in the sup-norm. We need a further auxiliary result. Proposition 4.33.

.1/ P t is a C0 -semigroup on L1 .Rd I L2 .G//.

.2/ If f 2 L1 .Rd M2 / \ C.Rd I L2 .G//, then P t f 2 Cb .Rd I L2 .G//, .3/ If f 2 L1 .Rd M2 / \ C.Rd I L2 .G//, then lim kP t f .x; / f .x; /kL2 .G/ D 0

t!0

uniformly in x on compact subsets of Rd . Proof. (1) Once we have shown that P t is bounded on L1 .Rd I L2 .G//, the semigroup property follows from the Markov property of G and W . To show boundedness, note that jP t f j P t jf j pointwise. Therefore it suffices to consider non-negative functions. First let f 2 L1 .Rd M2 /. We fix a not necessarily continuous path X W Œt; t ! Rd and define the bilinear form Z Rt hf; giX D f .Xt ; t /e t Xs s .%/ds g.X t ; t /d G . /: Q For two paths X and X, jhf; giX hf; giXQ j kf k1 kgk1 EG Œ.e

Rt

t Xs s .%/ds

e

Rt t

XQ s s .%/ds 2

/ : (4.4.18)

232


Gaussian integration with respect to the measure G then shows that X 7! hf; giX is continuous from L1 .Œt; t ; R/ to R. Now let X be a continuous path, and define Xs.n/

D

n1 X

X tj=n 1Œtj=n;t.j C1/=n/ .s/:

j Dn

Then jhf; giX .n/ j e c kf .Xt ; /kL2 .G/ kg.X t ; /kL2 .G/ for bounded f and g. X .n/ ! X uniformly on Œt; t as n ! 1, where c D R 2 =!.k/3 d k, and thus the above inequality remains valid when we .1=4/ Rd j.k/j O replace X .n/ by X . For f; g 2 L2 .Rd I L2 .G// the monotone convergence theorem implies jhf; giX j e c kf .Xt ; /kL2 .G/ kg.X t ; /kL2 .G/

(4.4.19)

for all X 2 X. Using this in (4.4.17) shows (1). (2) Choose x; y 2 Rd , f; g 2 L1 .Rd M2 /, and write fy .x; / D f .x C y; /. Then 2 kP t f .x; / P t f .x C y; /kL 2 .G/

D EG Œ.EW x ˝G Œe

Rt

0 Bs s .%/ds

f .B t ; t / e

Rt 0

.Bs Cy/ s .%/ds

fy .B t ; t //2

D EW x Œhf; f iB C hfy ; fy iBCy 2hf; fy iBC1¹t 0º y : Using (4.4.18), (4.4.19) and continuity of B 7! f .B; /, we see that each term in the last line above converges to hf; f iB as r ! 0. It follows that the integrand in the first line converges to zero pathwise, and the second claim follows by the Lebesgue dominated convergence theorem. (3) Define .Q t f /.x; / D EW x ˝G Œf .B t ; t /; .x; /: If f is bounded, a calculation similar to (4.4.18) shows kQ t f P t f kL1 .Rd IL2 .G// ! 0 as t ! 0. Therefore we only need to show that kQ t f f kL2 .G/ vanishes uniformly on compact sets as t ! 0. Write f t .x; / D f .x; t /. Note that f t is a function on Rd Y while f is a function on Rd M2 . By reversibility of W and G , and Schwarz inequality we get 2 2 kQ t f .x; / f .x; /kL 2 .G/ EW x Œkf t .B t ; / f0 .B0 ; /kL2 .G / :


233

Moreover, kf t .B t ; / f0 .B0 ; /kL2 .G / kf t .B t ; / f t .B0 ; /kL2 .G / C kf t .B0 ; / f0 .B0 ; /kL2 .G / D kf .B t ; / f .B0 ; /kL2 .G/ C kf t .B0 ; / f0 .B0 ; /kL2 .G / : It thus suffices to show that lim kf t .x; / f0 .x; /kL2 .G / D 0

t!0

(4.4.20)

uniformly in x on compact sets, and m lim EW x Œkf .B t ; / f .B0 ; /kL 2 .G/ D 0

t!0

(4.4.21)

uniformly on compact sets for m D 1; 2. For proving (4.4.20), assume the contrary. Then there exist bounded sequences .xn /n2N Rd , .tn /n2N RC with tn ! 0, and kf tn .xn ; / f0 .xn ; /kL2 .G / > ı for all n. By compactness, we may assume that xn converges to x 2 Rd . Then ı < kf tn .xn ; / f0 .xn ; /kL2 .G / kf tn .xn ; / f tn .x; /kL2 .G / C kf tn .x; / f0 .x; /kL2 .G / C kf0 .x; / f0 .xn ; /kL2 .G / D kf tn .x; / f0 .x; /kL2 .G / C 2kf .xn ; / f .x; /kL2 .G/ : We have assumed continuity of f . Thus we can choose n0 large enough such that kf .xn ; / f .x; /kL2 .G/ < ı=3, for all n > n0 . Then the above calculation shows that kf tn .x; / f0 .x; /kL2 .G / > ı=3 for all such n. Since kf tn .x; / f0 .x; /kL2 .G / D ke tn Hf f .x; / f .x; /kL2 .G /

(4.4.22)

must converge to zero by the strong continuity of the semigroup e tHf , this is a contradiction and thus (4.4.20) holds. For (4.4.21), note that x 7! f .x; / is uniformly continuous on compact sets. Thus for " > 0 we can choose ı > 0 such that kf .x; / f .x; Q /kL2 .G/ < "=2 whenever jx xj Q < ı. By the properties of Brownian motion there exists t0 > 0 such that for 0 t < t0 , m W x .jB t xj > ı/ < "=.2mC1 kf kL 1 .Rd IL2 .G// /: m Then EW x Œkf .x t ; / f .x0 ; /kL 2 .G/ < " for these t , uniformly in x. This proves (4.4.21).

234


Proof of Theorem 4.30. We first show that P t is a strongly continuous one parameter semigroup on C0 D C0 .Rd I L2 .G//. We use Proposition 4.33 (2) and (3) for an approximation and therefore must first show that L1 .Rd M2 / \ C0 is dense in C0 . Let f 2 C0 and take fR D .f ^ R/ _ R for R 0. L1 .M2 / is dense in L2 .G/, and thus for each x 2 Rd and all " > 0, Rx ."/ D inf¹R 0jkf .x; / fR .x; /kL2 .G/ "º < 1:

(4.4.23)

Rx ."/ is bounded on compact subsets of Rd . To see this, assume the contrary. Then there exists .xn /n2N Rd with xn ! x, and Rn D Rxn ."/ > n for all n. Choosing n0 large enough such that kf .xn ; / f .x; /kL2 .G/ < "=3 for all n > n0 , gives " D kf .xn ; / fRn .xn ; /kL2 .G/ kf .xn ; / f .x; /kL2 .G/ C kf .x; / fRn .x; /kL2 .G/ C kfRn .x; / fRn .xn ; /kL2 .G/ kf .x; / fRn .x; /kL2 .G/ C 2"=3 kf .x; / fn .x; /kL2 .G/ C 2"=3 for each n > n0 . This implies Rx ."=3/ D 1, in contradiction to (4.4.23). Thus Rx ."/ is bounded on compact sets. Since f 2 C0 , Rx ."/ D 0 for jxj large enough, and thus Rx ."/ is bounded on all Rd . Thus bounded functions are dense in C0 . Next we show that P t leaves C0 invariant. Let f 2 C0 . From (4.4.17) and (4.4.19) we have kP t f .x; /kL2 .G/ e c EW x Œkf .Bt ; /kL2 .G/ kf .B t ; /kL2 .G/ D e c EW x Œ1¹jB t jjxj=2º kf .Bt ; /kL2 .G/ kf .B t ; /kL2 .G/ C e c EW x Œ1¹jB t j>jxj=2º kf .Bt ; /kL2 .G/ kf .B t ; /kL2 .G/ : W 0 .jB t j R/ decays exponentially in R for all t , and thus the term in the second line above also decays exponentially since f 2 C0 L1 .Rd ; L2 .G//. The last line is bounded by a constant times supjyj>jxj=2 kf .x; /kL2 .G/ , and this term decays to zero as jxj ! 1, by the assumption f 2 C0 . In a similar way, we obtain strong continuity from Proposition 4.33 (3). Thus P t is a strongly continuous semigroup of bounded operators on C0 D C0 .RI L2 .G//. Now we show that P t is a strongly continuous one-parameter semigroup on T . Choose f 2 T . The measure G is invariant under the map 7! x for every x 2 Rd . Using this, and invariance of the full Wiener measure under translations B 7! B C x, we find .P t f /.x; / D EW 0 ˝G x Œe

Rt

D U.EW 0 ˝G Œe

0 Bs s .%/ds

Rt

B t U 1 f . t /

0 Bs s .%/ds

B t U 1 f . t //:

235


We conclude that for each f 2 T , also P t f 2 T . It is seen that L1 .Rd ; L2 .G//\T is dense in T . Strong continuity then follows directly from Proposition 4.33 (3) and (4.4.13). Finally, we show that the generator of P t is given by H , with H from (4.4.14). W HC0 for the generator in C0 , and HT for the generator in T . For f 2 D.HT / or f 2 D.HC0 /, we have fg 2 D.HC0 / for every g 2 C 2 .Rd ; C/ with compact support, and fg 2 D.HL2 /, where HL2 is the generator of P t as a semigroup on L2 .Rd ; L2 .G//. By the Feynman–Kac formula we know that HL2 fg D Hfg almost everywhere. Therefore HC0 fg D Hfg in L1 .Rd ; L2 .G//. The operators HT , HC0 and H are local in q in the sense that whenever f D g on an open set, then also HT f D HT g on this set, and the same is true for HC0 and H . Thus we can use a smooth partition of unity to conclude that HT f D HC0 f D Hf . H is symmetric in T , and is thus self-adjoint as the generator of a strongly continuous semigroup. This completes the proof of the theorem. We can now get some spectral information on the generator H of P t . Theorem 4.34 (Ground state of H ). Let H be the generator of the semigroup P t acting on T . Then inf Spec.H / is an eigenvalue of multiplicity one. The corresponding eigenfunction ‰ 2 T can be chosen strictly positive. Proof. For a bounded interval I R write Z Z SI D W .Bs B t ; js t j/dsdt: I

(4.4.24)

I

Using Proposition 4.29, we find h1; PT 1iT D EW 0 Œe SŒ0;T ; and

kPT 1k2T D EG Œ.EW 0 ˝G Œe

RT 0

Bs s .%/ 2

/ :

By a similar calculation as the one leading to equation (4.4.17), we obtain kPT 1k2T D EW 0 Œe SŒT;T :

(4.4.25)

This is a key formula. By assumption (4.4.3) there exists C% > 0 such that SŒT;T SŒT;0 C SŒ0;T C C% uniformly in the path. We use this in (4.4.25), apply the Markov property of Brownian motion in the resulting term and reverse time in one of the factors to get kPT 1k2T e C% .EW 0 Œe SŒ0;T /2 D e C% h1; PT 1i2T ;

236


and thus

.T / D

.1; PT 1/2T

e C% :

kPT 1k2T

(4.4.26)

Thus limT !1 .T / > 0 and hence inf Spec.H / is an element in the point spectrum by Proposition 6.8 in Chapter 6. Since e tH D P t is positivity improving, ‰ can be chosen strictly positive by the Perron–Frobenius theorem, and the multiplicity of inf Spec.H / is one. We denote by ‰ the eigenvector corresponding to inf Spec.H /, which we understand to be the strictly positive and normalized version. In the context of the Nelson model discussed below ‰ is the ground state of the dressed electron for total momentum zero, see Section 6.6. Next we consider the limit of the families .PTW /T >0 and .PT0 /T >0 . For a bounded interval I R recall that FI D .X t ; t 2 I /, and write E0 D inf Spec.H /. W to be the unique probability measure Definition 4.9 (Limit measure). We define P1 d on .C.RI R M2 /; F /, such that for every N > 0 and A 2 FŒN;N W P1 .A/ D e 2TE0 EP W Œ‰.BT ; T /e

RT T

Bs s .%/ds

0

‰.BT ; T /1A :

(4.4.27)

W is the path measure of a Markov process with By Theorems 4.30 and 4.34, P1 generator L acting as

Lf D

1 .H E0 /.‰f /: ‰

(4.4.28)

This is shown in the same way as in Theorem 3.106. W as T ! 1 in the topology of local convergence, i.e. Proposition 4.35. PTW ! P1 EP W Œf ! EP1 W Œf for every bounded, FŒt;t -measurable function f and every T t > 0.

Proof. Define ‰T D e TE0 PT 1. By Theorem 4.34 and the spectral theorem we have ‰T ! h1; ‰iT ‰

as T ! 1

(4.4.29)

in T and thus in L1 .Rd I L2 .G//. Choose f 2 FŒt;t and t < T . We have EP W Œf D T

Rt e 2tE0 EP W Œ‰T t .Bt ; t /e t Bs s .%/ds f ‰T t .B t ; t /: 2 0 k‰T kT (4.4.30)


237

(4.4.29) and the fact h1; ‰iT > 0 guarantee that ‰T t =k‰T kT ! ‰ in T . For bounded f 2 FŒ0;T , the map Q on L1 .Rd ; L2 .G// defined through

Z t .Qg/.x; / D EW x ˝G exp Bs s .%/ds fg.B t ; t / 0

is a bounded linear operator on L1 .Rd ; L2 .G//. This follows from the fact jQgj kf k1 P t jgj, and the boundedness of P t . We conclude T !1

Q.‰T t =k‰T kT / ! Q‰ in L1 .Rd I L2 .G//, and the claim follows. We are now ready to state and prove the main result of this section. Theorem 4.36 (Gibbs measure). The family .PT0 /T >0 converges to a probability 0 in the topology of local convergence. Moreover, if f 2 F measure P1 Œt;t satisfies EW 0 Œjf j < 1, then also EP1 0 Œjf j < 1, and in this case we have E 0 Œf ! PT EP1 0 Œf as T ! 1. Proof. The first statement follows from Proposition 4.35 and Proposition 4.29. All the other statements will be proved once we show that there exists C > 0 such that sup EP 0 Œjf j C EW 0 Œjf j

T >0

T

(4.4.31)

for all f 2 FŒt;t . To see (4.4.31), note that EP 0 Œjf j T

e 2C% E 0 Œe SŒT;t e SŒt;t e SŒt;T jf j; ZT W

where SŒa;b is defined in (4.4.24). By using the Markov property, stationarity of increments and time reversal invariance of two-sided Brownian motion it follows that the right-hand side above equals e 2C% EW 0 ŒRT .Bt /e SŒt;t f RT .B t /; where RT .x/ D p

1 h1; PT t 1iT EW x Œe SŒ0;T t D : kPT 1kT ZT

Therefore, RT .x/ is in fact independent of x and converges as T ! 1. Since the pair interaction potential W is bounded, it follows that also SŒt;t is uniformly bounded, and (4.4.31) holds.

238


4.4.2 Diffusive behaviour 0 exists. In this section we discuss By Theorem 4.36 we know that a limit measure P1 the peculiar property of this measure that on a large scale under this measure paths behave like Brownian motion, however, with a reduced diffusion constant. We use the same notations as in the previous section. Notice that for every g 2 C k .R/ it follows that g 2 C k .Rd ; L2 .G//, since the constant function 1 is square integrable with respect to G. We will not distinguish between these two meanings of g in the notation. Let L be as in (4.4.28).

Proposition 4.37. If g 2 C 2 .Rd /, then g 2 D.L/, ‰Lg 2 C.Rd I L2 .G//, and 1 Lg.x; / D x g.x/ C rx g.x/ rx ln ‰.x; /: 2 Proof. With ˛ 2 ¹1; : : : ; d º, @˛ D @x@˛ is the generator of the unitary group f 7! tx˛ f on T , and i @˛ is self-adjoint on T . By Proposition 4.31, Hf C V% is bounded from below on T by a 2 R. Thus E0 D h‰; H ‰iT D

d 1X h‰; @2˛ ‰iT C h‰; .Hf C V% /‰iT 2 ˛D1

d 1X k@˛ ‰k2T a: 2

(4.4.32)

˛D1

This shows @˛ ‰ 2 T . By .H E0 /‰ D 0 and .Hf C V% /g‰ D g.Hf C V% /‰, we find 1 .H E0 /g‰ D ‰x g rx g rx ‰; 2 and this completes the proof. Choosing g.x/ D h .x/ D x above gives .Lh / D rx ln ‰. Thus Lh 2 T . With j D U 1 . rx ln ‰/ 2 L2 .G/

(4.4.33)

we have L. x/ D j./ with .x; t / D x t , and Lh .B t ; t / D j. t /;

with

t D B t t :

Proposition 4.38. The random process . t / t2R is a reversible Markov process under P . Its reversible measure is given by .U 1 ‰/2 d G.

239


Proof. Take f; g 2 L2 .G/. Then jtsjE0 .‰Uf; Pjtsj .‰Ug//T : EP1 W Œf .s /g. t / D e

Thus the generator of the t -process is unitary equivalent to the operator L on the Hilbert space .T ; k k‰ /, where kf k‰ D k‰f kT . L is self-adjoint on this Hilbert space, L1 D 0, and k1k‰ D 1. This proves reversibility. We are interested in proving a central limit theorem for the process g.B t ; t / t2R D W , or equivalently the process . B / 0

B t under P1 t t0 under P1 . The fundamental tool will be the martingale central limit theorem. Write Z

Bt D Mt C

t

j.s /ds;

(4.4.34)

0

Rt with M t D B t 0 j.s /ds. .M t / t0 is a martingale. This follows from the following more general Theorem 4.39. Let .X t / t0 be a Markov process associated with semigroup P t D e tL , and f a continuous function from the state space of the process to the real numbers. Assume that f 2 D.L/, and that EŒf 2 .X t / < 1 and EŒ.Lf /2 .X t / < 1, for all t . Then Z t M t D f .X t / f .X0 / Lf .Xs /ds 0

is a martingale, and Z EŒM t2

2

t

2

D EŒf .X t / EŒf .X0 / 2

0

EŒf .Xs /.Lf /.Xs /ds:

Proof. We have almost surely

EŒM t jFs D Ms C E f .X t / f .X0 /

Z

t s

ˇ ˇ .Lf /.Xr /dr ˇˇ Fs :

(4.4.35)

Our aim is to show that the last term above is almost surely zero. We know that d Pr f .Xs / D Pr Lf .Xs / D .Lf /.XrCs /; dr where the equality is understood to hold when integrated against a bounded Fs measurable test function. Integrating over .s; t / we obtain Z

t s

.Lf /.Xr /dr D P ts f .Xs / f .Xs / D f .X t / f .Xs /;

240


in the same sense as above. This shows that the right-hand side of (4.4.35) is zero when integrated against an Fs -measurable function, thus .M t / t0 is a martingale. Write g.Xs / D .Lf /.Xs /. We have Z t Z t 2 2 2 ds drEŒg.Xs /g.Xr / EŒM t D EŒf .X t / C EŒf .X0 / C 2 0

Z 2EŒf .X0 /f .X t / C 2 Z 2

s

t

drEŒf .X0 /g.Xr / 0

t

drEŒf .X t /g.Xr /:

(4.4.36)

0

Rt Since EŒf .X0 / 0 g.Xr /dr D EŒf .X0 /.f .X t / f .X0 //, the second, fourth and fifth terms of (4.4.36) combine to give EŒf 2 .X0 /. Also, since Z t EŒg.Xs /g.Xr /dr D EŒg.Xs /.f .X t / f .Xs //; s

the third term of (4.4.36) decomposes into two expressions, one of which cancels the last term of (4.4.36), while the other gives the last term in the formula for EŒM t2 above. W -a.s. we now show that .M / Using that M0 D B0 D 0 P1 t t0 is a martingale. 2 Consider the decomposition (4.4.34). We have EP1 0 Œ. B t / < 1 for all t by Proposition 4.36. By (4.4.33), we have Lh D j , and 2 2 EP1 D k rx ‰kL2 .G/ D k rx ‰kT < 1; W Œj. t / D k rx ln ‰k 2 L .G/

where the last assertion follows from the proof of Proposition 4.37. Thus the conditions of Theorem 4.39 are satisfied, and .M t / t0 is a martingale. We show that also the second term of (4.4.34) is also a martingale up to a correction term that is negligible in the diffusive limit. We will need the following result. Theorem 4.40. Let .Y t / t0 be a Markov process with respect to a filtration F t , with state space A, where A is a measure space. Assume that .Y t / t0 is reversible with respect to a probability measure 0 , and that the reversible stationary process with invariant R measure 0 is ergodic. Let F W A ! R be a 0 square integrable function with F d0 D 0. Suppose in addition that F is in the domain of L1=2 , where L is the generator of the process Y t . Let Z t F .Ys /ds: Lt D 0

Then there exists a square integrable martingale .N t ; F t /, with stationary increments, such that 1 lim p sup jLs Ns j D 0 t!1 t 0st

241


in probability with respect to , where L0 D N0 D 0. Moreover, lim

t!1

1 E ŒjL t N t j2 D 0: t

The above theorem can be used to prove the following result. Proposition 4.41. There exists an F t -martingale .Z t / t0 such that 1 lim p

t!1

t

sup jBs Zs j D 0

0st

0 , where B D Z D 0. Moreover, in probability with respect to P1 0 0

1 E 0 ŒjB t Z t j2 D 0; t!1 t P1 lim

and lim

t!1

1 E 0 Œ. B t /2 D j j2 2h rx ‰; .H E0 /1 rx ‰iT ; t P1

(4.4.37)

for all 2 Rd . Proof. By Proposition 4.37, we have .Lh2 /.x; / D j j2 C 2h .x/. rx ln ‰.x; // D j j2 C 2h .x/.Lh /.x; /: By Theorem 4.39, it then follows that Z t 2 2 2 EP1 EP1 W .M t / D W Œ.Lh /.Bs ; s / 2h .Bs /.Lh /.Bs ; s /ds D j j t: 0

2 Next we calculate EP1 W Œ. B t / . By (4.4.34),

Z 2

EP1 W Œ. B t / D

2 EP1 W ŒM t

EP1 W

2

t

j.s /ds 0

Z t j.s /ds : C 2EP1 W . B t / 0

The third term in (4.4.38) is zero. This can be seen as follows. We have

Z t j.s /ds D EW 0 Œ. B t /I.B/; . B t / EP1 W 0

where

Z t Rt 0 s .%/ds I.B/ D EG ‰.0 /e j.s /ds ‰. t / : 0

(4.4.38)

242


Put BQ s D B ts B t . Then by the reversibility of G and the fact that G is invariant Q D I.X /. Moreover, XQ t D B t , under the constant shift by q t , we have I.X/ 0 0 Q Thus W -a.s. and W is invariant under the transformation B 7! B. EW 0 Œ. B t /I.X / D EW 0 Œ. B t /I.X / D 0: 2 2 From the above we know that EP1 W ŒM t D j j t . Denote by Q t the transition W . Then semigroup of P1

1 E W t P1

Z

2

t

j.s /ds 0

1 D t

Z

Z

t

ds 0

0

t

drh rx ln ‰; Qjrsj rx ln ‰i‰

t!1

! 2h rx ln ‰; L1 rx ln ‰i‰ D 2h rx ‰; .H E0 /1 rx ‰iT :

Note that the last quantity is automatically finite; this follows from (4.4.38). So we have shown (4.4.37), but since h rx ‰; .H E0 /1 rx ‰iT D hj; L1 j iL2 .‰2 G/ ; we have also seen that j 2 D.L1=2 /. We have already seen earlier in the proof that Pd j.s / is square integrable for all s. Moreover, EP1 W Œ t D lD1 l h‰; @l ‰iT D 0 since i @l is self-adjoint on T and ‰ is real-valued. Finally, s is a reversible Markov process with respect to F t by Proposition 4.38, and is ergodic, R t since P t is positivity improving. Thus we can apply Theorem 4.40 to see that 0 j.s /ds D M1 C R where M1 is a martingale and R is a process that disappears in the scaling limit in the sense specified in Theorem 4.40. Since the sum of the two F t -martingales Z t D M t C .M1 / t is again a martingale, the proof is complete. The last ingredient that we need is a version of the martingale central limit theorem. We first specify the topology for the convergence of processes that we need. Definition 4.10 (Skorokhod topology). Let ƒT be the set of all strictly increasing, continuous functions W Œ0; T ! Œ0; T such that .0/ D 0 and .T / D T . For bounded functions f; g W Œ0; T ! Œ0; T we define d.f; g/ D inf max¹k 1k1 ; kf g ı k1 º: 2ƒT

It is obvious that d is a metric on the set of bounded function on Œ0; T , and the induced topology is the Skorokhod topology. The space of bounded functions on Œ0; T with the Skorokhod topology is denoted by D.Œ0; T /.

243


Theorem 4.42 (Martingale convergence theorem). Let .Mn /n2N be a martingale with 2 D limn!1 n1 EŒMn2 < 1, and assume that .Mn /n2N has stationary increments. Then 1 lim p MŒtn D 2 B t ; n!1 n in the sense of weak convergence of measures on D.Œ0; T /, for all T > 0. Our main result is as follows. W , a funcTheorem 4.43 (Central limit theorem). For the process .B t / t0 under P1 tional central limit theorem holds, i.e. as n ! 1, the process t 7! p1n B tn , t 0, converges in distribution to Brownian motion with diffusion matrix D given by

Dik D ıik 2h@i ‰; .H E0 /1 @k ‰iT ;

i; k D 1; : : : ; d:

Proof. Let .Z t / t0 be the martingale from Proposition 4.41. By that proposition, the difference B t Z t vanishes in the diffusive limit. The martingale .Z t / t0 is a sum of the martingales from Theorems 4.39 and 4.40. Both of these martingales have stationary increments, the first one due to the fact that .B t / t0 itself has stationary increments. Since .B t / t0 has ergodic increments, so does .Z t / t0 . Using (4.4.37), we see that Theorem 4.42 applies for each coordinate direction, and also find the diffusion matrix.

Part II

Rigorous quantum field theory

Chapter 5

Free Euclidean quantum field and Ornstein–Uhlenbeck processes

5.1

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 an L2 .Q; d/ 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 2 L2 .Q; d/ with its Hamiltonian Hf one way is to changing e itH to e tH regarding it as the analytic continuation of t to i t . In this procedure the Minkowski metric turns into a Euclidean metric: t 2 C x2 ! t 2 C x2 , .t; x/ 2 R R3 . In the Lorentz covariant formulation of scalar quantum field theory, the so called two point function is given Z by .1; .f /.g/1/ D f .x/g.y/W .x y/dxdy with the Wightman distribution W .x y/ D .1; .x/.y/1/. Here x D .t; x/ and .x/ D e itH ..0; x//e itH . The Wightman distribution of the free field Hamiltonian is p Z 2 2 e it jkj C ikx 1 1 W ..t; x// D d k: p 2 .2/3 R3 jkj2 C 2

248

Chapter 5 Free Euclidean quantum field and Ornstein–Uhlenbeck processes

Then the field at time zero is given by Z Z 1 dk fO.k/g.k/ O .f /.g/d D : p 2 R3 .2/3 jkj2 C 2

(5.1.1)

Due to the singularities of the so called Feynman propagator .k02 jkj2 2 /1 ; 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 region 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 7! i t of W ..t; x// yields the Schwinger function Z 1 1 e itk0 ikx W ..i t; x// D d k; k D .k0 ; k/ 2 R R3 : 2 .2/4 R4 jkj2 C 2 C k02 Then the Euclidean time zero quantum field E .f / has covariance Z Z 1 dk E .f /E .g/dE D fO.k/g.k/ O : 2 2 R4 jkj C 2 C k02 QE To establish the connection between L2 .Q/ and the Euclidean space L2 .QE ; dE /, a family of operators ¹I t º t2R is introduced. The functional integral representation of e t.Hf CHI / with some perturbation HI can be also constructed by using the Trotter product formula, the Markov property of Is Is and the identity It Is D e jtsjHf , we have Z .F; e tH G/ D F0 G t e K dE ; (5.1.2) QE

where F0 D I0 F , G t D I t G and e K denotes an integral kernel. The right-hand side 0 of (5.1.2) is the integral over QE , e.g., Sreal .Rd C1 /, which call it functional integral in distinction to path integral. An alternative approach is based on a path measure constructed on C.R; M2 /, i.e., the set of M2 -valued continuous paths, where M2 is the dual of a Hilbert space MC2 . For path integral representations of the heat semigroup .f; e tH g/ in quantum mechanics a path measure is given on C.Œ0; 1/I Rd /. In quantum field theory to M2 -valued paths. In the finite mode ak D p the same procedure yields p .1= 2/.xk C @k / and ak D .1= 2/.xk @k /, k D 1; : : : ; n, and the free field Hamiltonian Hf becomes n q n p X X jkj2 C 2 Hf ! jkj2 C 2 ak ak D .k C jxk j2 1/: 2 kD1

kD1

249

Section 5.2 Boson Fock space

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 on C.RI Rd / is the harmonic oscillator on L2 .Rd /. This suggests that it might be possible to construct an infinite dimensional Ornstein–Uhlenbeck process . t / t2R on Y D C.R; M2 / 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 Z .F; e tH G/ D F . 0 /G. t /e K. / d G (5.1.3) Y

with a suitable kernel K. /.

5.2

Boson Fock space

5.2.1 Second quantization In this section we start the above outlined programme 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 ˝ns of n-fold symmetric tensor product defined through the symmetrization operator 1 X Sn .f1 ˝ ˝ fn / D f .1/ ˝ ˝ f .n/ ; n 1; (5.2.1) nŠ 2} n

where f1 ; : : : ; fn 2 W and }n denotes the permutation group of order n. Define .n/

Fb

.n/

D Fb .W / D ˝ns W D Sn .˝n W /;

where ˝0s W D C. The space Fb D Fb .W / D

1 M

.n/

Fb .W /;

(5.2.2)

(5.2.3)

nD0

L where 1 nD0 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 .n/ space of `2 -sequences .‰ .n/ /n2N such that ‰ .n/ 2 Fb and k‰k2Fb D

1 X

k‰ .n/ k2

.n/

Fb

nD0

< 1:

(5.2.4)

Fb is a Hilbert space endowed with the scalar product .‰; ˆ/Fb D

1 X

.‰ .n/ ; ˆ.n/ /F .n/ :

nD0

b

(5.2.5)

250


The vector b D .1; 0; 0; : : : /

(5.2.6) .n/

is called Fock vacuum. Write Pn for the projection from Fb onto Fb . 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 behaviour 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 7, the particles will be photons. If kˆkFb D 1, then .ˆ; Pn ˆ/Fb represents the probability of having exactly n bosons in the state described by ˆ. p Note that by (5.2.4) this probability must decay faster than 1= n for large n in order to have ˆ 2 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 2 W , and the annihilation operator by a.f /, both acting on Fb , defined by p .a .f /‰/.n/ D nSn .f ˝ ‰ .n1/ /; n 1; (5.2.7) .a .f /‰/.0/ D 0

(5.2.8)

with domain

°

D.a .f // D .‰

.n/

/n0

1 ˇX ˇ 2 Fb ˇ nkSn .f ˝ ‰ .n1/ /k2 nD1

.n/ Fb

± 0 (note the positive exponent), ‰ 2 D.e ˇN / implies that the probability of finding n bosons decays exponentially as n ! 1. We will use the following facts below. The superscript in a] indicates that either of the creation or annihilation operators is meant. Proposition 5.1 (Relative bounds). Let h be a positive self-adjoint operator, and f 2 D.h1=2 /, ‰ 2 D.d .h/1=2 /. Then ‰ 2 D.a] .f // and ka.f /‰k kh1=2 f kkd .h/1=2 ‰k;

ka .f /‰k kh

1=2

1=2

f kkd .h/

‰k C kf kk‰k:

In particular, D.d .h/1=2 / D.a] .f //, whenever f 2 D.h1=2 /.

(5.2.25) (5.2.26)

253


While we do not give the proof of this proposition, we will prove it for a special case below. Example 5.1. (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 2 W . For f 2 W , ka.f /‰k kf kkN 1=2 ‰k; ka .f /‰k kf k.kN 1=2 ‰k C k‰k/:

(5.2.27) (5.2.28)

(2) Another applicationpis considering a perturbation of d .h/ by adding the operator ˆ.f / D .1= 2/.a .f / C a.fN//. Let f 2 D.h1=2 /. Then d .h/ C ˛ˆ.f / is self-adjoint on D.d .h// for all ˛ 2 R. This follows from (5.2.25), (5.2.26) and the Kato–Rellich theorem. To obtain the commutation relations between a] .f / and d .h/, suppose that f 2 D.h1=2 / \ D.h/. Then Œd .h/; a .f /‰ D a .hf /‰;

Œd .h/; a.f /‰ D a.hf /‰;

(5.2.29)

for ‰ 2 D.d .h/3=2 / \ Fb;fin . By a limiting argument (5.2.29) can be extended to ‰ 2 D.d .h/3=2 /, and it is seen that a] .f / maps D.d .h/3=2 / into D.d .h//. In general a] .f / W D.d .h/nC1=2 / ! D.d .h/n / T n / \ D.hn1=2 / . In particular, a] .f / maps for all n 1, when f 2 1 D.h nD1 T1 n / into itself. D.d .h/ nD1 Take now W D L2 .Rd / and consider the boson Fock space Fb .L2 .Rd //. In this .n/ case, for n 2 N the space Fb can be identified with the set of symmetric functions on L2 .Rd n / through ˝ns L2 .Rd / Š ¹f 2 L2 .Rd n /jf .k1 ; : : : ; kn / D f .k .1/ ; : : : ; k .n/ /; 8 2 }n º: (5.2.30) The creation and annihilation operators are realized as Z p .n/ .a.f /‰/ .k1 ; : : : ; kn / D n C 1 f .k/‰ .nC1/ .k; k1 ; : : : ; kn /d k; Rd

.n/

.a .f /‰/

n 1 X .k1 ; : : : ; kn / D p f .kj /‰ .n1/ .k1 ; : : : ; kOj ; : : : ; kn /; n j D1

.a .f /‰/.0/ D 0:

n 0; (5.2.31) n 1; (5.2.32) (5.2.33)

254

Chapter 5 Free Euclidean quantum field and Ornstein–Uhlenbeck processes .n/

Here ‰ 2 Fb 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/ to write p .a.k/‰/.n/ .k1 ; : : : ; kn / D n C 1‰ .nC1/ .k; k1 ; : : : ; kn /; (5.2.34) n 1 X .a .k/‰/.n/ .k1 ; : : : ; kn / D p ı.k kj /‰ .n1/ .k1 ; : : : ; kOj ; : : : ; kn /; n j D1 (5.2.35)

and Œa.k/; a .k 0 / D ı.k k 0 /;

Œa.k/; a.k 0 / D 0 D Œa .k/; a .k 0 /:

(5.2.36)

Let ! W L2 .Rd / ! L2 .Rd / be the multiplication operator called dispersion relation given by q ! .k/ D jkj2 C 2 ; k 2 Rd ; (5.2.37) with 0. Here describes the boson mass. The second quantization of the dispersion relation is .n/

.d .! /‰/

.k1 ; : : : ; kn / D

n X

! .kj / ‰ .n/ .k1 ; : : : ; kn /:

(5.2.38)

j D1

Definition 5.1 (Free field Hamiltonian). The self-adjoint operator d .! / is called free field Hamiltonian on Fb .L2 .Rd // and we use the notation Hf D d .! /:

(5.2.39)

The spectrum of the free field Hamiltonian is Spec.Hf / D Œ0; 1/, with component Specp .Hf / D ¹0º, which is of single multiplicity with Hf b D 0. Then formally we may write the free field Hamiltonian as Z Hf D ! .k/a .k/a.k/d k: (5.2.40) 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 ! .k/ of a single boson, and integrated over all momenta. The commutation relations are ŒHf ; a.f / D a.! f /;

ŒHf ; a .f / D a .! f /

(5.2.41)

and in formal expression they can be written as Œd .! /; a.k/ D ! .k/a.k/;

Œd .! /; a .k/ D ! .k/a .k/:

(5.2.42)

255


The relative bound of a] .f / with respect to the free Hamiltonian Hf can be seen p from (5.2.43) and (5.2.44). If f = ! 2 L2 .Rd /, then p 1=2 ka.f /‰k kf = ! kkHf ‰k; p 1=2 ka .f /‰k kf = ! kkHf ‰k C kf kk‰k

(5.2.43) (5.2.44)

hold, following directly from the lemma below. Lemma 5.2. Let h W Rd ! C be measurable, and g 2 D.h/. Then for every ‰ 2 D.d .jhj2 /1=2 / we have ‰ 2 D.a] .hg//, and it follows that ka.hg/‰k2 kgk2 kd .jhj2 /1=2 ‰k;

(5.2.45)

ka .hg/‰k2 kgk2 kd .jhj2 /1=2 ‰k C khgk2 k‰k2 :

(5.2.46)

Proof. Let ‰ 2 .d .jhj2 //. First note that .‰

.n/

2

.n/

; .d .jhj /‰/

/D

n Z X nd iD1 R

j‰ .n/ .k1 ; : : : ; kn /j2 jh.ki /j2 d k1 d kn

Z

Dn

Rnd

j‰ .n/ .k1 ; : : : ; kn /j2 jh.k1 /j2 d k1 d kn :

Thus ˇZ ˇ2 ˇ ˇ .n/ ˇ k Dn d k2 d kn ˇ g.k1 /h.k1 /‰ .k1 ; : : : ; kn /d k1 ˇˇ R.n1/d Rd Z nkgk2 jh.k1 /j2 j‰ .n/ .k1 ; : : : ; kn /j2 d k1 d kn Z

.n1/ 2

k.a.hg/‰/

Rnd

D kgk2 .‰ .n/ ; .d .jhj2 /‰/.n/ /; and summation over n gives (5.2.45). By the closedness of both d .jhj2 / and a.hg/ we can extend to ‰ 2 D.d .jhj2 /1=2 /. The other inequality can be derived from the canonical commutation relation Œa.hg/; a .hg/ D khgk2 and (5.2.45).

5.2.2 Segal fields The creation and annihilation operators are not symmetric and do not commute. Roughly speaking, a creation operator corresponds to p1 .x C @x / and an annihi2

lation operator to p1 .x @x / in L2 .Rd /. We can, however, construct symmetric and 2 commutative operators by combining the two field operators and this leads to 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

256


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 5.2 (Segal field). The Segal field ˆ.f / on the boson Fock space Fb .W / is defined by 1 ˆ.f / D p .a .fN/ C a.f //; 2

f 2W;

(5.2.47)

f 2W:

(5.2.48)

and its conjugate momentum by i ….f / D p .a .fN/ a.f //; 2 Here fN denotes the complex conjugate of f . By the above definition both ˆ.f / and ….g/ are symmetric, however, not linear in f and g over C. Note that, in contrast, they are linear operators over R. It is straightforward to check that Œˆ.f /; ….g/ D i Re.f; g/;

(5.2.49)

and Œˆ.f /; ˆ.g/ D i Im.f; g/;

Œ….f /; ….g/ D i Im.f; g/:

(5.2.50)

In particular, for real-valued f and g the canonical commutation relations become Œˆ.f /; ….g/ D i.f; g/;

Œˆ.f /; ˆ.g/ D Œ….f /; ….g/ D 0:

(5.2.51)

Applying the inequalities (5.2.43) and (5.2.44) to h D 1, we see that Fb;fin is the set of analytic vectors of ˆ.f /, i.e., m X kˆ.f /n ‰kt n 0. Then K is essentially self-adjoint on D. By Nelson’s analytic vector theorem ˆ.f / and ….g/ are essentially self-adjoint on Fb;fin . We keep denoting the closures of ˆ.f /dFb;fin and ….g/dFb;fin by the same symbols.

257


5.2.3 Wick product Loosely speaking, the so-called Wick product Wa] .f1 / a] .fn /W 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, Wa.f1 /a .f2 /a.f3 /a .f4 /W D a .f2 /a .f4 /a.f1 /a.f3 /. Q Q Definition 5.3 (Wick product). The Wick product W niD1 ˆ.gi /W of niD1 ˆ.gi / is recursively defined by the equalities Wˆ.f /W D ˆ.f /; Wˆ.f /

n Y

ˆ.fi /W D ˆ.f /W

iD1

n Y

iD1

n Y 1X ˆ.fi /W .f; fj /W ˆ.fi /W: 2 j D1

i6Dj

By the above definition we have n=2

n

Wˆ.f / W D 2

n X mD0

! n a .f /m a.f /nm m

(5.2.52)

or n

Wˆ.f / W D

Œn=2 X kD0

k nŠ 1 n2k 2 ˆ.f / : kf k kŠ.n 2k/Š 4

(5.2.53)

Note that Wˆ.f1 / ˆ.fn /Wb D 2n=2 a .f1 / a .fn /b . From this the orthogonality property ! n m n Y Y X Y W ˆ.fi /Wb ; W ˆ.gi /Wb D ınm 2n=2 .gi ; f .i/ / (5.2.54) iD1

iD1

2Pn iD1

follows. Hence n ˇ °Y ± ˇ .n/ L:H: W ˆ.fi /Wb ˇfj 2 W ; j D 1; : : : ; n D Fb :

(5.2.55)

iD1

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

˛ˆ.f /

m X ˛n 2 2 Wˆ.f /n Wb D e .1=4/˛ kf k e ˛ˆ.f / b : Wb D s-lim m!1 nŠ

(5.2.56)

nD0

Hence for real-valued f and g, .b ; ˆ.f /b / D 0;

1 .b ; ˆ.f /ˆ.g/b / D .f; g/; 2

(5.2.57)

258


and .b ; e ˛ˆ.f / b /Fb D e .1=4/˛

2 kf

k2

:

(5.2.58)

The commutator Œˆ.f /; ˆ.g/ D 0, and (5.2.57)–(5.2.58) suggest that ˆ.f / with a real-valued test function f can thus be realized as a Gaussian random variable, as it will be further explored below.

5.3

Q-spaces

5.3.1 Gaussian random processes Now we define a family of Gaussian random variables indexed by a given real vector space E on a probability space .Q; †; /, and study L2 .Q/ D L2 .Q; †; /. Note that L2 .Q/ is a vector space over C. Our main goal here is to show unitary equivalence of L2 .Q/ and Fb .EC /, where EC is the complexification of E . Definition 5.4 (Gaussian random variables indexed by E ). We say that .f / is a family of Gaussian random variables on a probability space .Q; †; / indexed by a real vector space E whenever (1) W E 3 f 7! .f / is a map from E to a Gaussian random variable on .Q; †; / with zero mean and covariance Z 1 .f /.g/d D .f; g/E (5.3.1) 2 Q (2) .˛f C ˇg/ D ˛.f / C ˇ.g/, ˛; ˇ 2 R (3) † is the minimal -field generated by ¹.f /jf 2 E º. Property (3) in Definition 5.4 is expressed by saying that the -field † is full. The existence of such a family of Gaussian random variables will be discussed in the next section, here we first look at some properties. Define SQ D ¹F ..f1 /; : : : ; .fn // j F 2 S .Rn /; fj 2 E ; j D 1; : : : ; n; n 1º: (5.3.2) Lemma 5.4. The following statements are equivalent: (1) SQ is dense in L2 .Q/. (2) † is the minimal -field generated by ¹.f /jf 2 E º. Proof. .1/ ) .2/ is fairly standard and we omit the proof. To prove .2/ ) .1/, we regard SQ as the set of bounded multiplication operators on L2 .Q/. Hence the closure SQ taken in strong topology is a von Neumann algebra. It follows that SQ L1 .Q/, since a strongly convergent bounded operator is bounded. Let P be the set of

Section 5.3 Q-spaces

259

all projections in SQ . Every projection is the characteristic function of a measurable subset in Q. Let †P denote the family of measurable subsets of Q associated with P , satisfying †P †. It is easily seen by a limiting argument that .f / is measurable with respect to †P . Hence † D †P , as † is the minimal -field generated by ¹.f /jf 2 E º. Since a function in L1 .Q/ can be approximated by a finite linear sum of characteristic functions and † D †P , SQ D L1 .Q/ follows as sets of bounded operators. It is clear that L1 .Q/ D SQ 1 SQ dense in L2 .Q/, it follows that SQ

kkL2 .Q/

kkL2 .Q/

. Since L1 .Q/ is

D L2 .Q/.

We define Wick product in L2 .Q/ in the same way as in the case of the boson Fock space. Definition 5.5 (Wick product). The Wick product of fined by

Qn

iD1 .fi /

is recursively de-

W.f /W D .f /; W.f /

n Y

.fi /W D .f /W

iD1

n Y

iD1

n Y 1X .fi /W .f; fj /W .fi /W: 2 j D1

(5.3.3)

i6Dj

Since d is a Gaussian measure, any Wick-ordered polynomial W

n Y

.fi /W 2 L2 .Q; d/:

(5.3.4)

iD1

Definition 5.5 has the virtue of allowing explicit calculations, and we will need especially (5.3.3) later on. In particular, for fi ; gj 2 E , ! n m n Y Y X Y W .fi /W; W .gi /W D ımn 2n .fi ; g .i/ /: (5.3.5) iD1

2}n

iD1

iD1

Furthermore, m X ˛n 2 2 W.f /n W D e .1=4/˛ kf k e ˛.f / : m!1 nŠ

We ˛.f / W D s-lim

(5.3.6)

nD0

Write n ˇ °Y ± ˇ .fi /Wˇfi 2 E ; i D 1; : : : ; n [ ¹1º: L2n .Q/ D L:H: W iD1

Then L2n .Q/ L2 .Q/ and L2m .Q/ ? L2n .Q/ if n 6D m.

(5.3.7)

260


Proposition 5.5 (Wiener–Itô decomposition). We have that L2 .Q/ D

1 M

L2n .Q/:

(5.3.8)

nD0

L 2 2 i.f / 2 G , Proof. Set G D 1 nD0 Ln .Q/. It is clear that G L .Q/. By (5.3.6), e n which means that F ..f1 /; : : : ; .fn // 2 G for F 2 S .R /. Since SQ is dense in L2 .Q/ by Lemma 5.4, L2 .Q/ D G follows. The following result says that the family of Gaussian random variables .f / is unique up to unitary equivalence. Proposition 5.6 (Uniqueness of Gaussian random variables). Let .f / and 0 .f / be Gaussian random variables indexed by a real vector space E on .Q; †; / and .Q 0 ; †0 ; 0 /, respectively. Then there exists a unitary operator U W L2 .Q/ ! L2 .Q 0 / such that U1 D 1 and U.f /U 1 D 0 .f /. Proof. Define U W L2 .Q/ ! L2 .Q 0 / by U W.f1 / .fn /W D W 0 .f1 / 0 .fn /W. By Proposition 5.5, U can be extended to a unitary operator.

5.3.2 Wiener–Itô–Segal isomorphism Define the complexification EC of the real Hilbert space E by EC D ¹¹f; gºjf; g 2 E º

(5.3.9)

such that for 2 R, ¹f; gº D ¹ f; gº;

i ¹f; gº D ¹ g; f º;

¹f; gºC¹f 0 ; g 0 º D ¹f Cf 0 ; gCg 0 º:

The scalar product of EC is defined by .¹f; gº; ¹f 0 ; g 0 º/EC D .f; f 0 / C .g; g 0 / C i..f; g 0 / .g; f 0 //: With these definitions .EC ; .; /EC / is a Hilbert space over C. Proposition 5.7 (Wiener–Itô–Segal isomorphism). There exists a unitary operator W W Fb .EC / ! L2 .Q/ such that .1/ W b D 1, .n/

.2/ W Fb .EC / D L2n .Q/, 1 .3/ W ˆ.f /W D .f /, where ˆ.f / is a Segal field.

Section 5.3 Q-spaces

261

The unitary operator W is called Wiener–Itô–Segal isomorphism, and it allows to identify Fb .EC / with L2 .Q/. Proof. Define W W Fb .EC / ! L2 .Q/ by W W

n Y

ˆ.fi /Wb D W

iD1

n Y

.fi /W;

f1 ; : : : ; fn 2 E ;

(5.3.10)

iD1

W b D 1:

(5.3.11)

Q Notice that L:H:¹W niD1 ˆ.fi /Wb jfi 2 E ; i D 1; : : : ; n; n 1º [ ¹b º is dense in the Fock space Fb .EC / over the complexification of E . It is straightforward to check that W can be extended to a unitary operator from Fb .EC / to L2 .Q/ by Proposition 5.5, and the lemma follows. Let T W E ! E be a contraction operator. Then T can be linearly extended to 1 a contraction on EC . The operator W .T /W W L2 .Q/ ! L2 .Q/ is called the second quantization of T and is denoted by the same symbol .T /. It is seen directly that .T / acts as .T /W

n Y

.fi /W D W

iD1

n Y

.Tfi /W

(5.3.12)

iD1

and .T /1 D 1:

(5.3.13)

1 is also For a self-adjoint operator h, the differential second quantization W d .h/W denoted by the same symbol. Then

d .h/W.f1 / .fn /W D

n X

W.f1 / .hfj / .fn /W

(5.3.14)

j D1

with d .h/1 D 0

(5.3.15)

for fj 2 D.h/, j D 1; : : : ; n, follows. Proposition 5.8 (Positivity preserving). Let T be a contraction operator on a real Hilbert space E . Then .T / is positivity preserving on L2 .Q/. Proof. Note that .T /Wexp.˛.f //W D Wexp.˛.Tf //W for every ˛ 2 C. Thus 1

.T /e ˛.f / D We ˛.Tf / We 4 ˛

2 kf

k2

1

D e ˛.Tf / e 4 ˛

2 .f;.1T T /f

/

:

262


Hence for F ..f1 /; : : : ; .fn // 2 SQ , .T /F ..f1 /; : : : ; .fn // D .2/n=2

Z Rn

d k1 d kn FO .k1 ; : : : ; kn /

n n X 1 X exp .fi ; .1 T T /fj /ki kj exp i kj .Tfj / : 4

i;j D1

(5.3.16)

j D1

Since kT k 1, ¹.fi ; .1 T T /fj /ºi;j is positive semi-definite. Then the right-hand side of (5.3.16) is expressed by the convolution of F and a Gaussian kernel DT , i.e., .T /F ..f1 /; : : : ; .fn // D .2/n=2 .F DT /..Tf1 /; : : : ; .Tfn //: Thus F 0 implies that .T /F 0. Let ‰ 2 L2 .Q/ be a non-negative function. By Lemma 5.4 we can construct a sequence .Fn /n2N SQ such that Fn ! ‰ as n ! 1 in L2 .Q/ and Fn 0. Hence .T /‰ 0 follows.

5.3.3 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 ı.k02 k 2 2 /d k, k D .k0 ; k/ 2 R R3 , on the 4-dimensional space-time R4 supported on the light cone with mass is Lorentz-covariant and Z Z 1 dk d 3k D .2/ı.k02 jkj2 2 /d¹k0 >0º 3 .2/4 R3 2! .k/ .2/ R4 is a Lorentz invariant integral. On normalization, the Lorentz covariant scalar field is formally given by Z 1 .t; x/ D .a.k/e it! .k/Cikx C a .k/e it! .k/ikx /d k: (5.3.17) p d 2! .k/ R Then instead of a Segal field the conventional time-zero scalar field is obtained, p 1 e p p .a .fO= ! / C a.fO= ! //: 2

(5.3.18)

e e N Here fO.k/ D fO.k/. Note that fO D fO for real f , and (5.3.18) is linear in f . In Chapter 6 we will discuss the Nelson model of the scalar quantum field given by (5.3.18). Note that (5.3.18) can be generalized by using a covariance operator C . Then a quantum field can be defined in the form ˆ.C / satisfying 1 .ˆ.Cf /; ˆ.Cg// D .Cf; Cg/: 2

(5.3.19)

Section 5.4 Existence of Q-spaces

263

Let E ; E 0 be real Hilbert spaces and C W E ! E 0 be a linear operator such that Ran C is dense in E 0 , which we apply as covariance operator. Define C .f / D .Cf /:

(5.3.20)

For C .f / the Wick product can be defined in the same way as for .f /, i.e., WC .f /W D C .f / and WC .f /

n Y

C .fi /W D C .f /W

iD1

n Y

iD1

n Y 1X C .fi /W .Cf; Cfj /W C .fi /W: 2 j D1

i6Dj

The Wiener–Itô–Segal isomorphism W W L2 .QI d/ ! Fb .EC0 / acts as W W

n Y

C .fi /W D a .Cf1 / a .Cfn /b :

iD1

The map W can be extended by linearity and using the fact that RanC is dense.

5.4

Existence of Q-spaces

5.4.1 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; †; / on which we will be able to construct a family of Gaussian random variables indexed by a real Hilbert space E . Theorem 5.9. Let E be a real Hilbert space. Then there exists a probability space .Q; †; / and a family of Gaussian random variables ..f /; f 2 E / with mean zero and covariance Z 1 .f /.g/.d/ D .f; g/E : 2 Q Proof. We show that Q can be obtained as a countable product space and, roughly, the measure may be regarded as the infinite product of Gaussian distributions d D

1 Y iD1

2

1=2 e xi dxi

on

1 Y

R:

iD1

We establish this next. P P D R [ ¹1º be the one-point compactification of R, and set Q D 1 R. Let R nD1 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

264


Q depending on a finite number of variables. For F D F .x1 ; : : : ; xn / 2 Cfin .Q/ define ` W Cfin .Q/ ! C by Z Pn 2 n=2 `.F / D F .x1 ; : : : ; xn /e iD1 xi dx1 dxn : Rn

Since j`.F /j kF k1 , ` 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 ` can be uniquely extended to a bounded linear functional `Q on the Banach space Q / 0, for F 0. The Riesz–Markov theorem .C.Q/; k k1 /. It is clear that `.F Q /D furthermore implies that there exists a probability space .Q; †; / such that `.F R 1 Q F .q/d.q/. Let ¹en ºnD1 be a complete orthonormal system of a real Hilbert space E . For each en we define en .q/ D xn for q D ¹xk º1 2 Q. Note that kD1 Z Z 1 2 Q 2 / D 1=2 jen .q/j2 d.q/ D `.e x 2 e x dx D : n 2 Q R Since C.Q/ is dense in L2 .Q/ in k kL2 .Q/ norm, Cfin .Q/ is dense in L2 .Q/. Then the minimal ./º1 nD1 is † by Lemma 5.4. Expand f 2 E as P1 -field generated by ¹enP n f D nD1 ˛n en , and set n .f / D kD1 ˛k ek ./. It is seen that 2 km .f / n .f /kL 2 .Q/ D

1 2

n X

˛k2 ! 0

m; n ! 1:

kDmC1

R Thus .f / D s-limn!1 n .f / exists in L2 .Q/. The equality Q e i.f / d D R 2 e .1=4/kf k easily follows. From this we obtain Q .f /d D 0 and Z .f /.g/d D .1=2/.f; g/; Q

and thereby we have constructed the desired family of Gaussian random variables .f / indexed by E on .Q; †; /.

5.4.2 Bochner theorem and Minlos theorem P The Q space constructed in Theorem 5.9 was seen to be the countable product 1 nD1 R, however, the Gaussian random variables ¹.f /jf 2 E º, depend on the base of the given Hilbert space E , and therefore the construction is not canonical. 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 0 .Rd /. Let X be a random variable on .; F ; P /, and consider its characteristic function C.z/ D EP Œe izX ;

z 2 Rd :

(5.4.1)


265

Clearly, C.z/ has the properties .1/

n X

˛i ˛Nj C.zi zj / 0;

.2/ C is uniformly continuous;

i;j D1

.3/ C.0/ D 1: (5.4.2)

The converse statement is known as the Bochner theorem. Theorem 5.10 (Bochner). Suppose that function C W Rd ! C satisfies condition (5.4.2). Then there exists a probability space .Rd ; B.Rd /; B / such that Z C.z/ D e izx B .dx/: (5.4.3) Rd

Proof. Let C t .z/ D e tjzj implies that

2 =2

. It is seen that C t .z/C.z/ is positive definite, which

t .x/ D .2/ and

d=2

Z Rd

e izx C t .z/C.z/dz 0

Z Rd

t .x/dx D .2/d=2 C t .0/C.0/ D .2/d=2 :

Put t .dx/ D .2/d=2 t .x/dx. Thus t .dx/ is a probability measure on Rd with characteristic function C t .z/C.z/. Since lim t#0 C t .z/C.z/ D C.z/ pointwise and C.z/ is continuous, there exists a measure such that t is weakly convergent to and its characteristic function is C.z/. The Bochner theorem can be extended to infinite dimension, however, as it will be illustrated in Remark 5.1 below, the Hilbert space E is too small to support the corresponding measure . To replace E we will introduce a suitable nuclear space. Let E be a complete topological vector space with a countable family of norms ¹k kn º1 nD0 such that k k0 < k k1 < k k2 < , and there exists a scalar product .; /n generating these norms so that kf k2n D .f; f /n for each n. Define E D

1 \

En ;

nD0 kkn

and En is a Hilbert space with the scalar product .; /n . Identifying where En D E E0 with E0 gives rise to the nested inclusions E2 E1 E0 Š E0 E1 E2 : Furthermore, let E D

S1

nD0 En ,

and denote its norm by kxkn D sup 2En jx. /j.

266


Definition 5.6 (Nuclear space). E is a called a nuclear space whenever for any m > 0 there exists n > m such that the injection nm W En ! Em is a Hilbert–Schmidt operator. Example 5.2. Let Hosc D 12 . C jxj2 1/ be the harmonic oscillator on L2 .Rd / with purely discrete spectrum ¹n1 C C nd j nj 2 N [ ¹0º; j D 1; : : : ; d º. Using a suitable orthonormal basis ¹gi1 ;:::;id º, Hosc is diagonalized and .gi1;:::;id ;Hgi1;:::;id / D Pd .d C"/=2 is a Hilbert–Schmidt operator for j D1 ij . It is easily seen that .Hosc C 1/ any " > 0 and .d C"/

TrŒ.Hosc C 1/

D

1 X i1 D0

1 X d X id D0

.d C"/ .i C 1/ :

D1

Let .‰; ˆ/n D .‰; .Hosc C1/n ˆ/. Then k‰kn < k‰knC1 and denote the completion of L2 .Rd / with respect to k kn by Hn . Then HnC1 Hn , and it can be proven that 1 1 \ [ S D Hn ; S 0 D Hn : nD1

nD1

Let 1nC1 D 1 W HnC1 ! Hn be the identity map. Thus .Hosc C 1/1=2 1nC1 is unitary and so 1 D 1nC1 1nCDC1 W HnCDC1 ! Hn is a Hilbert–Schmidt operator for sufficiently large D since 1 D .Hosc C 1/D=2 .Hosc C 1/D=2 1 is a product of a Hilbert–Schmidt operator with a unitary operator. Theorem 5.11 (Minlos). Let E be a nuclear space, and suppose that C W E ! C satisfies .1/

n X

˛i ˛Nj C. i j / 0; .2/ C./ is continuous in E ;

i;j D1

Then there exists a probability space .E ; B; / such that Z C. / D e ihhx; ii d.x/; 2 E ; E

.3/ C.0/ D 1: (5.4.4)

(5.4.5)

where hhx; ii denotes the pair x 2 E and 2 E . Remark 5.1. Let E be a Hilbert space and C. / D exp..1=4/k k2 / for 2 E . Then C. / satisfies conditions (5.4.4) in Minlos’ theorem. Let ¹ n º1 nD1 be a complete orthonormal system of E . If a measure on E satisfying (5.4.5) existed, then Z 2 e izhhx; n ii d.x/ D e .1=4/z : (5.4.6) E


267

Since hhx; n ii D .x; n / ! 0 as n ! 1, the left-hand side of (5.4.6) converges to 1, with side. Moreover, from the fact that Pwhich is in contradiction P the right-hand C. nkD1 zk k / D exp .1=4/ nkD1 zk2 and the law of large numbers, it would follow that n 1X lim hhx; n ii2 D 1 n!1 n kD1

which is also a contradiction. Let Sreal .Rd / denote the set of real-valued, rapidly decreasing and infinitely differentiable functions on Rd . Sreal .Rd / is a nuclear space. Corollary 5.12. Suppose that Sreal .Rd / is dense in a real Hilbert space H . Then there exists a family of Gaussian random variables .f / indexed by H on a proba0 .Rd /; B; /. bility space .Sreal Proof. Write C. / D exp..1=4/k k2 /, for 2 Sreal .Rd /. Then C satisfies (5.4.4). 0 Thus the Minlos yields that a probability space .Sreal .Rd /; B; / exists such R theoremihh; ii that C. / D S 0 .Rd / e d./. Define real

. / D hh; ii; Then we have Z . /d D 0; 0 Sreal .Rd /

Z

0 .Rd / and 2 Sreal .Rd /: 2 Sreal

1 . /./d D . ; /; ; 2 Sreal .Rd /: 0 2 Sreal .Rd / (5.4.7)

Since B is the minimal -field generated by cylinder sets, B is the minimal -field generated by ¹. /; º. For f 2 H , there is a sequence . n /n2N Sreal .Rd / such that n ! f as n ! 1 in H . Thus Z 1 j. n / . m /j2 d D k n m k2H ! 0 0 2 Sreal .Rd / 0 2 d as n; m ! 1. Hence ¹. n /º1 nD1 is a Cauchy sequence in L .Sreal .R //. Define .f / D s-limn!1 . n /. Then ¹.f /jf 2 H º is a family of Gaussian random 0 .Rd /; B; / indexed by H . variables on .Sreal

Example 5.3. Suppose that B.f; g/ is a weakly continuous positive semi-definite quadratic form on Sreal .Rd /. Then by the Minlos can construct a measure R theorem we 0 .Rd / and .f / D hh; f ii such that i.f / d D e .1=4/B.f;f / . on Sreal e 0 S .Rd / real

268

5.5


Functional integration representation of Euclidean quantum fields

5.5.1 Basic results in Euclidean quantum field theory We use the notations kk˛

H˛ .Rd / D S .Rd /

Z ;

kf k2˛ D

Rd

jfO.k/j2 ! .k/2˛ d k;

˛ 2 R;

and the shorthands HM D Hreal;1=2 .Rd /;

HE D Hreal;1 .Rd C1 /:

(5.5.1)

Let .f / be a family of Gaussian random variables indexed by f 2 HM on a probability space .Q; †; / and E .F / by F 2 HE on .QE ; †E ; E /. One possible choice 0 0 of Q and QE is Sreal .Rd / and Sreal .Rd C1 /, respectively, however, we do not fix any particular space here. For notational convenience we write .f; g/1=2 and .f; g/1 for the scalar products in HM and HE , respectively. Note that .f / and E .F / are Gaussian random variables with mean zero and covariance Z Z 1 1 .f /.g/d D .f; g/1=2 ; E .F /E .G/d E D .F; G/1 : (5.5.2) 2 2 Q QE We use the identifications H1=2 .Rd / Š ŒHM C and L2 .Q/ Š Fb .H1=2 .Rd // by the Wiener–Itô–Segal isomorphism, as well as p 1 Q p 1 W .f /W D p .a .fO= ! / C a.fO= ! //: 2

(5.5.3)

We will define a family of transformations I t from L2 .Q/ to L2 .QE / through the second quantization of a specific transformation t from HM to HE . Define t W H M ! HE

(5.5.4)

t W f 7! ı t ˝ f:

(5.5.5)

by

Here ı t .x/ D ı.x t / is the delta function with mass at t . Note that for f 2 HM ,

2

2

2

ı t ˝ f .k; k0 / D ı t ˝ f .k; k0 / D ı t ˝ f .k; k0 /: Thus ı t ˝ f D ı t ˝ f , which implies that t preserves realness. Define !O D ! .i r/:

(5.5.6)

269

Section 5.5 Functional integration representation of Euclidean quantum fields

Lemma 5.13. It follows that s t D e jtsj!O ;

s; t 2 R:

(5.5.7)

In particular, t is isometry for each t 2 R. Proof. We see that .ı t ˝ f; ıs ˝ g/1 D D

1 Z

Z

Rd

Z

Rd

fO.k/g.k/d O k

fO.k/g.k/ O

e ik0 .st/ R

e jtsj! .k/ ! .k/

1 d k0 ! .k/2 C jk0 j2

dk

D .fO; e jtsj! g/ O 1=2 :

(5.5.8)

Then the lemma follows. Definition 5.7 (Contraction operators). (1) Let T 2 C .HM ! HM /. Define .T / 2 C .L2 .Q/ ! L2 .Q// by .T /W.f1 / .fn /W D W.Tf1 / .Tfn /W

(5.5.9)

with .T /1M D 1M , where 1M denotes the identity function in L2 .Q/. (2) Let T 2 C .HE ! HE /. Define E .T / 2 C .L2 .QE / ! L2 .QE // by (5.5.9) with replaced by E . (3) Let T 2 C .HM ! HE /. Define Int .T / 2 C .L2 .Q/ ! L2 .QE // by Int .T /W.f1 / .fn /W D WE .Tf1 / E .Tfn /W

(5.5.10)

with Int .T /1M D 1E , where 1E denotes the identity function in L2 .QE /. Definition 5.8 (Isometries). Define the family of isometries ¹I t º t2R by I t D Int . t / W L2 .Q/ ! L2 .QE /, i.e., I t 1M D 1E ;

I t W.f1 / .fn /W D WE .ı t ˝ f1 / E .ı t ˝ fn /W:

(5.5.11)

From the identity s t D e jtsj!O it follows that O

It Is D e jtsjHf ;

s; t 2 R;

1 Hf W is the free field Hamiltonian in L2 .Q/. where HO f D W

(5.5.12)

270


Proposition 5.14 (Functional integral representation for free field Hamiltonian). Let F; G 2 L2 .Q/ and t 0. Then Z F0 G t d E ; (5.5.13) .F; e tHf G/L2 .Q/ D QE

where F0 D I0 F and G t D I t G. In order to extend (5.5.13) to Hamiltonians of the form HO f C perturbation, we will use the Markov property of projections E t D I t It ;

t 2 R;

(5.5.14)

which will be proven below. The following argument shows why the Markov property is useful. Let HO D HO f C HI with some interaction HI . By the Trotter product formula we have O

O

e t H D s-lim .e .t=n/Hf e .t=n/HI /n : n!1

(5.5.15)

O

Substituting e jtsjHf D It Is into (5.5.15), we have O

e t H D s-lim I0 n!1

where

Qn

j D1 Tj

n Y

I tj=n e .tj=n/HI Itj=n I t ;

(5.5.16)

j D1

D T1 T2 Tn . Thus it is obtained that O

e t H D s-lim I0 n!1

n Y

E tj=n e .tj=n/HI .tj=n/ E tj=n I t

(5.5.17)

j D1

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 the Es in (5.5.17) can be disregarded, thus net result is .F; e

t HO

n Y G/ D lim F0 ; exp..tj=n/HI .tj=n// G t n!1

j D1

1 X D lim F0 ; exp .tj=n/HI .tj=n/ G t n!1

Z D

QE

j D1

Z t F0 G t exp HI .s/ds dE : 0

In the next section we investigate the Markov property of E t and establish the funcO tional integral representation of .F; e t H G/.

271


5.5.2 Markov property of projections Let O R. Put HE .O/ D ¹f 2 HE j supp f O Rd º

(5.5.18)

and the projection HE ! HE .O/ is denoted by eO . Let †O be the minimal -field generated by ¹E .f /jf 2 HE .O/º. Define EO D ¹ˆ 2 L2 .QE /jˆ is †O -measurableº:

(5.5.19)

Let e t D t t , t 2 R. Then ¹e t º t2R is a family of projections from HE to Ran. t /. Let † t , t 2 R, be the minimal -field generated by ¹E .f /jf 2 Ran.e t /º. Define E t D ¹ˆ 2 L2 .QE /jˆ is † t -measurableº;

t 2 R:

(5.5.20)

We will see below that F 2 EŒa;b can be characterized by supp F Œa; b Rd . Lemma 5.15. (1) HE .¹t º/ D Ran.e t / and any f 2 Ran.e t / can be expressed as f D ı t ˝ g for some g 2 HM . In particular, e¹tº D e t . kk1

(2) HE .Œa; b/ D L:H: ¹f 2 HE jf 2 Ran.e t /; a t bº

holds.

0 Proof. (1) The key fact is that any f 2 Sreal .Rd C1 / with support ¹t º Rd can be expressed as

f .x0 ; x/ D

n X

ı .j / .x0 t / ˝ gj .x/;

.x0 ; x/ 2 R Rd ;

j D0 0 where gj 2 Sreal .Rd / and ı .j / is the j th derivative of the delta function ı. Then n 1 X fO.k0 ; k/ D p .i k0 /j e ik0 t gOj .k/; 2 j D0

.k0 ; k/ 2 R Rd ;

and hence n D 0 and g0 2 HM is necessary and sufficient for f 2 HE . Then f 2 HE .¹t º/ if and only if there exists g 2 HM such that f D ı t ˝g, which gives that f 2 Ran.e t /. In particular, HE .¹t º/ Ran.e t /. The inclusion Ran.e t / HE .¹t º/ is trivial. (2) Let A1 D HE ..a; b//, and f 2 HE .Œa; b/. Then for 2 .1; 1/ we have f .x0 ; x/ D f . x0 C .1 /.a C b/=2; x/ 2 A1 , and kf f k1 ! 0 as ! 1. Hence A1 is dense in HE .Œa; b/. Let A2 D C 1 .Rd / \ AR1 , and define f" D " f , where " .x/ D .x="/="d , 2 C01 .Rd /, .x/ 0, .x/dx D 1 and supp ¹x 2 Rd jjxj 1º. Then

272


f" 2 A2 for sufficiently small " and kf" f k1 D 0 as " ! 0. Thus A2 is dense in HE .Œa; b/. Next we will show that A3 D C01 .Rd C1 / \ A1 is dense in HE .Œa; b/. In the case of > 0 let ´ 1; jxj n; n .x/ D n 2 C01 .Rd / and 0 n 1: 0; jxj > n C 1; Then for f 2 A2 we have fn D n f 2 A2 and kfn f k1 D 0 as n ! 1. Since A2 is dense in A3 , A3 is dense in HE .Œa; b/. For the case of D 0, let A4 D ¹f 2 C01 .Rd C1 / \ A1 j supp fO Rd C1 n ¹0ºº. It is similar to show that A4 is dense in HE .Œa; b/. Finally, we show that any f 2 A3 can be represented as a limit of finite linear sums of vectors in Ran.e t /; .a < t < b/. Let f 2 A3 and define fn .x0 ; x/ D

1 X j j 1 ı x0 f ;x : n n n

j D1

Then fOn .k0 ; k/ D where g.X; k/ D .2/

R d=2

ij k0 =n 1 X j e 1 g ;k ; p n n 2

j D1

f .X; x/e ikx dx. Note that

jfOn .k0 ; k/j cN =.jkj2 C 1/N for some N and fOn .k0 ; k/ ! fO.k0 ; k/ as n ! 1 for every k. Hence kfn f k1 ! 0 as n ! 1 by the Lebesgue dominated convergence theorem. Then (2) with 6D 0 follows. In the case of D 0, we use A3 replaced by A4 . Lemma 5.16. Let a b t c d . Then .1/ ea eb ec D ea ec , .2/ eŒa;b e t eŒc;d D eŒa;b eŒc;d , .3/ ec eb ea D ec ea , .4/ eŒc;d e t eŒa;b D eŒc;d eŒa;b . Proof. Since ea eb ec D a a b b c c D a e jabj!O e jbcj!O c D a e jacj!O c D a a c c D ea ec ;

273


(1) readily follows. Take f; g 2 HE . By Lemma 5.15 (2) it is clear that Nn X

eŒc;d f D s-lim

n!1

eŒa;b g D s-lim

fn ˛ ;

fn˛ 2 Ran.e tn˛ /;

tn˛ 2 Œc; d ;

˛D1 Nm X

m!1

fm ˇ ;

fmˇ 2 Ran.e tmˇ /;

tmˇ 2 Œa; b:

ˇ D1

Hence by (1) we have .eŒa;b e t eŒc;d f; g/ D

D

lim

m;n!1

lim

m;n!1

;Mm Nn X

.e t fn˛ ; gm˛ /

˛;ˇ D1 ;Mm Nn X

.fn˛ ; gm˛ / D .eŒa;b eŒc;d f; g/:

˛;ˇ D1

This gives (2). (3) and (4) are similarly proven. Define E t D I t It D E .e t /;

t 2 R;

(5.5.21)

and EO D E .eO /;

O R:

(5.5.22)

Notice that E¹tº D E t for t 2 R by Lemma 5.15. The following relationship holds between the projection EŒa;b and the set of †Œa;b -measurable functions. Proposition 5.17.

.1/ Ran.EŒa;b / D EŒa;b ,

.2/ EŒa;b E t EŒc;d D EŒa;b EŒc;d and EŒc;d E t EŒa;b D EŒc;d EŒa;b hold for a b t c d, .3/ EŒa;b EŒc;d D EŒc;d EŒa;b D EŒa;b if Œa; b Œc; d . Proof. Let¹gn º1 nD1 be a complete orthonormal system of HE .Œa; b/. Then Ran.EŒa;b / D L:H:¹WE .g1 /n1 E .gk /nk Wjgj 2 ¹gn º1 nD1 ; nj 0; j D 1; : : : ; k; k 0º; which implies that Ran.EŒa;b / EŒa;b . Noticing that Wexp.i E .f //W 2 Ran.EŒa;b /, we obtain exp.i E .f // 2 Ran.EŒa;b /. Thus F .E .f1 /; : : : ; E .fn // 2 Ran.EŒa;b /, for F 2 S .Rn / and f1 ; : : : ; fn 2 HE .Œa; b/. Since ¹F .E .f1 / E .fn //jF 2 Sreal .Rd /.Rn /; f1 ; : : : ; fn 2 HE .Œa; b/º

274


is dense in EŒa;b , we conclude that EŒa;b Ran.EŒa;b /. Hence (1) is obtained. (2) follows from (2) and (4) of Lemma 5.16. Since a †Œa;b -measurable function is also †Œc;d -measurable, (3) follows. Remark 5.2. (1) of Lemma 5.17 implies that EŒa;b is the projection to the set of †Œa;b -measurable functions in L2 .QE /. This is a conditional expectation, and we can write EŒa;b F as EŒF j†Œa;b . We also denote E t F by EŒF j† t . (2) of Lemma 5.17 is called Markov property of the family Es , s 2 R. Proposition 5.18 (Markov property). Let F 2 EsCt . Then EŒF j†.1;s D EŒF j†s :

(5.5.23)

Proof. It follows that EŒF j†.1;s D E.1;s EsCt F D E.1;s Es EsCt F from the Markov property. Since E.1;s Es D E.1;s E¹sº D E¹sº D Es by Proposition 5.17, we see that E.1;s Es EsCt F D Es EsCt F D Es F D EŒF j†s . Then the proposition follows.

5.5.3 Feynman–Kac–Nelson formula Consider the polynomial P .X / D a2n X 2n C a2n1 X 2n1 C C a1 X C a0

(5.5.24)

with a2n > 0. For f 2 HM define HI D WP ..f //W:

(5.5.25)

Note that HI is a polynomial of .f / of degree 2n and in the Schrödinger representation there exists ˛ > 0 such that HI > ˛. Define the self-adjoint operator P HI HP D HO f C

(5.5.26)

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 . Before going to state the functional integral representation of e tHP we give a corollary to Lemma 5.17. Corollary 5.19. Let F 2 EŒa;b , G 2 EŒc;d with a b t c d . Then .F; E t G/ D .F; G/. Proof. By Proposition 5.17 it is immediate that .F; E t G/ D .EŒa;b F; E t EŒc;d G/ D .EŒa;b F; EŒc;d G/ D .F; G/:

275


Now we can prove the functional integral representation of HP given by (5.5.26). Theorem 5.20 (Feynman–Kac–Nelson formula). Let F; G 2 L2 .Q/. Then Z t Z tHP .F; e G/ D F0 G t exp WP .E .ıs ˝ f //Wds d E ; (5.5.27) QE

0

where F0 D I t F and G t D I t G. Proof. By the Trotter product formula and the fact e jtsjHf D It Is , we have .F; e tHP G/ D lim .F; .e .t=n/Hf e .t=n/HI /n G/ n!1

D lim

n Y

F0 ;

n!1

n!1

iD1

D lim

.I ti=n e .t=n/HI Iti=n /G t

n Y

F0 ;

.E ti=n Ri E ti=n /G t ;

(5.5.28)

iD1

where Ri D exp..t =n/WP .E .ıit=n f //W/ and we used that Is exp.tHI /Is D Es exp.t WP .E .ıs ˝ f //W/Es

(5.5.29)

as operators. In fact, we can check that I t .f /W

n Y

° .fi /W D I t W.f /

iD1

n Y iD1

D WE .ı t ˝ f /

n Y iD1

D E .ı t ˝ f /I t W

n n ± Y 1X .fi /W C .f; fj /W .fi /W 2

1 E .ı t ˝ fi /W C 2

n Y

j D1

n X

i6Dj

.ı t ˝ f; ı t ˝ fj /W

j D1

n Y

.ı t ˝ fi /W

i6Dj

.fi /W:

iD1

Thus I t .f /It D E .ı t ˝ f /E t D E t E .ı t ˝ f /E t . Inductively it follows that It

n Y

n Y .fj / It D E t E .ı t ˝ fj / E t :

j D1

j D1

A limiting argument gives then (5.5.29). Notice that F0 ; D

n Y

.E ti=n Ri E ti=n /G t

iD1

F0 ; E t=n R1 E t=n .E2t=n R2 E2t=n / .E t Rn E t /G t : „ƒ‚… „ ƒ‚ … 2E¹0º

2EŒt =n;t

(5.5.30)

276


Then by Corollary 5.19 we can remove E t=n in (5.5.30). Furthermore, note that n Y .E ti=n Ri E ti=n /G t D R1 F0 ; E t=n .E2t=n R2 E2t=n / .E t Rn E t /G t : F0 ; „ƒ‚… „ ƒ‚ … iD1

2EŒ0;t =n

2EŒ2t =n;t

(5.5.31) Similarly, E t=n can be removed in (5.5.31), and since n Y .E ti=n Ri E ti=n /G t D R1 F0 ; E2t=n R2 E2t=n .E t Rn E t /G t ; F0 ; „ƒ‚… „ ƒ‚ … iD1

2EŒ0;t =n

2EŒ2t =n;t

(5.5.32) also E2t=n can be removed. Recursively, we can delete all Es in the rightmost side of (5.5.28). Thus .F; e tHP G/ D lim .F0 ; R1 Rn G t / n!1

D lim

n!1

Z D

5.6

QE

n X F0 ; exp .t =n/WP .E .ıjt=n ˝ f //W G t j D1

Z t F0 G t exp WP .E .ıs ˝ f //Wds d E : 0

Infinite dimensional Ornstein–Uhlenbeck process

5.6.1 Abstract theory of measures on Hilbert spaces In this section we consider an infinite dimensional Ornstein–Uhlenbeck (OU) process as a Hilbert space-valued random process . s /s2R on the set of continuous paths C.RI M2 /. As explained in Remark 5.1 it is not possible to construct a Gaussian measure on the Hilbert space K that the covariance form defines. Nevertheless, if the covariance is given by .A1 ; A1 /K for some self-adjoint operator A with a Hilbert–Schmidt inverse, i.e., Tr.A2 / < 1, then there exists a Gaussian measure on an enlarged Hilbert space. Definition 5.9 (Sazanov topology). Let K be a real separable Hilbert space. For a non-negative trace class operator B defines the seminorm p pB .f / D .f; Bf /: (5.6.1) The Sazanov topology on K is the locally convex topology defined by the family of seminorms ¹pB j B W non-negative trace class operator on Kº.

Section 5.6 Infinite dimensional Ornstein–Uhlenbeck process

277

Proposition 5.21 (Minlos–Sazanov). P Let K be a real separable Hilbert space. Suppose that C W K ! C satisfies ni;j D1 zi zNj C.fi fj / 0 for all zi ; zj 2 C and fi ; fj 2 K. Then there exists a finite Borel measure on K such that Z C.f / D e i. ;f /K d. /; (5.6.2) K

if and only if C./ is continuous in the Sazanov topology. Example 5.4. Let B 0 be a self-adjoint bounded operator and CB .f / D e .f;Bf / . It is well known that CB is continuous in the Sazanov topology if and only if B is trace-class. Therefore, whenever B is trace-class, there is a finite Borel measure B R on K such that K e i. ;f /K dB . / D e .f;Bf /K . Let A 0 be a self-adjoint operator on K such that A1 is a Hilbert–Schmidt operator. Define C 1 .A/ D

1 \

D.An /

(5.6.3)

nD0

and set kf kn D kAn=2 f kK . Also, by the completion Kn D C 1 .A/

kkn

(5.6.4)

defines a Hilbert space with the scalar product .f; g/n D .An=2 f; An=2 g/K for f; g 2 D.An=2 /. Then we have the sequence of Hilbert spaces KC2 KC1 K0 K1 K2 :

(5.6.5)

Under the identification K0 Š K0 we can also make Kn Š Kn ;

n 1:

(5.6.6)

Denote the dual pair between K2 and KC2 by hh ; f ii. An immediate corollary of the Minlos–Sazanov theorem is the following. Lemma 5.22. Suppose that L 0 is a bounded self-adjoint operator on K and ŒA1 ; L D 0. Then there exists a Borel probability measure mL on .K2 ; B.K2 // such that Z 1 e ihh ;f ii d mL . / D e 2 .f;Lf /0 : (5.6.7) K2

In particular, if L D 1, then there exists a probability measure m1 on K2 such that Z 1 2 e ihh ;f ii d m1 . / D e 2 kf k0 (5.6.8) K2

278


Proof. We show that A1 is a Hilbert–Schmidt operator also on K2 . To see this, note that for a complete orthonormal system ¹ei º1 iD1 of K consisting of eigenvectors of A, the set ¹Aei º1 is also a complete orthonormal system of K2 , and we put iD1 eQi D Aei . We have 1 1 X X .eQi ; A2 eQi /2 D .ei ; A2 ei /0 D Tr.A2 / < 1: iD1

iD1

Hence A1 is Hilbert–Schmidt operator on K2 . Moreover, since kLf k2 D kA1 Lf k0 D kLA1 f k0 kLkkf k2 ; L is bounded on K2 , so that LA1 is a Hilbert–Schmidt operator on K2 . Now by Proposition 5.21 and Example 5.4 there exists a probability measure mL on K2 such that Z 1 2 e i. ;f /2 d mL . / D e 2 .f;LA f /2 ; f 2 C 1 .A/: K2

Inserting A2 f into f above we have Z 1 e i. ;f / d mL . / D e 2 .f;Lf /0 ; K2

(5.6.9)

where we put . ; f / D . ; A2 f /2 ;

f 2 C 1 .A/;

2 K2 :

(5.6.10)

Thus we have the bound j. ; f /j k k2 kf k2 . From this bound we can extend . ; f / to f 2 KC2 by . ; f / D lim . ; fn /; n!1

(5.6.11)

where fn ! f in KC2 . . ; f / with f 2 KC2 also satisfies the bound j. ; f /j k k2 kf k2 , hence . ; f / D hh ; f ii. Moreover, for f 2 K2 there exists a sequence fn 2 C 1 .A/ such that fn ! f in both K0 and K2 since A is closed and C 1 .A/ is a core of A. Then (5.6.9) can be extended to f 2 KC2 by a limiting argument, and the lemma follows. We can extend Lemma 5.22 to f 2 K. Theorem 5.23 (Existence of a Gaussian measure on a Hilbert space). Suppose that L 0 is a bounded self-adjoint operator on K and ŒA1 ; L D 0. Then there exists a Borel probability measure mL on .K2 ; B.K2 // and a family of Gaussian random variables .f / indexed by f 2 K such that


.1/ K 3 f 7! .2/

279

.f / 2 R is linear

.f / D hh ; f ii when f 2 KC2

.3/ it satisfies Z ei

.f /

K2

1

d mL . / D e 2 .f;Lf /0 :

(5.6.12)

Proof. In the case of f 2 KC2 , we set .f / D hh ; f ii. Then (5.6.12) follows from Lemma 5.22. In the case of f 2 K n KC2 , .f / can be defined through a simple limiting procedure. By (5.6.7) it can be seen that Z .f /2 d mL . / D .f; Lf /0 ; f 2 KC2 : (5.6.13) K2

Since for f 2 K there exists a sequence fn 2 KC2 such that fn ! f strongly in K, we can define .f / by .f / D s-lim .fn / n!1

(5.6.14)

in L2 .K2 ; d mL /. Then the theorem follows by a simple limiting argument. Remark 5.3. Not that KC2 3 f 7! .f / 2 R is continuous with j .f /j k k2 kf k2 , but it is not as a map K 3 f 7! .f / 2 R.

5.6.2 Fock space as a function space In order to construct a functional integral representation of a model in quantum field theory we 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. ! W Rd ! R is measurable with 0 !.k/ D !.k/, and !.k/ D 0 only on a set of Lebesgue-measure zero (2) Lorentz-covariant space. ² M D f 2

ˇ ˇ

L2real .Rd /ˇˇkf

Z k2M

D

³ jfO.k/j2 dk < 1 : 2!.k/

We also define the scalar product on M by Z .f; g/M D fO.k/g.k/ O Rd

1 d k: 2!.k/

(5.6.15)

280


(3) Hilbert–Schmidt operator. A given positive self-adjoint operator D with Hilbert– p Schmidt inverse on M and such that !D 1 is bounded on M . Let Spec.D/ D 1 ¹ i º1 normalized eigenvectors such that Dei D i ei . Note iD1 and ¹ei ºiD1 P be the 2 2 that Tr.D / D 1 iD1 i < 1. Define Mn D C 1 .D/

kD n=2 kM

:

(5.6.16)

Remark 5.4. (1) Since the Lebesgue measure of the set ¹k 2 Rd j !.k/ D 0º is p p zero, the map 1= ! W f 7! f = ! is a self-adjoint multiplication operap p tor on L2 .Rd /. Hence the range of 1= ! is dense in L2 .Rd /; ¹f = !jf 2 p D.1= !/º is dense in L2 .Rd /. p (2) Let !.k/ D jkj2 C 2 , with 0. The positive self-adjoint operator .d C1/=2 1 1 1 T D C jkj2 C 2 2 2 2 is a specific case of D above, see Example 5.2. Indeed T 1 is a Hilbert– p Schmidt operator and !T 1 is bounded since the operator norm has the bound p k !. C jkj2 C 2 /.d C1/=2 k 1. p (3) The condition that !D 1 is bounded on M will be used in Lemma 5.30 below, where we will prove path continuity of the infinite dimensional Ornstein– Uhlenbeck process. Moreover this condition yields that the embedding W M 3 f 7! f 2 L2 .Rd / is bounded. It can be indeed seen that p p kf kL2 .Rd / D k 2!D 1 Df kL2 .Rd / k 2!Dkkf kM2 :

(5.6.17)

We identify MC2 with the topological dual of M2 ; M2 Š MC2 , and the pairing between MC2 and M2 is denoted by hh; ii.

Theorem 5.24 (Gaussian measure G on M2 ). Under the above set-up there exists a probability measure G on M2 and a family of Gaussian random variables .f / indexed by f 2 M such that (1) M 3 f 7! .f / 2 R is linear (2) .f / D hh ; f ii when f 2 MC2 (3) it follows that Z

1

M2

2

e i .f / d G D e 2 kf kM :

(5.6.18)

281


Proof. This is due to Theorem 5.23 with A D D, L D 1, K D M , and mL D G.

.f / D .f /

Now we discuss the unitary equivalence of the boson Fock space Fb .L2 .Rd // and 2 ; d G/. Let Z 1 .f / D .a .k/fO.k/ C a.k/fO.k//d k; f 2 M ; p 2!.k/

L2 .M

be a scalar field in Fb . Recall that b denotes the Fock vacuum. We see that Z 1 fO.k/g.k/ O ..f /b ; .g/b /Fb D d k D .f; g/M D EG Œ .f / .g/: d 2!.k/ R (5.6.19) Denote the complexification of M by MC . For f 2 MC , f D g C ih with g; h 2 M , define

.f / D .g/ C i .h/: Qn The Wick product of j D1 .fj / is inductively defined by W .f /W D .f /; W .f /

n Y

.fj /W D .f /W

j D1

n Y

j D1

Y 1X

.fj /W .f; fi /MC W

.fj /W: 2 n

n

iD1

j 6Di

Proposition 5.25 (Unitary equivalence). There exists a unitary operator W W Fb ! L2 .M2 ; d G/ such that .1/ W b D 1, Q Q .2/ W W niD1 .fi /Wb D W niD1 .fi /W; 1

.f /W D .f / for f 2 M as an operator. .3/ W

Proof. The proof is similar to that of Proposition 5.7. Define the linear operator W W Fb ! L2 .M2 ; d G/ by W b D 1;

W W

n Y

.fi /Wb D W

iD1

n Y

.fi /W:

(5.6.20)

iD1

By the commutation relations it follows that n Y .fi /Wb W iD1

Fb

n Y D W

.fi /W iD1

L2 .M2 ;d G/

Q Q and the linear hull of W niD1 .fi /Wb and W niD1 .fi /W are dense in Fb and L2 .M2 ; d G/, respectively. Then W can be extended to a unitary operator and the proposition follows.

282


Definition 5.10 (Free field Hamiltonian). The free field Hamiltonian HQ f is defined on L2 .M2 ; d G/ by 1 ; W Hf W

(5.6.21)

where Hf is the free Hamiltonian of Fb . In the previous section we defined the self-adjoint operator HP D HO f C WP ..f //W in L2 .Q/ and have given the functional integral representation of the semigroup e tHP by making use of Euclidean fields. The operator HQ P D HQ f C WP . .f //W

(5.6.22)

1 D H Q P . We can is the operator associated with HP in L2 .M2 ; d G/, W HP W Q

also construct a functional integral representation of e t HP in terms of an infinite dimensional Ornstein–Uhlenbeck process which will be investigated next.

5.6.3 Infinite dimensional Ornstein–Uhlenbeck-process The one-dimensional Ornstein–Uhlenbeck process has been discussed in Section 3.10 before, and we have seen that it is a stationary Gaussian Markov process defined on the probability space X D C.R; R/ with a measure characterized by having mean and covariance given by (3.10.59). The construction of an infinite dimensional Ornstein– Uhlenbeck process is similar to this. Denote the set of M2 -valued continuous functions on the real line R by Y D C.RI M2 /:

(5.6.23)

On Y we introduce the topology derived from the metric dist.f; g/ D

X .sup0jxjk kf .x/ g.x/kM2 / ^ 1 2k

k0

:

(5.6.24)

This metric induces the locally uniform topology, giving rise to the Borel -field B.Y/. The main theorem in this section is the following. Theorem 5.26. .1/ There exists a probability measure G on .Y; B.Y// and an M2 -valued continuous stochastic process R 3 s 7! s 2 M2 on the probability space .Y; B.Y/; G / such that 1 1 X 1 EG Œ. s ; t /M2 D .ei ; e jtsj!O ej /M ; 2 i j

(5.6.25)

i;j D1

where Spec.D/ D ¹ j º and ¹ej º is the normalized eigenvector such that Dej D j ej .

283


.2/ There exists a random process . s .f //s2R indexed by f 2 M on the probability space .Y; B.Y/; G / such that (a) linearity: M 3 f 7! s .f / 2 R is linear (b) boundedness: s .f / D hh s ; f ii for f 2 MC2 , in particular, j s .f /j k s kM2 kf kMC2 for f 2 MC2 (c) path continuity: (a) if f 2 MC2 , then R 3 s 7! s .f / 2 R is continuous almost surely (b) if f 2 M , then R 3 s 7! s .f / 2 L2 .E2 ; d G / is strongly continuous (d) characteristic function: s .f / is a Gaussian random process with respect to G with mean zero and EG Œe

i

Pn

j D1 sj

.fj /

n 1 X D exp .fi ; e jsi sj j!O fj /M : 2

(5.6.26)

i;j D1

We will prove Theorem 5.26 below. Definition 5.11 (Infinite dimensional Ornstein–Uhlenbeck process). The M2 valued stochastic process . s /s2R on the probability space .Y; B.Y/; G / given in Theorem 5.26 is called infinite dimensional Ornstein–Uhlenbeck process. In order to construct . s /s2R we introduce the Euclidean version of M , E , and a self-adjoint operator D. Let ˇ ² ³ Z ˇ jfO.k/j2 2 d C1 ˇ 2 E D f 2 Lreal .R /ˇkf kE D dk < 1 ; (5.6.27) !.k/2 C 2 where k D .k; / 2 Rd R. Let D be a positive self-adjoint operator with Hilbert– Schmidt inverse on E , and define En D C 1 .D/ by Theorem 5.23.

kD n=2 kE

. The next lemma follows

Lemma 5.27. There exists a Borel probability measure on .E2 ; B.E2 // and a Q / indexed by f 2 E with mean zero such family of Gaussian random variables .f that Q / 2 R is linear .1/ E 3 f 7! .f Q / D hh ; Q f iiE when f 2 EC2 , where hh ; Q f iiE denotes the pairing between .2/ .f EC2 and E2 .3/ it satisfies Z E2

Q Q D e 12 kf k2E : e i .f / d . /

(5.6.28)

284


Take g 2 M . We have already checked in (5.5.8) that ı t ˝ g 2 E and .ıs ˝ f; ı t ˝ g/E D .f; e jtsj!O g/M

(5.6.29)

for all f; g 2 K. For every t 2 R and g 2 M put Q t ˝ g/:

Qt .g/ D .ı

(5.6.30)

In particular, it follows that Z E2

Z E2

Q D 0;

Qs .f /d . /

(5.6.31)

Q D .f; e jtsj!O g/M :

Qs .f / Qt .g/d . /

(5.6.32)

Lemma 5.28. For every fixed t1 < t2 < < tn 2 R there exists a Gaussian measure n G t1 ;:::;tn on M2 D nkD1 M2 and a family of random variables Qt1 ;:::;tn .g/ indexed n by g 2 M such that .1/ g 3 M n 7! Qt ;:::;t .g/ 2 R is linear n

1

n , where hh; ii denotes the pairing .2/ Qt1 ;:::;tn .g/ D hh Qt1 ;:::;tn ; gii when g 2 MC2 n n between MC2 and M2

.3/ Z n M2

e

i Qt1 ;:::;tn .g/

Z d G t1 ;:::;tn D

ei

Pn

j D1

Qtj .gj /

Q d . /

E2

(5.6.33)

n . for all g D .g1 ; : : : ; gn / 2 MC2

Proof. By the definition of Qt we have Z

ei

Pn

j D1

Qtj .gj /

E2

Q D exp d . /

n 1 X .ı tj ˝ gj ; ı tl ˝ gl /E : 2

(5.6.34)

j;lD1

The quadratic form .f; g/ 7! Q t1 ;:::;tn .f; g/ D

n X

.ı tj ˝ fj ; ı tl ˝ gl /E

j;lD1

on M n M n satisfies jQ t1 ;:::;tn .f; g/j kf kM n kgkM n . Thus there exists a bounded operator L t1 ;:::;tn D L on M n such that Q t1 ;:::;tn .f; g/ D .f; Lg/M n . Hence we can write Z Pn Q Q D e 12 .g;Lg/M n : e i j D1 tj .gj / d . / (5.6.35) E2

285


Let Dn D ˚n D. Notice that .f; LDn1 g/

n X

D

.ı tj ˝ fj ; ı tl ˝ D

1

gl /E D

i;j D1

.ı tj ˝ D 1 fj ; ı tl ˝ gl /E

i;j D1

.Dn1 f; Lg/

D

n X

D

.f; Dn1 Lg/:

Thus ŒL; Dn1 D 0. By Theorem 5.23 there exists a measure G t1 ;:::;tn and a random variable Qt1 ;:::;tn .g/ such that Z 1 Q e i t1 ;:::;tn .g/ d G t1 ;:::;tn D e 2 .g;Lg/M n ; n M2

and hence (5.6.33) is obtained. Let J D RN be the index set and ƒ J be such that jƒj < 1, where jƒj D #ƒ. ƒ J ƒ Write M2 D ¹f W ƒ ! M2 º, let ƒ W M2 ! M2 be the projection defined by ƒ f D f dƒ , and define jJ j

1 J A D ¹ƒ .E/ 2 M2 j ƒ J; jƒj < 1; E 2 B.M2 /º:

(5.6.36)

J Now we construct a probability measure G on .M2 ; .A // by using the Kolmogorov extension theorem. J Lemma 5.29. There exists a probability measure G on .M2 ; .A //, an M2 valued stochastic process . t / t2R , and an R-valued stochastic process . s .f //s2R J ; .A /; G / such that with f 2 M on .M2

.1/ its covariance is EG Œ. s ; t /M2 D

1 1X 1 .e ; e jtsj!O ej /M ; 2 j 2 j D1 j

(5.6.37)

where ¹ j º and ¹ej º are given in Section 5.6.2. In particular, 1 EG Œk t k2M2 D Tr.D 2 /: 2

(5.6.38)

.2/ M 3 f 7! s .f / 2 R is linear, and s .f / D hh s ; f ii whenever f 2 M2 .3/ s .f / is a Gaussian random process with zero mean and EG Œe

i

Pn

j D1 sj

.fj /

n 1 X jsi sj j!O D exp .fi ; e fj /M : 2 i;j D1

(5.6.39)

286


Proof. Write eQj D D 1 ej D j1 ej . Then ¹eQj ºj1D1 forms a complete orthonormal system in MC2 . For every k1 ; : : : ; kn 2 J with kj D .tj ; ij /, and E1 En 2 B.Rn / put k1 ;:::;kn .E1 En / D E

n hY

i 1Ej . Qtj .eQij // :

(5.6.40)

j D1 n By Lemma 5.28 there exists a probability measure G t1 ;:::;tn on M2 such that

Z k1 ;:::;kn .E1 En / D

n Y n M2 j D1

1Ej . Qij /d G t1 ;:::;tn ;

(5.6.41)

j

where Qij D Qt1 ;:::;tn .0 ˚ eQij ˚ 0/. Since we can see that the family of set „ ƒ‚ … n

functions ¹ƒ ºƒJ;jƒj 0. A direct computation gives EG Œj t .g/ s .g/j4 D E Œj Qt .g/ Qs .g/j4 D 12.g; .1 e jtsj! /g/2M Z 2 1 e jtsj!.k/ 2 jg.k/j O D3 d k 3jt sj2 kgk4L2 .Rd / : !.k/

288


Notice that the embedding W MC2 !L2 .Rd / is bounded with the bound kgkL2 .Rd / p k 2!D 1 kkgkMC2 . From this it follows that EG Œj s .g/ t .g/j4 C kgk4MC2 with a constant C > 0. Since k s kM2 D

sup f 6D0;f 2MC2

j s .f /j ; kf kMC2

there exists f" 2 MC2 such that k s t kM2 " D

j s .f" / .f" /j ; kf" kMC2

we obtain that EG Œk s t k4M2 " C C jt sj2

(5.6.53)

for any " > 0. Hence (5.6.52) follows. Proof of Theorem 5.26. We denote the continuous version of . t / t2R by N D . Nt / t2R , and the image measure G ı N 1 on Y by the same symbol G . Thus we have constructed above the probability space .Y; B.Y/; G /. (1) and (2) of the theorem follow from Lemma 5.29 except for path continuity, which we show now. Whenever f 2 MC2 we see that s .f / D hh s ; f ii. Then s 7! s .f / is continuous almost surely since R 3 s 7! s 2 M2 is continuous. Next let f 2 M . In this case lim EG Œj s .f / t .f /j2 D lim .f; .1 e jtsj!O /f /M D 0:

s!t

s!t

This gives the desired result.

5.6.4 Markov property The stochastic process . s /s2R is stationary due to the form of its covariance as given by (5.6.37). Proposition 5.18 shows that the Euclidean quantum field has the Markov property. In this section we show that, moreover, . s /s2R is a Markov process. Q s ˝f /, Recall that ıs ˝f 2 E for f 2 M . We give the relationship of Qs .f / D .ı . s /s2R and the Euclidean field t .f /. Proposition 5.31. Let Fj be Borel measurable bounded functions on R, and fj 2 M . Then n n n i i i hY hY hY EG Fj . tj .fj // D E Fj . Qtj .fj // D EE Fj . tj .fj // : j D1

j D1

j D1

(5.6.54)

289


Proof. Let F 2 .Rn /. From (5.6.50) it follows that EG ŒF . s1 .f1 /; : : : ; sn .fn // Z Pn FL .k1 ; : : : ; kn /EG Œe i j D1 kj sj .fj / d k1 : : : d kn D .2/n=2 D .2/n=2

Z

Rd Rd

Rd Rd

FL .k1 ; : : : ; kn /E Œe i

Pn

j D1

kj Qsj .fj /

d k1 : : : d kn

D E ŒF . Qs1 .f1 /; : : : ; Qsn .fn //: The first equality can be completed by a limiting argument. The second equality is proven similarly. Lemma 5.32. Let E 0 E be a closed subspace, and P W E ! E 0 be the correspondQ /jf 2 E 0 /. Then for each ˛ 2 C ing orthogonal projection. Consider GP D . .f and f 2 E it follows that Q

Q

E Œe ˛ .f / jGP D e ˛ .Pf / e

˛2 2 kf

Pf k2E

:

(5.6.55)

Proof. We have that Q

Q

Q

Q

Q

E Œe ˛ .f / jGP D E Œe ˛ .Pf / e ˛ .f Pf / jGP D e ˛ .Pf / E Œe ˛ .f Pf / jGP ; (5.6.56) Q Q Pf / is independent of GP , since .Pf / is GP -measurable. Moreover, since .f we have Q

Q

Q

e ˛ .Pf / E Œe ˛ .f Pf / D e ˛ .Pf / e

˛2 2 kf

Pf k2E

:

(5.6.57)

For any interval I R write FI D .L.H. ¹ t .f /jt 2 I; f 2 M º/ : Theorem 5.33 (Markov property). Let F be a bounded FŒs;1/ -measurable function. Then it follows that EG ŒF jF.1;s D EG ŒF jF¹sº :

(5.6.58)

Proof. Denote by E.1;s the closed subspace of E generated by the linear hull of ¹ır ˝f 2 E jr s; f 2 M º, and by E¹sº the closed subspace generated by ¹ıs ˝f 2 E jf 2 M º. Write P.1;s and P¹sº for the corresponding projections. We claim that for every t s and every g 2 M , P.1;s .ı t ˝ g/ D ıs ˝ e .ts/!O g D P¹sº .ıs ˝ e .ts/!O g/:

(5.6.59)

290


Indeed, ıs ˝ e .ts/!O g 2 E.1;s , and for all r s, Z dk e jsrj!.k/ fO.k/.e .ts/!O g/O .k/ .ır ˝ f; ı t ˝ g/E D d 2!.k/ R D .ır ˝ f; ıs ˝ e .ts/!O g/E : From this we get .F; ı t ˝ g/E D .F; ıs ˝ e .ts/!O g/E ;

F 2 E.1;s ;

by linearity and approximation, and thus the first equality in (5.6.59) is shown. The second equality now follows from the fact that ıs ˝ e .ts/!O g is not only in E.1;s but even in E¹sº . By Lemma 5.32 we furthermore have 2 ˛ Q .1;s ı t ˝g/ ˛ Qt .g/ ˛ .P 2 E Œe k.1 P.1;s /ı t ˝ gkE ; jGP.1;s D e exp 2 and Proposition 5.31 and (5.6.56) give EG Œe

˛ t .g/

jF.1;s D e

˛ s .e .t s/!O g/

˛2 exp .g; .1 e 2.ts/!O /g/M 2

2 F¹sº

for t 0. Again by approximation, we find EG ŒF jF.1;s 2 F¹sº for all bounded FŒs;1/ -measurable functions. Thus (5.6.58) follows.

5.6.5 Regular conditional Gaussian probability measures The Gaussian measure G is a probability measure on .Y; B.Y// and 0 W Y ! M2 is a random variable. Since both Y D C.RI M2 / and M2 are Polish spaces, we can define the regular conditional probability measure G with 2 M2 by G ./ D G .j 0 D /:

(5.6.60)

In particular we can assume that G . 0 2 B/ D 1 .B/;

B 2 B.M2 /;

(5.6.61)

and G . 0 D / D 1;

G ı 01 -a.s. :

(5.6.62)

See Theorem 2.12 for (5.6.60)–(5.6.62).

We compute EG Œ t .f / and EG Œ t .f / s .g/ for later use. Note that EG Œe

Pn

j D1

tj .fj /

jF¹0º D EG Œe

Pn

j D1

tj .fj /

j. 0 /:

(5.6.63)

It is seen by Lemma 2.11 that the right-hand side of (5.6.63) is a function of 0 , h. 0 /, Pn

j D1 tj .fj / is defined by h. 0 / with 0 replaced by . and EG Œe

291


Theorem 5.34. Suppose that f; g 2 M and 2 M2 . Then it follows that EG Œ t .f / D .e jtj!O f /;

(5.6.64)

EG Œ t .f

/ s .g/ D .f; e jtsj!O g/M .f; e .jtjCjsj/!O g/M C .e jtj!O f / .e jsj!O g/: (5.6.65)

Proof. In a similar way to the proof of Theorem 5.33 we see that EG Œe

Pn

j D1

tj .fj /

jF¹rº D e

Pn

j D1 r .e

jtj rj! O fj /

1

e 2 Q;

(5.6.66)

where QD

n X

¹.fi ; e jti tj j!O fj /M .fi ; e jti rj!O e jtj rj!O fj /M º:

i;j D1

Let ˛; ˇ 2 R. Inserting ˛f and ˇg into f1 and f2 in (5.6.66), respectively, and setting n D 2 and r D 0 we have EG Œe ˛ t .f /Cˇ s .g/ j. 0 / D e ˛ 0 .e

jt j! Of

/Cˇ 0 .e jsj!O g/

1

e 2 Q;

(5.6.67)

where Q D ˛ 2 .kf k2M .f; e 2jtj!O f /M / C ˇ 2 .kgk2M .f; e 2jsj!O g/M / C 2˛ˇ..f; e jtsj!O g/M .f; e .jtjCjsj/!O g/M /: Hence

EG Œe ˛ t .f /Cˇ s .g/ D e ˛ .e

jt j! Of

/Cˇ .e jsj!O g/

1

e 2 Q:

(5.6.68)

(5.6.64) is derived through @˛ EG Œe ˛ t .f /Cˇ s .g/ ˛D0 , while formula (5.6.65) can be

obtained via @˛ @ˇ EG Œe ˛ t .f /Cˇ t .g/ ˛Dˇ D0 .

Using Theorem 5.34 we can also compute the covariance:

cov. t .f /I t .g// D EG Œ t .f / t .g/ EG Œ t .f /EG Œ t .g/ D .f; e jtsj!O g/M .f; e .jtjCjsj/!O g/M : In particular, cov. t .f /I t .g// is independent of .

(5.6.69)

292


5.6.6 Feynman–Kac–Nelson formula by path measures By making use of the path measure G , an alternative functional integral representation of .F; e tHP G/ in Theorem 5.20 can be given. Define f 2 M;

P WP . .f //W; HQ P D HQ f C

(5.6.70)

where we recall that HQ f denotes the free field Hamiltonian on L2 .M2 ; d G/. Theorem 5.35 (Feynman–Kac–Nelson formula). Let F; G 2 L2 .M2 ; d G/. Then Z t

Q WP . s .f //Wds : (5.6.71) .F; e t HP G/ D EG F . 0 /G. t / exp 0

Proof. Let F . / D P1 . .f1 /; : : : ; .fn // and G. / D P2 . .f1 /; : : : ; .fm // with a polynomial Pj . In the proof of Theorem 5.20 we have 1 1 1 .W F; e tHP W G/Fb D lim E ŒW F . Q0 /e

Pn1

Q

j D1 .tj=n/WP . tj=n .f

n!1

//W 1 W G. Qt /:

Q

1 Thus by the unitary equivalence W e tHP W D e t HP , and Proposition 5.31 we see that Q

.F; e t HP G/L2 .M2 ;d G/ D lim EG ŒF . 0 /e n!1

D EG ŒF . 0 /e

R

Pn1

j D1 .tj=n/WP . tj=n .f

WP . s .f //Wds

//W

G. t /

G. t /:

Here we used strong continuity of s 7! s .f /. By a limiting argument F; G can be extended to F; G 2 L2 .M2 ; d G/. Making use of the regular conditional probability measure G we can derive the Q functional integral representation of e t HP . Corollary 5.36. Let G 2 L2 .M2 ; d G/. Then Q

.e t HP G/. / D EG Œe

R

WP . s .f //Wds

G. t /;

a.s. 2 M2 :

Proof. The proof is straightforward. By Theorem 5.35 we have Z t

QP

t H G/ D EG F . /EG G. t / exp WP . s .f //Wds : .F; e 0

Since F is arbitrary, the corollary follows.

(5.6.72)

(5.6.73)

Chapter 6

The Nelson model by path measures

6.1

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. In this section we explain how the Nelson model can be derived from a Lagrangian, and we continue with a rigorous definition and analysis in the following sections. The Lagrangian density LN D LN .x; t /, .x; t / 2 R3 R, of the Nelson model is described by P C LN D i ‰ ‰

1 1 1 @j ‰ @j ‰ C @ @ 2 2 C ‰ ‰; 2m 2 2

(6.1.1)

where ‰ D ‰.x; t / is a complex scalar field describing a non-relativistic, spinless electron, and .x; t / is a neutral scalar field describing scalar bosons, 0 is the mass of bosons, and m > 0 that of the electron. Here @ @ D P P @j @j , @j D @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 ´ . C 2 /.x; t / D ‰ .x; t /‰.x; t /; (6.1.2) 1 i @t C 2m x ‰.x; t / D .x; t /‰.x; t /: The Hamiltonian density is derived from the Legendre transform of LN . The conjugate momenta are defined by ˆD

@LN D i ‰; P @‰

(6.1.3)

D

@LN P D : @P

(6.1.4)

Thus the Hamiltonian density HN D HN .x; t / is given by P C P LN HN D ˆ‰ D

1 1 j@x ‰j2 C .P 2 C .@x /2 C 2 2 / ‰ ‰ 2m 2

294

Chapter 6 The Nelson model by path measures

and the Hamiltonian ³ Z ² 1 1 P2 2 2 2 ‰ HN D x ‰ C . C .@x / C / ‰ ‰ dx: (6.1.5) 2m 2 R3 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 of particles and the number of particles is conserved. In order to obtain the quantized model the kinetic part is replaced as Z 1 1 x ‰dx ! ‰ 2m 2m R3 Z

and the interaction as

R3

‰ ‰dx ! .x/:

Adding an external potential V , the formal expression of the Nelson model is where

6.2

1 C V C Hf C .x/; 2m

1 Hf D 2

(6.1.6)

Z .P 2 C .@x /2 C 2 2 /dx:

The Nelson model in Fock space

6.2.1 Definition We now give a rigorous definition of the Nelson model. L2 .Rd / describes the state space of the particle, and the scalar boson field is defined by FN D Fb .L2 .Rd //;

(6.2.1)

i.e., the boson Fock space over L2 .Rd /. The joint state space is the tensor product HN D L2 .Rd / ˝ FN :

(6.2.2)

The free particle Hamiltonian is described by the Schrödinger operator 1 Hp D C V 2

(6.2.3)

acting in L2 .Rd /. The following are standing assumptions throughout this section.

295

Section 6.2 The Nelson model in Fock space

Assumption 6.1. The following conditions hold: Dispersion relation: q ! D !.k/ D jkj2 C 2 ;

0:

(6.2.4)

p Charge distribution: ' W Rd ! R, and '.k/ O D '.k/ O D '.k/, O '= O ! 2 L2 .Rd /, '=! O 2 L2 .Rd / External potential: V is Kato-decomposable and is such that Hp has a ground state ‰p with H ‰p D 0. Recall that by Theorem 3.54 the ground state ‰p of Hp is unique and it has a strictly positive version. We will choose this version of ‰p throughout. The free field Hamiltonian Hf D d .!/

(6.2.5)

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 a space of FN -valued L2 functions on Rd , ˇZ ² ³ Z ˚ ˇ d 2 ˇ HN Š FN dx D F W R ! FN ˇ kF .x/kFN dx < 1 : (6.2.6) Rd

Rd

HI .x/ is defined by the time-zero field p p 1 O ikx = !/ C a.e 'e O ikx = !/º HI .x/ D p ¹a .'e 2 d for x 2 R , where e '.k/ O D '.k/. O We will use the formal notation Z 1 ikx ikx HI .x/ D a .k/ C '.k/e O a.k//d k .'.k/e O p d 2!.k/ R

(6.2.7)

(6.2.8)

for convenience. Since '.k/ O D '.k/, O 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 of FN . We denote the self-adjoint extension of HI .x/ by HI .x/. The interaction HI is then defined by the self-adjoint operator Z ˚ HI D HI .x/dx: (6.2.9) Rd

acting as .HI ‰/.x/ D HI .x/‰.x/;

x 2 Rd ;

(6.2.10)

with domain D.HI / D ¹‰ 2 HN j‰.x/ 2 D.HI .x//; x 2 Rd º:

(6.2.11)

296


Definition 6.1 (Nelson Hamiltonian in Fock space). Under the conditions of Assumption 6.1 the operator HN D Hp ˝ 1 C 1 ˝ Hf C HI

(6.2.12)

in HN is the Nelson Hamiltonian. A first natural question about the Nelson Hamiltonian is whether it is self-adjoint. Proposition 6.1 (Self-adjointness). Under Assumption 6.1 .1/ H0 D Hp ˝ 1 C 1 ˝ Hf is self-adjoint on D.H0 / D D.Hp ˝ 1/ \ D.1 ˝ Hf /

(6.2.13)

and non-negative .2/ HN is self-adjoint on D.H0 / and bounded from below, furthermore, it is essentially self-adjoint on any core of H0 . (In fact, HI is infinitesimally small with respect to H0 .) Proof. H0 is self-adjoint and non-negative as it is the sum of two commuting, nonnegative self-adjoint operators. By Lemma 5.2, D.HI / D.H0 /, Proposition 5.2 gives p 1=2 kHI .x/‰kFN .2k'= O !k C k'k/k.H O ‰kFN f C 1/ for ‰ 2 D.Hf / and for every x 2 Rd . Thus for ˆ 2 D.1 ˝ Hf /, p kHI ˆkHN .2k'= O !k C k'k/k1 O ˝ .Hf C 1/1=2 ˆkHN : 1 /k‰k, the Since k1 ˝ .Hf C 1/1=2 ‰k k.H0 C 1/1=2 ‰k "kH0 ‰k C .1 C 4" Kato–Rellich theorem yields that HN is self-adjoint on D.H0 / and bounded from below.

6.2.2 Infrared and ultraviolet divergences We discuss here the divergences appearing at the two ends of the spectrum. First we consider the ultraviolet divergence. Since in quantum theory the electron is regarded as a point particle, the form factor is chosen as 'ph .x/ D ı.x/. This implies Z Rd

j'Oph .k/j2 dk D 1 !.k/

as 'Oph .k/ D .2/d=2 :

(6.2.14)

297

Section 6.2 The Nelson model in Fock space

This situation is ultraviolet divergence. In the mathematical description a more regular form factor 'O is imposed so that Z 2 j'.k/j O dk < 1 (6.2.15) Rd !.k/ and the field operator .'. x// is well-defined as an operator on FN . At the other end of the energy spectrum an infrared divergence is encountered. Suppose '.k/ O D .2/d=2 for jkj < ", with some " > 0. In this case Z 2 j'.k/j O d k D 1; d 3; (6.2.16) 3 jkj 0 be arbitrary. Since ‰gT ! ‰g strongly as T ! 1, there exists T0 > 0 such that for all T > T0 , .‰gT ; 1¹jxj2 >ƒº ‰gT / .‰g ; 1¹jxj2 >ƒº ‰g / C ":

(6.5.3)

Hence NT .jX0 j2 > ƒ/

sup .‰gT ; 1¹jxj2 >ƒº ‰gT / C .‰g ; 1¹jxj2 >ƒº ‰g / C ": (6.5.4)

0T T0

We estimate the first factor in the right-hand side above. Note that 1

.‰gT ; 1¹jxj2 >ƒº ‰gT / Since 1 2

Z

Z

T T

Z

T

ds T

D

W dt

Rd

EN0 Œ1¹jX0 j2 >ƒº e 2 EN0 Œe

1 2

RT

T

RT

ds

T

ds

RT T

RT T

dtW

dtW

:

(6.5.5)

2 j'.k/j O .e 2T !.k/ 1 C 2T !.k//d k aT C b 4!.k/3

with some a; b, together with (6.5.5) it gives .‰gT ; 1¹jxj2 >ƒº ‰gT /

EN0 Œ1¹jX0 j2 >ƒº e aT Cb e .aT Cb/

D .‰p ; 1¹jxj2 >ƒº ‰p /e 2.aT Cb/ : (6.5.6)

Thus sup0T T0 .‰gT ; 1¹jxj2 >ƒº ‰gT / ! 0 as ƒ ! 1 and (1) follows. To prove (2) it suffices to show that ENT ŒjXs X t j2n Djt sjn for some n 1 with a constant D. We have

! d X 2n X 2n ENT ŒjXs X t j2n D .1/k ENT Œ.Xs /2nk .X t /k k !

D

D1 kD0

d X 2n X 2n 2nk .ts/HN k CtHN T .1/k .e sHN ‰gT ; x e x e ‰g /: k

D1 kD0

(6.5.7)

311

Section 6.5 Ground state expectations

The last term can be represented by functional integration given by Theorem 6.3 in terms of Brownian motion B t and the Euclidean field E as ENT ŒjXs X t j2n Z R t s D dxEx ŒjB0 B ts j2n e 0 V .Br /dr Rd

R t s

.I0 e sHN ‰gT .B0 /; e E . 0 ır ˝'.Br /dr/ I ts e CtHN ‰gT .B ts // Z R t s dxEx ŒjB0 B ts j2n e 0 V .Br /dr Rd

R t s

ke sHN ‰gT .B0 /kke E .

0

ır ˝'.Br /dr/

k1 ke CtHN ‰gT .B ts /k;

where the norms with no subscript are L2 .Q/-norms, and those with subscript 1 are L1 .QE /-norms. We also note that R t s

ke E .

1

D e2

0

ır ˝'.Br /dr/

R t s 0

dr

R t s 0

R t s

k1 D .1; e E .

dr 0 W .Br Br 0 ;jrr 0 j/

0

ır ˝'.Br /dr/

e

t s 2

R

1/

R 2 2 j'.k/j O O d kC Rd j'.k/j dk Rd !.k/2 !.k/3

D C: (6.5.8)

Then by Schwarz inequality, ENT ŒjXs X t j2n

Z R t s C E jB0 B ts j2n dxe 0 V .Br Cx/dr ke sHN ‰gT .x/kke tHN ‰gT .B ts C x/k

Z 1=2 R t s 2n 2 0 V .Br Cx/dr sHN T 2 ke ‰g .x/k C E jB0 B ts j dxe Z 1=2 dxke tHN ‰gT .B ts C x/k2 4n 1=2

C.EŒjB0 B ts j /

Z R t s 2 0 V .Br Cx/dr sHN T 2 ke ‰g .x/k E dxe Z 1=2 dxke tHN ‰gT .B ts C x/k2

R t s 1=2 sH T N Djt sjn sup Ex Œe 2 0 V .Br /dr ke ‰g kHN ke tHN ‰gT kHN : x

Finally, noticing that ke sHN ‰gT kke tHN ‰gT k ! k‰g k2 as T ! 1, we complete (2).

312


We now establish an explicit formula for expectations .‰g ; .g ˝ L/‰g / of an operator g ˝ L as averages with respect to the probability measure N . In order to state our main theorem, we introduce some special elements of MC . For every T 2 Œ0; 1 and every path define Z T Z T C C jsj!O ikXs !.k/jsj 'T;X .x/ D e 'Xs .x/ds i.e. 'T;X .k/ D '.k/e O e ds

b

'T;X .x/ D

Z

0

0 T

0

b

e jsj!O 'Xs .x/ds

i.e. 'T;X .k/ D

where !O D !.i r/. We also define 'XC .x/ D 'X .x/ D

Z Z

1 0 0 1

Z

0

T

ikXs !.k/jsj '.k/e O e ds;

e jsj!O 'Xs .x/ds; e jsj!O 'Xs .x/ds:

˙ 2 M and It follows that 'T;X C C .'T;X ; 'T;X /MC

Z D

˙ k'T;X k2MC

Z

0

T

ds Z

T

Rd

0

W .X t Xs ; t s/dt;

(6.5.9)

2 j'.k/j O d k < 1: 2!.k/3

(6.5.10)

Write x;

P0

D N0x ˝ G ;

.x; / 2 Rd M2 :

(6.5.11)

Theorem 6.13 (Ground state expectations for bounded operators). Let L be a bounded operator on L2 .G/, and g 2 L1 .Rd / viewed as a multiplication operator. Then .‰g ; .g ˝ L/‰g /

C

D EN Œ.We .'X / W; LWe .'X / W/L2 .G/ g.X0 /e

R0 1

ds

R1 0

dtW .X t Xs ;ts/

:

(6.5.12)

Proof. By Theorem 6.2 we have .F; ‰gT / D

R 1 x; 0T s .'Xs /ds T LN .F; e 1/ D E ŒF .x;

/E Œe : p GN 0 P 0 ke T LN 1k ZT

1

Hence we arrive at ‰gT .x; /

!# " Z T 1 x;

Dp E

s .'Xs /ds exp ZT P0 0

(6.5.13)

313

Section 6.5 Ground state expectations x;

for almost every .x; / 2 Rd M2 . Notice that EP0 D ExN0 EG . We can compute RT

EG Œexp. 0 s .'Xs /ds/ by using that (see Theorem 5.34) "Z # T

EG

EG

D

C

s .'Xs /ds D .'T;X /;

0

Z 0

1 2

Z

2

T

s .'Xs /ds Z

T 0

Z

T

ds

dt

dk Rd

0

EG

"Z

#!2

T

0

s .'Xs /ds

2 j'.k/j O e ik.Xs X t / .e !.k/jtsj e !.k/.jtjCjsj/ /: 2!.k/

Now the integration with respect to G in (6.5.13) can be carried out with the result 1 ‰gT .x; / D p ZT C

1

ExN0 Œe .'T;X / e 2

RT 0

ds

RT 0

dt

R

2

Rd

'.k/j O d k j2!.k/ e ik.Xs X t / .e !.k/jt sj e !.k/.jt jCjsj/ /

: (6.5.14)

2

By the Wick ordering We .f / W D e .1=2/kf kM e .f / from (5.2.56), and then C

C

1

e .'T;X / D We .'T;X / We 2

RT 0

ds

RT 0

dt

R

2

Rd

'.k/j O d k j2!.k/ e ik.Xs X t / e !.k/.jt jCjsj/

:

(6.5.15)

Hence, by combining (6.5.14) and (6.5.15), R RT C 1 T 1 ExN0 ŒWe .'T;X / We 2 0 ds 0 W .Xs X t ;st/dt : ‰gT .x; / D p ZT

(6.5.16)

By the reflection symmetry of X t , we also have ‰gT .x; / D p

R R0 1 0 1 ExN0 ŒWe .'T;X / We 2 T ds T W .Xs X t ;st/dt : ZT

(6.5.17)

Now we write (6.5.17) for the left and (6.5.16) for the right entry of the scalar product .‰gT ; .1 ˝ L/‰gT /, use independence of Xs , s 0; and X t , t 0, and we add and R0 RT subtract the term T ds 0 W .Xs X t ; s t /dt in the exponent to obtain .‰gT ; .g ˝ L/‰gT /L2 .P0 /

(6.5.18) C

D ENT Œ.We .'T;X / W; LWe .'T;X / W/g.X0 /e

R0 T

ds

RT 0

W .X t Xs ;ts/dt

:

This is the finite T version of (6.5.12). It remains to take the limit T ! 1. On the left-hand side of (6.5.18) this is immediate since ‰gT ! ‰g in L2 .P0 / and L

314


is continuous. On the right-hand side, we already know that NT 0 ! N weakly with some subsequence. Reset T 0 to T . Thus it only remains to show that the integrand of (6.5.18) is uniformly convergent in the path. For the first factor we find ˙ O uniformly in T and paths, and ' ˙ ! 'c strongly that j' ˙ .k/j j'.k/j=!.k/

b

b

T;X

T;X

X C

.' / X We W in

C

.'T;X /

in MC as T ! 1 uniformly in paths. Thus We W ! L2 .G/ uniformly in path. Since the same argument applies to 'T;X and L is continuous, C

C

.We .'T;X / W; LWe .'T;X / W/ is uniformly convergent to .We .'X / W; LWe .'X / W/ as T ! 1. Moreover, the infrared regularity condition implies ˇZ ˇ Z Z T ˇ 0 ˇ 2 j'.k/j O ˇ ˇ ds W .Xs X t ; s t /dt ˇ dk < 1 (6.5.19) ˇ 3 ˇ ˇ T Rd 2!.k/ 0 and hence Z 0 Z ds T

0

Z

T

W .Xs X t ; s t /dt !

Z

0

1

W .Xs X t ; s t /dt

ds 1

0

(6.5.20) uniformly in path. Hence the integrand of (6.5.18) is uniformly convergent as T ! 1. Most operators of physical interest are, however, not bounded. Therefore we need to extend formula (6.5.12) to unbounded operators. Theorem 6.14 (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 Œjg.X0 /j2 kLWe .'X / Wk2L2 .G/ < 1:

(6.5.21)

Then ‰g 2 D.g ˝ L/, and (6.5.12) holds. Proof. Without loss of generality we can suppose that g is real-valued, and ´ g.x/; jg.x/j < M; gM .x/ D M; jgj M: Let LN D 1ŒN;N .L/. RThen gM ˝ LN is a bounded operator, hence (6.5.12) holds 2 =!.k/3 d k. Using Schwarz inequality, we have for gM ˝ LN . Let I D j'.k/j O k.gM ˝ LN /‰g k2

C

D EN Œ.We .'X / W; L2N We .'X / W/jgM .X0 /j2 e

e I EN ŒjgM .X0 /j2 kLN We .'X / WkkLN We

C

R0 1

C

.'X /

e I EN Œjg.X0 /j2 kLWe .'X / WkkLWe .'X / Wk;

ds

Wk

R1 0

dt W .X t Xs ;ts/

315


which is bounded according to (6.5.21). By the monotone convergence theorem, this shows that ‰g 2 D.g ˝ L/, and then we derive that .gM ˝ LN /‰g ! .g ˝ L/‰g as M; N ! 1. Hence .‰g ; .gM ˝ LN /‰g / ! .‰g ; .g ˝ L/‰g /

(6.5.22) ˙

as N; M ! 1. On the other hand, it follows from (6.5.21) that We .'X / W 2 D.L/ for N -almost all paths. From this we conclude C

C

.We .'X / W; LN We .'X / W/ ! .We .'X / W; LWe .'X / W/ for N almost all paths as N ! 1. We also have C

C

C

.We .'X / W; LN We .'X / W/ kWe .'X / WkkLN We .'X / Wk e I =2 kLWe .'X / Wk for all paths, and the right-hand side of the above is N -integrable. Thus the dominated convergence theorem implies C

EN Œ.We .'X / W; LN We .'X / W/gM .X0 /e C

! EN Œ.We .'X / W; LWe .'X / W/g.X0 /e

R0 1

R0 1

ds

R1

ds

R1

0

0

W .X t Xs ;ts/dt

W .X t Xs ;ts/dt

(6.5.23)

as M; N ! 1. By (6.5.22) and (6.5.23), the proof is complete.

6.5.2 Spatial decay of the ground state We have already seen that in many cases the ground state of Schrödinger operators decays exponentially. Our interest is now the spatial decay of the ground state ‰g of HN . Recall the class V upper defined in Definition 3.26. The key estimate is Lemma 6.15 (Carmona’s estimate). Let V 2 V upper . Then for any t; a > 0 and every 0 < ˛ < 1=2, there exist constants D1 ; D2 ; D3 > 0 such that Q

˛ a2 t

k‰g .x/kL2 .Q/ D1 e I e D2 kU kp t e EN t .D3 e 4

e tW1 C e tWa .x/ /k‰g kHN ; (6.5.24)

R R 2 !.k/3 d k, E 2 !.k/2 d k and W .x/ D eN D EN C d j'.k/j O O where I D Rd j'.k/j a R inf¹W .y/jjx yj < aº. Proof. Since ‰g D e tEN e tHN ‰g , the functional integral representation yields that Rt

‰g .x/ D Ex ŒI0 e E .

0

ıs ˝'.Bs /ds/

It e

Rt 0

V .Bs /ds

‰g .B t /:

(6.5.25)

316


From this it follows directly that Rt

k‰g .x/kL2 .Q/ ke E .

0

ıs ˝'.Bs /ds/

kL1 .Q/ Ex Œe R

Rt

Rt 0

V .Bs /ds

k‰g .B t /kL2 .Q/ : (6.5.26)

2

O !.k/ d k by (6.5.8), the lemma Since ke E . 0 ıs ˝'.Bs /ds/ kL1 .Q/ e I Ct Rd j'.k/j follows in a similar as Carmona’s estimate in Lemma 3.59 for Schrödinger operators. 2

From the above lemma we can draw results similar to Corollaries 3.60 and 3.61 for Schrödinger operators. For V D W U 2 V upper , define † D lim infjxj!1 V .x/. Notice that † D lim infjxj!1 W .x/, and † W1 holds. We only state the results. Corollary 6.16. Let V D W U 2 V upper . .1/ Suppose that W .x/ jxj2n outside a compact set K, for some n > 0 and

> 0. Take 0 < ˛ < 1=2. Then there exists a constant C1 > 0 such that ˛c k‰g .x/kL2 .Q/ C1 exp jxjnC1 k‰g kHN : (6.5.27) 16 where c D infx2Rd nK Wjxj=2 .x/=jxj2n . eN , † > W1 , and let 0 < ˇ < 1. Then .2/ Decaying potential: Suppose that † > E there exists a constant C2 > 0 such that ! eN / ˇ .† E k‰g .x/kL2 .Q/ C2 exp p p jxj k‰g kHN : (6.5.28) 8 2 † W1 .3/ Confining potential: Suppose that limjxj!1 W .x/ D 1. Then there exist constants C; ı > 0 such that k‰g .x/kL2 .Q/ C exp .ıjxj/ k‰g kHN :

(6.5.29)

6.5.3 Ground state expectation for second quantized operators Our next goal is to find similar expressions for the ground state expectation of second quantized operators. Theorem 6.17. Let L be a bounded self-adjoint operator on L2 .Rd / \ MC . Then ‰g 2 D..L// \ D.d .L// and

C

.‰g ; .L/‰g /L2 .P0 / D EN Œe .'X ;L'X /MC e .‰g ; d .L/‰g /L2 .P0 / D EN Œ.'X ; L'XC /MC :

R0 1

ds

R1 0

W .Xs X t ;st/dt

(6.5.30) (6.5.31)

317


Proof. Note that ˙

k.L/We .'X / Wk2L2 .G/ D e

C 2 kL'X kM

C

;

(6.5.32)

˙

kd .L/We .'X / Wk2L2 .G/ D .kL'X˙ k2MC C .'X˙ ; L'X˙ /2MC /e

˙ 2 k'X kM

C

;

(6.5.33)

R O 2 =2! 3 < 1. Since kLf kMC kLkkf kMC for all f 2 MC , and k'X˙ kMC j'j (6.5.21) is satisfied. By Theorem 6.14, the theorem follows. We now discuss some particular cases. Corollary 6.18 (Super-exponential decay of boson number). Let ˛ 2 C. Then ‰g 2 D.e ˛N / and .‰g ; e ˛N ‰g / D EN Œe .1e

˛/

R0 1

ds

R1 0

W .Xs X t ;st/dt

:

(6.5.34)

Proof. Since e ˛N D .e ˛ 1/, Theorem 6.17 gives the result. By Corollary 6.18 .‰g ; N n ‰g / D

dn .‰g ; e ˛N ‰g /d˛D0 : d˛ n

(6.5.35)

In particular,

Z .‰g ; N ‰g / D EN

Z

0

ds 1

1 0

W .Xs X t ; s t /dt :

Corollary 6.19 (Pull-through formula). We have Z 2 j'.k/j O .‰g ; N ‰g / D k.HN EN C !.k//1 e ikx ‰g k2 d k: Rd 2!.k/ Proof. Computing (6.5.36) we obtain Z 0 Z 1 Z dt ds .‰g ; N ‰g / D 1

Since

Z

Z

0

1

dt 1

Z

D

0 0

0

Z

1

ds.e ikx ‰g ; e .ts/.HN EN C!.k// e ikx ‰g /

0

D k.HN EN C !.k//1 e ikx ‰g k2 ; the corollary follows.

(6.5.37)

2 j'.k/j O e jtsj!.k/ EN Œe ik.Xs X t / d k: 2!.k/

dse jtsj!.k/ EN Œe ik.Xs X t /

dt 1

Rd

(6.5.36)

318


The formula (6.5.37) is called pull-through formula. It can also be obtained by the formal .HN EN /a.k/‰g D Œ.HN EN /; a.k/‰g D .!.k/ C ŒHI ; a.k//‰g and

Z

.‰g ; N ‰g / D

Z Rd

ka.k/‰g k2 d k D

Rd

k.HN EN C !.k//1 ŒHI ; a.k/‰g k2 d k:

p ikx = 2!.k/. The formula is useful to study the specO Note that ŒHI ; a.k/ D '.k/e tral properties of the Nelson model. The formal derivation mentioned above is justified, for instance, in Corollary 6.19. The expectation for the number operator N D d .1/ can be extended to more general operators of the form d .g/. Corollary 6.20. Let 0 g be such that Z Rd

jg.k/j

2 j'.k/j O d k < 1: 3 !.k/

(6.5.38)

Then ‰g 2 D.d .g// and Z .‰g ; d .g/‰g / D

dk Rd

2 j'.k/j O g.k/ 2!.k/

Z

Z

0

1

ds 1

0

dt e !.k/jtsj EN Œe ik.X t Xs / : (6.5.39)

In particular .1/–.3/ below follow. .1/ Pull-through formula: .‰g ; d .g/‰g / Z 2 j'.k/j O D k.HN EN C !.k//1 e ikx ‰g k2 d k: jg.k/j 2!.k/ Rd

(6.5.40)

.2/ Upper bound: 1 .‰g ; d .g/‰g / 2

Z Rd

jg.k/j

2 j'.k/j O d k: !.k/3

(6.5.41)

.3/ Lower bound: if ‰g 2 D.jxj ˝ 1/, then there exists C > 0 such that 1 2

Z Rd

jg.k/j

2 j'.k/j O .1 C jkj2 /d k .‰g ; d .g/‰g /: 3 !.k/

(6.5.42)

319


Proof. It can be checked that with I D

R

2 j'.k/j O Rd !.k/3 d k

k' ˙ k2

˙

kd .g/We .'X / Wk2L2 .G/ .kg'X˙ k2MC C .'X˙ ; g'X˙ /2MC /e X MC Z 2 j'.k/j O I .1 C I / e jg.k/j d k < 1: !.k/3 Rd Then (6.5.39) follows from Theorem 6.17. The pull-through formula can be proven in a similar way to the case involving N . The upper bound (6.5.41) follows directly from the representation (6.5.39). From 1 .jkj2 jxj2 /=2 cos k x we get jkj2 EN ŒX t2 C Xs2 2X t Xs 2 1 jkj2 kjxj‰g k2 :

EN Œcos.k .X t Xs // 1

The last inequality above follows from EN ŒXs X t D ke .jtsj=2/.HN EN / x‰g k2 0: Writing C D kjxj‰g k2 , we have .‰g ; d .g/‰g /

1 2

Z Rd

jg.k/j

2 j'.k/j O .1 C jkj2 /d k: !.k/3

This proves the lower bound. Corollary 6.21 (Infrared divergence). Let MQ Rd be a compact set and M D Rd n MQ . Suppose that d 3, ‰g 2 D.jxj ˝ 1/, 0 < < ƒ < 1, and 8 ˆ 0. Then there exists 0 < D such that

where I D

R

D n I In I e pn e ; nŠ nŠ

(6.5.47)

2 j'.k/j O Rd !.k/3 d k.

Proof. The right-hand side of (6.5.47) is obvious by Corollary 6.22. Note that Z 1 1 W .X Y; t / D .'X ; e r !O 'Y /dr 2 jtj and .'X ; e

r !O

Z

1 'Y / D p 2

1

0

re r =.4p/ 2 .'X ; e p!O 'Y /dp: 3=2 p 2

2 e r !O

Since is positivity preserving and ' 0, we see that W .Xs X t ; s t / 0 and hence IX 0. Then the left-hand side follows from 1 1 pn e I EN IXn e I .EN ŒIX /n : nŠ nŠ D is the expectation of the double integral above. Corollary 6.24. Let g 2 L1 .Rd /\L1 .Rd / be real-valued, and '=! O 3=2 2 L1 .Rd /. Then Z 0 Z 1 Z 1 '.k/ O '.l/ O .‰g ; d .g.i r//‰g / D ds dt d kd l p d=2 2.2/ !.k/!.l/ Rd Rd 1 0 L l/EN Œe i.lX t kXs / : e !.k/jtj!.l/jsj g.k

(6.5.48)

In particular, we have the upper bound 1 2 k'=! O 3=2 kL 1 kgkL1 : 2.2/d

.‰g ; d .g.i r//‰g /

(6.5.49)

Proof. In order to apply Theorem 6.17, we calculate 1 .'X ; g.i rk /'XC /MC D ..! 1=2 'X /_ ; g.! 1=2 'XC /_ /L2 .Rd / : 2 Also, .! 1=2 'XC /_ .x/

1 D .2/d=2

Z

Z

1

ds Rd

0

'.k/ O e ik.Xs Cx/ e !.k/jsj d k; p !.k/

and consequently 1 (6.5.50) D 2.2/d=2

Z

Z

0

ds 1

Z

1

dt 0

(6.5.50)

Rd Rd

'.k/ O '.l/ O d kd l p !.k/!.l/

L l/: e i.lX t kXs / e !.k/jtj!.l/jsj g.k

322


Let M Rd be a measurable set. Then .‰g ; d .1M .i rk //‰g / counts the expected number of bosons with position inside M . It is common to write Z d .1M .i rk // D a .x/a.x/dx: M

A special case of Corollary 6.24 is Corollary 6.25. Let M Rd be a compact set and write VolM D .‰g ; d .1M .i rk //‰g /

R M

dx. Then

1 2 k'=! O 3=2 kL 1 VolM: 2.2/d

(6.5.51)

In particular, .‰g ; d .1M .i rk //‰g / 1 2 k'=! O 3=2 kL 1 < 1: Vol M !1 Vol M 2.2/d lim

(6.5.52)

6.5.4 Ground state expectation for field operators For ˇ 2 R and g 2 MC consider .‰g ; e ˇ .g/ ‰g /L2 .P0 / , i.e., the moment generating function of the random variable 7! 0 .g/. Corollary 6.26 (Ground state expectations for field operators). Let ˇ 2 C. Then .‰g ; e ˇ .g/ ‰g /L2 .P0 / is finite and .‰g ; e ˇ .g/ ‰g /L2 .P0 / D EN Œe

ˇ2 2 Ig ˇIX

;

(6.5.53)

where Z Ig D

Rd

2 jg.k/j O d k; 2!.k/

Z IX D

dk Rd

O '.k/ O g.k/ 2!.k/

Z

1

dse !.k/jsj e ikXs :

1

Proof. Note that

C

.We .'X / W; e ˇ 0 .g/ We .'X / W/ 2 Z 0 Z 1 ˇ D exp Ig ˇIX exp ds W .X t Xs ; t s/dt : 2 1 0 The statement is then a direct consequence of Theorem 6.14. By this corollary .‰g ; .g/n ‰g /L2 .P0 / D

dn .‰g ; e ˇ .g/ ‰g /L2 .P0 / dˇ D0 dˇ n

for all n 2 N. (6.5.54)

323

Section 6.5 Ground state expectations 2=2

d n x 2=2 e . dx n

H3 .x/ D x 3 3x:

(6.5.55)

Let Hn be the Hermite polynomial of order n defined by Hn D .1/n e x For example, H0 .x/ D 1; We see that e

H1 .x/ D x;

H2 .x/ D x 2 1;

ˇ2 2 Ig ˇIX

can be expressed by an exponential generating function as p 1 X p ˇ2 .iˇ I g /n I ˇI g X D Hn .iIX = I g / : (6.5.56) e2 nŠ nD0

Hence we derive p d n ˇ2 Ig ˇIX e2 dˇ D0 D i n Hn .iIX = I g /I n=2 : n dˇ

(6.5.57)

In the next corollary we will look at the mean value and variance of the random variable 7! 0 .g/ for g 2 MC , using the results of Corollary 6.26. Corollary 6.27 (Functional integral representation of .‰g ; .g/n ‰g /). We have .1/ Average field strength: for n D 1 Z 1 Z '.k/ O g.k/ O dk .‰g ; .g/‰g / D dse !.k/jsj EN Œe ikXs : (6.5.58) 2!.k/ Rd 1 .2/ Field fluctuations: for n D 2 Z 2 jg.k/j O .‰g ; .g/2 ‰g / D dk (6.5.59) Rd 2!.k/

Z 2 Z 1 '.k/ O g.k/ O !.k/jsj ikXs C EN dk dse e : 2!.k/ Rd 1 .1/ By using the previous result and Schwarz inequality we find that Z 2 jg.k/j O 2 2 d k D .1; .g/2 1/.1; .g/1/2 : .‰g ; .g/ ‰g /.‰g ; .g/‰g / Rd 2!.k/

Remark 6.2.

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 Œe ikXs D

Z Rd

‰p .x/2 .x/2 e ikx dx;

R where .x/2 D ‰g2 . ; x/d G is the density of N with respect to N0 , and ‰p2 is the density of N0 with respect to Lebesgue measure and the square of the ground state of Hp .

324


.3/ Writing D ‰p2 2 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 .‰g ; .k/‰g /L2 .P0 / D

'.k/ O .k/ O d=2 .2/ ! 2 .k/

k 2 Rd ;

and .‰g ; .x/‰g /L2 .P0 / D . V! '/.x/

x 2 Rd ;

(6.5.60)

respectively. Here V! denotes the Fourier transform of 1=! 2 and is the Coulomb potential for massless bosons, i.e. for !.k/ D jkj. (6.5.60) is the classical field generated by a particle with position distribution .x/dx.

6.6

The translation invariant Nelson model

In this section we study the ground state of the translation invariant Nelson model. Translation invariant models will be revisited in Section 7.10. In the present section we only state the results without proofs. p Here we suppose that the dispersion relation is ! .k/ D jkj2 C 2 . With coupling constant g 2 R we write the Nelson Hamiltonian as HN D Hp C gHI C Hf : Furthermore, we suppose throughout this section that V D 0. Write Z Pf D k a .k/a.k/d k; D 1; : : : ; d; Rd

(6.6.1)

for the field momentum operator. The total momentum operator P , D 1; : : : ; d , on HN is defined by the sum of particle and field momenta, P D i rx ˝ 1 C 1 ˝ Pf :

(6.6.2)

It is straightforward to see that ŒHN ; P D 0;

D 1; : : : ; d;

(6.6.3)

as V D 0 is assumed. This has the implication that HN is decomposable on the P spectrum of the total momentum operator, Spec.P / D R. We denote dD1 Pf 2 by Pf2 .

325

Section 6.6 The translation invariant Nelson model

Definition 6.4 (Nelson Hamiltonian with fixed total momentum). For each p 2 Rd , the Nelson Hamiltonian with fixed total momentum is defined by 1 HN .p/ D .p Pf /2 C g.0/ C Hf 2

(6.6.4)

with domain D.HN .p// D D.Hf / \ D.Pf2 /, where p p 1 QO !//: O !/ C a.'= .0/ D p .a .'= 2 Here p 2 Rd is the total momentum. Define the unitary operator T W H D L2 .Rdx / ˝ FN ! L2 .Rpd / ˝ FN by Z T D .F ˝ 1/

˚

Rd

exp.ix Pf /dx;

(6.6.5)

with F denoting Fourier transformation from L2 .Rdx / to L2 .Rpd /. For ‰ 2 HN we see that Z 1 e ixp e ixPf ‰.x/dx: .T ‰/.p/ D p .2/d Rd Lemma 6.28. HN .p/ is a non-negative self-adjoint operator and Z T

˚ Rd

FN dp T

1

Z D HN ;

T

˚

Rd

HN .p/dp T 1 D HN :

(6.6.6)

This can be proven similarly to (7.6.11). We can construct the functional integral representation of e tHN .p/ in a similar way as that of e tHPF .p/ , which will be done in Section 7.6. Theorem 6.29 (Functional integral representation for Nelson Hamiltonian with fixed total momentum). If ‰; ˆ 2 FN , then .‰; e tHN .p/ ˆ/FN D E0 Œ.I0 ‰; e gE .L t / I t e id .ir/B t ˆ/e ipB t ;

(6.6.7)

where L t is an HE -valued integral given by Z Lt D

t 0

ıs ˝ '. Bs /ds:

(6.6.8)

The following result comes as an immediate corollary to the functional integral representation. Write EN .p/ D inf Spec.HN .p//.

326


Corollary 6.30 (Uniqueness of ground state). .1/ Let p D 0 and ‚ D e i. =2/N . Then ‚1 e tHN .0/ ‚ is positivity improving. .2/ The ground state of HN .0/ is unique whenever it exists. .3/ EN .0/ EN .p/ holds. Next we review some spectral properties of HN .p/. Let .p/ D inf Specess .HN .p// EN .p/:

(6.6.9)

.p/ D inf .EN .p k/ C ! .k// EN .p/:

(6.6.10)

It is known that k2Rd

It follows that .p/ > 0 for > 0 and sufficiently small jpj. From this follows the important fact that since !0 .0/ D 0, .p/ D 0 for D 0. In particular, HN .p/ has a ground state for sufficiently small jpj and strictly positive > 0. The question is the existence of a ground state of HN .p/ for D 0. Suppose now that 'O is rotation invariant. Then .p/ is also rotation invariant, i.e., .Rp/ D .p/ for all d d orthogonal matrices R. One of the criteria for the existence of a ground state of HN .p/ for D 0 is Lemma 6.31. If kN 1=2 ‰g .p/k < 1

(6.6.11)

uniformly in , then HN .p/ for D 0 has a ground state. To show the bound (6.6.11) a standard strategy is to apply the pull-through formula kN 1=2 ‰g .p/k2 D

g2 2

Z

2 O d k; .HN .p k/ C ! .k/ EN .p//1 p'.k/ .p/ ‰ g ! .k/ Rd (6.6.12)

where ‰g .p/ denotes the ground state of HN .p/ for > 0. A naive estimate from the pull-through formula yields kN

1=2

p O ! k2 g 2 k'= : ‰g .p/k 2 0 .jpj/ 2

(6.6.13)

Here 0 .jpj/ D .p/. As ! 0, however, 0 .jpj/ ! 0 and then the righthand side of (6.6.13) diverges. Hence the pull-through formula seems not to work. Nevertheless, functional integration gives an elegant solution for p D 0.

327

Section 6.6 The translation invariant Nelson model

Theorem 6.32 (Existence of ground state for zero total momentum). Let p D 0 and D 0. Suppose that 'O is rotation invariant and Z 1 2 j'.k/j O jgj2 2 d k : (6.6.14) 3 Rd !0 .k/ Then (6.6.11) holds. In particular, HN .0/ has a unique ground state for D 0. Let ‰gT .0/ D ke THN .0/ 1k1 e THN .0/ 1. It is known that ‰gT .0/ is strongly convergent to the ground state ‰g .0/ as T ! 1, since ‰g .0/ overlaps with 1. Using the Euclidean Green function associated with the Nelson model we see that .‰gT .0/; e ˇN ‰gT .0// D ET Œe g

2 .1e ˇ /

R0 T

ds

RT 0

W .Xs X t ;st/dt

;

(6.6.15)

where dT D

1 R T ds R T W .Xs X t ;st/dt T e T dW0 ZT

(6.6.16)

is a probability measure on X D C.RI Rd /, ZT denotes the normalizing constant R such that X dT D 1, and W is given by (6.3.16). It can be seen that the right-hand side of (6.6.15) can be analytically continued from Œ0; 1/ to the entire complex plane C with respect to ˇ. We can easily see a stronger statement below. Lemma 6.33. It follows that ‰gT 2 D.e CˇN / for all ˇ 2 C and (6.6.15) holds true for all ˇ 2 C. Proof of Theorem 6.32. By Lemma 6.33, .‰gT .0/; e ˇN ‰gT .0// is differentiable at ˇ D 0 and then d .‰ T .0/; e ˇN ‰gT .0//jˇ D0 dˇ g

Z 0 Z T 2 D g ET ds W dt :

.‰gT .0/; N ‰gT .0// D

T

From this we can estimate j.‰gT .0/; N ‰gT .0//j

g2 2

Z Rd

(6.6.17)

0

2 j'.k/j O d k: !0 .k/3

(6.6.18)

The right-hand side of (6.6.18) is independent of p and , and ‰gT .0/ is strongly convergent to ‰g .0/ as T ! 1. Hence we have Z 2 g2 j'.k/j O kN 1=2 ‰g .0/k2 lim sup kN 1=2 ‰gT .0/k d k: (6.6.19) 2 Rd !0 .k/3 t!1 Taking jgj2 as in (6.6.14), we get the desired results.

328

6.7


Infrared divergence

As seen above, in order to give a rigorous mathematical definition of the Nelson model an ultraviolet cut-off was necessary to avoid ultraviolet divergences. This was done by introducing a charge distribution ', making the coupling to the field sufficiently R regular. The total charge Rd '.x/dx D g 0 measures the strength of the interaction. '.x/ ! gı.x/ corresponds to the point charge limit which we will consider later. First we discuss the infrared divergence problem. Assumption 6.2. We assume !.k/ D jkj, and that V is of Kato-class and satisfies the bound V .x/ C jxj2ˇ

(6.7.1)

with some constant C > 0 and exponent ˇ > 0. In particular, we consider here the massless Nelson model. Under the conditions above Hp has a unique, strictly positive ground state ‰p at eigenvalue Ep , and ‰p .x/ e C jxj

ˇC1

R

(6.7.2)

2 !.k/3 d k < holds with a constant C . Under the infrared regular condition j'.k/j O 1 we have seen in the previous sections that HN has a unique, strictly positive ground state, and we have obtained detailed information of its properties. What we discuss here is whether there is any ground state in the same space when the infrared regular condition is not assumed.

Theorem 6.34 (Absence of ground state). Let d D 3, ' 0 but ' 6 0, and Assumption 6.2 hold. Then HN has no ground state. Remark 6.3. Under the assumption in Theorem 6.34, the infrared singularity Z 2 j'.k/j O dk D 1 (6.7.3) 3 R3 !.k/ holds since '.0/ O > 0. Recall that .T / D .‰p ˝ 1; ‰gT /2 . In order to prove Theorem 6.34 it suffices to show that lim .T / D 0

T !1

by Proposition 6.8. Recall that SŒa; b D potential of the Nelson model. Since

.T / D

Rb a

ds

(6.7.4) Rb a

dtW and W denotes the pair

.‰p ˝ 1; e T LN ‰p ˝ 1/2 ; .‰p ˝ 1; e 2T LN ‰p ˝ 1/

329

Section 6.7 Infrared divergence

we can express .T / by the P ./1 -process X D .X t / t2R associated with the selfadjoint operator Lp D U 1 .Hp E.Hp //U as .EN0 Œe SŒ0;T /2 : EN0 Œe SŒT;T

.T / D

(6.7.5)

Here we used the shift invariance of the process, i.e., EN0 Œe SŒT;T D EN0 Œe SŒ0;2T : Let NT be the probability measure given in (6.5.2) Lemma 6.35. It follows that

.T / ENT Œe

R0 T

ds

RT

W .X t Xs ;ts/dt

0

:

(6.7.6)

Proof. The numerator of (6.7.5) can be estimated by the Schwarz inequality with respect to d N0 and the reflection symmetry of the process as Z .EN0 Œe SŒ0;T /2

R3

Z D

R3

.ExN0 Œe SŒ0;T /2 d N0 .ExN0 Œe SŒ0;T /.ExN0 Œe SŒT;0 /d N0 :

Since Xs , s 0, and X t , t 0, are independent, we also see that Z .EN0 Œe

SŒ0;T 2

/

Rd

ExN0 Œe SŒ0;T CSŒT;0 d N0 D EN0 Œe SŒ0;T CSŒT;0 :

Since SŒ0; T C SŒT; 0 D SŒT; T

R0 T

ds

RT

R0

.EN0 Œe SŒ0;T /2 EN0 Œe SŒT;T

T

0 ds

W .X t Xs ; t s/dt , we have RT 0

W .X t Xs ;ts/dt

Thus R0

RT

EN0 Œe SŒT;T T ds 0 W .X t Xs ;ts/dt

.T / EN0 Œe SŒT;T D ENT Œe and the lemma follows.

R0

T

ds

RT 0

W .X t Xs ;ts/dt

;

:

330


The pair potential W .x; t / can be computed explicitly. The integral kernel of p e jtj is p 1 jt j e jtj .x; y/ D 2 : .jx yj2 C jt j2 /2 Hence W .x y; t s/ D D

1 2

Z

1 jtsj

d jT j.e ikx '; O e jT j! e iky '/ O Z

Z

1 4 2

du R3

dv R3

(6.7.7)

'.u/'.v/ > 0: jx y C u vj2 C jt sj2

To show that .T / ! 0 we first restrict to the set AT D ¹! 2 XjjX t .!/j T ; jt j T º with some < 1. We split up the right-hand side of (6.7.6) by restricting to AT and X n AT , and show that the corresponding expectations converge to zero separately. Lemma 6.36. It follows that lim ENT Œ1AT e

R0 T

ds

RT 0

W .X t Xs ;ts/dt

T !1

D 0:

(6.7.8)

Proof. By using the estimate jX t Xs C x yj2 C jt sj2 8T 2 C 2jx yj2 C jt sj2 ; on AT , and

Z

Z

0

T

ds T

0

2 a C T 2 =2 1 dt 2 log a C jt sj2 a2

we obtain by (6.7.7) that Z 0 Z T ds W .X t Xs ; t s/dt T

Z

D

0

0

Z

ds Z

T

dt Z

0

Z

T 0

dx Z

T

Z

R3

dy Z

R3

'.x/'.y/ jX t Xs C x yj2 C jt sj2

'.x/'.y/ C 2jx yj2 C jt sj2 R3 R3 T 0 ! Z Z 8T 2 C 2jx yj2 C T 2 =2 D dx dy'.x/'.y/ log : 8T 2 C 2jx yj2 R3 R3

ds

dt

dx

dy

8T 2

The right-hand side in the formula above diverges as T ! 1, since < 1. This gives (6.7.8).

331

Section 6.7 Infrared divergence

Lemma 6.37. It follows that lim ENT Œ1XnAT e

R0 T

ds

RT 0

W .X t Xs ;ts/dt

T !1

D 0:

(6.7.9)

Proof. Note that Z

Z

0

T

W .X t Xs ; jt sj/dt

ds T

0

T 2 k'=!k O : 2

(6.7.10)

We have O

.T / e .T =4/k'=!k

2

O e .T =4/k'=!k

2

EN0 Œ1XnAT e SŒT;T EN0 Œe SŒT;T .EN0 Œe SŒT;T /1=2 .EN0 Œ1XnAT /1=2 : EN0 Œe SŒT;T

(6.7.11)

Moreover, there exists a constant ı > 0 such that 2 2 T ık'=!k O jSŒT; T j T ık'=!k O :

Thus

.EN0 Œe 2SŒT;T /1=2 2 O : e 2T ık'=!k SŒT;T EN0 Œe

To complete the proof we argue that this exponential growth is balanced by the second factor of (6.7.11). The estimate p EN0 Œ1XnAT T a C T b exp.cT .ˇ C1/ / (6.7.12) is obtained in Lemma 6.38. Here a; b; c > 0. By choosing 1=.ˇ C 1/ < < 1, the right-hand side of (6.7.12) drops to zero as T ! 1. Now we are in the position to prove Theorem 6.34. Proof of Theorem 6.34. Since

.T / ENT Œe R0

R0

RT T ds 0 W .Xs ;X t ;ts/dt

D ENT Œ1AT F C ENT Œ1XnAT F ;

RT

where F D e T ds 0 W .Xs ;X t ;ts/dt , limT !1 ET Œ1AT D 0 by Lemma 6.35 and limT !1 ET Œ1XnAT D 0 by Lemma 6.36, the theorem follows. In order to prove (6.7.12) we make use of the path properties of the P ./1 -process .X t / t0 shown in Section 3.3.10. Note that EN0 Œ1XnAT D N0 . sup

t2ŒT;T

jX t j T /:

332


Lemma 6.38. (6.7.12) holds. Proof. Suppose that f 2 C 1 .Rd / is an even function, and 8 ˆ T º and by Proposition 3.111, EN0 Œ sup jf .Xs /j > T jsj 0;

.1/ There ex-

(6.8.7)

where the limit is in the uniform topology. .2/ The unitary operator e iGƒ strongly converges to e iG1 as ƒ ! 1, where Z g ˇ.k/ G1 D i p p .1 1 .k//.e ikx a.k/ e ikx a .k//d k; (6.8.8) 2 R3 jkj and then s-lim e t.HN Eƒ / D e iG1 e tHN e iG1 ; ren

ƒ!1

t > 0:

(6.8.9)

HNren is formally given by HNren D H0 g.p A C A p/ C

g2 2 2 .A C A C 2A A/; 2

(6.8.10)

Z 1 .1 1 .k// A D A.x/ D p kˇ.k/e ikx a.k/d k: p 3 2 R jkj p When jkj is large enough, jkjˇ.k/ 1=jkj3=2 . Notice that the test function of A is not an L2 function: and

1 ˇ1 .k/ D p kˇ.k/e ikx … L2 .R3k /: jkj In particular, we see that ŒA.x/; A .y/ D 1;

(6.8.11)

335

Section 6.8 Ultraviolet divergence

and 1 HNren 6D .p g.A C A //2 C V C Hf : 2

(6.8.12)

Therefore (6.8.10) is a formal description, however, we know that p 1=2 kA.x/‰k kˇ1 = !1 kkH1 ‰k;

(6.8.13)

where !1 .k/ D jkj for jkj 1 and D 1 for jkj < 1, and Z !1 .k/a .k/a.k/d k: H1 D R3

Note that p kˇ1 = !1 k D

Z

Z

jkj 1=2, Schwarz inequality applied to the latter integral gives Ex ŒjY" Y0 j2

Z Z T 00 x E 4c ."/ T

4c 00 ."/ 21

Z

t

Œt

T

T

Z

jB t Bs j

t Œt

2

Z

t

ds Œt

jt sj

2.1/

ds dt

Ex ŒjB t Bs j2 dsdt:

Since Ex ŒjB t Bs j2 D

Z R3

(6.8.41)

jxj2 … ts .x/dx < 1;

we obtain that Ex ŒjY" Y0 j2 ! 0 as " ! 0. This proves that there exists a subsequence "0 such that Y"0 ! Y0 almost surely. 0 is exponentially integrable, Lemma 6.43. For every ˛ 2 R the random variable ˛Sren i.e., 0

Ex Œe ˛Sren < 1:

(6.8.42)

0 D S 0 C Y C Z and the pathwise bounds Proof. Recall the decomposition Sren 0 0 2 0 2 0 we show that S2 cT and Z0 cT . To prove exponential integrability of ˛Sren x 2˛Y ˛Y0 is exponentially integrable. Consider E Œe 0 . Since 2˛Y0 can be represented as the stochastic integral Z T Y0 D ˆdB t ; (6.8.43) T

Rt

where ˆ D 2 Œt r'0 .B t Bs ; t s/ds, by an application of the Girsanov theorem we obtain the bound .Ex Œe 2˛Y0 /2 D .Ex Œe 2˛ Ex Œe 4˛ D Ex Œe 8˛ RT

RT T

RT T

R 2 T

ˆdB t 4˛ 2

RT T

ˆdB t 12 .4˛/2

T

jˆj2 dt

RT ˆdB t 12 .4˛/2 T jˆj2 dt

jˆj2 dtC4˛ 2

RT T

jˆj2 dt

RT T

jˆj2 dt

Ex Œe 8˛

2

RT T

/2 jˆj2 dt

:

Here Ex Œe 4˛ T D 1 is derived from the Girsanov theorem. In a similar way to (6.8.40) we have Z T jˆj2 dt 4c 21 Q; T

342


where c is the constant in Lemma 6.40, Z T Z ŒsC QD ds jB t Bs j2 dt T

s

and recall that we have chosen a suitable

1 2

< < 1. Then we have

.Ex Œe 2˛Y0 /2 Ex Œe 32c˛

2 21 Q

:

(6.8.44)

Set D 32c˛ 2 21 . By the Jensen inequality we have Z T ds x 2T R ŒsC jB t Bs j2 dt x Q s E Œe E Œe : T 2T

(6.8.45)

We split up the right-hand side of (6.8.45) by integrating over .T; T / and .T ; T /. Consider the first term. Since Œs C D s C , we have Ex Œe 2T

R ŒsC s

jB t Bs j2 dt

D Ex Œe 2T

R 0

jBsCt Bs j2 dt

D Ex Œe 2T

R 0

jB t j2 dt

:

Let < 1. Then 1=jxj2 R2 Lp .R3 / for p > d=2, thus 1=jxj2 is a Kato-class 2 potential and supx Ex Œe 2T 0 jB t j dt < 1. Hence Z T ds x 2T R ŒsC jB t Bs j2 dt s E Œe < 1: 2T T Next consider the second term. Note that Œs C D T . In the same way as above, Ex Œe 2T

RT s

jB t Bs j2 dt

D Ex Œe 2T D Ex Œe 2T

Hence

Z

T T

R T s 0

R T s 0

jBsCt Bs j2 dt jB t

j2 dt

:

ds x 2T R ŒsC jB t Bs j2 dt s E Œe 0 such that a.f / ! a.f / and a .f / ! a .f /. The scaled Nelson Hamiltonian with cutoff function (6.8.54) is then defined by HN .; ƒ/ D Hp C gHI .ƒ/ C 2 Hf :

(6.8.55)

We call the lim!1 in (6.8.55) weak coupling limit. Here we consider the asymptotic behaviour of e THN .;ƒ/ as ƒ; ! 1. Before going to a rigorous computation we give a heuristic description of the 2 asymptotic behaviour. First, note that e T Hf ! Pb as ! 1, with Pb being the projection onto the space spanned by the Fock vacuum. Then one can expect that e THN .;ƒ/ ! e TH1 .ƒ/ ˝ Pb as ! 1 with some self-adjoint operator H1 .ƒ/ in L2 .R3N /. It can indeed be seen that .f ˝ b ; e THN .;ƒ/ g ˝ b / Z RT RT 2 D dxEx Œf .B0 /g.BT /e .g =2/ 0 ds 0 W .X t Xs ;ts/dt ; R3

(6.8.56)

where W D

N X i;j D1

1 2.2/3

Z jkj 0, is a contraction from Lp to Lq if t is sufficiently large but it fails to be so for D 0. Lemma 7.7. Suppose that > 0 and let a b < t < c d , F 2 EŒa;b and G 2 EŒc;d . Take 1 r < 1, r < p and r < q. Suppose moreover that e 2 .cb/ .p=r 1/.q=r 1/ 1 and F 2 Lp .QE / and G 2 Lq .QE /. Then F G 2 Lr .QE / and kF Gkr kF kp kGkq . In particular, for r such that

r 2 1;

2

[

1 C e .cb/

2 1 e .cb/

;1 ;

we have kF Gkr kF k2 kGk2 . Proof. Let ´

FN

F; jF j < N D 0; jF j N

´ and

GN D

G; jGj < N 0; jGj N:

Then jFN jr 2 EŒa;b , jGN jr 2 EŒc;d and Z QE

jFN jr jGN jr dE D .EŒa;b jFN jr ; EŒc;d jGN jr /2 D .jFN jr ; E .eŒa;b eŒc;d /jGN jr /2 :

371

Section 7.2 The Pauli–Fierz model in non-relativistic QED

Note that eŒa;b eŒc;d satisfies keŒa;b eŒc;d k2 D keŒa;b eb ec eŒc;d k2 kjb jc k2 D ke jcbj! k2 e 2 .cb/ .p=r 1/.q=r 1/: Thus by the Hölder inequality, kFN GN krr kjFN jr kq=r kE .eŒa;b eŒc;d /jGN jr ks ;

(7.2.60)

where 1 D 1=s C r=q. Since keŒa;b eŒc;d k2 .p=r 1/.q=r 1/ D .p=r 1/.s 1/1 1; by Proposition 7.6 it is seen that kE .eŒa;b eŒc;d /jGN jr ks kjGN jr kp=r . Together with (7.2.60) this yields kFN GN kr kFN kq kGN kp kF kq kGkp :

(7.2.61)

Taking the limit N ! 1 on both sides of (7.2.61), by monotone convergence the lemma follows. An immediate consequence is Corollary 7.8. Let D 0, ˆ 2 L1 .QE / and F; G 2 L2 .Q/. Then .Ja F /.Jb G/ˆ 2 L1 .QE / and Z j.Ja F /.Jb G/ˆjdE kˆk1 kF k2 kGk2 (7.2.62) QE

for a ¤ b. If a 6D b, then Ja ˆJb is a bounded operator and kJa ˆJb k kˆk1 . 2 Proof. First assume > 0. Let a < b, and r D 1e .ba/ and s > 1 be such that 1=r C 1=s D 1, i.e., s D s./ D r=.r 1/. Note s./ # 1 as ! 0. Without loss of generality we can assume that ˆ is real-valued. Truncate ˆ by 8 ˆ ˆ > N; 0. In particular, e t HPF DPF D..1=2//. Proof. Suppose that F D f ˝ ˆ and G D g ˝ ‰, where f; g 2 C01 .R3 / and ˆ; ‰ 2 DQ . By functional integration we have Z O dxEŒ.J0 F .x/; e ieAE .K t / U J t G.B t C x//; .F; e t HPF G/ D R3

388

Chapter 7 The Pauli–Fierz model by path measures

where U DC

3 X

.i eAE .K / e 2 AE .K /2 2i eAE .K / r /

D1

and

Z K D

˚3 D1 Z

K D ˚3 D1

t 0 t 0

j t 'Q . Bs /dBs ;

p 'Q D .i k '= O !/_ ;

j t 'Q . Bs /dBs ;

p 2 'Q D .k '= O !/_ :

p 2 '= O ! 2 L2 .Rd / and Since we assumed that ! 3=2 'O 2 L2 .Rd /, it follows that k thus 'Q is well defined. Hence Z 1=2 2 t HO PF G/j kF k dx jE ŒkU J t G.B t C x/kj : (7.4.20) j.F; e R3

We estimate E ŒkU J t G.B t C x/k. By the BDG-type inequality again we have sup Ex Œk!O E K k2m C 0 t m k! .nC3/=2 'k O 2m

(7.4.21)

sup Ex Œk!O E K k2m C 00 t m k! .nC1/=2 'k O 2m

(7.4.22)

n=2

x2R3

and n=2

x2R3

with constants C 0 and C 00 . Let kf k˛ D k! ˛=2 f k and A D kf k1 C kf k0 ;

B D .kf k1 C kf k0 /.kgk1 C kgk2 /:

Combining the inequalities ka] .f /‰k Ak.Hrad C 1/1=2 ‰k; ka] .f /a] .g/‰k C1 Bk.Hrad C 1/1=2 ‰k C C2 Ak.Hrad C 1/‰k with (7.4.21) and (7.4.22), we can estimate the right-hand side of (7.4.20) and show (7.4.19). By a limiting argument, the lemma follows for F 2 D./ and G 2 DPF . Lemma 7.24. Take Assumption 7.1 and ! 3=2 'O 2 L2 .R3 /. Then HPF is essentially self-adjoint on DPF . O

Proof. By Lemmas 7.22 and 7.23 we obtain that j.HPF;0 F; e t HPF G/j CG kF k for O F; G 2 D.HPF;0 / D DPF with a constant CG depending on G. Then e t HPF leaves DPF invariant. This implies that HO PF is essentially self-adjoint on DPF and then so is HPF since HPF D HO PF on DPF .

389

Section 7.4 Applications of functional integral representations

Next we show self-adjointness of HPF on DPF without assuming ! 3=2 'O 2 L2 .Rd /. Lemma 7.25. Under Assumption 7.1 and with ! 3=2 'O 2 L2 .R3 /, HPF;0 .HO PF C z/1 is bounded for every z 2 C with Im z 6D 0. Proof. We prove that both of .HO PF C z/1 and Hrad .HO PF C z/1 are bounded. Note that HPF D HO PF on DPF . Let p D i r . In this proof all numbered constants are non-negative. We have kp 2 ‰k2 C1 .k.p eA /2 ‰k2 C kA .p eA /‰k2 C kA 2 ‰k2 /: We estimate kA .p eA /‰k2 as 2

kA .p eA /‰k C2

3 X

..p eA / ‰; .Hrad C 1/.p eA / ‰/

D1 2

D C2 ..p eA / ‰; .Hrad C 1/‰/ C C2

3 X

..p eA / ‰; eŒHrad ; A ‰/:

D1

Since kŒHrad ; A ‰k C20 k.Hrad C 1/1=2 ‰k, we have kp 2 ‰k2 C3 .k.p eA /2 ‰k2 C kHrad ‰k2 C k‰k2 /:

(7.4.23)

We note that 2 1 2 2 2 kHO PF ‰k2 D .p eA / ‰ 2 C kHrad ‰k C Re..p eA / ‰; Hrad ‰/: Since Re..p eA /2 ‰; Hrad ‰/ D

3 X

¹..p eA / ‰; Hrad .p eA / ‰/ C Re..p eA / ‰; eŒHrad ; A ‰/º

D1

3 X

Re..p eA / ‰; eŒHrad ; A ‰/

D1

"k.p eA /2 ‰k2 C4

1 1=2 .kHrad ‰k2 C k‰k2 / 4" 1=2

for any " > 0, by the trivial bound kHrad ‰k2 kHrad ‰k2 C > 0, we further have

1 2 4 k‰k

kHO PF ‰k2 C5 k.p eA /2 ‰k2 C C6 kHrad ‰k2 C7 k‰k2 :

for any (7.4.24)

390


Together with (7.4.23) this gives kp 2 ‰k2 C kHrad ‰k2 C8 .kHO PF ‰k2 C k‰k2 /:

(7.4.25)

Thus .HO PF C z/1 and Hrad .HO PF C z/1 are bounded. Theorem 7.26 (Self-adjointness). Under Assumption 7.1 .1/ HPF is self-adjoint on DPF and essentially self-adjoint on any core of HPF;0 .2/ the functional integral representation Z Rt .F; e tHPF G/ D dxEx Œe 0 V .Bs /ds .J0 F .B0 /; e ieAE .K t / J t G.B t // R3

(7.4.26) holds. Proof. First we assume that ! 3=2 'O 2 L2 .Rd /. Take V D 0. By Lemma 7.25, kHPF;0 ‰k C 0 k.HO PF C z/‰k

(7.4.27)

holds with some C 0 for all ‰ 2 D.HO PF /. Let ¹‰n º1 nD1 D.HPF;0 / be such that O ‰n ! ‰ and HPF ‰n ! ˆ for some ˆ in strong sense as n ! 1. Note that D.HO PF / D.HPF;0 / D DPF . Due to (7.4.27), .HPF;0 ‰n /n is a Cauchy sequence and then ‰ 2 D.HPF;0 / since HPF;0 is closed. In particular, HO PF is thus closed on DPF . This implies that HPF dDPF is closed and HPF is self-adjoint since we already proved in Lemma 7.24 that HPF dDPF is essentially self-adjoint and HO PF D HPF on DPF . Next let V ¤ 0. As mentioned in Remark 7.2, V is relatively bounded with respect to HO 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 ! 3=2 'O 2 L2 .Rd /. We divide 'O into 'O D 'Oƒ C 'Oƒ? ; where 'Oƒ D '1 O ƒ and 'Oƒ? D 'O 'Oƒ with the indicator function 1ƒ on jkj ƒ. Write A D Aƒ C Aƒ? , where Aƒ has the form factor 'Oƒ instead of ', O and Aƒ? does 'Oƒ? . Then HPF can be split off like HPF D HPF .ƒ/ C Hƒ? , where 1 HPF .ƒ/ D .p eAƒ /2 C V C Hrad ; 2

Hƒ? D eAƒ? .p eAƒ / C

e2 2 A ?: 2 ƒ

Since ! 3=2 'Oƒ 2 L2 .Rd /, HPF .ƒ/ is self-adjoint on DPF . We shall show below that Hƒ? is relatively bounded with respect to HPF .ƒ/ with a relative bound strictly

391


smaller than one for the large ƒ. The estimate is similar to the proof of Lemma 7.25. Since for any ı > 0, keAƒ? .p eAƒ /‰k ık.p eAƒ /2 ‰k C we have

kHƒ? ‰k ık.p eAƒ /2 ‰k C e 2

e2 kA 2? ‰k; 2ı ƒ

1 C 1 kAƒ2? ‰k: 2ı

Similarly to the estimate in the proof of Lemma 7.25 we have for any " > 0 and > 0, 2 1 .p eAƒ /2 ‰ C kHrad ‰k2 "k.p eAƒ /2 ‰k2 2 C1 1 2 2 (7.4.28) kHrad ‰k C k‰k kHPF .ƒ/‰k2 ; 4" 4 p p where C1 depends on k ! 'Oƒ k and k'Oƒ = !k, but is independent of k! 3=2 'Oƒ k. In particular, limƒ!1 C1 < 1. From this we have 1 4" C1 k.p eAƒ /2 ‰k2 C 1 C1 k‰k2 : kHrad ‰k2 kHPF .ƒ/‰k2 C 4 4" 16" (7.4.29) Let " be sufficiently small and choose such that 1 k.p eAƒ /2 ‰k2

C1 4"

> 0. Thus we obtain that

4 4C1 kHPF .ƒ/‰k2 C k‰k2 : 1 4" 16".1 4"/

Next we estimate kAƒ2? ‰k. We have kAƒ2? ‰k2 C2 .kHrad ‰k2 C k‰k2 /; where C2 depends on k! ˛=2 'Oƒ? k, ˛ D 2; 1; 0; 1. Thus limƒ!1 C2 D 0 and by (7.4.29), ! C2 2 2 kHPF .ƒ/‰k2 C C3 k‰k2 : kAƒ? ‰k 1 C1 4 Hence we are led to the bounds k.p eAƒ /2 ‰k2 C4 .kHPF .ƒ/‰k2 C k‰k2 / and kAƒ2? ‰k2 C5 .kHPF .ƒ/‰k2 C k‰k2 /, where limƒ!1 C4 < 1 and limƒ!1 C5 D 0. From the bound 1 2 C 1 C5 kHPF .ƒ/‰k C C6 k‰k kHƒ? ‰k ıC4 C e 2ı

392


it follows that for sufficiently large ƒ and small ı such that 1 C 1 C5 < 1; ıC4 C e 2 2ı Hƒ? is relatively bounded with respect to HPF .ƒ/ with a relative bound strictly smaller than 1. Thus HPF is self-adjoint on DPF . The inequality kHPF ‰k C k‰k C 00 .kHPF;0 ‰k C k‰k/

(7.4.30)

can be derived directly without effort with some constant C 00 . Let D be a core of HPF;0 . For any ‰ 2 D.HPF / D D.HPF;0 / there exists a sequence ¹‰n º1 nD1 D such that ‰n ! ‰ and HPF;0 ‰ ! HPF;0 ‰. This implies that .HPF ‰n /n is a Cauchy sequence by (7.4.30). Thus HPF ‰n converges to some ˆ as n ! 1. Since HPF is closed, ˆ D HPF ‰. Thus D is a core of HPF . Since (7.4.26) with e tHPF replaced by O e t HPF is verified, (7.4.26) holds since HO PF D HPF .

7.4.2 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. 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 .K t / appears, therefore intuition would say that the semigroup doe 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 T t D e itx , t 2 R, in L2 .R/ with respect to x. Although T t is not a positivity preserving operator, the operator F T t F 1 , obtained under Fourier transform F on L2 .R/, is a shift operator, i.e. .f; F T t F 1 g/ D .f; g. C t // 0 for non-negative functions f and g. This implies that F T t F 1 is a positivity preserving 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 5, in which we defined the Segal field ˆ.f / and its conjugate ….f /. Since for ˛ 2 R, e i˛N a .f /e i˛N D a .e i˛ f /; e i˛N a.f /e i˛N D a.e i˛ f /


393

in particular for ˛ D =2 we have e i. =2/N a .f /e i. =2/N D i a .f /; e i. =2/N a.f /e i. =2/N D i a.f /: Thus ˆ and … are related by the unitary operator e i. =2/N as e i. =2/N ˆ.f /e i. =2/N D ….f /:

(7.4.31)

(7.4.31) may suggest that e i. =2/N can be regarded as a Fourier transform on Fb . By the argument above we define the unitary operator Sˇ on L2 .Qˇ / by Sˇ D exp i Nˇ ; ˇ D 0; 1; (7.4.32) 2 where Nˇ denotes the number operator in L2 .Qˇ /. We simply write Sˇ for the unitary operator 1˝Sˇ on L2 .R3 /˝L2 .Qˇ / as long as there is no risk of confusion. We have …ˇ .f/ D S1 ˇ Aˇ .f/Sˇ

(7.4.33)

and write …0 D …, …1 D …E , S0 D S and S1 D SE . Clearly, …E coincides with the expression under (7.4.9). It is plain that ŒAˇ .f/; …ˇ .g/ D 2i qˇ .f; g/ and, importantly, e i…ˇ .g/ 1 D e qˇ .g;g/ e Aˇ .g/ 1:

(7.4.34)

Moreover, the Weyl relation e i…ˇ .g/ e iAˇ .f/ D e i2qˇ .f;g/ e iAˇ .f/ e i…ˇ .g/

(7.4.35)

holds. Lemma 7.27. The unitary operator e i….f/ is positivity preserving for any f 2 ˚3 L2real .R3 /. Proof. Let F; G 2 DQ (see (7.4.18)) such that Z Pn FO .k/e i j D1 kj A .fj / d k; F D .2/n=2 Rn Z P i jmD1 kj A .gj / O G D .2/m=2 G.k/e dk Rm

394


with positive F 2 S .Rn / and G 2 S .Rm /. Using the fact that .e iA .g1 / ; e i….f/ e iA .g2 / / D .e iA .g1 g2 / ; e i….f/ /e 2iq.g2 ;f/ ; it is directly seen that .F; e

i….f/

1 G/ D .2/nCm

Z

n X 0 O O d kd k F .k/G.k / exp i q.fj ; f/kj 0

RnCm

j D1

m X 1 0 q.gj ; f/kj exp q.f; f/ G .k; k 0 /; exp i 2 j D1

where n 1 X G .k; k / D exp 2 0

m X

q.ki fi

ki0 0 gi 0 ; kj fj

kj0 0 gj 0 /

:

i;j D1 i 0 ;j 0 D1

Hence .F; e i….f/ G/ 0. Since any nonnegative function can be approximated by functions in DQ , the lemma follows. ieAE .K t / S D e ie…E .K t / . Note the intertwining property J t S D SE J t and S1 E E e Using the functional integral representation (7.3.12), we have Z Rt 1 tHPF .F; S e SG/ D dxEx Œe 0 V .Bs /ds .J0 F .B0 /; e ie…E .K t / J t G.Bs //: R3

(7.4.36) Although Proposition 5.8 shows that e tHrad is positivity preserving, we can show a stronger statement below. Proposition 7.28 (Positivity improving for e tHrad ). The semigroup e tHrad , t > 0, is positivity improving. This proposition implies that Jt J0 is positivity improving. We show furthermore that Jt e ie…E .K t / J0 is positivity improving. The key factorization identity to achieving that is Jt e i…E .f/ J0 ‰ D cAB‰;

(7.4.37)

for ‰ 2 DQ , where c D e .qE .f;f/Cq.j0 f;j0 f// is a constant, A D Jt e AE .f/ J0 and B D e i….j0 f/ e A .j0 f/ . We will see below that there exists AM and BM such that A AM and B BM , moreover, AM is positivity improving and BM positivity preserving, which shows that the closure of cAB is also positivity improving, since cABdDQ cAM BM

395


and BM G 6D 0 if G 6D 0. First we show (7.4.37). Note that J0 G D G.AE .j0 g1 /; : : : ; AE .j0 gm //1: By the commutation relations, e i…E .f/ J0 G1 D G.AE .j0 g1 / C 2qE .j0 g1 ; f/; : : : ; AE .j0 gm / C 2qE .j0 gm ; f//e i…E .f/ 1: From e i…E .f/ 1 D e qE .f;f/ e AE .f/ 1 it then follows that e i…E .f/ J0 G1 D e qE .f;f/ e AE .f/ G.AE .j0 g1 / C 2qE .j0 g1 ; f/; : : : ; AE .j0 gm / C 2qE .j0 gm ; f//1 and from qE .j0 g; f/ D q.g; j0 f/, this is further D e qE .f;f/ e AE .f/ G.AE .j0 g1 / C 2q.g1 ; j0 f/; : : : ; AE .j0 gm / C 2q.gm ; j0 f//1: (7.4.38) Z 2 jkj O k0 ; h.k/ D e itk0 h.k/d jc p t 2 2 R jkj C jk0 j2 The right-hand side of (7.4.38) can be written as

Here

t 2 R:

e qE .f;f/ e AE .f/ J0 e i….j0 f/ Ge i….j0 f/ 1:

Using again e i….f/ 1 D e qE .f;f/ e A .f/ 1 and noting that G commutes with e A .j0 f/ , finally we have

e i…E .f/ J0 G1 D e qE .f;f/q.j0 f;j0 f/ e AE .f/ J0 e i….j0 f/ e A .j0 f/ G1: This gives (7.4.37). From this we have the operator identity

Jt e i…E .f/ J0 D e .qE .f;f/Cq.j0 f;j0 f// .Jt e AE .f/ J0 e i….j0 f/ e A .j0 f/ /dDQ :

Note that e AE .f/ and e A .j0 f/ are unbounded. Define the bounded functions obtained by the truncations ´ e AE .f/ ; e AE .f/ < M; AE .f/ .e /M D M; e AE .f/ M; ´ e A .j0 f/ ; e A .j0 f/ < M; A .j f/ 0 /M D .e M; e A .j0 f/ M:

(7.4.39)

396


Lemma 7.29. For t > 0, Jt .eAE .f/ /M J0 is positivity improving. Proof. It suffices to show that .F; Jt .eAE .f/ /M J0 G/ ¤ 0 for any non-negative F and G, but F 6 0 and G 6 0 since Jt .eAE .f/ /M J0 is positivity preserving. Suppose .F; Jt .eAE .f/ /M J0 G/ D ..e AE .f/ /M J t F; J0 G/ D 0

(7.4.40)

for some non-negative F and G. (7.4.40) implies that E .suppŒ.e AE .f/ /M J t F \ suppŒJ0 G/ D 0:

(7.4.41)

As AE .f/ 2 L2 .QE /, E .¹AE 2 QE j.e AE .f/ /M D 0º/ D 0, (7.4.41) implies that E .suppŒJ t F \ suppŒJ0 G/ D 0 and thus 0 D .J t F; J0 G/ D .F; e tHrad G/: Since e tHrad is positivity improving by Proposition 7.28, this is in contradiction with the assumption. Theorem 7.30 (Positivity improving for e tHPF ). The semigroup S1 e tHPF S is positivity improving for t > 0. Proof. Let f be real valued. Notice that if f g, then Tf T g for any positivity preserving operator T . For a non-negative G 2 DQ ,

Jt e i…E .f/ J0 G e .qE .f;f/Cq.j0 f;j0 f// Jt .e AE .f/ /M J0 e i….j0 f/ .e A .j0 f/ /M G D SM G: Since SM is bounded and any non-negative function can be approximated by a function in DQ , we have for an arbitrary non-negative G, Jt e i…E .f/ J0 G SM G:

(7.4.42)

Since Jt .e AE .f/ /M J0 is positivity improving by Lemma 7.30 and e i….j0 f/ .e A .j0 f/ /M is positivity preserving with e i….j0 f/ .e A .j0 f/ /M G 6 0 for G 0 but G 6 0, SM is positivity improving and so is Jt e i…E .f/ J0 . In particular, S1 Jt S is positivity improving as well. Let F; G 0, not identically vanishing. Note that Z Rt 1 tHPF .F; S e SG/ D dxEx Œe 0 V .Bs /ds .S1 Jt SF .B0 /; G.B t //: R3

(7.4.43) We show that (7.4.43) is strictly positive. Let DG D ¹x 2 R3 jG.x; / 6 0º, DF D ¹x 2 R3 jF .x; / 6 0º and DF G D ¹! 2 X jB0 .!/ 2 DF ; B t .!/ 2 DG º:

397


In order to complete the proof of the theorem it suffices to see that the measure of DF G X is positive as S1 J t S is positivity improving. We have Z

Z x

dxE Œ1DF G D

DF

dxEx Œ1DG .B t /

D .2 t /3=2

Z

Z dx

DF

e jxyj

2 =2t

dy > 0:

DG

Thus it is obtained that the right-hand side of (7.4.43) is strictly positive and the theorem follows. We proved that Se tHPF S1 is positivity improving. A direct consequence of this property is uniqueness of the ground state of HPF . Corollary 7.31 (Uniqueness of ground state). The ground state ‰g of HPF satisfies S1 ‰g > 0, and it is unique up to additive constants. Proof. It follows from Perron–Frobenius theorem (Theorem 3.54) and Theorem 7.30.

By Theorem 7.30 again we have .f ˝ 1; S‰g / 6D 0. This allows to derive an expression of the ground state energy inf Spec.HPF / from which some of its properties follow. Write E.e 2 / D inf Spec.HPF /. Theorem 7.32 (Concavity of ground state energy). The function e 2 7! E.e 2 / is monotonously increasing, continuous and concave. Proof. We know that HPF has a unique ground state ‰g for any value of the coupling constant e 2 R. Since 0 6D .f ˝ 1; S1 ‰g / D .f ˝ 1; ‰g / for any e 2 R and not identically zero f 0, we have 1 tHPF f ˝ 1/ E.e / D lim log.f ˝ 1; e t!1 t Z Rt 1 2 dxEx Œf .B0 /f .B t /e 0 V .Bs /ds e .e =2/qE .K t ;K t / : D lim log t!1 t R3 2

As e .e =2/qE .K t ;K t / is log-convex in e 2 , E./ is concave. Thus E.e 2 / is continuous on .0; 1/. Since E.e 2 / is continuous at e D 0, E 2 C.RC / and E.e 2 / can be R e2 expressed as E.e 2 / D 0 .t /dt with a suitable positive function .t /. This implies that E.e 2 / is monotonously increasing in e 2 . 2

398


7.4.3 Spatial decay of the ground state Similarly to the Nelson model in the previous chapter we can derive the spatial exponential decay of the ground state ‰g of the Pauli–Fierz Hamiltonian. Recall the class V upper defined in Definition 3.26, and let EPF D inf Spec.HPF /. We have the following Carmona-type estimate for the Pauli–Fierz Hamiltonian. Lemma 7.33. If V 2 V upper , then for any t; a > 0 and every 0 < ˛ < 1=2, there exist constants D1 ; D2 ; D3 > 0 such that ˛ a2 t

k‰g .x/kL2 .Q/ D1 e D2 kU kp t e EPF t .D3 e 4

e tW1 C e tWa .x/ /k‰g kHPF ; (7.4.44)

where Wa .x/ D inf¹W .y/jjx yj < aº. Proof. Since ‰g D e tEPF e tHPF ‰g , the functional integral representation yields ‰g .x/ D Ex ŒJ0 e ieAE .K t / J t e

Rt 0

V .Bs /ds

‰g .B t /:

(7.4.45)

k‰g .B t /kL2 .Q/ :

(7.4.46)

From this we directly obtain that k‰g .x/kL2 .Q/ Ex Œe

Rt 0

V .Bs /ds

Then the lemma follows in the similar way to Carmona’s estimate, Lemma 3.59. Remark 7.3.R In the Carmona estimate for the Nelson model the infrared regular con2 =!.k/3 d k < 1 is assumed and e I =4 appears in the upper dition I D R3 j'.k/j O bound of k‰g .x/k, see Lemma 6.15. For the Pauli–Fierz model, however, I < 1 is not needed and does not appear in the bound of k‰g .x/k. Furthermore, the ground state of the Pauli–Fierz Hamiltonian exists even in the case of I D 1. This lemma implies a similar result of decay of the ground state as for Schrödinger operators obtained in Corollaries 3.60 and 3.61. For V D W U 2 V upper , recall that † D lim infjxj!1 V .x/. We only state the results. Corollary 7.34 (Exponential decay of ground state). Let V D W U 2 V upper . .1/ Suppose that W .x/ jxj2n outside a compact set K, for some n > 0 and

> 0. Take 0 < ˛ < 1=2. Then there exists a constant C1 > 0 such that ˛c k‰g .x/kL2 .Q/ C1 exp jxjnC1 k‰g kHPF : (7.4.47) 16 where c D infx2R3 nK Wjxj=2 .x/=jxj2n .

399

Section 7.5 The Pauli–Fierz model with Kato class potential

.2/ Decaying potential: Suppose that † > EPF , † > W1 , and Let 0 < ˇ < 1. Then there exists a constant C2 > 0 such that ˇ .† EPF / k‰g .x/kL2 .Q/ C2 exp p p jxj k‰g kHPF : (7.4.48) 8 2 † W1 .3/ Confining potential: Suppose that limjxj!1 W .x/ D 1. Then there exist constants C; ı > 0 such that k‰g .x/kL2 .Q/ C exp .ıjxj/ k‰g kHPF :

7.5

(7.4.49)

The Pauli–Fierz model with Kato class potential

Now we consider the Pauli–Fierz model with Kato class potential V . 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 Z Rt .F; e tHPF G/ D dxEx Œe 0 V .Bs /ds .J0 F .x/; e ieAE .K t / J t G.B t //: R3

implies .e tHPF G/.x/ D Ex Œe

Rt 0

V .Bs /ds ieAE .K t / J0 e J t G.B t /:

(7.5.1)

Conversely, we shall show that a sufficient condition to define the right-hand side of (7.5.1) is that V is of Kato class. This idea used for Schrödinger operators with Kato class 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 t F / .x/ D Ex Œe

Rt 0

V .Br /dr ieAE .K t / J0 e J t F .B t /:

(7.5.2)

Lemma 7.35. Let V be a Kato decomposable potential. Then T t is bounded on HPF . Proof. Let F 2 HPF . We show that Z Rt 2 kT t F kHPF D dxkEx Œe 0 V .Br /dr J0 e ieAE .K t / J t F .B t /k2L2 .Q/ < 1: (7.5.3) R3

The left-hand side of (7.5.3) is bounded by Schwarz inequality, Z Rt kT t F k2HPF dxE0 Œe 2 0 V .Br Cx/dr E0 ŒkF .B t C x/k2 : R3

Since V is of Kato class we have supx2R3 E0 Œe 2

Rt 0

V .Br Cx/dr

kT t F k2HPF C kF k2HPF :

(7.5.4)

D C < 1, and thus (7.5.5)

400


In what follows we show that ¹T t W t 0º is a symmetric C0 -semigroup. To do that we introduce the time shift operator u t on L2 .R4 / by x D .x0 ; x/ 2 R R3 :

u t f .x/ D f .x0 t; x/;

(7.5.6)

It is straightforward that ut D ut and ut u t D 1. We denote the second quantization of u t denoted by U t D E .u t / which acts on L2 .QE / and is a unitary map. The relationship between js and u t is given next. Lemma 7.36. It follows that u t js D jsCt , for every t; s 2 R. Proof. In the position representation js is given by 1 js f .x/ D p .2/2

Z e

Ci.k0 .x0 s/Ckx/

R4

p !.k/

fO.k/d k: p !.k/2 C jk0 j2

Hence u t js f .x/ D jsCt f .x/. Lemma 7.36 implies the formula U t Js D JsCt :

(7.5.7)

Lemma 7.37. Let V 2 K.R3 /. Then Ts T t D TsCt holds for s; t 0. Proof. By the definition of T t we have Ts T t F D Ex Œe

Rs 0

V .Br /dr ieAE .Ks / J0 e Js EBs Œe

Rt 0

V .Br /dr ieAE .K t / J0 e J t F .B t /:

(7.5.8) D E U and J D U J By the formulae Js J0 D Js Js Us s s t s tCs , (7.5.8) is equal to

Ex Œe

Rs 0

V .Br /dr ieAE .Ks / J0 e Es EBs Œe

Rt 0

V .Br /dr

ieAE .K t / Us e Us J tCs F .B t /: (7.5.9)

e ieAE .K t / U . Since U is unitary, we have Now we compute Us s s

ieAE .K t / Us e Us D e ieAE .us K t /

as an operator. The test function of the exponent us K t is given by us K t D

3 Z M D1 0

t

jrCs '. Q Br /dBr :

401

Section 7.6 Translation invariant Pauli–Fierz model

Moreover by the Markov property of E t , t 2 R, we may neglect Es in (7.5.9), and by the Markov property of .B t / t0 we have Ts T t F D Ex Œe D Ex Œe

Rs 0

V .Br /dr ieAE .Ks / x J0 e E Œe

R sCt 0

R sCt s

V .Br /dr ieAE .KssCt /

e

V .Br /dr ieAE .KsCt / J0 e JsCt F .BsCt /

JsCt F .BsCt /jFs

D TsCt F;

where .F t / t0 denotes the natural filtration of .B t / t0 . Strong continuity of T t on HPF can be checked similarly as previously done, while T0 D 1 is trivial. Theorem 7.38. Let V 2 K.R3 /. Under Assumption 7.1 ¹T t W t 0º is a C0 semigroup. Kato By Theorem 7.38 there exists a self-adjoint operator HPF such that

T t D e tHPF ; Kato

t 0:

(7.5.10)

Definition 7.8 (Pauli–Fierz Hamiltonian for Kato-class potential). We call the selfKato Pauli–Fierz Hamiltonian for Kato-class potential V . adjoint operator HPF Proposition 7.39 (Diamagnetic inequality). Let V 2 K.R3 / and take Assumption 7.1. Then it follows that for F 2 HPF , je tHPF F j e t.K.0/CHrad / jF j; Kato

t 0;

(7.5.11)

where K.0/ is the Schrödinger operator with Kato-class potential. Proof. Since J0 and J t are positivity preserving, we have jJ0 F j J0 jF j, jJ t F j J t jF j and Rt Kato je tHPF F j Ex Œe 0 V .Br /dr J0 J t jF .B t /j: The right-hand side above is just e t.K.0/CHrad / jF j.

7.6

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 then R˚ to decompose HPF as HPF D R3 HPF .p/dp. Moreover, for each p 2 R3 , the fiber Hamiltonian HPF .p/ is self-adjoint on Frad . We will specify the exact form of HPF .p/

402


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 in the spinless case. As said, a standing assumption throughout this section is V D 0: Put Pf D

XZ

(7.6.1)

k a .k; j /a.k; j /d k;

D 1; 2; 3;

(7.6.2)

j D˙

which describes the field momentum. The total momentum operator P on HPF is defined by the sum of the momentum operator for the particle and that of field: P D i rx ˝ 1 C 1 ˝ Pf ;

D 1; 2; 3:

(7.6.3)

p j j ikx = !.k/. Since Recall that % .x/ D '.k/e O .k/e Œi rx ; a .%j ; j / D a .k %j ; j /; ŒPf ; a .%j ; j / D a .k %j ; j /; it follows ŒHPF ; P D 0;

D 1; 2; 3:

(7.6.4)

This leads to a decomposition HPF of the of the total momentum operator P3spectrum 2 Spec.P / D R, D 1; 2; 3. We denote D1 Pf by Pf2 . Definition 7.9 (Pauli–Fierz Hamiltonian with fixed total momentum). The Pauli– Fierz Hamiltonian with a fixed total momentum is 1 HPF .p/ D .p Pf eA.0//2 C Hrad ; 2

p 2 R3 ;

with domain D.HPF .p// D D.Hrad / \ D.Pf2 /; where A .0/ D A .x D 0/ is given by Z 1 X '.k/ O '.k/ O j A .0/ D p e .k/ a .k; j / p C a.k; j / p d k: 2 j D˙ R3 !.k/ !.k/ Here p 2 R3 is called total momentum.

403


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/ D HPF;0 .p/ C HPF;I .p/;

(7.6.5)

with 1 HPF;0 .p/ D Hrad C .p Pf /2 ; 2 HPF;I .p/ D e.p Pf / A.0/ C

(7.6.6) e2 A.0/ A.0/; 2

(7.6.7)

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 T W HPF D L2 .R3x / ˝ Frad ! L2 .Rp3 / ˝ Frad

(7.6.8)

by Z T D .F ˝ 1/

˚ R3

exp.ix Pf /dx;

(7.6.9)

with F denoting Fourier transformation from L2 .R3x / to L2 .Rp3 /. For ‰ 2 HPF , Z 1 .T ‰/.p/ D p e ixp e ixPf ‰.x/dx: 3 3 .2/ R Let M f .x/ D x f .x/, D 1; 2; 3, be the multiplication by x in L2 .R3 /. Write M D .M1 ; M2 ; M3 / and n ˇ °Y ± ˇ 1 D L:H: a .fi ; ji /PF ˇfi 2 C01 .R3 /; ji D ˙; i D 1; : : : ; n; n 1 [¹PF º Frad iD1 1 . We would like to define H .p/ for arbitrary values of O rad and D1 D C01 .R3 /˝F PF e 2 R and p 2 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 D .M ˝ 1 1 ˝ Pf e1 ˝ A.0//2 C 1 ˝ Hrad dD1 : 2

(7.6.10)

We have T HPF ˆ D LT ˆ for ˆ 2 D1 . Since by Theorem 7.26 D1 is a core of HPF , and L is closed, it can be seen that T maps D.HPF / onto D.L/ with T HPF T 1 D L. Note also that D.L/ D T D.HPF / D D..M ˝ 1 1 ˝ Pf /2 / \ D.1 ˝ Hrad /:

404


Theorem 7.40 (Self-adjointness). Under Assumption 7.1 HPF .p/ is a non-negative self-adjoint operator, and Z ˚ Z ˚ T Frad dp T 1 D HPF ; T HPF .p/dp T 1 D HPF : (7.6.11) R3

R3

Proof. Define the quadratic form Qp .‰; ˆ/ D

3 1X 1=2 1=2 ..p Pf eA.0// ‰; .p Pf eA.0// ˆ/ C .Hrad ‰; Hrad ˆ/ 2 D1

T 1=2 on Frad Frad , with form domain 3D1 D.Pf / \ D.A .0// \ D.Hrad /. Since Qp is densely defined and non-negative, there exists a positive self-adjoint operator L.p/ such that Qp .‰; ˆ/ D .L.p/1=2 ‰; L.p/1=2 ˆ/. Note that L.p/ D HPF .p/

(7.6.12)

R˚ on D.Hrad / \ D.Pf2 /. Define LQ D R3 L.p/dp. For ; 2 D1 , we have .T ; Q /. Hence T 1 LT D T 1 LT Q LT / D .T ; LT on D1 and then HPF D 1 Q on D1 . Since D1 is a core of HPF and LQ is self-adjoint, it follows that T T LT Q with maps D.HPF / onto D.L/ Q T HPF T 1 D L:

(7.6.13)

Next we prove self-adjointness of HPF .p/. The operator HPF;0 .p/ is self-adjoint on D.HPF;0 .p// D D.Pf2 / \ D.Hrad / and Z T

˚

R3

HPF;0 .p/dp T 1 D HPF;0 :

Thus we have for F 2 HPF such that .T F /.p/ D f .p/ˆ with f 2 C01 .R3 / and 1 ˆ 2 Frad , by the inequality kHPF;0 F k C k.HPF C 1/F k derived from Theorem 7.26 and the closed graph theorem, and (7.6.13), Z Z 2 2 2 jf .p/j kHPF;0 .p/ˆk dp C jf .p/j2 k.L.p/ C 1/ˆk2 dp: R3

R3

Hence kHPF;0 .p/ˆk C k.L.p/ C 1/ˆk

(7.6.14)

follows for almost every p 2 R3 , since f 2 C01 .Rd / is arbitrary. Since both sides of (7.6.14) are continuous in p, this inequality holds for all p 2 R3 . Thus HPF;0 .p/.L.p/C1/1 is bounded, and so is also HPF;0 .p/e tL.p/ , which implies that

405


e tL.p/ leaves D.Hrad / \ D.Pf2 / invariant. This means that L.p/ is essentially selfadjoint on D.Hrad / \ D.Pf2 / by Lemma 7.19. Moreover, (7.6.14) 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 kL.p/ˆk C 0 .kHPF;0 .p/ˆk C kˆk/;

ˆ 2 D.HPF;0 .p//;

it is clear that L.p/ is essentially self-adjoint on any core of HPF;0 .p/. By (7.6.12) 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. This way HPF .p/ becomes 1 HPF .p/ D .p d .i r/ A .0//2 C d .!.i r// 2

(7.6.15)

on L2 .Q/, where A .0/ D A .x D 0/ D .A1 .'/; Q A2 .'/; Q A3 .'//. Q We also use the same notation Pf and Hrad for d .i r/ and d .!.i r//, respectively, in what follows. The functional integral representation of e tHPF .p/ can be also constructed as an application of that of e tHPF . Theorem 7.41 (Functional integral representation for Pauli–Fierz Hamiltonian with fixed total momentum). Take Assumption 7.1 and let ‰; ˆ 2 L2 .Q/. Then .‰; e tHPF .p/ ˆ/ D E0 Œe ipB t .J0 ‰; e ieAE .K t / J t e iPf B t ˆ/L2 .QE / :

(7.6.16)

Proof. Write Fs D …s ˝ ‰ and Gr D …r ˝ ˆ, where …s .x/ D .2s/3=2 exp.jxj2 =2s/ R˚ is the heat kernel, and ‰; ˆ 2 L2 .Q/. By HPF D T . R3 HPF .p/dp/T 1 , we have .Fs ; e

tHPF i P

e

Z Gr / D

R3

dpe i p ..T Fs /.p/; e tHPF .p/ .T Gr /.p//

for any 2 R3 . Note that lims!0 .T Fs /.p/ D .2/3=2 ‰ strongly in L2 .Q/ for each p 2 R3 . Hence Z 1 lim .Fs ; e tHPF e i P Gr / D p dp.‰; e tHPF .p/ e i p .T Gr /.p//: 3 s!0 3 .2/ R (7.6.17)

406


By using the fact e i P Gr .x/ D …r .x /e iPf ˆ, we obtain by the functional integral representation in Theorem 7.14 that Z tHPF i P .Fs ; e e Gr / D dxEx Œ…s .x/…r .B t /.J0 ‰; e ieAE .K t / J t e iPf ˆ/: R3

Then it follows that by …s .x/ ! ı.x/ as s ! 0, lim .Fs ; e tHPF e i P Gr / D E0 Œ…r .B t /.J0 ‰; e ieAE .K t / J t e i Pf ˆ/:

s!0

(7.6.18) Combining (7.6.17) and (7.6.18) leads to Z 1 dpe i p .‰; e tHPF .p/ .T Gr /.p// p 3 3 .2/ R D E0 Œ…r .B t /.J0 ‰; e ieAE .K t / J t e i Pf ˆ/:

(7.6.19)

Since j.‰; e tHPF .p/ .T Gr /.p//j k.T Gr /.p/k and k.T Gr /./k 2 L2 .R3 /, we have .‰; e tHPF .p/ .T Gr /.p// 2 L2 .Rp3 /. Taking the inverse Fourier transform on both sides of (7.6.19) with respect to p gives .‰; e tHPF .p/ .T Gr /.p//

Z 1 Dp E0 d e ip …r .B t /.J0 ‰; e ieAE .K t / J t e i Pf ˆ/ .2/3 R3

(7.6.20)

for almost every p 2 R3 . Since both sides of (7.6.20) are continuous in p, the equality stays valid for all p 2 R3 . After taking r ! 0 on both sides we arrive at (7.6.16). By Theorem 7.41 the functional integral representation of e tHPF .p/ can formally be written as .‰; e tHPF .p/ ˆ/

Z Rt formal 0 i 0 .pd E .ir˝1/eAE .'.B Q s ///dBs D E J0 ‰e J t ˆdE : QE

Q In the exponent the expression pd E .i r ˝1/eAE .'.B s // appears suggestive of the Euclidean version of p d .i r/ eA .'/. Q Since intuitively e iPf B t is a shift operator on L2 .Q/, it can be expected to be a positivity preserving. Lemma 7.42. Let 2 R3 . Then e iPf is positivity preserving. In particular it follows that je iPf ‰j e iPf j‰j.

407


Proof. Let n=2

F .A .f1 /; : : : ; A .fn // D .2/

G.A .g1 /; : : : ; A .gm // D .2/m=2

Z FO .k/e i

Pn

Rn

Z

i O G.k/e

j D1

Pm

kj A .fj /

j D1

d k;

kj A .gj /

dk

Rm

with 0 F 2 S .Rn / and 0 G 2 S .Rm /. By .e iA .g1 / ; e iPf e iA .g2 / / D .e iA .g1 / ; e iA .g2 .C // /, we see that .F; e iPf G/ D .F .A .f1 /; : : : ; A .fn //; G.A .g1 . C //; : : : ; A .gm . C //// 0: Hence by a limiting argument, .F; e iPf G/ 0 for arbitrary F; G 2 L2 .Q/. An analogue of the diamagnetic inequality for e tHPF can be derived from the functional integral representation for e tHPF .p/ . Corollary 7.43 (Diamagnetic inequality). It follows that 2

j.‰; e tHPF .p/ ˆ/j .j‰j; e t.Pf

CHf /

jˆj/:

(7.6.21)

Proof. This follows from Lemma 7.42 and 2

j.‰; e tHPF .p/ ˆ/j EŒ.J0 j‰j; J t e iB t Pf jˆj D .j‰j; e t.Pf

CHf /

jˆj/:

From this diamagnetic inequality, however, we can only deduce the trivial energy comparison inequality inf Spec.Pf2 C Hf / inf Spec.HPF .p// D 0. However, combining the unitary transformation e i. =2/N with the functional integral representation (7.6.16), we obtain an interesting result. Denote E.p/ D inf Spec.HPF .p//. Corollary 7.44 (Positivity improving for e tHPF .0/ ). Under Assumption 7.1 we have .1/ Let p D 0 and S D e i. =2/N . Then S1 e tHPF .0/ S is positivity improving. .2/ The ground state of HPF .0/ is unique whenever it exists. .3/ E.0/ E.p/ holds. Proof. In the case of p D 0 we remark that e ipB t D 1. Then we have .‰; S1 e tHPF .0/ Sˆ/ D E0 Œ.J0 ‰; e ie…E .K t / J t e iPf B t ˆ/: Since J0 e ie…E .K t / J t is positivity improving and e iPf B t is positivity preserving by Lemma 7.42, (1) follows. (2) is implied by (1) and the Perron–Frobenius theorem. We have j.‰; S1 e tHPF .p/ Sˆ/j E0 Œ.J0 j‰j; e ie…E .K t / J t e iPf B t jˆj/ D .j‰j; S1 e tHPF .0/ Sjˆj/: This yields (3).

408

7.7


Path measure associated with the ground state

7.7.1 Path measures with double stochastic integrals The functional integral method for non-relativistic QED provides a new class of objects for probability theory. An infinite volume Gibbs measure on path space is derived from the Nelson model discussed in Chapter 6. The Pauli–Fierz model also yields a Gibbs 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 Gibbs measures with densities dependent on a double stochastic integral. Let ‰g be the ground state of HPF , which is unique and S1 ‰g > 0 by Corollary 7.31. 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 .‰g ; O‰g / for various operators O can be expressed in terms of averages of the Gibbs measure. Let M be a positive Borel measurable function on R3 . Write M D d .M.i r//

(7.7.1)

and consider O D e ˇ M in the discussion above. The strategy is to construct a sequence ¹‰gt º t>0 such that ‰gt ! ‰g as t ! 1. One candidate is ‰gt D ke tHPF .f ˝ 1/k1 e tHPF .f ˝ 1/

(7.7.2)

with f 0, since f ˝ 1 overlaps with ‰g by Corollary 7.31, i.e., .‰g ; f ˝ 1/ D .S1 ‰g ; f ˝ 1/ > 0: This gives lim .‰gt ; e ˇ M ‰gt / D .‰g ; e ˇ M ‰g /;

t!0

(7.7.3)

i.e., .f ˝ 1; e tHPF e ˇ M e tHPF .f ˝ 1// : t!0 .e tHPF .f ˝ 1/; e tHPF .f ˝ 1//

.‰g ; e ˇ M ‰g / D lim

(7.7.4)

To have a functional integral representation of the right-hand side of (7.7.4) we need to extend the functional integration .F; e tHPF e ˇ M e sHPF G/. It was a key step before that e tHrad could be decomposed as e jtsjHrad D Js J t leading to the functional integral representation of .F; e tHPF G/. Our goal is to extend this idea to M.

409

Section 7.7 Path measure associated with the ground state

Let I t W L2 .R4 / ! L2 .R5 / be defined by p M.k/ e ik1 t Itf .k; k1 / D p p fO.k/; .k; k1 / D .k; k0 ; k1 / 2 R3 R R: M.k/2 C jk1 j2 (7.7.5)

b

Then Is I t D e jtsj.M.ir/˝1/ ;

s; t 2 R;

(7.7.6)

follows. Here M.i r/ ˝ 1 is defined under the identification L2 .R4 // Š L2 .R3 / ˝ L2 .R/. Let .Q2 ; †2 ; 2L / be a probability space and A2 .f/ denote a Gaussian random variable indexed by f 2 3 L2 .R5 / with mean zero and covariance given by Z Z 1 Of.k; k1 / ı ? .k/Og.k; k1 /d kd k1 D q2 .f; g/: A2 .f/A2 .g/d2 D 5 2 Q2 R Define the contraction operator I t W L2 .QE / ! L2 .Q2 / by the second quantization of I t , i.e., I t D 12 .I t /: Then the factorization formula e jtsjd E .M.ir/˝1/ D I t Is ;

s; t 2 R;

(7.7.7)

follows. Theorem 7.45 (Euclidean Green functions). Let 0 D t0 < t1 < < tn D t and 0 D s0 < s1 < < sn D s. Then F;

n Y j D1

Z D

e .tj tj 1 /HPF e .sj sj 1 /M G

dxEx Œe

Rt 0

V .Bs /ds

R3

HPF

.I0 J0 F .B0 /; e ieAE .K t .s0 ;s1 ;:::;sn // Is J t G.B t //L2 .Q2 / ; (7.7.8)

where K t .s0 ; s1 ; : : : ; sn / D

3 X n Z M D1 j D1

tj tj 1

Isj 1 js '. Q Bs /dBs :

Proof. We give an outline of the proof. Let Tj D .tj tj 1 /=Mj . Substitute e

.tj tj 1 /HPF

D s-lim

Mj !1

lim lim Jtj 1 m !1 m !1 1

n

Mj Y

mi .JiTj .PTj =2mi /2 JiTj / J tj :

iD1

(7.7.9)

410


The left-hand side of (7.7.8) can be expressed as the limit of n Y 1 n n T .M1 ; m11 ; : : : ; mM ; : : : ; M ; m ; : : : ; m / D J F; G ; J n t i i t n 0 1 M1 1 iD1

(7.7.10) where j D

j j j .tj 1 ; tj ; m1 ; : : : ; mMj ; Mj /

j D

J tj e .sj sj 1 /M Jtj :

Mj Y D JiTj .P

j m Tj =2 i

iD1

m

/2

j i

JiTj ;

j

In other words, T converges to the left-hand side of (7.7.8) as mi ! 1, for i D 1; : : : ; n, j D 1; : : : ; Mj , Mj ! 1 for j D 1; : : : ; n and n ! 1. Note that J tj e .sj sj 1 /M Jtj D Isj 1 Isj E tj :

(7.7.11)

Inserting this identity on (7.7.10), and using the Markov properties of both J t and Is , we obtain the theorem. Next we construct a functional integral representation for other types of Green functions. Define L21 .Q/ D ¹ˆ.A .f1 /; : : : ; A .fn //jˆ 2 L1 .Rn /; fj 2 ˚3 L2 .R3 /; j D 1; : : : ; n; n 0º: Theorem 7.46 (Euclidean Green functions). Let 0 D t0 < t1 < < tn D t . Supj j pose that Fj D fj ˝ˆj .A .f1 /; : : : ; A .fnj // 2 L1 .R3 /˝L21 .Q/, j D 1; : : : ; n1 and F0 ; Fn 2 HPF . Then n Y e .tj tj 1 /HPF Fj F0 ;

(7.7.12)

j D1

Z D

h Rt n1 Y 0 V .Bs /ds ieAE .K t / FQj .B tj / J t Fn .B t / dxE e J0 F0 .B0 /; e

i

x

R3

j D1

L2 .Q2 /

;

j j where FQj .x/ D fj .x/ˆj .AE .j tj f1 /; : : : ; AE .j tj fnj //.

Proof. First assume that fj and ˆj , j D 1; : : : ; n, are sufficiently smooth functions with a compact support. Note that Js Fj Js D fj ˆj .AE .js f1 /; : : : ; AE .js fjnj // j

(7.7.13)

as a bounded operator. Thus substituting (7.7.9) and (7.7.13) on the left-hand side of (7.7.12), the theorem follows. For general Fj a limiting argument can be used.

411


As an application we show the path integral presentation of .‰g ; e ˇN ‰g /, where N is the boson number operator and ˇ > 0. Here and in what follows we write N for 1 ˝ N for notational convenience. Let .1; T 1/L2 .QE / D hT ivac :

(7.7.14)

Set M D 1 in (7.7.1). Then M D N and let I t W L2 .R4 / ! L2 .R5 / be the family of isometries defined in (7.7.5). Write Z t 3 Z 0 M Y D Y .t; ˇ/ D I0 js '. Q Bs /dBs C Iˇ js '. Q Bs /dBs ;

Z D Z.t / D

D1

t

3 Z M

t

D1 t

0

js '. Q Bs /dBs ;

where and in what follows in this section .B t / t2R denotes two-sided 3-dimensional Brownian motion on R. We note that Bs and B t are independent, for all s; t > 0. Let f 0 be fixed. Using Theorem 7.46 one can compute Rt R x Œf .B /f .B /e t V .Bs /ds he ieAE .Y / i dxE 3 t t vac .‰gt ; e ˇ M ‰gt / D RR : (7.7.15) Rt t V .Bs /ds he ieAE .Z/ i x vac R3 dxE Œf .Bt /f .B t /e Notice that he

ieAE .Y /

ivac

2 e D exp q2 .Y; Y / : 2

As q2 is a bilinear form, the identity q2 .Y; Y / D qE .Z; Z/ C 2.e ˇ 1/qE .Z.t;0/ ; Z.0;t/ /;

(7.7.16)

holds, where Z.S;T / D

3 Z M

T

D1 S

js '. Q Bs /dBs :

(7.7.17)

Definition 7.10. Define the family Z Rt 1 2 t V .xCBs /ds f .x C Bt /f .x C B t /e dx e .e =2/qE .Z;Z/ d W 0 ; d t D 3 Nt R (7.7.18) of probability measures on C.RI R3 /, where we recall that d W 0 is Wiener measure on C.RI R3 / such that W 0 .B0 D 0/ D 1. In Proposition 2.10 WQ 0 appears for W 0 . We use, however, the notation W 0 . In (7.7.18) N t denotes the denominator R of the right-hand side of (7.7.15), regarded as the normalizing constant such that C.RIR3 / d t D 1, and qE .Z; Z/ is independent of x.

412


Corollary 7.47. It follows that .‰g ; e

ˇN

Z ‰g / D lim

t!1 C.RIR3 /

e Ce

2 .1e ˇ /q .Z E .t;0/ ;Z.0;t / /

d t :

Proof. By (7.7.16) and (7.7.15) we have Z 2 ˇ .‰gt ; e ˇN ‰gt / D e e .1e /qE .Z.t;0/ ;Z.0;t / / d t :

(7.7.19)

C.RIR3 /

Since ‰gt ! ‰g as t ! 1, the corollary follows. The measure d t includes qE .Z; Z/ which has the formal expression formal

qE .Z; Z/ D

Z 3 X

t

; D1 t

Z dBs

t t

dBr W .Bs Br ; s r/;

(7.7.20)

where W is given by Z W .X; T / D

R3

? ı .k/

2 j'.k/j O e jT j!.k/ e ikX d k: 2!.k/

(7.7.21)

Moreover formal

qE .Z.t;0/ ; Z.0;t/ / D

Z 3 X ; D1 0

t

Z dBs

0

t

dBr W .Bs Br ; s r/:

(7.7.22)

(7.7.21) 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 (7.7.20), however, is not well defined, since the integrand is not admissible. In the next section we find a suitable expression of the integral.

7.7.2 Expression in terms of iterated stochastic integrals We define the iterated stochastic integral ST by 1 ST D 2

Z R3

2 j'.k/j O !.k/

Z

CT

R3

Z

T

T

2 j'.k/j O dk !.k/

e ikBs dBs ı ? .k/

Z

s T

! e !.k/.sr/ e ikBr dBr d k (7.7.23)

ST is a well-defined expression and we use it to replace (7.7.20). We will refer to (7.7.23) as the double stochastic integral with the diagonal removed.

413


Theorem 7.48 (Iterated stochastic integral). Suppose that 'O is rotation invariant. Then we have he ieA .Z/ ivac D e e

2S

T

:

(7.7.24)

Proof. We replace the time interval ŒT; T with Œ0; 2T by shift invariance, and reset 2T by T for notational convenience. Using the definition of Z and dominated convergence give he ieA .Z/ ivac D lim

n!1

D

n E X AE .jj '. Q Bj // Bj exp i e j D1

vac

;

where we set Bj D Bj T =n B.j 1/T =n and j D .j 1/T =n, j D 1; : : : ; n. Since AE .f/ is Gaussian, it can be computed as 2Z 2 e j'.k/j O dk he ieA .Z/ ivac D lim exp n!1 2 R3 2!.k/ n n XX jj l j!.k/ ik.B j B l / ? e e Bj ı .k/Bl : j D1 lD1

R

Now the integral Z

R3

d k above can be computed as

n X n 2 X j'.k/j O dk e jj l j!.k/ e ik.B j B l / Bj ı ? .k/Bl 3 2!.k/ R j D1 lD1

D

n X

Z Bj

j D1

C

n Z X j D1

dk R3

2 j'.k/j O ı ? .k/ Bj 2!.k/

(7.7.25)

1 jX 2 j'.k/j O j !.k/ ikB j ? dk e e Bj ı .k/ e l !.k/ e ikB l Bl : !.k/ R3 lD1 (7.7.26)

For the diagonal term (7.7.25) we note that Z Z 2 2 j'.k/j O 2 j'.k/j O ? ı .k/d k D ı dk 3 R3 2!.k/ R3 2!.k/ P P by rotation invariance of '. O As n ! 1, 3D1 jnD1 jBj j2 ! 3T almost surely. Thus we find Z Z n 2 2 X j'.k/j O j'.k/j O lim ı ? .k/ Bj D T dk Bj dk n!1 2!.k/ R3 R3 !.k/ j D1

414


almost surely. For the off-diagonal term (7.7.26), we start by noting that by the definition of the Itô integral for locally bounded functions f; g W R R3 ! R, ˇ2

Z t ˇ Z s ˇ ˇ E ds ˇˇf .s; Bs / g.r; Br /dBr ˇˇ 0 0 Z s Z t dsEŒjf .s; Bs /j2 drEŒjg.r; Br /j2 < 1: D 0

0

Hence the stochastic integral of .s/ D f .s; Bs / k 2 R3 , n X

lim

n!1

Z D

0

g.r; Br /dBr exists and for every

f .j ; Bj /Bj ı ? .k/

j D1

0

Rs

T

Z

?

f .s; Bs /dBs ı .k/

Z

j

g.r; Br /dBr 0

s

g.r; Br / dBr

(7.7.27)

0

strongly in L2 .X P ; d W x /. By the independence of increments of Brownian motion and the fact that 3D1 EŒ.Bj /2 D 3=n, EŒBj D 0, we can estimate the difference of (7.7.27) and the off-diagonal term (7.7.26) to get

X Z j 2 j n X E f .j ; Bj /Bj ı ? .k/ g.r; Br / dBr g.l ; Bl /Bl 0

j D1

lD1

ˇ2

ˇ Z j 3 n X ˇ 3 X X ˇˇ j 2 ˇ kf k1 E ˇ g.r; Br / dBr g.l ; Bl /Bl ˇ : n 0 D1 j D1

lD1

(7.7.28) Then the right-hand side above converges to zero as n ! 1. Combining (7.7.27) and (7.7.28) we find that for every k 2 R3 , n X

lim

n!1

Z D

j D1 t

0

j X f .j ; Bj /Bj ı ? .k/ g.l ; Bl /Bl

f .s; Bs /dBs ı ? .k/

lD1

Z

s

g.r; Br / dBr

(7.7.29)

0

strongly in L2 .X I d W x /. By putting f .t; x/ D e ikx e !.k/t , g.t; x/ D e ikx e !.k/t in (7.7.29), it follows that n X

lim

n!1

Z D

j D1 T

0

1 jX e j !.k/ e ikB j Bj ı ? .k/ e l !.k/ e ikB l Bl

e ikBs dBs ı ? .k/

Z 0

lD1 s

e !.k/.sr/ e ikBr dBr :

(7.7.30)

415


Hence

Z

lim (7.7.26) D

n!1

R3

2 j'.k/j O dk !.k/

Z

T

e

ikBs

0

?

dBs ı .k/

Z

s

e !.k/.sr/ e ikBr dBr :

0

Here is a concluding summary. Proposition 7.49. Let T be the family of probability measures on C.RI R3 / defined in Definition 7.10. Then the following identity holds: Z RT 1 2 O T V .Bs Cx/ ds dxf .BT C x/f .BT C x/e dT D e e ST d W 0 ; ZO T R3 where SOT is defined by ST with the diagonal part removed Z Z s 2 Z T 1 j'.k/j O SOT D dk e ikBs dBs ı ? .k/ e !.k/.sr/ e ikBr dBr 2 R3 !.k/ T T R and ZO T denotes the normalizing constant such that C.RIR3 / dT D 1. SOT may be symbolically written as Z Z 3 X O ST D dBs dBr 2 ; D1 ŒT;T n¹sDrº

R3

W .Bs Br ; s r/dsdr:

(7.7.31)

7.7.3 Weak convergence of path measures In this section we show tightness of the family of probability measures ¹ t º t0 on C.RI R3 /. Lemma 7.50. Let f1 ; : : : ; fn1 2 L1 .Rd /, 0 f 2 L2 .R3 /, and T D t0 t1 tn D T , Then the n-point Green function is expressed as .f ˝ 1; e .t1 t0 /HPF .f1 ˝ 1/ .fn1 ˝ 1/e .tn tn1 /HPF f ˝ 1/ .f ˝ 1; e 2TH f ˝ 1/ D ET

h n1 Y

i fj .B tj / :

j D1

Proof. By Theorem 7.46 we can directly see that .f ˝ 1; e .t1 t0 /HPF .f1 ˝ 1/ .fn1 ˝ 1/e .tn tn1 /HPF f ˝ 1/ Z n1 RT h i Y dxEx f .BT /f .BT / fj .B tj / e T V .Bs /ds he ieAE .Z/ ivac : D R3

j D1

By the definition of the measure T the lemma follows.

416


An immediate result of s-lim t!1 ‰gt D ‰g and Lemma 7.50 is as follows. Let ; 1 ; 2 2 L1 .R3 /. Then for t > s, lim ET Œ.B0 / D .‰g ; . ˝ 1/‰g /;

(7.7.32)

T !1

lim ET Œ1 .Bs /2 .B t / D .‰g ; .1 ˝ 1/e .ts/HPF .2 ˝ 1/‰g /e .ts/E.HPF / ; (7.7.33)

T !1

where E.HPF / D inf Spec.HPF / denotes the ground state energy of HPF . Theorem 7.51 (Tightness). Suppose that there exists the ground state of HPF and " !# Z sup Ex exp 4 x2R3

T

V .Bs /ds 0

0. Then there exists a subsequence T 0 such that T 0 has a weak limit as T 0 ! 1. Proof. The proof is a modification of that of Theorem 6.12. By the Prokhorov theorem, it suffices to show (1) and (2) below: .1/ limƒ!1 supT T .jB0 j2 > ƒ/ D 0, .2/ for any " > 0, limı#0 supT T .maxjtsj "/ D 0. We have T .jB0 j2 > ƒ/ D .‰gT ; .1¹jxj2 >ƒº ˝ 1/‰gT /: Note that ‰gT ! ‰g strongly as T ! 1. Set ˛.T / D .‰gT ; .1¹jxj2 >ƒº ˝ 1/‰gT /, and without loss of generality we can assume that T 1. Let " > 0 be given. Since as a bounded operator k1¹jxj2 >ƒº ˝ 1k 1, there exists T > 0 independent of ƒ such that, 8T > T :

(7.7.35)

˛.T / C .‰g ; .1¹jxj2 >ƒº ˝ 1/‰g / C ":

(7.7.36)

˛.T / .‰g ; .1¹jxj2 >ƒº ˝ 1/‰g / C " Thus sup ˛.T / 1T

sup

1T T

We shall estimate sup1T T ˛.T /. Let S D e i. =2/N . Since S1 e THPF S is positivity improving and S1 .f ˝ 1/ D f ˝ 1, we see that ke THPF .f ˝ 1/k D kS1 e THPF S.f ˝ 1/k > 0 for all T 0. Then inf1T T ke THPF .f ˝ 1/k > 0. Denote the left-hand side by c. Since T is independent of ƒ, c is also independent of ƒ. Thus ˛.T / c 2 .e THPF .f ˝ 1/; 1¹jxj2 >ƒº e THPF .f ˝ 1//:

417


By the functional integral representation and Schwarz inequality we have .e THPF .f ˝ 1/; 1¹jxj2 >ƒº e THPF .f ˝ 1// Z R 2T D dxEx Œf .B0 /f .B2T /1¹jxj2 >ƒº .BT /e 0 V .Bs /ds e S2T R3 Z 0 c dx.Ex Œf .B2T /2 /1=2 f .x/.Ex Œ1¹jxj2 >ƒº .BT //1=4 ; where S2T is given by (7.7.23) and c 0 D supx2Rd .Ex Œe 4 Schwarz inequality again we have

R 2T 0

V .Bs /ds /1=4 .

By

.e THPF .f ˝ 1/; 1¹jxj2 >ƒº e THPF .f ˝ 1// Z 1=2 c 0 kf k dx.Ex Œ1¹jxj2 >ƒº .BT //1=2 f .x/2 : R3

Since 1 T T , the bound Z dx.Ex Œ1¹jxj2 >ƒº .BT //1=2 f .x/2 R3

3=4

.2/

Z

Z 2

dxf .x/ R3

dye

jxyj2 =.2T /

R3

1=2 1¹jxj2 >ƒº .y/

is obtained. Denote the right-hand side above by Cf .ƒ/. Thus sup

1T T

˛.T / kf k

c 0 Cf .ƒ/1=2 : c2

(7.7.37)

Since c and c 0 are independent of ƒ and Cf .ƒ/ ! 0 as ƒ ! 1, we obtain c 0 Cf .ƒ/1=2 lim sup ˛.T / lim kf k C .‰g ; 1¹jxj2 >ƒº ‰g / C " D ": ƒ!1 T 1 ƒ!1 c2 Since " is arbitrary, (1) follows. (2) can be proven in the same way as Theorem 6.12. Denote the weak limit of the measure T 0 on C.RI R3 / by 1 . Using the functional integration representation of e tHPF we show in Corollary 7.34 that if V .x/ D nC1 jxj2n , then k‰g .x/kFrad C1 e C2 jxj , while if V .x/ D 1=jxj, then k‰g .x/kFrad C3 e C4 jxj for appropriate constants Cj . Corollary 7.52. Assume that k‰g .x/kFrad C e cjxj for some positive constants C; c and . Then Z e cjB0 j d1 < 1: (7.7.38)

C.RIR3 /

418


Proof. Let

´ e cjxj ; e cjxj m; m .x/ D m; e cjxj > m

be the truncated function of e cjxj . Then .‰g ; .m ˝ 1/‰g / D follows. By a limiting argument as m ! 1 (7.7.38) follows.

R

C.RIR3 / m .B0 /d1

Using this measure 1 the expectation he ˇN ivac can be formally represented as he ˇN ivac D E1 Œe e

2 .1e ˇ /

R0 1

dBs

R1 0

dBr W .Bs Br ;sr/

:

R0 Rt It is, however, not easy to control t dBs 0 dBr W .Bs Br ; s r/ as t ! 1. For the Nelson model the role of the double stochastic integral is taken by a double Riemann integral and then ˇZ 0 Z t ˇ Z ˇ 1 ˇ '.k/ O 2 ˇ ˇ ds drW .B B ; s r/ s r ˇ ˇ 2 3 !.k/3 d k: R t 0 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 .

7.8

Relativistic Pauli–Fierz model

7.8.1 Definition In p quantum mechanics the relativistic Schrödinger operator is defined by HR .a/ D .p a/2 C m2 m C V , and the functional integral representation of e tHR .a/ is given in Chapter 3. 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 q R HPF D .i r ˝ 1 eA/2 C m2 m C V ˝ 1 C 1 ˝ Hrad (7.8.1) on HPF as a self-adjoint operator. R has a difficulty caused by the term containing the A rigorous definition of HPF square root. Although a standard way to define .i r ˝ 1 eA/2 C m2 as a selfadjoint operator is to take the self-adjoint operator associated with the quadratic form F; G 7!

3 X

..i r eA/ F; .i r eA/ G/ C m2 .F; G/;

D1

we proceed instead by finding a core of .i r ˝ 1 eA/2 C m2 by using functional integration. In Section 7.4.1 we defined the self-adjoint operator HO PF .A / by the

419

Section 7.8 Relativistic Pauli–Fierz model

generator of a C0 -semigroup, and have seen that 1 HO PF .A / .i r eA /2 dDPF : 2 Let D D D./ \ C 1 .N /. Lemma 7.53. If ! 3=2 'O 2 L2 .R3 /, then 12 .i r eA /2 dD is essentially self-adjoint. Proof. By (7.3.18) we have .F; e

t HO PF .A /

G/ D

where we recall that L t D O

Z R3

dxEx Œ.F .B0 /; e ieA .L t / G.B t //;

Rt D1 0

L3

(7.8.2)

'. Q Bs /dBs . By using (7.8.2) we show that

e t HPF .A / leaves D invariant. Then the lemma follows. O First, it can be proven that e t HPF .A / D D./ in a similar manner to Lemma O 7.23 with Hrad replaced by N . To see that e t HPF .A / D C 1 .N /, take z 2 N and F; G 2 D.N z /. We have Z O dxEx Œ.N z F .B0 /; e ieA .L t / G.B t //: (7.8.3) .N z F; e t HPF .A / G/ D R3

Note that e ieA .L t / Ne ieA .L t / D N e….L t / C

e2 kL t k2 ; 2

(7.8.4)

where ….f / D i ŒN; A .f /, and thus O

.N z F; e t HPF .A / G/

z Z e2 D dxEx F .B0 /; e ieA .L t / N e….L t / C kL t k2 G.B t / : (7.8.5) 2 R3 By the BDG-type inequality 2z

Z t x E '. Q Bs /dBs 0

L2 .R3 /

.2z/Š z 2z t k'k O L 2 .R3 / ; 2z

and we obtain 2

z Z e2 x 2 2 z 2 dxE N e….L t / C kL t k G.B t / Cz k.N C 1/ Gk 3 2 R (7.8.6) with a constant Cz . A combination of (7.8.5) and (7.8.6) gives O

j.N z F; e t HPF .A / G/j Cz kF kk.N C 1/Gk: O

O

(7.8.7)

This implies e t HPF .A / C 1 .N / C 1 .N / and e t HPF .A / D D follows. Hence HO PF .A / is essentially self-adjoint on D.

420


We keep denoting the self-adjoint extension of HO PF .A /dD by the same symbol q HO PF .A / for simplicity, and 2HO PF .A / C m2 by the spectral resolution of HO PF .A /. Recall that .T t / t0 is the subordinator on a probability space .T ; BT ; / with p E0 Œe uT t D exp.t . 2u C m2 m//: Since

p 2 O O .F; e t. 2HPF .A /Cm m/ G/ D E0 Œ.F; e T t HPF .A / G/;

we immediately have that Lemma 7.54. The functional integral representation Z p 2 O dxEx;0 Œ.F .B0 /; e ieA .LT t / G.BT t // (7.8.8) .F; e t. 2HPF .A /Cm m/ G/ D R3

holds. From here the diamagnetic inequality p p 2 2 O j.F; e t. 2HPF .A /Cm m/ G/j .jF j; e t. Cm m/ jGj/

(7.8.9)

directly follows. In a similar way to Lemma 7.17 we also have p Lemma 7.55. .1/ If V is C m2 m-form bounded with relative bound a, then jV j is also HO PF .A /-form bounded with a relative bound smaller than a. p .2/ If V is relatively bounded with respect to C m2 m with relative bound a, then V is also relatively bounded with respect to HO PF .A / with a relative bound a. Definition 7.11 (Relativistic Pauli–Fierz Hamiltonian). Let ! 3=2 'O 2 L2 .R3 /, m > 0 andpsuppose that V D VC V is such that V is relatively form bounded with respect to C m2 m and D.VC / D./. Then we define the relativistic Pauli–Fierz Hamiltonian by q R P VC P Hrad : HPF D 2HO PF .A / C m2 m C P V C (7.8.10)

7.8.2 Functional integral representation Now we will construct the functional integral representation of e tHPF through the Trotter product formula. Fix t > 0, and let tj D tj=2n , j D 0; : : : ; 2n . Define an L2 .R4 /-valued stochastic process Sn on X T by R

2 Z X n

Sn

D

T tj

j D1 T tj 1

j tj 1 f . Bs /dBs ;

D 1; 2; 3;

(7.8.11)

421


where f 2 L2 .R3 / and S D T tj .

R T tj T tj 1

dBs D

RS T

dBs evaluated at T D T tj 1 and

Lemma 7.56. .Sn /n2N is a Cauchy sequence in L2 .X T ; W x ˝ / ˝ L2 .R4 /.

Proof. For simplicity, write Sn and T .k/ for Sn and Tkt=2nC1 , respectively. First we estimate Ex;0 ŒkSnC1 Sn k2 for each n. It is seen directly that Ex;0 ŒkSnC1 Sn k2 2n Z T .2j / i hX 0 ExW ŒkjT .2j 1/ f . Bs / jT .2j 2/ f . Bs /k2 ds : D E j D1 T .2j 1/

In a similar way to the proof of Lemma 7.11, we have Ex;0 ŒkSnC1 Sn k2 2n Z T .2j / hX i 0 2jT .2j 1/ T .2j 2/jk!f kkf kds E j D1 T .2j 1/

D

E0

2n hX

i 2jT .2j / T .2j 1/jjT .2j 1/ T .2j 2/jk!f kkf k :

j D1

By stationarity and independence of increments of .T t / t0 , we have n

x;0

E

2

ŒkSnC1 Sn k

2 X

2.E0 ŒT t=2nC1 /2 k!f kkf k:

j D1

Note that the distribution of T t is 1 t2 t 2 exp C m r C mt 1Œ0;1/ .r/ 2 r .2 r 3 /1=2 and m > 0. Thus E0 ŒjT t j C t with some C . Then Ex;0 ŒkSnC1 Sn k2

t2 2nC1

C 2 k!f kkf k;

and hence we obtain that .Ex;0 ŒkSm Sn k2 /1=2 .k!f kkf k/1=2

m X j DnC1

This implies that .Sn /n2N is a Cauchy sequence.

t 2j=2

C:

422


RT Definition 7.12. Define the L2 .R4 /-valued random process 0 t j.T 1 /s f .Bs /dBs x on the probability space .X T ; B.X / BT ; W ˝ / by the following strong limit of Sn : Z Tt j.T 1 /s f . Bs /dBs D s-lim Sn ; D 1; 2; 3: (7.8.12) n!1

0

Remark 7.4. The subordinator Œ0; 1/ 3 t 7! T t 2 Œ0; 1/ is monotonously increasing, but it is not injective. Thus the inverse .T 1 / t can not be defined. (7.8.12) is a formal description of the limit of Sn . Theorem 7.57 (Functional integral representation for relativistic Pauli–Fierz Hamiltonian). Let ! 3=2 'O 2 L2 .Rd /. Suppose p that V D VC V is such that V is relatively form bounded with respect to C m2 m and D.VC / D./. Then Z Rt rel R tHPF .F; e G/ D dxEx;0 Œe 0 V .BTs /ds .J0 F .B0 /; e ieAE .K t / J t G.BT t //; R3

(7.8.13) where K trel D

3 Z M D1 0

Tt

j.T 1 /s '. Q Bs /dBs :

(7.8.14)

Proof. We set V D 0 for simplicity. By the Trotter product formula, n /H O

.F; e tHPF G/ D lim .F; .e .t=2 R

n!1

PF .A

/ .t=2n /Hrad 2n

e

/ G/:

By the Markov property of E t D J J t the right-hand side above is equal to lim

n!1

where HO PF .AE tj=2n / D

n

J0 F;

q

.F; e tHPF G/ D lim

Jt G ;

.i r ˝ 1 eAE .j tj=2n '. x//2 C m2 m:

R

Z

n!1

dxEx;0 Œ.J0 F .B0 /; e ieAE .K t .n// J t G.BT t //;

3 X 2 Z M n

K t .n/ D

n /H O PF .AE tj=2n /

e .t=2

j D0

Thus we have

where

2 Y

T tj=2n

D1 j D1 T t .j 1/=2n

jt.j 1/=2n '. Q Bs /dBs :

423


R By Lemma 7.56 and a limiting argument the theorem follows for V D 0. When HPF has a bounded continuous V , the theorem can again be proven by the Trotter formula. Furthermore, it can be extended to V D VC V such that V is relatively form p bounded with respect to C m2 m and D.VC / D./, in a similar manner to Theorem 3.31.

By using this functional integral representation we can obtain similar results to those of HPF . Corollary 7.58 (Diamagnetic inequality). Under the assumptions of Theorem 7.57 R and with E.e/ D inf Spec.HPF / p

j.F; e tHPF G/j .jF j; e t. R

Cm2 mCHrad /

jGj/

(7.8.15)

follows. In particular, E.0/ E.e/. Corollary 7.59 (Positivity improving for e tHPF ). Under the assumptions of TheoR rem 7.57 Se tHPF S1 is positivity improving, where S D e i. =2/N . In particular, R the ground state of HPF is unique. R

Corollary 7.60p(Essential self-adjointness). Suppose that V is relatively bounded with respect to C m2 m with a relative bound strictly smaller p than 1, and asR is essentially self-adjoint on D. / \ D.H /. sume ! 3=2 'O 2 L2 .Rd /. Then HPF rad p R / D. / \ D.H /. It has been proven Proof. Let V D 0. Note that D.HPF rad p R above that D. / \ D.Hrad / is invariant under e tHPF in apsimilar way to LemR mas 7.22 and 7.23. Hence HPF is essentially self-adjoint on D. / \ D.Hrad / for V D 0. By the diamagnetic inequality q p kjV j. .i r eA /2 C m2 mCHrad Cz/1k kjV j. C m2 mCHrad Cz/1k for z withpIm z 6D 0. Therefore, V is also relatively bounded with respect to the operator .i r eA /2 C m2 m C Hrad with a relative bound strictly smaller than 1. Hence the corollary follows.

7.8.3 Translation invariant case In the case of the relativistic Pauli–Fierz Hamiltonian with V D 0, it can be seen R similarly to the non-relativistic translation invariant case that ŒHPF ; P D 0, where R P D i rx ˝ 1 C 1 ˝ Pf . This allows then a self-adjoint operator HPF .p/ such that Z ˚ R R HPF .p/dp D HPF : (7.8.16) R3

424


R The self-adjoint operator HPF .p/, p 2 R3 , is called relativistic Pauli–Fierz Hamiltonian with a fixed total momentum p. We can show similar results to those of HPF .p/ R by using the functional integral representation of e tHPF . We only state the results and leave their proof to the reader.

Theorem 7.61 (Functional integral representation for relativistic Pauli–Fierz model with fixed total momentum). Suppose .1/ and .2/ of Assumption 7.1 hold, and assume ! 3=2 'O 2 L2 .R3 /. Let ‰; ˆ 2 Q. Then .‰; e tHPF .p/ ˆ/ D E0;0 Œe ipBT t .J0 ‰; e ieAE .K t / J t e iPf BT t ˆ/: (7.8.17) T3 Using this we can show that the domain D1 D.Pf / \ D.Hrad / stays invariant rel

R

under e tHPF .p/ . This implies R

Corollary 7.62 (Essential self-adjointness). Under the assumptions of Theorem 7.61 T R HPF .p/ is essentially self-adjoint on 3D1 D.Pf / \ D.Hrad /. R .p//. Denote E.p/ D inf Spec.HPF

Corollary 7.63. Under the assumptions of Theorem 7.61 we have .1/ Diamagnetic inequality: p 2 R 2 j.‰; e tHPF .p/ ˆ/j .j‰j; e t. Pf Cm mCHrad / jˆj/:

(7.8.18)

.2/ Energy comparison inequality: E.0/ E.p/. Corollary 7.64 (Positivity improving for e tHPF .0/ ). Under the assumptions of TheR orem 7.61 and assuming p D 0 the operator S1 e tHPF .0/ S is positivity improving. R In particular, the ground state of HPF .0/ is unique whenever it exists. R

7.9

The Pauli–Fierz model with spin

7.9.1 Definition 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. In this section we choose a specific Q space instead of the abstract setup made previously. This will serve the purpose of drawing sufficient regularity of A with respect to 2 Q. Let S .R3Cˇ /, ˇ D 0; 1; 2; : : : , be the set of real-valued Schwartz test functions on R3Cˇ and put Sˇ D ˚3 Sreal .R3Cˇ /. Put Qˇ D Sˇ0 ;

(7.9.1)

425

Section 7.9 The Pauli–Fierz model with spin

with Sˇ0 denoting the dual space of Sˇ . Also, denote the pairing between elements of Qˇ and Sˇ by hh; fii 2 R, where 2 Qˇ and f 2 Sˇ . By the Bochner–Minlos Theorem there exists a probability space .Qˇ ; †ˇ ; ˇ / such that †ˇ is the smallest -field generated by ¹hh; fii j f 2 Sˇ º and hh; fii is a Gaussian random variable with mean zero and covariance Z hh; fiihh; giidˇ D qˇ .f; g/: (7.9.2) Qˇ

We can extend hh; fii to more general f. For any f D 0 and a constant C" , we know that the spin interaction part 2e B is infinitesimally small with respect to HPF . Then the proposition follows.

7.9.2 Symmetry and polarization In this section we take the Fock representation (7.9.14). When the form factor 'O and the external potential V are rotation invariant, i.e., '.Rk/ O D '.k/ O and V .Rx/ D S V .x/ for any R 2 O.3/, then HPF has the symmetry SU.2/ SOpart .3/ SOfield .3/ helicity;

(7.9.17)

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. Let R 2 SO.3/ and kO D k=jkj. Note that R kO D R k. Thus the orthogonal bases C O in R3 at Rk satisfy ¹e .Rk/; e .Rk/; R kº and ¹Re C .k/; Re .k/; R kº 1 0 C 10 1 0 e .Rk/ Re C .k/ cos #13 sin #13 0 @ e .Rk/ A D @ sin #13 cos #13 0 A @ Re .k/ A ; (7.9.18) O 0 0 1 Rk Rk 3

b

b

b

where 13 is the 33 unit matrix and # D #.R; k/ satisfies cos # D Re C .k/e C .Rk/. Let R D R.n; / 2 SO.3/ be the rotation around n 2 S2 D ¹k 2 R3 jjkj D 1º by

428


the angle 2 R, and det R D 1. Also, let `k D k .i rk / D .`k 1 ; `k 2 ; `k 3 /

(7.9.19)

be the triplet of angular momentum operators in L2 .R3k /. Then (7.9.18) can be rewritten as ! C ! C e .k/ R 0 in`k e .k/ i#†2 De ; (7.9.20) e 0 R e .k/ e .k/ where †2 D i

0 13 13 0

:

(7.9.21)

S we introduce coherent polarization vectors In order to discuss the symmetry of HPF in given directions. We make the following

Definition 7.14 (Coherent polarization vectors). The polarization vectors e ˙ are coherent polarization vectors in direction n 2 S2 whenever there exists z 2 Z such that for any 2 Œ0; 2/ and any k with kO ¤ n, ! C ! e C .Rk/ e .k/ R 0 cos.z/13 sin.z/13 D (7.9.22) 0 R sin.z/13 cos.z/13 e .Rk/ e .k/ or componentwise ! ! C .Rk/ e .Re C .k// cos.z/ sin.z/ D ; .Rk/ sin.z/ cos.z/ .Re .k// e

(7.9.23)

where R D R.n; /. Assumption 7.2. e ˙ are coherent polarization vectors in direction n 2 S2 . By Assumption 7.2 we have ! ! C .k/ e .Re C .k// exp ¹i .z†2 C n `k /º D : .Re .k// e .k/

(7.9.24)

Here is an example of polarization vectors satisfying Assumption 7.2. p Example 7.2. Let n3 D .0; 0; 1/ and Sk2 D ¹. 1 z 2 ; 0; z/ 2 S 2 j 1 z 1º. Take any polarization vectors on Sk2 : e ˙ .k/ for k 2 Sk2 . For k 2 S 2 n Sk2 , there exists

429


a unique hk 2 Sk2 and 0 < < 2 such that R.n3 ; /hk D k. Then we define e ˙ .k/ for k 2 S 2 n Sk2 by ! ! R.n3 ; /e C .hk / e C .k/ cos.z/13 sin.z/13 D : (7.9.25) sin.z/13 cos.z/13 e .k/ R.n3 ; /e .hk / It is readily checked that e ˙ satisfy (7.9.22) with .n3 ; z/ 2 S 2 Z. Example 7.3. Let n 2 S2 , and e C .k/ D kO n= sin #;

e .k/ D kO e C .k/;

(7.9.26)

where # D cos1 .kO n/. Then, since R D R.n; / satisfies that Rn D n and Ru Rv D R.u v/, e ˙ .k/ obeys (7.9.22) with .n; 0/ 2 S2 q Z, i.e., Re ˙ .k/ D e ˙ .Rk/. In particular, when .n; z/ D .n3 ; 0/ 2 S 2 Z, sin # D .k2 ; k1 ; 0/ e C .k/ D q ; k12 C k22

e .k/ D

k12 C k22 and

.k3 k1 ; k2 k3 ; k12 C k22 / : q jkj k12 C k22

Let H2 W L2 .R3 / ˚ L2 .R3 / ! L2 .R3 / ˚ L2 .R3 / be given by 0 1L2 .R3 / H2 D i : 1L2 .R3 / 0

(7.9.27)

(7.9.28)

Under Assumption 7.2 with a suitable z 2 Z we define Sf D d .zH2 /

(7.9.29)

Lf D .Lf;1 ; Lf;2 ; Lf;3 / D d .`k /:

(7.9.30)

and

Sf is called the helicity and Lf the angular momentum of the field. The helicity Sf can be formally written as Z Sf D i z.a .k; /a.k; C/ a .k; C/a.k; //d k: (7.9.31) For .n; z/ 2 S2 Z define Jf D Jf .n; z/ by Jf D n Lf C Sf

(7.9.32)

1 Jp D n `x C n : 2

(7.9.33)

and Jp D Jp .n/ by

430


Jp is the angular momentum plus spin for the particle. Write J D Jp ˝ 1 C 1 ˝ Jf :

(7.9.34)

Clearly, J D J.n; z/ is defined for each .n; z/ 2 S2 Z. S Proposition 7.66 (J-invariance of HPF ). If the polarization vectors are coherent in direction n, and 'O and V are rotation invariant, then S iJ S e iJ HPF e D HPF

for every 2 R.

Proof. Write a] . fg / for a] .f ˚ g/. Notice that for a rotation invariant f , e iJf a f e ikx

C e e

!! e iJf D a f e i.z†2 Cn`k / e ikx

C e e

Since the polarization vectors are coherent, we have by (7.9.24) !! !! 3 C C X iRkx .Re / ] ikR 1 x e D ; R a f e D a fe .Re / e

!! :

(7.9.35)

D1

where R D R.n; / D .R /1; 3 . By (7.9.35), we see that the field part transforms as e iJf Hrad e iJf D Hrad ; e iJf A .x/e iJf D .RA/ .R 1 x/; and the particle part as e in`x x e in`x D .Rx/ ; e in`x .i rx / e in`x D .R.i rx // ; e in.1=2/ e in.1=2/ D .R/ : Together with all the identities above we have 1 S iJ S e iJ HPF e D .R.i r/ eRA .R 1 Rx//2 C Hrad C V .Rx/ D HPF : 2 S From this proposition it is clear that HPF with coherent polarization vectors has the symmetry

SU.2/ SOpart .3/ SOfield .3/ helicity:

(7.9.36)

431


Denote the set of half integers by Z1=2 D ¹w=2jw 2 Zº. For each .n; z/ 2 S2 Z, notice that Spec.n .`x C .1=2/// D Z1=2 ; (7.9.37) Spec.n Lf / D Z; ´ Z; z 6D 0; Spec.Sf / D 0; z D 0:

(7.9.38) (7.9.39)

Thus for each .n; z/ 2 S2 Z, Spec.J/ D Z1=2

(7.9.40)

and we have the theorem below. Theorem 7.67. If the polarization vectors are coherent in direction n, and 'O and V S are rotation invariant, then HS and HPF can be decomposed as M M S S HS D HS .w/; HPF D HPF .w/: (7.9.41) w2Z1=2

w2Z1=2

Here HS .w/ is the subspace spanned by eigenvectors of J associated with eigenvalue S S w 2 Z1=2 and HPF .w/ D HPF dHS .w/ . Proof. This follows from Proposition 7.66 and the fact that Spec.J/ D 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 7.3 and Theorem 7.67, we have the corollary below. Corollary 7.68 (Fiber decomposition of Pauli–Fierz Hamiltonian with spin 1=2; general polarization). Suppose that 'O and V are rotation invariant, and e ˙ are coherent S ˙ polarization vectors. polarization vectors. Then HPF .˙ / is uniL Let beSarbitrary tary equivalent to w2Z1=2 HPF .e ˙ ; w/. By using the symmetries of the Pauli–Fierz Hamiltonian we can show the degenerS acy of ground states of HPF . Assume that V is rotation invariant and the polarization ˙ vectors e are given by .k2 ; k1 ; 0/ e C .k/ D q ; e .k/ D kO e C .k/: (7.9.42) k12 C k22 These are coherent in direction n3 and their helicity is zero. Let ƒ W R3 ! R3 be the flip defined by 1 0 1 0 k1 k1 ƒ @k2 A D @k2 A: k3 k3

432


Consider the unitary operators L2 .R3 I C 2 / ! L2 .R3 I C 2 / f uQ W g

!

! f ıƒ 7 ! ; gıƒ

f uW g

!

! f ı ƒ 7! : gıƒ

(7.9.43)

A computation gives ] 1

u

´

k ; D 1; 3; k u] D k ; D 2;

] 1

u

´ r ; D 1; 3; r u] D r ; D 2;

(7.9.44)

where u] D u or u. Q From this and e .k/ D

.k3 k1 ; k2 k3 ; k12 C k22 / ; q jkj k12 C k22

we have for rotation invariant f and g, u1

Cf e g e

!

8 C < ef ; D 1; 3; e g D :eCf ; D 2: e g

(7.9.45)

Then the second quantization .u/ of u induces the unitary operator on Frad and for rotation invariant 'O we obtain 1

.u/

´ A .ƒx/; A .x/.u/ D A .ƒx/;

D 1; 3; D 2;

(7.9.46)

and .u/1 Hrad .u/ D Hrad :

(7.9.47)

Next we consider the transformation on D .1 ; 2 ; 3 / given by ´ ; D 1; 3; 7 2 2 D ! ; D 2:

(7.9.48)

Under the identification L2 .R3 I C 2 / Š C 2 ˝ L2 .R3 /, we define D 2 ˝ uQ W L2 .R3 I C 2 / ! L2 .R3 I C 2 /. This satisfies ´ 1 . ˝ f / D

˝ f ı ƒ1 ; D 1; 3; ˝ f ı ƒ1 ; D 2;

(7.9.49)

433


and 1 . ˝ r / D ˝ r ;

D 1; 2; 3:

(7.9.50)

We finally define the unitary operator J W HS ! HS by J D ˝ .u/:

(7.9.51)

Combining (7.9.46)–(7.9.50), for rotation invariant 'O and V J 1 .i r A /J D .i r A /; J

1

J

Hrad J D Hrad ; 1

VJ DV

(7.9.52) (7.9.53) (7.9.54)

are obtained. From these relations we can show the theorem below. S Theorem 7.69 (Reflection symmetry of HPF ). If 'O and V are rotation invariant, and S S ˙ the polarization vectors e are given by (7.9.42), 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 D .`x;3 C 12 3 / ˝ 1 C 1 ˝ Lf;3 . It follows that J 1 JJ D J:

(7.9.55)

This implies that J maps HS .w/ onto HS .w/. Furthermore, S J 1 HPF J D

1 S . .i r eA//2 C V C Hrad D HPF : 2

S S Thus J 1 HPF .w/J D HPF .w/ follows.

An interesting application of Theorem 7.69 is to estimate the multiplicity of bound S states of HPF . Corollary 7.70 (Degeneracy of bound states). Suppose that V and 'O is rotation inS . Then M is an even number. variant. Let M be the multiplicity of eigenvalues of HPF In particular, whenever a ground states exists, it is degenerate. Proof. By the unitary equivalence we may the polarization vectors of L suppose that S S S HPF are given by (7.9.42). Thus HPF D w2Z1=2 HPF .w/. Let ‰ be a bound state S S of HPF with eigenvalue a. Thus ‰ is a bound state of HPF .w/ with some w 2 Z1=2 , S S S HPF .w/‰ D a‰. Theorem 7.69 implies HPF .w/ Š HPF .w/, and thus there exists S ˆ 2 HS .w/ such that HPF .w/ˆ D aˆ. Thus the multiplicity of a is even.

434


7.9.3 Functional integral representation As in the classical case investigated in Chapter 3, in order to construct the functional S integral representation of .F; e tHPF G/ with a scalar integrand we introduce a twoS valued spin variable 2 Z2 D ¹1; C1º D ¹1 ; 2 º and redefine HPF in order to reduce it to a scalar operator. Since 1 e B1 i B2 B3 S 2 HPF D .i r eA / C V C Hrad ; B3 2 2 B1 C i B2 our Hamiltonian can be regarded as an operator acting on L2 .R3 Z2 / ˝ L2 .Q/ and is defined by 1 Z2 2 .i r eA / C V C Hrad C Hd . / F . / C Hod . /F . / .HPF F /. / D 2 (7.9.56) F .C1/ with 2 Z2 for F D F .1/ 2 L2 .R3 Z2 / ˝ L2 .Q/, where each component is F .˙/ 2 L2 .R3 / L2 .Q/. Here Hd and Hod denote the diagonal resp. off-diagonal parts of the spin interaction explicitly given by Z ˚ Z ˚ Hd . / D Hd .x; /dx; Hod . / D Hod .x; /dx; (7.9.57) R3

R3

where e Hd .x; / D B3 .x/; 2

e Hod .x; / D .B1 .x/ i B2 .x// : 2

(7.9.58)

Here B .x/ is defined in (7.9.8). Now we define the Pauli–Fierz Hamiltonian with spin 1=2 on L2 .R3 Z2 / ˝ L2 .Q/ instead of HS . Furthermore, we treat A and B or Hd and Hod as not necessarily dependent vectors. This is the same idea as the one applied to HZ2 .a; b/ discussed in Chapter 3. Write K D L2 .R3 Z2 / ˝ L2 .Q/

(7.9.59)

KE D L2 .R3 Z2 / ˝ L2 .QE /:

(7.9.60)

and its Euclidean version

Definition 7.15 (Pauli–Fierz Hamiltonian with spin 1=2 on K ). We define the Pauli– Z2 Fierz Hamiltonian with spin 1=2 on K by HPF . Z2

The key idea of constructing a functional integral representation of e tHPF is to use the identity Z ˚ K Š L2 .R3 Z2 /d: (7.9.61) Q

435


In other words, we regard K as the set of L2 .R3 Z2 /-valued L2 functions on Q. We make the decomposition Z ˚ Z2 HPF D H./d C Hrad ; Q

where H./ is a self-adjoint operator on L2 .R3 Z2 /. We construct a functional inteZ2

gral representation of e tHPF through the functional integrals of e tH./ and e tHrad , and the Trotter product formula. In order to do that we will use the identity Z R˚ t Q H./d .F; e G/ D F ./; e tH./ G./ d Q

while we have already done this of .F ./; e tH./ G.//, 2 Q, in Chapter 3. First we make the fiber decomposition of the quantized radiation field A and the quantized magnetic field B on Q. For each 2 Q define Z ˚ Z ˚ A ./ D A .x; /dx; B ./ D B .x; /dx; R3

R3

where A .x; / D A .'. Q x/; /;

B .x; / D .rx A .'. Q x/; // :

(7.9.62)

Recall that A .'. Q x/; / D hh; ˚3 D1 ı '. Q x/ii. For each fiber 2 Q, define 2 3 the Hamiltonian H./ on L .R Z2 / by 1 2 .i r eA .// C V C Hd ./ F .x; / C Hod ./F .x; /: .H./F /.x; / D 2 (7.9.63) Here Hd ./ D

Z

˚ R3

Z Hd .x; ; /dx;

Hod ./ D

˚

R3

Hod .x; ; /dx;

(7.9.64)

where e Hd .x; ; / D B3 .x; /; 2

e Hod .x; ; / D .B1 .x; / i B2 .x; //: 2 (7.9.65)

Z2 in a similar manner To prevent the off-diagonal part Hod vanish we introduce HPF;" " as in HZ2 .a; b/ in Chapter 3 by 1 Z2 2 .HPF;" F /. / D .i r eA / C V C Hrad C Hd . / F . / 2

C ‰" .Hod . //F . /;

(7.9.66)

436


where ‰" .X / D X C" " .X / and " is the indicator function given by (3.7.30). Also, let H" ./ be the counterpart of H./ with Hod ./ replaced by ‰" .Hod .//. Recall that ˛ D .1/˛ and .q t / t0 D .B t ; N t / t0 is an .R3 Z2 /-valued stochastic process on X S; see Section 3.7.2. Lemma 7.71. If 'Q 2 C01 .R3 /, then for every 2 Q, H" ./ is self-adjoint on D..1=2// ˝ Z2 and for g 2 L2 .R3 Z2 /, .e tH" ./ g/.x; ˛ / D e t Ex;˛ Œe

Rt 0

V .Bs /ds Z t .;"/

e

g.q t /;

where Z Z t .; "/ D i e Z

0

t 0

A .'. Q Bs /; / dBs Z

t

Hd .Bs ; Ns ; /ds C

tC 0

W"; .Bs ; Ns /dNs

and W"; D log .‰" .Hod .x; ; /// : Q x/ii 2 Cb1 .R3x /, for Proof. As 'Q 2 C01 .R3 /, we have A ./ D hh; ˚3 D1 ı '. every 2 Q. Thus H" ./ is the Pauli operator with spin 1=2 with a smooth bounded vector potential A ./, and the off-diagonal part is perturbed by the bounded operator " " .Hod .//. Hence it is self-adjoint on D..1=2//˝Z2 and its functional integral representation follows by Theorem 3.86. Next we define the operator K" .A / on K through H" ./ and the constant fiber direct integral representation (7.9.61) of K . Take 'Q 2 C01 .R3 / and define the selfadjoint operator K" .A / on K by Z ˚ K" .A / D H" ./d; Q

that is, .K" .A /F /./ D H" ./F ./ with domain ˇZ ² ³ ˇ 2 ˇ D.K" .A // D F 2 K ˇ kH" ./F ./kL2 .R3 Z2 / d < 1 : Q

Then we are able to define the self-adjoint operator K" by P Hrad : K" D K" .A / C

(7.9.67)

In what follows we construct the functional integral representation of e tK" and show that Z2

e tK" D e tHPF;" :

(7.9.68)

437


Let L2fin .Qˇ / D by

S1

Lm

mD0 ¹

2 nD0 Ln .Qˇ /

K

1

L1

nDmC1 ¹0ºº and define the dense subspace

O L2fin .Q/ D C01 .R3 Z2 / ˝

(7.9.69)

and its Euclidean version by O L2fin .QE /: KE1 D C01 .R3 Z2 / ˝ Z2 Let 'Q 2 C01 .R3 /. It is seen that K" D HPF;" on K

as a self-adjoint operator since K K

1

as " ! 0 and K

1

1

1,

(7.9.70)

Z2 implying that K" D HPF;"

Z2 Z2 is a core of HPF;" . Moreover, HPF;" ! HPF on

Z2 is a common core of the sequence ¹HPF;" º"0 . Thus Z2

s-lim e tHPF;" D e tHPF ; "!0

whence Z2

.F; e tHPF G/ D lim.F; e tK" G/ "#0

(7.9.71)

follows. By (7.9.71) it suffices to construct a functional integral representation for the expressions at its right-hand side of (7.9.71) and then use a limiting procedure. Define the Euclidean version of Hd .x; / and Hod .x; / by e HE;d .x; ; s/ D BE3 .js '. Q x// ; 2 e HE;od .x; ; s/ D .BE1 .js '. Q x// i BE2 .js '. Q x///; 2

(7.9.72) (7.9.73)

where the Euclidean version of the quantized magnetic field is defined by Q x// D rx AE .˚3 D1 js '. Q x//: BE .js '.

(7.9.74)

To obtain a functional integral representation of e tK , we apply the Trotter product formula, i.e., e tK" D s-lim .e .t=n/K .A / e .t=n/Hrad /n : n!1

By the Markov property of Es D Js Js , s 2 R, it is reduced to s-lim J0

n!1

n1 Y

J ti=n e .t=n/K" .A / Jti=n J t :

iD0

From this we can construct a functional integral representation of e tK" . Before going to construct to a functional integral representation of e tK" , we need a functional integral representation of Js e tK" .A / Js .

438


Lemma 7.72. Let 'Q 2 C01 .R3 /, F; G 2 KE1 , F 2 EŒa;b and s … Œa; b. Suppose that VM D supx2R3 Ex Œe 2 .F; Js e tK" .A / Js G/ D e t

Rt 0

V .Bs /ds

X Z

3 ˛D1;2 R

< 1. Then

dx Ex;˛ Œe

Rt 0

V .Br /dr

.F .q0 /; e X t .";s/ Es G.q t //: (7.9.75)

Here Z X t ."; s/ D i e

0

Z

t

AE .js '. Q Br // dBr Z

t 0

HE;d .Br ; Nr ; s/dr C

tC 0

WE;" .Br ; Nr ; s/dNr ;

(7.9.76)

and WE;" .x; ; s/ D log. ‰" .HE;od .x; ; s///:

(7.9.77)

Proof. Notice that the right-hand side of (7.9.75) is bounded. Since F .x; / D Jl Jl F .x; / 2 Ran Jl for some l 2 Œa; b and Es G.B t ; N t / D Js Js G.B t ; N t / 2 Ran Js with l ¤ s, we obtain that F Es G e X t .";s/ 2 L1 .QE / and kF Es G e X t .";s/ k1 kF k2 kEs Gk2 ke X t .";s/ k1 by Corollary 7.8. Then it follows that

R t t 0;0 0 V .Br Cx/dr e jr:h:s: (7.9.75)j e E

X Z 3 ˛D1;2 R

dx kF .x; ˛ /k2 kG.B t C x; ˛CN t /k2 ke

X t .";s/

k1 :

(7.9.78) We will prove in Lemma 7.73 below that there exists a random variable Y D Y . /, 2 S, such that (1) ke X t .";s/ k21 Y , (2) Y is independent of .x; / 2 R3 Z2 , (3) Y is independent of B t , D 1; 2; 3, and (4) E0;0 ŒY 1=2 < 1. By (7.9.78), 1=2

jr:h:s: (7.9.75)j kGkkF kVM E0;0 ŒY 1=2 < 1;

(7.9.79)

where we used properties (1)–(4) above. Next we prove (7.9.75). Note that K" .A / is defined by a direct integral representation and then Z tK" .A / .Js F; e Js G/ D d..Js F /./; e tH" ./ .Js G/.//L2 .R3 Z2 / : Q

439


Thus by Lemma 7.71 we have .Js F; e tK" .A / Js G/

R Z X Z t dx Ex;˛ e 0 V .Br /dr d.Js F /.; q0 /e Zt .;"/ .Js G/.; q t / : D et 3 ˛D1;2 R

Q

Here we used Fubini’s theorem. Put Z t Z t Z t ."/ D i e A .'. Q Bs // dBs Hd .Bs ; Ns /ds 0 tC

Z C

0

W" .Bs ; Ns /dNs ;

0

with W" .x; / D log.‰" .Hod .x; ///. Pick F; G 2 KE1 . Given that Js F 2 L2 .Q/ and e Zt ."/ Js G.B t ; N t / 2 L2 .Q/, we rewrite as .Js F; e tK" .A / Js G/ Rt X Z t De dx Ex;˛ Œe 0 V .Br /dr .F .q0 /; Js e Zt ."/ Js G.q t //L2 .QE / : 3 ˛D1;2 R

We can directly see that Js e Zt ."/ Js D e X t .;s/ Es , leading to (7.9.75) for F; G 2 KE1 . By a limiting argument and the bound (7.9.79) the proof can be completed. Lemma 7.73. There exists a random variable Y D Y . / satisfying .1/–.4/ in the proof of Lemma 7.72. Proof. Note that ke X t .";s/ k21 ke

Rt 0

HE;d .Br ;Nr ;s/dr 2 k2 ke

Rt 0

jWE;" .Br ;Nr ;s/jdNr 2 k2 :

We estimate the right-hand side of this expression. Since BE .f / defined in (7.9.11) is linear in test function f , it follows that Z t Z e t HE;d .Br ; Nr ; s/dr D BE3 Nr js '. Q Br /dr ; 2 0 0 BE .f / is a Gaussian random variable with mean zero and covariance Z Z 1 2 fO.k/g.k/jkj O BE .f /BE .g/dE D ı .k/d k; 4 2 QE R

(7.9.80)

we have ke

Rt 0

HE;d .Br ;Nr ;s/dr 2 k2

c1 < 1;

(7.9.81)

440

Chapter 7 The Pauli–Fierz model by path measures 2

R

2 j'.k/j O 2 3 R3 !.k/ jkj d k/ and c1 is thus independent of .x; / 2 R Rt ke 0 jW" .Br ;Nr ;s/jdNr k22 . Write BE .t / D BE .j t '. Q B t //

where c1 D exp. e4 t 2

Z2 . Next consider for notational convenience. For each 2 S, there exists N D N. / 2 N and points of discontinuity of r 7! Nr . /, s1 D s1 . /; : : : ; sN D sN . / 2 .0; 1/, such that Z

t

0

jWE;" .Br ; Nr ; s/jdNr D

N X

WE;" .Bsj ; Nsj ; s/:

j D1

Since j‰" .w/j jwj C ", we have Rt

ke

0

jWE;" .Br ;Nr ;s/jdNr 2 k2

X

N q jej 1; exp 2 log p BE1 .si /2 C BE2 .si /2 C "2 1 2 2 iD1 where we used that ja C i bj C " hand side is computed as Rt

ke

0


p p 2 a2 C b 2 C "2 , a; b; " 2 R. Then the right-

jej p 2

2N X N

"2.N m/

mD0

X BE# BE# 1 2 ; „ ƒ‚ … 2

combm

m-fold

P where combm denotes summation over the 2m terms in the expansion of the product Qm 2 2 iD1 .BE1 .si / CBE2 .si / /, BE# denotes one of BE .si /, D 1; 2, i D 1; : : : ; N . Thus we obtain Rt

ke jej 2N where c2 . / D . p /

0


PN

2

is independent of .x; / 2

p

c2 . /;

2.N m/ 2m . 2/2m mŠ k mD0 " R3 Z2 and B t . Write

Y . / D c1 c2 . /:

(7.9.82)

p jkj'k O 2m . Note that c2 . /

(7.9.83)

Then 0;0

E

ŒY

1=2

e

1 e2 2 2 4 t k

1 p X jkj'k O 2 N D0

jej p 2

p p N X N "N m mŠ2m k jkj'k O m t e < 1: NŠ mD0

(7.9.84) This completes the proof of claims (1)–(4) above.

441


Theorem 7.74 (Functional integral representation for Pauli–Fierz Hamiltonian with Rt spin 1=2). Let VM D supx2R3 Ex Œe 2 0 V .Bs /ds < 1. Then for every t 0 and all F; G 2 K it follows that Z2

Z2

.F; e tHPF G/ D lim.F; e tHPF;" G/ "#0

and

X Z

Z2

.F; e tHPF;" G/ D e t

˛D1;2

R3

dx Ex;˛ Œe

Rt 0

V .Bs /ds

(7.9.85)

.J0 F .q0 /; e X t ."/ J t G.q t //: (7.9.86)

Here the exponent X t ."/ is given by Z t X t ."/ D i e AE .js '. Q Bs // dBs Z

0

0

(7.9.87)

Z

t

HE;d .Bs ; Ns ; s/ds C

or e X t ."/ D i eAE .K t / C BE3 .M t / C 2

tC

log.‰" .HE;od .Bs ; Ns ; s///dNs ;

0

Z

tC 0

log.‰" .HE;od .Bs ; Ns ; s///dNs ; (7.9.88)

Rt

L3 where K t D D1 Rt j '. Q B s /ds. 0 Ns s

Q 0 js '.

Bs /dBs , BE .f / is defined in (7.9.11) and M t D

Proof. Notice that BE .js f / is a Gaussian random variable with mean zero and covariance given by Z Z 1 2 BE .js f /BE .j t g/dE D ı .k/e jtsj!.k/ d k: fO.k/g.k/jkj O 2 R3 QE Then similarly to (7.9.79) we obtain 1=2

jr:h:s: (7.9.86)j kF kkGke t VM

X Z

dxEx;˛ ŒY 1=2 < C;

R3

˛D1;2

where Y is a random variable given by (7.9.83) and C is a constant independent of ". Z2

Since e tHPF;" ! e tHPF strongly as " ! 0, (7.9.85) follows. Write Z T XS;T ."; s/ D i e AE .js '. Q Br // dBr Z

S T

S

Z HE;d .Br ; Nr ; s/dr C

TC S

WE;" .Br ; Nr ; s/dNr :

442


Here WE;" .x; ; r/ D log.‰" .HE;od .x; ; s///. Now we turn to proving (7.9.86). " W K 1 ! K 1 by Take 'Q 2 C01 .R3 / and define S t;s E E " .S t;s G/.x; ˛ / D e t Ex;˛ Œe

Rt 0

V .Br /dr X0;t .";s/

e

G.q t /:

Then kEx;˛ Œe

Rt 0

V .Br /dr X0;t .;s/

e

1=2

G.q t /kK VM Ex;˛ ŒY 1=2 kGkK :

" has the property: It can be seen that S t;s 0

" S t"0 ;s 0 G.x; ˛ / D e tCt Ex;˛ Œe S t;s

R t Ct 0 0

V .Br /dr X0;t .";s/CX t;t Ct 0 .";s 0 /

e

G.q tCt 0 /: (7.9.89)

Note that for s1 sn , e X0;t1 .";s1 /CX t1 ;t1 Ct2 .";s2 /CCX t1 CCtn1 ;t1 CCtn .";sn / 2 EŒs1 ;sn

(7.9.90)

Z2

Since .F; e tHPF;" G/ D .F; e tK" G/, similarly to the proof of Theorem 5.20, by the Trotter product formula, (7.9.89), (7.9.90) and the Markov property of Es , s 2 R, we obtain that Z2

.F; e tHPF;" G/ n1 Y " S t=n;it=n D lim J0 F; Jt G n!1

D e t lim

n!1

iD0

X Z

3 ˛D1;2 R

R Z t dx Ex;˛ e 0 V .Br /dr

QE

n dE J0 F .x; /e X t ."/ J t G.q t / ; (7.9.91)

where the exponent X tn ."/ consists of three parts: X tn ."/ D Y tn .1/CY tn .2/CY tn .3; "/ and 0 1 3 Z t M Y tn .1/ D i eAE @ jn .s/ '. Q Bs /dBs A ; D1 0

Z Y tn .2/ D Z Y tn .3; "/ D

0

t

0

HE;d .Bs ; Ns ; n .s//ds;

t

WE;" .Bs ; Ns ; n .s//dNs :

443


Here n .s/ is the step function given by n X t .i 1/ 1.t.i1/=n;ti=n .s/: n .s/ D n iD1

We claim that r:h:s: (7.9.91) D e

t

X Z 3 ˛D1;2 R

dx E

x;˛

R Z t 0 V .Bs /ds e QE

dE J0 F .q0 /e

X t ."/

J t G.q t / (7.9.92)

by a limiting argument. The proof is thus complete. It only remains to show (7.9.92). Note that

R Z X Z t n dx Ex;˛ e 0 V .Bs /ds jJ0 F .q0 /j jJ t G.q t /j je X t ."/ e X t ."/ jdE 3 ˛D1;2 R

QE

x;˛

kGk E

X Z

dx e

2

Rt 0

V .Bs /ds

3 ˛D1;2 R

n kF .x; ˛ /k22 ke X t ."/

1=2

e X t ."/ k21 (7.9.93)

and by Schwarz inequality it follows that n

n

ke X t ."/ k21 .1; je Y t

.2/ 2

n

j 1/.1; je Y t

.3;"/ 2

j 1/:

We estimate the right-hand side above. It readily follows that 2 Z e 2 n 2 .1; e 2Y t .2/ 1/ exp t j'.k/j O jkjd k D c1 ; 4 R3

(7.9.94)

and in the similar way as in (7.9.82), for each 2 S, n

.1; je Y t

.3;"/ 2

j 1/ c2 . /

(7.9.95)

is shown with c2 . / given in (7.9.82). Thus we conclude that n

ke X t ."/ k21 < c. /;

(7.9.96)

where c. / D c1 c2 . / and Ex;˛ Œc 1=2 < 1. Similarly, ke X t ."/ k1 < C. /

(7.9.97)

and Ex;˛ ŒC 1=2 < 1 follows for a random variable C. /. Note that both c and C are independent of .x; / 2 R3 Z2 , B t and n. Thus by (7.9.93) and the Lebesgue

444


dominated convergence theorem, it suffices to show that for almost every 2 S, n e X t ."/ ! e X t ."/ as n ! 1 in L1 .QE /. Put 0 1 3 Z t M Y t .1/ D i eAE @ js '. Q Bs /dBs A ; D1 0

Z Y t .2/ D

t

Z Y t .3; "/ D

HE;d .Bs ; Ns ; s/ds;

0

tC

WE;" .Bs ; Ns ; s/dNs :

0

Then X t ."/ D Y t .1/ C Y t .2/ C Y t .3; "/. It follows directly n

n

e X t ."/ e X t ."/ D e„Y t

.1/ Y tn .2/ Y tn .3;"/

e

n

e Y t .1/ e Y t ƒ‚

e

.2/ Y tn .3;"/

e

…

DI

n

C e„Y t .1/ e Y t

.2/ Y tn .3;"/

n

e Y t .1/ e Y t .2/ e Y t ƒ‚

e

.3;"/

DII

n

C e„Y t .1/ e Y t .2/ e Y t

…

.3;"/

e Y t .1/ e Y t .2/ e Y t .3/ ƒ‚ …:

(7.9.98)

DIII

We estimate I; II and III. It is not difficult to see that lim kIk1 D 0;

n!1

lim kIIk1 D 0:

(7.9.99)

n!1

We deal with III. Since n

ke Y t .1/ e Y t .2/ e Y t

.3;"/

n

e Y t .1/ e Y t .2/ e Y t .3;"/ k1 ke Y t .2/ k2 ke Y t p

n

O , it is enough to show that e Y t and ke Y t .2/ k22 e 4.e=2/ t k jkj'k 2 n L .QE /. By the definition of Y t .3; "/ we have 2 2

e

Y tn .3;"/

D

n Y

Z

2

t.i1/=n

.3;"/

e Y t .3;"/ k2 ! e Y t .3;"/ in

ti=nC

exp

iD1

.3;"/

W" .Bs ; Ns ; t .i 1/=n/dNs :

For every 2 S there exists N D N. / 2 N such that at the points s1 D s1 . /; : : : ; sN D sN . /, t 7! N t . / is not continuous. For sufficiently large n the number of sk contained in the interval .t .i 1/=n; t i=n is at most one. Then by taking n large enough and putting .n.si /; n.si / C t =n for the interval containing si , we get n

e Yt

.3;"/

D

N Y

.‰" .HE;od .Bsi ; Nsi ; n.si ////:

iD1

(7.9.100)

445


Clearly, n.si / ! si as n ! 1. Since HE;od .Bsi ; Nsi ; n.si // strongly converges to HE;od .Bsi ; Nsi ; si / as n ! 1 in L2 .QE /, we have lim r:h:s: (7.9.100) D

n!1

N Y

.‰" .HE;od .Bsi ; Nsi ; si ///:

(7.9.101)

iD1 n

Since the right-hand side of (7.9.101) equals e Y t .3;"/ , limn!1 ke Y t 0 is plainly seen, and

.3;"/ e Y t .3;"/ k

lim kIIIk1 D 0:

n!1

2D

(7.9.102)

A combination of (7.9.99) and (7.9.102) implies (7.9.92). By using the functional integral representation we can estimate the ground state Z2 energy of HPF . Write Z2 /: E.A ; B1 ; B2 ; B3 / D inf Spec.HPF

(7.9.103)

For the spinless Pauli–Fierz Hamiltonian HPF we have inf Spec.HPF / D E.A ; 0; 0; 0/ and the diamagnetic inequality E.0; 0; 0; 0/ E.A ; 0; 0; 0/ was already seen. We Z2 extend this inequality to HPF . Define Z2 .0/F /. / D .Hp C Hrad C Hd . //F . / jHod . /jF . /; .HPF

where jej jHod . /j D 2

Z

˚q R3

B12 .x/ C B22 .x/dx

(7.9.104)

(7.9.105)

Z2 .0/ corresponds to is independent of 2 Z2 . HPF q 1 0 jej e 2 2 B B C B 3 1 2 A 2 Hp C Hrad @ q 2 jej e 2 2 B1 C B2 2 B3 2

acting in L2 .R3 I C 2 / ˝ L2 .Q/. Furthermore, to avoid zeroes of the off-diagonal part to occur we also define Z2 .HPF;" .0/F /. / D .Hp C Hrad C Hd . //F . / ‰" .jHod . /j/F . /: (7.9.106)

Since the spin interaction is infinitesimally small with respect to the free Hamiltonian Z2 Z2 Hp C Hrad , HPF .0/ and HPF;" .0/ are self-adjoint on DPF and bounded from below. Z2

The functional integral representation of e tHPF .0/ is given by Rt X Z Z ? tHPF2 .0/ t .F; e G/ D lim e dx Ex;˛ Œe 0 V .Bs /ds .J0 F .q0 /; e X t ."/ Jt G.q t //; "#0

3 ˛D1;2 R

446


where X t? ."/ D

Z

Z

t

0

HE;d .Bs ; Ns ; s/ds C

tC

log.‰" .jHE;od .Bs ; s/j//dNs : 0

(7.9.107) Corollary 7.75 (Diamagnetic inequality). It follows that Z2

Z2

j.F; e tHPF G/j .jF j; e tHPF

.0/

jGj/

(7.9.108)

and q max

.˛ˇ/D.123/;.231/;.312/

E.0;

B˛2 C Bˇ2 ; 0; B / E.A ; B1 ; B2 ; B3 /: (7.9.109) ?

Proof. Since j‰" .HE;od /j ‰" .jHE;od j/, je X t ."/ j e X t ."/ and jJ t Gj J t jGj as J t is positivity preserving, by the functional integral representation of e tHPF we have Z2

Z2

Z2

j.F; e tHPF G/j lim.jF j; e tHPF;" .0/ jGj/ D .jF j; e tHPF "#0

Thus (7.9.108) follows. From this, E.0; is obtained. We will show that

.0/

jGj/:

q B12 C B22 ; 0; B3 / E.A ; B1 ; B2 ; B3 /

E.A ; B1 ; B2 ; B3 / D E.A ; B2 ; B3 ; B1 / D E.A ; B3 ; B1 ; B2 /

(7.9.110)

by SU.2/-symmetry. Let R 2 O.3/ be such that 0

1 0 1 x1 x2 R @x2 A D @x3 A: x3 x1 Then there exists .n; /2S 2R such that R D R.n; /. Recall that R.n; / describes the rotation around n by angle . Hence we see that e in.1=2/ e in.1=2/ D Z2 Z2 by HPF .A ; B1 ; B2 ; B3 /. Thus we obtain that .R/ . Now we write HPF Z2 Z2 e in.1=2/ HPF .A ; B1 ; B2 ; B3 /e in.1=2/ D HPF .A ; B2 ; B3 ; B1 /

which implies the first equality in (7.9.110). The second one is proven in the same way.

447


7.9.4 Spin-boson model An interesting application of the functional integral representation of the Pauli–Fierz Hamiltonian with spin (Theorem 7.74) is the analysis of the spin-boson model. Let F D Fb .L2 .Rd //. Definition 7.16 (Spin-boson Hamiltonian). The spin-boson Hamiltonian is defined by HSB D "1 ˝ 1 C 1 ˝ Hf C 3 ˝ .h/

(7.9.111)

on C 2 ˝ F , where " 0 is a coupling constant and .h/ is a scalar field defined by Z O O .h/ D .a .k/h.k/ C a.k/h.k//d k: (7.9.112) Rd

O h= O p! 2 L2 .Rd /. Then HSB is self-adjoint on C 2 ˝ D.Hf /. HSB Suppose that h; is realized as Hf C .h/ " HSB D : " Hf .h/ By introducing Z2 , HSB transforms into HSB ‰. / D .Hf C .h//‰. / C "‰. /;

2 Z2 :

(7.9.113)

Theorem 7.76 (Functional integral representation of spin-boson model). Let ˆ; ‰ 2 C 2 ˝ F . Then Rt X O .ˆ; e tHSB ‰/ D e t E˛ Œ.J0 ˆ.N0 /; e E . 0 Ns js hds/ ."/N t J t ‰.N t //; ˛D1;2

.ˆ; e tHSB ‰/ D e t

X

.J0 ˆ.˛ /; e E .˛

Rt

O

0 js hds/

" 6D 0;

(7.9.114)

" D 0:

(7.9.115)

J t ‰.˛ //;

˛D1;2

Proof. Let " 6D 0. We see that HSB ‰. / D .Hf C .h//‰. / e i Clog " ‰. /; and .ˆ; e tHSB ‰/ Rt R tC X O E˛ Œ.J0 ˆ.N0 /; e 0 Ns E .js h/dsC 0 .i Clog "/dNs J t ‰.N t //: D et ˛D1;2

448


R tC Since 0 .i C log "/dNs D .i C log "/N t , (7.9.114) follows. Next we consider the case of " D 0. As " ! 0 on both sides of (7.9.114), the integrands on ¹ 2 T jN t . / 1º vanish and those on K D ¹ 2 T jN t . / D 0º stay different from zero. Moreover, note that Ns . / D 0, s t , for 2 K. Hence (7.9.115) is obtained by taking the limit. In the case of " D 0, the Hamiltonian HSB is diagonal. Hence the ground state of HSB with " D 0 is two-fold degenerate, provided that ground states exists. Thus we estimate the dimension of Ker.HSB inf Spec.HSB // for " 6D 0. The transformations X ! X , X D 1; 2; 3, and a] .f / ! a] .f / can be implemented by some unitary operators, and then HQ SB D "1 ˝ 1 C 1 ˝ Hf C 3 ˝ .f /

(7.9.116)

is unitary equivalent with HSB . Flipping the sign " ! " in (7.9.114), we have Rt X O Q E˛ Œ.J0 ˆ.˛ /; e E . 0 Ns js hds/ "N t J t ‰.N t //: .ˆ; e t HSB ‰/ D e t ˛D1;2

(7.9.117) Q

Corollary 7.77 (Uniqueness of ground state). Assume that " 6D 0. Then e t HSB , t > 0, is positivity improving. In particular, the ground state of HSB is unique.

7.9.5 Translation invariant case Finally, we study the translation invariant Pauli–Fierz Hamiltonian with spin 1=2, that S is, HPF with V 0. Definition 7.17 (Pauli–Fierz Hamiltonian with spin 1=2 and fixed total momentum). The Pauli–Fierz Hamiltonian with spin and fixed total momentum p is defined by 1 e S HPF .p/ D .p Pf eA.0//2 C Hrad B.0/; 2 2

p 2 R3 ;

(7.9.118)

S .p// D D.Hrad / \ D.Pf 2 /. with domain D.HPF S Theorem 7.78 (Self-adjointness). If Assumption 7.1 holds, then HPF .p/ is self-ad2 joint and essentially self-adjoint on any core of Hrad C Pf .

Proof. The proof is similar to that of Theorem 7.65. By (7.6.14) we have kHPF;0 .p/ˆk C k.HPF .p/ C 1/ˆk for ˆ 2 D.Hrad / \ D.Pf2 /, with some constant C . From this inequality we see that, 1=2 for any " > 0 there exists C > 0 such that kHrad ˆk "kHPF .p/ˆk C C" kˆk for

449

Section 7.9 The Pauli–Fierz model with spin 1=2

ˆ 2 D.Hrad / \ D.Pf2 /. Since k B.0/ˆk kHrad ˆk C b" kˆk, it follows that S B.0/ is infinitesimally small with respect to HPF .p/. Thus HPF .p/ is self-adjoint 2 on D.Hrad / \ D.Pf /. Essential self-adjointness follows from the trivial inequality S .p/ˆk C k.Hrad C Pf2 /ˆk C kˆk. kHPF As in the case of HPF .p/ we have 2

3

Z

˚

2

L .R I C / ˝ Frad D T and

R3

Z S DT HPF

˚

R3

C ˝ Frad dp T 1 2

S HPF .p/dp T 1 ;

(7.9.119)

(7.9.120)

S where T is defined in (7.6.9). In Q-representation HPF .p/ is rewritten as

1 e S HPF .p/ D .p Pf eA .0//2 C Hrad B.0/; 2 2

(7.9.121)

where we write d .i r/ and d .!.i r// as Pf and Hrad , respectively, and B.0/ D B.x D 0/ defined in (7.9.8). S Before going to construct the functional integral representation of e tHPF .p/ we S look at the symmetry properties of HPF .p/. Lemma 7.79. Let 'O be rotation invariant. Then for any choice of the polarization S S vectors HPF .p/ is unitary equivalent with HPF .R 1 p/, for all R 2 SO.3/. Proof. It is sufficient to show the lemma for an arbitrary R D R.m; / with .m; / 2 S 2 Œ0; 2/. For any polarization vectors e ˙ define hf D d ..R; /H2 /;

(7.9.122)

where .R; k/ D cos1 .e C .Rk/ Re C .k// and H2 in (7.9.28). Formally, XZ hf D .R; k/.a .k; /a.k; C/ a .k; C/a.k; //d k: (7.9.123) 3 j D˙ R

Thus we obtain that e ihf e imLf Hrad e imLf e ihf D Hrad ; e ihf e imLf Pf e imLf e ihf D .RPf / ; e ihf e imLf A .0/e imLf e ihf D .RA.0// ; e ihf e imLf B .0/e imLf e ihf D .RB.0// ; e im.1=2/ e im.1=2/ D .R/ :

450


From these identities it follows that S .p/e im..1=2/ CLf / e ihf e ihf e im..1=2/ CLf / HPF

1 1 D .p RPf eRA.0//2 C Hrad .R/ .RB.0// 2 2 S 1 D HPF .R p/;

(7.9.124)

proving the lemma. S Let E S .p; e 2 / D inf Spec.HPF .p//. A straightforward consequence of Lemma 7.79 is as follows.

Corollary 7.80 (Rotation invariance). Let 'O be rotation invariant. Then for every R 2 SO.3/ it follows that E S .Rp; e 2 / D E S .p; e 2 /. Theorem 7.81 (Fiber decomposition of Pauli–Fierz Hamiltonian with spin 1=2 and fixed total momentum). Suppose that polarization vectors are coherent in direction n 2 S2 and 'O is rotation invariant. Let p=jpj D n. Then the Hilbert space C 2 ˝ Frad S S and the self-adjoint operator HPF .p/ D HPF .jpjn/ are decomposed as C 2 ˝ Frad D

M w2Z1=2

Frad .w/;

S HPF .p/ D

M

S HPF .p; w/:

(7.9.125)

w2Z1=2

Here Frad .w/ is the subspace spanned by eigenvectors of 1 J0 D n C n Lf C Sf 2 S S associated with the eigenvalue w 2 Z1=2 and HPF .jpjn; w/ D HPF .jpjn/dFrad .w/ .

Proof. Let R D R.n; /. Since the polarization vectors are coherent in direction n, it follows that .R; k/ D z with some z 2 Z, and then hf D Sf . Since Rp D jpjRn D jpjn D p, we have from (7.9.124) S S S e iJ0 HPF .p/e iJ0 D HPF .R 1 p/ D HPF .p/;

2 R:

(7.9.126)

S .p/ is reduced by J0 and (7.9.125) follows. Hence HPF S From this theorem HPF .p/ with coherent polarization vectors in direction p=jpj and p 6D 0 has the symmetry

SU.2/ SOfield .3/ helicity:

(7.9.127)

451


S Let 'O be rotation invariant and HPF .p; e ˙ / be the Hamiltonian with coherent polarization vectors given in Example 7.3 with n D n3 D .0; 0; 1/, i.e.,

.k2 ; k1 ; 0/ e C .k/ D q ; k12 C k22

e .k/ D kO e C .k/:

(7.9.128)

S S HPF .p/ with arbitrary polarization vectors is unitary equivalent to HPF .p; e ˙ /, S S HPF .p/ Š HPF .p; e ˙ /:

Since there exists R such that Rp D jpjn3 , we have by Lemma 7.79, S S HPF .p; e ˙ / Š HPF .jpjn3 ; e ˙ /:

By (7.9.125) we obtain that S .jpjn3 ; e ˙ / D HPF

M

S HPF .jpjn3 ; e ˙ ; w/:

w2Z1=2

Then for rotation invariant ', O we have the unitary equivalence M S S .p/ Š HPF .jpjn3 ; e ˙ ; w/: HPF

(7.9.129)

w2Z1=2

We can show the theorem below in a similar way to Theorem 7.69. Theorem 7.82 (Reflection symmetry). Let 'O be rotation invariant and the coherent S S polarization vectors e ˙ be given by (7.9.128). Then HPF .p; w/ Š HPF .p; w/ for w 2 Z1=2 and p with .0; 0; 1/ D p=jpj. Proof. The proof is similar to that of Theorem 7.69. Let J0 D 2 ˝ .u/, where u is defined in (7.9.43). Thus we have ´ Lf ; D 2; J01 Lf J0 D Lf ; D 1; 3; ´ D 1; 3; Pf ; J01 Pf J0 D Pf ; D 2; ´ D 1; 3; A .0/; J01 A .0/J0 D A .0/; D 2; ´ ; D 1; 3; J01 J0 D ; D 2:

452


Since n D n3 D .0; 0; 1/ and p D jpjn3 , and the helicity of e ˙ is zero, we have J0 D 12 3 C Lf;3 and J01 J0 J0 D J0 :

(7.9.130)

We also see that J01 .p Pf eA/ J0 D .p Pf eA/ for D 1; 2; 3. From here it follows that S S .p/J0 D HPF .p/: J01 HPF

(7.9.131)

Note also that S S .p/ D HPF .p; w/; HPF

2 Frad .w/:

(7.9.132)

S .p; w/J0 D Let 2 Frad .w/. We have J0 2 Frad .w/ by (7.9.130), and J01 HPF S S S 1 1 J0 HPF .p/J0 by (7.9.132). (7.9.131) implies that J0 HPF .p/J0 D HPF .p/ D S HPF .p; w/. Thus S S J01 HPF .p; w/J0 D HPF .p; w/

follows. S Since the operators HPF .p/ with p 6D 0 and different polarization vectors from S (7.9.128) are unitary equivalent with HPF .n3 jpj/ with e ˙ in (7.9.128), we have

Corollary 7.83 (Degeneracy of bound states). Let 'O be rotation invariant and M be S .p/. Then M is an even number. In particular, the multiplicity of bound states of HPF whenever a ground states exists, it is degenerate. The argument extends to the Hamiltonian without spin 1=2. With a rotation invariant ', O similarly to (7.9.126) also HPF .p/ with p=jpj D n and coherent polarization vectors in direction n satisfies e iJf HPF .p/e iJf D HPF .p/; 2 R: Thus HPF .p/ with coherent polarization vectors in direction n D p=jpj has the symmetry SOfield .3/ helicity: Since .Jf / D Z, this gives rise to the decomposition M M 0 Frad .w/; HPF .p/ D HPF .p; w/ Frad D w2Z

(7.9.133)

w2Z

0 where Frad .w/ denotes the subspace spanned by eigenvectors at eigenvalue w 2 Z of Jf . By Corollary 7.44, as soon as p D 0 the ground state of HPF .p D 0/ is unique for any e 2 R. It is also known that HPF .p/ has the unique ground state

453


for sufficiently small jpj and jej. We may suppose that the polarization vectors of HPF .p/ is coherent in direction p=jpj, since HPF .p/ with different polarization vectors are unitary equivalent. Suppose that the ground state ‰g .p/ of HPF .p/ is unique. 0 Then ‰g .p/ must belong to some Frad .w/. The only possibility is z D 0, since HPF .jpjn; z/ Š HPF .jpjn; z/, z 2 Z, by Lemma 7.82. This implies Jf ‰g .p/ D 0. We now construct the functional integral representation for the translation invariant S Pauli–Fierz Hamiltonian with spin 1=2. As before, we transform HPF on C 2 ˝ Frad Z2 to HPF on `2 .Z2 / ˝ L2 .Q/ which is defined by Z2 .HPF .p/‰/. / D .HPF .p/ C Hd .0; //‰. / C Hod .0; /‰. /;

(7.9.134)

where e Hd .0; / D B3 .0/; 2 e Hod .; 0/ D .B1 .0/ i B2 .0// 2

(7.9.135) (7.9.136)

Z2 Z2 with 2 Z2 . Moreover, HPF;" .p/ is defined by HPF .p/ with Hod replaced by

Z2 ‰" .Hod /, which is introduced to avoid the off-diagonal part of HPF .p/ becoming sinS gular. The strategy of constructing the functional integral representation of e tHPF .p/ is similar to that of the spinless case.

Theorem 7.84 (Functional integral representation for Pauli–Fierz Hamiltonian with spin 1=2 and fixed total momentum). Let ˆ; ‰ 2 `2 .Z2 / ˝ L2 .Q/. We have Z2

Z2

lim.ˆ; e tHPF;" .p/ ‰/ D .ˆ; e tHPF "#0

.p/

‰/

(7.9.137)

and Z2

.ˆ; e tHPF;" .p/ ‰/ D e t

X

E0;˛ Œe ipB t .J0 ˆ.˛ /; e X t ."/ J t e id Pf B t ‰.N t //;

˛D1;2

(7.9.138) where the exponent X t ."/ is given in (7.9.87). Proof. The proof goes along the same lines as in Theorem 7.41. From Theorem 7.84 we can derive energy inequalities in a similar manner to Corollary 7.43 for the spinless case. Write Z2 E.p; A ; B1 ; B2 ; B3 / D inf Spec.HPF .p//;

(7.9.139)

454


Z2 and define HPF .p; 0/ by Z2 .HPF .p; 0/‰/. / D .HPF .p/ C Hd .0; //‰. / jHod .0; /j‰. /; (7.9.140)

where jHod .0; /j D

jej 2

p B1 .0/2 C B2 .0/2 . This corresponds to

1 .p Pf /2 C Hrad 2

jej e 2 B3 .0/ 2 p jej 2 C B .0/2 B .0/ 1 2 2

! p B1 .0/2 C B2 .0/2 2e B3 .0/ (7.9.141)

in C 2 ˝ L2 .Q/. Note that jHod .0; /j is independent of . We have the energy comparison inequality Corollary 7.85 (Diamagnetic inequality). It follows that Z2

j.ˆ; e tHPF

.p/

and

q max

.˛ˇ/D.123/;.231/;.312/

Z2

‰/j .jˆj; e tHPF

E.0; 0;

.p;0/

j‰j/

(7.9.142)

B˛2 C Bˇ2 ; 0; B / E.p; A ; B1 ; B2 ; B3 /: (7.9.143)

Proof. We have j.ˆ; e tHPF .p/ ‰/j e t lim S

"#0

X

?

E0;˛ Œ.J0 jˆ.˛ /j; e X t

."/

J t e iPf B t jˆ.N t /j/

2Z2

D r:h:s: (7.9.142); where X t? ."/ is given by (7.9.107). (7.9.143) is immediate from (7.9.142) and a similar argument to (7.9.109).

Chapter 8


Notes to the Preface Classic books of this early period include H. Poincaré: Les méthodes nouvelles de mécanique céleste (Gauthier-Villars, 1893, English translation: New Methods of Celestial Mechanics), 3 vols., see also C. Siegel and J. Moser: Vorlesungen über Himmelsmechanik, 1956 (English translation: Lectures on Celestial Mechanics, Springer, 1971), J. C. Maxwell: A Treatise on Electricity and Magnetism, Clarendon Press, Oxford, 1873 (reprinted by Dover 1954, 2003), and Theory of Heat, Longmans, 1872 (reprinted Dover 2001). A comprehensive account of methods of mathematical physics with a strong stress on PDE which stood as a standard work for generations is D. Hilbert and R. Courant: Methoden der mathematischen Physik, vols. 1–2, Springer, 1924, 1937 (with subsequent English translation). The intellectual biographies and autobiographies of some of the main protagonists of early quantum mechanics capture the excitement of those days and convey the spirit how these ideas were born. See, for instance, W. Heisenberg: Der Teil und das Ganze, Piper, 1969, D. Cassidy: Uncertainty: The Life and Science of Werner Heisenberg, W.H. Freeman, 1993, A. Pais: Niels Bohr’s Times: In Physics, Philosophy, and Polity, Oxford UP, 1994, G. Farmelo: The Strangest Man: The Hidden Life of Paul Dirac, Mystic of the Atom, Basic Books, 2009. For an account from a major figure in the development of stochastic analysis and Feynman–Kac formulae see R. Bhatia: A conversation with S.R.S. Varadhan, Math. Intelligencer 2008, pp. 24– 42. E. Wigner’s essay The Unreasonable Effectiveness of Mathematics in the Natural Sciences appeared in Commun. Pure Appl. Math., 13 (1960). He mentions the idea attributed to Galileo that the laws of nature are written in the language of mathematics. The concept of the “book of nature” is a topos that can be traced back to antiquity and the middle ages, see the early and illuminating account given by the great German literary scholar E. R. Curtius in Europäische Literatur und lateinisches Mittelalter, Francke, 1954, 350ff (especially the concept of “chiffre”), and more modern discussions in The Book of Nature, A. Vanderjagt, K. van Berkel (eds.), vol. 1: Antiquity and the Middle Ages, vol. 2: Early Modern and Modern History, Peeters, Leuven, 2005–2006. On the quoted passage of Feynman see R. Dijkgraaf, Rapporteur Talk: Mathematical structures, in: The Quantum Structure of Space and Time, Proceedings of the 23rd Solvay Conference on Physics (D. Gross, M. Henneaux, A. Sevrin, eds.), Word Scientific, 2007; p. 91. Apparently a mathematician has replied that the time by which mathematics would be set back is “precisely the week in which God

456


created the world”; a version of this anecdote names Mark Kac to be this mathematician.

Notes to Chapter 1 Feynman’s path integral method appeared first in the papers [161, 162, 163], see also [164]. Dirac’s contact transformation theory was published in [128, 129, 130]. For some historic background see [424]. Modern references on Feynman integrals include [3, 80, 86, 94, 126, 127, 407, 224, 209, 292, 313, 289, 492]. A first paper by Kac using analytic continuation of the time variable is [293], see [79, 372] for later developments. The disadvantage of Kac’s approach is that it deals with the diffusion equation with dissipation instead of the time-dependent Schrödinger equation, and therefore the Feynman–Kac formula gives no direct information on the quantum dynamics. However, the time-independent solutions of both equations are the same and coincide with the eigenfunctions of the Schrödinger operator, which justifies the use of the Feynman–Kac formula in quantum mechanics.

Notes to Chapter 2 2.1 The Scottish botanist Robert Brown is credited with the observation reported in 1827 that pollen grains suspended in water perform an irregular motion, however, he was preceded by more than a century by the Dutch natural scientist van Leeuwenhoek by a similar observation. Also less known is the fact that Brown continued his observations by replacing pollen with minerals and even a piece of the Sphinx. Brownian motion has been first modelled in terms of a random process by Bachelier [38] as early as 1900 in order to describe stock prices. Einstein [146] and von Smoluchowski [452] studied in 1905/06 the fluctuations of thermal motion of particles and related them with Brownian motion, which Perrin tested in 1909 experimentally, noting the rough shape of particle paths. Langevin proposed in 1908 an equation to describe the equation of motion of a particle performing Brownian motion. In 1923 Wiener constructed Brownian motion as a random process in a mathematically rigorous way, and Lévy made important early contributions in the study of some fine details of Brownian motion. For a history see [143] as well as [316, 378]. Brownian motion in many ways stands out as a random process having a distinguished role. It is a Gaussian process, a Markov process and a martingale. It is the only Lévy process (see Section 2.4) allowing a version with continuous paths. It is the scaling limit of simple random walk, and it is intimately related with the Laplace operator in that is its infinitesimal generator, and thus also with the heat equation. There are a number of equivalent ways of defining and constructing Brownian


457

motion, of which we have presented one and commented on others. The literature on Brownian motion is vast due to the fact that it is an object in the overlap zone of the fields and approaches mentioned above. A wealth of facts and explicit formulae can be found in [72] and the references therein, and we single out [350, 486] for further developments.

2.2 Apart from the original interest in physics, a major modern source of motivation for stochastic analysis are applications in financial mathematics. We single out [280, 284, 303, 378, 380, 406, 464] for fundamental facts on Brownian motion and general stochastic analysis. On a more specialized note, we refer to [411] for a general analysis and measure theory background, [25, 47, 141, 144, 171, 465] for probability theory discussed with an analytic point of view, [196, 319] for random processes, [137, 145, 67, 149] for Markov processes, [287] for Markov generators, [61, 288] for limit theorems, and [409] for continuous time martingales.

2.3 In 1944 Itô [282] proposed a notion of stochastic integral using the paths of Brownian motion as integrator, and he also established the chain rule for stochastic differentials, which is known today as the Itô formula. While the construction of the Itô integral departs from the classic notion of integral proceeding through approximate Riemann sums over finite divisions, it successfully copes with the difficulty that these sums do not stand a chance to converge pathwise since Brownian motion has paths of almost surely unbounded variation. However, by requiring less Itô has shown that convergence holds in L2 -sense (see Definition 3.10.36), which extends to convergence in probability (see Definition 2.23). Since the original ideas of Itô stochastic integration theory has further developed by using continuous martingales as integrators instead of Brownian motion. We refer to [226, 284, 303, 353, 354, 380, 406, 464, 409] for stochastic integration theory with Brownian or more general integrators, and to [101, 280, 380] for stochastic differential equations.

2.4 Lévy processes are random processes with stationary and independent increments, in many ways generalizing Brownian motion. In particular, they are Markov processes and semimartingales, with right continuous paths allowing left limits, which in general contain continuous pieces with jump discontinuities at random times. The generators of Lévy processes are in general non-local pseudo-differential operators. We refer [11, 52, 51, 45, 423, 426] for general results on Lévy processes, to [136, 318] for their fluctuation theory, and to the three volume set [287] which is a comprehen-

458


sive study emphasizing the context of semigroups and generators. More specifically, see [11, Th. 2.3.5] for Proposition 2.45 and [11, Th. 2.4.16] for Theorem 2.47. The Itô formula for semimartingales (Proposition 2.48) is taken from [280, Chapter II, Section 5]. Subordinators are one-dimensional Lévy processes with non-decreasing paths. We refer to [51, 11, 426] for the relationship between Bernstein functions and subordinators.

Notes to Chapter 3 3.1 There is a large literature on quantum mechanics from a mathematical point of view. The series [400, 401, 402, 403] is well established and [305] is classic, all of them containing fundamental discussions on the spectral analysis of Schrödinger operators. We also refer to [99, 121, 165, 263, 330, 473, 471, 222, 120, 65] for various aspects of rigorous quantum mechanics. [443, 444] are concise but stimulating and up-todate accounts of the spectral property of Schrödinger operators. The substantial paper [441] and the monograph [438] are seminal discussing Schrödinger semigroups and very appealing applications of the Feynman–Kac formula to the spectral analysis of Schrödinger operators. Proposition 3.9 on the criterion for the compactness of operators is from [403, Th. XIII.64], and Proposition 3.10 from [403, XIII.4, Corollary 2]. The Kato–Rellich theorem is due to [405], and a first application to atomic Hamiltonians is [304]. Kato’s inequality is due to [306]. The operator version of Kato’s inequality, called abstract Kato’s inequality,, is considered in [125, 240, 437, 434]. Proposition 3.16 is taken from [400, Th. VIII.5], and the KLMN theorem from [401, Th. X.17]. For Theorem 3.18 see [305, Th. VIII.3.11, Th. VIII.3.13 with Supplementary notes to Chapter VIII, 5 (p. 575)]. Rellich’s criterion first appeared in [404]. The Trotter product formula (3.28) is a powerful tool in the construction of the functional integral representation of Schrödinger-type semigroups. The original paper is [475]. [307, 308] extend the Trotter product formula to the case of a quadratic form P B for A C B in (3.28). Proposition 3.29 is due to [309], where the Trotter sum A C formula is further extended to non-linear semigroups generated by sub-differentials of convex functions. [369, 370] use trace norm limits in the Trotter formula instead of the strong limits, assuming that e t.A CP B/ PAB is trace class (here PAB denotes projection onto D.A/\D.B/). Operator norm convergence of products in the Trotter formula is obtained by [277] with an error bound k.e .t=n/A e .t=n/B /n e t.ACB/ k D O.n1=2 /, as n ! 1. Here it is assumed that A C B is essentially self-adjoint on D.A/ \ D.B/. That this error bound is the best possible is shown in [470]. In the case where A C B is moreover self-adjoint on D.A/ \ D.B/, the error term can be improved to O.n1 / [278]. Furthermore, [68] shows norm convergence in the Trotter product formula on Banach spaces.


459

3.2 For the classic Feynman–Kac formula see [86, 119, 292, 438]. Kac has originally considered positive potentials, however, this strong restriction has been gradually weakened (see Section 3.3 below). There is a large and important body of research on Feynman–Kac semigroups for Schrödinger operators on bounded domains, which we cannot address in this book for lack of space. The analysis in this case involves tools of potential theory, stopping times and killed processes. The earliest papers establishing a link between Brownian motion and harmonic functions are [138, 139, 297, 298]. Further developments include the sequence of papers [272]; see [67, 117, 140, 433] for monographic treatments, and the modern [393] presenting a unifying analytic-probabilistic approach. In the context of processes generated by Schrödinger operators a seminal paper is [2], and a book length discussion of the subject is [94]. For the Schrödinger operator H D .1=2/ C V the Dirichlet boundary value problem for a given bounded domain D Rd with smooth enough boundary @D is ´ H D 0 in D D on @D: Let .B t / t0 be standard Brownian motion, and consider its first exit time D D inf¹t > 0j B t … Dº from D. By the definition B t 2 D for t < D and BD 2 @D if D < 1. It is possible to prove that D < 1 W -a.s. Then the Feynman–Kac formula becomes R D .x/ D Ex Œe 0 V .B t /dt .BD / and thus it offers a probabilistic representation of the solution to the Dirichlet problem. In this case the spectrum of the Laplace operator is discrete consisting of eigenvalues 0 < 0 .D/ < 1 .D/ < . (a) Isoperimetric inequalities: One line of research is addressing the question how the various properties of the spectrum depends on geometric properties of D. Let D be a convex domain and V be a convex function. In this case also the Schrödinger operator has a purely discrete spectrum 0 < 0 .D; V / < 1 .D; V / < . Suppose without restricting generality that 0 2 D, and let B denote a ball centered in the origin such that jDj D jBj, where the bars denote Lebesgue measure. Then it can be shown that the first exit time of Brownian motion is maximized by the ball, i.e., supx2D W x . D > t / W 0 . B > t /, for all t > 0. This is an example of an isoperimetric inequality. Since lim t!0 1t log W x . D > t / D 0 .D/, the above inequality implies the Faber–Krahn inequality 0 .B/ 0 .D/, which is an isoperimetric property of the eigenvalues of the Dirichlet Laplacian [39, 89], see also [382]. The Faber–Krahn inequality has been similarly shown for Schrödinger operators for convex domains [392, 230, 231, 116], see the conditions on V in the referred papers.

460


Another isoperimetric inequality concerns the ratio of consecutive eigenvalues, originating from the Payne–Pólya–Weinberger conjecture [386] which originally stated that for planar domains 1 .D/= 0 .D/ is maximized when D is chosen to be a disk. The conjecture has been proven to hold in any dimension in [26, 27]. (b) Spectral gaps: The difference ƒ.D/ D 1 .D/ 0 .D/ is called fundamental gap for the Dirichlet Laplacian, and a similar object can be defined for the Dirichlet Schrödinger operator. Based on specific cases in which the eigenvalues can be com3 2 puted it has been conjectured [479] that ƒ.D/ 2d 2 , where d is the diameter of D, provided both D and V are convex. An upper bound can be obtained as follows. Since by the Payne–Pólya–Weinberger conjecture 1 .D/= 0 .D/ 1 .B/= 0 .B/, it follows that ƒ.D/ ƒ.B/. 0 .D/= 0 .B//, in which the second factor at the righthand side can be further estimated by using the Faber–Krahn inequality. For a discussion see [28] and the references therein. For Schrödinger operators with convex V the fundamental gap conjecture was proved recently in [8]. (c) Harnack inequalities: In its classical form, the boundary Harnack inequality states that for any open sets K K Rd with K K , there exists C > 0 such that for all functions f harmonic and non-negative on K we have f .x/ Cf .y/, for all x; y 2 K. In other words, just by knowing that a non-negative f solves f D 0 we can already conclude that infx2K f .x/ C1 supx2K f .x/, where C depends on K and K but not on f . A version of this inequality for Schrödinger operators was first proved by [476] using analytic methods. [2] realized that the Feynman–Kac formula can be used to obtain these results. They used the fact that when stopped at the boundary of K , a Feynman–Kac type formula just gives the solution of the Dirichlet problem for H and K . Further references include [490, 491, 233, 94]. Further interesting aspects include heat kernel estimates [108, 109, 42, 391, 207, 208], functional inequalities [421], intrinsic ultracontractivity [110, 111, 40], and others. For the general theory of how the Feynman–Kac formula can be used to study various classes of PDE we also refer to [144, 170, 47, 48, 466].

3.3 Kato-class has been introduced in [306]. Simon [441] studies Schrödinger semigroups with Kato-class potentials in great detail. In particular, here it is proven that etH is a bounded operator from Lp to Lq for 1 p q 1. A natural question is how the norm of this operator depends on t for large t . It is shown in [439] that ˛ D .1=t / ln ketH kp;q is independent of p and q; it turned out [440] that the same ˛ is obtained by taking pointwise limits of images of non-negative functions. Since then there has been a considerable activity on the topic, see [108, 109, 316, 364, 365, 391, 488, 489], as well as [317, 469, 467]. Yet another, more recent, line of research is concerned with cases when the potential V is replaced by more singular objects


461

such as distributions, or when Brownian motion is replaced by a Bessel process. For research in this direction see [415, 414, 416, 412, 413] and the references therein.

3.4 Theorem 3.44 can be improved by showing that e tK .x; y/ is even jointly continuous in x; y and t [441]. The proof of Lemma 3.47 is due to [98]. Estimates of the number of negative eigenvalues of Schrödinger operators are due to [327, 328], where the Birman–Schwinger principle and the Feynman–Kac formula are used. The proof of Lemma 3.51 is taken from [445] and [438, Ths. 8.1 and 8.2]. The Lieb– Thirring inequality [327, 332, 333] can be extended to a wide class of Schrödinger operators, including those with spin and magnetic field, we refer to [329] and references therein. Proposition 3.52 is [403, Th. XIII.52]. The Perron–Frobenius theorem first appeared in [387, 172] for matrices, and it was applied to quantum field theory by [200, 211, 212]. The Klauder phenomenon is discussed in [151, 150, 157, 435]. Proposition 3.56 is proven by [300]. Due to the Klauder phenomenon one can construct examples of Schrödinger operators with degenerate ground states. Exponential decay in L2 sense, i.e., ke ajxj k < 1 for eigenvectors is studied in [1], where also the exponential decay of eigenvectors of the divergence form @ A .x/@ C V is considered. Pointwise exponential decay of eigenvectors of Schrödinger operators is investigated by [83], and by p [84] for the relativistic Schrödinger operator. In the massless relativistic case, i.e., C V , eigenvectors decay polynomially. In the case of the divergence form @ A .x/@ C V with uniform elliptic bound c jA .x/j C and some regularity condition on A , the pointwise exponential decay of eigenvectors also can be proven in a similar way to Carmona’s estimates with respect to the diffusion process associated with the divergence form. An example how far functional integration and related tools can be used to draw information on the spectral properties of a pseudo-differential operator is [342].

3.5 References on Schrödinger operators with magnetic field include [29, 30, 31]. Selfadjointness of the Schrödinger operator with magnetic field H.a/ is established in [325]. The alternative proof of the Feynman–Kato–Itô formula of the Schrödinger operator with magnetic field H.a/ is taken from [438, Section V], where ra D 0 is assumed. The applications of functional integration with magnetic field and the diamagnetic inequality are from [147, 148, 379, 76, 169, 271].

3.6 For results on the spectrum of the relativistic Schrödinger operator see [104, 105, 239, 326]. The Feynman–Kac formula for relativistic Schrödinger operators HR .a/ with

462


a D 0 is studied [112, 84]. The p case of non-zero vector potential is discussed in [115, 254]. The function f .x/ D x 2 C m2 m is an example of Bernstein functions. The relativistic Schrödinger operator is then realized as f . 12 .i ra/2 /CV . A Feynman– Kac-type formula of a general Schrödinger operator of the form ‰. 12 .i r a/2 /CV with Bernstein function ‰ defined on positive real line is constructed by [254]. The relativistic Schrödinger operator can be also defined through the Weyl quantization. This is given by H w D hw .x; D/ C V .x/, where “ xCy w i.xy/

; u.y/.2/d dyd ; .h u/.x/ D e h 2 p with symbol h.x; / D j a.x/j2 C m2 . [367] shows that in general .hw .x; D//2 6D .i r a/2 C m2 : The Feynman–Kac formula of e tH is given by [275]. w

3.7 In quantum mechanics the notion of spin is introduced as an internal degree of freedom of electrons. Spin plays a central role in the statistics of electrons. Another aspect of the influence of spin is in the behavior of electrons in a magnetic field. The Feynman–Kac formula for the Schrödinger operator with spin is discussed in [114, 113, 179, 180, 254]. The diamagnetic inequality (3.7.36) is due to [114, 254]. Paramagnetism of Schrödinger operators with spin 1=2 is also an interesting subject. It has been conjectured in [266] that paramagnetism is universal, but a counterexample has been given in [32], while paramagnetism in the classical limit is discussed in [265]. For a 2-dimensional space-time Dirac operator, [273, 274, 276] proposes a path integral representation in terms of a S 0 .R2 I M2 .C//-valued countably additive measure on C.Œ0; t I R/. In particular, in [276], the measure is concentrated on paths having differential coefficients equal to “˙1 speed of light” in every finite time interval except at finitely many instants of time. This relates with the Zitterbewegung of Dirac particles. A similar result is also given by [113] with respect to a Poisson process. Furthermore, [270, 268, 269, 336] treat quantum probability and derive a non-commutative Feynman–Kac formula.

3.8 The Feynman–Kac type formula for relativistic Schrödinger operators with spin is due to [115, 254], where the combination of a stopping time (subordinator) and a Poisson process is applied. In [254] a Feynman–Kac formula for Schrödinger operators of the form ‰. 12 . .i r a//2 / C V is given for arbitrary Bernstein functions ‰,

463


and an energy comparison inequality is obtained. The Schrödinger operator with spin can be transformed to the operator on L2 .R3 Z2 /. Also operators on L2 .R3 ¹e 2 i ; e 2 i=2 ; : : : ; e 2 i=p º/ can be considered.

3.9 This material is taken from [443].

3.10 P ./1 processes have an interesting property deriving from connections with statistical mechanics and Gibbs measures. Formally such a process is the stationary solution of the SPDE 1 XP .u; t / D u X.u; t / .@V /.X.u; t // C W ; 2

(8.0.1)

where W now is space-time white noise. This was shown rigorously for a restricted class of potentials V by [286]. An interesting application appears in [229], where the SPDE above is used in an MCMC algorithm for sampling the conditional P ./1 measure in various situations. In the context of SPDE the result of [58] is also of interest: when the potential grows sufficiently rapidly, it implies uniqueness of those stationary measures of the SPDE above that are Gibbs measures, i.e. satisfy the DLR equations. It is expected, but not proven, that all stationary measures of the above SPDE satisfy the DLR equations. The P ./1 process associated with the divergence form d X @i aij .x/@j C V .x/ i;j D1

with a confining potential can be constructed, and also that of C V .x/. Refer to e.g., [192]. Moreover for the case of a non-local operator, the P ./1 process associated with a fractional Schrödinger operator can be constructed. We refer to [410]. Lemma 3.112 is taken from [192], which is a modification of [311]. For intrinsic ultracontractivity properties see [110, 108, 111], and for asymptotic intrinsic ultracontractivity [299].

Notes to Chapter 4 4.1 The seminal papers introducing Gibbs measures for lattice spin systems are [131, 132] and [321]. Further details on the prescribed conditional probability kernels (called specifications) can be found in [133, 168] and [396, 205, 453]. Surveys and books on Gibbs measures include the early [419], covering also classical continuous and

464


quantum lattice systems, and [454, 183, 455, 320, 184, 395, 397, 418, 399]. In [396] a general framework is developed, while the approach of [281] is more from the perspective of the variational principle in which equilibrium states appear as tangents to the free energy functional. Reference [185] is a highly authoritative comprehensive monograph with a strong stress on DLR theory, covering a wealth of material. [348] is a systematic text on cluster expansion methods applied to Gibbs measures. Other relevant books on rigorous statistical mechanics include [217] focusing more on Ising models, [447] giving an early account of contour techniques, and [442] of lattice gases. A reappraisal of Gibbsian theory was presented in the book-length paper [478], especially in the context of renormalization group transformations. For Gibbs measures on lattice spin systems with non-compact space existence of a Gibbs measure is harder to prove: see the superstability conditions in [417], tempered Gibbs measures [322, 87], [341, 340, 337] for cases when the potential in not uniformly but pointwise or almost surely summable, and [186] for Gibbs measures indexed by subsets of an uncountable index set. For other approaches to and applications of Gibbs measures on path space(s) we refer to [5] and the references therein.

4.2 Some of the basic notions, such as local uniform domination, are adapted from discrete models, see [185]. The discussion of uniqueness of Gibbs measures follows [58]. The case ˛ D 1 in the uniqueness criterion suggests that a maximal configuration space on which is the unique Gibbs measure, a set consisting of exponentially growing paths is closer to truth than X0 , however, establishing this is beyond the method of our proof. In [286] sets of exponentially growing paths as sets of uniqueness for the Gibbs measure were obtained, however, assuming convexity of the function V .x/ jxj2 for some > 0, therefore the case ˛ < 1 is not covered by his results. The advantage of our method is that it is not limited to polynomials, nor even to continuous or semibounded potentials, in particular, local singularities and other perturbations of V in the above examples will not alter the conclusions. This corresponds to the intuition that only the behaviour of the potential at infinity should determine whether uniqueness of the Gibbs measure holds on a given set. The existence proof for Gibbs measures with pair potential comes from [55]. For further literature on Gibbs measures with pair interaction potential we refer to [381, 343, 234, 338, 59, 55]. Densities dependent on double stochastic integrals are treated in [219, 339, 56], the first two papers using techniques of rough paths, and processes with jump discontinuities in [299].

4.3 There are several versions of cluster expansion method. A systematic monograph is [348] and useful reviews are [156, 77]. For the general combinatorial scheme see


465

[314, 155, 394]. We refer to [73, 358, 134, 389, 78, 49] for improving on convergence, however, these method depend on the specific models at hand.

4.4 The presentation here follows [60]. The functional central limit theorem in Theorem 4.42 follows from [237]. The above theorem still contains the possibility that the diffusion matrix is zero, which would imply subdiffusive behaviour of Brownian paths. There are two independent proofs of ruling this out, one in [60, 456] using ideas of [74], and another in [218]. Both require substantial additional techniques to the material of this book, and therefore they are omitted in our presentation.

Notes to Chapter 5 5.1 A mathematical introduction to quantum field theory is given in [17, 24, 204, 223, 436, 463]. References [285, 388, 482] treat quantum field theory from the point of view of particle physics. Useful texts on quantum field theory include [63, 70, 71, 167, 238, 368, 429, 420]. [164] is a monumental book on quantum field theory using path integrals. We also refer to [4, 86, 103, 225, 407, 427, 468] for texts in this spirit. [159] offers a concise summary of fermion functional integrals. Geometric and topological discussions on quantum field theory are [160, 182, 428, 462, 481]. Standard texts on QED are [236, 291, 301, 355, 425, 487]. Quantum field theory in curved space-time is discussed in [62, 43, 178]. [50, 349] covers spectral methods in infinite dimensional analysis.

5.2 Fock space was introduced in [166]. The second quantization d was first treated by Cook [96], where self-adjointness of the field operator ˆ.f / and its conjugate momentum ….f / was shown. Nelson’s analytic vector theorem (Proposition 5.3) appeared in [371]. Wick product was introduced in [484]. The number operator bounds (5.2.27)–(5.2.28) 1/m=2 TQ .N C 1/n=2 k R can be generalized to k.NQC m kW kL2 , where T D W .k1 ; : : : ; km ; p1 ; : : : ; pm / j D1 a .kj / niD1 a.pi /d kdp, see [197].

5.3 Segal [430, 431] introduced Q space, which linked the theory of free quantum fields with probability. Using this expression the spectrum of the P ./2 Hamiltonian has

466


been studied, for instance, in the series of papers [198, 200, 201, 199, 202, 203] by Glimm and Jaffe.

5.4 Theorem 5.9 is taken from [436, Section I.2]. Minlos’s theorem first appeared in [356].

5.5 Nelson emphasized the functional integral approach to quantum field theory in [376], and the fundamental results concerning the Markov property in quantum field theory were established in [377, 375]. We also refer to [436] for for further aspects of Euclidean quantum field theory. Nelson’s Euclidean quantum field and its lattice approximation due to [221] changed the object studied in quantum field theory before.

5.6 As explained in this section, it is impossible to construct a Gaussian measure on a real Hilbert space H with the given covariance. Instead of this we consider the triplet MC2 M M2 under the identification M D M , and construct the process t 7! t 2 M2 such that t .f / D hh t ; f ii is a Gaussian process for f 2 MC2 . Similar results appear in [82], where path continuity is considered. [390] treats the Ornstein–Uhlenbeck semigroup in an infinite dimensional L2 setting. [290] is a comprehensive study of Gaussian measures on a Hilbert space; we also refer to [135] for non-Gaussian measures. [100, 302, 398] treat infinite dimensional stochastic equations. For the Minlos–Sazanov theorem (Theorem 5.21) we refer to [302, Th. 2.3.4].

Notes to Chapter 6 6.1 In 1964 E. Nelson [373] demonstrated the mathematical existence of a meson theory with non-relativistic nucleons. 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 Z Z 1 H D .2m/ .x/./ .x/dx C Hf C g .x/HI .x/ .x/dx; Rd

Rd

L where is a non-relativistic complex scalar field on H D 1 nD0 Hn . The restriction H dHn 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.

467


6.2 As is explained in this section, IR divergence and UV divergence are a ubiquitous difficulty not only in the Nelson model but in general the general theory of quantum fields. In 1952 Miyatake [361] and van Hove [480] found that the ground state of a Hamiltonian in Fock space weakly converges to zero when the UV cutoff removed. Shale [432] developed mathematical methods to study both IR divergence and UV divergence. In [373] Nelson defined a self-adjoint operator with UV cutoff on the tensor product of Hilbert spaces, L2 .Rd / ˝ F . The solution of the Schrödinger equation diverges as the cutoffs tends to infinity, but the divergence amounts merely to a complex infinite phase shift due to the self-energy of the particles. A canonical transformation, which is implemented by a unitary operator when the IR cutoff is introduced, separates the divergent self-energy term. It is then shown that, after removing the UV cutoff and subtracting an infinite constant, a limit Hamiltonian exists. The infrared problem in the Nelson model was first studied by Fröhlich [173, 174]. We also refer to [81, 213, 232, 264, 310, 451, 450, 449] for earlier results on the Nelson model.

6.3 The Feynman–Kac formula of the Nelson Hamiltonian in terms of a combination of a diffusion process X t and a M2 -valued stochastic process t is due to [57]. Here the ground state transformation of the Nelson Hamiltonian is applied. A similar Feynman–Kac formula, with no quantized field, is discussed in [334]. In [191, 192] the functional integral representation of the Nelson model defined on a static Lorentz manifold is constructed. There the diffusion process X t is replaced by a process associated with the divergence form

3 X ; D1

1 1 r A .x/r C V; c.x/ c.x/

p and the dispersion relation ! D C m2 expressed in the position representation is replaced by the pseudo-differential operator !a;m

3 X 1=2 D r a .x/r C m2 .x/ : ; D1

1

Note that, importantly, the mass becomes position-dependent, m.x/. It is then not ˛ '.k/ as jkj ! 1. Moreover, it is not clear how to estimate the decay rate of !a;m straightforward to show existence or absence of the ground state of the Nelson model on a static Lorentz manifold by operator theory. Instead of operator theory one can apply the functional integral method introduced in Chapter 6 to show absence of ground state.

468


6.4 For the massive case the existence of the ground state of the translation invariant Nelson model is studied in [212, 174]. In the massless case Bach, Fröhlich and Sigal [35, 36] 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. Spohn [460] and Gérard [188, 190] proved the existence of the ground state for arbitrary values of the coupling constant. Functional integral methods are used in [460], and operator theory in [188]. Theorem 6.6 is due to [460]. The external potential V in [460, 188] is confining. For non-confining potentials we refer to [422, 241]. 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 nonFock representation is used, see [19, 344]. The existence of a ground state without UV cutoff is due to [6] for the massive case and [245] for the massless case. In a series of papers [191, 193, 192], the Nelson model on a static Lorenz manifold is considered. Here the static Lorentz manifold is a 4-dimensional pseudo-Riemannian manifold P with line element ds 2 D g00 .x/dt ˝ dt i;j ij .x/dx i ˝ dx j , where g00 > 0 is time-independent and D . ij / denotes a 3-dimensional Riemannian metric. In particular, the infrared problem is studied, and criteria for existence or absence of ground states are shown. We also refer to [123, 122, 124, 187, 189, 181, 194].

6.5 This material is taken from [57]. 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. (6.5.44) mathematically validates the first half of this intuition. The second half, the absence of ground state, will be discussed in Section 6.6. The pull-through formula is quite useful to estimate .‰g ; N ‰g /, which is used in e.g. [6, 36, 37, 173, 188, 245]. Superexponential decay of the number of bosons in the Nelson model is also studied in [213] using operator theory.

6.6 The translation invariant Nelson model has been studied in e.g. [9, 81, 173, 174, 213, 362, 363, 357]. It is also related with the polaron model. [195] is a useful summary of various versions of polaron models. The path integral approach to this model is developed in [252, 457]. (6.6.10) is due to [174, 457]. Proposition 6.6.11 is shown in [174].

6.7

R 2 =!.k/3 d k D 1, non-existence of the In the IR singular case, i.e., when j'.k/j O ground state of the Nelson model is proven in [22, 123, 244, 345, 344]. Theorem 6.34 is due to [345, 344].

469


6.8 The existence of the Nelson Hamiltonian without UV cutoff (Theorem 6.39) has been proven by Nelson [373] by using the so called Gross transform [210, 323]. Nelson [374] also showed that by using functional integration for the Hamiltonian with the potential jxj2 . The bounds (6.8.14)–(6.8.15) are presented in [6, 245]. The functional integral representation of the Nelson model without UV cutoff (Theorem 6.45) is due to [220, 374]. Davies considered a weak coupling limit of the form Hp C ƒ.f / C ƒ2 Hf in [106, 107]. In [247] the weak coupling limit with a simultaneous removal of UV cutoff is studied by using functional integrals. Related works on the scaling limit in quantum field theory are [16, 18, 142, 384, 472]. In particular, in [16] a general criterion of convergence in the scaling limit is found.

Notes to Chapter 7 7.1 In 1938 the Pauli–Fierz model was introduced in [385], where the generation of infrared photons was studied. See also [66] for the infrared divergence. Bethe [54] and Welton [483] proposed a theoretical interpretation of the Lamb shift by a variant of the Pauli–Fierz model. We refer to [95, 355] for details, and to [461] for a summary of recent progress of the spectral analysis of the Pauli–Fierz and related models.

7.2 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 D 0 can be diagonalized, i.e., HPF can be approximated by the 3 C 3L dimensional harmonic oscillator 2 1 X X 1 '.l/Q O C .Pl Pl C Ql ! 2 .l/Ql !.l//: pC l 2m 2 L

L

l

l

Here ŒPl ; Ql 0 D i ıl l 0 is satisfied. Then it can be shown that 1

dip

HPF Š

2.m C

2/ e 2 d d1 k'=!k O

p 2 ˝ 1 C 1 ˝ Hf C g;

where g is a constant. In the series of papers [12, 13, 14, 15] Arai investigated the spectrum of the Pauli–Fierz model in the dipole approximation. We also refer to [64, 459] for the scattering theory.

470


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 proved by Bach, Fröhlich and Sigal [37] for weak couplings and no infrared cutoff. Griesemer, Lieb and Loss [206], Lieb and Loss [331] 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 [44]). 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. A basic assumption is the existence of a ground state for the zero coupling Hamiltonian, i.e. that Hp has a ground state with positive spectral gap. [259] shows that for sufficiently large coupling constants the Pauli–Fierz Hamiltonian in the dipole approximation has a ground state even when H has no ground state. This phenomenon is called enhanced binding. Enhanced binding for the Pauli–Fierz Hamiltonian is also studied in [23, 88, 93, 228, 258, 262]. Hypercontractivity was introduced by Nelson in [377], where also Proposition 7.6 was proved. We also refer to [446].

7.3 Functional integral representations of the Pauli–Fierz model are discussed in [158, 176, 224, 246, 461].R We emphasize that the exponent in the Feynman–Kac formula t i eAE .K t / D i e 0 AEs .'. Q Bs // dBs depends on time s explicitly, in contrast to the classical case. Gross [214] discusses a Q-space representation of a Proca field, which is a model of massive quantum electrodynamics. The diamagnetic inequality (Theorem 7.15) with a quantized radiation field is obtained in [29, 246].

7.4 Essential self-adjointness of H C V such that e tH is positivity preserving is studied in [154, 312]. Self-adjointness of the Pauli–Fierz Hamiltonian is established by [37] for weak couplings, and by [248, 250] for arbitrary values of coupling constants. Theorem 7.26 is taken from [250]. An alternative proof is given in [235]. An infinite dimensional version of the Perron–Frobenius theorem is due to [200, 211, 212]. Proposition 7.28 is taken from [153, 157]. 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 2 K ! u C v 2 K, v 2 K and a 0 ! av 2 K and K \ K D ;. 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 [212, 157]. Theorem 7.30 is due to [249]. We also refer to [34, 251, 347, 448, 200] for the uniqueness of the ground state of the Pauli–Fierz Hamiltonian. The spatial exponential decay of eigenvectors of the Pauli–Fierz model is discussed in [37, 206, 242].

471


7.5 This material is taken from [242]. e tHPF is also positivity improving, and the ground Kato state of HPF decays exponentially. These facts are proved in the same way as for HPF . Moreover, in [242] Pauli–Fierz Hamiltonians with general cutoff functions are defined through Feynman–Kac type formulae. Kato

7.6 This material is taken from [252]. The idea to construct the functional integral representation comes from [457]; see also [7, 346]. Chen [90] shows the existence of the ground state of HPF .p/ when p D 0. The effective mass meff of the Pauli–Fierz Hamiltonian is defined by m1eff D 13 p E.p/dpD0 , where E.p/ denotes the ground state energy of HPF .p/ [216, 215]. When a sharp infrared cutoff is introduced in HPF , E.p/ is differentiable with respect to p for sufficiently small coupling constants. It is, however, not clear that E./ is twice differentiable in a neighborhood of p D 0, when no infrared cutoff is introduced. The infrared problem of the effective mass is studied in [33, 92, 91, 227]. The functional integral approach of the effective mass is [60, 456]. The asymptotic behaviour of effective mass with respect to the ultraviolet cutoff is studied in [261, 255].

7.7 This R Rmaterial is taken from [56]. The formal double stochastic integral of the type dt dsW .B t Bs ; t s/ appears in R. Feynman [163, p. 443]. The mathematical derivation is done in [249, 374, 461]. See also [97, 360].

7.8 This material is taken from [253], where also the relativistic Pauli–Fierz model with spin 1=2 is studied. The spectrum of the relativistic Pauli–Fierz model is studied in [175, 257, 352, 359].

7.9 A basic reference for the material of this section is [256]. The degeneracy of ground states for weak enough coupling is proven in [251, 260]. The uniqueness of the ground state of the translation invariant spinless Pauli–Fierz Hamiltonian is also proven in [260]. Results related to Theorem 7.76 can be found in [267]. Spohn [458] treats the model by functional integration. Arai and Hirokawa [20, 21] define a generalized spin-model and prove existence and uniqueness of the ground state. The general reference for the spin-boson model is [324]. The functional integral representations of the spin-boson model follows directly from results in [256]; see also [53]. The vacuum

472


expectation .˛ ˝ ; e tHSB ˇ ˝ / is given in terms of a Poisson process through the Feynman–Kac formula. This was also studied operator-theoretically in [243]. The electronic part of the Pauli–Fierz Hamiltonian is non-relativistic, 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 [383]. [485] is an axiomatic approach. See, furthermore, [46] for the Itô–Clifford integral, [10] for the fermionic Itô formula, [335] for fermion martingales, [408] for fermion path integrals, and [315] for functional integration of the Euclidean Dirac field.

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[491] Z. Zhao. Green function for Schrödinger operator and conditioned Feynman–Kac gauge. J. Math. Anal. Appl., 116:309–334, 1986. [492] J. Zinn-Justin. Path Integrals in Quantum Mechanics. Oxford University Press, 2010.

Index

absence of ground state, 328 abstract Kato’s inequality, 458 adapted process, 23 adjoint operator, 76 analytic vector, 256 angular momentum, 427 angular momentum of the field, 429 annihilation operator, 250, 253, 359 Bernstein function, 69, 462 characterization, 70 Bernstein functions of the Laplacian, 148 bipolaron model, 193 Birman–Schwinger kernel, 115, 116 Birman–Schwinger principle, 114 Bochner’s theorem, 54, 265 Borel measurable function, 11 Borel -field, 11 boson Fock space, 249 boson mass, 254 boson number distribution, 320 boundary condition, 192, 197 bounded operator, 72 Brownian bridge, 49, 50 Brownian bridge measure, 32 Brownian motion, 15 d -dimensional, 15 moment, 18 over R, 20 standard, 15 Brownian path, 16 Burkholder–Davis–Gundy inequality, 45, 384, 419 C0 -semigroup, 89 generator, 89 on C1 .Rd /, 58 on L2 .Rd /, 59 càdlàg paths, 56 Calkin algebra, 81

Cameron–Martin formula, 52 canonical commutation relation, 251, 256 Carmona’s estimate, 128 divergence form, 461 Nelson model, 315 Pauli–Fierz model, 398 Cauchy distribution, 54 central limit theorem, 243 chain, 220 Chapman–Kolmogorov identity, 25 charge distribution, 295 closable operator, 73 closed operator, 73 cluster, 221 cluster expansion, 217 cluster representation, 222 coherent polarization vectors, 428 compact operator, 80, 88 V 1=2 . C m2 /1 V 1=2 , 114 compensated Poisson random measure, 63 complexification, 260 compound Poisson process, 64 conditional expectation, 21 confining potential, 87, 128 conjugate momentum, 256, 392 constant drift, 64 continuity equation, 357 continuous version, 16, 39, 176 infinite dimensional Ornstein–Uhlenbeck process, 287 contour, 220 contraction operator, 73, 269 convergence in distribution, 13 convergence in probability, 13 convergent almost surely, 12 coordinate process, 16 core, 73 correlation function associated with a cluster, 222

500 Coulomb gauge, 355, 357, 361 counting measure, 61, 62, 152 creation operator, 250, 253, 359 cylinder set, 15 decaying potential, 88, 129 diamagnetic inequality Pauli–Fierz Hamiltonian, 381 fixed total momentum, 407 Kato-decomposable potential, 401 Pauli–Fierz Hamiltonian with spin 1=2, 446 fixed total momentum, 454 relativistic Pauli–Fierz Hamiltonian, 423 fixed total momentum, 424 relativistic Schrödinger operator, 148 Kato-decomposable potential, 150 spin 1=2, 165 Schrödinger operator, 142 Kato-decomposable potential, 143 spin 1=2, 161 differential second quantization, 251, 261 diffusion matrix, 56 diffusion process, 46, 175 diffusion term, 41 dipole approximation, 469 Dirac equation, 354 Dirac operator 2-dimensional space-time, 462 Dirichlet principle, 182, 332 dispersion relation, 254, 295 distribution, 12 DLR equation, 197, 463 Dobrushin–Lanford–Ruelle construction, 191 dominated convergence theorem, 13 Doob’s h-transform, 171 double stochastic integral, 194, 408, 471 drift term, 41, 56 drift transformation, 52 Dynkin’s formula, 48 effective mass, 471 embedded eigenvalue, 79, 298 energy comparison inequality, 407 relativistic, 424

Index spin 1=2, 454 enhanced binding, 470 equation of motion, 357 ergodic map, 204 ergodic theorem, 204 Euclidean Green functions, 409, 410 Euler-Lagrange equation, 293 existence of Gibbs measure, 217, 219 existence of ground state, 303, 352 any coupling strength, 309, 352 criterion, 304 zero total momentum, 326, 471 expectation, 11 exponential decay of the ground state Nelson model, 316 Pauli–Fierz model, 398 Schrödinger operators, 126 external potential, 195 Feller transition kernel, 30, 58 Feynman propagator, 248 Feynman–Kac formula e E.ir/ , 122 Kato-decomposable potential, 111 non-commutative, 462 P ./1 -process, 180 relativistic Schrödinger operator singular external potential and vector potential, 148 spin 1=2, 163 vector potential, 144 Schrödinger operator singular external potential, 96 singular external potential and vector potential, 142 smooth external potential, 93 spin 1=2, 154 spin, singular external potential and vector potential, 159 unbounded semigroup, 168 Feynman–Kac semigroup, 95 intrinsic, 170 Feynman–Kac–Itô formula, 133 Feynman–Kac–Nelson formula Euclidean field, 275 path measure, 292 fiber decomposition

501

Index Pauli–Fierz Hamiltonian with spin 1=2 fixed total momentum, 450 general polarization, 431 fiber Hamiltonian, 435 field momentum operator, 324, 402 field strength tensor, 353 filtered space, 23 filtration, 23 natural, 23 finite particle subspace, 251, 359 finite volume Gibbs measure, 227 Fock vacuum, 250 form domain, 85 fractional Laplacian, 80 relativistic, 80 fractional Schrödinger operator, 112, 194 free field Hamiltonian in Fock space, 254 in function space, 282 Nelson model, 295 Pauli–Fierz model, 361 full, 365 full Wiener measure, 32 functional integral representation free field Hamiltonian, 270 Nelson Hamiltonian Euclidean field and Brownian motion, 301 infinite dimensional Ornstein– Uhlenbeck process and P ./1 process, 300 Nelson Hamiltonian with a fixed total momentum, 325 Nelson Hamiltonian without ultraviolet cutoff, 343 non-interacting Pauli–Fierz Hamiltonian, 375 Pauli–Fierz Hamiltonian fixed total momentum, 405 general external potentials, 382 Pauli–Fierz Hamiltonian with spin 1=2, 441 fixed total momentum, 453 Pauli–Fierz Hamiltonian, 377 relativistic Pauli–Fierz Hamiltonian, 422

fixed total momentum, 424 spin-boson Hamiltonian, 447 gamma matrix, 353 Gaussian distribution, 54 Gaussian measure on a Hilbert space, 278, 280 Gaussian random variable, 12, 186 indexed by real Hilbert space, 258, 267, 283, 365 multivariate, 12 standard, 12 generator ˛-stable process, 60 Brownian motion, 31 Brownian motion with drift, 60 C0 -semigroup, 89 Itô diffusion, 47, 48 Lévy process, 59 Poisson process, 60 Gibbs measure, 196 boundary condition, 197 existence, 217, 219 finite volume, 196, 227 pair interaction potential, 195 partition function, 197 reference measure, 195 sharp boundary condition, 198 stochastic boundary condition, 199 uniqueness, 225 Girsanov theorem, 51, 341, 343 Gross transform, 333 ground state, 121, 235, 352 concavity, 397 degenerate, 126 existence, 303, 352 multiplicity, 121 ground state expectation bounded operators, 312 field operators, 322 second quantized operators, 316 unbounded operators, 314 ground state transform, 170, 185, 299 Hölder continuity, 19 Hardy–Littlewood–Sobolev inequality, 80, 115 harmonic oscillator, 49, 89, 184, 204, 266

502 fermionic, 153 Harnack inequality, 460 heat kernel, 16, 90 helicity, 427, 429 Hermitian polynomial, 323 Hilbert space-valued Gaussian random process, 283 stochastic integral, 372 Hille–Yoshida theorem, 59, 91 hydrogen atom, 89 hypercontractivity, 171, 370 identically distributed, 12 image measure, 12 independent events, 14 random variables, 14 sub- -fields, 14 infinite dimensional Ornstein–Uhlenbeck process, 227, 283 continuous version, 287 infinitely divisible process, 53 infinitesimally small, 82 infrared divergence, 297, 319 infrared regularity, 297, 328 infrared singularity, 297 initial distribution, 174 Markov process, 26 integral kernel, 3 . C /1 , 101, 114 e tE.ir/ , 122 intensity measure, 62 interaction energy between field and particle, 295 intersection local time, 193 intertwining property, 368 intrinsic ultracontractivity, 206, 218 inverse Gaussian subordinator, 68 Itô bridge measure, 202 Itô diffusion, 47 Itô formula Brownian motion, 42 Itô process, 43 product, 43 rules, 43 semimartingale, 65, 153 Itô integral, 37

Index Itô isometry, 36, 373 Itô measure, 202 Itô process, 41 Iterated stochastic integral, 413 jump process, 152 Kato’s inequality, 84 abstract, 458 Kato–Rellich theorem, 82, 296 Kato-class, 98, 206, 399 relativistic, 149 Kato-decomposable, 295 relativistic, 149 Khasminskii’s lemma, 106 Klauder phenomenon, 126 KLMN theorem, 85 Kolmogorov consistency relation, 17 Kolmogorov extension theorem, 17 ˇ Kolmogorov–Centsov theorem, 18, 287 Lévy measure, 55 Bernstein functions, 70 Lévy process, 56, 152 characteristics, 56 generator, 59 Lévy symbol, 56 Lévy–Itô decomposition, 64 Lévy–Khintchine formula, 55 Lagrangian Nelson model, 293 Pauli–Fierz model, 357 QED, 353 Langevin equation, 49 Laplacian, 79, 80 law of the iterated logarithm, 19 Legendre transform, 293, 354 Lieb–Thirring inequality, 113 linear SDE, 48 local convergence of measures, 199, 236, 237 local martingale, 25 locally uniform topology on a continuous path space, 15 locally uniformly dominated probability measures, 200 longitudinal component, 353 Lorentz covariant quantum fields, 262

503

Index Lorentz gauge, 355 lower bound of ground state, 131 Lp –Lq boundedness, 108 Schrödinger operator with vector potentials and Kato decomposable potentials, 143 second quantization, 370 Lp -convergence, 12 magnetic field, 151 Markov process, 25, 236 generator, 31 stationary, 27, 47, 58 Markov property, 274, 369 infinite dimensional Ornstein– Uhlenbeck process, 289 strong, 29 martingale, 23, 38 inequality, 24 local, 25 martingale convergence theorem, 243 Maxwell equations, 354, 356 measurable, 11 Mehler’s formula, 184 Minlos theorem, 264, 266 Minlos–Sazanov theorem, 277 modulus of continuity, 14 moment generating function, 322 monotone convergence theorem for forms, 86 multiplication operator, 79 natural filtration, 23 Nelson Hamiltonian, 296 fixed total momentum, 325 in Fock space, 296 in function space, 299 Nelson model, 193, 466 massive, 298 massless, 297 static Lorentz manifold, 467, 468 Nelson’s analytic vector theorem, 256, 295 non-commutative Feynman–Kac formula, 462 nowhere differentiability, 19 nuclear space, 266

number operator, 252 Ornstein–Uhlenbeck process, 49, 184, 204 infinite dimensional, 227, 283 Markov property, 289 p-variation, 20 pair interaction potential, 195, 218 pair potential Nelson model, 302 Pauli–Fierz model, 412 regularized, 336 partition function, 197 path, 15 Pauli–Fierz Hamiltonian dipole approximation, 469 fixed total momentum, 402 in Fock space, 362 in function space, 368 Kato-class potential, 401 relativistic, 420 fixed total momentum, 424 singular external potential, 381 spin 1=2, 426, 434 fixed total momentum, 448 Pauli–Fierz model, 194 Perron–Frobenius theorem, 123, 124, 303, 397, 407, 470 P ./1 -process, 175, 193, 202, 463 divergence form, 463 Poisson distribution, 54 Poisson equation, 355 Poisson process, 62, 152 Poisson random measure, 61 polarization vectors, 360 coherent, 428 polaron model, 193, 468 Polish space, 14, 23 positivity improving, 121, 392 free field Hamiltonian, 394 Nelson Hamiltonian, 303 zero total momentum, 326 Pauli–Fierz Hamiltonian, 396 zero total momentum, 407 relativistic Pauli–Fierz Hamiltonian, 423 zero total momentum, 424 spin-boson Hamiltonian, 448

504 positivity preserving, 121, 261, 321 predictable, 65 predictable -field, 65 probability space, 11 probability transition kernel, 25, 57 diffusion process, 47 Lévy process, 58 Proca field, 470 product formula, 67 product Itô formula, 43 semimartingale, 67 Prokhorov theorem, 14, 310, 416 pull-through formula, 318, 326 quadratic form, 85 semibounded, 85 symmetric, 85 quantized magnetic filed, 426 quantized radiation field, 360, 425 quantum probability, 462 random measure, 61 random process, 14 random variable, 11 reference measure, 195 reflection symmetry, 19 Pauli–Fierz Hamiltonian with spin 1=2, 433 Pauli-Fierz Hamiltonian with a fixed total momentum, 451 regular conditional probability measure, 22, 290 regularized interaction, 336 relative bound, 82 relative bounds for annihilation and creation operators, 252, 255 relative compactness of probability measures, 13 relatively compact, 81 relatively form bounded, 86 relativistic fractional Laplacian, 80 relativistic Kato-class, 149 relativistic Kato-decomposable, 149 relativistic Pauli–Fierz Hamiltonian, 420 fixed total momentum, 424 relativistic Schrödinger operator singular external potential and vector potential, 147

Index spin 1=2, 162 vector potential, 143 vector potential and relativistic Katodecomposable potential, 149 Rellich’s criterion, 87 renormalized regularized interaction, 340 resolvent, 74 resolvent set, 74 Riesz–Markov theorm, 264 Riesz–Thorin theorem, 108 rough paths, 464 rules of Itô differential calculus, 43 sample space, 11 Sazanov topology, 276 scalar field, 262, 294 time-zero, 262 Schrödinger operator, 81 Kato-decomposable potential, 112 relativistic singular external potential and vector potential, 147 spin 1=2, 162 vector potential, 143 vector potential and relativistic Kato-decomposable potential, 149 singular external potential and vector potential, 138, 142 spin 1=2, 151 spin 1=2, singular potential and vector potential, 158 vector potential, 132 vector potential and Kato-decomposable potential, 143 Schrödinger semigroup, 92 second quantization, 251, 254, 261, 366 spectrum, 252 Segal field, 256 self-adjoint operator, 76 essential, 76 positive, 77 self-similarity, 19 semibounded quadratic form, 85 semigroup asymptotically ultracontractive, 171 intrinsically ultracontractive, 171 ultracontractive, 171

505

Index semimartingale, 65 sharp boundary condition, 198 shift invariance, 175 Skorokhod topology, 242 spatial homogenuity, 19 spectral gap, 206 spectral resolution, 78 spectral theorem, 78 spectrum, 74 continuous, 75 discrete, 79 essential, 79 point, 75 residual, 75 second quantization, 252 spin-boson model, 447 stable process, 57 stable subordinator, 68 Stark Hamiltonian, 166 static Lorentz manifold, 468 stochastic boundary condition, 199, 209 stochastic continuity, 56 stochastic differential equation, 46 solution, 46 stochastic integral, 37 Hilbert space-valued, 372 stochastic partial differential equation, 463 Stone’s theorem, 91 semigroup version, 91 Stone–Weierstrass theorem, 264 stopping time, 24 Stratonovich integral, 41, 44, 133 strong convergence, 73 strong law of large numbers, 19 strong Markov property, 29 strong resolvent convergence, 78 strongly continuous semigroup, 89 Stummel-class, 104 subordinator, 67 characterization, 67 inverse Gaussian, 68 stable, 68 super-quadratic potential, 208 supercontractivity, 171

superexponential decay of boson number, 317 symmetric operator, 76 symmetric quadratic form, 85 thermodynamic limit, 191 tightness, 14, 310, 416 time inversion, 19 time reversibility, 19 time-zero scalar field, 262 total momentum, 325, 402 total momentum operator, 324, 402 transversal delta function, 364 transversal wave, 359 Trotter product formula, 95, 270, 275 turbulent fluids, 194 Tychonoff theorem, 263 ultracontractivity, 171 ultraviolet divergence, 297 unbounded operator, 72 uniform convergence, 73 uniform resolvent convergence, 78 uniqueness of Gibbs measure, 225 unitary operator, 77 version of random process, 16 wave equation, 355 weak convergence linear operators, 73 probability measures, 13 weak coupling limit, 344, 345 Weyl quantization, 462 white noise, 463 Wick exponential, 44 Wick product, 259 Wiener measure, 15, 16 conditional, 32 full, 32 Wiener process, 15 Wiener–Itô decomposition, 260 Wiener–Itô–Segal isomorphism, 261 Wightman distribution, 247 Yukawa potential, 350

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