Feynman-Kac-Type Theorems and Gibbs Measures on Path Space (eBook)

lt;!doctype html public "-//w3c//dtd html 4.0 transitional//en">

József Lőrinczi, Loughborough University, UK; Fumio Hiroshima, University of Kyushu, Fukuoka, Japan; Volker Betz, University of Warwick, Coventry, UK.

Preface 6
Contents 8
I Feynman–Kac-type theorems and Gibbs measures 14
1 Heuristics and history 16
1.1 Feynman path integrals and Feynman–Kac formulae 16
1.2 Plan and scope 20
2 Probabilistic preliminaries 24
2.1 An invitation to Brownian motion 24
2.2 Martingale and Markov properties 34
2.2.1 Martingale property 34
2.2.2 Markov property 38
2.2.3 Feller transition kernels and generators 42
2.2.4 Conditional Wiener measure 45
2.3 Basics of stochastic calculus 46
2.3.1 The classical integral and its extensions 46
2.3.2 Stochastic integrals 47
2.3.3 Itô formula 55
2.3.4 Stochastic differential equations and diffusions 59
2.3.5 Girsanov theorem and Cameron–Martin formula 63
2.4 Lévy processes 66
2.4.1 Lévy process and Lévy–Khintchine formula 66
2.4.2 Markov property of Lévy processes 70
2.4.3 Random measures and Lévy–Itô decomposition 74
2.4.4 Itô formula for semimartingales 77
2.4.5 Subordinators 80
2.4.6 Bernstein functions 82
3 Feynman–Kac formulae 72
3.1 Schrödinger semigroups 84
3.1.1 Schrödinger equation and path integral solutions 84
3.1.2 Linear operators and their spectra 85
3.1.3 Spectral resolution 91
3.1.4 Compact operators 93
3.1.5 Schrödinger operators 94
3.1.6 Schrödinger operators by quadratic forms 98
3.1.7 Confining potential and decaying potential 100
3.1.8 Strongly continuous operator semigroups 102
3.2 Feynman-Kac formula for external potentials 106
3.2.1 Bounded smooth external potentials 106
3.2.2 Derivation through the Trotter product formula 108
3.3 Feynman-Kac formula for Kato-class potentials 110
3.3.1 Kato-class potentials 110
3.3.2 Feynman-Kac formula for Kato-decomposable potentials 121
3.4 Properties of Schrödinger operators and semigroups 125
3.4.1 Kernel of the Schrödinger semigroup 125
3.4.2 Number of eigenfunctions with negative eigenvalues 126
3.4.3 Positivity improving and uniqueness of ground state 133
3.4.4 Degenerate ground state and Klauder phenomenon 137
3.4.5 Exponential decay of the eigenfunctions 139
3.5 Feynman-Kac-Itô formula for magnetic field 144
3.5.1 Feynman-Kac-Itô formula 144
3.5.2 Alternate proof of the Feynman-Kac-Itô formula 148
3.5.3 Extension to singular external potentials and vector potentials 151
3.5.4 Kato-class potentials and Lp-Lq boundedness 155
3.6 Feynman-Kac formula for relativistic Schrödinger operators 156
3.6.1 Relativistic Schrödinger operator 156
3.6.2 Relativistic Kato-class potentials and Lp-Lq boundedness 162
3.7 Feynman-Kac formula for Schrödinger operator with spin 163
3.7.1 Schrödinger operator with spin 163
3.7.2 A jump process 165
3.7.3 Feynman-Kac formula for the jump process 167
3.7.4 Extension to singular potentials and vector potentials 170
3.8 Feynman-Kac formula for relativistic Schrödinger operator with spin 175
3.9 Feynman-Kac formula for unbounded semigroups and Stark effect 179
3.10 Ground state transform and related diffusions 183
3.10.1 Ground state transform and the intrinsic semigroup 183
3.10.2 Feynman-Kac formula for P(f)1-processes 187
3.10.3 Dirichlet principle 194
3.10.4 Mehler’s formula 197
4 Gibbs measures associated with Feynman-Kac semigroups 203
4.1 Gibbs measures on path space 203
4.1.1 From Feynman-Kac formulae to Gibbs measures 203
4.1.2 Definitions and basic facts 207
4.2 Existence and uniqueness by direct methods 214
4.2.1 External potentials: existence 214
4.2.2 Uniqueness 217
4.2.3 Gibbs measure for pair interaction potentials 221
4.3 Existence and properties by cluster expansion 230
4.3.1 Cluster representation 230
4.3.2 Basic estimates and convergence of cluster expansion 236
4.3.3 Further properties of the Gibbs measure 237
4.4 Gibbs measures with no external potential 239
4.4.1 Gibbs measure 239
4.4.2 Diffusive behaviour 251
II Rigorous quantumfield theory 258
5 Free Euclidean quantum field and Ornstein–Uhlenbeck processes 260
5.1 Background 260
5.2 Boson Fock space 262
5.2.1 Second quantization 262
5.2.2 Segal fields 268
5.2.3 Wick product 270
5.3 Q-spaces 271
5.3.1 Gaussian random processes 271
5.3.2 Wiener–Itô–Segal isomorphism 273
5.3.3 Lorentz covariant quantum fields 275
5.4 Existence of Q-spaces 276
5.4.1 Countable product spaces 276
5.4.2 Bochner theorem and Minlos theorem 277
5.5 Functional integration representation of Euclidean quantum fields 281
5.5.1 Basic results in Euclidean quantum field theory 281
5.5.2 Markov property of projections 284
5.5.3 Feynman–Kac–Nelson formula 287
5.6 Infinite dimensional Ornstein–Uhlenbeck process 289
5.6.1 Abstract theory of measures on Hilbert spaces 289
5.6.2 Fock space as a function space 292
5.6.3 Infinite dimensional Ornstein–Uhlenbeck-process 295
5.6.4 Markov property 301
5.6.5 Regular conditional Gaussian probability measures 303
5.6.6 Feynman–Kac–Nelson formula by path measures 305
6 The Nelson model by path measures 306
6.2 The Nelson model in Fock space 307
6.2.1 Definition 307
6.2.2 Infrared and ultraviolet divergences 309
6.2.3 Embedded eigenvalues 311
6.3 The Nelson model in function space 311
6.4 Existence and uniqueness of the ground state 316
6.5 Ground state expectations 322
6.5.1 General theorems 322
6.5.2 Spatial decay of the ground state 328
6.5.3 Ground state expectation for second quantized operators 329
6.5.4 Ground state expectation for field operators 335
6.6 The translation invariant Nelson model 337
6.7 Infrared divergence 341
6.8 Ultraviolet divergence 346
6.8.1 Energy renormalization 346
6.8.2 Regularized interaction 348
6.8.3 Removal of the ultraviolet cutoff 352
6.8.4 Weak coupling limit and removal of ultraviolet cutoff 357
7 The Pauli–Fierz model by path measures 364
7.1 Preliminaries 364
7.1.1 Introduction 364
7.1.2 Lagrangian QED 365
7.1.3 Classical variant of non-relativistic QED 369
7.2 The Pauli–Fierz model in non-relativistic QED 372
7.2.1 The Pauli–Fierz model in Fock space 372
7.2.2 The Pauli–Fierz model in function space 376
7.2.3 Markov property 382
7.3 Functional integral representation for the Pauli–Fierz Hamiltonian 385
7.3.1 Hilbert space-valued stochastic integrals 385
7.3.2 Functional integral representation 388
7.3.3 Extension to general external potential 394
7.4 Applications of functional integral representations 395
7.4.1 Self-adjointness of the Pauli–Fierz Hamiltonian 395
7.4.2 Positivity improving and uniqueness of the ground state 405
7.4.3 Spatial decay of the ground state 411
7.5 The Pauli–Fierz model with Kato class potential 412
7.6 Translation invariant Pauli–Fierz model 414
7.7 Path measure associated with the ground state 421
7.7.1 Path measures with double stochastic integrals 421
7.7.2 Expression in terms of iterated stochastic integrals 425
7.7.3 Weak convergence of path measures 428
7.8 Relativistic Pauli–Fierz model 431
7.8.1 Definition 431
7.8.2 Functional integral representation 433
7.8.3 Translation invariant case 436
7.9 The Pauli–Fierz model with spin 437
7.9.1 Definition 437
7.9.2 Symmetry and polarization 440
7.9.3 Functional integral representation 447
7.9.4 Spin-boson model 460
7.9.5 Translation invariant case 461
8 Notes and References 468
Bibliography 486
Index 512

lt;P>"This book is more or less self-contained, and is useful to both beginners and experts." Mathematical Reviews

"The book is targeted to advanced students and researchers. The reader will profit from many proofs spelled out in detail." Zentralblatt für Mathematik