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| Title: | Feynman-Kac-type theorems and Gibbs measures on path space with applications to rigorous quantum field theory |
|---|---|
| From: |
József Lörinczi ; Fumio Hiroshima ; Volker Betz
|
| Person: |
Lőrinczi, József
1966- Verfasser aut Hiroshima, Fumio Betz, Volker 1972- |
| Main Authors: | , , |
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
de Gruyter
2011
|
| Series: | De Gruyter studies in mathematics
34 |
| Subjects: | |
| Online Access: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018638432&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Physical Description: | XI, 505 S. 240 mm x 170 mm |
| ISBN: | 9783110201482 |
Staff View
MARC
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| 100 | 1 | |a Lőrinczi, József |d 1966- |e Verfasser |0 (DE-588)173059767 |4 aut | |
| 245 | 1 | 0 | |a Feynman-Kac-type theorems and Gibbs measures on path space |b with applications to rigorous quantum field theory |c József Lörinczi ; Fumio Hiroshima ; Volker Betz |
| 264 | 1 | |a Berlin [u.a.] |b de Gruyter |c 2011 | |
| 300 | |a XI, 505 S. |c 240 mm x 170 mm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a De Gruyter studies in mathematics |v 34 | |
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| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-018638432 | |
Record in the Search Index
| _version_ | 1819367168189399040 |
|---|---|
| adam_text | Contents
Preface
I Feynman-Kac-type theorems and Gibbs measures
1
1
Heuristics and history
3
1.1
Feynman path integrals and Feynman-Kac formulae
......... 3
1.2
Plan and scope
............................. 7
2
Probabilistic preliminaries
11
2.1
An invitation to Brownian motion
................... 11
2.2
Martingale and Markov properties
................... 21
2.2.1
Martingale property
...................... 21
2.2.2
Markov property
........................ 25
2.2.3
Feller transition kernels and generators
............ 29
2.2.4
Conditional Wiener measure
.................. 32
2.3
Basics of stochastic calculus
...................... 33
2.3.1
The classical integral and its extensions
............ 33
2.3.2
Stochastic integrals
....................... 34
2.3.3
Ito
formula
........................... 42
2.3.4
Stochastic differential equations and diffusions
........ 46
2.3.5
Girsanov theorem and Cameron-Martin formula
....... 50
2.4
Levy processes
............................. 53
2.4.1
Levy process and
Lévy-Khintchine
formula
.......... 53
2.4.2
Markov property of Levy processes
.............. 57
2.4.3
Random measures and
Lévy-Itô
decomposition
........ 61
2.4.4
Ito
formula for semimartingales
................ 64
2.4.5
Subordinators
.......................... 67
2.4.6
Bernstein functions
....................... 69
3
Feynman-Kac formulae
71
3.1 Schrödinger
semigroups
........................ 71
3.1.1 Schrödinger
equation and path integral solutions
....... 71
3.1.2
Linear operators and their spectra
............... 72
3.1.3
Spectral resolution
....................... 78
3.1.4
Compact operators
....................... 80
viii Contents
3.1.5 Schrödinger Operators ..................... 81
3.1.6 Schrödinger
operators
by quadratic
f
orms...........
85
3.1.7
Confining
potential
and decaying
potential
.......... 87
3.1.8
Strongly continuous operator semigroups
........... 89
3.2
Feynman-Kac formula for external potentials
............. 93
3.2.1
Bounded smooth external potentials
.............. 93
3.2.2
Derivation through the Trotter product formula
........ 95
3.3
Feynman-Kac formula for Kato-class potentials
............ 97
3.3.1
Kato-class potentials
...................... 97
3.3.2
Feynman-Kac formula for Kato-decomposable potentials
. . 108
3.4
Properties of
Schrödinger
operators and semigroups
.......... 112
3.4.1
Kernel of the
Schrödinger
semigroup
............. 112
3.4.2
Number of eigenfunctions with negative eigenvalues
..... 113
3.4.3
Positivity
improving and uniqueness of ground state
..... 120
3.4.4
Degenerate ground state and Klauder phenomenon
...... 124
3.4.5
Exponential decay of the eigenfunctions
............ 126
3.5
Feynman-Kac-Itô
formula for magnetic field
............. 131
3.5.1
Feynman-Kac-Itô
formula
................... 131
3.5.2
Alternate proof of the
Feynman-Kac-Itô
formula
....... 135
3.5.3
Extension to singular external potentials and vector potentials
138
3.5.4
Kato-class potentials and Lp—Lq boundedness
........ 142
3.6
Feynman-Kac formula for relativistic
Schrödinger
operators
..... 143
3.6.1
Relativistic
Schrödinger
operator
............... 143
3.6.2
Relativistic Kato-class potentials and Lp-Lq boundedness
. . 149
3.7
Feynman-Kac formula for
Schrödinger
operator with spin
...... 150
3.7.1
Schrödinger
operator with spin
................. 150
3.7.2
A jump process
......................... 152
3.7.3
Feynman-Kac formula for the jump process
.......... 154
3.7.4
Extension to singular potentials and vector potentials
..... 157
3.8
Feynman-Kac formula for relativistic
Schrödinger
operator with spin
162
3.9
Feynman-Kac formula for unbounded semigroups and Stark effect
. . 166
3.10
Ground state transform and related diffusions
............. 170
3.10.1
Ground state transform and the intrinsic semigroup
...... 170
3.10.2
Feynman-Kac formula for
Р(ф)
-processes
.......... 174
3.10.3
Dirichlet principle
....................... 181
3.10.4
Mehler s formula
........................ 184
4
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
Contents_______________________________________________________________
їх
4.2
Existence
and uniqueness by
direct
methods
..............201
4.2.1
External potentials: existence
................. 201
4.2.2
Uniqueness
........................... 204
4.2.3
Gibbs measure for pair interaction potentials
......... 208
4.3
Existence and properties by cluster expansion
............. 217
4.3.1
Cluster representation
..................... 217
4.3.2
Basic estimates and convergence of cluster expansion
. .... 223
4.3.3
Further properties of the Gibbs measure
............ 224
4.4
Gibbs measures with no external potential
............... 226
4.4.1
Gibbs measure
.........................226
4.4.2
Diffusive behaviour
.......................238
II Rigorous quantum field theory
245
5
Free Euclidean quantum field and Ornstein-Uhlenbeck processes
247
5.1
Background
............................... 247
5.2
Boson Fock space
............................ 249
5.2.1
Second quantization
...................... 249
5.2.2
Segal fields
........................... 255
5.2.3
Wick product
.......................... 257
5.3
^-spaces
................................ 258
5.3.1
Gaussian random processes
.................. 258
5.3.2
Wiener-Itô—
Segal isomorphism
................ 260
5.3.3
Lorentz
covariant quantum fields
................ 262
5.4
Existence of ^-spaces
......................... 263
5.4.1
Countable product spaces
................... 263
5.4.2
Bochner theorem and
Minios
theorem
............. 264
5.5
Functional integration representation of Euclidean quantum fields
. . 268
5.5.1
Basic results in Euclidean quantum field theory
........ 268
5.5.2
Markov property of projections
................ 271
5.5.3
Feynman-Kac-Nelson formula
................ 274
5.6
Infinite dimensional Ornstein-Uhlenbeck process
........... 276
5.6.1
Abstract theory of measures on
Hubert
spaces
......... 276
5.6.2
Fock space as a function space
................. 279
5.6.3
Infinite dimensional Ornstein-Uhlenbeck-process
....... 282
5.6.4
Markov property
........................ 288
5.6.5
Regular conditional Gaussian probability measures
...... 290
5.6.6
Feynman-Kac-Nelson formula by path measures
.,..,,, 29?,
6
The Nelson model by path measures
293
6.1
Preliminaries
.............................. 293
χ
Contents
6.2
The Nelson model in Fock space
.................... 294
6.2.1
Definition
............................ 294
6.2.2
Infrared and ultraviolet divergences
.............. 296
6.2.3
Embedded eigenvalues
..................... 298
6.3
The Nelson model in function space
.................. 298
6.4
Existence and uniqueness of the ground state
............. 303
6.5
Ground state expectations
........................ 309
6.5.1
General theorems
........................ 309
6.5.2
Spatial decay of the ground state
................ 315
6.5.3
Ground state expectation for second quantized operators
... 316
6.5.4
Ground state expectation for field operators
.......... 322
6.6
The translation invariant Nelson model
................. 324
6.7
Infrared divergence
........................... 328
6.8
Ultraviolet divergence
.......................... 333
6.8.1
Energy renormalization
..................... 333
6.8.2
Regularized interaction
..................... 335
6.8.3
Removal of the ultraviolet cutoff
................ 339
6.8.4
Weak coupling limit and removal of ultraviolet cutoff
.... 344
7
The Pauli-Fierz model by path measures
351
7.1
Preliminaries
.............................. 351
7.1.1
Introduction
........................... 351
7.1.2
Lagrangian QED
........................ 352
7.1.3
Classical variant of non-relativistic QED
........... 356
7.2
The Pauli-Fierz model in non-relativistic QED
............ 359
7.2.1
The Pauli-Fierz model in Fock space
............. 359
7.2.2
The Pauli-Fierz model in function space
............ 363
7.2.3
Markov property
........................ 369
7.3
Functional integral representation for the Pauli-Fierz Hamiltonian
. . 372
7.3.1
Hubert space-valued stochastic integrals
............ 372
7.3.2
Functional integral representation
............... 375
7.3.3
Extension to general external potential
............. 381
7.4
Applications of functional integral representations
........... 382
7.4.1
Self-adjointness of the Pauli-Fierz Hamiltonian
........ 382
7.4.2
Positivity
improving and uniqueness of the ground state
. . . 392
7.4.3
Spatial decay of the ground state
................ 398
7.5
The Pauli-Fierz model with
Kato
class potential
............ 399
7.6
Translation invariant Pauli-Fierz model
................ 401
7.7
Path measure associated with the ground state
............. 408
7.7.1
Path measures with double stochastic integrals
........ 408
7.7.2
Expression in terms of iterated stochastic integrals
...... 412
7.7.3
Weak convergence of path measures
.............. 415
Contents_______________________________________________________________xi
7.8 Relativistic Pauli-Fierz
model
..................... 418
7.8.1 Definition............................ 418
7.8.2
Functional
integral
representation
............... 420
7.8.3 Translation invariant
case
.................... 423
7.9 The Pauli-Fierz
model with spin....................
424
7.9.1 Definition............................ 424
7.9.2
Symmetry and polarization
................... 427
7.9.3
Functional integral representation
............... 434
7.9.4
Spin-boson model
....................... 447
7.9.5
Translation invariant case
.................... 448
8
Notes and References
455
Bibliography
473
Index
499
|
| any_adam_object | 1 |
| author | Lőrinczi, József 1966- Hiroshima, Fumio Betz, Volker 1972- |
| author_GND | (DE-588)173059767 (DE-588)1016708270 (DE-588)123809568 |
| author_facet | Lőrinczi, József 1966- Hiroshima, Fumio Betz, Volker 1972- |
| author_role | aut aut aut |
| author_sort | Lőrinczi, József 1966- |
| author_variant | j l jl f h fh v b vb |
| building | Verbundindex |
| bvnumber | BV035778875 |
| classification_rvk | SK 820 SK 910 |
| ctrlnum | (OCoLC)634496548 (DE-599)DNB988967189 |
| dewey-full | 515.724 530.143 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis 530 - Physics |
| dewey-raw | 515.724 530.143 |
| dewey-search | 515.724 530.143 |
| dewey-sort | 3515.724 |
| dewey-tens | 510 - Mathematics 530 - Physics |
| discipline | Physik Mathematik |
| format | Book |
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| id | DE-604.BV035778875 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-20T13:59:59Z |
| institution | BVB |
| isbn | 9783110201482 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-018638432 |
| oclc_num | 634496548 |
| open_access_boolean | |
| owner | DE-384 DE-824 DE-83 DE-739 DE-19 DE-BY-UBM DE-11 |
| owner_facet | DE-384 DE-824 DE-83 DE-739 DE-19 DE-BY-UBM DE-11 |
| physical | XI, 505 S. 240 mm x 170 mm |
| publishDate | 2011 |
| publishDateSearch | 2011 |
| publishDateSort | 2011 |
| publisher | de Gruyter |
| record_format | marc |
| series | De Gruyter studies in mathematics |
| series2 | De Gruyter studies in mathematics |
| spellingShingle | Lőrinczi, József 1966- Hiroshima, Fumio Betz, Volker 1972- Feynman-Kac-type theorems and Gibbs measures on path space with applications to rigorous quantum field theory De Gruyter studies in mathematics Stochastische Analysis (DE-588)4132272-1 gnd Pfadintegral (DE-588)4173973-5 gnd Gibbs-Maß (DE-588)4157328-6 gnd Selbstadjungierter Operator (DE-588)4180810-1 gnd Feynman-Kac-Formel (DE-588)4820124-8 gnd Quantenfeldtheorie (DE-588)4047984-5 gnd |
| subject_GND | (DE-588)4132272-1 (DE-588)4173973-5 (DE-588)4157328-6 (DE-588)4180810-1 (DE-588)4820124-8 (DE-588)4047984-5 |
| title | Feynman-Kac-type theorems and Gibbs measures on path space with applications to rigorous quantum field theory |
| title_auth | Feynman-Kac-type theorems and Gibbs measures on path space with applications to rigorous quantum field theory |
| title_exact_search | Feynman-Kac-type theorems and Gibbs measures on path space with applications to rigorous quantum field theory |
| title_full | Feynman-Kac-type theorems and Gibbs measures on path space with applications to rigorous quantum field theory József Lörinczi ; Fumio Hiroshima ; Volker Betz |
| title_fullStr | Feynman-Kac-type theorems and Gibbs measures on path space with applications to rigorous quantum field theory József Lörinczi ; Fumio Hiroshima ; Volker Betz |
| title_full_unstemmed | Feynman-Kac-type theorems and Gibbs measures on path space with applications to rigorous quantum field theory József Lörinczi ; Fumio Hiroshima ; Volker Betz |
| title_short | Feynman-Kac-type theorems and Gibbs measures on path space |
| title_sort | feynman kac type theorems and gibbs measures on path space with applications to rigorous quantum field theory |
| title_sub | with applications to rigorous quantum field theory |
| topic | Stochastische Analysis (DE-588)4132272-1 gnd Pfadintegral (DE-588)4173973-5 gnd Gibbs-Maß (DE-588)4157328-6 gnd Selbstadjungierter Operator (DE-588)4180810-1 gnd Feynman-Kac-Formel (DE-588)4820124-8 gnd Quantenfeldtheorie (DE-588)4047984-5 gnd |
| topic_facet | Stochastische Analysis Pfadintegral Gibbs-Maß Selbstadjungierter Operator Feynman-Kac-Formel Quantenfeldtheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018638432&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000005407 |
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