Path integral approach to quantum physics: an introduction
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | German English |
| Published: |
Berlin [u.a.]
Springer
1996
|
| Edition: | Study ed., 1. ed., 2. printing |
| Series: | Texts and monographs in physics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 387 S. graph. Darst. |
| ISBN: | 3540611061 9783540611066 |
Staff View
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Record in the Search Index
| _version_ | 1804125254035963904 |
|---|---|
| adam_text | GERT ROEPSTORFF
PATH INTEGRAL APPROACH
TO QUANTUM PHYSICS
AN INTRODUCTION
WITH 26 FIGURE
S
SPRINGER
CONTENTS
1. BROWNIAN MOTION
1
1.1 THE ONE-DIMENSIONAL RANDOM WALK 2
1.2 MULTIDIMENSIONAL RANDOM WALK 6
1.3 GENERATING FUNCTIONS 10
1.3.1 RETUR
N OR ESCAPE? 11
1.4 THE CONTINUUM LIMIT 13
1.5 IMAGINARY TIME 15
1.6 THE WIENER PROCESS 20
1.6.1 THE ANALYSIS OF RANDOM PATH
S 20
1.6.2 MULTIDIMENSIONAL GAUSSIAN MEASURES 24
1.6.3 INCREMENTS 27
1.7 EXPECTATION VALUES 29
1.8 TH
E ORNSTEIN-UHLENBECK PROCESS 33
1.8.1 THE OSCILLATOR PROCESS 36
2. THE FEYNMAN-KAC FORMULA
39
2.1 THE CONDITIONAL WIENER MEASURE 40
2.1.1 TH
E PAT
H INTEGRAL 41
2.1.2 THE STOCHASTIC FORMULATION 45
2.2 TH
E INTEGRAL EQUATION METHOD 48
2.2.1 STOCHASTIC REPRESENTATION OF OPERATOR NORMS ..
. 51
2.2.2 STOCHASTIC REPRESENTATION OF GREEN S FUNCTIONS . 53
2.3 THE LIE-TROTTER PRODUCT METHOD 54
2.3.1 THE LIE-TROTTER PRODUCT FORMULA 55
2.3.2 MISCELLANEOUS REMARKS AND RESULTS 57
2.3.3 SEVERAL PARTICLES WITH DIFFERENT MASSES 61
2.4 THE BROWNIAN TUBE 62
2.5 THE GOLDEN-THOMPSON-SYMAHZIK BOUND 66
2.6 HAMILTONIANS AND THEIR ASSOCIATED PROCESSES 74
2.6.1 CORRELATION FUNCTIONS 75
2.6.2 THE OSCILLATOR PROCESS REVISITED 76
2.6.3 NONLINEAR TRANSFORMATIONS OF TIME 78
2.6.4 THE PERTURBED HARMONIC OSCILLATOR 79
2.7 THE THERMODYNAMICAL FORMALISM 81
2.8 A CASE STUDY: TH
E HARMONIC SPIN CHAIN 85
2.8.1 THE INVERTED HARMONIC OSCILLATOR 88
X CONTENTS
- 2.9 THE REFLECTION PRINCIPLE 90
2.9.1 REFLECTION GROUPS OF ORDER TWO 91
2.9.2 REFLECTION GROUPS OF INFINITE ORDER 94
2.10 FEYNMAN VERSUS WIENER INTEGRALS 101
2.10.1 SUMMING OVER HISTORIES IN YYCONFIGURATION SPACE . 102
2.10.2 THE METHOD OF STATIONARY PHASE 104
2.10.3 SUMMING OVER HISTORIES IN PHASE SPACE 105
2.10.4 THE FEYNMAN INTEGRAND AS A HIDA DISTRIBUTION . 107
3. THE BROWNIAN BRIDGE
109
3.1 THE CANONICAL SCALING OF BROWNIAN PATH
S 109
3.1.1 THE PROCESS
X
T
112
3.1.2 RESCALING OF PAT
H INTEGRALS 113
3.1.3 THE STOCHASTIC INTEGRAL WITH RESPECT
T
O TH
E BROWNIAN BRIDGE 114
3.2 BOUNDS ON TH
E TRANSITION AMPLITUDE 115
3.2.1 DEFINING A SUBSET OF PATH
S 115
3.2.2 THE SEMICLASSICAL APPROXIMATION 117
3.2.3 BOUNDS ON TH
E FUNCTIONAL
$(V)
118
3.2.4 CONVEXITY OF TH
E FUNCTIONAL
$(V)
120
3.3 VARIATIONAL PRINCIPLES 122
3.3.1 THE MEAN POSITION OF A PAT
H 125
3.4 BOUND STATES 126
3.4.1 MOMENT INEQUALITIES FOR EIGENVALUES 135
3.5 MONTE CARLO CALCULATION OF PAT
H INTEGRALS 142
4. FOURIER DECOMPOSITION
150
4.1 RANDOM FOURIER COEFFICIENTS 150
4.1.1 FOURIER ANALYSIS OF TIME INTEGRALS 151
4.2 THE WIGNER-KIRKWOOD EXPANSION
OF TH
E EFFECTIVE POTENTIAL 154
4.3 COUPLED SYSTEMS 157
4.3.1 OPEN SYSTEMS 159
4.4 THE DRIVEN HARMONIC OSCILLATOR 161
4.4.1 FROM TIME INTEGRALS T
O SUMS 162
4.4.2 FROM SUMS BACK T
O TIME INTEGRALS 162
4.5 OSCILLATING ELECTRIC FIELDS 166
4.5.1 POISSON STATISTICS 167
5. THE LINEAR-COUPLING THEORY OF BOSONS
170
5.1 PAT
H INTEGRALS FOR BOSONS 170
5.1.1 THE PARTIA
L TRACE AND IT
S EVALUATION 172
5.2 A RANDOM POTENTIAL FOR TH
E ELECTRON 175
5.3 THE POLARON PROBLEM 177
5.3.1 THE LIMIT
L -*
OO,
B
- 0 179
CONTENTS XI
5.3.2 THE FREE ENERGY OF TH
E POLARON 182
5.3.3 BOUNDS ON TH
E POLARON FREE ENERGY 183
5.3.4 PEKAR S LARGE-COUPLING RESULT 185
5.4 THE FIELD THEORY OF TH
E POLARON MODEL 186
6. MAGNETIC FIELDS
192
6.1 HEURISTIC CONSIDERATIONS 192
6.2 ITO INTEGRALS 195
6.2.1 THE FEYNMAN-KAC-ITO FORMULA 197
6.2.2 THE SEMICLASSICAL APPROXIMATION 199
6.3 THE CONSTANT MAGNETIC FIELD 201
6.3.1 A BRIEF DISCUSSION OF TH
E RESULT 204
6.4 DIAMAGNETISM OF ELECTRONS IN A SOLID 205
6.5 MAGNETIC FLUX LINES 208
6.5.1 WINDING NUMBERS 209
6.5.2 SPECTRAL DECOMPOSITION 210
6.5.3 IMAGINARY TIMES 212
7. EUCLIDEAN FIELD THEORY
215
7.1 WHA
T IS A EUCLIDEAN FIELD? 216
7.2 TH
E EUCLIDEAN TWO-POINT FUNCTION 218
7.3 TH
E EUCLIDEAN FREE FIELD 222
7.3.1 THE N-POINT FUNCTIONS 222
7.3.2 THE STOCHASTIC INTERPRETATION 225
7.4 GAUSSIAN FUNCTIONAL INTEGRALS 227
7.5 BASIC POSTULATES 233
7.5.1 THE HAMILTONIAN 236
7.5.2 THE FREE FIELD REVISITED = 238
8. FIELD THEORY ON A LATTICE
242
8.1 THE LATTICE VERSION OF TH
E SCALAR FIELD 242
8.2 THE EUCLIDEAN PROPAGATOR ON TH
E LATTICE 245
8.2.1 THE FOURIER REPRESENTATION 245
8.2.2 RANDOM PATH
S ON A LATTICE 250
8.3 THE VARIATIONAL PRINCIPLE 252
8.3.1 THE CASE OF A DISCRETE CONFIGURATION SPACE ...
. 252
8.3.2 THE DETERMINISTIC LIMIT 254
8.3.3 CONTINUOUS CONFIGURATION SPACE 255.
8.3.4 TH
E CLASSICAL LIMIT 257
8.3.5 FLUCTUATIONS AROUND TH
E CLASSICAL SOLUTION 258
8.4 TH
E EFFECTIVE ACTION 260
8.5 THE EFFECTIVE POTENTIAL 265
8.5.1 SPONTANEOUS BREAKDOWN OF SYMMETRY 266
8.5.2 ORDER PARAMETERS , 267
8.6 THE GINZBURG-LANDAU EQUATIONS 268
XII CONTENTS
8.7 TH
E MEAN-FIELD APPROXIMATION 272
8.7.1 THE CURIE-WEISS APPROXIMATION
OF TH
E ISING MODEL 274
8.7.2 THE ISING SPIN LIMIT OF TH
E NEUTRAL SCALAR FIELD 277
8.8 THE GAUSSIAN APPROXIMATION 278
8.8.1 A CASE STUDY 278
9. THE QUANTIZATION OF GAUGE THEORIES
281
9.1 TH
E EUCLIDEAN VERSION OF MAXWELL THEORY 281
9.1.1 THE CLASSICAL SITUATION
(H
= 0) 282
9.1.2 GAUGE FIXING 285
9.1.3 THE QUANTIZED SITUATION
(H
0) 287
9.2 NON-ABELIAN GAUGE THEORIES: PRELIMINARIES 289
9.3 TH
E FADDEEV-POPOV QUANTIZATION 292
9.3.1 DIVISION BY
Q
295
9.3.2 FADDEEV-POPOV GHOSTS 297
9.4 GAUGE THEORIES ON A LATTICE 300
9.5 WEGNER-WILSON LOOPS 306
9.5.1 THE STATIC APPROXIMATION IN MINKOWSKIAN
FIELD THEORY 306
9.5.2 LOOP VARIABLES IN EUCLIDEAN QED 308
9.5.3 AREA LAW OR PERIMETER LAW? 310
9.6 THE
SU(N)
HIGGS MODEL 312
10. FERMIONS
316
10.1 THE DIRAC FIELD IN MINKOWSKI SPACE 316
10.2 THE EUCLIDEAN DIRAC FIELD 319
10.2.1 EXTERNAL VECTOR POTENTIALS 324
10.3 GRASSMANN ALGEBRAS 326
10.3.1 WHEN
E
IS A FUNCTION SPACE 329
10.4 FORMAL DERIVATIVES 331
10.5 FORMAL INTEGRATION 334
10.5.1 INTEGRALS IN
A(E)
334
10.5.2 INTEGRALS IN
A(E F)
336
10.5.3 INTEGRALS OF TH
E EXPONENTIAL TYPE 337
10.5.4 THE FOURIER-LAPLACE TRANSFORMATION 339
10.6 FUNCTIONAL INTEGRALS OF QED 342
10.7 THE
SU(N)
GAUGE THEORY WITH FERMIONS 346
APPENDICES
A LIST OF SYMBOLS AND GLOSSARY 349
B FREQUENTLY USED GAUSSIAN PROCESSES 357
C JENSEN S INEQUALITY 360
D A TABLE OF PAT
H INTEGRALS 362
CONTENTS XIII
REFERENCES 369
INDEX 383
|
| any_adam_object | 1 |
| author | Roepstorff, Gert 1937- |
| author_GND | (DE-588)172334942 |
| author_facet | Roepstorff, Gert 1937- |
| author_role | aut |
| author_sort | Roepstorff, Gert 1937- |
| author_variant | g r gr |
| building | Verbundindex |
| bvnumber | BV010772917 |
| classification_rvk | UK 4500 |
| classification_tum | PHY 020f PHY 013f |
| ctrlnum | (OCoLC)75670981 (DE-599)BVBBV010772917 |
| dewey-full | 530.12 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 530 - Physics |
| dewey-raw | 530.12 |
| dewey-search | 530.12 |
| dewey-sort | 3530.12 |
| dewey-tens | 530 - Physics |
| discipline | Physik |
| edition | Study ed., 1. ed., 2. printing |
| format | Book |
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| id | DE-604.BV010772917 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T17:58:38Z |
| institution | BVB |
| isbn | 3540611061 9783540611066 |
| language | German English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007194809 |
| oclc_num | 75670981 |
| open_access_boolean | |
| owner | DE-20 DE-703 DE-19 DE-BY-UBM DE-384 DE-29T DE-355 DE-BY-UBR DE-634 DE-83 |
| owner_facet | DE-20 DE-703 DE-19 DE-BY-UBM DE-384 DE-29T DE-355 DE-BY-UBR DE-634 DE-83 |
| physical | XIII, 387 S. graph. Darst. |
| publishDate | 1996 |
| publishDateSearch | 1996 |
| publishDateSort | 1996 |
| publisher | Springer |
| record_format | marc |
| series2 | Texts and monographs in physics |
| spelling | Roepstorff, Gert 1937- Verfasser (DE-588)172334942 aut Pfadintegrale in der Quantenphysik Path integral approach to quantum physics an introduction Gert Roepstorff Study ed., 1. ed., 2. printing Berlin [u.a.] Springer 1996 XIII, 387 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Texts and monographs in physics Quantenfeldtheorie (DE-588)4047984-5 gnd rswk-swf Quantenphysik (DE-588)4266670-3 gnd rswk-swf Quantentheorie (DE-588)4047992-4 gnd rswk-swf Pfadintegral (DE-588)4173973-5 gnd rswk-swf Quantenmechanik (DE-588)4047989-4 gnd rswk-swf Pfadintegral (DE-588)4173973-5 s Quantenphysik (DE-588)4266670-3 s DE-604 Quantentheorie (DE-588)4047992-4 s 1\p DE-604 Quantenmechanik (DE-588)4047989-4 s 2\p DE-604 Quantenfeldtheorie (DE-588)4047984-5 s 3\p DE-604 DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007194809&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Roepstorff, Gert 1937- Path integral approach to quantum physics an introduction Quantenfeldtheorie (DE-588)4047984-5 gnd Quantenphysik (DE-588)4266670-3 gnd Quantentheorie (DE-588)4047992-4 gnd Pfadintegral (DE-588)4173973-5 gnd Quantenmechanik (DE-588)4047989-4 gnd |
| subject_GND | (DE-588)4047984-5 (DE-588)4266670-3 (DE-588)4047992-4 (DE-588)4173973-5 (DE-588)4047989-4 |
| title | Path integral approach to quantum physics an introduction |
| title_alt | Pfadintegrale in der Quantenphysik |
| title_auth | Path integral approach to quantum physics an introduction |
| title_exact_search | Path integral approach to quantum physics an introduction |
| title_full | Path integral approach to quantum physics an introduction Gert Roepstorff |
| title_fullStr | Path integral approach to quantum physics an introduction Gert Roepstorff |
| title_full_unstemmed | Path integral approach to quantum physics an introduction Gert Roepstorff |
| title_short | Path integral approach to quantum physics |
| title_sort | path integral approach to quantum physics an introduction |
| title_sub | an introduction |
| topic | Quantenfeldtheorie (DE-588)4047984-5 gnd Quantenphysik (DE-588)4266670-3 gnd Quantentheorie (DE-588)4047992-4 gnd Pfadintegral (DE-588)4173973-5 gnd Quantenmechanik (DE-588)4047989-4 gnd |
| topic_facet | Quantenfeldtheorie Quantenphysik Quantentheorie Pfadintegral Quantenmechanik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007194809&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT roepstorffgert pfadintegraleinderquantenphysik AT roepstorffgert pathintegralapproachtoquantumphysicsanintroduction |