Random walks, critical phenomena, and triviality in quantum field theory:
Gespeichert in:
| Beteiligte Personen: | , , |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Berlin ; Heidelberg ; New York ; London ; Paris ; Tokyo ; Hong Kong ; Barcelona ; Budapest
Springer
1992
|
| Schriftenreihe: | Texts and monographs in physics
|
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=019107951&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XIII, 444 S. |
| ISBN: | 3540543589 0387543589 |
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Datensatz im Suchindex
| _version_ | 1819276304601579520 |
|---|---|
| adam_text | CONTENTS
PART I CRITICAL PHENOMENA, QUANTUM FIELD THEORY, RANDOM WALKS
AND RANDOM SURFACES: SOME PERSPECTIVES
1. GENERAL INTRODUCTION 3
1.1 PHENOMENOLOGY OF PHASE TRANSITIONS
AND CRITICAL PHENOMENA 3
1.2 MULTICRITICAL POINTS 11
1.3 SPIN SYSTEMS, QUANTUM FIELD THEORY AND RANDOM WALKS:
AN OVERVIEW 16
1.4 CLASSICAL LATTICE SPIN SYSTEMS 17
1.5 RELATIVISTIC QUANTUM FIELD THEORY AND EUCLIDEAN FIELD THEORY 21
1.6 SOME RANDOM-WALK MODELS 32
2. PHASE TRANSITIONS AND CRITICAL POINTS IN CLASSICAL SPIN SYSTEMS:
A BRIEF SURVEY 37
2.1 EXISTENCE OF PHASE TRANSITIONS 37
2.2 EXISTENCE OF CRITICAL POINTS 47
2.3 CRITICAL EXPONENTS IN SOME EXACTLY SOLVABLE MODELS
AND STANDARD APPROXIMATIONS 47
2.4 BOUNDS ON CRITICAL EXPONENTS 51
3. SCALE TRANSFORMATIONS AND SCALING (CONTINUUM) LIMITS
IN LATTICE SPIN SYSTEMS 53
4. CONSTRUCTION OF SCALING LIMITS: THE RENORMALIZATION GROUP 59
4.1 BLOCK SPIN TRANSFORMATIONS 59
4.2 FIXED POINTS OF BLOCK SPIN TRANSFORMATIONS,
STABLE AND UNSTABLE MANIFOLDS, CRITICAL EXPONENTS 61
4.3 A BRIEF SKETCH OF THE TREE EXPANSION 68
4.4 RIGOROUS USES OF BLOCK SPIN TRANSFORMATIONS 76
5. RANDOM WALKS AS EUCLIDEAN FIELD THEORY (EFT) 79
5.1 DEFINITION OF THE MODEL AND STATEMENT OF RESULTS 79
5.2 HEURISTIC RENORMALIZATION-GROUP ARGUMENT 81
5.3 PROOF OF LOWER BOUND 85
HTTP://D-NB.INFO/911477179
X
CONTENTS
5.4 PROOF OF UPPER BOUND 89
5.4.1 EXPECTED NUMBER OF INTERSECTIONS 90
5.4.2 RIGOROUS RENORMILIZATION-GROUP ARGUMENT 91
5.4.3 DIRECT INCLUSION-EXCLUSION ARGUMENT 94
6. EFT AS A GAS OF RANDOM WALKS WITH HARD-CORE INTERACTIONS 101
6.1 THE SYMANZIK-BFS RANDOM-WALK REPRESENTATION 101
6.2 CONSEQUENCES FOR TRIVIALITY 106
6.3 CONSEQUENCES FOR NONTRIVIALITY 112
7. RANDOM-SURFACE MODELS 117
7.1 CONTINUUM RANDOM-SURFACE ACTIONS 118
7.2 RANDOM-SURFACE MODELS IN 121
7.2.1 BASIC DEFINITIONS 121
7.2.2 A MEAN-FIELD THEORY FOR RANDOM-SURFACE MODELS 124
7.2.3. THE PLANAR RANDOM-SURFACE MODEL 128
7.3 TYPICAL PHENOMENA IN RANDOM-SURFACE THEORY 132
7.4 RANDOMLY TRIANGULATED RANDOM-SURFACE MODELS 135
7.4.1 DEFINITION OF THE MODEL 135
7.4.2 PROPERTIES OF THE MODEL 138
7.5 RANDOM-MATRIX MODELS 142
7.5.1 MATRIX FIELD THEORIES 142
7.5.2 RANDOM-MATRIX MODELS AND RANDOM TRIANGULATIONS:
PURE GRAVITY 148
7.5.3 RANDOM-MATRIX MODELS FOR GRAVITY COUPLED
TO MATTER FIELDS: POTTS SPINS 151
7.5.4 RANDOM-MATRIX MODELS FOR GRAVITY COUPLED
TO MATTER FIELDS: POLYAKOV STRING 153
7.6 THE TOPOLOGICAL EXPANSION: NON-PERTURBATIVE RESULTS
FOR THE PURE GRAVITY CASE 155
7.6.1 SIMPLE VS. DOUBLE SCALING LIMIT 155
7.6.2 THE MATHEMATICAL TOOLBOX 157
7.6.3 THE SIMPLE SCALING LIMIT 163
7.6.4 THE DOUBLE SCALING LIMIT 165
7.6.5 OTHER APPROACHES TO THE DOUBLE SCALING LIMIT 170
7.6.6 PERTURBATION OF THE STRING EQUATION AND KDV FLOW ... 174
7.6.7 EPILOGUE 176
PART II RANDOM-WALK MODELS AND RANDOM-WALK REPRESENTATIONS
OF CLASSICAL LATTICE SPIN SYSTEMS
8. INTRODUCTION 181
9. RANDOM-WALK MODELS IN THE ABSENCE OF MAGNETIC FIELD 189
9.1 GENERAL DEFINITIONS 189
CONTENTS XI
9.2 EXAMPLES 192
9.2.1 POLYMER-CHAIN MODELS 192
9.2.2 BFS REPRESENTATION FOR CONTINUOUS SPIN SYSTEMS 195
9.2.3 THE ARW REPRESENTATION FOR THE ISING MODEL 201
10. RANDOM-WALK MODELS IN THE PRESENCE OF A MAGNETIC FIELD 205
10.1 GENERAL DEFINITIONS 205
10.2 EXAMPLES 207
10.2.1 POLYMER-CHAIN MODELS 207
10.2.2 BABY POLYMER-CHAIN MODELS 208
10.2.3 BFS RANDOM-WALK MODELS 208
10.2.4 ARW MODEL 210
11. FACTORIZATION AND DIFFERENTIATION OF THE WEIGHTS 213
11.1 INEQUALITIES INVOLVING THE PARTITION OF A FAMILY OF WALKS 213
11.1.1 REPULSIVENESS (OR REPULSIVENESS ON THE AVERAGE ) .... 213
11.1.2 ATTRACTIVENESS (OR NONINTERACTION)
BETWEEN NONOVERLAPPING (OR COMPATIBLE) WALKS 218
11.2 INEQUALITIES INVOLVING THE SPLITTING OF A WALK 220
11.3 DIFFERENTIATION OF THE WEIGHTS WITH RESPECT TO J OR H 223
11.3.1 DIFFERENTIATION WITH RESPECT TO J 223
11.3.2 DIFFERENTIATION WITH RESPECT TO H 226
12. CORRELATION INEQUALITIES: A SURVEY OF RESULTS 229
12.1 GAUSSIAN UPPER BOUNDS 230
12.2 TRUNCATED FOUR-POINT FUNCTION IN ZERO MAGNETIC FIELD:
LEBOWITZ AND AIZENMAN-FROHLICH INEQUALITIES 231
12.2.1 UPPER BOUND (LEBOWITZ INEQUALITY) 231
12.2.2 NONTRIVIAL LOWER BOUNDS
(AIZENMAN-FROHLICH INEQUALITIES) 232
12.2.3 ONCE-IMPROVED AIZENMAN-FROHLICH INEQUALITY
(AIZENMAN-GRAHAM INEQUALITY) 236
12.2.4 TWICE-IMPROVED AIZENMAN-FROHLICH INEQUALITY 240
12.3 INEQUALITIES INVOLVING INFINITELY MANY ORDERS
IN THE EXPANSION PARAMETER (NON-GAUSSIAN UPPER BOUNDS) .. 241
12.3.1 GENERAL SETUP FOR THE DILUTION TRICK 242
12.3.2 UPPER BOUND ON THE TRUNCATED FOUR-POINT FUNCTION
FOR THE INTERSECTION PROPERTIES
OF ORDINARY RANDOM WALKS (IPORW MODEL) 246
12.3.3 UPPER BOUND ON THE TRUNCATED FOUR-POINT FUNCTION
FOR THE SAW 249
12.3.4 UPPER BOUND ON THE TRUNCATED FOUR-POINT FUNCTION
FOR THE ARW (ISING) MODEL 252
12.4 THE KERNEL K(X,Y) AND THE TRUNCATED TWO-POINT FUNCTION
IN NONZERO MAGNETIC FIELD 253
12.4.1 THE KERNEL K(X,Y) AND THE MAGNETIZATION 253
XII
CONTENTS
12.4.2 THE KERNEL K(X,Y) AND THE TWO-POINT FUNCTION 254
12.4.3 MONOTONICITY OF K(X,Y) IN H,
AND ANOTHER UPPER BOUND ON K 254
12.4.4 THE KERNEL K(X,Y) AND THE TRUNCATED TWO-POINT
FUNCTION [LOWER BOUND ON K(X,Y) 254
12.4.5 NON-TRIVIAL UPPER BOUND ON K(X, Y) 257
12.4.6 ONCE-IMPROVED UPPER BOUND ON K(X,Y) 258
12.5 TRUNCATED THREE-POINT FUNCTION 260
12.6 TRUNCATED GREEN FUNCTIONS OF HIGHER ORDER 263
12.7 RELATIONSHIP BETWEEN THE TRUNCATED GREEN FUNCTIONS,
THE TRUNCATED KERNELS AND THE TRUNCATED WEIGHTS 265
12.7.1 ALGEBRAIC DEFINITION OF TRUNCATION 265
12 7.2 RELATION BETWEEN TRUNCATION IN DIFFERENT ALGEBRAS .... 267
12 7.3 RELATIONS BETWEEN THE TRUNCATION
FOR GREEN FUNCTIONS, KERNELS AND WEIGHTS 268
PART III CONSEQUENCES FOR CRITICAL PHENOMENA AND QUANTUM FIELD THEORY
13. BACKGROUND MATERIAL 275
13.1 MODELS TO BE CONSIDERED 275
13.2 CRITICAL EXPONENTS 278
13.3 SUMMARY OF CORRELATION INEQUALITIES 283
13.4 REFLECTION POSITIVITY, SPECTRAL REPRESENTATIONS
AND INFRARED BOUNDS 288
14. INEQUALITIES FOR CRITICAL EXPONENTS 297
14.1 SYMMETRIC REGIME 299
14.1.1 UPPER BOUND ON THE BUBBLE DIAGRAM
(CRITICAL EXPONENT B) 299
14.1.2 LOWER BOUND ON THE BUBBLE DIAGRAM
(CRITICAL EXPONENT B) 303
14.1.3 UPPER BOUND ON THE SPECIFIC HEAT
(CRITICAL EXPONENT A) 304
14.1.4 BOUNDS ON THE SUSCEPTIBILITY
(CRITICAL EXPONENT
7)
304
11.1.5 UPPER BOUNDS ON THE RENORMALIZED COUPLING
CONSTANT (CRITICAL EXPONENT A4) 306
14.1.6 NON-GAUSSIAN LOWER BOUND ON THE RENORMALIZED
COUPLING CONSTANT (CRITICAL EXPONENT A4) 309
14.2 EXTRAPOLATION PRINCIPLES 310
14.2.1 REVIEW OF LANDAU-GINZBURG THEORY 311
14.2.2 CONSEQUENCES OF THE GHS INEQUALITY 320
14.2.3 CONSEQUENCES OF THE ABF INEQUALITY 328
14.3 NON-SYMMETRIC REGIME: STANDARD APPROACHES
TO THE CRITICAL POINT 335
CONTENTS XIII
14.3.1 UPPER BOUND ON THE TRUNCATED BUBBLE DIAGRAM
(CRITICAL EXPONENTS B AND B
C
) 337
14.3.2 LOWER BOUND ON THE TRUNCATED BUBBLE DIAGRAM
(CRITICAL EXPONENTS B AND B
C
) 338
14.3.3 UPPER BOUND ON THE SPECIFIC HEAT
(CRITICAL EXPONENTS A! AND A
C
) 338
14.3.4 BOUNDS ON THE SUSPCEPTIBILITY (CRITICAL EXPONENT
7 )
.. 339
14.3.5 OTHER HYPERSCALING INEQUALITIES 340
14.4 NON-SYMMETRIC REGIME: BOUNDS IN A FULL NEIGHBORHOOD
OF THE CRITICAL POINT 341
14.4.1 LOWER BOUNDS ON THE MAGNETIZATION 341
14.4.2 UPPER BOUNDS ON THE MAGNETIZATION 346
14.5 HORNS AND ALL THAT 351
14.5.1 THE FUNDAMENTAL QUANTITIES 351
14.5.2 HORNS AND THE SUSCEPTIBILITY 352
14.6 GENERAL CONSIDERATIONS ON EXTRAPOLATION PRINCIPLES 354
14.6.1 EXTRAPOLATION PRINCIPLES
FOR A SINGLE ORDINARY DIFFERENTIAL EQUATION 355
14.6.2 CONSERVATION LAWS. CHARACTERISTICS 357
14.6.3 SUB- AND SUPRA-CONSERVATION LAWS.
EXTRAPOLATION PRINCIPLES AND COMPARISON THEOREMS ... 359
15. CONTINUUM LIMITS 367
15.1 GENERALITIES ON CONTINUUM LIMITS 367
15.1.1 WHAT IS A CONTINUUM LIMIT? WHAT IS TRIVIALITY? 367
15.1.2 PRECISE MATHEMATICAL FORMULATION I:
THE CONTINUUM LIMIT 371
15.1.3 PRECISE MATHEMATICAL FORMULATION II:
NECESSARY CONDITIONS
FOR NON-BORING LIMIT 375
15.1.4 PRECISE MATHEMATICAL FORMULATION III:
CONVERGENCE OF SUMMED QUANTITIES 380
15.1.5 SOME CRITERIA FOR TRIVIALITY 385
15.1.6 SUMMARY OF WHAT IS KNOWN 387
15.2 TRIVIALITY OF P (D 4) IN THE SYMMETRIC PHASE 390
15.3 NON-SYMMETRIC REGIME FOR D 4 399
15.3.1 CASES OF NON-EXISTENCE OF BROKEN-SYMMETRY PHASE . . . 399
15.3.2 PARTIAL RESULTS ON TRIVIALITY 401
REFERENCES 405
INDEX 431
|
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| dewey-ones | 530 - Physics |
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| dewey-sort | 3530.143 |
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| id | DE-604.BV025860226 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-20T14:08:57Z |
| institution | BVB |
| isbn | 3540543589 0387543589 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-019107951 |
| oclc_num | 246778264 |
| open_access_boolean | |
| owner | DE-11 |
| owner_facet | DE-11 |
| physical | XIII, 444 S. |
| publishDate | 1992 |
| publishDateSearch | 1992 |
| publishDateSort | 1992 |
| publisher | Springer |
| record_format | marc |
| series2 | Texts and monographs in physics |
| spellingShingle | Fernández, Roberto Fröhlich, Jürg Sokal, Alan D. 1955- Random walks, critical phenomena, and triviality in quantum field theory Statistische Mechanik (DE-588)4056999-8 gnd Kritisches Phänomen (DE-588)4165788-3 gnd Gleichgewicht (DE-588)4121372-5 gnd Quantenfeldtheorie (DE-588)4047984-5 gnd Irrfahrtsproblem (DE-588)4162442-7 gnd |
| subject_GND | (DE-588)4056999-8 (DE-588)4165788-3 (DE-588)4121372-5 (DE-588)4047984-5 (DE-588)4162442-7 |
| title | Random walks, critical phenomena, and triviality in quantum field theory |
| title_auth | Random walks, critical phenomena, and triviality in quantum field theory |
| title_exact_search | Random walks, critical phenomena, and triviality in quantum field theory |
| title_full | Random walks, critical phenomena, and triviality in quantum field theory R. Fernández ; J. Fröhlich ; A. D. Sokal |
| title_fullStr | Random walks, critical phenomena, and triviality in quantum field theory R. Fernández ; J. Fröhlich ; A. D. Sokal |
| title_full_unstemmed | Random walks, critical phenomena, and triviality in quantum field theory R. Fernández ; J. Fröhlich ; A. D. Sokal |
| title_short | Random walks, critical phenomena, and triviality in quantum field theory |
| title_sort | random walks critical phenomena and triviality in quantum field theory |
| topic | Statistische Mechanik (DE-588)4056999-8 gnd Kritisches Phänomen (DE-588)4165788-3 gnd Gleichgewicht (DE-588)4121372-5 gnd Quantenfeldtheorie (DE-588)4047984-5 gnd Irrfahrtsproblem (DE-588)4162442-7 gnd |
| topic_facet | Statistische Mechanik Kritisches Phänomen Gleichgewicht Quantenfeldtheorie Irrfahrtsproblem |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=019107951&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT fernandezroberto randomwalkscriticalphenomenaandtrivialityinquantumfieldtheory AT frohlichjurg randomwalkscriticalphenomenaandtrivialityinquantumfieldtheory AT sokalaland randomwalkscriticalphenomenaandtrivialityinquantumfieldtheory |