Phase transitions and renormalization group:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Oxford [u.a.]
Oxford Univ. Press
2007
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| Ausgabe: | 1. publ. |
| Schriftenreihe: | Oxford graduate texts
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Hier auch später erschienene, unveränderte Nachdrucke |
| Beschreibung: | XII, 452 S. |
| ISBN: | 9780199227198 9780199665167 |
Internformat
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| 100 | 1 | |a Zinn-Justin, Jean |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Phase transitions and renormalization group |c Jean Zinn-Justin |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Oxford [u.a.] |b Oxford Univ. Press |c 2007 | |
| 300 | |a XII, 452 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Oxford graduate texts | |
| 500 | |a Hier auch später erschienene, unveränderte Nachdrucke | ||
| 650 | 4 | |a Phase transformations (Statistical physics) | |
| 650 | 4 | |a Renormalization (Physics) | |
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Datensatz im Suchindex
| _version_ | 1804136773962432512 |
|---|---|
| adam_text | Contents
1 Quantum
field theory and the renormaiization group
.........1
1.1
Quantum electrodynamics: A quantum field theory
.........3
1.2
Quantum electrodynamics: The problem of infinities
........4
1.3
Renormaiization
........................7
1.4
Quantum field theory and the renormaiization group
........9
1.5
A triumph of QFT: The Standard Model
.............10
1.6
Critical phenomena: Other infinities
...............12
1.7
Kadanoff and Wilson s renormaiization group
...........14
1.8
Effective quantum field theories
.................16
2
Gaussian expectation values. Steepest descent method
........19
2.1
Generating functions
...................... 19
2.2
Gaussian expectation values. Wick s theorem
........... 20
2.3
Perturbed Gaussian measure. Connected contributions
....... 24
2.4
Feynman diagrams. Connected contributions
............ 25
2.5
Expectation values. Generating function.
Cumulants
........ 28
2.6
Steepest descent method
.................... 31
2.7
Steepest descent method: Several variables, generating functions
... 37
Exercises
............................. 40
3
Universality and the continuum limit
................. 45
3.1
Central limit theorem of probabilities
...............45
3.2
Universality and fixed points of transformations
..........54
3.3
Random walk and Brownian motion
...............59
3.4
Random walk: Additional remarks
................71
3.5
Brownian motion and path integrals
...............72
Exercises
.............................75
4
Classical statistical physics: One dimension
..............79
4.1
Nearest-neighbour interactions. Transfer matrix
.......... 80
4.2
Correlation functions
...................... 83
4.3
Thermodynamic limit
...................... 85
4.4
Connected functions and cluster properties
............ 88
4.5
Statistical models: Simple examples
............... 90
4.6
The Gaussian model
...................... 92
x
Contents
4.7
Gaussian model: The continuum limit
...............98
4.8
More general models: The continuum limit
........... 102
Exercises
............................ 104
5
Continuum limit and path integrals
................
Ill
5.1
Gaussian path integrals
....................
Ill
5.2
Gaussian correlations. Wick s theorem
............. 118
5.3
Perturbed Gaussian measure
.................. 118
5.4
Perturbative calculations: Examples
.............. 120
Exercises
............................ 124
6
Ferromagnetic systems. Correlation functions
........... 127
6.1
Ferromagnetic systems: Definition
............... 127
6.2
Correlation functions. Fourier representation
........... 133
6.3
Legendre transformation and vertex functions
.......... 137
6.4
Legendre transformation and steepest descent method
....... 142
6.5
Two- and four-point vertex functions
.............. 143
Exercises
............................ 145
7
Phase transitions: Generalities and examples
............ 147
7.1
Infinite temperature or independent spins
............ 150
7.2
Phase transitions in infinite dimension
............. 153
7.3
Universality in infinite space dimension
............. 158
7.4
Transformations, fixed points and universality
.......... 161
7.5
Finite-range interactions in finite dimension
........... 163
7.6
Ising model: Transfer matrix
.................. 166
7.7
Continuous symmetries and transfer matrix
........... 171
7.8
Continuous symmetries and
Goldstone
modes
.......... 173
Exercises
............................ 175
8
Quasi-Gaussian approximation: Universality, critical dimension
.... 179
8.1
Short-range two-spin interactions
................ 181
8.2
The Gaussian model: Two-point function
............ 183
8.3
Gaussian model and random walk
............... 188
8.4
Gaussian model and field integral
................ 190
8.5
Quasi-Gaussian approximation
................. 194
8.6
The two-point function: Universality
.............. 196
8.7
Quasi-Gaussian approximation and Landau s theory
....... 199
8.8
Continuous symmetries and
Goldstone
modes
.......... 200
8.9
Corrections to the quasi-Gaussian approximation
......... 202
8.10
Mean-field approximation and corrections
........... 207
8.11
Tricritical points
...................... 211
Exercises
............................ 212
9
Renormalization group: General formulation
............ 217
9.1
Statistical field theory. Landau s Hamiltonian
.......... 218
9.2
Connected correlation functions. Vertex functions
........ 220
9.3
Renormalization group: General idea
.............. 222
9.4
Hamiltonian flow: Fixed points, stability
............ 226
9.5
The Gaussian fixed point
................... 231
Contents xi
9.6 Eigen-perturbations: General
analysis
.............. 234
9.7
A non-Gaussian fixed point: The
ε
-expansion ..........
237
9.8
Eigenvalues and dimensions of local polynomials
......... 241
10
Perturbative renormalization group: Explicit calculations
...... 243
10.1
Critical Hamiltonian and perturbative expansion
........ 243
10.2
Feynman diagrams at one-loop order
.............. 246
10.3
Fixed point and critical behaviour
............... 248
10.4
Critical domain
....................... 254
10.5
Models with O(N) orthogonal symmetry
............ 258
10.6
Renormalization group near dimension
4 ............ 259
10.7
Universal quantities: Numerical results
............. 262
11
Renormalization group: iV-component fields
............ 267
11.1
Renormalization group: General remarks
............ 268
11.2
Gradient flow
........................ 269
11.3
Model with cubic anisotropy
................. 272
11.4
Explicit general expressions: RG analysis
............ 276
11.5
Exercise: General model with two parameters
.......... 281
Exercises
............................ 284
12
Statistical field theory: Perturbative expansion
.......... 285
12.1
Generating
funcţionale
.................... 285
12.2
Gaussian field theory. Wick s theorem
............. 287
12.3
Perturbative expansion
.................... 289
12.4
Loop expansion
....................... 296
12.5
Dimensional continuation and regularization
.......... 299
Exercises
............................ 306
13
The
σ4
field theory near dimension
4............... 307
13.1
Effective Hamiltonian. Renormalization
............ 308
13.2
Renormalization group equations
............... 313
13.3
Solution of RGE: The
ε
-expansion...............
316
13.4
Effective and renormalized interactions
............. 323
13.5
The critical domain above Tc
................. 324
14
The O(N) symmetric
(φ2)2
field theory in the large
N
limit
.... 329
14.1
Algebraic preliminaries
.................... 330
14.2
Integration over the field
φ:
The determinant
.......... 331
14.3
The limit
N -»·
oo: The critical domain
............ 335
14.4
The
(φ2)2
field theory for
N ->
oo
............... 337
14.5
Singular part of the free energy and equation of state
...... 340
14.6
The
(λλ)
and
(φ2φ2)
two-point functions
............ 343
14.7
Renormalization group and corrections to scaling
........ 345
14.8
The 1/N expansion
..................... 348
14.9
The exponent
η
at order 1/N
................. 350
14.10
The non-linear
σ
-model
...................
351
15
The non-linear
σ
-model
.....................
353
15.1
The non-linear
σ
-model
on the lattice
............. 353
15.2
Low-temperature expansion
.................. 355
xii Contents
15.3 Formal
continuum limit
................... 360
15.4 Regularization ....................... 361
15.5
Zero-momentum or
IR
divergences
............... 362
15.6
Renormalization group
.................... 363
15.7
Solution of the RGE. Fixed points
............... 368
15.8
Correlation functions: Scaling form
.............. 370
15.9
The critical domain: Critical exponents
............ 372
15.10
Dimension
2 ........................ 373
15.11
The
(φ2)2
field theory at low temperature
........... 377
16
Functional renormalization group
................. 381
16.1
Partial field integration and effective Hamiltonian
........ 381
16.2
High-momentum mode integration and RGE
.......... 390
16.3
Perturbative solution:
φ4
theory
................ 396
16.4
RGE: Standard form
..................... 399
16.5
Dimension
4 ........................ 402
16.6
Fixed point:
ε
-expansion ...................
409
16.7
Local stability of the fixed point
................ 411
Appendix
............................ 417
Al
Technical results
....................... 417
A2 Fourier transformation: Decay and regularity
.......... 421
A3
Phase transitions: General remarks
............... 426
A4
Í/N
expansion: Calculations
.................. 431
A5 Functional renormalization group: Complements
......... 433
Bibliography
........................... 441
Index
............................... 447
|
| adam_txt |
Contents
1 Quantum
field theory and the renormaiization group
.1
1.1
Quantum electrodynamics: A quantum field theory
.3
1.2
Quantum electrodynamics: The problem of infinities
.4
1.3
Renormaiization
.7
1.4
Quantum field theory and the renormaiization group
.9
1.5
A triumph of QFT: The Standard Model
.10
1.6
Critical phenomena: Other infinities
.12
1.7
Kadanoff and Wilson's renormaiization group
.14
1.8
Effective quantum field theories
.16
2
Gaussian expectation values. Steepest descent method
.19
2.1
Generating functions
. 19
2.2
Gaussian expectation values. Wick's theorem
. 20
2.3
Perturbed Gaussian measure. Connected contributions
. 24
2.4
Feynman diagrams. Connected contributions
. 25
2.5
Expectation values. Generating function.
Cumulants
. 28
2.6
Steepest descent method
. 31
2.7
Steepest descent method: Several variables, generating functions
. 37
Exercises
. 40
3
Universality and the continuum limit
. 45
3.1
Central limit theorem of probabilities
.45
3.2
Universality and fixed points of transformations
.54
3.3
Random walk and Brownian motion
.59
3.4
Random walk: Additional remarks
.71
3.5
Brownian motion and path integrals
.72
Exercises
.75
4
Classical statistical physics: One dimension
.79
4.1
Nearest-neighbour interactions. Transfer matrix
. 80
4.2
Correlation functions
. 83
4.3
Thermodynamic limit
. 85
4.4
Connected functions and cluster properties
. 88
4.5
Statistical models: Simple examples
. 90
4.6
The Gaussian model
. 92
x
Contents
4.7
Gaussian model: The continuum limit
.98
4.8
More general models: The continuum limit
. 102
Exercises
. 104
5
Continuum limit and path integrals
.
Ill
5.1
Gaussian path integrals
.
Ill
5.2
Gaussian correlations. Wick's theorem
. 118
5.3
Perturbed Gaussian measure
. 118
5.4
Perturbative calculations: Examples
. 120
Exercises
. 124
6
Ferromagnetic systems. Correlation functions
. 127
6.1
Ferromagnetic systems: Definition
. 127
6.2
Correlation functions. Fourier representation
. 133
6.3
Legendre transformation and vertex functions
. 137
6.4
Legendre transformation and steepest descent method
. 142
6.5
Two- and four-point vertex functions
. 143
Exercises
. 145
7
Phase transitions: Generalities and examples
. 147
7.1
Infinite temperature or independent spins
. 150
7.2
Phase transitions in infinite dimension
. 153
7.3
Universality in infinite space dimension
. 158
7.4
Transformations, fixed points and universality
. 161
7.5
Finite-range interactions in finite dimension
. 163
7.6
Ising model: Transfer matrix
. 166
7.7
Continuous symmetries and transfer matrix
. 171
7.8
Continuous symmetries and
Goldstone
modes
. 173
Exercises
. 175
8
Quasi-Gaussian approximation: Universality, critical dimension
. 179
8.1
Short-range two-spin interactions
. 181
8.2
The Gaussian model: Two-point function
. 183
8.3
Gaussian model and random walk
. 188
8.4
Gaussian model and field integral
. 190
8.5
Quasi-Gaussian approximation
. 194
8.6
The two-point function: Universality
. 196
8.7
Quasi-Gaussian approximation and Landau's theory
. 199
8.8
Continuous symmetries and
Goldstone
modes
. 200
8.9
Corrections to the quasi-Gaussian approximation
. 202
8.10
Mean-field approximation and corrections
. 207
8.11
Tricritical points
. 211
Exercises
. 212
9
Renormalization group: General formulation
. 217
9.1
Statistical field theory. Landau's Hamiltonian
. 218
9.2
Connected correlation functions. Vertex functions
. 220
9.3
Renormalization group: General idea
. 222
9.4
Hamiltonian flow: Fixed points, stability
. 226
9.5
The Gaussian fixed point
. 231
Contents xi
9.6 Eigen-perturbations: General
analysis
. 234
9.7
A non-Gaussian fixed point: The
ε
-expansion .
237
9.8
Eigenvalues and dimensions of local polynomials
. 241
10
Perturbative renormalization group: Explicit calculations
. 243
10.1
Critical Hamiltonian and perturbative expansion
. 243
10.2
Feynman diagrams at one-loop order
. 246
10.3
Fixed point and critical behaviour
. 248
10.4
Critical domain
. 254
10.5
Models with O(N) orthogonal symmetry
. 258
10.6
Renormalization group near dimension
4 . 259
10.7
Universal quantities: Numerical results
. 262
11
Renormalization group: iV-component fields
. 267
11.1
Renormalization group: General remarks
. 268
11.2
Gradient flow
. 269
11.3
Model with cubic anisotropy
. 272
11.4
Explicit general expressions: RG analysis
. 276
11.5
Exercise: General model with two parameters
. 281
Exercises
. 284
12
Statistical field theory: Perturbative expansion
. 285
12.1
Generating
funcţionale
. 285
12.2
Gaussian field theory. Wick's theorem
. 287
12.3
Perturbative expansion
. 289
12.4
Loop expansion
. 296
12.5
Dimensional continuation and regularization
. 299
Exercises
. 306
13
The
σ4
field theory near dimension
4. 307
13.1
Effective Hamiltonian. Renormalization
. 308
13.2
Renormalization group equations
. 313
13.3
Solution of RGE: The
ε
-expansion.
316
13.4
Effective and renormalized interactions
. 323
13.5
The critical domain above Tc
. 324
14
The O(N) symmetric
(φ2)2
field theory in the large
N
limit
. 329
14.1
Algebraic preliminaries
. 330
14.2
Integration over the field
φ:
The determinant
. 331
14.3
The limit
N -»·
oo: The critical domain
. 335
14.4
The
(φ2)2
field theory for
N ->
oo
. 337
14.5
Singular part of the free energy and equation of state
. 340
14.6
The
(λλ)
and
(φ2φ2)
two-point functions
. 343
14.7
Renormalization group and corrections to scaling
. 345
14.8
The 1/N expansion
. 348
14.9
The exponent
η
at order 1/N
. 350
14.10
The non-linear
σ
-model
.
351
15
The non-linear
σ
-model
.
353
15.1
The non-linear
σ
-model
on the lattice
. 353
15.2
Low-temperature expansion
. 355
xii Contents
15.3 Formal
continuum limit
. 360
15.4 Regularization . 361
15.5
Zero-momentum or
IR
divergences
. 362
15.6
Renormalization group
. 363
15.7
Solution of the RGE. Fixed points
. 368
15.8
Correlation functions: Scaling form
. 370
15.9
The critical domain: Critical exponents
. 372
15.10
Dimension
2 . 373
15.11
The
(φ2)2
field theory at low temperature
. 377
16
Functional renormalization group
. 381
16.1
Partial field integration and effective Hamiltonian
. 381
16.2
High-momentum mode integration and RGE
. 390
16.3
Perturbative solution:
φ4
theory
. 396
16.4
RGE: Standard form
. 399
16.5
Dimension
4 . 402
16.6
Fixed point:
ε
-expansion .
409
16.7
Local stability of the fixed point
. 411
Appendix
. 417
Al
Technical results
. 417
A2 Fourier transformation: Decay and regularity
. 421
A3
Phase transitions: General remarks
. 426
A4
Í/N
expansion: Calculations
. 431
A5 Functional renormalization group: Complements
. 433
Bibliography
. 441
Index
. 447 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Zinn-Justin, Jean |
| author_facet | Zinn-Justin, Jean |
| author_role | aut |
| author_sort | Zinn-Justin, Jean |
| author_variant | j z j jzj |
| building | Verbundindex |
| bvnumber | BV022616040 |
| callnumber-first | Q - Science |
| callnumber-label | QC175 |
| callnumber-raw | QC175.16.P5 |
| callnumber-search | QC175.16.P5 |
| callnumber-sort | QC 3175.16 P5 |
| callnumber-subject | QC - Physics |
| classification_rvk | UO 4020 |
| classification_tum | PHY 025f PHY 065f PHY 023f |
| ctrlnum | (OCoLC)255563633 (DE-599)BVBBV022616040 |
| dewey-full | 530.414 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 530 - Physics |
| dewey-raw | 530.414 |
| dewey-search | 530.414 |
| dewey-sort | 3530.414 |
| dewey-tens | 530 - Physics |
| discipline | Physik |
| discipline_str_mv | Physik |
| edition | 1. publ. |
| format | Book |
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| id | DE-604.BV022616040 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T18:18:11Z |
| indexdate | 2024-07-09T21:01:44Z |
| institution | BVB |
| isbn | 9780199227198 9780199665167 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015822176 |
| oclc_num | 255563633 |
| open_access_boolean | |
| owner | DE-29T DE-703 DE-91G DE-BY-TUM DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-188 DE-20 |
| owner_facet | DE-29T DE-703 DE-91G DE-BY-TUM DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-188 DE-20 |
| physical | XII, 452 S. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Oxford Univ. Press |
| record_format | marc |
| series2 | Oxford graduate texts |
| spelling | Zinn-Justin, Jean Verfasser aut Phase transitions and renormalization group Jean Zinn-Justin 1. publ. Oxford [u.a.] Oxford Univ. Press 2007 XII, 452 S. txt rdacontent n rdamedia nc rdacarrier Oxford graduate texts Hier auch später erschienene, unveränderte Nachdrucke Phase transformations (Statistical physics) Renormalization (Physics) Phasenumwandlung (DE-588)4132140-6 gnd rswk-swf Renormierungsgruppe (DE-588)4177773-6 gnd rswk-swf Quantenfeldtheorie (DE-588)4047984-5 gnd rswk-swf Renormierungsgruppe (DE-588)4177773-6 s Phasenumwandlung (DE-588)4132140-6 s DE-604 Quantenfeldtheorie (DE-588)4047984-5 s Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015822176&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Zinn-Justin, Jean Phase transitions and renormalization group Phase transformations (Statistical physics) Renormalization (Physics) Phasenumwandlung (DE-588)4132140-6 gnd Renormierungsgruppe (DE-588)4177773-6 gnd Quantenfeldtheorie (DE-588)4047984-5 gnd |
| subject_GND | (DE-588)4132140-6 (DE-588)4177773-6 (DE-588)4047984-5 |
| title | Phase transitions and renormalization group |
| title_auth | Phase transitions and renormalization group |
| title_exact_search | Phase transitions and renormalization group |
| title_exact_search_txtP | Phase transitions and renormalization group |
| title_full | Phase transitions and renormalization group Jean Zinn-Justin |
| title_fullStr | Phase transitions and renormalization group Jean Zinn-Justin |
| title_full_unstemmed | Phase transitions and renormalization group Jean Zinn-Justin |
| title_short | Phase transitions and renormalization group |
| title_sort | phase transitions and renormalization group |
| topic | Phase transformations (Statistical physics) Renormalization (Physics) Phasenumwandlung (DE-588)4132140-6 gnd Renormierungsgruppe (DE-588)4177773-6 gnd Quantenfeldtheorie (DE-588)4047984-5 gnd |
| topic_facet | Phase transformations (Statistical physics) Renormalization (Physics) Phasenumwandlung Renormierungsgruppe Quantenfeldtheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015822176&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT zinnjustinjean phasetransitionsandrenormalizationgroup |