Functional integration: action and symmetries
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2006
|
| Edition: | 1. publ. |
| Series: | Cambridge monographs on mathematical physics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | XX, 456 S. graph. Darst. |
| ISBN: | 9780521866965 0521866960 |
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MARC
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| 245 | 1 | 0 | |a Functional integration |b action and symmetries |c P. Cartier & C. DeWitt-Morette |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2006 | |
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Record in the Search Index
| _version_ | 1807953571464871936 |
|---|---|
| adam_text |
Contents
Acknowledgements page
xi
List of symbols, conventions, and formulary
xv
PART I THE PHYSICAL AND MATHEMATICAL
ENVIRONMENT
1
The physical and mathematical environment
3
A: An inheritance from physics
3
1.1
The beginning
3
1.2
Integrals over function spaces
6
1.3
The operator formalism
б
1.4
A few titles
7
B: A toolkit from analysis
9
1.5
A tutorial in Lebesgue integration
9
1.6
Stochastic processes and promeasures
15
1.7
Fourier transformation and
prodistributions
19
С:
Feynman's integral versus Kac's integral
23
1.8
Planck's
blackbody
radiation law
23
1.9
Imaginary time and inverse temperature
26
1.10
Feynman's integral versus Kac's integral
27
1.11
Hamiltonian versus lagrangian
29
References
31
PART II QUANTUM MECHANICS
2
First lesson:
gaussian
integrals
35
2.1
Gaussiane
in M.
35
2.2
Gaussiane
in RD
35
2.3
Gaussiane
on a Banach space
38
vi
Contents
2.4
Variances and covariances
42
2.5
Scaling and coarse-graining
46
References
55
3
Selected examples
56
3.1
The Wiener measure and brownian paths
57
3.2
Canonical gaussians in I? and L2'1
59
3.3
The forced harmonic oscillator
63
3.4
Phase-space path integrals
73
References
76
4
Semiclassical expansion; WKB
78
4.1
Introduction
78
4.2
The WKB approximation
80
4.3
An example: the anharmonic oscillator
88
4.4
Incompatibility with analytic continuation
92
4.5
Physical interpretation of the WKB approximation
93
References
94
5
Semiclassical expansion; beyond WKB
96
5.1
Introduction
96
5.2
Constants of the motion
100
5.3
Caustics
101
5.4
Glory scattering
104
5.5
Tunneling
106
References 111
6
Quantum dynamics: path integrals and
the operator formalism
114
6.1
Physical dimensions and expansions
114
6.2
A free particle
115
6.3
Particles in a scalar potential V
118
6.4
Particles in a vector potential A
126
6.5
Matrix elements and kernels
129
References
130
PART III METHODS FROM DIFFERENTIAL
GEOMETRY
7
Symmetries
135
7.1
Groups of transformations. Dynamical vector fields
135
7.2
A basic theorem
137
Contents
vii
7.3
The group of transformations on a frame bundle
139
7.4
Symplectic manifolds
141
References
144
8
Homotopy
146
8.1
An example: quantizing a spinning top
146
8.2
Propagators on SO(3) and SU(2)
147
8.3
The homotopy theorem for path integration
150
8.4
Systems of indistinguishable particles. Anyons
151
8.5
A simple model of the Aharanov-Bohm effect
152
References
156
9 Grassmann
analysis: basics
157
9.1
Introduction
157
9.2
A compendium of
Grassmann
analysis
158
9.3
Berezin integration
164
9.4
Forms and densities
168
References
173
10 Grassmann
analysis: applications
175
10.1
The
Euler Poincaré
characteristic
175
10.2
Supersymmetric quantum field theory
183
10.3
The Dirac operator and Dirac matrices
186
References
189
11
Volume elements, divergences, gradients
191
11.1
Introduction. Divergences
191
11.2
Comparing volume elements
197
11.3
Integration by parts
202
References
210
PART IV NON-GAUSSIAN APPLICATIONS
12
Poisson
processes in physics
215
12.1
The telegraph equation
215
12.2
Klein-Gordon and Dirac equations
220
12.3
Two-state systems interacting with their environment
225
References
231
13
A mathematical theory of
Poisson
processes
233
13.1
Poisson
stochastic processes
234
13.2
Spaces of
Poisson
paths
241
viii Contents
13.3
Stochastic solutions of differential equations
251
13.4
Differential equations: explicit solutions
262
References
266
14
The first exit time; energy problems
268
14.1
Introduction: fixed-energy Green's function
268
14.2
The path integral for a fixed-energy amplitude
272
14.3
Periodic and quasiperiodic orbits
276
14.4
Intrinsic and tuned times of a process
281
References
284
PART V PROBLEMS IN QUANTUM FIELD THEORY
15
Renormalization
1:
an introduction
289
15.1
Introduction
289
15.2
Prom paths to fields
291
15.3
Green's example
297
15.4
Dimensional regularization
300
References
307
16
Renormalization
2:
scaling
308
16.1
The renormalization group
308
16.2
The
λφΑ
system
314
References
323
17
Renormalization
3:
combinatorics, contributed
by
Markus Berg 324
17.1
Introduction
324
17.2
Background
325
17.3
Graph summary
327
17.4
The grafting operator
328
17.5
Lie algebra
331
17.6
Other operations
338
17.7
Renormalization
339
17.8
A three-loop example
342
17.9
Renormalization-group flows and nonrenormalizable
theories
344
17.10
Conclusion
345
References
351
18
Volume elements in quantum field theory,
contributed by
Bryce
DeWitt 355
18.1
Introduction
355
Contents ix
18.2
Cases in which equation
(18.3)
is exact
357
18.3
Loop expansions
358
References
364
PART VI PROJECTS
19
Projects
367
19.1
Gaussian integrals
· 367
19.2
Semiclassical expansions
370
19.3
Homotopy
371
19.4 Grassmann
analysis
373
19.5
Volume elements, divergences, gradients
376
19.6
Poisson
processes
379
19.7
Renormalization
380
APPENDICES
Appendix A Forward and backward integrals.
Spaces of pointed paths
387
Appendix
В
Product integrals
391
Appendix
С
A compendium of
gaussian
integrals
395
Appendix
D
Wick calculus,
contributed by Alexander
Wurm 399
Appendix
E
The Jacobi operator
404
Appendix
F
Change of variables of integration
415
Appendix
G
Analytic properties of covariances
422
Appendix
H Feynman's
checkerboard
432
Bibliography
437
Index
451
FUNCTIONAL INTEGRATION
Functional integration successfully entered physics as path integrals in the
1942
Ph.D. dissertation of Richard P. Feynman, but it made no sense at all as a math¬
ematical definition.
Cartier
and DeWitt-Morette have created, in this book, a
new approach to functional integration. The close collaboration between a math¬
ematician and a physicist brings a unique perspective to this topic. The book is
self-contained: mathematical ideas are introduced, developed, generalized, and
applied. In the authors' hands, functional integration is shown to be a robust,
user-friendly, and multi-purpose tool that can be applied to a great variety of
situations, for example systems of indistinguishable particles, caustics-analysis,
superanalysis,
and
non-gaussian
integrals. Problems in quantum field theory are
also considered. In the final part the authors outline topics that can profitably
be pursued using material already presented.
Pierre Cartier
is a mathematician with an extraordinarily wide range
of interests and expertise. He has been called
"un homme de la Renaissance."
He is Emeritus Director of Research at the Centre National
de la
Recherche
Scientifique,
France, and a long-term visitor of the
Institut des Hautes
Etudes
Scientifiques.
From
1981
to
1989,
he was a senior researcher at the
Ecole Poly¬
technique de
Paris, and, between
1988
and
1997,
held a professorship at the
Ecole
Normale Supérieure.
He is a member of the
Société Mathématique de
France,
the American Mathematical Society, and the Vietnamese Mathematical Society.
Cécile DeWitt-Morette
is the Jane and Roland
Blumberg
Centen¬
nial Professor in Physics,
Emerita,
at the University of Texas at Austin. She
is a member of the American and European Physical Societies, and
a Mem¬
bre
d'Honneur de la Société Française de Physique.
DeWitt-Morette's interest in
functional integration began in
1948.
In
F. J.
Dyson's words, "she was the first of
the younger generation to grasp the full scope and power of the Feynman path
integral approach in physics." She is co-author with Yvonne Choquet-Bruhat
of the two-volume book Analysis, Manifolds and Physics, a standard text first
published in
1977,
which is now in its seventh edition. She is the author of
100
publications in various areas of theoretical physics and has edited
28
books.
She has lectured, worldwide, in many institutions and summer schools on topics
related to functional integration. |
| adam_txt |
Contents
Acknowledgements page
xi
List of symbols, conventions, and formulary
xv
PART I THE PHYSICAL AND MATHEMATICAL
ENVIRONMENT
1
The physical and mathematical environment
3
A: An inheritance from physics
3
1.1
The beginning
3
1.2
Integrals over function spaces
6
1.3
The operator formalism
б
1.4
A few titles
7
B: A toolkit from analysis
9
1.5
A tutorial in Lebesgue integration
9
1.6
Stochastic processes and promeasures
15
1.7
Fourier transformation and
prodistributions
19
С:
Feynman's integral versus Kac's integral
23
1.8
Planck's
blackbody
radiation law
23
1.9
Imaginary time and inverse temperature
26
1.10
Feynman's integral versus Kac's integral
27
1.11
Hamiltonian versus lagrangian
29
References
31
PART II QUANTUM MECHANICS
2
First lesson:
gaussian
integrals
35
2.1
Gaussiane
in M.
35
2.2
Gaussiane
in RD
35
2.3
Gaussiane
on a Banach space
38
vi
Contents
2.4
Variances and covariances
42
2.5
Scaling and coarse-graining
46
References
55
3
Selected examples
56
3.1
The Wiener measure and brownian paths
57
3.2
Canonical gaussians in I? and L2'1
59
3.3
The forced harmonic oscillator
63
3.4
Phase-space path integrals
73
References
76
4
Semiclassical expansion; WKB
78
4.1
Introduction
78
4.2
The WKB approximation
80
4.3
An example: the anharmonic oscillator
88
4.4
Incompatibility with analytic continuation
92
4.5
Physical interpretation of the WKB approximation
93
References
94
5
Semiclassical expansion; beyond WKB
96
5.1
Introduction
96
5.2
Constants of the motion
100
5.3
Caustics
101
5.4
Glory scattering
104
5.5
Tunneling
106
References 111
6
Quantum dynamics: path integrals and
the operator formalism
114
6.1
Physical dimensions and expansions
114
6.2
A free particle
115
6.3
Particles in a scalar potential V
118
6.4
Particles in a vector potential A
126
6.5
Matrix elements and kernels
129
References
130
PART III METHODS FROM DIFFERENTIAL
GEOMETRY
7
Symmetries
135
7.1
Groups of transformations. Dynamical vector fields
135
7.2
A basic theorem
137
Contents
vii
7.3
The group of transformations on a frame bundle
139
7.4
Symplectic manifolds
141
References
144
8
Homotopy
146
8.1
An example: quantizing a spinning top
146
8.2
Propagators on SO(3) and SU(2)
147
8.3
The homotopy theorem for path integration
150
8.4
Systems of indistinguishable particles. Anyons
151
8.5
A simple model of the Aharanov-Bohm effect
152
References
156
9 Grassmann
analysis: basics
157
9.1
Introduction
157
9.2
A compendium of
Grassmann
analysis
158
9.3
Berezin integration
164
9.4
Forms and densities
168
References
173
10 Grassmann
analysis: applications
175
10.1
The
Euler Poincaré
characteristic
175
10.2
Supersymmetric quantum field theory
183
10.3
The Dirac operator and Dirac matrices
186
References
189
11
Volume elements, divergences, gradients
191
11.1
Introduction. Divergences
191
11.2
Comparing volume elements
197
11.3
Integration by parts
202
References
210
PART IV NON-GAUSSIAN APPLICATIONS
12
Poisson
processes in physics
215
12.1
The telegraph equation
215
12.2
Klein-Gordon and Dirac equations
220
12.3
Two-state systems interacting with their environment
225
References
231
13
A mathematical theory of
Poisson
processes
233
13.1
Poisson
stochastic processes
234
13.2
Spaces of
Poisson
paths
241
viii Contents
13.3
Stochastic solutions of differential equations
251
13.4
Differential equations: explicit solutions
262
References
266
14
The first exit time; energy problems
268
14.1
Introduction: fixed-energy Green's function
268
14.2
The path integral for a fixed-energy amplitude
272
14.3
Periodic and quasiperiodic orbits
276
14.4
Intrinsic and tuned times of a process
281
References
284
PART V PROBLEMS IN QUANTUM FIELD THEORY
15
Renormalization
1:
an introduction
289
15.1
Introduction
289
15.2
Prom paths to fields
291
15.3
Green's example
297
15.4
Dimensional regularization
300
References
307
16
Renormalization
2:
scaling
308
16.1
The renormalization group
308
16.2
The
λφΑ
system
314
References
323
17
Renormalization
3:
combinatorics, contributed
by
Markus Berg 324
17.1
Introduction
324
17.2
Background
325
17.3
Graph summary
327
17.4
The grafting operator
328
17.5
Lie algebra
331
17.6
Other operations
338
17.7
Renormalization
339
17.8
A three-loop example
342
17.9
Renormalization-group flows and nonrenormalizable
theories
344
17.10
Conclusion
345
References
351
18
Volume elements in quantum field theory,
contributed by
Bryce
DeWitt 355
18.1
Introduction
355
Contents ix
18.2
Cases in which equation
(18.3)
is exact
357
18.3
Loop expansions
358
References
364
PART VI PROJECTS
19
Projects
367
19.1
Gaussian integrals
· 367
19.2
Semiclassical expansions
370
19.3
Homotopy
371
19.4 Grassmann
analysis
373
19.5
Volume elements, divergences, gradients
376
19.6
Poisson
processes
379
19.7
Renormalization
380
APPENDICES
Appendix A Forward and backward integrals.
Spaces of pointed paths
387
Appendix
В
Product integrals
391
Appendix
С
A compendium of
gaussian
integrals
395
Appendix
D
Wick calculus,
contributed by Alexander
Wurm 399
Appendix
E
The Jacobi operator
404
Appendix
F
Change of variables of integration
415
Appendix
G
Analytic properties of covariances
422
Appendix
H Feynman's
checkerboard
432
Bibliography
437
Index
451
FUNCTIONAL INTEGRATION
Functional integration successfully entered physics as path integrals in the
1942
Ph.D. dissertation of Richard P. Feynman, but it made no sense at all as a math¬
ematical definition.
Cartier
and DeWitt-Morette have created, in this book, a
new approach to functional integration. The close collaboration between a math¬
ematician and a physicist brings a unique perspective to this topic. The book is
self-contained: mathematical ideas are introduced, developed, generalized, and
applied. In the authors' hands, functional integration is shown to be a robust,
user-friendly, and multi-purpose tool that can be applied to a great variety of
situations, for example systems of indistinguishable particles, caustics-analysis,
superanalysis,
and
non-gaussian
integrals. Problems in quantum field theory are
also considered. In the final part the authors outline topics that can profitably
be pursued using material already presented.
Pierre Cartier
is a mathematician with an extraordinarily wide range
of interests and expertise. He has been called
"un homme de la Renaissance."
He is Emeritus Director of Research at the Centre National
de la
Recherche
Scientifique,
France, and a long-term visitor of the
Institut des Hautes
Etudes
Scientifiques.
From
1981
to
1989,
he was a senior researcher at the
Ecole Poly¬
technique de
Paris, and, between
1988
and
1997,
held a professorship at the
Ecole
Normale Supérieure.
He is a member of the
Société Mathématique de
France,
the American Mathematical Society, and the Vietnamese Mathematical Society.
Cécile DeWitt-Morette
is the Jane and Roland
Blumberg
Centen¬
nial Professor in Physics,
Emerita,
at the University of Texas at Austin. She
is a member of the American and European Physical Societies, and
a Mem¬
bre
d'Honneur de la Société Française de Physique.
DeWitt-Morette's interest in
functional integration began in
1948.
In
F. J.
Dyson's words, "she was the first of
the younger generation to grasp the full scope and power of the Feynman path
integral approach in physics." She is co-author with Yvonne Choquet-Bruhat
of the two-volume book Analysis, Manifolds and Physics, a standard text first
published in
1977,
which is now in its seventh edition. She is the author of
100
publications in various areas of theoretical physics and has edited
28
books.
She has lectured, worldwide, in many institutions and summer schools on topics
related to functional integration. |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Cartier, Pierre 1932-2024 DeWitt-Morette, Cécile 1922-2017 |
| author_GND | (DE-588)141948256 (DE-588)141948361 |
| author_facet | Cartier, Pierre 1932-2024 DeWitt-Morette, Cécile 1922-2017 |
| author_role | aut aut |
| author_sort | Cartier, Pierre 1932-2024 |
| author_variant | p c pc c d m cdm |
| building | Verbundindex |
| bvnumber | BV022254209 |
| callnumber-first | Q - Science |
| callnumber-label | QC20 |
| callnumber-raw | QC20.7.F85 |
| callnumber-search | QC20.7.F85 |
| callnumber-sort | QC 220.7 F85 |
| callnumber-subject | QC - Physics |
| classification_rvk | UK 1200 UK 4500 |
| classification_tum | MAT 460f MAT 390f |
| ctrlnum | (OCoLC)70764880 (DE-599)BVBBV022254209 |
| dewey-full | 530.1557 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 530 - Physics |
| dewey-raw | 530.1557 |
| dewey-search | 530.1557 |
| dewey-sort | 3530.1557 |
| dewey-tens | 530 - Physics |
| discipline | Physik Mathematik |
| discipline_str_mv | Physik Mathematik |
| edition | 1. publ. |
| format | Book |
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| id | DE-604.BV022254209 |
| illustrated | Illustrated |
| index_date | 2024-07-02T16:40:31Z |
| indexdate | 2024-08-21T00:08:06Z |
| institution | BVB |
| isbn | 9780521866965 0521866960 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015464954 |
| oclc_num | 70764880 |
| open_access_boolean | |
| owner | DE-91G DE-BY-TUM DE-29T DE-703 DE-19 DE-BY-UBM |
| owner_facet | DE-91G DE-BY-TUM DE-29T DE-703 DE-19 DE-BY-UBM |
| physical | XX, 456 S. graph. Darst. |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| series2 | Cambridge monographs on mathematical physics |
| spelling | Cartier, Pierre 1932-2024 Verfasser (DE-588)141948256 aut Functional integration action and symmetries P. Cartier & C. DeWitt-Morette 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2006 XX, 456 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Cambridge monographs on mathematical physics Functionaalintegratie gtt Integration, Functional Funktionalintegration (DE-588)4155674-4 gnd rswk-swf Funktionalintegration (DE-588)4155674-4 s DE-604 DeWitt-Morette, Cécile 1922-2017 Verfasser (DE-588)141948361 aut Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015464954&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015464954&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Cartier, Pierre 1932-2024 DeWitt-Morette, Cécile 1922-2017 Functional integration action and symmetries Functionaalintegratie gtt Integration, Functional Funktionalintegration (DE-588)4155674-4 gnd |
| subject_GND | (DE-588)4155674-4 |
| title | Functional integration action and symmetries |
| title_auth | Functional integration action and symmetries |
| title_exact_search | Functional integration action and symmetries |
| title_exact_search_txtP | Functional integration action and symmetries |
| title_full | Functional integration action and symmetries P. Cartier & C. DeWitt-Morette |
| title_fullStr | Functional integration action and symmetries P. Cartier & C. DeWitt-Morette |
| title_full_unstemmed | Functional integration action and symmetries P. Cartier & C. DeWitt-Morette |
| title_short | Functional integration |
| title_sort | functional integration action and symmetries |
| title_sub | action and symmetries |
| topic | Functionaalintegratie gtt Integration, Functional Funktionalintegration (DE-588)4155674-4 gnd |
| topic_facet | Functionaalintegratie Integration, Functional Funktionalintegration |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015464954&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015464954&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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