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Functional Integration: Action and Symmetries [Mīkstie vāki]

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(Institut des Hautes Études Scientifiques, France), (University of Texas, Austin)
  • Formāts: Paperback / softback, 480 pages, height x width x depth: 244x170x25 mm, weight: 760 g, Worked examples or Exercises
  • Sērija : Cambridge Monographs on Mathematical Physics
  • Izdošanas datums: 10-Jun-2010
  • Izdevniecība: Cambridge University Press
  • ISBN-10: 0521143578
  • ISBN-13: 9780521143578
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Functional Integration: Action and SymmetriesPalielināt
  • Formāts: Paperback / softback, 480 pages, height x width x depth: 244x170x25 mm, weight: 760 g, Worked examples or Exercises
  • Sērija : Cambridge Monographs on Mathematical Physics
  • Izdošanas datums: 10-Jun-2010
  • Izdevniecība: Cambridge University Press
  • ISBN-10: 0521143578
  • ISBN-13: 9780521143578
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9780521143578.html
In this text, Cartier and DeWitt-Morette, using their complementary interests and expertise, successfully condense and apply the essentials of Functional Integration to a great variety of systems, showing this mathematically elusive technique to be a robust, user friendly and multipurpose tool.

Functional integration successfully entered physics as path integrals in the 1942 PhD dissertation of Richard P. Feynman, but it made no sense at all as a mathematical definition. Cartier and DeWitt-Morette have created, in this book, a fresh approach to functional integration. The book is self-contained: mathematical ideas are introduced, developed, generalised 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; Aharonov–Bohm systems; supersymmetry; non-gaussian integrals. Problems in quantum field theory are also considered. In the final part the authors outline topics that can be profitably pursued using material already presented.

Recenzijas

Review of the hardback: 'will be helpful for those mathematicians who are interested in physical applications of the general theory of measure (theory of integrals) and for the physicists who are interested in mathematically rigorous formulations of complicated problems in quantum physics.' Zentralblatt MATH Review of the hardback: 'for someone who is interested in the mathematical foundations or merely curious to see some of the deep insight of two true experts, Cartier and DeWitt-Morette's book is well worth reading.' Physics Today

Papildus informācija

The powerful tool of functional integration is widely applied and shown to be user-friendly and mathematically robust.
Acknowledgements xi
List of symbols, conventions, and formulary
xv
PART I THE PHYSICAL AND MATHEMATICAL ENVIRONMENT
1 The physical and mathematical environment
3(32)
A An inheritance from physics
3(1)
1.1 The beginning
3(3)
1.2 Integrals over function spaces
6(1)
1.3 The operator formalism
6(1)
1.4 A few titles
7(2)
B A toolkit from analysis
9(1)
1.5 A tutorial in Lebesgue integration
9(6)
1.6 Stochastic processes and promeasures
15(4)
1.7 Fourier transformation and prodistributions
19(4)
C Feynman's integral versus Kac's integral
23(1)
1.8 Planck's blackbody radiation law
23(3)
1.9 Imaginary time and inverse temperature
26(1)
1.10 Feynman's integral versus Kac's integral
27(2)
1.11 Hamiltonian versus lagrangian
29(6)
References
31(4)
PART II QUANTUM MECHANICS
2 First lesson: gaussian integrals
35(21)
2.1 Gaussians in R
35(1)
2.2 Gaussians in RD
35(3)
2.3 Gaussians on a Banach space
38(4)
2.4 Variances and covariances
42(4)
2.5 Scaling and coarse-graining
46(10)
References
55(1)
3 Selected examples
56(22)
3.1 The Wiener measure and brownian paths
57(2)
3.2 Canonical gaussians in L2 and L2, 1
59(4)
3.3 The forced harmonic oscillator
63(10)
3.4 Phase-space path integrals
73(5)
References
76(2)
4 Semiclassical expansion; WKB
78(18)
4.1 Introduction
78(2)
4.2 The WKB approximation
80(8)
4.3 An example: the anharmonic oscillator
88(4)
4.4 Incompatibility with analytic continuation
92(1)
4.5 Physical interpretation of the WKB approximation
93(3)
References
94(2)
5 Semiclassical expansion; beyond WKB
96(18)
5.1 Introduction
96(4)
5.2 Constants of the motion
100(1)
5.3 Caustics
101(3)
5.4 Glory scattering
104(2)
5.5 Tunneling
106(8)
References
111(3)
6 Quantum dynamics: path integrals and the operator formalism
114(21)
6.1 Physical dimensions and expansions
114(1)
6.2 A free particle
115(3)
6.3 Particles in a scalar potential V
118(8)
6.4 Particles in a vector potential A
126(3)
6.5 Matrix elements and kernels
129(6)
References
130(5)
PART III METHODS FROM DIFFERENTIAL GEOMETRY
7 Symmetries
135(11)
7.1 Groups of transformations. Dynamical vector fields
135(2)
7.2 A basic theorem
137(2)
7.3 The group of transformations on a frame bundle
139(2)
7.4 Symplectic manifolds
141(5)
References
144(2)
8 Homotopy
146(11)
8.1 An example: quantizing a spinning top
146(1)
8.2 Propagators on SO(3) and SU(2)
147(3)
8.3 The homotopy theorem for path integration
150(1)
8.4 Systems of indistinguishable particles. Anyons
151(1)
8.5 A simple model of the Aharanov-Bohm effect
152(5)
References
156(1)
9 Grassmann analysis: basics
157(18)
9.1 Introduction
157(1)
9.2 A compendium of Grassmann analysis
158(6)
9.3 Berezin integration
164(4)
9.4 Forms and densities
168(7)
References
173(2)
10 Grassmann analysis: applications
175(16)
10.1 The Euler-Poincare characteristic
175(8)
10.2 Supersymmetric quantum field theory
183(3)
10.3 The Dirac operator and Dirac matrices
186(5)
References
189(2)
11 Volume elements, divergences, gradients
191(24)
11.1 Introduction. Divergences
191(6)
11.2 Comparing volume elements
197(5)
11.3 Integration by parts
202(13)
References
210(5)
PART IV NON-GAUSSIAN APPLICATIONS
12 Poisson processes in physics
215(18)
12.1 The telegraph equation
215(5)
12.2 Klein-Gordon and Dirac equations
220(5)
12.3 Two-state systems interacting with their environment
225(8)
References
231(2)
13 A mathematical theory of Poisson processes
233(35)
13.1 Poisson stochastic processes
234(7)
13.2 Spaces of Poisson paths
241(10)
13.3 Stochastic solutions of differential equations
251(11)
13.4 Differential equations: explicit solutions
262(6)
References
266(2)
14 The first exit time; energy problems
268(21)
14.1 Introduction: fixed-energy Green's function
268(4)
14.2 The path integral for a fixed-energy amplitude
272(4)
14.3 Periodic and quasiperiodic orbits
276(5)
14.4 Intrinsic and tuned times of a process
281(8)
References
284(5)
PART V PROBLEMS IN QUANTUM FIELD THEORY
15 Renormalization 1 an introduction
289(19)
15.1 Introduction
289(2)
15.2 From paths to fields
291(6)
15.3 Green's example
297(3)
15.4 Dimensional regularization
300(8)
References
307(1)
16 Renormalization 2 scaling
308(16)
16.1 The renormalization group
308(6)
16.2 The λø4 system
314(10)
References
323(1)
17 Renormalization 3 combinatorics, contributed
324(31)
Markus Berg
17.1 Introduction
324(1)
17.2 Background
325(2)
17.3 Graph summary
327(1)
17.4 The grafting operator
328(3)
17.5 Lie algebra
331(7)
17.6 Other operations
338(1)
17.7 Renormalization
339(3)
17.8 A three-loop example
342(2)
17.9 Renormalization-group flows and nonrenormalizable theories
344(1)
17.10 Conclusion
345(10)
References
351(4)
18 Volume elements in quantum field theory, contributed
355(12)
Bryce DeWitt
18.1 Introduction
355(2)
18.2 Cases in which equation (18.3) is exact
357(1)
18.3 Loop expansions
358(9)
References
364(3)
PART VI PROJECTS
19 Projects
367(20)
19.1 Gaussian integrals
367(3)
19.2 Semiclassical expansions
370(1)
19.3 Homotopy
371(2)
19.4 Grassmann analysis
373(3)
19.5 Volume elements, divergences, gradients
376(3)
19.6 Poisson processes
379(1)
19.7 Renormalization
380(7)
APPENDICES
Appendix A Forward and backward integrals. Spaces of pointed paths
387(4)
Appendix B Product integrals
391(4)
Appendix C A compendium of gaussian integrals
395(4)
Appendix D Wick calculus, contributed
399(5)
Alexander Wurm
Appendix E The Jacobi operator
404(11)
Appendix F Change of variables of integration
415(7)
Appendix G Analytic properties of covariances
422(10)
Appendix H Feynman's checkerboard
432(5)
Bibliography 437(14)
Index 451
Emeritus Director of Research, Center National de la Recherche Scientifique, France. Member of Societe Francaise de Mathematiques and American Mathematical Society. Jane and Roland Blumberg Centennial Professor in Physics, Emerita, University of Texas at Austin. Member of American and European Physical Societies.