The main theme of this book is the "path integral technique" and its applications to constructive methods of quantum physics. The central topic is probabilistic foundations of the Feynman-Kac formula. Starting with main examples of Gaussian processes (the Brownian motion, the oscillatory process, and the Brownian bridge), the author presents four different proofs of the Feynman-Kac formula. Also included is a simple exposition of stochastic Itô calculus and its applications, in particular to the Hamiltonian of a particle in a magnetic field (the Feynman-Kac-Itô formula).
Among other topics discussed are the probabilistic approach to the bound of the number of ground states of correlation inequalities (the Birman-Schwinger principle, Lieb's formula, etc.), the calculation of asymptotics for functional integrals of Laplace type (the theory of Donsker-Varadhan) and applications, scattering theory, the theory of crushed ice, and the Wiener sausage.
Written with great care and containing many highly illuminating examples, this classic book is highly recommended to anyone interested in applications of functional integration to quantum physics. It can also serve as a textbook for a course in functional integration.
Author(s): Barry Simon
Series: Pure and Applied Mathematics, a Series of Monographs and Textbooks, 86
Publisher: Academic Press Inc
Year: 1979
Language: English
Pages: 311
Functional Integration and Quantum Physics......Page 4
Copyright Page......Page 5
Contents......Page 6
Preface......Page 8
List of Symbols......Page 10
1. Introduction......Page 14
2. Construction of Gaussian Processes......Page 21
3. Some Fundamental Tools of Probability Theory......Page 30
4. The Wiener Process, the Oscillator Process, and the Brownian Bridge......Page 45
5. Regularity Properties—1......Page 56
6. The Feynman–Kac Formula......Page 61
7. Regularity and Recurrence Properties—2......Page 73
8. The Birman–Schwinger Kernel and Lieb's Formula......Page 101
9. Phase Space Bounds......Page 106
10. The Classical Limit......Page 118
11. Recurrence and Weak Coupling......Page 127
12. Correlation Inequalities......Page 132
13. Other inequalities: Log Concavity, Symmetric Rearrangement, Conditioning, Hypercontractivity......Page 149
14. ltô’s Integral......Page 161
15. Schrödinger Operators with Magnetic Fields......Page 172
16. Introduction to Stochastic Calculus......Page 183
17. Donsker's Theorem......Page 187
18. Laplace's Method in Function Space......Page 194
19. Introduction to the Donsker-Varadhan Theory......Page 211
20. Perturbation Theory for the Ground State Energy......Page 224
21. Dirichlet Boundaries and Decoupling Singularities in Scattering Theory......Page 237
22. Crushed Ice and the Wiener Sausage......Page 244
23. The Statistical Mechanics of Charged Particles with Positive Definite Interactions......Page 258
24. An Introduction to Euclidean Quantum Field Theory......Page 265
25. Properties of Eigenfunctions, Wave Packets, and Green's Functions......Page 271
26. Inverse Problems and the Feynman–Kac Formula......Page 285
References......Page 292
Index......Page 306
