Functional integration and quantum physics, Volume 86 (Pure and Applied Mathematics)

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. 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 It is fairly well known that one of Hilbert’s famous list of problems is that of developing an axiomatic theory of mathematical probability theory (this problem could be said to have been solved by Khintchine, Kolmogorov, and
Levy), and also among the list is the “axiomatization of physics.” What is not so well known is that these are two parts of one and the same problem, namely, the sixth, and that the axiomatics of probability are discussed in the context of the foundations of statistical mechanics. Although Hilbert could not have known it when he formulated his problems, probability theory is also central to the foundations of quantum theory. In this book, I wish to describe a very different interface between probability and mathematical physics, namely, the use of certain notions of integration in function spaces as technical tools in quantum physics. Although Nelson has proposed some connection between these notions and foundational questions, we shall deal solely with their use to answer a variety of questions in
conventional quantum theory. It is fairly well known that one of Hilbert's famous list of problems is that of developing an axiomatic theory of mathematical probability theory (this problem could be said to have been solved by Khintchine, Kolmogorov, and Levy), and also among the list is the "axiomatization of physics.” What is not so well known is that these are two parts of one and the same problem, namely, the sixth, and that the axiomatics of probability are discussed in the context of the foundations of statistical mechanics. Although Hilbert could not have known it when he formulated his problems, probability theory is also central to the foundations of quantum theory. In this book, I wish to describe a very different interface between probability and mathematical physics, namely, the use of certain notions of integration in function spaces as technical tools in quantum physics. Although Nelson has proposed some connection between these notions and foundational questions, we shall deal solely with their use to answer a variety of questions in conventional quantum theory. The main theme of this classic book is the path integral technique and its applications to constructive methods of quantum physics. The central topic is probabilistic foundations of Feynman-Kac formula. Starting with main examples of Gaussian processes, the author presents four different proofs of the Feynman-Kac formula