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Functional Integration and Quantum Physics
1st Edition - September 28, 1979
Editor: Barry Simon
eBook ISBN:
9 7 8 - 0 - 0 8 - 0 8 7 4 0 2 - 9
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… Read more
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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, andLevy), 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 inconventional quantum theory.
Preface
List of Symbols
Chapter 1: Introduction
1 Introduction
2 Construction of Gaussian Processes
3 Some Fundamental Tools of Probability Theory
Chapter 2: The Basic Processes
4. The Wiener Process, the Oscillator Process, and the Brownian Bridge
5. Regularity Properties—1
6. The Feynman–Kac Formula
7. Regularity and Recurrence Properties—2
Chapter 3: Bound State Problems
8 The Birman–Schwinger Kernel and Lieb’s Formula
9 Phase Space Bounds
10 The Classical Limit
11 Recurrence and Weak Coupling
Chapter 4: Inequalities
12 Correlation Inequalities
13 Other Inequalities: Log Concavity, Symmetric Rearrangement, Conditioning, Hypercontractivity
Chapter 5: Magnetic Fields and Stochastic Integrals
14 Itô′s Integral
15 Schrödinger Operators with Magnetic Fields
16 Introduction to Stochastic Calculus
Chapter 6: Asymptotics
17. Donsker’s Theorem
18. Laplace’s Method in Function Space
19. Introduction to the Donsker-Varadhan Theory
Chapter 7: Other Topics
20 Perturbation Theory for the Ground State Energy
21 Dirichlet Boundaries and Decoupling Singularities in Scattering Theory
22 Crushed Ice and the Wiener Sausage
23 The Statistical Mechanics of Charged Particles with Positive Definite Interactions
24 An Introduction to Euclidean Quantum Field Theory
25 Properties of Eigenfunctions, Wave Packets, and Green’s Functions
26 Inverse Problems and the Feynman–Kac Formula
References
Index
Pure and Applied Mathematics
A Series of Monographs and Textbooks
- No. of pages: 295
- Language: English
- Published: September 28, 1979
- Imprint: Academic Press
- eBook ISBN: 9780080874029
BS
Barry Simon
Affiliations and expertise
Department of Mathematics and Physics, Princeton University