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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

Immediately download your ebook while waiting for your print delivery. No promo code is needed.

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

Affiliations and expertise

Department of Mathematics and Physics, Princeton University