Functional Integration and Quantum Physics

Functional Integration and Quantum Physics

1st Edition - September 28, 1979

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  • Editor: Barry Simon
  • eBook ISBN: 9780080874029

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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.

Table of Contents

  • 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



    Pure and Applied Mathematics

    A Series of Monographs and Textbooks

Product details

  • No. of pages: 295
  • Language: English
  • Copyright: © Academic Press 1979
  • Published: September 28, 1979
  • Imprint: Academic Press
  • eBook ISBN: 9780080874029

About the Series Editor

Barry Simon

Affiliations and Expertise

Department of Mathematics and Physics, Princeton University

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