Path Integral Approach to Quantum Physics An Introduction

This book has been written twice. After having written and published it in German in 1990, I started allover again and rewrote the whole story for an English speaking audience. During the first round I received encouraging words and critical remarks from students and colleagues alike which have helped to sustain me the second time around. In the preface the author usually states that his or her book resulted from a course that he or she gave at some university. I cannot claim that the present book is any exception to the rule. But I expanded and remodelled the original material which circulated as a manuscript so that the printed version would follow a more stringent and coherent architectural plan. In doing so I have concentrated on the conceptual problems inherent in the path integral formalism rather than on certain highly specialized techniques used in applications. Nevertheless, I have also included those methods that are of fundamental interest and have treated specific problems mainly to illustrate them.

1. Brownian Motion.- 1.1 The One-Dimensional Random Walk.- 1.2 Multidimensional Random Walk.- 1.3 Generating Functions.- 1.4 The Continuum Limit.- 1.5 Imaginary Time.- 1.6 The Wiener Process.- 1.7 Expectation Values.- 1.8 The Ornstein-Uhlenbeck Process.- 2. The Feynman-Kac Formula.- 2.1 The Conditional Wiener Measure.- 2.2 The Integral Equation Method.- 2.3 The Lie-Trotter Product Method.- 2.4 The Brownian Tube.- 2.5 The Golden-Thompson-Symanzik Bound.- 2.6 Hamiltonians and Their Associated Processes.- 2.7 The Thermodynamical Formalism.- 2.8 A Case Study: the Harmonic Spin Chain.- 2.9 The Reflection Principle.- 2.10 Feynman Versus Wiener Integrals.- 3. The Brownian Bridge.- 3.1 The Canonical Scaling of Brownian Paths.- 3.2 Bounds on the Transition Amplitude.- 3.3 Variational Principles.- 3.4 Bound States.- 3.5 Monte Carlo Calculation of Path Integrals.- 4. Fourier Decomposition.- 4.1 Random Fourier Coefficients.- 4.2 The Wigner-Kirkwood Expansion of the Effective Potential.- 4.3 Coupled Systems.- 4.4 The Driven Harmonic Oscillator.- 4.5 Oscillating Electric Fields.- 5. The Linear-Coupling Theory of Bosons.- 5.1 Path Integrals for Bosons.- 5.2 A Random Potential for the Electron.- 5.3 The Polaron Problem.- 5.4 The Field Theory of the Polaron Model.- 6. Magnetic Fields.- 6.1 Heuristic Considerations.- 6.2 Itô Integrals.- 6.3 The Constant Magnetic Field.- 6.4 Diamagnetism of Electrons in a Solid.- 6.5 Magnetic Flux Lines.- 7. Euclidean Field Theory.- 7.1 What Is a Euclidean Field?.- 7.2 The Euclidean Two-Point Function.- 7.3 The Euclidean Free Field.- 7.4 Gaussian Functional Integrals.- 7.5 Basic Postulates.- 8. Field Theory on a Lattice.- 8.1 The Lattice Version of the Scalar Field.- 8.2 The Euclidean Propagator on the Lattice.- 8.3 The Variational Principle.- 8.4 TheEffective Action.- 8.5 The Effective Potential.- 8.6 The Ginzburg-Landau Equations.- 8.7 The Mean-Field Approximation.- 8.8 The Gaussian Approximation.- 9. The Quantization of Gauge Theories.- 9.1 The Euclidean Version of Maxwell Theory.- 9.2 Non-Abelian Gauge Theories: Preliminaries.- 9.3 The Faddeev-Popov Quantization.- 9.4 Gauge Theories on a Lattice.- 9.5 Wegner-Wilson Loops.- 9.6 The SU(n) Higgs Model.- 10. Fermions.- 10.1 The Dirac Field in Minkowski Space.- 10.2 The Euclidean Dirac Field.- 10.3 Grassmann Algebras.- 10.4 Formal Derivatives.- 10.5 Formal Integration.- 10.6 Functional Integrals of QED.- 10.7 The SU(n) Gauge Theory with Fermions.- Appendices.- A List of Symbols and Glossary.- B Frequently Used Gaussian Processes.- C Jensen’s Inequality.- D A Table of Path Integrals.- References.