Path Integral Approach to Quantum Physics

Specifically designed to introduce graduate students to the functional integration method in contemporary physics as painlessly as possible, the book concentrates on the conceptual problems inherent in the path integral formalism. Throughout, the striking interplay between stochastic processes, statistical physics and quantum mechanics comes to the fore, and all the methods of fundamental interest are generously illustrated by important physical examples.

1. Brownian Motion.- 1.1 The One-Dimensional Random Walk.- 1.2 Multidimensional Random Walk.- 1.3 Generating Functions.- 1.3.1 Return or Escape?.- 1.4 The Continuum Limit.- 1.5 Imaginary Time.- 1.6 The Wiener Process.- 1.6.1 The Analysis of Random Paths.- 1.6.2 Multidimensional Gaussian Measures.- 1.6.3 Increments.- 1.7 Expectation Values.- 1.8 The Ornstein-Uhlenbeck Process.- 1.8.1 The Oscillator Process.- 2. The Feynman-Kac Formula.- 2.1 The Conditional Wiener Measure.- 2.1.1 The Path Integral.- 2.1.2 The Stochastic Formulation.- 2.2 The Integral Equation Method.- 2.2.1 Stochastic Representation of Operator Norms.- 2.2.2 Stochastic Representation of Green’s Functions.- 2.3 The Lie-Trotter Product Method.- 2.3.1 The Lie-Trotter Product Formula.- 2.3.2 Miscellaneous Remarks and Results.- 2.3.3 Several Particles with Different Masses.- 2.4 The Brownian Tube.- 2.5 The Golden-Thompson-Symanzik Bound.- 2.6 Hamiltonians and Their Associated Processes.- 2.6.1 Correlation Functions.- 2.6.2 The Oscillator Process Revisited.- 2.6.3 Nonlinear Transformations of Time.- 2.6.4 The Perturbed Harmonic Oscillator.- 2.7 The Thermodynamical Formalism.- 2.8 A Case Study: the Harmonic Spin Chain.- 2.8.1 The Inverted Harmonic Oscillator.- 2.9 The Reflection Principle.- 2.9.1 Reflection Groups of Order Two.- 2.9.2 Reflection Groups of Infinite Order.- 2.10 Feynman Versus Wiener Integrals.- 2.10.1 Summing over Histories in Configuration Space.- 2.10.2 The Method of Stationary Phase.- 2.10.3 Summing over Histories in Phase Space.- 2.10.4 The Feynman Integrand as a Hida Distribution.- 3. The Brownian Bridge.- 3.1 The Canonical Scaling of Brownian Paths.- 3.1.1 The Process X?.- 3.1.2 Rescaling of Path Integrals.- 3.1.3 The Stochastic Integral with Respect to the Brownian Bridge.- 3.2 Bounds on the Transition Amplitude.- 3.2.1 Defining a Subset of Paths.- 3.2.2 The Semiclassical Approximation.- 3.2.3 Bounds on the Functional ?(V).- 3.2.4 Convexity of the Functional ?(V).- 3.3 Variational Principles.- 3.3.1 The Mean Position of a Path.- 3.4 Bound States.- 3.4.1 Moment Inequalities for Eigenvalues.- 3.5 Monte Carlo Calculation of Path Integrals.- 4. Fourier Decomposition.- 4.1 Random Fourier Coefficients.- 4.1.1 Fourier Analysis of Time Integrals.- 4.2 The Wigner-Kirkwood Expansion of the Effective Potential.- 4.3 Coupled Systems.- 4.3.1 Open Systems.- 4.4 The Driven Harmonic Oscillator.- 4.4.1 From Time Integrals to Sums.- 4.4.2 From Sums Back to Time Integrals.- 4.5 Oscillating Electric Fields.- 4.5.1 Poisson Statistics.- 5. The Linear-Coupling Theory of Bosons.- 5.1 Path Integrals for Bosons.- 5.1.1 The Partial Trace and Its Evaluation.- 5.2 A Random Potential for the Electron.- 5.3 The Polaron Problem.- 5.3.1 The Limit L ? ?, b ?.- 5.3.2 The Free Energy of the Polaron.- 5.3.3 Bounds on the Polaxon Free Energy.- 5.3.4 Pekar’s Large-Coupling Result.- 5.4 The Field Theory of the Polaron Model.- 6. Magnetic Fields.- 6.1 Heuristic Considerations.- 6.2 Itô Integrals.- 6.2.1 The Feynman-Kac-Itô Formula.- 6.2.2 The Semiclassical Approximation.- 6.3 The Constant Magnetic Field.- 6.3.1 A Brief Discussion of the Result.- 6.4 Diamagnetism of Electrons in a Solid.- 6.5 Magnetic Flux Lines.- 6.5.1 Winding Numbers.- 6.5.2 Spectral Decomposition.- 6.5.3 Imaginary Times.- 7. Euclidean Field Theory.- 7.1 What Is a Euclidean Field?.- 7.2 The Euclidean Two-Point Function.- 7.3 The Euclidean Free Field.- 7.3.1 The n-Point Functions.- 7.3.2 The Stochastic Interpretation.- 7.4 Gaussian Functional Integrals.- 7.5 Basic Postulates.- 7.5.1 The Hamiltonian.- 7.5.2 The Free Field Revisited.- 8. Field Theory on a Lattice.- 8.1 The Lattice Version of the Scalar Field.- 8.2 The Euclidean Propagator on the Lattice.- 8.2.1 The Fourier Representation.- 8.2.2 Random Paths on a Lattice.- 8.3 The Variational Principle.- 8.3.1 The Case of a Discrete Configuration Space.- 8.3.2 The Deterministic Limit.- 8.3.3 Continuous Configuration Space.- 8.3.4 The Classical Limit.- 8.3.5 Fluctuations Around the Classical Solution.- 8.4 The Effective Action.- 8.5 The Effective Potential.- 8.5.1 Spontaneous Breakdown of Symmetry.- 8.5.2 Order Parameters.- 8.6 The Ginzburg-Landau Equations.- 8.7 The Mean-Field Approximation.- 8.7.1 The Curie-Weiss Approximation of the Ising Model.- 8.7.2 The Ising Spin Limit of the Neutral Scalar Field.- 8.8 The Gaussian Approximation.- 8.8.1 A Case Study.- 9. The Quantization of Gauge Theories.- 9.1 The Euclidean Version of Maxwell Theory.- 9.1.1 The Classical Situation (h? = 0).- 9.1.2 Gauge Fixing.- 9.1.3 The Quantized Situation (h? > 0).- 9.2 Non-Abelian Gauge Theories: Preliminaries.- 9.3 The Faddeev-Popov Quantization.- 9.3.1 Division by |G|.- 9.3.2 Faddeev-Popov Ghosts.- 9.4 Gauge Theories on a Lattice.- 9.5 Wegner-Wilson Loops.- 9.5.1 The Static Approximation in Minkowskian Field Theory.- 9.5.2 Loop Variables in Euclidean QED.- 9.5.3 Area Law or Perimeter Law?.- 9.6 The SU(n) Higgs Model.- 10. Fermions.- 10.1 The Dirac Field in Minkowski Space.- 10.2 The Euclidean Dirac Field.- 10.2.1 External Vector Potentials.- 10.3 Grassmann Algebras.- 10.3.1 When E Is a Function Space.- 10.4 Formal Derivatives.- 10.5 Formal Integration.- 10.5.1 Integrals in A(E).- 10.5.2 Integrals in A(E ? F).- 10.5.3 Integrals of the Exponential Type.- 10.5.4 The Fourier-Laplace Transformation.- 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.