Path Integral Approach to Quantum Physics: An Introduction - Tapa blanda

Sinopsis

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 Fluctuatio

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