Preface page xv Acknowledgements xviii 1 Synopsis 1 Part one Fundamental principles 5 2 The mathematical structure of quantum mechanics 7 3 Operators 30 4 The Feynman path integral 61 5 Hamiltonian mechanics 80 6 Path integral quantization 105 Part two Stochastic processes 123 7 Stochastic systems 125 Part three Discrete degrees of freedom 159 8 Ising model 161 9 Ising model: magnetic field 180 10 Fermions 198 Part four Quadratic path integrals 223 11 Simple harmonic oscillator 225 12 Gaussian path integrals 251 Part five Action with acceleration 271 13 Acceleration Lagrangian 273 14 Pseudo-Hermitian Euclidean Hamiltonian 294 15 Non-Hermitian Hamiltonian: Jordan blocks 330 Part six Nonlinear path integrals 349 16 The quartic potential: instantons 351 17 Compact degrees of freedom 385 18 Conclusions 405 References 409 Index 413