Path Integral Methods

The last two decades have seen a consistent growth in the significance of path integrals. They are useful in quantum as well as statistical mechanics, providing more intuitive and easier methods of handling equations than more commonly-known treatments. However, the method is still greeted with skepticism. Where has the operator ordering problem gone? Do polar coordinates work in PI, despite the fact that they cannot exist in the canonical operator formalism? Is it possible to obtain a quantum mechanics on the sphere by inserting a delta function into the unconstrained formula? This book addresses and solves these fundamental issues, and covers more recent topics, such as the lattice fermion problem in quantum theory and the Monte Carlo method in statistical mechanics. Contents......Page 6 1.1 The Time Evolution Operator and Trotter's Formula......Page 8 1.2 Operator Ordering in the Path Integral......Page 11 1.3 The Euclidean Technique and Trace Formula......Page 16 1.4 Coherent States and the Path Integral......Page 20 1.5 Fermionic Path Integral......Page 23 2.1 Harmonic Oscillators......Page 33 2.2 Perturbation Theory......Page 47 2.3 The WKB Approximation......Page 54 2.4 Instantons and the Tunnelling Phenomena......Page 61 3.1 Change of Variables under the Path Integral......Page 73 3.2 Constrained Systems......Page 90 3.3 Multiply Connected Spaces......Page 110 3.4 Periodic Coherent States and Path Integral for Spin......Page 126 3.5 The WKB-exactness in Some Models......Page 137 4.1 The Path Integral in Field Theory......Page 151 4.2 Auxiliary Field Method......Page 169 5.1 Introduction......Page 175 5.2 Basic Ideas......Page 176 5.3 Quantum Effect and Importance Sampling Method......Page 181 5.4 Applications to Quantum Spin Systems......Page 182 5.5 Effectiveness in the Future......Page 186 6.1 Introduction......Page 188 6.2 Generalization of Trotter's Formula......Page 191 6.3 General Theory of Higher Order Decomposition......Page 195 6.4 Fractal Path Integral......Page 205 7.1 Stochastic Processes......Page 208 7.2 Molecular Collisions and Chemical Reactions......Page 212 7.3 The Tunnelling Effect in Condensed Matter Physics......Page 213 References......Page 215 Index......Page 220