Path Integrals in Quantum Physics

These lectures aim at giving graduate students an introduction to and a working knowledge of path integral methods in a wide variety of fields in physics. Consequently, the lecture notes are organized in three main parts dealing with non-relativistic quantum mechanics, many-body physics and field theory. In the first part the basic concepts of path integrals are developed in the usual heuristic, non-mathematical way followed by standard examples and special applications including numerical evaluation of (euclidean) path integrals by Monte-Carlo methods with a program for the anharmonic oscillator. The second part deals with the application of path integrals in statistical mechanics and many-body problems treating the polaron problem, dissipative quantum systems, path integrals over ordinary and Grassmannian coherent states and perturbation theory for both bosons and fermions. Again a simple Fortran program is included for illustrating the use of strong-coupling methods. Finally, in the third part path integrals in relativistic quantum field theory are discussed. Standard topics like the generating functional for Green functions, perturbative expansions, effective actions and quantization of gauge theories are treated as well as special applications (the worldline formalism, spin in relativistic path integrals and the derivation of anomalies by path integral methods). The last section tries to give a simple introduction into lattice (gauge) field theory including a numerical example which can be run on a PC. The set of problems which accompanied the lectures is also included in the present notes.