Summary

This comprehensive textbook on the theory and applications of path integrals is the first to contain the solutions to a number of non-trivial path integrals, most notably of the Coulomb system. It has become possible by finding a consistent formulation of path integrals in spaces with curvature and torsion, presented here also for the first time. Special emphasis is given to stability problems of path fluctuations in the presence of singular potentials such as centrifugal and angular barriers. The limitations of Feynman's time slicing procedure are exhibited and a new path integral formula is found which avoids the frequent danger of path collapse. The physically important applications to tunneling problems are analyzed in detail. Their relevance to superconductivity and the large-order behavior of perturbation expansions is demonstrated. The path integral description of equilibrium thermodynamics is presented, and an extension to non-equilibrium processes is given. Much attention is paid to path integrals in spaces with topological restrictions. Their applications to entanglement problems in polymer physics and their relevance to particle statistics are discussed, also to the recently popular phenomenon of fractional statistics.