Path Integrals and Anomalies in Curved Space

Path integrals provide a powerful method for describing quantum phenomena, first introduced in physics by Dirac and Feynman. This book introduces the quantum mechanics of particles that move in curved space by employing the path integral method and uses this formalism to compute anomalies in quantum field theories. The authors start by deriving path integrals for particles moving in curved space (onedimensional nonlinear sigma models), and their supersymmetric generalizations. Coherent states are used for fermionic particles. They then discuss the regularization schemes essential to constructing and computing these path integrals. In the second part of the book, the authors apply these methods to discuss and calculate anomalies in quantum field theory, with and without gravity. Anomalies constitute one of the most important aspects of quantum field theory; requiring that there are no anomalies provides an enormous constraint in the search for physical theories of elementary particles, quantum gravity, and string theories. In particular, the authors include explicit calculations of the gravitational anomalies, reviewing the seminal work of Alvarez-Gaumé and Witten in an original way. An advanced text for researchers and graduate students of quantum field theory in curved spaces and string theory; the first part can also be used as an introduction to path integrals in quantum mechanics.