Harmonic analysis and propagators on homogeneous spaces

Abstract

The techniques of harmonic analysis of homogeneous spaces are reviewed, and applied to the theory of propagators. The spectral geometry of homogeneous and, in particular, of symmetric spaces is considered, with explicit calculations of the heat kernel and the zeta function. Several topics relevant to physical applications are discussed, including the Schwinger-DeWitt expansion, the exactness of the WKB approximation in curved spaces, the connection between free motion on symmetric spaces and quantum integrable systems, and finite-temperature quantum field theories in higher dimensions. The paper contains some new results of both mathematical and physical interest; e.g., explicit formulas for the scalar degeneracies of the Laplacian on a compact symmetric space, exact forms of the zeta function on the symmetric spaces of rank one, extension of the finite-temperature formalism to spinor fields in higher-dimensional static spacetimes, and Casimir energy calculations in even dimensions.