Heat kernels in quantum field theory

ABSTRACT: In quantum field theory arise the problems of vacuum polarization - the particle creation by gravitational or electromagnetic fields. These background (classical) fields define the space as a Riemannian manifold with a vector bundle. Such problems require covariant methods best achieved in the effective action approach. The effective action, which is the generating functional of the Green functions for the loop expansion, can be derived from the kernel of the heat equation. Therefore, for a quantum field theory defined by its elliptic differential operator, the problem boils down to finding approximate schemes for calculating the heat kernel. In this talk, I focus on the covariant perturbation theory, while some other techniques are compared. Relevant physical applications are also reviewed.