Heat kernel expansion in the background field formalism

Heat kernel expansion and background field formalism represent the combination of two calculational methods within the functional approach to quantum field theory. This approach implies construction of generating functionals for matrix elements and expectation values of physical observables. These are functionals of arbitrary external sources or the mean field of a generic configuration -- the background field. Exact calculation of quantum effects on a generic background is impossible. However, a special integral (proper time) representation for the Green's function of the wave operator -- the propagator of the theory -- and its expansion in the ultraviolet and infrared limits of respectively short and late proper time parameter allow one to construct approximations which are valid on generic background fields. Current progress of quantum field theory, its renormalization properties, model building in unification of fundamental physical interactions and QFT applications in high energy physics, gravitation and cosmology critically rely on efficiency of the heat kernel expansion and background field formalism.