Asymptotic heat kernels in quantum field theory

Asymptotic expansions were first introduced by Henri Poincar ́e in 1886. This paper describes their application to the semi-classical evaluation of amplitudes in quantum field theory with boundaries. By using zeta-function regularization, the conformal anomaly for a massless spin-1/2 field in flat Euclidean backgrounds with boundary is obtained on imposing locally supersymmetric boundary conditions. The quantization program for gauge fields and gravitation in the presence of boundaries is then introduced by focusing on conformal anomalies for higher-spin fields. The conditions under which the covariant Schwinger-DeWitt and the non-covariant, mode-by-mode analysis of quantum amplitudes agree are described.