Heat Kernel as the Basis for the Functional Integration in Quantum Field Theory

Abstract

The only way to get numbers from the functional integral in QFT is to define it as a series of Gaussian integrals, which are well defined in terms of the functional determinants and the Green functions, and finally, by stansard procedure, in terms of the heat kernel. Thus the heat kernel is the basis for the calculation of the functional integral in QFT. In our recent papers (see I. G. Avramidi, J. Math. Phys. 37 (1996) 374; 36 (1995) 5055) we studied the heat kernel in quantum gravity and gauge theories for strong slowly varying background fields and proposed a new covariant purely algebraic approach based on taking into account only a finite number of low-order covariant derivatives of the background fields. The explicit manifestly covariant closed formulas for the heat kernel expressed purely in terms of curvature invariants are obtained.