Operator regularization and one-loop Green’s functions

In this paper we present an alternate way of computing amplitudes in quantum field theory in the context of background-field quantization. We concentrate mainly on one-loop effects. The Feynman diagrams of the usual perturbation series are avoided by first performing the functional integration and then using a perturbative expansion due to Schwinger. In this approach we regulate operators rather than the initial Lagrangian. To one-loop order our scheme reduces to a perturbative expansion of the well-known ζ function associated with the superdeterminant of an operator. This technique preserves all symmetries present in the initial theory and does not lead to any explicit divergences as the regulating parameter approaches its limiting value. For illustration, we apply our approach to a toy (${\phi }^3$ ${)}_6$ scalar theory, to Yang-Mills theory in the covariant gauge, and to quantum electrodynamics. This method reproduces the usual axial anomaly in the three-point functions VVA and AAA. Operator regularization is used in a dimensionally regulated theory reproducing the usual results obtained in the dimensionally regulated Feynman-diagram approach. An outline of how operator regularization is applied beyond one-loop order is provided. Other possible applications of operator regularization are discussed.