Propagators for relativistic systems with non-Abelian interactions

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Abstract
The relativistic evolution of a system of particles in the proper-time Schwinger–DeWitt formalism is investigated. For a class of interactions that can be represented as Fourier transforms of bounded complex matrix-valued measures, a Dyson series representation of the propagator is obtained. This class of interactions is non-Abelian and includes both external electromagnetic and Yang–Mills fields. The study of the relativistic problem is facilitated by embedding the original quantum evolution into a larger class of evolution problems that result if one makes an analytic continuation of the metric tensor gμν. This continuation is chosen so that the extended propagator shares (for all signatures of gμν ) the Gaussian decay properties typical of heat kernels. Estimates of the nth-order Dyson iterate kernels are found that ensure the absolute convergence of the perturbation series. In this fashion a number of analytic and smoothness properties of the propagator are determined. In particular, it is demonstrated that the convergent Dyson series representation constructs a fundamental solution of the equations of motion.