Schwinger--DeWitt expansion for the heat kernel of nonminimal operators in causal theories

TL;DR

The paper develops a covariant, pseudo-differential framework to extend the Schwinger–DeWitt heat-kernel expansion to nonminimal operators in causal theories by mapping e^{−τH} to a minimal-operator kernel e^{−τ′F} with a local curvature expansion. A finite-range subtraction scheme ensures finiteness of the coefficient functions and controls infrared issues, while a perturbative, noncommutative-algebra approach builds the full heat kernel from the minimal-operator data. The method is illustrated with Proca (degenerate symbol) and nondegenerate vector operators, showing that nondegenerate cases yield smooth heat kernels and that degenerate cases may require distributional treatments beyond the Gilkey–Seeley framework. The approach offers manifest covariance, a clear perturbative calculus in curvatures, and potential for computer-algebra implementation, with clear avenues for extension to higher-derivative operators and more general background structures.

Abstract

We suggest a systematic calculational scheme for heat kernels of covariant nonminimal operators in causal theories whose characteristic surfaces are null with respect to a generic metric. The calculational formalism is based on a pseudodifferential operator calculus which allows one to build a linear operator map from the heat kernel of the minimal operator to the nonminimal one. This map is realized as a local expansion in powers of spacetime curvature, dimensional background fields, and their covariant derivatives with the coefficients -- the functions of the Synge world function and its derivatives. Finiteness of these functions, determined by multiple proper time integrals, is achieved by a special subtraction procedure which is an important part of the calculational scheme. We illustrate this technique on the examples of the vector Proca model and the vector field operator with a nondegenerate principal symbol. We also discuss smoothness properties of heat kernels of nonminimal operators in connection with the nondegenerate nature of their operator symbols.