Pseudodifferential calculus in Schwinger--DeWitt formalism: UV and IR parts

TL;DR

The paper develops a framework to obtain off-diagonal expansions for kernels of operator functions on curved backgrounds by separating UV and IR data. It extends the DeWitt heat-kernel formalism via a Laplace-transform approach to produce basis kernels that capture UV terms term-by-term, while treating IR contributions through large-$\tau$ behavior and regularization. Two IR regularization strategies are explored: analytic continuation and a mass regulator, with a detailed discussion of how each affects UV/IR contributions and their physical interpretation. The Bessel–Clifford function serves as a toy model to illustrate the UV/IR split, and the concept of off-diagonal functoriality ensures the same HaMiDeW coefficients appear across a broad class of operator functions.

Abstract

We consider expansions for the kernels of operator functions of second-order minimal operators on a curved background. We show that the terms of these expansions originate in the ultraviolet or infrared regions. We propose a systematic approach to obtaining ultraviolet terms using term-by-term integration of the DeWitt expansion of the heat kernel. We discuss two methods for regularizing infrared divergences arising at intermediate computational steps -- using analytic continuation and introducing a mass term -- and the relationship between them.