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

TL;DR

This paper develops a systematic approach to analyze the ultraviolet and infrared contributions in the heat kernel expansion of second-order operators on curved backgrounds, addressing regularization of infrared divergences.

Contribution

It introduces a method for extracting UV terms via term-by-term integration of the DeWitt expansion and compares two IR regularization techniques: analytic continuation and mass introduction.

Findings

UV and IR contributions originate from different expansion regions

A systematic method for UV term extraction is proposed

Infrared divergences can be regularized by two equivalent methods

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.