Functorial properties of Schwinger-DeWitt expansion and Mellin-Barnes representation

TL;DR

This paper develops a functorial expansion for integral kernels of functions of differential operators on curved spacetime, separating geometric data from function-specific information, and computes these for certain operator functions using Mellin-Barnes integrals.

Contribution

It introduces an off-diagonal functorial expansion for kernels of operator functions, distinguishing geometric and functional data, and computes explicit forms for specific operator functions.

Findings

The expansion separates geometric data from function data.

Explicit Mellin-Barnes integral representations are derived.

The approach addresses regularization and physical interpretation issues.

Abstract

We consider integral kernels for functions $f(\hat F)$ of a minimal second-order differential operator $\hat F(\nabla)$ on a curved spacetime. We show that they can be expanded in a functional series, analogous to the DeWitt expansion for the heat kernel, by integrating the latter term-by-term. This procedure leads to a separation of two types of data: all information about the bundle geometry and the operator $\hat F(\nabla)$ is still contained in the standard HaMiDeW coefficients $\hat a_k[F | x,x']$ (we call this property ``off-diagonal functoriality''), while information about the function $f$ is encoded in some new scalar functions $\mathbb{B}_\alpha[f | \sigma]$ and $\mathbb{W}_\alpha[f | \sigma, m^2]$, which we call basis and complete massive kernels, respectively. These objects are calculated for operator functions of the form $\exp(-\tau\hat F^\nu)/(\hat F^\mu + \lambda)$ as multiple Mellin--Barnes integrals. The article also discusses subtle issues such as the validity of the term-by-term integration, the regularization of IR divergent integrals, and the physical interpretation of the resulting expansions.