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

Andrei O. Barvinsky, Alexey E. Kalugin, Władysław Wachowski

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 $\mathbb{B}_α[f | σ]$ and $\mathbb{W}_α[f | σ, m^2]$, which we call basis and complete massive kernels, respectively. These objects are calculated for operator functions of the form $\exp(-τ\hat F^ν)/(\hat F^μ+ λ)$ 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.

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

Abstract

We consider integral kernels for functions of a minimal second-order differential operator 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 is still contained in the standard HaMiDeW coefficients (we call this property ``off-diagonal functoriality''), while information about the function is encoded in some new scalar functions and , which we call basis and complete massive kernels, respectively. These objects are calculated for operator functions of the form 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.

Paper Structure

This paper contains 28 sections, 142 equations.

Table of Contents

  1. Introduction
  2. Operator functions, their kernels, and integral transforms
  3. The simplest examples
  4. Basis kernels for $\hat{F}^{-\mu}$ and $\exp(-\tau\hat{F}^\nu)$
  5. Complementary kernels for $\hat{F}^{-\mu}$ and $\exp({-\tau\hat{F}^\nu})$
  6. Complete massive kernels for $\hat{F}^{-\mu}$
  7. More complex examples
  8. Basis kernels for $\exp(-\tau\hat{F}^\nu)/\hat{F}^\mu$
  9. Complete kernels for $\exp(-\tau\hat{F}^\nu)/\hat{F}^\mu$
  10. Complete kernels for $1/(\hat{F}^\mu + \lambda)$
  11. The operator function $\exp(-\tau F^\nu)/(F^\mu + \lambda)$
  12. The resonant case
  13. Regularization via analytic continuation
  14. Bare poles for $\hat{F}^{-\mu}$.
  15. Pinches for $\exp(-\tau\hat{F})/\hat{F}^\mu$.
  16. ...and 13 more sections