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

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 ${\hat{a}}_k[F|x,x^{\prime }]$ (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.

arXiv comment22 pages

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Cited by in corpus (1)

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