On the Stress-Energy Tensor of Quantum Fields in Curved Spacetimes - Comparison of Different Regularization Schemes and Symmetry of the Hadamard/Seeley-DeWitt Coefficients

TL;DR

This paper reviews rigorous results on regularizing the stress-energy tensor of quantum fields in curved spacetimes, focusing on symmetry properties and equivalences of different regularization schemes in various geometric settings.

Contribution

It establishes the symmetry of Hadamard/Seeley-DeWitt coefficients and proves the equivalence of local zeta-function and Hadamard point-splitting methods in smooth spacetimes.

Findings

Symmetry of Hadamard/Seeley-DeWitt coefficients in smooth spacetimes

Equivalence of local ζ-function and Hadamard point-splitting in static spacetimes

Equivalence of DeWitt-Schwinger and Hadamard point-splitting procedures

Abstract

We review a few rigorous and partly unpublished results on the regularisation of the stress-energy in quantum field theory on curved spacetimes: 1) the symmetry of the Hadamard/Seeley-DeWitt coefficients in smooth Riemannian and Lorentzian spacetimes 2) the equivalence of the local $\zeta$-function and the Hadamard-point-splitting procedure in smooth static spacetimes 3) the equivalence of the DeWitt-Schwinger- and the Hadamard-point-splitting procedure in smooth Riemannian and Lorentzian spacetimes.