On the stress-energy tensor of quantum fields in curved spacetimes-comparison of different regularization schemes and symmetry of the Hadamard/Seeley-DeWitt coefficients

We review a few rigorous and partly unpublished results on the regularization of the stress-energy in quantum field theory on curved spacetimes: (1) the symmetry of the Hadamard/Seeley-DeWitt coefficients in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 and (3) the equivalence of the DeWitt-Schwinger- and the Hadamard-point-splitting procedure in smooth Riemannian and Lorentzian spacetimes. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'.