ζ-function regularization and one-loop renormalization of field fluctuations in curved space-time

The naive ζ-function approach to regularize and renormalize the fluctuations of a quantum field in a curved background by taking the value of ζ(1) is improved. The proposed method produces finite quantities without the need of further subtractions and finite scale-parameterized counterterms at most also in the presence of a pole at the value s=1 of ζ(s). This method is employed to obtain a general expression for the trace of the stress tensor of a non-conformal invariant scalar field and checked on some examples. In particular, the fluctuations of a quantum scalar field are computed in the closed Einstein universe and similar ultrastatic manifolds for a massive field with a generally non-conformal coupling with the curvature.