Direct ζ-function approach and renormalization of one-loop stress tensors in curved spacetimes

A method which uses a generalized ζ-function to compute the renormalized (Euclidean) stress tensor of a quantum field propagating in a curved background is presented. The method does not use point-splitting procedures or off-diagonal ζ functions but employs an analytic continuation of a generalized ζ-function. The starting point of the method is the direct computation of the functional derivatives of the Euclidean one-loop effective action with respect to the background metric. It is proved that the method, when available, gives rise to a conserved stress tensor and, in the case of a massless conformally coupled field, produces the conformal anomaly formula directly. Moreover, it is proved that the obtained stress tensor agrees with statistical mechanics in the case of a finite-temperature theory. The renormalization procedure is controlled by the structure of the poles of the stress-tensor ζ function. The infinite renormalization is automatic due to a “magic” cancellation of two poles. The remaining finite renormalization involves locally geometrical terms arising by a certain residue. Such terms are also conserved and thus represent just a finite renormalization of the geometric part of the Einstein gravitation equations (customary generalized through high-order curvature terms). Finally, the method is checked in two particular cases finding a perfect agreement with other approaches. PACS number(s): 04.62.+v e-mail: [email protected]