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

A method which uses a generalized tensorial ζ function to compute the renormalized stress tensor of a quantum field propagating in a (static) 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 proven 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 proven 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 equations (customary generalized through high-order curvature terms). The method is checked in several particular cases finding a perfect agreement with other approaches. First the method is checked in the case of a conformally coupled massless field in the static Einstein universe where all hypotheses initially requested by the method hold true. Second, dropping the hypothesis of a closed manifold, the method is checked in the open static Einstein universe. Finally, the method is checked for a massless scalar field in the presence of a conical singularity in the Euclidean manifold (i.e., Rindler spacetimes, large mass black hole manifold, cosmic string manifold). Concerning the last case in particular, the method is proven to give rise to the stress tensor already got by the point-splitting approach for every coupling with the curvature regardless of the presence of the singular curvature. Comments on the measure employed in the path integral, the use of the optical manifold and the different approaches to renormalize the Hamiltonian are made.