DIRECT ZETA-FUNCTION APPROACH AND RENORMALIZATION OF ONE-LOOP STRESS TENSORS IN CURVED SPACETIMES

A method which uses a generalized tensorial zeta function to compute t he renormalized stress tensor of a quantum field propagating in a (sta tic) curved background is presented. The method does not use point-spl itting procedures or off-diagonal zeta functions but employs an analyt ic continuation of a generalized zeta 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 co upled 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 renormaliza tion procedure is controlled by the structure of the poles of the stre ss-tensor zeta function. The infinite renormalization is automatic due to a ''magic'' cancellation of two poles. The remaining finite renorm alization involves locally geometrical terms arising by a certain resi due. Such terms are also conserved and thus represent just a finite re normalization of the geometric part of the Einstein equations (customa ry generalized through high-order curvature terms). The method is chec ked 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 hypot heses initially requested by the method hold true. Second, dropping th e hypothesis of a closed manifold, the method is checked in the open s tatic Einstein universe. Finally, the method is checked for a massless scalar held in the presence of a conical singularity in the Euclidean manifold (i.e., Rindler spacetimes, large mass black hole manifold, c osmic string manifold). Concerning the last case in particular, the me thod is proven to give rise to the stress tensor already got by the po int-splitting approach for every coupling with the curvature regardles s of the presence of the singular curvature. Comments on the measure e mployed in the path integral, the use of the optical manifold and the different approaches to renormalize the Hamiltonian are made. [S0556-2 821(97)02724-0].