Hadamard Regularization and Conformal Transformation

Abstract

We investigate the conformal transformation of the renormalized stress tensor of conformally invariant scalar fields in the framework of the Hadamard regularization, We find that the renorma.l· ized stress tensor consists of three parts. The first is the simple scale transformation of the renormal· ized stress tensor, the second comes from the scale transformation of the logarithmic term in the Green's function and the third consists of curvature tensors, the conformal factor and their covariant derivatives. The method is applied to Friedmann universes and Schwarzschild black hole. § 1. Introduction Conformally invariant scalar field is one of the most interesting models to study the properties of quantized fields in curved spacetime. Conformal invariance of field equations helps us to construct mode solutions and the Green's functions. The conformal behaviour of the renormalized stress tensor is, however, not so trivial because we execute some regularization procedures to obtain a finite renormalized stress tensor and they break the conformal invariance and give the trace anomaly. Nevertheless in evaluating the renormalized stress tensor in concrete spacetimes, a' conformal transformation technique seems to be important. The de Sitter invariant vacuum state corresponds to the conformal vacuum with closed and fiat coordinatiza­ tion of de Sitter space.!) For the two-dimensional Schwarzschild black hole the static thermal state (Israel-Hartle-Hawking state)2) is equivalent to the conformal vacuum. Moreover Page has succeeded in obtaining an approximate renormalized stress tensor, in the Schwarzschild geometry with a conformal transformation technique. 3 ) He defined the optical metric which is conformally related to the static Einstein metric and where the trace anomaly of the renorma:lized stress tensor vanishes. In the optical metric the Gaussian approximation of the thermal propagator is very good. 4 ) He chose the renormalized stress tensor in the optical metric to have a thermal form and obtained the physical renormalized stress tensor by conformal transformation. His result remarkably agreed with numerical work by Howard and Candelas. 5 ) His method was improved by Zannias, Brown and Ottewill (Brown-Ottewill-Page-Zannias (BOPZ) Ansatz).6),7) These results suggest that more explicit investigation of the behaviour of the renormalized stress tensor under conformal tninsformation i~ impor­ tant to understand the behaviour of quantized fields in curved spacetime. There are two approaches to study the behaviour of the renormalized stress­ tensor under conformal transformation. The first one i~ to examine the conformal behaviour of the effective action. This method was first applied to the conformally fiat cases by Brown and Cassidy.S) For general cases the conformal transformation of the effective action was studied with dimensional regularization by Brown and OttewilF) and with zeta function regularization by Dowker.