Derivation of the Stress Tensor Operator in Two-Dimentional Space-Times Based on Wald's Axioms

Abstract

We give a derivation of the renormalized stress tensor operator of conformally invariant fields in two-dimensional space-times. It is based on axioms and simpler than that of Davies or Waldo Especially it clarifies which of the axioms are needed to obtain the trace anomaly, and a defect is pointed out in the arguments of Davies and Waldo The technique used in the derivation is applied to the problem of the moving mirror in two-dimensional space-time. The expectation value of the stress tensor operator of a quantum field is frequent­ ly used in the study of the quatum field theory in curved space-time. The operator TplI, which is given by replacing the field by a field operator in a classical expression for the stress tensor, is singular. Therefore one needs a renormalized or operator Tt~Y which gives physical, finite expectation value, and many renormaliza­ tion schemes have been developed. For a conformally invariant field in two­ dimensional space-times, by using the point-splitting renormalization, Davies, Fulling and Unruh found a formula (referred to as the D-F-U formula hereafter) which gives the expectation value of Tt~Y for an arbitrary geometry and an arbitrary vacuum. 1 ),2) It has been used in many investigations, and rederived with other renormalization schemes. 2 )-4) Especially, Davies 3 ) showed that the formula can be obtained without an ad hoc prescription of discarding infinity from the assumption that is a non-vanishing local quantity, and Wald 4 ) gave another derivation based on four assumptions 5 ) about physical properties of Tt~Y. derivation is preferable because his axioms are so physically reasonable as to be called Wald's axioms, and, by using the fifth axiom in addition it is applicable to the case of a conformally invariant field in four-dimensional conformally flat space-times. Restricted to the case of two-dimensional space-times, however, it is unnecessarily involved. On the other hand, Davies' derivation is simple, but his assumption seems to need a proof. In this paper we reconstruct the derivation of the D-F-U formula as concisely as possible. We proceed along the same way as Davies giving a proof of his assumption. The basic assumptions we will use are axioms and the properties of confor­ mally invariant fields. They are summarized in § 2. Our derivation which is given in § 3 clarifies which of the five axioms of Wald.are needed at each step. Especially to show that the trace anomaly is proportional to the scalar curvature (see theorem I and its corollary) the conservation law is not necessary in contrast to the argument of Wald. 4 ) Moreover a defect in the discussions of Davies and Wald is pointed out,