Comments on the Stress-Energy Tensor Operator in Curved Spacetime

TL;DR

This paper extends the definition of the stress-energy tensor operator to curved spacetimes using Wick products, ensuring conservation, covariance, and independence from arbitrary scales, with a focus on Hadamard states and local geometry.

Contribution

It generalizes Hollands and Wald's technique to define a conserved, covariant stress-energy tensor operator in curved spacetimes, incorporating local Wick products and scale independence.

Findings

The operator reduces to the classical form on solutions.

The averaged tensor is obtained via an improved point-splitting method.

The approach aligns with the local $z$-function method.

Abstract

Hollands and Wald's technique based on *-algebras of Wick products of field operators is strightforwardly generalized to define the stress-energy tensor operator in curved globally hyperbolic spacetimes. In particular, the locality and covariance requirement is generalized to Wick products of differentiated quantum fields. The proposed stress-energy tensor operator is conserved and reduces to the classical form if field operators are replaced by classical fields satisfying the equation of motion. The definition is based on the existence of convenient counterterms given by certain local Wick products of differentiated fields. They are independent from the arbitrary length scale (and any quantum state) and they classically vanish on solutions of field equation. The averaged stress-energy tensor with respect to Hadamard quantum states can be obtained by an improved point-splitting renormalization procedure which makes use of the nonambiguous part of the Hadamard parametrix only that is completely determined by the local geometry and the parameters which appear in the Klein-Gordon operator. The averaged stress-energy tensor also coincides with that found by employing the local $\zeta$-function approach.