Comments on the Stress-Energy Tensor Operator in Curved Spacetime
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The technique based on a *-algebra of Wick products of field operators in curved spacetime, in the local covariant version proposed by Hollands and Wald, is strightforwardly generalized in order 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. Within the proposed formalism, there is room to accomplish all of the physical requirements provided that known problems concerning the conservation of the stress-energy tensor are assumed to be related to the interface between the quantum and classical formalism. The proposed stress-energy tensor operator turns out to be 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. These terms are independent from the arbitrary length scale (and any quantum state) and they classically vanish on solutions of the Klein-Gordon equation. Considering the averaged stress-energy tensor with respect to Hadamard quantum states, the presented definition turns out to be equivalent to an improved point-splitting renormalization procedure which makes use of the nonambiguous part of the Hadamard parametrix only that is determined by the local geometry and the parameters which appear in the Klein-Gordon operator. In particular, no extra added-by-hand term g αβQ and no arbitrary smooth part of the Hadamard parametrix (generated by some arbitrary smooth term ``ω 0 '') are involved. The averaged stress-energy tensor obtained by the point-splitting procedure also coincides with that found by employing the local ζ-function approach whenever that technique can be implemented.
KeywordsField Operator Classical Form Curve Spacetime Classical Field Local Geometry
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