Abstract
In the point-splitting prescription for renormalizing the stress-energy tensor of a scalar field in curved spacetime, it is assumed that the anticommutator expectation valueG(x, x′)=〈ø(x)ø(x′)+ø(x′)ø(x)〉 has a singularity of the Hadamard form asx→x′. We prove here that ifG(x,x′) has the Hadamard singularity structure in an open neighborhood of a Cauchy surface, then it does so everywhere, i.e., Cauchy evolution preserves the Hadamard singularity structure. In particular, in a spacetime which is flat below a Cauchy surface, for the “in” vacuum stateG(x,x′) is of the Hadamard form everywhere, and thus the point-splitting prescription in this case has been rigorously shown to give meaningful, finite answers.
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Communicated by R. Geroch
Supported in part by NSF grants PHY 77-01432 to Texas A&M University and PHY 76-81102 to the University of Chicago, and by the Alfred P. Sloan Foundation
NSF Predoctoral Fellow
Sloan Foundation Fellow
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Fulling, S.A., Sweeny, M. & Wald, R.M. Singularity structure of the two-point function in quantum field theory in curved spacetime. Commun.Math. Phys. 63, 257–264 (1978). https://doi.org/10.1007/BF01196934
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DOI: https://doi.org/10.1007/BF01196934