Abstract
This paper describes a technique for regularizing quadratic path integrals on a curved background spacetime. One forms a generalized zeta function from the eigenvalues of the differential operator that appears in the action integral. The zeta function is a meromorphic function and its gradient at the origin is defined to be the determinant of the operator. This technique agrees with dimensional regularization where one generalises ton dimensions by adding extra flat dimensions. The generalized zeta function can be expressed as a Mellin transform of the kernel of the heat equation which describes diffusion over the four dimensional spacetime manifold in a fith dimension of parameter time. Using the asymptotic expansion for the heat kernel, one can deduce the behaviour of the path integral under scale transformations of the background metric. This suggests that there may be a natural cut off in the integral over all black hole background metrics. By functionally differentiating the path integral one obtains an energy momentum tensor which is finite even on the horizon of a black hole. This energy momentum tensor has an anomalous trace.
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References
DeWitt, B.S.: Dynamical theory of groups and fields in relativity, groups and topology (eds. C. M. DeWitt and B. S. DeWitt). New York: Gordon and Breach 1964
DeWitt, B.S.: Phys. Rep.19C, 295 (1975)
McKean, H.P., Singer, J.M.: J. Diff. Geo.5, 233–249 (1971)
Gilkey, P.B.: The index theorem and the heat equation. Boston: Publish or Perish 1974
Candelas, P., Raine, D.J.: Phys. Rev. D12, 965–974 (1975)
Drummond, I.T.: Nucl. Phys.94B, 115–144 (1975)
Capper, D., Duff, M.: Nuovo Cimento23A, 173 (1974)
Duff, M., Deser, S., Isham, C.J.: Nucl. Phys.111B, 45 (1976)
Brown, L.S.: Stress tensor trace anomaly in a gravitational metric: scalar field. University of Washington, Preprint (1976)
Brown, L.S., Cassidy, J.P.: Stress tensor trace anomaly in a gravitational metric: General theory, Maxwell field. University of Washington, Preprint (1976)
Dowker, J.S., Critchley, R.: Phys. Rev. D13, 3224 (1976)
Dowker, J.S., Critchley, R.: The stress tensor conformal anomaly for scalar and spinor fields. University of Manchester, Preprint (1976)
Gibbons, G.W., Hawking, S.W.: Action integrals and partition functions in quantum gravity. Phys. Rev. D (to be published)
Manor, Y.: Complex Riemannian sections. University of Cambridge, Preprint (1977)
Feynman, R. P.: Magic without magic, (eds. J. A. Wheeler and J. Klaunder). San Francisco: W. H. Freeman 1972
DeWitt, B.S.: Phys. Rev.162, 1195–1239 (1967)
Fadeev, L.D., Popov, V.N.: Usp. Fiz. Nauk111, 427–450 (1973) [English translation in Sov. Phys. Usp.16, 777–788 (1974)]
Seeley, R.T.: Amer. Math. Soc. Proc. Symp. Pure Math.10, 288–307 (1967)
Ray, D.B., Singer, I.M.: Advances in Math.7, 145–210 (1971)
Gilkey, P.B.: Advanc. Math.15, 334–360 (1975)
'tHooft, G.: Phys. Rev. Letters37, 8–11 (1976)
'tHooft, G.: Computation of the quantum effects due to a four dimensional pseudoparticle. Harvard University, Preprint
Hartle, J.B., Hawking, S.W.: Phys. Rev. D13, 2188–2203 (1976)
Adler, S., Lieverman, J., Ng, N.J.: Regularization of the stress-energy tensor for vector and scalar particles. Propagating in a general background metric. IAS Preprint (1976)
Fulling, S.A., Christensen, S.: Trace anomalies and the Hawking effect. Kings College London, Preprint (1976)
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Communicated by R. Geroch
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Hawking, S.W. Zeta function regularization of path integrals in curved spacetime. Commun.Math. Phys. 55, 133–148 (1977). https://doi.org/10.1007/BF01626516
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DOI: https://doi.org/10.1007/BF01626516