Communications in Mathematical Physics

, Volume 55, Issue 2, pp 133–148 | Cite as

Zeta function regularization of path integrals in curved spacetime

  • S. W. Hawking
Article

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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    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 1964Google Scholar
  2. 2.
    DeWitt, B.S.: Phys. Rep.19C, 295 (1975)Google Scholar
  3. 3.
    McKean, H.P., Singer, J.M.: J. Diff. Geo.5, 233–249 (1971)Google Scholar
  4. 4.
    Gilkey, P.B.: The index theorem and the heat equation. Boston: Publish or Perish 1974Google Scholar
  5. 5.
    Candelas, P., Raine, D.J.: Phys. Rev. D12, 965–974 (1975)Google Scholar
  6. 6.
    Drummond, I.T.: Nucl. Phys.94B, 115–144 (1975)Google Scholar
  7. 7.
    Capper, D., Duff, M.: Nuovo Cimento23A, 173 (1974)Google Scholar
  8. 8.
    Duff, M., Deser, S., Isham, C.J.: Nucl. Phys.111B, 45 (1976)Google Scholar
  9. 9.
    Brown, L.S.: Stress tensor trace anomaly in a gravitational metric: scalar field. University of Washington, Preprint (1976)Google Scholar
  10. 10.
    Brown, L.S., Cassidy, J.P.: Stress tensor trace anomaly in a gravitational metric: General theory, Maxwell field. University of Washington, Preprint (1976)Google Scholar
  11. 11.
    Dowker, J.S., Critchley, R.: Phys. Rev. D13, 3224 (1976)Google Scholar
  12. 12.
    Dowker, J.S., Critchley, R.: The stress tensor conformal anomaly for scalar and spinor fields. University of Manchester, Preprint (1976)Google Scholar
  13. 13.
    Gibbons, G.W., Hawking, S.W.: Action integrals and partition functions in quantum gravity. Phys. Rev. D (to be published)Google Scholar
  14. 14.
    Manor, Y.: Complex Riemannian sections. University of Cambridge, Preprint (1977)Google Scholar
  15. 15.
    Feynman, R. P.: Magic without magic, (eds. J. A. Wheeler and J. Klaunder). San Francisco: W. H. Freeman 1972Google Scholar
  16. 16.
    DeWitt, B.S.: Phys. Rev.162, 1195–1239 (1967)Google Scholar
  17. 17.
    Fadeev, L.D., Popov, V.N.: Usp. Fiz. Nauk111, 427–450 (1973) [English translation in Sov. Phys. Usp.16, 777–788 (1974)]Google Scholar
  18. 18.
    Seeley, R.T.: Amer. Math. Soc. Proc. Symp. Pure Math.10, 288–307 (1967)Google Scholar
  19. 19.
    Ray, D.B., Singer, I.M.: Advances in Math.7, 145–210 (1971)Google Scholar
  20. 20.
    Gilkey, P.B.: Advanc. Math.15, 334–360 (1975)Google Scholar
  21. 21.
    'tHooft, G.: Phys. Rev. Letters37, 8–11 (1976)Google Scholar
  22. 22.
    'tHooft, G.: Computation of the quantum effects due to a four dimensional pseudoparticle. Harvard University, PreprintGoogle Scholar
  23. 23.
    Hartle, J.B., Hawking, S.W.: Phys. Rev. D13, 2188–2203 (1976)Google Scholar
  24. 24.
    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)Google Scholar
  25. 25.
    Fulling, S.A., Christensen, S.: Trace anomalies and the Hawking effect. Kings College London, Preprint (1976)Google Scholar

Copyright information

© Springer-Verlag 1977

Authors and Affiliations

  • S. W. Hawking
    • 1
  1. 1.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeCambridgeEngland

Personalised recommendations