Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Propagator, sewing rules, and vacuum amplitude for the Polyakov point particle with ghosts


We apply techniques developed for strings to the case of the spinless point particle. The Polyakov path integral with ghosts is used to obtain the propagator and one-loop vacuum amplitude. The propagator is shown to correspond to the Green's function for the BRST field theory in Siegel gauge. The reparametrization invariance of the Polyakov path integral is shown to lead automatically to the correct “trace log” result for the one-loop diagram, despite the fact that “naive sewing” of the ends of a propagator would give an incorrect answer. This type of failure of “naive sewing” is identical to that found in the string case. The present treatment provides, in the simplified context of the point particle, a pedagogical introduction to Polyakov path integral methods with and without ghosts.

This is a preview of subscription content, log in to check access.


  1. Abers, E. S., and Lee, B. W. (1973).Physics Reports,9C, 1.

  2. Brink, L., and Nielsen, H. B. (1973).Physics Letters,45B, 332.

  3. Brown, L. S., (1977).Physical Review D,15, 1469.

  4. Candelas, P., and Raine, D. J. (1977).Physical Review D,15, 1494.

  5. Chaudhuri, S., Kawai, H., and Tye, S.-H. H. (1987).Physical Review D,36, 1148.

  6. Cohen, A., Moore, G., Nelson, P., and Polchinski, J. (1986).Nuclear Physics B,267, 143.

  7. DeWitt, B. (1964). InRelativity, Groups, and Topology, Gordon and Breach.

  8. DeWitt, B. (1984).Supermanifolds, Cambridge University Press.

  9. D'Hoker, E. and Phong, D. H. (1988). The geometry of string perturbation theory. Princeton University preprint PUPT-1039.

  10. Dirac, P. A. M. (1958).The Principles of Quantum Mechanics, 4th ed. (rev.), Oxford University Press.

  11. Faddeev, L. D., and Popov, V. N. (1967).Physics Letters,175B, 145.

  12. Feynman, R. P., and Hibbs, A. R. (1965).Quantum Mechanics and Path Integrals, McGraw-Hill.

  13. Green, M. B., Schwarz, J. H., and Witten, E. (1987).Superstring Theory, Vol. 1, Cambridge University Press.

  14. Govaerts, J. (1988). The Nambu-Goto string: Its phase-space path integral, CERN preprint, TH 4950/88.

  15. Hawking, S. W. (1977).Communications in Mathematical Physics,55, 133.

  16. Henty, J. C., Townsend, P. K., and Howe, P. S. (1988). Quantum mechanics of the relativistic spinning particle DAMPT preprint.

  17. Kaku, M. (1988).Introduction to Superstrings, Springer-Verlag.

  18. Mannheim, P. (1986).Physics Letters,166B, 191.

  19. Moore, G., and Nelson, P. (1986).Nuclear Physics B,266, 58.

  20. Ordóñez, C. R., Rubin, M. A., and Zucchini, R. (1987a). Path integral with ghosts for the bosonic string propagator,Journal of Physics A (to appear).

  21. Ordóñez, C. R., Rubin, M. A., and Zucchini, R. (1987b). InSuperstrings, K. T. Mahanthappa and P. G. O. Freund, eds., Plenum Press.

  22. Ordóñez, C. R., Rubin, M. A., and Zucchini, R. (1987c). Polyakov path integrals with ghosts: Closed strings and one-loop amplitudes,Physics Letters B (to appear).

  23. Ordóñez, C. R., Rubin, M. A., and Zwanziger, D. (1988). Modular invariance and stochastic quantization. Rockefeller University preprint RU88/B1/38.

  24. Polchinski, J. (1986).Communications in Mathematical Physics,104, 104.

  25. Polyakov, A. M. (1981).Physics Letters,103B, 207, 211.

  26. Ramond, P. (1981).Field Theory, Benjamin/Cummings.

  27. Ryder, L. H. (1985).Quantum Field Theory, Cambridge University Press.

  28. Schutz, B. F. (1980).Geometrical Methods of Mathematical Physics, Cambridge University Press.

  29. Shapiro, J. A. (1972).Physical Review D,5, 1945.

  30. Siegel, W. (1986). InUnified String Theories, M. Green and D. Gross, eds., World Scientific.

  31. Wald, R. M. (1984).General Relativity, University of Chicago Press.

  32. Weinberg, S. (1972).Gravitation and Cosmology, Wiley.

  33. Weinberg, S. (1987). Covariant path integral approach to string theory, Lectures at the 3rd Jerusalem Winter School of Theoretical Physics, University of Texas preprint UTTG-17-87.

  34. Witten, E. (1986).Nuclear Physics B,268, 253.

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Giannakis, I., Ordóñez, C.R., Rubin, M.A. et al. Propagator, sewing rules, and vacuum amplitude for the Polyakov point particle with ghosts. Int J Theor Phys 28, 3–25 (1989).

Download citation


  • Field Theory
  • Quantum Field Theory
  • Ghost
  • Integral Method
  • Point Particle