Wiener Measure Regularization for Quantum Mechanical Path Integrals

  • J. R. Klauder
Conference paper

Abstract

The problems associated with a regularization of quantum mechanical path integrals using continuous-time (as opposed to discrete-time) schemes are examined. All such proposals insert regularizing Wiener measures and consider the limit as the diffusion constant diverges as the final step. Two unsuccessful approaches in the Schrödinger representation are reviewed before a fairly complete treatment of the successful coherent-state representation approach is presented. Not only does the coherent-state approach provide a rigorous continuous-time regularization scheme for quantum mechanical path integrals but it also offers a natural and physically appealing formulation that is covariant under classical canonical transformations.

Keywords

Covariance Plague 

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References

  1. 1.
    See, e.g., R.P. Feynman: Rev. Mod. Phys. 20. 367 (1948)MathSciNetADSCrossRefGoogle Scholar
  2. J. Tarski; Ann. Inst. H. Poincaré 17, 313 (1972)MathSciNetGoogle Scholar
  3. A. Truman: J. Math. Phys. 17, 1852 (1976)MathSciNetADSCrossRefGoogle Scholar
  4. V.P. Maslov, A.M. Chebotarev: Theor. Math. Phys. 28, 793 (1976)CrossRefGoogle Scholar
  5. F.A. Berezin: Sov. Phys. Usp. 23, 763 (1980)ADSCrossRefGoogle Scholar
  6. See also R.P. Feynman and A.R. Hibbs: Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965)MATHGoogle Scholar
  7. L. Schulman: Techniques and Applications of Path Integration (Wiley, New York, 1981)MATHGoogle Scholar
  8. 2.
    E. Nelson: J. Math. Phys. 5, 332 (1964)ADSMATHCrossRefGoogle Scholar
  9. 3.
    S.F. Edwards, Y.V. Gulyaev: Proc. Roy. Soc. (London) A279, 229 (1964). A careful and thorough discussion of lattice regularization of path integrals in non-Cartesian coordinates is given by M. Böhm and G. Junker: “Path Integration over Compact and Non-compact Rotation Groups”, Universität Würzburg preprint, December 1986MathSciNetADSGoogle Scholar
  10. 4.
    J.R. Klauder, C.B. Lang, P. Salomonson, and B.-S. Skagerstam: Z. Phys. C-Particles and Fields 26, 149 (1984)ADSCrossRefGoogle Scholar
  11. 5.
    See, e.g., C.B. Lang, C. Rebbi, P. Salomonson, B.-S. Skagerstam: Phys. Lett. 101B, 173 (1981)ADSGoogle Scholar
  12. G.’t Hooft: Phys. Lett. 109B, 474 (1982)ADSGoogle Scholar
  13. 6.
    K. Ito and H. McKean: Diffusion Processes and Their Sample Paths (Springer-Verlag, Berlin and New York, 1965)MATHGoogle Scholar
  14. B. Simon: Functional Integration and QuantumPhysics (Academic Press, New York, 1979)MATHGoogle Scholar
  15. 7.
    J.R. Klauder: in Progress in Quantum /Field Theory, eds. H. Ezawa, S. Kamefuchi (North-Holland, Amsterdam, 1986), 1986), p.31Google Scholar
  16. 8.
    I.M. Gel’fand and A.M. Yaglom: J. Math. Phys. I, 48 (1960)CrossRefGoogle Scholar
  17. 9.
    R.H. Cameron: J. Anal. Math. 10, 287 (1962/63)CrossRefGoogle Scholar
  18. 10.
    K. Ito: in Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, 1961, Vol.II., p.227Google Scholar
  19. For a significantly improved treatment see K. Ito: in Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, 1966, Vol.II, part 1, p.145Google Scholar
  20. 11.
    See e.g., V. Bargmann: Commun. Pure and Appl. Math. 14, 187 (1961)MathSciNetMATHCrossRefGoogle Scholar
  21. J. McKenna and J.R. Klauder: J. Math. Phys. 5, 878 (1964)MathSciNetADSMATHCrossRefGoogle Scholar
  22. 12.
    J.R. Klauder: Ann. Phys. (N.Y.) 11, 123 (1960); see also Acta Physica Austriaca, Suppl XXII, 3 (1980)MathSciNetADSMATHCrossRefGoogle Scholar
  23. 13.
    See e.g., R. Shankar: Phys. Rev. Lett. 45, 1088 (1980)MathSciNetADSCrossRefGoogle Scholar
  24. 14.
    J.R. Klauder: in Path Integrals, and Their Applications in Quantum, Statistical, and Solid State Physics, eds. G.J. Papadopoulos and J.T. Devreese (Plenum Pub. Corp., 1978), p.5Google Scholar
  25. 15.
    J.R. Klauder: Acta Physica Austriaca, Suppl. XXII, 3 (1980)MathSciNetGoogle Scholar
  26. 16.
    I. Daubechies and J.R. Klauder: J. Math. Phys. 26, 2239 (1985)MathSciNetADSMATHCrossRefGoogle Scholar
  27. For a related (but weaker) result see: J.R. Klauder and I. Daubechies: Phys. Rev. Lett. 52, 1161 (1984)MathSciNetADSMATHCrossRefGoogle Scholar
  28. For very limited results with ordered symbols using Wiener measures with drift, see, I. Daubechies and J.R. Klauder: J. Math. Phys. 23 1806 (1982)MathSciNetADSCrossRefGoogle Scholar
  29. 17.
    J.R. Klauder and E.C.G. Sudarshan: Fundamentals of Quantum Optics (W.A. Benjamin Inc., New York, 1968)Google Scholar
  30. 18.
    J.R. Klauder and B.-S. Skagerstam: Coherent States (World Scientific, Singapore, 1985)MATHGoogle Scholar
  31. 19.
    I. Daubechies, J.R. Klauder, and T. Paul: J. Math. Phys. 28, 85 (1987)MathSciNetADSMATHCrossRefGoogle Scholar
  32. 20.
    J.R. Klauder: “Coherent-State Path Integrals for Unitary Group Representations”, to be publishedGoogle Scholar
  33. 21.
    I. Bakas and H. La Roche: J. Phys. A: Math. Gen. 19, 2513 (1986)ADSMATHCrossRefGoogle Scholar
  34. 22.
    See, e.g., M.B. Halpern: “Schwinger-Dyson Formulation of Coordinate-Invariant Regularization,” UCB-PTH-86/28 preprintGoogle Scholar
  35. Z. Bern, M.B. Halpern, L. Sadun, and C. Taubes: Phys. Lett. 165B, 151 (1985)ADSGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • J. R. Klauder
    • 1
  1. 1.AT&T Bell LaboratoriesMurray HillUSA

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