No Access Submitted: 08 January 1985 Accepted: 03 May 1985 Published Online: 04 June 1998
J. Math. Phys. 26, 2239 (1985); https://doi.org/10.1063/1.526803
more...View Contributors
  • Ingrid Daubechies
  • John R. Klauder
The coherent‐state representation of quantum‐mechanical propagators as well‐defined phase‐space path integrals involving Wiener measure on continuous phase‐space paths in the limit that the diffusion constant diverges is formulated and proved. This construction covers a wide class of self‐adjoint Hamiltonians, including all those which are polynomials in the Heisenberg operators; in fact, this method also applies to maximal symmetric Hamiltonians that do not possess a self‐adjoint extension. This construction also leads to a natural covariance of the path integral under canonical transformations. An entirely parallel discussion for spin variables leads to the representation of the propagator for an arbitrary spin‐operator Hamiltonian as well‐defined path integrals involving Wiener measure on the unit sphere, again in the limit that the diffusion constant diverges.
  1. 1. A representative sample of references is the following: I. M. Gel’fand and A. M. Yaglom, J. Math. Phys. 1, 48 (1960); Google ScholarScitation, ISI
    D. G. Babbitt, J. Math. Phys. 4, 36 (1963); , Google ScholarScitation, ISI
    E. Nelson, J. Math. Phys. 5, 332 (1964); , Google ScholarScitation, ISI
    K. Itô, Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability (California U.P., Berkeley, 1967), Vol. 2, Part 1, pp. 145–161; , Google Scholar
    J. Tarski, Ann. Inst. H. Poincaré 17, 313 (1972); Google Scholar
    K. Gawedzki, Rep. Math. Phys. 6, 327 (1974); Google ScholarCrossref
    A. Truman, J. Math. Phys. 17, 1852 (1976); , Google ScholarScitation, ISI
    S. A. Albeverio and R. J. Hoegh‐Krohn, Mathematical Theory of Feynman Path Integrals (Springer, Berlin, 1976); , Google Scholar
    V. P. Maslov and A. M. Chebotarev, Theor. Math. Phys. 28, 793 (1976); Google ScholarCrossref
    C. DeWitt‐Morette, A. Maheshwari, and B. Nelson, Phys. Rep. 50, 255 (1979); , Google ScholarCrossref, ISI
    D. Fujiwara, Duke Math. J. 47, 559 (1980); , Google ScholarCrossref
    P. Combe, R. Hoegh‐Krohn, R. Rodriguez, and M. Sirugue, Commun. Math. Phys. 77, 269 (1980); , Google ScholarCrossref
    F. A. Berezin, Sov. Phys. Usp. 23, 763 (1980); , Google ScholarCrossref
    T. Ichinose, Proc. Jpn. Acad. Ser. A 58, 290 (1982); , Google ScholarCrossref
    I. Daubechies and J. R. Klauder, J. Math. Phys. 23, 1806 (1982). , Google ScholarScitation
  2. 2. E. Nelson, J. Math. Phys. 5, 332 (1964). Google ScholarScitation, ISI
  3. 3. See. e.g., L. Shulman, Techniques and Applications of Path Integration (Wiley, New York, 1981). Google Scholar
  4. 4. An announcement of our results appears in J. R. Klauder and I. Daubechies, Phys. Rev. 52, 1161 (1984). Google Scholar
  5. 5. J. R. Klauder, in Path Integrals, edited by George J. Papadopoulos and J. T. Devreese (Plenum, New York, 1978). Google Scholar
  6. 6. R. H. Cameron, J. Anal. Math. 10, 287 (1962/63). Google ScholarCrossref
  7. 7. E. Lieb, Commun. Math. Phys. 31, 327 (1973). Google ScholarCrossref, ISI
  8. 8. See. e.g, E. J. McShane, Stochastic Calculus and Stochastic Models (Academic, New York, 1974). Google Scholar
  9. 9. V. Bargmann, Commun. Pure Appl. Math. 14, 187 (1961). Google ScholarCrossref, ISI
  10. 10. J. E. Moyal, Proc. Camb. Philos. Soc. 45, 99 (1949); Google ScholarCrossref, ISI
    M. S. Bartlett and J. E. Moyal, Proc. Camb. Philos. Soc. 45, 545 (1949). , Google ScholarCrossref
  11. 11. I. Daubechies and J. R. Klauder, J. Math. Phys. 23, 1806 (1982). Google ScholarScitation
  12. 12. S. Girvin and T. Jach, Phys. Rev. B 29, 5617 (1984). Google ScholarCrossref
  13. 13. T. Kato, Perturbation Theory for Linear Operators (Springer, New York, 1966). Google Scholar
  14. 14. I. Daubechies and J. R. Klauder, Lett. Math. Phys. 7, 229 (1983). Google ScholarCrossref
  15. 15. J. R. Klauder, J. Math. Phys. 23, 1797 (1982). Google ScholarScitation
  16. 16. See, e.g., I. M. Gel’fand, R. A. Minlos, and Z. Ya. Shapiro, Representations of the Rotation and Lorentz Groups and their Applications (Pergamon, New York, 1983); Google Scholar
    or N. Ja. Vilenkin, Special Functions and the Theory of Group Representations (A. M. S., Providence, RI, 1968), Vol. 22. Google Scholar
  17. 17. P. Chernoff, J. Funct. Anal. 2, 238 (1968). Google ScholarCrossref
  1. © 1985 American Institute of Physics.