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Mathematische Zeitschrift

, Volume 221, Issue 1, pp 307–335 | Cite as

Feynman path integrals on phase space and the metaplectic representation

  • Joel Robbin
  • Dietmar Salamon
Article

Keywords

Phase Space Heisenberg Group Morse Index Maslov Index Fourier Integral Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    S. Albevario and R. Hoegh-Krohn, Feynman path integrals and the corresponding method of stationary phase, inFeynman Path Integrals, ed. S. Albevario et al, Springer Lecture Notes in Physics106 (1978) 3–57.Google Scholar
  2. 2.
    M.F. Atiyah,The Geometry and Physics of Knots, Cambridge University Press, 1990.Google Scholar
  3. 3.
    S. Albeverio and R.J. Hoegh-Krohn,Mathematical Theory of Feynman Path Integrals, Springer Lecture Notes in Math523, 1976.Google Scholar
  4. 4.
    S. Albeverio and R.J. Hoegh-Krohn, Oscillatory integrals and the method of staionary phase in infinitely many dimensions with applications to the classical limit of quantum mechanics I., Inventiones Math.40 (1977) 59–106.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    S. Albeverio and Z. Breźniak, Finite dimensional approximations approach to oscillatory integrals and stationary phase in infinite dimensions, J. Funct. Anal.113 (1993) 177–244.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    S. Albeverio and Z. Breźniak, Feynman path integrals as infinite dimensional oscillatory integrals: some new developments, preprint.Google Scholar
  7. 7.
    S. Albeverio, A.M. Boutet de Monvel Berthier, Z. Brzeźniak, Stationary phase in infinite dimensions by finite approximations: applications to the Schroedinger equation, preprint (1992).Google Scholar
  8. 8.
    S. Albeverio, A.M. Boutet de Monvel Berthier, Z. Brzeźniak, The trace formula for Schrödinger operators from infinite dimensional oscillatory integrals, preprint (1993).Google Scholar
  9. 9.
    C. Conley and E. Zehnder, The Birkhoff-Lewis fixed point theorem and a conjecture of V.I. Arnold,Invent. Math. 73 (1983), 33–49.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    C.C. Conley and E. Zehnder, Morse-type index theory for flows and periodic solutions of Hamiltonian equations,Commun. Pure Appl. Math. 37 (1984), 207–253.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    P.A.M. Dirac, The Lagrangian in quantum mechanics, Phys. Zeitschr. d. Sovyetunion3 (1933) 64–72. (Reprinted in [32].)zbMATHGoogle Scholar
  12. 12.
    I. Daubechies and J.R. Klauder, Quantum mechanical path integrals with Wiener measure for all polynomial Hamiltonians,Phys. Rev Letters 52 (1984) 1161;J. Math. Physics 26 (1985) 2239–2256.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    I. Daubechies, J.R. Klauder, and T. Paul, Wiener measures for path integrals with affine kinematic variables,J. Math. Physics 28 (1987) 85–102.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    J.J. Duistermaat, On the Morse index in variational calculus,Advances in Mathematics 21 (1976), 173–195.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    R.P. Feynman, Space-time approach to non-relativistic quantum mechanics,Rev. Mod. Physics 20 (1948), 367–387. (Reprinted in [32].)CrossRefMathSciNetGoogle Scholar
  16. 16.
    D. Elworthy and A. Truman, Feynman maps, Cameron-Martin formulae and anharmonic oscillators, Ann. Inst. Henri Poincaré (Physique théorique,41 (1984) 115–142.zbMATHMathSciNetGoogle Scholar
  17. 17.
    R.P. Feynman and A.R. Hibbs,Quantum Mechanics and Path Integrals, McGraw-Hill, 1965.Google Scholar
  18. 18.
    D. Fujiwara, Remarks on the convergence of Feynman path integrals, Duke Math. J,47 (1980) 559–600.zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    V. Guillemin and S. Sternberg,Geometric Asymptotics, AMS Math Surveys14, 1977.Google Scholar
  20. 20.
    V. Guillemin and S. Sternberg,Symplectic Techniques in Physics, Cambridge University Press, 1984.Google Scholar
  21. 21.
    L. Hörmander, Fourier integral operators I,Acta Math. 127 (1971), 79–183.zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    J.B. Keller, Corrected Bohr-Sommerfeld quantum conditions for nonseparable systems,Annals of Physics 4 (1958), 180–188.zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    B. Kostant, Quantization and unitary representations, inModern Analysis and its Applications Springer Lecture Notes in Math170 (1970) 87–207.Google Scholar
  24. 24.
    J. Leray,Lagrangian Analysis and Quantum Mechanics, MIT press, 1981.Google Scholar
  25. 25.
    J.W. Robbin, and D.A. Salamon, A Maslov index for paths, Preprint 1992.Google Scholar
  26. 26.
    J.W. Robbin, and D.A. Salamon, Phase functions and path integrals,Proceedings of a conference on Symplectic Geometry, edited by D. Salamon, to appear.Google Scholar
  27. 27.
    J.W. Robbin, and D.A. Salamon, The spectral flow and the Maslov index, Preprint 1992.Google Scholar
  28. 28.
    J. Rezende, The method of stationary phase for oscillatory integrals on Hilbert spaces, Communications in Math. Physics,101 (1985) 187–206.zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    J. Rezende Quantum systems with time-dependent harmonic part and the Morse index, J. Math. Physics25 (1984) 3264–3269.CrossRefMathSciNetGoogle Scholar
  30. 30.
    P.L. Robinson and J.H. Rawnsley,The metaplectic representation, Mp C structures and geometric quantization, Memoirs of the American Mathematical Society81, 1989.Google Scholar
  31. 31.
    D. Salamon and E. Zehnder, Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, to appear in Comm. Pure Appl. Math.Google Scholar
  32. 32.
    J. Schwinger (Ed.),Selected papers in Quantum Electrodynamics, Dover Publications, 1958.Google Scholar
  33. 33.
    I.E. Segal, Foundations of the theory of dynamical systems of infinitely many degrees of freedom (I),Mat. Fys. Medd. Danske Vid. Selsk. 31 (1959) 1–39.Google Scholar
  34. 34.
    I.E. Segal, Transforms for operators and symplectic automorphisms over a locally compact abelian group,Math. Scand. 13 (1963) 31–43.zbMATHMathSciNetGoogle Scholar
  35. 35.
    D. Shale, Linear symmetries of free boson fields,Trans. AMS 103 (1962), 149–167.zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    B. Simon,Functional Integration and Quantum Physics, Academic Press, 1979.Google Scholar
  37. 37.
    J.M. Souriau,Structures des Systemes Dynamiques, Dunod, Paris, 1970.Google Scholar
  38. 38.
    J.M. Souriau, Construction explicite de l’indice de Maslov.Group Theoretical Methods in Physics Springer Lecture Notes in Physics50 (1975), 117–148.MathSciNetCrossRefGoogle Scholar
  39. 39.
    A: Weil, Sur certaine groupes d’operateurs unitaires,Acta math 111 (1964), 143–211.zbMATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    A. Weinstein,Lectures on Symplectic Manifolds, AMS Reg. Conf. Ser. Math.29, 1977.Google Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Joel Robbin
    • 1
  • Dietmar Salamon
    • 2
  1. 1.Mathematics DepartmentUniversity of WisconsinMadisonUSA
  2. 2.Mathematics InstituteUniversity of WarwickCoventryGreat Britain

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