Feynman path integrals on phase space and the metaplectic representation

Mathematische Zeitschrift

Volumn 221, Issue 1, 1996, Pages 307-335

  Robbin, Joel ^a   Salamon, Dietmar ^b  

^a UNIVERSITY OF WISCONSIN MADISON   (United States)

^b UNIVERSITY OF WARWICK   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 52449145013     PISSN: 00255874     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (41)

1. S. Albevario and R. Hoegh-Krohn, Feynman path integrals and the corresponding method of stationary phase, in Feynman Path Integrals, ed. S. Albevario et al. Springer Lecture Notes in Physics 106 (1978) 3-57.
2. M.F. Atiyah, The Geometry and Physics of Knots, Cambridge University Press, 1990.
3. S. Albeverio and R.J. Hoegh-Krohn, Mathematical Theory of Feynman Path Integrals, Springer Lecture Notes in Math 523, 1976.
4. S. Albeveno 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.
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.
6. S. Albeverio and Z. Breźniak, Feynman path integrals as infinite dimensional oscillatory integrals: some new developments, preprint.
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).
8. S. Albeverio, A.M. Boutet de Monvel Berthier, Z. Brzeźniak, The trace formula for Schrödinger operators from infinite dimensional oscillatory integrals, preprint (1993).
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.
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.
11. P.A.M. Dirac, The Lagrangian in quantum mechanics, Phys. Zeitschr. d. Sovyetunion 3 (1933) 64-72. (Reprinted in [32].)
12. I. Daubechies and J.R. Klauder, Quantum mechanical path integrals with Wiener measure for all polynomial Hamiltomans. Phys. Rev Letters 52 (1984) 1161; J. Math. Physics 26 (1985) 2239-2256.
13. I. Daubechies and J.R. Klauder, Quantum mechanical path integrals with Wiener measure for all polynomial Hamiltomans. Phys. Rev Letters 52 (1984) 1161; J. Math. Physics 26 (1985) 2239-2256.
14. I. Daubechies, J.R. Klauder, and T. Paul, Wiener measures for path integrals with affine kinematic variables, J. Math. Physics 28 (1987) 85-102.
15. J.J. Duistermaat, On the Morse index in variational calculus, Advances in Mathematics 21 (1976), 173-195.
16. R.P. Feynman, Space-time approach to non-relativistic quantum mechanics, Rev. Mod. Physics 20 (1948), 367-387. (Reprinted in [32].)
17. D. Elworthy and A. Truman, Feynman maps, Cameron-Martin formulae and anharmonic oscillators. Ann. Inst. Henri Poincaré (Physique théorique, 41 (1984) 115-142.
18. R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, 1965.
19. D. Fujiwara, Remarks on the convergence of Feynman path integrals, Duke Math. J, 47 (1980) 559-600.
20. V. Guillemin and S. Sternberg, Geometric Asymptotics, AMS Math Surveys 14, 1977.
21. V. Guillemin and S. Sternberg, Symplectic Techniques in Physics, Cambridge University Press, 1984.
22. L. Hörmander, Fourier integral operators I, Acta Math. 127 (1971), 79-183.
23. J.B. Keller, Corrected Bohr-Sommerfeld quantum conditions for nonseparable systems, Annals of Physics 4 (1958), 180-188.
24. B. Kostant, Quantization and unitary representations, in Modern Analysis and its Applications Springer Lecture Notes in Math 170 (1970) 87-207.
25. J. Leray, Lagrangian Analysis and Quantum Mechanics, MIT press, 1981.
26. J.W. Robbin, and D.A. Salamon, A Maslov index for paths, Preprint 1992.
27. 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.
28. J.W. Robbin, and D.A. Salamon, The spectral flow and the Maslov index, Preprint 1992.
29. J. Rezende, The method of stationary phase for oscillatory integrals on Hilbert spaces, Communications in Math. Physics, 101 (1985) 187-206.
30. J. Rezende Quantum systems with time-dependent harmonic part and the Morse index, J. Math. Physics 25 (1984) 3264-3269.
31. C structures and geometric quantization, Memoirs of the American Mathematical Society 81, 1989.
32. D. Salamon and E. Zehnder, Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, to appear in Comm. Pure Appl. Math.
33. J. Schwinger (Ed.), Selected papers in Quantum Electrodynamics, Dover Publications, 1958.
34. 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.
35. I.E. Segal, Transforms for operators and symplectic automorphisms over a locally compact abelian group, Math. Scand. 13 (1963) 31-43.
36. D. Shale, Linear symmetries of free boson fields, Trans. AMS 103 (1962), 149-167.
37. B. Simon, Functional Integration and Quantum Physics, Academic Press, 1979.
38. J.M. Souriau, Structures des Systemes Dynamiques, Dunod, Paris, 1970.
39. J.M. Souriau, Construction explicite de l'indice de Maslov. Group Theoretical Methods in Physics Springer Lecture Notes in Physics 50 (1975), 117-148.
40. A: Weil, Sur certaine groupes d'operateurs unitaires, Acta math 111 (1964), 143-211.
41. A. Weinstein, Lectures on Symplectic Manifolds, AMS Reg. Conf. Ser. Math. 29, 1977.