Functional integration

Journal of Mathematical Physics

Volumn 41, Issue 6, 2000, Pages 4154-4187

  Cartier, Pierre ^a   DeWitt Morette, Cécile ^b  

^a ECOLE NORMALE SUPÉRIEURE   (France)

^b UNIVERSITY OF TEXAS AT AUSTIN   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034390239     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.533338     Document Type: Article

References (68)

1. We have not been able to locate the reference to this well known sentence.
2. J. C. Baez, I. E. Segal, and Z. Zhou, Introduction to Algebraic and Constructive Quantum Field Theory (Princeton University Press, Princeton, NJ, 1992).
3. J. Glimm and A. Jaffe, Quantum Physics: A Functional Integral Point of View, 2nd ed. (Springer-Verlag, New York, 1987).
4. B. Simon, Functional Integration and Quantum Physics (Academic, New York, 1979).
5. S. Bochner, Harmonic Analysis and the Theory of Probability (University of California Press, Berkeley, 1960).
6. I. E. Segal, "Distributions in Hilbert space and canonical systems of operators," Trans. Am. Math. Soc. 88, 12-42 (1958).
7. P. Malliavin, Stochastic Analysis (Springer-Verlag, Berlin, 1997).
8. T. Hida, H.-H. Kuo, J. Potthof, and L. Streit, White Noise, An Infinite Dimensional Calculus (Kluwer Academic, Dordrecht, 1993).
9. Section II is based on the article "Functional integration; a semi-historical perspective," by one of us (C. DeW.) in Symposia Gaussiana, Conf. A, edited by Behara, Fritsch, and Lintz (Walter de Gruyter, Berlin, 1995).
10. R. P. Feynman, "The Principle of Least Action in Quantum Mechanics," Princeton University Publication No. 2948, p. 73 (1942).
11. See Ref. 10, Eq. (24).
12. See Ref. 10, p. 72.
13. R. P. Feynman, "The development of the space-time view of quantum electrodynamics," Phys. Today No. 8, 31-44 (1966).
14. Path integration is referred to as "she" not because integration is feminine in French but because Feynman, when accepting the Nobel Prize, referred to it as a lady: "So what happened to the old theory that I fell in love with as a youth? Well, I would say it's become an old lady who has very little attractive left in her, and the young today will not have their hearts pound when they look at her anymore. But, we can say the best we can for any old woman, that she has been a very good mother and has given birth to some very good children. And I thank the Swedish Academy of Sciences for complimenting one of them. Thank you." (Feynman, 1966, p. 44).
15. According to subsequent accounts, only four or five physicists appreciated path integration from the very beginning. See, for instance, Ref. 13; Dyson, 1989; and Mehra, 1988.
16. F. J. Dyson, "Feynman at Cornell," Phys. Today 42 (2), 32-38 (1989).
17. J. Mehra, "My Last Encounter with Richard P. Feynman," memorial talk delivered on 24 February 1988, Cornell University.
18. F. J. Dyson, "The radiation theories of Tomonaga, Schwinger, and Feynman," Phys. Rev. 75, 486-502 (1949).
19. J. Gleick, Genius (New York, 1992) pp. 269-270. (See the interoffice memorandum from Dyson to J. Robert Oppenheimer in the early Fall of 1948 and the interoffice reply a couple of months later "Nolo contendre. R.O.")
20. The semiclassical approximation of a path integral (Morette, 1951) had been constructed but had not yet proved its usefulness.
21. C. Morette, "On the definition and approximation of Feynman's path integral," Phys. Rev. 81, 848-852 (1951).
22. See, for instance, the state of the art in G. t'Hooft and M. Veltman, "Diagrammar," CERN preprint 78-9, 1-114 (1973), subsequently expanded into M. Veltman, Diagrammatica: the path to Feynman rules (Cambridge University Press, Cambridge, 1994).
23. See, for instance, the state of the art in G. t'Hooft and M. Veltman, "Diagrammar," CERN preprint 78-9, 1-114 (1973), subsequently expanded into M. Veltman, Diagrammatica: the path to Feynman rules (Cambridge University Press, Cambridge, 1994).
24. 2/2C+W(q)}dT. Hence the Hamiltonian and not the Lagrangian!
25. M. Kac, Probability and Related Topics in Physical Sciences Vol. I, 167 (Interscience, New York, 1957-1959).
26. Feynman, Letter to Cécile DeWitt-Morette, 10 December 1971.
27. F. J. Dyson, "Missed opportunities," Bull. Am. Math. Soc. 78 (5), 635-652 (1972).
28. See for instance Cameron, 1960, p. 126; but see also Cameron and Storvick, 1980.
29. See for instance Cameron, 1960, p. 126; but see also Cameron and Storvick, 1980.
30. R. H. Cameron, "A family of integrals serving to connect the Wiener and Feynman integrals," J. Math. Phys. 39, 126-140 (1960).
31. R. H. Cameron and D. A. Storvick, Some Banach Algebras of Analytic Feynman Integrable Functionals, Springer-Verlag Lecture Notes in Mathematics, Vol. 798 (Springer-Verlag, Berlin, 1980).
32. E.g., references to sum over histories sprinkle J. A. Wheeler's discussions at the 1957 Chapel Hill conference as a possible approach to quantum gravity. Recorded in "On the Role of Gravitation in Physics," Chapel Hill conference, Astia Document No. AD 11 81 80. Wright Air Development Center (1957).
33. B. S. DeWitt, "Dynamical Theory of Groups and Fields," in Relativity Groups and Topology, edited by C. DeWitt and B. DeWitt (Gordon and Breach, New York, 1963-1964), pp. 585-820, developed further in Ref. 32.
34. B. S. DeWitt, "The Spacetime Approach to Quantum Field Theory," in Relativity, Groups and Topology II, edited by B. DeWitt and R. Stora (North-Holland, Amsterdam, 1983-1984), pp. 382-738.
35. J. Ginibre, "Some Applications of Functional Integration in Statistical Mechanics," in Statistical Mechanics and Quantum Field Theory, edited by C. DeWitt and R. Stora (Gordon and Breach, New York, 1971), pp. 327-427.
36. Functional Integration, Basics find Applications, edited by C. DeWitt-Morette, P. Cartier, and A. Folacci (Plenum, New York, 1997).
37. M. G. G. Laidlaw and C. DeWitt-Morette, "Feynman functional integrals for systems of indistinguishable particles," Phys. Rev. D 3, 1375-1378 (1971).
38. L. Alvarez-Gaumé, "Supersymmetry and Index Theorem," Supersymmetry, Proceedings of the 1984 NATO School at Bonn, edited by K. Dietz, R. Flume, G. V. Gehlen, and V. Rittenberg (1984).
39. E. Witten, "Quantum field theory and the Jones polynomial," Commun. Math. Phys. 121, 351-399 (1989).
40. For the convenience of the reader interested in studying B. DeWitt's work, we introduce in this section the author's condensed notation, including the different labelling for right and left derivatives as required for fermion fields [see, e.g., Eqs. (III.8)].
41. A. Mostafazadeh, "Supersymmetry and the Atiyah-Singer index theorem. I. Peierls Brackets, Green's functions, and a proof of the Index theorem via Gaussian superdeterminants," J. Math. Phys. 35, 1095-1124 (1994); 35, 1125-1138 (1994). (The scalar curvature factor in the Schrödinger equation.)
42. A. Mostafazadeh, "Supersymmetry and the Atiyah-Singer index theorem. I. Peierls Brackets, Green's functions, and a proof of the Index theorem via Gaussian superdeterminants," J. Math. Phys. 35, 1095-1124 (1994); 35, 1125-1138 (1994). (The scalar curvature factor in the Schrödinger equation.)
43. C. Morette DeWitt, "Feymnan's path integral; definition without limiting procedure," Commun. Math. Phys. 28, 47-67 (1972).
44. C. DeWitt-Morette, "Feynman path integrals I. Linear and affine techniques. II. The Feynman Green function," Commun. Math. Phys. 37, 63-81 (1973).
45. N. Bourbaki, Intégration, (Hermann, Paris, 1969), Chap. IX.
46. P. Krée, "Introduction aux théories des distributions en dimension infinie," Bull. Soc. Math. France 46, 143-162 (1976), and references therein, in particular, Seminar P. Lelong, Springer-Verlag Lecture Notes in Mathematics, Vols. 410 and 474 (Springer-Verlag, Berlin, 1972-1974).
47. P. Krée, "Introduction aux théories des distributions en dimension infinie," Bull. Soc. Math. France 46, 143-162 (1976), and references therein, in particular, Seminar P. Lelong, Springer-Verlag Lecture Notes in Mathematics, Vols. 410 and 474 (Springer-Verlag, Berlin, 1972-1974).
48. C. DeWitt-Morette, A. Maheswari, and B. Nelson, "Path integration in non-relativistic quantum mechanics," Phys. Rep. 50, 266-372 (1979).
49. P. Cartier and C. DeWitt-Morette, "A new perspective on functional integration," J. Math. Phys. 36, 2237-2312 (1995). (Located on the World Wide Web at http://godel.ph.utexas.edu/Center/Papers.html and http://babbage.sissa.it/list/funct-an/9602.)
50. C. DeWitt-Morette, "Feynman path integrals. From the Prodistribution Definition to the Calculation of Glory Scattering," in Stochastic Methods and Computer Techniques in Quantum Dynamics, edited by H. Mitter and L. Pittner, Acta Phys. Austriaca, Suppl. 26, 101-170 (1984), Reviewed in Zentralblatt für Mathematik 1985.
51. P. Cartier and C. DeWitt-Morette, "Physics on and near caustics," NATO-ASI Proceedings, "Functional Integration: Basics and Applications" (Carǵese 1996, Plenum, NY. 1997).
52. P. Cartier and C. DeWitt-Morette, "Physics on and near caustics. A simpler version," RCTP (Jagna, January 1998).
53. See Kit Foong, "Functional integration and wave propagation," in Ref. 34, pp. 97-180.
54. S. A. Albeverio and R. J. Hoegh-Krohn, Mathematical Theory of Feynman Path Integrals, Springer-Verlag Lecture Notes in Mathematics, Vol. 523 (Springer-Verlag, Berlin, 1974-1976).
55. b].
56. K. D. Elworthy, Stochastic Differential Equations on Manifolds (Cambridge University Press, Cambridge, MA, 1982).
57. 4 in 4 - ∈ dimensions," Commun. Math. Phys. 198, 111-156 (1998).
58. C. Morette, "On the definition and approximation of Feynman's path integral," Phys. Rev. 81, 848-852 (1951).
59. C. DeWitt-Morette, K. D. Elworthy, B. L. Nelson, and G. S. Sammelmann, "A stochastic scheme for constructing solutions of the Schrödinger equation," Ann. Inst. Henri Poincaré, Sect. A 32, 327-341 (1980).
60. B. S. DeWitt, Supermanifolds, 2nd ed. (Cambridge University Press, Cambridge, MA, 1992), pp. 281 and 334.
61. See Ref. 45, pp. 2260-2263.
62. See Ref. 56, pp. 358.
63. b, because G is boundary-adapted to paths with two fixed points.
64. L. S. Schulman, "Introduction to the Path Integral," in Path Summation: Achievements and Goals, edited by S. Lundquist, A. A. Ranfagni, V. Sa-yakanit, and L. S. Schulman (World Scientific, Singapore, 1987-1988), pp. 3-46.
65. C. Grosche and F. Steiner, Handbook of Feynman Path Integrals (Springer-Verlag, Berlin, 1998).
66. D. S. Freed and R. E. Gompf, "Computer calculation of Witten's three-manifold invariant," Commun. Math. Phys. 147, 563-604 (1992).
67. L. H. Kauffman, "Witten's Integral and the Kontsevich Integral" (preprint).
68. M. Blau and G. Thompson, "Localization and diagonalizatioi: A review of functional inlegral techniques for low-dimensional gauge theories and topological field theories," J. Math. Phys. 36, 2192-2236 (1995).