A Rigorous Mathematical Foundation of Functional Integration

  • P. Cartier
  • C. DeWitt-Morette
Part of the NATO ASI Series book series (NSSB, volume 361)


Due to the growing interest in path integrals as a valuable tool in theoretical physics, a rigorous mathematical foundation of functional integration is needed more than ever. In these lectures we will present a new approach that is in part a synthesis of what has been accomplished over the past decades and in part an extension of functional integration to a larger class of ftinctionals. After a discussion of integration theory and its shortcomings in infinite-dimensional, non-compact spaces, we will present the new approach in detail and give examples of its application.


Quadratic Form Volume Element Dirac Equation Radon Measure Polish Space 
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  1. [1]
    Albeverio, S. A. and Høegh-Krohn, R. J., Mathematical Theory of Feynman Path Integrals, Springer Verlag Lecture Notes in Mathematics 523 (1976).Google Scholar
  2. [2]
    Bourbaki, N., “ Intégration, chapitre 9”, Masson, Paris (1982).Google Scholar
  3. [3]
    Cartier, P. and DeWitt-Morette, C., “Intégration fonctionnelle; éléments d’axiomatique,” C. R. Acad. Sci. Paris 316, 733–738 (1993).zbMATHGoogle Scholar
  4. [4]
    Cartier, P. and DeWitt-Morette, C., “A new perspective on functional integration,” J. Math. Phys. 36, 2237–2312 (1995).MathSciNetADSzbMATHCrossRefGoogle Scholar
  5. [5]
    Cartier, P., “A New Perspective on Path Integrals: Rigorous Mathematical Foundations,” Lectures at the University of Munich Spring 1996, Notes taken by Lang, J., preprint (1996).Google Scholar
  6. [6]
    Choquet-Bruhat, Y. and DeWitt-Morette, C., “Supplement to Analysis, Manifolds and Physics,” Armadillo preprint, Center for Relativity, University of Texas, Austin, TX 78712 (1994).Google Scholar
  7. [7]
    Dautray, R. and Lions, J.-L., Mathematical Analysis and Numerical Methods for Science and Technology, Springer Verlag (1988).Google Scholar
  8. [8]
    DeWitt-Morette, C., “Feynman’s Path Integral: Definition without Limiting Procedure,” Commun. Math. Phys. 28, 47–67 (1972).MathSciNetADSCrossRefGoogle Scholar
  9. [9]
    DeWitt-Morette, C., “Feynman Path Integrals: 1. Linear and Affine Techniques, II. The Feynman-Green Function,” Commun. Math. Phys. 37, 63–81 (1974).MathSciNetADSzbMATHCrossRefGoogle Scholar
  10. [10]
    DeWitt-Morette, C., Maheshwari, A. and Nelson, B., “Path Integration in Non-Relativistic Quantum Mechanics,” Phys. Rep 50, 266–372 (1979).MathSciNetCrossRefGoogle Scholar
  11. [11]
    DeWitt-Morette, C., “Path integration at the crossroad of stochastic and differential calculus,” pp. 166-70, in Gauge Theory and Gravitation Eds. K. Kikkawa, N. Nakanishi, and H. Nariai, Springer Verlag Lecture Notes in Physics 176 (1983).Google Scholar
  12. [12]
    DeWitt-Morette, C., “Quantum Mechanics in curved spacetimes; stochastic processes on frame bundles,” pp.49–87 in Quantum Mechanics in Curved Space—Time Eds. J. Audretsch and V. de Sabbata, Plenum Press, New York (1990).CrossRefGoogle Scholar
  13. [13]
    Eells, J. and Elworthy, K. D., “Wiener integration on certain manifolds,” pp. 67-94, in Problems in non-linear analysis, C.I.M. EIV (1971).Google Scholar
  14. [14]
    Elworthy, K. D., Stochastic Differential Equations on Manifolds, Cambridge University Press, Cambridge U. K. (1982).zbMATHCrossRefGoogle Scholar
  15. [15]
    Hida, T., Kuo, H.-H., Potthoff, J. and Streit, L., White Noise — an Infinite Dimensional Calculus, Kluwer-Academic, New York (1993).zbMATHGoogle Scholar
  16. [16]
    Mizrahi, M. M., Ph. D. Thesis, University of Texas at Austin (1975).Google Scholar
  17. [17]
    Mizrahi, M. M., “The semiclassical expansion of the anharmonic-oscillator propagator,” J. Math. Phys. 20, 844–855 (1979).MathSciNetADSzbMATHCrossRefGoogle Scholar
  18. [18]
    Reed, M. and Simon, B., Functional Analysis (Methods of Modern Mathematical Physics I), Academic Press, New York (1980).zbMATHGoogle Scholar
  19. [19]
    Young, A. and DeWitt-Morette, C., “Time substitutions in stochastic Processes as a Tool in Path Integration,” Ann. of Phys. 69, 140–166 (1986).MathSciNetADSCrossRefGoogle Scholar
  20. [20]
    Wurm, A., Master Thesis, University of Texas at Austin (1995).Google Scholar

References for Appendix

  1. [1]
    Feynman, R. P. and Vernon, F. L., “The Theory of a General Quantum System Interacting with a Linear Dissipative System,” Ann. Phys. 24, 118–173 (1963).MathSciNetADSCrossRefGoogle Scholar
  2. [2]
    Gaveau, B., Jacobson, T., Kac, M. and Schulman, L. S., “Relativistic Extension of the Analogy between Quantum Mechanics and Brownian Motion,” Phys. Rev. Lett. 53, 419–422 (1984).MathSciNetADSCrossRefGoogle Scholar
  3. [3]
    Kac, M., “A Stochastic Model Related to the Telegrapher’s Equation,” Rocky Mountain J. of Math. 4, 497–509 (1974). Reprinted from the Magnolia Petroleum Company Colloquium Lectures in the Pure and Applied Sciences, No 2. October 1956, “Some Stochastic Problems in Physics and Mathematics ”.zbMATHCrossRefGoogle Scholar
  4. [4]
    Leggett, A., Chakravarty, S., Dorsey, A. T., Fisher, M. P. A., Garg, A. and Zwerger, W., “Dynamics of the Dissipative Two-state System,” Rev. Mod. Phys. 59, 1–85 (1987).ADSCrossRefGoogle Scholar
  5. [5]
    Nelson, E., “Feynman Integrals and the Schrödinger Equation,” J. Math. Phys. 5, 332–43 (1964).ADSzbMATHCrossRefGoogle Scholar
  6. [6]
    Niu, Q., “Quantum Coherence of a Narrow-Band Particle Interacting with Phonons and Static Disorder,” J. Stat. Phys. 65, 317–361 (1991).ADSzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • P. Cartier
    • 1
  • C. DeWitt-Morette
    • 2
  1. 1.Ecole Normale SupérieureParisFrance
  2. 2.Department of Physics and Center for RelativityUniversity of Texas at AustinAustinUSA

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