Hostname: page-component-7dd5485656-npwhs Total loading time: 0 Render date: 2025-10-30T18:42:46.760Z Has data issue: false hasContentIssue false
  • English
  • Français

The Feynman integral of quadratic potentials depending on n time variables

Published online by Cambridge University Press:  22 January 2016

Chull Park  and
David Skoug
Chull Park
Affiliation:
Department of Mathematics and Statistics, Miami University, Oxford, OH 45056
David Skoug
Affiliation:
Department of Mathematics and Statistics, University of Nebraska, Lincoln, NE 68588-0323
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let C1[0, T] denote (one-parameter) Wiener space; that is the space of continuous functions x on [0, T] such that x(0) = 0. In a recent expository essay [21], Nelson calls attention to some functions on Wiener space which were discussed in the book of Feynman and Hibbs [13, section 3-10] and in Feynman’s original paper [12, section 13]. These functions have the form

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 110 , June 1988 , pp. 151 - 162
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1988

References

[1] Albeverio, S. and Høegh-Krohn, R., Mathematical theory of Feynman path integrals, Springer Lecture Notes in Mathematics, Berlin, 5;23 (1976).CrossRefGoogle Scholar
[2] Albeverio, S. and Høegh-Krohn, R., Oscillatory integrals and the method of stationary phase in infinitely many dimensions, with applications to the classical limit of quantum mechanics, I, Invent. Math., 40 (1977), 59–106.Google Scholar
[3] Albeverio, S. and Høegh-Krohn, R., Feynman path integrals and the corresponding method of stationary phase, in Feynman Path Integrals, Marseille, 1978, Springer Lecture Notes in Physics, 106 (1979), 3–57.Google Scholar
[4] Berkson, E. and Gillespie, T. A., Absolutely continuous functions of two variables and well-bounded operators, J. London Math. Soc, 30 (1984), 305–321.CrossRefGoogle Scholar
[5] Cameron, R. H. and Storvick, D. A., An operator valued function space integral applied to multiple integrals of functions of class Li, Nagoya Math. J., 51 (1973), 91–122.Google Scholar
[6] Cameron, R. H. and Storvick, D. A., Some Banach algebras of analytic Feynman integrable functionals, in Ana lytic Functions, Kozubnik, 1979, Springer Lecture Notes in Mathematics, Berlin, 798 (1980), 18–67.Google Scholar
[7] Chang, K. S., Scale-invariant measurability in Yeh-Wiener space, J. of Korean Math. Soc, 19 (1982), 61–67.Google Scholar
[8] Chang, K. S., Johnson, G. W. and Skoug, D. L., The Feynman integral of quadratic potentials depending on two time variables, Pacific J. Math., 122 (1986), 11–33.Google Scholar
[9] Chang, K. S., Johnson, G. W. and Skoug, D. L., Functions in the Fresnel class, Proc. Amer. Math. Soc, 100 (1987), 309–318.CrossRefGoogle Scholar
[10] Chang, K. S., Johnson, G. W. and Skoug, D. L., Functions in the Banach algebra S(y)9 J. of Korean Math. Soc, 24 (1987), 151–158.Google Scholar
[11] Elworthy, D. and Truman, A., Feynman maps, Cameron-Martin formulae and anharmonic oscillators, Ann. Inst. H. Poincaré, 41 (1984), 115–142.Google Scholar
[12] Feynman, R. J., Space-time approach to non-relativistic quantum mechanics, Rev. Mod. Phys., 20 (1948), 367–387.CrossRefGoogle Scholar
[13] Feynman, R. J. and Hibbs, A. R., Quantum Mechanics and Path Integrals, McGraw-Hill, New York, 1965.Google Scholar
[14] Halmos, P., Measure Theory, Van-Nostrand, Princeton, New Jersey, 1950.Google Scholar
[15] Johnson, G. W., The equivalence of two approaches to the Feynman integral, J. Math. Phys., 23 (1982), 2090–2096.Google Scholar
[16] Johnson, G. W. and Skoug, D. L., A Banach algebra of Feynman integrable functionals with applications to an integral equation formally equivalent to Schroedinger’s equation, J. Funct. Anal., 12 (1973), 129–152.Google Scholar
[17] Johnson, G. W. and Skoug, D. L., Scale-invariant measurability in Wiener space, Pacific J. Math., 83 (1979), 157–176.Google Scholar
[18] Johnson, G. W. and Skoug, D. L., Notes on the Feynman integral, I, Pacific J. Math., 93 (1981), 313–324.CrossRefGoogle Scholar
[19] Johnson, G. W. and Skoug, D. L., Notes on the Feynman integral, II, J. Funct. Anal., 41 (1981), 277–289.Google Scholar
[20] Johnson, G. W. and Skoug, D. L., Notes on the Feynman integral, III, The Schroedinger equation, Pacific J. Math., 105 (1983), 321–358.Google Scholar
[21] Nelson, E., The use of the Wiener process in quantum theory, preprint intended for inclusion in Vol. III of the Collected Works of Norbert Wiener, edited by Masani, P., M.I.T. Press.Google Scholar