Abstract
A method is given for the derivation of a path integral representation of the Green's function solutionP of equations∂P/∂t=L P,L being some Liouville operator. The method is applied to general diffusion processes.
Feynman's path integral representation of the Schrödinger equation and Stratonovich's path integral representation of multivariate Markovian processes are obtained as special cases if the metric of the general diffusion process is flat. For curved phase spaces our result is a nontrivial generalization and new. New applications e.g. to quantized motion in general relativity, to transport processes in inhomogeneous systems, or to nonlinear non-equilibrium thermodynamics are made possible. We expect applications to be fruitfull in all cases where (continuous) macroscopic transport processes in Riemann geometries have to be considered.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Stratovonich, R.L.: Topics in the Theory of Random Noise, Vol. 1, New York: Gordon and Breach 1963
Onsager, L., Machlup, S.: Phys. Rev.91, 1505 (1953);91, 1512 (1953)
Hashitsume, N.: Progr. Theor. Phys.8, 461 (1952);15, 369 (1956); Tisza, L., Manning, I.: Phys. Rev.105, 1695 (1957)
Feynman, R.P., Hibbs, A.R.: Quantum Mechanics and Phath Integrals, New York: Mc. Graw Hill 1965
Graham, R., to be published
Martin, P.C., Siggia, E.D., Rose, H.A.: Phys. Rev. A8, 423 (1973); Deker, U., Haake, F.: Phys. Rev. A11, 2043 (1975); A12, 1629 (1975)
Janssen, H.K.: Physik B23, 377 (1976)
Stratonovich, R.L.: Sel. Transl. Math. Stat. Prob.10, 273 (1971) (translation of russian paper 1962)
Graham, R.: Phys. Rev. Letters38, Jan. 10 (1977)
Bach, A., Dürr, D., Stawicki, B.: preprint 1976
Graham, R.: Z. Physik, to be published
cf. Weinberg, S.: Gravitation and Cosmology, New York: John Wiley 1972
Wiener, N.: Acta Math.55, 117 (1930)
Uhlenbeck, G.E., Ornstein, L.S.: Phys. Rev.36, 823 (1930)
Graham, R.: Springer Tracts in Modern Physics, Vol. 66, New York: Springer 1973
Graham, R.: in Fluctuations, Instabilities and Phase Transitions (ed. Riste) New York: Plenum 1976
Horsthemke, W., Bach, A.: Z. Physik B22, 189 (1975)
Horsthemke, W.: private comunication 1975
Hänggi, P., Nettel, S.J.: Feynman Path-Integral Formulation for the Fokker-Planck Equation, preprint 1976
Feynmann, R.P.: Phys. Rev.84, 108 (1951)
ref. 20, p. 127, first paragraph
Haken, H.: Z. Physik B24, 321 (1976)
Dekker, H.: On the Functional Integral for Generalized Wiener Processes and Nonequilibrium Phenomena, preprint 1976
Horner, H.: private communication 1976
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Graham, R. Path integral formulation of general diffusion processes. Z Physik B 26, 281–290 (1977). https://doi.org/10.1007/BF01312935
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01312935