Abstract
A path integral representation is obtained for the stochastic partial differential equation of Schrödinger type arising in the theory of open quantum systems subject to continuous nondemolition measurement and filtering, known as the a posteriori or Belavkin equation. The result is established by means of Fresnel-type integrals over paths in configuration space. This is achieved by modifying the classical action functional in the expression for the amplitude along each path by means of a stochastic Itô integral. This modification can be regarded as an extension of Menski's path integral formula for a quantum system subject to continuous measurement to the case of the stochastic Schrödinger equation.
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References
Albeverio, S. A. and HØegh-Krohn, R. J.: Mathematical Theory of Feynman Path Integrals, Lecture Notes in Math. 523, Springer-Verlag, Berlin, 1976.
Albeverio, S. A., Kolokol'tsov, V. N. and Smolyanov, O. G.: Representations of the Belavkin quantum measurement equation by the Menski functional formula, to appear in CR Acad. Sci. Paris, série 1, Math.
Belavkin, V. P.:A new wave equation for a continuous nondemolition measurement, Phys. Lett. A 140 (1989), 355-359.
Belavkin, V. P.: A posterior Schrödinger equation for continuous nondemolition measurement, J. Math. Phys. 31 (1990), 2930-2934.
Belavkin, V. P. and Smolyanov, O. G.: Feynman path integral corresponding the stochastic Schrödinger equation (in Russian), to appear in Dokl. Akad. Nauk.
Dawson, D. A. and Papanicolaou, G. C.: A random wave process, Appl. Math. Optim. 12 (1984), 97-114.
Diósi, L.: Continuous quantum measurement and Itô formalism, Phys. Lett. A 129 (1988), 419- 423.
Doss, H.: Sur une résolution stochastique de l'équation de Schrödinger à coefficients analytiques, Comm. Math. Phys. 73 (1980), 247-264.
Gizin, N.: Quantum measurements and stochastic processes, Phys. Rev. Lett. 52 (1984), 1657- 1660.
Hudson, R. L. and Parthasarathy, K. R.: Quantum Itô's formula and stochastic evolutions, Comm. Math. Phys. 93 (1984), 301-323.
Menski, M. B.: Continuous Quantum Measurements and Path Integrals, Institute of Physics, Bristol, 1993.
Pearle, P.: Combining stochastic dynamical state-vector reduction with spontaneous localization, Phys. Rev. A 39 (1989), 2277-2289.
Truman, A. and Zhao, H. Z.: The stochastic Hamilton-Jacobi equation, stochastic heat equations, and stochastic Schrödinger equations, in: A. Truman, K. D. Elworthy, and I. M. Davies (eds), Stochastic Analysis and Applications, Fifth Gregynog Symposium, Gregynog, Wales, 9-14 July 1995, World Scientific, Singapore, 1996.
Truman, A. and Zhao, H. Z.: On stochastic diffusion equations and stochastic Burgers' equations, J. Math. Phys. 37 (1996), 283-307.
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ZASTAWNIAK, T.J. Fresnel Type Path Integral for the Stochastic Schrödinger Equation. Letters in Mathematical Physics 41, 93–99 (1997). https://doi.org/10.1023/A:1007375114656
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DOI: https://doi.org/10.1023/A:1007375114656