Abstract
The survey is devoted to generalizations of the modern theory of measurement and the probabilistic interpretation of quantum-mechanical quantities. The relation between the quasidispersion and dispersion of indirect measurements is discussed. An example is presented of a dynamical system with random parameters averaging with respect to which is equivalent to averaging of an appropriate pseudodifferential operator relative to a certain quantum-mechanical function of state.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literature cited
V. P. Belavkin, “On generalized Heisenberg indeterminacy relations and effective measurements in quantum systems,” Teor. Mat. Fiz.,26, No. 3, 316–329 (1976).
V. P. Belavkin, “Operational theory of stochastic processes,” in: VII All-Union Conference on Coding and Transmission of Information, Part I, Moscow-Vilnius (1978), pp. 23–28.
V. V. Voevodin, Computational Foundations of Linear Algebra, Nauka, Moscow (1977).
P. A. Dirac, Principles of Quantum Mechanics, Oxford Univ. Press (1958).
L. V. Kantorovich, B. Z. Vulikh, and A. G. Pinsker, Functional Analysis in Semiordered Spaces [in Russian], Gostekhizdat (1950).
A. N. Kolmogorov, Basic Concepts of Probability Theory [in Russian], Nauka, Moscow.
L. D. Landau and E. M. Lifshits, Statistical Physics [in Russian], Nauka, Moscow (1964).
V. P. Maslov, Operator Methods [in Russian], Nauka, Moscow (1973).
V. P. Maslov, Complex Markov Chains and Feynman's Path Integral for Nonlinear Equations [in Russian], Nauka, Moscow (1975).
V. P. Maslov and A. M. Chebotarev, “Jump-type processes and their applications in quantum mechanics,” Itogi Nauki i Tekhniki, VINITI AN SSSR, Ser. Teor. Veroyatn. Mat. Stat. Teor. Kibern.,18 (1978), pp. 1–78.
Yu. V. Prokhorov and Yu. A. Rozanov, Probability Theory. Basic Concepts. Limit Theorems. Stochastic Processes [in Russian], 2nd ed., Nauka, Moscow (1973).
R. Richtmeyer and K. Morton, Difference Methods for Solving Boundary Value Problems [Russian translation], Mir, Moscow (1972).
S. G. Krein (ed.), Functional Analysis [in Russian], 2nd ed., Nauka, Moscow (1972).
A. S. Kholevo, Probabilistic and Statistical Aspects of Quantum Theory [in Russian], Nauka, Moscow (1980).
Ya. I. Khurgin and V. P. Yakovlev, Finite Functions in Physics and Technology [in Russian], Nauka, Moscow (1971).
G. G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Wiley (1972).
F. Bayeh, “Deformation theory and quantization. I, II,” Ann. Fis., New York,111, No. 1, 61–151 (1978).
E. B. Davies and J. T. Lewis, “An operational approach to quantum probability,” Commun. Math. Phys.,17, No. 3, 239–260 (1970).
K. B. Mielnik, “Theory of filters,” Commun. Math. Phys.,15, No. 1, 1–46 (1969).
P. Sharan, “Star-product representation of path integrals,” Phys. Rev.,20, No. 2, 414–413 (1979).
T. B. Smith, “Semiclassical approximation in the Weyl picture by path summation,” J. Phys. Math. Gen.,A11, No. 11, 2179–2190 (1978).
Additional information
Translated from Itogi Nauki i Tekhniki, Seriya Teoriya Veroyatnostei, Matematicheskaya Statistika, Teoreticheskaya Kibernetika, Vol. 19, pp. 55–84, 1982.
Rights and permissions
About this article
Cite this article
Maslov, V.P. The Kolmogorov-Feller equation and the probabilistic model of quantum mechanics. J Math Sci 23, 2534–2553 (1983). https://doi.org/10.1007/BF01084703
Issue Date:
DOI: https://doi.org/10.1007/BF01084703