zbMATH — the first resource for mathematics

Stochastic theories and deterministic differential equations. (English) Zbl 1201.81084
Summary: We discuss the concept of “hydrodynamic” stochastic theory, which is not based on the traditional Markovian concept. A Wigner function developed for friction is used for the study of operators in quantum physics, and for the construction of a quantum equation with friction. We compare this theory with the quantum theory, the Liouville process, and the Ornstein-Uhlenbeck process. Analytical and numerical examples are presented and compared.
81T08 Constructive quantum field theory
81T10 Model quantum field theories
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
Full Text: DOI EuDML
[1] J. L. Doob, “Stochastic processes Additional book information,” John Wiley & Sons, New York, NY, USA, 1953. · Zbl 0053.26802
[2] H. M. Taylor and S. Karlin, An Introduction to Stochastic Modeling, Academic Press, New York, NY, USA, 3rd edition, 1998. · Zbl 0946.60002
[3] A. A. Markov, “Extension de la loi de grands nombres aux énévenements dependants les uns de autres,” Bulletin de La Société Physico-Mathématique de Kasan, vol. 15, pp. 135-156, 1906.
[4] W. Feller, “Markov processes and semi-groups,” in Introduction to Probability Theory and Its Applications, vol. 2, chapter 5, John Wiley & Sons, New York, NY, USA, 2nd edition, 1971. · Zbl 0219.60003
[5] P. Garbaczewski, “On the statistical origins of the de Broglie-Bohm quantum potential: brownian motion in a field of force as Bernstein diffusion,” Physics Letters A, vol. 178, no. 1-2, pp. 7-10, 1993. · doi:10.1016/0375-9601(93)90718-F
[6] K. Huang, Statistical Mechanics, John Wiley & Sons, New York, NY, USA, 1963.
[7] S. Chandrasekhar, L. S. Ornstein, M. C. Wang, G. E. Uhlenbeck, and J. L. Dood, Selected Papers on Noise and Stochastic Processes, N. Wax, Ed., Dover, New York, NY, USA, 1954. · Zbl 0059.11903
[8] P. Garbaczewski, J. R. Klauder, and R. Olkiewicz, “Schrödinger problem, Lévy processes, and noise in relativistic quantum mechanics,” Physical Review E, vol. 51, no. 5, pp. 4114-4131, 1995. · doi:10.1103/PhysRevE.51.4114
[9] R. Czopnik and P. Garbaczewski, “Frictionless random dynamics: hydrodynamical formalism,” Physica A, vol. 317, no. 3-4, pp. 449-471, 2003. · Zbl 1005.70014 · doi:10.1016/S0378-4371(02)01343-2
[10] P. Garbaczewski, “Differential entropy and dynamics of uncertainty,” Journal of Statistical Physics, vol. 123, no. 2, pp. 315-355, 2006. · Zbl 1124.82014 · doi:10.1007/s10955-006-9058-2
[11] E. Nelson, “Derivation of the Schrödinger equation from Newtonian mechanics,” Physical Review, vol. 150, no. 4, pp. 1079-1085, 1966. · doi:10.1103/PhysRev.150.1079
[12] E. Nelson, Dynamical Theories of Brownian Motion, Princeton University Press, Princeton, NJ, USA, 1967. · Zbl 0165.58502
[13] E. Nelson, Quantum Fluctuations, Princeton Series in Physics, Princeton University Press, Princeton, NJ, USA, 1985. · Zbl 0563.60001
[14] G. Kaniadakis, “Statistical origin of quantum mechanics Physica A,” Statistical Mechanics and its Applications, vol. 307, no. 1-2, pp. 172-184, 2002. · Zbl 0994.81054 · doi:10.1016/S0378-4371(01)00626-4
[15] E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Physical Review, vol. 40, no. 5, pp. 749-759, 1932. · Zbl 0004.38201 · doi:10.1103/PhysRev.40.749
[16] L. Cohen, “Generalized phase-space distribution functions,” Journal of Mathematical Physics, vol. 7, pp. 781-786, 1966. · doi:10.1063/1.1931206
[17] R. F. O’Connell and E. P. Wigner, “Quantum-mechanical distribution functions: conditions for uniqueness,” Physics Letters A, vol. 83, no. 4, pp. 145-148, 1981. · doi:10.1016/0375-9601(81)90870-7
[18] M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Physics Reports, vol. 106, no. 3, pp. 121-167, 1984. · doi:10.1016/0370-1573(84)90160-1
[19] H.-W. Lee, “Theory and application of the quantum phase-space distribution functions,” Physics Report, vol. 259, no. 3, pp. 147-211, 1995. · doi:10.1016/0370-1573(95)00007-4
[20] L. Cohen and P. Loughlin, “Generalized Wigner distributions, moments and conditional correspondence rules,” Journal of Modern Optics, vol. 49, no. 3-4, pp. 539-560, 2001. · Zbl 1019.81030 · doi:10.1080/09500340110087642
[21] Q. S. Li, G. M. Wei, and L. Q. Lü, “Relationship between the Wigner function and the probability density function in quantum phase space representation,” Physical Review A, vol. 70, no. 2, Article ID 022105, 2004. · Zbl 1227.81215 · doi:10.1103/PhysRevA.70.022105
[22] J. F. Moxnes and K. Hausken, “A non-linear Schrödinger equation used to describe friction,” Annales de la Fondation Louis de Broglie, vol. 30, no. 3-4, pp. 309-324, 2005.
[23] N. P. Landsman, “Algebraic theory of superselection sectors and the measurement problem in quantum mechanics,” International Journal of Modern Physics A, vol. 6, no. 30, pp. 5349-5371, 1991. · Zbl 0809.46089 · doi:10.1142/S0217751X91002513
[24] M. Ozawa, “Cat paradox for C\ast -dynamical systems,” Progress of Theoretical Physics, vol. 88, no. 6, pp. 1051-1064, 1992. · doi:10.1143/PTP.88.1051
[25] J. S. Bell, “Towards an exact quantum mechanics,” in Themes in Contemporary Physics, S. Deser and R. J. Finkelstein, Eds., vol. 226 of NATO ASI Series B, World Scientific, Singapore, 1989.
[26] J. S. Bell, “Against measurement,” in Sixty-Two Years of Uncertainty: Historical, Philosophical and Physical Inquiries into the Foundations of Quantum Mechanics, A. I. Miller, Ed., vol. 226 of NATO ASI Series B, Plenum Press, New York, NY, USA, 1990.
[27] R. Haag, “Events, histories, irreversibility and the concept of events,” Communications in Mathematical Physics, vol. 132, no. 1, pp. 245-251, 1990. · Zbl 0709.53549 · doi:10.1007/BF02278010
[28] P. Blanchard and A. Jadczyk, “On the interaction between classical and quantum systems,” Physics Letters A, vol. 165, pp. 157-164, 1993.
[29] R. Haag, “Events, histories, irreversibility,” in Quantum Control and Measurement, H. Ezawa and Y. Murayama, Eds., Elsevier, North Holland, Amsterdam, 1993.
[30] S. Machida and M. Namiki, “Theory of measurement in quantum mechanics,” Progress of Theoretical Physics, vol. 63, pp. 1457-1473, 1833-1847, 1980.
[31] H. Araki, “A remark on Machida-Namiki theory of measurement,” Progress of Theoretical Physics, vol. 64, no. 3, pp. 719-730, 1980. · Zbl 1097.81503 · doi:10.1143/PTP.64.719
[32] H. Araki, “A continuous super selection rule as a model of classical measuring apparatus in quantum mechanics,” in Fundamental Aspects of Quantum Theory, V. Goring and A. Frigerio, Eds., vol. 144 of NATO Science Series B, Plenum Press, New York, NY, USA, 1986.
[33] R. Olkiewicz, “Some mathematical problems related to classical-quantum interactions,” Reviews in Mathematical Physics, vol. 9, no. 6, pp. 719-747, 1997. · Zbl 0965.81007 · doi:10.1142/S0129055X97000269
[34] G. C. Ghirardi, A. Rimini, and T. Weber, “Unified dynamics for microscopic and macroscopic systems,” Physical Review D, vol. 34, no. 2, pp. 470-491, 1986. · Zbl 1222.82047 · doi:10.1103/PhysRevD.34.470
[35] H. J. Efinger, “A nonlinear unitary framework for quantum state reduction: a phenomenological approach,” Tech. Rep. 2005-03, Universitat Salzburg, June 2005.
[36] A. Bassi, E. Ippoliti, and S. Adler, “Towards quantum superpositions of a mirror: an exact open systems analysis,” Journal of Physics A, vol. 38, no. 12, pp. 2715-2727, 2005. · Zbl 1067.81004 · doi:10.1088/0305-4470/38/12/013
[37] C. W. Gardiner and P. Zoller, A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Application to Quantum Optics, Springer Series in Synergetics, Springer, Berlin, Germany, 3rd edition, 2004. · Zbl 1072.81002
[38] J. Heinrichs, “Probability distributions for second-order processes driven by Gaussian noise,” Physical Review E, vol. 47, no. 5, pp. 3007-3012, 1993. · doi:10.1103/PhysRevE.47.3007
[39] J. F. Moxnes and K. Hausken, “Introducing randomness into first order and second order deterministic differential equations,” Advances in Mathematical Physics. In press. · Zbl 1209.60040 · doi:10.1155/2010/509326 · eudml:227694
[40] U. Leonhard, Measuring the Quantum State of Light, Cambridge University Press, Cambridge, UK, 1997. · Zbl 0901.65079
[41] W. P. Schleich , Quantum Optics in Phase Space, Wiley-VCH, Berlin, Germany, 2001. · Zbl 0961.81136
[42] M. O. Schully and S. Zubairy, Quantum Optics, Cambridge University Press, Cambridge, UK, 1997.
[43] H. Margenau and L. Cohen, “Probabilities in quantum mechanics,” in Quantum Theory and Reality, M. Bunge, Ed., vol. 2, Springer, Berlin, Germany, 1967.
[44] H. Margenau and R. N. Hill, “Correlation between measurements in quantum theory,” Progress of Theoretical Physics, vol. 20, pp. 722-738, 1961.
[45] J. F. Moxnes and K. Hausken, “Uncertainty relations and the operator problem in quantum mechanics,” Fondation Louis de Broglie, vol. 30, no. 1, pp. 11-30, 2006. · Zbl 1329.81059
[46] S. Weinberg, “Testing quantum mechanics,” Annals of Physics, vol. 194, no. 2, pp. 336-386, 1989. · doi:10.1016/0003-4916(89)90276-5
[47] H.-D. Doebner and G. A. Goldin, “On a general nonlinear Schrödinger equation admitting diffusion currents,” Physics Letters A, vol. 162, no. 5, pp. 397-401, 1992. · doi:10.1016/0375-9601(92)90061-P
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.