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Stochastic Dynamics and the Semiclassical Limit of Quantum Mechanics

  • G. Jona-Lasinio

Abstract

We outline a new approach to tunnelling problems in quantum mechanics based on the so called stochastic quantization. The starting point of the method is the Observation that in the limit \(\frac{\hbar }{m} \to o\)one can approximate the stochastic process underlying the quantum motion with a Markov chain making transitions at random times from one classical equilibrium position to another. This picture is suggested by the theory of small random perturbations of dynamical systems developed by Ventzel and Freidlin which provides the natural mathematical ideas for our purpose.

As an example of application we discuss the instanton problem for the double well anharmonic oscillator. The flexibility and wide applicability of the present approach is pointed out.

Keywords

Semiclassical Limit Anharmonic Oscillator Drift Term Stochastic Quantization Strong Markov Property 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    A.M. Polyakov, Nucl.Phys. B121, 429 (1977);CrossRefADSMathSciNetGoogle Scholar
  2. [1a]
    G. t’ Hooft, Phys.Rev.Lett. 37, 8 (1976)CrossRefADSGoogle Scholar
  3. [1b]
    G. t’ Hooft Phys.Rev. D14, 3432 (1976).CrossRefGoogle Scholar
  4. [2]
    For a clear exposition of the functional integral approach see S. Colemann, “The Uses of Instantons”, Lectures at the 1977 International School “Ettore Majorana”.Google Scholar
  5. [3]
    E. Nelson, Phys.Rev. 150, 1079 (1966)CrossRefADSGoogle Scholar
  6. [3a]
    E. Nelson “Dynamical Theories of Brownian Motion”, Princeton 1967.MATHGoogle Scholar
  7. [4]
    F. Guerra, P. Ruggiero, Phys.Rev.Lett. 31 1022 (1973).CrossRefADSGoogle Scholar
  8. [5]
    R.Z. Khasminski, “Stability of Systems of Differential Equations under Random Perturbations of their Parameters”, Moscow 1969 (in Russian).Google Scholar
  9. [6]
    A.D. Ventzel, M.I. Freidlin, Uspehi Math.Nauk 25, 3 (1970)Google Scholar
  10. [6a]
    A.D. Ventzel, M.I. Freidlin (English translation Russian Mathematical Surveys 25, 1 (1970)).CrossRefADSGoogle Scholar
  11. [7]
    D. Dürr, A. Bach, Comm.Math.Phys. 60, 153 (1978).CrossRefMATHADSMathSciNetGoogle Scholar
  12. [8]
    I.I. Gihman, A.V. Skorohod, “Stochastic Differential Equations”, Springer Verlag, Berlin 1972, pag. 109.MATHGoogle Scholar
  13. [9]
    S. Albeverio, R. Høegh-Krohn, in “Les Méthodes Mathématiques de la Théorie Quantique des Champs” CNRS, Marseille 1975, pag. 11–59.Google Scholar
  14. [10]
    K. Yasue, Phys.Rev.Lett. 40, 665 (1978)CrossRefADSGoogle Scholar
  15. [10a]
    K. Yasue Phys.Rev. D18, 532 (1978);ADSMathSciNetGoogle Scholar
  16. [10b]
    D.L. Weaver, Phys.Rev.Lett. 40, 1473 (1978).CrossRefADSGoogle Scholar

Copyright information

© Springer-Verlag/Wien 1980

Authors and Affiliations

  • G. Jona-Lasinio
    • 1
  1. 1.Istituto di Fisica dell’, Gruppo G.N.S.MUniversità — RomaRomaItaly

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