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Quantum Stochastic Calculus in Fock Space: A Review

  • R. L. Hudson
Part of the NATO ASI Series book series (NSSB, volume 144)

Abstract

A survey is given of the recently developed theory of quantum stocha-stic calculus in Boson Fock space, together with its applications.

Keywords

Brownian Motion Stochastic Differential Equation Quantum Dynamical Semigroup Unitary Dilation Gauge Process 
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.
    D B Applebaum and R L Hudson, Fermion Ito’s formula and stochastic evolutions, Commun. Math. Phys. 96, 473 (1984).MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. 2.
    A Barchielli and G Lupieri, Quantum stochastic calculus, operation valued stochastic processes and continual measurements in quantum mechanics, J. Math. Phys. 26,2222 (1985).MathSciNetADSzbMATHCrossRefGoogle Scholar
  3. 3.
    C Barnett, R F Streater and I Wilde, The Ito-Clifford integral, J. Func. Anal. 48, 172 (1982).MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    A Frigerio, Covariant Markov dilations of quantum dynamical semigroups, preprint.Google Scholar
  5. 5.
    R L Hudson, P D F Ion and K R Parthasarathy, Time-orthogonal unitary dilations and noncommutative Feynman-Kac formulae, Commun. Math. Phys. 83, 761 (1982).MathSciNetCrossRefGoogle Scholar
  6. 6.
    R L Hudson and J M Lindsay, Uses of non-Fock quantum Brownian motion and a quantum martingale representation theorem, in “Quantum Probability and Applications II”, ed. L Accardi and W von Waldenfels, Springer LNM 1136 (1985).Google Scholar
  7. 7.
    R L Hudson and J M Lindsay, A noncommutative martingale representation theorem for non-Fock quantum Brownian motion, J. Func. Anal. 61, 202 (1985).MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    R L Hudson, J M Lindsay and K R Parthasarathy, Stochastic integral representation of some quantum martingales in Fock space, to appear in proceedings of Warwick symposium on stochastic differential equation, ed. D Elworthy.Google Scholar
  9. 9.
    R L Hudson, J M Lindsay and K R Parthasarathy, Quantum stochastic unitary evolutions on Fock space, preprint.Google Scholar
  10. 10.
    R L Hudson and K R Parthasarathy, Quantum Ito’s formula and stochastic evolutions, Commun. Math. Phys. 93, 301 (1984).MathSciNetADSzbMATHCrossRefGoogle Scholar
  11. 11.
    R L Hudson and K R Parthasarathy, Stochastic dilations of uniformly continuous completely positive semigroups, Acta Applicandae Math. 2, 353 (1984).MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    R L Hudson and K R Parthasarathy, Unification of Fermion and Boson stochastic calculus, to appear in Commun. Math. Phys..Google Scholar
  13. 13.
    N Ikeda and S Watanabe, “Stochastic differential equations and diffusion processes”, North Holland, (1981).zbMATHGoogle Scholar
  14. 14.
    G Lindblad, On the generators of quantum dynamical semigroups, Commun. Math. Phys. 48, 119 (1976).MathSciNetADSzbMATHCrossRefGoogle Scholar
  15. 15.
    H Maassen, Quantum Markov processes on Fock space described by integral kernels, in “Quantum Probability and Applications II”, ed. L Accardi and W von Waldenfels, Springer LNM 1136 (1985).Google Scholar
  16. 16.
    P A Meyer, “ELements de Probabilités Quantiques”, Exposés I à IV, Institut de Mathématique, Université Louis Pasteur, Strasbourg (1985).Google Scholar
  17. 17.
    K R Parthasarathy and K B Sinha, Stochastic integral representation of bounded quantum martingales in Fock space, preprint.Google Scholar
  18. 18.
    K R Parthasarathy, A remark on the integration of Schrödinger’s equation using quantum Ito’s formula, Lett. Math. Phys. 8, 227 (1984).MathSciNetADSzbMATHCrossRefGoogle Scholar
  19. 19.
    I E Segal, Tensor algebras over Hilbert spaces I, Trans. Amer. Math. Soc. 81, 106 (1956).MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1986

Authors and Affiliations

  • R. L. Hudson
    • 1
  1. 1.Mathematics DepartmentNottingham UniversityNottinghamUK

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