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The fermion stochastic calculus I

Part of the Lecture Notes in Mathematics book series (LNM,volume 1158)

Keywords

  • Brownian Motion
  • Stochastic Differential Equation
  • Conditional Expectation
  • Quantum Probability
  • Wick Product

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References

  1. C. Barnett, R.F. Streater and I.F. Wilde. The Ito-Clifford Integral. Jour. Funct. Anal. 48, 172–212 (1982).

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. R.F. Streater. The damped oscillator with quantum noise. J. Phys. A 15, 1477–1486 (1982).

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. I.R. Senitzky, Phys. Rev. 119, 670 (1960); ibid A3 421 (1970).

    CrossRef  MathSciNet  Google Scholar 

  4. M. Lax. Phys. Rev. 145, 111–129 (1965).

    MathSciNet  Google Scholar 

  5. E. Nelson, Dynamical Theories of Brownian Motion. Princeton Lecture Notes, Princeton University Press.

    Google Scholar 

  6. F. Guerra and P. Ruggiero: New interpretation of the Euclidean Markov field in the framework of physical space-time. Phys. Rev. Lett. 31, 1022 (1972).

    CrossRef  Google Scholar 

  7. G. Parisi and Y. Wu. Scientia Sinica 24, 483 (1981).

    MathSciNet  Google Scholar 

  8. R.F. Streater, Current Commutation Relations and Continuous Tensor Products. Nuovo Cimento 53, 487 (1968). R.F. Streater, Current Commutation Relations, Continuous Tensor Products and Infinitely Divisible Group Representations; in Local Quantum Theory (R. Jost, Ed.), Academic Press 1969.

    CrossRef  MATH  Google Scholar 

  9. T. Hida; Brownian Motion. Springer Verlag, N.Y., Berlin, Heidelberg 1980. R.L. Hudson and R.F. Streater. Examples of quantum martingales. Phys. Lett. 85A, 64–66 (1981). R.L. Hudson and R.F. Streater. Ito's rule is the chain rule with Wick ordering. Phys. Lett. 86A, 277–279 (1981). r.L. Hudson and R.F. Streater, Non-commutative martingales and stochastic integrals in Fock space; in Lecture Notes in Physics 173, Springer 1981.

    CrossRef  MATH  Google Scholar 

  10. D. Mathon and R.F. Streater. Infinitely Divisible Representations of Clifford Algebras. Z. für Wakr. verw. Geb. 20, 308–316 (1971).

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. H. Umegaki, Conditional Expectation in an Operator Algebra. II. Tohuku Math. J. 8, 86–100 (1956).

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. D.E. Evans and J.T. Lewis. Commun. Dublin Institute for Advanced Studies A, 24 (1977).

    Google Scholar 

  13. I.E. Segal, A non-commutative extension of abstract integration. Ann. of Math. 57, 401–457 (1953). Ibid, 58, 595–596 (1953). E. Nelson. Notes on non-commutative integration. Jour. Functl. Anal. 15, 103–116 (1974).

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. J. Dixmier, Formes linéares sur un anneau d'opérateurs. Bull. Soc. Math. France 81, 9–39 (1953).

    MathSciNet  MATH  Google Scholar 

  15. C. Barnett, R.F. Streater and I.F. Wilde. The Ito-Clifford Integral III: Markov property of solutions to stochastic differential equations. Commun. Math. Phys. 89, 13–17 (1983).

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. C. Barnett, R.F. Streater and I.F. Wilde. The Ito-Clifford Integral II, Stochastic differential equations. J. Lond. Math. Soc. (2), 27, 373–384 (1983).

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. R.F. Streater, Quantum Stochastic Integrals, Acta Physica Austriaca, Suppl.XXVI, 53–74 (1984).

    MathSciNet  MATH  Google Scholar 

  18. C. Barnett, Supermartingales on semi-finite von Neumann algebras. J. Lond. Math. Soc. (2), 24, 175–181 (1981).

    CrossRef  MATH  Google Scholar 

  19. C. Barnett, R.F. Streater and I.F. Wilde. The Ito-Clifford Integral IV: A Radon-Nikodym Theorem and Bracket Processes. J. Operator Theory, 11, 255–271 (1984).

    MathSciNet  MATH  Google Scholar 

  20. R.G. Bartle. A general bilinear vector integral. Studia Math. 15, 337–352 (1956).

    MathSciNet  MATH  Google Scholar 

  21. C. Barnett, R.F. Streater and I.F. Wilde. Stochastic integrals in an arbitrary probability gauge space. Math. Proc. Camb. Phil. Soc. 94, 541 (1983).

    CrossRef  MathSciNet  MATH  Google Scholar 

  22. C. Barnett, R.F. Streater and I.F. Wilde. Quasi-free quantum stochastic integrals for the CAR and CCR. J. Functl. Anal. 52, 17–47 (1983).

    MathSciNet  MATH  Google Scholar 

  23. R.F. Streater. Quantum Stochastic Processes. Rome II Conference on Quantum Probability (L. Accardi, A. Frigerio and V. Gorini, Eds.).

    Google Scholar 

  24. C. Barnett, R.F. Streater and I.F. Wilde. Quantum Stochastic Integrals under Standing Hypotheses; submitted to J. Lond. Math. Soc.

    Google Scholar 

  25. E.J. McShane. Stochastic calculus and stochastic models. Academic Press, N.Y. 1974.

    MATH  Google Scholar 

  26. D. Stroock and V.S. Varadhan, Multidimensional Diffusion Processes (Springer), 1979.

    Google Scholar 

  27. H. Hasegawa and R.F. Streater. Stochastic Schrödinger and Heisenberg equations: a martingale problem in quantum stochastic processes. J. Phys. A. L697–L703 (1983).

    Google Scholar 

  28. C. Barnett and T. Lyons. "Stopping non-commutative processes", Imperial College preprint, 1984.

    Google Scholar 

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© 1986 Springer-Verlag

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Streater, R.F. (1986). The fermion stochastic calculus I. In: Albeverio, S.A., Blanchard, P., Streit, L. (eds) Stochastic Processes — Mathematics and Physics. Lecture Notes in Mathematics, vol 1158. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0080223

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  • DOI: https://doi.org/10.1007/BFb0080223

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