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On a Stochastic Trotter Formula with Application to Spontaneous Localization Models

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Abstract

We consider the relation between so called continuous localization models—i.e. non-linear stochastic Schrödinger evolutions—and the discrete GRW-model of wave function collapse. The former can be understood as scaling limit of the GRW process. The proof relies on a stochastic Trotter formula, which is of interest in its own right. Our Trotter formula also allows to complement results on existence theory of stochastic Schrödinger evolutions by Holevo and Mora/Rebolledo.

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References

  1. Bassi, A.: Collapse models: analysis of the free particle dynamics. J. Phys. A 38(14), 3173–3192 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bassi, A., Dürr, D., Kolb, M.: On the long time behavior of free stochastic Schrödinger evolutions. Rev. Math. Phys. 22(1), 55–89 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bassi, A., Ghirardi, G.C.: Dynamical reduction models. Phys. Rep. 379(5–6), 257–426 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  4. Belavkin, V.P., Staszewski, P.: A quantum particle undergoing continuous observation. Phys. Lett. A 140(7–8), 359–362 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  5. Bell, J.S.: Speakable and Unspeakable in Quantum Mechanics. Cambridge University Press, Cambridge (1987). Collected papers on quantum philosophy

    Google Scholar 

  6. Breuer, H.-P., Petruccione, F.: The Theory of Open Quantum Systems. Oxford University Press, New York (2002)

    MATH  Google Scholar 

  7. Diósi, L.: Continuous quantum measurement and Itô formalism. Phys. Lett. A 129(8–9), 419–423 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  8. Diósi, L.: Models for universal reduction of macroscopic quantum fluctuations. Phys. Rev. A 40(3), 1165–1174 (1989)

    Article  ADS  Google Scholar 

  9. Dürr, D., Hinrichs, G., Kolb, M.: Work in progress

  10. Ghirardi, G.C., Rimini, A., Weber, T.: Unified dynamics for microscopic and macroscopic systems. Phys. Rev. D 34(2), 470–491 (1986)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Ghirardi, G.C., Pearle, P., Rimini, A.: Markov processes in Hilbert space and continuous spontaneous localization of systems of identical particles. Phys. Rev. A 42(1), 78–89 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  12. Gough, J.E., Obrezkov, O.O., Smolyanov, O.G.: Randomized Hamiltonian Feynman integrals and stochastic Schrödinger-Itô equations. Izv. Ross. Akad. Nauk Ser. Mat. 69(6), 3–20 (2005)

    MathSciNet  Google Scholar 

  13. Hackenbroch, W., Thalmaier, A.: Stochastische Analysis. Mathematische Leitfäden. Teubner, Stuttgart (1994). Eine Einführung in die Theorie der stetigen Semimartingale

    Google Scholar 

  14. Holevo, A.S.: On dissipative stochastic equations in a Hilbert space. Probab. Theory Relat. Fields 104(4), 483–500 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  15. Huang, M.-J.: Commutators and invariant domains for Schrödinger propagators. Pac. J. Math. 175(1), 83–91 (1996)

    Google Scholar 

  16. Joos, E., Zeh, H.D., Kiefer, C., Giulini, D., Kupsch, J., Stamatescu, I.-O.: Decoherence and the Appearance of a Classical World in Quantum Theory, 2nd edn. Springer, Berlin (2003)

    Google Scholar 

  17. Kato, T., Masuda, K.: Trotter’s product formula for nonlinear semigroups generated by the subdifferentials of convex functionals. J. Math. Soc. Jpn. 30(1), 169–178 (1978)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  18. Kolokol’tsov, V.N.: Localization and analytic properties of the solutions of the simplest quantum filtering equation. Rev. Math. Phys. 10(6), 801–828 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kurtz, Th.G.: A random Trotter product formula. Proc. Am. Math. Soc. 35, 147–154 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  20. Lindblad, G.: On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48(2), 119–130 (1976)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  21. Mora, C.M., Rebolledo, R.: Regularity of solutions to linear stochastic Schrödinger equations. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10(2), 237–259 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  22. Mora, C.M., Rebolledo, R.: Basic properties of nonlinear stochastic Schrödinger equations driven by Brownian motions. Ann. Appl. Probab. 18(2), 591–619 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  23. Penrose, R.: On gravity’s role in quantum state reduction. Gen. Relativ. Gravit. 28(5), 581–600 (1996)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  24. Protter, Ph.E.: Stochastic Integration and Differential Equations. Stochastic Modelling and Applied Probability, vol. 21, 2nd edn. Springer, Berlin (2005). Version 2.1. Corrected third printing

    Google Scholar 

  25. Shigekawa, I.: Stochastic Analysis. Translations of Mathematical Monographs, vol. 224. Am. Math. Soc., Providence (2004). Translated from the 1998 Japanese original by the author, Iwanami Series in Modern Mathematics

    MATH  Google Scholar 

  26. Tumulka, R.: The point processes of the GRW theory of wave function collapse. Rev. Math. Phys. 21(2), 155–227 (2009)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Mathematisches Institut, Ludwig-Maximilians Universität, Theresienstr. 39, 80333, München, Germany

    Detlef Dürr & Günter Hinrichs

  2. Department of Statistics, Oxford University, 1 South Parks Road, Oxford, OX1 3TG, UK

    Martin Kolb

Authors
  1. Detlef Dürr
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  2. Günter Hinrichs
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  3. Martin Kolb
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Correspondence to Günter Hinrichs.

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Dürr, D., Hinrichs, G. & Kolb, M. On a Stochastic Trotter Formula with Application to Spontaneous Localization Models. J Stat Phys 143, 1096–1119 (2011). https://doi.org/10.1007/s10955-011-0235-6

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  • DOI: https://doi.org/10.1007/s10955-011-0235-6

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