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Quantum Hamiltonians and Stochastic Jumps

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Abstract

With many Hamiltonians one can naturally associate a |Ψ|2-distributed Markov process. For nonrelativistic quantum mechanics, this process is in fact deterministic, and is known as Bohmian mechanics. For the Hamiltonian of a quantum field theory, it is typically a jump process on the configuration space of a variable number of particles. We define these processes for regularized quantum field theories, thereby generalizing previous work of John S. Bell [3] and of ourselves [11]. We introduce a formula expressing the jump rates in terms of the interaction Hamiltonian, and establish a condition for finiteness of the rates.

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Authors and Affiliations

  1. Mathematisches Institut der Universität München, Theresienstraße 39, 80333, München, Germany

    Detlef Dürr

  2. Departments of Mathematics and Physics, Hill Center, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, NJ, 08854-8019, USA

    Sheldon Goldstein

  3. Dipartimento di Fisica and INFN sezione di Genova,  , Via Dodecaneso 33, 16146, Genova, Italy

    Roderich Tumulka & Nino Zangh

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  1. Detlef Dürr
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  2. Sheldon Goldstein
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  3. Roderich Tumulka
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  4. Nino Zangh
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Correspondence to Detlef Dürr.

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Communicated by A. Connes

Acknowledgement We thank Ovidiu Costin, Avraham Soffer, and James Taylor of Rutgers University, Stefan Teufel of Technische Universität München, and Gianni Cassinelli and Alessandro Toigo of Università di Genova for helpful discussions. R.T. gratefully acknowledges support by the German National Science Foundation (DFG). N.Z. gratefully acknowledges support by INFN and DFG. Finally, we appreciate the hospitality that some of us have enjoyed, on more than one occasion, at the Mathematisches Institut of Ludwig-Maximilians-Universität München, at the Dipartimento di Fisica of Università di Genova, and at the Mathematics Department of Rutgers University.

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Dürr, D., Goldstein, S., Tumulka, R. et al. Quantum Hamiltonians and Stochastic Jumps. Commun. Math. Phys. 254, 129–166 (2005). https://doi.org/10.1007/s00220-004-1242-0

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