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The Feynman–Kac Representation and Dobrushin–Lanford–Ruelle States of a Quantum Bose-Gas

Author

Listed:
  • Yuri Suhov

    (Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 OWS, UK
    Mathematics Department, Penn State University, University Park, State College, PA 16802, USA)

  • Mark Kelbert

    (Laboratory of Stochastic Analysis, National Research University the Higher School of Economics, 101000 Moscow, Russia)

  • Izabella Stuhl

    (Mathematics Department, Penn State University, University Park, State College, PA 16802, USA)

Abstract

This paper focuses on infinite-volume bosonic states for a quantum particle system (a quantum gas) in R d . The kinetic energy part of the Hamiltonian is the standard Laplacian (with a boundary condition at the border of a ‘box’). The particles interact with each other through a two-body finite-range potential depending on the distance between them and featuring a hard core of diameter a > 0 . We introduce a class of so-called FK-DLR functionals containing all limiting Gibbs states of the system. As a justification of this concept, we prove that for d = 2 , any FK-DLR functional is shift-invariant, regardless of whether it is unique or not. This yields a quantum analog of results previously achieved by Richthammer.

Suggested Citation

  • Yuri Suhov & Mark Kelbert & Izabella Stuhl, 2020. "The Feynman–Kac Representation and Dobrushin–Lanford–Ruelle States of a Quantum Bose-Gas," Mathematics, MDPI, vol. 8(10), pages 1-41, October.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:10:p:1683-:d:422606
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    References listed on IDEAS

    as
    1. Richthammer, Thomas, 2009. "Translation invariance of two-dimensional Gibbsian systems of particles with internal degrees of freedom," Stochastic Processes and their Applications, Elsevier, vol. 119(3), pages 700-736, March.
    2. Richthammer, Thomas, 2005. "Two-dimensional Gibbsian point processes with continuous spin symmetries," Stochastic Processes and their Applications, Elsevier, vol. 115(5), pages 827-848, May.
    3. Mark Kelbert & Yurii Suhov, 2013. "A Quantum Mermin-Wagner Theorem for a Generalized Hubbard Model," Advances in Mathematical Physics, Hindawi, vol. 2013, pages 1-20, September.
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    1. Richthammer, Thomas, 2009. "Translation invariance of two-dimensional Gibbsian systems of particles with internal degrees of freedom," Stochastic Processes and their Applications, Elsevier, vol. 119(3), pages 700-736, March.
    2. Fiedler, Michael & Richthammer, Thomas, 2021. "A lower bound on the displacement of particles in 2D Gibbsian particle systems," Stochastic Processes and their Applications, Elsevier, vol. 132(C), pages 1-32.

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