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

Yuri Suhov, Mark Kelbert and Izabella Stuhl
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Yuri Suhov: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 OWS, UK
Izabella Stuhl: Mathematics Department, Penn State University, University Park, State College, PA 16802, USA

Mathematics, 2020, vol. 8, issue 10, 1-41

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.

Keywords: bosonic quantum system; Hamiltonian; Laplacian; two-body interaction; finite-range potential; hard core; Fock space; FK-representation; density matrix; Gibbs state; reduced density matrix; thermodynamic limit; FK-DLR equations (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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