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

• Main Authors: Yuri Suhov, Mark Kelbert, Izabella Stuhl
• Format: Article
• Language: English
• Published: MDPI AG 2020-10-01
• Series: Mathematics
• Subjects: bosonic quantum system; Hamiltonian; Laplacian; two-body interaction; finite-range potential; hard core
• Online Access: https://www.mdpi.com/2227-7390/8/10/1683

This paper focuses on infinite-volume bosonic states for a quantum particle system (a quantum gas) in <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup></semantics></math></inline-formula>. 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 <inline-formula><math display="inline"><semantics><mrow><mi>a</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. 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 <inline-formula><math display="inline"><semantics><mrow><mi>d</mi><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula>, 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.