A Note on the Resolvent Algebra and Functional Integral Approach to the van Hove Model

TL;DR

This paper reviews the operator-algebraic and functional integral methods applied to the van Hove model, focusing on cutoff removal, ground state existence, and Bose--Einstein condensation, without presenting new results.

Contribution

It compiles computational notes on the van Hove model, emphasizing the removal of cutoffs and state existence from multiple mathematical perspectives.

Findings

Infrared and ultraviolet cutoffs can be removed in the van Hove model.

Ground states and β-KMS states exist in the infinite-volume, finite-temperature case.

Bose--Einstein condensation can occur in the model.

Abstract

This paper is a collection of the author's computational notes on the van Hove model and contains no essentially new results. We discuss, from both the operator-algebraic perspective via the Weyl algebra and the resolvent algebra and the functional integral approach, the removal of infrared and ultraviolet cutoffs and the existence of the ground state and $\beta$-KMS states in the case of a point source. In the infinite-volume system at finite temperature, Bose--Einstein condensation can arise.