On the thermodynamic limit of interacting fermions in the continuum

TL;DR

This paper develops a mathematical framework for analyzing the dynamics of interacting non-relativistic fermions in continuous space, extending algebraic methods to ensure well-defined time evolution and continuity.

Contribution

It introduces an extended algebraic setting for fermion dynamics, ensuring continuous time evolution and paving the way for constructing equilibrium states.

Findings

Identified an algebra extension where dynamics acts as *-automorphisms.

Proved strong continuity of the dynamics in a dense subalgebra.

Outlined potential for constructing KMS states in future work.

Abstract

We study the dynamics of non-relativistic fermions in $\mathbb R^d$ interacting through a pair potential. Employing methods developed by Buchholz in the framework of resolvent algebras, we identify an extension of the CAR algebra where the dynamics acts as a group of *-automorphisms, which are continuous in time in all sectors for fixed particle numbers. In addition, we identify a suitable dense subalgebra where the time evolution is also strongly continuous. Finally, we briefly discuss how this framework could be used to construct KMS states in the future.