Classical dynamics of infinite particle systems in an operator algebraic framework

TL;DR

This paper develops a C*-algebraic framework for classical infinite particle systems with harmonic oscillators, accommodating various complexities like irregular lattices and varying particle properties, and establishes their dynamical properties.

Contribution

It introduces a novel operator algebraic approach using the commutative resolvent algebra to model classical infinite particle dynamics, extending previous quantum algebraic methods.

Findings

Constructed C*-dynamical systems for classical infinite particles.

Showed the commutative resolvent algebra is time-stable under dynamics.

Established strong continuity of the dynamics on a sub-algebra.

Abstract

We construct C*-dynamical systems for the dynamics of classical infinite particle systems describing harmonic oscillators interacting with arbitrarily many neighbors on lattices, as well on more general structures. Our approach allows particles with varying masses, varying frequencies, irregularly placed lattice sites and varying interactions subject to a simple summability constraint. A key role is played by the commutative resolvent algebra, which is a C*-algebra of bounded continuous functions on an infinite dimensional vector space, and in a strong sense the classical limit of the Buchholz--Grundling resolvent algebra, which suggests that quantum analogs of our results are likely to exist. For a general class of Hamiltonians, we show that the commutative resolvent algebra is time-stable, and admits a time-stable sub-algebra on which the dynamics is strongly continuous, therefore obtaining a C*-dynamical system.