Lieb-Robinson bounds in classical oscillating lattice systems

TL;DR

This paper establishes Lieb-Robinson bounds for classical harmonic oscillator systems on various structures and constructs a global dynamical system on a classical analog of a C*-algebra, advancing understanding of classical lattice dynamics.

Contribution

It proves Lieb-Robinson bounds for classical oscillating systems and develops a classical analog of the resolvent algebra, bridging quantum and classical dynamical frameworks.

Findings

Lieb-Robinson bounds hold for classical harmonic oscillators with many neighbors

Existence of a global dynamical system on the classical resolvent algebra

Extension of quantum algebra concepts to classical systems

Abstract

The aim of this paper is two-fold. First, we prove the existence of Lieb-Robinson bounds for classical particle systems describing harmonic oscillators interacting with arbitrarily many neighbors, both on lattices and on more general structures. Second, we prove the existence of a global dynamical system on the commutative resolvent algebra, a C*-algebra of bounded continuous functions on an infinite dimensional vector space, which serves as the classical analog of the Buchholz--Grundling resolvent algebra.