Conservation laws in the continuum $1/r^2$ systems

TL;DR

This paper investigates the conservation laws in classical and quantum continuum $1/r^2$ systems, introducing new integrals of motion and establishing relationships between classical and quantum integrals.

Contribution

It introduces new integrals of motion for both classical and quantum $1/r^2$ systems and relates them to existing integrals, advancing understanding of their conservation laws.

Findings

Classical integrals relate algebraically to each other.

New quantum integrals are constructed diagrammatically.

Relationships between classical and quantum integrals are explicitly shown.

Abstract

We study the conservation laws of both the classical and the quantum mechanical continuum $1/r^2$ type systems. For the classical case, we introduce new integrals of motion along the recent ideas of Shastry and Sutherland (SS), supplementing the usual integrals of motion constructed much earlier by Moser. We show by explicit construction that one set of integrals can be related algebraically to the other. The difference of these two sets of integrals then gives rise to yet another complete set of integrals of motion. For the quantum case, we first need to resum the integrals proposed by Calogero, Marchioro and Ragnisco. We give a diagrammatic construction scheme for these new integrals, which are the quantum analogues of the classical traces. Again we show that there is a relationship between these new integrals and the quantum integrals of SS by explicit construction.