Quantum Mechanics on the Circle and W_{1+\INFTY}

TL;DR

This paper explores the algebraic structures of quantum observables for a particle on a circle, identifying W_{1+ finite} and super W_\infty as universal enveloping algebras of Euclidean algebras.

Contribution

It establishes a mathematical correspondence between W_{1+ finite} algebra and quantum observables on a circle, extending to superalgebras.

Findings

W_{1+ finite} is the universal enveloping algebra of the Euclidean algebra in two dimensions.

Super W_\infty algebra corresponds to the super-Euclidean algebra.

Provides a mathematical framework linking algebraic structures to quantum systems on a circle.

Abstract

The algebra W_{1+\infty} with central charge c=0 can be identified with the algebra of quantum observables of a particle moving on a circle. Mathematically, it is the universal enveloping algebra of the Euclidean algebra in two dimensions. Similarly, the super W_\infty algebra is found to be the universal enveloping algebra of the super-Euclidean algebra in two dimensions.