[Paper Review] Quantum Mechanics on the Circle and W_{1+\INFTY}

Roberto Floreanini, Roberto Percacci+1 more|arXiv (Cornell University)|Nov 25, 1991

Matrix Theory and Algorithms4 references4 citations

TL;DR

This paper establishes a direct correspondence between the quantum mechanics of a particle on a circle and the W_{1+∞} algebra with central charge c=0, identifying the algebra as the universal enveloping algebra of the 2D Euclidean algebra. It further shows that the super W_∞ algebra arises as the universal enveloping algebra of the super-Euclidean algebra in two dimensions, providing a physical realization of these infinite-dimensional symmetry algebras through quantum systems on compactified spatial dimensions.

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.

Motivation & Objective

• To establish a physical realization of the W_{1+∞} algebra with central charge c=0 through quantum mechanics on a circle.
• To identify the algebra of observables for a particle constrained to move on a circle as isomorphic to W_{1+∞} with c=0.
• To extend the construction to supersymmetric systems, identifying the super W_∞ algebra as the universal enveloping algebra of the super-Euclidean algebra in two dimensions.
• To provide a geometric and physical interpretation of infinite-dimensional algebras in terms of quantum systems on compact manifolds.
• To clarify the mathematical structure of W_{1+∞} and its superextension via quantum mechanical models with rotational symmetry.

Proposed method

• Analyzes the quantum mechanical system of a particle moving on a one-dimensional circle (S^1), focusing on its symmetry algebra.
• Identifies the algebra of quantum observables—specifically, the algebra generated by position and momentum operators under periodic boundary conditions—as isomorphic to W_{1+∞} with c=0.
• Demonstrates that this algebra is equivalent to the universal enveloping algebra of the 2D Euclidean Lie algebra (ISO(2)).
• Extends the analysis to supersymmetric systems by introducing fermionic degrees of freedom, leading to the super-W_∞ algebra.
• Shows that the super-W_∞ algebra arises as the universal enveloping algebra of the super-Euclidean algebra in two dimensions.
• Uses representation theory and algebraic structure analysis to confirm the isomorphism between the observable algebra and the infinite-dimensional symmetry algebra.

Experimental results

Research questions

• RQ1 Can the W_{1+∞} algebra with central charge c=0 be physically realized as the algebra of observables for a quantum system?
• RQ2 What is the precise algebraic structure of the quantum observables for a particle on a circle, and how does it relate to known infinite-dimensional algebras?
• RQ3 How does the construction generalize to supersymmetric systems, and what algebraic structure emerges?
• RQ4 Is there a direct correspondence between the universal enveloping algebra of the 2D Euclidean algebra and the W_{1+∞} algebra?
• RQ5 Can the super-W_∞ algebra be derived as the universal enveloping algebra of the super-Euclidean algebra in two dimensions?

Key findings

• The algebra of quantum observables for a particle on a circle is isomorphic to W_{1+∞} with central charge c=0.
• This W_{1+∞} algebra is identified as the universal enveloping algebra of the 2D Euclidean Lie algebra (ISO(2)).
• The super W_∞ algebra is shown to be the universal enveloping algebra of the super-Euclidean algebra in two dimensions.
• The construction provides a physical realization of the W_{1+∞} algebra through a simple quantum mechanical system.
• The central charge c=0 is essential in establishing the isomorphism between the observable algebra and W_{1+∞}.
• The results establish a direct link between compact spatial geometry (S^1) and infinite-dimensional symmetry algebras in quantum field theory.