Chiral Quantization on a Group Manifold

TL;DR

This paper explores the classical and quantum aspects of chiral sectors on a group manifold, specifically SU(2), revealing a non-commutative quantum sphere and a complex exchange algebra, with implications for the structure of the Hilbert space.

Contribution

It provides a detailed classical analysis and geometric quantization of chiral sectors on SU(2), introducing a non-commutative quantum sphere and a novel quartic exchange algebra.

Findings

Quantum sphere replaces classical $S^3$ relation.

Quantum exchange algebra is quartic, not quadratic.

Hilbert space structure differs from direct quantization.

Abstract

The phase space of a particle on a group manifold can be split in left and right sectors, in close analogy with the chiral sectors in Wess Zumino Witten models. We perform a classical analysis of the sectors, and the geometric quantization in the case of $SU(2)$. The quadratic relation, classically identifying $SU(2)$ as the sphere $S^3$, is replaced quantum mechanically by a similar condition on non-commutative operators ('quantum sphere'). The resulting quantum exchange algebra of the chiral group variables is quartic, not quadratic. The fusion of the sectors leads to a Hilbert space that is subtly different from the one obtained by a more direct (un--split) quantization.