Quantisation of a particle moving on a group manifold

TL;DR

This paper investigates the quantisation of a massless particle on a group manifold, revealing a connection to quantum groups and providing classical symplectic structures that could lead to a Lagrangian formulation.

Contribution

It introduces a method to factorise the theory's symmetry, and proposes a deformation to realise the quantum group $U_t(SL(2))$ from the classical model.

Findings

Classical symplectic structures derived for the particle on a group manifold.

A deformation of the quantised theory to realise $U_t(SL(2))$ quantum group.

Potential for a Lagrangian formulation of the quantum group symmetry.

Abstract

The Hilbert space of a free massless particle moving on a group manifold is studied in details using canonical quantisation. While the simplest model is invariant under a global symmetry, $G \times G$, there is a very natural way to ``factorise" the theory so that only one copy of the global symmetry is preserved. In the case of $G=SU(2)$, a simple deformation of the quantised theory is proposed to give a realisation of the quantum group, $U_t(SL(2))$. The symplectic structures of the corresponding classical theory is derived. This can be used, in principle, to obtain a Lagrangian formulation for the $U_t(SL(2))$ symmetry.