Quantisation of a Particle Moving on a Group Manifold

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 × 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, Ut(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 Ut(SL(2)) symmetry.