$SU(2)$ particle sigma model: the role of non-point symmetries in global quantization

TL;DR

This paper presents a group-theoretical quantization of a particle on the $SU(2)$ manifold, emphasizing the importance of non-point symmetries and their impact on quantum commutators and measure, considering the topology of the configuration space.

Contribution

It introduces a novel quantization approach using non-point symmetries for a particle on $SU(2)$, highlighting their influence on quantum structures and topology.

Findings

Non-point symmetries affect quantum commutators.

The measure in Hilbert space relates to the topology of $S^3$.

Quantization on momentum space is also discussed.

Abstract

In this paper we achieve the quantization of a particle moving on the $SU(2)$ group manifold, that is, the three-dimensional sphere $S^{3}$, by using group-theoretical methods. For this purpose, a fundamental role is played by contact, non-point symmetries, i.e., symmetries that leave the Poincar\'e-Cartan form semi-invariant at the classical level, although not necessarily the Lagrangian. Special attention is paid to the role played by the basic quantum commutators, which depart from the canonical, Heisenberg-Weyl ones, as well as the relationship between the integration measure in the Hilbert space of the system and the non-trivial topology of the configuration space. Also, the quantization on momentum space is briefly outlined.