# $SU(2)$-particle sigma model: Momentum-space quantization of a particle on the sphere $S^3$

## TL;DR

This paper develops a momentum-space quantization method for a particle on the $S^3$ sphere, revealing unique features in the scalar product and Fourier transform due to the non-trivial topology of the phase space.

## Contribution

It introduces a symmetry-based, non-canonical momentum-space quantization for a particle on $S^3$, highlighting deviations from standard Fourier analysis.

## Key findings

- Scalar product has a non-trivial kernel due to topology
- Momentum space forms an irreducible representation of the symmetry group
- Fourier transform links two irreducible representations

## Abstract

We perform the momentum-space quantization of a spin-less particle moving on the $SU(2)$ group manifold, that is, the three-dimensional sphere $S^{3}$, by using a non-canonical method entirely based on symmetry grounds. To achieve this task, non-standard (contact) symmetries are required as already shown in a previous article where the configuration-space quantization was given. The Hilbert space in the momentum space representation turns out to be made of a subset of (oscillatory) solutions of the Helmholtz equation in four dimensions. The most relevant result is the fact that both the scalar product and the generalized Fourier transform between configuration and momentum spaces deviate notably from the naively expected expressions, the former exhibiting now a non-trivial kernel, under a double integral, traced back to the non-trivial topology of the phase space, even though the momentum space as such is flat. In addition, momentum space itself appears directly as the carrier space of an irreducible representation of the symmetry group, and the Fourier transform as the unitary equivalence between two unitary irreducible representations.