SU(2)-particle sigma model: Momentumspace quantization of a particle on the sphere S3

DOI: 
10.1088/1751-8121/ab661d
Publication date: 
02/06/2020
Main author: 
Guerrero J.
IAA authors: 
Aldaya, V.
Authors: 
Guerrero J., López-Ruiz F.F., Aldaya V.
Journal: 
Journal of Physics A: Mathematical and Theoretical
Refereed: 
Yes
Publication type: 
Article
Volume: 
53.0
Pages: 
145301
Number: 
145301
Abstract: 
We perform the momentum-space quantization of a spin-less particle moving on the 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 the 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. © 2020 IOP Publishing Ltd.
Database: 
SCOPUS
ADS
URL: 
https://ui.adsabs.harvard.edu/#abs/2020JPhA...53n5301G/abstract
ADS Bibcode: 
2020JPhA...53n5301G
Keywords: 
Contact symmetries; Helmholtz equation; Momentum-space quantization; Non trivial topology; Non-canonical quantization; Particle on the sphere; Sigma model