SU(2)-particle sigma model: momentum-space quantization of a particle on the sphere S-3
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...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Estado: | Versión aceptada para publicación |
| Fecha de publicación: | 2020 |
| País: | España |
| Institución: | Consejo Superior de Investigaciones Científicas (CSIC) |
| Repositorio: | DIGITAL.CSIC. Repositorio Institucional del CSIC |
| OAI Identifier: | oai:digital.csic.es:10261/207222 |
| Acceso en línea: | http://hdl.handle.net/10261/207222 |
| Access Level: | acceso abierto |
| Palabra clave: | Sigma model Contact symmetries Non-canonical quantization Particle on the sphere Momentum-space quantization Non trivial topology Helmholtz equation |
| Sumario: | 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 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. |
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