Weyl-Heisenberg covariant quantization for the discrete torus
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910762141745152 |
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| author | Murenzi, Romain Zlotak, Aidan Gazeau, Jean Pierre |
| author_facet | Murenzi, Romain Zlotak, Aidan Gazeau, Jean Pierre |
| contents | Covariant integral quantization is implemented for systems whose phase space is $Z_{d} \times Z_{d}$, i.e., for systems moving on the discrete periodic set $Z_d= \{0,1,\dotsc d-1$ mod$ d\}$. The symmetry group of this phase space is the periodic discrete version of the Weyl-Heisenberg group, namely the central extension of the abelian group $Z_d \times Z_d$. In this regard, the phase space is viewed as the left coset of the group with its center. The non-trivial unitary irreducible representation of this group, as acting on $L^2(Z_{N})$, is square integrable on the phase phase. We derive the corresponding covariant integral quantizations from (weight) functions on the phase space, and display their phase space portrait. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_18521 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weyl-Heisenberg covariant quantization for the discrete torus Murenzi, Romain Zlotak, Aidan Gazeau, Jean Pierre Quantum Physics Mathematical Physics Covariant integral quantization is implemented for systems whose phase space is $Z_{d} \times Z_{d}$, i.e., for systems moving on the discrete periodic set $Z_d= \{0,1,\dotsc d-1$ mod$ d\}$. The symmetry group of this phase space is the periodic discrete version of the Weyl-Heisenberg group, namely the central extension of the abelian group $Z_d \times Z_d$. In this regard, the phase space is viewed as the left coset of the group with its center. The non-trivial unitary irreducible representation of this group, as acting on $L^2(Z_{N})$, is square integrable on the phase phase. We derive the corresponding covariant integral quantizations from (weight) functions on the phase space, and display their phase space portrait. |
| title | Weyl-Heisenberg covariant quantization for the discrete torus |
| topic | Quantum Physics Mathematical Physics |
| url | https://arxiv.org/abs/2412.18521 |