The Heisenberg-Weyl-parity group its coherent states and a unified Wigner-Weyl function
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914567402029056 |
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| author | Vourdas, A. |
| author_facet | Vourdas, A. |
| contents | The Heisenberg-Weyl group $HW(d)$ related to a $d$-dimensional Hilbert space $H(d)$, is enlarged into the Heisenberg-Weyl-parity group $HWP(d)$ that incorporates parity transformations. It consists of $2d^3$ elements, of which $d^3$ elements belong to the $HW(d)$ subgroup, and extra $d^3$ elements which are related through a Fourier transform with the former ones. It is shown that $HWP(d)$ is a generalised version of the dihedral group. The properties of operators that combine displacements and parity, are discussed. $HWP(d)$ is shown to be a solvable group, and commutators of its elements perform displacement and parity transformations of quantum states, along loops in the discrete phase space.$2d^2$ coherent states related to the $HWP(d)$ group are introduced, which consist of $d^2$ coherent states related to the $HW(d)$ subgroup, and extra $d^2$ coherent states which are related through a Fourier transform with the former ones. In noisy cases, expansion of an arbitrary state in terms of the $2d^2$ coherent states with Bargmann coefficients, is advantageous in comparison to expansion in terms of the $d^2$ coherent states related to $HW(d)$. One of the consequences of the $HWP(d)$ group, is a natural unification of the Wigner and Weyl functions. The properties of the unified Wigner-Weyl function are discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_14820 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Heisenberg-Weyl-parity group its coherent states and a unified Wigner-Weyl function Vourdas, A. Quantum Physics The Heisenberg-Weyl group $HW(d)$ related to a $d$-dimensional Hilbert space $H(d)$, is enlarged into the Heisenberg-Weyl-parity group $HWP(d)$ that incorporates parity transformations. It consists of $2d^3$ elements, of which $d^3$ elements belong to the $HW(d)$ subgroup, and extra $d^3$ elements which are related through a Fourier transform with the former ones. It is shown that $HWP(d)$ is a generalised version of the dihedral group. The properties of operators that combine displacements and parity, are discussed. $HWP(d)$ is shown to be a solvable group, and commutators of its elements perform displacement and parity transformations of quantum states, along loops in the discrete phase space.$2d^2$ coherent states related to the $HWP(d)$ group are introduced, which consist of $d^2$ coherent states related to the $HW(d)$ subgroup, and extra $d^2$ coherent states which are related through a Fourier transform with the former ones. In noisy cases, expansion of an arbitrary state in terms of the $2d^2$ coherent states with Bargmann coefficients, is advantageous in comparison to expansion in terms of the $d^2$ coherent states related to $HW(d)$. One of the consequences of the $HWP(d)$ group, is a natural unification of the Wigner and Weyl functions. The properties of the unified Wigner-Weyl function are discussed. |
| title | The Heisenberg-Weyl-parity group its coherent states and a unified Wigner-Weyl function |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2605.14820 |