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arXiv:2210.04263 · International Journal of Theoretical Physics

Complete Set of Unitary Irreps of Discrete Heisenberg Group $$ HW_{2^{s}}$$

Emmanuel G. Floratos, I. Tsohantjis

Published Jan 18, 2025• 1 citation
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Abstract

Abstract Following the method of induced group representations of Wigner-Mackay, the explicit construction of all the unitary irreducible representations of the discrete finite Heisenberg-Weyl group $$HW_{2^{s}}$$ H W 2 s over the discrete phase space lattice $$\mathbb {Z}_{2^{s}}\otimes $$ Z 2 s ⊗ $$\mathbb {Z}_{2^{s}}$$ Z 2 s is presented. We explicitly determine their characters and their fusion rules. We discuss possible physical applications for finite quantum mechanics and quantum computation.

DOI
10.1007/s10773-025-05888-9

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Abstract Following the method of induced group representations of Wigner-Mackay, the explicit construction of all the unitary irreducible representations of the discrete finite Heisenberg-Weyl group HW2sHW_{2^{s}}HW2s​ H W 2 s over the discrete phase space lattice Z2s⊗\mathbb {Z}_{2^{s}}\otimes Z2s​⊗ Z 2 s ⊗ Z2s\mathbb {Z}_{2^{s}}Z2s​ Z 2 s is presented. We explicitly determine their characters and their fusion rules. We discuss possible physical applications for finite quantum mechanics and quantum computation.
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Abstract We present a self-consistent theoretical framework for finite-dimensional discrete phase spaces that leads us to establish a well-grounded mapping scheme between Schwinger unitary operators and generators of the special unitary group . This general mathematical construction provides a sound pathway to the formulation of a genuinely discrete Wigner function for arbitrary quantum systems described by finite-dimensional state vector spaces. To illustrate our results, we obtain a general discrete Wigner function for the group and apply this to the study of a particular three-level system. Moreover, we also discuss possible extensions to the discrete Husimi and Glauber–Sudarshan functions, as well as future investigations on multipartite quantum states.
In this work, the version of Wigner transforms and Wigner functions on discrete systems are formulated. Using frame quantization, the star-product of phase space functions are also computed for a particular finite group. Specifically, the Wigner transform, Wigner function and star-product are computed for the dicyclic group Dien. Frames are the discrete version of coherent states. In the literature, quasiprobability distributions are essential in the study of the phase space representations of quantum mechanics, and are naturally associated to frame quantizations. For this work, the unitary irreducible representations of a finite group play an essential role in the construction.
A classical theorem of Stone and von Neumann says that the Schrödinger representation is, up to unitary equivalences, the only irreducible representation of the Heisenberg group on the Hilbert space of square-integrable functions on configuration space. Using the Wigner-Moyal transform we construct an irreducible representation of the Heisenberg group on a certain Hilbert space of square-integrable functions defined on phase space. This allows us to extend the usual Weyl calculus into a phase-space calculus and leads us to a quantum mechanics in phase space, equivalent to standard quantum mechanics. We also briefly discuss the extension of metaplectic operators to phase space and the probabilistic interpretation of the solutions of the phase space Schrödinger equation