Weyl-Heisenberg covariant quantization for the discrete torus

2024/12/24 by Murenzi, Romain, Zlotak, Aidan, Gazeau, Jean Pierre
#FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Physics (quant-ph)

paper · doi:10.48550/arxiv.2412.18521

Abstract

Covariant integral quantization is implemented for systems whose phase space is Z_d × 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 × 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 L²(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.

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