Weyl-Heisenberg covariant quantization for the discrete torus

TL;DR

This paper develops a covariant integral quantization method for systems on a discrete phase space modeled by the finite group Z_d, extending Weyl-Heisenberg symmetry to discrete settings and providing phase space representations.

Contribution

It introduces a covariant integral quantization framework for discrete phase spaces using the Weyl-Heisenberg group, with explicit construction and phase space portraits.

Findings

Quantization scheme for discrete phase space Z_d x Z_d

Representation of the Weyl-Heisenberg group on L^2(Z_N)

Explicit phase space portraits of quantized systems

Abstract

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.