Weyl-Heisenberg covariant quantization for the discrete torus

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.