Weyl-Heisenberg covariant quantization for the discrete torus

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 $Z_d=\{0,1,\dots d-1$ 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.

27 pages, 3 figures