Phase space methods for finite quantum systems

Finite quantum systems are considered and the Heisenberg-Weyl group associated with discrete displacements in the Z(d) × Z(d) phase space is studied. The SL(2, Z(d)) group associated with discrete Bogoliubov transformations is also considered, and the corresponding operators are explicitly constructed. When d is a power of a prime, the Z(d) is a Galois field and stronger results can be proved.