Unitary Representation of Symplectic Group for Phase Point Operators on Discrete Phase Space

The phase point operator $\mathrm{\Delta }(q,p)$ is the quantum mechanical counterpart of the classical phase point $(q,p)$. The discrete form of $\mathrm{\Delta }(q,p)$ was formulated for an odd number of lattice points by Cohendet et al. and for an even number of lattice points by Leonhardt. Both versions have symplectic covariance, which is of fundamental importance in quantum mechanics. However, an explicit form of the projective representation of the symplectic group that appears in the covariance relation is not yet known. We show in this paper the existence and uniqueness of the representation, and describe a method to construct it using the Euclidean algorithm.