Extended Cahill-Glauber formalism for finite dimensional spaces: I. Fundamentals

The Cahill-Glauber approach for quantum mechanics on phase-space is extended to the finite dimensional case through the use of discrete coherent states. All properties and features of the continuous formalism are appropriately generalized. The continuum results are promptly recovered as a limiting case. The Jacobi Theta functions are shown to have a prominent role in the context.