Quantum mechanics as a matrix symplectic geometry

Abstract

The main purpose of this work is to describe the quantum analog of the usual classical symplectic geometry and then to formulate quantum mechanics as a noncommutative symplectic geometry. First, we describe a discrete Weyl-Schwinger realization of the Heisenberg group and we develop a discrete version of the Weyl-Wigner-Moyal formalism. We also study the continuous limit and the case of higher degrees of freedom. In analogy with the classical case, we present the noncommutative (quantum) symplectic geometry associated with the matrix algebraMN(C) generated by the Schwinger matrices.