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

TL;DR

This paper establishes the existence and uniqueness of a symplectic group representation for phase point operators in discrete quantum phase space, providing a method to explicitly construct it, which is fundamental for quantum mechanics applications.

Contribution

It introduces a novel explicit construction of the projective representation of the symplectic group for discrete phase space operators, filling a key gap in the theoretical framework.

Findings

Proves existence and uniqueness of the symplectic group representation.

Provides a construction method using the Euclidean algorithm.

Enhances understanding of symplectic covariance in discrete quantum systems.

Abstract

The phase point operator $\Delta(q,p)$ is the quantum mechanical counterpart of the classical phase point $(q,p)$. The discrete form of $\Delta(q,p)$ 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.