Finite Linear Spaces, Plane Geometries, Hilbert spaces and Finite Phase Space

Abstract

Finite plane geometry is associated with finite dimensional Hilbert space. The association allows mapping of q-number Hilbert space observables to the c-number formalism of quantum mechanics in phase space. The mapped entities reflect geometrically based line-point interrelation. Particularly simple formulas are involved when use is made of mutually unbiased bases (MUB) representations for the Hilbert space entries. The geometry specifies a point-line interrelation. Thus underpinning d-dimensional Hilbert space operators (resp. states) with geometrical points leads to operators termed "line operators" underpinned by the geometrical lines. These "line operators", $\hat{L}_j;$ (j designates the line) form a complete orthogonal basis for Hilbert space operators. The representation of Hilbert space operators in terms of these operators form the phase space representation of the d-dimensional Hilbert space. The "line operators" (resp. "line states") are studied in detail. The paper aims at self sufficiency and to this end all relevant notions are explained herewith.