Noncommutative Coordinate Picture of the Quantum Phase Space

TL;DR

This paper presents a noncommutative geometric framework for quantum phase space by representing the observable algebra as functions on the projective Hilbert space with a noncommutative product, linking symplectic geometry and quantum mechanics.

Contribution

It introduces an isomorphic representation of quantum observable algebra as a noncommutative geometry based on explicit coordinates, extending the Schrödinger and Heisenberg formalisms.

Findings

Establishes a coordinate map from phase space to noncommutative geometry.

Shows the observable algebra as formal functions of position and momentum operators.

Provides a geometric interpretation of quantum mechanics in noncommutative terms.

Abstract

We illustrate an isomorphic representation of the observable algebra for quantum mechanics in terms of the functions on the projective Hilbert space, and its Hilbert space analog, with a noncommutative product in terms of explicit coordinates and discuss the physical and dynamical picture. The isomorphism is then used as a base for the translation of the differential symplectic geometry of the infinite dimensional manifolds onto the observable algebra as a noncommutative geometry. Hence, we obtain the latter from the physical theory itself. We have essentially an extended formalism of the Schrodinger versus Heisenberg picture which we describe mathematically as like a coordinate map from the phase space, for which we have presented argument to be seen as the quantum model of the physical space, to the noncommutative geometry coordinated by the six position and momentum operators. The observable algebra is taken essentially as an algebra of formal functions on the latter operators. The work formulates the intuitive idea that the noncommutative geometry can be seen as an alternative, noncommutative coordinate, picture of familiar quantum phase space, at least so long as the symplectic geometry is concerned.