Noncommutative Coordinate Picture of the Quantum (Phase) Space

We illustrate an isomorphic description of the observable algebra for quantum mechanics in terms of functions on the projective Hilbert space, and its Hilbert space analog, with a noncommutative product with explicit coordinates and discuss the physical and dynamical picture. The isomorphism is then used as a base to essentially translate the differential symplectic geometry of the infinite dimensional manifolds onto the observable algebra as a noncommutative geometry, hence obtaining the latter from the physical theory itself. We have essentially an extended formalism of the Schrodinger versus Heisenberg picture which we try to describe mathematically as a coordinate map from the phase space, which we have presented argument to be seen as the quantum model of the physical space, to the noncommutative geometry as coordinated by the six position and momentum operators. The observable algebra is taken as an algebra of functions on the latter operators. We advocate the idea that the noncommutative geometry can be seen as an alternative, noncommutative coordinate, picture of quantum (phase) space. Issues about the kind of noncommutative geometry obtained are also explored.