A geometric picture of quantum mechanics with noncommutative values for observables

We present here what we consider a new picture of quantum mechanics with the position and momentum observables as coordinates of the usual quantum phase space of a single particle, which also serves as the model of the physical space. To minimize the mathematics involved, we stick here to the Hilbert space picture of the phase space. We argue that a quantum observable should be seen as taking noncommutative values each of which can equivalently be represented by an infinite number of real numbers. The six noncommutative values of the position and momentum observables hence serve as an alternative system of coordinates for the Hilbert space. The values can, at least in principle, be experimentally determined. This can be seen as a complete resolution of the Einstein–Bohr debate that Einstein would probably be happy with. What we have is a solid noncommutative geometrical picture of the physical space, or spacetime, beyond the Newtonian and Einsteinian framework that is sure to be relevant to Nature directly coupled with the idea that each physical quantity should better be seen as having a value beyond what can be represented by a number. We finish by sketching some implications of the results for the physics and mathematics of quantum spacetime in general.