Quantum mechanics in curved space(time) with a noncommutative geometric perspective

Abstract

We have previously presented a version of the Weak Equivalence Principle for a quantum particle as an exact analog of the classical case, based on the Heisenberg picture analysis of free particle motion. Here, we take that to a full formalism of quantum mechanics in a generic curved space(time). Our basic perspective is to take seriously the noncommutative symplectic geometry corresponding to the quantum observable algebra. Particle position coordinate transformations and a nontrivial metric assigning an invariant inner product to vectors, and covectors, are implemented accordingly. That allows an analog to the classical picture of the phase space as the cotangent bundle. The mass-independent quantum geodesic equations as equations of free particle motion under a generic metric as a quantum observable are obtained from an invariant Hamiltonian. Hermiticity of momentum observables is to be taken as reference frame dependent. Our results have a big contrast to the alternative obtained based on the Schrödinger wavefunction representation which we argue to be less appealing. Hence, the work points to a very different approach to quantum gravity, plausibly with a quantum Einstein equation suggested.