Equivalence Principle for Quantum Mechanics in the Heisenberg Picture

Abstract: We present an exact quantum observable analog of the weak equivalence principle for a `relativistic' quantum particle. The quantum geodesic equations are obtained from Heisenberg equations of motion as an exact analog of a fully covariant classical Hamiltonian evolution picture, with the proper identification of the canonical momentum variables as $p_\mu$, rather than $p^\mu$. We discuss the meaning of the equations in relation to projective measurements as well as equations with solution curves as ones in the noncommutative geometric picture of spacetime, and a plausible approach to quantum gravity as a theory about quantum observables as physical quantities including the notion of quantum coordinate transformation.