Equivalence Principle for Quantum Mechanics in the Heisenberg Picture

TL;DR

This paper formulates an exact quantum analog of the weak equivalence principle within the Heisenberg picture, deriving quantum geodesic equations and exploring implications for quantum gravity and measurement.

Contribution

It introduces a covariant quantum geodesic equation framework using Heisenberg equations, emphasizing the role of canonical momentum and quantum observables in a noncommutative spacetime.

Findings

Derived quantum geodesic equations from Heisenberg motion

Identified proper canonical momentum as p_mu in covariant formulation

Proposed a quantum gravity approach based on quantum observables

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.