Quantum geodesics in quantum mechanics

TL;DR

This paper develops a quantum geometric framework where Schrödinger and Klein-Gordon equations are interpreted as quantum geodesics within a noncommutative differential calculus on extended phase space, linking quantum dynamics to geometric structures.

Contribution

It introduces a noncommutative differential calculus and quantum connection that recasts quantum evolution equations as geodesics, unifying quantum mechanics and geometry.

Findings

Schrödinger's equation as a quantum geodesic on extended phase space

Formulation of Klein-Gordon equation as quantum geodesic with proper time

Application to relativistic particles and atomic systems

Abstract

We show that the standard Heisenberg algebra of quantum mechanics admits a noncommutative differential calculus $\Omega^1$ depending on the Hamiltonian $p^2/2m + V(x)$, and a flat quantum connection $\nabla$ with torsion such that a previous quantum-geometric formulation of flow along autoparallel curves (or `geodesics') is exactly Schr\"odinger's equation. The connection $\nabla$ preserves a generalised `skew metric' given by the canonical symplectic structure lifted to a certain rank (0,2) tensor on the extended phase space where we adjoin a time variable. We also apply the same approach to the Klein Gordon equation on Minkowski spacetime with a background electromagnetic field, formulating quantum `geodesics' on the relativistic Heisenberg algebra with proper time for the external geodesic parameter. Examples include a relativistic free particle wave packet and a hydrogen-like atom.