Quantum Geodesics

TL;DR

This paper extends classical differential geometry to quantum and gravitational fields using a five-dimensional framework, deriving geodesic equations that unify classical and quantum trajectories.

Contribution

It introduces a quantum Kaluza-Klein metric linking Weyl and Kaluza theories with quantum mechanics, establishing a covariant basis for quantum state geodesics.

Findings

Null geodesics describe valid classical and quantum trajectories.

Trajectories are tangent to the probability density four vector.

A covariant framework for quantum geodesic motion is established.

Abstract

Classical methods of differential geometry are used to construct equations of motion for particles in quantum, electrodynamic and gravitational fields. For a five dimensional geometrical system, the equivalence principle can be extended. Local transformations generate the effects of electromagnetic and quantum fields. A combination of five dimensional coordinate transformations and internal conformal transformations leads to a quantum Kaluza-Klein metric. The theories of Weyl and Kaluza can be interrelated when charged particle quantum mechanics is included. Measurements of trajectories are made relative to an observers' space that is defined by the motion of neutral particles. It is shown that a preferred set of null geodesics describe valid classical and quantum trajectories. These are tangent to the probability density four vector. This construction establishes a generally covariant basis for geodesic motion of quantum states.