Equations of motion in General Relativity and Quantum Mechanics

TL;DR

This paper explores the connection between General Relativity and Quantum Mechanics by extending previous work to arbitrary curves, deriving the Dirac equation from geometric principles, and linking geodesic motion to statistical distributions.

Contribution

It extends the relationship between metrics and quantum equations to arbitrary curves, deriving the Dirac equation from geometric and Hamiltonian frameworks.

Findings

Dirac equation derived from Lie derivative of momentum.

Equations of motion linked to Hamilton-Jacobi formalism.

Maxwell-Boltzmann distribution shown to follow from geodesic motion.

Abstract

In a previous article a relationship was established between the linearized metrics of General Relativity associated with geodesics and the Dirac Equation of quantum mechanics. In this paper the extension of that result to arbitrary curves is investigated. The Dirac equation is derived and shown to be related to the Lie derivative of the momentum along the curve. In addition,the equations of motion are derived from the Hamilton-Jacobi equation associated with the metric and the wave equation associated with the Hamiltonian is then shown not to commute with the Dirac operator. Finally, the Maxwell-Boltzmann distribution is shown to be a consequence of geodesic motion.