Quantum equations of motion and the geometrical imperative II: relativistic

TL;DR

This paper derives the Dirac equation for electrons using a novel approach based on extracting the square root of the Minkowski metric with Dirac matrices, linking geometry and relativistic quantum mechanics.

Contribution

It introduces a new method to derive the Dirac equation by using a square root of the Minkowski metric, connecting geometric operators with relativistic quantum equations.

Findings

Derivation of the Dirac equation from geometric principles

Operator acts on Dirac bi-spinors and transforms vectors between observers

Provides a new perspective on relativistic quantum mechanics

Abstract

We extract the square root of the Minkowski metric using Dirac/Clifford matrices. The resulting $4\times 4$ operator $d{\bf S}$ that represents the square root, can be used to transform four vectors between relatively moving observers. This effects the usual Lorentz transformation. In addition it acts on a Dirac bi-spinor. The operator is essentially a Hamiltonian that can be used to write an equation of motion for a relativistic spinor. This turns out to be the Dirac equation for electrons in standard form. We believe that is is a new approach to familiar results.