Geometric Correlation between Dirac Equation and Yang-mills Equation/ Maxwell Equation

TL;DR

This paper explores the geometric relationship between Dirac, Yang-Mills, and Maxwell equations in Minkowski space, introducing hyperbolic units and establishing connections between particle equations and mass relations.

Contribution

It introduces a hyperbolic imaginary unit in Minkowski space and derives a geometric correspondence linking Dirac, Yang-Mills, and Maxwell equations, revealing new insights into particle mass and interactions.

Findings

Derived Klein-Gordon and Schrödinger equations from Dirac equations.

Established geometric relations between rest mass and electromagnetic mass.

Elucidated the connection between Dirac equations and Yang-Mills equations.

Abstract

The problem about geometric correspondence of Dirac particle and contain quality item of Yang-Mills equation has always not been solved.This paper introduced the hyperbolic imaginary unit in Minkowski space, established a classes of Dirac wave equations with t'Hooft matrices.In lightlike region of Minkowski space,we can discuss the hermitian conjugate transformation of Dirac positive particle and antiparticle, find the space-time corresponding points of Dirac particle,and draw Feynman clip-art though the geometrical relation between timelike region and lightlike region.The coupling of motion equation of Dirac positive particle and antiparticle can get Klein-Gordon equation, when it reach classical approximate we can get Schrodinger equation,and this illustrated that p meson or m meson may be composite particle. Using the relation of timelike region and lightlike region in Minkowski momentum space to renormalize the rest mass of particles,we can describe the geometric relation between rest mass and electromagnetic mass of particles. Then, we can elicit the Yang-Mills equation with electromagnetic mass through four Dirac wave equations with the hermitian conjugate transformation relation, and further launch the common forms of Maxwell equations.