General Form of Color Charge of the Quark

TL;DR

This paper derives the general form of the time-dependent color charges of a quark in SU(3) Yang-Mills theory, addressing the non-conservation of color charge unlike electric charge in Maxwell theory.

Contribution

It introduces a novel derivation of the general form of quark color charges in SU(3) Yang-Mills theory, highlighting their time dependence.

Findings

Derived the general form of eight time-dependent color charges.

Showed that quark color charge is not conserved and varies with time.

Provided mathematical framework for understanding quark color dynamics.

Abstract

In Maxwell theory the constant electric charge e of the electron is consistent with the continuity equation $\partial_\mu j^\mu(x)=0$ where $j^\mu(x)$ is the current density of the electron where the repeated indices $\mu=0,1,2,3$ are summed. However, in Yang-Mills theory the Yang-Mills color current density $j^{\mu a}(x)$ of the quark satisfies the equation $D_\mu[A]j^{\mu a}(x)=0$ which is not a continuity equation ($\partial_\mu j^{\mu a}(x)\neq 0$) which implies that the color charge of the quark is not constant where a=1,2,...,8 are the color indices. Since the charge of a point particle is obtained from the zero ($\mu =0$) component of a corresponding current density by integrating over the entire (physically) allowed volume, the color charge $q^a(t)$ of the quark in Yang-Mills theory is time dependent. In this paper we derive the general form of eight time dependent fundamental color charges $q^a(t)$ of the quark in Yang-Mills theory in SU(3) where a=1,2,...,8.