General Form of the Color Potential Produced by Color Charges of the Quark

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Constant electric charge e satisfies the continuity equation ∂μjμ(x)= 0 where jμ(x) is the current density of the electron. However, the Yang-Mills color current density jμᵃ(x) of the quark satisfies the equation Dμ[A] jμᵃ(x)= 0 which is not a continuity equation (∂μjμᵃ(x)≠ 0) which implies that a color charge qᵃ(t) of the quark is not constant but it is time dependent where a=1,2,...8 are color indices. In this paper we derive general form of color potential produced by color charges of the quark. We find that the general form of the color potential produced by the color charges of the quark at rest is given by Φᵃ(x) =A₀ᵃ(t, x) =qᵇ(t-r/c)/r exp[g∫ dr Q(t-r/c)/r] -1g ∫ dr Q(t-r/c)/r ab where dr integration is an indefinite integration, Qab(τ₀)=fabdqᵈ(τ₀), r=| x- X(τ₀)|, τ₀=t-r/c is the retarded time, c is the speed of light, X(τ₀) is the position of the quark at the retarded time and the repeated color indices b,d(=1,2,...8) are summed. For constant color charge qᵃ we reproduce the Coulomb-like potential Φᵃ(x)=qᵃ/r which is consistent with the Maxwell theory where constant electric charge e produces the Coulomb potential Φ(x)=e/r.