Classical algebraic chromodynamics

I develop an extension of the usual equations of $\mathrm{SU}\mathrm{}\left(n\right)\mathrm{}$ chromodynamics which permits the consistent introduction of classical, noncommuting quark source charges. The extension involves adding a singlet gluon, giving a $\mathrm{U}\mathrm{}\left(n\right)\mathrm{}$-based theory with outer product $P^a(u, v)=\left(\frac{1}{2}\right)(d^{\mathrm{abc}}+i{f}^{\mathrm{abc}})(u^b{v}^c-v^b{u}^c)$ which obeys the Jacobi identity, inner product $S(u, v)=\left(\frac{1}{2}\right)(u^a{v}^a+v^a{u}^a)$, and with the $n^2$ gluon fields elevated to algebraic fields over the quark color charge $C^{*}$ algebra. I show that provided the color charge algebra satisfies the condition $S\left(P\right(u, v), w)=S(u, P(v, w\left)\right)$ for all elements $u$, $v$, $w$ of the algebra, all the standard derivations of Lagrangian chromodynamics continue to hold in the algebraic chromodynamics case. I analyze in detail the color charge algebra in the two-particle ($qq, q\overline{q}, \overline{\mathrm{qq}}$) case and show that the above consistency condition is satisfied for the following unique (and, interstingly, asymmetric) choice of quark and antiquark charges: $Q_q^a={\xi }^a$ $Q_{\overline{q}}^a={\overline{\xi }}^a+{\delta }^{a0\mathrm{}}{\left(\frac{n}{2}\right)}^{\frac{3}{2}}1$, with ${\xi }^a{\xi }^b=\left(\frac{1}{2}\right)(d^{\mathrm{abc}}+i{f}^{\mathrm{abc}}){\xi }^c$, ${\overline{\xi }}^a{\overline{\xi }}^b=-\left(\frac{1}{2}\right)(d^{\mathrm{abc}}-i{f}^{\mathrm{abc}}){\overline{\xi }}^c$. The algebraic structure of the two-particle