Integration of source-charge constraints in quantum chromodynamics with fixed quark and antiquark sources

Abstract

I study the problem of satisfying the source-charge constraints in operator-symmetrized quantum chromodynamics (QCD) with static sources. I show that the color-charge algebras generated by the QCD outer product $P^A\mathrm{}(u,v)=\left(\frac{i}{2}\right)f^{\mathrm{ABC}}(u^B{v}^C+v^C{u}^B)$ can always be put in the form $P^A(w_a,w_b)=i{C}_{\mathrm{abc}}{w}_c$, with structure constants $C_{\mathrm{abc}}$ which are totally antisymmetric, but which do not in general satisfy the Jacobi identity. However, total antisymmetry of the $C$'s is enough for the corresponding overlying classical field equations to be derivable from a Lagrangian and to possess a conserved stress-energy tensor, involving a finite number of undetermined constants of integration. I postulate conditions for determining the integration constants when the sources are in a color-singlet state, and use them to fix the overlying classical theory in the $q\overline{q}$ and the $\mathrm{qqq}\left(\overline{q}\overline{q}\overline{q}\right)$ cases.