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 ${P}^{A}(u,v)=(\frac{i}{2}){f}^{\mathrm{ABC}}({u}^{B}{v}^{C}+{v}^{C}{u}^{B})$ 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 $\mathrm{qqq}(\overline{q}\overline{q}\overline{q})$ cases.