Derivation of a quark-confinement equation in the Hamiltonian formalism of gauge field theories

Abstract

Using the Hamiltonian formalism of gauge field theories I derive a quark-confining wave equation for a gauge-invariant amplitude describing a system of a quark and an antiquark connected by a linear electric flux. I obtain an exact potential in the form of the Brillouin-Wigner series, which I make finite and well defined by introducing a finite radius to the tube of the electric flux as a dynamical parameter, to be determined from the stability of the solution. The potential is shown to vanish at the origin and become linear at a large distance. The confinement solution is compared with the normal positronium solution to the wave equation with the Coulomb potential, and the latter is shown the stabler of the two for $\frac{g_r^{}{}_{}{}^2}{4\pi }<2\mathrm{}$ where $g_r$ is the renormalized coupling constant. For $\frac{g_r^{}{}_{}{}^2}{4\pi }>2\mathrm{}$, the confinement solution is the only possible one. Essential differences between Abelian and non-Abelian gauge fields are pointed out. A possibility of $g_r$ being replaced by $g_{\mathrm{eff}}$ in the sense of asymptotic freedom is pointed out also.