We study bound states of Abelian gauge theory in $D=1+1$ dimensions using an equal-time, Poincaré-covariant framework. The normalization of the linear confining potential is determined by a boundary condition in the solution of Gauss’ law for the instantaneous $A^0$ field. As in the case of the Dirac equation, the norm of the relativistic fermion-antifermion ($f\overline{f}$) wave functions gives inclusive particle densities. However, while the Dirac spectrum is known to be continuous we find that regular $f\overline{f}$ solutions exist only for discrete bound-state masses. The $f\overline{f}$ wave functions are consistent with the parton picture when the kinetic energy of the fermions is large compared to the binding potential. We verify that the electromagnetic form factors of the bound states are gauge invariant and calculate the parton distributions from the transition form factors in the Bjorken limit. For relativistic states we find a large sea contribution at low $x_{Bj}$. Since the potential is independent of the gauge coupling the bound states may serve as “Born terms” in a perturbative expansion, in analogy to the usual plane wave in and out states.

• Received 3 January 2013

DOI:https://doi.org/10.1103/PhysRevD.87.065021