Poincaré- and gauge-invariant two-dimensional quantum chromodynamics

Abstract

We present a canonical formulation of two-dimensional quantum chromodynamics in the axial or Coulomb gauge ${A}_{1}^{a}=0$. For consistency with the Lagrange equations of motion, the Hamiltonian must include a nontrivial dynamical background electric field term. This breaks translation invariance in the gauge-noninvariant sector. We argue, however, that for the purpose of calculating gauge-invariant physical quantities one can consider the naive theory defined by the Feynman rules without the background electric field. We show that the naive theory has an anomalous Poincar\'e algebra due to its non-Abelian character; the theory is Lorentz invariant only in the color-singlet sector. Because of this fact the quark propagator has a noncovariant pole, and the $i\ensuremath{\epsilon}$ prescription is different from the naive one. In the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit (where $N$ is the number of colors) we can set up a two-component Bethe-Salpeter equation in the color-singlet sector to determine the spectrum of the theory. The resulting equation has an obvious interpretation in terms of forward- and backward-moving strings, and leads to the same spectrum of bound states as that obtained by 't Hooft in the light-cone gauge.