  We consider $\text{SU}(2)$ quantum chromodynamics in $1+1$ dimensions with a
single quark in the spin $J$ representation of the gauge group and study the
theory in the large $J$ limit where the gauge coupling $g^2 \to 0$ and $J \to
\infty$ with $\lambda = g^2 J^2$ fixed. We work with a Dirac spinor field for
arbitrary $J$, and with a Majorana spinor for integer $J$ since the integer
spin representations of $\text{SU}(2)$ are real, and analyse the two cases
separately.
  The theory is reformulated in terms of global color non-singlet fermion
bilocal operators which satisfy a $W_\infty \times \text{U}(2J+1)$ algebra. In
the large $J$ limit, the dynamics of the bilocal fields is captured by
fluctuations along a particular coadjoint orbit of the $W_\infty$ algebra. We
show that the global colour-singlet sector of the bilocal field fluctuations
satisfy the same integral equation for meson wavefunctions that appears in the
't Hooft model. For Majorana spinors in the integer spin $J$ representation,
the Majorana condition projects out half of the meson spectrum, as a result of
which the linear spacing of the asymptotic meson spectrum for Majorana fermions
is double that of Dirac fermions. The Majorana condition also projects out the
zero mass bound state that is present for the Dirac quark at zero quark mass.
  We also consider the formulation of the model in terms of local charge
densities and compute the quark spectral function in the large $J$ limit: we
see evidence for the absence of a pole in the quark propagator.
