  We consider two dimensional QCD with the spatial dimension compactified to a
circle. We show that the states in the theory consist of interacting strings
that wind around the circle and derive the Hamiltonian for this theory in the
large $N$ limit, complete with interactions. Mapping the winding states into
momentum states, we express this Hamiltonian in terms of a continuous field.
For a $U(N)$ gauge group with a background source of Wilson loops, we recover
the collective field Hamiltonian found by Das and Jevicki for the $c=1$ matrix
model, except the spatial coordinate is on a circle. We then proceed to show
that two dimensional QCD with a $U(N)$ gauge group can be reduced to a one-
dimensional unitary matrix model and is hence equivalent to a theory of $N$
free nonrelativistic fermions on a circle. A similar result is true for the
group $SU(N)$, but the fermions must be modded out by the center of mass
coordinate.
