  By studying the pure Yang-Mills theory on a circle, as well as an adjoint
scalar coupled to the gauge field on a circle, we propose a quenching
prescription in which the combination of the spatial component of the gauge
field and $P$ is treated as a dynamic variable. Averaging over momentum is not
necessary, therefore the usual ultraviolet cut-off is eliminated. We then apply
this prescription to study the large $N$ two dimensional supersymmetric gauge
theory. An one dimensional supersymmetric matrix model is obtained. It is not
known whether this model can be solved exactly. However, an extended model with
one more complex fermion is exactly solvable, with $N=1$ supersymmetry as
Parisi-Sourlas supersymmetry. The exact solvability may have some implications
for the $N=1$ quenched model.
