We construct an approximation to field theories on the noncommutative torus
based on soliton projections and partial isometries which together form a
matrix algebra of functions on the sum of two circles. The matrix quantum
mechanics is applied to the perturbative dynamics of scalar field theory, to
tachyon dynamics in string field theory, and to the Hamiltonian dynamics of
noncommutative gauge theory in two dimensions. We also describe the adiabatic
dynamics of solitons on the noncommutative torus and compare various classes of
noncommutative solitons on the torus and the plane.Comment: 70 pages, 4 figures; v2: References added and update