Abstract
We obtain the exact nonperturbative solution of a scalar field theory defined on a space with noncommuting position and momentum coordinates. The model describes nonlocally interacting charged particles in a background magnetic field. It is an exactly solvable quantum field theory which has nontrivial interactions only when it is defined with a finite ultraviolet cutoff. We propose that small perturbations of this theory can produce solvable models with renormalizable interactions.
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