  We show that a class of matrix theories can be understood as an extension of
quantum field theory which has non-local interactions. This reformulation is
based on the Wigner-Weyl transformation, and the interactions take the form of
Moyal product on a doubled geometry. We recover local dynamics on the spacetime
as a low-energy limit. This framework opens up the possibility for studying
novel high-energy phenomena, including the unification of gauge and geometric
symmetries in a gauge theory.
