  We present a first-quantized formulation of the quadratic non-commutative
field theory in the background of abelian (gauge) field. Even in this simple
case the Hamiltonian of a propagating particle depends non-trivially on the
momentum (since external fields depend on location of the Landau orbit) so that
one can not integrate out momentum to obtain a local theory in the second order
formalism. The cases of scalar and spinning particles are considered. A
representation for exact propagators is found and the result is applied to
description of the Schwinger-type processes (pair-production in homogenous
external field).
