  Field theories on the plane wave background are considered. We discuss that
for such field theories one can only form 1+1 dimensional freely propagating
wave packets. We analyze tree level four point functions of scalar field theory
as well as scalars coupled to gauge fields in detail and show that these four
point functions are well-behaved so that they can be interpreted as S-matrix
elements for 2 particle $\to$ 2 particle scattering amplitudes. Therefore, at
least classically, field theories on the plane wave background have S-matrix
formulation.
