  In order to better understand quantum field theory we present some toy models
on finite dimensional Hilbert spaces. We discuss how these models converge to a
discrete spacetime version of quantum field theory. We first define toy
fermion, boson and mixed fermion-boson free quantum fields. The free fields are
employed to form what we consider to be the simplest examples of interacting
fields which we then use to construct Hamiltonian densities. Scattering
operators are defined in terms of these densities and are employed to construct
scattering probabilities. Although our definition of a scattering operator is
not the usual "second-quantization" definition we believe it is simpler and
more natural. Emphasis is placed on finding the eigenstructures of toy quantum
fields.
