  In constrast to discretized space-time approximations to continuum quantum
field theories, discretized velocity space approximations to continuum quantum
field theories are investigated. A four-momentum operator is given in terms of
bare fermion-antifermion-boson creation and annihilation operators with
discrete indices. In continuum quantum field theories the fermion-antifermion
creation and annihilation operators appear as bilinears in the four-momentum
operator and generate a unitary algebra. When the number of modes range over
only a finite number of values, the algebra is that associated with the Lie
algebra of U(2N). By keeping N finite (but arbitrary) problems due to an
infinite Lorentz volume and to the creation of infinite numbers of bare
fermion-antifermion pairs are avoided. But even with a finite number of modes,
it is still possible to create an infinite number of bare bosons. We show how
the full boson algebra arises as the contraction limit of another unitary
algebra that restricts the number of bare bosons in any mode to be finite.
Generic properties of finite mode Hamiltonians are investigated, as are several
simple models to see the rate of convergence of the boson contraction; the
possibility of fine tuning the bare strong coupling constant is also briefly
discussed.
