  We develop a new systematic approach to quantum field theory that is designed
to lead to physical states that rapidly converge in an expansion in
free-particle Fock-space sectors. To make this possible, we use light-front
field theory to isolate vacuum effects, and we place a smooth cutoff on the
Hamiltonian to force its free-state matrix elements to quickly decrease as the
difference of the free masses of the states increases. The cutoff violates a
number of physical principles of light-front field theory, including Lorentz
covariance and gauge covariance. This means that the operators in the
Hamiltonian are not required to respect these physical principles. However, by
requiring the Hamiltonian to produce cutoff-independent physical quantities and
by requiring it to respect the unviolated physical principles of the theory, we
are able to derive recursion relations that define the Hamiltonian to all
orders in perturbation theory in terms of the fundamental parameters of the
field theory. We present two applications of this method. First we work in
massless phi-cubed theory in six dimensions. We derive the recursion relations
that determine the Hamiltonian and demonstrate how they are used by computing
and analyzing some of its second- and third-order matrix elements. Then we
apply our method to pure-glue quantum chromodynamics. After deriving the
recursion relations for this theory, we use them to calculate to second order
the part of the Hamiltonian that is required to compute the spectrum. We
diagonalize the Hamiltonian using basis-function expansions for the gluons'
color, spin, and momentum degrees of freedom. We analyze our results for the
spectrum, compare them to recent lattice results, and discuss the various
sources of error in our calculation.
