The most obvious application of this numerical approach to
quantum field theory is the first-principles study of the physics
of QCD. There are a variety of low energy properties of the
strongly interacting particles, for example mass ratios and
specific weak decay amplitudes, that have been very well measured
for decades and should now be computed using this fundamental
approach. Such calculations provide tests of the underlying
theory and confidence in our control of numerical errors. The
present state-of-the-art in these calculations may be discerned
from the proceedings of the annual lattice meeting
.
Although much progress has been made over the past decade,
present calculations contain systemic errors on perhaps the
5-10% level from finite lattice spacing effects and possibly
uncontrolled errors coming from working with quark masses which
are too heavy and the frequent neglect of the computationally
demanding 
factor in the above statistical
weight. The new machine at Tsukuba, CP-PACS and our new QCDSP
machine represent a 10-100 fold improvement over present
resources and should make significant progress in reducing these
errors.
Perhaps even more important than providing concrete numerical
evidence for the validity of QCD, these numerical methods offer
the possibility of predicting a variety of strong interaction
effects that are either important in their own right or required
to extract fundamental physical parameters from experiment. Two
classic examples are the study of the QCD phase transition and
calculation of the quark masses. Major experimental efforts at
the Brookhaven and CERN accelerator laboratories use heavy ion
collisions to create a very short-lived high temperature region
in which the quark/anti-quark condensate mentioned above, melts,
producing a new, chirally symmetric state of matter. Lattice QCD
calculations are predicting with increasing confidence the
temperature of the transition to this new quark-gluon plasma, its
equation of state and the latent heat of the transition. Because
the quarks are confined, their masses can be computed only
indirectly from the masses of the bound states in which the
quarks appear. These underlying quark masses are of great
fundamental importance, being one of the few intrinsic properties
known of these structure-less particles and possibly holding
clues to their origin in some more fundamental theory. Although a
variety of analytical methods have been developed to compute
these masses, all are inherently uncertain with errors perhaps as
large as 100%. Lattice calculations have already improved on
these results, reducing errors to the 
level.
A third use of this numerical approach to relativistic quantum field theory is more speculative in nature. Given the dominant role played by the non-linear interactions in QCD, it is natural to wonder if other strongly-coupled field theories might also exhibit unusual properties, revealing a dynamics very different from what might be naively guessed based on a simple linearization of the theory. This suggests a type of experimental computational physics where one simulates a variety of potentially interesting theories in the hope of discovering new, possibly useful behavior. By varying some of the elements of QCD, for example the dimension of the Yang Mills group or the number of species of light quarks, we may gain a deeper understanding of the physics of QCD. We may also discover new behavior that could suggest the form for new theories of matter on the next higher scale of energy. At present there is no successful theory explaining the observed families of quarks and leptons, their masses or the pattern of weak, electromagnetic and strong interactions that they experience. New, candidate theories would certainly be of interest.