  A new non-perturbative approach to quantum field theory --- D-theory --- is
proposed, in which continuous classical fields are replaced by discrete
quantized variables which undergo dimensional reduction. The 2-d classical O(3)
model emerges from the (2+1)-d quantum Heisenberg model formulated in terms of
quantum spins. Dimensional reduction is demonstrated explicitly by simulating
correlation lengths up to 350,000 lattice spacings using a loop cluster
algorithm. In the framework of D-theory, gauge theories are formulated in terms
of quantum links --- the gauge analogs of quantum spins. Quantum links are
parallel transporter matrices whose elements are non-commuting operators. They
can be expressed as bilinears of anticommuting fermion constituents. In quantum
link models dimensional reduction to four dimensions occurs, due to the
presence of a 5-d Coulomb phase, whose existence is confirmed by detailed
simulations using standard lattice gauge theory. Using Shamir's variant of
Kaplan's fermion proposal, in quantum link QCD quarks appear as edge states of
a 5-d slab. This naturally protects their chiral symmetries without
fine-tuning. The first efficient cluster algorithm for a gauge theory with a
continuous gauge group is formulated for the U(1) quantum link model. Improved
estimators for Wilson loops are constructed, and dimensional reduction to
ordinary lattice QED is verified numerically.
