  We construct lattice gauge theories in which the elements of the link
matrices are represented by non-commuting operators acting in a Hilbert space.
These quantum link models are related to ordinary lattice gauge theories in the
same way as quantum spin models are related to ordinary classical spin systems.
Here U(1) and SU(2) quantum link models are constructed explicitly. As
Hamiltonian theories quantum link models are nonrelativistic gauge theories
with potential applications in condensed matter physics. When formulated with a
fifth Euclidean dimension, universality arguments suggest that dimensional
reduction to four dimensions occurs. Hence, quantum link models are also
reformulations of ordinary quantum field theories and are applicable to
particle physics, for example to QCD. The configuration space of quantum link
models is discrete and hence their numerical treatment should be simpler than
that of ordinary lattice gauge theories with a continuous configuration space.
