  It has long been recognized that lattice gauge theory formulations, when
applied to general relativity, conflict with the invariance of the theory under
diffeomorphisms. Additionally, the traditional lattice field theory approach
consists in fixing the gauge in a Euclidean action, which does not appear
appropriate for general relativity. We analyze discrete lattice general
relativity and develop a canonical formalism that allows to treat constrained
theories in Lorentzian signature space-times. The presence of the lattice
introduces a ``dynamical gauge'' fixing that makes the quantization of the
theories conceptually clear, albeit computationally involved. Among other
issues the problem of a consistent algebra of constraints is automatically
solved in our approach. The approach works successfully in other field theories
as well, including topological theories like BF theory. We discuss a simple
cosmological application that exhibits the quantum elimination of the
singularity at the big bang.
