  We discuss the canonical quantization of systems formulated on discrete
space-times. We start by analyzing the quantization of simple mechanical
systems with discrete time. The quantization becomes challenging when the
systems have anholonomic constraints. We propose a new canonical formulation
and quantization for such systems in terms of discrete canonical
transformations. This allows to construct, for the first time, a canonical
formulation for general constrained mechanical systems with discrete time. We
extend the analysis to gauge field theories on the lattice. We consider a
complete canonical formulation, starting from a discrete action, for lattice
Yang--Mills theory discretized in space and Maxwell theory discretized in space
and time. After completing the treatment, the results can be shown to coincide
with the results of the traditional transfer matrix method. We then apply the
method to BF theory, yielding the first lattice treatment for such a theory
ever. The framework presented deals directly with the Lorentzian signature
without requiring an Euclidean rotation. The whole discussion is framed in such
a way as to provide a formalism that would allow a consistent, well defined,
canonical formulation and quantization of discrete general relativity, which we
will discuss in a forthcoming paper.
