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We discuss the canonical quantization of systems formulated on discrete spacetimes. We start by analysing 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 us 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 a Euclidean rotation. The whole discussion is framed in such a way so 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.

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Classical and Quantum Gravity

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