This work builds on an existing model of discrete canonical evolution and applies it to the case of a linear dynamical system, i.e., a finite-dimensional system with vector configuration space and linear equations of motion. The system is assumed to evolve in discrete time steps. The most distinctive feature of the model is that the equations of motion can be irregular. After an analysis of the arising constraints and the symplectic form, we introduce adjusted coordinates on the phase space, which uncover its internal structure and result in a trivial form of the Hamiltonian evolution map. For illustration, the formalism is applied to the example of a massless scalar field on a two-dimensional spacetime lattice.

The classical spacetime is usually described by a differentiable manifold with infinitely many degrees of freedom. Occasionally though, it is useful to consider an approximation whose number of degrees of freedom is finite. There are several discrete models of spacetime like that, some of which have been used to build a (simplified) representation of a black hole. We will shortly revisit these discrete black hole models. Then we limit ourselves to one particular case and show how it can be inhabited by quantum matter fields. It is suggested that the field dynamics should be described by the framework of discrete canonical evolution, and we point out some of the most significant implications of this approach.