  Discrete canonical evolution is a key tool for understanding the dynamics in
discrete models of spacetime, in particular those represented by a triangular
Regge lattice. We consider a finite-dimensional system whose evolution is
realized by a series of discrete-time evolution steps governed by Hamiltonian
equations of motion that are linear in the canonical coordinates. The evolution
is allowed to be irregular, which produces constraints as well as
non-uniqueness of solutions. We provide two independent and fundamentally
different approaches to canonical quantization of this system, including
detailed description of the evolution maps, measurement and path integrals. It
is argued that some irregular discrete systems may be most naturally described
by a non-unitary quantum evolution. The formalism is then applied to a simple
yet physically relevant model of massless scalar field on a two-dimensional
spacetime lattice.
