Analytical solutions of the Dirac Quantum Cellular Automata

The thesis focuses on the analytical and the numerical study of the Weyl and Dirac Quantum Cellular Automata (QCAs). A QCA describes the unitary local evolution of a denumerable collection of quantum systems in mutual interaction. The Weyl QCA can be derived under the assumptions of linearity, unitarity, locality, homogeneity and isotropy and one can show that it recovers the relativistic evolution of the Weyl equation in the limit of small wave-vectors. Under these assumptions a QCA can be described in terms of a set of so-called transition matrices. In the specific case of the Weyl automaton the transition matrices generate a semigroup allowing for the explicit computation of the propagator in position space as a sum over paths. The computation of the propagator is simplified by encoding the lattice paths as binary strings, so that the problems reduces to a combinatorial problem for the binary strings. The thesis also addresses the Thirring QCA, featuring an on-site interaction. We provide a starting point for the study of the perturbation theory in this case.