Path-integral solution of the one-dimensional Dirac quantum cellular automaton

•Discrete path integral formulation of linear QCAs in terms of transition matrices.•Derivation of the analytical solution for the one dimensional Dirac QCA.•Solution given in terms of Jacobi polynomials versus the arbitrary mass parameter.•The discrete paths and the transition matrices of the Dirac QCA are binary encoded.•Paths are grouped in equivalence classes according to their overall transition matrix.

Quantum cellular automata, which describe the discrete and exactly causal unitary evolution of a lattice of quantum systems, have been recently considered as a fundamental approach to quantum field theory and a linear automaton for the Dirac equation in one dimension has been derived. In the linear case a quantum cellular automaton is isomorphic to a quantum walk and its evolution is conveniently formulated in terms of transition matrices. The semigroup structure of the matrices leads to a new kind of discrete path-integral, different from the well known Feynman checkerboard one, that is solved analytically in terms of Jacobi polynomials of the arbitrary mass parameter.