  Quantum cellular automata have been recently considered as a fundamental
approach to quantum field theory, resorting to a precise automaton, linear in
the field, for the Dirac equation in one dimension. In such linear case a
quantum automaton is isomorphic to a quantum walk, and a convenient formulation
can be given in terms of transition matrices, leading to a new kind of discrete
path integral that we solve analytically in terms of Jacobi polynomials versus
the arbitrary mass parameter.
