Path-integral solution of the one-dimensional Dirac quantum cellular   automaton
  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. 
Giacomo Mauro D'Ariano, Nicola Mosco, Paolo Perinotti, Alessandro   Tosini
quant-ph hep-lat math-ph math.MP