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.