Simple derivation of the Weyl and Dirac quantum cellular automata

We consider quantum cellular automata on a body-centered cubic lattice and provide a simple derivation of the only two homogenous, local, isotropic, and unitary two-dimensional automata [G. M. D'Ariano and P. Perinotti, Phys. Rev. A 90, 062106 (2014)]. Our derivation relies on the notion of Gram matrix and emphasizes the link between the transition matrices that characterize the automata and the body-centered cubic lattice: The transition matrices essentially are the matrix representation of the vertices of the lattice's primitive cell. As expected, the dynamics of these two automata reduce to the Weyl equation in the limit of small wave vectors and continuous time. We also briefly examine the four-dimensional case, where we find two one-parameter families of automata that reduce to the Dirac equation in a suitable limit.

Figure 1

The primitive cell of the body-centered cubic lattice can be seen as a central vertex surrounded by a tetrahedron and its dual. (a) Graphical representation of the primitive cell of the body-centered cubic lattice with a central vertex and its eight neighboring vertices (thick and dashed vectors). (b) Graphical representation of the tetrahedron generated by the four vertices in $S_{+}$ (thick vectors). (c) Graphical representation of the dual tetrahedron generated by the four vertices in $S_{-}$ (dashed vectors).