Dirac quantum walks with conserved angular momentum

A quantum walk (QW) simulating the flat $$(1 + 2)$$ D Dirac equation on a spatial polar grid is constructed. Because fermions are represented by spinors, which do not constitute a representation of the rotation group $${\mathrm{SO}}(3)$$ , but rather of its double cover $${\mathrm{SU}}(2)$$ , the QW can only be defined globally on an extended spacetime where the polar angle extends from 0 to $$4 \pi $$ . The coupling of the QW with arbitrary electromagnetic fields is also presented. Finally, the cylindrical relativistic Landau levels of the Dirac equation are computed explicitly and simulated by the QW.