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 representation of rotation group $${\mathrm{SO}}(3)$$ , but rather its double cover $${\mathrm{SU}}(2)$$ QW can only be defined globally an extended spacetime where angle extends from 0 to $$4 \pi $$ . The coupling with arbitrary electromagnetic fields also presented. Finally, cylindrical relativistic Landau levels computed explicitly and simulated QW.