  Describing a particle in an external electromagnetic field is a basic task of
quantum mechanics. The standard scheme for this is known as "minimal coupling",
and consists of replacing the momentum operators in the Hamiltonian by modified
ones with an added vector potential. In lattice systems it is not so clear how
to do this, because there is no continuous translation symmetry, and hence
there are no momenta. Moreover, when time is also discrete, as in quantum walk
systems, there is no Hamiltonian, only a unitary step operator. We present a
unified framework of gauge theory for such discrete systems, keeping a close
analogy to the continuum case. In particular, we show how to implement minimal
coupling in a way that automatically guarantees unitary dynamics. The scheme
works in any lattice dimension, for any number of internal degree of freedom,
for walks that allow jumps to a finite neighborhood rather than to nearest
neighbours, is naturally gauge invariant, and prepares possible extensions to
non-abelian gauge groups.
