  If the structure of spacetime is discrete, then Lorentz symmetry should only
be an approximation, valid at long length scales. At finite lattice spacings
there will be small corrections to the Dirac evolution that could in principle
be experimentally detected. In particular, the lattice structure should be
reflected in a modification of the free-particle dispersion relation. We show
that these can produce a surprisingly large phase shift between the two arms of
an asymmetrical interferometer. This method could be employed to test any model
that predicts a direction-dependent dispersion relation. Here, we calculate the
size of this phase shift for a particular model, the 3D quantum walk on the
body-centered cubic lattice, which has been shown to give rise to the Dirac
equation in the continuum limit. Though the details of this model will affect
the size of the shift, its magnitude is set largely by dimensional analysis, so
there is reason to believe that other models would yield similar results. We
find that, with current technology, a modest-sized neutron interferometer could
put strong bounds on the size of the lattice spacing. This discreteness could
possibly be detected even for lattice spacings at the Planck scale by a
suitably scaled-up experiment.
