Detection of discrete spacetime by matter interferometry

Abstract: 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.