Relativistic wave equations from quantum walks

Quantum walks are unitary analogues of classical random walks. We examine the quantum walk on the 3D body-centered lattice, and show that a set of natural symmetry assumptions lead, in the long wavelength limit, to its wave functions becoming solutions to the Dirac equation. These assumptions require at least a four-dimensional internal space. Taking this as a model of a particle propagating in discrete spacetime, we show that the discreteness could be detected in non-parallel matter interferometers. We also look at the problem of generalizing to the many-body case, by replacing the quantum walk with a quantum cellular automaton that gives the same evolution as the quantum walk for the single-particle sector. This automaton approaches a quantum field theory in the long-wavelength limit.