It has been shown that certain quantum walks give rise to relativistic wave equations, such as the Dirac and Weyl equations, in their long-wavelength limits. This intriguing result raises the question of whether something similar can happen in the multi-particle case. We construct a one-dimensional quantum cellular automaton (QCA) model which matches the quantum walk in the single particle case, and which approaches the quantum field theory of free fermions in the long-wavelength limit. However, we show that this class of constructions does not generalize to higher spatial dimensions in any straightforward way, and that no construction with similar properties is possible in two or more spatial dimensions. This rules out the most common approaches based on QCAs. We suggest possible methods to overcome this barrier while retaining locality.

28 pages, preprint format, small changes from first version. To appear in Physical Review A