Quantum Fields from Quantum Cellular Automata

It has been shown that quantum walks on a lattice can give rise to relativistic wave equations like the Dirac equation in the long wavelength limit. To go to multiple particles and describe a quantum field theory in discrete spacetime, a quantum cellular automaton (QCA) is a natural choice. But can a QCA, with purely local dynamics, give rise to a fermionic field theory at long wavelengths? A straightforward QCA construction can work in one spatial dimension, but we prove that this same approach cannot succeed in two or three dimensions. To evade this no-go result, we construct a QCA in a different way, confining distinguishable particles in a totally antisymmetrized subspace, and show that this recovers Dirac field theory at long wavelengths. We show that an analogous construction for bosons can produce Maxwell's theory, and discuss the issues in going from free to interacting fields.