DTQWs are very simple models of quantum transport on a spacetime lattice which are unitary and ultralocal (the evolution operator is strictly local on the spatial lattice, i.e., there is an in-built finite speed of propagation). In the limit to the continuum, they reproduce classical fermionic propagation (Dirac equation), which can be coupled to external Yang-Mills gauge fields (both Abelian and non-Abelian), and external relativistic gravitational ones (curved spacetime). The classical dynamics for the Yang-Mills gauge fields is under study; preliminary results exist, both in the Abelian and in the non-Abelian case. Action principles for these DTQWs, both in flat and curved discrete spacetimes, have been suggested. The connections between the lattice models of field theory that DTQWs are, and standard LGT, have started to be studied in detail recently. When it comes to quantising the fields, the free case is well-known, and corresponds to Quantum Cellular Automata (QCA). A first interacting (1+1)D QCA has been proposed, which reproduces phenomenology of the Schwinger model of quantum electrodynamics. A recent work shows how putting temporal noise on the internal-space parameters of the DTQW yields, in the continuum limit, via entanglement to the position, a very simple model of quantum relativistic diffusion.
Daniel G. Figueroa