Quantum Cellular Automata as Quantum Fields: Interactions, Locality, and Negative Energy States

It has been shown that quantum walks (QWs) on a lattice can give rise to relativistic wave equations like the Dirac equation for fermions and the Klein-Gordon or Maxwell equations for bosons in the long wavelength limit. In the many-particle case, quantum cellular automata (QCAs) can produce free quantum field theories (fermionic or bosonic) in the long-wavelength limit, at the cost of allowing the local subsystems to become high-dimensional. However, adding a local interaction between bosonic and fermionic QCAs reveals a tension between locality (in the strong sense of QCAs) and positive energy: QCAs generically have both positive and negative energy solutions, and local interactions will in general excite both, making the ``vacuum'' unstable to cascades of positive- and negative-energy particle creation. We discuss the source of this conflict, and possible methods to overcome it, as well as the relationship between the strong notion of locality in QCAs and weaker notions in quantum field theories.