Interacting quantum fields from quantum cellular automata

Quantum cellular automata (QCAs), consisting of of local subsystems on a lattice that evolve by unitary interactions with their nearest neighbors, can give rise to quantum field theories in the long-wavelength limit; these QFTs can include both fermionic and bosonic free fields. The QCAs are strongly local, depending only on nearest-neighbor interactions. However, adding interactions between QCAs--for example, a fermionic QCA coupled to a bosonic QCA--raises many subtle issues. Systems with time-reversal symmetry will in general exhibit both positive-energy and negative-energy solutions. In free fields these do not mix; but adding interactions can couple positive- and negative-energy solutions, so that (for example) a cascade of particle-antiparticle pairs could spontaneously arise, balanced by the emission of negative-energy bosons. We show that for reasonable types of coupling, a truly local interaction must always allow the excitation of negative-energy states. However, the amplitude for such negative-energy transitions can be suppressed exponentially by increasing the (finite) range of the interaction, to the point where it can be made arbitrarily small. We will demonstrate this behavior in a 1D spatial lattice, and show how to generalize it to 2 and 3 spatial dimensions.