  In a series of recent papers it has been shown how free quantum field theory
can be derived without using mechanical primitives (including space-time,
special relativity, quantization rules, etc.), but only considering the easiest
quantum algorithm encompassing a countable set of quantum systems whose network
of interactions satisfies the simple principles of unitarity, homogeneity,
locality, and isotropy. This has opened the route to extending the axiomatic
information-theoretic derivation of the quantum theory of abstract systems to
include quantum field theory. The inherent discrete nature of the informational
axiomatization leads to an extension of quantum field theory to a quantum
cellular automata theory, where the usual field theory is recovered in a regime
where the discrete structure of the automata cannot be probed. A simple
heuristic argument sets the scale of discreteness to the Planck scale, and the
customary physical regime where discreteness is not visible is the relativistic
one of small wavevectors. In this paper we provide a thorough derivation from
principles that in the most general case the graph of the quantum cellular
automaton is the Cayley graph of a finitely presented group, and showing how
for the case corresponding to Euclidean emergent space (where the group resorts
to an Abelian one) the automata leads to Weyl, Dirac and Maxwell field dynamics
in the relativistic limit. We conclude with some perspectives towards the more
general scenario of non-linear automata for interacting quantum field theory.
