  A class of fermionic quantum field theories with interactions is shown to be
equivalent to probabilistic cellular automata, namely cellular automata with a
probability distribution for the initial states. Probabilistic cellular
automata on a one-dimensional lattice are equivalent to two - dimensional
quantum field theories for fermions. They can be viewed as generalized Ising
models on a square lattice and therefore as classical statistical systems. As
quantum field theories they are quantum systems. Thus quantum mechanics emerges
from classical statistics. As an explicit example for an interacting fermionic
quantum field theory we describe a type of discretized Thirring model as a
cellular automaton. The updating rule of the automaton is encoded in the step
evolution operator that can be expressed in terms of fermionic annihilation and
creation operators. The complex structure of quantum mechanics is associated to
particle -- hole transformations. The naive continuum limit exhibits Lorentz
symmetry. We exploit the equivalence to quantum field theory in order to show
how quantum concepts as wave functions, density matrix, non-commuting operators
for observables and similarity transformations are convenient and useful
concepts for the description of probabilistic cellular automata.
