  An Ising-type classical statistical model is shown to describe quantum
fermions. For a suitable time-evolution law for the probability distribution of
the Ising-spins our model describes a quantum field theory for Dirac spinors in
external electromagnetic fields, corresponding to a mean field approximation to
quantum electrodynamics. All quantum features for the motion of an arbitrary
number of electrons and positrons, including the characteristic interference
effects for two-fermion states, are described by the classical statistical
model. For one-particle states in the non-relativistic approximation we derive
the Schr\"odinger equation for a particle in a potential from the time
evolution law for the probability distribution of the Ising-spins. Thus all
characteristic quantum features, as interference in a double slit experiment,
tunneling or discrete energy levels for stationary states, are derived from a
classical statistical ensemble. Concerning the particle-wave-duality of quantum
mechanics, the discreteness of particles is traced back to the discreteness of
occupation numbers or Ising-spins, while the continuity of the wave function
reflects the continuity of the probability distribution for the Ising-spins.
