  Any local relativistic quantum field theory of Dirac-Weyl fermions conserves
CPT. Here we examine whether a simple nonlocal field theory can violate CPT. We
construct a new relativistic field theory of fermions, which we call
``homeotic'', which is nonlocal but causal and Lorentz invariant. The free
homeotic theory is in fact equivalent to free Dirac theory. We show that a
homeotic theory with a suitable nonlocal four-fermion interaction is causal and
as a result has a well-defined perturbative S-matrix. By coupling a
right-handed homeotic fermion to a left-handed Dirac-Weyl fermion, we obtain a
causal theory of CPT-violating neutrino oscillations.
