  We show that Bose-Einstein condensates in a honeycomb optical lattice are
described by a nonlinear Dirac equation in the long wavelength, mean field
limit. Unlike nonlinear Dirac equations posited by particle theorists, which
are designed to preserve the principle of relativity, i.e., Poincar\'e
covariance, the nonlinear Dirac equation for Bose-Einstein condensates breaks
this symmetry. We present a rigorous derivation of the nonlinear Dirac equation
from first principles. We provide a thorough discussion of all symmetries
broken and maintained.
