Boundary conditions for a massless Dirac fermion in 2+1 dimensions where the
space is a half-plane are discussed in detail. It is argued that linear
boundary conditions that leave the Hamiltonian Hermitian generically break $C$
$P$ and $T$ symmetries as well as Lorentz and conformal symmetry. We show that
there is essentially one special case where a single species of fermion has
$CPT$ and the full Poincare and conformal symmetry of the boundary. We show
that, with doubled fermions, there is a second special case which respects
$CPT$ but still violates Lorentz and conformal symmetry. This second special
case is essentially the unique boundary condition where the Dirac operator has
fermion zero mode edge states. We discuss how the edge states lead to exotic
representations of scale, phase and translation symmetries and how imposing a
symmetry requirement leads to edge ferromagnetism of the system. We prove that
the exotic ferromagnetic representations are indeed carried by the ground
states of the system perturbed by a class of interaction Hamiltonians which
includes the non-relativistic Coulomb interaction.
