Scale Invariance at the Edge

Some aspects of the theory of fermions living on three dimensional spacetime with a flat co-dimension one boundary are discussed, particularly a case where the boundary condition preserves scale and translation invariance but violates the residual Lorentz and conformal symmetries. This case is interesting because the Dirac equation has normalizable stationary edge states which must be taken into account in the quantization. We show that a consequence of the edge states for quantization of the Dirac field is that there are no states of the Dirac field theory which are simultaneously scale, C and P invariant even when these are good symmetries of the theory. The scale invariant states of the Dirac field contain either a nonzero scale covariant momentum density or a U(1) charge density concentrated near the edge.