On the Implementation of Constraints through Projection Operators

Quantum constraints of the typeQ|ψphys〉 = 0 can be straightforwardly implemented in cases where Q is a self-adjoint operator for which zero is an eigenvalue. In that case, the physical Hilbert space is obtained by projecting onto the kernel of Q, i.e. Hphys = ker Q = ker Q . It is, however, nontrivial to identify and project onto Hphys when zero is not in the point spectrum but instead is in the continuous spectrum of Q, because then ker Q = ∅. Here, we observe that the topology of the underlying Hilbert space can be harmlessly modified in the direction perpendicular to the constraint surface in such a way that Q becomes non-self-adjoint. This procedure then allows us to conveniently obtain Hphys as the proper Hilbert subspace Hphys = ker Q ∗ on which one can project as usual. In the simplest case, the necessary change of topology amounts to passing from an L2 Hilbert space to a Sobolev space.