On the implementation of constraints through projection operators

Quantum constraints of the type Q \ psi (phys) = 0 can be straightforwardly implemented in cases where I! 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. H-phys = ker Q = ker Q*. It is, however, non-trivial to identify and project onto H-phys when zero is not in the point spectrum but instead is in the continuous spectrum of Q, because then ker Q = 0. Here, we observe that the topology of the underlying Hilbert space can be harmlessly modified, namely, loosely speaking, in the direction perpendicular to the constraint surface. Consequently, e becomes non-self-adjoint, which then allows us to conveniently obtain H-phys as the proper Hilbert subspace H-phys = ker Q* on which one can project as usual. In the simplest case, the necessary change of topology amounts to passing from an L-2 Hilbert space to a Sobolev space.