On the implementation of constraints through projection operators

Quantum constraints of the type Q |ψ phys ⟩ = 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. 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 = ∅. 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, Q 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.