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 pro
jecting 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 poi
nt spectrum but instead is in the continuous spectrum of Q, because then ke
r Q = 0.
Here, we observe that the topology of the underlying Hilbert space can be h
armlessly modified, namely, loosely speaking, in the direction perpendicula
r to the constraint surface. Consequently, e becomes non-self-adjoint, whic
h then allows us to conveniently obtain H-phys as the proper Hilbert subspa
ce 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 sp
ace to a Sobolev space.