There is an argument which is often put forward against the
requirement of self-adjointness of the hamiltonian constraint
H
: Let
H
be self-adjoint and
O
be any operator of the form
O
=[H,
A], where
A
is any operator (many operators that we do not expect could
vanish have this form). Then the expectation value of
O
vanishes on physical states
from
. The mistake in this argument is easily detected by replacing
H
with a nonrelativistic free particle quantum hamiltonian,
A
with
x, and
with an eigenstate of the momentum: The error is to neglect the
infinities generated by the use of generalized states.