Operator ordering for generally covariant systems

An essential aspect of a generally covariant system is the invariance of its action under reparametrizations; this means that the label that parametrizes the trajectories of the system is not the time but a physically irrelevant parameter. As a consequence, the system is constrained to remain on the hypersurface of the phase space where the Hamiltonian is null. In fact, since the “evolution” generated by the Hamiltonian can be regarded as a reparametrization of the classical trajectory, then the Hamiltonian behaves like a generator of a gauge transformation of the system; so the Hamiltonian is a first class constraint. Besides the Hamiltonian constraintHo associated with the reparametrization invariance, the system can exhibit additional gauge invariance generated by first class constraints Ha linear and homogeneous in the momenta, telling that some canonical variables are not genuine degrees of freedom but mere spurious variables devoid of physical meaning. The observables are not sensitive to the values of these spurious variables, nor to the choice of the parametrization. The super-Hamiltonian and super-momenta constraints of General Relativity are an example of such a set of first class constraints.[1,2] According to Dirac’s method, the gauge invariance is preserved at the quantum level by including in the Hilbert space only those states that are