Constraint quantization of parametrized relativistic gauge systems in curved spacetimes

The Dirac constraint quantization of a finite-dimensional relativistic gauge system with a quadratic super-Hamiltonian and linear supermomenta is investigated as a model for quantizing generally covariant field theories (such as the Einstein theory of gravitation). It is shown that the constraints can be geometrically factor ordered in such a way that their commutators do not produce more constraints. The ensuing quantum theory is invariant under all relevant transformations of the classical theory (point transformations in phase space, mixing of the supermomentum constraints, their adjoinment to the super-Hamiltonian, and scaling of the super-Hamiltonian). Moreover, it yields the same results—namely, the Klein-Gordon equation and the associated (indefinite) inner product—as those obtained by first eliminating the gauge degrees of freedom and then quantizing the ensuing physical theory.