Dirac came up with a general theory of constraints, including second-class constraints. To quantize such systems, he first computed the Dirac bracket classically, and only then "promoted" the classical Dirac bracket into a commutator. However, this leads to operator ordering ambiguities over and above what already exists for the Poisson bracket.
Is there any more direct way of coming up with a quantum Dirac bracket and Hamiltonian operator which resolves such ambiguities? If the system is symmetrical, symmetry considerations can often single out the "correct" quantization, but what about asymmetrical systems?
Ideally, the correct quantum prescription ought to match what we get from using path integrals.
Similarly, if the second-class constraint is linear, the quantum commutator and Hamiltonian is also straightforward. However, even relatively simple models can have subtleties. Consider the toy model $$L=\frac{\dot{q}^2}{2} -f[q]\frac{F^2}{2}$$ where F is an auxiliary variable and $f$ is a function of $q$. The classical second-class constraints are given by $p_F \approx 0$ and $fF\approx 0$. Using path integrals as a guide, $$\int \mathcal{D}q\, \mathcal{D}F\, \exp\left( \frac{i\dot{q}^2}{2} - \frac{ifF^2}{2} \right) \propto \int f^{-1/2}\mathcal{D}q\, \exp\left( \frac{i\dot{q}^2}{2} \right),$$ which corresponds to the following operator ordering for the Hamiltonian: $$\widehat{H} = \frac{1}{2} \widehat{f}^{1/4}\widehat{p}\widehat{f}^{-1/2}\widehat{p}\widehat{f}^{1/4}.$$
But what about the generic nonlinear case?
On another note, is there a formalism where we can impose the Dirac bracket after quantization, rather than before?
