Commutators and the Parametric Nonrelativistic Particle?

[Angryphysicist]

OK, I'm a wee bit sleep deprived and cannot recollect some facts about the Dirac quantization of gauge theories. With the quantization of the parametrized nonrelativistics particle, do we still change the Poisson bracket into commutators?

More specifically, for the non-relativistic particle, would the following hold:
$$i\hbar\dot{x} = [x,H]$$

If not how can I find the velocity operator for the parametrized nonrelativistic particle?

Thanks for all the help!

[edit]: I suppose my real question is: Can I still use the Heisenberg picture with the Dirac Quantization of First Class Constrained gauge systems?

 

[azntoon]

Yes, you can still use the Heisenberg picture with the Dirac quantization of first-class constrained gauge systems. The requirement is that the Poisson bracket of the constraints must be a linear combination of the constraints themselves. In this case, the Poisson bracket should be replaced with the commutator of the constraint operators. However, the velocity operator for the parametrized non-relativistic particle cannot be written in the same way as it would for an unconstrained system, since the acceleration operator is no longer directly related to the Hamiltonian. To find the velocity operator, you will need to use the equations of motion for the system along with the constraint equations.