Calculation of Commutation in constraint analysis

Question

During analysis the constraint from a theory, suppose my canonical Hamiltonian is $$H_c=P^A\dot{A}+P^B\dot{B}-L$$ where $P^A=\frac{\partial L}{\partial \dot A}$ and $P^B=\frac{\partial L}{\partial \dot B}$ In this case does the commutation of $[P^A,P^B]=0$?

Answer 1

If there are not second class constraints, then yes. In general, see http://en.wikipedia.org/wiki/Dirac_bracket

The result is $$ \left[P^A,P^B \right]=i\hbar\{P^A,P^B\}_{DB}=-i\hbar\sum _{j,k}\{P^A,\phi_j\}_{PB}\,\left( \{\phi_j,\phi_k\}_{PB}\right)^{-1}\,\{\phi _k,P^B\}_{PB} $$ where $\phi _i$ is a second class constraint, and DB and PB stand for Dirac and Poisson brackets, respectively.