After computing the Hamiltonian constraint and the momentum constraint in general relativity the Hamiltonian constraint is turned into an operator equation and solved in a manner similar to a Schrodinger equation (Wheeler-de Witt equation). However, the momentum constraints are actually second-class and in order to convert the Hamiltonian constraint into a Schrodinger-like equation I believe that the Dirac brackets must be computed first as the commutation relations between the canonical variables might change after the Dirac procedure. But I have not seen this approach taken anywhere? The Wheeler-de Witt equation is used without the Dirac procedure. How is this correct? Can somebody please comment on this?
General Relativity – Dirac Procedure for Wheeler De Witt Equation Explained
constrained-dynamicsgeneral-relativityhamiltonian-formalismquantization
Best Answer
For what it's worth, generically in the bulk (i.e. away from the boundary), there are $4$ primary constraints, $1$ Hamiltonian constraint, and $3$ momentum constraints. These form a Poisson algebra of $4+1+3=8$ first-class constraints. There are no second-class constraints per se in this basic Hamiltonian formulation, cf. Ref. 1.
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