General Relativity – Dirac Procedure for Wheeler De Witt Equation Explained

constrained-dynamicsgeneral-relativityhamiltonian-formalismquantization

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?

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.

References:

  1. K. Sundermeyer, Constrained Dynamics, Lecture Notes in Physics 169, 1982; section IX.2 eq. (IX.2.13).
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