Dirac procedure for Wheeler De Witt equation

Question

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?

Answer 1

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).

Answer 2

As stated in another response, I think this comes down to a flawed premise, specifically that "the momentum constraints are second-class". They are not. While they are "secondary constraints" to borrow the old verbiage (i.e. they are created on the second iteration of the Dirac procedure), but they are definitely first-class constraints, not second-class.

Note on the physical degrees of freedom of gravity: If they weren't, the physical degrees of freedom of gravitational waves (2) would come out wrong. It has 2 physical degrees of freedom in 4-D (this is a bit backwards, but for reference this is what allows the graviton to sit inside of the massless spin-2 unitary representation of the Poincare group).

Note on seeing this via the ADM formalism: While there are many ways of seeing this, let's take the Dirac procedure you're working with now. This counting comes from the fact that there are initially 20 phase space degrees of freedom (10 for spacetime metric and also 10 for their conjugate momenta) and 8 first-class constraints, which results in $20 - 2\times 8 = 4$ phase space degrees of freedom or 2 physical degrees of freedom (we conventionally count in configuration space, so whatever the remaining phase space is divided by 2). But if it was 2 first-class constraints (-4 phase space) and 6 second-class constraints (-6 phase space), you'd end up with 10 phase space dimensions or 5 physical degrees of freedom. (Which is what happens for massive gravity/massive gravitons, but that's a different story. To answer another question that cropped up --like all gauge systems, matter couplings must not introduce new degrees of freedom because changing the number of degrees of freedom, e.g. breaking first-class constraints into second-class constraints means that you broke the gauge symmetry. But this is exactly what happens in massive theories.)

The first round of the Dirac analysis starts because lapse and the shift lack a conjugate momenta, i.e. the primary constraints are the absence of the momenta (4 constraints) and so its a constrained Hamiltonian system. In E&M, this is analogous to the voltage $A_0$ lacking a conjugate momenta, which sets it up as a constrained system and was why Dirac created this procedure.

Anyways, after the second round of the Dirac procedure, you get your secondary constraints, which are the ones that correspond to conservation of stress-energy/momentum (Fun fact: the secondary first-class constraints are typically the generators of your gauge symmetry). Doing this by hand is obviously as good deal of work. If you do the work though, you will see that all 8 of those constraints are first-class (i.e. commute under the Poisson bracket), and critically this ensures the correct degrees of freedom of the graviton/gravitational wave.

Note on implications for Wheeler-de Witt: Whilst technically there are 7 other constraints in the "full" Hamiltonian, I think these are "safely" ignored. In any case, there are more complex ways of quantizing the theory, making all of the constraints manifest via Batalin-Vilkovisky formalism and the Zinn-Justin master equation. These are needed for perturbative calculations that extend past a semi-classical approximation. But there's other interesting questions, too, like if we really should be generating conjugate momenta via the (time coordinate) Lie derivative of the fields. This gets challenged if you try to, for instance, quantize in the Einstein-Cartan formalism which is necessary for coupling GR to fermions.