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3.9 The Wheeler-DeWitt equation

The approaches to quantization of Sections  3.1, 3.2, 3.3, 3.4, and  3.5 share an important feature: All are “reduced phase space” quantizations, quantum theories based on the true physical degrees of freedom of the classical theory. That is, the classical constraints have been solved before quantizing, eliminating classically redundant “gauge” degrees of freedom. In Dirac’s approach to quantization  [112113114], in contrast, one quantizes the entire space of degrees of freedom of classical theory, and only then imposes the constraints. States are initially determined from the full classical phase space; in the ADM formulation of quantum gravity, for instance, they are functionals Y[gij] of the full spatial metric. The constraints then act as operators on this auxiliary Hilbert space; the physical Hilbert space consists of those states that are annihilated by the constraints, with a suitable new inner product, acted on by physical operators that commute with the constraints. For gravity, in particular, the Hamiltonian constraint acting on states leads to a functional differential equation, the Wheeler-DeWitt equation  [110276] .

In the first order formalism, it is straightforward to show that Dirac quantization is equivalent to the Chern-Simons quantum theory we have already seen. Details can be found in Chapter 8 of  [81Jump To The Next Citation Point], but the basic argument is fairly clear: At least for /\ = 0, the first order constraints coming from Equations (15View Equation, 16View Equation) are at most linear in the momenta, and are thus uncomplicated to solve.

In the second order formalism, matters become considerably more complicated  [73] . We begin with a wave function Y[gij], upon which we wish to impose the constraints (30View Equation), with momenta acting as functional derivatives,

ij -d-- p = - idgij. (73)
The first difficulty is that we are no longer allowed to choose a nice time-slicing such as York time; that would be a form of gauge-fixing, and is not permitted in Dirac quantization. We can still decompose the spatial metric and momentum as in Equation (31View Equation), but only up to a spatial diffeomorphism, which depends on an undetermined vector field i Y appearing in the momentum ij p   [206] . The momentum constraint fixes Y i in terms of the scale factor c, but it does so nonlocally. As a consequence, the Hamiltonian constraint becomes a nonlocal functional differential equation, and very little is understood about its solutions, even for the simplest case of the torus universe. Further complications come from the fact that the inner product on the space of solutions of the Wheeler-DeWitt equation must be gauge-fixed  [282144] ; again, little is understood about the resulting Hilbert space.

In view of the difficulty in finding exact solutions to the Wheeler-DeWitt equation, it is natural to look for perturbative methods, for example an expansion in powers of Newton’s constant G . One can solve the momentum constraints order by order by insisting that each term depend only on (nonlocal) spatially diffeomorphism-invariant quantities. Such an expansion has been studied by Banks, Fischler, and Susskind for the physically trivial topology R × S2   [42], following much earlier work by Leutwyler  [176] . Even in this simple case, computations quickly become extremely difficult. Other attempts have been made  [47Jump To The Next Citation Point186] to write a discrete version of the Wheeler-DeWitt equation in the Ponzano-Regge formalism of Section  3.6 . This approach has the advantage that the spatial diffeomorphisms have already been largely eliminated, removing the main source of nonlocality discussed above. The Wheeler-DeWitt-like equation in  [47] has been shown to agree with the the Ponzano-Regge model.

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