For spatial surfaces of genus , the complexity of the constraint (32) seems to make this approach to quantization impractical [207] . A perturbative expression for may still exist, though, as discussed in [217, 218], and the Gauss map has been proposed as a useful tool [226] .

For genus one, on the other hand, a full quantization is possible. The classical Hamiltonian (44) becomes, up to operator ordering ambiguities,

where is the ordinary scalar Laplacian for the constant negative curvature Poincaré metric on moduli space, and one chooses the positive square root in order to have a Hamiltonian that is bounded below. This Laplacian is invariant under the modular transformations (50), and its invariant eigenfunctions, the weight zero Maass forms, have been studied extensively by mathematicians (see, for example, [167]). The behavior of the corresponding wave functions has been explored by Puzio [227], who argues that they are well-behaved and nonsingular at the boundaries of moduli space. Such behavior is relevant to the question of how quantum gravity handles singularities: The degeneration of the torus geometry at the big bang, described in Section 2.8, corresponds to an approach to the boundary of moduli space, and Puzio’s results suggest that the classical singularity may be better-behaved in the quantum theory.A related form of quantization comes from reexpressing the moduli space for the torus as a quotient space [193, 273] . Here, the symmetric space describes the transverse traceless deformations of the spatial metric, while is the modular group. As Waldron has observed [273], this makes it possible to reinterpret the quantum mechanical problem as that of a fictitious free particle, with mass proportional to , moving in a quotient space of the (flat) three-dimensional Milne universe. With a suitable choice of coordinates, though, the problem again reduces to that of understanding the Hamiltonian (53) and the corresponding Maass forms.

While the choice (53) of operator ordering is not unique, the number of alternatives is smaller than one might expect. The key restriction is diffeomorphism invariance: The eigenfunctions of the Hamiltonian should transform under a one-dimensional unitary representation of the mapping class group (50). The representation theory of this group is well-understood [120, 184] ; one finds that the possible Hamiltonians are all of the form (53), but with replaced by

the Maass Laplacian acting on automorphic forms of weight . It has been suggested in [68] that the choice is most natural from the point of view of Chern-Simons quantization. Note that when written in terms of the momentum , the operators differ from each other by terms of order , as one would expect for operator ordering ambiguities. Nevertheless, the choice of ordering may have dramatic effects on the physics, since the spectra of the various Maass Laplacians are quite different.This ordering ambiguity may be viewed as arising from the structure of the classical phase space. The torus moduli space is not a manifold, but rather has orbifold singularities, and quantization on an orbifold is generally not unique. Since the space of solutions of the Einstein equations in 3+1 dimensions has a similar orbifold structure [163], we might expect a similar ambiguity in realistic (3+1)-dimensional quantum gravity.

The quantization described here is an example of what KuchaĆ has called an “internal Schrödinger interpretation” [173] . It appears to be self-consistent, and like ordinary quantum mechanics, it is guaranteed to have the correct classical limit on the reduced phase space of Section 2.4 . The principal drawback is that the method relies on a classical choice of time coordinate, which occurs as part of the gauge-fixing needed to solve the constraints. In particular, the analysis of Section 2.4 required that we choose the York time-slicing from the start; a different choice might lead to a different quantum theory, as it is known to do in quantum field theory [258] . In other words, it is not clear that this approach to quantum gravity preserves general covariance.

The problem may be rephrased as a statement about the kinds of questions we can ask in this quantum theory. The model naturally allows us to compute the transition amplitude between the spatial geometry of a time slice of constant mean curvature and the geometry of a later slice of constant mean curvature . Indeed, such amplitudes are given explicitly in [118], where it is shown that they are peaked around the classical trajectory. But it is far less clear how to ask for transition amplitudes between other spatial slices, on which is not constant. Such questions would seem to require a different classical time-slicing, and thus a different - and perhaps inequivalent - quantum theory.

We will eventually find a possible way out of this difficulty in Section 3.4 . As a first step, we next turn to an alternative approach to quantization, one that starts from the first order formalism.

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