Quantum gravity in 2+1 dimensions offers a possible answer to this dilemma. Note first that the problem is already present classically. A geometric structure determines a spacetime, and must contain within it all of the dynamics of that spacetime. On the other hand, the basic data that fix the geometric structure - the transition functions, or, often, the holonomies - have no obvious dynamics. In principle, the classical answer is simple:
This procedure can be understood as a concrete realization of the isomorphism described in Section 3.3 between the phase space and the space of classical solutions, with the Cauchy surface fixed by the choice of time-slicing.
For the simple case of the torus universe, these steps can be transcribed almost directly to the quantum theories. Equations (42, 43, 44) become definitions of operators,[236, 237] .
From this point of view, we should think of Chern-Simons/covariant canonical quantization as a sort of Heisenberg picture, with time-independent states and “time”-dependent operators. To obtain the corresponding Schrödinger picture, we proceed as in ordinary quantum mechanics: We diagonalize , obtaining a transition matrix that allows us to transform between representations [68, 88] . The resulting “time”-dependent wave functions obey a Schrödinger equation of the form (52, 53), but with the Laplacian in replaced by the weight Maass Laplacian of Equation (54). In , it has been shown that these wave functions are peaked around the correct classical trajectories. (Different operator orderings in Equation (62) give different weight Laplacians  .)
As a useful byproduct, this analysis allows us to solve the problem of the poorly-behaved action of the modular group discussed at the end of Section 3.2 [88, 89] . If we start with a reduced phase space wave function and use the transition matrix to determine a Chern-Simons wave function , we find, indeed, that is not modular invariant. Instead, though, the entire Hilbert space of Chern-Simons wave functions splits into “fundamental regions,” orthogonal subspaces that transform into each other under the action of the modular group. Any one of these fundamental regions is equivalent to any other, and each is equivalent to the Hilbert space arising from reduced phase space quantization. Moreover, matrix elements of any modular invariant function vanish unless they are taken between states in the same fundamental region. Modular invariance thus takes a slightly unexpected form, but can still be imposed by restricting the theory to a single fundamental region of the Hilbert space.
We can also begin to address the problem raised at the end of Section 3.1, the limited and slicing-dependent range of questions one can ask in reduced phase space quantization. The operators (62) introduced here on the covariant canonical Hilbert space were obtained from a particular classical time-slicing, and answer questions about spatial geometry in that slicing. In principle, however, we can choose any other slicing, with a new time coordinate , and determine the corresponding operators , , and . The operator ordering of such operators will, of course, be ambiguous, though one might hope that the action of the modular group might again restrict the choices. But such an ambiguity need not be seen as a problem with the theory; rather, it is merely a statement that many different quantum operators can have the same classical limit, and that ultimately experiment must decide which operator we are really observing.
There is, to be sure, a danger that the “Schrödinger pictures” coming from different time-slicings may not be consistent. Suppose, for example, that we choose two slicings that agree on an initial and a final slice and , but disagree in between. If we start with an initial wave function on , we must check that the Hamiltonians coming from the different slicings evolve us to the same final wave function on . For field theories, even in flat spacetime, this will not always happen  . For (2+1)-dimensional gravity, on the other hand, there is evidence that one can always find operator orderings of the Hamiltonians that ensure consistent evolution  . If this ultimately turns out not to be the case, however, it may simply mean that we should treat the covariant canonical picture as fundamental, and discard the Schrödinger pictures of time-dependent states.
|http://www.livingreviews.org/lrr-2005-1||© Max Planck Society and