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,
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
[118], 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
[70]
.)
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
[258]
. 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
[95]
. 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.
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