In contrast to the reduced phase space
quantization of the preceding Section
3.1, our understanding of the quantum Chern-Simons gravity depends
strongly on the sign of the cosmological constant. For
, the relevant gauge group is
or its cover
. This is the most poorly understood case; an explicit
quantization of the algebra holonomies exists for genus one (see
below) and genus two
[214], but more general results do not yet exist.
For
, the relevant gauge group is
or its cover
, a complex gauge group whose Chern-Simons theory is somewhat
better understood
[280
, 155
, 43]
. As noted in Section
2.3, the Poisson brackets for this theory are related to the quantum
double of the Lorentz group, and Buffenoir et al. have used
this structure to write down an explicit quantization
[61]
. As far as I know, the relationship between this work, which is
based on a Hamiltonian formalism and combinatorial quantization,
and that of Witten and Hayashi
[280, 155], which is based on geometric quantization, has not yet been
explored.
For
, the relevant gauge group is
, the (2+1)-dimensional Poincaré group, or its universal cover.
Here there is again a connection to the quantum double of the
Lorentz group, which has been used in
[37, 36, 201]
to explore the quantum theory, although largely in the context
of point particles. In this case, one has the nice feature that
the phase space has a natural cotangent bundle structure,
allowing us to immediately identify the holonomies of the spin
connection
as generalized positions, and their derivatives as generalized
momenta. This provides a direct link to the loop variables of
Ashtekar, Rovelli, and Smolin
[26
, 29
],
As in reduced phase space quantization, matters
simplify considerably for the torus universe
. Let us again focus on the case
. A complete - in fact, overcomplete - set of observables is
given by the traces (47
) of the holonomies, and our goal is to quantize the
algebra (48
). To do so, we proceed as follows:
The resulting algebra is defined by the relations
withWe must also implement the action of the
modular group (50) on the operators
. One can find an action preserving the algebraic
relations (57
), corresponding to a particular factor ordering of the classical
modular group. The Nelson-Picken quantization (61
) admits a similar modular group action.
For a full quantum theory, of course, one needs
not only an abstract operator algebra, but a Hilbert space upon
which the operators act. For the
universe, Equation (60
) suggests that a natural choice is to take wave functions to be
square integrable functions of the
. There is a potential difficulty here, however: The modular
group does not act properly discontinuously on this configuration
space. This means that the quotient of this space by the modular
group is badly behaved; in fact, there are no nonconstant modular
invariant functions of the
[182, 143, 221]
. We shall return to this problem in Section
3.4
.
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