Let us restrict ourselves to invertible spatial
metrics, and attempt to quantize the algebra of loop variables
and
. For the torus universe, it is not hard to show that such a
quantization simply reproduces the theory we already obtained in
the Chern-Simons formulation (see, for example, Section 7.2
of
[81
]). So far, there is nothing new here.
There is another way to look at the operator
algebra of the operators
and
, however, which leads to a new approach, the loop
representation. Up to now, we have been thinking of the operators
as a set of functions of the triad and spin connection, indexed
by loops
. Our wave functions are thus functionals of the “configuration
space” variable
, or, more precisely, functions on the moduli space of flat
or
connections on
. But we could equally well view the
operators as functions of loops - or, in 2+1 dimensions,
homotopy classes
of loops - indexed by
and
. Wave functions would then be functions of loops or sets of
loops. This change of viewpoint is rather like the decision in
ordinary quantum mechanics to view a wave function
as a function on momentum space, indexed by
, rather than a function on position space, indexed by
.
The loop representation is complicated by the
existence of Mandelstam identities
[190]
among holonomies of loops, but for the case of the torus
universe, a complete, explicit description of the states is again
possible
[26, 29
]
. The simplest construction begins with a vacuum state
annihilated by every operator
, and treats the
as “creation operators.” Since any homotopy class
of loops on the torus is completely characterized by a pair of
winding numbers
, one can write these states as
. The action
Observe now that the loop variables
depend only on the “configuration space” variable
. We can thus relate the loop representation to the Chern-Simons
representation by simultaneously diagonalizing these operators,
obtaining wave functions that are functions of the
holonomies alone. For the torus universe, this “loop transform”
can be obtained explicitly
[26, 29, 192
], and written as a simple integral transform.
The properties of this transform depend on the
holonomies, that is, the eigenvalues of
. For simplicity, let us take the generator
in Equation (56
) to be in the two-dimensional representation of
. In the “timelike sector,” in which the traces of the two
holonomies are both less than two, the loop transform is a simple
Fourier transformation, and Chern-Simons and loop quantization
are equivalent.
Unfortunately, though, this is not the
physically relevant sector: It does not correspond to a geometric
structure with spacelike
slices. For a physically interesting geometry, one must go to
the “spacelike sector,” in which the traces of the holonomies are
both greater than two. In this sector, the transform is
not
very well-behaved: In fact, a dense set of Chern-Simons states
transforms to zero
[192
]
. The loop representation thus appears to be rather drastically
different from the Chern-Simons formulation.
The problems in the physical sector can be
traced back to the fact that
is a noncompact group. There have been two proposals for an
escape from this dilemma. One is to start with a different dense
set of Chern-Simons states that transform faithfully, determine
the inner product and the action of the
operators on the resulting loop states, and then form the Cauchy
completion to define the Hilbert space in the loop
representation
[192]
. This is a consistent procedure, but many of the resulting
states in the Cauchy completion are no longer functions of loops
in any clear sense; they correspond instead to “extended
loops”
[136], whose geometrical interpretation is not entirely clear. A
second possibility is to change the integration measure in the
loop transform to make various integrals converge better
[30]
. Such a choice introduces order
corrections to the action of the
operators, and one must be careful that the algebra remains
consistent. This is possible, but at some cost - the inner
products between loop states become considerably more complex, as
does the action of the mapping class group - and it is not
obvious that there is a canonical choice of the new measure and
algebra.
A third possibility is suggested by recent work
on spin networks for noncompact groups
[129, 130
]
. This new technology essentially allows one to consider
holonomies (56
) that lie in infinite-dimensional unitary representations of the
Lorentz group, with a finite inner product defined by appropriate
gauge-fixing. The quantities
and
can be represented as Hermitian operators on this space of
holonomies (or on a larger space of spin networks). At this
writing, implications of this approach for the loop transform in
2+1 dimensions have not yet been investigated.
Finally, I should briefly mention the role of
spin networks in (2+1)-dimensional quantum geometry. In the
(3+1)-dimensional theory, loop states have been largely
superseded by spin network states, states characterized by a
graph
with edges labeled by representations and vertices labeled by
intertwiners
[239]
. Such states can be defined in 2+1 dimensions as well, and there
has been some interesting recent work on their role as
“kinematic” states
[130
]
. In 2+1 dimensions, however, the full constraints imply that
such states have their support on flat connections, and only
holonomies around noncontractible curves describe nontrivial
physics. An interesting step toward projecting out the physical
states has recently been taken in
[216], in the context of Euclidean quantum gravity; the ultimate
effect is to reduce spin network states to loop states of the
sort we have considered above. A better understanding of the
relationship to the gauge-fixing procedure of
[129, 131
]
would be valuable.
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