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

then gives a representation of the loop algebra.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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