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3.2 Chern-Simons quantization

As we saw in Section  2.3, (2+1)-dimensional general relativity in first order form can be rewritten as a Chern-Simons theory. For compact gauge groups, the quantization of Chern-Simons theory is well understood  [27824324411634] . For noncompact groups such as those that appear in gravity, much less is known, though there has been some promising work  [280Jump To The Next Citation Point155Jump To The Next Citation Point43Jump To The Next Citation Point150] . Nevertheless, interesting progress can be made, especially in the simple case of a manifold with topology 2 R × T .

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 /\ < 0, the relevant gauge group is SO(2, 2) or its cover SL(2, R) × SL(2, R) . 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 /\ > 0, the relevant gauge group is SO(3, 1) or its cover SL(2,C), a complex gauge group whose Chern-Simons theory is somewhat better understood  [280Jump To The Next Citation Point155Jump To The Next Citation Point43] . 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  [280155], which is based on geometric quantization, has not yet been explored.

For /\ = 0, the relevant gauge group is ISO(2, 1), 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  [3736201] 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 w as generalized positions, and their derivatives as generalized momenta. This provides a direct link to the loop variables of Ashtekar, Rovelli, and Smolin  [26Jump To The Next Citation Point29Jump To The Next Citation Point],

integral 0 1- 1 a T [g] = 2 tr r0[g, x], T [g] = tr{r0[g, x(s)]e (g(s))Ja}, (55) g
(i ntegral ) r [g, x] = P exp waJ (56) 0 g a
is the SL(2, R) holonomy of the spin connection and 1 T [g] can be expressed as a derivative of 0 T [g] along a path in the space of flat connections  [81Jump To The Next Citation Point] . Note that the generator J may, in principle, be in any representation of SL(2, R), and that the trace in Equation (55View Equation) may depend on the choice of representation. I will return to the resulting quantum theory, loop quantization, in Section  3.5 .

As in reduced phase space quantization, matters simplify considerably for the torus universe 2 R × T . Let us again focus on the case /\ < 0 . A complete - in fact, overcomplete - set of observables is given by the traces (47View Equation) of the holonomies, and our goal is to quantize the algebra (48View Equation). To do so, we proceed as follows:

  1. We replace the classical Poisson brackets {, } by commutators [, ], with the rule [x, y] = xy- yx = ih{x, y} .
  2. On the right hand side of Equation (48View Equation), we replace the product by the symmetrized product, xy --> 1(xy + yx) 2 .

The resulting algebra is defined by the relations

^ ±^ ± ±ih ^± ^± ±ih ^± R 1R 2 e - R2 R1 e = ± 2isin hR 12 and cyclical permutations, (57)
with tan h = - h/(8l) . The algebra (57View Equation) is not a Lie algebra, but it is related to the Lie algebra of the quantum group U (sl(2)) q with q = exp 4ih   [211] . Classically, the observables R ± 1, R± 2, and R± 12 are not independent; in the quantum theory, the corresponding statement is that the quantities
( ) F^± = 1 - tan2h - e± 2ih (^R±1 )2 + (R^1±2)2 - e± 2ih(^R2±)2 + 2e ±ih coshR^1±R^2±R^1±2 (58)
commute with the holonomies, and can be consistently set to zero. In terms of the parameters r±a of Equation (38View Equation), the algebra can be represented by  [8786]
^r±1 ^r±2 (r^±1 + ^r±2 ) ^R1± = sech cosh --, ^R±2 = sech cosh---, ^R±12 = sech cosh ---------, (59) 2 2 2
± ± + - [^r1 ,^r2 ] = ± 8ih, [^ra ,^rb ] = 0. (60)
For /\ small, these commutators differ from the naive quantization of the classical brackets (45View Equation) by terms of order h3 . An alternative quantization, also differing by terms of order h3, works directly with the holonomy matrices (38View Equation), imposing a quantum-group-like quantization condition  [208]
r+[g1]r+[g2] = qr+[g2]r+[g1], q = e-ih/4l, (61)
with a similar condition for r- .

We must also implement the action of the modular group (50View Equation) on the operators ^± R a . One can find an action preserving the algebraic relations (57View Equation), corresponding to a particular factor ordering of the classical modular group. The Nelson-Picken quantization (61View Equation) 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 2 R × T universe, Equation (60View Equation) suggests that a natural choice is to take wave functions to be square integrable functions of the r±2 . 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 ± r2   [182143221] . We shall return to this problem in Section  3.4 .

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