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3.6 Lattice methods I: Ponzano-Regge and spin foams

A long-standing approach to quantum gravity in 3+1 dimensions has been to look for discrete approximations to the path integral  [179Jump To The Next Citation Point232] : quantized Regge calculus  [231], for example, or sums over random triangulations  [8Jump To The Next Citation Point] . The basic idea is that although the full “sum over geometries” may be impossible to evaluate, a sum over appropriately discretized geometries might give a good approximation, perhaps becoming extremely good near a phase transition at which lattice spacing can go to zero. When applied to 2+1 dimensions, such methods have the added feature of sometimes being exact: Since geometries satisfying the constraints have constant or zero curvature, a discrete “approximation” may give a complete description.

Regge calculus in 2+1 dimensions begins with a triangulated three-manifold, consisting of a collection of flat simplices joined along one-dimensional edges. The curvature of such a manifold is concentrated entirely at the edges. For a simplicial three-manifold with Riemannian signature, composed of simplices with edges of length le, Regge’s form of the Einstein-Hilbert action is

sum IRegge = 2 dele, (65) edges:e
where de is the conical deficit angle at the edge labeled by the index e . A similar expression exists for Lorentzian signature, although the definition of the deficit angle is a bit more complicated  [48Jump To The Next Citation Point] .

The first hint that (2+1)-dimensional gravity might be exceptional came from the observation by Ponzano and Regge  [225] that the Regge action in 2+1 dimensions can be re-expressed in terms of Wigner-Racah 6j -symbols. (See  [90Jump To The Next Citation Point] for more about these quantities.) Consider first a single tetrahedron with edge lengths li = 12(ji + 12), where the ji are integers or half-integers. Ponzano and Regge noticed, and Roberts later proved rigorously  [234], that in the limit of large j,

( ) sum 6 { } 1 { [ ( p )] [ ( p)]} exp pi ji j1 j2 j3 ~ V~ ----- exp i IRegge + -- + exp - i IRegge + -- , (66) i=1 j4 j5 j6 6pV 4 4
where { j j j } 1 2 3 j4 j5 j6 is a 6j -symbol, IRegge is the Regge action (65View Equation) for the tetrahedron, and V is its volume. For a manifold made of a collection of such tetrahedra, the full Regge action will occur in a product of such 6j -symbols. This suggests that the (2+1)-dimensional path integral, which is essentially a sum over geometries of terms of the form exp( -iIRegge), might be expressible as a sum of such products. Ponzano and Regge’s specific proposal, developed by Hasslacher and Perry  [153] and modified by Ooguri  [219Jump To The Next Citation Point] to account for boundaries, was the following:

Consider a three-manifold M with boundary @M, with a given triangulation D of @M . Choose a triangulation of M that agrees with the triangulation of the boundary. Label interior edges of tetrahedra by integers or half-integers xi and exterior (boundary) edges by ji, and for a given tetrahedron t, let ji(t) denote the spins that color its (interior and exterior) edges. Then

sum ( prod V~ ------- prod ZD[{ji}] = lim (-1)2ji 2ji + 1 /\(L) -1 L-->o o xe<L ext.edges:i in{t.vertices }) prod prod sum 6 j(t) j1(t) j2(t) j3(t) × (2xl + 1) (- 1) i=1 i j (t) j (t) j (t) , (67) int.edges:l tetra:t 4 5 6
where “int” and “ext” mean “interior” and “exterior” and
sum 2 /\(L) = (2j + 1) (68) j<L
is a regularization factor that controls divergences in the sum over interior lengths. With this weighting, identities among 6j -symbols may be used to show that the amplitude is invariant under refinement - that is, subdivision of a tetrahedron into four smaller tetrahedra - suggesting that we are dealing with a “topological” theory that does not depend on the choice of triangulation. This is, of course, what one would hope for, based on the classical characteristics of (2+1)-dimensional gravity.

The “topological” feature of the Ponzano-Regge model was made more precise by Turaev and Viro  [262], who discovered an improved regularization, based on the technology of quantum groups. The “spins” j in Equation (67View Equation) can be viewed as labeling representations of SU(2) . If these are replaced by representations of the quantum group Uq(sl(2)) (“quantum SU(2) ”), with ( ) q = exp 2pi k+2, k (- Z, the number of such representations is finite, and the sum over interior edge lengths is automatically cut off. With appropriate substitutions (e.g., “quantum” 6j -symbols  [90]), the Ponzano-Regge amplitude (67View Equation) becomes well-defined without any regularization.

The construction of physical states as appropriate functions of boundary edge lengths is described in Section 11.2 of  [81Jump To The Next Citation Point] . The resulting amplitudes can be computed for simple topologies  [162161], and have several key features:

Although it has not been universally appreciated, the existence of a divergence in the sum (67View Equation) - regulated either by an explicit cut-off or by quantum group tricks - is rather mysterious, given the absence of local excitations and the general well-behavedness of gravity in three dimensions. This mystery may have recently been solved by Freidel and Louapre  [131], who show that a residual piece of the diffeomorphism symmetry has not been factored out of the Ponzano-Regge action. Because of this symmetry, the sum (67View Equation) overcounts physical configurations, and the regulator /\(L) is simply the remaining gauge volume. Freidel and Louapre further show that the symmetry can instead be gauge-fixed, leading to a sum over a restricted and considerably simplified class of “collapsed” triangulations.

While the mathematics of Ponzano-Regge and Turaev-Viro models has been studied extensively, so far only a bit of attention has been given to the “traditional” issues of quantum gravity. A few numerical investigations of the Ponzano-Regge path integral have been undertaken  [151], but the evidence of a continuum limit is thus far inconclusive. The model has been used to study conditional probabilities and the emergence of quasiclassical behavior in quantum gravity  [223], but the cut-off dependence of these results makes their physical significance unclear. In an interesting recent paper, Colosi et al. have investigated the dynamics of a single tetrahedron  [241Jump To The Next Citation Point], showing that a quantum description of the evolution can be given in terms of a boundary amplitude.

A number of observables, whose expectation values generally give topological information about the spacetime or about knots within spacetime, have been discussed in  [2354261139] . With a few exceptions, though, work in this area has remained largely mathematical in nature; fairly little is understood about the physics of these observables, although some are probably related to length spectra  [46] and perhaps volumes  [12764], and others are almost certainly connected to scattering amplitudes for test particles.

The Ponzano-Regge and Turaev-Viro models are examples of “spin foam” models  [35222Jump To The Next Citation Point], that is, a model based on simplicial complexes with faces, edges, and vertices labeled by group representations and intertwiners. A key question is whether one can extend such models to Lorentzian signature. It has been known for several years how to generalize the Ponzano-Regge action for a single tetrahedron  [48101191], and recently considerable progress has been made in constructing Lorentzian spin foam models  [222125Jump To The Next Citation Point102] .

Probably the most elegant derivation of a Lorentzian spin foam description starts with the first-order action (14View Equation), with /\ = 0, for a triangulated manifold  [125Jump To The Next Citation Point128] . One can rewrite the action in terms of a set of discrete variables: a Lie algebra element ea corresponding to the integral of e along the edge a of a tetrahedron in the triangulation, and a holonomy ga of the connection w around the edge. The path integral then becomes an integral over these variables. As in the continuum path integral of Section  3.10, the integral over the ea produces a delta function d(ga) for each edge. This translates back to the geometric statement that the constraints require the connection w to be flat, and thus to have trivial holonomy around a contractible curve surrounding an edge.

For the Euclidean Ponzano-Regge action, g (- SU(2), and the key trick is now to use the Plancherel formula to express each d(ga) as a sum over the characters of finite-dimensional representations of SU(2) . Fairly straightforward arguments then permit an exact evaluation of the remaining integrals over the ga, reproducing the 6j symbols in the Ponzano-Regge action. To obtain a Lorentzian version, one must replace SU(2) by SO(2, 1) . The corresponding Plancherel formula involves a sum over both the (continuous) principle series of representations of SO(2, 1) and the discrete series. Consequently, edges may now be labeled either by discrete or continuous spins. Similar methods may be used for supergravity  [177] .

The resulting rather complicated expression for the partition function may be found in  [125] . The appearance of both continuous and discrete labels has a nice physical interpretation  [130Jump To The Next Citation Point] : Continuous representations describe spacelike edges, and seem to imply a continuous length spectrum, while discrete representations label timelike edges, and suggest discrete time. These results should probably not yet be considered conclusive, since they require operators that do not commute with all of the constraints, but they are certainly suggestive.

While spin foam models ordinarily assume a fixed spacetime topology, recent work has suggested a method for summing over all topologies as well, thus allowing quantum fluctuations of spacetime topology  [132Jump To The Next Citation Point] . These results will be discussed in Section  3.11 . Methods from 2+1 dimensions have also been generalized to higher dimensions, leading to new insights into the construction of spin foams.


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