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
, Regge’s form of the Einstein-Hilbert action is
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
-symbols. (See
[90
]
for more about these quantities.) Consider first a single
tetrahedron with edge lengths
, where the
are integers or half-integers. Ponzano and Regge noticed, and
Roberts later proved rigorously
[234], that in the limit of large
,
Consider a three-manifold
with boundary
, with a given triangulation
of
. Choose a triangulation of
that agrees with the triangulation of the boundary. Label
interior edges of tetrahedra by integers or half-integers
and exterior (boundary) edges by
, and for a given tetrahedron
, let
denote the spins that color its (interior and exterior) edges.
Then
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”
in Equation (67
) can be viewed as labeling representations of
. If these are replaced by representations of the quantum group
(“quantum
”), with
,
, the number of such representations is finite, and the sum over
interior edge lengths is automatically cut off. With appropriate
substitutions (e.g., “quantum”
-symbols
[90]), the Ponzano-Regge amplitude (67
) 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
[81]
. The resulting amplitudes can be computed for simple
topologies
[162, 161], and have several key features:
Although it has not been universally
appreciated, the existence of a divergence in the sum (67) - 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 (67
) overcounts physical configurations, and the regulator
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
[241], 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 [23, 54, 261, 139] . 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 [127, 64], 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
[35, 222], 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
[48, 101, 191], and recently considerable progress has been made in
constructing Lorentzian spin foam models
[222, 125
, 102]
.
Probably the most elegant derivation of a
Lorentzian spin foam description starts with the first-order
action (14), with
, for a triangulated manifold
[125
, 128]
. One can rewrite the action in terms of a set of discrete
variables: a Lie algebra element
corresponding to the integral of
along the edge
of a tetrahedron in the triangulation, and a holonomy
of the connection
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
produces a delta function
for each edge. This translates back to the geometric statement
that the constraints require the connection
to be flat, and thus to have trivial holonomy around a
contractible curve surrounding an edge.
For the Euclidean Ponzano-Regge action,
, and the key trick is now to use the Plancherel formula to
express each
as a sum over the characters of finite-dimensional
representations of
. Fairly straightforward arguments then permit an exact
evaluation of the remaining integrals over the
, reproducing the
symbols in the Ponzano-Regge action. To obtain a Lorentzian
version, one must replace
by
. The corresponding Plancherel formula involves a sum over both
the (continuous) principle series of representations of
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
[130]
: 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
[132]
. 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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