The simplest path integral approach to
(2+1)-dimensional quantum gravity is the phase space path
integral, in which the action is written in the ADM form (29,
30
), and the spatial metric
and momentum
are treated as independent integration variables. The lapse
and shift
appear as Lagrange multipliers, and the integrals over these
quantities yield delta functionals for the constraints
and
. One might therefore expect the result to be equivalent to the
canonical quantization of Section
3.1, in which the constraints are set to zero and solved for the
physical degrees of freedom. This is indeed true, as shown
in
[77, 245]
for spatially closed universes and
[63]
for geometries with point particles. The main subtlety comes
from the appearance of many different determinants, arising from
gauge-fixing and from the delta functionals, which must be shown
to cancel. The phase space path integral for the first order
formulation similarly reproduces the corresponding canonically
quantized theory.
It is perhaps more interesting to look at the
covariant metric path integral, in which one starts with the
ordinary Einstein-Hilbert action and gauge-fixes the full
(2+1)-dimensional diffeomorphism group. This approach does not
require a topology
, and could potentially describe topology-changing amplitudes.
Unfortunately, very little is yet understood about this approach.
Section 9.2 of
[81]
describes a partial gauge-fixing, which takes advantage of the
fact that every metric on a three-manifold is conformal to one of
constant scalar curvature. But while this leads to some
simplification, we are still left with an infinite-dimensional
integral about which very little can yet be said.
By far the most useful results in the path
integral approach to (2+1)-dimensional quantum gravity have come
from the covariant first-order action (14). The path integral for this action was first fully analyzed in
two seminal papers by Witten
[277, 279
], who showed that it reduced to a ratio of determinants that has
an elegant topological interpretation as the analytic or
Ray-Singer torsion
[230]
. The partition function for a closed three-manifold with
takes the form
Although it was originally derived for closed
manifolds, Equation (74) can be extended to manifolds with boundary in a straightforward
manner. The path integral then gives a transition function that
depends on specified boundary data - most simply, the induced
spin connection
, with some additional restrictions on the normal component of
and the triad
[283, 84
]
. For a manifold with the topology
, the results agree with those of covariant canonical
quantization: The transition amplitude between two surfaces with
prescribed spin connections is nonzero only if the holonomies
agree.
But the path integral can also give transition
amplitudes between states on surfaces
and
with different topologies. If we demand that the initial and
final surfaces be nondegenerate and spacelike, their topologies
are severely restricted: Amano and Higuchi have shown that
and
must have equal Euler numbers
[6
]
. For such manifolds, concrete computations can exploit the
topological invariance of the Ray-Singer torsion. Carlip and
Cosgrove
[84], for example, explicitly compute amplitudes for a transition
between a genus three surface and a pair of genus two
surfaces.
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