where the
so
(3,1)-valued spin-connection
(with curvature
R) and the vierbein
are considered as independent variables. The important feature
of (1
) is that
is a gauge potential, and that the action - in addition to its
diffeomorphism-invariance - is invariant under local frame
rotations. Variation with respect to
leads to the metricity condition,
which can be solved to yield the unique torsion-free spin
connection
A
=
A
[e] compatible with the
. There are some obvious differences with usual gauge theories:
i) the action (1
) is linear instead of quadratic in the curvature two-form
of
A, and ii) it contains additional fields
. Substituting the solution to (2
) into the action, one obtains
, where
R
denotes the four-dimensional curvature scalar. This expression
coincides with the usual Einstein action
only for
.
Most of the lattice gauge formulations I will discuss below
share some common features. The lattice geometry is hypercubic,
defining a natural global coordinate system for labelling the
lattice sites and edges. The gauge group
G
is
SO
(3,1) or its ``Euclideanized'' form
SO
(4), or a larger group containing it as a subgroup or via a
contraction limit. Local curvature terms are represented by (the
traces of)
G
-valued Wilson holonomies
around lattice plaquettes. The vierbeins are either considered
as additional fields or identified with part of the connection
variables. The symmetry group of the lattice Lagrangian is a
subgroup of the gauge group
G, and does not contain any translation generators that appear
when
G
is the Poincaré group.
When discretizing conformal gravity (where G = SO (5,1)) or higher-derivative gravity in first-order form, the metricity condition on the connection has to be imposed by hand. This leads to technical complications in the evaluation of the functional integral.
The diffeomorphism invariance of the continuum theory is broken on the lattice; only the local gauge invariances can be preserved exactly. The reparametrization invariance re-emerges only at the linearized level, i.e. when considering small perturbations about flat space.
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Discrete Approaches to Quantum Gravity in Four Dimensions
Renate Loll http://www.livingreviews.org/lrr-1998-13 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |