Feynman Path Integral Formulation

190 6 Lattice Regularized Quantum Gravity
parallel transports < R >= 0, but < RR −1 >≠ 0, which would require, for a nonvanishing
lowest order contribution to the Wilson loop, that the loop at least be tiled
by elementary action contributions from Eqs. (6.38) or (6.39), thus forming a minimal
surface spanning the loop. Then, in close analogy to the Yang-Mills case of
Eq. (3.145) (see for example Peskin and Schroeder, 1995), the leading contribution
to the gravitational Wilson loop would be expected to follow an area law,
W(C) ∼ const.k A(C) ∼ exp(−A(C)/ξ 2 ) , (6.67)
where A(C) is the minimal physical area spanned by the near-planar loop C, and
ξ = 1/ √ |lnk| the gravitational correlation length at small k. Thus for a close-tocircular
loop of perimeter P one would use A(C) ≈ P 2 /4π.
The rapid decay of the gravitational Wilson loop as a function of the area is seen
here simply as a general and direct consequence of the disorder in the fluctuations
of the parallel transport matrices R(s,s ′ ) at strong coupling. It should then be clear
from the above discussion that the gravitational Wilson loop provides in a sense a
measure of the magnitude of the large-scale, averaged curvature, where the latter is
most suitably defined by the process of parallel-transporting test vectors around very
large loops, and is therefore, from the above expression, computed to be of the order
R ∼ 1/ξ 2 , at least in the small k limit. A direct calculation of the Wilson loop should
therefore Provide, among other things, a direct insight into whether the manifold is
de Sitter or anti-de Sitter at large distances. More details on the lattice construction
of the gravitational Wilson loop, the various issues that arise in its precise definition,
and a sample calculation in the strong coupling limit of lattice gravity, can be found
in (Hamber and Williams, 2007).
Finally we note that the definition of the gravitational Wilson loop is based on a
surface with a given boundary C, in the simplest case the minimal surface spanning
the loop. It is possible though to consider other surfaces built out of elementary
parallel transport loops. An example of such a generic closed surface tiled with
elementary parallel transport polygons (here chosen for illustrative purposes to be
triangles) will be given later in Fig. 7.13.
Later similar surfaces will arise naturally in the context of the strong coupling
(small k) expansion for gravity, as well as in the high dimension (large d) expansion.
6.9 Lattice Regularized Path Integral
As the edge lengths l ij play the role of the continuum metric g μν (x), one would
expect the discrete measure to involve an integration over the squared edge lengths
(Hamber, 1984; Hartle, 1984; 1986; Hamber and Williams, 1999). Indeed the induced
metric at a simplex is related to the squared edge lengths within that simplex,
via the expression for the invariant line element ds 2 = g μν dx μ dx ν . After choosing
coordinates along the edges emanating from a vertex, the relation between metric
perturbations and squared edge length variations for a given simplex based at 0 in d