Feynman Path Integral Formulation

260 7 Analytical Lattice Expansion Methods
(
{ ∫
})
1
W(C) ∼ Tr (U C + ε I 4 ) exp
2
R·· μν A μν
C
. (7.125)
S(C)
Next, as is standard in simplicial gravity, we consider the lattice analog of a background
manifold with constant or near-constant large scale curvature,
so that here in our case we can set
R μνλσ = 1 3 λ (g μν g λσ − g μλ g νσ )
R μνλσ R μνλσ = 8 3 λ 2 , (7.126)
R α βμν = ¯RU α β U μν , (7.127)
where ¯R is some average curvature over the loop, and the U’s here will be taken to
coincide with U C . The trace of the product of (U C + ε I 4 ) with this expression gives
Tr( ¯RU 2 C A C )=− 2 ¯RA C , (7.128)
where one has used U μν A μν
C
= 2A C (the choice of direction for the bivectors will
be such that the latter is true for all loops). This is to be compared with the linear
term from the other exponential expression, −A C /ξ 2 . Thus the average curvature is
computed to be of the order
¯R ∼ 1/ξ 2 , (7.129)
at least in the small k = 1/8πG limit. An equivalent way of phrasing the last result
is that 1/ξ 2 should be identified, up to a constant of proportionality, with the scaled
cosmological constant λ, with the latter being regarded as a measure of the intrinsic
curvature of the vacuum. We see therefore that a direct calculation of the Wilson
loop for gravity provides an insight into whether the manifold is De Sitter or anti-
De Sitter at large distances.
7.6 Discrete Gravity in the Large-d Limit
In the large-d limit the geometric expressions for volume, areas and angles simplify
considerably, and as will be shown below one can obtain a number of interesting
results for lattice gravity. These can then be compared to earlier investigations of
continuum Einstein gravity in the same limit (Strominger, 1981).
Here we will consider a general simplicial lattice in d dimensions, made out
of a collection of flat d-simplices glued together at their common faces so as to
constitute a triangulation of a smooth continuum manifold, such as the d-torus or the
surface of a sphere. Each simplex is endowed with d +1 vertices, and its geometry is