Feynman Path Integral Formulation

194 6 Lattice Regularized Quantum Gravity
The r.h.s. of this equation contains precisely the expression appearing in the continuum
supermetric of Eq. (2.14), for the specific choice of the parameter λ = −2.
One is led therefore to the identification
G ij (l 2 )=− d! ∑
s
1 ∂ 2 V 2 (s)
V (s) ∂li 2 ∂l 2 j
, (6.86)
and therefore for the norm
{
‖δl 2 ‖ 2 = ∑ V (s) − d!
s V 2 (s) ∑ ij
∂ 2 V 2 (s)
∂l 2 i
∂l 2 j
δl 2 i δl 2 j
}
. (6.87)
One could be tempted at this point to write down a lattice measure, in parallel with
Eq. (2.16), and write
∫ ∫ √
[dl 2 ]= detG ij(ω′) (l 2 )dli 2 (6.88)
with
∏
i
G ij(ω′) (l 2 )=− d! ∑
s
1 ∂ 2 V 2 (s)
[V (s)] 1+ω′ ∂li 2 ∂l 2 j
, (6.89)
where one has allowed for a parameter ω ′ , possibly different from zero, interpolating
between apparently equally acceptable measures. The reasoning here is that,
as in the continuum, different edge length measures, here parametrized by ω’, are
obtained, depending on whether the local volume factor V (s) is included in the supermetric
or not.
One rather undesirable, and puzzling, feature of the lattice measure of Eq. (6.88)
is that in general it is non-local, in spite of the fact that the original continuum
measure of Eq. (2.18) is completely local (although it is clear that for some special
choices of ω ′ and d, one does recover a local measure; thus in two dimensions and
for ω ′ = −1 one obtains again the simple result ∫ [dl 2 ]= ∫ ∞
0 ∏ i dli 2 ). Unfortunately
irrespective of the value chosen for ω ′ , one can show (Hamber and Williams, 1999)
that the measure of Eq. (6.88) disagrees with the continuum measure of Eq. (2.18)
already to lowest order in the weak field expansion, and does not therefore describe
an acceptable lattice measure.
The lattice action for pure four-dimensional Euclidean gravity contains a cosmological
constant and Regge scalar curvature term
I latt = λ 0 ∑
h
V h (l 2 ) − k∑δ h (l 2 )A h (l 2 ) , (6.90)
h
with k = 1/(8πG), as well as possibly higher derivative terms. It only couples edges
which belong either to the same simplex or to a set of neighboring simplices, and
can therefore be considered as local, just like the continuum action, and leads to the
regularized lattice functional integral