Feynman Path Integral Formulation

206 6 Lattice Regularized Quantum Gravity
are always positive due to the enforcement of the triangle inequalities. Thus from
a practical point of view the barycentric volume subdivision is the simplest to deal
with.
φ 3
l 2
3
A 23
l 5
A 24
4
φ 4
A 13
φ 1
1
A 11
A 12
l 1
A 22
l 4
2
l 3
φ 2
Fig. 6.16 More dual areas appearing in the scalar field action.
The scalar action of Eq. (6.134) has a very natural form: it involves the squared
difference of fields at neighboring points divided by their invariant distances (φ i −
φ j )/l ij , weighted by the appropriate space-time volume element V (d)
ij
associated
with the lattice link ij. This suggests that one could just as well define the scalar
fields on the vertices of the dual lattice, and write
(
I(l 2 ,φ) = 1 2 ∑ V rs
(d) φr − φ
) s 2
, (6.137)
l rs
with l rs the length of the edge connecting the dual lattice vertices r and s, and consequently
V rs
(d) the spacetime volume fraction associated with the dual lattice edge
rs. One would expect both forms to be equivalent in the continuum limit.
Continuing on with the two-dimensional case, mass and curvature terms such as
the ones appearing in Eq. (6.124) can be added to the action, so that the total scalar
lattice action contribution becomes
I = 1 2
∑
A ij
( φi − φ j
l ij
) 2
+
1
2 ∑A i (m 2 + ξ R i )φi 2 . (6.138)
i
The term containing the discrete analog of the scalar curvature involves the quantity
A i R i ≡ ∑ δ h ∼ √ gR . (6.139)
h⊃i
In the above expression for the scalar action, A ij is the area associated with the
edge l ij , while A i is associated with the site i. Again there is more than one way to