Feynman Path Integral Formulation

182 6 Lattice Regularized Quantum Gravity
χ =
d
∑
i=0
(−1) i N i , (6.48)
where N i is the number of simplices of dimension i. Also it should be noted that
in two dimensions the compact action I com = ∑ p sinδ p does not satisfy the Gauss-
Bonnet relation.
In three dimensions the Regge lattice action reads
I R = −k
∑
edges h
l h δ h , (6.49)
where δ h is the deficit angle around the edge labeled by h. Variation with respect to
an edge length l h gives two terms, of which only the term involving the variation of
the edge is non-zero
δ I R = −k δl h · δ h . (6.50)
∑
edges h
In fact it was shown by Regge that for any d > 2 the term involving the variation
of the deficit angle does not contribute to the equations of motion (just as in the
continuum the variation of the Ricci tensor does not contribute to the equations of
motion either). Therefore in three dimensions the lattice equations of motion, in the
absence of sources and cosmological constant term, reduce to
δ h = 0 , (6.51)
implying that all deficit angles have to vanish, i.e. a flat space.
In four dimensions variation of I R with respect to the edge lengths gives the
simplicial analogue of Einstein’s field equations, whose derivation is again, as mentioned,
simplified by the fact that the contribution from the variation of the deficit
angle is zero
δ I R = δ(A h ) · δ h . (6.52)
∑
triangles h
In the discrete case the field equations reduce therefore to
∂V s
λ 0 ∑
s
∂l ij
− k ∑
h
δ h
∂A h
∂l ij
= 0 , (6.53)
and the derivatives can then be worked out for example from Eq. (6.5). Alternatively,
a rather convenient and compact expression can be given (Hartle, 1984) for the
derivative of the squared volume V 2 n of an arbitrary n-simplex with respect to one of
its squared edge lengths
∂V 2 n
∂l 2 ij
= 1 n 2 ω n−1 · ω ′ n−1 , (6.54)