Feynman Path Integral Formulation

136 4 Hamiltonian and Wheeler-DeWitt Equation
for discretization are possible. One could discretize the theory from the very beginning,
while it it is still formulated in terms of an action, and introduce for it lapse
and shift functions, extrinsic and intrinsic discrete curvatures, etc. Alternatively one
could try to discretize the continuum Wheeler-deWitt equation directly, a procedure
that makes sense in the lattice formulation, as these equations are still given in
terms of geometric objects, for which the Regge theory is well suited. It is the latter
approach which we will outline here.
The starting point for the following discussion is therefore the Wheeler-DeWitt
equation for pure gravity in the absence of matter, Eq. (4.72),
{
−(16πG) 2 δ 2
G ij,kl (x)
δg ij (x)δg kl (x) − √ g(x) ( 3 R(x) − 2λ )} Ψ[g ij (x)] = 0 ,
(4.168)
and the diffeomorphism constraint of Eq. (4.73),
{
}
δ
2ig ij (x)∇ k (x) Ψ[g ij (x)] = 0 . (4.169)
δg jk (x)
Note that there is a constraint on the state |Ψ〉 at every x, each of the form
Ĥ(x)|Ψ〉 = 0 and Ĥ i (x)|Ψ〉 = 0.
On a simplicial lattice (see Sect. 6.1 for a discussion of the lattice formulation
for gravity) one knows that deformations of the squared edge lengths are linearly
related to deformations of the induced metric, within a given simplex s and based
on a vertex 0, as discussed elsewhere here, for example in Eq. (6.3),
g ij (s) = 1 (
2 l
2
0i + l0 2 j − lij
2 )
. (4.170)
Note that in the following discussion only edges and volumes along the spatial direction
are involved. Furthermore, one can introduce in a natural way a lattice analog
of the DeWitt supermetric of Eq. (4.66), by writing
‖δl 2 ‖ 2 = ∑
ij
G ij (l 2 ) δl 2 i δl 2 j , (4.171)
with the quantity G ij (l 2 ) suitably defined on the space of squared edge lengths
(Lund and Regge, 1974; Hartle, Miller and Williams, 1997). Through a straightforward
exercise of varying the squared volume of a given simplex s in d dimensions
V 2 (s) = ( 1
d!
) 2
detgij [l 2 (s)] , (4.172)
to quadratic order in the metric (on the r.h.s.), or in the squared edge lengths (on the
l.h.s.), one finds the identity
1
V (l 2 ) ∑ ij
∂ 2 V 2 (l 2 )
∂l 2 i ∂l2 j
δl 2 i δl 2 j = 1 d!
√ [ ]
det(g ij ) g ij g kl δg ij δg kl − g ij g kl δg jk δg li
.
(4.173)