Page 45 - Nature Of Space And Time

of the coordinates x i . This implies that the wave function has to obey four functional
di erential equations. Three of these equations are called the momentum constraints.

Momentum Constraint Equation

@
=0
@h ij
;j

They express the fact that the wave function should be the same for di erent 3 metrics
h ij that can be obtained from each other by transformations of the coordinates x i .The
fourth equation is called the Wheeler-DeWitt equation.

Wheeler - DeWitt Equation

2
@ 1 3
G ijkl − h 2 R = 0
@h ij @h kl

It corresponds to the independence of the wave function on . One can think of it as the
Schrodinger equation for the universe. But there is no time derivative term because the
wave function does not depend on time explicitly.

In order to estimate the wave function of the universe, one can use the saddle point
approximation to the path integral as in the case of black holes. One nds a Euclidean
metric g 0 on the manifold M + that satis es the eld equations and induces the metric
h ij on the boundary . One can then expand the action in a power series around the
background metric g 0.
1
I[g]= I[g 0]+ gI 2 g + :::
2

As before the term linear in the perturbations vanishes. The quadratic term can be re-
garded as giving the contribution of gravitons on the background and the higher order
terms as interactions between the gravitons. These can be ignored when the radius of
curvature of the background is large compared to the Planck scale. Therefore

1
≈ 1 e −I[g o ]
(det I 2) 2

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