Feynman Path Integral Formulation

114 4 Hamiltonian and Wheeler-DeWitt Equation
constant conformal displacements in the three-metric δg ij (x)=δΩ 2 g ij (x) (Giulini,
1995).
4.7 Wheeler-DeWitt Equation
Within the framework of the previous construction, a transition from the classical to
the quantum description of gravity is obtained by promoting g ij , π ij , H and H a to
quantum operators, with ĝ ij and ˆπ ij satisfying canonical commutation relations. In
particular the classical constraints now select a physical vacuum state |Ψ〉, such that
in the source free case
Ĥ |Ψ〉 = 0 Ĥ i |Ψ〉 = 0 , (4.68)
and in the presence of sources more generally
ˆT |Ψ〉 = 0 ˆT i |Ψ〉 = 0 . (4.69)
As in ordinary non-relativistic quantum mechanics, one can choose different representations
for the canonically conjugate operators ĝ ij and ˆπ ij . In the functional
position representation one sets
ĝ ij (x) → g ij (x) ˆπ ij (x) →−i¯h · 16πG ·
δ
δg ij (x) . (4.70)
In further developing the analogy with standard non-relativistic quantum mechanics,
one notices that in this picture the quantum states become wave functionals of the
three-metric g ij (x),
|Ψ〉 →Ψ[g ij (x)] . (4.71)
The two quantum constraint equations in Eq. (4.69) become the Wheeler-DeWitt
equation
δ
{−16πG 2
· G ij,kl − 1 √ (
g
3
R − 2λ ) }
+ Ĥ φ Ψ[g ij (x)] = 0 ,
δg ij δg kl 16πG
(4.72)
and the diffeomorphism, or momentum, constraint
{
}
δ
2ig ij ∇ k + Ĥ φ i Ψ[g ij (x)] = 0 . (4.73)
δg jk
The last constraint implies that the gradient of Ψ on the superspace of g ij ’s and
φ’s is zero along those directions that correspond to gauge transformations, i.e. diffeomorphisms
on the three dimensional manifold whose points are labeled by the
coordinates x.
A number of basic issues need to be addressed before one can gain a full and
consistent understanding of the dynamical content of the theory. These include the