Feynman Path Integral Formulation

4.8 Semiclassical Expansion of the Wheeler-DeWitt Equation 115
problem of operator ordering in the above equations (in particular regarding the π 2
term, which classically can be written in a number of different and equivalent ways),
a discussion of what is meant by the time variable, and how it can be suitably defined
in concrete models, for example in cosmological applications. In addition one needs
to be specific about a suitable Hilbert space, which entails at some point a specific
choice for the inner product of wave functionals over the space Σ (and thus a notion
of self-adjointness for operators), for example in the Schrödinger form
∫
〈Ψ|Φ〉 = dμ[g] Ψ ∗ [g ij ]Φ[g ij ] , (4.74)
Σ
where dμ[g] is some appropriate measure over the three-metric g. The latter does
not seem to be the only choice, since a Klein Gordon inner product could be used
instead, which is not positive definite.
Another peculiar property of the Wheeler-DeWitt equation, and which distinguishes
it from the usual Schrödinger equation HΨ = i¯h∂ t Ψ, is the absence of an
explicit time coordinate. As a result the r.h.s. term of the Schrödinger equation is
here entirely absent. The reason is of course diffeomorphism invariance of the underlying
theory, which expresses now the fundamental quantum equations in terms
of fields g ij , and not coordinates. As a result the Wheeler-DeWitt equation contains
no explicit time evolution parameter, a problem that is usually referred to as
the problem of time (see for example Kucha˘r, 1992). Nevertheless in some cases
it seems possible to assign the interpretation of “time coordinate” to some specific
variable entering the Wheeler-DeWitt equation, such as the overall spatial volume
or the magnitude of some scalar field. But in general a consistent and unambiguous
prescription does not seem to be known yet.
4.8 Semiclassical Expansion of the Wheeler-DeWitt Equation
The simplest approach to finding solutions to the Wheeler-DeWitt equations, besides
working with the linearized theory, is to expand around the classical theory.
One writes a (WKB-type) ansatz for the wave functional Ψ,
Ψ[g ij (x)] = exp { i
16πG S[g ij] } Φ[g ij (x)] , (4.75)
where the action function S[g ij ] is a solution of the Hamilton-Jacobi equations for
classical gravity (Peres, 1962),
δS
G ij,kl
δg ij
δS
δg kl
− √ g ( 3 R − 2λ ) = 0
2ig ij ∇ k
δS
δg jk
= 0 , (4.76)