Feynman Path Integral Formulation

4.8 Semiclassical Expansion of the Wheeler-DeWitt Equation 115problem of operator ordering in the above equations (in particular regarding the π 2term, 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 definedin concrete models, for example in cosmological applications. In addition one needsto be specific about a suitable Hilbert space, which entails at some point a specificchoice for the inner product of wave functionals over the space Σ (and thus a notionof self-adjointness for operators), for example in the Schrödinger form∫〈Ψ|Φ〉 = dμ[g] Ψ ∗ [g ij ]Φ[g ij ] , (4.74)Σwhere dμ[g] is some appropriate measure over the three-metric g. The latter doesnot seem to be the only choice, since a Klein Gordon inner product could be usedinstead, which is not positive definite.Another peculiar property of the Wheeler-DeWitt equation, and which distinguishesit from the usual Schrödinger equation HΨ = i¯h∂ t Ψ, is the absence of anexplicit time coordinate. As a result the r.h.s. term of the Schrödinger equation ishere entirely absent. The reason is of course diffeomorphism invariance of the underlyingtheory, which expresses now the fundamental quantum equations in termsof fields g ij , and not coordinates. As a result the Wheeler-DeWitt equation containsno explicit time evolution parameter, a problem that is usually referred to asthe problem of time (see for example Kucha˘r, 1992). Nevertheless in some casesit seems possible to assign the interpretation of “time coordinate” to some specificvariable entering the Wheeler-DeWitt equation, such as the overall spatial volumeor the magnitude of some scalar field. But in general a consistent and unambiguousprescription does not seem to be known yet.4.8 Semiclassical Expansion of the Wheeler-DeWitt EquationThe simplest approach to finding solutions to the Wheeler-DeWitt equations, besidesworking 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 { i16πG S[g ij] } Φ[g ij (x)] , (4.75)where the action function S[g ij ] is a solution of the Hamilton-Jacobi equations forclassical gravity (Peres, 1962),δSG ij,klδg ijδSδg kl− √ g ( 3 R − 2λ ) = 02ig ij ∇ kδSδg jk= 0 , (4.76)