Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 977on Σ. If M is simply–connected, which we will assume, the surface Σwill divide M into two parts M + and M − [Hawking and Israel (1979);Hawking and Penrose (1996)],Probability of h ij = Ψ + (h ij ) × Ψ − (h ij ), where∫Ψ + (h ij ) =d[g] exp(−A[g]).metrics on M + thatinduce h ij on ΣIn this case, the probability for Σ to have the metric h ij can be factorized.It is the product of two wave functions Ψ + and Ψ − . These are given bypath integrals over all metrics on M + and M − respectively, that inducethe given three metric h ij on Σ. In most cases, the two wave functions willbe equal and we will drop the superscripts + and +. Ψ is called the wavefunction of the universe. If there are matter fields φ, the wave functionwill also depend on their values φ 0 on Σ. But it will not depend explicitlyon time because there is no preferred time coordinate in a closed universe.The no boundary proposal implies that the wave function of the universe isgiven by a path integral over fields on a compact manifold M + whose onlyboundary is the surface Σ. The path integral is taken over all metrics andmatter fields on M + that agree with the metric h ij and matter fields φ 0 onΣ.One can describe the position of the surface Σ by a function τ of threecoordinates x i on Σ. But the wave function defined by the path integralcannot depend on τ or on the choice of the coordinates x i . This impliesthat the wave function Ψ has to obey four functional differential equations.Three of these equations are called the momentum constraint One candescribe the position of the surface Σ by a function τ of three coordinatesx i on Σ. But the wave function defined by the path integral cannot dependon τ or on the choice of the coordinates x i . This implies that the wavefunction Ψ has to obey four functional differential equations. ( Three of∂Ψthese equations are called the momentum constraint equation:∂h ij);j =0. They express the fact that the wave function should be the same fordifferent 3 metrics h ij that can be get from each other by transformationsof the coordinates x i . The fourth equation is called the Wheeler–DeWittequation∂(G 2)ijkl − h 1 2 3 R Ψ = 0.∂h ij ∂h klIt corresponds to the independence of the wave function on τ. One can