Quantum Gravity

THE GEOMETRODYNAMICAL WAVE FUNCTION 147π 0 (D F (S 3 )) = 0 ,so no θ-structure is available in the cosmologically interesting case S 3 .Aθstructureis present, for example, in the case of ‘Wheeler’s wormhole’, that is,for Σ = S 1 × S 2 .Inthatcase,π 0 (D F (S 1 × S 2 )) = Z 2 ⊕ Z 2 ,where Z 2 = {−1, 1}. Also for the three-torus S 1 ×S 1 ×S 1 , one has a non-vanishingπ 0 , but the expression is more complicated. Therefore, θ-sectors in quantum gravityare associated with the disconnectedness of Diff Σ. In the asymptotically flatcase, something interesting may occur in addition (Friedman and Sorkin 1980).If one allows for rotations at infinity, one can get half-integer spin states in casethat a 2π-rotation acts non-trivially, that is, if one cannot communicate a rotationby 2π at ∞ to the whole interior of space. An example of a manifold thatallows such states is Σ = R 3 #T 3 .5.3.2 WKB approximationAn important approximation in quantum mechanics is the WKB approximation.On a formal level, this can also be performed for equations (5.18) and (5.19).For this purpose, one makes the ansatz) iΨ[h ab ]=C[h ab ]exp( S[h ab] , (5.29)where C[h ab ] is a ‘slowly varying amplitude’ and S[h ab ] is a ‘rapidly varyingphase’ (an ‘eikonal’ like in geometrical optics). This corresponds top ab −→δS ,δh abwhich is the classical relation for the canonical momentum, and from (5.18) and(5.19) one finds the approximate equations√δS δS h16πG G abcd −δh ab δh cd 16πG ( (3) R − 2Λ) = 0 , (5.30)δSD a =0. (5.31)δh abIn the presence of matter one has additional terms. Equation (5.30) is theHamilton–Jacobi equation for the gravitational field (Peres 1962). Equation(5.31) expresses again the fact that S[h ab ] is invariant under coordinate transformations.One can show that (5.30) and (5.31) are fully equivalent to the classicalEinstein field equations (Gerlach 1969)—this is one of the ‘six routes to geometrodynamics’(Misner et al. 1973). This route again shows how the dynamical lawsfollow from the laws of the instant (Kuchař 1993).