Quantum Gravity

THE SEMICLASSICAL APPROXIMATION 167Various approximations can now be performed. In a first step, one can assumethat the ‘heavy’ part is approximately insensitive to changes in the ‘light’ part.This enables one to neglect the off-diagonal parts in (5.129), leading to[ ( ) 21 ∂2M i ∂Q − A nn(Q) + V (Q)+E n (Q)]χ n (Q) =Eχ n (Q) , (5.131)where E n (Q) ≡ ɛ nn (Q). For real ψ n , the connection vanishes, A nn =0.Otherwise,it leads to a geometric phase (‘Berry phase’); cf. Berry (1984). We shallneglect the connection in the following. 19In a second step, one can perform a standard semiclassical (WKB) approximationfor the heavy part through the ansatzχ n (Q) =C n (Q)e iMSn(Q)/ . (5.132)This is inserted into (5.131). For the Q-derivative, one gets∂ 2 χ n∂Q 2χ n= ∂2 C n∂Q 2 + 2iM C n ( M−∂C n∂Q∂S n∂Q) 2 ( ∂Sn∂Q) 2χ n + iM χ nC n∂ 2 S n∂Q 2 χ n . (5.133)Assuming M to be large corresponds to neglecting derivatives of C n and secondderivatives of S n (the usual assumptions for WKB). One then has∂ 2 χ n∂Q 2≈− ( M) 2 ( ) 2 ∂Snχ n . (5.134)∂QThe classical momentum is then given byP n = M ∂S n∂Q ≈and (5.131) becomes the Hamilton–Jacobi equation,iχ n∂χ n∂Q , (5.135)H cl ≡ P 2 n2M + V (Q)+E n(Q) =E. (5.136)Since E n (Q) =〈ψ n |h|ψ n 〉, this corresponds, in the gravitational context, to thesemiclassical Einstein equations discussedinSection1.2,wheretheexpectationvalue of the energy–momentum tensor appears.19 An intriguing idea would be to derive the connection in gauge theories along these lines.