Kiefer C. Quantum gravity

THE SEMICLASSICAL APPROXIMATION 167
Various approximations can now be performed. In a first step, one can assume
that 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
[ ( ) 2
1 ∂
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 shall
neglect the connection in the following. 19
In a second step, one can perform a standard semiclassical (WKB) approximation
for the heavy part through the ansatz
χ 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 2
χ n
= ∂2 C n
∂Q 2 + 2iM C n
( M
−
∂C n
∂Q
∂S n
∂Q
) 2 ( ∂Sn
∂Q
) 2
χ n + iM
χ n
C n
∂ 2 S n
∂Q 2 χ n . (5.133)
Assuming M to be large corresponds to neglecting derivatives of C n and second
derivatives of S n (the usual assumptions for WKB). One then has
∂ 2 χ n
∂Q 2
≈− ( M
) 2 ( ) 2 ∂Sn
χ n . (5.134)
∂Q
The classical momentum is then given by
P n = M ∂S n
∂Q ≈
and (5.131) becomes the Hamilton–Jacobi equation,
iχ n
∂χ n
∂Q , (5.135)
H cl ≡ P 2 n
2M + V (Q)+E n(Q) =E. (5.136)
Since E n (Q) =〈ψ n |h|ψ n 〉, this corresponds, in the gravitational context, to the
semiclassical Einstein equations discussedinSection1.2,wheretheexpectation
value of the energy–momentum tensor appears.
19 An intriguing idea would be to derive the connection in gauge theories along these lines.