Kiefer C. Quantum gravity

THE SEMICLASSICAL APPROXIMATION 175
the Wheeler–DeWitt equations at one loop and which are analogous to (5.143)
in the quantum-mechanical example,
D i (D i µ P 2 )=U λ µλ P 2 , (5.175)
D i µ ≡ ∂Hg µ
∂p i
∣ ∣∣∣p
= ∂S/∂q
, (5.176)
with the generators Dµ i here evaluated at the Hamilton–Jacobi values of the
canonical momenta. The solution of this equation turns out to be a particular
generalization of the Pauli–van Vleck–Morette formula—the determinant calculated
on the subspace of non-degeneracy for the matrix
S ik ′ = ∂2 S(q, q ′ )
∂q i ∂q k′ . (5.177)
This matrix has the generators D i µ as zero-eigenvalue eigenvectors (Barvinsky
and Krykhtin 1993). An invariant algorithm of calculating this determinant is
equivalent to the Faddeev–Popov gauge-fixing procedure; cf. Section 2.2.3. It
consists in introducing a ‘gauge-breaking’ term to the matrix (5.177),
F ik ′ = S ik ′ + φ µ i c µνφ ν k ′ , (5.178)
formed with the aid of the gauge-fixing matrix c µν and two sets of arbitrary
covectors (of ‘gauge conditions’) φ µ i and φ ν k at the points q and ′ q′ , respectively.
They satisfy invertibility conditions for ‘Faddeev–Popov operators’ at these two
points,
J µ ν = φ µ i Di ν , J ≡ detJ µ ν ≠0, J ′µ ν
= φ µ i ′Di′ ν , J ′ ≡ detJ ′µ ν ≠0. (5.179)
In terms of these objects, the pre-exponential factor solving the continuity equations
(5.175) is given by
P =
[ detF ik ′
JJ ′ det c µν
] 1/2
, (5.180)
which is independent of the gauge fixing. This finishes the discussion of the
Born–Oppenheimer scheme at the highest level of approximation.
5.4.3 Quantum-gravitational correction terms
We shall now proceed to perform the semiclassical expansion for solutions to the
Wheeler–DeWitt equations. Since we are interested in giving an interpretation in
terms of Feynman diagrams, we shall not consider wave functionals but—as in the
last part of the last subsection— two-point solutions (‘propagators’). Due to the
absence of an external time parameter in the full theory, such two-point functions
play more the role of energy Green functions than ordinary propagators; see