Kiefer C. Quantum gravity

176 QUANTUM GEOMETRODYNAMICS
Section 5.3.4. In the semiclassical limit, however, a background time parameter
is available, with respect to which Feynman ‘propagators’ can be formulated.
Let us, therefore, look for a two-point solution of the Wheeler–DeWitt equations
(i.e. Wheeler–DeWitt equation and momentum constraints) in the form of
the ansatz
ˆK(q + ,q − )=P (q + ,q − )e im2 P S(q +,q − )Û(q+
,q − ) , (5.181)
where we denote as above by a hat, the operators acting in the Hilbert space
of matter fields. Here, S(q + ,q − ) satisfies (5.174), and P (q + ,q − ) is the preexponential
factor (5.180). Substituting this ansatz into the system of the Wheeler–DeWitt
equations and taking into account the Hamilton–Jacobi equations
and the continuity equations for P (q + ,q − ), we get for the ‘evolution’ operator
Û(q + ,q − ) the equations
iD⊥ k D kÛ = Ĥm ⊥ Û − 1
2m 2 P −1 G mn
⊥ D mD n (P Û) ,
P
(5.182)
iDaD k k Û = Ĥm a Û , (5.183)
where all the derivatives are understood as acting on the argument q + . Evaluating
this operator at the classical extremal q + → q(t),
Û(t) =Û(q(t),q −) , (5.184)
where q(t) satisfies the canonical equations of motion corresponding to S(q + ,q − ),
one easily obtains the quasi-evolutionary equation
with the effective matter Hamiltonian
˙ q i = N µ ∇ i µ , (5.185)
i ∂ ∂tÛ(t) =Ĥeff Û(t) (5.186)
Ĥ eff = Ĥm − 1
2m 2 NG mn
⊥ D m D n [ P Û ]P −1 Û −1 . (5.187)
P
(Recall that we use a condensed notation and that this equation is, in fact, an
integral equation.) The first term on the right-hand side is the Hamiltonian of
matter fields at the gravitational background of (q, N)-variables,
Ĥ m = N µ Ĥ m µ . (5.188)
The second term involves the operator Û itself in a non-linear way and contributes
only at order m −2
P
of the expansion. Thus, (5.186) is not a true linear