Kiefer C. Quantum gravity

178 QUANTUM GEOMETRODYNAMICS
technical and can be found in Barvinsky and Kiefer (1998). Here we shall only
quote the main steps and include a brief discussion of the results.
The following quantities play a role in the discussion. First, we introduce a
collective notation for the full set of Lagrangian gravitational variables, which
includes both the spatial metric as well as lapse and shift functions,
g a ≡ (q i (t),N µ (t)) . (5.193)
This comprises the space–time metric. (Recall that q i (t) stands for the threemetric
h ab (x,t).) Next, the second functional derivatives of the gravitational
action with respect to the space–time metric is denoted by
δ 2 S[g]
S ab ≡
δg a (t)δg b (t ′ ) . (5.194)
Since S ab is not invertible, one must add gauge-fixing terms similar to (5.178).
This leads to an operator F ab . The ‘graviton propagator’ D bc is then defined as
its inverse via
F ab D bc = δa c . (5.195)
We also need the components of the Wronskian operator obtained from the
gravitational Lagrangian L g ,
→
W ib (d/dt) δg b (t) =−δ ∂Lg (q, ˙q,N)
∂ ˙q i . (5.196)
With the help of the ‘graviton propagator’, one can define
ˆt a (t) =− 1
m 2 P
∫ t+
t −
dt ′ D ab (t, t ′ ) ˆT b (t ′ ) ≡− 1
m 2 P
D ab ˆTb , (5.197)
where ˆT b is the condensed notation for the energy–momentum tensor of the
matter field. The quantity ˆt a (t) obeys the linearized Einstein equations with
source ˆT b and can thus be interpreted as the gravitational potential generated
by the back reaction of quantum matter on the gravitational background.
The first correction term in (5.191)—the contribution of quantum matter—is
found to read
− 1
2m 2 G mn (D m D n Û 0 )Û 0 −1
P
= 1 ( →W
2 m2 P G mn T ma ˆt a W → nb ˆt
b)
+ i 2 Dab w abc (t + ) ˆt c
− i (
2 Gmn ( W → ma D ac )( W → nb D bd ) S cde ˆt e + 1 )
m 2 Ŝ mat
cd . (5.198)
P
The resulting three terms can be given a Feynman diagrammatic representation
with different structure. Note that because of (5.197) all terms are of the same