Kiefer C. Quantum gravity

170 QUANTUM GEOMETRODYNAMICS
an expansion of the Dirac equation must be employed). A more general case is
the Klein–Gordon equation coupled to gravity and the electromagnetic field. This
leads to additional relativistic correction terms (Lämmerzahl 1995).
A major difference of the Klein–Gordon example to the first example is the
indefinite structure of the kinetic term (d’Alembertian instead of Laplacian).
Therefore, on the full level, the conserved inner product is the Klein–Gordon
one (cf. Section 5.2.2). In order c 0 of the approximation, one obtains from this
inner product the standard Schrödinger inner product as an approximation. The
next order yields corrections to the Schrödinger inner product proportional to
c −2 . Does this mean that unitarity is violated at this order? Not necessarily. In
the case of the Klein–Gordon equation in external gravitational and electromagnetic
fields, one can make a (t-dependent!) redefinition of wave functions and
Hamiltonian to arrive at a conserved Schrödinger inner product with respect to
which the Hamiltonian is Hermitian (Lämmerzahl 1995).
5.4.2 Derivation of the Schrödinger equation
Similar to the discussion of the examples in the last subsection one can perform
a semiclassical (‘Born–Oppenheimer’) approximation for the Wheeler–DeWitt
equation and the momentum constraints. In this way one can recover approximately
the limit of ordinary quantum field theory in an external gravitational
background. This is done in the Schrödinger picture, so this limit emerges through
the functional Schrödinger equation, not the quantum-mechanical Schrödinger
equation as in the last subsection. In the following, we shall mainly follow, with
elaborations, the presentation in Barvinsky and Kiefer (1998); see also Kiefer
(1994) and references therein.
The starting point is the Wheeler–DeWitt equation (5.18) and the momentum
constraint (5.19). Taking into account non-gravitational degrees of freedom, these
equations can be written in the following form:
{
− 1 δ 2
2m 2 G abcd − 2m 2 P
P
δh ab δh cd
{
− 2 i h abD c
√
h (3) R + Ĥm ⊥
δ
+
δh Ĥm a
bc
}
|Ψ[h ab ]〉 =0, (5.150)
}
|Ψ[h ab ]〉 =0. (5.151)
Here, m 2 P =(32πG)−1 , =1,Λ=0,andĤm ⊥ and Ĥm a denote the contributions
from non-gravitational fields. To be concrete, we think about the presence of a
scalar field. The notation |Ψ[h ab ]〉 means: Ψ is a wave functional with respect to
the three-metric h ab and a state in the standard Hilbert space referring to the
scalar field (bra- and ket-notation).
The situation is now formally similar to the example discussed in the previous
subsection. One of the main differences is the presence of the momentum constraints
(5.151), which have no analogue in the quantum-mechanical example.
Comparing (5.150) with (5.123), one notes the following formal correspondence
between the terms: