Kiefer C. Quantum gravity

172 QUANTUM GEOMETRODYNAMICS
∫
∂
∂t |ψ(t)〉 = d 3 x ḣab(x,t) δ
δh ab (x) |ψ[h ab]〉 . (5.160)
This, then, leads to the functional Schrödinger equation for quantized matter
fields in the chosen external classical gravitational field,
i ∂ ∂t |ψ(t)〉 = Ĥm |ψ(t)〉 ,
∫
Ĥ m ≡ d 3 {N(x)Ĥm }
x
⊥ (x)+N a (x)Ĥm a (x) . (5.161)
Here, Ĥ m is the matter-field Hamiltonian in the Schrödinger picture, parametrically
depending on (generally non-static) metric coefficients of the curved
space–time background. This equation is the analogue of (5.140) in the quantummechanical
example. (The back-reaction terms have again been absorbed into the
phase of |ψ(t)〉.) The standard concept of time in quantum theory thus emerges
only in a semiclassical approximation—the Wheeler–DeWitt equation itself is
‘timeless’. 20
Such a derivation of quantum field theory from the Wheeler–DeWitt equations
dates back, on the level of cosmological models, to DeWitt (1967a). It was
later performed by Lapchinsky and Rubakov (1979) for generic gravitational
systems and discussed in various contexts in Banks (1985), Halliwell and Hawking
(1985), Hartle (1987), Kiefer (1987), Barvinsky (1989), Brout and Venturi
(1989), Singh and Padmanabhan (1989), Parentani (2000), and others; see also
Anderson (2006a, b) for a general discussion. Although performed on a formal
level only, this derivation yields an important bridge connecting the full theory
of quantum gravity with the limit of quantum field theory in an external
space–time; it lies behind many cosmological applications.
This ‘Born–Oppenheimer type of approach’ is also well suited for the calculation
of quantum-gravitational correction terms to the Schrödinger equation
(5.161). This will be discussed in the next subsection. As a preparation it is,
however, most appropriate to introduce again a condensed, so-called ‘DeWitt’,
notation; cf. Section 2.2. We introduce the notation
q i = h ab (x) , p i = p ab (x) , (5.162)
in which the condensed index i =(ab, x) includes both discrete tensor indices
and three-dimensional spatial coordinates x. In this way, the situation is formally
the same as for a finite-dimensional model. A similar notation can be introduced
for the constraints,
Hµ g (q, p) =(Hg ⊥ (x), Hg a (x)) , Hm µ (q, ϕ, p ϕ)=(H⊥ m (x), Hm a (x)) . (5.163)
The index µ enumerates the superhamiltonian and supermomenta of the theory
as well as their spatial coordinates, µ → (µ, x). In this notation, the functional
20 An attempt to extrapolate the standard interpretational framework of quantum theory
into the ‘timeless realm’ is the use of ‘evolving constants’ in the Heisenberg picture by Rovelli
(1991b).