Kiefer C. Quantum gravity

246 QUANTUM COSMOLOGY
8.1.2 Quantization of a Friedmann universe
As an example, we shall treat in some detail the case of a closed Friedmann
universe with a massive scalar field; cf. Kiefer (1988) and Halliwell (1991). Classically,
the model is thus characterized by the scale factor a(t) and the homogeneous
field φ(t)withmassm. In the quantum theory, the classical time parameter
t is absent, and the system is fully characterized by a wave function ψ(a, φ). The
ansatz for the classical line element is
ds 2 = −N 2 (t)dt 2 + a 2 (t)dΩ 2 3 , (8.1)
where dΩ 2 3 =dχ2 +sin 2 χ(dθ 2 +sin 2 θdϕ 2 ) is the standard line element on S 3 .A
special foliation has thus been chosen in order to capture the symmetries of this
model. For this reason no shift vector appears, only the lapse function N. The
latter naturally occurs in combination with dt, expressing the classical invariance
under reparametrizations of the time parameter; see Chapter 3.
The three-metric h ab is fully specified by the scale factor a. The second fundamental
form, cf. (4.48), reads here
K ab = 1 ∂h ab
= ȧ
2N ∂t aN h ab . (8.2)
Its trace is
K ≡ K ab h ab = 3ȧ
Na , (8.3)
which is thus proportional to the Hubble parameter ȧ/a. Itsinverse,K −1 ,isfor
this reason called ‘extrinsic time’; cf. Section 5.2.
Since the model fulfils the symmetric criticality principle, we can insert the
ansatz (8.1) directly into the Einstein–Hilbert action (1.1) and derive the Euler–
Lagrange equations from the reduced action. For the surface term in (1.1), one
obtains
Inserting
∫
1
8πG
d 3 x √ hK = 3 ∫
8πG
d 3 x √ h ȧ
Na .
√
h d 3 x = a 3 sin 2 χ sin θ dχdθdϕ ,
one finds that this surface term is cancelled by a term that appears after partial
integration from the first term in (1.1). This is how the general surface term is
constructed. More explicitly, one has for the curvature scalar:
(
R = 6
N 2 − Ṅȧ
Na + ä [ȧ ] ) 2
a + + 6 a a 2 .
Partial integration of the second term in the parentheses cancels both the surface
term and the first term and changes the sign of the third term.