Kiefer C. Quantum gravity

LOOP QUANTUM COSMOLOGY 273
constructed via a truncation of the classical phase space of GR to spatially
homogeneous situations, which is then quantized by using the methods and results
of loop quantum gravity (Chapter 6). Features such as the quantization
of geometric operators are thereby transferred to the truncated models. In the
present section we restrict ourselves to the simplest case of Friedmann universes;
anisotropic models as well as inhomogeneous situations can also be addressed
(Bojowald 2005).
We consider the model of a Friedmann universe containing a scalar field;
cf. Section 8.1.2. Instead of the original variables a and p a we shall use new
canonical variables which result from the truncation of the general canonical
variables holonomy and triad to the homogeneous and isotropic model. How
this truncation is performed in a mathematically clean way is shown in detail
in Bojowald (2005) and the references therein. From the triad one obtains the
single variable ˜p, while the holonomy leads to the single variable ˜c. Howare
they defined? We shall in the following assume a Friedmann universe with finite
spatial volume V 0 and allow it to be either positively curved (k =1)orflat
(k = 0). The new variables are then obtained from the ones in Section 8.1.2 by
where from (8.9) we have
|˜p| = a 2 , ˜c = k + βȧ , (8.79)
ȧ = − 4πG Np a
,
3V 0 a
and β is the Barbero–Immirzi parameter introduced in Section 4.3.1. The Poisson
bracket between the new variables reads
{˜c, ˜p} = 8πGβ .
3V 0
It is convenient to absorb the volume V 0 into the canonical variables by the
substitution
˜p = V −2/3
0 p, ˜c = V −1/3
0 c,
leading to
{c, p} = 8πGβ . (8.80)
3
We note that p has the physical dimension of a length squared, while c is dimensionless.
The sign of p reflects the orientation of the triad. Both orientations
are thus present in the formalism, a feature that will play a central role in the
quantum theory.
The Hamiltonian constraint (8.10) can easily be rewritten in terms of the new
variables. If the lapse function is chosen as N = 1, it reads (using the identity
k 2 = k)
H = − 3 ( ) (c − k)
2 √|p|
8πG β 2 + k 2 + Hm ≈ 0 , (8.81)