Quantum Gravity

LOOP QUANTUM COSMOLOGY 273constructed via a truncation of the classical phase space of GR to spatiallyhomogeneous situations, which is then quantized by using the methods and resultsof loop quantum gravity (Chapter 6). Features such as the quantizationof geometric operators are thereby transferred to the truncated models. In thepresent 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 newcanonical variables which result from the truncation of the general canonicalvariables holonomy and triad to the homogeneous and isotropic model. Howthis truncation is performed in a mathematically clean way is shown in detailin Bojowald (2005) and the references therein. From the triad one obtains thesingle variable ˜p, while the holonomy leads to the single variable ˜c. Howarethey defined? We shall in the following assume a Friedmann universe with finitespatial 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 bywhere from (8.9) we have|˜p| = a 2 , ˜c = k + βȧ , (8.79)ȧ = − 4πG Np a,3V 0 aand β is the Barbero–Immirzi parameter introduced in Section 4.3.1. The Poissonbracket between the new variables reads{˜c, ˜p} = 8πGβ .3V 0It is convenient to absorb the volume V 0 into the canonical variables by thesubstitution˜p = V −2/30 p, ˜c = V −1/30 c,leading to{c, p} = 8πGβ . (8.80)3We 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 orientationsare thus present in the formalism, a feature that will play a central role in thequantum theory.The Hamiltonian constraint (8.10) can easily be rewritten in terms of the newvariables. If the lapse function is chosen as N = 1, it reads (using the identityk 2 = k)H = − 3 ( ) (c − k)2 √|p|8πG β 2 + k 2 + Hm ≈ 0 , (8.81)