Kiefer C. Quantum gravity

272 QUANTUM COSMOLOGY
8.3.5 Symmetric initial condition
This condition has been proposed by Conradi and Zeh (1991); see also Conradi
(1992). For the wave-packet solutions of Section 8.1, we had to demand that ψ
goes to zero for α →∞. Otherwise, the packet would not reflect the behaviour
of a classically recollapsing universe. But what about the behaviour at α →−∞
(a → 0)? Consider again the model of a massive scalar field in a Friedmann universe
given by the Wheeler–DeWitt equation (8.25). The potential term vanishes
in the limit α →−∞, so the solutions which are exponentially decreasing for
large α become constant in this limit. With regard to finding normalizable solutions,
it would be ideal if there were a reflecting potential also at α →−∞.One
can add for this purpose in an ad hoc manner a repulsive (negative) potential
that would be of relevance only in the Planck regime. The application of loop
quantum gravity to cosmology as discussed in the next section can lead to the
occurrence of such a respulsive potential. It is also imaginable that it results from
a unification of interactions. One can, for example, choose the ‘Planck potential’
V P (α) =−C 2 e −2α , (8.78)
where C is a real constant. Neglecting as in the previous subsections the φ-
derivatives (corresponding to the slow-roll approximation), one can thereby select
a solution to the Wheeler–DeWitt equation ((8.25) supplemented by V P )
that decreases exponentially towards α →−∞. This implements also DeWitt’s
boundary condition that ψ → 0forα →−∞(Section 8.3.1).
The ‘symmetric initial condition’ (SIC) now states that the full wave function
depends for α →−∞only on α; cf. Conradi and Zeh (1991). In other words,
it is a particular superposition (not an ensemble) of all excited states of Φ and
the three-metric, that is, these degrees of freedom are completely absent in the
wave function. This is analogous to the symmetric vacuum state in field theory
before the symmetry breaking into the ‘false’ vacuum (Zeh 2001). In both cases,
the actual symmetry breaking will occur through decoherence (Section 10.1).
The resulting wave function coincides approximately with the no-boundary wave
function. Like there, the higher multipoles enter the semiclassical Friedmann
regime in their ground state. The SIC is also well suited for a discussion of the
arrow of time and the dynamical origin of irreversibility; cf. Section 10.2.
8.4 Loop quantum cosmology
8.4.1 Classical variables
The cosmological applications of quantum gravity discussed above mainly make
use of the geometrodynamical variables. However, quantum cosmology can also
be treated using the loop variables discussed in Chapter 6. The ensuing framework
of loop quantum cosmology was introduced by Martin Bojowald; see Bojowald
(2005) for a detailed review and references. The mathematical structure
of loop quantum cosmology is presented in Ashtekar et al. (2003). Similarly
to the minisuperspace approach discussed above, loop quantum cosmology is