Quantum Gravity

252 QUANTUM COSMOLOGYWhereas in the above example it is possible to construct wave packets followingthe classical trajectory without dispersion, this is actually not possiblein the general case. Already for m ≠ 0, one finds that the demand for ψ → 0as a →∞is in conflict with the existence of narrow wave packets all along theclassical trajectory (Kiefer 1988). Consider the Wheeler–DeWitt equation (8.16)for Λ = 0,( )∂2∂α 2 − ∂2∂φ 2 − e4α +e 6α m 2 φ 2 ψ(α, φ) =0. (8.25)Within a Born–Oppenheimer type of approximation, one can make the followingansatz for the wave packet,ψ(a, φ) = ∑ (√ )c n (α)ϕ n me3αφ , (8.26)nwhere the ϕ n denote the usual eigenfunctions of the harmonic oscillator. From(8.25), one gets the following effective potentials for the c n (α),V n (α) = 1 2(−e 4α +(2n +1)me 3α) . (8.27)These potentials become negative for large enough α. In the classical theory, thismeans that trajectories are drawn into the region with negative V n and are reflected.In the quantum theory, it means that the wave function is a combinationof exponentially increasing or decreasing solutions. In order to fulfil the ‘finalcondition’ ψ → 0fora →∞, the exponentially decreasing solution has to bechosen. The n-dependent reflection expressed by (8.27) leads to an unavoidablespreading of the wave packet (Kiefer 1988). This means that the semiclassicalapproximation does not hold all along the expanding and recollapsing part ofthe classical trajectory. How, then, does classical behaviour emerge? The answeris provided by adding other degrees of freedom (Section 8.2). They can act asa kind of environment for the minisuperspace variables a and φ and force themto behave classically. This process of decoherence is discussed in Section 10.1.It has also to be emphasized that wave packets are here always understood ascorresponding to branches of the full wave function (representing quasiclassicalhistories), but not to the full wave function itself.Another example is a closed Friedmann universe with a non-minimally coupledscalar field. In the general case, the Wheeler–DeWitt equation can becomeelliptic instead of hyperbolic for a certain range of field values; see Kiefer (1989).This would modify the ‘initial-value problem’ in quantum cosmology. No ellipticregion occurs, for example, in the simplest case—the case of a conformallycoupled field φ. Choosing units such that 8πG = 1 and performing a field redefinition√2πaφφ → χ = ,6one obtains the following Wheeler–DeWitt equation (Zeh 1988; Kiefer 1990):