Quantum Gravity

DECOHERENCE AND THE QUANTUM UNIVERSE 317while in the vicinity of the diagonal one finds)| ˜D(t|φ, φ ′ )|∼exp(− π232 (H − H′ ) 2 t 2 e 4Ht , (10.39)∼ exp(− π2 H 4 a 2 )(a − a ′ ) 2 , Ht ≫ 1 . (10.40)8These expressions exhibit a rapid disappearance of non-diagonal elements duringthe inflationary evolution. The universe thus assumes classical properties at theonset of inflation. This justifies the use of classical cosmology since then.The decohering influence of fermionic degrees of freedom has to be treatedseparately (Barvinsky et al. 1999b). It turns out that they are less efficient inproducing decoherence. In the massless case, for example, their influence is fullyabsent.The above analysis of decoherence was based on the state (10.9). One might,however, start with a quantum state which is a superposition of many semiclassicalcomponents, that is, many components of the form exp(iS0 k ), where eachS0 k is a solution of the Hamilton–Jacobi equation for a and φ. Decoherence betweendifferent such semiclassical branches has also been the subject of intenseinvestigation (Halliwell 1989; Kiefer 1992). The important point is that decoherencebetween different branches is usually weaker than the above discusseddecoherence within one branch. Moreover, it usually follows from the presenceof decoherence within one branch. In the special case of a superposition of (10.9)with its complex conjugate, one can immediately recognize that decoherence betweenthe semiclassical components is smaller than within one component: in theexpression (10.20) for the decoherence factor, the term Ω n +Ω ′∗ n in the denominatoris replaced by Ω n +Ω ′ n . Therefore, the imaginary parts of the frequencyfunctions add up instead of partially cancelling each other and (10.20) becomessmaller. One also finds that the decoherence factor is equal to one for vanishingexpansion of the semiclassical universe (Kiefer 1992).We note that the decoherence between the exp(iS 0 ) and exp(−iS 0 )componentscan be interpreted as a symmetry breaking in analogy to the case of sugarmolecules (Joos et al. 2003). There, the Hamiltonian is invariant under space reflections,but the state of the sugar molecules exhibits chirality. Here, the Hamiltonianin the Wheeler–DeWitt equation is invariant under complex conjugation,while the ‘actual states’ (i.e. one decohering WKB component in the total superposition)are of the form exp(iS 0 ) and are thus intrinsically complex. It istherefore not surprising that the recovery of the classical world follows only forcomplex states, in spite of the real nature of the Wheeler–DeWitt equation (seein this context Barbour 1993). Since this is a prerequisite for the derivation of theSchrödinger equation, one might even say that time (the WKB time parameterin the Schrödinger equation) arises from symmetry breaking.The above considerations thus lead to the following picture. The universewas essentially ‘quantum’ at the onset of inflation. Mainly due to bosonic fields,decoherence set in and led to the emergence of many ‘quasi-classical branches’