Quantum Gravity

ARROW OF TIME 323these variables (‘modes’) again by {f n }, one has for the total Hamiltonian in theWheeler–DeWitt equation, an expression of the form (cf. Zeh 2001)Ĥ = ∂2∂α 2 + ∑ n(− ∂2∂f 2 n)+ V n (α, f n ) + V int (α, {f n }) , (10.43)where the last term describes the interaction between the modes (assumed to besmall), and the V n describe the interaction of the mode f n with the scale factorα. Both terms have, in fact, the property that they vanish for α →−∞.Itis,therefore, possible to impose in this limit a separating solution of ĤΨ =0,Ψ α→−∞−→ψ(α) ∏ nχ n (f n ) , (10.44)that is, a solution of lacking entanglement. If this is taken as an ‘initial condition’,the Wheeler–DeWitt equation automatically—through the occurrence ofthe potentials in (10.43)—leads to a wave function which for increasing α becomesentangled between α and all modes. This, then, leads to an increase oflocal entropy, that is, an increase of the entropy which is connected with thesubset of ‘relevant’ degrees of freedom. Calling the latter {y i }, one hasS(α, {y i })=−k B tr(ρ ln ρ) , (10.45)where ρ is the reduced density matrix corresponding to α and {y i }. It is obtainedby tracing out all irrelevant degrees of freedom in the full wave function. Entropythus increases with increasing scale factor—this would be the gravitational arrowof time. It is also the arrow of time that is connected with decoherence. It is,therefore, the root for both the quantum mechanical and the thermodynamicalarrow of time. Quantum gravity could thus really yield the master arrow, theformal reason being the asymmetric appearance of α in the Wheeler–DeWittequation: the potential goes to zero near the big bang, but becomes highly nontrivialfor increasing size of the universe. It is an interesting question whethera boundary condition of the form (10.44) would automatically result from oneof the proposals discussed in Section 8.3. The symmetric initial condition (Section8.3.5) is an example where this can be achieved—in fact, this condition wastailored for this purpose. It is expected that one can apply such an analysis alsoto loop quantum gravity and to string theory.In the case of a classically recollapsing universe, the boundary condition(10.44) has interesting consequences: since it is formulated at α →∞,increasingentropy is always correlated with increasing α, that is, increasing size of the universe;cf. also Fig. 8.1. Consequently, the arrow of time formally reverses near theclassical turning point (Kiefer and Zeh 1995). It turns out that this region is fullyquantum, so no paradox arises; it just means that there are many quasi-classicalcomponents of the wave function, each describing a universe that is experiencedfrom within as expanding. All these components interfere destructively near the