Quantum Gravity

168 QUANTUM GEOMETRODYNAMICSOne can now introduce a time coordinate t via the Hamilton equations ofmotion,ddt P n = − ∂∂Q H cl = − ∂∂Q (V (Q)+E n(Q)) ,ddt Q =∂∂P nH cl = P nM . (5.137)In fact, the very definition of t depends on n, and we call it therefore t n in thefollowing. Since it arises from the WKB approximation (5.132), it is called WKBtime (Zeh 1988). The last term in (5.130) can then be written as−2Mχ n∂χ n∂Q∂ψ n∂Q ≈−i∂S n ∂ψ n∂Q ∂Q ≡−i∂ψ n. (5.138)∂t nThis means that ψ n is evaluated along a particular classical trajectory of the‘heavy’ variable, ψ n (Q(t n ),q) ≡ ψ n (t n ,q). Assuming slow variation of ψ n withrespect to Q, one can neglect the term proportional to ∂ 2 ψ n /∂Q 2 in (5.130).Also using (5.136), one is then left with∑[χ n h(q, t n ) − E n (t n ) − i ∂ ]ψ n (t n ,q)=0. (5.139)∂t nnThis equation still describes a coupling between ‘heavy’ and ‘light’ part.In a third and last step one can assume that instead of the whole sum (5.124)only one component is available, that is, one has—up to an (adiabatic) dependenceof ψ on Q—a factorizing state,χ(Q)ψ(q, Q) .This lack of entanglement can of course only arise in certain situations and mustbe dynamically justified (through decoherence; cf. Chapter 10). If it happens,and after absorbing E n (t) into a redefinition of ψ (yielding only a phase), onegets from (5.139):i ∂ψ = hψ , (5.140)∂tthat is, just the Schrödinger equation. The ‘heavy’ system acts as a ‘clock’ anddefines the time with respect to which the ‘light’ system evolves. Therefore,a time-dependent Schrödinger equation has arisen for one of the subsystems,although the full Schrödinger equation is of a stationary form; cf. Mott (1931).Considering the terms with order M in (5.133), one finds an equation for theC n ,2 ∂C n∂Qor in the case of one component only,∂S n∂Q + ∂2 S n∂Q 2 C n =0, (5.141)