Quantum Gravity

136 QUANTUM GEOMETRODYNAMICSBell (1987). 2 For an operator corresponding to a classical observable satisfying{O, H µ }≈0, one would expect that in the quantum theory the relation[Ô, Ĥµ]Ψ = 0 (5.8)holds. For operators ˆF with [ ˆF,Ĥµ]Ψ ≠ 0 one would have Ĥµ( ˆF Ψ) ≠0.Thisissometimes interpreted as meaning that the ‘measurement’ of the quantity beingrelated to this operator leads to a state that is no longer annihilated by theconstraints, ‘throwing one out’ of the solution space. This would, however, onlybe a problem for a ‘collapse’ interpretation of quantum gravity, an interpretationthat seems to be highly unlikely to hold in quantum gravity; see Chapter 10.Since the classical Hamiltonian and diffeomorphism constraints differ fromeach other in their interpretation (Chapter 4), the same should hold for theirquantum versions (5.6) and (5.7). This is, in fact, the case and will be discussedin detail in this chapter. The distinction between ‘observables’ and ‘perennials’,see (4.82) and (4.83), thus applies also to the quantum case.The sixth (and last) step concerns the role of the physical Hilbert space (cf.also step 3). Do the observables have to be represented in some Hilbert space?If yes, which one? It can certainly not be the auxiliary space F, but it is unclearwhether it is F 0 or only F phys ⊂F 0 . Moreover, it may turn out that only aconstruction with rigged Hilbert spaces (Gel’fand triples) is possible; this is infact the case in the loop representation (Section 6.1.2).A general method to deal with the construction of a physical Hilbert spacein the quantization of constrained systems is the group averaging procedure. Ithas been shown that this method works and yields a unique F phys at least forfinite dimensional compact Lie groups; cf. Giulini and Marolf (1999). (As for anextension to non-compact Lie groups, see Louko (2006).) The situation for GR,where the constraint algebra is not a Lie algebra at all, remains unclear.To represent all perennials by self-adjoint operators in Hilbert space wouldbe contradictory: be ˆF and Ĝ self-adjoint perennials, then the product ˆF Ĝ isagain a perennial, but no longer self-adjoint, since(ˆF Ĝ) †= Ĝ † ˆF † =Ĝ ˆFin general≠ ˆF Ĝ.Since, moreover, the fundamental variables h ab and p cd are not perennials, onemight, at this stage, forget about this notion. The ‘problem of Hilbert space’ isintimately connected with the ‘problem of time’ in quantum gravity, to whichwe shall now turn.5.2 The problem of timeThe concepts of time in GR and quantum theory differ drastically from eachother. As already remarked in Section 1.1, time in quantum theory is an external2 These are quantities that are subject to decoherence; see Chapter 10.