Approaches to Quantum Gravity

Loop quantum gravity 241equivalence. Every element ∈ H Kin can be thought of as the collection of “Fouriercoefficients” ( ˆ(λ)) λ∈R + where (λ) ∈ H ⊕ (λ) and∫ HKin = dμ(λ) < Psi(λ), ˆ Psi ˆ′ (λ) > H ⊕ (λ) . (13.7)The point of the Fourier representation (13.6) is of course that it is adapted to ̂M,namely ̂M acts diagonally: ̂M( ˆ(λ)) = (λ ˆ(λ)). It follows that the physical Hilbertspace is given byThree remarks are in order.H Phys = H ⊕ (0). (13.8)1. While the representative μ is irrelevant, the representative N is crucial and requiresfurther physical input. For instance, if the point zero is of measure zero (lies entirelyin the continuous spectrum of ̂M) then we may choose the representative N suchthat N(0) = 0 which would mean that the physical Hilbert space is trivial. This iscertainly not what one wants. The input required is that we want an irreducible representationof the algebra of Dirac observables (gauge invariant functions), whichare automatically fibre preserving, on H Phys . This can be shown to drasticallyreduce the freedom in the choice of N.2. It may happen that the spectrum of ̂M does not contain the point zero at all inwhich case the physical Hilbert space again would be trivial. This can be the consequenceof an anomaly. In this case it turns out to be physically correct to replacêM with ̂M ′ := ̂M − min(spec(̂M))1 provided that the “normal ordering constant” isfinite and vanishes in the classical limit, that is, lim → 0 min(spec(̂M)) = 0, so that̂M ′ is a valid quantisation of M. Finiteness and the question whether ̂M is denselydefined at all crucially depends on the choice of K IJ .3. To see how an anomaly may arise, especially in the case of structure functions,suppose that Ĉ I , fˆK IJ are symmetric operators. Then the classical relation{C I , C J }= f K IJ C K is replaced by the quantum relation[Ĉ I , Ĉ J ]=i( ˆ f IJ K= i ˆ f IJ KĈ K + Ĉ K ˆ f IJ K )/2Ĉ K +− 22[Ĉ K , ˆ f IJ K ]i(13.9)where the symmetric ordering on the right hand side is a consequence of theantisymmetry of the commutator. It follows that any (generalised) solution ofĈ I = 0 for all I automatically satisfies also 2 ([Ĉ K , ˆ f IJ K ]/(i)) = 0for all I, J. However, the classical limit of that operator is 2 {C K , f IJ K } whichmight be non-vanishing, not even on the constraint surface. This means that thephysical Hilbert space is constrained more than the physical phase space and thusis not a proper quantisation of the classical system. We see in particular that in