Approaches to Quantum Gravity

Loop quantum gravity 247U(ϕ)T n = T ϕ(n) , ϕ((γ, D)) = (ϕ(γ ), D). We define the (generalised) knot classes[n] :={ϕ(n); ϕ ∈ Diff()} and with it the distributionsl [n] (T n ′) := χ [n] (n ′ ) = δ [n],[n ′ ] (13.17)where χ B denotes the characteristic function of the set B. The solution space consistsof the linear span of the distributions (13.17) which can be given a Hilbertspace structure H Diff by completing it in the scalar product〈l [n] , l [n ′ ]〉 Diff := l [n ′ ](T n ). (13.18)Let us now turn to the final Hamiltonian or Wheeler–DeWitt constraint which interms of A, E takes the formC =|det(E)| −1/2 Tr([(1 + ι 2 )[K a , K b ]−F ab ][E a , E b ]) (13.19)where ιK (A, E) = A−Ɣ(E) and F is the curvature of A. It is obvious that (13.19)presents a challenge for the representation H Kin because it is a non-polynomialfunction of the unsmeared functions E which become operator valued distributions.∫Indeed, in order to define the smeared Hamiltonian constraint C(α) =d3 xαC we must proceed entirely differently from the Gauss or spatial diffeomorphismconstraint because it does not generate a Lie algebra due to the structurefunctions involved. One can proceed as follows: one point splits (regularises) theconstraint (13.19), thus arriving at a well defined operator Ĉ ɛ (N) and then takesthe limit ɛ → 0 in a suitable operator topology. The operator topology that naturallysuggests itself is a weak topology based on the space H Diff viewed as a spaceof linear functionals over (a dense subspace of ) H Kin . It turns out that the limitexists in this topology precisely due to spatial diffeomorphism invariance of thedistributions l [n] . In a technically precise sense, the group Diff() swallows theultraviolet regulator because in a background independent framework there is nomeaning to the notion of “short” distance behaviour. One can also show that thecommutator [Ĉ(N), Ĉ(N ′ )] is non-vanishing but that its dual 12 annihilates H Diff .As one can show [9], also the classical Poisson bracket {C(N), C(N ′ )} vanisheson the constraint surface defined by the spatial diffeomorphism constraint, hencewe get a consistent constraint algebra. However, the disadvantage of this procedure[25; 26; 27; 28; 29; 30; 31; 32] is that one does not have access to a physical innerproduct.The more elegant solution uses the Master constraint technique outlined in theprevious subsection. Recall the relation {C( ⃗N), C(N)} ∝C( ⃗N[N]) which says12 Given a Hilbert space H with dense subspace on which a operator A is defined together with its adjoint,the dual A ′ on the space ∗ of linear functionals l on is defined by (A ′ l)[ f ]:=l(A † f ) for all f ∈ .