Approaches to Quantum Gravity

240 T. Thiemannspectrum of all the Ĉ I , there is no non-trivial solution ∈ H Kin to the system ofquantum constraint equations Ĉ I = 0 for all I which is the quantum analogue ofthe classical system of constraint equations C I = 0 for all I (because this wouldmean that is a common zero eigenvector). For instance, the operator id/dx onL 2 (R, dx) has spectrum R but none of the formal “eigenvectors” exp(−ikx) witheigenvalue k is normalizable. Thus, the solution to the constraints has to be understooddifferently, namely in a generalised sense. This comes at the price that the solutionsmust be given a new Hilbert space inner product with respect to which they arenormalisable.We will now present a method to solve all the constraints and to construct an innerproduct induced from that of H Kin in a single stroke, see [12] and [13] formoredetails. Consider the Master constraintM := ∑ IJC I K IJ C J (13.5)where K IJ is a positive definite matrix which may depend non-trivially on the phasespace and which decays sufficiently fast so that M is globally defined and differentiableon M. It is called the Master constraint because obviously M = 0 ⇔ C I =0 ∀I . The concrete choice of K IJ is further guided by possible symmetry propertiesthat M is supposed to have and by the requirement that the corresponding Masterconstraint operator ̂M is densely defined on H Kin . As a first check, consider the casethat the point zero is only contained in the point spectrum of every Ĉ I and definêM := ∑ I K I Ĉ † I ĈI where K I > 0 are positive numbers. Obviously, Ĉ I = 0 for allI implies ̂M = 0. Conversely, if ̂M = 0 then 0 =< ,̂M >= ∑ I K I ||Ĉ I || 2implies Ĉ I = 0 for all I . Hence, in the simplest case, the single Master constraintcontains the same information as the system of all constraints.Let us now consider the general case and assume that ̂M has been quantised as apositive self-adjoint operator on H Kin . 4 Then it is a well known fact that the Hilbertspace H Kin is unitarily equivalent to a direct integral of Hilbert spaces subordinate tôM, that is,H Kin∼ =∫ ⊕R + dμ(λ) H ⊕ (λ) =: H ⊕ μ,N . (13.6)Here the Hilbert spaces H ⊕ (λ) are induced from H Kin and by the choice of themeasure μ and come with their own inner product. One can show that the measureclass [μ] and the function class [N], where N(λ) = dim(H ⊕ (λ)) is the multiplicityof the “eigenvalue” λ, are unique 5 and in turn determine ̂M uniquely up to unitary4 Notice that ̂M is naturally quantised as a positive operator and that every positive operator has a natural selfadjointextension, the so-called Friedrichs extension [14].5 Two measures are equivalent if they have the same measure zero sets. Two measurable functions are equivalentif they agree up to measure zero sets.