Quantum Gravity

184 QUANTUM GRAVITY WITH CONNECTIONS AND LOOPSconstraints have not been considered at this stage, this level corresponds to havingin Chapter 5 states Ψ[h ab ] before imposing the constraints H a Ψ=0=H ⊥ Ψ.A candidate for an inner product on A/G would be∫〈Ψ 1 |Ψ 2 〉 = Dµ[A] Ψ ∗ 1[A]Ψ 2 [A] ; (6.10)A/Gcf. (5.20). The main problem is: can one construct a suitable measure Dµ[A]in a rigorous way? The obstacles are that the configuration space A/G is bothnon-linear and infinite-dimensional. Such a measure has been constructed; seeAshtekar et al. (1994). In the construction process, it was necessary to extendthe configuration space to its closure A/G. This space is much bigger than theclassical configuration space of smooth field configurations, since it contains distributionalanalogues of gauge-equivalent connections.The occurrence of distributional configurations can also be understood froma path-integral point of view where one sums over (mostly) non-differentiableconfigurations. In field theory, an imprint of this is left on the boundary configurationwhich shows up as the argument of the wave functional. Although classicalconfigurations form a set of measure zero in the space of all configurations, theynevertheless possess physical significance, since one can construct semiclassicalstates that are concentrated on them. Moreover, for the measurement of fieldvariables, one would not expect much difference to the case of having smoothfield configurations only, since only measurable functions count, and these are integralsof field configurations; see Bohr and Rosenfeld (1933). In a sense, the looprepresentation to be discussed in the following implements ‘smeared versions’ ofthe variables A and E.6.1.2 Loop representationInstead of considering wave functionals defined on the space of connections, Ψ[A],one can use states defined on the space of loops α a (s), Ψ[α]; cf. Section 4.3.3.This is possible due to the availability of the measure on A/G, and the statesare obtained by the transformation (‘loop transform’)∫Ψ[α] = Dµ[A] T [α]Ψ[A] , (6.11)A/Gwhere T [α] was defined in (4.144). This corresponds to the usual Fourier transformin quantum mechanics,∫1˜ψ(p) =d 3 x e −ipx/ ψ(x) . (6.12)(2π) 3/2 R 3The plane wave corresponds to T [α] ≡ Ψ α [A]. We shall refer to the latter as‘loop states’. In the loop approach to quantum gravity they can be taken to bethe basis states (Rovelli and Smolin 1990). The prevalent opinion nowadays is