Approaches to Quantum Gravity

Loop quantum gravity 245will play the role of labelling smearing functions. Furthermore, let k be an su(2)valued, smooth function on . We define holonomy and flux functions∫∫A(s) := P exp( A), E k (S) := Tr(k ∗ E) (13.13)pwhere ∗E is the metric independent, pseudo-two form dual to E. Now consideran arbitrary, finite collection of paths s. Their union forms a finite 10 graph γ andwe may compose the paths s from the edges of the resulting graph. We now calla function cylindrical over a graph provided it is a complex valued function of theA(s) where s runs through the edges of the graph. The cylindrical functions form anAbelean ∗ -algebra which we denote by Cyl. Next, denote by Y k,S the Hamiltonianvector field of E k (S).ThenP is defined as the Lie algebra of cylindrical functionsf and vector fields v equipped with the following Lie bracket [( f,v),(f ′ ,v ′ )]=(v[ f ′ ]−v ′ [ f ], [v, v ′ ]). The most important building block in that algebra isY k,S [A(s)] ={E k,S , A(s)} =ικ A(s 1 )k(s ∩ S)A(s 2 ) (13.14)where s = s 1 ◦s 2 and we have assumed that s ∩ S is precisely one point, the generalcase being similar.13.3.1.1 The quantum algebra A and its representationsThe corresponding A is defined by formally following the procedure of section13.2. We now consider its representation theory. Since we are dealing with afield theory the representation theory of A will be very rich so we have to downsizeit by imposing additional physical requirements. The natural requirement isthat the representation derives from a state invariant under the automorphisms ofthe bundle P. Locally these automorphisms can be identified with the semidirectproduct G := G ⋊ Diff() of local SU(2) gauge transformations and spatial diffeomorphisms.The requirement of G-invariance is natural because both groupsare generated canonically, that is by the exponential of the respective Hamiltonianvector fields, from the Gauss constraint and spatial diffeomorphism constraintrespectively.ForinstancewehavewithC() = ∫ ∫ d3 x j C j and C( ⃗N) =d3 xN a C a thatα g (A(s)) := exp(X C() ) · A(s) = g (b(s)A(s)g ( f (s))α ϕ (A(s)) := exp(X N ⃗ C( ⃗N) ) · A(s) = A(ϕ ⃗N(s)), (13.15)where g = exp() and ϕN ⃗ is the diffeomorphism defined by the integralcurves of the vector field ⃗N. Here, b(s), f (s) respectively denote beginning andfinal point of the path s and X F denotes the Hamiltonian vector field of F.S10 Technically paths and surfaces must be semi-analytic and compactly supported in order for that to be true[1; 2; 21] but we will not go into these details here.