Quantum Gravity

196 QUANTUM GRAVITY WITH CONNECTIONS AND LOOPSÚ´¡µ¡ Fig. 6.5. An elementary tetrahedron ∆.One can make a triangulation (called ‘Tri’) of Σ into elementary tetrahedra ∆and choose one vertex, v(∆), for each ∆; see Fig. 6.5. Calling e i (∆), i =1, 2, 3,the three edges of ∆ meeting at v(∆), one can consider the loopα ij (∆) ≡ e −1j (∆) ◦ a ij (∆) ◦ e i (∆) ,where a ij (∆) connects the other vertices different from v(∆). Thiemann (1996)could then show that one can get the correct Euclidean Hamiltonian (6.41) inthe limit where all tetrahedra ∆ shrink to their base points v(∆): considerHETri [N] = ∑∆∈TriHE ∆ [N] , (6.42)withHE ∆ [N] ≡ 1()12πβ N (v(∆)) ɛijk tr U αij(∆)U ek (∆){U −1e k (∆) ,V} , (6.43)where U ... denotes the holonomies along the corresponding loops and edges. Usinglim U α ij(∆) =1+ 1∆→v(∆)2 F abe a i (∆)eb j (∆) ,lim U e k (∆) =1+A a e a k(∆) ,∆→v(∆)the expression (6.42) tends to (6.41) in this limit.The quantum operator corresponding to (6.42) is then defined by replacingV → ˆV and by substituting Poisson brackets with commutators,ĤETri ∆∈TriĤE ∆ [N] , (6.44)withĤ ∆ E[N] ≡−i)12πβ N (v(∆)) ɛijk tr(Ûαij(∆)Ûe k (∆){Û −1e k (∆) ,V} . (6.45)