Quantum Gravity

CANONICAL GRAVITY WITH CONNECTIONS AND LOOPS 131U[A, α](s) ∈ SU(2) , U[A, α](0) = I ,dds U[A, α](s) − GA a(α(s)) ˙α a (s)U[A, α](s) =0, (4.142)where ˙α a (s) ≡ dα a /ds (tangential vector to the curve) and U[A, α](s) isashorthandfor U[A, α](0,s). The formal solution for the holonomy reads( ∫ ) ( ∫ s)U[A, α](0,s)=P exp G A ≡Pexp G d˜s ˙α a A i a(α(˜s))τ i . (4.143)αHere, P denotes path ordering which is necessary because the A are matrices(like in Yang–Mills theories). One has for example for s =1,( ∫ 1)P exp G ds A(α(s)) ≡ U[A, α]= I + G0∫ 10∫ 1ds A(α(s)) + G 2 ds0∫ s00dt A(α(t))A(α(s)) + ... .We note that the A i a can be reconstructed uniquely if all holonomies are known(Giles 1981).The holonomy is not yet gauge invariant with respect to SU(2)-transformations.Under g ∈ SU(2)ittransformsasU[A, α] → U g [A, α] =gU[A, α]g −1 .Gauge invariance is achieved after performing the trace, thus arriving at the‘Wilson loop’ known, for example, from lattice gauge theories,One can also defineT [α] =trU[A, α] . (4.144)T a [α](s) =tr[U[A, α](s, s)E a (s)] , (4.145)where E a is inserted at the point s of the loop. Analogously one can define higher‘loop observables’,T a1...aN [α](s 1 ,...,s n ) ,by inserting E a at the corresponding points described by the s-values. Theseloop observables obey a closed Poisson algebra called the loop algebra. One has,for example,{T [α], T a [β](s)} =∆ a [α, β(s)] ( T [α#β] −T[α#β −1 ] ) , (4.146)where∫∆ a [α, x] =ds ˙α a (s)δ(α(s),x) , (4.147)and β −1 denotes the loop β with the reversed direction. The right-hand side of(4.147) is only non-vanishing if α and β have an intersection at a point P; α#β