Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 717a (linear combination of) multi–loopsT [α] |Ψ〉 = |α〉|Ψ〉.Higher order loop operators are expressed in terms of the elementary‘grasp’ operation. Consider first the operator T a (s)[α], with one hand inthe point α(s). The operator annihilates all loop states that do not crossthe point α(s). Acting on a loop state |β〉, it givesT a (s)[α] |β〉 = l 2 0 ∆ a [β, α(s)] [ |α#β〉 − |α#β −1 〉 ] , (4.201)where we have introduced the elementary length l 0 byl 2 0 = G = 16πG Newtonc 3 = 16π l 2 P lanck (4.202)and ∆ a and # were defined above. This action extends by linearity, continuityand by the Leibniz rule to products and linear combinations of loopstates, and to the full H. In particular, it is not difficult to compute itsaction on a spin network state [DePietri and Rovelli (1996)]. Higher orderloop operators act similarly. It is easy to verify that these operators providea representation of the classical Poisson loop algebra.All the operators in the theory are then constructed in terms of thesebasics loop operators, in the same way in which in conventional QFT oneconstructs all operators, including the Hamiltonian, in terms of creationand annihilation operators. The construction of the composite operatorsrequires the development of regularization techniques that can be used inthe absence of a background metric.4.13.4.8 Loop v.s. Connection RepresentationImagine we want to quantize the one dimensional harmonic oscillator. Wecan consider the Hilbert space of square integrable functions ψ(x) on thereal line, and express the momentum and the Hamiltonian as differentialoperators. Denote the eigenstates of the Hamiltonian as ψ n (x) = 〈x|n〉. Itis well known that the theory can be expressed entirely in algebraic form interms of the states |n〉. In doing so, all elementary operators are algebraic:ˆx|n〉 = √ 12(|n − 1〉 + (n + 1)|n + 1〉), ˆp|n〉 = √ −i2(|n − 1〉 − (n + 1)|n + 1〉).Similarly, in quantum gravity we can directly construct the quantum theoryin the spin–network (or loop) basis, without ever mentioning functionals ofthe connections. This representation of the theory is denoted the looprepresentation.