Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 7114.13.4.4 Loop Quantum GravityThe kinematic of a quantum theory is defined by an algebra of ‘elementary’operators (such as x and id/dx, or creation and annihilation operators)on a Hilbert space H. The physical interpretation of the theory is based onthe connection between these operators and classical variables, and on theinterpretation of H as the space of the quantum states. The dynamics isgoverned by a Hamiltonian, or, as in general relativity, by a set of quantumconstraints, constructed in terms of the elementary operators. To assurethat the quantum Heisenberg equations have the correct classical limit,the algebra of the elementary operator has to be isomorphic to the Poissonalgebra of the elementary observables. This yields the heuristic quantizationrule: ‘promote Poisson brackets to commutators’. In other words, definethe quantum theory as a linear representation of the Poisson algebra formedby the elementary observables. The kinematics of the quantum theory isdefined by a unitary representation of the loop algebra.We can start à la Schrödinger, by expressing quantum states by meansof the amplitude of the connection, namely by means of functionals Ψ(A) ofthe (smooth) connection. These functionals form a linear space, which wepromote to a Hilbert space by defining a inner product. To define the innerproduct, we choose a particular set of states, which we denote ‘cylindricalstates’ and begin by defining the scalar product between these.Pick a graph Γ, say with n links, denoted γ 1 . . . γ n , immersed in themanifold M. For technical reasons, we require the links to be analytic.Let U i (A) = U γi , i = 1, . . . , n be the parallel transport operator of theconnection A along γ i . U i (A) is an element of SU(2). Pick a functionf(g 1 . . . g n ) on [SU(2)] n . The graph Γ and the function f determine afunctional of the connection as followsψ Γ,f (A) = f(U 1 (A), . . . , U n (A)), (4.192)(these states are called cylindrical states because they were previously introducedas cylindrical functions for the definition of a cylindrical measure).Notice that we can always ‘enlarge the graph’, in the sense that if Γ is asubgraph of Γ ′ we can writeψ Γ,f (A) = ψ Γ′ ,f ′(A), (4.193)by simply choosing f ′ independent from the U i ’s of the links which are inΓ ′ but not in Γ. Thus, given any two cylindrical functions, we can alwaysview them as having the same graph (formed by the union of the two