Loops, Knots, Gauge Theories and Quantum Gravity 2026, 22 - Loop Representation

The loop transform involves a functional integral in the space of connections, ignoring gauge transformations. It was originally assumed and its effects were studied a posteriori (p. 76). The correctness of the results are judged by (self-)consistency. Through the Wilson loop's function as a projection operator, the definition of "cylindrical measures" emerges, reducing the infinite-dim. integral to a finite set of integrals over the gauge group. The classical configuration space of a scalar field ϕ in flat spacetime satisfying Klein-Gordon is the set of all smooth field configurations on a spatial manifold that falls off at infinity. Quantum states for the theory are functions on the space of classical configurations. Using the inner product through an integral requires a suitable measure μ. The simplest function following from these definitions are the functionals F_f(ϕ) = ∫d³xf(x)ϕ(x) (Eq. 3.94), requiring that f(x) have regularity and fulfills a good falloff conditions, so that the integral is well defined. A classical configuration allows a projection of ϕ, which yields a set of n numbers F_ei(ϕ) (Eq. 3.95). If the theory is non-Abelian, the configuration space is non-linear, which requires using the holonomy property of associating to the functionals. Given a fixed finite set of independent loops, define the value of the holonomies as g(A) = G(H_A(β₁),...) where G is a function defined on n copies of the gauge group (3.5.1).

The alternative approach for the loop representation arrives again from the Ashtekar variables expressed through various tensors noted as T. T(γ) are the Wilson loops, and the other Ts consist of breaking up the holonomy at points x_i, inserting an electric field and continuing the holonomy until returning to the basepoint. The Ts with electric fields inserted T^k behave as multivector densities on their indices a_i at points of the manifold x_i. The Wilson loops contain enough information to construct any gauge invariant function of the connection. The general SU(2) case can be written in a compact form using Poisson brackets. For higher order Ts, Ts of arbitrarily large order are required to close the algebra, the resulting algebra is not closed, or requires some "artificial" completion. The T operators in SU(2) as they act on wavefunctions of loops T(η)Ψ(γ) = Ψ(η ○ γ) + Ψ(η ○ γ^-1) (Eq. 3.113) are difficult to regularize. The quantum theory is the loop representation, since the T operators in connection representation canbe transformed into operators in the loop representation.

Both methods of arriving at loop representation are equally good (3.5.2).

The wavefunctions in the loop representation are weighted products of Wilson loops, and they appear in the transform as wavefunctions are symmetric switch of arguments. Require that Ψ(γ ○ η) = Ψ(η ○ γ)) (3.5.3)