The inverse loop transform
Thomas Thiemann
TL;DR
The paper addresses reconstructing a measure on the compact gauge-history space ${\overline {{\cal A}/{\cal G}}}$ from its loop-transform, the characteristic functional, by developing an inverse Fourier transform for compact groups. It introduces loop- and edge-network bases to express cylindrical measures and proves an inverse Fourier transform theorem using the heat-kernel as an approximate identity, enabling the explicit computation of finite joint distributions via $\frac{d\mu_\gamma}{d\mu_{0,\gamma}}=\sum_{\vec{\pi},\pi} \chi(\gamma,\vec{\pi},\pi) T_{\gamma,\vec{\pi},\pi}$. The framework is illustrated with examples including the vacuum measure, orbit-sum knot measures, and two-dimensional Yang–Mills characteristic functionals, yielding concrete densities and 1D distributions in terms of heat kernels with area. This provides a constructive bridge between abstract measure-theoretic descriptions in quantum gauge theory and computable discrete data, with potential impact on quantum gravity and diffeomorphism-invariant formulations.
Abstract
The loop transform in quantum gauge field theory can be recognized as the Fourier transform (or characteristic functional) of a measure on the space of generalized connections modulo gauge transformations. Since this space is a compact Hausdorff space, conversely, we know from the Riesz-Markov theorem that every positive linear functional on the space of continuous functions thereon qualifies as the loop transform of a regular Borel measure on the moduli space. In the present article we show how one can compute the finite joint distributions of a given characteristic functional, that is, we derive the inverse loop transform.