Loops, Knots, Gauge Theories and Quantum Gravity 2026, 25 - Extended Loop Representation
Applying the loop holonomy to the formalism, the abelian matrix HA(γ) is extended to include a divergence-free vector density X on the 3-manifold (Eqs. 4.128 - 129), which is passed onto the wave function in its loop coordinate representation (Eq. 4.130). These become functionals of X, which redefines the field operators and Wilson loop, so the quantum Hamiltonian becomes
(Eq. 4.134). This identifies the dual of X P to A; and X to E. This is in agreement with having the loops trace the lines electric flux (p. 110). The vacuum in terms of loops is a singular divergent quantity that requires regularization, but in this representation, it's automatically well defined (p. 111). From this representation, a canonical classical theory arises, which can be generalized to non-Abelian cases. As a consequence, the general loop action is
(Eq. 4.140)