Loops, Knots, Gauge Theories and Quantum Gravity 2026, 33 - Constraints in T Algebra

The classical differomorphism and Hamiltonian constraints in terms of the T operators emerges from the limit of a T¹ operator. Its contribution to the holonomy of drops out because of the tracelessness of the triad. The T variables are gauge invariant, so the contribution from the holonomy reduces to the identity in the limit. The shrinking loop procedure for the split constraint recovers the usual constraint in its limit. The antisymmetric part of T^ab yields a regularized expression for the Hamiltonian constraint (Eqs. 8.16-17). The quantum version of the constraints emerges from the SU(2) Yang-Mills method. The fact that the GR connection is now complex, means that its conjugate might be complicated if expressed by the reality conditions. The regular transform then doesn't suffice. Assuming that the gauge field A is real, then the ordering operator behaves relatively simply if applied to Ψ(γ). It can be evaluated on a Wilson loop like a calculation in the connection representation (8.3). The diffeomorphism constraint acts on functions of loops by infinitesimally deforming the loop along a vector, as if it existed in a spatial manifold on which a diffeomorphism is perfomed along the same vector. If the wavefunction is invariant under deformations of the loop argument, its loop representation annihilates. They then only depend on the knot class of the loop. This trivializes the diffeomorphism constraint (8.4.1). The solutions to the Hamiltonian constraint requires a regularization, which tends to be difficult to combine with diffeomorphism invariance. When performing the regularization, it's standard practice to add diffeomorphism invariance afterward. If the loop argument of Ψ in the argument of the wavefunction resolves in a linear combination of Ψ with arguments with a different knot class (Eq. 8.39), the Hamiltonian provides a different, safe approach, since the change of topology of the loop in such equations imply that the loop derivative is not generally well defined on function invariant under diffeomorphisms. So far, Hamiltonian constraints in the loop representation are based on appending an infinitesimal loop to the knot. That means it has formally includes a vanishing actions at points where loops are smooth and non-intersecting. Since the Hamiltonian itself doesn't change the number of intersections of the loop, it could be considered well defined, but this poses a problem when considering Mandelstam identities (8.4.2). The Hamiltonian's Dirac-delta component can be point-split to include a usual symmetric regulator f_ε(y - z), which can be considered some Gaussian (Eq. 8.48) (or some other regulator). The background metric determines the distance y - z. Through the point-splitting, the path appears in the expression of the regularized constraint to not close a loop. The regulator determines the form of the leading action of the Hamiltonian through resolving the integral. Given a finite limit, this comes out to about the same. For a smooth loop with a finite number of kinks and no intersections, the action of this diffeomorphism on the loop state is the same as that of the Hamiltonian in the regularized limit, which differs from the connection representation in that the kinks do not pose a problem (p. 206). If the loop has intersections, the process is not dissimilar to the one for kinks, though there are four possible contributions (the four lines extending from the point of intersection). An orientation convention has to be determined a priori once again. The double intersection, each case composes of the four edges contributes in a way visible in the scenario chosen. If there are kinks at the intersection, the crossing term does not occur and the result is background dependent. The action of the regularized Hamiltonian in loop space is only non-trivial at points where the loops have intersections.