Loops, Knots, Gauge Theories and Quantum Gravity 2026, 34 - Inclusion of Matter
The loop representation for quantum gravity naturally accommodates the inclusion of matter. The Yang-Mills case couples to 4-Dirac spinors, which complicates into the staggered fermion techniques. It's more natural to couple the gravitational case to uncharged spinning particles, i.e. 2-Weyl fermions. The Lagrangian for general relativity coupled to Weyl fermions in self-dual first order variables is
(Eq. 9.2). The caonical decomposition introduces a an isomorphism in the spinor spaces that casts the formalism in terms of SU(2) spinors. The theory has the same constraints as usual general relativity but the constraints are appropriately modified to generate the corresponding transformations in the fermionic variables (p. 211, Eq. 9.3 - 5). The usual T variables constructed from the connection tensor define objects of a closed algebra under Poisson brackets (Eq. 9.6-10). The Grassmanian characteristics of the spinor field factor heavily into the tensor identities, and also their Poisson brackets. The diffeomorphism and Hamiltonian constraints are written purely by in terms of the T variables. The algebra has a quantum representation in terms of the operators acting on a space of wavefunctions of loops and open paths (p. 213). The new contributions of the fermionic parts can be isolated by letting the Hamiltonian act on a state dependent on a single path, that may or may not self-intersect.
(Eq. 9.18) where Day is the Mandelstam covariant derivative. It can be applied directly on the open path (Eq. 9.19). The geometric meaning of the Weyl part translates the end of the open paths in the direction of the tangent vector at that point. The usual regularization and renormalization are required after this (9.2). A mixed loop representation that associates some loops with the connection of certain theories emerges from coupling the the loop representation to the connections. The unification aspects of physical theories is affirmed through substituting the Maxwell theory with a Yang-Mills field. This is aided by the Gauss laws for both gauge groups in question appear separately, so U(1) invariance and SU(2) invariance can be addressed separately (p. 215). The constraints also take simple forms (Eqs. 9.24-25), and a loop representation based on a single loop for U(2) symmetry can be constructed. In this unified case, there is no distinction between the 2-loops and the Mandelstam identities. This idea is not yet fully explored and features some uncertainties (p. 217)