Loops, Knots, Gauge Theories and Quantum Gravity 2026, 30 - Inclusion of Fermions

Fermions introduce open paths to loops. On a lattice, fermions can be modeled to interact with the gauge fields. The action of a gauge field interacting with a charged fermion is given through the 4-Dirac spinor

(Eq. 6.101). The Hamiltonian of the action uses the Dirac spinors (and their conjugates) as the canonical variables. In loop representations, they become holonomies along open paths with fermions at the ends. The resulting energy depends on the configuration variable and conjugate momentum, so it can't be used to expand the wavefunctions in the connection representation. Solving this problem depends heavily on the gauge group in question. For lattice gauge theories, the staggered fermion technique can be applied. In simplified context, assume the Dirac spinors have only two components, which can be group valued, but this is irrelevant for the discussion, so the group index is dropped. The massless Dirac equation and the Dirac matrices combine to a discretized equations

(Eqs. 6.105 - 106). The continuum equation solves to plane waves with eigenvalue problem k₀ψ = kαψ, implying the dispersion relation k₀ = ±k₁. The discretized equation solves to discrete plane waves through, so |k₁| = πm/Na, which corresponds to a Brillouin zone of |k| ≤ π/a. The up components lie at the even sites and down components at the odd sites. This staggers the lattice position of the Dirac fermion components, which splits into two discrete solutions with two discrete fields, which can be used to express the Dirac equation (Eqs. 6.108 - 111).