Loops, Knots, Gauge Theories and Quantum Gravity 2026, 23 - Abelian and Classical Maxwell Theory

A subgroup of an Abelian group of loops suffices to formulate a Maxwell theory. The elements compose as κ = γ ○ η ○ γ^-1 ○ η^-1, where γ and η are commutators, i.e. they can be composed of an arbitrary number of loops. They form a normal subgroup, written L_comm, implying the quotient group through the equivalence relation γ ~ η ⇔ γ ○ η^-1 ∈ L_comm, written L_Abel = L/L_comm. The representation of the Abelian subgroup requires matrices H(γ), which behave Abelian, as the loops do (Eq. 4.5), which happen to be related to the Wilson loops (Eq. 4.7). By writing out the infinitesimal deformation, it turns out the loop derivatives are not path dependent, but just point dependent (Eq. 4.9 - 10), so the loop derivatives commute (4.1).

The classical canonical Maxwell theory can be written in terms of the canonical pair of the electric field and magnetic guage field with the classical Hamiltonian. The Gauss law can be solved by only considering transverse electric fields, so the canonical theory can also be written using transverse fields only, utilizing the transverse Dirac delta. The Hamiltonian written in terms of the basic generalized variables adopts the form of an infinite collection of harmonic oscillators (Eq. 4.23), and from it follow the creation/destruction operators directly (4.2).