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An Induced Wilson Action from Record-Channel Networks: Gauge Invariance and a Measured Character Expansion for SU(2) and SU(3) Link Variables

Abraham J. Letter

Published Aug 5, 2026• 0 citations
Publisher

Abstract

A network of non-equilibrium information channels, each carrying an internal-symmetry rotation on its transported content, induces on those rotations the structure of a lattice gauge theory. In a two-arm interferometer built from the exactly solvable correlated-injection channel we establish four properties by direct computation: the link acts as a Wilson line; induced scalars are gauge invariant to machine precision; the functionals depend on the link configuration only through the conjugacy class of the loop holonomy; and the induced action is reproduced by a low-order character expansion of Wilson form. Results hold unchanged for SU(2) and SU(3). Keywords: lattice gauge theory, Wilson action, gauge invariance, character expansion, holonomy, emergent gauge symmetry, non-equilibrium steady states, interferometry, hep-lat, hep-th, cond-mat.stat-mech

DOI
10.5281/zenodo.21536216

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A network of non-equilibrium information channels, each carrying an internal-symmetry rotation on its transported content, induces on those rotations the structure of a lattice gauge theory. In a two-arm interferometer built from the exactly solvable correlated-injection channel we establish four properties by direct computation: the link acts as a Wilson line; induced scalars are gauge invariant to machine precision; the functionals depend on the link configuration only through the conjugacy class of the loop holonomy; and the induced action is reproduced by a low-order character expansion of Wilson form. Results hold unchanged for SU(2) and SU(3). Keywords: lattice gauge theory, Wilson action, gauge invariance, character expansion, holonomy, emergent gauge symmetry, non-equilibrium steady states, interferometry, hep-lat, hep-th, cond-mat.stat-mech
We take up three questions left open by the induced-Wilson-action companion: quantum dynamical links, composition across a network, and coupling flow. A record channel decohering a quantum link superselects the link's holonomy relative to the boundary frames — not, as first conjectured and then disproved, its conjugacy class. Whether the link is merely superselected or driven to a preferred value is decided by the bath spectrum: a flat-band renewal bath superselects without pinning, a thermal matter bath pins. The induced action's harmonic content decays by an affine law, and fermion back-action threads a chiral connection through link space. Keywords: quantum link models, lattice gauge theory, superselection rules, decoherence, holonomy, open quantum systems, Berry connection, emergent gauge symmetry, quant-ph, hep-lat, hep-th
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We exploit the spin network properties of the magnetic eigenstates of SU(2) Hamiltonian lattice gauge theory and use the Wilson loop operators to obtain a wide class of new identities amongst 3nj Wigner coefficients. We also show that the topological ground states of the SU(2) toric code Hamiltonian lead to Wigner 3nj identities with non-trivial phases. The method is very general and involves only the eigenvalue equations of any gauge invariant operator and their solutions. Therefore, it can be extended to any higher dimensional spin networks as well as larger SU(N) groups.
A spin-orbit Hamiltonian with an effective gauge structure carries two distinct loop objects that are routinely conflated: an energy-independent Wilson holonomy, which organizes interference and internal spin transport, and an energy-dependent monodromy, which quantizes the spectrum. We show that cleanly separating these objects supplies a precise, computable bridge between the loop/holonomy representation of gauge theories and condensed-matter spin-orbit transport. The construction maps a spin-orbit Hamiltonian to an effective U(1)U(1)U(1) plus internal non-Abelian connection, reduces it to a first-order transport problem, and reads physical predictions from holonomy, monodromy, curvature, and eigenphase data. Two rings make the separation explicit. For a Dirac (graphene) ring with Rashba coupling and Aharonov-Bohm flux, the total holonomy factorizes exactly into a commuting U(1)U(1)U(1) flux phase times an internal spin/pseudospin holonomy, and the spectrum follows from a holonomy-eigenvalue condition. For a Rashba-Dresselhaus ring, the internal SU(2)SU(2)SU(2) transport is genuinely non-Abelian away from the α=±βα=\pmβα=±β pure-gauge locus, where curvature controls path ordering; spectral quantization then requires an explicit first-order reduction obtained by phase-space doubling of the second-order Schrödinger problem. A non-Abelian Stokes formulation and Magnus expansion serve as ordering diagnostics rather than spectral tools. Spin-network ideas enter only as historical geometric motivation, not as a dynamical import into spintronics.