Gauge invariance (U(1), SU(2), SU(3))

David Barbeau, Independent Researcher

david@bigbadaboom.ca | www.bigbadaboom.ca

April 26, 2026

License: arXiv.org perpetual, non-exclusive license 1.0. Non-commercial use (e.g., education, videos) encouraged with attribution to David Barbeau. Commercial use requires permission.

The C.O.R.E. Lagrangian (derived in the corrected action) is fully U(1) gauge invariant. The electromagnetic sector is

\mathcal{L}_{\rm EM} = \frac12 \varepsilon(\Phi) E^2 - \frac{1}{2\mu(\Phi)} B^2, \tag{1}

where \(\mathbf{E} = -\partial_t\mathbf{A} - \nabla A_0\) and \(\mathbf{B} = \nabla\times\mathbf{A}\) are gauge-invariant combinations. Under the local U(1) transformation

A_\mu \to A_\mu + \partial_\mu \Lambda(x), \tag{2}

both \(E\) and \(B\) (and therefore \(\mathcal{L}_{\rm EM}\)) are unchanged. The vacuum response \(\varepsilon(\Phi)\), \(\mu(\Phi)\) depends only on the scalar gravitational potential \(\Phi\) (CUGE) and commutes with the gauge transformation. The impedance remains strictly invariant:

Z_0 = \sqrt{\frac{\mu(\Phi)}{\varepsilon(\Phi)}} = \sqrt{\frac{\mu_0}{\varepsilon_0}} \quad (\text{dimensionless}). \tag{3}

Phase continuity \(\phi = \mathbf{k}\cdot\mathbf{r} - \omega t\) (REFORM) is the physical origin of this U(1) symmetry in the eikonal limit. Variation w.r.t. \(\mathbf{A}\) recovers the macroscopic Maxwell equations in the responsive medium, all in SI base units with dimensionless \(n \equiv \sqrt{\varepsilon_r \mu_r}\).

SU(2) (weak) and SU(3) (strong) sectors are unaddressed: the framework is classical EM + vacuum response (no non-Abelian gauge fields or Yang–Mills terms appear in any C.O.R.E. document). These remain open extensions.

Status: U(1) closed and variationally consistent. Non-Abelian gauge structure is outside the current classical scope.