Appendix G — Derivation 7: Gauge Symmetry from Anchoring Invariance

Appendix G — Derivation 7: Gauge Symmetry as Anchoring-Cost Invariance

(Foundations v1.0 • 2025-06-10)

The quadratic anchoring density from Foundations §1

E = \frac{1}{2} γ (\partial_{t} Φ)^{2} n - \frac{1}{2} α | \nabla Φ |^{2} n - \frac{1}{2} β n, n \equiv ρ^{2},

remains form-invariant under local internal phase rotations once one
promotes ordinary derivatives to covariant derivatives. The resulting
connections $A_{i}$ or $A_{i}^{a}$ are not new degrees of freedom; they are
geometric shorthand for phase gradients already present in $Φ$ .

1 U(1) phase invariance

1.1 Global shift (trivial)

$Φ \to Φ + θ_{0}$ leaves $\nabla Φ$ unchanged ⇒
$E$ unchanged.

1.2 Local shift

$Φ (x) \to Φ + θ (x)$ .
Then $| \nabla Φ |^{2} \mapsto | \nabla Φ + \nabla θ |^{2}$ , raising the
cost unless we define

A_{i} \equiv \partial_{i} θ, D_{i} Φ \equiv \partial_{i} Φ + A_{i} .

With $| \nabla Φ |^{2} \to | D_{i} Φ |^{2}$ the density is identical in
form: anchoring cost is gauge-invariant.

2 Non-Abelian extension (SU(N))

For an internal matrix phase
$U (x) = \exp [i θ^{a} (x) T^{a}]$ the colour cost

E_{col} = \frac{1}{2} α n Tr [(\partial_{i} U^{†}) (\partial_{i} U)] (see Appendix C).

A local rotation $V (x)$ sends $U \to V U$ .
Define

A_{i} \equiv V^{- 1} \partial_{i} V = - i (\partial_{i} θ^{a}) T^{a}, D_{i} U \equiv \partial_{i} U + A_{i} U .

Then $E_{col}$ keeps the same functional form—SU(N)
gauge symmetry arises directly as anchoring-cost invariance.

3 Charge & current from phase continuity

Vary the action under $Φ \to Φ + θ (x, t)$ :

δ S = \int d^{4} x [- \partial_{t} (γ n \partial_{t} Φ) - \nabla \cdot (α n \nabla Φ)] θ .

Arbitrariness of θ gives the continuity equation

\underset{J^{0}}{\underset{⏟}{\partial_{t} (γ n \partial_{t} Φ)}} + \underset{\nabla \cdot J}{\underset{⏟}{\nabla \cdot (α n \nabla Φ)}} = 0,

so charge and current are built from phase flow; no extra Noether
postulate is needed.

4 SU(2) & SU(3) examples

SU(2) – Doublet envelope $(ψ_{1}, ψ_{2})^{⊤}$ with balanced
densities: cost invariant under weak-isospin rotations.
SU(3) – Triplet caustic sheets (Appendix W) give colour structure;
cost penalises isolated colour ⇒ confinement.

Key take-aways

Gauge fields are phase-derivative geometries, not independent
dynamical entities.
Covariant derivatives arise when enforcing local anchoring-cost
invariance.
Charge conservation is automatic from phase continuity.
Higher groups introduce no new universal constants; everything
still rests on $(α, β, γ)$ .

Appendix F - Thermodynamic Structure | [Index](./Appendix Master) | Appendix H - Particle Statistics