Appendix G — Derivation 7: Gauge Symmetry from Anchoring Invariance

Appendix G — Derivation 7: Gauge symmetry as anchoring-cost invariance

For a scalar coherence mode

ψ (x, t) = ρ e^{i ϕ}, n \equiv ρ^{2},

the quadratic cost density is

\begin{matrix} (0.1) & E = \frac{1}{2} γ (\partial_{t} ϕ)^{2} n + \frac{1}{2} α (\nabla ϕ)^{2} n + \frac{1}{2} β n . \end{matrix}

ϕ \to ϕ + θ_{0}, θ_{0} = const.

$\nabla ϕ$ unchanged ⇒ $E$ unchanged.

ϕ (x) \to ϕ^{'} = ϕ + θ (x) .

The gradient term becomes

(\nabla ϕ^{'})^{2} = (\nabla ϕ + \nabla θ)^{2} .

Cost increases unless the medium supplies a compensating twist.
Define a compensating field

A_{i} \equiv \partial_{i} θ

and a covariant derivative

D_{i} ϕ = \partial_{i} ϕ + A_{i} .

With this replacement

(\nabla ϕ + \nabla θ)^{2} \to (D_{i} ϕ)^{2},

so the density (0.1) is form-invariant:

E [ϕ] = E [ϕ + θ, A_{i} = \partial_{i} θ] .

▶ Gauge principle emerges as anchoring invariance;
(A_i) is not independent but a derived object.

Let

Ψ (x) = ρ U (x) Ψ_{0}, U (x) \in SU (N) .

Quadratic colour cost (Appendix-QCD)

\begin{matrix} (2.1) & E_{c o l} = \frac{1}{2} α n Tr [(\partial_{i} U^{†}) (\partial_{i} U)] . \end{matrix}

U \to U^{'} = V (x) U, V (x) = \exp [i θ^{a} (x) T^{a}] .

Write
$A_{i} = V^{- 1} \partial_{i} V = - i (\partial_{i} θ^{a}) T^{a}$
and define the colour-covariant derivative

\begin{matrix} (2.2) & D_{i} U \equiv \partial_{i} U + A_{i} U . \end{matrix}

Then

Tr [(\partial_{i} U^{' †}) (\partial_{i} U^{'})] = Tr [(D_{i} U)^{†} (D_{i} U)],

so (2.1) is invariant.

▶ Gauge bosons $A_{i}^{a}$ are again just phase-derivative fields; no new constants are introduced.

Take the variation of the action under an infinitesimal $θ (x, t)$ :

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

Setting (\delta S=0) for arbitrary θ gives

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

i.e. charge conservation with
$J^{0}$ and $J$ constructed from phase flow—no additional Noether theorem required.

SU(2) internal doublet $Ψ = (ψ_{1}, ψ_{2})^{⊤}$ .
Balanced anchoring (equal $n_{1}, n_{2}$ ) ⇒ cost invariant under any internal rotation; mixing angle plays the role of weak isospin.
SU(3) triplet anchors compete via quartic term in QCD appendix.
Only colour-neutral composites minimise global cost ⇒ confinement emerges as energy penalty for isolated colour modes.

Gauge fields = phase-derivative geometries; not independent dynamical dofs.
Covariant derivatives (1) & (2.2) appear naturally when demanding anchoring invariance under local internal rotations.
Charge/current conservation follows directly from phase continuity, no extra Noether machinery.
Higher groups (SU(2), SU(3)) are still parameter-free; the substrate constants ${\tilde{α}, β, γ}$ stay untouched.

Appendix F | [Index](./Appendix Master) | Appendix H