Appendix G — Derivation 7: Gauge Symmetry from Anchoring Invariance

Overview

In field theory, gauge symmetry refers to transformations of internal variables (like phase) that leave the Lagrangian invariant. Each symmetry group (U(1), SU(2), SU(3)) gives rise to conserved charges and gauge bosons.

In modal dynamics, there are no fields or particles.
But gauge symmetry still emerges—as invariance of the anchoring cost under specific deformations of a mode’s internal coherence structure.

1. Anchoring Cost Recap

Each mode is described by a coherence function:

ψ (x, t) = ρ (x, t) e^{i ϕ (x, t)}

The anchoring cost is:

C [ψ] = \int (γ | \partial_{t} ψ |^{2} + α | \nabla ψ |^{2} + β | ψ |^{2}) d^{3} x

If this cost remains invariant under a transformation of $ψ (x)$ , the transformation is a structural symmetry of the coherence medium.

2. U(1) Symmetry: Global and Local Phase

Let:

ψ (x) \to ψ^{'} (x) = ψ (x) e^{i θ (x)}

If $θ$ is constant (global phase), invariance is trivial.
If $θ (x)$ varies smoothly, the cost becomes:

| \nabla ψ^{'} |^{2} = | \nabla ψ + i ψ \nabla θ |^{2}

This introduces an extra term unless the coherence medium adapts structurally to accommodate the new gradient.

In traditional QFT, this adaptation requires a gauge field.
In modal dynamics, it requires compensating phase redistribution to preserve coherence cost.

If a transformation can be undone structurally—i.e., through internal re-anchoring—then it is a true modal symmetry.

3. SU(2) Symmetry: Internal Basis Rotation

Consider a mode with two coherent components:

Ψ (x) = (\begin{matrix} ψ_{1} (x) \\ ψ_{2} (x) \end{matrix})

The total cost is the sum:

C [Ψ] = \sum_{i = 1}^{2} C [ψ_{i}]

If we apply a unitary transformation:

Ψ \to Ψ^{'} = U (x) Ψ, U \in SU (2)

then the internal components mix. The cost remains unchanged if the coherence medium supports interchangeable anchoring between the two internal modes.

This is equivalent to chiral rotation in particle theory.
The symmetry is structural, not imposed—it depends on modal overlap and anchoring balance.

4. SU(3) Symmetry: Triple-Mode Interference

For three interacting modes:

Ψ (x) = (\begin{matrix} ψ_{1} (x) \\ ψ_{2} (x) \\ ψ_{3} (x) \end{matrix})

Transformations in SU(3) preserve total cost if:

The coherence field supports balanced triple anchoring
Interference terms $Γ (ψ_{i}, ψ_{j})$ remain invariant under $U \in SU (3)$

This structure underlies colour confinement and modal saturation:

Only globally balanced states are coherence-stable
Isolated colour modes decohere due to anchoring asymmetry

This replaces field lines and gluon exchange with structural anchoring invariance.

5. Conservation Without Noether

In classical physics, Noether's theorem links symmetry to conservation laws.

In modal dynamics:

There are no conserved charges as primitives
Instead, structural constraints preserve coherence class under symmetry-respecting evolution
Conservation arises from the topology and stability of anchoring, not from symmetry imposition

Examples:

Phase winding $\to$ charge invariance
Mode rotation $\to$ spin preservation
Structural balance $\to$ SU(3) colour neutrality

Conclusion

Gauge symmetry is not a rule to impose.
It is a reflection of what internal coherence transformations the medium can structurally tolerate without incurring additional cost.

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