Gauge Symmetries

Gauge symmetry is one of the pillars of the Standard Model. It defines how particles interact, which charges exist, and what kinds of forces are allowed. In quantum field theory, gauge invariance leads directly to the existence of force-mediating fields.

But in modal dynamics, there are no particles and no fields.
Instead, gauge symmetries arise from the structure of anchoring itself.

Anchoring Cost Invariance

The coherence action for a mode is:

S [ψ] = \int (- γ | \partial_{t} ψ |^{2} + α | \nabla ψ |^{2} + β | ψ |^{2}) d^{4} x

This action must remain stationary under structural evolution. That means the cost of anchoring must be invariant under certain transformations of $ψ$ .

The simplest case is local phase rotation:

ψ (x) \to ψ (x) e^{i θ (x)}

If the coherence action remains invariant under this transformation, we recover the analogue of U(1) gauge symmetry—but it is not a symmetry of a field. It is a freedom in how a mode’s anchoring configuration can be reshaped without increasing cost.

Emergent Gauge Structures

The different gauge groups in the Standard Model—U(1), SU(2), SU(3)—correspond in modal terms to different classes of structural invariance under transformation:

U(1) — invariance under local phase rotation (electromagnetic symmetry)
SU(2) — invariance under coherence-preserving transformations of a two-component mode (e.g. chiral rotation or mode polarity)
SU(3) — invariance of triple-mode coherence structures with mutual anchoring balance

Each group arises from a deeper rule:

If a transformation leaves the anchoring cost unchanged, it is permitted.

This replaces Noether’s theorem.
There is no need to derive conserved charges from symmetries—conservation arises because cost-preserving deformations do not disrupt phase structure.

Field-Free Gauge Invariance

In standard gauge theory:

Symmetries require fields to “restore invariance” under local transformation.
Interactions are carried by bosonic force mediators.

In modal dynamics:

No such field is needed.
Invariance is structural: the mode’s coherence configuration itself is compatible with multiple internal reshaping options.
Apparent “forces” are bias responses to incompatible anchoring—not field transmission.

(See Appendix G — Gauge Symmetry from Anchoring Invariance.)

Gauge invariance is not a principle to be imposed.
It is a natural consequence of coherence-preserving structure.