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Helicity bundles, rotations and Berry curvature — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)Open item page →

Latitude-loop Berry holonomy

Physical scope

A constructed path in the conditional classical Maxwell-Lorentz vacuum helicity model's smooth source-free nonzero-Fourier-mode regime, with no extra approximation. The loop, solid angle, and phase are dimensionless geometric quantities; no experimental measurement, material medium, quantization, or photon position operator is asserted.

Example

Fix $σ \in {+ 1, - 1}$ and $0 < θ_{0} < π$ . Let $γ (t) = (θ_{0}, 2 π t)$ for $0 \leq t \leq 1$ be the latitude loop in the direction sphere, and set $Ω : = \int_{0}^{2 π} \int_{0}^{θ_{0}} \sin θ d θ d ϕ = 2 π (1 - \cos θ_{0})$ , the outward solid angle of its north cap. The spherical gauge gives ${H o l}_{γ} = \exp (i 2 π σ \cos θ_{0})$ . The north-pole regular gauge $v_{N} = e^{i σ ϕ} v_{σ}$ has $A_{N} = σ (\cos θ - 1) d ϕ$ and is smooth on the cap; its loop integral and the cap flux are both $- σ Ω$ . Hence ${H o l}_{γ} = \exp (- i σ Ω) = \exp (i 2 π σ \cos θ_{0})$ .

Example references

SO(2) helicity character and geometric phase

Facts & Assumptions

Given: A fixed $σ \in {+ 1, - 1}$ and a latitude parameter $0 < θ_{0} < π$ .

[F1]

SO(2) helicity character and geometric phase states that in a single unit frame defined on a neighborhood of the loop, ${H o l}_{γ} = \exp (i \oint_{γ} A)$ . For its Stokes clause, when $γ$ is piecewise smooth with associated finite smooth singular path chain $c_{γ}$ , a finite smooth singular $2$ -chain $Σ$ in one unit-frame domain satisfying $\partial Σ = c_{γ}$ gives ${H o l}_{γ} = \exp (i \int_{Σ} F_{σ})$ .

[F2]

Curvature and the obstruction to a global polarization basis gives $F_{σ} = - σ \sin θ d θ \land d ϕ$ , and SO(2) helicity character and geometric phase gives $A_{σ} = σ \cos θ d ϕ$ and its holonomy rule.

Verification

1.1F2algebra

On the punctured cap, the phase change $v_{N} = e^{i σ ϕ} v_{σ}$ gives $A_{N} = A_{σ} - σ d ϕ = σ (\cos θ - 1) d ϕ$ . Put $x = \sin θ \cos ϕ$ , $y = \sin θ \sin ϕ$ , and $z = \cos θ$ . Since $d ϕ = (x d y - y d x) / (1 - z^{2})$ , this is $A_{N} = - σ (x d y - y d x) / (1 + z)$ , which extends smoothly across the north pole because $1 + z > 0$ there. The rephased frame itself also extends: for $σ = + 1$ its components are $\frac{1}{2 \sqrt{2} (1 + z)} ((1 + z)^{2} - (x + i y)^{2}, i ((1 + z)^{2} + (x + i y)^{2}), - 2 (x + i y) (1 + z))^{T}$ , and for $σ = - 1$ they are the complex conjugates. These formulas have nonzero denominator throughout the north patch.

2.1F1F2step 1.1algebra

Along the loop, $\oint_{γ} A_{N} = \int_{0}^{2 π} σ (\cos θ_{0} - 1) d ϕ = - σ Ω$ , so [F1] gives ${H o l}_{γ} = e^{- i σ Ω}$ . Direct integration of the global curvature over the north cap gives $\int_{Σ_{N}} F_{σ} = - σ \int_{0}^{2 π} \int_{0}^{θ_{0}} \sin θ d θ d ϕ = - σ Ω$ as well. Thus the cap-flux expression agrees without using the singular spherical gauge at its pole.

3.1F1F2step 2.1algebra∎

In the spherical gauge, $\oint_{γ} A_{σ} = 2 π σ \cos θ_{0} = - σ Ω + 2 π σ$ . Since $e^{i 2 π σ} = 1$ for the integer helicities specified by [F1], exponentiation gives the same holonomy in both gauges.

CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)Open item page →

The polar helicity frame is singular while its projector is smooth

Physical scope

Conditional classical Maxwell-Lorentz vacuum model in the smooth source-free nonzero-Fourier-mode regime stated by the supplier, with no extra approximation. The example distinguishes a coordinate-dependent phase frame from the smooth helicity projection; it makes no empirical claim, quantization assertion, or claim of a global nonvanishing polarization basis.

Statement refuted

For the helicity line on $S^{2}$ , the fact that the spherical unit frame $v_{σ} = (e_{θ} + i σ e_{ϕ}) / \sqrt{2}$ is undefined at the coordinate poles implies that the projection family $P_{σ} (n)$ is itself undefined or nonsmooth there.

Statement references

Helicity eigenlines over the direction sphere

Facts & Assumptions

Given: The direction sphere $S^{2}$ , $σ \in {+ 1, - 1}$ , and the projector and line bundle of Helicity eigenlines over the direction sphere.

[F1]

Helicity eigenlines over the direction sphere defines $P_{T} (n) z = z - n (n \cdot z)$ and $P_{σ} (n) = \frac{1}{2} (P_{T} (n) + σ i n \times)$ for every $n \in S^{2}$ and proves that its range is a smooth line bundle with local smooth unit frames.

[F2]

Curvature and the obstruction to a global polarization basis proves that the line has no globally smooth unit frame. This is an obstruction to one global phase basis, not a singularity of the projection family.

Counterexample

Given: The facts [F1] and [F2].

1.1F1algebra

Since $n \cdot n = 1$ , the formula in [F1] is equivalently $P_{σ} (n) = \frac{1}{2} (I - n n^{T} + σ i N_{n})$ , where $N_{n} z = n \times z$ . Each matrix entry is a polynomial in the real coordinates of $n$ , so $n \mapsto P_{σ} (n)$ is smooth on all of $S^{2}$ , including both poles.

2.1F1F2step 1.1algebra∎

The local formula for $v_{σ}$ uses $ϕ$ , which is not a coordinate at either pole, while the projection matrix from step 1.1 is defined and smooth there. By [F1], projecting a fixed vector with nonzero projection at a given point and normalizing it gives a smooth local unit frame near that point. Thus local polarization sections remain smooth; [F2] rules out only a single nowhere-zero unit frame on the entire sphere. The displayed false implication is refuted.

Helicity bundles, rotations and Berry curvature — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

5 · Examples, experiments and counterexamples

Latitude-loop Berry holonomy

Example

Example references

Facts & Assumptions

Verification

The polar helicity frame is singular while its projector is smooth

Statement refuted

Statement references

Facts & Assumptions

Counterexample

Sources