DefinitionDefinition: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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Helicity eigenlines over the direction sphere

Physical scope

Conditional classical Maxwell-Lorentz vacuum model in the smooth source-free nonzero-Fourier-mode regime stated by the supplier. The construction is the direction-sphere geometry of those modes, with no additional approximation. Directions, unit polarization vectors, the helicity sigma, and angular coordinates are dimensionless; the Berry connection and holonomy are dimensionless. There is no quantization, photon position operator, material medium, measurement, or empirical claim.

Definition

For $u, v \in C^{3}$ , extend the real Euclidean dot and right-handed cross products complex-bilinearly, with coordinate formulas $u \cdot v : = \sum_{j = 1}^{3} u_{j} v_{j}, u \times v : = (u_{2} v_{3} - u_{3} v_{2}, u_{3} v_{1} - u_{1} v_{3}, u_{1} v_{2} - u_{2} v_{1}) .$ The dot product in these formulas uses no complex conjugation; it differs from the Hermitian product $⟨ u, v ⟩_{H} : = \sum_{j = 1}^{3} \overline{u_{j}} v_{j}$ . Write $∥ z ∥_{H} : = \sqrt{⟨ z, z ⟩_{H}}$ for the associated nonnegative norm on $C^{3}$ .

Let $S^{2} : = {n \in R^{3} : n \cdot n = 1}$ with the smooth structure given by the six graph charts solving $n_{j} = \pm \sqrt{1 - \sum_{k \neq j} n_{k}^{2}}$ on the hemispheres $\pm n_{j} > 0$ . Orient it outward using the ordered spatial basis $x^{1}, x^{2}, x^{3}$ . Its tangent plane is $T_{n} S^{2} = {X \in R^{3} : n \cdot X = 0}$ . Physically, $n = k / ∣ k ∣$ for a nonzero spatial wavevector $k$ in the SI Fourier description, where $k$ has units $m^{- 1}$ ; hence $n$ is dimensionless and the radial magnitude is discarded. For $σ \in {+ 1, - 1}$ , set $C_{n} z : = i n \times z$ , $P_{T} (n) z : = z - n (n \cdot z)$ , and $P_{σ} (n) : = \frac{1}{2} (P_{T} (n) + σ C_{n})$ on $C^{3}$ . Define the helicity fibre $L_{σ} (n) : = range P_{σ} (n)$ and the total space $L_{σ} : = {(n, z) : n \in S^{2}, z \in L_{σ} (n)}$ , projected to $S^{2}$ by $(n, z) \mapsto n$ .

For an open set $U \subseteq S^{2}$ , a unit frame of $L_{σ}$ over $U$ is a smooth map $v : U \to C^{3}$ such that $v (n) \in L_{σ} (n)$ and $⟨ v (n), v (n) ⟩_{H} = 1$ for every $n \in U$ . Since each fibre is a complex line, this vector spans that fibre. A local unit frame is such a map on an open neighborhood in $S^{2}$ ; a global unit frame is one defined on all of $S^{2}$ .

Fibrewise addition and scalar multiplication are inherited from $C^{3}$ . A unit frame $v$ gives the local trivialization $(n, a) \mapsto (n, a v (n))$ from $U \times C$ to $L_{σ} ∣_{U}$ , with inverse $(n, z) \mapsto (n, ⟨ v (n), z ⟩_{H})$ . The construction in the Proof supplies such a frame near every $n$ , so $L_{σ}$ is a smooth rank-one complex vector subbundle of the product bundle $S^{2} \times C^{3} \to S^{2}$ , whose projection is $(n, z) \mapsto n$ .

On a spherical chart with $0 < θ < π$ and a smooth longitude branch $ϕ$ , write $n = (\sin θ \cos ϕ, \sin θ \sin ϕ, \cos θ)$ , $e_{θ} = (\cos θ \cos ϕ, \cos θ \sin ϕ, - \sin θ)$ , and $e_{ϕ} = (- \sin ϕ, \cos ϕ, 0)$ . The local unit helicity frame is $v_{σ} : = (e_{θ} + i σ e_{ϕ}) / \sqrt{2}$ . For any unit frame $v$ over an open set $U$ , write $v^{†} d v : = \sum_{j = 1}^{3} \overline{v_{j}} d v_{j}$ , where $d v_{j}$ is the exterior derivative of its $j$ th component. Define its Berry connection one-form by $A : = i v^{†} d v$ . A smooth phase change $v^{'} = e^{i χ} v$ , with real smooth $χ$ , changes this local form to $A^{'} = A - d χ$ . The local Berry curvature is the real two-form $F : = d A$ on $U$ ; no global gluing assertion is made here.

Definition references

Exact helicity projectors in vacuum

Classical transverse Fourier amplitudes

Lorentzian Hodge star and periods

The cross product in $R^{3}$

The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$

Oriented smooth manifolds and oriented charts

A smooth differential $k$ -form

The wedge product of differential forms

The exterior derivative by the invariant vector-field formula

Facts & Assumptions

Given: The oriented unit direction sphere, its standard Hermitian space $C^{3}$ , and one fixed $σ \in {+ 1, - 1}$ .

[F1]

Exact helicity projectors in vacuum defines $C_{n} = i n \times$ , $P_{T} z = z - n (n \cdot z)$ , and $P_{σ} = \frac{1}{2} (P_{T} + σ C_{n})$ for unit directions; it proves that $P_{σ}$ is smooth and is an orthogonal rank-one projection onto the $σ$ helicity eigenspace.

[F2]

Classical transverse Fourier amplitudes defines $k$ as a spatial wavevector with SI units $m^{- 1}$ and uses nonzero $k$ for its transverse Fourier amplitudes; normalizing $k$ gives the dimensionless direction $n$ .

[F3]

Lorentzian Hodge star and periods fixes the ordered coframe $d x^{0}, d x^{1}, d x^{2}, d x^{3}$ and volume $d x^{0} \land d x^{1} \land d x^{2} \land d x^{3}$ ; restricting to the spatial subspace gives the ordered orientation used here.

[F4]

The cross product in $R^{3}$ fixes the coordinate formula for the right-handed cross product; The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ fixes the real Euclidean dot product and norm.

[F5]

Oriented smooth manifolds and oriented charts defines an orientation as a smoothly chosen orientation ray; the outward normal and fixed spatial orientation specify that ray here.

[F6]

A smooth differential $k$ -form defines smooth $k$ -forms as smooth sections of $⋀^{k} T^{*} S^{2}$ ; The wedge product of differential forms defines their wedge product pointwise, and The exterior derivative by the invariant vector-field formula defines $d$ .

Proof

1.1F1F2F3F5givenconstruct

The six graph charts in the Definition are smooth coordinate charts on open hemispheres and cover $S^{2}$ ; their square-root transition formulas are smooth on overlaps. The outward tangent orientation ray is smooth by the nonvanishing normal $n$ , with ambient spatial orientation fixed in [F3] and orientation-ray convention [F5]. By [F1], each $P_{σ} (n)$ is a smooth rank-one projection, so each fibre is a complex line. [F2] identifies the physical direction index with a nonzero SI wavevector normalized to unit length. At any $n_{0}$ , choose a fixed $w \in C^{3}$ with $P_{σ} (n_{0}) w \neq 0$ , which exists because the projection has nonzero range. Continuity makes $P_{σ} (n) w \neq 0$ on a neighborhood of $n_{0}$ , and $P_{σ} (n) w / ∥ P_{σ} (n) w ∥$ is a smooth unit local section there. This proves local triviality without choosing a frame family over the whole sphere.

1.2F1F4algebra

The coordinate vectors satisfy $n \cdot e_{θ} = n \cdot e_{ϕ} = e_{θ} \cdot e_{ϕ} = 0$ , $∥ e_{θ} ∥ = ∥ e_{ϕ} ∥ = 1$ , and $n \times e_{θ} = e_{ϕ}$ , $n \times e_{ϕ} = - e_{θ}$ , by the coordinate formulas in [F4]. Hence $v_{σ}^{†} v_{σ} = 1$ , $P_{T} v_{σ} = v_{σ}$ , and $C_{n} v_{σ} = σ v_{σ}$ . Substitution gives $P_{σ} v_{σ} = v_{σ}$ , so this is a unit local frame of the stated fibre.

2.1F1F6step 1.1step 1.2algebra

For a smooth section $s$ on a chart, define $\nabla s : = P_{σ} d s$ . Since $P_{σ} s = s$ , this is again $L_{σ}$ -valued. For every smooth scalar $f$ , the product rule gives $\nabla (f s) = P_{σ} (d f s + f d s) = d f s + f \nabla s$ , so $\nabla$ is a connection on the line bundle. In a unit frame $v$ , writing $s = a v$ gives $\nabla s = (d a + a v^{†} d v) v = (d a - i A a) v$ because $A = i v^{†} d v$ .

3.1givenF6step 2.1algebra∎

Differentiating $⟨ v, v ⟩_{H} = 1$ gives $2 Re (\sum_{j} \overline{v_{j}} d v_{j}) = 0$ , so $A = i \sum_{j} \overline{v_{j}} d v_{j}$ is real-valued. Under $v^{'} = e^{i χ} v$ , direct differentiation gives $v^{' †} d v^{'} = i d χ + v^{†} d v$ , hence $A^{'} = i v^{' †} d v^{'} = A - d χ$ . This verifies the declared local gauge transformation and completes the definition of the local connection and curvature conventions.

Helicity eigenlines over the direction sphere

Definition

Definition references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources