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

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Paraxial orbital winding in its exact scalar domain

Physical scope

Conditional identity for a smooth, finite-transverse-norm scalar paraxial field in the specified angular-spectrum model, at a fixed propagation plane. The dimensionless scalar rotation generator is not identified with torque, total electromagnetic angular momentum, a photon angular momentum, or a general exact-vector spin/orbital decomposition. No empirical claim is made.

Statement

Fix a propagation plane $z$ and let $u_{z} \in C^{\infty} (R^{2}; C) \cap L^{2} (R^{2}; C)$ be a scalar field in the paraxial model. Assume $u_{z}$ is nonzero as an $L^{2}$ equivalence class. For $r > 0$ write $u_{z} (r \cos ϕ, r \sin ϕ) = f (r) e^{i ℓ ϕ}$ , where $ℓ \in Z$ and $ϕ$ is read modulo $2 π$ . Require a smooth axis extension; explicitly, assume there are $ϵ > 0$ and a smooth complex-valued $g : (- ϵ, ϵ) \to C$ such that $f (r) = r^{∣ ℓ ∣} g (r^{2})$ for $0 < r < \sqrt{ϵ}$ . Define $L_{z}^{s c} : = - i (- y \partial_{x} + x \partial_{y})$ and use the first-variable-linear $L^{2}$ pairing $⟨ v, w ⟩_{2} : = \int_{R^{2}} v (x, y) \overline{w (x, y)} d^{2} x .$ Then the normalized scalar angular-generator expectation is $\frac{⟨ L_{z}^{s c} u_{z}, u_{z} ⟩_{2}}{⟨ u_{z}, u_{z} ⟩_{2}} = ℓ .$

Statement references

Forward angular-spectrum beam

The complex exponential by its power series

Real and imaginary parts, complex conjugation, and modulus

Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$

The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$

The complex exponential is entire and its complex derivative is itself

Facts & Assumptions

Given: A fixed plane $z$ , the stated smooth $L^{2}$ scalar field $u_{z}$ with nonzero $L^{2}$ class and the stated integer-winding form.

[F1]

The paraxial scalar field and its SI field interpretation are those of the angular-spectrum Definition. Forward angular-spectrum beam Framework-local SI units, the vacuum constants and the exact/measured status of their values

[F2]

The real partial derivatives are those on the fixed laboratory spatial slice, and complex exponentiation satisfies $\exp^{'} = \exp$ . Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ The complex exponential is entire and its complex derivative is itself

[F3]

The field uses the complex exponential and conjugation/modulus operations defined on all complex numbers. The complex exponential by its power series Real and imaginary parts, complex conjugation, and modulus

[F4]

On each local polar-coordinate chart with $r > 0$ , the smooth map $(r, ϕ) \mapsto (x, y) = (r \cos ϕ, r \sin ϕ)$ has $\partial_{ϕ} x = - y$ and $\partial_{ϕ} y = x$ . The chain rule therefore gives $\partial_{ϕ} u_{z} = - y \partial_{x} u_{z} + x \partial_{y} u_{z}$ there. The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$

Proof

1.1givenF1F2F3F4algebra

For $r > 0$ , [F4] identifies the angular vector field $- y \partial_{x} + x \partial_{y}$ with $\partial_{ϕ}$ . By [F2]–[F3], differentiating the integer winding factor gives $L_{z}^{s c} u_{z} = - i \partial_{ϕ} (f e^{i ℓ ϕ}) = ℓ f e^{i ℓ ϕ} = ℓ u_{z}$ . The required axis form makes $u_{z}$ smooth at the origin: for $ℓ \geq 0$ its angular factor is $(x + i y)^{ℓ}$ , and for $ℓ < 0$ it is $(x - i y)^{∣ ℓ ∣}$ , each multiplied by the smooth radial factor $g (x^{2} + y^{2})$ . The rotation vector field vanishes at the origin, so the same identity holds there as well.

2.1step 1.1givenalgebra∎

Step 1.1 gives $L_{z}^{s c} u_{z} = ℓ u_{z}$ everywhere, hence $L_{z}^{s c} u_{z} \in L^{2}$ because $u_{z} \in L^{2} (R^{2}; C)$ . Therefore $⟨ L_{z}^{s c} u_{z}, u_{z} ⟩_{2} = \int_{R^{2}} ℓ ∣ u_{z} ∣^{2} d^{2} x = ℓ ⟨ u_{z}, u_{z} ⟩_{2}$ ; the denominator is positive since $u_{z}$ is nonzero. Division proves the claimed identity.

Remarks

The axis hypothesis is stated as a smooth extension condition; the bound $f (r) = O (r^{∣ ℓ ∣})$ alone would not guarantee smoothness. No radial integration by parts is used. The scalar generator is dimensionless and does not by itself determine a torque or a photon angular momentum.

Paraxial orbital winding in its exact scalar domain

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources