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Axial spin approaches helicity with a quantitative cone bound

Statement

Assume the Axiom of Choice and Countable Choice, $ℏ > 0$ , and fix $0 < κ_{-} < κ_{+} < \infty$ and $0 < θ_{\max} < π / 2$ . Let $f$ be a normalized helicity- $σ$ packet in the cone $κ_{-} \leq ∣ k ∣ \leq κ_{+}$ , $0 \leq θ (k) \leq θ_{\max}$ , with $σ \in {+ 1, - 1}$ . For the helicity-projected operator $S_{z}^{'}$ on the transverse one-photon Hilbert space, $⟨ f, S_{z}^{'} f ⟩_{ν} = ℏ σ \int_{M} \cos θ (k) ∣ f (k) ∣^{2} d ν (k), ∣ ⟨ f, S_{z}^{'} f ⟩_{ν} - ℏ σ ∣ \leq ℏ (1 - \cos θ_{\max}) \leq \frac{ℏ θ_{\max}^{}}{2} .$ For the exact north-gauge vortex packet of winding $ℓ$ constructed in the preceding lemma, $J_{z}$ has eigenvalue $(ℓ + σ) ℏ$ , and $L_{z}^{'} = J_{z} - S_{z}^{'}$ therefore satisfies $⟨ f, L_{z}^{'} f ⟩_{ν} = ℓ ℏ + ℏ σ (1 - \int_{M} \cos θ (k) ∣ f (k) ∣^{2} d ν (k)), ∣ ⟨ f, L_{z}^{'} f ⟩_{ν} - ℓ ℏ ∣ \leq \frac{ℏ θ_{\max}^{}}{2} .$ The integer $ℓ$ is the coefficient winding in the chosen north gauge; neither formula asserts a global gauge-independent orbital label.

Statement references

Forward cone and regular north-patch helicity gauge

Finite-bandwidth vortex packets are rotation eigenvectors

Projected photon spin and orbital operators

Second-quantized total rotations and helicity counts

Transverse rotations preserve helicity and generate total angular momentum

Facts & Assumptions

Given: The Axiom of Choice, $ℏ > 0$ , the normalized transverse helicity packet, the polar-angle support condition, and the operator conventions stated above.

[A1]

Assume the Axiom of Choice and Countable Choice. The Axiom of Choice The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F1]

The cone definition gives a packet with $∥ f ∥_{ν} = 1$ , $P_{σ} f = f$ almost everywhere, and support in $0 \leq θ \leq θ_{\max}$ . Forward cone and regular north-patch helicity gauge

[F2]

For every normalized one-photon helicity vector, $⟨ f, S_{z}^{'} f ⟩_{ν} = ℏ σ \int_{M} \cos θ (k) ∣ f (k) ∣^{2} d ν (k)$ . Second-quantized total rotations and helicity counts

[F3]

$S_{z}^{'}$ is bounded and self-adjoint, and $L_{z}^{'} = J_{z} - S_{z}^{'}$ on the exact domain $D (J_{z})$ . Projected photon spin and orbital operators

[F4]

The rotation convention is $U (R_{z} (t)) = e^{- i t J_{z} / ℏ}$ , with $J_{z}$ the self-adjoint Stone generator. Transverse rotations preserve helicity and generate total angular momentum

[F5]

The preceding lemma constructs a normalized vortex packet in $D (J_{z})$ with $J_{z} f = (ℓ + σ) ℏ f$ . Finite-bandwidth vortex packets are rotation eigenvectors

[F6]

Sine is $1$ -Lipschitz, so $∣ \sin u ∣ \leq ∣ u ∣$ . Sine and cosine are $1$ -Lipschitz on $R$

[F7]

Cosine is decreasing on $[0, π]$ and takes values in $[- 1, 1]$ . Signs, monotonicity intervals, and ranges of sine and cosine

[F8]

The cosine addition formula and $\sin^{2} u + \cos^{2} u = 1$ imply $1 - \cos x = 2 \sin^{2} (x / 2)$ . The addition formulas for sine and cosine Parity and the Pythagorean identity for sine and cosine

Proof

Given: [A1], [F1]–[F8], and the Statement hypotheses.

1.1A1F2F7givenalgebra

Under the Choice assumptions [A1], [F2] gives $⟨ f, S_{z}^{'} f ⟩_{ν} = ℏ σ \int_{M} \cos θ d ρ$ , where $d ρ = ∣ f ∣^{2} d ν$ is a probability measure because $∥ f ∥_{ν} = 1$ . Since $∣ σ ∣ = 1$ , $∣ ⟨ f, S_{z}^{'} f ⟩_{ν} - ℏ σ ∣ = ℏ (1 - \int_{M} \cos θ d ρ)$ ; the right side is nonnegative by the range bound in [F7].

2.1F1F7step 1.1algebra

On the support, $0 \leq θ \leq θ_{\max} < π / 2$ . The monotonicity in [F7] gives $0 \leq 1 - \cos θ \leq 1 - \cos θ_{\max}$ pointwise. Integrating against the probability measure $ρ$ in step 1.1 proves the first upper bound.

3.1F6F8step 2.1algebra

By [F8], $1 - \cos θ_{\max} = 2 \sin^{2} (θ_{\max} / 2)$ . Applying [F6] with $u = θ_{\max} / 2$ gives $2 \sin^{2} (θ_{\max} / 2) \leq 2 (θ_{\max} / 2)^{2} = θ_{\max}^{} / 2$ . Combining this with step 2.1 proves the spin estimate.

4.1A1F2F3F4F5step 3.1algebra∎

Under [A1], the preceding lemma [F5] gives $J_{z} f = (ℓ + σ) ℏ f$ and $f \in D (J_{z})$ in the rotation convention [F4]. By [F3], $L_{z}^{'} = J_{z} - S_{z}^{'}$ on that domain, so $⟨ f, L_{z}^{'} f ⟩_{ν} = (ℓ + σ) ℏ - ⟨ f, S_{z}^{'} f ⟩_{ν}$ . Substituting [F2] gives the displayed exact orbital expectation; subtracting $ℓ ℏ$ and applying step 3.1 gives its stated bound.

Remarks

These are conditional Hilbert-space deductions for the stated one-photon model. The gauge winding is local, and the expectation identities do not identify the projected operators with an apparatus-specific orbital-angular-momentum measurement.

Axial spin approaches helicity with a quantitative cone bound

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources