Editing Openai/6897769e-4ee4-800f-aba5-69cca34f701c (section)

=== Key mathematical fact (used without proof here; it’s standard): the Dirac operator on the 2-sphere has eigenmodes whose angular quantum numbers are half-integers. Concretely, eigenmodes satisfy ===

\slashedDS2 Φj,m(Ω)  =  λj Φj,m(Ω),j=12,32,52,…,m=−j,…,+j,\slashed{D}_{S^2}\,\Phi_{j,m}(\Omega) \;=\; \lambda_j\,\Phi_{j,m}(\Omega),
\qquad j=\tfrac12,\tfrac32,\tfrac52,\ldots,\quad m=-j,\dots,+j,\slashedDS2Φj,m(Ω)=λjΦj,m(Ω),j=21,23,25,…,m=−j,…,+j,
so the total angular momentum labels jjj are half-integers.

Because Ψ\PsiΨ is a spinor on the sphere, under a geometric spatial rotation by 2π2\pi2π the spinor picks up a sign:

Ψ(φ+2π)  =  − Ψ(φ).\Psi(\varphi+2\pi) \;=\; -\,\Psi(\varphi).Ψ(φ+2π)=−Ψ(φ).
Equivalently the spinor is single-valued only after a 4π4\pi4π geometric rotation. This is the antiperiodic condition on the geometric circle coordinate φ\varphiφ, which directly yields half-integer LzL_zLz eigenvalues:

L^zΨ  =  (m) (h/2π) Ψ,m∈Z+12.\hat L_z\Psi \;=\; (m)\,(h/2\pi)\,\Psi,\qquad m\in\mathbb{Z}+\tfrac12.L^zΨ=(m)(h/2π)Ψ,m∈Z+21.
Thus the boundary Dirac action on S2S^2S2 gives a natural origin for half-integer angular momentum: the spinor fields on the 2-D spherical shell are sections of a spin bundle and their eigenmodes are half-integer.

(If you want a compact derivation of the Dirac spectrum on the sphere we can add it — but the main point is: Dirac on S2S^2S2 ⇒ j∈Z+1/2j\in \mathbb{Z}+1/2j∈Z+1/2.)