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

=== Now the spin / antiperiodic boundary condition question. ===
* If a fermionic (spin-½) field ψ\psiψ is transported around a closed path parameterized by an angular coordinate φ\varphiφ with φ→φ+2π\varphi\to\varphi+2\piφ→φ+2π (for example a great-circle on Σ\SigmaΣ), then the spinor transforms under a rotation by angle 2π2\pi2π in 3D physical space by the spin representation:

ψ↦e−i(2π) n^⋅Sψ\psi \mapsto e^{-i(2\pi)\,\hat{\mathbf{n}}\cdot\mathbf{S}}\psiψ↦e−i(2π)n^⋅Sψ
For spin-½, n^⋅S=12σn\hat{\mathbf{n}}\cdot\mathbf{S} = \tfrac12 \sigma_nn^⋅S=21σn and so

e−i2π(1/2)=e−iπ=−1.e^{-i 2\pi (1/2)} = e^{-i\pi} = -1.e−i2π(1/2)=e−iπ=−1.
Therefore parallel-transporting a spinor around a full 2π2\pi2π rotation produces a minus sign: ψ(φ+2π)=−ψ(φ)\psi(\varphi+2\pi) = -\psi(\varphi)ψ(φ+2π)=−ψ(φ). That is the standard geometric origin of antiperiodic boundary conditions for fermions under a single full rotation.
* More generally, the total phase accumulated by the charged spinor going around a closed loop is the product of the spin rotation phase (the −1-1−1 from 2π2\pi2π for spin-½) and the gauge holonomy (Wilson loop)

exp⁡(ie∮Aa dxa).\exp\Big( i e \oint A_a\,dx^a \Big).exp(ie∮Aadxa).
If the gauge holonomy supplies an additional minus sign (for example if the enclosed magnetic flux is half a flux quantum, modulo Dirac quantization caveats), then the total monodromy can be ±1 depending on geometry and flux. On S2S^2S2 a nontrivial gauge patching (Dirac monopole at the center) can produce nontrivial holonomy.
* Bottom line: antiperiodic boundary conditions are geometrically natural for spin-½ fields on closed loops (great circles) because a spatial 2π2\pi2π rotation gives the minus sign. If you place the electron spinor on the 2-sphere surface and examine loops that correspond to rotations about an axis, antiperiodic BCs arise automatically from spin geometry.

(Technical note: a fully rigorous global statement about spinors on S2S^2S2 requires attention to the spin structure. S2S^2S2 is orientable but the global classification of spin structures becomes important if you demand a globally well-defined spinor; in many physical situations one uses stereographic charts and accepts the local antiperiodic behavior around closed loops. If you want, I can add the topological details about w2w_2w2 and spin structures on S2S^2S2 — but it doesn’t change the physical point: fermions pick up a −1-1−1 under 2π2\pi2π rotation.)