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

=== Varying StotS_{\rm tot}Stot: ===

Variation w.r.t. ψˉ\bar\psiψˉ (on the surface) gives the surface Dirac equation (gauge-covariant)

ih2π γaDaψ−msψ+⋯=0i\frac{h}{2\pi}\,\gamma^a D_a\psi - m_s \psi + \cdots =0i2πhγaDaψ−msψ+⋯=0
— photon absorption/emission on the surface appears through AaA_aAa inside DaD_aDa.

Variation w.r.t. AμA_\muAμ (bulk) yields Maxwell’s equations with sources, the surface current appearing as a 3D delta concentrated on ∂M\partial M∂M:

∇νFνμ=μ0(Jbulkμ+Jsurfμ δ∂M).\nabla_\nu F^{\nu\mu} = \mu_0 \big( J^\mu_{\rm bulk} + J^\mu_{\rm surf}\, \delta_{\partial M} \big).∇νFνμ=μ0(Jbulkμ+Jsurfμδ∂M).
Antiperiodic (half-integer rotation) condition:
* A spinor, under a  full 2π2\pi2π rotation, picks up sign −1-1−1. On the round azimuthal circle parameterized by ϕ\phiϕ on the 2-sphere, this means the allowed spinor modes have ψ(ϕ+2π)=−ψ(ϕ)\psi(\phi+2\pi)=-\psi(\phi)ψ(ϕ+2π)=−ψ(ϕ). That antiperiodicity is what enforces half-integer quantization for angular momentum modes on the surface. Concretely, when you expand ψ(ϕ)\psi(\phi)ψ(ϕ) in eigenmodes eimϕe^{i m \phi}eimϕ, antiperiodicity forces m∈Z+12m\in\mathbb{Z}+\tfrac12m∈Z+21 (half-integer), hence half-spin behavior.

(In a rigorous treatment one uses the spin connection on the surface and the Spin(2)Spin(2)Spin(2) double cover of rotations; the minus sign is the usual spinor monodromy. For our geometric picture it suffices that imposing single-valuedness of the underlying electron wave-function relative to rotations requires that spinor boundary fields be antiperiodic around the azimuthal loop.)