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

=== Write a minimal action for a 2-component spinor Ψ(t,Ω)\Psi(t,\Omega)Ψ(t,Ω) on the spherical boundary Sr2S^2_rSr2. Use the unit 2-sphere metric scaled by rrr. The (real) action is ===

Sb  =  ∫dt∫Sr2dΩ r2  { i h2π Ψ†∂tΨ  −  v  Ψ†(\slashedDS2)Ψ  −  V(Ω) Ψ†Ψ },S_{\rm b} \;=\; \int dt \int_{S^2_r} d\Omega\, r^2
\;\Big\{\, i\,\frac{h}{2\pi}\,\Psi^\dagger \partial_t \Psi \;-\; v\;\Psi^\dagger \big(\slashed{D}_{S^2}\big)\Psi
\;-\; V(\Omega)\,\Psi^\dagger\Psi \,\Big\},Sb=∫dt∫Sr2dΩr2{i2πhΨ†∂tΨ−vΨ†(\slashedDS2)Ψ−V(Ω)Ψ†Ψ},
where
* dΩ=sin⁡θ dθ dφd\Omega=\sin\theta\,d\theta\,d\varphidΩ=sinθdθdφ is the angular measure,
* vvv is a characteristic velocity scale (a conversion so the Dirac operator has units of energy),
* \slashedDS2\slashed{D}_{S^2}\slashedDS2 is the (2-dimensional) Dirac operator on the sphere including the spin connection (it acts on spinor fields), and
* V(Ω)V(\Omega)V(Ω) is any geometry/potential term (e.g. coupling to local surface charge density).

Remarks: the factor h/2πh/2\pih/2π appears explicitly in the time-derivative term as the canonical action quantum. This keeps the geometry of angular action (circumference 2πr2\pi r2πr) visible.

Varying SbS_{\rm b}Sb with respect to Ψ†\Psi^\daggerΨ† gives the boundary Dirac equation

ih2π ∂tΨ  =  v \slashedDS2Ψ+V(Ω)Ψ.i\frac{h}{2\pi}\,\partial_t\Psi \;=\; v\,\slashed{D}_{S^2}\Psi + V(\Omega)\Psi.i2πh∂tΨ=v\slashedDS2Ψ+V(Ω)Ψ.
We now analyze the angular eigenmodes.