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

=== We work in 3+1 bulk spacetime with coordinates xμx^\muxμ. A spherical electron shell sits at radius r=r0r=r_0r=r0. The bulk electromagnetic action is ===

SEMbulk  =  −ε04∫d4x  FμνFμν,Fμν=∂μAν−∂νAμ.S_{\rm EM}^{\rm bulk}
\;=\;
-\frac{\varepsilon_0}{4}\int d^4x \; F_{\mu\nu}F^{\mu\nu},
\qquad
F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu.SEMbulk=−4ε0∫d4xFμνFμν,Fμν=∂μAν−∂νAμ.
Put a dynamical spinor field Ψ(t,Ω)\Psi(t,\Omega)Ψ(t,Ω) on the spherical 2-D boundary Sr02S^2_{r_0}Sr02 (with coordinates (t,θ,φ)(t,\theta,\varphi)(t,θ,φ)). Use the boundary Dirac action (schematic)

Sb=∫dt ⁣∫Sr02 ⁣r02dΩ  [i (h/2π) Ψ†DtΨ−v Ψ†\slashedDS2Ψ−V(Ω) Ψ†Ψ],S_{\rm b}=\int dt\!\int_{S^2_{r_0}}\! r_0^2 d\Omega\;\Big[
i\,(h/2\pi)\,\Psi^\dagger D_t\Psi
* v\,\Psi^\dagger \slashed D_{S^2}\Psi
* V(\Omega)\,\Psi^\dagger\Psi
\Big],Sb=∫dt∫Sr02r02dΩ[i(h/2π)Ψ†DtΨ−vΨ†\slashedDS2Ψ−V(Ω)Ψ†Ψ],
where the covariant derivative includes electromagnetic minimal coupling:

Dμ  =  ∂μ+i q(h/2π) Aμ(on the boundary).D_\mu \;=\; \partial_\mu + i\,\frac{q}{(h/2\pi)}\,A_\mu
\qquad\text{(on the boundary).}Dμ=∂μ+i(h/2π)qAμ(on the boundary).
(Using h/2πh/2\pih/2π explicitly: the usual factor iq/ℏi q/\hbariq/ℏ is here iq/(h/2π)i q/(h/2\pi)iq/(h/2π).)

Add the surface–bulk interaction term (minimal coupling of boundary current to bulk field):

Sint  =  ∫Sr02 ⁣d3σ  Aμ(x) jsurfμ(x)  =  ∫dt ⁣∫Sr02 ⁣r02dΩ  Aμ(t,Ω) jsurfμ(t,Ω),S_{\rm int}
\;=\;
\int_{S^2_{r_0}}\! d^3\sigma\; A_\mu(x)\, j^{\mu}_{\rm surf}(x)
\;=\;
\int dt\!\int_{S^2_{r_0}}\! r_0^2 d\Omega\; A_\mu(t,\Omega)\, j^\mu_{\rm surf}(t,\Omega),Sint=∫Sr02d3σAμ(x)jsurfμ(x)=∫dt∫Sr02r02dΩAμ(t,Ω)jsurfμ(t,Ω),
with the dynamical surface current obtained by Noether’s procedure (global U(1) phase symmetry of Ψ\PsiΨ):

jsurfμ  =  q ΨˉγμΨ(on the 2+1 boundary).j^\mu_{\rm surf}
\;=\;
q\,\bar\Psi\gamma^\mu\Psi \quad\text{(on the 2+1 boundary)}.jsurfμ=qΨˉγμΨ(on the 2+1 boundary).
(In 2+1 language the index μ\muμ runs over (t,θ,φ)(t,\theta,\varphi)(t,θ,φ) intrinsic to the shell. When embedded into 3+1, the surface current is a delta-supported source.)

Total action is S=SEMbulk+Sb+Sint+…S=S_{\rm EM}^{\rm bulk}+S_{\rm b}+S_{\rm int}+\dotsS=SEMbulk+Sb+Sint+… (plus any bulk charges).