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

==== • A single spherical electron shell sits at radius r=r0r=r_0r=r0. ====
• Bulk EM potential Aμ(x)A_\mu(x)Aμ(x) with field-strength Fμν=∂μAν−∂νAμF_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\muFμν=∂μAν−∂νAμ. Bulk Maxwell action is unchanged.
• Boundary spinor Ψ(t,Ω)\Psi(t,\Omega)Ψ(t,Ω) lives on the shell; Noether U(1) gives a surface current jsurfα(t,Ω)j^\alpha_{\rm surf}(t,\Omega)jsurfα(t,Ω) (indices α\alphaα run over shell coordinates (t,θ,φ)(t,\theta,\varphi)(t,θ,φ)). That surface current is what appears in the JμAμJ_\mu A^\muJμAμ coupling:

Sint=∫Sr02d3σ  Aα jsurfα.S_{\rm int}=\int_{S^2_{r_0}} d^3\sigma\;A_\alpha\,j^\alpha_{\rm surf}.Sint=∫Sr02d3σAαjsurfα.
• In 3+1 bulk coordinates the surface source appears as a delta layer:

Jμ(x)  =  Jbulkμ(x)+δ(r−r0) jsurfμ(t,Ω).J^\mu(x) \;=\; J^\mu_{\rm bulk}(x) + \delta(r-r_0)\,j^\mu_{\rm surf}(t,\Omega).Jμ(x)=Jbulkμ(x)+δ(r−r0)jsurfμ(t,Ω).
This is the source we plug into Maxwell’s equations.

Maxwell (differential form, SI):

∇⋅E=ρε0,∇×B=μ0J+μ0ε0∂tE\nabla\cdot\mathbf{E}=\frac{\rho}{\varepsilon_0},\qquad
\nabla\times\mathbf{B}=\mu_0\mathbf{J}+\mu_0\varepsilon_0\partial_t\mathbf{E}∇⋅E=ε0ρ,∇×B=μ0J+μ0ε0∂tE
and the homogeneous pair

∇⋅B=0,∇×E=−∂tB.\nabla\cdot\mathbf{B}=0,\qquad \nabla\times\mathbf{E}=-\partial_t\mathbf{B}.∇⋅B=0,∇×E=−∂tB.