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

==== ### ====
* The electron’s probability shell sits at radius r=r0r=r_0r=r0. We model the shell with a complex scalar field ϕ(t,θ,φ)\phi(t,\theta,\varphi)ϕ(t,θ,φ) living on the 2-sphere S2(r0)S^2(r_0)S2(r0).
* Electromagnetic field Aμ(x)A_\mu(x)Aμ(x) lives in the 3-D space (bulk). The shell interacts with the tangential components At,Aθ,AφA_t,A_\theta,A_\varphiAt,Aθ,Aφ evaluated at r=r0r=r_0r=r0.
* We keep the quantum of action explicit: ℏ≡h2π\displaystyle \hbar\equiv\frac{h}{2\pi}ℏ≡2πh.
* Units: I will write equations in a mixed but transparent way — where necessary I remind how to add SI factors (ε₀, μ₀, c). The structure is valid in natural units (c = 1) and can be converted.

===== Write the total action as =====

S=Sbulk EM[A]  +  Sshell[ϕ,A∣r=r0],S = S_{\rm bulk\,EM}[A] \;+\; S_{\rm shell}[\phi,A|_{r=r_0}],S=SbulkEM[A]+Sshell[ϕ,A∣r=r0],
with the surface action

Sshell  =  ℏ∫dt  ∫S2(r0) ⁣dΩ r02  Lshell(ϕ,Dμϕ;Aμ)S_{\rm shell} \;=\; \hbar \int dt \; \int_{S^2(r_0)} \! d\Omega\, r_0^2 \; \mathcal{L}_{\rm shell}(\phi,D_\mu\phi;A_\mu)Sshell=ℏ∫dt∫S2(r0)dΩr02Lshell(ϕ,Dμϕ;Aμ)
where dΩ=sin⁡θ dθ dφd\Omega=\sin\theta\,d\theta\,d\varphidΩ=sinθdθdφ, and the covariant derivative (minimal coupling) on the shell is

Dtϕ=∂tϕ+iqAt(t,θ,φ),Daϕ=∇aϕ+iqAa(t,θ,φ),D_t\phi = \partial_t\phi + i q A_t(t,\theta,\varphi),\qquad
D_a\phi = \nabla_a\phi + i q A_a(t,\theta,\varphi),Dtϕ=∂tϕ+iqAt(t,θ,φ),Daϕ=∇aϕ+iqAa(t,θ,φ),
with a∈{θ,φ}a\in\{\theta,\varphi\}a∈{θ,φ} the tangential indices and ∇a\nabla_a∇a the covariant derivative on the sphere. The factor ℏ\hbarℏ outside encodes the action quantum visibly.

A convenient choice for the shell Lagrangian density is

Lshell=12c2  ∣Dtϕ∣2−12r02  ∣∇ ⁣Ωϕ−iqr0AΩ ϕ∣2−12ms2∣ϕ∣2  +  ⋯\mathcal{L}_{\rm shell}
= \frac{1}{2c^{2}}\;|D_t\phi|^{2}
* \frac{1}{2r_0^{2}}\;| \nabla_{\!\Omega}\phi - i q r_0 A_{\Omega}\,\phi |^{2}
* \frac{1}{2} m_s^{2} |\phi|^{2} \;+\; \cdotsLshell=2c21∣Dtϕ∣2−2r021∣∇Ωϕ−iqr0AΩϕ∣2−21ms2∣ϕ∣2+⋯
where ∇ ⁣Ω\nabla_{\!\Omega}∇Ω is the angular gradient and AΩA_{\Omega}AΩ denotes the tangential components Aθ,AφA_\theta,A_\varphiAθ,Aφ. msm_sms is an effective shell mass parameter (from a quadratic potential) controlling how easily the shell field localizes.

: 

===== Vary SshellS_{\rm shell}Sshell with respect to ϕ∗\phi^*ϕ∗ to get the shell EOM (local on the 2-sphere): =====

1c2 DtDtϕ−1r02  ΔΩϕ+ms2 ϕ  +  (terms from AΩ)  =  0\frac{1}{c^2}\,D_t D_t \phi - \frac{1}{r_0^{2}}\;\Delta_{\Omega}\phi + m_s^{2}\,\phi
\;+\; (\text{terms from }A_\Omega) \;=\; 0c21DtDtϕ−r021ΔΩϕ+ms2ϕ+(terms from AΩ)=0
where ΔΩ\Delta_{\Omega}ΔΩ is the Laplace–Beltrami operator on the unit sphere and the AΩA_\OmegaAΩ terms are the usual minimal-coupling contributions.

This is the wave/oscillator equation for modes on the spherical shell.

===== The bulk EM Lagrangian is the usual =====

LEM=−14μ0FμνFμν,Fμν=∂μAν−∂νAμ.\mathcal{L}_{\rm EM}=-\frac{1}{4\mu_0}F^{\mu\nu}F_{\mu\nu},
\qquad F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu.LEM=−4μ01FμνFμν,Fμν=∂μAν−∂νAμ.
Varying the total action w.r.t. AμA_\muAμ gives Maxwell’s equations in the bulk with a surface current contribution from the shell:

∂νFνμ(x)  =  μ0 Jshellμ(t,θ,φ)  δ(r−r0)\partial_\nu F^{\nu\mu}(x) \;=\; \mu_0\, J^\mu_{\rm shell}(t,\theta,\varphi)\; \delta(r-r_0)∂νFνμ(x)=μ0Jshellμ(t,θ,φ)δ(r−r0)
where the shell current 4-vector (localized at r0r_0r0) is

  Jshellμ(t,θ,φ)  =  ℏ  iq  [ ϕ∗ Dμϕ  −  ϕ (Dμϕ)∗ ]∣r=r0  \boxed{\;
J^\mu_{\rm shell}(t,\theta,\varphi)
\;=\; \hbar \; i q \; \big[\,\phi^''\,D^\mu\phi \;-\; \phi\, (D^\mu\phi)^''\,\big]\Big|_{r=r_0}
\;}Jshellμ(t,θ,φ)=ℏiq[ϕ∗Dμϕ−ϕ(Dμϕ)∗]r=r0
Interpretation: JshellμJ^\mu_{\rm shell}Jshellμ is nonzero only on the sphere; its timelike component is the surface charge density and tangential components give the surface current density.

: 

===== Integrate Maxwell’s equations across the thin shell at r0r_0r0. Standard results (SI units) give: =====

Gauss law (radial electric jump):

ε0 (Erout−Erin)  =  σ(θ,φ,t)\varepsilon_{0}\,\big( E_r^{\text{out}} - E_r^{\text{in}} \big)
\;=\; \sigma(\theta,\varphi,t)ε0(Erout−Erin)=σ(θ,φ,t)
where the surface charge density is

σ(θ,φ,t)  =  Jshellt(t,θ,φ)\sigma(\theta,\varphi,t) \;=\; J^t_{\rm shell}(t,\theta,\varphi)σ(θ,φ,t)=Jshellt(t,θ,φ)
(modulo the ccc factor if you keep four-current normalization Jμ=(cρ,J)J^\mu=(c\rho,\mathbf{J})Jμ=(cρ,J)).

Ampère / tangential magnetic conditions (surface current):

r^×(Hout−Hin)  =  K(θ,φ,t)\hat r \times \big( \mathbf{H}^{\text{out}} - \mathbf{H}^{\text{in}} \big)
\;=\; \mathbf{K}(\theta,\varphi,t)r^×(Hout−Hin)=K(θ,φ,t)
where the surface current K\mathbf{K}K (tangential) is given by the spatial components of JshellμJ^\mu_{\rm shell}Jshellμ restricted to the sphere.

So localized shell fields produce discontinuities in the radial/tangential components of E and H consistent with the surface charge and current computed from ϕ\phiϕ.

===== Expand the surface charge density in spherical harmonics: =====

σ(θ,φ,t)  =  ∑ℓ,mσℓm(t) Yℓm(θ,φ).\sigma(\theta,\varphi,t) \;=\; \sum_{\ell,m} \sigma_{\ell m}(t)\, Y_{\ell m}(\theta,\varphi).σ(θ,φ,t)=ℓ,m∑σℓm(t)Yℓm(θ,φ).
Each σℓm\sigma_{\ell m}σℓm produces outside electromagnetic potentials that fall like r−(ℓ+1)r^{-(\ell+1)}r−(ℓ+1) (multipole fields). Thus a given shell mode (ℓ,m)(\ell,m)(ℓ,m) will preferentially excite external EM multipoles of order ℓ\ellℓ. This is the mechanism that couples shell oscillation modes to radiation and external fields.