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

=== Assistant: Wonderful — let’s do the classic pillbox calculation step-by-step, but with the QAT story in the foreground so the phys… ===
Wonderful — let’s do the classic pillbox calculation step-by-step, but with the QAT story in the foreground so the physics stays intuitive. I’ll keep the math direct and the geometry explicit (use h/2πh/2\pih/2π where relevant). At the end I’ll summarize how this ties to Faraday and the Third Law in QAT language.

==== • 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.

==== Take a tiny cylindrical pillbox that straddles the spherical shell at a chosen point: ====
* Cylinder axis aligned with the local normal n^\hat nn^ (radial direction).
* Top face just outside the shell (out''), bottom face just inside (in'').
* Let face area be AAA and radial thickness h→0h\to0h→0. The curved side area is proportional to hhh and will vanish as h→0h\to0h→0.

We integrate Maxwell’s equations over this pillbox and then take the limit h→0h\to0h→0.

==== Integrate ∇⋅E=ρ/ε0\nabla\cdot\mathbf{E}=\rho/\varepsilon_0∇⋅E=ρ/ε0 over the pillbox volume VVV: ====

∫V(∇⋅E) dV=1ε0∫Vρ dV.\int_V (\nabla\cdot\mathbf{E})\,dV = \frac{1}{\varepsilon_0}\int_V \rho\,dV.∫V(∇⋅E)dV=ε01∫VρdV.
Left side → flux through faces (curved side vanishes as h→0h\to0h→0):

Eout ⁣⋅n^ A−Ein ⁣⋅n^ A=1ε0∫Vρ dV.\mathbf{E}_{\rm out}\!\cdot \hat n\,A - \mathbf{E}_{\rm in}\!\cdot \hat n\,A = \frac{1}{\varepsilon_0}\int_V \rho\,dV.Eout⋅n^A−Ein⋅n^A=ε01∫VρdV.
The volume integral of ρ\rhoρ picks up the delta layer:

∫Vρ dV=∫V[ρbulk+δ(r−r0)σ(Ω,t)]dV⟶σ(Ω,t) A(assuming no bulk charge inside pillbox).\int_V \rho\,dV = \int_V \big[\rho_{\rm bulk} + \delta(r-r_0)\sigma(\Omega,t)\big] dV
\longrightarrow \sigma(\Omega,t)\,A \quad\text{(assuming no bulk charge inside pillbox).}∫VρdV=∫V[ρbulk+δ(r−r0)σ(Ω,t)]dV⟶σ(Ω,t)A(assuming no bulk charge inside pillbox).
Therefore, dividing by AAA:

  n^ ⁣⋅(Eout−Ein)  =  σ(θ,φ,t)ε0  (Gauss jump)\boxed{\;\hat n\!\cdot(\mathbf{E}_{\rm out}-\mathbf{E}_{\rm in}) \;=\; \frac{\sigma(\theta,\varphi,t)}{\varepsilon_0}\; } \tag{Gauss jump}n^⋅(Eout−Ein)=ε0σ(θ,φ,t)(Gauss jump)
This is the exact relation: change in normal EEE equals surface charge density (ledger entry: surface charge).

QAT interpretation: σ\sigmaσ is the surface-charge density produced by absorbed photon energy changing the boundary occupation. It is precisely the quantity recorded in the surface ledger.

==== Integrate ∇×B=μ0J+μ0ε0∂tE\nabla\times\mathbf{B}=\mu_0\mathbf{J}+\mu_0\varepsilon_0\partial_t\mathbf{E}∇×B=μ0J+μ0ε0∂tE over the pillbox volume and apply Stokes’ theorem to the tiny loop that runs around the pillbox edge. More simply, take the tangential components and integrate across the small height: ====

In the limit h→0h\to0h→0 the dominant contribution is from the tangential integral across the top and bottom faces. One obtains

n^×(Bout−Bin)=μ0K∥(θ,φ,t)+μ0ε0n^×(Eout−Ein)⋅(order h).\hat n\times(\mathbf{B}_{\rm out}-\mathbf{B}_{\rm in}) = \mu_0 \mathbf{K}_\parallel(\theta,\varphi,t) + \mu_0\varepsilon_0 \hat n\times \big(\mathbf{E}_{\rm out}-\mathbf{E}_{\rm in}\big) \cdot(\text{order }h).n^×(Bout−Bin)=μ0K∥(θ,φ,t)+μ0ε0n^×(Eout−Ein)⋅(order h).
The displacement term is order hhh and drops out as h→0h\to0h→0. Hence,

  n^×(Bout−Bin)  =  μ0 j∥(θ,φ,t)  (Ampeˋre jump)\boxed{\;\hat n\times(\mathbf{B}_{\rm out}-\mathbf{B}_{\rm in}) \;=\; \mu_0\,\mathbf{j}_\parallel(\theta,\varphi,t)\; } \tag{Ampère jump}n^×(Bout−Bin)=μ0j∥(θ,φ,t)(Ampeˋre jump)
where j∥\mathbf{j}_\parallelj∥ is the tangential surface current (the spatial part of jsurfμj^\mu_{\rm surf}jsurfμ).

QAT interpretation: j∥\mathbf{j}_\parallelj∥ is produced by charge motion along the spherical boundary after absorption (decoherence and redistribution); it is the agent that radiates (emission) and rearranges local field lines.

==== The 4D conservation law ∂μJμ=0\partial_\mu J^\mu=0∂μJμ=0 with Jμ=Jbulkμ+δ(r−r0)jsurfμJ^\mu = J^\mu_{\rm bulk} + \delta(r-r_0) j^\mu_{\rm surf}Jμ=Jbulkμ+δ(r−r0)jsurfμ implies, after integrating across the thin radial interval, the surface continuity equation: ====

  ∂tσ(θ,φ,t)+∇∥ ⁣⋅j∥(θ,φ,t)  =  − n^ ⁣⋅Jbulk∣r=r0  (Surface continuity)\boxed{\; \partial_t \sigma(\theta,\varphi,t) + \nabla_{\parallel}\!\cdot \mathbf{j}_\parallel(\theta,\varphi,t) \;=\; -\,\hat n\!\cdot \mathbf{J}_{\rm bulk}\big|_{r=r_0}\; } \tag{Surface continuity}∂tσ(θ,φ,t)+∇∥⋅j∥(θ,φ,t)=−n^⋅Jbulkr=r0(Surface continuity)
Right-hand side is the normal bulk current flux into the shell (positive for absorption). This is the accounting equation: change of surface charge = net tangential outflow + normal inflow.

QAT read: an arriving photon contributes a normal flux term n^⋅Jbulk\hat n\cdot\mathbf{J}_{\rm bulk}n^⋅Jbulk which increases σ\sigmaσ (absorption); later, surface currents j∥\mathbf{j}_\parallelj∥ can redistribute or radiate energy away (emission).

==== Start from Poynting theorem in differential form: ====

∂tuem+∇⋅S=−E⋅Jbulk.\partial_t u_{\rm em} + \nabla\cdot\mathbf{S} = -\mathbf{E}\cdot\mathbf{J}_{\rm bulk}.∂tuem+∇⋅S=−E⋅Jbulk.
Integrate over the pillbox and take h→0h\to0h→0. The integral of ∇⋅S\nabla\cdot\mathbf{S}∇⋅S becomes the net flux through the top and bottom faces:

n^⋅Sout−n^⋅Sin=−∫VE⋅J dV−∫V∂tuem dV.\hat n\cdot\mathbf{S}_{\rm out} - \hat n\cdot\mathbf{S}_{\rm in} = -\int_V \mathbf{E}\cdot\mathbf{J}\,dV - \int_V \partial_t u_{\rm em}\,dV.n^⋅Sout−n^⋅Sin=−∫VE⋅JdV−∫V∂tuemdV.
In the thin-shell limit, define usurfu_{\rm surf}usurf as the surface energy density stored in the shell degrees of freedom. Then the balance reduces to the surface energy equation (we wrote before):

  ∂tusurf=−n^⋅S∣r=r0−E∥⋅j∥+Qint  (Surface energy)\boxed{\;\partial_t u_{\rm surf} = -\hat n\cdot\mathbf{S}\big|_{r=r_0} - \mathbf{E}_\parallel\cdot\mathbf{j}_\parallel + \mathcal{Q}_{\rm int}\;} \tag{Surface energy}∂tusurf=−n^⋅Sr=r0−E∥⋅j∥+Qint(Surface energy)
This is the ledger: Poynting flux into the shell plus work by tangential fields changes the shell’s internal energy (absorption); emission is the reverse.

==== Imagine an incoming photon wavefront strikes the shell locally: ====
# Poynting flux n^⋅S>0\hat n\cdot\mathbf{S}>0n^⋅S>0 into the pillbox increases usurfu_{\rm surf}usurf.
# The continuity equation records a positive normal flux −n^⋅Jbulk∣r0-\hat n\cdot\mathbf{J}_{\rm bulk}\big|_{r_0}−n^⋅Jbulkr0 which increases σ\sigmaσ.
# The Gauss jump (3) updates the normal EEE accordingly; the Ampère jump (4) relates any induced surface current to a change in tangential BBB.
# Later, the shell may re-emit: surface currents arrange so that outgoing Poynting flux is negative (energy leaves), and σ\sigmaσ falls consistent with the continuity equation. Every step follows the pillbox equations — Maxwell is never altered, only the localized source terms (the ledger entries) change.

==== • Faraday: flux changes (local ∂tB\partial_t\mathbf{B}∂tB produced by surface current changes) induce tangential electric fields around loops on the sphere (Faraday’s law). The pillbox tells you exactly how a tiny patch’s surface current and charge change the field-lines passing through that patch — the precise, quantitative realization of Faraday’s lines of force. ====
• Third Law (QAT read): if microscopic transitions freeze (no normal flux, n^⋅Jbulk→0\hat n\cdot\mathbf{J}_{\rm bulk}\to0n^⋅Jbulk→0, and j∥→0\mathbf{j}_\parallel\to0j∥→0), then ∂tσ=0\partial_t\sigma=0∂tσ=0 and ∂tusurf=0\partial_t u_{\rm surf}=0∂tusurf=0. The pillbox equations then reduce to static jump conditions and no Poynting exchange — the dynamical ledger stops being written. That’s the “locking” picture.

==== Pillbox jump/ledger equations (exact): ====

  n^ ⁣⋅(Eout−Ein)=σε0  ,n^×(Bout−Bin)=μ0j∥  \boxed{\;\hat n\!\cdot(\mathbf{E}_{\rm out}-\mathbf{E}_{\rm in})=\frac{\sigma}{\varepsilon_0}\;,\qquad
\hat n\times(\mathbf{B}_{\rm out}-\mathbf{B}_{\rm in})=\mu_0\mathbf{j}_\parallel\;}n^⋅(Eout−Ein)=ε0σ,n^×(Bout−Bin)=μ0j∥
  ∂tσ+∇∥ ⁣⋅j∥=− n^ ⁣⋅Jbulk∣r=r0  \boxed{\;\partial_t\sigma + \nabla_{\parallel}\!\cdot\mathbf{j}_\parallel = -\,\hat n\!\cdot\mathbf{J}_{\rm bulk}\big|_{r=r_0}\;}∂tσ+∇∥⋅j∥=−n^⋅Jbulkr=r0
  ∂tusurf=−n^ ⁣⋅S∣r=r0−E∥⋅j∥+Qint  \boxed{\;\partial_t u_{\rm surf} = -\hat n\!\cdot\mathbf{S}\big|_{r=r_0} - \mathbf{E}_\parallel\cdot\mathbf{j}_\parallel + \mathcal{Q}_{\rm int}\;}∂tusurf=−n^⋅Sr=r0−E∥⋅j∥+Qint
Each equation follows from integrating Maxwell (and Poynting) over the little pillbox that straddles the shell; the delta-supported surface current jsurfμj^\mu_{\rm surf}jsurfμ comes from the JμAμJ_\mu A^\muJμAμ boundary coupling (Noether current of the boundary action). Maxwell’s differential equations keep their form — QAT’s content sits inside the surface source terms and the dynamical rules that update σ\sigmaσ, j∥\mathbf{j}_\parallelj∥, and usurfu_{\rm surf}usurf.

If you’d like next I can:
• write this derivation as a LaTeX snippet you can paste into your notes or video description, or
• make a simple labeled diagram (pillbox crossing a spherical shell with arrows showing n^⋅S\hat n\cdot\mathbf{S}n^⋅S, j∥\mathbf{j}_\parallelj∥, normal flux) suitable for a slide.

Which do you prefer?