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

==== We have now a standard field-theory coupling, but you asked how your geometry shows up: ====
# Explicit ℏ\hbarℏ (i.e., h/2πh/2\pih/2π) — it appears in the field normalization and therefore in the expression for JμJ^\muJμ. Keeping ℏ2\hbar^2ℏ2 visible makes the unit of action part of the current strength (QAT emphasis).
# Spherical surface 4πr24\pi r^24πr2 — this appears when (and only when) you go from the local field density Jμ(x)J^\mu(x)Jμ(x) to global, surface-integrated quantities (Gauss’s law style): - The charge enclosed by a spherical surface SSS of radius rrr is Qenc(t)=∫SE⋅dA  ε0=∫Vρ dV,Q_{\text{enc}}(t) = \int_{S} \mathbf{E}\cdot d\mathbf{A} \; \varepsilon_0 = \int_{V} \rho\,dV,Qenc(t)=∫SE⋅dAε0=∫VρdV, where dA=r2sin⁡θ dθ dϕ r^d\mathbf{A}=r^2\sin\theta\,d\theta\,d\phi\,\hat{r}dA=r2sinθdθdϕr^ and the surface area factor 4πr24\pi r^24πr2 appears explicitly. - In QAT terms: if the electron’s probability surface is the natural place where ϕ\phiϕ interacts, then the effective surface charge density and the observed macroscopic field will naturally scale with 4πr24\pi r^24πr2. Converting the density J0(x)J^0(x)J0(x) to a surface-quantity picks up the geometric 4πr24\pi r^24πr2.
# How to encode spherical-surface dynamics directly into the Lagrangian — earlier we used an action weighted by 4πr24\pi r^24πr2 when deriving the radial wave equation. If you prefer the electromagnetic coupling to be surface-localized, you can write an action term concentrated on the spherical shell (a delta function or boundary term at radius r0r_0r0), e.g.:

Sshell=ℏ∫dt∫S(r0)dA  Lshell(ϕ,Aμ),S_{\text{shell}} = \hbar \int dt \int_{S(r_0)} dA \; \mathcal{L}_{\text{shell}}(\phi, A_\mu),Sshell=ℏ∫dt∫S(r0)dALshell(ϕ,Aμ),
which would yield currents strongly localized on that surface. That is a deliberate modeling choice (useful to make the sphere central), but it is nonstandard vs. the usual volumetric field theory. Both are mathematically manageable; choose depending on which physical picture you want as fundamental.