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

=== The boundary Lagrangian Lbdy(ψ,Aa) \mathcal{L}_{\rm bdy}(\psi, A_a)Lbdy(ψ,Aa) is invariant under the local U(1) gauge transformation ===

ψ(y)↦eiα(y)ψ(y),Aa↦Aa−∂aα.\psi(y)\mapsto e^{i\alpha(y)}\psi(y),\qquad A_a\mapsto A_a - \partial_a\alpha.ψ(y)↦eiα(y)ψ(y),Aa↦Aa−∂aα.
From Noether’s theorem (or direct variation with respect to α\alphaα), one obtains the boundary current conservation equation (including exchange with the bulk):

  ∇aja  =  −nμJbulkμ∣Σ    \boxed{\; \nabla_a j^a \;=\; - n_\mu J^\mu_{\rm bulk}\big|_\Sigma \; \;}∇aja=−nμJbulkμΣ
which is the same continuity relation written more covariantly: the covariant divergence of the surface current equals the inward flux from the bulk.

So the Noether current ja=e ψˉγaψj^a=e\,\bar\psi\gamma^a\psija=eψˉγaψ is the “bookkeeper” for charge on the electron sphere: gauge symmetry → conserved current, and the boundary equation shows explicitly how emission/absorption from the bulk makes the boundary occupancy (and thus the surface charge/current) change.

This is the precise, local statement of “photon ΔE=hf\Delta E = h fΔE=hf depositing energy/charge on the electron surface and thereby incrementing the future moment-by-moment” in QAT language.