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

==== Start with the complex scalar field ϕ\phiϕ minimally coupled to the electromagnetic 4-potential AμA_\muAμ. We keep the Planck constant explicit as h/2πh/2\pih/2π. ====

Write ℏ≡h2π\hbar \equiv \dfrac{h}{2\pi}ℏ≡2πh when convenient for compactness, but I’ll show the h/2πh/2\pih/2π form in key places.

Lagrangian density (flat space, metric signature +,−,−,−+,-,-,-+,−,−,−):

L=Lϕ+LEM+Lint,\mathcal{L} = \mathcal{L}_{\phi} + \mathcal{L}_{\text{EM}} + \mathcal{L}_{\text{int}},L=Lϕ+LEM+Lint,
with

Lϕ  =  ℏ2 (Dμϕ)∗ (Dμϕ)−V(ϕ),Dμ≡∂μ+iqAμ,\mathcal{L}_{\phi} \;=\; \hbar^{2}\, (\mathcal{D}^\mu \phi)^{*}\,(\mathcal{D}_\mu \phi) - V(\phi),
\qquad
\mathcal{D}_\mu \equiv \partial_\mu + i q A_\mu,Lϕ=ℏ2(Dμϕ)∗(Dμϕ)−V(ϕ),Dμ≡∂μ+iqAμ,
LEM  =  −14μ0FμνFμν,Fμν≡∂μAν−∂νAμ,\mathcal{L}_{\text{EM}} \;=\; -\frac{1}{4\mu_0} F^{\mu\nu}F_{\mu\nu},
\qquad
F_{\mu\nu}\equiv \partial_\mu A_\nu-\partial_\nu A_\mu,LEM=−4μ01FμνFμν,Fμν≡∂μAν−∂νAμ,
Lint  is already contained in minimal coupling via Dμ.\mathcal{L}_{\text{int}} \;\text{is already contained in minimal coupling via }\mathcal{D}_\mu.Lintis already contained in minimal coupling via Dμ.
Notes:
* qqq is the charge coupling (electron charge magnitude).
* ℏ=h2π\hbar=\dfrac{h}{2\pi}ℏ=2πh appears explicitly multiplying the kinetic term so the quantum of action is visible in the field normalization (QAT preference).
* V(ϕ)V(\phi)V(ϕ) can encode symmetry-breaking dynamics (we keep it general for now).