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

==== Vary the total action with respect to AμA_\muAμ. The Euler–Lagrange equation for AμA_\muAμ gives: ====

∂ν(∂L∂(∂νAμ))−∂L∂Aμ=0.\partial_\nu\left(\frac{\partial \mathcal{L}}{\partial(\partial_\nu A_\mu)}\right) - \frac{\partial \mathcal{L}}{\partial A_\mu} = 0.∂ν(∂(∂νAμ)∂L)−∂Aμ∂L=0.
Compute terms:
* From LEM:  ∂LEM∂(∂νAμ)=−1μ0Fνμ\mathcal{L}_{\text{EM}}: \; \dfrac{\partial \mathcal{L}_{\text{EM}}}{\partial(\partial_\nu A_\mu)} = -\dfrac{1}{\mu_0}F^{\nu\mu}LEM:∂(∂νAμ)∂LEM=−μ01Fνμ. So ∂ν(−1μ0Fνμ)\partial_\nu(-\frac{1}{\mu_0}F^{\nu\mu})∂ν(−μ01Fνμ) contributes −1μ0∂νFνμ-\frac{1}{\mu_0}\partial_\nu F^{\nu\mu}−μ01∂νFνμ.
'' From Lϕ\mathcal{L}_{\phi}Lϕ (through minimal coupling): ∂Lϕ∂Aμ=ℏ2 iq[ϕ∗(Dμϕ)−ϕ(Dμϕ)∗]\dfrac{\partial \mathcal{L}_{\phi}}{\partial A_\mu} = \hbar^{2}\, i q \left[ \phi^''(\mathcal{D}^\mu \phi) - \phi(\mathcal{D}^\mu \phi)^* \right]∂Aμ∂Lϕ=ℏ2iq[ϕ∗(Dμϕ)−ϕ(Dμϕ)∗]. This is exactly the Noether current computed earlier (up to the ℏ2\hbar^{2}ℏ2 factor).

Putting these together:

−1μ0 ∂νFνμ−ℏ2iq[ϕ∗(Dμϕ)−ϕ(Dμϕ)∗]=0.-\frac{1}{\mu_0}\,\partial_\nu F^{\nu\mu} - \hbar^{2} i q \left[ \phi^''(\mathcal{D}^\mu \phi) - \phi(\mathcal{D}^\mu \phi)^'' \right] = 0.−μ01∂νFνμ−ℏ2iq[ϕ∗(Dμϕ)−ϕ(Dμϕ)∗]=0.
Rearrange and define the QAT current JμJ^\muJμ:

  ∂νFνμ  =  μ0 Jμ  withJμ≡μ0−1 ℏ2iq[ϕ∗(Dμϕ)−ϕ(Dμϕ)∗].\boxed{\;
\partial_\nu F^{\nu\mu} \;=\; \mu_0\, J^\mu
\;}
\qquad\text{with}\qquad
J^\mu \equiv \mu_0^{-1}\,\hbar^{2} i q \left[ \phi^''(\mathcal{D}^\mu \phi) - \phi(\mathcal{D}^\mu \phi)^'' \right].∂νFνμ=μ0JμwithJμ≡μ0−1ℏ2iq[ϕ∗(Dμϕ)−ϕ(Dμϕ)∗].
You can absorb constants or choose units (SI vs. natural) as needed; the important point is the structural form: Maxwell’s equations with source JμJ^\muJμ where JμJ^\muJμ is the Noether current of the QAT scalar field.