I'm taking at look at QED foundations, and started thinking about how it relates to Dirac's Equation.
Dirac spinors are invariant under a global phase transformation $\psi(x)\mapsto e^{i\alpha}\psi(x)$. That's simple the conservation of the fermion's charge. If we want to transition to an interacting theory we must then impose local gauge symmetry: \begin{equation}\psi(x)\mapsto e^{i\alpha(x)}\psi(x).\end{equation}
The derivative $\partial_\mu\psi$ from Dirac's equation obviously does not transform covariantly. We could then define a gauge field $A_\mu(x)$ s.t the derivative: \begin{equation}\nabla_\mu=\partial_\mu+ieA_\mu(x)\end{equation}
Is covariant. That is, $A_\mu$ transforms like: \begin{equation}A_\mu(x)\mapsto A_\mu(x)-\frac{1}{e}\partial_\mu\alpha(x).\end{equation}
That's simply the gauge transformation for electromagnetism, and $A_\mu$ is the 4-potential. So we kinda of naturally arrive at QED from $U(1)$ local symmetry of the Dirac Field.
Thing is, I haven't seem this approach anywhere, and, given its simplicity, I assume it's either blantly wrong or basically a useless line of reasoning.
Any help will be appreciated.
