Can I derive the point charge Lagrangian $$ L = -\frac{mc^2}{\gamma} - \frac{1}{c} J_{\mu} A'^{\mu}\tag{1} $$ from the Abelian Yang-Mills Lagrangian $$L = \int d^3x [ - \frac{1}{16 \pi} F_{\mu \nu} F^{\mu \nu} - \frac{1}{c} J'_{\mu} A^{\mu}] ~~? \tag{2} $$
I recognized that one is the Lagrangian for a point charge and the other is the Lagrangian for the EM fields, and I know that each is derived independently using Lorentz invariance, unit analysis and static limits (see Jackson sections 12.1A-B, 12.7) but surely there must be a relationship between the two?
The Lagrangian for a point electric charge located at $\vec{r}$ is, according to Jackson's "Classical Electrodynamics" equation 12.12, $L = -m c^2 \sqrt{1-\frac{v^2}{c^2}} + \frac{e}{c} \vec{v} \cdot \vec{A}' - e A'_0 = -\frac{mc^2}{\gamma} - \frac{1}{c} J_{\mu} A'^{\mu}$, where $J_{\mu} = e \delta(\vec{r}) \cdot u_{\mu}$ and $u_{\mu} = \gamma(c,\vec{v})$. In other words, this is the Lagrangian for an electron of mass $m$ and four-current $J_{\mu}$ interacting with external electromagnetic fields $A'_{\mu}$. The associated action is $S = \int L dt = \int \gamma L d \tau$.
The Lagrangian density for the electromagnetic field $A_{\mu}$, according to Jackson equation 12.85 is $\mathcal{L} = -\frac{1}{16 \pi} F_{\mu \nu} F^{\mu \nu} - \frac{1}{c} J'_{\mu} A_{\nu}$, where $J'_{\mu}$ is the current density associated with external field sources/charges. In other words, this is the Lagrangian describing the dynamics of the electromagnetic field $A_{\mu}$ interacting with external field sources described by $J'_{\mu}$. The associated action is $S = \int \mathcal{L} d^4 x = \int L dt = \int \gamma L d\tau$.
My attempt:
I need to convert Lagrangian (2) from describing the dynamical fields $A_{\mu}$ interacting with external charge distributions $J'_{\nu}$ to something which describes the dynamics of a charge distribution $J_{\mu}$ (which generates field $A_{\mu}$) interacting with external field $A'_{\nu}$ (which is generated by $J'_{\nu}$).
I integrate by parts the $-F_{\mu \nu}^2$ term in equation (2) to obtain $A_{\mu} (\partial_{\nu} F^{\nu \mu}) + \partial_{\mu}(A_{\nu} F^{\nu \mu})$. According to the Euler-Lagrange equation $\frac{c}{4\pi} \partial_{\nu} F^{\nu \mu}=J'^{\mu}$, i.e. the change in the curvature $\partial_{\nu} F^{\nu \mu}$ results from the presence of the external current-density $J'^{\mu}$, analogous to how an acceleration results from an external force.
I believe the $\partial_{\mu}(A_{\nu} F^{\nu \mu})$ term should yield something like the electric flux through the boundary of the space for field $A_{\mu}$, which is just the charge, allowing me to derive a $J_{\mu}$ term (not a $J'_{\nu}$ term). But expanding this term gives $\partial_{\mu}(A_{\nu} F^{\nu \mu}) = \partial_0(-A_i E^i) - \partial_j(-A_0 E^j) + \varepsilon^{kij} \partial_j (A_i B_k)$.
But I get stuck here, and cannot seem to derive either a $J_{\mu}$ term or a mass term like $-m c^2 / \gamma$. Does anyone have any pointers for me? I'd be much obliged.
Note:
I have read the following and my question is not a duplicate of either, as I know the two Lagrangians above are derived independently using Lorentz invariance, unit analysis, and known static limits: Deriving Lagrangian density for electromagnetic field, How is the electromagnetic Lagrangian derived?
