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.


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?

  • 1
    $\begingroup$ I'm not so sure about "surely there must be a relation". One is the Lagrangian of a point particle in a background electric field, the other the Lagrangian of the field itself. The dynamical variables are completely different. Without better justification, it is unclear to me why you would expect to be able to derive one from the other. $\endgroup$ – ACuriousMind Jul 29 '16 at 12:29
  • 1
    $\begingroup$ At the level of eqs. of motion, the question (v2) seems to essentially be asking if the Lorentz force can be derived from Maxwell's equations. $\endgroup$ – Qmechanic Jul 29 '16 at 15:00
  • $\begingroup$ @Qmechanic: Thank you for your comment and edits. I am truly interested in the Lagrangians themselves, not the equations of motion or the relationship between the equations of motion. I am also not interested in deriving the Lorentz force from Maxwell's equations. $\endgroup$ – R.C. Jul 29 '16 at 18:45
  • $\begingroup$ @ACuriousMind: Thanks for your interest. As I point out in line 4 of my question statement, I know that the Lagrangians describe the dynamics of different objects. I am interested in how the two Lagrangians are related to one another, even in a schematic way. For lack a of a better term, I should be able to at least put them into context with one another and find a point of connection between the two since (1) describes a charge distribution interacting with background fields and (2) describes fields interacting with a charge distribution. $\endgroup$ – R.C. Jul 29 '16 at 18:45
  • $\begingroup$ At the level of actions, for the opposite question (v2), see also physics.stackexchange.com/q/55291/2451 $\endgroup$ – Qmechanic Jul 29 '16 at 18:56

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