I am trying to calculate the Lagrangian of a charged particle in background gauge field evaluaed on-shell.

Let $A^{\mu}(x)$ be a gauge field. The action of a charged particle in this background gauge field is


The equation of motion is


Now I want to plug the solution to the above equation of motion back into the action.

Is there any way to evaluate the on-shell action as a functional of $A^{\mu}$ without solving the equation?

  • $\begingroup$ Have you tried integrating by parts and solving the equation of motion for $\frac{d^2x^\beta}{d\tau^2}$ and then inserting that in the action that you just integrated by parts. This should at the very least simplify the action a lot (0?). $\endgroup$ – Michael Aug 12 '18 at 1:58
  • $\begingroup$ @Michael Could you be specific? Which term do you integrate by part? $\endgroup$ – Libertarian Monarchist Bot Aug 12 '18 at 9:51

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.