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$$S[x(\tau),A(x)]=\int - m\ ds - \int d^4x\ A_{\mu}(x)j^{\mu}(x) -\frac{1}{4}\int d^4x F^{\mu\nu}(x)F_{\mu\nu}(x) $$ with $$ds= \sqrt{\eta_{\mu\nu}\dot{x}^{\mu}(\tau) \ \dot{x}^{\nu}(\tau) }d\tau$$ and $$j^{\mu}(x)=e \int d\tau \ \dot{x}^{\mu}(\tau)\delta^{(4)}(x-x(\tau))$$ ,where $$x(\tau)$$ is the parametrization of the world line, $$\eta_{\mu\nu}$$ the lorentz metric and $$F_{\mu\nu} = \partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$$ the field strength tensor. Now by variation of this, we get the usual equations of motion $$\partial_{\mu}F^{\mu\nu}=j^{\mu}$$ and $$-m\frac{d}{d\tau}\frac{\dot{x}_{\mu}}{\sqrt{\dot{x}^2}}=eF_{\mu\nu}\dot{x}^{\nu}$$. My confusion now comes from the appearence of the same Field strength tensor $$F_{\mu\nu}$$ in the equations of motion for the electron as well as being the object that is determined only by the electrons world line. Therefore I am asking, if in principle the electrons field is also having an influence on itself in classical electrodynamics, since only according to the given action it should. As given by Feynman (http://www.feynmanlectures.caltech.edu/II_26.html) the field of an uniformly moving electron diverges at its position, if we consider it to be a point charge. I would like to know if there is some kind of classical treatment for this issue and maybe someone can lift my confusion. If there is some explanation from the modern point of view I would also be glad to hear it.

Thank you

Lev