The (classical, relativistic) Lagrangian for electrodynamics contains the field energy density -F(adsbygoogle = window.adsbygoogle || []).push({}); _{μν}F^{μν}/4 and the interaction term -A_{μ}j^{μ}. I understand the maths of that - for one thing, the equations of motion turn out right if you plug this into the Euler Lagrange equantion.

Now I recall having learned that you can explain the forces between charged particles solely with the field energy: pushing 2 electrons together increases field energy because it goes with the square of the field strength, and pushing an electron and a positron together decreases field energy. If this is true, why do we need the interaction term at all? What am I missing?

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# I Interaction term in EM Lagrangian

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