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Is there an electric current in Born-Infeld theory analogous to the Maxwell current that satisfies a continuity equation?

The treatment of currents and U(1) local gauge in Born-Infeld is identical to Maxwell theory, it is just that the "free" electromagnetic Lagrangian is different. I.e. the current is the Euler-Lagrange equation and it identically vanishes. (more e.g. here)

Electrodynamics of Maxwell-Lorentz is generally non-linear too, but the charge conservation holds anyway. Any theory (construction) without conservation of charge is in contradiction with experiment and cannot be considered seriously.

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