Lagrangian for the Electrodynamic Field

We now have to construct the interaction term between the dynamical variables, i.e., the gauge field and the sources of the electrodynamical field, i.e., charged particles giving rise to a charge-current density cf. Eq. (3.161). Again, the contribution of this interaction term to the action S has to be Lorentz and gauge invariant, and the simplest choice is therefore given by 

This interaction Lagrangian density may depend explicitly on the space-time coordinates x and the 4-velocity u via the charge-current density T. However, as far as only the equation of motion for the electrodynamic field is concerned they do not represent dynamical variables. Lorentz invariance of this interaction term is obvious, and gauge invariance of the corresponding action is a direct consequence of the continuity equation for the charge-current density f, cf. Eq. (3.162), 

The Lagrangian density em for the electromagnetic field is therefore given as the sum of the kinetic term 3 and the interaction term 

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