Classical electrodynamics as a gauge theory

Analysis of the classical Dirac theory shows a similar inconsistency under local phase transformations, such that tjr(x) - eie/ X) li(x), corresponding to the local infinitesimal transformation, for x 0, 

For the Dirac field in an externally determined Maxwell field, the Lagrangian density including a renormalized mass term is 

The field interaction is expressed in terms of the covariant derivative 

This is gauge-covariant only if the gauge transformation of A/( and the phase transformation of ifr are combined, such that 

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