The Dirac Equation ("free Dirac") is a relativistic Equation of Motion (EoM) for a free ($V=0$) Spin $1/2$ particle (like an electron).

The free Dirac equation is invariant under global phase transformations (which can be easily seen).

The free Dirac equation is not invariant under local phase transformations. In order to make it invariant people change the normal partial derivative into a covariant derivative which introduces a four-vector-potential into the equation. Why is this a legitimate step? The introduction of a potential does not apply for a free particle so the situation this adapted equation describes is totally different from the original (free) Dirac equation.

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    $\begingroup$ It is a legitimate step because the resulting equation nicely agrees with experiments. This is the only reason. $\endgroup$ – AccidentalFourierTransform May 21 '16 at 14:36
  • $\begingroup$ Why should the equation for the electromagnetically interacting Dirac field not be different from the free Dirac equation? $\endgroup$ – Robin Ekman May 21 '16 at 15:16
  • $\begingroup$ @Robin Since we wanted local phase transformation invariance for free Dirac I don't understand how changing the physical context (by introducing an additional potential) solves this in a reasonable way. $\endgroup$ – Thomas Elliot May 21 '16 at 16:23
  • $\begingroup$ @AccidentalFourierTransform So it is a matter of opinion/frame of reference if there is a potential or not? $\endgroup$ – Thomas Elliot May 21 '16 at 16:24
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    $\begingroup$ @ThomasElliot There is a wonderful chapter explaining clearly all about local phase invariance in general, and for the Dirac eq. in particular, here: staff.science.uu.nl/~wit00103/ftip/Ch11.pdf $\endgroup$ – udrv May 21 '16 at 23:26

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