Editing Dirac equation (section)

=== Lagrangian formulation ===
Both the Dirac equation and the Adjoint Dirac equation can be obtained from (varying) the action with a specific Lagrangian density that is given by:
<math display="block">\mathcal{L} = i\hbar c\overline{\psi}\gamma^{\mu}\partial_{\mu}\psi - mc^2\overline{\psi}\psi</math>

If one varies this with respect to <math>\psi</math> one gets the adjoint Dirac equation. Meanwhile, if one varies this with respect to <math>\bar\psi</math> one gets the Dirac equation.

In natural units and with the slash notation, the action is then
{{Equation box 1
|title='''Dirac Action'''
|indent=:
|equation = <math>S = \int d^4x\,\bar\psi\,(i\partial\!\!\!\big / - m)\,\psi</math>
|border
|border colour =#50C878
|background colour = #ECFCF4
}}

For this action, the conserved current <math>J^\mu</math> above arises as the conserved current corresponding to the global <math>\text{U}(1)</math> symmetry through [[Noether's theorem]] for field theory. Gauging this field theory by changing the symmetry to a local, spacetime point dependent one gives gauge symmetry (really, gauge redundancy). The resultant theory is [[quantum electrodynamics]] or QED. See below for a more detailed discussion.