Editing Dirac equation (section)

=== Gauging the symmetry ===
{{See also|Quantum electrodynamics}}
If we 'promote' the global symmetry, parametrised by the constant <math>\alpha</math>, to a local symmetry, parametrised by a function <math>\alpha:\mathbb{R}^{1,3} \to \mathbb{R}</math>, or equivalently <math>e^{i\alpha}: \mathbb{R}^{1,3} \to \text{U}(1),</math> the Dirac equation is no longer invariant: there is a residual derivative of <math>\alpha(x)</math>.

The fix proceeds as in [[scalar electrodynamics]]: the partial derivative is promoted to a covariant derivative <math>D_\mu</math>
<math display="block">D_\mu \psi = \partial_\mu \psi + i e A_\mu\psi,</math>
<math display="block">D_\mu \bar\psi = \partial_\mu \bar\psi - i e A_\mu\bar\psi.</math>
The covariant derivative depends on the field being acted on. The newly introduced <math>A_\mu</math> is the 4-vector potential from electrodynamics, but also can be viewed as a <math>\text{U}(1)</math> [[gauge field]] (which, mathematically, is defined as a <math>\text{U}(1)</math> [[Principal connection|connection]]).

The transformation law under gauge transformations for <math>A_\mu</math> is then the usual
<math display="block">A_\mu(x) \mapsto A_\mu(x) + \frac{1}{e}\partial_\mu\alpha(x)</math>
but can also be derived by asking that covariant derivatives transform under a gauge transformation as
<math display="block">D_\mu\psi(x) \mapsto e^{i\alpha(x)}D_\mu\psi(x),</math>
<math display="block">D_\mu\bar\psi(x) \mapsto e^{-i\alpha(x)}D_\mu\bar\psi(x).</math>
We then obtain a gauge-invariant Dirac action by promoting the partial derivative to a covariant one:
<math display="block">S = \int d^4x\,\bar\psi\,(iD\!\!\!\!\big / - m)\,\psi = \int d^4x\,\bar\psi\,(i\gamma^\mu D_\mu - m)\,\psi.</math>
The final step needed to write down a gauge-invariant Lagrangian is to add a Maxwell Lagrangian term,
<math display="block">S_{\text{Maxwell}} = \int d^4x\,\left[-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}\right].</math>
Putting these together gives
{{Equation box 1
|title='''QED Action'''
|indent=:
|equation = <math>S_{\text{QED}} = \int d^4x\,\left[-\frac{1}{4}F^{\mu\nu}F_{\mu\nu} + \bar\psi\,(iD\!\!\!\!\big / - m)\,\psi\right]</math>
|border
|border colour =#50C878
|background colour = #ECFCF4
}}
Expanding out the covariant derivative allows the action to be written in a second useful form:
<math display="block">S_{\text{QED}} = \int d^4x\,\left[-\frac{1}{4}F^{\mu\nu}F_{\mu\nu} + \bar\psi\,(i\partial\!\!\!\big / - m)\,\psi - eJ^\mu A_\mu\right]</math>