Editing Quantum electrodynamics (section)

=== QED action ===
Mathematically, QED is an [[abelian group|abelian]] [[gauge theory]] with the symmetry group [[U(1)]], defined on [[Minkowski space]] (flat spacetime). The [[gauge field]], which mediates the interaction between the charged [[Spin (physics)|spin-1/2]] [[field (physics)|field]]s, is the [[electromagnetic field]].
The QED [[Lagrangian (field theory)|Lagrangian]] for a spin-1/2 field interacting with the electromagnetic field in natural units gives rise to the action<ref name=Peskin>{{cite book | last1 =Peskin | first1 =Michael | last2 =Schroeder | first2 =Daniel | title =An introduction to quantum field theory | publisher =Westview Press | edition =Reprint | date =1995 | isbn =978-0201503975 | url-access =registration | url =https://archive.org/details/introductiontoqu0000pesk }}</ref>{{rp|78}}
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|title='''QED Action'''
|indent=:
|equation = <math>S_{\text{QED}} = \int d^4x\,\left[-\frac{1}{4}F^{\mu\nu}F_{\mu\nu} + \bar\psi\,(i\gamma^\mu D_\mu - m)\,\psi\right]</math>
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where
*<math> \gamma^\mu </math> are [[Dirac matrices]].
*<math>\psi</math> a [[bispinor]] [[field (physics)|field]] of [[spin-1/2]] particles (e.g. [[electron]]–[[positron]] field).
*<math>\bar\psi\equiv\psi^\dagger\gamma^0</math>, called "psi-bar", is sometimes referred to as the [[Dirac adjoint]].
*<math>D_\mu \equiv \partial_\mu+ieA_\mu+ieB_\mu </math> is the [[gauge covariant derivative]].
**''e'' is the [[Fine-structure constant|coupling constant]], equal to the [[electric charge]] of the bispinor field.
**<math>A_\mu</math> is the [[Lorentz covariance|covariant]] [[four-potential]] of the electromagnetic field generated by the electron itself. It is also known as a gauge field or a <math>\text{U}(1)</math> connection.
**<math>B_\mu</math> is the external field imposed by external source.
*''m'' is the mass of the electron or positron.
*<math>F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu </math> is the [[electromagnetic field tensor]]. This is also known as the curvature of the gauge field.

Expanding the covariant derivative reveals a second useful form of the Lagrangian (external field <math>B_\mu</math> set to zero for simplicity)
:<math>\mathcal{L} = - \frac{1}{4}F_{\mu\nu}F^{\mu\nu} + \bar\psi(i\gamma^\mu \partial_\mu - m)\psi - ej^\mu A_\mu</math>
where <math>j^\mu</math> is the conserved <math>\text{U}(1)</math> current arising from Noether's theorem. It is written
:<math>j^\mu = \bar\psi\gamma^\mu\psi.</math>