Editing Quantum electrodynamics (section)

=== Equations of motion ===
Expanding the covariant derivative in the Lagrangian gives

:<math>\mathcal{L} = - \frac{1}{4}F_{\mu\nu}F^{\mu\nu} + i \bar\psi \gamma^\mu \partial_\mu \psi - e\bar{\psi}\gamma^\mu A_\mu \psi -m \bar{\psi} \psi </math>
:<math> = - \frac{1}{4}F_{\mu\nu}F^{\mu\nu} + i \bar\psi \gamma^\mu \partial_\mu \psi -m \bar{\psi} \psi - ej^\mu A_\mu .</math>
For simplicity, <math>B_\mu</math> has been set to zero, with no loss of generality. Alternatively, we can absorb <math>B_\mu</math> into a new gauge field <math>A'_\mu = A_\mu + B_\mu</math> and relabel the new field as <math>A_\mu.</math>

From this Lagrangian, the equations of motion for the <math>\psi</math> and <math>A_\mu</math> fields can be obtained.

==== Equation of motion for ψ ====
These arise most straightforwardly by considering the Euler-Lagrange equation for <math>\bar\psi</math>. Since the Lagrangian contains no <math>\partial_\mu\bar\psi</math> terms, we immediately get
:<math>\frac{\partial \mathcal{L}}{\partial(\partial_\mu \bar\psi)} = 0</math>
so the equation of motion can be written
<math>(i\gamma^\mu\partial_\mu-m)\psi = e\gamma^\mu A_\mu\psi.</math>

==== Equation of motion for A<sub>μ</sub> ====
* Using the Euler–Lagrange equation for the <math>A_\mu</math> field,
{{NumBlk2||<math display="block"> \partial_\nu \left( \frac{\partial \mathcal{L}}{\partial ( \partial_\nu A_\mu )} \right) - \frac{\partial \mathcal{L}}{\partial A_\mu} = 0,</math>|3}}

the derivatives this time are
<math display="block">\partial_\nu \left( \frac{\partial \mathcal{L}}{\partial ( \partial_\nu A_\mu )} \right) = \partial_\nu \left( \partial^\mu A^\nu - \partial^\nu A^\mu \right),</math>
<math display="block">\frac{\partial \mathcal{L}}{\partial A_\mu} = -e\bar{\psi} \gamma^\mu \psi.</math>

Substituting back into ({{EquationNote|3}}) leads to

:<math>\partial_\mu F^{\mu\nu} = e\bar\psi \gamma^\nu \psi</math>
which can be written in terms of the <math>\text{U}(1)</math> current <math>j^\mu</math> as
{{Equation box 1
|indent =:
|equation = <math>\partial_\mu F^{\mu \nu} = e j^\nu.</math>
|cellpadding
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

Now, if we impose the [[Lorenz gauge condition]]
<math display="block">\partial_\mu A^\mu = 0,</math>
the equations reduce to
<math display="block">\Box A^\mu = ej^\mu,</math>
which is a [[wave equation]] for the four-potential, the QED version of the classical [[Maxwell equations]] in the [[Lorenz gauge]]. (The square represents the [[wave operator]], <math>\Box = \partial_\mu \partial^\mu</math>.)