Editing Quantum electrodynamics (section)

===Feynman diagrams===
Despite the conceptual clarity of the Feynman approach to QED, almost no early textbooks follow him in their presentation. When performing calculations, it is much easier to work with the [[Fourier transform]]s of the [[propagator]]s. Experimental tests of quantum electrodynamics are typically scattering experiments. In scattering theory, particles' [[Momentum|momenta]] rather than their positions are considered, and it is convenient to think of particles as being created or annihilated when they interact. Feynman diagrams then ''look'' the same, but the lines have different interpretations. The electron line represents an electron with a given energy and momentum, with a similar interpretation of the photon line. A vertex diagram represents the annihilation of one electron and the creation of another together with the absorption or creation of a photon, each having specified energies and momenta.

Using [[Wick's theorem]] on the terms of the Dyson series, all the terms of the [[S-matrix]] for quantum electrodynamics can be computed through the technique of [[Feynman diagrams]]. In this case, rules for drawing are the following<ref name=Peskin/>{{rp|801–802}}

[[Image:qed rules.jpg|488px|center]]

[[Image:qed2e.jpg|488px|center]]

To these rules we must add a further one for closed loops that implies an integration on momenta <math display="inline">\int d^4p/(2\pi)^4</math>, since these internal ("virtual") particles are not constrained to any specific energy–momentum, even that usually required by special relativity (see [[Propagator#Propagators in Feynman diagrams|Propagator]] for details). The signature of the metric <math>\eta_{\mu \nu }</math> is <math>{\rm diag}(+---)</math>.

From them, computations of [[probability amplitude]]s are straightforwardly given. An example is [[Compton scattering]], with an [[electron]] and a [[photon]] undergoing [[elastic scattering]]. Feynman diagrams are in this case<ref name=Peskin/>{{rp|158–159}}

[[Image:compton qed.jpg|300px|center]]

and so we are able to get the corresponding amplitude at the first order of a [[Perturbation theory (quantum mechanics)|perturbation series]] for the [[S-matrix]]:

<math display="block">M_{fi} = (ie)^2 \overline{u}(\vec{p}', s')\epsilon\!\!\!/\,'(\vec{k}',\lambda')^* \frac{p\!\!\!/ + k\!\!\!/ + m_e}
{(p + k)^2 - m^2_e} \epsilon\!\!\!/(\vec{k}, \lambda) u(\vec{p}, s) + (ie)^2\overline{u}(\vec{p}', s')\epsilon\!\!\!/(\vec{k},\lambda) \frac{p\!\!\!/ - k\!\!\!/' + m_e}{(p - k')^2 - m^2_e} \epsilon\!\!\!/\,'(\vec{k}', \lambda')^* u(\vec{p}, s),</math>

from which we can compute the [[Cross section (physics)|cross section]] for this scattering.