Editing Propagator (section)

==== Spin {{frac|1|2}} ====
If the particle possesses [[Spin (physics)|spin]] then its propagator is in general somewhat more complicated, as it will involve the particle's spin or polarization indices. The differential equation satisfied by the propagator for a spin {{frac|1|2}} particle is given by<ref>{{harvnb|Greiner|Reinhardt|2008|loc=Ch.2}}</ref>

:<math>(i\not\nabla' - m)S_F(x', x) = I_4\delta^4(x'-x),</math>

where {{math|''I''<sub>4</sub>}} is the unit matrix in four dimensions, and employing the [[Feynman slash notation]]. This is the Dirac equation for a delta function source in spacetime. Using the momentum representation,
<math display="block">S_F(x', x) = \int\frac{d^4p}{(2\pi)^4}\exp{\left[-ip \cdot(x'-x)\right]}\tilde S_F(p),</math>
the equation becomes

: <math>
\begin{align}
& (i \not \nabla' - m)\int\frac{d^4p}{(2\pi)^4}\tilde S_F(p)\exp{\left[-ip \cdot(x'-x)\right]} \\[6pt]
= {} & \int\frac{d^4p}{(2\pi)^4}(\not p - m)\tilde S_F(p)\exp{\left[-ip \cdot(x'-x)\right]} \\[6pt]
= {} & \int\frac{d^4p}{(2\pi)^4}I_4\exp{\left[-ip \cdot(x'-x)\right]} \\[6pt]
= {} & I_4\delta^4(x'-x),
\end{align}
</math>

where on the right-hand side an integral representation of the four-dimensional delta function is used. Thus

:<math>(\not p - m I_4)\tilde S_F(p) = I_4.</math>

By multiplying from the left with
<math display="block">(\not p + m)</math>
(dropping unit matrices from the notation) and using properties of the [[gamma matrices]],
<math display="block">\begin{align}
\not p \not p & = \tfrac{1}{2}(\not p \not p + \not p \not p) \\[6pt]
& = \tfrac{1}{2}(\gamma_\mu p^\mu \gamma_\nu p^\nu + \gamma_\nu p^\nu \gamma_\mu p^\mu) \\[6pt]
& = \tfrac{1}{2}(\gamma_\mu  \gamma_\nu + \gamma_\nu\gamma_\mu)p^\mu p^\nu \\[6pt]
& = g_{\mu\nu}p^\mu p^\nu = p_\nu p^\nu  = p^2,
\end{align}</math>

the momentum-space propagator used in Feynman diagrams for a [[Dirac equation|Dirac]] field representing the [[electron]] in [[quantum electrodynamics]] is found to have form

:<math> \tilde{S}_F(p) = \frac{(\not p + m)}{p^2 - m^2 + i \varepsilon} = \frac{(\gamma^\mu p_\mu + m)}{p^2 - m^2 + i \varepsilon}.</math>

The {{math|''iε''}} downstairs is a prescription for how to handle the poles in the complex {{math|''p''<sub>0</sub>}}-plane. It automatically yields the [[Feynman propagator|Feynman contour of integration]] by shifting the poles appropriately. It is sometimes written

:<math>\tilde{S}_F(p) = {1 \over \gamma^\mu p_\mu - m + i\varepsilon} = {1 \over \not p - m + i\varepsilon} </math>

for short. It should be remembered that this expression is just shorthand notation for {{math|(''γ''<sub>''μ''</sub>''p''<sup>''μ''</sup> − ''m'')<sup>−1</sup>}}. "One over matrix" is otherwise nonsensical. In position space one has
<math display="block">S_F(x-y) = \int \frac{d^4 p}{(2\pi)^4} \, e^{-i p \cdot (x-y)} \frac{\gamma^\mu p_\mu + m}{p^2 - m^2 + i \varepsilon} = \left( \frac{\gamma^\mu (x-y)_\mu}{|x-y|^5} + \frac{m}{|x-y|^3} \right) J_1(m |x-y|).</math>

This is related to the Feynman propagator by

:<math>S_F(x-y) = (i \not \partial + m) G_F(x-y)</math>

where <math>\not \partial := \gamma^\mu \partial_\mu</math>.