= Derivation of the quantum electrodynamics Lagrangian
= Derivation of the QED Lagrangian
{synonym}
{title2}
Like the rest of the <Standard Model Lagrangian>, this can be split into two parts:
* <spacetime symmetry>: reaches the <derivation of the Dirac equation>, but has no interactions
* add the <U(1)> <internal symmetry> to add interactions, which reaches the full equation
\Video[https://www.youtube.com/watch?v=IFRyN3fQMO8]
{title=Deriving the <qED Lagrangian> by <Dietterich Labs> (2018)}
{description=
As mentioned at the start of the video, he starts with the <Dirac equation> Lagrangian derived in a previous video. It has nothing to do with <electromagnetism> specifically.
He notes that that <Dirac Lagrangian>, besides being globally <Lorentz invariant>, it also also has a global <U(1)> invariance.
However, it does not have a local invariance if the <U(1)> transformation depends on the point in spacetime.
He doesn't mention it, but I think this is highly desirable, because in general <local symmetries of the Lagrangian imply conserved currents>, and in this case we want conservation of charges.
To fix that, he adds an extra <gauge field> $A_\mu$ (a field of $4 \times 4$ matrices) to the regular derivative, and the resulting derivative has a fancy name: the <covariant derivative>.
Then finally he notes that this <gauge field> he had to add has to transform exactly like the <electromagnetic four-potential>!
So he uses that as the gauge, and also adds in the <Maxwell Lagrangian> in the same go. It is kind of a guess, but it is a natural guess, and it turns out to be correct.
https://www.youtube.com/watch?v=IFRyN3fQMO8&lc=UgzNGkLXdwcSl7z8Lap4AaABAg
}