= Local symmetries of the Lagrangian imply conserved currents
TODO. I think this is the key point. Notably, <U(1)> symmetry implies <charge conservation>.
More precisely, each <generator of a Lie algebra>[generator of the corresponding Lie algebra] leads to one separate conserved current, such that a single symmetry can lead to multiple conserved currents.
This is basically the <local symmetry> version of <Noether's theorem>.
Then to maintain charge conservation, we have to maintain <local symmetry>, which in turn means we have to add a <gauge field> as shown at <video Deriving the qED Lagrangian by Dietterich Labs (2018)>.
Forces can then be seen as kind of a side effect of this.
Bibliography:
* https://photonics101.com/relativistic-electrodynamics/gauge-invariance-action-charge-conservation\#show-solution has a good explanation of the Gauge transformation. TODO how does that relate to <U(1)> symmetry?
* https://physics.stackexchange.com/questions/57901/noether-theorem-gauge-symmetry-and-conservation-of-charge