TODO. I think this is the key point. Notably, symmetry implies charge conservation.
More precisely, each generator of the corresponding Lie algebra leads to one separate conserved current, such that a single symmetry can lead to multiple conserved currents.
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.
- 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 symmetry?