TODO find/create decent answer.
I think the best answer is something along:
- local symmetries of the Lagrangian imply conserved currents. gives conserved charges.
- OK now. We want a local symmetry. And we also want:Given all of that, the most obvious and direct thing we reach a guess at the quantum electrodynamics Lagrangian is Video 96. "Deriving the qED Lagrangian by Dietterich Labs (2018)"
A basic non-precise intuition is that a good model of reality is that electrons do not "interact with one another directly via the electromagnetic field".
A better model happens to be the quantum field theory view that the electromagnetic field interacts with the photon field but not directly with itself, and then the photon field interacts with parts of the electromagnetic field further away.
From Video 94. "Lorenzo Sadun on the "Yang-Mills and Mass Gap" Millennium problem": https://www.youtube.com/watch?v=pCQ9GIqpGBI&t=1663s mentions this idea first came about from Hermann Weyl. https://youtu.be/pCQ9GIqpGBI?t=2827 mentions that in that case the curvature is given by the electromagnetic tensor.