First of all, I do not have any problems concerning what symmetries are or how to describe them. However, I do not have any knowledge concerning how the reasoning for quantum field theory and thus the standard model works. I hope it is still appropriate to ask such a question this early.
What concerns me is a statement I now have heard numerous times and which goes along these lines:
Electromagnetism is built upon a $U(1)$ Symmetry. If we consider other symmetries, we end up with other forces, for instance, if we consider $SU(2)\times U(1)$, we get the electroweak interaction.
Assuming this statement were true, I imagine something like the following to be done:
- Consider some mathematical framework along the lines of “Configuration space + Function of the latter + Axioms”
- Postulate that said function has a $U(1)$ symmetry
- End up with Maxwell's equations (or the corresponding Lagrangian or something equivalent to that)
I cannot imagine any process along these lines though. How can one postulate a symmetry and find physical laws? Hasn't it always been the other way around? That seems like complete magic to me!