# How does the electroweak interaction and QCD form $SU(2)$ and $SU(3)$?

I'm trying to get a foothold into quantum field theory from a mathematical background. I see the use of $$SU(2)$$ and $$SU(3)$$ in gauge theory and wonder the following questions to help me bring QFT closer to things I'm more familiar with.

Wikipedia states that:

Sheldon Glashow, Abdus Salam,[1][2] and Steven Weinberg[3] were awarded the 1979 Nobel Prize in Physics for their contributions to the unification of the weak and electromagnetic interaction between elementary particles, known as the Weinberg–Salam theory.

1. How did we arrive to the conclusion that the electroweak interaction and QCD form $$SU(2)$$ and $$SU(3)$$ (ie. what we observed to believe that they form a "real" Lie group or a topological space that is compact and simply connected)?

Wikipedia also states that:

Every field theory of particle physics is based on certain symmetries of nature whose existence is deduced from observations.

1. Did our knowledge of more "concrete" equations helped in verified these symmetries? Are there cases where these symmetries are formed from these "concrete" equations?

Lastly, looking at the formation of Gauge theory I understand we can't derive Maxwell equations from $$SU(3)$$. Yet, these more classical equations are considered under the same umbrella of "Quantum chromodynamics".

1. What are the reasons to think that QFT is a generalisation of more "concrete" results such as the Maxwell equations? eg. are there common results that QCD and Maxwell equations predict? (I will assume in that case that their relationship would be more entangled) - I'm not sure if that is a philosophical question of what consists a generalisation and if that's the case I apologise but I am very interested in your opinions on this.
• Maybe you should read the eightfold way,en.wikipedia.org/wiki/Eightfold_way_(physics) to see how the experimentally observed symmetries led to the need for the SU groups. Mar 6, 2021 at 10:10
• Do you understand quantum mechanics? Classical electromagnetism? Mar 6, 2021 at 10:23
• @annav Thanks for the suggestion. There is some interesting information in the "Historical Development" section. It is short and I'd be very interested in learning more but it is a good start. The "Flavor symmetry" section is useful.It drops a hint that elements of SU(3) are automorphisms of a vector space (ie. all possible quantum states that you get from flavour-rotating a proton). Mar 6, 2021 at 10:47
• @MitchellPorter I'd say I'm familiar with elementary quantum mechanics concepts but have not any formal education in them nor I'm familiar with the experimental verification of the concepts. I think the mathematics of Maxwell equations help me understand them well, surely better than either elementary or advanced QM. Mar 6, 2021 at 10:54
• as for Maxwell equatiions, and QFT maybe this blog article by Motl will help see the coonnection motls.blogspot.com/2011/11/… Mar 6, 2021 at 11:54