My questions about QFT and the standard model

[planck999]

TL;DR

My questions about qft and standart model

First of all, I guess classifying particles as excitations of fields stem from the fact that fields can be decomposed into plane waves which are particles in some sense and since they are infinite we have infinitely many of them. Like, any continious electric field can be decomposed into plane waves, for which those plane waves are em waves which are in turn photons when they behave like particles. In the same wave they said all elementary particles are plane waves of some field. Like electron being a dirac plane wave.

I also learned from schwartz that any fundemental particle is characterized by spin and mass, and it's lagrangian and equations of motion are determined from spin. Gauge freedom stems from the extra degrees of freedom in the tensors we embed them into while there are less freedom for given spin, 2j+1 degrees of freedom namely.

I also understood the non-abelian gauge theories of su2 and su3.

Now my questions are:
how does obtaining form of lagrangian just from the spin work? And why is gauge invariance necessary? I can do the calculations but my intuition is missing. And, how did parity experiment of wu, while it does explain the left handedness, lead to a su2 symmetry and the concept of doublets? I can understand the maths of them very well but I don't get why did that experiment result in the concept of doublets and an su2 symmetry?

Also, I didn't quite understand what the degrees of freedom are. Are they linearly independent components of a field? And about the local symmetries, why did the original inventors promoted global symmetries into local ones? Does every global symmetry which dictates consevration of charge imply a local symmetry too?

Textbooks don't build the intuition very well, they give all tools to calculate but even getting the idea of why particles are excitations of fields was too consuming.

To summarize:
1- Why do we need local symmetries, and do we need one for each global continious symmetry?
2- How did they come up with su2 and su3 symmetries?
3- How is form of a lagrangian determined from the spin of the particle alone?
4- What actually is the degrees of freedom of a given particle?

 

[Matterwave]

This is a very long question which covers quite a bit of the QFT machinery. It might be more productive to focus on one specific question.

For example, ultimately why does nature follow $SU(3)\times SU(2)\times U(1)$ symmetry is unknown.