>weak isospin is 3-dimensional >only the third component

first explain why you are posting oppai piggu when loli is clearly superior

also, linear algebra stuff explains all dem matrices n shiet

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>>7921131
>[math]SU(N)[/math] has dimension [math]N^2 -1 [/math]
>all components matter
>[math]SU(N)[/math] must have irreducible representations in [math]GL(N,\mathbb{C})[/math]
Any more questions?

>>7921153
Oops, meant [math]SL(n,\mathbb{C})[/math]

>>7921153
That misses the point of the question. Why does SU(2) describe weak isospin instead of SU(3) or SU(1) or any other scalar or 3x3 set for that matter?

What do those two components represent? Is it some superposition of chirality? Would that mean the pauli matricies represent superpositions of superpositions?

>>7921169
"Isospin" is the conserved current obtained from [math]SU(2)[/math] symmetry a la Noether. It was first experimentally observed that conservation of lepton number was satisfied in all scattering and decaying events, which means a rotation of the lepton and the neutrino parts of the lepton doublet Hilbert space [math]\mathcal{H}= \mathcal{H}_{e} \otimes \mathcal{H}_{\nu_e}[/math] must leave the action (or the Lagrangian density) invariant. A rotation of the doublet [math]U \in SU(2)[/math], so naturally physicists imposed gauge group [math]SU(2)[/math] as a symmetry group of the Lagrangian.

>What do those two components represent?
[math]W^{1,2}[/math] are the gauge fields, which can be combined into ladder operators [math]W^{\pm}[/math], which creates and destroys the charged W bosons that mediates beta decay.

[math]SU(3)[/math] arise in QCD due to the conservation of quak triplet [math]\left[u,d,s\right][/math]

>>7921203
hold on a sec, I'm not a physicist but something's not right here

I thought the three components SU(3) in QCD referred to the three color charges. That's what every source I've seen has said.

If that's wrong than I don't really think I can accept your answer, not that I understand the rest of it

>>7921225
Sorry I must've confused you by throwing in SU(3) and QCD. That part isn't pertinent in understand weak theory.
>three components SU(3) in QCD referred to the three color charges
There are 8 components to the Lie algebra of [math]SU(3)[/math], and the components of the gauge fields (which is a connection 1-form [math]\mathcal{A} \in \Lambda^{1}(P) \otimes \mathfrak{su}(3)[/math]) give you the 8 "colours" currents of the quarks, which have a total of 6 flavours (known so far). I think you've confused flavour with colour.

The triplet [math]\left[u,d,s\right][/math] was experimentally observed to have their total number conserved, which arises from the conservation of the colour currents.

>>7921238
Can you just explain your answer a bit more simply for someone who's knowledge of mathematics pretty much ends at Linear algebra 1?

SU(2) has three matrices that represent the W1, W2, and W3 bosons. I get you can multiply those matrices with certain vectors to represent the bosons interacting with a fermion. I get that. Everything else is meaningless to me

>>7921261
Read Griffith's. If you haven't done any actual linear algebra, group theory and PDE it's meaningless to be talking about quantum mechanics.

>>7921265
By ends at, I mean I've done linear algebra 1 and that's it. I've learned a bit of group theory from wikipedia but my knowledge is patchy

If I ask some yes or no questions, can I just ask you a string of questions an you tell me at what point the answer is no?

>>7921225
Those are two different symmetries. They just both involve the group SU(3).

>>7921272
No that's unproductive.
I suggest you read at least Griffith's QM book and then go from there. Wikipedia is not a good source for you yet.

>>7921169
>What do those two components represent?
The left-handed parts of the up and down quarks are two parts of an SU(2) doublet. Same with the left-handed parts of the electron and neutrino. And similarly for the second and third generations. The right handed parts of the up quark and down quark are each an SU(2) singlet on their own.

>>7921291
This is the simple answer I was looking for. Thank you.

However, there is one point of clarification: is this implying that particles are sort of in a superposition of flavors?

>>7921303
SU(2) is spontaneously broken by the Higgs field. The Higgs field is another SU(2) doublet. The "Mexican hat" potential of the Higgs field causes it to have a nonzero value at its lowest energy state. There's an interaction between the Higgs field, the field that gives you the right-handed part of the up quark, and the field that gives you the left-handed parts. Through this interaction the value of the Higgs field determines which component of the doublet interacts with the right-handed up. And through another interaction, it makes the other part of the doublet interact with the right-handed down. And these interactions create the masses for the up and down quarks. Similarly for the electron and neutrino, except that there's no right-handed neutrino as far as we've detected (but see the seesaw mechanism).

To answer directly, superpositions of flavors exist but interactions with the Higgs field creates mass eigenstates which are the particles we observe.

>>7921354
So the flavor is random but the energy doesn't change?

>>7921272
you can read a bit on spin and groups here

https://boards.fireden.net/sci/thread/7837947/#q7869720

then go back up, since there is the same material, but expressed in even more autistic terms.

>>7921131
>only the third component matters
>>7921169
What do those two components represent?

The other anon is spot on and is explaining everything perfectly, but I think you still might be confused about this.
The third component is the "important" one because that is the one we have chosen to diagonalize. Its eigenstates are the ones you're familiar with and describe the state of the lepton (before spontaneous symmetry breaking). In SU(2), if we choose to diagonalize the first or second component of the Lie algebra, we will find that the similarity transformation will change the other two components such that the lie algebra is identical. The state will change too, but it is identical to preforming a rotation of the state onto the other coordinate.
The mathematics doesn't change. using the third component of su(2) as the diagonal is just common practice in physics.

Alternatively, we could choose a direction along the axis of travel or some external interacting field (like a magnetic field for the spin SU(2) group) to be the one we diagonalize.

You might benefit from watching Susskind's lectures on the Higgs. His math leaves a lot to be desired but if you listen closely he explains things well.