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I am confused when it comes to the relation between some arbitrary gauge transformation/ internal symmetry and the different symmetries considered. I will illustrate this by considering the $SU(2)$ transformations.

From my understanding, when we consider a lagrangian density in this case, the fermion fields are in the fundamental representation, the doublet, while the corresponding gauge fields are in the adjoint representation.

I came across the concept of "isovector", which was framed as "the triplet (3-dimensional) representation of $SU(2)$ Isospin"

Now my question is as to why we add "Isospin" in the end?

I searched for it and I found also other symmetries for $SU(2)$ such as:

$SU(2)_L$ denotes weak isospin symmetry in electroweak theory.

$SU(2)_{isospin}$ refers to isospin (strong interaction) symmetry between up/down quarks (nucleons, pions).

There's also spin $SU(2)$ (rotational symmetry in space).

I have the following questions regarding this topic:

  1. Let's take the first case of $SU(2)_L$. When we say that it denotes weak isospin symmetry in electroweak theory, we are essentially saying that the electroweak lagrangian under this specific group transformation is invariant, and the conserved charge is the weak isospin?

  2. For the isospin in Wikipedia is said : "Isospin is regarded as a symmetry of the strong interaction under the action of the Lie group $SU(2)$, the two states being the up flavour and down flavour." If isospin is a symmetry of the strong interaction, in other words the conserved charge of the strong interaction lagrangian, how is it possible that we tie it with $SU(2)$ transformations and not $SU(3)$ transformations, which are related to QCD, the strong interaction?

  3. Can someone explain to me the idea how we define all these groups of $SU(2)$ and how would things differ for $SU(3)$?

Qmechanic
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imbAF
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1 Answers1

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As you indicated, the group SU(2) is featured in the space-like rotation symmetries of fermions, and the Lorentz group action on them, but you indicated you are comfortable with its logical distinctions from the two isospins.

Strong isospin, SU(2), may organize fermions, (u,d), (p,n), which are in the doublet representation. The Strange baryons, however, $(\Sigma^+,\Sigma^0, \Sigma^-)$, are in the 3d representation, so then the isotriplet representation,

But it also applies to bosons, e.g. in the triplet representation $(\pi^+, \pi^0, \pi^-)$, so then an isotriplet. But also the isodoublet $(K^+, K^0)$.

It is not associated with chirality, and is completely independent/disjoint of the color SU(3) of QCD. Hadrons are color singlets, in the 1d representation of SU(3), but all quarks, including the above isodoublet (u,d), are in the color-triplet representation of color SU(3).

Weak isospin, a completely different SU(2), organizes Left-chiral quark doublets, heavy or light, so, (u,d),(c,s), (t,b); but all Right-chiral quarks are weak isosinglets, so invisible to weak isospin. As you indicated, before EW SSB, the original weak SU(2) gauge bosons, $W^i$ are a weak isotriplet.

All antifermions of the above are in the conjugate doublet representation, as covered by several questions of this site.

Cosmas Zachos
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