What Does Isospin Symmetry Mean in Particle Physics?

[Hymne]

Hello!
I have difficulty trying with the interpretation of:
$$(\frac{1}{2}) \otimes (\frac{1}{2}) \otimes (\frac{1}{2}) = (\frac{3}{2}) \otimes (\frac{1}{2}) \otimes (\frac{1}{2})$$
If I´m right this is the SU(2) irrep for isospin but what does this mean?
Proton = Neutron?

 

[fzero]

Hello!
I have difficulty trying with the interpretation of:
$$(\frac{1}{2}) \otimes (\frac{1}{2}) \otimes (\frac{1}{2}) = (\frac{3}{2}) \otimes (\frac{1}{2}) \otimes (\frac{1}{2})$$
If I´m right this is the SU(2) irrep for isospin but what does this mean?
Proton = Neutron?

The equation is incorrect as written. The objects appearing above are representations of $$SU(2)$$. These are characterized by their spin $$j$$ and have dimension $$2j+1$$. For instance, the $$j=1/2$$ representation has dimension 2. For each dimension, there is a unique irreducible representation. By irreducible, we mean that we can't write the matrix generators of the representation in block-diagonal form. It is common to use the dimension rather than the spin to label irreducible representations of $$SU(2)$$. So instead of writing $$\tfrac{1}{2}$$, we would write $$\mathbf{2}$$.

Now the expression of the LHS contains the tensor product of 3 spin 1/2 representations. The states in this object have 3 spinor indices $$t_{abc}$$, but are actually reducible, so the usual question is what irreducible representations appear when we reduce the product representation. To work this out, it's best to work in steps. In terms of the dimensions, the product is $$\mathbf{2}\otimes \mathbf{2}\otimes \mathbf{2}$$. We'll first consider part of this, $$\mathbf{2}\otimes \mathbf{2}$$. Here we take two spin 1/2 states and make a tensor $$t_{ab}$$. This is a 2x2 matrix in spin space, which we can break up into a symmetric part and an antisymmetric part. The symmetric part has (2)(3)/2 = 3 components, while the antisymmetric part has 1 component. We can therefore identify the antisymmetric part with the trivial $$j=0$$, one-dimensional irrep, while the symmetric part is a $$j=1$$, three-dimensional vector irrep. So we would write

$$\mathbf{2}\otimes \mathbf{2}=\mathbf{1}\oplus\mathbf{3}.$$

Note that the RHS has a sum of irreps, not a product.

Now, one could have obtained this from usual angular momentum addition rules. We have $$j_1=1/2$$ and $$j_2=1/2$$ and when we add these we are allowed to obtain the representations $$j_1+j_2$$, $$j_1+j_2-1$$, ... $$|j_1-j_2|$$. But for $$j_1=j_2=1/2$$, we only have two possibilities: $$j_1+j_2 =1$$ and $$|j_1-j_2|=0$$, as we found above.

Now we can go back to the original product and write it as

$$\mathbf{2}\otimes\mathbf{2}\otimes \mathbf{2}=\mathbf{2}\otimes(\mathbf{1}\oplus\mathbf{3}).$$

The tensor product distributes over the sum, so we can consider $$\mathbf{2}\otimes\mathbb{1}$$ and $$\mathbf{2}\otimes\mathbb{3}$$ separately. The first is simply $$\mathbf{2}\otimes\mathbf{1}=\mathbf{2}$$, since the spin 0 part doesn't change anything. For the second, we are adding $$j_1=1/2$$ and $$j_2=1$$, so we obtain a spin 3/2 and a spin 1/2 state, so $$\mathbf{2}\otimes\mathbf{3}=\mathbf{2}\oplus\mathbf{4}$$. So finally we obtain


$$\mathbf{2}\otimes\mathbf{2}\otimes \mathbf{2}=\mathbf{4}\oplus \mathbf{2}\oplus\mathbf{2}.$$

A consistency check is that the dimension on the LHS, 2x2x2 = 8, matches the dimension on the RHS, 4+2+2=8. We also see that the original expression is wrong because it is the sum, not product of irreps that should appear on the RHS.

In order to explain a physical interpretation for this formula in terms of isospin, we need to define what objects are carrying the isospin on the LHS. Are they quarks, mesons, baryons, etc? Without more background, the only thing that I can comment on is your "Proton = Neutron" question. It is safe to say that it does not mean that. In the theory of isospin, the proton and neutron form a doublet, that is a $$j=1/2$$ irrep, where the proton has $$m_j=1/2$$ and the neutron has $$m_j=-1/2$$. So isospin is a quantum number that actually distinguishes the proton from the neutron.

 

[Hymne]

Thaaanks for the answer! Very pedagogical!
That is enough for me actually
Have an awesome spring!