Consider pion states composed of ##q \bar q## pairs where ##q \in \left\{u,d \right\}## transforms under an ##SU(2)## isospin flavour symmetry. These bound states transform in the tensor product ##R_1 \otimes R_2## of two representations ##(R_1, R_2)## of ##SU(2)##. Take ##R_2## as the fundamental representation of isospin with generators ##I^i = \sigma^i/2## and ##R_1## is the conjugate fundamental with generators ##-(\sigma^{i})^*/2##. If the third component of isospin is ##I_{\pm}^{R_1 \otimes R_2} = \frac{1}{2} \left( \sigma_1^{R_1 \otimes R_2} \pm i \sigma_2^{R_1 \otimes R_2}\right)## I can try and form a representation of this operator using the standard Pauli matrices. Take ##|\pi^+ \rangle = |u\rangle |\bar d \rangle \equiv |u \rangle \otimes | \bar d \rangle \equiv |u \bar d \rangle##(adsbygoogle = window.adsbygoogle || []).push({});

Then ##I_{+}^{R_1 \otimes R_2} |u \bar d \rangle = \frac{1}{2} \left( \sigma_1^{R_1 \otimes R_2} \pm i \sigma_2^{R_1 \otimes R_2}\right)|u \bar d \rangle = \frac{1}{2}\left( \sigma_1^{R_1} |\bar d\rangle \otimes \text{Id} |u \rangle + \text{Id} |\bar d \rangle \otimes \sigma_1^{R_2} |u \rangle \pm i(1 \leftrightarrow 2)\right)##

1)My first question is if I take ##|u \rangle \rightarrow (1,0), |\bar d \rangle = (0,1)## then I have $$I_+^{R_1 \otimes R_2} |\pi^+ \rangle = I_+^{R_1 \otimes R_2} |u \bar d \rangle $$ This is equal to, $$ \frac{1}{2} \left( \sigma_1^{R_1} | \bar d \rangle \otimes \text{Id}_{2\times 2} |u \rangle + \text{Id}_{2 \times 2} |\bar d \rangle \otimes \sigma_1^{R_2}|u \rangle + i(1 \rightarrow 2) \right)$$ Inputting relevant matrices for sigma_i I get the zero vector as I should since pi^+ is the state of greatest weight and I am applying the raising operator. I just wondered would I get this result independent of the choice of what I take for ##| u \rangle ## and ##|d \bar \rangle## basis vectors? As long as the choice for u and d transforming in fundamental and ubar and dbar transforming in conjugate fundamental are linearly independent?

2) Alternatively, I could just construct the representations for ##I_{\pm}^{R_1 \otimes R_2}## and I would end up with ##4 \times 4## matrices. But what would the ##4 \times 1## objects that these operators act on represent? Would a generic vector be something like ##(u, d, \bar u, \bar d)## so for example I would write ##u = (1,0,0,0)## and ##\bar d = (0,0,0,1)## for example?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# I Raising and lowering operators for a composite isospin SU(2)

Have something to add?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**