Hi everyone!(adsbygoogle = window.adsbygoogle || []).push({});

I would like to ask you a very basic question on the decomposition [tex]3\otimes\bar 3=1\oplus 8[/tex] of su(3) representation.

Suppose I have a tensor that transforms under the 8 representation (the adjoint rep), of the form [tex]O^{y}_{x}[/tex]

where upper index belongs to the $\bar 3$ rep and the lower ones to the 3 rep (x,y=1,2,3).

Now, I know that under $T_a$ (an element of the Lie algebra) this tensor transforms as

[tex](T_a O)^y_x\equiv (T_a)_k^y O_x^k-(T_a)_x^k O_k^y[/tex]

And that's by definition.

But it is also true that, for any s-dimensional representation of a Lie algrebra I should have

[tex][T_a,\mathcal O^{s}_i]=(T_a^{s})_{ij}\mathcal O_j[/tex]

where -s≤ i ≤s, as an operatorial equation.

This means that if we take the an operator that transforms under the adjoint rep (s=8) we should get

[tex][T_a,\mathcal O^{adj}_i]=(T_a^{\text{adj}})_{ij}\mathcal O_j[/tex]

where

[tex](T_a^{\text{adj}})_{ij}\equiv -if^{aij}[/tex]

are the matrix element of the adjoint representation of the su(3) Lie algebra, and the f^{abc}'s are the structure constant.

My problem is that I can't explicitly constuct the correnspondence between the tensorial approach and the operatorial one, that is

[tex](T_a)_k^y O_x^k-(T_a)_x^k O_k^y\leftrightarrow (T_a^{\text{adj}})_{ij}\mathcal O_j=-if^{aij}\mathcal O_j [/tex]

In other words, I cannot find an identification

[tex] O^x_y\leftrightarrow \mathcal O_i[/tex]

where again (x,y=1,2,3), (i=1,8) that could make the algebra strcture constants appear into

[tex](T_a)_k^y O_x^k-(T_a)_x^k O_k^y[/tex]

to check that the tensor actually transforms in the adjoint representation.

I know that this may seem a rather cumbersome question, but it is in fact a ground question in group theory.

Please tell me if I have not been clear enough!!

Thanks for your time!

**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!

# Tensor techniques in $3\otimes\bar 3$ representation of su(3)

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