Commutation Relations and Symmetries for SU(2)

[Astrofiend]

Homework Statement

I'm working through a bit of group theory (specifically SU(2) commutation relations). I have a question a bout symmetries in the SU(2) group. It's something I'm trying to work through in my lecture notes for particle physics, but it's a bit of a mathsy question so I thought I'd post it here where the maths gurus play!

I was showing that the matrices

$$<br /> \sigma_1 = \left( \begin{array}{ccc}<br /> 0 & -1/2 & 0\\<br /> -1/2 & 0 & 1/2\\<br /> 0 & 1/2 & 0\end{array} \right)\]<br />$$

$$<br /> \sigma_2 = \left( \begin{array}{ccc}<br /> 0 & 1/2i & 0\\<br /> -1/2i & 0 & -1/2i\\<br /> 0 & 1/2i & 0\end{array} \right)\]<br />$$

$$<br /> \sigma_3 = \left( \begin{array}{ccc}<br /> 1/2 & 0 & 0\\<br /> 0 & 0 & 0\\<br /> 0 & 0 & -1/2\end{array} \right)\]<br />$$

commute with via the relation

$$[I_i , I_j] = i \epsilon_i_j_k I_k$$

where $$\epsilon_i_j_k$$ takes it's usual meaning as the Levi-Civita symbol.

I was just going to work through each example and verify that they each satisfied the commutation relation, but somebody said to me in passing the other day that you can use symmetry arguments such that you only need to verify a few cases, with the rest following from these symmetry arguments.

I have been trying to work out what they were talking about for ages now, but in vain!

Does anybody have any ideas? Any suggestion of what these symmetries might be and how they eliminate the need to verify every last case would give relief to my aching brain! It's teasing me...

 

[foxjwill]

Maybe they were referencing the fact that the commutator is anti-symmetric? Because that means that you only actually have three cases.

 

[jacobrhcp]

maybe that they are all hermitian up to a factor of i, so maybe if you calculate the top half you can just fill in the bottom half triangle of the matrix?

*is also working through SU(3) for particle physics*

EDIT2: nevermind what I said before was stupid.

 

[Astrofiend]

Hmmm - maybe there's an error in the notes? I'll take a look, check back and let you know what I come up with...

Thanks for the replies!

> It may indeed be that because the commutator is antisymmetric we only have to test three cases. I'll think about that one a little more...

Cheers!