How to calculate Pauli matrix commutators

[physiker99]

I'm completely lost and need some advice on how to continue. I need to prove the 1st line on the link
http://upload.wikimedia.org/math/0/f/8/0f873eaca989ffa1af9a323c6e62f3ed.png

 

[fzero]

Do you know how to write down the Pauli matrices? Do you know what the commutator [A,B] and anticommutator {A,B} stand for? How about the Levi-Civita symbol $$\epsilon_{abc}$$?

 

[physiker99]

hey, I know how what Pauli matrices are, and I know [a, b], but I don't know {a, b}. I know Levi-Civita as well. I tried to do matrix multiplication [a,b]=a*b-b*a, but that did not lead me anywhere.

As matrices, I wrote down arbitrary values such as a=[a1,a2],[a3,a4] and b=[b1,b2],[b3,b4]

 

[fzero]

The anti-commutator is just {A,B} = AB+BA. I'm not sure why you're using arbitrary matrices when the $$\sigma_a$$ are the known Pauli matrices. Just compute $$[\sigma_1,\sigma_2]$$, etc. and you should see more progress.

 

[Pengwuino]

Wait, so are you asking how to do the first line or the second?

What the first line tells you is that when you do the commutator of 2 pauli matrices, you should get $$2i\epsilon_{abc}\sigma_c$$.

What do you get when you find the commutator of $$[\sigma_1 ,\sigma_2]$$? What does the right hand side tell you? I'll give you a hint, $$\epsilon_{12c}$$ is non-zero when c is equal to what?

For the anti-commutators, it's the same idea. You plug in various values for 'a' and 'b' and see what you get and see if the pattern you see corresponds to what the answer should be.