# How to prove $(α·σ)(β·σ) = α·β +iα×β·σ$ (where, $α$ and $β$ are 3 dimensional vectors and $σ$ represents Pauli matrices)?

I tried to evaluate the LHS first and obtained the first term of RHS easily. Then i tried to use the commutation relations of $$\mathrm{SU}(2)$$ group to proceed further to obtain the second term of the LHS from remaining quantities but got stuck there. Can somebody show how it's done?

• WP. – Cosmas Zachos May 17 at 10:28
• Unless you know a smarter way, try writing out all of the components. – my2cts May 17 at 11:40

## 2 Answers

The commutation relation $$[\sigma_i,\sigma_j] = 2i\epsilon_{ijk}\sigma_k$$ is strictly insufficient for what you want to achieve, and it needs to be supplemented with the anticommutation relations between the Pauli matrices, $$\{\sigma_i,\sigma_j \} = 2\delta_{ij}.$$ As the Wikipedia page points out, adding those two will get you an expression for the individual products, $$\sigma_i\sigma_j = \delta_{ij} + i\epsilon_{ijk}\sigma_k,$$ and you can then contract this with the two relevant vectors on either side to get the expression you're after.

Write the expression in index notation. Use commutation relation of Pauli matrices to commute them. Than use anti-commutation relation to get them back in the same order as what you started with but with opposite sign. Now you can rearrange to get the desired answer.

Hope this helps.