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

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.