# Representation of the Lorentz group using matrices of $SL(2,\mathbb{C})$

There is a correspondence between the Lorentz group and the group $$SL(2,\mathbb{C})$$. To each Lorentz transformation $$\Lambda$$ we can associate two matrices $$\pm A(\Lambda) \in SL(2,\mathbb{C})$$ such that $$A \sigma_{\nu} A^{\dagger} = \Lambda^{\mu}_{\ \ \nu} \ \sigma_{\nu},$$ where $$\sigma_{\nu}$$ ($$\nu = 1,2,3$$) are the Pauli matrices. I was trying to prove that the matrices $$\{A(\Lambda)\}$$ forms a representation of the Lorentz group, and I would like if my proof is correct.

Since the product of two Lorentz transformations $$\Lambda_{1}$$ and $$\Lambda_{2}$$ is a Lorentz transformation ($$\Lambda$$), using the relation above, we have $$A(\Lambda) \sigma_{\nu} A^{\dagger}(\Lambda) = \Lambda^{\mu}_{\ \ \nu} \ \sigma_{\nu} = \Lambda^{\mu}_{1 \ \alpha} \Lambda^{\alpha}_{2 \ \nu} \ \sigma_{\nu} \\ = (\Lambda^{\mu}_{1 \ \alpha}\sigma_{\nu}) \Lambda^{\alpha}_{2 \ \nu}$$ (using that $$\Lambda_{1}$$ also satisfy the first equation) $$= A(\Lambda_{1}) \sigma_{\alpha} A^{\dagger}(\Lambda_{1})\Lambda^{\alpha}_{2 \ \nu} \\ = A(\Lambda_{1}) (\Lambda^{\alpha}_{2 \ \nu} \sigma_{\alpha}) A^{\dagger}(\Lambda_{1}) \\ = A(\Lambda_{1}) A(\Lambda_{2}) \sigma_{\nu} A^{\dagger}(\Lambda_{2}) A^{\dagger}(\Lambda_{1}).$$ So, in the end I get $$A(\Lambda) \sigma_{\nu} A^{\dagger}(\Lambda) = A(\Lambda_{1}) A(\Lambda_{2}) \sigma_{\nu} (A(\Lambda_{1}) A(\Lambda_{2}))^{\dagger}.$$ Can I conclude from this that $$A(\Lambda) = A(\Lambda_{1})A(\Lambda_{2})$$?