Proving the Isomorphism Property of the Spinor Map in SL(2,C) and SO(3,1)

[gentsagree]

In the context of the homomorphism between SL(2,C) and SO(3,1), I have that

$$\textbf{x}=\overline{\sigma}_{\mu}x^{\mu}$$

$$x^{\mu}=\frac{1}{2}tr(\sigma^{\mu}\textbf{x})$$

give the explicit form of the isomorphism, where $\textbf{x}$ is a 2x2 matrix of SL(2,C) and $x^{\mu}$ a 4-vector of SO(3,1).

Considering the linear map (the spinor map)

$$\textbf{x}\rightarrow\textbf{x}'=A\textbf{x}A^{\dagger}$$

one can show that the 4-vectors on the SO(3,1) side are also linearly related by

$$x'^{\mu}=\phi(A)^{\mu}_{\nu}x^{\nu}$$

where it is easy to show that

$$\phi(A)^{\mu}_{\nu}=\frac{1}{2}tr(\sigma^{\mu}A\overline{\sigma}_{\nu}A^{\dagger})$$

I understand all this, but I want to prove that $\phi(AB)=\phi(A)\phi(B)$. How would I go about doing this? I tried a few things but not very successfully.

 

[dextercioby]

I can't copy paste the proof here, I can only tell you where to find it: Mueller-Kirsten + Wiedemann's <Supersymmetry> (WS, 1987), pages 66 and 67.

 

[gentsagree]

Thanks a lot dextercioby, the book is really helpful!