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Showing a group homomorphism

  1. Oct 18, 2014 #1
    In the context of the homomorphism between SL(2,C) and SO(3,1), I have that

    [tex]\textbf{x}=\overline{\sigma}_{\mu}x^{\mu}[/tex]

    [tex]x^{\mu}=\frac{1}{2}tr(\sigma^{\mu}\textbf{x})[/tex]

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

    Considering the linear map (the spinor map)

    [tex]\textbf{x}\rightarrow\textbf{x}'=A\textbf{x}A^{\dagger}[/tex]

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

    [tex]x'^{\mu}=\phi(A)^{\mu}_{\nu}x^{\nu}[/tex]

    where it is easy to show that

    [tex]\phi(A)^{\mu}_{\nu}=\frac{1}{2}tr(\sigma^{\mu}A\overline{\sigma}_{\nu}A^{\dagger})[/tex]

    I understand all this, but I want to prove that [itex]\phi(AB)=\phi(A)\phi(B)[/itex]. How would I go about doing this? I tried a few things but not very successfully.
     
  2. jcsd
  3. Oct 18, 2014 #2

    dextercioby

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    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.
     
  4. Oct 20, 2014 #3
    Thanks a lot dextercioby, the book is really helpful!
     
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