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Matrix equation in SL(2,C)

  1. May 21, 2009 #1
    1. The problem statement, all variables and given/known data

    Solve the equation [tex]A_{k}k_{0}A_{k}^{\dagger}=k[/tex] in SL(2,C) where [tex]k_0[/tex] corresponds to the unit vector [tex]\{0,0,1\}[/tex] and [tex]k[/tex] is an arbitrary vector, i.e.:

    [tex]k0=
    \left( \begin{array}{cc}
    2 & 0 \\
    0 & 0 \\
    \end{array} \right)
    [/tex]

    [tex]k=
    \left( \begin{array}{cc}
    1+n_3 & n_- \\
    n_+ & 1-n_3 \\
    \end{array} \right)
    [/tex]



    2. Relevant equations

    If I try to solve for
    [tex]A_k=
    \left( \begin{array}{cc}
    a & b \\
    c & d \\
    \end{array} \right)
    [/tex]

    this gives (where [tex]a*[/tex] is the conjugate of [tex]a[/tex]):
    [tex]A_{k}k_{0}A_{k}^{\dagger}=
    \left( \begin{array}{cc}
    2aa* & 2ac* \\
    2ca* & 2cc* \\
    \end{array} \right)
    [/tex]



    3. The attempt at a solution

    So this gives conditions on [tex]\{a,c\}[/tex] but can [tex]\{b,c\}[/tex] be arbitrary ? How do I solve this equation and obtain the expression of [tex]A_k[/tex] involving only [tex]n_+, n_-[/tex] and [tex]n_3[/tex] ?

    Thanks a lot for your help!
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
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