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Pauli Matrices and Structure Constants

  1. Jul 16, 2008 #1
    Hey folks,

    I am trying to generate the Pauli matrices and am using the following formula taken from http://en.wikipedia.org/wiki/SU(3)

    "In the adjoint representation the generators are represented by (n^2-1)×(n^2-1) matrices whose elements are defined by the structure constants"

    [tex](T_a)_{jk} = -if_{ajk}[/tex]

    ok - I'm fine up to here. Now it says, "For SU(2), the generators T, in the defining representation, are proportional to the Pauli matrices, via:"

    [tex]T_a=\frac{\sigma_a}{2}[/tex]

    So here is my problem. I am assuming that j and k run from 1:2. This way T_a is a 2x2 matrix. But lets try this for the first Pauli matrix:

    [tex](T_1)_{11} = -if_{111}=0[/tex]
    [tex](T_1)_{12} = -if_{112}=0[/tex]
    [tex](T_1)_{21} = -if_{121}=0[/tex]
    [tex](T_1)_{22} = -if_{122}=0[/tex]

    [tex]
    \sigma_{1} = \left(\begin{array}{cc}0 & 0\\0 & 0\end{array}\right)[/tex]

    Clearly I am doing something wrong...but what?????
     
  2. jcsd
  3. Jul 27, 2008 #2
    first of all here's a friendly advise : one might not want to truly learn something from wikipedia. democracy has its own problems.

    anyways, please note that the smallest non-trivial matrix representation of dim n[tex]^{2}[/tex]-1 is for n=2, and the dim is 3. Pauli matrices are 2d reps. So you see all that follows this point need to be re-thought.


    cheers
     
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