- #1
robousy
- 334
- 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 let's 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?
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 let's 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?
Last edited by a moderator: