Pauli Matrices and Structure Constants

1. Jul 16, 2008

robousy

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 [Broken])

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

$$(T_a)_{jk} = -if_{ajk}$$

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:"

$$T_a=\frac{\sigma_a}{2}$$

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:

$$(T_1)_{11} = -if_{111}=0$$
$$(T_1)_{12} = -if_{112}=0$$
$$(T_1)_{21} = -if_{121}=0$$
$$(T_1)_{22} = -if_{122}=0$$

$$\sigma_{1} = \left(\begin{array}{cc}0 & 0\\0 & 0\end{array}\right)$$

Clearly I am doing something wrong...but what?????

Last edited by a moderator: May 3, 2017
2. Jul 27, 2008

vkroom

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$$^{2}$$-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