- #1

Tony3

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- TL;DR Summary
- I need to find a closed form of the anticommutators of the Pauli matrices in the spin-1 representation.

The Pauli matrices of the spin-1 representation are given by: ##T_{1}=\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}##, ##T_{2}=\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & -i & 0 \\ i & 0 & -i \\ 0 & i & 0 \end{pmatrix}## and ##T_{3}=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix}##. I need to find what ##\left\{T_{i},T_{j}\right\}## is equal to.

Doing some calculations, I found that ##\left\{T_{1},T_{2}\right\}=i\begin{pmatrix} 0 & 0 & -1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}##, ##\left\{T_{1},T_{3}\right\}=\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & -1 \\ 0 & -1 & 0 \end{pmatrix}##,##\left\{T_{2},T_{3}\right\}=\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & -i & 0 \\ i & 0 & i \\ 0 & -i & 0 \end{pmatrix}##.

Is there a general relation that I can derive from these special relations? I think that I am close, but I can't quite see it.

Doing some calculations, I found that ##\left\{T_{1},T_{2}\right\}=i\begin{pmatrix} 0 & 0 & -1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}##, ##\left\{T_{1},T_{3}\right\}=\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & -1 \\ 0 & -1 & 0 \end{pmatrix}##,##\left\{T_{2},T_{3}\right\}=\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & -i & 0 \\ i & 0 & i \\ 0 & -i & 0 \end{pmatrix}##.

Is there a general relation that I can derive from these special relations? I think that I am close, but I can't quite see it.