- #1

James1238765

- 120

- 8

The 8 gluon fields of SU(3) can be represented (generated) by the 8 Gel-Mann matrices:

$$ \lambda_1 =

\begin{bmatrix}

0 & 1 & 0 \\

1 & 0 & 0 \\

0 & 0 & 0

\end{bmatrix} , \lambda_2 =

\begin{bmatrix}

0 & -i & 0 \\

i & 0 & 0 \\

0 & 0 & 0

\end{bmatrix} , \lambda_3 =

\begin{bmatrix}

1 & 0 & 0 \\

0 & -1 & 0 \\

0 & 0 & 0

\end{bmatrix} $$

$$\lambda_4 =

\begin{bmatrix}

0 & 0 & 1 \\

0 & 0 & 0 \\

1 & 0 & 0

\end{bmatrix}, \lambda_5 =

\begin{bmatrix}

0 & 0 & -i \\

0 & 0 & 0 \\

i & 0 & 0

\end{bmatrix} , \lambda_6 =

\begin{bmatrix}

0 & 0 & 0 \\

0 & 0 & 1 \\

0 & 1 & 0

\end{bmatrix} $$

$$\lambda_7 =

\begin{bmatrix}

0 & 0 & 0 \\

0 & 0 & -i \\

0 & i & 0

\end{bmatrix} , \lambda_8 =

\begin{bmatrix}

\frac{1}{\sqrt3} & 0 & 0 \\

0 & \frac{1}{\sqrt3} & 0 \\

0 & 0 & -\frac{2}{\sqrt3}

\end{bmatrix}

$$

While I have seen many derivations for the Gel-Mann matrices, I have not seen a demonstration of the basic usage of these matrices.

Suppose we have a "red gluon". Is this to be represented by the column vector

$$ \vec{red} = \begin{bmatrix}

1 \\

0 \\

0

\end{bmatrix}?$$

Then, if we would like to calculate the inverse of this "red gluon", do we multiply this column vector representation against one of the Gel-mann matrices, such as ##\lambda_1 \vec {red} ##:

$$

\begin{bmatrix}

0 & 1 & 0 \\

1 & 0 & 0 \\

0 & 0 & 0

\end{bmatrix}

\begin{bmatrix}

1 \\

0 \\

0

\end{bmatrix} =

\begin{bmatrix}

0 \\

1 \\

0

\end{bmatrix}$$

What does ##\lambda_1 \vec {red} = \vec {green} ## mean in this representation?

$$ \lambda_1 =

\begin{bmatrix}

0 & 1 & 0 \\

1 & 0 & 0 \\

0 & 0 & 0

\end{bmatrix} , \lambda_2 =

\begin{bmatrix}

0 & -i & 0 \\

i & 0 & 0 \\

0 & 0 & 0

\end{bmatrix} , \lambda_3 =

\begin{bmatrix}

1 & 0 & 0 \\

0 & -1 & 0 \\

0 & 0 & 0

\end{bmatrix} $$

$$\lambda_4 =

\begin{bmatrix}

0 & 0 & 1 \\

0 & 0 & 0 \\

1 & 0 & 0

\end{bmatrix}, \lambda_5 =

\begin{bmatrix}

0 & 0 & -i \\

0 & 0 & 0 \\

i & 0 & 0

\end{bmatrix} , \lambda_6 =

\begin{bmatrix}

0 & 0 & 0 \\

0 & 0 & 1 \\

0 & 1 & 0

\end{bmatrix} $$

$$\lambda_7 =

\begin{bmatrix}

0 & 0 & 0 \\

0 & 0 & -i \\

0 & i & 0

\end{bmatrix} , \lambda_8 =

\begin{bmatrix}

\frac{1}{\sqrt3} & 0 & 0 \\

0 & \frac{1}{\sqrt3} & 0 \\

0 & 0 & -\frac{2}{\sqrt3}

\end{bmatrix}

$$

While I have seen many derivations for the Gel-Mann matrices, I have not seen a demonstration of the basic usage of these matrices.

Suppose we have a "red gluon". Is this to be represented by the column vector

$$ \vec{red} = \begin{bmatrix}

1 \\

0 \\

0

\end{bmatrix}?$$

Then, if we would like to calculate the inverse of this "red gluon", do we multiply this column vector representation against one of the Gel-mann matrices, such as ##\lambda_1 \vec {red} ##:

$$

\begin{bmatrix}

0 & 1 & 0 \\

1 & 0 & 0 \\

0 & 0 & 0

\end{bmatrix}

\begin{bmatrix}

1 \\

0 \\

0

\end{bmatrix} =

\begin{bmatrix}

0 \\

1 \\

0

\end{bmatrix}$$

What does ##\lambda_1 \vec {red} = \vec {green} ## mean in this representation?

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