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In Tales of 1001 Gluons by Stefan Weinzierl, in the end of page 36, (163), $\sigma_{A \dot B}^{\mu} = (1, -\sigma_x, -\sigma_y, -\sigma_z)$. It seems that $\sigma_{A \dot B}^{\mu} = (1, -\sigma_x, -\sigma_y, -\sigma_z)$ is a two by four matrix. Why it is said that $\sigma_{A \dot B}^{\mu} = (1, -\sigma_x, -\sigma_y, -\sigma_z)$ is 4-dimensional? What is matrix $\sigma_{A \dot B}^{\mu}$ explicitly?

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Each matrix $\sigma^{\mu}$, for $\mu = 0,1,2,3$, is a $2 \times 2$ matrix.

This set of four $2 \times 2$ matrices is called the four-dimensional sigma-matrices because they appear in the algebra involving spinors in four space-time dimensions. It does not mean that they are $4 \times 4$ matrices.

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