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In scattering amplitudes, page 9, equation (2.6), (2.7), $\not{p}$ (in the Dirac equation (2.4)) is as follows:

\begin{align} \not{p} = \left( \begin{matrix} 0 & p_{a\dot{b}} \\ p^{\dot{a}b} & 0 \end{matrix} \right), \end{align} where $p_{a\dot{b}} = \left( \begin{matrix} -p_0+p_3 & p_1-ip_2 \\ p_1+ip_2 & -p_0-p_3 \end{matrix} \right)$.

But on the other hand, in the article, \begin{align} \not{p} = \left( \begin{matrix} E & \sigma \cdot \vec p \\ -\sigma \cdot \vec p & -E \end{matrix} \right), \end{align} where $E$ is the $2 \times 2$ identity matrix. Why these two expression of $\not{p}$ are different?

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In the first reference you are quoting the chiral basis with $$\gamma^0=\begin{pmatrix} 0 & \mathbf{1}_{2\times 2} \\ \mathbf{1}_{2\times 2} & 0 \end{pmatrix} $$ is used while the second reference uses the (standard) Dirac basis $$\gamma^0=\begin{pmatrix} \mathbf{1}_{2\times 2} & 0 \\ 0 & -\mathbf{1}_{2\times 2} \end{pmatrix} $$ see for example the corresponding Wikipedia article.

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