In A. Zee's book Quantum Field Theory in a Nutshell (2nd edition), Chapter N.2, page 486, the momentum $p$ is written as a $2\times 2$ matrix:

$$ p_{\alpha\dot{\alpha}} = p_{\mu} (\sigma^{\mu})_{\alpha\dot{\alpha}} = (p_0I - p_i\sigma^i)_{\alpha\dot{\alpha}} = \begin{pmatrix} (p^0 - p^3) && -(p^1 - ip^2) \\ -(p^1 + ip^2) && -(p^0 + p^3) \end{pmatrix}_{\alpha\dot{\alpha}} $$ Given two vectors $p$ and $q$, their scalar product is given by $$ p\cdot q = \varepsilon^{\alpha\beta}\varepsilon^{\dot{\alpha}\dot{\beta}} p_{\alpha\dot{\alpha}}q_{\beta\dot{\beta}} $$ In E. Witten's article arXiv:hep-th/0312171, the same formula can also be found above Eq.(2.7) in page 5. However, I checked explicitly that it might be not valid $$ \begin{split} \varepsilon^{\alpha\beta}\varepsilon^{\dot{\alpha}\dot{\beta}} p_{\alpha\dot{\alpha}}q_{\beta\dot{\beta}} &= \varepsilon^{12}\varepsilon^{\dot{1}\dot{2}}p_{1\dot{1}}q_{2\dot{2}} + \varepsilon^{12}\varepsilon^{\dot{2}\dot{1}}p_{1\dot{2}}q_{2\dot{1}} + \varepsilon^{21}\varepsilon^{\dot{1}\dot{2}}p_{2\dot{1}}q_{1\dot{2}} + \varepsilon^{21}\varepsilon^{\dot{2}\dot{1}}p_{2\dot{2}}q_{1\dot{1}} \\ &= 2(p^0q^0 - p^1q^1 - p^2q^2 - p^3q^3) \end{split} $$ which differs with the above formula in a factor of $2$. This is only a simple exercise, but I don't know whether they use a different summation convention. Any clarifications will be appreciated.Thanks a lot!

In A. Zee's book Quantum Field Theory in a Nutshell (2nd edition), Chapter N.2, page 486, the momentum $p$ is written as a $2\times 2$ matrix:

$$ p_{\alpha\dot{\alpha}} = p_{\mu} (\sigma^{\mu})_{\alpha\dot{\alpha}} = (p_0I - p_i\sigma^i)_{\alpha\dot{\alpha}} = \begin{pmatrix} (p^0 - p^3) && -(p^1 - ip^2) \\ -(p^1 + ip^2) && -(p^0 + p^3) \end{pmatrix}_{\alpha\dot{\alpha}} $$ Given two vectors $p$ and $q$, their scalar product is given by $$ p\cdot q = \varepsilon^{\alpha\beta}\varepsilon^{\dot{\alpha}\dot{\beta}} p_{\alpha\dot{\alpha}}q_{\beta\dot{\beta}} $$ In E. Witten's article arXiv:hep-th/0312171, the same formula can also be found above Eq.(2.7) in page 5. However, I checked explicitly that it might be not valid $$ \begin{split} \varepsilon^{\alpha\beta}\varepsilon^{\dot{\alpha}\dot{\beta}} p_{\alpha\dot{\alpha}}q_{\beta\dot{\beta}} &= \varepsilon^{12}\varepsilon^{\dot{1}\dot{2}}p_{1\dot{1}}q_{2\dot{2}} + \varepsilon^{12}\varepsilon^{\dot{2}\dot{1}}p_{1\dot{2}}q_{2\dot{1}} + \varepsilon^{21}\varepsilon^{\dot{1}\dot{2}}p_{2\dot{1}}q_{1\dot{2}} + \varepsilon^{21}\varepsilon^{\dot{2}\dot{1}}p_{2\dot{2}}q_{1\dot{1}} \\ &= 2(p^0q^0 - p^1q^1 - p^2q^2 - p^3q^3) \end{split} $$ which differs with the above formula in a factor of $2$. This is only a simple exercise, but I don't know whether they use a different summation convention. And why there is a factor of 2 difference? Thanks a lot!