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!

This post imported from StackExchange Physics at 2014-12-09 15:07 (UTC), posted by SE-user soliton