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Consider the 2-components Weyl spinors with the following scalar product \begin{equation}\tag{1} \langle \, \phi, \, \psi \, \rangle = \phi^{\top} \, \sigma_y \: \phi, \end{equation} where $\sigma_y$ is the second Pauli matrix. To each spinor, we could associate a light-like four-vector of components \begin{equation}\tag{2} n_\psi^a = \psi^{\dagger} \sigma^a \, \psi = \psi_i^{*} \, \sigma_{ij}^a \, \psi_j, \end{equation} where $\sigma^0 = 1$, $\sigma^1 = \sigma_x$, $\sigma^2 = \sigma_y$ and $\sigma^3 = \sigma_z$ are the Pauli matrices.

Now, using explicit components $\phi = (a, \, b)$ and $\psi = (c, \, d)$ (where $a$, $b$, $c$ and $d$ are complex numbers), it is possible to prove the following formula by brute force (it's not too painfull, just a bit long to do): \begin{equation}\tag{3} |\langle \, \phi, \, \psi \, \rangle |^2 = \frac{1}{2} \: \eta_{ab} \, n_\phi^a \, n_\psi^b, \end{equation} where $\eta = (1, -1, -1, -1)$ is the Minkowski metric of flat spacetime.

I'm wondering if there's a simpler proof of (3), using only properties of the Pauli matrices. Maybe there's another identity related to $\eta_{ab} \, \sigma_{ij}^a \, \sigma_{pq}^b$?

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