# How to demonstrate that Fierz-like identity for 2-components Weyl spinors? [duplicate]

Consider the 2-components Weyl spinors with the following scalar product $$$$\tag{1} \langle \, \phi, \, \psi \, \rangle = \phi^{\top} \, \sigma_y \: \phi,$$$$ where $$\sigma_y$$ is the second Pauli matrix. To each spinor, we could associate a light-like four-vector of components $$$$\tag{2} n_\psi^a = \psi^{\dagger} \sigma^a \, \psi = \psi_i^{*} \, \sigma_{ij}^a \, \psi_j,$$$$ where $$\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): $$$$\tag{3} |\langle \, \phi, \, \psi \, \rangle |^2 = \frac{1}{2} \: \eta_{ab} \, n_\phi^a \, n_\psi^b,$$$$ where $$\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 $$\eta_{ab} \, \sigma_{ij}^a \, \sigma_{pq}^b$$?