A very powerful technique for expressing amplitudes of scattering processes in quantum field theory, the harmonious marriage of special relativity and quantum mechanics, is through spinor helicity, in which every quantity is related to the eigenstates of the spin- \(1 / 2 \hat{S}_{z}\) operator. This is exceptionally convenient, because eigenstates of spin- \(1 / 2\) are just two-component spinors! What could be simpler.

In this problem, we will just study one identity that is often exploited in this business, called the Schouten identity. For four spin-1/2 states \(|\psiangle,|hoangle,|\chiangle,|\etaangle\), it states that

\[\begin{equation*}\langle\psi \mid hoangle\langle\chi \mid \etaangle=\langle\psi \mid \etaangle\langle\chi \mid hoangle+(\langleho|)^{*} i \sigma_{2}|\etaangle\langle\psi| i \sigma_{2}(|\chiangle)^{*},\tag{8.151}\end{equation*}\]

where \(\sigma_{2}\) is the second Pauli spin matrix. Prove this equality. Note that the complex conjugation acts only on a single bra or ket in the final term.