Show that Majorana equation implies Klein-Gordon equation

Question

Show that the Majorana equation

$i \bar{\sigma}\cdot\partial\chi -im\sigma^2\chi^* = 0$

for 2-component spinors $\chi$ implies the Klein-Gordon equation

$(\partial^2+m^2)\chi$.

This is part of an Exercise from Peskin and Shroeder.

Here $\bar{\sigma} = (1_{2\times2},-\sigma^1, -\sigma^2, -\sigma^3 )$, with $\sigma^i$ being the Pauli matrices. And $^*$ denotes the complex conjugation.

Supposedly this can be solved by complex conjugating the Majorana equation, followed by an elimination of $\chi^*$. However I'm stuck at this calculation.

Complex conjugating gives

$-i\bar{\sigma}^*\partial\chi^* -im\sigma^2\chi = 0$. And from the Marjorana equation we have that

$\chi^* = \frac{1}{m}\sigma^2\bar{\sigma}\cdot\partial\chi$. Substituting this in the former equation gives and multiplying by $mi\sigma^2$ gives

$\sigma^2\bar{\sigma}^*\partial\sigma^2\bar{\sigma}\cdot\partial\chi +m^2\chi = 0$.

Are my steps correct, and how to finish the calculation?

Answer 1

Starting from your last equation, $$\sigma^2(\bar{\sigma}^\mu)^\ast \sigma^2\bar{\sigma}^\nu \partial_\mu \partial_\nu \chi + m^2 \chi=0, $$ use $\sigma^2 (\bar{\sigma}^\mu)^\ast \sigma^2= \sigma^\mu=(\mathbf{1},\vec{\sigma})$ (c.f. eqs. (3.38) and (3.41) in Peskin-Schroeder) and $\partial_\mu \partial_\nu \chi= \partial_\nu \partial_\mu \chi$, arriving at$$\frac{1}{2}(\sigma^\mu \bar{\sigma}^\nu +\sigma^\nu \bar{\sigma}^\mu)\partial_\mu \partial_\nu \chi +m^2 \chi=0.$$ Convince yourself that the relation $$ \sigma^\mu \bar{\sigma}^\nu+ \sigma^\nu \bar{\sigma}^\mu = 2 \eta^{\mu \nu} \mathbf{1}$$ holds, leading to the desired result $$(\square+m^2)\chi=0.$$