- #1
diracsgrandgrandson
- 6
- 1
- Homework Statement
- Peskin and Schroeder Exercise 3.4 Majorana Fermions part b wants me to show that the variation of the action ##S## with respect to ##\chi^\dagger## gives the Majorana equation.
- Relevant Equations
- The action is given by: $$S = \int d^4 x \left[ \chi^\dagger i \sigma \cdot \partial \chi + \frac{im}{2} \left( \chi^T \sigma^2 \chi - \chi^\dagger \sigma^2 \chi^* \right) \right]$$
The Majorana equation is $$i \bar{\sigma} \cdot \partial \chi - im \sigma^2 \chi^* = 0$$
I am stuck at the final part where one is supposed to show that the derivative of the second term of the action gives the mass term in the Majorana equation. For $$\chi^T\sigma^2\chi = -(\chi^\dagger\sigma^2\chi^*)^*$$ we get $$\frac{\delta}{\delta\chi^\dagger}(\chi^\dagger\sigma^2\chi^*)^*$$ which is supposed to give $$\sigma^2\chi^*.$$ I don't see how. Suppose $$f(\chi) = \chi^*,$$ and now $$\frac{d}{d\chi}f(\chi) = \frac{d\chi^*}{d\chi}$$ which would be zero due to the field and its complex conjugate being zero.