# Lagrangian density for Majorana particles

The Majorana fermion satisfies the equation $$(i\tilde\gamma^\mu\partial_\mu -m)\psi=0,$$ where the matrices $\tilde\gamma^\mu$ satisfy the standard Dirac relations $$\{\tilde\gamma^\mu,\tilde\gamma^\nu\}=2\eta^{\mu\nu},$$ but they are purely imaginary, in the contrary to the usual Dirac matrices.

Question. How to get the Majorana equation from a Lagrangian density?

Actually in Execise 3.4 to Ch. 3 in the book "An introduction to QFT" by Peskin and Schroeder it is mentioned that it is possible to construct such a Lagrangian density if one considers the field $\psi$ taking values in Grassmann numbers. I am wondering whether Grassmann numbers are really necessary here and whether there exists a more standard approach to get the equations from a Lagrangian (like in the case of the usual Dirac equation).

• see Lagrangian and grassmann numbers (also, Srednicki discusses the Majorana Lagrangian in chapter 36) – AccidentalFourierTransform Jan 21 '17 at 12:25
• The equation You wrote is not Majorana equation, it's just the Dirac equation written in particular basis of Dirac matrices (the so-called Majorana basis). – Name YYY Jan 21 '17 at 17:33
• @NameYYY Majorana fermions are (always?) taken to satisfy the Majorana condition $\psi=\psi^c$, in which case the Majorana equation becomes identical to the Dirac equation. (this is similar to the case of the Klein-Gordon field: the real and complex fields satisfy the same equation). – AccidentalFourierTransform Jan 21 '17 at 18:18