# Axiom approach for majorana fermions

This is the usual way of introducing majorana operators. First we have $N$ fermionic modes. The corresponding operators satisfy the commutation relations

$$\{c_i, c_j \}= \{c_i^\dagger, c_j^\dagger \} =0,$$

$$\{\ c_i , c_j^\dagger \}=\delta_{ij} .$$

Then we introduce the $2N$ majorana operators

$$\gamma_{2j - 1} = c_j + c_j^\dagger,$$

$$\gamma_{2j} = -i (c_j - c_j^\dagger) .$$

These operators satisfy the conditions

$$\gamma_l \gamma_m + \gamma_m \gamma_l = 0, \quad l \neq m ,$$

$$\gamma_l \gamma_l = 1 .$$

The question is, can we start directly from these relations and derive the consequence of the algebra? In this way, the number of operators can be odd too.

• You are just splitting the complex Dirac representation of the Lorentz algebra into real subrepresentations here, so you can as well take only one of these subrepresentations initially. An odd number of Majoranas is for example used in the seesaw mechanism with right-handed sterile neutrinos. – ACuriousMind Feb 10 '15 at 14:01
• Also, related regarding the difference between Weyl, Majorana and Dirac fermions: physics.stackexchange.com/q/103005/50583 – ACuriousMind Feb 10 '15 at 14:02