I know that
1) Majorana fermions are the antiparticals of its own and the Dirac particals are not anti particals of their own
2) Weyl particals are the half charged Dirac particals.
But how can i understand these concepts using DFT methods
Dirac fermions couple directly to the electromagnetic field, Majorana fermions don't. Weyl fermions don't carry half the charge of Dirac fermions-that's wrong. They have definite chirality, or helicity, that's all. So they're Dirac fermions, appropriately projected, using chiral projectors.
(Under certain conditions Majorana fermions can have definite chirality, i.e., also, be Weyl fermions.)
Fermions can be described through the Dirac, Weyl or Majorana representations. According which representation you chose, you are able to study different properties of fermions. For instance if you chose the Dirac representation of electrons, you will be able to describe electromagnetic properties of them. Instead of this, if you chose the Weyl representation, you will be able to describe weak interaction of electrons. So you can write down the Lagrangian density in the representation you want and you can quantify with the method you like.
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