# Why do or don't neutrinos have antiparticles?

This was inspired by this question. According to Wikipedia, a Majorana neutrino must be its own antiparticle, while a Dirac neutrino cannot be its own antiparticle. Why is this true?

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Because the spinor of the Majorana neutrino is an eigenstate of the charge conjugation operator. This is different from the case of a Dirac spinor that will change under the effect of the same operator.

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But why can't a Dirac spinor be an eigenstate of the charge conjugation operator? This is only a partial answer. –  Peter Shor Dec 2 '11 at 15:56
This depends on the way the Dirac equation is formulated. In order to produce a Majorana spinor, you need to perform an unitary transformation on the $\gamma$s matrices of the Dirac equation. This will change the behavior of your solution under charge conjugation, that is given by the $\gamma_2$ matrix. –  Jon Dec 2 '11 at 16:00

In crude terms I think it amounts to the following:

Consider, for instance, a (Dirac) fermion creation operator: $c_j^\dagger$. A Majorana fermion is somehow the "real" Part of a Dirac fermion:

$m_j = c_j + c_j^\dagger$

(conventions on normalization differs). Hence a Majorana fermion transforms into itself under charge conjugation.

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This is not good as an answer either, because you can then take a pair of Majorana spinors and make a Dirac spinor out of them, and the charge conjugation eigenstates are combinations of the two. There are many different charge conjugations possible in noninteracting theories, the interesting restrictions come when when you make particles charged (hence the name "charge conjugation"). I have a hard time answering this question, because I find the idea of Dirac neutrinos laughably absurd, it is so obvious that neutrinos are Majorana. Also in (3+1)d, Majorana equals Weyl. –  Ron Maimon Dec 9 '11 at 5:37
@Ron yes but as you said you need two Majorana fermions to make a Dirac one –  lcv Dec 12 '11 at 20:05