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If I well understood a Dirac description for fermions is :

##\Psi_{D}=\Psi_{L}+\Psi_{R}## where ##\Psi_{L}## is the left-chiral spinor and ##\Psi_{R}## the right-chiral spinor.

Each spinor, ##\Psi_{L} ## and ##\Psi_{R}## has 2 components cotrresponding to the particle and antiparticle :

Q1 : Can we write ##\Psi_{L}=(\nu_{L},\bar{\nu}_{R}) ##? and ##\Psi_{R}=(\nu_{R},\bar{\nu}_{L})## ?

Majorana description

The Majorana condition is ##\Psi_{L}=\Psi_{L}^{c}## and ##\Psi_{R}=\Psi_{R}^{c}##.

Q2: is it right ?

If yes, ##\Psi_{M}=\Psi_{L}+\Psi_{R}## can be written as ##\Psi_{M}=\Psi_{L}+\Psi_{R}^{c}=\Psi_{L}+(\Psi_{L})^{c}##

with ##\Psi_{L}=(\nu_{L},\bar{\nu}_{R}) ## and ##(\Psi_{L})^{c}=((\nu_{L})^{c},(\bar{\nu}_{R})^{c}) =(\nu_{R}^{c},\bar{\nu}_{L}^{c})=(\nu_{R},\bar{\nu}_{L})##

So the Majorana field describes the 4 states of the neutrino (##\nu_{L},\bar{\nu}_{R},\nu_{R},\bar{\nu}_{L}##)

Q3: In such notation what is the difference between ##\nu_{L}^{c}## and ##\bar{\nu}_{L} ## ? Majorana condition ##\nu_{L}^{c} = \nu_{L}## can be also written as : ##\bar{\nu}_{L} = \nu_{L}## ???

I realise I am lost between antiparticle notation ##\bar{\nu}## and charge conjugate ##\nu^{c}##

Can you help me ?

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# Dirac and Majorana spinors for neutrinos

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