Dirac and Majorana spinors for neutrinos

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Dirac description
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 realize I am lost between antiparticle notation $\bar{\nu}$ and charge conjugate $\nu^{c}$
Can you help me ?

 

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[Avodyne]

Your questions cannot be answered without careful explanation of the notation. Are you using a particular text?

I recommend Srednicki's text (draft version available free from his web page) for a detailed explanation of Weyl, Dirac, and Majorana fields.

 

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Thank you, this text is very useful.
But I am still not sure to understand the difference between the fields component and the particle.
For example for the Dirac field where $\Psi_{D}=\Psi_{L}+\Psi_{R}$, the four components of the field $\Psi_{D}$ are $\nu_{L},\bar{\nu}_{R},\nu_{R},\bar{\nu}_{L}$ or is it something completely different ?
Thank you...

 

[samalkhaiat]

Dirac description
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

No, $\Psi_{L}$ and $\Psi_{R}$ are 4-component spinors just like $\Psi_{D}$. Why don’t you work it out yourself? In Dirac representation, you have $$\gamma_{5} = \left( \begin{array}{cc} 0_{2 \times 2} & I_{2 \times 2} \\ I_{2 \times 2} & 0_{2 \times 2} \end{array} \right) .$$ So, if you write $\Psi_{D} = \left( \chi , \phi \right)^{T}$, with $\chi = \left( \psi_{1} , \psi_{2} \right)^{T}$ and $\phi = \left( \psi_{3} , \psi_{4} \right)^{T}$, you find $$\Psi_{L} = \frac{1}{2} \left( \begin{array}{c} \chi - \phi \\ \phi - \chi \end{array} \right) , \ \ \ \Psi_{R} = \frac{1}{2} \left( \begin{array}{c} \chi + \phi \\ \phi + \chi \end{array} \right) .$$

corresponding to the particle and antiparticle

Wrong again, Dirac equation has 4 independent solutions, each (massive) solution is given by 4-component spinor. Two of these solutions (spin up: $u^{(1)} (p)$ & spin down: $u^{(2)} (p)$) represent a (massive) fermion and the other two solutions ( $v^{( 1 , 2 )} (p)$ ) describe the corresponding anti-fermion. The same is true for massless fermions except that the Dirac equation splits into two decoupled equations for the 2-component spinors $\chi$ and $\phi$ : $$E \chi = - ( \sigma \cdot p ) \chi ,$$ $$E \phi = ( \sigma \cdot p ) \phi .$$ Each one of these equations has 2 independent solutions: one (2-spinor) for $E = | p |$ ( $\nu_{L}$, if you like), and the other 2-spinor solution is for $E = - | p |$ ( i.e., $\bar{\nu}_{R}$ ) . So, when we say that $\chi$ describes $\nu_{L}$ and $\bar{\nu}_{R}$, this DOSE NOT mean that $\nu_{L}$ is the first component of $\chi$ and $\bar{\nu}_{R}$ is the second component. No, $\nu_{L}$ and $\bar{\nu}_{R}$ are 2 independent solutions of the SAME equation and each (massless particle) is described by 2-component spinor. Indeed, in the Weyl (chiral) representation, you find $\Psi_{L} = ( \chi , 0 )^{T}$ and $\Psi_{R} = ( 0 , \phi )^{T}$.

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

What does it mean to pair together 2 independent solutions of the same equation? Is $\Psi_{D} ( p ) = \left( u ( p ) , v ( p ) \right)$? Does the object $( e^{-} , e^{+} )$ make any mathematical sense? No, $\Psi_{D} = u^{(1)} , u^{(2)} , v^{(1)} , v^{(2)}$, these are four independent 4-component spinors solutions of the Dirac equation.

Sam