In many questions like What is chirality?, Confusion with chirality eigenstates and What is the relation between the Higgs field and chirality?; and also here http://www.quantumdiaries.org/2011/06/19/helicity-chirality-mass-and-the-higgs/ there is this explanation of chirality and fermions:
The (4-component) Dirac spinor $\psi$ is in the $(1/2,0)+(0,1/2)$ representation of the Lorentz Group where $(1/2,0)$ represents a Left-Handed (2-component) Weyl Spinor and $(0,1/2)$ a Right-Handed (2-component) Weyl Spinor. In those links they say that components with diferent chirality should be interpretated as different particles. So instead of left and right electron and left and right positron, they talk of left electron, right anti-positron and their respective antiparticles right anti-electron and left positron.
On the other hand, a Dirac Spinor in the chiral representation is $\psi =( \begin{smallmatrix} \phi_L \\ \phi_R \end{smallmatrix})$ and its conjugate is $\psi^{\dagger}=(\phi^{\dagger}_L\ \ \phi^{\dagger}_R)$. My question is: which of this components is each tipe of particle? I'm also confused with the mass term $\phi^{\dagger}_L\phi_R + \phi^{\dagger}_R\phi_L$ in this new interpretation.
