# Massless neutrinos and Chirality

The massless neutrinos can be represented by two component Weyl spinors. Then how does one say that it is an eigenstate of the chirality operator $\gamma^5$, which is a $4\times 4$ matrix and can act on $4\times 1$ column?

• Simply think of a Weyl Spinor as a Dirac spinor where the other two components are set to zero. Equivalently, a Weyl spinor (of chirality +1 belongs to the two-dimensional subspace with eigenvalue 1 under the action of the projection operator $P_+=\frac{(1-\gamma^5)}2$. Feb 7, 2014 at 5:51
• @suresh Write this as an answer :) Feb 7, 2014 at 9:53

Simply think of a Weyl Spinor as a Dirac spinor where the other two components are set to zero. Equivalently, a Weyl spinor (of chirality +1) belongs to the two-dimensional subspace with eigenvalue 1 under the action of the projection operator $P_+=\frac{(1-\gamma^5)}2$. For negative chirality, use the other projection operator, i.e., $P_-=\frac{(1+\gamma^5)}2$.