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?
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5$\begingroup$ 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 $P_+=\frac{(1-\gamma^5)}2$. $\endgroup$ – suresh Feb 7 '14 at 5:51
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1$\begingroup$ @suresh Write this as an answer :) $\endgroup$ – Neuneck Feb 7 '14 at 9:53
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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., $P_-=\frac{(1+\gamma^5)}2$.
This is in response to @neuneck's suggestion though I don't understand the motivation for it. :-)
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$\begingroup$ It's a full answer to the question, that's why! +1 $\endgroup$ – Neuneck Feb 7 '14 at 12:54
