 Solution

Direct multiplication of the gamma matrices in the chiral representation givesFor example ${\sigma }^1{\sigma }^2{\sigma }^3=i{I}_2$ by the Pauli matrix multiplication law, and the block multiplications give the displayed signs. Every ${\gamma }^{\mu }$ is off-diagonal in this representation, whereas the chirality matrix has opposite diagonal blocks. Therefore ${\gamma }^5{\gamma }^{\mu }+{\gamma }^{\mu }{\gamma }^5=0$ for all $\mu$.

The left-handed chirality condition makes the plane-wave coefficient $\lambda (p)=\left(\genfrac{}{}{0ex}{}{\chi }{0}\right)$. With $p\cdot x=E{x}^0-\mathbf{p}\cdot \mathbf{x}$, the Dirac equation givesSince $(\mathbf{\sigma }\cdot \mathbf{p}{)}^2=|\mathbf{p}{|}^2{I}_2$, its determinant is $E^2-|\mathbf{p}{|}^2$. A nonzero spinor therefore exists exactly whenFor nonzero momentum, put $r=|\mathbf{p}|$ and write its direction as $(\sin \vartheta \cos \phi ,\sin \vartheta \sin \phi ,\cos \vartheta )$. Normalized eigenvectors of $\mathbf{\sigma }\cdot \hat{\mathbf{p}}$ areThe general nonzero solution is $\lambda (p)=C\left(\genfrac{}{}{0ex}{}{{\chi }_{-}}{0}\right)$ for $E=r$ and $\lambda (p)=C\left(\genfrac{}{}{0ex}{}{{\chi }_{+}}{0}\right)$ for $E=-r$, with arbitrary nonzero complex normalization $C$. At $p=0$ any constant two-component $\chi$ solves the equation, but the momentum direction and helicity are undefined. For the physical positive-energy branch this is the left-handed massless plane-wave spinor.

A rotation by angle $\alpha$ about $\hat{\mathbf{p}}$ acts on the two-component spinor asThe plane-wave argument is unchanged because the rotation fixes $\mathbf{p}$, and $S(R){\chi }_{-}=e^{i\alpha /2}{\chi }_{-}$. Thus the positive-energy solution has helicity $h=-1/2$, where a helicity-$h$ state has rotation phase $e^{-i\alpha h}$. The negative-energy coefficient has the opposite spinor phase.

In a quantized complex left-handed Weyl field, the positive-frequency term annihilates particles and the negative-frequency term creates antiparticles. Reinterpreting negative energy also reverses its spatial momentum; for the positive physical momentum the two spinor coefficients have the same left-chiral kernel. Because one multiplies an annihilation operator and the other a creation operator, the corresponding particle and antiparticle state phases are conjugate. The helicity content of a quantized Weyl field is consequentlyThere is one particle spin state at each nonzero momentum, rather than the two helicities supplied by a full massless Dirac field. The right-handed Weyl field interchanges these signs. This is the massless relation between chirality and helicity; it does not identify them for a massive field.