Past exam of the groups and representations course of the University of Oxford / 2025 / 3 / b / Solution

Choose the chirality operatorA Weyl spinor is a Dirac spinor restricted to one eigenspace of $\Gamma$. The chiral projectors are $P_R=(I+\Gamma )/2$ and $P_L=(I-\Gamma )/2$. ConsequentlyThe two independent Weyl spinors each contain one complex component. Since $[S(\theta ),\Gamma ]=0$, each chirality is preserved by every connected Lorentz transformation, with ${\psi }_R\mapsto e^{\theta /2}{\psi }_R$ and ${\psi }_L\mapsto e^{-\theta /2}{\psi }_L$. A spatial reflection exchanges the two chiralities.

The left/right names depend on the sign convention for $\Gamma$. Choosing ${\gamma }_0{\gamma }_1=-{\sigma }_3$ as the chirality operator exchanges the labels $L$ and $R$; the two invariant one-component representations are the same.