Past exam of the particle physics course of the University of Oxford / 2017 / 1 / c / Solution

In the supplied Dirac representation, ${\gamma }^5$ exchanges the upper and lower blocks. ThereforeDefine $\eta =[I-\mathbf{\sigma }\cdot \mathbf{p}/(E+m)]\chi$. Acting with this projector givesFor completeness, its two independent components areThe equal-and-opposite block structure verifies ${\gamma }^5{\psi }_L=-{\psi }_L$. Projection isolates a chiral component of the massive Dirac solution; that component alone generally does not satisfy an independent massive Dirac equation, since the mass couples both chiralities.