 Solution

The two Weyl spinors transform in the inequivalent left- and right-chiral representations of the proper Lorentz group. They are selected by $P_{L,R}=(1\mp {\gamma }^5)/2$ and have different boost transformation laws; parity exchanges the two. A massive Dirac equation couples them, whereas in the massless limit they can propagate independently.

For the particular $u$ printed in this paper, transforming its supplied Dirac ${\gamma }^5$ givesThus the upper block is right chiral and the lower block left chiral in this convention. Many references use the opposite block order; neither convention changes physics, but they must not be mixed. Chirality is not generally the same as helicity for a massive electron. The weak charged current selects the left-chiral electron component, reducing to negative helicity at high energy.