Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 301 / 4 / a

In the Weyl representation of the gamma matrices,A Lorentz transformation acts as ${\psi }^{\prime }(x^{\prime })=S(\Lambda )\psi (x)$ with $S(\Lambda )=\exp [-\frac{i}{4}{\omega }_{\mu \nu }{\sigma }^{\mu \nu }]$. In this basis a boost is block diagonal and gives $u_L\mapsto e^{\mathbf{\chi }\cdot \mathbf{\sigma }/2}{u}_L$ and $u_R\mapsto e^{-\mathbf{\chi }\cdot \mathbf{\sigma }/2}{u}_R$.

For boosts, the two exponentials cancel in $u_L^{†}{u}_R$; for rotations, the unitary rotation and its inverse cancel. Hence this bilinear is a Lorentz scalar. The Pauli matrices obey the identityshows that $i{\sigma }^2{u}_L^{\ast }$ acquires the right-handed boost matrix and the usual rotation matrix, so it is right-handed.