Past exam of the groups and representations course of the University of Oxford / 2018 / 3 / e / Solution

The spinor basis in c is compatible with the supplied gamma matrices. The rotation generator $r(J)$ is anti-Hermitian and commutes with ${\gamma }_0=-i{\sigma }_1$. The two boost generators are Hermitian matrices and anticommute with ${\gamma }_0$. It follows separately for all three generators, and hence for every real $T\in \mathfrak{so}(1,2)$, thatFor the infinitesimal Lorentz transformation, $\delta \psi =r(T)\psi$ and $\delta {\psi }^{†}={\psi }^{†}r(T{)}^{†}$. Hence the Dirac adjoint transforms contragrediently:The two contributions to the scalar variation cancel:This proves the spinor invariant in three spacetime dimensions has infinitesimal Lorentz invariance. Real transformation parameters matter because the Hermitian adjoint conjugates coefficients; this is not an invariance under arbitrary complexified parameters. For ordinary commuting spinor components, the bilinear is purely imaginary in this mostly-plus convention, and multiplying it by $i$ gives a real invariant. Its reality is separate from its invariance.