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

All gamma matrices in this basis are real, and the Lorentz transformation $S(\theta )$ is real. Therefore complex conjugation commutes with the connected Spinor representation of the Lorentz group. With a suitable phase convention, charge conjugation is simply ${\psi }^c={\psi }^{\ast }$. The Majorana spinor condition ${\psi }^c=\psi$ becomesThe reality constraint is preserved by Lorentz transformations because $(S{\psi }_M{)}^{\ast }=S{\psi }_M$. For fermion fields, real components mean self-conjugate Grassmann variables or self-adjoint field components, rather than ordinary commuting real numbers.

Moreover, $\Gamma ={\sigma }_3$ is real, so charge conjugation commutes with the chiral projectors. The Majorana spinor and Weyl spinor conditions can therefore be imposed simultaneously. The Majorana-Weyl spinors areThe independent real component counts areThese are component counts before imposing an equation of motion. In particular the simultaneous reality and chirality constraints are compatible in this spacetime dimension, unlike the familiar four-dimensional case.