 Spinor invariant in three spacetime dimensions

For ${\gamma }_0=-i{\sigma }_1$, ${\gamma }_1={\sigma }_2$ and ${\gamma }_2={\sigma }_3$, choose $r(K_1)={\sigma }_3/2$, $r(K_2)=-{\sigma }_2/2$ and $r(J)=-i{\sigma }_1/2$. These satisfy the three-dimensional Lorentz algebra brackets and $r(T{)}^{†}{\gamma }_0+{\gamma }_{0}r(T)=0$ for real $T$. Consequently $\delta \psi =r(T)\psi$ preserves the Dirac adjoint bilinear. Its matrix is anti-Hermitian in this signature convention, so $i{\psi }^{†}{\gamma }_0\psi$ is real for ordinary commuting components.