= Spinor invariant in three spacetime dimensions
{title2=$\delta(\psi^\dagger\gamma_0\psi)=0$}
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 $r(T)^\dagger\gamma_0+\gamma_0r(T)=0$ for real $T$. Consequently $\delta\psi=r(T)\psi$ preserves the <Dirac adjoint> bilinear. Its matrix is anti-Hermitian in this signature convention, so $i\psi^\dagger\gamma_0\psi$ is real for ordinary commuting components.
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