= Lichnerowicz spinor-square formula
{c}
{title2=$\not\nabla^{\,2}=\nabla^2-R/4$}
With $\{\gamma^\mu,\gamma^\nu\}=2g^{\mu\nu}$ and <spin> <curvature> $[\nabla_\mu,\nabla_\nu]=R_{\mu\nu ab}\gamma^{ab}/4$, the <Dirac equation> operator squares to the <rough Laplacian> minus $R/4$. In the <curvature> contraction the four-gamma term vanishes by the <first Bianchi identity>, the two-gamma terms vanish by Ricci symmetry, and the scalar term is $-2R$. In Riemannian signature $i\not\nabla$ is the <self-adjoint> convention and its square is $-\nabla^2+R/4$.
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