= Torsion correction to the Dirac square
{title2=$\not\nabla_T^2=\nabla_T^2-\tfrac12\gamma^{\mu\nu}T^\rho{}_{\mu\nu}\nabla^T_\rho+\tfrac18\gamma^{\mu\nu}R^T_{\mu\nu ab}\gamma^{ab}$}
For a <metric-compatible connection> with <torsion tensor> $T^\rho{}_{\mu\nu}=\Gamma^\rho_{\mu\nu}-\Gamma^\rho_{\nu\mu}$, expand the square using Clifford multiplication and the full affine second derivative. Its antisymmetric part is spin curvature minus $T^\rho{}_{\mu\nu}\nabla_\rho$. This gives the displayed formula. The curvature need not have the torsion-free pair symmetry or <first Bianchi identity>, so its Clifford contraction need not reduce to a scalar $-R/4$. General torsion produces additional first-order and nonscalar curvature couplings.
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