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 Incoming links: Curved spacetime gamma matrices

e_{I}^{a}

of tangent vectors together with inverse coframe

e_{a}^{I}

, satisfying

g_{ab} e_{I}^{a} e_{J}^{b} = η_{I J}

and

g_{ab} = e_{a}^{I} e_{b}^{J} η_{I J}

. Capital indices label the locally orthonormal Lorentz frame; lower-case indices label spacetime coordinates. Changing the frame by a local Lorentz transformation leaves the metric unchanged. A spinor description uses a compatible local spin frame and the torsion-free spin connection.

Let fixed flat matrices satisfy

{Γ_{I}, Γ_{J}} = 2 η_{I J} 1

γ_{a} = e_{a}^{I} Γ_{I}

and

γ^{a} = e_{I}^{a} Γ^{I}

. Therefore

{γ_{a}, γ_{b}} = e_{a}^{I} e_{b}^{J} {Γ_{I}, Γ_{J}} = 2 g_{ab} 1 .

The identity matrix is four by four for a four-dimensional Dirac spinor. The combined tensor/spin covariant derivative makes

\nabla_{a} γ_{b} = 0

, the Clifford version of the tetrad postulate.

Fix the curvature slots by

R_{ab c d} = g_{a e} R^{e}_{b c d}

, with

[\nabla_{c}, \nabla_{d}] v^{a} = R^{a}_{b c d} v^{b}

, and

R = g^{a c} g^{b d} R_{ab c d}

. The torsion-free spin connection then has

[\nabla_{a}, \nabla_{b}] ψ = \frac{1}{4} R_{ab c d} γ^{c} γ^{d} ψ

. These choices fix the signs in the requested identities.

Write

γ^{ab} = γ^{[a} γ^{b]}

and similarly for four indices. Clifford anticommutation gives

γ^{a} γ^{b} γ^{c} γ^{d} = γ^{ab c d} + g^{ab} γ^{c d} - g^{a c} γ^{b d} + g^{a d} γ^{b c} + g^{b c} γ^{a d} - g^{b d} γ^{a c} + g^{c d} γ^{ab} + (g^{ab} g^{c d} - g^{a c} g^{b d} + g^{a d} g^{b c}) 1 .

On contraction with the Riemann curvature tensor, the four-form term vanishes by the first Bianchi identity. The first and last two-form terms vanish by pair antisymmetry; the remaining two-form terms are symmetric Ricci contractions against antisymmetric gamma matrices and vanish too. The scalar terms are zero, minus R and minus R respectively. Thus the Riemann tensor Clifford contraction is

R_{ab c d} γ^{a} γ^{b} γ^{c} γ^{d} = - 2 R 1 .

This algebraic result is independent of Lorentz signature when the contraction convention is held fixed.

Apply the Dirac operator twice, using covariant constancy of the gamma matrices. Define the rough wave operator by

□ = g^{ab} \nabla_{a} \nabla_{b}

, with the full connection acting on the intermediate covector-spinor derivative. Splitting the product into symmetric and antisymmetric parts gives

(γ^{a} \nabla_{a})^{2} ψ = □ ψ + \frac{1}{2} γ^{ab} [\nabla_{a}, \nabla_{b}] ψ = □ ψ + \frac{1}{8} R_{ab c d} γ^{a} γ^{b} γ^{c} γ^{d} ψ = (□ - R /4) ψ .

Consequently the Lichnerowicz spinor-square formula and the assumed massless Dirac equation imply

(□ - R /4) ψ = 0

. Torsion would modify the commutator and introduce extra terms, so the torsion-free connection is part of this argument.