quantum field theory - Dirac Equation in General Relativity

Dirac equation for the massless fermions in curved spase time is $γ^ae^μ_aD_μΨ=0$, where $e^μ_a$ are the tetrads. I have to show that Dirac spinors obey the following equation: $$(−D_μD^μ+\frac{1}{4}R)Ψ=0\qquad(1)$$

where $R$ is the Ricci scalar.

I already know that $[D_\mu,D_\nu]A^\rho={{R_{\mu\nu}}^\rho}_\sigma A^\sigma$, but a key point is to know what $[D_\mu,D_\nu]\Psi$ is.

($D_μΨ=∂_μΨ+A^{ab}_μΣ_{ab}$ is the covariant derivative of the spinor field and $Σ_{ab}$ the Lorentz generators involving gamma matrices).

Answer

Denoting by $\gamma^a$ the Minkowski space gamma matrices with respect to the Lorentz tetrad $\{e^a\}$, and covariant derivative $D_a$, then the gammas are covariantly constant.

Start with the massless Dirac equation $$\gamma^{b}D_{b}\Psi = 0$$

Act again with the Dirac operator $$\gamma^{a}D_{a}\gamma^{b}D_{b}\Psi=0$$ So, since $D$ annihilates $\gamma$ $$\gamma^{a}\gamma^{b}D_{a}D_{b}\Psi = 0$$ so $$\frac{1}{2}\{\gamma^{a},\gamma^{b}\}D_{a}D_{b}\Psi + \frac{1}{2}\gamma^{a}\gamma^{b}[D_a,D_b]\Psi = 0 \ \ (1)$$ But $$\{\gamma^{a},\gamma^{b}\}=2\eta^{ab}$$ and $$[D_a,D_b]\Psi = {\mathcal{R}_{ab}}\Psi$$ Where ${\mathcal{R}}_{ab}$ is the spin-curvature (antisymmetric in a and b). ${\mathcal{R}}_{ab}$ satisfies the identity $$-\gamma^b{\mathcal{R}}_{ab} = {\mathcal{R}}_{ab}\gamma^b = \frac{1}{2}\gamma^b R_{ab}$$ where $R_{ab}$ is the Ricci tensor (in the Lorentz tetrad). so (1) becomes $$[D^aD_a+\frac{1}{4}\gamma^a\gamma^bR_{ab}]\Psi = 0$$ i.e. $$[D^aD_a-\frac{1}{4}R]\Psi = 0$$