Gauge transformation of Dirac equation in curved spacetime
For a research project, I tried to explicitly show the invariance of the Dirac equation in curved space time under $\operatorname{Spin}(1,3) \cong SL(2,\mathbb C)$-gauge transformations.
I'm working with the spin-connection $\omega_{\mu}{}^{I}{}_{J}$ in the $\mathfrak{so}(1,3)$-matrix representation. Let's focus on a single Weyl spinor $\psi \in \mathbb C^2$. The equation I used for the gauge covariant derivative was
$$D \psi = \partial_\mu \psi - \omega_\mu{}^I{}_K \frac{1}{4} \eta^{KJ} \sigma_{[I} \bar \sigma_{J]} \psi$$
where $\sigma_I$ and $\bar \sigma_J$ come from the Dirac gamma matrices $$ \gamma^I= \begin{pmatrix} 0 & \sigma ^I\\ \bar \sigma^I&0 \end{pmatrix} $$
For the gauge transformations with $\mathcal M: \to SL(2,\mathbb C)$ I used $$ \tilde \psi(x) = S(x) \psi(x)$$ $$ \omega_\mu{}^{\tilde I}{}_{\tilde J}(x) = \Lambda^{\tilde I}{}_{I}[S(x)] \Lambda^{-1}{}^{ J}{}_{\tilde J}[S(x)] \omega_\mu{}^{I}{}_{J}(x) + \Lambda^{\tilde I}{}_{K}[S(x)] \partial_\mu \Lambda^{-1}{}^{ J}{}_{\tilde J}[S(x)]\,,$$ where the Lorentz transformation is implicitly defined by $$ \Lambda^{\tilde I}{}_{I}[S] \sigma_{\tilde I} = S \sigma_I S^\dagger\,$$ $$\Lambda^{\tilde I}{}_I[S] \bar \sigma_{\tilde I} = S^\dagger{}^{-1} \bar \sigma_I S^{-1}\,\tag{*}$$
Failed proof
I tried to prove the gauge invariance as follows:
\begin{align} \tilde D_\mu S \psi &= \partial_\mu (S \psi ) - \omega_\mu{}^{\tilde I}{}_{\tilde K} \frac{1}{4} \eta^{\tilde K \tilde J} \sigma_{[\tilde I}\bar \sigma_{\tilde J]} S \psi \nonumber\\ &= \partial_\mu (S \psi) - \Lambda^{\tilde I}{}_{I} \Lambda^{-1}{}^{ K}{}_{\tilde K} \omega_\mu{}^{I}{}_{K} \frac{1}{4} \eta^{\tilde K \tilde J} \sigma_{[\tilde I}\bar \sigma_{\tilde J]} S\psi % \nonumber\\ &- \Lambda^{\tilde I}{}_{L} \partial_\mu \Lambda^{-1}{}^{ L}{}_{\tilde K} \frac{1}{4} \eta^{\tilde K \tilde J} \sigma_{[\tilde I}\bar \sigma_{\tilde J]} S \psi\nonumber\\ &= (\partial_\mu S) \psi + S (\partial_\mu \psi)- \Lambda^{\tilde I}{}_{I} \eta^{KJ} \Lambda^{ \tilde J}{}_{J} \omega_\mu{}^{I}{}_{K} \frac{1}{4} \sigma_{[\tilde I}\bar \sigma_{\tilde J]} S\psi % \nonumber\\ &+ (\partial_\mu \Lambda^{\tilde I}{}_{L}) \eta^{LJ} \Lambda^{ \tilde J }{}_{J} \frac{1}{4} \sigma_{[\tilde I}\bar \sigma_{\tilde J]} S \psi\nonumber\\ &= (\partial_\mu S) \psi + S (\partial_\mu \psi)\nonumber\\ &- \eta^{KJ} \omega_\mu{}^{I}{}_{K} \frac{1}{8} \Lambda^{\tilde I}{}_{I} \sigma_{\tilde I} \Lambda^{ \tilde J}{}_{J}\bar \sigma_{\tilde J} S\psi \nonumber\\ % &+ \eta^{KJ} \omega_\mu{}^{I}{}_{K} \frac{1}{8} \Lambda^{ \tilde J}{}_{J} \sigma_{\tilde J} \Lambda^{\tilde I}{}_{I}\bar \sigma_{\tilde I} S\psi \nonumber\\ % % &+ \eta^{LJ} \frac{1}{8} \partial_\mu ( \Lambda^{\tilde I}{}_{L} \sigma_{\tilde I}) \Lambda^{ \tilde J }{}_{J}\bar \sigma_{\tilde J} S \psi\nonumber\\ &- \eta^{LJ} \frac{1}{8} \Lambda^{ \tilde J }{}_{J} \sigma_{\tilde J} \partial_\mu (\Lambda^{\tilde I}{}_{L}\bar\sigma_{\tilde I}) S\psi\nonumber\\ \end{align} Using $(*)$ I simplified to \begin{align} \tilde D_\mu S \psi &= (\partial_\mu S) \psi + S (\partial_\mu \psi)\nonumber\\ &- \eta^{KJ} \omega_\mu{}^{I}{}_{K} \frac{1}{4} S \sigma_{[ I} \bar \sigma_{ J]} \psi \nonumber\\ % % &+ \eta^{LJ} \frac{1}{8} \partial_\mu ( S \sigma_{L} S^\dagger) S^\dagger{}^{-1} \bar \sigma_{ J} S^{-1} S \psi\nonumber\\ &- \eta^{LJ} \frac{1}{8} S\sigma_{ J} S^\dagger \partial_\mu ( S^\dagger{}^ {-1} \bar \sigma_{L} S^{-1}) S\psi\nonumber\\ % % &= (\partial_\mu S) \psi+ S D_\mu \psi\nonumber\\ &+ \eta^{LJ} \frac{1}{8} (\partial_\mu S) \sigma_{L} S^\dagger S^\dagger{}^{-1} \bar \sigma_{ J} S^{-1} S \psi\nonumber\\ &+ \eta^{LJ} \frac{1}{8} S \sigma_{L}( \partial_\mu S^\dagger) S^\dagger{}^{-1} \bar \sigma_{ J} S^{-1} S \psi\nonumber\\ &- \eta^{LJ} \frac{1}{8} S\sigma_{ L} S^\dagger ( \partial_\mu S^\dagger{}^ {-1}) \bar \sigma_{J} S^{-1} S\psi\nonumber\\ &- \eta^{LJ} \frac{1}{8} S\sigma_{ J} S^\dagger S^\dagger{}^ {-1} \bar \sigma_{L} (\partial_\mu S^{-1}) S\psi\nonumber\\ &= (\partial_\mu S) \psi+ S D_\mu \psi\nonumber\\ &+ \eta^{LJ} \frac{1}{8} (\partial_\mu S) \sigma_{L} \bar \sigma_{ J} \psi\nonumber\\ &+ \eta^{LJ} \frac{1}{8} S\sigma_{ J} \bar \sigma_{L} S^{-1} (\partial_\mu S)\psi\,.\tag{**} \end{align} Then I used $\eta^{LJ} \sigma_{ J} \bar \sigma_{L} = - 4 \mathbb I$ to obtain \begin{align} \label{eq:stament_for_psi} \tilde D_\mu S \psi &= (\partial_\mu S) \psi+ S D_\mu \psi\nonumber\\ &- \frac{1}{2} (\partial_\mu S \psi)\nonumber\\ &- \frac{1}{2} (\partial_\mu S\psi)\nonumber\\ &=S D_\mu \psi\,. \end{align}
Unfortunately, I realized that I did a sign mistake when deriving the last line of $(**)$. Therefore, the holewhole calculation seems to fail.
Does anyone know, how to modify the Weyl equation in curved spacetime such that the transformation behaviour of $D\psi$ can be verified. Note, that I am interested in non-infinitesimal gauge transformations.
I looked much into the literature, but I couldn't find any solution.
Convention: I'm using $(-+++)$-signature following Wess and Bagger
Remark: I don't think this question is a duplicate. I am not asking just for the Dirac equation in curved spacetime. What I am looking for is a definition of the spin-covariant derivative consistent with the gauge transformations of the spin-connection.
