= Solution
For the stated <matrix exponential> parametrization in the connected <Proper orthochronous Lorentz group>, use the corresponding <Spinor representation of the Lorentz group>:
$$
\boxed{\psi'_\alpha(x)=D(\Lambda)_\alpha{}^\beta\psi_\beta(\Lambda^{-1}x),\qquad D(\Lambda)=\exp\!\left(\frac12\Omega_{\rho\sigma}S^{\rho\sigma}\right).}
$$
The spacetime argument is inverse-transformed because this is an active transformation of the field at a fixed coordinate $x$. Equivalently, with $x'=\Lambda x$, $\psi'(x')=D(\Lambda)\psi(x)$. The identity $[S^{\mu\nu},\gamma^\rho]=\eta^{\nu\rho}\gamma^\mu-\eta^{\mu\rho}\gamma^\nu$ integrates to $D^{-1}\gamma^\mu D=\Lambda^\mu{}_\nu\gamma^\nu$, which ensures covariance of the <Dirac equation>.
Strictly, $D$ depends on a lift to the <Spin group>, not just on the final <Lorentz transformation>: two lifts differ by sign. The specified generator exponential chooses a lift along its continuous path from the identity. This qualification is essential for the full-rotation comparison in part (g).
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