# What's the difference between Lorentz transformation properties of Hermitian and Dirac adjoint lepton doublets?

If the lepton doublet transforms like a left-handed Weyl spinor under Lorentz transformations, $$L \longrightarrow exp\left[\frac{1}{2}(i\theta_j\sigma^j - \beta_j\sigma^j)\right]L = \Lambda_{sL}L,$$ then does the Hermitian adjoint transform like $$L^\dagger \longrightarrow (\Lambda_{sL}L)^\dagger = L^\dagger\Lambda_{sL}^\dagger = L^\dagger exp\left[-\frac{1}{2}(i\theta_j\sigma^j + \beta_j\sigma^j)\right] = L^\dagger\Lambda_{sR}^{-1}?$$ How does the Dirac adjoint lepton then transform? I've tried $$\overline{L} \longrightarrow (\Lambda_{sL}L)^\dagger\gamma^0 = L^\dagger\Lambda_{sR}^{-1}\gamma^0.$$ $$\Lambda_{sR}^{-1}$$ and $$\gamma^0$$ aren't the same dimensions, so do they commute? Do I then get $$\overline{L} \longrightarrow \overline{L}\Lambda_{sR}^{-1},$$ such that theres no difference between the transformation properties of Hermitian adjoint and Dirac adjoint lepton doublets?

$$(\Lambda)^\dagger \neq \Lambda^{-1}$$ instead $$\gamma^0(\Lambda)^\dagger \gamma^0 = \Lambda^{-1}$$
• Isn't that only true for the whole $\Lambda_s = exp[i\theta_{\mu\nu}S^{\mu\nu}]$? Commented Mar 14, 2023 at 15:09