I know, from my professor notes, that a general field operator can transform under Lorentz as $$U(\Lambda)\hat{\mathcal{O}}^r(x)U^\dagger(\Lambda)=\hat{\cal{O}}^{r'} {M(\Lambda)_{r'}}^r$$ where $M(\Lambda_1\Lambda_2)=M(\Lambda_1)M(\Lambda_2)$. Where does this relation come from?

Then I have to show that $$U(\Lambda)\hat{\tau}_{\mu\nu}U^\dagger(\Lambda)=\hat{\tau}_{\mu'\nu'} {\Lambda^{\mu'}}_{\mu}{\Lambda^{\nu'}}_\nu$$ namely, this transforms as a tensor.

I started at the infinitesimal level where $${\Lambda^\mu}_\nu=\delta^\mu_\nu-\omega^\mu_\nu$$ and $$U(\Lambda)=\mathbf{1}-\frac{1}{2}i\omega_{\mu\nu} \hat{M}^{\mu\nu}$$ I don't know how to proceed with this calculation. I tried substituting these two relation for the infinitesimal level but I can't reach any good result. Thanks for your help.

I know, from my professor notes, that a general field operator can transform under Lorentz as $$U(\Lambda)\hat{\mathcal{O}}^r(x)U^\dagger(\Lambda)=\hat{\cal{O}}^{r'} {M(\Lambda)_{r'}}^r$$ where $M(\Lambda_1\Lambda_2)=M(\Lambda_1)M(\Lambda_2)$. Where does this relation come from?

Then I have to show that $$U(\Lambda)\hat{\tau}_{\mu\nu}U^\dagger(\Lambda)=\hat{\tau}_{\mu'\nu'} {\Lambda^{\mu'}}_{\mu}{\Lambda^{\nu'}}_\nu$$ namely, this transforms as a tensor.

I started at the infinitesimal level where $${\Lambda^\mu}_\nu=\delta^\mu_\nu-\omega^\mu_\nu$$ and $$U(\Lambda)=\mathbf{1}-\frac{1}{2}i\omega_{\mu\nu} \hat{M}^{\mu\nu}$$ I don't know how to proceed with this calculation. I tried substituting these two relation for the infinitesimal level but I can't reach any good result.

I know, from my professor notes, that a general field operator can transform under Lorentz as $$U(\Lambda)\hat{\mathcal{O}}^r(x)U^\dagger(\Lambda)=\hat{\cal{O}}^{r'} {M(\Lambda)_{r'}}^r$$ where $M(\Lambda_1\Lambda_2)=M(\Lambda_1)M(\Lambda_2)$. From where does this relation come from?

Then I have to show that $$U(\Lambda)\hat{\tau}_{\mu\nu}U^\dagger(\Lambda)=\hat{\tau}_{\mu'\nu'} {\Lambda^{\mu'}}_{\mu}{\Lambda^{\nu'}}_\nu$$ namely, this transforms as a tensor.

I started at the infinitesimal level where $${\Lambda^\mu}_\nu=\delta^\mu_\nu-\omega^\mu_\nu$$ and $$U(\Lambda)=\mathbf{1}-\frac{1}{2}i\omega_{\mu\nu} \hat{M}^{\mu\nu}$$ I don't know how to proceed with this calculation. I tried substituting these two relation for the infinitesimal level but I can't reach any good result. Thanks for your help.

I know, from my professor notes, that a general field operator can transform under Lorentz as $$U(\Lambda)\hat{\mathcal{O}}^r(x)U^\dagger(\Lambda)=\hat{\cal{O}}^{r'} {M(\Lambda)_{r'}}^r$$ where $M(\Lambda_1\Lambda_2)=M(\Lambda_1)M(\Lambda_2)$. Where does this relation come from?

Then I have to show that $$U(\Lambda)\hat{\tau}_{\mu\nu}U^\dagger(\Lambda)=\hat{\tau}_{\mu'\nu'} {\Lambda^{\mu'}}_{\mu}{\Lambda^{\nu'}}_\nu$$ namely, this transforms as a tensor.

I started at the infinitesimal level where $${\Lambda^\mu}_\nu=\delta^\mu_\nu-\omega^\mu_\nu$$ and $$U(\Lambda)=\mathbf{1}-\frac{1}{2}i\omega_{\mu\nu} \hat{M}^{\mu\nu}$$ I don't know how to proceed with this calculation. I tried substituting these two relation for the infinitesimal level but I can't reach any good result. Thanks for your help.