# Spinor and vector representation matrices commutation relation

To show Lorentz invariance of Dirac equation P&S section 3.2 swap $$\Lambda$$ and $$S(\Lambda)$$ as both matrices commute. But why is it true? For example taking $${\cal J}^{01}=\left(\begin{matrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\\ \end{matrix}\right)\,,$$ and $$S^{03}=\left(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1\\ \end{matrix}\right)\,,$$ $$[{\cal J}^{01},S^{03}]\neq0$$. What am I missing?

$$({\cal J}^{\mu\nu})^{\rho}{}_{\sigma}$$ acts in a 4-vector representation of the Lorentz group while $$(S^{\mu\nu})^a{}_b$$ acts in the Dirac spinor representation, i.e. they don't live in the same representation despite both happen to be given by $$4\times4$$ matrices.
• $\uparrow$ Yes. – Qmechanic Feb 18 at 13:37