 Solution

It is essential to distinguish a finite-dimensional spinor matrix from its action on a family of operator components. Take the Pauli matrices and ${\bar{\sigma }}^{\mu \nu }$ in the supplied convention, with $\eta =\mathrm{diag}(1,-1,-1,-1)$. They giveIf both displayed transformations are read literally as left multiplication on the column ${\bar{Q}}^{\dot{\alpha }}$, expanding the operator conjugation and comparing its linear term gives the formal answer $[M^{\mu \nu },\bar{Q}]=+{\bar{\sigma }}^{\mu \nu }\bar{Q}$.

There is a sign inconsistency in that literal reading. With the usual angular momentum commutation relations, coefficient matrices $R_i$ in $[J_i,q_a]=(R_i{)}_{ab}{q}_b$ must satisfyIndeed, the Jacobi identity gives $[J_i,[J_j,q]]-[J_j,[J_i,q]]=(R_j{R}_i-R_i{R}_j)q$, whereas $[[J_i,J_j],q]=i{ϵ}_{ijk}{R}_{k}q$. The formal $R_i=+{\sigma }_i/2$ violates this identity. For example it would give $[J_3,q_1]=q_1/2$, $[J_{+},q_1]=q_2$ and $[J_3,q_2]=-q_2/2$; the two sides of the Jacobi identity for $J_3,J_{+},q_1$ are then $-q_2/2$ and $3q_2/2$.

A consistent canonical column-operator convention uses the inverse adjoint action for the displayed spinor transformation, equivalently the contragredient operator-component prescription. It givesThis is the convention used for the physical states below. Reversing the adjoint action is the minimum needed sign correction in the literal column reading; a dual row notation can encode the same correction by its index contractions. The formal printed-sign coefficients are also recorded below, so the distinction is explicit rather than silently changing a convention. The representation matrices themselves remain $J_i={\sigma }_i/2$ and $K_i=i{\sigma }_i/2$ in Question 3: the minus sign here concerns commutators of operator components.