In p. 119 (as shown below) of Weinberg's field theory book 1, he wants to illuminate that the following commutation relations will give some restrictions on the Hamiltonian $H$ of the interacting system.

$$

[H_0,S] = [\mathbf P_0,S] = [\mathbf J_0,S] = [\mathbf K_0,S] = 0

$$

He finds that the first three commutation relations require two restrictions:

$$

H=H_0+V, \quad \mathbf P = \mathbf P_0, \quad \mathbf J = \mathbf J_0,

$$

$$

[V,\mathbf P_0] = [V, \mathbf J_0]=0.

$$

The last commutation relations require four restrictions:

\begin{align}

H &= H_0+V \\

\mathbf K &= \mathbf K_0 +\mathbf W,\\

\mathbf [K_0,V] &= -[\mathbf W,H],

\end{align}

and $W$ is smooth.

I know that we can construct $H$ satisfying the first group requirement easily. My question is how do we construct $H$ satisfying the second group requirement: $K=K_0+W$ and $W$ must satisfy the condition $[K_0,V]=−[W,H]$?

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