On the generator K of boost transformation in interacting system

Question

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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Comments

Please replace the ''image''s by latex formulas!

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Thank you, Mr. Neumaier. 

Answer 1

The $W$ with the properties postulated on the page you quote is constructed in (3.5.7) on p. 145 and in (7.4.20) on p. 317.

Comments

Thanks a lot! I will read it carefully later on.