# On the generator K of boost transformation in interacting system

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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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edited Dec 11, 2017

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

Thank you, Mr. Neumaier.

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.
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