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The commutation relation between H and $L_i$ is $[H,L_i]$. I want to show the equation of motion. Is this right: $ [H,L_i] = -i \frac{dL_i}{dt}?$

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I am not too sure what you mean by continuity equation, but, if you're talking about the equation of motion for an operator in the Heisenberg picture then, yes, that is correct (so long as the operator does not explicitly change with time).

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