In Schwartz's QFT, there is a problem (3.2) on p. 42 asking to calculate the conserved current Kμνρ associated with the Lorentz transformation xμ→Λμνxν. It's expression is
Kμνρ=Tμ[ρxν]
where, the square brackets imply the antisymmetrization.
One can define the conserved quantities Qj=∫d3x K00j,
which induce the boosts. Being conserved, one means that dQjdt=0.
However, Schwartz also stated that dQjdt=0 is consistent with i∂Qj∂t=[Qj,H] !
Now, let us look at the definition of Qj more carefully. You will find out that Qj is a function of t only. So ∂Qj∂t=dQjdt. But [Qj,H]≠0! What is going on?
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