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In QFT the evolution of momentum and field operators is given by $∂_0φ=i[H,φ]$ and $∂_0\pi=i[H,\pi]$.

Is it possible to derive these equations from the basic operator commutation relations or are they postulated?

Note: this is a follow-up to Canonical quantization of quantum field

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See the "Heisenberg picture". This is just the Schrodinger equation in a different point of view. –  Ron Maimon May 3 '12 at 5:33
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The basic canonical commutation relations are equal time relations, which carry no information about evolution. In the operator formalism, the Heisenberg equations of motion are postulated as evolution equations.

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