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Dirac in his book fundamental of quantum mechanic used the following derivation:

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Is this a derivation of the canonical commutation relations (CCR) in quantum mechanics?

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It is not a derivation per se, more like a suggestive argument. Dirac is assuming that the usual Poisson bracket on the algebra of functions can be replaced by a bracket $\{\cdot,\cdot\}$ on the algebra of operators that satisfies a non-commutative Leibniz rule
$$\{\hat{a}\hat{b},\hat{c}\}~=~\hat{a}\{\hat{b},\hat{c}\}+\{\hat{a},\hat{c}\}\hat{b}.$$ In other words, $\{\cdot,\cdot\}$ is a non-commutative Poisson structure, just like the commutator $[\cdot,\cdot]$. It is perhaps not too surprising that consistency then suggests that $\{\cdot,\cdot\}$ and $[\cdot,\cdot]$ are proportional.

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