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I don't understand canonical quantization. In passing from classical to quantum, one replaces the Poisson brackets with the commutators. I don't really understand this.

How can we generally show that in the classical limit, the quantum theory that we obtained using this operation leads to the classical theory that we started with?

Can there be a quantum theory that has the same classical limit but that can't be obtained using this prescription?

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    $\begingroup$ Note that canonical quantization is not quite as straightforward as you describe. For example, note that in promoting classical expressions, like the Hamiltonian, to quantum ones, one has, in general, to choose an operator ordering in the quantum expressions. This, I think, just adds more confusion, but it's probably a good thing to realize that canonical quantization is not nearly as subtlety-free as it is made to seem in some physics books. $\endgroup$ – joshphysics Jun 21 '13 at 18:05
  • $\begingroup$ Related: physics.stackexchange.com/q/19770/2451 $\endgroup$ – Qmechanic Jun 21 '13 at 18:39
  • $\begingroup$ Maybe the Dirac presentation will be an introduction into his way of thinking why he did so. youtube.com/watch?v=vwYs8tTLZ24 $\endgroup$ – Vladimir Kalitvianski Jun 21 '13 at 21:21

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