- 1,170
- 2
Should I in any way find this intuitive? Apart from the fact that the idea of a commutation relation resembles the idea of a poisson bracket for operators I can't see how I should find it intuitive.
Perhaps the intuitive view of commutation relations is the fact that nature does not allow the simultaneous measurement of two conjugate quantities. This happens because commutation relations imply some uncertainty principle.Should I in any way find this intuitive? Apart from the fact that the idea of a commutation relation resembles the idea of a poisson bracket for operators I can't see how I should find it intuitive.
The relation between commutators and poisson brackets is their role in the algebra and geometry of phase space. Geometric transformations that preserve the local phase space volume ("smyplectimorphisms") can be described as a lie group with a lie algebra of infinitesimal transformations. In classical mechanics the lie algebra is created by the poisson bracket as the product between two algebra elements. In quantum theory the same transformations have the commutator as the product of the lie algebra. So if you construct the phase space transformations you automatically arrive at both the canonical poisson bracket and the canonical commutator.Should I in any way find this intuitive? Apart from the fact that the idea of a commutation relation resembles the idea of a poisson bracket for operators I can't see how I should find it intuitive.