The Role of Commutators and Poisson Brackets in Phase Space Geometry

[aaaa202]

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.

 

[andrien]

you can not,the resemblance came from the mind of dirac.

 

[cosmic dust]

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.

 

[Jazzdude]

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.

The reason why you have to get the commutator is that translations on a function space (which are a special symplectimorphism) are generated by the derivative. And the derivative does not commute with the coordinate it refers to. That results in the noncommutativity in quantum theory, just from the same principles of symplectic geometry that underly classical mechanics.