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[tex]\frac{d A}{dt} = \{ A , H \}+ \frac{\partial A}{\partial t}[/tex]

and I say [tex] \{ A, H \} \rightarrow [A,H] [/tex] what single fundamental ASSUMPTION am I changing about my reality? I've read about symplectic manifolds and moyal brackets and such but what intuitively/physically is occurring in this transition?

I would like to be able to make a statement like:

In classical mechanics we assumed our universe has property BLAH, however, in reality it has property BLARG. In our equation we had our classical poisson bracket which represented THIS about our system, however, in light of our new assumptions we can CALCULATE that it must be THIS, which is called the commutator.

With all the capitalized stuff filled in.

Thanks for the help