Then there can be a plausible way to extend directly this "classical" dynamics to quaternion extension of Poisson bracket, by assuming the dynamics as element of the type: r [member of] H [conjunction] H of the type: r = ai [conjunction] j + bi [conjunction] k + cj [conjunction] k, from which we can define Poisson bracket on H.
The one-to-one correspondence between classical and quantum wave interpretation actually can be expected not only in the context of Feynman's derivation of Maxwell equations from Lorentz force, but also from known exact correspondence between commutation relation and Poisson bracket [3,5].
The one-to-one correspondence between classical and quantum wave interpretation asserted here actually can be expected not only in the context of Feynman's derivation of Maxwell equations from Lorentz force, but also from known exact correspondence between commutation relation and Poisson bracket [3,6].
In particular, if [omega] comes from the inverse of a symplectic form on M, then a (non-singular) Poisson bracket
follows (thus 3 applies).
was obtained in  The modified (noncommutative) Poisson bracket is now given by
Novel Moyal-Yang-Fedosov-Kontsevich star products deformations of the Noncommutative Poisson Brackets are employed to construct star product deformations of scalar field theories.
In particular, Noncommutative p-brane actions, for even p + 1 = 2n-dimensional world-volumes, were written explicitly  in terms of the novel Moyal-Yang (Fedosov-Kontsevich) star product deformations [11, 12] of the Noncommutative Nambu Poisson Brackets (NCNPB) that are associated with the noncommuting world-volume coordinates [q.
The canonical formalism is essential in quantization program since it leads directly to Poisson bracket relations among conjugate variables.
Let us now show that on a generic curved (torsion-free) manifolds the Poisson brackets are conserved.
Presenters discuss current research on various aspects of noncommutative birational geometry and related topics, focusing primarily on the structure and representations of quantum groups and algebras, braided algebras, rational series in free groups, Poisson brackets
on free algebras, and related problems in combinatories.
Once you get through Lagrangians, Hamiltonians and Poisson brackets
, you'll just have to grasp gauge symmetries and vector potentials.
Using several times the Leibniz rule for the Poisson brackets