Distortion of the Poisson Bracket by the Noncommutative Planck Constants

Details Distortion of the Poisson Bracket by the Noncommutative Planck Constants Distortion of the Poisson Bracket by the Noncommutative Planck Constants

2009-10-15

arxiv.org/abs/0910.2949v2

Commun. Math. Phys. 304, 369 -- 393 (2011)

Physics

Mathematical Physics

Scientific paper

In this paper we introduce a kind of "noncommutative neighbourhood" of a semiclassical parameter corresponding to the Planck constant. This construction is defined as a certain filtered and graded algebra with an infinite number of generators indexed by planar binary leaf-labelled trees. The associated graded algebra (the classical shadow) is interpreted as a "distortion" of the algebra of classical observables of a physical system. It is proven that there exists a q-analogue of the Weyl quantization, where q is a matrix of formal variables, which induces a nontrivial noncommutative analogue of a Poisson bracket on the classical shadow.

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