Distortion of the Poisson bracket by the noncommutative Planck constantsDistortion of the Poisson bracket by the noncommutative Planck constants
Faculty of Sciences. Mathematics and Computer Science

Fundamental Mathematics

article

2011Heidelberg, 2011

Mathematics

Physics

Communications in mathematical physics. - Heidelberg

304(2011):2, p. 369-393

0010-3616

000290229600003

E

English (eng)

University of Antwerp

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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