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
Communications in mathematical physics. - Heidelberg
304(2011):2, p. 369-393
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.