Distortion of the Poisson Bracket by the Noncommutative Planck Constants
Mathematical Physics
2011-06-20 v2 math.MP
Quantum Algebra
Quantum Physics
Abstract
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.
Keywords
Cite
@article{arxiv.0910.2949,
title = {Distortion of the Poisson Bracket by the Noncommutative Planck Constants},
author = {Artur E. Ruuge and Freddy van Oystaeyen},
journal= {arXiv preprint arXiv:0910.2949},
year = {2011}
}