Distortion of the Poisson Bracket by the Noncommutative Planck Constants

TL;DR

This paper introduces a noncommutative framework for semiclassical parameters, creating a distorted classical observable algebra and establishing a q-analogue of Weyl quantization that induces a noncommutative Poisson bracket.

Contribution

It develops a novel noncommutative neighborhood of the Planck constant and constructs a q-analogue of Weyl quantization with a noncommutative Poisson bracket.

Findings

Existence of a noncommutative neighborhood of the Planck constant.

Construction of a q-analogue of Weyl quantization.

Induction of a noncommutative Poisson bracket on the classical shadow.

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.