Dynamical origin of the $\star_\theta$-noncommutativity in field theory from quantum mechanics

TL;DR

This paper demonstrates how extending the Heisenberg algebra within the Weyl-Wigner-Groenewold-Moyal framework results in a deformed product of variables, linking quantum mechanics noncommutativity to field theory algebra deformation.

Contribution

It introduces an extended Heisenberg algebra that connects quantum mechanical noncommutativity with noncommutative field theory via algebra deformation.

Findings

Deformed product of classical variables derived from extended Heisenberg algebra

Relation established between quantum mechanics and field theory noncommutativity

Framework unifies operator space and algebra deformation approaches

Abstract

We show that introducing an extended Heisenberg algebra in the context of the Weyl-Wigner-Groenewold-Moyal formalism leads to a deformed product of the classical dynamical variables that is inherited to the level of quantum field theory, and that allows us to relate the operator space noncommutativity in quantum mechanics to the quantum group inspired algebra deformation noncommutativity in field theory.