Tensor Coordinates in Noncommutative Mechanics

TL;DR

This paper develops a classical mechanics framework incorporating tensor coordinates that, upon quantization, yields a noncommutative quantum theory with rotational symmetry and novel features.

Contribution

It introduces a classical formulation with tensor coordinates and noncommutativity parameters, ensuring rotational invariance and providing a foundation for noncommutative quantum mechanics.

Findings

Classical theory with tensor coordinates and noncommutative parameters.

Quantum theory exhibits noncommutativity with rotational symmetry.

Framework maintains invariance under SO(D) group.

Abstract

A consistent classical mechanics formulation is presented in such a way that, under quantization, it gives a noncommutative quantum theory with interesting new features. The Dirac formalism for constrained Hamiltonian systems is strongly used, and the object of noncommutativity ${\mathbf \theta}^{ij}$ plays a fundamental rule as an independent quantity. The presented classical theory, as its quantum counterpart, is naturally invariant under the rotation group $SO(D)$.