Noncommutative oscillators from a Hopf algebra twist deformation. A first principles derivation

TL;DR

This paper derives noncommutative oscillators using a Hopf algebra twist, ensuring consistent quantization, symmetry properties, and spectrum calculation, with specific focus on rotational invariance in different dimensions.

Contribution

It provides a first-principles derivation of noncommutative oscillators via Hopf algebra twist deformation, clarifying quantization and symmetry issues.

Findings

Spectrum of single-particle Hamiltonians computed

Multi-particle Hamiltonians fixed by Hopf algebra coproduct

Rotational invariance preserved in 2D, broken to so(2) in 3D

Abstract

Noncommutative oscillators are first-quantized through an abelian Drinfel'd twist deformation of a Hopf algebra and its action on a module. Several important and subtle issues making possible the quantization are solved. The spectrum of the single-particle Hamiltonians is computed. The multi-particle Hamiltonians are fixed, unambiguously, by the Hopf algebra coproduct. The symmetry under particle exchange is guaranteed. In d=2 dimensions the rotational invariance is preserved, while in d=3 the so(3) rotational invariance is broken down to an so(2) invariance.