Twist Deformation of Rotationally Invariant Quantum Mechanics

TL;DR

This paper explores how non-commutative quantum mechanics in three dimensions can be deformed using an abelian Drinfeld twist, affecting rotational symmetry and operator algebra, with applications to harmonic, anharmonic, and Coulomb systems.

Contribution

It introduces a twist deformation framework for 3D non-commutative quantum mechanics that preserves Hopf algebra structure and analyzes the resulting anomalies in rotational invariance.

Findings

Deformed angular momenta close the so(3) algebra.

Anomalies vanish as deformation parameter approaches zero.

Explicit deformation and anomaly analysis for harmonic, anharmonic, and Coulomb potentials.

Abstract

Non-commutative Quantum Mechanics in 3D is investigated in the framework of the abelian Drinfeld twist which deforms a given Hopf algebra while preserving its Hopf algebra structure. Composite operators (of coordinates and momenta) entering the Hamiltonian have to be reinterpreted as primitive elements of a dynamical Lie algebra which could be either finite (for the harmonic oscillator) or infinite (in the general case). The deformed brackets of the deformed angular momenta close the so(3) algebra. On the other hand, undeformed rotationally invariant operators can become, under deformation, anomalous (the anomaly vanishes when the deformation parameter goes to zero). The deformed operators, Taylor-expanded in the deformation parameter, can be selected to minimize the anomaly. We present the deformations (and their anomalies) of undeformed rotationally-invariant operators corresponding to the harmonic oscillator (quadratic potential), the anharmonic oscillator (quartic potential) and the Coulomb potential.