Inverse square problem and so(2,1) symmetry in noncommutative space

TL;DR

This paper investigates the effects of noncommutativity on the inverse square potential in quantum mechanics, revealing symmetry breaking and algebraic modifications in noncommutative space.

Contribution

It analyzes how noncommutative geometry alters the so(2,1) symmetry and algebraic structure in the quantum inverse square potential problem.

Findings

Noncommutativity breaks the original so(2,1) symmetry.

The algebra in noncommutative space is not closed.

The commutative limit recovers the standard so(2,1) algebra.

Abstract

We study the quantum mechanics of a system with inverse square potential in noncommutative space. Both the coordinates and momentums are considered to be noncommutative, which breaks the original so(2,1) symmetry. The energy levels and eigenfunctions are obtained. The generators of the so(2,1) algebra are also studied in noncommutative phase space and the commutators are calculated, which shows that the so(2,1) algebra obtained in noncommutative space is not closed. However the commutative limit \Theta,\bar{\Theta}\to 0 for the algebra smoothly goes to the standard so(2,1) algebra.