Spin Non-commutativity and the Three-Dimensional Harmonic Oscillator

TL;DR

This paper investigates a three-dimensional harmonic oscillator incorporating spin non-commutativity, revealing an infinitely degenerate ground state with spontaneous symmetry breaking, while conserving total angular momentum, using supersymmetric quantum mechanics methods.

Contribution

It introduces a novel model of a harmonic oscillator with spin non-commutativity and computes its ground state exactly, highlighting spontaneous symmetry breaking.

Findings

Ground state is infinitely degenerate.

Spontaneous broken symmetry observed.

Total angular momentum remains conserved.

Abstract

A three-dimensional harmonic oscillator with spin non-commutativity in the phase space is considered. The system has a regular symplectic structure and by using supersymmetric quantum mechanics techniques, the ground state is calculated exactly. We find that this state is infinitely degenerate and it has explicit spontaneous broken symmetry. Analyzing the Heisenberg equations, we show that the total angular momentum is conserved.