Four dimensional quantum oscillator and magnetic-monopole with U(1) dynamical group

TL;DR

This paper demonstrates that a 4D quantum harmonic oscillator can model a charged particle in a magnetic monopole field, revealing hidden U(1) symmetry and deriving the system's spectrum through group theoretical methods.

Contribution

It introduces a transformation linking the 4D harmonic oscillator to monopole dynamics and uncovers the U(1) symmetry and dynamical group structure of the system.

Findings

Identified U(1) symmetry in the magnetic monopole system.

Derived the dynamical U(1)*U(1) group for the 4D oscillator.

Calculated the energy spectrum explicitly.

Abstract

By using of an appropriate transformation, it was shown that the quantum system of 4 dimensional simple harmonic oscillator can describe the motion of a charged particle in the presence of a magnetic monopole field. It was shown that the Dirac magnetic monopole has the hidden algebra of U(1) symmetry and by reducing of the dimensions of space, the U(1)*U(1) dynamical group for 4D harmonic oscillator quantum system was obtained. Using the group representation and based on explicit solution of the obtained differential equation, the spectrum of system was calculated.