On the role of Schwinger's SU(2) generators for Simple Harmonic Oscillator in 2D Moyal plane

摘要

The Hilbert-Schmidt operator formulation of non-commutative quantum mechanicsin 2D Moyal plane is shown to allow one to construct Schwinger's SU(2)generators. Using this the SU(2) symmetry aspect of both commutative andnon-commutative harmonic oscillator are studied and compared. Particularly, inthe non-commutative case we demonstrate the existence of a critical point inthe parameter space of mass and angular frequency where there is a manifestSU(2) symmetry for a unphysical harmonic oscillator Hamiltonian built out ofcommuting (unphysical yet covariantly transforming under SU(2)) position likeobservable. The existence of this critical point is shown to be a novel aspectin non-commutative harmonic oscillator, which is exploited to obtain thespectrum and the observable mass and angular frequency parameters of thephysical oscillator-which is generically different from the bare parametersoccurring in the Hamiltonian. Finally, we show that a Zeeman term in theHamiltonian of non-commutative physical harmonic oscillator, is solelyresponsible for both SU(2) and time reversal symmetry breaking.