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Abstract : "The Hilbert-Schmidt operator formulation of non-commutative quantum mechanics in the 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 and non-commutative harmonic oscillator is studied and compared. Particularly, in the non-commutative case we demonstrate the existence of a critical point in the parameter space of mass (μ) and angular frequency (ω) where there is a manifest SU (2) symmetry for an unphysical harmonic oscillator Hamiltonian built out of commuting (unphysical yet covariantly transforming under SU (2)) position like observable, so that the ground state of this unphysical oscillator coincides with the “vacuum” of the quantum Hilbert space. The existence of this critical point is shown to be a novel aspect in the non-commutative harmonic oscillator, which is exploited to obtain the spectrum and the observable mass (μ) and angular frequency (ω) parameters of the physical oscillator which are generically different from the corresponding bare parameters occurring in the original Hamiltonian. For generic values of μ and ω, the ground state of the harmonic oscillator is shown to be somewhat analogous to the squeezed coherent state with respect to the above-mentioned “vacuum” state. Finally, we show that a Zeeman term in the original Hamiltonian of non-commutative physical harmonic oscillator, is solely responsible for both SU (2) and time reversal symmetry breaking."
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