4. Yossi Bachar, Lawrence P. Horwitz, and Igal Aharonovich, The invariant Berry phase of the quantum relativistic four-dimensional harmonic oscillator

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Volume 30: Pages 21-27, 2017


The invariant Berry phase of the quantum relativistic four-dimensional harmonic oscillator


Yossi Bachar,1,a) Lawrence P. Horwitz,1,2,3 and Igal Aharonovich1


1Department of Physics, Bar-Ilan University, Ramat-Gan 5290001, Israel

2School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv 69978, Israel

3Department of Physics, Ariel University in the Shomron, Ariel 40700, Israel


We review our previous work on the existence of Lorentz invariant Berry phases generated in the Stueckleberg–Horwitz–Piron manifestly covariant quantum theory (SHP), also reviewed here, by a perturbed four dimensional harmonic oscillator. These phases are associated with a fractional perturbation of the azimuthal symmetry of the oscillator. They are computed numerically by using time independent perturbation theory and the definition of the Berry phase generalized to the framework of the SHP relativistic quantum theory. In this paper, we give an explicit proof of the Lorentz invariance.


Nous démontrons, dans la théorie manifestement covariante quantique de Stueckelberg-Horwitz-Piron (SHP), l’existence de phases Berry invariantes de Lorentz générées par un oscillateur harmonique quatre-dimensionnel perturbé. Ces phases sont associées à une perturbation fractionnelle de la symétrie azimutale de l’oscillateur. Elles sont calculées numériquement à l’aide de la théorie des perturbations indépendates du temps et de la définition de la phase de Berry généralisée au cadre de la théorie quantique relativiste SHP. Ici, nous notons une preuve explicite de la invariance.


Key words: Berry Phase; Relativistic Quantum Mechanics; Induced Representation; 4D Relativistic Harmonic Oscillator.

Received: July 20, 2016; Accepted: November 28, 2016; Published Online: January 4, 2017


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