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Symmetry 2016, 8(7), 55; doi:10.3390/sym8070055

Entangled Harmonic Oscillators and Space-Time Entanglement

Department of Physics, Middle East Technical University, 06800 Ankara, Turkey
Center for Fundamental Physics, University of Maryland College Park, College Park, MD 20742, USA
Department of Radiology, New York University School of Medicine, New York, NY 10016, USA
Author to whom correspondence should be addressed.
Academic Editor: Sergei D. Odintsov
Received: 26 February 2016 / Revised: 23 May 2016 / Accepted: 20 June 2016 / Published: 28 June 2016
(This article belongs to the Special Issue Harmonic Oscillators In Modern Physics)
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The mathematical basis for the Gaussian entanglement is discussed in detail, as well as its implications in the internal space-time structure of relativistic extended particles. It is shown that the Gaussian entanglement shares the same set of mathematical formulas with the harmonic oscillator in the Lorentz-covariant world. It is thus possible to transfer the concept of entanglement to the Lorentz-covariant picture of the bound state, which requires both space and time separations between two constituent particles. These space and time variables become entangled as the bound state moves with a relativistic speed. It is shown also that our inability to measure the time-separation variable leads to an entanglement entropy together with a rise in the temperature of the bound state. As was noted by Paul A. M. Dirac in 1963, the system of two oscillators contains the symmetries of the O ( 3 , 2 ) de Sitter group containing two O ( 3 , 1 ) Lorentz groups as its subgroups. Dirac noted also that the system contains the symmetry of the S p ( 4 ) group, which serves as the basic language for two-mode squeezed states. Since the S p ( 4 ) symmetry contains both rotations and squeezes, one interesting case is the combination of rotation and squeeze, resulting in a shear. While the current literature is mostly on the entanglement based on squeeze along the normal coordinates, the shear transformation is an interesting future possibility. The mathematical issues on this problem are clarified. View Full-Text
Keywords: Gaussian entanglement; two coupled harmonic oscillators; coupled Lorentz groups; space-time separation; Wigner’s little groups; O(3, 2) group; Dirac’s generators for two coupled oscillators Gaussian entanglement; two coupled harmonic oscillators; coupled Lorentz groups; space-time separation; Wigner’s little groups; O(3, 2) group; Dirac’s generators for two coupled oscillators

Figure 1

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Başkal, S.; Kim, Y.S.; Noz, M.E. Entangled Harmonic Oscillators and Space-Time Entanglement. Symmetry 2016, 8, 55.

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