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
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
de Sitter group containing two
Lorentz groups as its subgroups. Dirac noted also that the system contains the symmetry of the
group, which serves as the basic language for two-mode squeezed states. Since the
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.
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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MDPI and ACS Style
Başkal, S.; Kim, Y.S.; Noz, M.E. Entangled Harmonic Oscillators and Space-Time Entanglement. Symmetry 2016, 8, 55.
Başkal S, Kim YS, Noz ME. Entangled Harmonic Oscillators and Space-Time Entanglement. Symmetry. 2016; 8(7):55.
Başkal, Sibel; Kim, Young S.; Noz, Marilyn E. 2016. "Entangled Harmonic Oscillators and Space-Time Entanglement." Symmetry 8, no. 7: 55.
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