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Poincaré Symmetry from Heisenberg’s Uncertainty Relations

1
Department of Physics, Middle East Technical University, 06800 Ankara, Turkey
2
Center for Fundamental Physics, University of Maryland, College Park, MD 20742, USA
3
Department of Radiology, New York University, New York, NY 10016, USA
*
Author to whom correspondence should be addressed.
Symmetry 2019, 11(3), 409; https://doi.org/10.3390/sym11030409
Received: 6 March 2019 / Revised: 16 March 2019 / Accepted: 18 March 2019 / Published: 20 March 2019
It is noted that the single-variable Heisenberg commutation relation contains the symmetry of the S p ( 2 ) group which is isomorphic to the Lorentz group applicable to one time-like dimension and two space-like dimensions, known as the S O ( 2 , 1 ) group. According to Paul A. M. Dirac, from the uncertainty commutation relations for two variables, it possible to construct the de Sitter group S O ( 3 , 2 ) , namely the Lorentz group applicable to three space-like variables and two time-like variables. By contracting one of the time-like variables in S O ( 3 , 2 ) , it is possible to construct the inhomogeneous Lorentz group I S O ( 3 , 1 ) which serves as the fundamental symmetry group for quantum mechanics and quantum field theory in the Lorentz-covariant world. This I S O ( 3 , 1 ) group is commonly known as the Poincaré group. View Full-Text
Keywords: Poincaré symmetry from uncertainty relations; one symmetry for quantum mechanics; special relativity Poincaré symmetry from uncertainty relations; one symmetry for quantum mechanics; special relativity
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Başkal, S.; Kim, Y.S.; Noz, M.E. Poincaré Symmetry from Heisenberg’s Uncertainty Relations. Symmetry 2019, 11, 409.

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