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Einstein’s E = mc2 Derivable 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.
Quantum Reports 2019, 1(2), 236-251; https://doi.org/10.3390/quantum1020021
Received: 12 September 2019 / Revised: 22 October 2019 / Accepted: 7 November 2019 / Published: 9 November 2019
Heisenberg’s uncertainty relation can be written in terms of the step-up and step-down operators in the harmonic oscillator representation. 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 O ( 2 , 1 ) group. This group has three independent generators. The one-dimensional step-up and step-down operators can be combined into one two-by-two Hermitian matrix which contains three independent operators. If we use a two-variable Heisenberg commutation relation, the two pairs of independent step-up, step-down operators can be combined into a four-by-four block-diagonal Hermitian matrix with six independent parameters. It is then possible to add one off-diagonal two-by-two matrix and its Hermitian conjugate to complete the four-by-four Hermitian matrix. This off-diagonal matrix has four independent generators. There are thus ten independent generators. It is then shown that these ten generators can be linearly combined to the ten generators for Dirac’s two oscillator system leading to the group isomorphic to the de Sitter group O ( 3 , 2 ) , which can then be contracted to the inhomogeneous Lorentz group with four translation generators corresponding to the four-momentum in the Lorentz-covariant world. This Lorentz-covariant four-momentum is known as Einstein’s E = m c 2 . View Full-Text
Keywords: E = mc2 from Heisenberg’s uncertainty relations; one symmetry for quantum mechanics and special relativity E = mc2 from Heisenberg’s uncertainty relations; one symmetry for quantum mechanics and special relativity
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Başkal, S.; Kim, Y.S.; Noz, M.E. Einstein’s E = mc2 Derivable from Heisenberg’s Uncertainty Relations. Quantum Reports 2019, 1, 236-251.

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