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arXiv · 1903.05348

Poincar\'e Symmetry from Heisenberg's Uncertainty Relations

Sibel Baskal·Young S. Kim·Marilyn E. Noz
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Abstract

It is noted that the single-variable Heisenberg commutation relation contains the symmetry of the $Sp(2)$ 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. According to Paul A. M. Dirac, from the uncertainty commutation relations for two variables, it possible to construct the de Sitter group $O(3,2)$, namely the Lorentz group applicable to three space-like variables and two time-like variables. By contracting one of the time-like variables in $O(3,2)$, it is possible, to construct the inhomogeneous Lorentz group $IO(3,1)$ which serves as the fundamental symmetry group for quantum mechanics and quantum field theory in the Lorentz covariant world. This $IO(3,1)$ group is commonly known as the Poincar\'e group.

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Sibel Baskal, Young S. Kim, Marilyn E. Noz. 2019-03-13. Poincar\'e Symmetry from Heisenberg's Uncertainty Relations. https://doi.org/10.3390/sym11030409

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