2019/03/21 by Sibel Baskal, Young S. Kim, Marilyn E. Noz
Physics and Astronomy · Mathematics · #quant-ph #math-ph #math.MP
paper · pdf · doi:10.3390/sym11030409
published as Symmetry, vol.11, pages 49:1-9 (2019) · 10 pages, 1 figure
arxiv created 2019/03/21 · arxiv updated 2019/03/22
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 Poincaré group.