Poincare Symmetry from Heisenberg's Uncertainty Relations

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 SO(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 SO(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 SO(3,2), it is possible to construct the inhomogeneous Lorentz group ISO(3,1) which serves as the fundamental symmetry group for quantum mechanics and quantum field theory in the Lorentz-covariant world. This ISO(3,1) group is commonly known as the Poincare group.