The main properties of the Poincaré group are reviewed, in particular the Wigner boosts needed for the rest-frame instant form of dynamics.

Given an isolated system of either free or interacting particles and the associated realization of the ten conserved Poincaré generators its total conserved time-like 4-momentum defines its inertial rest-frame as the 3+1 splitting whose space-like 3-spaces (named Wigner 3-spaces) are orthogonal to it and whose inertial observer is the Fokker–Pryce 4-center of inertia. There is a discussion of the problem of the relativistic center of mass based on the fact that the 4-center functions “only” of the Poincaré generators of the isolated system are the following three non-local quantities: the non-canonical covariant Fokker–Pryce 4-center of inertia, the canonical non-covariant Newton–Wigner 4-center of mass and the non-canonical non-covariant Mőller 4-center of energy. At the Hamiltonian level one is able to express the canonical world-lines of the particles and their momenta in terms of the Jacobi variables of the external Newton–Wigner center of mass (a non-local non-covariant non-measurable quantity) and of Wigner-covariant relative 3-coordinates and 3-momenta inside the Wigner 3-spaces. This solves the problem of the elimination of relative times in relativistic bound states and to formulate a consistent Wigner-covariant relativistic quantum mechanics of point particles. The non-relativistic limit gives the Hamilton–Jacobi description of the system after the separation of Newtonian center of mass. Finally there is the definition of the non-inertial rest-frames whose 3-spaces are orthogonal to the total 4-momentum of the isolated system at spatial infinity.

We find the Lie algebra of the Lorentz group and then extend it to the Poincaré group, the group of symmetries of flat space. We then point out that, as SU(2) is the universal cover of SO(3), for the Lorentz group SO(3,1) the universal cover is SL(2,C).We then use Wigner's method, using the little group in four dimensions, to find massive and massless representations of the Lorentz and Poincaré groups. We thus find various possible fields, corresponding to these representations. We end by explaining how SL(2,C) is the universal cover of SO(3,1).