Chapter 20: The Poincaré Group

This chapter is devoted to the study of some aspects and examples of classical canonical realizations of the Poincaré group, or as it is also called, the inhomogeneous Lorentz group. For brevity, we write P for this group; for the homogeneous Lorentz group we use the abbreviation HLG or sometimes SO(3, 1) as we did in Chapter 15. In passing over from the Galilei group, which describes the invariance properties of the equations of motion of Newtonian mechanics, to P, what is retained is the idea that there is a privileged class of reference frames in nature, the inertial frames, in which the laws of particle mechanics are especially simple. In particular a particle “not acted on by any forces” follows a straight-line trajectory in any one of these frames. As in the case of G, two inertial frames in special relativity theory must be related to one another by a space-time translation, or by a spatial rotation, or by the fact that one of them moves with a uniform velocity relative to the other, or by a combination of all these. What distinguishes P from G is the way in which we relate the spacetime coordinates assigned to an event by two observers in two different inertial frames. In the case of G this relation is the one that is “reasonable” and “obvious” from the stand point of Newtonian mechanics (and in fact from common, everyday experience). In the case of P we have, on the other hand, the Lorentz transformation formulae discussed in detail in Chapter 15. These different formulae for the way the space-time coordinates of events change in going from one inertial frame to another give different structures to the group of all possible changes of inertial frame, leading to G in the one case and P in the other. The transformation formulae in the case of P contain the velocity of light in vacuum, c, in an essential manner; one can, in a physically meaningful way, think of G as the limiting form of P as c → ∞. (Of course, c is a fixed real number in any given system of units and never really goes to infinity; what is meant is that in specific situations if certain characteristic velocities are neglected in comparison with c, in a consistent way, then P is essentially replaced by G). For the most part, we assume that our system of units is such that c can be set equal to unity, so that the writing becomes easier…