The mass-centre in the restricted theory of relativity and its connexion with the quantum theory of elementary particles

Abstract

The Newtonian definition of the mass-centre can be generalized to the restricted theory of relativity in several ways. Three in particular lead to fairly simple expressions in terms of instantaneous variables for quite general systems. Of these only one is independent of the frame in which it is defined. It suffers from the disadvantage that its components do not commute (in classical mechanics, do not have zero Poisson brackets), and are therefore unsuitable as generalized co-ordinates in mechanics. Of the other two, one is particularly simply defined, and the other has commuting co-ordinates. The Poisson brackets can be derived from quite general considerations because the various mass-centres are expressible in terms of integrals of the energy-momentum tensor which are directly connected with the infinitesimal operators of the group of Lorentz transformations. The definitions are readily applicable to a single particle in theories, such as are current for elementary particles, where a co-ordinate observable does not exist, but an energy-momentum tensor does, and furnish the nearest approach possible to such observables. They are applied to electrons, particles of spin 0 and $\hslash $ (scalar- and vector-meson theories), and to photons.