Lie algebra of the Poincaré group and the relativistic center-of-mass variables of a composite system

Abstract

We show that when we consider the consequences of the Lie algebra of the Poincar\'e group only to a given order in $\frac{1}{{c}^{2}}$ in the construction of the relativistic theories of the composite system, we should take into account all the nine commutator relations in the Lie algebra even though in general only six of them are considered to be independent. When this is done we get an important constraint on the possible forms of the Lorentz boost operator $\stackrel{\ensuremath{\rightarrow}}{\mathrm{K}}$ and hence on the relativistic relation between the constituent and the center-of-mass variables of the composite system.