Coupling of Internal and Space-Time Symmetries

Abstract

When Lorentz invariance is ignored, internal symmetries which are violated in a given way, and which are independent of the space-time symmetries, can be rewritten in terms of a larger group which contains the internal group and the time-translation group in a coupled (noncommuting) way. This rewriting can be chosen so that the noncommutativity of the time-translation and internal groups splits the mass degeneracy of internal multiplets, i.e., accounts for what was previously called "violation" of the internal group. This procedure is illustrated explicitly for the SU(3) baryon octet with octet symmetry violation. When Lorentz invariance is required, the coupling of internal and space-time symmetries becomes more difficult. A global and more general proof is given of McGlinn's result that in a larger group $\mathcal{G}$ whose generators are those of the Poincar\'e (i.e., inhomogeneous Lorentz) group and an internal group, for certain internal groups the commutativity of the (homogeneous) Lorentz and internal subgroups implies the commutativity of the space-time translation and internal subgroups. In addition, it is shown that in such a group $\mathcal{G}$, which has an internal group for which the commutativity of the Lorentz and internal subgroups does not imply the commutativity of the translation and internal subgroups, the internal multiplets still remain degenerate in mass.