Eugene Wigner and Translational Symmetries

Abstract

As Einstein's $E = mc^{2}$ unifies the energy-momentum relation for massive and massless particles, Wigner's little group unifies their internal space-time symmetries. It is pointed out that translational symmetries play essential roles both in formulating the problem, and in deriving the conclusions. The input translational symmetry is the space-time translation, and the output translational symmetry is built in the E(2)-like little group for massless particles. The translational degrees of freedom in this little group are gauge degrees of freedom. It should be noted that a number of condensed matter physicists played important supporting and supplementary roles in Wigner's unification of internal space-time symmetries of massive and massless particles.