Little groups, the quark model and gauge transformations

Two modern applications of Wigner's original work on the Poincaré group are discussed. The little group for a particle with non-zero mass is O(3), and that for a massless particle is isomorphic to E(2). First, we use the covariant harmonic-oscillator model to illustrate that the basic symmetry of the space-time separation between the quarks inside a hadron is isomorphic to that of O(3). Second, we give a realization of the E(2) group which described both the rotation and gauge transformation of the photon wave function.