Revisiting the Poincaré and Little Groups with Physical Applications

This paper elucidates the Lorentz group, a fundamental subgroup of the Poincaré group. The orbits and little groups associated with the Lorentz group are described in detail, along with their corresponding properties. The Poincaré group is presented. Another of the fundamental aspects of the Poincaré group is Wigner's little groups obtained from this group. An in-depth discussion both for the cases of massive and massless relativistic particles within the context of little groups is given. Our examination extends to the properties of various special groups associated with the Poincaré group. Applications of these groups are elaborated by physical examples taken from high-energy physics and optics from both classical and quantum domains. Specifically, covariant harmonic oscillators including entangled states, proton form factors, and the parton picture as proposed by Feynman are discussed. In this context, laser cavities and shear states are also addressed. We lay out the underlying mathematics that connects these apparently disparate realms of physics.