The Poincaré Group and its Representations

In chapter 19, we saw that the Euclidean group E(3) has infinite dimensional irreducible unitary representations on the state space of a quantum free particle. The free particle Hamiltonian plays the role of a Casimir operator: to get irreducible representations, one fixes the eigenvalue of the Hamiltonian (the energy), and then the representation is on the space of solutions to the Schrodinger equation with this energy. There is also a second Casimir operator, with integral eigenvalue the helicity, which further characterizes irreducible representations.