Representations of relativistic particles of arbitrary spin in Poincar\'e, Lorentz, and Euclidean covariant formulations of relativistic quantum mechanics

Relativistic treatments of quantum mechanical systems are important for understanding hadronic structure and dynamics at sub-nucleon distance scales. Hadronic states in different inertial reference frames are needed to compute current matrix elements that probe hadronic structure and dynamics. Relativistic invariance is an important consideration as the resolution of the probe is increased. Many different treatments of relativistic dynamics are used in practice, including Poincar\'e covariant methods, Lorentz covariant methods, Euclidean covariant methods and methods based on quantum fields. Wave functions are typically matrix elements of interacting relativistic states in a basis of non-interacting relativistic states. The purpose of this work is to develop the relation between these different representations of relativistic states that are used in different applications from a unified point of view, starting with positive mass irreducible representations of the Poincar\'e group.