Representations of relativistic particles of arbitrary spin in Poincaré, Lorentz, and Euclidean covariant formulations of relativistic quantum mechanics

Background: Relativistic treatments of quantum mechanical systems are important for understanding hadronic structure and dynamics at subnucleon scales. Relativistic invariance of a quantum system means that there is an underlying unitary representation of the Poincaré group. This is equivalent to the requirement that the quantum observables (probabilities, expectation values, and ensemble averages) for equivalent measurements performed in different inertial reference frames are identical. Different representations are used in practice, including Poincaré covariant forms of dynamics, representations based on Lorentz covariant wave functions, Euclidean covariant representations, and representations generated by Lorentz covariant fields.

Purpose: The purpose of this work is to illustrate the relation between the different equivalent representations of states in relativistic quantum mechanics.

Method: The starting point is a description of a particle of mass $m$ and spin $j$ using irreducible representations of the Poincaré group. Since any unitary representation of the Poincaré group can be decomposed into a direct integral of irreducible representations, these are the basic building blocks of any relativistically invariant quantum theory. The equivalence is established by constructing equivalent Lorentz covariant irreducible representations from Poincaré covariant irreducible representations and constructing equivalent Euclidean covariant irreducible representations from Lorentz covariant irreducible representations.

Results: Equivalent descriptions for positive mass representations of arbitrary spin are presented in each of these frameworks. Dynamical realizations of the different representations are briefly discussed.

Conclusion: Poincaré covariant, Lorentz covariant, and Euclidean covariant realizations of relativistic dynamics are shown to be equivalent by explicitly relating the positive-mass positive-energy irreducible representations of the Poincaré group that appear in the direct integral.