2018/09/30 by W. N. Polyzou
Mathematics · Physics and Astronomy · #Classical mechanics #Covariant transformation #Euclidean geometry #Four-momentum #Four-vector #Geometry #Group (periodic table) #Hadron #Lorentz covariance #Lorentz group #Lorentz transformation #Mathematical physics #Mathematics #Nuclear physics research studies #Particle physics theoretical and experimental studies #Physics #Poincaré group #Quantum #Quantum Chromodynamics and Particle Interactions #Quantum dynamics #Quantum mechanics #Relativistic dynamics #Relativistic mechanics #Relativistic particle #Relativistic quantum chemistry #Relativistic quantum mechanics #Relativistic speed #Relativistic wave equations #Theoretical physics #Theory of relativity #Wave function #hep-th #nucl-th
paper · pdf · doi:10.1103/physrevc.99.025202
published as Phys. Rev. C 99, 025202 (2019) · 44 pages - revision includes new section on dynamics
openalex created_date 2018/10/05 · arxiv created 2019/01/16 · openalex publication_date 2019/02/07 · arxiv updated 2019/02/13 · openalex updated_date 2026/08/05
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 Poincar'e 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 Poincar'e 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 Poincar'e group. Since any unitary representation of the Poincar'e 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 Poincar'e 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: Poincar'e 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 Poincar'e group that appear in the direct integral.