It is usually assumed that the appearance of accidental degeneracy in the energy levels of a given Hamiltonian is due to a symmetry group. By considering the elementary problem of a rotator with spin-orbit coupling, when the strength of the latter is equal to the inverse of the moment of inertia, we find that this assumption does not explain the degeneracy of all the levels of the Hamiltonian. Thus the relation between accidental degeneracies and symmetry group merits further probing.

Barut showed us how it is possible to get a Poincaré invariant n-body equation with a single time. Starting from the Barut equation for n-free particles, we show how to generalize it when they interact through Dirac oscillators with different frequencies. We then particularize the problem to n=2 and consider the particle-antiparticle system whose frequencies are respectively ω and −ω. We indicate how the resulting equation can be solved by perturbation theory, though the spectrum and its comparison with that of (...) the mesons will be given in another publication. (shrink)

The Hamiltonian for n relativistic electrons without interaction but in a Coulomb potential is well known. If in this Hamiltonian we take r ′ u =r′, P ′ u =P′ with u=1,2,..., n, we obtain a one-body problem in a Coulomb field, but the appearance of n of the α u , u=1,..., n, each of which corresponds to spin $\tfrac{1}{2}$ , indicates that we may have spins up to (n/2). We analyze this last problem first by denoting the 4×4 (...) matrices α, β as direct products of 2×2 matrices which correspond to the ordinary spin, and a new concept, also related to the SU(2) group, which we call sign spin. In this new notation our problem depends on the sixteen generators of a U(4) group reduced along the chain Û(2)⊗Ŭ(2) sub-groups associated with the ordinary and sign spins. We now make a change of variables in our Hamiltonian so a term ε related to the frequency ω of an oscillator, which will be our variational parameter, appears in it, and later construct the full states of the problem with a harmonic oscillator of frequency 1 and ordinary and sign spin parts. Finally we obtain the matrix representation of our Hamiltonian with respect to the states mentioned and discuss the energy spectra of the problem where the partition {h} representing the irrep of U(4) and j the total angular momentum, take the values {h}=[1], j= $\tfrac{1}{2}$ ; {h}=[11], j=0; {h}=[2], j=0. (shrink)

The Hamiltonian for n relativistic electrons without interaction but in a Coulomb potential is well known. If in this Hamiltonian we take r′u=r′, P′u=P′ with u=1,2,..., n, we obtain a one-body problem in a Coulomb field, but the appearance of n of the αu, u=1,..., n, each of which corresponds to spin\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\tfrac{1}{2}$$ \end{document}, indicates that we may have spins up to (n/2). We analyze this last problem first by denoting the (...) 4×4 matrices α, β as direct products of 2×2 matrices which correspond to the ordinary spin, and a new concept, also related to the SU(2) group, which we call sign spin. In this new notation our problem depends on the sixteen generators of a U(4) group reduced along the chain Û(2)⊗Ŭ(2) sub-groups associated with the ordinary and sign spins. We now make a change of variables in our Hamiltonian so a term ε related to the frequency ω of an oscillator, which will be our variational parameter, appears in it, and later construct the full states of the problem with a harmonic oscillator of frequency 1 and ordinary and sign spin parts. Finally we obtain the matrix representation of our Hamiltonian with respect to the states mentioned and discuss the energy spectra of the problem where the partition {h} representing the irrep of U(4) and j the total angular momentum, take the values {h}=[1], j=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\tfrac{1}{2}$$ \end{document}; {h}=[11], j=0; {h}=[2], j=0. (shrink)