The linear potential and harmonic oscillator in relativistic quantum mechanics

It is a nontrivial problem to formulate a Poincare invariant quantum theory, that describes the binding of two particles in a confining potential. Four attempts at such theories are discussed and subsequently used to calculate the spectrum of two particles, which are bound in an harmonic oscillator potential or in a linear potential. These theories are described by the following equations 1. The so called "Relativistic Schrodinger equation". 2. The Klein-Gordon equation. 3. The Dirac equation. 4. RQM (Relativistic Quantum Mechanics), the author's private theory, which is of the "quasiparticle" type. For each of these theories the Regge trajectories are calculated, both for the linear and for the harmonic potential. Since in RQM the interaction potential is the carrier, not only of energy, but also of momentum and hence of angular momentum, the Regge slopes differ from their usual values. Along the way it is shown how confining potentials can be handled in a theory which is formulated in the momentum representation, in spite of the fact that their Fourier transforms do not exist. For other quasiparticle theories the spectrum of the relativistic harmonic oscillator has not been calculated.