The quantum relativistic two-body bound state. I. The spectrum

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Abstract
In the framework of a manifestly covariant quantum theory on space‐time, it is shown that the ground state mass of a relativistic two‐body system with O(3,1) symmetric potential is lower when represented by a wave function with support in an O(2,1) invariant subspace of the spacelike region. The wave functions for the relativistic bound states are obtained explicitly. Coulomb type binding, the harmonic oscillator, and the relativistic square well are treated as examples. The mass spectrum is determined by a differential equation in the invariant spacelike interval ρ, which can be put into correspondence with the radial part of a nonrelativistic Schrödinger equation with potential of the same form, where r is replaced by ρ. In the case that the binding is small compared to the particle masses, the mass spectrum (bounded below) is well‐approximated by the results of the nonrelativistic theory. The eigenfunctions transform under the full Lorentz group as elements of an induced representation with O(2,1) little group. This representation is studied in a succeeding paper.