A cluster-separable Born approximation for the 3D reduction of the three-fermion Bethe-Salpeter equation

We perform a 3D reduction of the two-fermion Bethe-Salpeter equation based on Sazdjian's explicitly covariant propagator, combined with a covariant substitute of the projector on the positive-energy free states. We use this combination in the two fermions in an external potential and in the three-fermion problems. The covariance of the two-fermion propagators insures the covariance of the two-body equations obtained by switching off the external potential, or by switching off all interactions between any pair of two fermions and the third one, even if the series giving the 3D potential is limited to the Born term or more generally truncated. The covariant substitute of the positive energy projector preserves the equations against continuum dissolution without breaking the covariance.Comment: 21 pages, 1 figure This article has been deeply modified after refereeing. The presentation has been improved and examples have been added. Three subsections have been added (transition matrix elements, two-body limits, covariant Salpeter's equation). submitted to Journal of Physics

3D reduction of the N-body Bethe-Salpeter equation

We perform a 3D reduction of the two-fermion Bethe-Salpeter equation, by series expansion around a positive-energy instantaneous approximation of the Bethe-Salpeter kernel, followed by another series expansion at the 3D level in order to get a manifestly hermitian 3D potential. It turns out that this potential does not depend on the choice of the starting approximation of the kernel anymore, and can be written in a very compact form. This result can also be obtained directly by starting with an approximation of the free propagator, based on integrals in the relative energies instead of the more usual delta-constraint. Furthermore, the method can be generalized to a system of N particles, consisting in any combination of bosons and fermions. As an example, we write the 3D equation for systems of two or three fermions exchanging photons, in Feynman or Coulomb's gauge.Comment: 22 pages, 3 figures in one single ps file. In the first revision, the self-energy corrections to the propagator have been taken into account. The three figures were gathered in a single ps file instead of three eps. In this second revision (after submission to Nuclear Physics A and refereeing) some explanations have been added, plus a new subsection about the scattering of a particle by a bound stat

Supersymmetry and superalgebra for the two-body system with a Dirac oscillator interaction

Some years ago, one of the authors~(MM) revived a concept to which he gave the name of single-particle Dirac oscillator, while another~(CQ) showed that it corresponds to a realization of supersymmetric quantum mechanics. The Dirac oscillator in its one- and many-body versions has had a great number of applications. Recently, it included the analytic expression for the eigenstates and eigenvalues of a two-particle system with a new type of Dirac oscillator interaction of frequency~$\omega$. By considering the latter together with its partner corresponding to the replacement of~$\omega$ by~$-\omega$, we are able to get a supersymmetric formulation of the problem and find the superalgebra that explains its degeneracy.Comment: 21 pages, LaTeX, 1 figure (can be obtained from the authors), to appear in J. Phys.

Bethe-Salpeter equation: 3D reductions, heavy mass limits and abnormal solutions

We show that the 3D reductions of the Bethe-Salpeter equation have the same bound state spectrum as the original equation, with the possible exception of some solutions for which the corresponding 3D wave function vanishes. The abnormal solutions of the Bethe-Salpeter equation (corresponding to excitations in the relative time-energy degree of freedom), when they exist, are recovered in the 3D reductions via a complicated dependence of the final potential on the total energy. We know however that the one-body (or one high mass) limit of some 3D reductions of the exact Bethe-Salpeter equation leads to a compact 3D equation (by a mutual cancellation of the ladder and crossed graph contributions), which does not exhibit this kind of dependence on the total energy anymore. We conclude that the exact Bethe-Salpeter equation has no abnormal solution at this limit, or has only solutions for which our 3D wave function vanishes. This is in contrast with the results of the ladder approximation, where no such cancellation occurs. We draw the same conclusions for the static model, which we obtain by letting the mass of the lighter particle go also to infinity. These results support Wick's conjecture that the abnormal solutions are a spurious consequence of the ladder approximation.Comment: 11 pages Latex, 1 figure Postscript. Submitted to Journal of Physics