Instantaneous Bethe-Salpeter equation: Utmost analytic approach - art. no.056002

The Bethe-Salpeter formalism in the instantaneous approximation for the int eraction kernel entering into the Bethe-Salpeter equation represents a reas onable framework for the description of bound states within relativistic qu antum field theory. In contrast to its further simplifications (such as, fo r instance, the so-called reduced Salpeter equation), it allows also the co nsideration of bound states composed of "light" constituents. Every eigenva lue equation with solutions in some linear space may be (approximately) sol ved by conversion into an equivalent matrix eigenvalue problem. We demonstr ate that the matrices arising in these representations of the instantaneous Bethe-Salpeter equation may be found, at least for a wide class of interac tions, in an entirely algebraic manner. The advantages of having the involv ed matrices explicitly, i.e., not "contaminated" by errors induced by numer ical computations, at one's disposal are obvious: problems such as, for ins tance, questions of the stability of eigenvalues may be analyzed more rigor ously; furthermore, for small matrix sizes the eigenvalues may even be calc ulated analytically.