The instantaneous Bethe-Salpeter equation, derived from the general Bethe-Salpeter formalism by assuming that the involved interaction kernel is instantaneous, represents the most promising framework for the description of hadrons as bound states of quarks from first quantum-field-theoretic principles, that is, quantum chromodynamics. Here, by extending a previous analysis confined to the case of bound-state constituents with vanishing masses, we demonstrate that the instantaneous Bethe-Salpeter equation for bound-state constituents with (definitely) nonvanishing masses may be converted into an eigenvalue problem for an explicitly-more precisely, algebraically-known matrix, at least, for a rather wide class of interactions between these bound-state constituents. The advantages of the explicit knowledge of this matrix representation are self-evident.
The Bethe-Salpeter formalism in the instantaneous approximation for the interaction kernel entering into the Bethe-Salpeter equation represents a reasonable framework for the description of bound states within relativistic quantum field theory. In contrast to its further simplifications (like, for instance, the so-called reduced Salpeter equation), it allows also the consideration of bound states composed of "light" constituents. Every eigenvalue equation with solutions in some linear space may be (approximately) solved by conversion into an equivalent matrix eigenvalue problem. We demonstrate that the matrices arising in these representations of the instantaneous Bethe-Salpeter equation may be found, at least for a wide class of interactions, in an entirely algebraic manner. The advantages of having the involved matrices explicitly, i.e., not "contaminated" by errors induced by numerical computations, at one's disposal are obvious: problems like, for instance, questions of the stability of eigenvalues may be analyzed more rigorously; furthermore, for small matrix sizes the eigenvalues may even be calculated analytically.
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