Brief History of Bethe-Salpeter equation

The Bethe-Salpeter equation was first announced at the March Meeting of the American Physical Society, in 1951 and was published in the APS proceedings (Ref.1) of the Physical Review, with the name Relativistic Equation for Bound-State Problems.

Then, in December 15th, 1951, all details of the Bethe-Salpeter equation were published in a second paper, with the same title as the Ref.1, but with the inversed-alphabetical order of authors.

However, in October 15, 1951, the Bethe-Salpeter equation was proven by Gell-Mann and Low, within the rigour of Quantum Field Theory known in the fifties, in the paper Bound States in Quantum Field Theory.
The Bethe-Salpeter equation is a direct application of the Feynman rules, and a resummation of the infinite set of diagrams by using integral equation. Even if a priori the Bethe-Salpeter equation can be written down in any field theory, and for any subsector of the Fock space of constituents in a bound system, it is only in the late seventies that first applications of BSE in semiconductors appear, being up to that days essentially used (with success) in QED and particle physics.

The first application of the BSE was in 1980, when Hanke and Sham calculated the particle-hole response function of bulk silicon, using a linear combination of atomic orbital basis, with a semiempirical band structure fitted to optical experiments. This work lead for the first time to a calculated absorption spectrum of bulk silicon in qualitative agreement with experiment.

The ab initio approaches used today for solution of the BSE mostly follow the scheme introduced by Onida et. al. in 1995 for the spectrum of the cluster Na₄ and that of Albrecht et. al. in 1997 for the optical gap of LiO₂. This ab initio approach has been applied again to the calculation of optical absorption of bulk silicon (Albrecht et. al., 1998) showing that the BSE used without any empirical parameter can yield absorption spectra of continuum excitons in bulk semiconductors in quantitative agreement with experiment. At the same moment, the works of Benedict et. al. and Rohlfing et. al. showed that the BSE approach can also be used to successfully describe bound excitons in insulators and semiconducors, respectively.
See also the Review of Modern Physics of Onida et. al. and references therein.