Relativistic Many-Body Theory pp 199-210 | Cite as

# The Bethe–Salpeter Equation

## Abstract

In this chapter, we discuss the *Bethe–Salpeter equation* and its relation to the procedure we have developed so far. We start by summarizing the original derivations of the equation by Bethe and Salpeter and by Gell-Mann and Low, which represented the first rigorous covariant treatments of the *bound-state* problem. We demonstrate that this field-theoretical treatment is completely compatible with the presentation made here. The treatments of Bethe and Salpeter and of Gell-Mann and Low concern the single-reference situation, while our procedure is more general. We, later in this chapter, extend the Bethe–Salpeter equation to the multireference case, which will lead to what we refer to as the *Bethe–Salpeter–Bloch equation* in analogy with corresponding equation in MBPT.

## Keywords

Bloch Equation Dyson Equation Original Derivation Irreducible Graph Simple Atomic System## References

- 1.Boldwin, G.T., Yennie, D.R., Gregorio, M.A.:
*Recoil effects in the hyperfine structure of QED bound states*. Rev. Mod. Phys.**57**, 723–82 (1985)Google Scholar - 2.Caswell, W.E., Lepage, G.P.:
*Reduction of the Bethe-Salpeter equation to an equivalent Schrödinger equation, with applications*. Phys. Rev. A**18**, 810–19 (1978)Google Scholar - 3.Connell, J.H.:
*QED test of a Bethe-Salpeter solution method*. Phys. Rev. D**43**, 1393–1402 (1991)Google Scholar - 4.Cutkosky, R.E.:
*Solutions of the Bethe-Salpeter equation*. Phys. Rev.**96**, 1135–41 (1954)Google Scholar - 5.Douglas, M.H., Kroll, N.M.:
*Quantum Electrodynamical Corrections to the Fine Structure of Helium*. Ann. Phys. (N.Y.)**82**, 89–155 (1974)Google Scholar - 6.Dyson, F.J.:
*The Wave Function of a Relativistic System*. Phys. Rev.**91**, 1543–50 (1953)Google Scholar - 7.Feynman, R.P.:
*Space-Time Approach to Quantum Electrodynamics*. Phys. Rev.**76**, 769–88 (1949)Google Scholar - 8.
- 9.Gell-Mann, M., Low, F.:
*Bound States in Quantum Field Theory*. Phys. Rev.**84**, 350–54 (1951)Google Scholar - 10.Gross, F.:
*Three-dimensionsl Covariant Integral Equations for Low-Energy Systems*. Phys. Rev.**186**, 1448–62 (1969)Google Scholar - 11.Grotch, H., Yennie, D.R.:
*Effective Potential Model for Calculating Nuclear Corrections to the Eenergy Levels of Hydrogen*. Rev. Mod. Phys.**41**, 350–74 (1969)Google Scholar - 12.Lippmann, B.A., Schwinger, J.:
*Variational Principles for Scattering Processes. I*. Phys. Rev.**79**, 469–80 (1950)Google Scholar - 13.Nakanishi, N.:
*Normalization condition and normal and abnormal solutions of Bethe-Salpeter equation*. Phys. Rev.**138**, B1182 (1965)Google Scholar - 14.Namyslowski, J.M.:
*The Relativistic Bound State Wave Function*. in*Light-Front Quantization and Non-Perturbative QCD*, J.P. Vary and F. Wolz, eds. (International Institute of Theoretical and Applied Physics, Ames) (1997)Google Scholar - 15.Salpeter, E.E., Bethe, H.A.:
*A Relativistic Equation for Bound-State Problems*. Phys. Rev.**84**, 1232–42 (1951)Google Scholar - 16.Sazdjian, H.:
*Relativistiv wave equations for the dynamics of two interacting particles*. Phys. Rev. D**33**, 3401–24 (1987)Google Scholar - 17.Sazdjian, H.:
*The connection of two-particle relativistic quantum mechanics with the Bethe-Salpeter equation*. J. Math. Phys.**28**, 2618–38 (1987)Google Scholar - 18.Schwinger, J.:
*On quantum electrodynamics and the magnetic moment of the electron*. Phys. Rev.**73**, 416 (1948)Google Scholar - 19.Schwinger, J.:
*Quantum electrodynamics I. A covariant formulation*. Phys. Rev.**74**, 1439 (1948)Google Scholar - 20.Sucher, J.:
*Energy Levels of the Two-Electron Atom to Order α*^{3}*Ry; Ionization Energy of Helium*. Phys. Rev.**109**, 1010–11 (1957)Google Scholar - 21.Sucher, J.: Ph.D. thesis, Columbia University (1958). Univ. Microfilm Internat., Ann Arbor, MichiganGoogle Scholar
- 22.Todorov, I.T.:
*Quasipotential Equation Corresponding to the relativistic Eikonal Approximation*. Phys. Rev. D**3**, 2351–56 (1971)Google Scholar - 23.Tomanaga, S.:
*On Infinite Field Reactions in Quantum Field Theory*. Phys. Rev.**74**, 224–25 (1948)Google Scholar - 24.Wick, G.C.:
*Properties of Bethe-Salpeter Wave Functions*. Phys. Rev.**96**, 1124–34 (1954)Google Scholar - 25.Zhang, T.:
*Corrections to O(α*^{7}(ln*α)mc*^{2}*) fine-structure splittings and O(α*^{6}(ln*α)mc*^{2}*) energy levels in helium*. Phys. Rev. A**54**, 1252–1312 (1996)Google Scholar - 26.Zhang, T., Drake, G.W.F.:
*A rigorous treatment of O(α*^{6}*mc*^{2}*) QED corrections to the fine structure splitting of helium*. J. Phys. B**27**, L311–16 (1994)Google Scholar