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The Bethe–Salpeter Equation

  • Ingvar LindgrenEmail author
Chapter
Part of the Springer Series on Atomic, Optical, and Plasma Physics book series (SSAOPP, volume 63)

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 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. 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. 3.
    Connell, J.H.: QED test of a Bethe-Salpeter solution method. Phys. Rev. D 43, 1393–1402 (1991)Google Scholar
  4. 4.
    Cutkosky, R.E.: Solutions of the Bethe-Salpeter equation. Phys. Rev. 96, 1135–41 (1954)Google Scholar
  5. 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. 6.
    Dyson, F.J.: The Wave Function of a Relativistic System. Phys. Rev. 91, 1543–50 (1953)Google Scholar
  7. 7.
    Feynman, R.P.: Space-Time Approach to Quantum Electrodynamics. Phys. Rev. 76, 769–88 (1949)Google Scholar
  8. 8.
    Feynman, R.P.: The Theory of Positrons. Phys. Rev. 76, 749–59 (1949)Google Scholar
  9. 9.
    Gell-Mann, M., Low, F.: Bound States in Quantum Field Theory. Phys. Rev. 84, 350–54 (1951)Google Scholar
  10. 10.
    Gross, F.: Three-dimensionsl Covariant Integral Equations for Low-Energy Systems. Phys. Rev. 186, 1448–62 (1969)Google Scholar
  11. 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. 12.
    Lippmann, B.A., Schwinger, J.: Variational Principles for Scattering Processes. I. Phys. Rev. 79, 469–80 (1950)Google Scholar
  13. 13.
    Nakanishi, N.: Normalization condition and normal and abnormal solutions of Bethe-Salpeter equation. Phys. Rev. 138, B1182 (1965)Google Scholar
  14. 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. 15.
    Salpeter, E.E., Bethe, H.A.: A Relativistic Equation for Bound-State Problems. Phys. Rev. 84, 1232–42 (1951)Google Scholar
  16. 16.
    Sazdjian, H.: Relativistiv wave equations for the dynamics of two interacting particles. Phys. Rev. D 33, 3401–24 (1987)Google Scholar
  17. 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. 18.
    Schwinger, J.: On quantum electrodynamics and the magnetic moment of the electron. Phys. Rev. 73, 416 (1948)Google Scholar
  19. 19.
    Schwinger, J.: Quantum electrodynamics I. A covariant formulation. Phys. Rev. 74, 1439 (1948)Google Scholar
  20. 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. 21.
    Sucher, J.: Ph.D. thesis, Columbia University (1958). Univ. Microfilm Internat., Ann Arbor, MichiganGoogle Scholar
  22. 22.
    Todorov, I.T.: Quasipotential Equation Corresponding to the relativistic Eikonal Approximation. Phys. Rev. D 3, 2351–56 (1971)Google Scholar
  23. 23.
    Tomanaga, S.: On Infinite Field Reactions in Quantum Field Theory. Phys. Rev. 74, 224–25 (1948)Google Scholar
  24. 24.
    Wick, G.C.: Properties of Bethe-Salpeter Wave Functions. Phys. Rev. 96, 1124–34 (1954)Google Scholar
  25. 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. 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

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of PhysicsUniversity of GothenburgGöteborgSweden

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