Bethe-Salpeter equation

The Bethe-Salpeter equation[1][2] describes the bound states of a two-body (particles) quantum mechanical system.

Examples of two-particle systems described by the Bethe-Salpeter equation are the positronium, bound state of an electron-positron pair; and, in condensed matter physics, the exciton, bound state of an electron-hole pair.

Because the particle pair described by the Bethe-Salpeter equation is in a bound state, the particles can interact infinitely often. Concurrently, since the number of interactions can be arbitrary, the number of possible Feynman diagrams will quickly exceed feasible calculations.

Even for simple systems such as the positronium, the equation cannot be solved exactly although the equation's formulation can in principle be formulated exactly. Fortunately, a classification of the states can be achieved without the need for an exact solution. If one of the particles is significantly more massive than the other, the problem is considerably simplified as one solves the Dirac equation for the lighter particle under the external potential of the heavier particle.

References

1. ^ H. Bethe, E. Salpeter. Physical Review, vol.82 (1951), pp.309.

2. ^ H. Bethe, E. Salpeter, "A Relativistic Equation for Bound-State Problems". Physical Review, vol. 84 (1951), pp.1232. DOI:10.1103/PhysRev.84.1232.