# Bethe–Salpeter Equation

The Bethe–Salpeter equation, named after Hans Bethe and Edwin Salpeter, describes the bound states of a two-body (particles) quantum field theoretical system in a relativistically covariant formalism. The equation was actually first published in 1950 at the end of a paper by Yoichiro Nambu, but without derivation.

Due to its generality and its application in many branches of theoretical physics, the Bethe–Salpeter equation appears in many different forms. One form, that is quite often used in high energy physics is

where Γ is the Bethe–Salpeter amplitude, K the interaction and S the propagators of the two participating particles.

In quantum theory, bound states are objects that live for an infinite time (otherwise they are called resonances), thus the constituents interact infinitely many times. By summing up all possible interactions, that can occur between the two constituents, infinitely many times, the Bethe–Salpeter equation is a tool to calculate properties of bound states and its solution, the Bethe–Salpeter amplitude, is a description of the bound state under consideration.

As it can be derived via identifying bound-states with poles in the S-matrix, it can be connected to the quantum theoretical description of scattering processes and Green's functions.

The Bethe–Salpeter equation is a general quantum field theoretical tool, thus applications for it can be found in any quantum field theory. Some examples are positronium, bound state of an electron–positron pair, excitons, bound state of an electron–hole pair and meson as quark-antiquark bound-state.

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.