Bethe–Salpeter equation

The Bethe -Salpeter equation (after Hans Bethe and Edwin Salpeter 1951) describes the bound states of a quantum- field-theoretical two-body system.

Since the Bethe -Salpeter equation finds its application in many areas of theoretical physics, there are also different spellings. A form as it is often used in particle physics is,

Where Γ is the solution of the Bethe -Salpeter equation, the Bethe -Salpeter amplitude, K, the interaction core and S respectively the propagators of the particles that form the bound state ( hereinafter referred to as constituent ).

In a quantum theory of bound states are stable, that is, they exist infinitely long and so their constituents can infinitely often interact with each other. The Bethe -Salpeter equation describes these states by infinitely often iterated any interactions that may happen between the two constituents. Your solution, the Bethe -Salpeter amplitude describing the bound state, eg in the local, or momentum space.

Possible applications of the Bethe -Salpeter equation for the hydrogen atom, positronium, excitons and mesons

Derivation

A derivation of the Bethe -Salpeter equation is based on the fact that bound states poles in the Green's functions of the theory.

For this purpose, one starts with the Dyson equation for the 4- point functions

Wherein the 4-point Green function of the propagators and interaction core containing all two-particle interactions irreducible.

With the help of the so-called Bethe -Salpeter wave functions, which may be regarded as a transition amplitude of the two constituents in the bound state, one can, in the vicinity of the bound state pole, attach the Green's function as

The total momentum of the system is and the mass of the bound form. For this approach has a pole which corresponds exactly to the mass shell condition for relativistic momentum.

You go with this approach in the Dyson equation above, one obtains

Where you just put, both sides are dominated by your residuals and obtained

The is already a form of Bethe -Salpeter equation. Often the Bethe -Salpeter amplitude Γ are now introduced as

Which the above form of the Bethe -Salpeter equation is obtained:

Approximations

Since the Bethe -Salpeter equation includes all possible interactions between the two constituents of a complete solution rarely ( if ever) possible and in practical calculations approximations are necessary.

• One way is to assume one of the particles to be much heavier than the other, and then to solve the Dirac equation of ( the light ) particle in a potential.
• Will you really, in contrast to above, solve the Bethe -Salpeter equation, one has to model the interaction core. In quantum field theories interactions are described by particle exchange. The simplest assumption about the interaction core is now that it is precisely from the exchange of these force particles (eg photons in quantum electrodynamics, quantum chromodynamics gluons ) between the two constituents, which is then repeated indefinitely. Because the corresponding Feynman diagram of a circuit similar to this approximation is called the ladder approximation of the Bethe -Salpeter equation.