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. 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, infinitely many times, all possible interactions that can occur between the two constituents, the Bethe–Salpeter equation is a tool to calculate properties of bound states. 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 mesons (as quark-antiquark bound-state). Even for simple systems such as the positronium, the equation cannot be solved exactly, although in principle it can be formulated exactly. 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. The starting point for the derivation of the Bethe–Salpeter equation is the two-particle (or four point) Dyson equation in momentum space, where 'G' is the two-particle Green function ⟨ Ω | ϕ 1 ϕ 2 ϕ 3 ϕ 4 | Ω ⟩ {displaystyle langle Omega |phi _{1},phi _{2},phi _{3},phi _{4}|Omega angle } , 'S' are the free propagators and 'K' is an interaction kernel, which contains all possible interaction between the two particles. The crucial step is now, to assume that bound states appear as poles in the Green function. One assumes, that two particles come together and form a bound state with mass 'M', this bound state propagates freely, and then the bound state splits in its two constituents again. Therefore, one introduces the Bethe–Salpeter wave function Ψ = ⟨ Ω | ϕ 1 ϕ 2 | ψ ⟩ {displaystyle Psi =langle Omega |phi _{1},phi _{2}|psi angle } , which is a transition amplitude of two constituents ϕ i {displaystyle phi _{i}} into a bound state ψ {displaystyle psi } , and then makes an ansatz for the Green function in the vicinity of the pole as