Bethe–Salpeter equation

The Bethe–Salpeter equation (BSE, named after Hans Bethe and Edwin Salpeter) is an integral equation, the solution of which describes the structure of a relativistic two-body (particles) bound state in a covariant formalism quantum field theory (QFT). The equation was first published in 1950 at the end of a paper by Yoichiro Nambu, but without derivation.

Due to its common application in several branches of theoretical physics, the Bethe–Salpeter equation appears in many forms. One form often used in high energy physics is

${\displaystyle \Gamma (P,p)=\int \!{\frac {d^{4}k}{(2\pi )^{4}}}\;K(P,p,k)\,S(k-{\tfrac {P}{2}})\,\Gamma (P,k)\,S(k+{\tfrac {P}{2}})}$

where ${\displaystyle \Gamma }$ is the Bethe–Salpeter amplitude (BSA), ${\displaystyle K}$ the Green's function representing the interaction and ${\displaystyle S}$ the dressed propagators of the two constituent particles.

In quantum theory, bound states are composite physical systems with lifetime significantly longer than the time scale of the interaction breaking their structure (otherwise the physical systems under consideration are called resonances), thus allowing ample time for constituents to interact. By accounting all possible interactions that can occur between the two constituents, the BSE is a tool to calculate properties of deep-bound states. The BSA as Its solution encodes the structure of the bound state under consideration.

As it can be derived via identifying bound-states with poles in the S-matrix of the 4-point function involving the constituent particles, the equation is related to the quantum-field description of scattering processes applying Green's functions.

As a general-purpose tool the applications of the BSE can be found in most quantum field theories. Examples include positronium (bound state of an electron–positron pair), excitons (bound states of an electron–hole pairs), and mesons (as quark-antiquark bound states).

Even for simple systems such as the positronium, the equation cannot be solved exactly under quantum electrodynamics (QED), despite its exact formulation. A reduction of the equation can be achieved without the exact solution. In the case where particle-pair production can be ignored, if one of the two fermion constituent is significantly more massive than the other, the system is simplified into the Dirac equation for the light particle under the external potential of the heavy one.