BetheSalpeter

The Bethe-Salpeter equation is a relativistic framework for describing bound states of two interacting particles in quantum field theory. Named for Hans Bethe and Edwin Salpeter, it emerges from the study of the two-particle Green’s function and Dyson–Schwinger formalisms. The central object is the Bethe-Salpeter amplitude, a wavefunction-like quantity that encodes the internal structure of the bound state as a function of the total momentum and the relative motion of its constituents.

In general, the equation relates the Bethe-Salpeter amplitude to the constituent propagators and an interaction kernel that sums an infinite series of interaction diagrams. The kernel carries the effective interaction between the two particles, and the amplitude depends on the total momentum P of the bound state and the relative momentum p. Solving the equation yields the bound-state mass as an eigenvalue and provides the corresponding amplitude, which can be used to compute observables such as decay constants and transition form factors.

Applications of the Bethe-Salpeter framework span multiple fields. In quantum electrodynamics, it describes bound states like positronium. In quantum chromodynamics, it is employed to study mesons and other hadronic bound states, often using truncations such as the rainbow-ladder approximation to make calculations tractable. In solid-state physics, a related Bethe-Salpeter equation describes excitons and optical properties of semiconductors, capturing electron-hole interactions beyond mean-field theories.

Variants and practical considerations include formulations in Minkowski versus Euclidean space, instantaneous or quasipotential reductions, and the impact of truncations on covariance and accuracy. The Bethe-Salpeter approach remains a central tool for connecting microscopic interactions to observable bound-state phenomena in relativistic quantum systems.