Bethe-Salpeter approach for relativistic positronium in a strong magnetic field

Abstract

We study the electron-positron system in a strong, constant, and homogeneous magnetic field using the differential Bethe-Salpeter equation in the ladder approximation. We derive the fully relativistic two-dimensional form that the four-dimensional Bethe-Salpeter equation takes in the limit of an asymptotically strong magnetic field. A maximum value for the magnetic field is determined, which provides the full compensation of the positronium rest mass by the binding energy in the maximum symmetry state and the vanishing of the energy gap separating the electron-positron system from the vacuum. The compensation becomes possible owing to the falling-to-the-center phenomenon that occurs in a strong magnetic field because of the dimensional reduction. The solution to the Bethe-Salpeter equation corresponding to the vanishing energy momentum of the electron-positron system is obtained.