Normalization Condition for the Bethe-Salpeter Wavefunction and a Formal Solution to the Bethe-Salpeter Equation

Abstract

By the use of an inhomogeneous Bethe‐Salpeter equation, a normalization condition for the Bethe‐Salpeter wavefunction is obtained. This condition requires the normalization integral to be positive. A formal solution is obtained in the ladder approximation, and convergence of the normalization integral is proved by the use of this solution. This solution is also used to prove a dispersion relation for the vertex function of the compound particle and to give an approximate solution The positiveness of the normalization integral is proved in the nonrelativistic limit. The bound state of nucleon and antinucleon is studied in the ladder‐chain approximation and it is found that the normalization condition gives a finite wavefunction in spite of divergency of the normalization integral.