Properties of Bethe-Salpeter Wave Functions

Abstract

A boundary condition at $t=\pm \infty$ ($t$ being the "relative" time variable) is obtained for the four-dimensional wave function of a two-body system in a bound state. It is shown that this condition implies that the wave function can be continued analytically to complex values of the "relative time" variable; similarly the wave function in momentum space can be continued analytically to complex values of the "relative energy" variable $p_0$. In particular one is allowed to consider the wave function for purely imaginary values of $t$, or respectively $p_0$, i.e., for real values of $x_4=ict$ and $p_4=i{p}_0$. A wave equation satisfied by this function is obtained by rotation of the integration path in the complex plane of the variable $p_0$, and it is further shown that the formulation of the eigenvalue problem in terms of this equation presents several advantages in that many of the ordinary mathematical methods become available.