Properties of Bethe-Salpeter Wave Functions

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.

In an especially simple case ("ladder approximation" equation for two spinless particles bound by a scalar field of zero rest mass) an integral representation method is presented which allows one to reduce the problem exactly (and for arbitrary values of the total energy of the bound state) to an eigenvalue problem of the Sturm-Liouville type. A complete set of solutions for this problem is obtained in the subsequent paper by Cutkosky.