Properties of BetheSalpeter Wave Functions
Abstract
A boundary condition at t=+/∞ (t being the "relative" time variable) is obtained for the fourdimensional wave function of a twobody 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}=ip_{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 SturmLiouville type. A complete set of solutions for this problem is obtained in the subsequent paper by Cutkosky.
 Publication:

Physical Review
 Pub Date:
 November 1954
 DOI:
 10.1103/PhysRev.96.1124
 Bibcode:
 1954PhRv...96.1124W