ar X iv : h ep - p h / 98 03 30 7 v 3 1 1 M ar 2 00 9 Wick - Cutkosky model : an introduction

A general pedagogical survey of the basics of the Bethe-Salpeter equation and in particular Wick-Cutkosky model is given in great technical details. The Bethe-Salpeter approach to the relativistic two-body bound state problem assumes that all needed information about a bound state |B > is contained in the B-S amplitude Φ(x 1 , x 2 ; P B) =< 0|T φ 1 (x 1)φ 2 (x 2)|B > (1) and in its conjugate ¯ Φ(x 1 , x 2 ; P B) =< B|T φ + 1 (x 1)φ + 2 (x 2)|0 >. (2) Here P B is the bound state 4-momentum and field operators are in the Heisen-berg representation. Note, that < B|T φ + 1 (x 1)φ + 2 (x 2)|0 >=< 0| ¯ T φ 1 (x 1)φ 2 (x 2)|B > * ¯ T being antichronological operator. So we can say that ¯ Φ is obtained from Φ through time reversal. It is clear that instead of individual x 1 , x 2 coordinates more useful are variables which characterize two particle relative motion and the motion of the system as a whole. The coordinate corresponding to the relative motion is of course x = x 1 − x 2. Then any linear independent combination X = η 1 x 1 + η 2 x 2 can