A method to define the Green function of a composite particle and its vertex function for the effective coupling with its constituent particles is proposed, and applied to Zachariasen's soluble model. The eigenvalue equation introduces a cutoff mass, and suppresses the high energy behavior of various quantities. The results are compared with the dispersion-theoretical approach, and the position in bound state problems occupied by the S-matrix theory ifi discussed from the standpoint of Green's function. The dispersion-theoretical calculation of the propagator of bound state seems to be meaningless, but the results are as follows: For the model the Z factor of the bound state vanishes only in the sense =-1 , if the cutoff mentioned above is not taken into account, and such a property is not essential to bound state problems. Further, the selfenergy of the state is infinite. The problem of elementarity also is discussed from a formal point of view.From the S-matrix-theoretical point of view, the dispersion technique was successful to get the energy of the composite system and its effective coupling constant as the position of the corresponding pole and its residue, respectively. 6 ) Trying to attack the problems outside the frame of S-matrix theory, however, we can give no definite answer, yet. For instance, the vertex function of the effective coupling between a composite particle and its constituents also can be calculated by the dispersion technique if the composite system is approximated by an elementary particle, to be sure,7) but what does the off-shell quantity calculated by such a method mean? Further, it was proposed 8 ) that the Z-factor of a bound state is zero if it is calculated by regarding the composite system as an elementary particle. There seems to be something in such a conjecture, if one reminds that for elementary particles the Z factor has the meaning of the probability for an elemen~ at Mount Allison University on June 20, 2015 http://ptp.oxfordjournals.org/ Downloaded from at Mount Allison University on June 20, 2015 http://ptp.oxfordjournals.org/ Downloaded from *) The most general form of X is given by X=ax1 + (1-a)xz with any arbitrary constant a, but, in the following, we assume a=l/2 for simplicity. at Mount Allison University on June 20, 2015 http://ptp.oxfordjournals.org/ Downloaded from *) Go(p, X, y) = ( -zYC2n-)-4~d \teiQ(x-Y)[( ~ --q y -J-m 2 -ic rl[( ~-+q r +rn 2 -icf 1 .**) The correspondence between our notation and the one used by Thirring 9 ) is given by Go(s) = iLI 2 (s).
