On the Construction of Potential in Field Theory

Abstract

The general formal construction of potential in the field theory is given. It is constructed so as to conserve the normalization of the wave function, irrespective of the switch-off or -on of the potential. The potential is anaiyzed into its normal part and probability operator. The probability operator is shown to be related to the probability of the system staying in the same state as it does when the interaction is switched off.From the probability character of the probability operator, we can get a measure of the applicable region of the power series expansion of potential in the coupling constant. The intense meson cloud around the nucleon gives rise to strong singularities in the normal part of the potential, but, on the other hand, the probability of the nucleon to be bare becomes much smalier, and thus it turns out that the actual potential has a lesser singularity. Both potentials derived previously by Taketani-Machida-Onuma1l (T.M.O.) and its corrected form by Brueckner-Watson2l (B.W.) are shown to be two extreme cases in handling this probability operator, but the latter one is more appropriate qualitatively in many cases. The reasonable way of constructing the potential is given in the following paragraphs and the justification of this argument will be illustrated in a separate paper for the pseudo-scalar meson theory~l § 1. General remark There are various ways to get the field theoretical potential, most of which give the same results in the so-called static approximation. The method of using the canonical transformation was first applied to this probelm by M¢ller and Rosenfeld in a general form and developed by many authors 4l • This leads necessarily to the Taketani-Machida-6numa 1 ) potential, as shown by Nishijima 4l • Several authors have also attempted to derive the potential from the scattering matrix; usually, however, this method has some very ambiguous points.Especially, the derivation of the potential needs the off-energy-shell matrix elements of the scattering matrix; But usually only the diagonal-energy-shell matrix elements are used expecting that the off-and diagonal-elements are not so different in the adiabatic limittt. The third one is due to Tamm and Dancoff6) and formulated in a com'

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