Old Problems and New Hopes in $S$-Matrix Theory

1) The problem of constructing a convenient ser of axioms for the /S-matrix directly in terms of the scattering variables and without recourse to conventional field theory is still receiving wide attention by theoretiticians in recent years. The ingredients which go into these attempts are well-known. There is relativistic invariance, unitarity, and finally some degree of analyticity. The hope is that by combining them according to some recipe one obtains the *S-matrix. The trouble with this program is that really no one seems to know how to combine the ingredients in the proper order; moreover there is wide disagreement as to what analyticity really means. It is on this last point that I wish to concentrate the discussion. Obviously Lorentz invariance and unitarity have quite an ambiguous meaning. Every attempt toward 5-matrix theory usually starts with the rather ambitious goal of doing away with conventional field theory. The basic philosophy is opposite to that of constructive field theory whose remarkable progress in recent years has overshadowed any alternative approach. My personal belief however, is that the possibility of different approaches originates from a basic inadequacy of the theory and that ultimately field theory and ^-matrix theory will be unified. Secondly the constructive approach is still facing tremendous difficulties and is still far removed from any physical application; but basically I feel that committing ourselves to any particular philosophy at this early stage is an essentially unsound strategy.