F -74941 Annecy Le Vieux Cedex

I try to describe the extremely fruitful interaction I had with Harry Lehmann and the results which came out of the analyticity unitarity programme, especially the proof of the Froissart bound, which, with recent and future measurements of total cross-sections and real parts, remains topical. Dedication I dedicate this paper to Marie-Noëlle Fontaine, the last of the many papers she typed so skillfully for me, wishing her a happy retirement. CERN-TH/2000-138 LAPTH-794/2000 May 2000 To appear in a volume of Communications in Mathematical Physics, dedicated to the memory of Harry Lehmann. URA 1436 du CNRS, associée à l’Université de Savoie. My first meeting with Harry Lehmann was not with his person but with the famous paper of the trio Lehmann-Symanzik-Zimmermann, LSZ [1], the importance of which everybody in the Theory group of Maurice Lévy at Ecole Normale realized immediately. In spite of the fact that I did not know German (I still don’t) I read it, Nuovo Cimento in one hand, dictionary in the other hand (I am a “corrected” left hander). Later Harry visited the Ecole Normale in person and I was immediately impressed. That was the time where there was a wave of interest into what is an unstable particle and Lehmann and Lévy were some of the people involved. I remember also quite vividly our meeting at the La Jolla Conference in 1961 which I attended, coming from CERN. It was, as I realized a posteriori, a very important conference, for physicists and for people (some of the people I met there became my very best friends). I remember that Marcel Froissart gave a talk on his famous Froissart bound [2] on the total cross-section, σt < c log s) , s square of the centre-of-mass energy, and Harry with his very meticulous mind found out that some of the estimates of Froissart were not quite correct, though this did not affect the result (some year later, I published a sum rule on pion-nucleon scattering and Harry discovered a very well hidden mistake. I was very impressed). Anyway we were both admirative of the achievement of Froissart and for me it was a decisive turning point, since I left almost completely for many years potentials and the Schrödinger equation for the study of high-energy scattering and high-energy bounds. The Froissart bound was derived from a combination of the Mandelstam representation [3] where the scattering amplitude is the boundary value of an analytic function of two variables, which implies automatically dispersion relations proved from field theory [4] in one variable as well as the Lehmann ellipse [5] which is probably the most celebrated result of Harry, a fundamental result presented in 10 small pages of Nuovo Cimento (compare with the incredibly lengthy papers on what I would call “rigorous atomic physics” which appeared during the last 15 years!). The trouble with the Mandelstam representation is that nobody was ever able to prove it even in perturbation theory (through some wrong proofs were published!). Both Harry and I were anxious to obtain high-energy bounds with minimal assumptions. A step in this direction was made by Greenberg and Low [6] who used the Lehmann Ellipse to derive a bound on the total cross-section where (log s) was replaced by s(log s). Myself, I realized that the whole Mandelstam representation was not needed to get the Froissart bound and that it was sufficient to replace the Lehmann ellipse by a larger one [7]. Later, in Princeton, Y.S. Jin (a former student of Harry) and I found a way to control the growth of the scattering amplitude for unphysical momentum transfer using positivity [8] but at the time we made no progress on the derivation of the Froissart bound. In the autumn of 1965 I was visiting IHES (Institut des Hautes Etudes Scientifiques) and Harry was there. He attracted my attention on a paper by Nakanishi which contained the claim that the Lehmann Ellipse could be enlarged by using results from perturbative field theory, leading to the obtention of the Froissart bound. As I shall explain later, we tried to make sense of the paper of Nakanishi [9] but in the end could not. Nevertheless it started again my interest in the subject, and after a visit to Cambridge where I learnt that the Nakanishi perturbative domain of analyticity [10] had been obtained independently and in a simpler way by T.T. Wu [11], I came back to CERN and finally succeeded, using positivity properties not terribly different from those I had used with Jin, to enlarge the Ellipse without using perturbation and prove the Froissart bound from first principles. Some-