In Retrospect: a Personal View on Moshé Flato's Scientific Legacy

In retrospect: A personal view on Moshe´ Flato’s scientific legacy Daniel Sternheimer When a talented sculptor is confronted with a block of marble, he is using tools that come from centuries of tradition and the training which he received from his masters. But from the first stroke he already has, in the back of his mind, the design of the sculpture which he wants to extract from that block. Creating a masterpiece is a long and painful process, interspersed with surprises due to the irregularities of the physical properties of the noble material he is using, and the design continues to evolve with the artist’s maturing and subject to external influences. Only gradually does the shape emerge, at first visible to experts, then to the general public. Beyond the technicalities, however important they may be, Moshé’s scientific life was a perfect illustration of this idea. His masterpiece is not finished. Much polishing remains to be done, new elements still need to be added to the main themes which he conceived, and entirely new themes are now beginning to emerge from them. But a very partial view of what he had in mind can already be described. This is what I shall attempt here, based on my experience of having, for about 35 years, participated with others in the elaboration of his ideas. All those who have been close to him know that he used to call me affectionately an “idiot” more often than he used that expression for anybody else; this usually happened when my formulation of one of his ideas was not to his taste and needed to be corrected, either because he did not like some of the ingredients I was adding or when, while striving for too precise a formulation, I obscured an essential point which he had purposedly stated ambiguously. François Mitterrand used to say, paraphrasing the Cardinal de Retz, “On ne sort de l’ambigu¨ıtéqu’àson détriment”. In his case, this was smart politics, in Moshé’s, using seemingly ambiguous statements was the only way by which he could transmit the deep feeling which he had without the distortion of too precise, confining words. I hope that now, with him not around, my formulation will be faithful enough to the main ideas which he developed and often initiated. My purpose here is to show how, in retrospect, one can look at the genesis and evolution of ideas and concepts in some of the fields where Moshé made an impact that has only begun to be fully recognized. I shall refrain from entering into technicalities, because they are covered in part in the present volume and in recent review papers, and in order to make the ideas understandable to a general, scientifically educated audience. Towards the end of the 19th century, physics seemed to have achieved a com- plete understanding of the world, with classical mechanics describing the motion of rigid bodies, electromagnetism waves and charged particles, and the Lorentz force their interactions. It was only a plateau because then the ‘deformation dae- [31] G. Dito and D. Sternheimer (eds.), Conférence MoshéFlato 1999, Vol. 1, 31 – 41. c 2000 Kluwer Academic Publishers. Printed in the Netherlands. 32 mon’ struck. In 1881 an American physicist, Albert Michelson, came to Berlin and tried to observe the movement of the earth in the ‘ether’ by measuring the velocity of light in different directions, using the interferometer which he had developed. He found no differences, which did not conform to classical mechanics. In such a case, in Europe, the first reaction was and remains – as Charles Schulz wrote in a ‘Peanuts’ cartoon –, “my mind is made up, don’t bother me with facts”. That is how the first observation of neutral currents at the CERN, ten years before their theoretical prediction, was deemed a mistake and never published. However facts and some American scientists are stubborn. In 1887, with Edward Morley, Michel- son performed a more refined experiment in Cleveland and the fact was proved. So there was a paradox, because the speed of light is a limit. The so-called black- body radiation provided another paradox until in 1900, as a last resort, Max Planck proposed the quantum hypothesis, the energy of light is not emitted continuously but in quanta, proportional to its frequency. He wrote h for the proportionality constant which bears his name. At the same time the notion of symmetry started to appear in mathematics and in physics, developments in both sciences being very closely interrelated all through the twentieth century. In physics, equations and quantities are usually invariant under some transformation groups. The theory of the latter was system- atized by Sophus Lie and developed further by many, among them Elie´ Cartan and Henri Poincaré, to remain within that time range and in mathematics. Emphasizing this invariance property has many advantages, but we shall not elaborate on that aspect here because it has been developed in many books. In any case, this is a concept that Moshélearned from his teacher Giulio Racah, and first applied to molecular spectroscopy in his celebrated M.Sc. thesis, completed in 1960 under Racah but only published in 1965 [13], and in his unpublished Ph.D. thesis on applications of group theory to nuclear physics, a problem involving the restriction of representations from unitary to symplectic groups, which Moshéalso completed under Racah before coming to France in 1963 but never defended. As a conse- quence of the expertise which he gained in these applications of group theory to physics, he took with more than a grain of salt the quite naive applications that were fashionable among particle physicists in the sixties and, in a much more elaborate manner, still influence the present interpretation of experimental data. This can already be seen in his 1965 French D.Sc. thesis [14]. He never hesitated to express these views, which many considered blasphemous, in public, and eventually his outspokenness obliged him to seek “scientific asylum” in the French mathematical community while staying in contact with physicists around the world who dared and could afford to hold independent opinions. The notion of symmetry guided Moshéthroughout his scientific career, both as a mathematician and as a theoretical physicist, and led him to make a number of significant contributions. Even on the mathematical side, the motivation for the study of symmetry comes from physics and the mathematical theory has physical applications. In this connection I can mention two directions which his work, in 33 collaboration with (among others) his D.Sc. student and close friend since 1968, Jacques Simon, took. The first is the theory of analytic vectors in Lie group representations [25], suggested by the problems that arise in passing from Lie algebras to Lie groups. The second is the cohomological study of nonlinear representations [24] of covariance groups of nonlinear partial differential equations, which leads to important mathematical developments with nontrivial physical consequences such as the “tour de force” on Maxwell-Dirac equations [26], the very basis of classical electrodynamics. A lesser known, third direction, is the notion of supersymmetry which one can find already in an embryonic form – close to the Wess-Zumino Poincarésupersymmetry – in his 1970 paper [18]. In 1905 Einstein solved the first paradox that we described above when he 3 4 µ ¡ Ê ¡ Ê showed, in our terminology, that the Galilei invariance group SO´3 (space rotations, addition of velocities and space-time translations) of Newtonian me- 4 ; µ ¡ Ê chanics had to be deformed into the Poincarégroup SO´3 1 , where velocities do not add linearly, of relativistic mechanics. That same year Einstein contributed, in addition to a third basic paper, on the Brownian motion, to the solution of the second paradox by his theory of the photoelectric effect. Incidentally, that theory was the main contribution recognized by the Nobel Prize in Physics which was awarded to him only in 1922. Revolutionary ideas take time to be recognized. Around 1920, an “agrégéd’histoire”, Prince Louis de Broglie, was introduced to the photoelectric effect, together with the Planck-Einstein relations and the theory of relativity, in the laboratory of his much older brother, Maurice duc de Broglie. This led him, in 1923, to his discovery of the duality of waves and particles, which he described in his celebrated 1925 thesis. After Davisson and Germer’s observation of the diffraction of electrons in 1927 confirmed his theory, Louis de Broglie was awarded the Nobel Prize in 1929. Moshéhad been sent to Louis de Broglie when he came to Paris in October 1963. There was immediate sympathy between these two very different but pro- found personalities. Moshéused to say that Louis de Broglie was an extraordinarily well-adjusted schizophrenic, and that without pejorative connotation, because he could see single physical phenomena as dualities. The wave-particle duality is one example, but there are others, like his theory of fusion, in particular, his short- lived neutrino theory of light. As de Broglie formulated that theory, with photons composed of two neutrinos, it was soon ‘shot down’, and justly so, but the general idea was vindicated, in a way, by Moshéwhen he showed in 1988 [19] that it was possible to formulate quantum electrodynamics with photons composed of two scalar singletons. The latter ‘particles’, in fact, massless particles in a 2+1 dimensional space-time at infinity in the anti-de Sitter universe, are just one step below neutrinos (at the limit of unitarity) in the diagram of representations of the ; µ anti-de Sitter group SO´3 2 , which is the last possible deformation of the Poincaré group in the category of Lie groups when, due to a tiny negative curvature in the microworld, space-time translations do not add linearly.

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