Ettore Majorana Centennial and Neutrino Legacy S. Esposito Dipartimento di Scienze Fisiche, University of Naples Federico II and I.N.F.N. Sezione di Napoli In the world there are various categories of scientists: people of secondary or tertiary standing, who do their best but do not go very far. There are also those of high standing, who come to discoveries of great importance. But then there are geniuses like Galileo and Newton. Well, Ettore was one of them. Majorana had what no one else in the world has... Enrico Fermi

The family background Quirino Majorana was a renowned experimental physicist who was president of the Italian Physical Society. Ettores father was an engineer who founded the first telephone company in Sicily and who went on to become chief inspector of the Ministry of Communications. E. Majorana was born on 5 August 1906 in Catania, Sicily, to Fabio Majorana and Dorina Corso, fourth of five sons. He had a rich scientific, technological and political heritage..Three of his uncles were chancellors of the University of Catania and members of the Italian parliament

Academic studies Giovanni Gentile jr, Emilio Segrè, Enrico Volterra, Giovanni Enriques and others were some of his friends and colleagues. In 1923 he joined the Faculty of Engineering at the University of Rome, where he excelled.

In 1927 O.M.Corbino, the director of the Institute of Physics at Rome launched a famous appeal to the students of the engineering faculty to entice the most brilliant young minds into studying physics. Segrè and his friend Amaldi rose to the challenge, joining Fermi and Rasettis group and telling them of Ettores exceptional gifts. After some encouragement from Segrè and Amaldi, Majorana eventually decided to meet Fermi in the autumn of that year. The pair immediately started talking about the statistical model of atoms that Fermi was working on, later to be known as the Thomas–Fermi model. The model involves a complicated non-linear differential equation. The analytical solution of the equation was then unknown, but Fermi had managed to obtain a Numerical table of approximate values for it. Majorana carefully followed what Fermi said and, after asking a few questions, left the institute. The following morning he returned to Fermis office and asked for a closer look at the numerical table so that he could compare it with an analogous table he had drawn up the previous evening. Once he had established agreement between the two tables, Majorana noted that Fermis table was correct and left the institute with no further comment. Fermi passed an examination...

What did Majorana in that night? He first transformed the TF equation into an Abel equation, with a very original method that can be used for a large class of differetial equation. To the Abel equation, known theorems on the existence and uniqueness of the solution may be applied... Then, he transformed again the TF equation into another first-order differential equation, whose series solution is explicitly given in terms of only one quadrature. From this solution, Majorana obtained a table of numerical values as accurate as (at least) that of Fermi.

First studies in Physics As if satisfied that Fermi had passed his examination, Majorana decided to leave Engineering and join the Fermi group of the Via Panisperna boys. Majorana made substantial theoretical contributions to the groups research, and in 1928 – while still an undergraduate – published his first paper, in which he calculated the splitting of some spectroscopic terma in Gd, U and Cs due to the spin of electrons. It is one among the first successfull applications of the Dirac equation... (1) Sullo sdoppiamento dei termini Roentgen ottici a causa dellelettrone rotante e sulla intensità delle righe del Cesio, in collaboration with Giovanni Gentile jr.: Rend. Acc. Lincei, 8 (1928) At the end of the same year, Fermi invited Majorana to give a talk at the General Meeting of the Italian Physical Society on some applications of the Thomas-Fermi model. Then on 6 July 1929, Majorana graduated with a master degree in Physics; his dissertation was titled The quantum theory of radioactive nuclei..

Other published papers in In 1931 he published two articles (2), (4) on the chemical bond of molecules and two more papers (30, (5) on spectroscopy. In (3) Majorana anticipated results later obtained by a collaborator of Goudmsith in 1934 on the Auger effect in helium. (2) Sulla formazione dello ione molecolare di He: Nuovo Cimento 8 (1931) (3) I presunti termini anomali dellElio: Nuovo Cimento, 8 (1931) (4) Reazione pseudopolare fra atomi di Idrogeno: Rend. Acc. Lincei, 13 (1931) (5) Teoria dei tripletti P incompleti: Nuovo Cimento, 8 (1931) (6) Atomi orientati in un campo magnetico variabile: Nuovo Cimento, 9 (1932) In 1932, stimulated by Segrè, Majorana published an important paper on the non-adiabatic spin-flip of atoms in a magnetic field, which was extended by Nobel laureate Rabi in 1937 and by Bloch and Rabi in This paper contains an independent derivation of the well-known Landau-Zener formula (1932). It also introduces a mathematical tool for representing spherical functions (Majorana sphere) rediscovered only in recent times.

But.the most important paper of 1932 is that concerning a relativistic field theory of particles with arbitrary spin, where Majorana introduced for the first time the unitary infinite-dimensional representation of the Lorentz group, anticipating works by Nobel laureates Wigner (in 1938) and Dirac (in 1945). (7) Teoria relativistica di particelle con momento intrinseco arbitrario: Nuovo Cimento,. 9 (1932) The representations of the Lorentz group are, except for the identity representation, essentially not unitary, i.e., they cannot be converted into unitary representations by some transformation. The reason for this is that the Lorentz group is an open group. However, in contrast to what happens for closed groups, open groups may have irreducible representations (even unitary) in infinite dimensions. In what follows, we shall give two classes of such representations for the Lorentz group, each of them composed of a continuous infinity of unitary representations.

1933: a theory with the neutral proton In March 1932, after that Chadwick announced the discovery of the neutron, Majorana revealed to his friends and colleagues in Rome that he had built a theory of light nuclei based on the concept of exchange forces. Although incouraged by Fermi, he did not publish his theory, which was independently elaborated by Heisenberg. But... According to Heisenberg, the underlying nuclear forces should be interpreted in terms of nucleons exchanging spinless electrons (implicitly: n = p + e)... Instead, in Majoranas view, the neutron was pictured as a neutral proton, and the erroneous experimental consequences of the Heisenberg model were quickly recognized by Heisenberg himself and others. In many several occasions, the Nobel lauerate Heisenberg only marginally mentioned his own contribution, while annoyingly pointing out that of Majorana... (8) Über die Kerntheorie: Zeits. Phys. 82 (1933) ; Sulla teoria dei nuclei: Ricerca Scientifica, 4 (1933)

The most famous paper on the neutrino In 1937, probably after being invited by Fermi to compete for a full professorship, Majorana published (but the theory was elaborated some years before) what was to become his most famous paper, in which he introduced the so-called Majorana neutrino hypothesis. The problem: The Dirac theory is symmetric with respect to the electron and the positron, but the field quantization method (used in order to cancel divergencies) doesnt. This problem may be solved with a generalization of the Jordan-Wigner method. The cancellation of infinite constants is required by the symmetrization of the theory, which is already implicit in the adopted form of the variational principle. (9) Teoria simmetrica dellelettrone e del positrone: Nuovo Cimento 14 (1937)

The Majorana neutrino hypothesis was revolutionary because it argued that the antimatter partner of a given matter particle could be the particle itself. This was in direct contradiction to what Dirac had successfully assumed in order to solve the problem of negative energy states in Quantum Field Theory. Majorana is conscious that, for charged particles, the advantage is purely formal, but... the situation may be different for neutral particles... With unprecedented farsightedness, Majorana proposed that the neutrino, which had just been postulated by Pauli and Fermi, could be such a particle

Unpublished researches The largest part of the Majoranas work was left unpublished... Master thesis 5 Notebooks (Volumetti) 18 Booklets (Quaderni) 12 folders with spare papers Lecture notes for the course on Theoretical Physics at the University of Naples

Just very few examples... Anticipating Feynman Q.E.D.... In an attempt to find a relation between fundamental constants, Majorana gave an interpretaion of the electromagnetic interaction in terms of particle exchange: the space around charged particles is quantized, and two electrons interact between them by means of the exchange of particles from one to another. Anticipating Fano quasi-stationary states... Majorana was the first to study Nuclear Physics in Rome (see also master thesis). In the study of (α,p) reactions on light nuclei, he generalized the Gamov model with the introduction of quasi-stationary states in order to describe energy states composed of continuous and discrete terms. Generalization of the Thomas-Fermi model to ions and molecules and its applications...

Anticipating Feynman path integral approach to QM... In some notes (probably prepared for a seminar at the University of Naples), Majorana gave a physical interpretation of Quantum Mechanics which anticipated of several years the Feynman approach in terms of path integral, independently of the underlying mathematical formulation. and probably more... differently from what happens in Classical Mechanics for the single solutions of the dynamical equations, in general it is no longer true that S will be distinct from S. We can realize this easily by representing S with a set of classical solutions, as seen above; it then suffices that S includes, for any given solution, even the other one obtained from that solution by applying a symmetry property of the motions of the systems, in order that S results to be identical to S. redundant counting in the integration measure in gauge theories?

However, even in the case of standard or well-known topics, they were never faced off in an obvious way: Group-theoretical description of Quantum Mechanics in terms of symmetries... Relativity... Radiation theory... His writings are a goldmine of seminal new physical and mathematical ideas and suggestions, all still quite stimulating and useful for present-day research.

Epilogue Able at the same time to develop audacious hypothesis and criticize acutely his work and that of others; very skilled calculating man, a deep-routed mathematician that never loses the very essence of the physical problem behind the veil of numbers and algorithms, Ettore Majorana has at the highest level that rare collection of abilities which form the theoretical physicist of very first-rank. Indeed, in the few years during which his activity has been carried out, until now, he has been able to outclass the attention of scholars from all over the world, who recognized, in his works, the stamp of one of the greatest mind of our times and the promise of further conquests. Enrico Fermi

Published articles (1) Sullo sdoppiamento dei termini Roentgen ottici a causa dellelettrone rotante e sulla intensità delle righe del Cesio, in collaboration with Giovanni Gentile jr.: Rendiconti Accademia Lincei, vol. 8, pp (1928). (2) Sulla formazione dello ione molecolare di He: Nuovo Cimento, vol. 8, pp (1931). (3) I presunti termini anomali dellElio: Nuovo Cimento, vol. 8, pp (1931). (4) Reazione pseudopolare fra atomi di Idrogeno: Rendiconti Accademia Lincei, vol. 13, pp (1931). (5) Teoria dei tripletti P incompleti: Nuovo Cimento, vol. 8, pp (1931). (6) Atomi orientati in un campo magnetico variabile: Nuovo Cimento, vol. 9, pp (1932). (7) Teoria relativistica di particelle con momento intrinseco arbitrario: Nuovo Cimento, vol. 9, pp (1932). (8) Über die Kerntheorie: Zeitschrift für Physik, vol. 82, pp (1933); Sulla teoria dei nuclei: La Ricerca Scientifica, vol. 4 (1), pp (1933). (9) Teoria simmetrica dellelettrone e del positrone: Nuovo Cimento, vol. 14, pp (1937). (10) Il valore delle leggi statistiche nella fisica e nelle scienze sociali, (posthumous, edited by G. Gentile jr.): Scientia, vol. 36, pp (1942).

References E. Recami, Il caso Majorana (Di Renzo, Rome, 2004) S. Esposito, Fleeting genius, Physics World 19 (2006) 34 S. Esposito, E. Majorana jr, A. van der Merwe, E. Recami, Ettore Majorana: Notes on Theoretical Physics (Kluwer-Springer, New York, 2003) E. Di Grezia, S. Esposito, Fermi, Majorana and the statistical model of atoms, Found. Phys. 34 (2004) 1431 S. Esposito, Majorana solution of the Thomas-Fermi equation, Am. J. Phys. 70 (2002) 852 S. Esposito, Again on Majorana and the Thomas-Fermi model: a comment to physics/ , arXiv:physics/ S. Esposito, Majorana transformation for differential equations, Int. J. Theor. Phys. 41 (2002) 2417 E. Majorana, Lezioni di Fisica teorica, edited by S. Esposito (Bibliopolis, Naples, 2006) A. Drago, S. Esposito, Ettore Majoranas course on Theoretical Physics: the Moreno Lecture Notes, arXiv:physics/ , to be published in Physics in Perspective A. De Gregorio, S. Esposito, Teaching Theoretical Physics: the cases of Enrico Fermi and Ettore Majorana, arXiv:physics/ A. Drago, S. Esposito, Following Weyl on Quantum Mechanics: the contribution of Ettore Majorana, Found. Phys. 34 (2004) 871 S. Esposito, A peculiar lecture by Ettore Majorana, Eur. J. Phys. 27 (2006) 1147 S. Esposito, Majorana and the path-integral approach to Quantum Mechanics, arXiv:physics/ ; to be published in the Annales de la fondation Louis De Broglie A. Drago, S. Esposito, A logical analysis of Majorana papers on Theoretical Physics, Electron, J, Theor. Phys.3 (2006) 249 S. Esposito, Four variation on Theoretical Physics by Ettore Majorana, Electron, J, Theor. Phys.3 (2006) 265 S. Esposito, Un manoscritto inedito in francese di Ettore Majorana, arXiiv:physics/