1 Introduction

Who was Ettore Majorana? Probably, the best way to introduce the character is to resort to the words of those who knew him closely.

On Monday 28 March 1938, at the Royal Institute of Physics in Rome, Enrico Fermi was in the laboratory with a student of his, Giuseppe Cocconi, when a phone call arrived from Naples. According to Cocconi [1], Fermi visibly blanched, and the young student promptly asked him what happened, receiving as an answer that Ettore Majorana has disappeared from Naples. However, Cocconi plainly manifested to his teacher that he did not know Majorana, so that Fermi described who Ettore Majorana was in a few words that Cocconi still clearly remembered years later

You see, 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 [and here Cocconi got the impression that Fermi put himself in this category]. 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 [1].

This seems a particularly exaggerated statement, especially for Fermi, who was certainly always very balanced in his judgments. In the following we will then briefly discuss the life and work of the Italian scientist.

2 The Family Background and the First Meeting with Fermi

Ettore MajoranaFootnote 1 was born on the 5th of August 1906 in Catania, Sicily, to Fabio Majorana and Dorina Corso, being the 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, while his father was “only” an engineer, who founded the first telephone company in Sicily and then became chief inspector of the Ministry of Communications. His uncle Quirino was, instead, a renowned experimental physicist, who was president of the Italian Physical Society for many years. In 1921 his family moved to the capital, Rome, and Ettore completed the last years of high school there; then he joined the Faculty of Engineering at the University of Rome, where he excelled.

In 1927, the director of the Institute of Physics in Rome—Orso Mario Corbino—launched a famous appeal to engineering students to suggest that the best ones accept the challenge of moving on to Physics, where extraordinary opportunities were then opening up with the arrival of Fermi to the chair of Theoretical Physics (the first one established in Italy). Emilio Segré, and then Edoardo Amaldi, accepted the challenge, and also tried to convince Majorana to switch to Physics as well. They succeeded, however, only to convince him to go and see Fermi to talk to him. This first meeting was told to us by direct witnesses, that is Amaldi, Segré and also Franco Rasetti, with almost identical and very enlightening detailed descriptions. Here we quote, for example, Amaldi’s version.

Fermi was then working on the statistical model later known as the Thomas-Fermi model. The discussion with Majorana soon turned to the research taking place at the Institute, and Fermi gave a broad outline of the model and showed Majorana reprints of his recent works on the subject, in particular the table showing the numerical values of the so-called Fermi universal potential.

Majorana listened with interest and, after having asked for some explanations, left without giving any indication of his thoughts or intentions. The next day, towards the end of the morning, he again came into Fermi’s office and asked him without more ado to show him the table which he had seen for a few moments the day before. Holding this table in his hand, he took from his pocket a piece of paper on which he had worked out a similar table at home in the last twenty-four hours, transforming, as far as Segré remembers, the second-order Thomas-Fermi non-linear differential equation into a Riccati equation, which he had then integrated numerically. He compared the two tables and, having noted that they agreed, said that Fermi’s table was correct: he then went out of the office and left the Institute. A few days later he switched over to physics and began to attend the Institute regularly [3].

Evidently, Fermi had passed the exam... And then Majorana, reassured by the fact that good things were being done there, finally decided to move on to Physics.

This seems just an amusing anecdote, likely somewhat exaggerated by his friends in their recollections, but, interestingly enough, several years ago we did succeed to reconstruct what Majorana effectively did in that night [4], since he reported the whole calculations in his personal notebooks, now kept in Pisa [5]. In those few hours, Majorana first transformed the Thomas-Fermi equation into an Abel equation, rather than a Riccati one,Footnote 2 but, surprisingly, he did not solve such equation, since this passage served just to prove the existence and uniqueness of its solution. Note that the corresponding theorem was proved by mathematicians only in 1934, while no analytic solution of the Thomas-Fermi equation is (or rather, was) known. Then, Majorana came back to the original Thomas-Fermi equation and transformed it into another, first order differential equation, which he solved by series in terms of only one quadrature. From such solution he finally obtained the table of numerical values mentioned in the account before (and reported in [5]). We note that, according to what is known, Thomas worked for about one month to obtain the approximate numerical solution of his equation, while Fermi took about one week (indeed, his reformulation is cleverer). Interestingly enough, Majorana took just a few hours to find the analytic solution, from which to derive the numerical values.

3 Working in the Fermi’s Group: The Visible Side

After some time since the first meeting with Fermi, Majorana joined the Fermi group in Rome and did make substantial contribution to it, starting from few months after the event just told, when he published (while still undergraduate) his first paper in collaboration with his friend Giovannino Gentile. We will come back to it soon. In December of that same year, he was also invited by Fermi to give a talk at the General Meeting of SIF, the Italian Physical Society, on some application of the Thomas-Fermi model (again reported in his notebooks). And then he finally graduated in July 1929, with a dissertation on the quantum theory of radioactive nuclei, with the first paper on nuclear physics appeared in Italy.

The articles published by Majorana account to just nine, plus one posthumously published in 1942 by Gentile, but each of them is a small (or great) masterpiece. Let’s take a quick browse to try and appreciate something about them, while we refer the interested reader to Refs. [2, 6].

The first published paper on the splitting of the Roentgen and optical terms caused by the spinning electron and on the intensity of the cesium line is the only paper written in collaboration. It contains probably the very first application of the Dirac equation (the Dirac paper appeared, indeed, just few months earlier), which the authors exploit to deduce corrections to the Fermi universal potential. When their predictions are compared with the current experimental results, the agreement is within 5%. Probably not a very striking result, but note that a better approximation came only in 1997.

The second paper is about molecular spectroscopy and concerns the formation of the helium molecular ion; it introduces a generalization of the Heitler-London method inspired by group theory—a topic particularly liked by Majorana, who was among the three or four physicists in the world at that time to use it. The same qualitative results were obtained independently few months after by Linus Pauling, while we have to wait for two years for the same quantitative results. It is also interesting to note that Majorana predictions are now in better agreement with experiments than current theoretical estimates.

The paper N.3 on the presumed anomalous terms of helium is again on atomic spectroscopy (it was the dominant activity of the Fermi group in those years), and again Majorana employed group and symmetry arguments to obtain predictions confirmed only in 1994. The same results were rediscovered some years later by Ta-You Wu, a student of Samuel Goudsmith, who however did not employ symmetry properties, so that he initially made a mistake about spectroscopic assignments, which was corrected only ten years later.

The 4th published paper regarding a pseudopolar reaction of hydrogen atoms contains (in some sense) a revolutionary generalization of the Heitler-London theory, since Majorana introduced ionic structures in homopolar bonds—two apparently conflicting concepts, i.e., ionic bond vs covalent bond. Only decades later the same was started to do for Hartree-Fock eigenfunctions, and such ionic structures are now known as “Majorana structures”. They were first acknowledged by chemical physicists in 2006, the year of the centenary of the birth of Majorana (those scientists participated to a famous conference in Catania, where they discovered what Majorana did).

The paper N.5 on the theory of incomplete \(P^\prime \) triplets introduces what Majorana termed spontaneous ionization (later known as autoionization, according to Shenstone), i.e., a process taking place in the optical range that is completely equivalent to the Auger effect already known in X-ray emission. The generalization of the existing Wentzel theory allowed Majorana to make predictions about controversial levels that were experimentally identified only in 1955, while the complete validation of his theory came even later, in 1970. Interesting enough, Majorana did not publish all his work on the subject. In particular, from his personal notebooks we know that he introduced the so-called quasi-stationary states, that is mixing between a discrete level and a continuum, not only in atomic physics (this was “officially” done by Ugo Fano in 1935, while some other results were discovered only in 1961), but also in the theory of nuclear reactions, with what should be termed Majorana-Fano-Feshbach resonances. Similar phenomena have been very recently discovered also in high-\(T_c\) superconductivity and in ultracold gases.

The sixth paper about oriented atoms in a variable magnetic field is a real gem, with a number of both physical and mathematical results. It concerns non-adiabatic spin flip, and reports what is known as Landau-Zener probability. It is interesting to note that its derivation in modern books follows closely that reported clearer by Majorana, rather than that by Landau, Zener or Stuckelberg. “Mathematical” results concern the reduction of a system with arbitrary angular momentum to that of a spin 1/2 one, discovered only much later, and, in particular, the introduction of a Majorana sphere for representing the states of such a system, which is a generalization of what done only later in 1948 by Bloch for a special case (and known as Bloch sphere). The representation with the Majorana sphere has proved particularly relevant in recent years also in quantum computing, just to quote an example. Instead, “physical” results contained in the Majorana paper served, first of all, for nuclear magnetic resonance (done later by Rabi), the Majorana-Brossel effect, the realization of Bose-Einstein condensation, and so on.

The paper N.7, on a relativistic theory of particles with arbitrary intrinsic angular, is certainly the most important one by Majorana, where he practically founded the theoretical particle physics, based on variational principles and group theory methods. Equally certain, however, is the fact that this is Majorana’s most complicated article: just think of the fact, for example, that it took Pauli and his student Fierz quite a few years to understand it, as evidenced by their correspondence. Majorana discovered here the infinite-dimensional representations of the Lorentz group, only several years later rediscovered by Wigner. Thanks to the intervention by Amaldi, the paper was rediscovered by Fradkin in 1966 and, from that moment on, a large number of works appeared dealing with that theory—including papers by Barut, Sudarshan and even Dirac—till recent times, when Sagnotti recognized that the paper anticipated both Regge’s idea and its eventual realization in the Veneziano amplitude, that is string theory.

The best-known article during Majorana’s life is, on the other hand, paper N. 8 on nuclear theory, where he gave the first, consistent theory of nuclei as composed by protons and neutrons. When the news of Chadwick’s discovery of the neutron reached Rome in March 1932, according to Amaldi, Segré, etc. Majorana promptly elaborated his theory, which he discussed also with Fermi. He soon realized the relevance of Majorana’s results, so that he obviously urged him to publish the theory. Majorana, however, according to his hypercritical view, considered his work incomplete—this is not true, if we look at his personal notebooks—and refused to publish it. When several months later, in July, a paper by Heisenberg appeared reporting the first nuclear theory, Fermi again urged Majorana to publish his theory, and again Majorana refused to do, noting that “Heisenberg has already done everything, even too much”... What did that mean? According to Heisenberg theory, the most stable light nucleus should be deuterium, while it was well known that it was rather helium, i.e., alpha particles. This came from the fact that, while in both theories the forces responsible for the nuclear binding were quantum mechanical exchange forces, Heisenberg considered the exchange of both position and charge of what we term nucleons: practically, he incorrectly imagined the neutron as a compound state of a proton and an electron, exchanging an electron when interacting with another proton, for example. Instead, Majorana correctly considered the neutron not as a compound object, but rather as a “neutral proton”, according to his own wording, so that he based nuclear binding just on the exchange of only position, and just this directly prevented nuclear collapse at short distances, while predicted the stability of the alpha particles, rather than deuterons. Fermi failed to convince Majorana to publish his work, but succeeded to convince him to go to Leipzig (at the beginning of 1933) to work with Heisenberg, and he finally managed to persuade Majorana to publish his correct theory, that is paper N.8. Interesting enough, in all subsequent conferences and congresses, Heisenberg always emphasized Majorana’s contribution rather than his own. Paper N.8 also contains two other remarkable things: that is, the first application of the Thomas-Fermi model to nuclei and the anticipation of the Yukawa interaction potential.

The last paper (N.9) published by Majorana is, instead, his best-known paper now—that containing the famous Majorana neutrino hypothesis. It was originally aimed at providing a reformulation of the Dirac theory that was completely symmetric between electron and positron components. Starting again from a variational principle, Majorana succeeded to obtain a real wave equation, which he properly quantized, also automatically obtaining a cancellation of some divergencies, which were instead only artificially removed in the Dirac theory. Such real equation allowed novel solutions for neutral particles, not limited to neutrinos: Majorana particles are in general spin 1/2 particles that coincide with their antiparticles, again a revolutionary idea, especially for those times. Majorana neutrinos are responsible—if any—of the neutrinoless double beta decay, as envisaged by Furry shortly after; but here the experimental research has not yet given a definite result. Also, we note that the phenomenon of neutrino oscillations (suggested by Bruno Pontecorvo, another former member of the Fermi group in Rome) firstly appeared as neutrino-antineutrino oscillations (rather than flavor oscillations), which are allowed only if neutrinos are Majorana particles, as also assumed in the first phenomenological theory of neutrino oscillations of 1969. Flavor oscillations were directly observed in 1998, while the experimental hunting of Majorana neutrinos still continues. However, Majorana particles appear also in other fields, such as for example in supersymmetry, where they play a key role in the unification of the fundamental interactions, in dark matter, and so on. Even here, the research continues. Also, they appear as Majorana fermions in condensed matter systems: ferromagnetism, superconductivity, topological insulators, also in connection with quantum computing, etc.

Finally, in the list of papers published by Majorana, a tenth paper is included that was published in 1942 by his friend Gentile, although it was written in 1936 upon request of his uncle Giuseppe. It is quite a peculiar paper, on the value of statistical laws in physics and in social sciences, aimed at applying an approach to society within the framework of statistical physics, then by reversing the economic paradigm just developed in those years in terms of a strict determinism, in analogy rather with celestial mechanics. Interestingly, it is the first paper on complex systems, where even this very term appears for the first time. This program was later realized, for example, in the Black and Scholes model of option pricing, for which the two scholars received the Nobel Prize and, among the other things, it served in 1997 to found a new discipline, that is, econophysics. It has been studied also by sociologists, epistemologists, and so on.

4 The Invisible Side: Unpublished Researches

What we have just briefly told is only a small part of what Majorana achieved in the Fermi’s group. Unpublished notes [5, 7] account for thousands and thousands of pages, now kept in Pisa, and, in order to give an idea of what contained in such pages, in the following we will mention just few examples [6].

At the end of the 19201920ss, Majorana developed what was later known as the Pauli-Weisskopf theory of scalar electrodynamics, that is a consistent quantum field theory even for particles without spin 1/2. As recalled by Weisskopf,

it was a tremendous fun in working out something that, at that time, was quite unexpected: that one can get pair creation and pair annihilation without a Dirac equation, also for particles without spin [8].

However, the Majorana theory is also a bit more general than the Pauli-Weisskopf one, even in some applications, as revealed, for example, by the use of general sets of plane waves in the expansion of the field variables, or the adoption of four (for matter particles) plus four (for photons) instead of four plus two operators describing the quanta of the appropriate fields. Majorana also anticipated, in other pages, the typical reasoning of Feynman quantum electrodynamics, where electromagnetic interaction is mediated by particle exchange. Indeed, in an attempt to find a relation between fundamental constants, Majorana gave an interpretation 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.

Quantum electrodynamics was a subject much studied by Majorana. In some notes written in French, following a track left by Heisenberg, Majorana applied the formalism of second quantization to the Dirac’s hole theory, obtaining the general expression for the QED hamiltonian in terms of anticommuting holes quantities.

In his notebooks we also find some generalizations of the Thomas-Fermi model mentioned earlier, with a number of applications to ions and even molecules. And, as recalled above, a detailed study is performed of quasi-stationary states in nuclear physics, aimed at generalizing the Gamov model describing the nuclear alpha decay.

Also, in some notes (probably for a seminar at the University of Naples) Majorana provided a physical interpretation of quantum mechanics which anticipated the Feynman approach in terms of path integrals, independently of the underlying mathematical formulation.

Finally, we will focus on a very interesting alternative theory of ferromagnetism. This follows just the same guidelines of the Heisenberg theory of 1928, that is the framework of the statistical Ising model, although Majorana considered more general wavefunctions written in terms of Slater determinants, in order to include from the beginning the constraints imposed by the Pauli exclusion principle. However, interestingly enough, the Majorana theory is different from (and clearer than) the original Heisenberg one. For example, the partition function of the system is obtained by solving a differential equation, rather than following an algebraic method, and Majorana went well beyond the usual Gaussian approximation, with terms in the series expansion up to order ten! Of course, the theory predicts a non-vanishing critical temperature, as in Heisenberg, but, contrary to what happens in this, Majorana’s theory also predicts that ferromagnetism develops only in systems in more than one dimension, thus anticipating a later key result by Peierls. Furthermore, we also find several applications of the theory to different geometric arrangements of the elementary magnets.

5 Teaching Theoretical Physics

Few words about Majorana as a teacher, rather than a scientist, are finally required for a complete picture of the character. Majorana, indeed, did intend to teach theoretical physics. In 1932 he became a libero docente, something analogous to the English lecturer or German privatdozent, and, as discovered recently [9], in the subsequent years he also proposed repeatedly several courses at the University of Rome: Mathematical methods of quantum mechanics in 1933, Mathematical methods of atomic physics in 1935 and Quantum electrodynamics in 1936. However, none of these courses was effectively delivered, and this probably explains why testimonies like Amaldi and Segré forgot them. By giving a look at the syllabuses of such courses, it is also easily to realize why these courses were not taught: they were extremely advanced for those times and, actually, in Italy just Fermi was practically able to take them.

Majorana finally succeeded in teaching theoretical physics some time later. Indeed, in 1937 a novel competition was opened in Italy for a chair in Theoretical Physics at the University of Palermo, and Majorana was among the applicants, including Gian Carlo Wick, Giulio Racah, Gentile, and others. The judging committee, chaired by Fermi, as soon as it took office analyzed the papers presented by the applicants, and immediately stopped its work, by proposing to the Minister of Education to appoint Majorana as professor of Theoretical Physics for his high and well-deserved repute in one of the Kingdom Universities independently of the competition. After few days, the Minister accepted this proposal, and Majorana obtained the chair of Theoretical Physics at the University of Naples, the second one in Italy after Fermi’s. He moved to Naples and in January 1938 he started the course, which immediately revealed itself as the most advanced one in Italy, even with respect to the Fermi course in Rome [9].

In this first introductory lecture I will briefly discuss the aims of modern physics and the significance of its methods, with particular emphasis on their most unexpected and original aspects with respect to classical physics. Atomic physics, which will be the main subject of my discussion, despite its important and numerous practical applications—together with those of a wider and perhaps revolutionary impact that the future may have in store –, is first of all a science of immense speculative interest for the depth of its investigation that really reaches the extreme roots of natural facts [10].

In addition to quantum mechanics, Majorana also included some lectures on the theory of relativity [10] (this was the first time in Italy for a physics course). It was, then, really a modern course on quantum mechanics, following more or less the same lines as in today courses. The first part of it developed in strict analogy with the Fermi course in Rome (1927–28) followed by Majorana, where the phenomenology of atomic physics and the Bohr-Sommerfeld quantum theory are discussed. Instead the second part accounted for classical radiation theory, the theory of relativity and some quantum phenomena, such as the photoelectric effect, Thomson scattering, Compton effects and the Franck-Hertz experiment. Finally, the third part dealt with Quantum Mechanics, both in the Schrödinger and Heisenberg formulations; here Majorana adopted an approach different from Fermi’s, in analogy with the courses proposed as libero docente mentioned above.

6 Conclusions

Unfortunately, Majorana gave only 21 lectures in Naples, since on March 25, a Friday without lectures, he went to the Institute of Physics leaving a folder with his lecture notes to one of his students. Then he came back to his hotel, which he left in the late afternoon for probably embarking on the ferry to Palermo: from that moment on, we have no more certain news about him.

We will finish this survey with quoting once more Fermi, when he wrote to the Italian Prime Minister Mussolini in order to intensify searches about Majorana’s disappearance.

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 looses 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 minds of our times and the promise of further conquests.

Beyond the rhetoric used in this formal letter, the “further conquests” alluded here are very well evidenced by the large number of Majorana terms that can be found in today literature, from which we report a certainly incomplete list:

Majorana condition, Majorana electron, Majorana equation, Majorana fermion, Majorana field, Majorana formula, Majorana hole, Majorana infinite-component equation, Majorana inversion, Majorana lagrangian, Majorana mass term, Majorana matrix, Majorana neutrino, Majorana nuclear forces, Majorana representation, Majorana sphere, Majorana spinor, Majorana structure, Majorana theorem, Majorana transition, Majorana zero-mode, Majorana-Brossel effect, Majoron, Majorino, ...