On Saturday March 26, 1938 the director of the Institute of Physics at the University of Naples in Italy, Antonio Carrelli, received a mysterious telegram. It had been sent the previous day from the Sicilian capital Palermo, some 300 km across the Tyrrhenian Sea, and read: “Don't worry. A letter will follow. Majorana.” That same Saturday, Ettore Majorana – who had just been appointed as full professor of theoretical physics at the university at the age of 31 – had not turned up to give his three-weekly lecture on theoretical physics. By Sunday the promised letter had reached Carrelli. In it Majorana wrote that he had abandoned his suicidal intentions and would return to Naples, but it revealed no hint of where the illustrious physicist might be. The picture was quickly becoming clear: Majorana had disappeared.

Worried by these circumstances, Carrelli called his friend Enrico Fermi in Rome, who immediately realized the seriousness of the situation. Fermi was working in his laboratory with the young physicist Giuseppe Cocconi at the time. In order to give him an idea of the seriousness of the loss to the community of physicists caused by Majorana's disappearance, Fermi told Cocconi:

Fortunes and misfortunes of a genius

Physicists working in several areas of research know quite well the name of Ettore Majorana, since it is currently associated with fundamental concepts like Majorana neutrinos in particle physics and cosmology or Majorana fermions in condensed matter physics. For non-specialists, the name of Ettore Majorana is usually intimately related to the fact that he disappeared rather mysteriously in 1938 and was never seen again.

From January to March 1938 – as we have seen in Chapter 1 – Majorana delivered his only lectures on quantum mechanics at the University of Naples, where he obtained a position as a full professor of theoretical physics. He prepared a set of lecture notes [26] for his students (see Section 13.5), and, among the original manuscripts of these lectures, we find some additional spare papers that cannot be considered as notes for academic lectures, even for an advanced course such as that taught by Majorana. Instead, as suggested in Refs. [328, 329], they probably refer to a seminar or conference held at the University of Naples, whose main topic was likely the theoretical interpretation of the molecular bonding in the framework of quantum mechanics; the notes were prepared by Majorana for his own personal use.

Scientific interest in this dissertation, however, focuses not on its main topic, nor on the calculations (which are practically absent from the manuscript), but rather on the interpretation given by Majorana to key concepts of the quantum theory, i.e. the concept of quantum state and the direct application of the quantum theory to the particular case of molecular bonding. The manuscript indeed discloses a peculiar cleverness on the part of Majorana in referring to this pivotal argument of quantum mechanics, and also reveals the concept was at least ten years ahead of its time. This link had been already noted earlier by Nicola Cabibbo [330], who saw in the Majorana manuscript a vague and approximate anticipation of the idea underlying the Feynman interpretation of quantum mechanics in terms of path integrals. A more analytic study has revealed, however, some intriguing surprises, upon which we will focus here.

In addition to the historical interest of Majorana's manuscript, we stress also the particularly powerful didactic method used by him on the given subject: his original presentation of the quantum-mechanical problems is exceedingly useful in teaching now those or related topics. For this reason, in the Appendix we will report the entire text (translated from the Italian) as prepared by Majorana.

At the end of 1920s, the activities of the Fermi group in Rome focused almost exclusively on atomic physics and related subjects, as did the other major research centers throughout Europe. After having obtained several significant results in atomic and molecular spectroscopy, around 1930 some people in the group realized that such a field could no longer offer any great prospects, and Fermi himself predicted that the interest would shift from the study of the external parts of the atom to its nucleus. However, no general consensus was reached initially inside the group, and the spectroscopic activities continued, along with the acquisition of theoretical knowledge and experimental technologies required by nuclear physics, until 1934, when the Fermi group discovered the radioactivity induced by neutrons and the important properties of slow neutrons [231].

Majorana also actively participated in scientific discussions within the Rome group [10, 11], but, as we have seen in Section 2.6, we are left with only one published paper of his on nuclear physics topics (that is, paper(s) N.8), dealing with the Heisenberg–Majorana exchange forces in nuclei, while nothing was published of his previous works on these matters. Remarkably, Majorana was the first in Rome to study nuclear physics; in 1929 (on July 6) he defended his degree thesis on “The quantum theory of radioactive nuclei,” and his studies on such topics continued for several years, independent of the main line of research carried out by the Fermi group. In this chapter we will focus on these (unpublished) studies [232], performed around 1929–30, devoted mainly to nuclear reactions induced by α particles.

Probing the atomic nucleus with α particles

The first steps of nuclear physics were taken in the realm of radioactive phenomena, among which those involving α emission were readily recognized to induce a profound modification of the atomic nucleus by virtue of the large mass of the emitted particles.

Around 1933–34, Majorana was invited to a conference in Russia [1, 320]:

The conference mentioned in this letter was either the First All-Union Conference on Nuclear Physics, held in September 1933 at the Leningrad Physico-Technical Institute, or, more likely, the International Conference of Theoretical Physics that took place in May 1934 in Kharkov [321]. The uncertainty derives from the peculiar fact that, contrary to what is stated in the letter, and for some unknown reason, Majorana did not participate in any conferences in Russia. However, according to the reconstruction in Ref. [321], we probably have the (incomplete) text that Majorana prepared for such a conference, which deals with quantum electrodynamics, a frontier research topic tackled by many theoretical physicists of the time.

In these notes we find only a sketch of the quantum theory developed by Majorana, but it is nevertheless quite interesting due to the particular physical interpretation behind it, so we will give an account of it in this chapter. Majorana repeatedly studied quantum (and classical) electrodynamics and related applications from different points of view, and some of the material discussed in the notes for the conference is just a re-elaboration of previous studies that can be found in his personal notebooks. As a matter of fact, Majorana obtained several original, though never published results, only some of which were independently discovered by others.

The relevance of what is discussed in this chapter, however, does not lie mainly in Majorana's actual findings, as was the case for earlier chapters, but rather in the original methods developed and the continuous attention paid by him to the physical interpretation of the theory, which is particularly intriguing when dealing with quantum electrodynamics. This attitude was, nevertheless, an intrinsic characteristic of Majorana, for whom theory was inextricably related to phenomenology.

The first meeting between Majorana and Fermi, as recalled in Chapter 1, focused on the statistical model of atoms, known as the Thomas–Fermi model. Fermi developed the idea of applying his statistical method (the Fermi–Dirac statistics) to the completely degenerate state of electrons in an atom, in order to evaluate the effective potential acting on the electrons themselves. He was aware of the fact that, since the number of electrons involved is usually only moderately large, the results obtained would not present a very high accuracy, but the method turned out to be very simple, and gave an easy-to-use expression for the screening of the Coulomb potential accounted for by electrons as a whole. This was one of Fermi's favorite issues, and the main activity of the Fermi group at the Institute of Physics in Rome from 1928 to 1932 was essentially devoted to this subject.

As testified by a number of his notes [17] (and as announced, very briefly, at a conference [203]), Majorana repeatedly studied this same subject, contributing some very important results in both atomic and molecular physics [204, 205, 206]. In the following we shall see that the amusing anecdote recounted in Chapter 1 was only the tip of an iceberg that Amaldi, Segrè, and Rasetti were allowed to see (even without a full understanding of it), since – true to his style – Majorana did not publish anything on the subject.

Fermi universal potential

The main idea of the statistical model of atoms is to consider the electrons around the atomic nucleus as a gas of particles at absolute zero temperature, all obeying the Pauli exclusion principle.

Thomas–Fermi equation

The limiting case of the Fermi–Dirac statistics for strong degeneracy applies to such a gas, and Thomas [207] and Fermi [208] independently realized that its potential energy −eV satisfied a second-order inhomogeneous differential equation with a term proportional to V3/2:

This equation allowed the evaluation of the potential inside an atom with atomic number Z, using boundary conditions such that for the radius r → 0 the potential becomes the Coulomb field of the nucleus, V(r) → Ze/r, while for r → ∞ it vanishes: V(r) → 0.

At the October 1930 Solvay conference in Paris, Pauli made a comprehensive review of magnetism, including the intriguing results obtained by Heisenberg [246], Bloch [247, 248], and Slater [249] on the phenomenon of ferromagnetism. That meeting was also attended by Fermi, who, very likely, spread the novel results obtained by those scholars within his group in Rome, and Majorana, used as he was to spending time studying papers published by Heisenberg (and a few other authors), probably became (or was already) aware of his theory of ferromagnetism, in which exchange forces again play a key role in the understanding of the phenomenon. Fermi and his close collaborators never worked or published anything on this subject, but several people visiting the Institute of Physics in Rome in the early 1930s did, including, primarily, Bethe [250], who introduced the famous Bethe ansatz for the case of two interacting spin waves. Other minor (to some extent) contributions came from Inglis [251, 252, 253], who visited Rome in 1932, and from Majorana's friend Gentile [254], who provided a thorough justification of Bloch's results on elementary groups (Uebergangsgebiet) of aligned spins inside a ferromagnet [255].

As a matter of fact, a theory of ferromagnetism analogous to Heisenberg's can be found among Majorana's personal research notes: it was probably developed by Majorana at the end of the 1920s or early in the 1930s, but was never published by the author, so it remained unknown until recently [256]. As in Heisenberg's theory, the framework of Majorana's theory is that of the statistical Ising model [257], where the forces responsible for ferromagnetism are derived from the quantum-mechanical exchange interactions. However, Majorana's theory differs substantially from Heisenberg's in the methods employed and in the results obtained, though it describes successfully the main features of ferromagnetism. The key equation for the spontaneous mean magnetization and the expression for the Curie temperature deduced by Majorana are different from those deduced in the Heisenberg theory (and in the original phenomenological Weiss theory), and the novel theory presents also a peculiar prediction about the practical realization of ferromagnetism which avoids arbitrary assumptions based purely on experimental observations.

Majorana was involved in studying helium on at least two occasions, as we have seen in Chapter 2, namely for his papers N.2 and N.3 published in 1931. However, in his unpublished personal study [17] and research [18] notes we find several additional interesting results and methods directly related to the basic two-electron problem, dating back to 1928–9, some of which are apparently preliminary studies for his published papers (especially paper N.3). The theoretical contributions and numerical calculations, including empirical relations, contained in those notes were largely deduced by making recourse to novel methods not yet in the literature (both of that time and in present day studies), and while part of those numerical results were (and are) inaccurate when compared with the experimental data, the novel methods are nevertheless quite useful in the frontier research related to atomic and nuclear physics [169].

A long-lasting success for quantum mechanics

Following the discovery of the atomic nucleus [170], in 1913 Bohr succeeded [171, 172, 173] in explaining the energy levels of the hydrogen atom in terms of quantization of the action for the classical Kepler orbits. Numerous attempts were then explored to explain the ground state of helium by quantizing different two-electron periodic orbits in a similar manner, but without success. For example, Bohr first discussed a simple model where both electrons in the helium atom move along the same circular orbit and are located at the opposite ends of a diameter [171, 172, 173]. This followed the 1904 proof by the Japanese physicist Hantaro Nagaoka [174, 175] (obviously in the framework of classical mechanics) that such motion is mechanically stable (for sufficiently large attractive forces) and realizes the lowest possible energy. In general, the attempts to quantize the helium atom in the Bohr–Sommerfeld theory [176, 177] were always based on the assumptions that the ground state is related to a single periodic orbit of the electron pair, and that the electrons move on symmetric orbits with equal radii at all times.

Among the very few books owned by Majorana (fewer than 30 volumes) we find Weyl's Gruppentheorie und Quantenmechanik in its first German edition (1928) [16]. Indeed, as we have already recalled, the clear group-theoretical approach pioneered by Weyl greatly influenced the scientific thought and work of Majorana. Group theory was revealed to be the appropriate mathematical framework for treating symmetries in quantum mechanics, as established in the books by Weyl [16], Wigner [263], and van der Waerden [264], but, despite its recognized relevance, the group-theoretical description of quantum mechanics in terms of symmetries was ignored by almost all theoretical physicists of the time (see Section 7.1), and only in recent times has it been included in physics textbooks. As a matter of fact, although almost every physicist had a copy of Weyl's book in the 1950s, the extensive use of group theory in physics research started, during those years, only in nuclear and particle physics.

Remarkably, such indifference did not apply at all to Majorana: the brilliant works on group theory and its applications of that mathematician, as well as those of Wigner, left an unambiguous and fruitful impression on the work of our author. Although this impression is recognizable on any technical page of the present book, in this chapter we will discuss in some detail several studies performed by Majorana on this subject, ranging from exquisitely mathematical results to genuinely physical applications, as was customary for him. Most of these results are not original, as Majorana reproduced what had been obtained by others (mainly by Weyl), but it is nevertheless interesting to see how Majorana obtained those results, and how he used group theory in order to obtain new physical results. As explained in Ref. [265], Majorana can be considered a faithful follower (probably, the only follower) of Weyl's thinking who went well beyond the path taken by Weyl himself, the main difference being in the approach adopted: mathematical for Weyl, physical for Majorana.