What are Majorana particles? They are massive fermions that are their own antiparticles. In this chapter we will concentrate on spin-1/2 Majorana particles, though fermions of higher spin can also be of Majorana nature. Obviously, Majorana particles must be genuinely neutral, i.e. they cannot possess any conserved charge-like quantum number that would allow one to discriminate between the particle and its antiparticle. In particular, they must be electrically neutral. Among the known spin-1/2 particles, only neutrinos can be of Majorana nature. Another known quasi-stable neutral fermion, the neutron, has non-zero magnetic moment, which disqualifies it from being a Majorana particle: the antineutron exists, and its magnetic moment is opposite to that of the neutron.

Neutrinos are exactly massless in the original version of the Standard Model of electroweak interaction [314], and are massive Majorana particles in most of its extensions. Although massive Dirac neutrinos are also a possibility, most economical and natural models of neutrino mass lead to Majorana neutrinos. Since only massive neutrinos can oscillate, interest in the possibility of neutrinos being Majorana particles rose significantly after the first hints of neutrino oscillations obtained in solar and atmospheric neutrino experiments. This interest increased greatly after the oscillations were firmly established in the experiments with solar, atmospheric, accelerator, and reactor neutrinos [394, 395, 396]. In addition to being the simplest and most economical possibility, Majorana neutrinos have two important added bonuses: they can explain the smallness of the neutrino mass in a very natural way through the so-called seesaw mechanism, and they can account for the observed baryon asymmetry of the Universe through “baryogenesis via leptogenesis.” We shall discuss both here.

In the limit of vanishingly small mass, the difference between Dirac and Majorana fermions disappears. Therefore the observed smallness of the neutrino mass makes it very difficult to discriminate between different types of massive neutrinos, and it is not currently known if neutrinos are Majorana or Dirac particles. The most promising means of finding this out is through the experiments on neutrinoless double β-decay. Such experiments are currently being conducted in a number of laboratories.

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.