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arXiv:physics/0512174

Una lezione particolare di Ettore Majorana

Salvatore Esposito

Published Dec 19, 2005• 1 citation
arXivHTMLPDF

Abstract

We give, for the first time, the critical complete text of a manuscript by Ettore Majorana, conserved in the same file of the lecture notes for his course on Theoretical Physics at the University of Naples. He probably refers to a seminar delivered in that University. Some passages reveal a physical interpretation of the Quantum Mechanics which anticipates of several years the Feynman approach in terms of path integrals, independently of the underlying mathematical formulation.

DOI
10.48550/arxiv.physics/0512174

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We give, for the first time, the critical complete text of a manuscript by Ettore Majorana, conserved in the same file of the lecture notes for his course on Theoretical Physics at the University of Naples. He probably refers to a seminar delivered in that University. Some passages reveal a physical interpretation of the Quantum Mechanics which anticipates of several years the Feynman approach in terms of path integrals, independently of the underlying mathematical formulation.
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.
This year is the centenary of the birth of Ettore Majorana, one of the major Italian physicists of all times. In this note we briefly sketch a few biographical details about Ettore Majorana and introduce and discuss the main points of Majorana's 10th article. In his article Majorana explicitly considers quantum mechanics as an irreducible statistical theory because the theory is not able to describe the time evolution of a single particle or atom in a precise environment at a deterministic level. This lack of determinism at the level of an elementary physical system motivated him to suggest a formal analogy between statistical laws observed in physics and in the social sciences. We hope the occasion of the centenary of the birth of Ettore Majorana will be useful to remember and to reconsider not only his exceptional achievements in theoretical physics but also his fresh and original views on the role of statistical laws in physics and in other disciplines such as the social sciences.
The physicists working in several areas of research know quite well the name of Ettore Majorana, since it is currently associated to fundamental concepts like {\it Majorana neutrinos} in particle physics and cosmology or {\it Majorana fermions} in condensed matter physics. But, probably, very few is known about other substantial contributions of that ingenious scholar, and even less about his personal background. For non specialists, instead, the name of Ettore Majorana is usually intimately related to the fact that he disappeared rather mysteriously on March 26, 1938, just seventy years ago, and was never seen again. The life and the work of this Italian scientist is the object of the present review, which will also offer a summary of the main results achieved in recent times by the historical and scientific researches on his work.
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.