Scientific Papers of Ettore Majorana: A New Expanded Edition 3030235084, 9783030235086

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Luisa Cifarelli Editor

Scientific Papers of Ettore Majorana A New Expanded Edition Second Edition

Società Italiana di Fisica

Scientific Papers of Ettore Majorana

Luisa Cifarelli Editor

Scientific Papers of Ettore Majorana A New Expanded Edition Second Edition

Editor Luisa Cifarelli Department of Physics University of Bologna Bologna, Italy

The portraits and photographs of Ettore Majorana are published by kind permission of the Majorana family (reproduction is not permitted). ISBN 978-3-030-23508-6 ISBN 978-3-030-23509-3 (eBook) https://doi.org/10.1007/978-3-030-23509-3 Jointly published with SIF, Bologna, Italy 1st edition: © SIF, 2006 2nd edition: © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

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Preface

Ettore Majorana’s scientific papers, despite being very few, constitute a heritage of undeniable value and extraordinary scientific significance since they lay the foundation for research fields still topical to date. It is for this reason that they are of interest not only for experts in the history of science but also for scientists and researchers in all different fields of physics. Unlike the previous edition of this volume (SIF-Springer 2006, reprinted in 2018), Majorana’s papers are reproduced here only in their English translation to give once and for all the right international reach to his work, originally written in Italian or German. As in the previous edition, each paper is followed by a commentary from an expert in the specific subject. Moreover, this edition has been enriched by adding an introductory paper on the genius of Majorana by A. Zichichi and two contributions on today’s major issues concerning Majorana’s predictions in different fields by A. Bettini and G. Benedek, respectively. New biographical notes on Majorana written by F. Guerra and N. Robotti complete the volume. The first edition of this volume was published on the occasion of the Centenary of Ettore Majorana’s birth as a tribute to the great scientist. By republishing this volume on the heels of the 80th anniversary of his ill-fated disappearance, the Italian Physical Society, confident of the interest and appreciation of a wide scientific community, wishes to reaffirm the enormous impact of the work of this Italian genius that everybody acknowledges the world over. Bologna, December 2019

Luisa Cifarelli SIF President

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Contents

The Genius of Ettore Majorana . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Antonino Zichichi

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On the Splitting of the Roentgen and Optical Terms Caused by the Electron Rotation and on the Intensity of the Cesium Lines . . . . . . . . . . . . . . . . Giovanni Gentile Jr. and Ettore Majorana

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Majorana dr Ettore: Search for a General Expression of Rydberg Corrections, Valid for Neutral Atoms or Positive Ions . . . . . . . . . . . . . . . . . . . . . . . Ettore Majorana

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On the Formation of Molecular Helium Ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ettore Majorana

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On the Possible Anomalous Terms of Helium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ettore Majorana

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Pseudopolar Reaction of Hydrogen Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ettore Majorana

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Theory of the Incomplete P  Triplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ettore Majorana

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Oriented Atoms in a Variable Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ettore Majorana

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Relativistic Theory of Particles with Arbitrary Intrinsic Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ettore Majorana

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On Nuclear Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Ettore Majorana A Symmetric Theory of Electrons and Positrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Ettore Majorana

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Contents

The Value of Statistical Laws in Physics and Social Sciences . . . . . . . . . . . . . . . 129 Ettore Majorana Are Neutrinos Completely Neutral Particles? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Alessandro Bettini Majorana Fermions in Condensed Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Giorgio Benedek Biographical Notes on Ettore Majorana . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Francesco Guerra and Nadia Robotti

The Genius of Ettore Majorana Antonino Zichichi

1 Leonardo Sciascia’s Idea This great Sicilian writer was convinced that Ettore Majorana (Fig. 1) decided to disappear because he foresaw that nuclear forces would lead to nuclear explosives (a million times more powerful than conventional bombs) like those that would destroy Hiroshima and Nagasaki. He came to visit me at Erice where we discussed this topic for several days. I tried to change his mind, but there was no hope. Sciascia was too absorbed by an idea that, for a writer, was simply too appealing. In retrospect, after years of reflection on our meetings, I believe that one of my assertions about Majorana’s genius actually corroborated Sciascia’s idea. At one point in our conversations I assured Sciascia that it would have been nearly impossible—given the state of physics in those days—for a physicist to foresee that a heavy nucleus could be broken to trigger the chain reaction of nuclear fission. Impossible for what Fermi called first-rank physicists, those who were making important inventions and discoveries, I suggested, but not for geniuses like Ettore Majorana. May be this information convinced Sciascia that his idea about Majorana was not just probable, but actually true. A truth that his disappearance only further corroborated.

From the book by A. Zichichi “Ettore Majorana—His genius and long-lasting legacy, 1906–2006” (Società Italiana di Fisica, Bologna) 2007. A. Zichichi () Department of Physics and Astronomy, University of Bologna, Italy INFN, Bologna, Italy Enrico Fermi Historical Museum and Study and Research Centre, Rome, Italy CERN, Geneva, Switzerland e-mail: [email protected] © Springer Nature Switzerland AG 2020 L. Cifarelli (ed.), Scientific Papers of Ettore Majorana, https://doi.org/10.1007/978-3-030-23509-3_1

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Fig. 1 Ettore Majorana’s photo taken from his university card dated 3rd November 1923

There are also those who think that his disappearance was related to spiritual faith, and that Majorana retreated to a monastery. This perspective on Majorana as a believer comes from Monsignor Francesco Riccieri, the confessor of Ettore. I met him when he came from Catania to Trapani as Bishop. Remarking on his disappearance, Riccieri told me that Ettore had experienced “mystical crises” and that in his opinion, suicide in the sea was to be excluded. Bound by the sanctity of confessional, he could tell me no more. After the establishment of the Erice Centre, which bears Majorana’s name, I had the privilege of meeting Ettore’s entire family. No one ever believed it was suicide. Ettore was an enthusiastic and devout Catholic and, moreover, he withdrew his savings from the bank a week before his disappearance. The hypothesis shared by his family and others who had the privilege to know him (Laura Fermi was one of the few) is that he retired to a monastery.

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2 Enrico Fermi: Few Others in the World Could Match Majorana’s Deep Understanding of the Physics of the Time When he disappeared, Enrico Fermi said to his wife: “Ettore was too intelligent. If he has decided to disappear, no one will be able to find him. Nevertheless, we have to consider all possibilities”; in fact, he even tried to get Mussolini himself to support the search. On that occasion, Fermi said: “There are several categories of scientists in the world; those of second or third rank do their best but never get very far. Then there is the first rank, those who make important discoveries, fundamental to scientific progress. But then there are the geniuses, like Galilei and Newton. Majorana was one of these” (Rome 1938). A genius, however, who looked on his own work as completely banal; once a problem was solved, he did his best to leave no trace of his own brilliance. This can be witnessed in the stories of the “neutron” discovery (half of our weight comes from neutrons) and the hypothesis of the “neutrinos” that bear his name; we share below two testimonies, one by Emilio Segrè and Gian Carlo Wick (on the neutron) and the other by Bruno Pontecorvo (on neutrinos). Majorana’s comprehension of the physics of his time, according to Enrico Fermi, had a profundity and completeness that few others in the world could match. The proof of this statement is the content of my attempt to illustrate Majorana’s scientific work. In the early thirties of the last century, the great novelty was the Dirac equation, which will be illustrated in Sect. 5. This unexpected equation could finally explain why the electron could not be a scalar particle and had to be a particle with spin 1/2 (in Planck’s units: h), the reason being relativistic invariance. The same equation gave as a consequence of the existence of a particle the existence of its antiparticle, thus generating the “annihilation”, i.e. the destruction of both the particle and its antiparticle. We will see the enormous consequences of this new phenomenon. Ettore Majorana, in his 1932 paper [1] (see Sect. 7) demonstrated that relativity allows any value for the intrinsic angular momentum of a particle. There is no privilege for spin 1/2. Concerning the necessity for the existence of the antiparticle state, given the existence of a particle, Majorana discovered [2] that a particle with spin 1/2 can be identical to its antiparticle. We know today that it is not the privilege of spin-1/2 particles to have their antiparticle and that relativity allows any value for the spin. However, for the physicists of the time, these were topics of great concern. The Dirac equation was the starting point of the most elaborated description of all electromagnetic phenomena, now called quantum electrodynamics (QED). We also know that the fundamental particles are of two types: spin 1/2 and spin 1. The spin-1/2 particles (quarks and leptons) are the building blocks of our world. The spin-1 particles are the “glues”, i.e. the quanta of the gauge fields. We do understand the reason why the gauge fields must have spin 1: this is because the fundamental forces of nature originate from a basic principle called local gauge invariance. This principle dictates that the energy density must remain the same if we change something in the mathematical structure that describes the given

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fundamental forces of nature. For example, if the mathematical structure of the given force is described by a group such as SU (3) (this is the case for the strong force acting between quarks and gluons; the number 3 refers to the number of complex “intrinsic” dimensions where the group exists) we can operate changes obeying the mathematics of the group SU (3), and the physics must remain the same. By requiring that the physics must remain the same for changes in other “intrinsic” dimensions, 2, and 1, where other symmetry groups SU (2) and U (1) exist, we get the weak and the electromagnetic forces. It took three quarters of a century to discover that these two forces originate from a mixing between the SU (2) and the U (1) gauge forces. The changes in the “intrinsic” dimensions 3, 2, 1 can be made at any point in space-time; this is the meaning of “local” in the gauge invariance. This locality produces the spin 1 for the quantum of the three gauge forces SU (3), SU (2) and U (1), and spin 2 for the gravitational force, because here the “gauge” invariance refers to the Poincaré symmetry group, which exists in Lorentz spacetime dimensions, not in the “intrinsic” dimensions where the symmetry groups SU (3), SU (2) and U (1) exist. Since all fundamental forces (electromagnetic, weak, strong and gravitational) originate from a local gauge invariance, we understand why the quanta of these forces must have spin 1 and 2. The reason why the building blocks are all with spin 1/2 remains to be understood. What Majorana proved about the Dirac equation was correct: neither the spin of the electron nor the existence of its antiparticle was a “privilege” of spin-1/2 particles. In fact there is no single-particle relativistic quantum theory of the sort which Dirac initially was looking for. The combination of relativity and quantum mechanics inevitably leads to theories with unlimited numbers of particles. We do not know why the Standard Model needs only spin-1/2 and spin-1 particles, plus spin-0 particles associated with imaginary masses. But we know that the Dirac equation led physics to discover that a particle can annihilate with its own antiparticle, thus “annihilation” must exist. In fact the existence of the antielectron (positron) implies that an electron can annihilate with a positron, with the result that their mass-energy becomes a (virtual) high-energy photon, governed by QED. This photon can also transform into a pair of electron-positron, still governed by QED. But now think of a photon that can also transform into a “particle-antiparticle” such as quark-antiquark or lepton-antilepton, or (W+ W− ) pair. Quark-antiquark pairs are governed by the laws of all subnuclear forces, the strong, QCD (quantum chromodynamics), the electromagnetic, QED, and the weak forces, QFD (quantum flavour dynamics); (W+ W− ) and lepton-antilepton pairs are governed by the laws of QED and QFD. Each of these “particle-antiparticle” pairs can annihilate and form a photon again. The annihilation process allows these three forces, QED, QCD and QFD, to be present in the virtual effects. Without the existence of “annihilation” these processes could not occur, and the problem of the renormalization of the gauge forces (with or without spontaneous symmetry breaking) would never have been conceived. Had the renormalization problem not been solved—as was the case in the early 1970s, by the 1999 Nobel prize winners Gerard ’t Hooft and Martinus Veltman—we would not have the Standard Model, with its many precise quantitative predictions

The Genius of Ettore Majorana

5

that have been experimentally validated in labs all over the world. The roots of the Standard Model are in the Dirac equation. Majorana was fascinated by the “annihilation”, but he could not agree with the physics foundations that were considered to be at the origin of the “privileged” spin-1/2 particles. Let me emphasize the importance of the concept of ‘annihilation’ in the development of modern physics. In fact, the existence of the antielectron (or “positron” as it has become known) implied that when a particle (of any type) collided with its antiparticle they would annihilate each other, releasing their rest energy as high-energy photons (or other gauge bosons). In the case of a process described purely by QED, a gamma-ray photon can create an electron-positron pair, which can transform itself back into a photon. This process, called “vacuum polarization”, was the first virtual effect to have been theoretically predicted. The simplest one (see later) came from an experimental discovery. The first physicist to compute the vacuum-polarization effects in the hydrogen atom was Victor Weisskopf. He predicted that the 2p1/2 level in a hydrogen atom should be very slightly higher in energy than the 2s1/2 level, by some 17 MHz. However, in 1947, Willis Lamb and Robert Rutherford discovered that the 2p1/2 level was in fact lower than the 2s1/2 level by some 1000 ± 100 MHz. It was this experimental discovery, now called the Lamb shift, that prompted all theorists, including Weisskopf, Hans Bethe, Julian Schwinger and Richard Feynman, to compute the very simple radiative process in which an electron emits and then absorbs a photon. The “vacuum polarization” is not as simple. Nevertheless, had it not been for the discovery of the positron—and therefore the existence of electronpositron pairs and of their annihilation—no one would have imagined that such simple virtual effects as the one producing the “Lamb shift”, could exist in nature. And without “virtual effects”, the gauge couplings would not change with energy (in physics jargon this is called “running”), no correlation could exist between the different forces and, ultimately, no grand unification of all the fundamental forces and no Standard Model. Of course—and fortunately for us—there are sound reasons to believe that there is a lot of new physics beyond the Standard Model. The conclusion is that Majorana was right: the electron spin 1/2 was not a consequence of relativistic invariance, and the concept of antiparticle was not the privilege of spin-1/2 particles. Nevertheless it is the conceptual existence of particle-antiparticle pairs that sparked the new process called “annihilation”, with its far-reaching consequences, which led physics to the Standard Model and Beyond. This took three quarters of a century to be achieved, but it did not start as an equation deprived of immediate successes. Using his equation, Dirac was able to compute the “fine structure” of the hydrogen atom, i.e. the very small energy difference between states that differ only in their total angular momentum, in excellent agreement with experimental data. We will see in Sect. 5 that Dirac was able to show that the gyromagnetic ratio, the famous g factor, had to be 2, as wanted by the experimental data. The discovery of the antielectron came as a totally unexpected blessing to the “prediction” of the “hole” in the “Dirac sea”, with all consequences on positive- and negative-energy solutions of the Dirac equation. Despite these formidable successes,

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we now know that there is no relativistic quantum theory of a single particle the sort that Dirac was looking for initially. As already said, combining relativity and quantum mechanics leads to theories with unlimited numbers of particles. In such theories, the “true dynamical variables” on which the wave function depends are not the position of one particle or several particles, but “fields”, like the electromagnetic field of Maxwell. Particles are quanta—bundles of energy and momentum—of these fields. A photon is a quantum of the electromagnetic field, with spin 1, while an electron is a quantum of the electron field, with spin 1/2. So why did Dirac’s equation work so well? Because the equation for the “electron field operator” is mathematically the “same” as Dirac’s equation for the “wave function”. Therefore the results of the calculation turn out to be the same as Dirac’s. This does not diminish the value of Dirac’s impact on the development of new physics. Let me just mention an example related to the group where Majorana was working. In 1932 Enrico Fermi constructed a theory of radiative decays (beta decays), including the neutron decay, by exporting the concepts of QFT (quantum field theory) far from their origin. Neutron decay corresponds to the destruction of a neutron with the creation of a proton, plus a pair of an electron and an antineutrino. Thus, there exist processes which involve the creation and destruction of protons, neutrons, electrons and neutrinos. Since destruction of a particle means creation of an antiparticle, and destruction of an antiparticle means creation of a particle, none of these processes could have been imagined without the existence of “annihilation” between a particle and its antiparticle. To sum up, the “annihilation” was the seed for “virtual” physics, the “running” of the gauge couplings, the correlation between the fundamental forces and their “unification”: in other words, this totally unexpected phenomenon, born with the discovery of the Dirac equation, led physics to the triumph of the Standard Model. Majorana’s papers [1, 2] were both in the “turmoil” of these fundamental developments. Memories of this man had nearly faded when, in 1962, the International School of Physics was established in Geneva, with a branch in Erice. It was the first of the 123 schools that now compose the Centre for Scientific Culture that bears Majorana’s name. The next testimony we turn to is that of an illustrious exponent of twentieth century Physics, Robert Oppenheimer.

3 Recollections by Robert Oppenheimer After suffering heavy repercussions of his opposition to the development of weapons even stronger than those that destroyed Hiroshima and Nagasaki, Oppenheimer decided to get back to physics by visiting the biggest laboratories at the frontiers of scientific knowledge. This is how he came to CERN, the largest European Laboratory for Subnuclear Physics. At a ceremony organized for the presentation of the Erice School dedicated to Ettore Majorana (Fig. 2), many illustrious physicists

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Fig. 2 The courtyard of the Ettore Majorana Centre in Erice, Sicily

participated. I myself—at the time very young—was entrusted the task of speaking about the Majorana neutrinos. Oppenheimer (Fig. 3) wanted to voice his appreciation for how the Erice School and the Centre for Scientific Culture had been named. He knew the exceptional contributions Majorana made to physics from the papers he had read. This much, any physicist could do at any time. What would have remained unknown is the episode he told me as a testimony of Fermi’s exceptional esteem of “Ettore”. He recounted the following episode from the time when the Manhattan Project was being carried out. The Project, in the course of just 4 years, transformed the scientific discovery of nuclear fission (heavy atomic nuclei can be broken to produce enormous quantities of energy) into a weapon of war. There were three critical turning points during this Project. During the executive meeting convened to address the first of these crises, Fermi turned to Wigner and said: “If only Ettore were here”. The Project seemed to have reached a dead end in the second crisis, during which Fermi exclaimed once more: “This calls for Ettore!”. Other than the Project Director himself (Oppenheimer), three people were in attendance at these meetings: two scientists (Fermi and Wigner) and a general of the US armed forces. Wigner worked with nuclear forces, like Ettore Majorana. After the “top-secret” meeting, the general asked the great Professor Wigner who this “Ettore” was, and Wigner replied: “Majorana”. The general asked where Ettore was, so that he could try to bring him to America. Wigner replied: “Unfortunately, he disappeared many years ago”.

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Fig. 3 Robert Oppenheimer during an interview at the CERN Library (July 1962)

4 The Discovery of the Neutron—Recollections by Emilio Segrè and Gian Carlo Wick And now a testimony by Emilio Segrè and Gian Carlo Wick on the discovery of that omnipresent particle: the neutron. By the end of the second decade of last century, physics had identified three fundamental particles: the photon (quantum of light), the electron (needed to make atoms) and the proton (essential component of the atomic nucleus). These three particles alone, however, left the atomic nucleus shrouded in mystery: no one could understand how multiple protons could stick together in a single atomic nucleus. Every proton has an electric charge, and like charges push away from one another. A fourth particle was needed, heavy like the proton but without electric charge, the neutron, and a new force had to exist, the nuclear force, acting between protons and neutrons. But no one knew this yet. Here we will try to explain, in simple terms, what was known in that era about particles, which we present as “things”. Only three types exist: doves (photons), motorcycles (electrons) and trucks (protons). The doves—in our example—are white, the motorcycles red and the trucks green. We are substituting “colour” for electrical charge. Protons are electrically charged (green trucks) with a sign opposite to that of the electrons (red motorcycles). Photons are neutral (white doves). A single dove, even flying at very high velocity, could never move a parked truck. It would require a second truck in motion to move a stationary one. As we know, doves weigh very little, motorcycles are fairly light (relative to trucks), and trucks are very heavy. If a truck is moved from its parking space, then something must have moved it. This is what Frédéric Joliot

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and Irène Curie discovered. A neutral particle enters matter and expels a proton. Since the “thing” that enters into matter has no colour, their conclusion was that it must necessarily be a dove, because it is the only known “thing” that has no colour. Ettore Majorana had a different explanation, as Emilio Segrè and Gian Carlo Wick recounted on different occasions, including during their visits to Erice. Segrè and Wick were enthusiasts for what the School and the Centre had become in only a few years, all under the name of the young physicist that Fermi considered a genius alongside Galilei and Newton. Majorana explained to Fermi why that particle had to be as heavy as a proton, even while electrically neutral. To move a truck from its parking space requires something as heavy as the truck itself. Not a dove, which is far too light, and not a motorcycle because it has a colour. It must be a truck with no colour; white like the doves, but heavy like the green trucks. A fourth “thing”, therefore, must exist: a white truck. So was born the correct interpretation of what the Joliot-Curie discovered in France: the existence of a particle that is as heavy as a proton but without electric charge. This particle is the indispensable neutron. Without neutrons, atomic nuclei could not exist. Fermi told Majorana to publish his interpretation of the French discovery right away. Ettore, true to his belief that everything that can be understood is banal, did not bother to do so. The discovery of the “neutron” is in fact justly attributed to Chadwick (1932) for his beryllium experiments. Next we turn to the testimony of Bruno Pontecorvo on the neutrinos of Majorana.

5 The Majorana “Neutrinos”—Recollections by Bruno Pontecorvo—The Majorana Discovery on the Dirac γ-Matrices Today, Majorana is particularly well known for his ideas about neutrinos. Bruno Pontecorvo (Fig. 4), the “father” of neutrino oscillations, recalls the origin of Majorana neutrinos in the following way: Dirac discovers his famous equation describing the evolution of the electron (in our body there are billions and billions of “electrons”). Majorana goes to Fermi to point out a fundamental detail: “I have found a representation where all Dirac γ -matrices are real. In this representation it is possible to have a real spinor, which describes a particle identical to its antiparticle”. This means that neutrino and antineutrino are identical particles. The starting point is the Dirac equation, which, as we will see later, corresponds to a system of four coupled differential equations. How these equations are related to each other is described by the so-called γ -matrices whose “representation” reported on page 15 was found by Dirac.

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Fig. 4 Bruno Pontecorvo talking with the author in Rome (September 1978)

This “representation” is responsible for the existence of the antiparticle property, once the particle is given. Majorana discovered that the γ -matrices could have another totally different “representation” where these matrices are all “real”. The consequences are remarkable, since, in this case, we have a spin-1/2 particle identical to its antiparticle. For the benefit of the reader we report here the Majorana discovery about his γ -matrix representation: ⎛

0

⎜ ⎜1 γ0 = ⎜ ⎜0 ⎝ 0 ⎛ 0 ⎜ ⎜0 γ2 = ⎜ ⎜1 ⎝ 0

−1

0

0

0

0

0

0

−1

0

1

0

0

0

0

1

0

0

⎞

⎟ 0⎟ ⎟, 1⎟ ⎠ 0 ⎞ 0 ⎟ 1⎟ ⎟, 0⎟ ⎠ 0

⎛

0

⎜ ⎜1 γ1 = ⎜ ⎜0 ⎝ 0 ⎛ 1 ⎜ ⎜0 γ3 = ⎜ ⎜0 ⎝ 0

1

0

0

0

0

0

0

−1

0

0

0

−1

0

0

−1

0

0

⎞

⎟ 0⎟ ⎟, −1 ⎟ ⎠ 0 ⎞ 0 ⎟ 0⎟ ⎟. 0⎟ ⎠ 1

In order to understand the value of Majorana’s discovery concerning particles with mass, spin 1/2, but zero charge, it is necessary to know the deep meaning of the

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Dirac equation, which is shown in a synthetic form in the Dirac Lecture Hall at the Blackett Institute in Erice. Let me say a few words of introduction. During the past decades, thousands of scientists have been in the Dirac Lecture Hall. Very many fellows have repeatedly asked me the same question. Question: Why in the Aula Magna is there the Dirac equation, and not Einstein’s: E = mc2 ? Answer: Because the Dirac equation (i ∂/ + m)ψ = 0

(1)

is the one I like most. Why? Because its origin, its consequences, its impact on human intelligence overpass everybody’s imagination, as I will try to explain.

5.1 The Origin Dirac was fascinated by the discovery of Lorentz who found that the electromagnetic phenomena, described by the four Maxwell equations, obey an incredible invariance law, now called Lorentz invariance. The key feature of this invariance is related to a basic property of space and time: if we choose the space to be real, the time must be imaginary, and vice versa. Contrary to what Kant had imagined, space and time cannot be both “real” and “absolute”. The “absolute” quantities, called “relativistic invariants” can either be “space-like” or “time-like”. The world we are familiar with is a “time-like” world, where the sequence of past and future remains the same: no matter the motion of the observer, Napoleon will come after Caesar. There is also a “space-like” world, where the sequence of past and future, including the simultaneity of events, depends on the observer. The old belief that space and time are totally independent is over. No one can isolate space from time. Whatever happens in the world, it is described by a sequence of space-time events. Not of space and time but of space-time, united and inseparable. The young Dirac realized that no one had been able to describe the evolution of the first example of “elementary particle”, the electron (discovered by J.J. Thomson in 1897), in such a way as to obey the Lorentz condition, i.e. space and time united and drastically different: one real, the other imaginary.

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The most successful description of the evolution of the electron in space and time was the celebrated Schrödinger equation, where the charge e, the electromagnetic potential Aμ , the mass m, the derivative with respect to the space coordinate ∂ ∂x and with respect to time ∂ , ∂t were all present, including the concept of “wave function” whose square was the “probability” for the “electron” to be in a given configuration state. The Lorentz invariance was not there. The Schrödinger equation describes the evolution in space and time of a numerical quantity, called “wave function”, whose square at any position and time gives the probability, at that time, of finding a particle at that location in space. How the “wave function” changes with time and space are not treated in the same way. The rate of change with position is controlled by a second-order derivative, i.e. the rate of change with position of the rate of change of the wave function with position. But the rate of change with time, of the same function, is computed at the first order, i.e. the rate of change of the wave function with time. The second order would be to compute the rate of change with time of the rate of change of the wave function with time. These two ways of describing the evolution of the wave function in time (first order) and in space (second order) was in conflict with the condition of putting space and time in a perfectly symmetric way, as requested by relativistic invariance. Dirac knew that there was an equation, which described the evolution in space and time of a wave function, where the derivatives versus time and space were both of second order. In this equation, discovered by Klein and Gordon, space and time were treated in a symmetric way, as requested by relativity. But the KleinGordon equation gave positive and negative probabilities, negative probability being nonsense. In 1934, this difficulty was shown by Pauli and Weisskopf [3] to be overcome, since the Klein-Gordon “wave function” φ should not be treated as a “wave function” describing a single particle, but as an operator in a field equation describing a field of relativistic massive particles having positive and negative electric charges. Pauli and Weisskopf concluded that positive and negative values should not be attributed to probabilities, but to the net charge densities at any point in space-time. Let us return to Dirac and his struggle to overcome the difficulties existing with the Schrödinger and Klein-Gordon equations.

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Dirac wanted an equation where time and space were treated in a symmetric way, at the first order in the derivative, and obeying the principle that the probability must be positive. Once all the conditions were fulfilled, Dirac discovered that the particle needs an intrinsic angular momentum of 1/2 in units of Planck’s constant. The two equations existing before Dirac (Schrödinger (can be extended to have spin, but remains non relativistic) and Klein-Gordon (relativistic but no spin)) were both having problems. And the big question was to understand why the electron was not a scalar particle.

5.2 The Dirac Equation Corresponds to Four Coupled Equations Once Lorentz invariance is imposed, the result is that in order to describe the evolution in space-time of the electron, you need four coupled equations. The Dirac Eq. (1) corresponds to the following set of equations: ⎛

i∂0 + m

⎜ ⎜ 0 ⎜ ⎜ i(∂ − ∂ ) 3 ⎝ 1 ∂2

0

−i(∂1 + ∂3 )

i∂0 + m

−∂2

−∂3

−i∂0 + m

i(∂1 − ∂3 )

0

∂2

⎞⎛

ψe− ↑ (x)

⎞

⎟⎜ ⎟ −i(∂1 − ∂3 ) ⎟ ⎜ ψe− ↓ (x) ⎟ ⎟⎜ ⎟ ⎟ ⎜ ψ + ↑ (x) ⎟ = 0; 0 ⎠⎝ e ⎠ ψe+ ↓ (x) −i∂0 + m

the wave function that appears in Eq. (1), ψ(x), is made up of four components, and the electron cannot be a scalar particle: it must be a particle with spin 1/2. In the four pieces of ψ(x), ⎛

ψe− ↑ (x)

⎞

⎜ ⎟ ⎜ ψe− ↓ (x) ⎟ ⎟ ψ(x) ≡ ⎜ ⎜ ψ + ↑ (x) ⎟ , ⎝ e ⎠ ψe+ ↓ (x) each component is a function whose values depend on space and time, as indicated by the argument (x). The four components correspond to the following four possible states: electron with spin up, ψe− ↑ (x); electron with spin down, ψe− ↓ (x); positron with spin up, ψe+ ↑ (x); positron with spin down ψe+ ↓ (x). The totally unexpected result was the need for the existence of the electron antiparticle, called positron, e+ : a particle with the same mass, same spin, but opposite electric charge. This “antiparticle” had no experimental support. But in favour of Dirac

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there was another property of the electron. The study of atomic spectra was giving experimental results indicating that the electron, in addition to its spin, has another unexpected property. The electron behaved as if it was a tiny magnet. The magnetic properties required the electron to be like a spinning sphere, but it had to rotate at an extraordinarily rapid rate. So rapid that at its surface the rotation corresponded to a speed higher than that of light. The model of the spinning electron had been worked out by two Dutch graduate students, Samuel Goudsmit and George Uhlenbeck, who wanted to explain the experimental data of atomic spectra. Eminent physicists were sceptical about this model, and Wolfgang Pauli tried to dissuade them from publishing their paper since the model they proposed had a quantitative mismatch in the gyromagnetic ratio, the so-called g factor, i.e. the ratio of the magnetic moment divided by the angular momentum. The electron orbiting around a nucleus has an angular momentum. The same electron, since it is electrically charged, in its orbital motion produces a magnetic field. The ratio of this magnetic field to the angular momentum corresponds to the value g = 1. The problem was to understand why intrinsic rotation (spin) produces a magnetic field that is twice stronger than the one produced by the same electron when it is orbiting in an atom: this is the meaning of g = 2 and g = 1, respectively. In order to agree with the results from atomic spectra, Goudsmit and Uhlenbeck postulated g = 2. The situation was indeed very complicated. Not only could no one explain why the electron’s intrinsic rotation (called spin) had a value of 1/2 the smallest orbital angular momenta, which was 1 (in units of Planck’s constant). This unexpected result was coupled with the value g=2 for the intrinsic magnetic moment, divided by the intrinsic angular momentum. Dirac finds with his equation that the intrinsic angular momentum of the electron is 1/2h and the gyromagnetic ratio g = 2. In his celebrated 1928 paper, Dirac [4] simply says: “The magnetic moment is just that assumed in the spinning electron model”. In order to get this formidable result, Dirac needed to introduce in his equation the interaction of the electron with an electromagnetic field; Eq. (1) thus becomes: 

 ∂ γ μ i μ − eAμ (x) + m ψ(x) = 0. ∂x

(2)

This equation, as is the case for the free electron, corresponds to a system of four coupled equations shown below:

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⎛

15

i∂0 −eA0 +m

⎜ ⎜ 0 ⎜ ⎜ ⎝i(∂1 −∂3 )−e(A1 −A3 ) ∂2 +ieA2

⎛

ψe− ↑ (x)

∂2 +ieA2

⎞

0

−i(∂1 +∂3 )+e(A1 +A3 )

i∂0 −eA0 +m

−∂2 −ieA2

− i(∂1 −∂3 )+e(A1 −A3 )⎟

−∂3 −ieA2

−i∂0 −eA0 +m

0

i(∂1 −∂3 )−e(A1 −A3 )

0

−i∂0 +eA0 +m

⎟ ⎟ ⎟ ⎠

⎞

⎜ ⎟ ⎜ ψe− ↓ (x) ⎟ ⎜ ⎟ = 0. ×⎜ ⎟ + ↑ (x) ψ ⎝ e ⎠ ψe+ ↓ (x)

5.3 The Great Novelty: The Dirac γ -Matrices The Dirac equations for a free electron (1) and for an electron interacting with an electromagnetic field (2) correspond, each, to four coupled equations, the coupling being described by the so-called γ -matrices. These γ -matrices are the unexpected novelty discovered by Dirac in his attempt to describe the evolution of an elementary particle having spin 1/2, charge e, and mass m. Dirac found the following representation for the γ -matrices: ⎛

1

0

0

1

0

0

−1

0

0

0

0 ⎜ ⎜0 γ1 = ⎜ ⎜0 ⎝

0

0

0

−1

1

0

1

0

0

⎜ ⎜0 γ =⎜ ⎜0 ⎝ 0

⎛

⎛

0 ⎜ ⎜0 γ2 = ⎜ ⎜0 ⎝ i ⎛ 0 ⎜ ⎜0 γ3 = ⎜ ⎜1 ⎝ 0

0

0

0

0

−i

−i

0

0

0

0

−1

0

0

0

0

−1

0

⎞

⎟ 0⎟ ⎟, 0⎟ ⎠ −1 ⎞ −1 ⎟ 0⎟ ⎟, 0⎟ ⎠ 0 ⎞ i ⎟ 0⎟ ⎟, 0⎟ ⎠ 0 ⎞ 0 ⎟ 1⎟ ⎟. 0⎟ ⎠ 0

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Notice that the γ -matrices represent the correlations that exist between the four coupled Dirac equations, synthetically expressed in terms of a function ψ called spinor, which is composed of four parts. The fact that these correlation matrices γ 0 , γ 1 , γ 2 , γ 3 are of vectorial nature, thus being of the type γ μ , allows the construction of the celebrated scalar product with ∂μ : μ γ μ ∂μ = ∂/ , where ∂μ =

∂ . ∂x μ

When we write the four equations in terms of the unique equation (i ∂/ + m)ψ = 0, we do make use of the fact that the γ -matrices are four vectors. The symbol ∂ slashed, ∂/, was introduced by Feynman: ∂/ =

μ

γμ

∂ . ∂x μ

The properties of the Dirac γ -matrices are the source of the so much wanted properties of particles with spin 1/2, mass = 0 and charge = 0. What happens if the charge is zero? Here comes the great discovery of Majorana, now known as the Majorana representation of the γ -matrices (recall Pontecorvo’s testimony). This representation of the γ -matrices is responsible for the existence of particles with spin 1/2, identical to their antiparticles: the Majorana neutrinos prove that it is not a privilege of spin-1/2 particles to have antiparticles. The Majorana representation of γ -matrices is not limited to the case D = 4 of our familiar four-dimensional space-time (s = 3, t = 1). In fact the Majorana spinors exist in many space-time dimensions, provided that appropriate conditions are satisfied. These conditions are the number of space-time dimensions D =s+t and the so-called “signature parameter” ρ = s − t. For the case of our space-time: ⎧ ⎨D = s + t = 3 + 1 = 4, ⎩ρ = s − t = 3 − 1 = 2. Majorana spinors exist for even and odd numbers of space-time dimensions. If

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17

D even,

ρ = 0, 2, 6 modulo 8

and if ρ = 1, 7 modulo 8.

D odd,

For a conventional space-time signature ρ = D − 2, Majorana spinors exist for D = 2, 3, 4, 8, 9 modulo 8. This has enormous consequences for the construction of chiral superstring theories in D = 10 space-time dimensions, as illustrated by L. Andrianopoli and S. Ferrara [5], bringing the Majorana spinors to the most advanced frontier of our physics knowledge. For example, the quantum of the gravitational field, the graviton, has as supersymmetric pattern the gravitino, which is a Majorana spinor, i.e. a particle with mass, spin 3/2 and whose antiparticle is identical to it. Returning to Dirac, his equation needs “four” components to describe the evolution in space and time of the simplest of particles, the electron. Majorana jotted down a new equation: for a chargeless particle like the “neutrino”, which is similar to the electron except for its lack of charge, only two components are needed to describe its movement in space-time. “Brilliant”—said Fermi—“Write it up and publish it”. Remembering what happened with the “neutron” discovery, Fermi wrote the article himself and submitted the work, under Ettore Majorana’s name, to the prestigious scientific journal “Il Nuovo Cimento” [2]. Without Fermi’s initiative, we would know nothing about the Majorana spinors and the Majorana neutrinos. A few words to illustrate why the new particle, proposed by Pauli to avoid the violation of energy conservation in β-decay, and named by Enrico Fermi “neutrino”, attracted so much attention. A few years before, Enrico Fermi had given a rigorous formulation of the weak interactions [6], taking for granted the existence of the neutrino. The fact that a spin-1/2 particle without charge could relativistically be described by a spinor with only two components was indeed very interesting. There was another way of reducing the number of spinor components to two; this had been discovered by Weyl in 1929 [7]. The Dirac equation describes the space-time evolution of a particle with spin, mass and charge. Herman Weyl discovered that, if the mass is zero, the four coupled Dirac equations split into two pairs. Each pair needs a spinor with only two components, called ψ+

and

ψ− .

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The original Dirac spinor is the sum of these two spinors ψDirac = ψ+ + ψ− . Any spinor in even space-time dimensions may be decomposed as ψ = ψ+ +ψ− . The interest of this decomposition is that it corresponds to two different “chirality” states, obtained with the complex projection operator P± ≡ 1/2(1 ± iγ5 ), which produces P± ψ = ψ± . Notice that (P± )∗ = P∓ , and therefore (ψ± )∗ = ψ∓ . The discovery of Weyl implies that, in the Dirac equation, when m = 0 the corresponding particles with spin 1/2 can have either positive or negative “chirality”. This paper by Weyl, published in 1929 [7], was ignored for more than a quarter of a century since space inversion (parity operator) reverses chirality and the weak interactions were supposed not to break the law of parity conservation (the symmetry between left and right). In the middle fifties, it was discovered that parity conservation was violated in the Fermi forces [8–11], and that only left-handed (negative chirality) neutrinos and right-handed (positive chirality) antineutrinos appear to be coupled to the Fermi forces. The parity objection against the Weyl discovery turned to dust. The physics of the Fermi forces appears to be such that the two chirality states correspond to “particle” and “antiparticle” states. It is as if the property of “particle” and “antiparticle” were linked to the property of “chirality”. The origin of all this is that when m = 0 in the Dirac equation, the Lagrangian becomes invariant under the γ5 rotations, thus acquiring a new global invariance due to the existence of the γ5 matrix: γ5 ≡ γ 0 γ 1 γ 2 γ 3 . To sum up: we have seen that if a particle with spin 1/2 is massless, it can only exist in two different “chirality” states (Weyl). If a particle with spin 1/2 has mass, but zero charge, the particle and its antiparticle may be the same (Majorana).

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And thus the problem arises: What happens if a spin-1/2 particle has zero mass (Weyl) and zero charge (Majorana). Can Majorana-Weyl spinors exist? In other words, can a neutrino exist with zero mass and be identical to its antineutrino? The answer is no, in our four-dimensional space-time. In fact, the Weyl condition is that the two spinors are ψ+

and

ψ− ,

∗ is not equal to the spinor ψ , as but that the antispinor ψ+ + ∗ = ψ− . ψ+ ∗ , is not compatible with the Majorana Therefore the anti-Weyl spinor, ψ± condition ∗ = ψ± , ψ±

which established the equivalence between a particle and its antiparticle. The existence of spinors with particle-antiparticle equivalence (Majorana) and zero mass (Weyl), i.e. Majorana-Weyl spinors, is allowed in 2, 6, modulo 8 spacetime dimensions. As mentioned before, this is the case of chiral superstring theories in D = 10 space-time dimensions (see the paper by Andrianopoli and Ferrara [5]). In other words the existence of spinor particles with particle-antiparticle equivalence (Majorana) and zero mass (Weyl) is allowed in the D = 10 space-time dimensions. As was already remarked, the gravitino is a Majorana spinor with mass. To sum up: in 4 dimensions, a spinor cannot be both Weyl (m = 0 and q = q) ¯ and Majorana (m = 0 and q = q). ¯ In 10 dimensions, it can be both. In fact, a Dirac spinor (m = 0; q = q) ¯ in 10 dimensions has 32 degrees of freedom, while a Weyl (m = 0) or a Majorana (q = q) ¯ spinor has 16 degrees of freedom. A Majorana-Weyl (m = 0; q = q) ¯ spinor has only 8 degrees of freedom. This 8 exactly matches the number of transverse modes of a vector in 10 dimensions. This equality, 8 and 8 degrees of freedom, is used to construct supersymmetric theories in 10 dimensions, where the number of fermionic degrees of freedom must be equal to the number of bosonic ones.

6 The First Course of the Subnuclear Physics School (1963): John Bell on the Dirac and Majorana Neutrinos The great John Bell conducted a rigorous comparison of Dirac’s and Majorana’s “neutrinos” in the first year of the Erice Subnuclear Physics School (Fig. 5). The detailed version of it can be found in [12]. Since then, progress in physics has led us

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A. Zichichi

Fig. 5 John Stewart Bell at Erice (1963) lecturing on Dirac and Majorana neutrinos

to the most formidable synthesis of scientific thought of all times, that we physicists call the “Standard Model”. This Model has already pushed the frontiers of physics well beyond what the Model itself first promised, so that the present goal has come to be known as the SM&B: Standard Model and Beyond [12]. Going back to the neutrinos of our SM&B, we know today that there exist three types of neutrinos. The first controls the combustion of the Sun’s nuclear motor and keeps our Star from overheating. One of the dreams of today’s physicists is to prove the existence of Majorana’s hypothetical neutral particles, which are needed in the Grand Unification Theory. This is something that no one could have imagined in those years. And no one could have imagined the three conceptual bases needed for the SM&B, as we will discuss in the next section.

7 The First Step to Relativistically Describe Particles with Arbitrary Spin In 1932 the study of particles with arbitrary spin [1] was considered at the level of a pure mathematical curiosity.

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21

This paper attracted the interest of mathematically oriented theoretical physicists over many decades and up to now, as discussed by Y. Nambu [13]. The paper remained quasi-unknown in the area of physics, despite its being full of remarkable new ideas. In this paper in fact, there are the first hints of supersymmetry, of the spin-mass correlation, and of spontaneous symmetry breaking: three fundamental conceptual bases of what we now call the SM&B. First hints mean that our conceptual understanding of the fundamental laws of nature were already in Majorana’s attempts to describe particles with arbitrary spins in a relativistic invariant way. Majorana starts—as he correctly stated—with the simplest representation of the Lorentz group, which is infinite-dimensional. In this representation the states with integer (bosons) and semi-integer (fermions) spins are treated on equal ground. In other words, the relativistic description of particle states allows bosons and fermions to exist on equal grounds. These two fundamental sets of states (bosons and fermions) are the first hint of supersymmetry. Another remarkable novelty is the correlation between spin and mass. The eigenvalues of the masses are given by a relation of the type:  m=



m0 J+

1 2

,

where m0 is a given constant and J is the spin. The mass decreases with increasing spin, the opposite of what would appear, many decades later, in the study of the strong interactions between baryons and mesons (now known as Chew-FrautschiGribov-Regge trajectories). In this remarkable paper—as a consequence of the description of particle states with arbitrary spins—there is also the existence of imaginary mass eigenvalues. We know today that the only way to introduce real masses—without destroying the theoretical description of nature—is the spontaneous symmetry breaking (SSB) mechanism. But SSB could not exist without imaginary masses. Today, three quarters of a century later, what was considered in 1932 a purely mathematical curiosity represents a powerful source of incredibly new ideas, as those three mentioned earlier. There is a further development, which this paper contributed to: the formidable relation between spin and statistics, which was to lead to the discovery of another invariance law, valid for all quantized Relativistic Field Theories, the celebrated PCT theorem. Majorana’s paper shows first of all that the relativistic description of a particle state allows the existence of integer and semi-integer spin values. But it was already known that the electron must obey the Pauli exclusion principle and that the electron has semi-integer spin. Thus the problem arises of understanding if the Pauli principle is valid for all semi-integer spins. If this were the case, it would be necessary to find which properties characterize these two classes of particles, now known as “fermions” (semi-integer spin) and “bosons” (integer spin). The first of these properties are of a statistical nature, governing groups of identical fermions

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and groups of identical bosons. We now know that a fundamental distinction exists and that the bases for the statistical laws governing fermions and bosons are the anticommutation relations for fermions and the commutation relations for bosons. The “spin-statistics” theorem has an interesting and long history, whose main actors are some of the most distinguished theorists of the twentieth century. The first contribution to the study of the correlation between spin and statistics comes from Markus Fierz, with a paper, Über die Relativistische Theorie Kräfterfreier Teilchen mit Beliebigem Spin, where the case of general spin for free fields is investigated [14]. A year later, Wolfgang Pauli comes in with his paper On the Connection Between Spin and Statistics [15]. The first proofs, obtained using only the general properties of relativistic QFT, which include also the microscopic causality (also known as local commutativity), are due to G. Lüders and B. Zumino, Connection Between Spin and Statistics [16], and to N. Burgoyne, On the Connection Between Spin and Statistics [17]. Another important contribution to the clarification of the connection between spin and statistics came in 1961, with G.F. Dell’Antonio: On the Connection Between Spin and Statistics [18]. The correlation between spin and statistics had important consequences in understanding the relativistic description of QFTs, whose invariance properties ended in the celebrated P CT theorem. It certainly cannot be accidental that the first suggestion for the existence of such an invariance law, called P CT , came from the same fellows who were engaged in the study of the “spin-statistics” theorem: G. Lüders and B. Zumino. These two outstanding theoretical physicists suggested that if a relativistic QFT obeys the space inversion invariance law, called parity, P , it must also be invariant for the product of charge conjugation (particle-antiparticle) and time inversion, CT . It is in this form that it was proved by G. Lüders in 1954, in the paper On the Equivalence of Invariance under Time Reversal and under Particle Antiparticle conjugation for Relativistic Field Theories [19]. A year later, Pauli proved that the P CT invariance is a universal law, valid for all relativistic QFTs, Exclusion Principle, Lorentz Group and Reflection of Space-Time and Charge [20]. This paper closes a cycle started by Pauli in 1940, with his work on spin and statistics, where he proved already what is now considered the “classical” P CT invariance, since it was derived using free non-interacting fields. The validity of P CT invariance for QFTs was obtained by Julian Schwinger (a great admirer of Ettore Majorana) in 1951, with his work On the Theory of Quantized Fields I [21]. It is interesting to see what Arthur Wightman, another Ettore Majorana’s enthusiastic supporter, wrote about this Schwinger paper in his book PCT, Spin and Statistics, and All That [22]: “Readers of this paper did not generally recognize that it stated or proved the PCT theorem”. Something similar to those who, reading Majorana’s paper on arbitrary spins, have not found the imprint of the original ideas that we have discussed in the present short review.

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23

8 The Centennial of the Birth of a Genius—A Homage by the International Scientific Community The year 2006 marked the hundredth anniversary of the birth of Ettore Majorana, Enrico Fermi’s young student whom, on the occasion of his mysterious disappearance during a boat trip from Palermo to Naples, he referred to as “a genius of the order of Galilei and Newton”. On that occasion the President of the Sicilian Government and the Mayor of Erice have decided to launch many initiatives intended to make known not only Majorana’s contributions to the advancement of physics, but also the tribute expressed for decades in the unbending determination of the international scientific community. Through the International School of Subnuclear Physics, since 1963, this community has striven to provide the most prestigious protagonists of the most advanced frontiers of Galilean Science today with the best qualified new talents from all over the world, unrestricted by any ideological, political or racial barriers. Eminent figures of twentieth century physics, who all had great admiration for the genius of Ettore Majorana, through their participation in the Erice School of Subnuclear Physics, have made this school “the most prestigious post-university institution in the world” (these are the words of Isidor I. Rabi in Erice, July 1975): Gilberto Bernardini, Patrick M.S. Blackett, Richard H. Dalitz, Paul A.M. Dirac,

Fig. 6 The tower of thought “Piersanti Mattarella”, EPS historic site (2016)

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A. Zichichi

Enrico Fermi, Richard P. Feynman, Robert Hofstadter, Gunnar Källen, Giuseppe P.S. Occhialini, Wolfang Pauli, Bruno Pontecorvo, Isidor I. Rabi, Bruno Rossi, Julian S. Schwinger, Bruno Touschek, Victor F. Weisskopf, Eugene P. Wigner. In the past, our scientific community has proposed to dedicate—in the mythical City of Venus—streets, squares, courts, cloisters, institutes, and lecture halls to these illustrious physicists, in recognition of their link with the activities of the Erice School of Subnuclear Physics (Fig. 6). The President of Sicily and the Mayor of Erice have decided—on the occasion of the Majorana Centenary—to make official these dedications to our fellows whose inventions and discoveries have carried modern physics into an era of scientific glory. Again on the occasion of Majorana Centenary Celebration a clay bust of Ettore Majorana, made by the sculptor Giuseppe Ducrot, was unveiled (Fig. 7). From this bust nine bronze casts were made to be located at the following institutions: at the Ettore Majorana Centre in Erice; at the Enrico Fermi Centre in Rome; at the City of Science in Naples (it is in fact at the University of Naples that Ettore Majorana was appointed “Professore per chiara fama”); in Bologna, at the Museum of the Physics Institute, under the auspices of the Academy of Sciences of the oldest university in the world; at the “Laboratori Nazionali del Sud” in Catania (the city where Ettore Majorana was born). In Catania a square has been recently dedicated to Ettore Majorana and a statue erected. Another two bronze casts are located, Fig. 7 Clay bust of Ettore Majorana by sculptor Giuseppe Ducrot

The Genius of Ettore Majorana

25

respectively, at CERN and at Gran Sasso, where (at CERN) a crucial experiment to establish the effective nature of the third neutrino1 has been implemented and where (at Gran Sasso) the search for neutrinoless double-beta decay to establish whether neutrinos are Majorana or Dirac particles is being performed. A bronze cast has been installed in Dubna at JINR, the laboratory where the father of neutrino oscillations, Bruno Pontecorvo, was working. Finally the ninth bronze cast is located at the Chinese Center for Advanced Science and Technology (CCAST), founded by the Enrico Fermi pupil, Tsung Dao Lee. These nine prestigious centers are all related to Majorana. Actually on the occasion of the Majorana Centenary Tsung Dao Lee, and Rudolf Mössbauer, two great admirers of the genius of Ettore Majorana, have been awarded the “Ettore Majorana” Gold Medal.

Bibliography 1. Majorana, E.: Teoria relativistica di particelle con momento intrinseco arbitrario. Nuovo Cimento 9, 335 (1932) 2. Majorana, E.: Teoria Simmetrica dell’Elettrone e del Positrone. Nuovo Cimento 14, 171 (1937) 3. Pauli, W., Weisskopf, V.F.: Über die Quantisierung der skalaren relativistischen Wellengleichung. Helv. Phys. Acta. 7, 709 (1934) 4. Dirac, P.A.M.: The quantum theory of the electron. Proc. R. Soc. A. 117, 610 (1928) 5. Andrianopoli, L., Ferrara, S.: Majorana Spinors, their interactions and their role in supersymmetry. In: Zichichi, A. (ed). The Glorious Days of Physics and Erice, vol. I. World Scientific, Singapore (2007) 6. Fermi, E.: Tentativo di una Teoria dei raggi β. Nuovo Cimento 11, 1 (1934). Versuch Einer Theorie der Beta-Strahlen. I. Zeit für Phys. 88, 161 (1934). English translation “Attempt at a Theory of Beta-Rays”, published in Kabir P. K. (ed.). “The Development of Weak Interaction Theory”. Gordon and Breach, New York (1963), pp. 1–21 7. Weyl, H.: Elektron und gravitation. I. Zeit für Phys. 56, 330 (1929). English translation “Electron and gravitation”, published in “The Dawning of Gauge Theory”. Lochlainn O’Raifeartaigh, Princeton, Princeton University Press (1997), pp. 121–44 8. Lee, T.D., Yang, C.N.: Question of parity conservation in weak interactions. Phys. Rev. 104, 254 (1956) 9. Wu, C.S., Ambler, E., Hayward, R.W., Hoppes, D.D.: Experimental test of parity conservation in beta decay. Phys. Rev. 105, 1413 (1957) 10. Garwin, R., Lederman, L., Weinrich, M.: Observation of the failure of conservation of parity and charge conjugation in meson decays: the magnetic moment of the free Muon. Phys. Rev. 105, 1415 (1957). 11. Friedman, J.J., Telegdi, V.L.: Nuclear emulsion evidence for parity non-conservation in the decay chain π + μ+ e+ . Phys. Rev. 105, 1681 (1957) 12. Bell, J.: Strong, electromagnetic and weak interactions. In: Zichichi, A. (ed.) Proceedings of the 1963 International School of Physics “Ettore Majorana”. Benjamin, New York (1964) 13. Nambu, Y.: “Majorana’s infinite component wave equation. In: Zichichi, A. (ed.) The Glorious Days of Physics. World Scientific, Singapore (2007)

1 This

third neutrino was named heavy-lepton neutrino, νHL , when originally proposed.

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14. Fierz, M.: Über die Relativistische Theorie Kräfterfreier Teilchen mit Beliebigem Spin. Helv. Phys. Acta. 12, 3 (1939) 15. Pauli, W.: On the connection between spin and statistics. Phys. Rev. 58, 716 (1940) 16. Lüders, G., Zumino, B.: Connection between spin and statistics. Phys. Rev. 110, 1450 (1958) 17. Burgoyne, N.: On the connection between spin and statistics. Nuovo Cimento 8, 807 (1958) 18. Dell’Antonio, G.F.: On the connection between spin and statistics. Ann. Phys. 16, 153 (1961) 19. Lüders, G.: On the equivalence of invariance under time reversal and under particle-antiparticle conjugation for relativistic field theories. Dansk. Mat. Fys. Medd. 28, 5 (1954) 20. Pauli, W. (ed.): Exclusion principle, Lorentz group and reflection of space-time and charge. In: Niels Bohr and the Development of Physics. Pergamon Press, London (1955) 21. Schwinger, J.: On the theory of quantized fields I. Phys. Rev. 82, 914 (1951) 22. Wightman, A.: PCT, Spin and Statistics, and All That. Benjamin, New York (1964)

On the Splitting of the Roentgen and Optical Terms Caused by the Electron Rotation and on the Intensity of the Cesium Lines∗ Giovanni Gentile Jr. and Ettore Majorana

1. The purpose of this paper is to show that Fermi’s potential allows one to determine a priori and with very good approximation all the energy levels of heavy atoms. This also allows one to calculate with remarkable accuracy, considering its statistical character, the splitting of the various terms. This is of great importance considering that one could not apply Sommerfeld’s relativistic formula to these splittings, as the phenomenon goes well beyond the scheme of the fine-structure theory. Indeed it is well known that one has to use the assumption of the rotating electron which by now has lost its hypothetical character and appears to be well founded on a solid theoretical basis as Dirac’s last paper1 has shown. Our calculations will be applied to the Roentgen levels of the 3M term of gadolinium (Z = 64) and of uranium (Z = 92) and, in the optical case, to the P terms of cesium (Z = 55). The electrostatic potential inside an atom with charge Z can be written in the r form V = Ze r ϕ( μ ), where ϕ is a numerical function in general smaller than one representing the screening effect of the other electrons. More precisely, close to the nucleus where the screening effect is minimum, ϕ = 1; as r increases ϕ decreases until, for r = ∞ and for neutral atoms, ϕ = 0. Clearly the value of ϕ depends on the average distributions of the electrons around the nucleus. This electron cloud obeys Pauli’s principle and thus, applying Fermi’s statistics to this special degenerate gas, we get another relation between the potential and the charge density. Using Poisson’s equation Fermi2 obtained the following differential equation:

∗ Received

by the Academy on July 24, 1928; presented by the member O. M. Corbino. Translated from Sullo sdoppiamento dei termini Roentgen e ottici a causa dell’elettrone rotante e sulle intensità delle righe del cesio, “Rendiconti dell’Accademia dei Lincei”, vol. 8, 1928, pp. 229– 233, by P. Radicati di Brozolo. 1 Dirac, P.A.M.: Proc. R. Soc. (Lond) A. 117, 610; 118, 351 (1928) 2 Fermi, E.: Z Physik. 48, 73 (1928) © Springer Nature Switzerland AG 2020 L. Cifarelli (ed.), Scientific Papers of Ettore Majorana, https://doi.org/10.1007/978-3-030-23509-3_2

27

28

G. Gentile Jr. and E. Majorana

d 2ϕ ϕ 3/2 r = √ , where: x = ; 2 μ dx x

μ=

32/3 h2 . 213/3 π 4/3 me2 Z 1/3

If we consider a given electron we can suppose3 that the other Z −1 electrons are distributed as in a neutral atom with number Z − 1. The potential acting upon the electron is therefore V = re [1 + (Z − 1)ϕ( μr )]; obviously in this way we neglect, in the case of an internal electron, the consequences of Pauli’s principle, which is not a statistical principle but a rigourous exclusion principle; in the case of an external electron we neglect instead the polarisation generated in the rest of the atom. In the first case the error is minimal while in the second case there are more serious uncertainties arising from the periodical system which produces regular oscillations of all the superficial atomic properties around the average; this average behaviour is the only one that shows up in our statistical treatment. 2. The general Schrödinger equation is4 2 ψ +

  e2 e2 8π 2 m E + + (Z − 1)ϕ ψ = 0. r r h2

If k denotes the azimuthal quantum, the wave function ψ splits into the product of a spherical function of order k times a function of the radius that we shall write in the form χ /r, where χ satisfies the equation   d 2χ k(k + 1) 1 + (Z − 1)ϕ + ε χ = − a x dx 2 x2 with a=

8π 2 m 2 e μ; h2

ε=−

8π 2 m 2 μ E. h2

When the radial quantum is 1 (there are no nodes in the function χ ) (see Table I5 ) the equation can easily be integrated numerically. Close to zero we have used a series expansion for the function ϕ. For the term 3d of gadolinium, ε = 4.29 and therefore the energy of the term is (in Rydberg) −E = 86.3 in very good agreement with observation (86.6). We notice that if we take into account the relativistic correction, the energy of the term decreases so that its distance from the lower term is lower by half than the distance between the real terms. In a simplified theory the splitting should be calculated 3 Fermi,

E.: Rend. Acc. Lincei. 7, 726 (1928) is the actual notation used in the original, not corresponding to the standard one today. (Note of the Editor.) 5 Note added by the Editor in E. Amaldi, La vita e le opere di Ettore Majorana. Accademia dei Lincei, Roma, (1966). 4 2

On the Splitting of the Roentgen and Optical Terms Caused by the Electron. . .

29

on the basis of the interaction energy of the electron’s magnetic moment with the average value of the virtual magnetic field acting on the electron. In first approximation Dirac’s theory gives  h2

5 E = 2 8π 2 m

ψ ψ¯ ·

1 ∂v r ∂r dS

mc2

.

For gadolinium we find E = 2.20 R in good agreement with experiment (2.4 R). The same calculation for uranium gives: −E = 258 R (also this in good agreement with the experimental value 255) and E = 11.7 instead of 12.96. 3. The calculation of the 6p term of cesium, perfectly analogous to the previous one, leads to an eigenfunction whose numerical values are reported herein (see table II6 ). For this term we find the value: n = 24,600 cm−1 to be compared with the experimental values of the doublet: n1 = 19,674 n2 = 20,228; using the splitting formula recalled above with a coefficient corresponding to a different azimuth quantum we get: n = 1020 cm−1 instead of the experimental value: n = 554 cm−1 . The difference between theory and experiment is well explained by the statistical approximations, the most important of which is the one depending upon the position of the element in the periodical system. More precisely since cesium is an alkaline metal, the ionic core has the compact structure of the rare gases, so that the effective charge for the optical electron tends very rapidly to 1. It is also not surprising that the splitting is much larger than the splitting between the energies. Intuitively this can be explained by the classical Bohr-Sommerfeld model: indeed it is easy to see that all the very eccentric orbits with the same azimuthal quantum numbers have 6 Note

added by the Editor in E. Amaldi, op. cit.

30

G. Gentile Jr. and E. Majorana

approximately the same perihelion distance close to the nucleus so that they can be approximated by a unique orbit with the same motion. It is essentially in this region where the splitting is produced which is approximately inversely proportional to the orbit’s period (that is the time interval between two passages at the perihelion). If, as it happens in our case, the departure from the Newtonian potential is large, even at a distance not too far from the aphelium a small departure from the orbit causes a huge change in the revolution period.7 In any case a precise calculation shows that our interpretation is correct. Indeed let us suppose that we modify the statistical potential so as to make it agree with the experimental one. This can be achieved in infinitely many different ways, provided the correct potential always falls between the statistical one and the Newtonian limit. In this case one finds that there exists an upper limit to the splitting calculated on the basis of the corresponding eigenfunction which is s n = 750 cm−1 . This corresponds to passing abruptly from the statistical potential to the Newtonian one at a distance of approximately 2.2 Å from the nucleus. If, more realistically, one puts vs − v = e−kϕ , vs − vn one finds an almost perfect agreement with experiment. Finally we have calculated the ratio of the intensities of the first two absorption lines. If we denote by ψ0 the eigenfunction corresponding to the fundamental term 6s and by ψ1 and ψ1 the eigenfunctions corresponding to the terms 6p and 7p the ratio of the intensities is ⎡

⎤2 ¯ 1 x 3 dx ψ ψ 0 ⎥ ⎢ i1 ⎥ .  =⎢ ⎦ ⎣ i2  3 ψ1 ψ¯ 1 x dx The eigenfunctions (see Table III8 ) have been determined with the statistical9 potential up to the distance r = 2.2 Å and with the Newtonian potential for larger distances. Under these conditions one obtains, as we have said, the experimental 7 In

general in the classical model there is a normal rosetta-like motion both for non-excited and for excited levels with small azimuth quantum. On the contrary, for certain highly excited levels with large azimuth quantum the Bohr-Sommerfeld orbit splits into two different orbits. One of these reaches the interior of the atom, the other instead is mostly exterior. The model then loses its intuitive meaning. 8 Note added by the Editor in E. Amaldi, op. cit. 9 Note that in the original text of “Rendiconti dell’Accademia dei Lincei” it is erroneously printed “statico” (“static”), but in the original manuscript it is clearly written “statistico” (“statistical”). (Note of the Editor in E. Amaldi, op. cit.)

On the Splitting of the Roentgen and Optical Terms Caused by the Electron. . .

31

value for the term 6p. It is remarkable that under the same conditions also the theoretical value of the term 7p agrees almost exactly with experiment. For simplicity we have not calculated theoretically the term 6s; instead the eigenfunction ψ0 has been constructed from infinity up to a distance close to the nucleus starting from the experimental value. The non-normalized functions χ are given in the table. The ratio between the intensities is i1 = 125. i2 The splitting of the term 7p calculated from the eigenfunction ψ1  is s n = 220 cm−1 , which should be considered as an upper limit. This is to be compared with the experimental value n = 181. The agreement this time is, for obvious reasons, much better.

Table I

Table II

Table III

x

χ64

χ92

x

χ55 a

x

χ0

χ1

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

0.174 0.603 0.925 1.031 0.976 0.838 0.675 0.521 0.389 0.284

2.138 6.840 8.265 7.231 5.482 3.739 2.396 1.515 0.940 0.570

0 0.1 0.2 0.4 0.6 0.8 1 1.5 2 3 4 6 8 10 12 16 20 25 30 35 40 50

0 0.0519 0.1030 0.0627 −0.0593 −0.1442 −0.1587 0.0019 0.1831 0.2024 −0.0374 −0.3990 −0.3764 −0.1508 0.1276 0.5942 0.8503 0.9281 0.8474 0.7012 0.5462 0.2990

4 5 6 7 8 9 10 11 12 14 16 18 20 25 30 35 40 50 60 70 80 100

−0.091 −0.390 −0.513 −0.477 −0.336 −0.141 0.072 0.279 0.466 0.757 0.931 1.000 1.005 0.898 0.717 0.535 0.377 0.171 0.071 0.027 0.010 0.001

−0.038 −0.269 −3.099 −0.426 −0.376 −0.276 −0.147 −0.007 0.134 0.391 0.596 0.740 0.840 0.967 0.962 0.876 0.749 0.484 0.282 0.153 0.079 0.019

a The

lower index in χ55 has been added by the Editor in E. Amaldi, op. cit.

χ1 −0.039 −0.270 −0.399 −0.425 −0.373 −0.270 −0.139 0.044 0.147 0.406 0.601 0.723 0.788 0.781 0.571 0.234 −0.154 −0.820 −1.227 −1.326 −1.221 −0.791

32

E. ARIMONDO

Comment on: “On the splitting of the Roentgen and optical terms caused by the electron rotation and on the intensity of the cesium lines”. The first scientific paper of Ettore Majorana, presented to the Accademia dei Lincei and published in the Rendiconti of that Academy in 1928, is joint work with Giovanni Gentile Jr., a junior professor of the Physics Institute in Rome. At that time Majorana was still an undergraduate physics student. His doctoral thesis under the supervision of E. Fermi was defended in 1929. This paper is an early application to outstanding problems of atomic spectroscopy, using the statistical model of atomic structure introduced by Fermi in a series of papers between 1927 and 1928.10 That framework has long been known as the Thomas-Fermi model, because the same essential idea was independently and simultaneously developed by L. H. Thomas at Cambridge University.11 This model simplifies the complex problem of calculating the atomic structure of multielectron atoms. For each electron, an effective potential in its Schrödinger equation is approximated by a central field which accounts for the clustering of all other electrons centered around the nucleus. This effective potential is calculated through the statistical approach derived by Fermi, which is described on page 28, top, with the function ϕ derived from the Poisson equation including the charge density associated with the other electrons. The Schrödinger equation with the effective potential to be solved for the external electron is written down in the middle of page 28. Fermi applied this statistical approach to derive the ionization energies of several atomic species, producing good agreement with known experimental results. Here, Gentile and Majorana (G&M) applied that approach to derive the ionization energy of an electron in the 3d orbit of gadolinium and uranium, again in good agreement with experimental values. Furthermore, G&M also calculate the fine-structure splitting of different atomic states of gadolinium, uranium and cesium. As concerns both the ionization potential and the spin-orbit splitting in gadolinium and uranium, the levels treated are designated as Roentgen terms; today they would be called X-ray transitions, because their spectroscopic studies are performed in the X-ray region. From a classical standpoint, the spin-orbit splitting is produced by the coupling between the magnetic moment of the electron spin (“the electron rotation”), which was introduced in 1925 by G. E. Uhlenbeck and S. Goudsmit,12 and the magnetic field produced by the electron motion around the nucleus. However G&M derive that splitting within the quantum mechanics approach formulated by P. A. M. Dirac. They apply first-order perturbation theory to the determination of the splitting of atomic energy levels. The equation on the top of page 29 is derived from the Dirac theory of ref. (1) of the original G&M paper. In that equation the potential

10 Fermi, E.: Rend. Lincei. 6, 602 (1927); 7(342), 726 (1928); Z Phys. 48, 73 (1928); 49, 550 (1928). 11 Thomas, L.H.: Proc. Camb. Phylos. Soc. 23, 542 (1927). 12 Uhlenbeck, G.E., Goudsmit, S.: Naturwiss 13, 953 (1925); Nature (London) 117, 264 (1926).

Comment on: On the Splitting of the Roentgen and Optical Terms Caused by. . .

33

within the integral is defined as v, while the Dirac paper reports the potential as V . The v quantity is not defined in G&M, though in Fermi’s papers v and V differ by a constant, and are otherwise equivalent in the determination of the spin-orbit splitting. The G&M formula suggests that the statistical Thomas-Fermi effective potential should be used for the determination of the splitting. However, the largest contribution to the spin-orbit splitting arises from the electron wave function near the nucleus, and there the effective potential reduces to the unshielded Coulomb potential. For the ionization potential of gadolinium and uranium, the theoretical analysis reproduces experimental values within a few percent and the spin-orbit splitting within 20%. However for cesium 6P the agreement is not that good, only 20% for the ionization potential and 50% for the spin-orbit splitting. The authors claim that the reason for the difference is that for cesium, the effective potential experiences a strong radial dependence in the region near the nucleus, where the potential reduces to a screened Coulomb potential. Furthermore, because several eigenfunctions of the external electron have a similar spatial distribution, the first-order perturbation is not sufficiently accurate. At this point the authors make an additional brilliant intuitive step into the analysis of the cesium data. They modify the cesium effective potential in order to obtain good agreement for the spin-orbit splitting. More precisely, they introduce a new effective potential derived from the Fermi statistical model, and from the central Coulombian potential as described in the second equation on page 30. Using that effective potential they claim to obtain a very good agreement with the experimental value of the spin-orbit splitting in the 6P and 7P cesium states. The authors extended their spectroscopic analysis, focusing their attention on the ratio of the transition probability for the optical transitions from the 6S ground state to the two upper P states. They did not compare their value to experiments, but their result agrees with the presently accepted value within 5%. The interest in oscillator strengths expressed in this paper was linked to the research performed in Pisa by the group of L. Puccianti, where E. Fermi obtained his physics degree,13 and also to investigations performed by E. Fermi and F. Rasetti in Florence few years before. Fermi himself subsequently14 investigated the issue of anomalous intensity ratios for optical transitions in his publication of 1930, where he explained the anomalous ratio measured by Rasetti for the two components of the S-P doublets in cesium, that work being a masterpiece of atomic spectroscopy. Fermi used the G&M 6P /7P intensity ratio to derive his estimate for the 6p doublet intensity. Fermi reported the G&M results at a restricted conference held in Leipzig in 1928 under the chairmanship of P. Debye. As described by F. Rasetti in the book of Fermi’s collected papers, that was considered by Fermi as a great honor to report to a select international audience the results of the work performed in Rome. Fermi decided to review the statistical model of the atom and its applications to various

13 At

that time Italian Universities offered one degree only in all subjects. E.: Z Phys. 59, 680 (1930).

14 Fermi,

34

E. ARIMONDO

problems by him and his collaborators. In a resumé of Fermi’s lecture, published in Leipzig in 1928,15 the G&M spin-orbit splittings and intensity ratios are discussed. The G&M work is infrequently cited, apart from Fermi’s citations. However, in 1933, T.-Y. Wu, completing his PhD at the University of Michigan under the direction of S. Goudsmit, examined theoretically the quantum defects of heavy atoms and compared his results to those of G&M.16 In 1997 P. S. Lee and T.-Y. Wu17 re-examined the Thomas-Fermi statistical potential for neutral atoms and produced a better approximation for the effective potential, which they claimed to provide a better accuracy for the G&M numerical analyses. EA is grateful to C. W. Clark for carefully reading the English translations of the original paper and of his comment, and for useful suggestions.

ENNIO ARIMONDO NIST, Gaithersburg, MD (USA)

15 Fermi,

E.: In: Falkenhagen, H. (ed). Quantentheorie und Chemie. Leipzig (1928); reprinted in Collected papers, vol. 1. The University of Chicago Press (1961). 16 Wu, T-Y.: Phys. Rev. 44, 727 (1933). 17 Lee, P.S., Wu, T-Y. Chin. J. Phys. 35, 742 (1997).

Majorana dr Ettore: Search for a General Expression of Rydberg Corrections, Valid for Neutral Atoms or Positive Ions∗ Ettore Majorana

It is an application, the Author says, of the statistical method devised by Fermi. In the interior of an atom with number Z, n times ionized, the potential can be put in the form V =

Ze ϕ(x) + C, r

where x is the distance measured in units1 μ = 0.47Z

− 13



Z−n Z−n−1

2 3

10−8 cm,

ϕ obeys a well known differential equation and the boundary conditions ϕ(0) = 1, −x0 ϕ  (x0 ) =

n+1 being ϕ(x0 ) = 0, Z

and C, which is the potential at the boundary of the ion, has the value C=

(n + 1)e . μx0

∗ Talk given at the 22nd General Meeting of the Italian Physical Society, Rome, 29 December 1928. Translated from Majorana dr. Ettore: Ricerca di un’espressione generale delle correzioni di Rydberg, valevole per atomi neutri o ionizzati positivamente, “Il Nuovo Cimento”, vol. 6 (1929) XIV–XVI, by F. Guerra and N. Robotti. 1 Note that in the original text of “Il Nuovo Cimento” it is erroneously printed 0.47z− 13 instead of 1 0.47Z − 3 . (Note of the Editor.)

© Springer Nature Switzerland AG 2020 L. Cifarelli (ed.), Scientific Papers of Ettore Majorana, https://doi.org/10.1007/978-3-030-23509-3_3

35

36

E. Majorana

In the formula above, one does not consider the local potential, but the mean effective potential acting on some electron in any given point in space. The two potentials, which to the first approximation are identical, are to be kept distinct, as has been now tacitly understood, in the second approximation, in order to take into account that the elementary charge of an electron is not vanishing, but has a finite value. As a matter of fact, one cannot proceed to the second approximation in a rigorous way, but, in the case of an isolated atom, one can imagine some quite satisfactory methods. Among these, the simplest one leads to the expression mentioned before. If the potential at the boundary of the atom is known, one can derive the highest potential energy for an electron: U = −Ce. By sorting the electrons in s, p, d, . . . electrons, according to the azimuthal quantum number, since all electrons which are present must occupy the states of lower energy, then the same highest energy will be shared by the most external electron among the s electrons, and the most external electron among the p or d or f electrons. It is immediately seen that this highest energy must be equal to U . Then, if lk is the total quantum number, for example of the most external electron with azimuthal quantum number k, it will be connected to the Rydberg correction, denoted by ck for terms of azimuthal number k, through the relation Rh = −U. (lk − ck )2 The total quantum number lk of the most external electron can be calculated through the Pauli principle, provided one knows the number Nk of electrons with azimuthal number k, which are present in the atom. It is possible to evaluate this number, as has been shown by Fermi in his well-known paper on the statistical explanation of the periodical system. In order to avoid excessive calculations, it is supposed, as an approximation, that the distribution, along the quantum numbers, of the Z − n electrons, which are present in the Z atom, n times ionized, is the same as for a neutral atom, with atomic number Z − n, provided n/Z is small. Then, the announced expression of the Rydberg corrections is obtained  ck = k + (Z − n)

1 3

 0.42n 0.565



k2 2

(Z − n) 3

 − 0.665



x0 (n + 1) 1

2

Z 3 (Z − n − 1) 3

,

where k is the azimuthal quantum number increased of 12 and  is a function already examined or calculated by Fermi. The numerical constants are not experimental constants, but transcendent numbers, simple algebraic expressions of π , written here in decimal form, for the sake of brevity. By putting n = 0, the formulae relative to neutral atoms are obtained. The agreement with the experiments is quite good: in general, especially for the s terms, one obtains values slightly lower than the

Majorana dr Ettore: Search for a General Expression of Rydberg Corrections,. . .

37

observed ones. But the difference can be attributed mostly to the phenomenon of polarization. The Author then mentions an attempt of statistical evaluation of the chemical bond effect on the Röntgen spectra. The examined substances are the simple elements 13-14-15-16, respectively Al, Si, Ph, S, and their oxygen compounds. In these compounds, these elements behave as trivalent, tetravalent, pentavalent, or hexavalent, respectively. This means, for example, that a molecule of sulfuric anhydride should be considered schematically as composed by one six times positively ionized S atom, and by three two times negatively ionized O atoms. It is easily understood that this schematization is excessive. In fact, the hexavalent S ion would not leave its neighbors undisturbed. It is to be understood for the valence electrons that, even though they pass in first approximation under control of the oxygen, as a matter of fact, they maintain more or less close relations with the atom of sulfur (perhaps together with the valence electrons of oxygen). It is possible to think of an effective ionization, which would produce in the isolated ion the same variation of the Röntgen terms. This is what the Author did. The calculation has been performed in the second approximation with special care, by exploiting the measurements done by Erik Bäcklin in 1925, about the shift toward high frequencies for the line Kα , when one goes from the simple element to the compound. The calculation gives for the effective ionization respectively2 Z = 13 2

3

14 2.7 4 15 3.4 5 16 4.2 6. Researches performed until now are still too meager, in order to appreciate these results in their full value. However, one thing seems to be definitely acquired, i.e. that the analysis of the deep terms is bound to give interesting indications on the structure of the molecules. The Author thanks prof. Fermi, who has been generous in advice and suggestions about the new applications of the statistical method, explained here. The statistical method has thrown considerable light on atomic physics. Its fecundity, far from being exhausted, is still waiting to be challenged in research fields, which are more extensive and richer in promises.

2 In

“Il Nuovo Cimento” aside the value “5” in the table, there appears, for a presumable typo, also the value “0.7”. (Note of the Editor.)

38

F. GUERRA and N. ROBOTTI

Comment on: “Ettore Majorana on the Thomas-Fermi statistical model for atoms and ions. The communication at the meeting of the Italian Physical Society (Rome, December 1928)”. On occasion of the XXII General Meeting of the Italian Physical Society, held in Rome at the Physical Institute of the Royal University from December 28th to 30th, 1928, during the session of December 29th, the young Ettore Majorana delivered a communication with title “Ricerca di un’espressione generale delle correzioni di Rydberg, valevole per atomi neutri o ionizzati positivamente” (“Search for a general expression of Rydberg corrections, valid for neutral atoms or positive ions”).3 There he reports about his original results on the Thomas-Fermi statistical model for atoms and ions, obtained during the year 1928, while he was still a student at the School of Engineering. The communication is very detailed, and appears in the records of the session, regularly published in the journal of the Italian Physical Society, “Il Nuovo Cimento”. It was also listed in the German “Jahrbuch über die Fortschritte der Mathematik” 55, 1183 (1929). Now it appears also in the electronic archives of the Jahrbuch. It is our purpose here to describe the scientific atmosphere in which the research presented in the communication was carried out and to put in the right historical perspective Majorana contribution to the field. In this reconstruction, we will rely also on original documents, kept in the Majorana Archive of the Domus Galilaeana in Pisa, and in the Archives of the University of Rome “La Sapienza”. During the session of December 4th, 1927, at the Accademia dei Lincei, in Rome, Orso Mario Corbino, member of the Academy and also director of the Royal Institute of Physics of the University of Rome, presented a note4 by Enrico Fermi, with the title “Un metodo statistico per la determinazione di alcune proprietà dell’atomo,” (“A statistical method for the determination of some properties of the atom”).5 Fermi ideas are very simple and very powerful. In principle, according to quantum mechanics, an atom should be described by the full Schrödinger equation. However, due to the large number of variables involved, for practical applications, this equation was hard to control when the number of electrons is large. Fermi treatment is based instead on a kind of electrostatic effective mean-field potential, produced by the nucleus and all the electrons. The electrons are considered as a completely degenerate statistical ensemble, obeying Fermi statistics, under the local influence of this electrostatic potential. Fermi statistical distribution gives the electron density as a function of the potential in each point in space. On the other hand, the potential satisfies the Poisson equation, with sources given by the nucleus and the electron mean distribution. This nonlinear problem is easily solved

3 Majorana,

E.: Ricerca di un’espressione generale delle correzioni di Rydberg, valevole per atomi neutri o ionizzati positivamente. Nuovo Cimento 6, XIV–XVI (1929). 4 Fermi, E.: Un metodo statistico per la determinazione di alcune proprietà dell’atomo. Rend. Accad. Lincei. 6, 602–607 (1927) (n. 43 in ref. 5). 5 Fermi, E.: Note e Memorie (Collected Papers), vol. I, Italia 1921–1938. Roma/Chicago/Londra: Accademia Nazionale dei Lincei, The University of Chicago Press (1962).

Comment on: Majorana dr Ettore: Search for a General Expression of Rydberg. . .

39

by relying on the numerical integration of a second-order differential equation, with appropriate boundary conditions, for the so-called Fermi function, describing the screening effect of the electrons on the Coulomb potential of the nucleus. The Fermi function depends only on the distance from the nucleus. Therefore, we have a drastic reduction in the number of variables. Apparently Fermi was unaware that an essentially equivalent scheme had been presented by Llewellen Hilleth Thomas in the paper6 sent to the Cambridge Philosophical Society on November 6th and read on November 22nd, 1926. However, Fermi program was much more ambitious, aiming at a systematic exploration of atomic properties by using the statistical model. In particular, the spectroscopic levels of atoms could be calculated through a very simple scheme, by considering an approximate model, where a single electron (the optical electron) would obey Schrödinger equation under the influence of a properly defined effective potential, depending on the Fermi function. In fact, already during the first half of the year 1928 a series of papers appeared in Rome, where various atomic problems were considered. Fermi himself made applications to the periodic system of elements, giving a way to understand the formation of the anomalous groups of elements, as the rare-earth elements, and evaluated Rydberg correction for the spectroscopic sterms. Then Franco Rasetti7 calculated the M3 spectroscopic Röntgen terms for a series of elements. Also Ettore Majorana, participated to these efforts, as we will see, by collaborating with Giovanni Gentile jr.8 All results, obtained by Fermi and his associates in the first half of 1928, were summarized by Fermi in a very detailed review,9 based on his report at the conference Leipziger Tagung, held in Leipzig on June 17–24, 1928, where he was invited by Peter Debye, with a letter still kept in the Archives of the University of Rome. In the Fermi statistical model, long, but affordable, calculations give numerical results in reasonable agreement with the experimental findings. Fermi extended this scheme to the case of positive ions, with a given ionization number, in his paper.10 As mentioned before, the involvement of Ettore Majorana with Fermi statistical model started very early. Ettore Majorana had enrolled in the 2 year preparatory course for Engineers in Rome in Fall 1923. From the records kept in the archives of the University of Rome, we can follow his academic career as student (position number 10447). After obtaining the 2-year diploma in 1925, he continued his studies in the 3-year School of Engineering. He took regularly all requested examinations during the academic years 1925–1926, and 1926–1927. He was 6 Thomas,

L.H.: The calculation of atomic fields. Proc. Camb. Philos. Soc. 23, 542–548 (1927). F.: Eine statistische Berechnung der M-Röntgenterme. Z Phys. 48, 546–549 (1928). 8 Gentile, G., Majorana, E.: Sullo sdoppiamento dei termini Roentgen e ottici a causa dell’elettrone rotante e sulle intensità delle righe del cesio. Rend. Accad. Lincei. 8, 229–233 (1928). 9 Fermi, E.: Über die Anwendung der statistischen Methode auf die Probleme des Atombaues. In: Falkenhagen, H. (ed.) Quantentheorie und Chemie (Leipziger Vorträge 1928), pp. 95–111. S. Hirzel, Leipzig (1928) (n. 49 in ref. 5). 10 Fermi, E.: Sul calcolo degli spettri degli ioni. Mem. Accad. Italia. I (Fis.), 149–156 (1930) (n. 63 in ref. 5). 7 Rasetti,

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expected to finish the examinations in 1928 and to obtain the doctoral degree as Engineer, with the submission of a Thesis. However, during 1928 Majorana, still as a student of Engineering, took only one examination. This was “Theoretical Physics”, passed with a marking of 100/100 “cum laude”, on June 5th, 1928. The course “Theoretical Physics” was the course given by Enrico Fermi at the Faculty of Sciences. According to the rules, it was possible, for a student of Engineering, to take courses held in different Faculties. It is clear that the interests of Majorana had moved toward Physics at the beginning of 1928. It was only in Fall 1928 that Majorana asked to move officially to the Faculty of Sciences. The Faculty gave his assent during the session of November 19th, 1928. Then Majorana took the requested examinations in Physics, and obtained his doctoral degree, on July 6th, 1929, with full marks and “laude”. Starting from the beginning of 1928, when he initiated his contacts with Fermi, in few months, he acquired a very deep knowledge of the structure of the recently established Fermi statistical model. In fact, in his notebook “Volume II”, kept at the Domus Galilaeana in Pisa, reporting at the beginning the date of April 23rd, 1928, we can find, among other things, a very clever calculation of the values of the Fermi function, an evaluation of the infra-atomic potential without using the statistical method, some applications of Fermi potential with the calculation of the atomic ground state energy, and the statistical curve of the fundamental terms in neutral atoms. Along the program of applications of the statistical model, at the beginning of 1928, Ettore Majorana began a fruitful collaboration with Giovanni Gentile jr, who, after his graduation in Pisa in November 1927, received from Corbino a 6 month temporary appointment as assistant in Rome. Their results were published in the already mentioned joint paper8 . In this paper, by using Fermi method, they calculate the energy and the level splitting, due to the electron spin, for the 3d spectroscopic Röntgen term of gadolinium (Z = 64), and uranium (Z = 92), and for the 6p optical term of caesium (Z = 55). Moreover, they calculate also the intensity ratio for the first two absorption lines of caesium. The calculation of the level splitting of spectroscopic terms is very interesting. In fact, to this purpose the authors exploit a perturbation formula given by Dirac only few months before. The results of this paper were well received. In fact, Fermi makes reference to them in his Leipzig report (See 9 ), and heavily exploits the expression given for the intensity ratios in his subsequent papers. An early draft of this joint paper with Giovanni Gentile jr is kept in the Majorana Archives in Pisa. Handwritten parts produced by the two coworkers are easily recognizable. The part written by Majorana is extremely interesting, because it contains some deep considerations about the limits of the statistical model, due to polarization effects. After the joint paper with Gentile, Ettore Majorana in 1928 continued alone his research on the statistical model for the atom, with great scientific autonomy and effectiveness. In his December 1928 communication, Majorana presented results about his improvement of the Fermi model, and its extension to positive ions. The paper of

Comment on: Majorana dr Ettore: Search for a General Expression of Rydberg. . .

41

course is in Italian, and is written in the form of a record of the session, but is very clear and detailed. The very essence of Majorana improvement can be easily recognized. In the original Fermi formulation, the statistical distribution of electrons is ruled by the microscopic local electric field, i.e. the electric field acting on some hypothetical vanishing test charge, while in Majorana it is ruled by the effective field, acting on some electron in any given point in space. Majorana correctly remarks that these two electric fields, which to the first approximation are identical, are to be kept distinct in a more refined “second approximation”, in order to take into account that the elementary charge of an electron is not vanishing. From a physical point of view, in Fermi atoms each electron is interacting also with itself, because the repulsion between electrons is described through the overall electric density. In Majorana improvement this self-interaction is avoided, through a very simple approximate average argument, giving the relation between the microscopic and the effective potentials. Of course, the Majorana effective field acting on the optical electron is slightly different with respect to the Fermi expression. Moreover, in the Majorana formulation neutral atoms (and ions) have a finite radius. It must be also appreciated that the communication gives the first treatment of positive ions in the scientific literature, in the frame of the statistical model. Moreover, it is very simple to realize that Majorana improvement allows for stable negative ions of charge one (in this case with an electronic density extending to infinity). Fermi scheme1 0 for positive ions comes more than 1 year later, and does not take into account Majorana improvement. The communication continues with a very elegant expression, in closed form, of the Rydberg corrections to the spectral lines. Here, Majorana not only exploits the new improved proposed scheme, but follows a method radically different from the method introduced by Fermi (See 9 ). In the second part of the communication, Majorana gives a preliminary account about an attempt of statistical evaluation of the effect of chemical bonds on the deep Röntgen spectral lines. He considers the elements Al, Si, Ph, S, and their molecular compounds with oxygen. The displacement of the spectral lines in the compounds is interpreted in terms of the lines of the simple elements with some effective even noninteger ionization, in the frame of his improved theory. He performs an elaborate second-order calculation, and exploits the available experimental values for the displacement of the deep Röntgen spectral lines, going from the simple element, to its oxygen compound. The results are summarized in a table. For the elements Z = 13, 14, 15, 16, with chemical valences given by 3, 4, 5, 6, respectively, he finds that the displacement of the spectral lines for the compounds can be interpreted by attributing to the isolated atoms an effective ionization with values 2, 2.7, 3.4, 4.2, respectively. These extremely interesting ideas could suggest further applications even in present-day research. After the December 1928 communication, at the beginning of 1929, the activity of the young Majorana shifted toward nuclear physics. In fact, he obtained his doctoral degree in Physics by presenting a research Thesis with the title “Sulla

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meccanica dei nuclei radioattivi” (“On mechanics of radio-active nuclei”), under Fermi supervision. The December 1928 communication did not receive any mention, neither in the numerous publications of Fermi and his associates on the subject, nor in any of the further reconstructions of the life and scientific activity of Ettore Majorana. The Majorana proposal for the improvement of the effective potential in the statistical model apparently was not accepted by Enrico Fermi for years, and forgotten. However, it was finally exploited, without reference, in the 1934 monumental conclusive paper by Fermi and Amaldi on the statistical model for atoms.11 In fact, the Fermi-Amaldi paper is based, as enphasized by the authors, on an “improved” statistical model. The main improvement, concerning the effective potential acting on the optical electron, is exactly what was proposed by Majorana, more than 5 years before. After 77 years, the December 1928 communication to the General Meeting of the Italian Physical Society has been brought to the attention of the scientific community with a note12 posted on arXiv.com, and now it finally appears here in this Volume among the scientific publications of Ettore Majorana. FRANCESCO GUERRA Università di Roma “La Sapienza”, Rome, Italy NADIA ROBOTTI Università di Genova, Genoa, Italy

E., Amaldi, E.: Le orbite ∞s degli elementi. Mem. Accad. Italia. 6(Fis.), 119–149 (1934) (n. 82 in ref. 5). 12 Guerra, F., Robotti, N.: A forgotten publication of Ettore Majorana on the improvement of the Thomas-Fermi statistical model, available on http://arXiv.com/physics/0511222. 11 Fermi,

On the Formation of Molecular Helium Ion Ettore Majorana

Summary: The stability of the ion He+ 2 (In “Il Nuovo Cimento” it is printed He2 instead of He+ 2 . (Note of the Editor, see also E. Amaldi, op. cit.)) can be studied even quantitatively with the method of Heitler and London. The results agree with the available experimental data. It appears that this compound is analogous to H2 even though the external electron is in a different state and the internal electrons do not only have the effect of screening the nuclear charges but instead contribute effectively to the molecular construction. The observation of Weizel and Pestel,1 Curtis and Harvey2 and of others of the band spectrum of helium have proved that some molecular constants (oscillation quantum, moment of inertia) tend, for the higher states of the optical electron, to certain limits that can naturally be attributed to the corresponding quantities of the ion He+ 2 . This proves the possible existence, at least in the ground state, of this ion to which we can3 assign the configuration (1sσ )2 2pσ 2 . Since, according to recent conclusions by W. Weizel,4 it is plausible to believe that the neutral molecule is formed from an atom in the ground state and one in an excited state, we can conclude that the ion dissociates either by high energy oscillation quanta or by an adiabatic increase of the distance between the nuclei if, as we intend to do, we consider them as fixed in a neutral atom and in an ionized one, both in the ground state.5 In this paper we want to study the reaction He + He+ from the energy point

Translated from Sulla formazione dello ione molecolare di elio, “Il Nuovo Cimento”, vol. 8, 1931, pp. 22–28, by P. Radicati di Brozolo. 1 Weizel, W., Pestel, E.: Z Physik. 56, 97 (1929); see also ref. 3. 2 Curtis, W.E., Harvey, A.: Proc. R. Soc. Lond. A 125, 484 (1929). 3 Weizel, W.: Z Physik. 56, 727 (1929). 4 Weizel, W.: Z Physik. 59, 320 (1930). 5 Cfr. Hund, F.: Z Physik. 63, 719 (1930). © Springer Nature Switzerland AG 2020 L. Cifarelli (ed.), Scientific Papers of Ettore Majorana, https://doi.org/10.1007/978-3-030-23509-3_4

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of view and prove that such a reaction may lead to the formation of the molecular ion. The first order of approximation leads, both for the equilibrium distance of the nuclei and for the oscillation quanta, to values in fairly good agreement with the experimental data. The method we will follow is the one that has been originally applied by Heitler and London6 to the study of the hydrogen molecule. We shall assume that the electronic eigenfunctions of the molecule are linear combinations of the eigenfunctions belonging to the separate atoms and we shall use them to evaluate the average value of the interaction between the two atoms. However since the two nuclei have the same charge whereas only one of the atoms is ionized, the problem as we will show is mechanically rather different from the one discussed by Heitler and London and in general from the problem that one encounters in the normal theory of the omeopolar valence.7 1. Let us suppose that the distance between the two atoms is large and let us for the moment disregard the fact that the state in which the first atom is neutral and the second ionized and the state in which the first atom is ionized and the second neutral have the same energy and should therefore be considered together in a perturbation calculation. If we consider the first atom neutral and the second ionized, the problem is similar to that of the reaction8 He + H; thus, since He is in a closed electronic shell, there is a unique mode of reaction. This means that the interaction between the two atoms does not arise from a separation of the terms (which satisfy Pauli’s principle); moreover the resonance forces are certainly repulsive. It is true that the polarization attractive forces are in our case different from those acting between neutral atoms; indeed whereas the first forces arise from a potential decreasing at large distances like R −4 , the second instead vanish more rapidly since they depend upon a potential that varies as R −6 . However the polarizability of the neutral He atom is very small and therefore by no means the polarization forces can alone give rise to a stable binding. To explain the chemical affinity between He and He+ we must instead abandon the condition stated at the beginning and let the neutral atom free to share an electron with the ionized one and thus take its place. The net effect is to split the term resulting from the union of the two atoms, almost without changing its average value. The splitting thus depends not upon the resonance of the electrons but rather on the behaviour of the eigenfunctions under reflection with respect to the center of the molecule. The eigenfunctions may be unchanged under the above spatial inversions in which case we call them even, or may change sign, in which case we call them odd. We could perhaps speak of resonance of the nuclei, but only metaphorically and not in the proper meaning which is different, as here we consider only the electronic eigenfunctions. The splitting of the term originating

6 Heitler,

W., London, F.: Z Physik. 44, 455 (1927). F.: Z Physik. 46, 455 (1927); 50, 24 (1928). Heitler, W.: Z Physik. 46, 47 (1927); 47, 835 (1928). For a comprehensive review see, e.g., Heitler, W.: Phys. Z. 31, 185 (1930). E. A. Hylleraas in the study of the reaction between differently excited hydrogen atoms has found further degeneration deriving from the equality of the nuclei. Z Physik. 51, 150 (1928). 8 Cfr. Gentile, G.: Z Physik. 63, 795 (1930). 7 London,

On the Formation of Molecular Helium Ion

45

from its even or odd parity is greater by an order of magnitude than the energy due to the repulsive valence forces. One of the two modes of reaction thus corresponds to repulsion and does not give rise to chemical binding whereas the other gives rise to an attractive force and leads to the formation of a molecular ion. We can recognize the two modes of reaction with the following considerations. Since we are dealing with three electrons we can require that two of them, which we call electron 1 and electron 2, have parallel spins. Therefore their eigenfunction must be antisymmetric in the coordinates of the first two electrons in such a way that when the nuclei are very far apart one of the electrons is certainly close to the nucleus (a) and the other close to the other nucleus (b), and they can only exchange their place simultaneously. In this case the eigenfunction in its dependence on the first two electrons changes sign. It thus follows that the electrons 1 and 2 give rise together to an odd term described by an eigenfunction that changes sign under spatial inversion. However the only individual states to be considered are those arising from the states 1s of the separate atoms, i.e. 1sσ and 2pσ , the first being even and the second odd, and the electrons 1 and 2 are thus an electron 1sσ and a 2pσ one (or viceversa). To the third electron we can assign an even eigenfunction (1sσ ) or an odd one (2pσ ), respectively resulting from the sum or the difference of the eigenfunctions of the separate atoms. In the first case we obtain the overall configuration (1sσ )2 2pσ 2 , i.e. the configuration that belongs to the ground state of He+ 2 and actually corresponds to an attraction between the two atoms. In the second case we would find instead the configuration 1sσ (2pσ )2 2  that belongs to an excited level of He+ 2 which is probably unstable and gives rise to the other mode of reaction, i.e. that leading to repulsion. We thus conclude that the essential cause of the instability of He+ 2 is the same which gives rise to the stability of the molecular ion of hydrogen. 2. Let us call  and ϕ the eigenfunctions of the neutral and ionized atom (a) and similarly  and ψ those of atom (b). The six unperturbed eigenfunctions of the separate atoms, obtained by permutations of the electrons and exchange of the nuclei, are the following: ⎧ ⎪ B1 = ψ1 23 , ⎪ ⎨A1 = ϕ1 23 , (1) A2 = ϕ2 31 , B2 = ψ2 31 , ⎪ ⎪ ⎩A = ϕ  , B =ψ  . 3

3

12

3

3

12

The interaction of the atoms splits the six-time–degenerate term by separating the correct zeroth approximation eigenfunctions according to the symmetry characters9 of the electrons and according to their behaviour under spatial inversion. If we denote with + the even terms and with − the odd ones, the only symmetries are the following: (123)+ , (123)− , (123)+ , (123)− .

9 Hund,

F.: Z Physik. 43, 788 (1927).

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Correspondingly one obtains four terms: the first two are singlets and the last two doublets arising from a hidden degeneracy. Pauli’s principle leads to exclude the first two as they are symmetric in the three electrons; the only remaining ones are the third term which, because of its parity, corresponds to the configuration 1sσ (2pσ )2 2 , and the fourth which is for us the most interesting as it leads 2 2 to the formation of He+ 2 and therefore to the configuration (1sσ ) 2pσ . The eigenfunctions of the four terms are, apart from a normalization factor: ⎧ + ⎪ ⎪ ⎪(123) ⎪ ⎨(123)− ⎪(123)+ ⎪ ⎪ ⎪ ⎩ (123)−

y1 = A1 + A2 + A3 + B1 + B2 + B3 , y2 = A1 + A2 + A3 − B1 − B2 − B3 , y3 = A1 − A2 + B1 − B2 ,

(2)

y4 = A1 − A2 − B1 + B2 ,

where we have chosen for the third and fourth term the eigenfunction antisymmetric in 1 and 2. The perturbation function is of course different for each of the unperturbed states (1) as the electrons do not appear symmetrically in the Hamiltonian of the ideally separated atoms. We thus obtain six different Hamiltonians depending on the configuration of the electrons when the atoms are separate, and only one, symmetric, when the atoms are united. In the configuration A1 the perturbation energy is for instance expressed by H =

e2 e2 2e2 2e2 2e2 4e2 + + − − − , R r12 r13 r1b r2a r3a

where R, r12 , r13 , r1b , r2a , r3a , are the distances of the nucleus and of the electron of the first atom from the nucleus and from the electrons of the second. The variation of a generic eigenvalue is given in first approximation by the symbolic expression  Ei = 

y¯i Hyi dτ y¯i yi dτ

(i = 1, 2, 3, 4).

(3)

One should notice that H acts differently, as said, on the various terms which build up every yi according to (2). Explicitly the result (3) is ⎧ I0 + 2I1 + 2I2 + I3 ⎪ E1 = , ⎪ ⎪ ⎪ 1 + 2S1 + 2S2 + S3 ⎪ ⎪ ⎪ ⎪ I0 − 2I1 + 2I2 − I3 ⎪ ⎪ , ⎨E2 = 1 − 2S1 + 2S2 − S3 I0 − I1 − I2 + I3 ⎪ ⎪ ⎪ E3 = , ⎪ ⎪ 1 − S1 − S2 + S3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩E4 = I0 + I1 − I2 − I3 , 1 + S1 − S2 − S3

(4)

On the Formation of Molecular Helium Ion

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where  S1 =

 B2 A1 dτ ;

S2 =

 I0 =

 A2 A1 dτ ;

S3 =

 A1 H A1 dτ ;

I1 =

B1 A1 dτ ;

(5)

 B2 H A1 dτ ;

I2 =

A2 H A1 dτ ;

(6)

 I3 =

B1 H A1 dτ .

If the nuclei are sufficiently (but not exceedingly) distant all the integrals I are negative and the S positive and for the order of magnitude we have: −I1 > −I2 > −I3 , S1 > S2 > S3 . In the relations (4) we can disregard the differences between the denominators, which for large distances are close to one, as well as I3 . Then the main contributions to the energy come from the “electrostatic energy” I0 , which is common to all terms, including those not obeying Pauli’s principle, and can be neglected owing to its small value, and from combinations of the “exchange energy” I2 , corresponding to valence forces, and the larger symmetry energy I1 , arising from the behaviour of each term under spatial inversion. Considering the solutions y3 10 and y4 , the only ones that have a physical meaning, and taking into account the order of magnitude and the sign of I1 and I2 , we indeed find that y3 10 gives rise to repulsion whereas y4 leads to formation of a molecule. From now on we will consider only y4 . 3. The evaluation of E4 as a function of the distance, i.e. the determination of the so-called potential curve of the molecule He+ 2 , requires the evaluation of the integrals (5) and (6). The eigenfunction of the neutral atom of helium in its ground state has been calculated numerically with great accuracy but does not have a simple analytical expression.11 Therefore we need to use rather simplified unperturbed eigenfunctions. For example, we could assume, as commonly done, for the helium ground state the product of two hydrogen-like eigenfunctions with an effective Z equal to 1.6 ÷ 1.7, depending on the criterium used for the evaluation; if we want 5 to optimize (minimize) the average energy, we must then set Z = 2 − 16 = 1.6875; if instead we want that the diamagnetic constant be in agreement with the experimental value and at the same time with the value provided by accurate theoretic calculations, (see 11 ) we must set Z = 1.60. However more serious uncertainties are inherent to the nature of the method adopted, which can only 10 In

“Il Nuovo Cimento” it is erroneously printed y8 . (Note of the Editor, see also E. Amaldi, op. cit.) 11 Slater, J.C.: Phys. Rev. 32, 349 (1928); Hylleraas, E.: Z Physik. 54, 347 (1929).

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lead to a first and rather rough approximation. Indeed Heitler and London’s method is inaccurate when the atoms are very far apart not only because it neglects the polarization forces12 , that in our case predominate, but also because it leads to resonance forces that are wrong both in the order of magnitude and in the sign. For example, in the case of the hydrogen molecule discussed by Heitler and London, the exchange energy is positive for very large distances because Sugiura’s integral13 is larger than all the others. This would suggest that the antisymmetric solution is deeper than the symmetric one. This however must be excluded on the basis of very general theorems (see 6 ) and the apparent contradiction simply indicates the roughness of the method. For distances of the order of the equilibrium distance it is plausible that the perturbed eigenfunctions are considerably different from the unperturbed ones, so that the first approximation is nothing but an indicative value. For these reasons we have used a model that may appear too crude and that could only be improved with long calculations of doubtful significance. We have therefore assumed for the eigenfunctions of each single atom, hydrogen-like functions corresponding to the same Z = 1.8, both for the neutral and the ionized atom. The integrals (5) and (6) become then well-known elementary integrals, one of them being the one already mentioned given by Sugiura. We thus find that the potential curve has a minimum when the distance between the nuclei is d = 1.16 to be compared with the experimental value (see 2 ) d = 1.087 Å. It is likely that a more exact calculation of the integrals (5) and (6) can improve the already remarkable agreement between the two values. The corresponding energy is Emin = −1.41 V = −32,500 cal. In this respect there are no experimental data and we must consider this value of the chemical affinity as a lower limit. Including all the errors of the method under the expression “polarization forces”, we find that for very distant nuclei these depend on the potential −α

e2 , 2R 4

where α = 0.20 · 10−24 is the polarizability of neutral helium. If we assume this expression to be still approximately valid for the (true) equilibrium distance, which is rather implausible, we find that the chemical affinity is presumably −E = 2.4 V = 55,000 cal.

12 Eisenschitz, 13 Sugiura,

R., London, F.: Z Physik. 60, 491 (1930) Y.: Z Physik. 45, 484 (1927).

On the Formation of Molecular Helium Ion

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Finally we can calculate the initial oscillation quantum from:  1 n= 2π c

1 Mr



d 2E dR 2

, 0 2

where Mr = 12 MHe = 3.30 · 10−24 is the reduced mass of the oscillator and ( ddRE2 )0 is the second derivative of the potential curve at the equilibrium point. Since the 2 calculation gives ( ddRE2 )0 = 3.03 · 105 erg/cm2 , we find n = 1610 cm−1 which is, quite by chance, in perfect agreement with the experimentally determined value, (see 2,3 ) n = 1628 cm−1 , of the first oscillation quantum. I very much thank Professor E. Fermi who has given me some precious suggestions and help and Dr G. Gentile for the interest he has shown in this work.

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Comment on: “On the formation of molecular helium ion”. The second paper published by Ettore Majorana in 1931 concerns the question of the chemical bond that at that time was beginning to be posed. Indeed this paper appears only 4 years after the publication of a famous work by Heitler and London (1927) on the formation of the H2 molecule, essentially the first quantummechanical description on the chemical bond. Although the approach used by Majorana, as he himself stated, is similar to Heitler and London’s method, his study on the molecular ion He+ 2 is quite more intriguing than the case of H2 . Indeed the construction of the quantum states results more complicated either for the larger number of electrons or for the requirements of the Pauli exclusion principle. The expected number of terms is N ! (N being the number of electrons), therefore they are six in the case of He+ 2 (see Eq. (1) of the original paper). Majorana feels that the right approach is to consider the interaction between a helium atom and its ion (He + He+ ). To him it appears clear that a classical picture solely based on polarization forces is inadequate to explain the chemical affinity. The true reason, as Majorana guesses, has to be found in the quantum character of the electrons, and, in particular, in their indistinguishability. Majorana says: “To explain the chemical affinity between He and He+ we must instead abandon the condition stated at the beginning and let the neutral atom free to share an electron with the ionized one and thus take its place”. The idea of the exchange is important, not only for this fundamental problem of the chemical bond: it will be applied in a completely different context concerning nuclear forces (exchange forces). Majorana translates his concept in a quantummechanical language by constructing appropriate eigenfunctions in accordance with the symmetry properties of the investigated system. Majorana starts with a simple description in terms of two non-interacting (far away) species: He and He+ . The wave functions are those for a neutral helium atom and its ion. When the two nuclei approach each other it becomes necessary to take into account their reciprocal interaction. Under these conditions the quantum-mechanical rules play a crucial role because the total electronic wave function must show a definite symmetry with respect to the midpoint of the internuclear line. As for the case of H2 two molecular states are similarly originated: the state (1sσ )2 2pσ 2  corresponding 22 to the bonding molecular orbital of He+ 2 and the state 1sσ (2pσ )  which, in contrast, has no minimum and is repulsive at all internuclear distances. It is interesting to note that, although the average energy of these two states is quite close to the unperturbed value, the energy gap is “greater by an order of magnitude than the energy due to the repulsive valence forces”. In order to calculate the equilibrium distance, the corresponding energy difference, and the oscillation frequency, Majorana makes use of the variational principle. With regard to this point, it is impressive how Majorana handles the wave functions searching for easier but at the same time physically meaningful expressions. For instance, he makes use of hydrogenoid functions but introduces a screening effect by means of an effective nuclear charge. Majorana’s works are usually strongly connected to experimental observations—also the motivations that inspired this

Comment on: On the Formation of Molecular Helium Ion

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paper originate from a not well interpreted experimental phenomenon (the band structure in the helium emission spectrum)—and his final calculations are compared scrupulously to very accurate experimental data available in the literature. ANTONIO SASSO Università di Napoli, Naples, Italy

On the Possible Anomalous Terms of Helium Ettore Majorana

Summary: According to P. Gerald Krüger two new lines of helium originate from the anomalous terms (2p)2 3 P012 and (2s)2 1 S0 . Energy calculation together with stability considerations favour the interpretation of the first line. The explanation of the second line is almost certainly erroneous for energy reasons and because it appears improbable that this line should be attributed to atomic helium. A remarkable paper by Krüger1 has led to the discovery of two new lines of helium, λ = 320.4 Å and λ = 357.5 Å, which do not arise from combinations of known terms. The Author interprets the first line as a transition from the normal term 1s2p 3 P012 to a primed term 2p2p 3 P012 . From the latter there would arise a true group pp similar to those that are known in Zn, Cd and Hg; the resolution within the group is difficult because the levels are very close. For the second line the Author suggests the transition 1s2s 1 S0 -2s2s 1 S0 . Both these suggested new terms 2p2p 3 P012 and 2s2s 1 S0 have energy higher than the normal limit for He. It is therefore energetically possible that atoms undergo spontaneous ionization (Auger effect) with emission of an electron having the appropriate kinetic energy whereas the other electron falls into the orbit 1s. However, for the Auger effect to occur, it is not always sufficient that the energy of one term lie in the continuous spectrum leading to a reduction of the lifetime of the quantum state and to an uncertainty in the energy. In some cases symmetry considerations may forbid transitions from negative terms (i.e. terms higher than the ionization potential) to terms in the continuous spectrum. This may happen if to conserve energy we must require that they correspond to a free electron and to an ion in a definite state.

Translated from I presunti termini anomali dell’elio, “Il Nuovo Cimento”, vol. 8, 1931, pp. 78–83, by P. Radicati di Brozolo. 1 Gerald Krüger, P.: Phys. Rev. 36, 855 (1930). © Springer Nature Switzerland AG 2020 L. Cifarelli (ed.), Scientific Papers of Ettore Majorana, https://doi.org/10.1007/978-3-030-23509-3_5

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This is the case for the term 2p2p 3 P012 of helium and in general for the largest part of anomalous negative terms that have been observed till now and whose interpretation is not controversial. On the contrary the term 2s2s 1 S0 postulated by Krüger would give rise to an easily observable Auger effect whose order of magnitude, estimated by Wentzel2 and in particular by Fues,3 would make the widening of the terms perfectly observable or, which is equivalent, would make it extremely difficult to identify spectroscopically the latter. However, the argument against the observability of the term 2s2s 1 S0 of helium is not totally convincing. In reality the two terms 2s2s 1 S0 and 2p2p 1 S0 with the same symmetry do not exist separately in helium because their mutual interaction is of the same order of magnitude as their separation. The situation would remain the same in any heliumlike atom even for Z → ∞. Combining the two levels 2s2s 1 S0 and 2p2p 1 S0 , we obtain two terms (in the following denoted by X and Y ). The lower one (X), coming predominantly from 2s2s 1 S0 will be denoted with the same symbol. The interaction of the two electrons in this state is abnormally small because even in the zeroth approximation (for Z → ∞) they tend to remain diametrically opposite with respect to the nucleus. It is quite possible that in this case the Auger effect be much smaller than the effect calculated by Fues for the non-existing term 2s2s 1 S0 . Against the interpretation suggested by Krüger for the line λ = 375.5 there are problems of a different nature. Indeed the following considerations tend to exclude that this line should be attributed to atomic helium, since the transition formula is excluded by energy considerations and the few others that one can envisage and are compatible with the line’s frequency are unlikely for several reasons. On the other hand, none of these objections contradicts the interpretation of the line λ = 320.4 as due to the transition 1s2p 3 P012 -2p2p 3 P012 . Also the theoretical value of the energy of 2p2p 3 P012 that can easily be calculated with an error of 1% is in perfect agreement with the experimental observation. 1. If we neglect the interaction and the relativistic corrections and we disregard the spin variables, from two orbits with total quantum number 2, we obtain in a Coulomb field a term with a 16-fold degeneracy, since each electron can be in the orbit 2s or 2pm (m = 1, 0, −1). The perturbation matrix due to the interaction can be split into matrices of degree 1 and 2 as we can a priori separate states with different symmetry. We can thus reasonably build as combinations of unperturbed eigenfunctions those of the states 2s2p 3 P , 2s2p 1 P , 2p2p 1 D, 2p2p 3 P , 2p2p 1 S, 2s2s 1 S for all the allowed values of the magnetic quantum. All these states, with the exception of the last two, have at least some different symmetry characters arising from the different behaviour of the states under the exchange of the electrons (singlets and triplets), or under space rotations (azimuthal quantum) or axial rotations (magnetic quantum), or under reflection in the center of mass (even or odd terms).

2 Wentzel, 3 Fues,

G.: Z Physik. 43, 524 (1927). E.: Z Physik. 43, 726 (1927).

On the Possible Anomalous Terms of Helium

55

The last two terms 2p2p 1 S and 2s2s 1 S have instead the same symmetry. Correspondingly, we can get the correct eigenfunctions and the first-approximation eigenvalues from a quadratic equation for the first states X and Y that are linear combinations of 2s2s 1 S and 2p2p 1 S; the eigenvalues of the other terms can instead be expressed from the elements of the perturbation matrix. For an atom with charge number Z with only two electrons in the above specified 2 2 W orbits, the energy in the zeroth approximation is Rh = −2 Z22 = − Z2 ; in the first 2

W = − Z2 + aZ since the interaction increases as Z. approximation it will be Rh The second approximation can be evaluated with the method of the variation of the unit of length4 that is here equivalent to assuming hydrogen-like eigenfunctions with an effective Z ∗ . With this method we obtain in general for the terms under 2 W consideration instead of Rh = − Z2 + aZ the more correct expression

Z2 a2 W =− + aZ − , Rh 2 2

(1)

which corresponds to Z ∗ = Z − a. The values of a and the terms calculated according to formula (1) for helium corresponding to the doubly ionized atom are: a √ ⎧ 47 + 241 ⎪ 1 ⎪ ⎪ Y S ⎪ ⎪ 128 ⎪ ⎪ ⎪ ⎪ 49 ⎪ ⎪ 2s2p 1 P ⎪ ⎪ ⎪ 128 ⎪ ⎪ ⎪ ⎪ 237 ⎪ ⎪ ⎨ 2p2p 1 D 640 ⎪ 21 ⎪ ⎪ 2p2p 3 P ⎪ ⎪ 64 ⎪ ⎪ ⎪ ⎪ 17 ⎪ ⎪ 3 ⎪ 2s2p P ⎪ ⎪ 64 ⎪ ⎪ √ ⎪ ⎪ ⎪ 47 − 241 ⎪ 1 ⎩ X S 128

v cm−1 125,300 143,500 145,700

(2)

153,300 165,000 168,800.

Before discussing the accuracy of this method, we will first discuss the stability of the terms because unstable terms do not belong to well-defined energy levels. The terms of complex atoms can be divided into two classes that are by now experimentally well known. Indeed for radiative transitions between terms belonging to the same class there is a selection rule L = ±1 whereas for transitions between different classes the selection rule is L = 0 (this in the case of normal coupling is Laporte selection rule). Terms belonging to the first class (that Wigner calls 4 Cfr.

Fock, V.: Z Physik. 63, 855 (1930).

56

E. Majorana

normal5 ) are even or odd (i.e. they do not or they do change sign under reflection in the nucleus) for even or odd L. Terms in the second class (reflected terms according to Wigner) are even when the azimuthal quantum is odd or odd for even azimuthal quantum. In hydrogen or in atoms with many electrons, if all but one are in s orbits, all terms are, according to the above definition, normal. If the terms listed in table (2) are unstable they give rise to spontaneous transitions to states with an electron in a hyperbolic orbit and another electron in the ground state 1s, hence to states with normal symmetry. Since, when there is no radiation, the symmetry characters are unchangeable as they depend on constants of the motion, we conclude that the only stable terms listed in (2) are the reflected ones, whereas the normal terms are unstable for Auger effect at least in the non-relativistic approximation. The only reflected term among those in the table is 2p2p 3 P which is even with odd azimuthal quantum. It is therefore the only one with well-defined energy provided, of course, we disregard the coupling with the radiation field; all the other terms have a width presumably of the order of some hundreds cm−1 . 2. The precision of formula (1) is different for the various terms we have considered. For the term 2p2p 3 P , which is the lowest among the reflected ones, the determination of the energy by variational methods is a problem of absolute minimum when the approximating functions have the right symmetry. It follows that the value of the term we consider that appears in (2) represents a lower limit and we can assume that its error does not differ appreciably from the error that one finds if we calculate the ground term 1s1s 1 S with the same method. Indeed in both cases the two electrons are in equivalent orbits with zero radial number. Moreover, the relative value of the interaction compared with the value of the term and consequently the relative correction due to the variation of the unit of length differ but little in the two cases. With this method we find that the value of the W ground state is − Rh = 729 128 = 5.695, whereas the empirical value and the theoretical W = 5.807, the difference being less than 2%. If one obtained by Hylleraas is − Rh we assume in our case an identical relative error, the energy of the term 2p2p 3 P is 156,350 cm−1 below the limit of He+ . This figure is very close to the experimental one if the interpretation of the line λ = 320.4 is correct (156,000 cm−1 ). The determination of the other terms listed in (2) is not a problem of absolute minimum since there exist an infinite number of lower states and an infinite number of higher states with the same symmetry. This may lead to a partial compensation of the errors. Moreover, these terms are “obtuse”, i.e. their energy is not exactly defined and their calculation, as the one I attempted for the term X that I will recall later on, is meaningful only if we do not push the approximation too far. I believe that the errors on the values that are listed in (2) beside that for 2p2p 3 P do not exceed a few thousands cm−1 in the case of the deepest ones (X and 2s2p 3 P ) and several thousands in the case of the highest ones. To attribute the line λ = 357.5 to helium is thus rather problematic. According to the interpretation suggested

5 Wigner,

E.: Z Physik. 43, 624 (1927).

On the Possible Anomalous Terms of Helium

57

by Krüger the term 2s2s 1 S, or better the term X, should be about 191,000 cm−1 below the limit of He+ which looks to be too far from the theoretical value.6 To examine the other possible interpretations, let us note that this line, because of its position, should arise from the combination of an ordinary term of neutral helium with an anomalous one. On the other hand, the discrete spectrum of neutral helium, except for the ground state which cannot come into play, falls in the range between 439,000 and 470,000 cm−1 below the limit of He+ . Since the frequency of the line under discussion is about 280,000 cm−1 , it should arise from an anomalous term lying between 159,000 and 197,000 cm−1 below the same limit. The only terms satisfying this condition among those that we have mentioned (and there is no need to take into account other terms arising from different orbits which are certainly too small) are the term X and the term 2s2p 3 P . It is very unlikely that the line λ = 357.5 be a combination of the term 2s2p 3 P with a normal term; indeed the last one should have at least a total quantum equal to 4 and it would be very strange that its combination with deeper terms had not been observed; I just mention this possible interpretation for the unlikely case that this line be due to atomic helium. Finally, the term X could explain the line λ = 357.5 by combining with a normal term of total quantum 3; the most probable, as it is the only that satisfies the selection rules, would be X-1s3p 1 P . In this case the value of the term X should be approximately 171,300 cm−1 which is not too far from the one in table (2). Let us assume that the eigenfunction of the state X be [α + β(r1 + r2 ) + γ r1 r2 + δ(x1 x2 + y1 y2 + z1 z2 )]e−ε(r1 +r2 ) which coincides with the unperturbed eigenfunction for some values of the constants. If we determine these constants by the variational method, the value of the term X is approximately 168,500 cm−1 which is not too different from the one calculated using formula (1), even though the number of parameters is considerably larger, but still less compatible with the assumption 1s3p 1 P -X. The latter is rather unlikely because of the absence of the line 1s2p 1 P -X. The other possible interpretations that have not been discussed are even less acceptable. We thus conclude that the attribution of the line λ = 357.5 to helium is at the moment still doubtful in spite of the experimental evidence. In any case it cannot be accepted without further investigation.

6A

different evaluation referred by the author does not seem to be correct.

58

M. INGUSCIO and F. MINARDI

Comment on: “On the possible anomalous terms of helium”. In this paper, appeared in “Il Nuovo Cimento” in 1931, E. Majorana deals with the calculation of certain doubly excited levels of helium. The motivation was given by claims of P. G. Krüger, who, a year earlier, observed new emission lines from a helium discharge and ascribed them to the helium spectrum.7 According to Krüger, these lines at wavelengths of 32.04 and 35.75 nm were due to transitions involving doubly excited helium levels, namely to the 1s2p 3 P −2p2p 3 P and 1s2s 1 S −2s2s 1 S transitions. Majorana takes into account all helium levels generated by combining two idrogenoid orbits with principal quantum number n = 2 and includes the mutual electrons repulsion as a perturbation. The levels are ordered based on the electrons total angular momentum, total spin and parity: 2s2s 1 S, 2s2p 3 P , 2p2p 3 P , 2p2p 1 D, 2s2p 1 P , 2p2p 1 S. Actually, as Majorana points out, the first and the last in the list, having exactly the same symmetries, are mixed by the electrons Coulomb interaction and give rise to two new terms, called, respectively, X and Y in the paper. The energies of all levels are evaluated by a variational perturbative approach and the expected error is estimated from that in calculating the ground-state energy, precisely found by Hylleraas. Majorana was well aware that all doubly excited helium levels lie above the continuum, i.e. have an energy higher than the ionized state formed by the ground-state He+ ion and a free electron. As a consequence, the doubly excited states can undergo spontaneous ionization, also known as Auger effect. Generally, the levels subject to the Auger effect are so highly unstable, hence undetermined in energy, that they cannot give rise to radiative transitions with well-defined wavelengths, such as those observed by Krüger. Relying on symmetry arguments, Majorana deduces that, among the above listed levels, only the 2p2p 3 P is properly stable. This level is stable because it is a “reflected” state, meaning that, upon a parity transformation, its wave function is multiplied by a factor (−1)L+1 with L denoting the orbital angular momentum, while for “normal” levels the wave function acquires a factor (−1)L . In the absence of radiation, transitions from “reflected” to “normal” states, such as the ionized state, are forbidden. As for the 2s2s 1 S (X) state, he speculates that, although unstable, its lifetime could be much longer than the others, and thus Krüger’s assignment, involving the 2s2s 1 S (X) level for the line at 35.75 nm, should not be readily rejected without further inquiry. To judge Krüger’s assignment, Majorana compares the wave numbers of the two lines with his calculations. Majorana deems correct the identification of the 32.04 nm line, while unlikely that of 35.75 nm line: the former line implies that the term 2p2p 3 P lies 156,000 cm−1 below the energy of the bare nucleus He++ , in good agreement with Majorana’s value of 156,350 cm−1 . On the contrary, the line at 35.75 nm would put the 2s2s 1 S (X) energy 191,000 cm−1 below the He++ threshold, i.e. too far from the calculated value of 168,800 cm−1 . Unable to find other suitable terms, E. Majorana doubts that the 35.75 nm line belongs to the helium spectrum altogether. 7 Krüger,

P.G.: Phys. Rev. 36, 855 (1930).

Comment on: On the Possible Anomalous Terms of Helium

59

It is worth noticing how Majorana emphasizes the symmetry properties of the investigated levels upon rotations, exchange of electrons and parity. Such an approach, while commonplace in modern physics, was unusual at that time. Indeed, in the introduction of his fundamental book Gruppentheorie und Quantenmechanik (1931), H. Weyl remarks that the use of the group representations in physics was just dawning. Published in Italian, this work of Majorana’s seems to have gone largely unnoticed if a few years later Fender et al.,8 Ta-You Wu9 and Wilson10 separately attack the same issue with no reference to Majorana. Fender et al. and Wilson agree with Majorana on rejecting Krüger’s identification of the 35.75 nm, while Wu, even in the presence of a 3.4% discrepancy, backs Krüger’s assignment with “a partial support”.11 Later on, doubly excited helium levels have been studied by several authors, among those U. Fano who, in this context, elaborated his celebrated theory of the lineshapes of a discrete state coupled to a continuum.12 Recently, E. Lindroth revaluated the energies of all levels considered by Majorana, finding values systematically below those of Majorana, with differences ranging from 1% to 9%.13 The energy widths found by Lindroth show that Majorana was right in presuming the 2s2s 1 S (X) state to be longer lived than the other levels, but it is still too wide to yield sharp transition lines. Retrospectively we clearly see that, in the intervening 75 years, Majorana’s interest in the helium spectrum has been shared by many. In 1929, a fundamental contribution was given by E. A. Hylleraas who introduced a specific set of trial wave functions that proved very successful for variational calculations.14 A vigorous drive to improve upon Hylleraas’ work came in the 1960s by C. L. Pekeris and co-workers, who exploited the newly available computers.15 Another leap forward was carried out by G. W. F. Drake,16 who increased the precision of Schrödinger eigenvalues up to 10−15 . Nowadays, far from being exhausted, the interest for helium levels has shifted. Since the Schrödinger equation is solved for all practical purposes, the atomic theory takes now into account relativistic, recoil and radiative corrections. The challenging task is first to derive the perturbation operators in powers of the fine-structure 8 Fender,

F.G., Vinti, J.P.: Phys. Rev. 46, 77 (1934). T.-Y.: Phys. Rev. 46, 239 (1934) 10 Wilson, W.S.: Phys. Rev. 48, 536 (1935). 11 One year later, in 1935, Ta-You Wu found a 19% mistake in his calculation. Once corrected the mistake, the partial support to Krüger’s assignment resulted untenable. It is somewhat ironic that, as U. Fano pointed out in a conversation with E. Amaldi, Wu’s work has been frequently quoted while Majorana’s almost never. 12 Fano, U.: Phys. Rev. 124, 1866 (1961). 13 Lindroth, E.: Phys. Rev. 49, 4473 (1994). 14 Hylleraas, E.A.: Z Phys. 54, 347 (1929). 15 Accad, Y., Pekeris, C.L., Schiff, B.: Phys. Rev. 4, 516 (1971). 16 Drake, G.W.F.: In: Drake, G.W.F. (ed.) Atomic, molecular and optical physics handbook, p. 154. AIP Press, New York (1996), and reference therein. 9 Wu,

60

M. INGUSCIO and F. MINARDI

(FS) constant α and of the electron-to-nucleus mass ratio m/M, then to evaluate the operators on non-relativistic wave functions. Calculated levels can be checked by laser spectroscopy experiments, that on helium have reached precisions up to 0.01 ppb. Another important chapter in helium spectroscopy is the investigation of FS splittings. Because many contributions cancel out, the FS separations are less affected by unknown relativistic, recoil and QED corrections. Since, for instance, the FS splittings of the 1s2p 3 P state have been measured with a precision of 0.03 ppm, these could be used to determine α once the theory is sufficiently well established.17 We know how Majorana was cautious, even reluctant, in publishing his works. The influence he bears on physics with only nine papers highlights not only the depth of his insight but also his foresight in choosing his topics. Rephrasing A. Schawlow’s remark about diatomic molecules being molecules “with an atom too many”, one could think that the helium atom has an electron too many since it cannot be solved exactly like hydrogen. However, here lies the very reason of the interest that helium aroused in E. Majorana and continues to arouse among researchers, theoreticians and experimentalists alike. MASSIMO INGUSCIO Università di Firenze, Florence, Italy FRANCESCO MINARDI Università di Firenze, Florence, Italy

17 Mohr,

P.J., Taylor, B.: Rev. Mod. Phys. 77, 1 (2002).

Pseudopolar Reaction of Hydrogen Atoms∗ Ettore Majorana

Anomalous terms with both the electrons excited are known since a long time to occur in atoms with two valence electrons. In particular, the following are well known in numerous neutral or ionized atoms: 2p2p 3 P012 , 2p2p 1 D, 2p2p 1 S. According to a recent interpretation1 the X term of the hydrogen molecule is formally analogous to these terms and should be precisely assigned to the configuration (2pσ )21 g .2 The analogy, however, breaks down in regard to the energies: whereas in atoms the frequency of the line 2p2p → 1s2p is of the same order of magnitude as the frequency of the line 1s2p → 1s2s, the X term is instead relatively deep, slightly above the normal term 1sσ 2pσ 1 u 3 with which it intensely combines in the infrared region; but the second one is in turn much higher than the ground state (1sσ )21 g (ca. 12 volts). The problem is then to justify theoretically the abnormal energy level of such an anomalous term and even to justify its existence. Weizel, in his attempt to solve the problem, provided a rather questionable evaluation based on dubious analogies. We have attacked the problem directly and our calculations seem to confirm Weizel’s assumption about the existence of a deep term (2pσ )21 g , although the theoretical equilibrium distance between the two nuclei is in better agreement with the term K (according to Weizel 2pσ 3pσ 1 g ) than with the term X. On the other hand, to consider such a state as a state with two excited electrons has purely formal meaning. In reality, to designate such terms with the states of

∗ Presented

by the member O. M. Corbino at the meeting held on January 4, 1931. Translated from Reazione pseudopolare fra atomi di idrogeno, “Rendiconti dell’Accademia dei Lincei”, vol. 13, 1931, pp. 58–61, by P. Radicati di Brozolo. 1 Weizel, W.: Z. Physik. 65, 456 (1930). 2 The lower indices g and u are shorthand notations for “gerade” and “ungerade”, which are the German for “even” and “odd”. (Note of the Editor in E. Amaldi, op. cit.) 3 In “Rendiconti dell’Accademia dei Lincei” it is erroneously printed  , here and in the following, n in place of u . (Note of the Editor, see also E. Amaldi, op. cit.) © Springer Nature Switzerland AG 2020 L. Cifarelli (ed.), Scientific Papers of Ettore Majorana, https://doi.org/10.1007/978-3-030-23509-3_6

61

62

E. Majorana

the single electrons, though it may be convenient for their numbering and for the identification of those symmetry characters that are not affected by the interaction, does not allow by itself to draw reliable conclusions on the explicit form of the eigenfunctions. The situation is very different from the one of central fields where it is generally possible to neglect the interdependence of the electron motions (polarization) without losing sight of the essentials. The term (2pσ )21 g we have to deal with can be thought of as partially formed by the union H+ + H− . This does not mean, however, that it is a polar compound since, because of the equality of the constituents, the electric moment changes sign with a high frequency (exchange frequency) and therefore cannot be observed. It is in this sense that we speak of a pseudopolar compound. All that has been said so far holds true only in rough approximation. For a more accurate, even though still rather schematic, description, one needs to consider together with the reaction H+ + H− also the other, H + H, the sole one studied by Heitler and London.4 Then if we only consider the resulting even terms, we find that: (1) in second approximation, with respect to Heitler and London’s method, the ground state is (1sσ )21 g ; (2) the anomalous state is (2pσ )21 g . The first comes predominantly from H + H, the second predominantly from H+ + H− . 1. Let us divide the configuration space into four regions: aa, ab, ba, bb, according as to whether each of the electrons is closer to nucleus a, or to nucleus b, and let us disregard for the moment the interaction. The four possibilities are equally represented in the state (2pσ )21 g ; the eigenfunction is, say, positive in aa and bb and negative in the other two cases. The interaction increases the probability to find the system in aa and bb, whereas it decreases that of ab and ba. This anomalous behaviour can be easily understood by noticing that the state (2pσ )21 g must be orthogonal to the ground state. In this state the regions ab and ba are predominantly represented, if the nuclei are sufficiently distant, as indeed expected from the applicability of Heitler and London’s method. We can then assume with some approximation that the eigenfunction of H− + H+ symmetrized with respect to the nuclei belongs to (2pσ )21 g . However, it is easy to recognize that this approximation is insufficient by observing that the nodal surfaces are completely lost. These nodal surfaces reappear if we include the union H + H of two neutral atoms in the perturbative calculation. Let us denote by 12 and 12 the electronic eigenfunctions of H− for the electrons close to nucleus a and nucleus b, respectively, and by ϕ or ψ the eigenfunction of the neutral atom a or b. With these functions we can construct as a combination of the configurations H− +H+ , H+ +H− , H+H (the last being double due to the elctrons resonance) two even eigenfunctions belonging to the singlet system:  y1 = 12 + 12 y2 = ϕ1 ψ2 + ϕ2 ψ1 .

4 Heitler,

W., London, F.: Z. Physik. 44, 455 (1927).

(1)

Pseudopolar Reaction of Hydrogen Atoms

63

We disregard the other two states arising from the same configurations but having different symmetry; they are the odd state of the triplets 1sσ 2pσ 3 u which is unstable and has already been considered by Heitler and London and, with some approximation, the 1sσ 2pσ 1 u state that belongs to the singlets but is equally odd. The states y1 and y2 that appear in (1) are not orthogonal but the ground state (1sσ )21 g and the anomalous state (2pσ )21 g should result from their orthogonal combinations. The secular equation for determining the eigenvalues is5 :    I0 + I1 − (1 + S)E 2M − 2QE =0   2M − 2QE L0 + L1 − (1 + R)E 

(2)

where, if the eigenfunctions are real:  ⎧ ⎪ ⎪ I = ϕ1 ψ2 H ϕ1 ψ2 dτ 0 ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ ⎪ ⎪I1 = ϕ2 ψ1 H ϕ1 ψ2 dτ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ ⎪ L0 = 12 H 12 dτ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ ⎪ = 12 H 12 dτ L ⎨ 1  ⎪ ⎪ M = ϕ1 ψ2 H 12 dτ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ S = ϕ1 ψ1 ϕ2 ψ2 dτ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ R = 12 12 dτ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎩Q = ϕ ψ  dτ 1 2

(3)

12

If, for example, we set the energy of the separate neutral atoms equal to zero, we can consider H as a perturbation; the difference between the energies of H+ + H− and of H + H naturally appears then as a perturbation when H is applied to 12 or to 12 . 2. Some of the integrals (3) can be found in Heitler and London (see 4 ) and Sugiura6 ; to evaluate the other integrals we have to find an approximate expression for 12 . This eigenfunction, describing the ion H− , is not exactly known but its eigenvalue, which is related to the electronic affinity of hydrogen, has

5 In “Rendiconti dell’Accademia dei Lincei” the term “=

of the Editor, see also E. Amaldi, op. cit.) 6 Sugiura, Y.: Z. Physik. 45, 484 (1927).

0”of Eq. (2) is erroneously missing. (Note

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E. Majorana

been calculated with great precision by several Authors.7,8,9 This simplifies the evaluation of the integrals (3) since H reduces to a finite operator. For 12 we can approximately take the product 1 2 of two eigenfunctions depending on the single electrons. The best solution in the sense of the minimum energy is then, as is well known, the one given by Harthree’s method which in our case is, with great approximation,

r r c −0.29 a1 −1.68 a1 0 −e 0 e 1 = r1 and similarly for 2 . The whole calculation, using this eigenfunction, though feasible, is too cumbersome; we have therefore preferred to use the simpler expression 1 = c e

− 11 16

r1 a0

that has already been used by Hylleraas in his theory of solid lithium hydride.10 Using the approximate eigenfunction 12 and the conventional meaning for H 12 , we obtain for M two different values:   and M = 12 H ϕ1 ψ2 dτ. M = ϕ1 ψ2 H 12 dτ We prefer the first expression that is simpler and possibly more exact; however since the second does not allow simple calculations, we cannot control this statement except for the limiting cases. The largest of the eigenvalues of (2) that belongs to (2pσ )21 g reaches its minimum at a distance that is in disagreement with the equilibrium distance of the term X (approximately 2 Å instead of 1.35). However, the energy is 7.5 volts above the energy of the neutral atoms when they are separated and in the ground state, i.e. approximately 27,000 cm−1 below the normal limit of H2 (the experimental value for the X term being 22,000 cm−1 ). This result is even too favourable as, with the method we followed, we could have expected a value considerably smaller than the true one. We cannot definitely exclude that Weizel’s interpretation is wrong and that the term (2pσ )21 g , which is certainly stable and relatively deep, is, instead of the X term, rather the K term or some other term not yet observed. Nevertheless it is perhaps possible to attribute the error in the determination of the equilibrium point and the too favourable result in the calculation of the energy to the use for H− of a non-properly correct eigenfunction. A quantitative evaluation is difficult but it is plausible that such an approximation tends to produce errors compatible with the discrepancies ascertained between calculation and experiment.

7 Bethe,

H.: Z. Physik. 57, 815 (1929). E.A.: Z. Physik. 60, 624 (1930) 9 Starodubroski, P.: Z. Physik. 65, 806 (1930) 10 Hylleraas, E.A.: Z. Physik. 63, 771 (1930). 8 Hylleraas,

Comment on: Pseudopolar Reaction of Hydrogen Atoms

65

Comment on: “Pseudopolar reaction of hydrogen atoms”. This paper, no. 4 of the Majorana production, is “technically” similar to paper no. 2 on the chemical bond of the He+ 2 ion. Although Majorana is a theoretician, his interest always arouses from considerations of experimental kind. Hence, also the motivation of this work is not a purely academic exercise, but it is aimed to explain an unexplained phenomenon observed in the spectrum of the H2 molecule (the so-called X term). While for an atomic system the frequency corresponding to the transition 2p 2p − 1s 2p (involving two excited optical electrons) is quite close to that of the transition 1s 2p − 1s 2s (where only one optical electron is excited) this behaviour is no longer valid for the analogous molecular system as observed in the spectrum of H2 where the excited (2pσ )2 1 g state decays toward the state 1sσ 2pσ 1 u in the infrared spectral region (anomalous X term). As already done in paper no. 2, Majorana makes use of the concept based on the resonance force. His approach assumes the molecule formation to be the result of the interaction between two ions, H+ and H− (pseudopolar binding), and considers the possibility that one electron can jump from one nucleus to the other at a given frequency. Coherently with this idea, Majorana chooses combinations of electronic eigenfunctions of the (H+ + H− ) (where one electron can oscillate between the two protons) and (H + H) (where instead each electron is associated to one proton) systems and selects two even functions which give rise to bound states. In such a way he finds the ground state (1sσ )2 1 g (already contained in the theory of Heitler and London) and the excited state (2pσ )2 1 g . To reach this result once again Majorana demonstrates his capability to remarkably master the use of wave functions (see, for instance, the discussion on the configuration space for the electron density distribution, the comment on the nodal surfaces, and the motivation for selecting the singlet states). Moreover, in order to obtain quantitative results he manages to overcome the limits imposed by analytical non-integrability using, with artfulness, appropriate simplifications. As the starting point of his study is connected with experimental facts, similarly the data reported in the literature are always used as a feedback for his calculations. In this respect it is worth to underline the style of Majorana in making this comparison. In the presence of a good agreement with the experimental data Majorana does not hesitate to almost diminish its importance by commenting: “. . . it is perhaps possible to attribute. . . the too favourable result in the calculation of the energy to the use for H − of a non properly correct eigenfunction.” Similar comments are also present at the end of paper no. 2 and denote an intellectual honesty and, at the same time, a spirit inclined to astonishment which, especially nowadays, are quite rare virtues. ANTONIO SASSO Università di Napoli, Naples, Italy

Theory of the Incomplete P  Triplets Ettore Majorana

Summary: Most of the anomalous triplets above the limit of the normal series are stable if relativistic corrections are neglected. The presence of intrinsic magnetic moments produces a small instability that only in exceptional cases can have a quantitative importance. The necessary conditions for such a case to occur are satisfied by the component j = 2 of the anomalous triplets of Zn, Cd and Hg. Experimentally the component 3 P2 of these triplets seems absent or weak. Five groups of terms, each one with six lines, are known in the calcium spectrum that can be attributed to combinations of the term 1s2p 3 P012 (current empirical  that can be ordered as a series. However numbers) with a succession of triplets 3 P012 + the limit of this series corresponds to Ca with the external electron in the orbit 3d and not in the fundamental orbit 1s as happens in normal series. This fact, together with some other indications, suggests to attribute to the P  triplets of calcium the configuration 3dnd 3 P012 (n = 3, 4, 5, . . .). These states therefore correspond to excitation of two electrons. What we want to point out, apropos of these anomalous terms of calcium, is that only some of them are above the ionization potential in the continuous spectrum and nevertheless are stable, i.e. they do not give rise to an appreciable spontaneous ionization. There is nothing strange in this as it would be easy to prove that in the non-relativistic approximation such terms have a symmetry character which rigorously forbids transitions to the continuous spectrum. In other elements however, in particular in Zn, Cd and Hg, P  triplets are known above the ionization potential that should be stable for the same reasons; however the component P2 is absent, whereas P1 and P0 have been observed. These terms of Zn, Cd and Hg are analogous to those of calcium mentioned above, even though they arise from different configurations (pp instead of dd). Here we want to study

Translated from Teoria dei tripletti P incompleti , “Il Nuovo Cimento”, vol. 8, 1931, pp. 107–113, by P. Radicati di Brozolo. © Springer Nature Switzerland AG 2020 L. Cifarelli (ed.), Scientific Papers of Ettore Majorana, https://doi.org/10.1007/978-3-030-23509-3_7

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why in this case the component P2 is absent. Foote, Takamine and Chenault1 have suggested, only for the case of Cd and to explain some anomalies in the intensities, that the term P2 exists but is accidentally so close to P1 to make it impossible to separate the two. This looks to me difficult to believe both because it violently contradicts the rules of the normal coupling and also because it is disproved by the lack of analogous terms in Zn and Hg. I also want to point out that Sawyer2 has suggested that for Zn, at exactly the same position of the non-existing lines 3 P  − 3 P and 3 P  − 3 P , two lines 1 D − 3 P and 1 D − 3 P exist and that both 2 1 2 2 2 1 2 2 are weaker and of a different aspect from those of the remaining four lines of the group P P  (here 1 D2 is one of the states arising from two orbits 2p). According to the present theory, the lines of Sawyer are instead precisely the lines 3 P2 − 3 P2 and 3 P  − 3 P that complete the P P  group. These lines are weaker since the term 3 P  1 2 2 is unstable because of the Auger effect; this is due to spin-orbit interaction. Instead the lines that arise from term 1 D2 should not be observable because the instability of this term is much larger, as it is not contrasted by its symmetry characters even if we disregard the electron magnetic moment. 1. Without electron rotation, from two equivalent orbits 2p we obtain the terms 1 D, 3 P and 1 S (the multiplicity indices only denote the symmetry or antisymmetry of the center-of-mass eigenfunction). Their separation depends, in the first approximation, upon a single parameter and can be calculated with Slater’s method.3 One finds that 3 P is the deepest state, 1 S the highest and the distance 1 S − 1 D is 3/2 of the distance 1 D − 3 P . The order of magnitude of the absolute value of the separation can be estimated by using approximate eigenfunctions. The results of this method are in good agreement with the observed values, for example for the deep terms of silicon. However in some cases the first approximation is insufficient, as for example for magnesium, where the term (2p)2 3 P012 is known and, if Sawyer’s (see 2 ) interpretation is correct, also the (2p)2 1 D2 one. The term 1 D2 is actually higher than 3 P012 as it follows from Slater’s method but the separation is only a few hundred cm−1 and not several thousand as the method would predict. The anomaly can perhaps be attributed to the predominant influence of the continuous spectrum and in particular to the influence of the “virtual state” (3d)2 1 D2 on the term (2p)2 1 D2 . The same reasoning could apply to the terms (2p)2 3 P012 and (2p)2 1 D2 of Zn, Cd and Hg. In this case however the terms are above the ionization limit and, if we disregard the magnetic moment of the electron, only the first one is stable because it is a reflected term,4 whereas the states in the continuous spectrum belonging to the same energy have a normal symmetry. The term (2p)2 1 D2 is instead a “virtual state” in the sense of Beck5 and its energy, which is not exactly defined, is slightly greater than that of (2p)2 3 P012 . However, the interaction between the intrinsic momentum and the orbital one of 1 Foote,

P.D., Takamine, T., Chenault, R.L.: Phys. Rev. 26, 174 (1925). R.A.: J. Opt. Soc. Am. 13, 431 (1926). 3 Slater, J.C.: Phys. Rev. 34, 1293 (1929). 4 Wigner, E.: Z Physik. 43, 624 (1927). 5 Beck, G.: Z Physik. 62, 331 (1930). 2 Sawyer,

Theory of the Incomplete P  Triplets

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each of the two electrons establishes a mutual influence between the virtual state (2p)2 1 D2 and the component of (2p)2 3 P012 that belongs to the same value of the internal quantum (j = 2). As a consequence also 3 P2 becomes unstable and can give rise to a spontaneous ionization which means a transition without radiation from triplets to singlets. The closeness of a virtual state with appropriate symmetry characters is necessary only if the Auger effect has to acquire a sizable importance; but even when that is absent, as is the case for the component 3 P1 , or when it is sufficiently far away, as the virtual state (2p)2 1 S0 for the component 3 P0 , we can expect a weak instability which can explain the anomalies of the intensities that one observes in the remaining lines of the group P P  . Let us then premise a few general considerations on the influence of the intrinsic magnetic moment in all the states 1 D2 3 P2 3 P1 3 P0 1 S0 arising from two orbits 2p. For sufficiently high atomic numbers, we can neglect the magnetic interaction between the two electrons and only take into account the interaction between the constant electric field arising from the remaining atomic cloud and the magnetic moment of the moving electrons which is the so-called spin-orbit coupling. The perturbation is thus described by the Hamiltonian H = f (r1 )(s1 , M1 ) + f (r2 )(s2 , M2 ),

(1)

where s1 , s2 and M1 , M2 are the intrinsic and orbital momenta of the two electrons measured in units of h/2π and f (r) is a function of the radius proportional to 1r dV dr . Let us choose the coordinate system so that the unperturbed energy be diagonal and let us further suppose that the unperturbed eigenfunctions can, with sufficient approximation, be expressed as linear combinations of products of the eigenfunctions of the single electrons; these functions represent individual stationary states in an appropriate central field. Then the matrix of H can be conveniently represented as a sum of products of the matrices f (r1 ) and f (r2 ), that can be easily evaluated through quadratures, times the matrices (s1 , M1 ) and (s2 , M2 ). In this approximation the latter are constant matrices, i.e. that link only states with equal energies, because so are the matrices s1 , M1 , s2 and M2 , separately. However the functions that depend upon the coordinates of a single electron, as f (r1 ), s1 , M1 etc. are not physical quantities. We can formally consider them as such and express them by matrices only if we renounce Pauli’s principle and include also complete non-antisymmetrical eigenfunctions. In the case of only two electrons the resulting complication is not too great because we need to consider only one symmetry character that does not follow Pauli’s principle, i.e. that of symmetrical eigenfunctions. For example we should further add to the terms already considered the following terms arising from two orbits 2p: we shall write them in brackets to recall that they do not satisfy Pauli’s principle (3 D123 ) (1 P1 ) (3 S1 ). For each individual value of j all terms are listed below: ⎧ ⎪ j ⎪ ⎪ ⎪ ⎨j ⎪ j ⎪ ⎪ ⎪ ⎩ j

= 3 : (3 D3 ) = 2 : 1 D2 ,3 P2 , (3 D2 ) = 1 : 3 P1 , (3 D1 ), (1 P1 ), (3 S1 ) = 0 : 3 P2 , 1 S0 .

(2)

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Each of these terms is (2j + 1) times degenerate. Therefore we naturally have all together 36 states since each electron can be in 3 (orbital degeneration) ×2 (intrinsic degeneration) = 6 different states with no effect of the exclusion principle. The matrices (s1 , M1 ) and (s2 , M2 ), that in first approximation interconnect only the states of (2), can be constructed directly but we will not enter the details of the calculation. If we refer the rows and columns to the various states in the order in which these appear in (2) the matrices are: j =3

j =2

j =1

1 (s1 , M1 ) = (s2 , M2 ) = 2   √  √     1 1 3 3   0  √ − √  √ √    0 2 2 2 2 2 2 2 2   √  √      1  1 1 1 3 3     (s2 , M2 ) =  √ (s1 , M1 ) =  √ −  4 4  4   2 2 2 2 4 √ √    √ √   3 1  1  3 3 − √3  −    √ − 4 −4  2 2 2 4 2 2   √  1 1  5 1  − √ √ − √    4 4 3 2 2 3   √ √   5  √ − 3 √5 0   4   (s1 , M1 ) =  4 3 √ 2 6   1 5 1   √ 0 √  √  2 2 2 6 6     1 1 − √ 0 √ 0   3 6  √  1  − 1 − √5 − √  4  4 3 2 2  √  5 5 3 − √ − √  4  4 3 2 6 (s2 , M2 ) =  √  5 1 − √ 0 √   2 2 2 6   1 1  √ 0 √  3 6

j =0

   1 1  − √   2 2 . (s1 , M1 ) = (s2 , M2 ) =    √1 0    2

 1  √  3   0    1  √  6   0 

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These equations are valid for each allowed value of the magnetic quantum. We obtain in this way the following expressions of H ψ for the components of the triplets P  : ⎧ √ f (r1 ) − f (r2 ) 3 f (r1 ) + f (r2 ) 3 f (r1 ) + f (r2 ) 1 ⎪ ⎪ H 3 P2 = P2 + D2 + 3 D2 √ ⎪ ⎪ 4 4 ⎪ 2 2 ⎪ ⎪ ⎪ ⎪ √ f (r1 ) − f (r2 ) 3 f (r1 ) + f (r2 ) 3 ⎪ 3 ⎪ ⎪ P1 + 5 D1 + √ ⎨H P1 = − 4 4 3 ⎪ f (r1 ) − f (r2 ) 3 f (r1 ) − f (r2 ) 3 ⎪ ⎪ + P1 − S1 √ √ ⎪ ⎪ ⎪ 3 2 2 ⎪ ⎪ ⎪ ⎪ f (r1 ) + f (r2 ) 3 f (r1 ) + f (r2 ) 1 ⎪ ⎪ P0 + S0 . √ ⎩H 3 P0 = − 2 2 (3) According to Wentzel’s formula6 the squares of the coefficients in the expansion of H ψ for states in the continuous spectrum having the same energy as ψ are proportional, in absolute value and in first approximation, to the probability of spontaneous ionization. The various terms in the second members of (3) are even functions and those that belong to L = l are orthogonal to the states of the continuous spectrum having the same energy as the triplet P  , because those states come from one electron 1s and one in a hyperbolic orbit and can be even for L = 0, 2, 4, . . . , or odd for L = 1, 3, 5, . . . The other terms in the right-hand sides of (3) are usually almost orthogonal to the same states in the continuous spectrum because of the different angular dependence. They would be exactly orthogonal if one were to neglect the polarization. Furthermore, they all have a common f (r) factor which is of the order of the relativistic effects; since the Auger effect depends upon the square of the relevant components of H , we can conclude that the effect is usually very weak for the terms we are considering. The situation is different if a virtual state is present with almost the same energy, as 1 D2 for the component 3 P2 . Also 1 S0 is a virtual state that could influence the stability of 3 P0 ; however it is too far and the distance is increased by relativistic effects. Coming back to 1 D2 this state could be described as an eigenfunction whose time 2π i 2π i t dependence is of the form e− h E0 t e− 2T = e− h Et , where E = E0 + 4πhiT is a complex eigenvalue. This expression for ψ is valid of course only if both electrons are in a region of the order of the atomic dimensions. Let us now suppose that the uncertainty a = 4πhT of the 1 D2 7 energy and the distance d between E0 and the energy of 3 P2 (including the diagonal term of the perturbation H ) are small compared to the (negative) value of the term. Let us further define = 32 f (r), with the average value f (r) calculated on an orbit 2p, which represents the difference between the energies of the extreme components of the normal triplet 1s2p, 3 P012 6 Wentzel,

G.: Z Physik. 43, 524 (1927) that in the original text of “Il Nuovo Cimento” it is erroneously printed 3 P2 . (Note of the Editor.)

7 Note

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or, which is the same in first approximation, the separation of the anomalous triplet (2p)2 3 P012 . Let us further suppose that also be small in the same sense. Then one can use a very simple formalism whose justification can easily be based on the properties of those periodic solutions that belong to a narrow region of the continuous spectrum in terms of which the arbitrarily limited virtual state 1 D2 can in practice be expanded. The study of the influence of 1 D2 on 3 P2 is thus reduced to an ordinary perturbation problem and one has to determine the (complex) eigenvalues and the (non-orthogonal) eigenvectors of the matrix:  √   2   2 d − ai  3 . √   2 3P  2 0  3

1D

The absolute value of the imaginary coefficient of the perturbed eigenvalue of is a measure of the energy uncertainty (half-width) of this state. If we divide it by h/4π we obtain the probability√per unit time of spontaneous ionization. If is sufficiently small compared to d 2 + a 2 , which is not necessarily true, the 2 half-width of the term 3 P2 is 29 d 2 +a 2 a.8 When a → 0 this formula has a simple meaning: it says that to obtain the probability of ionization of 3 P2 , it is enough to multiply the probability for 1 D2 by the contribution of 1 D2 in the perturbed 3 P2 , according to common perturbation theory. For large values of a, which means for great instability of 1 D2 , the instability induced in 3 P2 decreases again after having reached a maximum: this should be interpreted in the sense that an exceedingly unstable state 1 D2 ceases to be “a virtual state” whose presence is in any case necessary for the Auger effect of the term (2p)2 3 P2 to become quantitatively relevant.

3P 2

8 Note

that in the original text of “Il Nuovo Cimento” it is erroneously printed the Editor.)

2 2 2 9 d 2 +a 2 a .

(Note of

Comment on: Theory of the Incomplete P  Triplets

73

Comment on: “Theory of the incomplete P  triplets”. This paper, published in 1931 while Majorana was in Rome after receiving his doctoral degree, deals with an atomic spectroscopy problem, the characterization of spectra of different atoms with two electrons in the outer shell. The study was stimulated by experimental observations published in 1925 by P. Foote et al. of the National Bureau of Standards of the US Department of Commerce, Washington DC (the predecessor of the US National Institute of Standards and Technology), see Ref. (1) of the original paper. The experimental finding was that some predicted lines in the absorption spectra of Hg, Cd and Zn atoms were missing. More precisely, of the three lines expected for transitions to the lowest p2 3 P0,1,2 triplet levels, those associated with the 3 P2 level were not observed in any of these spectra. Majorana presented a theoretical explanation of those results introducing a new process he designates as spontaneous ionization. At that time, deviations of energy levels and line intensities from the simple formulas for ordinary atomic series were already known for line series belonging to atoms (or ions) with several electrons in the outer shell. These deviations are known as perturbations. The perturbations are produced by a resonance process where two (or more) energy levels are nearly degenerate, and their wave functions are mixed up by the interactions between electrons. As a consequence, the new mixed wave functions are the solutions of the Schrödinger equation, with new energy eigenvalues and new transition probabilities for absorption of light. Majorana introduced a new kind of perturbation, proposing that the p2 3 P0,1,2 levels are modified by the interaction with levels in the continuum. Thus, level mixing takes place between discrete and continuum levels. Owing to the perturbation the discrete levels acquire features ordinarily seen only in continuum transitions: large natural linewidths, line asymmetries and intensity anomalies in emission or absorption of radiation. For instance, the mixed discrete level may decay through a radiationless transition process, which converts the excited atom into a free electron and a positive ion. This decay constitutes the “spontaneous ionization” process identified by Majorana. It is equivalent to the Auger process then known in X-ray emissions, which is mentioned at the end of Majorana’s paper. A similar process, discussed by R. de L. Kronig a few years earlier,9 is the radiationless dissociation of a molecule when the sum of its vibrational and electronic energies exceeds the energy necessary for dissociation in a lower energy electronic configuration. The spontaneous ionization process in optical spectra of atoms was introduced independently in the same year by A. G. Shenstone of Princeton University in a publication in The Physical Review.10 Shenstone analyzed a 3 P20 level of mercury that had been recently discovered by T. Takamine and T. Suga.11 and confirmed

9 de

L Kronig, R.: Z Phys. 50, 347 (1928). A.G.: Phys. Rev. 38, 873 (1931). 11 Takamine, T., Suga, T.: Sci. Pap. Inst. Phys. Chem. Res. Tokyo. 13, 1 (1930). 10 Shenstone,

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by F. Paschen.12 Shenstone identified the same process as Majorana, but called it auto-ionization, a name which has since become standard in the literature of atomic spectroscopy (usually as autoionization). The attention of Shenstone was really concentrated on copper whose spectrum is unique with autoionization being the rule rather than the exception, as stated in his detailed report of 1936.13 In an important later work in 1935,14 H. Beutler of the University of Berlin published a detailed investigation of the absorption spectra of noble gases for levels above the ionization limit. In that study, Beutler ascribes the observed strong asymmetric modulations of the absorption lines to the autoionization process, and refers to Kronig’s and Shenstone’s previous work, but not to Majorana’s. The Theory of Atomic Spectra, published by E. U. Condon and G. H. Shortley in 1935,15 often called the “bible” of atomic spectroscopy, recognizes the simultaneous and independent contributions of Majorana and Shenstone in identifying the autoionization concept yet. In the same year, Beutler’s work caught the attention of E. Fermi in Rome, who suggested to a junior associate, U. Fano, that he find a specific explanation for the line shapes seen by Beutler. In fact, as described by Fano himself, the hypothesis of autoionization alone does not provide a description of the asymmetrically broadened lines. Fano soon produced a theoretical analysis of the mixing of a discrete level with a continuum published in 1935.16 This work, and Fano’s more complete analysis published in 1961,17 introduced the Beutler-Fano autoionization profile, a lineshape formula that has found wide applicability in many branches of physics. Fano’s work at NIST in the 1960s, and the contemporaneous development in experimental techniques for extreme ultraviolet spectroscopy, again mainly at NIST, elevated the Beutler-Fano lineshapes to a frontier research topic in atomic physics, as described in a short report by C. W. Clark in 2001.18 Autoionization has played an important role in the progress of spectroscopy, because it is observed in a large variety of atomic and molecular spectra, and very different results are obtained for the energy and probabilities of the mixed levels. The 1931 Majorana analysis does not calculate the autoionization absorption spectra. Moreover, instead of describing the continuum through a continuous distribution of states as in the Fano analyses, Majorana mixes a discrete level having a negligible decay rate with a single level that simulates the continuum through its large linewidth, denoted by the quantity “a” introduced at the bottom of page 71. Furthermore, Majorana does not derive the transition probability for the absorption

12 Paschen,

F.: Ann. Phys. (Leipzig) 6, 47 (1930). A.G.: Philos. Trans. R. Soc. Lond. 235, 195 (1936). 14 Beutler, H.: Z Phys. 93, 177 (1935). 15 Condon, E.U., Shortley, G.H.: Theory of Atomic Spectra. Cambridge University Press, Cambridge (1935). 16 Fano, U.: Nuovo Cimento 12, 154 (1935). 17 Fano, U.: Phys. Rev. 124, 1866 (1961). 18 Clark, C.W.: In: Lide, D.R. (ed.) A Century of Excellence in Measurements, Standards, and Technology, p. 116. NIST Special Publication 958, Washington, DC (2001). 13 Shenstone,

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process. Even if the description by Majorana is not complete, the diagonalization of the perturbation matrix he writes down on the last page of his paper would produce an expression for the absorption lineshape having the characteristic sawthooth profile of autoionization. Such a lineshape derivation was reported by B. W. Shore in 1968.19 The mixing of discrete and continuum states introduced by Majorana and leading to autoionization is complex. Autoionization mixing may occur whenever a discrete level above the ionization limit is embedded in a continuum with the same parity and angular momentum. For the case of the p2 3 P2 levels of Hg, Cd and Zn, no 3 P continuum of even parity is available for the autoionization mixing. Therefore, 2 Majorana argued that the electron coupling is sufficiently removed from the strict LS Russell-Saunders case, so that the discrete levels share p2 3 P2 and sd 1 D2 characteristics. Thus, the discrete level acquires the singlet character necessary to autoionize readily into the sd 1 D2 continuum. As pointed out by Majorana, this complicated double interaction mixing, between discrete states and between discrete and continuum states, is not strictly required to produce the spontaneous ionization. However, in the absence of a singlet admixture in the triplet state, the autoionization mixing would not have been large enough to explain the disappearance of the lines associated with the p2 3 P2 level. In their 1935 book, Condon and Shortley stated that the Majorana argument is not entirely convincing, because a similar autoionization process should apply also to the p2 3 P0 level. Therefore, for a long time the spectroscopic assignments and the autoionization scheme proposed by Majorana remained under scrutiny. The story of the missing lines ended in 1955 when W. R. S. Garton and A. Rajaratnam.20 were able to identify the weak autoionizationbroadened absorption lines of Zn terminating on the p2 3 P2 level. In 1970, W. C. Martin and V. Kaufman.21 pointed out the correctness of Majorana’s spectroscopic assignments. Finally, between 1986 and 1988 research groups in Orsay and Caen measured the Cd22 and Zn23 autoionization linewidths using optogalvanic detection and produced a precise derivation of the perturbation mixing for the discrete and continuum levels. Therefore, all the lines belonging to the p2 configuration suffer perturbations by autoionization, large or small, following precisely the scheme predicted by Majorana.

19 Shore,

B.W.: Phys. Rev. 171, 43 (1968). W.R.S., Rajaratnam, A.: Proc. Phys. Soc. A 68, 1107 (1955). 21 Martin, W.C., Kaufman, V.: J. Opt. Soc. Am. 60, 1096 (1970). 22 Aymar, M., Luc-Koening, E., Chantepie, M., Cojan, J.L., Landais, J., Laniepce, B.: J. Phys. B At. Mol. Phys. 19, 3881 (1986). 23 Chantepie, M., Cheron, B., Cojan, J.L., Landais, J., Laniepce, B., Aymar. M.: J. Phys. B At. Mol. Phys. 21, 1379 (1988). 20 Garton,

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In conclusion, the Majorana paper contains three remarkable steps for the progress of atomic spectroscopy: (i) an identification of spectral lines in Hg, Cd and Zn; (ii) a treatment of the electron coupling for a case other than the LS RussellSaunders case; (iii) the introduction of the autoionization process. EA is grateful to W. C. CLARK for carefully reading the English translation of the original paper and of his comment, and for useful suggestions.

ENNIO ARIMONDO NIST, Gaithersburg, MD (USA)

Oriented Atoms in a Variable Magnetic Field Ettore Majorana

Summary: The author calculates the probability of non-adiabatic processes when an oriented atomic beam passes close to a point where the magnetic field vanishes. As is well known, an oriented atom in a slowly varying magnetic field follows adiabatically the direction, assumed to be variable, of the field. This is the cause of a phenomenon that has recently been observed: if a molecular beam emerging from a Stern-Gerlach experiment is passed through another Stern-Gerlach experiment, one does not observe any further splitting of the beam. The reason is the following: all the atoms have the same orientation since all of them have followed exactly the stray field inevitably existing in the region between the pole-pieces which are meant to produce the orientation of the beam and those which should test this orientation after a certain distance. However, Phipps has undertaken some experiments to observe a non-adiabatic variation of the field in this region; this requires that the field be sufficiently weak and the variation of its direction sufficiently fast so that its rotational frequency be comparable with the Larmor frequency. Since it is difficult to reduce the intensity of the field below a few gauss,1 it is necessary, with a velocity of the beam of the order of 105 cm/s, that the direction of the field changes appreciably over a distance of a fraction of a millimeter. These are, therefore, very delicate experiments that have not yet produced any decisive result. The problem has been discussed theoretically by Güttinger2 in the case of a uniformly rotating field with constant intensity. In this paper we will suppose instead that the molecular beam passes close to a point where the magnetic field vanishes.

Translated from Atomi orientati in campo magnetico variabile, “Il Nuovo Cimento”, vol. 9, 1932, pp. 43–50, by P. Radicati di Brozolo. 1 Phipps, T.E., Stern, O.: Z Physik. 73, 185 (1932). 2 Güttinger, P.: Z Physik. 73, 169 (1932). © Springer Nature Switzerland AG 2020 L. Cifarelli (ed.), Scientific Papers of Ettore Majorana, https://doi.org/10.1007/978-3-030-23509-3_8

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This case is of special importance because, if the beam were to pass exactly by a point of zero field, all the atoms would invert their orientation. On the other hand, we cannot obtain a zero field in a point on the trajectory of the molecular beam except by several trials, for example by using two orthogonal auxiliary fields that can be regulated independently; this makes it difficult to perform the experiment until fast means to detect the beam are available. Nevertheless we anticipate the discussion to better clarify the nature of the dynamical problem arising from the rotation of a magnetic atom in an arbitrarily varying magnetic field. Our calculations will show that both the classical and the quantum-mechanical treatment require the integration of the same differential equations. It follows that when the classic solution is known, as in the case of a uniformly rotating field with constant angular velocity—which is the problem treated by Güttinger—, the quantum solution can be immediately derived. For the problem that we will study later, namely the passing close to a point of vanishing field with a slowly varying gradient, the quantum treatment is mathematically more convenient. Also in this case to obtain the general solution it is enough to solve the simplest case j = 1/2. 1. A rotational state of an atom with internal quantum j = 1/2 can be represented as a linear combination of two orthogonal states ψ1/2 and ψ−1/2 with projection ±1/2 in the z-direction: ψ = C1/2 ψ1/2 + C−1/2 ψ−1/2 . The state is therefore essentially defined by the ratio

C−1/2 C1/2 .

If the phases of ψ1/2 and ψ−1/2 3 are chosen so as to obtain the normal representation of angular momenta, the state ψ can be represented, as is known, in an invariant way by a point P on a unit sphere whose spherical coordinates ϑ and ϕ are defined by4 C−1/2 ϑ tang eiϕ = . 2 C1/2 If O is the center of the sphere, the vector radius OP defines the direction along which the momentum in the state ψ has value 1/2. In the case j = 1/2 the most h in generic rotational state then corresponds to oriented atoms with momentum 12 2π an arbitrary direction. The probability of agreement for two states represented by the points P and P  is given by W (P , P  ) = cos2

1 α, 2

where α is the angle P OP  . The probability vanishes, i.e. the two states are orthogonal, when P and P  are diametrically opposite. For j > 1/2 there is in 3 In

“Il Nuovo Cimento” it is erroneusly printed ψ1/2 . (Note of the Editor, see also E. Amaldi, op. cit.) 4 In “Il Nuovo Cimento” it is erroneously printed tang ϑ here and in the following Eq. (3). (Note of z the Editor, see also E. Amaldi, op. cit.)

Oriented Atoms in a Variable Magnetic Field

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general no direction along which the atom is oriented, i.e. with a value determined by angular momentum. In spite of this, an intrinsic geometric representation similar to the previous one is still possible. The only difference is that every state, rather than being represented by a single point, is represented by 2j points on the unit sphere. Indeed let us consider a generic state: ψ = Cj ψj + Cj −1 ψj −1 + · · · + C−j ψ−j and let ζ1 , ζ2 , . . . , ζ2j be the roots of the equation a0 ζ 2j + a1 ζ 2j −1 + · · · + a2j = 0 ,

(1)

Cj − r ar = (−1)r √ . (2j − r)!r!

(2)

where

The state ψ can then be represented by the points P1 , P2 , . . . , P2j on the unit sphere where the spherical coordinates ϑs , ϕs of Ps are given by tang

ϑs iϕs e = ζs . 2

(3)

It is not difficult to verify that this geometric representation is independent of the choice of the coordinate system. The distribution of the representative points is a priori arbitrary but becomes particularly simple in the case of oriented atoms. To an oriented state with angular momentum component m in the direction OP, there actually correspond j + m points that coincide in P and j − m in the point diametrically opposite to P , as if each of the representative points indicated the h direction of a little gyroscope with angular momentum 12 2π . The probability of agreement for two oriented states, one with momentum m in the direction OP, the other with momentum m in the direction OP  forming with OP an angle α, is given by ! α "4j (j + m)!(j − m)!(j + m )!(j − m )! W (P , P  ; m, m ) = cos 2 ⎡ ⎤2  2r−m+m (−1)r tang α2 ⎣ ⎦ r r!(r − m + m )!(j + m − r)!(j − m − r)!

(4)

0

which is obviously symmetric in m and m . Let us now consider the rotation of the atom in a magnetic field H (t) varying arbitrarily with time; hence let us suppose that the atom has a magnetic moment −gj μ0 so that for a sufficiently weak field we can assume that the Hamiltonian is gμ0 (j, H ). The equations of motion are therefore:

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 # Hx − iHy 2π i ˙ gμ0 mHz Cm + (j + m)(j − m + 1) Cm−1 + Cm = − h 2  # Hx + iHy Cm+1 . + (j + m + 1)(j − m) 2 Using relation (2) we obtain a˙ r = −

 Hx − iHy 2π i gμ0 (j − r)Hz ar − (r + 1) ar+1 − h 2  Hx + iHy −(2j − r + 1) ar−1 . 2

Setting the time derivative of the left-hand side of (1) equal to zero, we easily find that the time derivative of a generic root ζi is: ζ˙i =



Hx − iHy 2 Hx + iHy 2π i gμ0 Hz ζi + ζi − . h 2 2

The time derivatives of ϑi , ϕi relative to the point Pi of which ζi is the stereographic projection on the complex plane x + iy follow from Eq. (3):  2π gμ0 Hy cos ϕi − Hx sin ϕi , h

Hx cos ϕi + Hy sin ϕi 2π . gμ0 Hz − ϕ˙ i = h tang ϑi ϑ˙ i =

These equations mean that each of the representative points on the unit sphere precedes around the field direction with frequency g · o, where o is the Larmor frequency. This is what would happen in classical mechanics if any of the radii h OPi denoted the direction of a gyroscope with self-momentum j 2π and magnetic moment −gj μ0 . It is actually possible to prove that the validity of this result is implicit in the invariance of the geometric representation. The converse is also true. From what has been said so far, it follows that the relative positions of the 2j representative points are invariant in time; as a consequence, if the atom is initially oriented with angular momentum component m in the direction of the field, at time t it will still be oriented with momentum m along a direction forming an angle α(t) with the field. Knowing the rotation angle α(t) that can be calculated both with classical and quantum mechanics, and is independent of j and m, we can calculate with (4) the probability that the angular momentum component in the direction of the field takes at time t the generic value m . 2. Let us now suppose that a beam of oriented atoms passes close to a point Q where Hx = Hy = Hz = 0. Close to Q the field components are linear functions of the Cartesian coordinates x, y, z; in a coordinate system centred on a moving

Oriented Atoms in a Variable Magnetic Field

81

atom whose motion we can assume with large approximation to be rectilinear and uniform, the components of the field acting on the atom are then linear functions of time. Let the x-axis be in the same direction of the field when it reaches the minimum intensity, which will in general happen not too far from Q and the zaxis in the direction opposite to the time derivative of the magnetic field, which is obviously orthogonal to the previous one. Let us further suppose that the origin of time coincides with the instant when the field is minimum. The components of the field acting on the atom at a given time t will be: Hx = A ;

Hy = 0 ;

Hz = −Ct .

If j = 1/2 let us set ψ = ξ ψ1/2 + ηψ−1/2 5 ; the equations of motion will then be  πi gμ0 − Ctξ + Aη , h  πi η˙ = − gμ0 Aξ + Ctη . h ξ˙ = −

To work with dimensionless variables let us introduce a new time measure6 : $ π gμ0 C · t τ= 2h and the numerical quantity k=

2πgμ0 A2 hC

which gives the ratio of the atom precession frequency to the rotation frequency of the field direction when this ratio reaches its minimum value, i.e. for τ = 0. If v is the vapour-beam velocity at distance d from point Q close to which the field 2 gradient is of the order of G gauss per cm, k 7 will be of the order of 107 G dv . For example for v = 105 and G ∼ 1, k will be ∼100 d 2 and therefore k ∼ 1 for d = 1 mm. With the above new variables we obtain: √  dξ = −i − 2τ ξ + kη , dτ

“Il Nuovo Cimento” 12 and − 12 are erronously printed in line. (Note of the Editor, see also E. Amaldi, op. cit.) 6 In “Il Nuovo Cimento” it is erroneously printed π . (Note of the Editor, see also E. Amaldi, op. zh cit.) 7 In “Il Nuovo Cimento” it is erroneously printed K, here and also in the 12th line of p. 84. (Note of the Editor, see also E. Amaldi, op. cit.) 5 In

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√ dη = −i k ξ + 2τ η . dτ These equations can be simplified by setting: η = e−iτ g

2

2

ξ = eiτ f ; from which it follows:

⎧ √ df 2 ⎪ ⎪ = −i ke−2iτ g ⎨ dτ √ ⎪ dg 2 ⎪ ⎩ = −i ke2iτ f. dτ

(5)

d 2f df + kf = 0. + hiτ 2 dτ dτ

(6)

Eliminating g, we obtain:

We can derive an integral representation of the solutions of this differential equation that allows to determine the asymptotic expression for large positive or negative values of τ . This is indeed what we need since we assume that for τ = −∞ the atom is oriented with respect to the field and we want to determine its orientation for τ → ∞. Since everything reduces to the calculation of the angle α defined above, it is enough to consider only one solution of Eq. (6). This is given by √

f (τ ) =

k e−kπ /8 √ 2(1 + i) π

 s (k/4i)−1 e(s

2 /8i)+sτ

ds

with logs assuming its principal value, and the boundary condition   k/4i (s 2 /8i)+sτ e =0 s C

being satisfied if the integration path has the form shown in the figure for the two cases τ < 0 e τ > 0.

Oriented Atoms in a Variable Magnetic Field

83

The negative part of the real axis is a line of discontinuity for the function to be integrated and cannot be crossed. Moreover, to evaluate the asymptotic expressions, the integration paths are drawn through the saddle point s = −4iτ in the direction of maximum slope. For τ → −∞ the whole integral comes from the vicinity of the point s = −4iτ . With the substitution s = −4iτ + (1 − i)p we can easily calculate the first terms in the asymptotic expansion of f and, with the help of the first Eq. (5), of g. Neglecting the terms that tend to zero we find τ → −∞;

f = 0;

g = (−4τ )k/4i .

For τ → ∞ the asymptotic expression of the integral comes partly from the neighbourhoods of the saddle point s = −4iτ , and partly from the branching point O. In this case the result is

k kπ 1 − i √ −k/4i −kπ/8  ke e sinh f =− √ 4 4i 2 π τ → ∞: g = (4τ )k/4i e−kπ/4 . Since for real a 8   (ai) =

8 See,

$

π , a sinh(π a)

e.g., Whittaker, E.T. and Watson, G.N. Modern Analysis, 4th edition, p. 259.

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or keeping in mind the constancy of |f 2 | + |g 2 | we find the following expressions for |f | and |g|: τ → −∞: τ → ∞:

|f | = 0; |g| = 1 # |g| = e−kπ/4 . |f | = 1 − e−kπ/2 ;

For τ → −∞ the field is directed along the z-axis and therefore at the beginning h in the direction of the field. of the phenomenon the momentum of the atom is − 12 2π For τ → ∞ the field is instead directed along −z, hence the limiting value of |g 2 | provides the probability that the atom reverses its orientation, i.e. that, once it has h passed in the vicinity of Q it acquires a momentum 12 2π in the direction of the field. This probability is therefore

1 1 W − , 2 2



= e−kπ/2 .

In this way, 21% of the atoms turn over for k = 1 and 4.3% for k = 2. The general solution of the problem for an arbitrary j and for transitions from m to m can be obtained from Eq. (4) with the appropriate value of the rotation angle α. Since in our case

1 1 α = sin2 , W − , 2 2 2 by comparison with the previous expression, we obtain α = 2 arcsin e−kπ/4 .

Comment on: Oriented Atoms in a Variable Magnetic Field

85

Comment on: “Oriented atoms in a variable magnetic field”. In this work Majorana evaluates the probability of spin-flip for atoms in a polarized beam in the presence of a varying magnetic field. It is a quantitative study of the non-adiabatic situation. The problem, “particularly important” to Majorana, is what happens close to a zero of the magnetic field: “all the atoms would invert their spin orientation”. The problem had been proposed to Majorana by E. Segré, as E. Amaldi recalls.9 Indeed E. Segré, together with R. Frisch,10 were setting up an experiment to generalize the famous Stern and Gerlach work on spatial quantization. The latter observations had been done in static magnetic gradient and adiabatic conditions. The goal of the new experiment was to measure the final state of an atomic magnetic moment initially prepared in a definite state in the presence of a rapidly varying field. In Segré and Frisch apparatus the non-adiabatic transition was induced by having the atomic beam close to a zero of the magnetic field. Their measurements were explained using Majorana’s quantitative predictions. In his paper Majorana shows how to interpret the total effect of a varying magnetic field on an object with a given angular momentum and a component m along the z-axis in terms of a sudden rotation of the angular momentum itself. By a rigorous solution of the time-dependent Schrödinger equation, Majorana obtains the rotation angle showing that there is a dependence on the gyromagnetic factor but not on the initial m value. As a consequence, after the rotation the system is not in a well-defined quantum state with respect to the original field direction but has to be described as wavepacket composed by a superposition of states with different m . In the paper the probability amplitude for a transition between m and m is explicitly calculated. Majorana immediately singles out two characteristic frequencies of the problem: Larmor precession frequency of the atomic dipole moment and the frequency of rotation of the magnetic field as seen by the atom. When the two frequencies become comparable the atom has a high probability of changing its magnetic substate, i.e. to undergo a spin-flip. Majorana demonstrates that the spin-flip probability depends on the ratio k between the two frequencies and is given by e−kπ/2 . As E. Amaldi points out “the problem was really solved by Majorana with extreme elegance and conciseness for the case J = 1/2”. The generalization to the case of any J was done by F. Bloch e I. I. Rabi who, in their celebrated work of 1945,11 cite the work of Majorana as seminal for the solution of the problem. Majorana’s treatment for spinflips is used in quantum mechanics textbooks12 and has recently been rediscovered

9 Amaldi,

E.: Ettore Majorana, man and scientist. In: Zichichi, A. (ed.) Strong and Weak Interactions, Present Problems, International School of Physics Ettore Majorana, Erice, 19th June– 4th July 1966. Academic, New York/London (1966); and E. Amaldi Op. Cit. 10 Frisch, R., Segrè, E.: Ricerche sulla quantizzazione spaziale. Nuovo Cimento. 10, 78 (1933); Über die Einstellung der Richtungsquantelung. Z Phys. 80, 610 (1933). 11 Bloch, F., Rabi, I.I. Atoms in variable magnetic fields. Rev. Mod. Phys. 17, 237 (1945). 12 Landau, L.D., Lifshitz, E.M.: Quantum Mechanics: Non-relativistic Theory. Nauka, Moscow (1974)/Pergamon Press, Oxford (1977).

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in atomic and molecular collision physics at low energy13 ; it is also widely applied outside atomic and molecular physics, e.g. to manipulate polarized beams in neutron physics. Let us discuss here two experimental fields were Majorana’s work has found an application and is still quite useful: namely double-resonance spectroscopy and Bose-Einstein condensation. In the early 1950s in atomic physics double-resonance spectroscopy (optics+radiofrequency) has been developed to investigate atomic structures. In particular, Brossel and Bitter14 have studied the polarization state of fluorescence from the lowest 3 P1 level of mercury excited with polarized light in order to populate only the sublevel m = 0. In the experiment the atoms are subjected to a magnetic field to separate Zeeman sublevels m = −1, 0, +1. A radiofrequency radiation is used to induce transition between m = 0 and m = +1 or m = −1. As a consequence, the polarization state of the fluorescence radiation is altered. The observed lineshape has a peculiar behaviour, indeed by increasing the radiofrequency power one observes a doubling of line with an evident central minimum. Brossel and Bitter interpret such behaviour as “Majorana transitions” and use Majorana’s formula to perfectly explain the experimental data. Nowadays such lineshape is known as “Majorana-Brossel”. The explanation of the experiment is still of great interest and has been treated by C. Cohen-Tannoudji in his course at Collège de France in 2003.15 He interprets the lineshape as a consequence of the quantum interference with a three-photon process where the atoms from sublevel m = 0, before making a transition to sublevel m = +1, undergo a transition to and from sublevel m = −1. Coming back to the problem of the spin-flip of atoms moving in a magnetic quadrupole field, it is interesting to note that the understanding of this effect has made the experimental realization of Bose-Einstein condensation in an atomic gas possible. In order to observe condensation, atoms must be cooled to the microkelvin regime and this has been achieved by using magnetic traps for atoms oriented and pre-cooled with laser radiation. The final cooling towards condensation is performed by forced evaporation of the hottest atoms from the magnetic trap and by the consequent thermalization of the remaining atoms. To this aim collision processes are fundamental requiring a large atomic density. By moving in a magnetic trap the atoms are exposed to a varying magnetic field; the coldest they are the slowest they move around the magnetic-field minimum. In the first attempts quadrupole magnetic traps were used; i.e. the configuration Majorana has studied in his paper. When the atoms undergo a spin-flip they are no longer confined and Majorana’s

13 Di

Giacomo, F., Nikitin, E.E.: The Majorana formula and the Landau-Zener-Stuckelberg treatment of the avoided crossing problem. Phys. Usp. 48, 515 (2005) 14 Brossel, J., Bitter, F.: A new double resonance method for investigating atomic energy levels. Application to Hg 3 P1 . Phys. Rev. 86, 308 (1952) see also: Brossel J. Thesis, Faculté des Sciences de l’Université de Paris (1952) 15 Cohen-Tannoudji, C.: Lessons available online at http://www.phys.ens.fr/cours/\college-defrance/.

Comment on: Oriented Atoms in a Variable Magnetic Field

τ

87

τ

Fig. 1 Atoms polarized and confined with magnetic field gradients on the route to Bose-Einstein condensation. The graph on the left refers to a quadrupole trap where there are losses due to Majorana spin-flip. The graph on the right refers to a trap specifically designed to avoid spin-flips18

formula allows a quantitative analysis in terms of the magnetic field gradient and of the atomic temperature. The zero in the magnetic field is a sort of “hole” that reduces the atomic density thus preventing the evaporative cooling. E. A. Cornell, C. E. Wieman and W. Ketterle, who have first realized Bose-Einstein condensation in 1995, were perfectly aware of the problem. Indeed in their Nobel Lectures16,17 they often mention the “Majorana hole”. The problem to be solved was to create traps with a non-zero minimum. The first solution came from the Boulder group using a quite sophisticate trap where the minimum was rotated (TOP) and the parameters were chosen using the quantitative expression for the spin-flip. It is interesting to report what Cornell and Wieman write: “This zero represents a hole in the trap, a site at which atoms can undergo Majorana transitions and thus escape from the trap. . . . the TOP design worked well, and the samples were cooled far colder, in fact too cold . . . ”. Ketterle, on the other hand, solved the problem by using a focused laser beam to “plug the hole”. Nowadays, having completely understood the problem, it is possible to use different static configurations with a non-zero minimum. Among the others we here cite the one used in the very town—Catania—of Majorana by F. S. Cataliotti where the atoms are trapped on a microelectronic circuit. Figure 1 shows the temporal behaviour of the atomic number trapped at a temperature of a few microkelvin in the presence of a zero (left) and of a situation where the “hole” is plugged by adding a magnetic field (right): as can be observed by removing Majorana spin-flip the atomic-trap lifetime is significantly enhanced.18

16 Cornell,

E.A., Wieman, C.E.: Bose-Einstein condensation in a dilute gas. Rev. Mod. Phys. 74, 875 (2002) 17 Ketterle, W.: When atoms behave as waves. Rev. Mod. Phys. 74, 1173 (2002) 18 Courtesy of F. S. Cataliotti, Università di Catania (2006).

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Once the condensate is realized, spin-flip transitions, this time induced in a controlled fashion, can be used for further developments in atomic physics at ultralow temperatures. Recently, X. Chen in Beijing University19 has used Majorana’s theoretical predictions to develop a new method to extract coherent atoms from a condensate and realize a pulsed atom laser. MASSIMO INGUSCIO Università di Firenze, Florence, Italy

19 Ma, X., et al.: Population oscillation of the multicomponent spinor BEC induced by nonadiabatic

transitions. Phys. Rev. A. 73, 013624 (2006).

Relativistic Theory of Particles with Arbitrary Intrinsic Angular Momentum Ettore Majorana

Summary: The author establishes wave equations for particles having arbitrary given intrinsic angular momentum. Such equations are linear in the energy and relativistically invariant. As is well known, Dirac’s theory of the electron makes use of a four-component wave function. When slow movements are considered, two of the components acquire negligible values, while the remaining two, at least in a first approximation, satisfy the Schrödinger equation. h In a similar fashion, a particle having intrinsic angular momentum s 2π (s = 1 3 0, 2 , 1, 2 , . . .) is described in quantum mechanics by a set of 2s + 1 wave functions which separately satisfy the Schrödinger equation. Of course, such a representation is valid as long as the relativistic effects are neglected, and this is allowed for a particle moving with speed much smaller than the speed of light. A different case in which the elementary theory retains its validity is, of course, the one in which the velocity of the particle is comparable with c but remains approximately constant in direction and magnitude. In fact, this case can be reduced to the study of slow movements by means of a suitable choice of the frame of reference. On the other hand, the following situation cannot be easily dealt with by means of the nonrelativistic Schrödinger equation: a particle with speed which retains an almost constant value within fairly large regions in the space-time continuum, but for which the speed is also slowly varying between very different extremum values from one such region to another, as an effect of weak external fields. A relativistic generalization of the preceding theory should satisfy the following hierarchy of conditions, as it becomes more and more accurate:

Translated from Teoria relativistica di particelle con momento intrinseco arbitrario, “Il Nuovo Cimento”, vol. 9, 1932, pp. 335–344, by C. A. Orzalesi in Technical Report no. 792, 1968, produced at, and currently available through, the University of Maryland. (Reproduced with kind permission of C. A. Orzalesi.) © Springer Nature Switzerland AG 2020 L. Cifarelli (ed.), Scientific Papers of Ettore Majorana, https://doi.org/10.1007/978-3-030-23509-3_9

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(a) The theory should allow the study of particles having an almost constant velocity (in direction and magnitude), and the results should be equivalent to those given by the nonrelativistic theory; however, there should be no need to specify any particular reference frame. (b) The theory should allow the study of processes where the speed of the particles is slowly varying, but within arbitrarily separated limits, under the effect of weak external fields. (c) The theory should retain its validity in general, even when the velocities of the particles vary arbitrarily. It is likely that a rigorous theory satisfying condition (c) might be incompatible with the present-day quantum scheme. [For instance,] Dirac’s theory of the electron has largely proved its fruitfulness in the study of genuine relativistic phenomena, e.g., scattering of hard γ -rays; however, this theory certainly satisfies condition (c) only incompletely as is shown by the well-known difficulties coming from the transitions to states having negative energy. On the contrary, it is probably true that a theory satisfying condition (b), and only partially satisfying condition (c), should not meet essential difficulties since its physical content might be essentially the same as that which justifies the Schrödinger equation, The most remarkable example of this type of generalization is provided precisely by Dirac’s theory. However, since this theory can be applied to particles with intrinsic [angular] momentum s = 1/2, I have investigated equations formally similar to the ones by Dirac, although considerably more involved; these equations allow us to consider particles with arbitrary (and, in particular, zero) angular momentum. According to Dirac, the wave equation of a material particle in the absence of external fields must have the following form: 

 W + (α, p) − βmc ψ = 0. c

(1)

Equations of this kind present a difficulty in principle. Indeed, the operator β 1 has to transform as the time component of a four-vector, and thus β cannot be simply a multiple of the unit matrix, but must have at least two different eigenvalues, say β1 and β2 . However, this implies that the energy of the particle at rest, obtained from Eq. (1) by taking p = 0, shall have at least two different values, i.e. β1 mc2 and β2 mc2 . According to Dirac’s equations, the allowed values of the mass at rest are, as well known, +m and −m; from this it follows by relativistic invariance that for#each value of p the energy can acquire two values differing in sign: W = ± m2 c 4 + c 2 p 2 . As a matter of fact, the indeterminacy in the sign of the energy can be eliminated by using equations of the type (1), only if the wave function has infinitely many components that cannot be split into finite tensors or spinors. “Il Nuovo Cimento” it is erroneously printed −1 instead of β. (Note of the Editor, see also E. Amaldi, op. cit.)

1 In

Relativistic Theory of Particles with Arbitrary Intrinsic Angular Momentum

91

1. Equation (1) can be derived from the following variational principle:  δ

ψ˜



 W + (α, p) − βmc ψdV dt = 0 c

(2)

(one of the conditions imposed by relativistic invariance is, of course, that the form ˜ ψβψ has to be invariant). If now we require the energy at rest to be always positive, all the eigenvalues of ˜ β have to be positive so that the form ψβψ will be positive definite. By means of a nonunitarity transformation ψ → ϕ it is then possible to reduce the expression at hand to the form unity: ˜ ψβψ = ϕϕ. ˜

(3)

By substituting in Eq. (2) ψ with its expression in terms of ϕ one obtains:  δ

  W ϕ˜ γ0 + (γ , p) − mc ϕdV dt = 0, c

(4)

from which the equations equivalent to Eq. (1): follow:   W γ0 + (γ , p) − mc ϕ = 0. c

(5)

We have now to determine the transformation law of ϕ under a Lorentz rotation, as well as the expressions for the matrices γ0 , γx , γy , γz , in such a way as to respect the relativistic invariance of the variational principle (4), and thus to have an invariant integrand function in Eq. (4). We begin by establishing the transformation law of ϕ and we note, first of all, that the invariance of ϕϕ ˜ implies that we must restrict ourselves to unitary transformations. Moreover, in order to avoid exaggerated complications, we will give the transformation law only for infinitesimal Lorentz transformations, since any finite transformation can be obtained by integration of the former ones. We introduce the infinitesimal transformations in the variables ct, x, y, z;  ⎧ 0 ⎪  ⎪ ⎪  ⎪ ⎪ 0 ⎪ ⎪  ⎪ S = x ⎪  ⎪ ⎪ 0 ⎪ ⎪  ⎪ ⎪ 0 ⎨  ⎪ 0 ⎪ ⎪  ⎪ ⎪  ⎪ ⎪ 1 ⎪ ⎪  ⎪ T = x ⎪  ⎪ ⎪ 0 ⎪ ⎪  ⎩ 0

0 0 0 0 1 0 0 0

  0 0 0     0 0 0 0  ; S = y   0 −1 0 0    0 −1 1 0   0 0 0 0     0 0 0 0  ; T = y   0 0 1 0    0 0 0 0 0

0 0 0 0 1 0 0 0

  0 0     0 1  ; S = z   0 0    0 0   0 0     0 0  ; T = z   0 0    1 0

0

0

0 −1 1

0

0

0

0

0

0

0

0

0

0

0

 0  0 ; 0  0  1  0 . 0  0

(6)

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We also define 

ax = iSx ;

ay = iSy ;

bx = −iTx ;

by = −iTy ;

az = iSz ; bz = −iTz .

(7)

The operators a and b must be Hermitian operators in a unitary representation, and vice versa; furthermore, in order for the infinitesimal transformations to be integrable, they must satisfy certain relationships under commutation, as can be deduced from Eqs. (6) and (7): ⎧ ⎪ ax , ay ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎨ax , bx a ,b ⎪ x y ⎪ ⎪ ⎪ ⎪ ax , bz ⎪ ⎪ ⎩b , b x y

= iaz =0 = ibz

(8)

= −iby = −iaz

the remaining relations can be obtained by cyclic permutations of x, y, z. The simplest solution of Eqs. (8) by means of Hermitian operators is given by the following infinite matrices, where the diagonal elements are labelled by two indices j and m; we have to distinguish two possibilities according to the assumption j = 1/2, 3/2, 5/2, . . . ; m = j, j − 1, . . . , −j , or j = 0, 1, 2, . . . ; m = j, j − 1, . . . , −j : ⎧   √ ⎪ j, max − iay j, m + 1 = (j + m + 1)(j − m) ⎪ ⎪ ⎪ ⎪    √ ⎪ ⎪ ⎪ j, max + iay j, m − 1 = (j + m)(j − m + 1) ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ j, m|az |j, m = m ⎪ ⎪ ⎪ ⎪    ⎪ 1√ ⎪ ⎪ j, mbx − iby j + 1, m + 1 = − (j + m + 1)(j + m + 2) ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎨ j, mb − ib j − 1, m + 1 = 1 √(j − m)(j − m − 1) x y 2 ⎪   ⎪ 1√ ⎪ ⎪ j, mbx + iby j + 1, m − 1 = (j − m + 1)(j − m + 2) ⎪ ⎪ 2 ⎪ ⎪ ⎪    ⎪ 1√ ⎪ ⎪ j, mbx + iby j − 1, m − 1 = − (j + m)(j + m − 1) ⎪ ⎪ 2 ⎪ ⎪ ⎪  ⎪ 1√ ⎪ ⎪ j, m|bz |j + 1, m = (j + m + 1)(j − m + 1) ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪  √ ⎪ ⎩ j, m|bz |j − 1, m = 1 (j + m)(j − m). 2

(9)

Relativistic Theory of Particles with Arbitrary Intrinsic Angular Momentum

93

If we assume that, by reflection with respect to the origin, the ϕj,m either remain unchanged or change in sign as j varies, b turns out to be a polar vector while a has axial properties. The entities to which a and b apply will be called infinite tensors (or spinors) of zero index, for integer (respectively, half-integer) j . The nomenclature of “zero index” comes from the fact that the invariant Z = ax bx + ay by + az bz

(10)

vanishes. More general infinite spinors or tensors can be introduced for any value of Z. A simple way to obtain the spinors is as follows. Let us consider a general solution ψ(q, t) of the Dirac equation with no external field and transform it relativistically: ψ(q, t) → ψ  (q, t).

(11)

Thus, the transformation in the space variables: ψ(q, 0) → ψ  (q, 0)

(12)

is unitary. Now, if instead of general functions ψ(q, 0) we consider only those belonging to a fixed eigenvalue z0 so of the operator (10), which has a spectrum extending from −∞ to +∞, we obtain functions which transform under (12) as infinite spinors, each function appearing twice. In the representation (12), the operators ax and bx have the following form: ax =

1 2π  ypz − zpy + σx h 2

bx =

i 2π H x + αx h c 2

and similarly for ay , az , by , bz . 2. We have now to determine the operators γ0 , γx , γy , γz in such a way as to make Eq. (4) invariant. Since we consider only unitary transformations, these operators transform in the same way as the Hermitian forms related to them; thus, in order for the integrand fraction in (4) to be invariant, it is necessary that the operators in question form a covariant vector (γ0 , γx , γy , γz ∼ ct, −x, −y, −z). The interpretation of ϕγ ˜ 0 ϕ and −ϕγ ˜ ϕ as charge and current densities is immediate. The γ operators must satisfy the following commutation relations:

94

E. Majorana

⎧ ⎪ (γ0 , ax ) = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪(γ0 , bx ) = iγx ⎪ ⎪ ⎪ ⎪(γx , ax ) = 0 ⎪ ⎪ ⎪ ⎨γ , a = iγ x

y

z

⎪ (γx , az ) = −iγy ⎪ ⎪ ⎪ ⎪ ⎪(γx , bx ) = iγ0 ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ γx , by = 0 ⎪ ⎪ ⎪ ⎩(γ , b ) = 0 z z

(13)

and the others obtained by cyclic permutation of x, y z. As can be easily checked, the commutation relations (13), determine γ0 , γx , γy , γz to within a constant factor. One finds: ⎧ 1 ⎪ ⎪γ0 = j + ⎪ ⎪ 2 ⎪ ⎪    ⎪ i√ ⎪  ⎪ γ j, m − iγy j + 1, m + 1 = − (j + m + 1)(j + m + 2) ⎪ x ⎪ 2 ⎪ ⎪ ⎪    ⎪ i√ ⎪ ⎪ (j − m)(j − m − 1) ⎪ j, mγx − iγy j − 1, m + 1 = − ⎪ 2 ⎪ ⎪ ⎨   i√ (j − m + 1)(j − m + 2) j, mγx + iγy j + 1, m − 1 = (14) 2 ⎪ ⎪ ⎪   ⎪ i√ ⎪ ⎪ j, mγx + iγy j − 1, m − 1 = (j + m)(j + m − 1) ⎪ ⎪ 2 ⎪ ⎪ ⎪  ⎪ i√ ⎪ ⎪ (j + m + 1)(j − m + 1) j, m|γz |j + 1, m = ⎪ ⎪ 2 ⎪ ⎪ ⎪  ⎪ i√ ⎪ ⎩ j, m|γz |j − 1, m = − (j + m)(j − m). 2 The omitted matrix elements of γx , γy , γz vanish. It should be noticed that the Hermitian form ϕγ0 ϕ is positive definite, as the physical interpretation requires. We now want to translate the equations written in the form (5) into the form of Eq. (5). For this, it suffices to write ψj,m ϕj,m = % , j + 12

(15)

since then the form related to γ0 reduces to the unit form. In this way we obtain equations having the desired form 

 W + (α, p) − βmc ψ = 0, c

(16)

Relativistic Theory of Particles with Arbitrary Intrinsic Angular Momentum

where β =

1 j + 21

95

and the non-vanishing components of αx , αy , αz are, given as

follows: & ⎧    ⎪ i' (j + m + 1)(j + m + 2) ⎪   ⎪ ! "! " j, m αx − iαy j + 1, m + 1 = − ' ( ⎪ ⎪ 2 ⎪ j + 12 j + 32 ⎪ ⎪ ⎪ ⎪ & ⎪ ⎪    ⎪ i' (j − m)(j − m − 1) ⎪   ⎪ "! " j, m αx − iαy j − 1, m + 1 = − ' ( ! ⎪ ⎪ 2 ⎪ ⎪ j − 12 j + 12 ⎪ ⎪ ⎪ & ⎪ ⎪    ⎪ i' (j − m + 1)(j − m + 2) ⎪   ⎪ ! "! " j, m ax + iαy j + 1, m − 1 = ' ⎪ ( ⎪ ⎪ 2 ⎨ j + 12 j + 32 &   ⎪  i' (j + m)(j + m − 1) ⎪   ⎪ "! " j, m αx + iαy j − 1, m − 1 = ' ⎪ ( ! ⎪ ⎪ 2 ⎪ j − 12 j + 12 ⎪ ⎪ ⎪ & ⎪ ⎪ ⎪  (j + m + 1)(j − m + 1) i' ⎪ ⎪ ! "! " j, m|αz |j + 1, m = ' ⎪ ( ⎪ ⎪ 1 3 2 ⎪ j + j + ⎪ 2 2 ⎪ ⎪ & ⎪ ⎪ ' ⎪  (j + m)(j − m) i ⎪ ⎪ "! ". j, m|αz |j − 1, m = − ' ⎪ (! ⎪ ⎪ 2 ⎩ j−1 j+1 2

(17)

2

In looking for solutions of Eq. (16) corresponding to plane waves with positive mass, one finds all those which can be derived by means of a relativistic transformation from a zero-momentum plane wave. For these, the energy is given by W0 =

mc2 j+

1 2

(18)

.

For half-integer values of j we thus obtain states corresponding to the values m, m/2, m/3, . . . , of the mass, while for integer j one has 2m, 2m/3, 2m/5, . . .. It should be emphasized that particles having different masses also have different intrinsic angular momentum, the latter having a determined value only in the system where the particle is at rest. If we consider the set of all states belonging to the value m1 of the rest mass, as s+ 2

is realized in nature, all other states having no significance, we obtain an invariant theory for particles of angular momentum s; in the absence of an external field, this theory can be regarded as satisfactory. One can easily verify that in the case of slow movements and for particles having intrinsic angular momentum s, only the functions ψs,m are appreciably different from zero and satisfy the Schrödinger equation with mass M = m1 ; the functions ψs+1,m and ψs−1,m are then of order s+ 2

v/c, while ψs+2,m , and ψs−2,m are of order v 2 /c2 , and so on.

96

E. Majorana

In this way we obtain only two equations; the one suitable for the description of particles with noninteger angular momentum and the other pertinent to zero or integer angular momentum. Besides the states pertinent to positive values of the mass, there are other states for which the energy is related to the momentum by a relation of the following type: % W = ± c2 p 2 − k 2 c4 ;

(19)

such states exist for all positive values of k but only for p ≥ kc, and can be regarded as pertaining to the imaginary value ik of the mass. The “spin” functions belonging to plane waves with p = 0 have a particularly simple expression in the case of particles with no intrinsic angular momentum if px = py = 0, pz = p. Apart from a normalization factor for these functions, one finds ⎧ 



⎪ 1 η−j j ⎪ ⎨ψj,0 = j+ i 2 ε (j = 0, 1, 2, . . .) (20) ⎪ ⎪ ⎩ψ =0 for m = 0, j,m

where p , ε= Mc

# η=

M 2 c2 + p 2 Mc

(21)

and M = 2m is the mass at rest. 3. We want now to discuss briefly the introduction of the electromagnetic field into Eq. (16). The simplest way to perform the transition from the field equations without external field to those with an external field is to substitute for W and p, the quantities W − eϕ and p − ec A, respectively, e being the charge of the particle and ϕ and A being the scalar and vector potentials. However, other possibilities are also open. For instance, one can add other invariant terms, analogous to those introduced by Pauli2 in the theory of the magnetic neutron. Those additional terms contain as a factor the field forces instead of the electromagnetic potentials and thus do not destroy the invariance of the field equations coming from the indeterminacy of the potentials. This artifice allow us to ascribe an arbitrarily fixed magnetic moment to particles having a non-vanishing angular momentum. For instance, in the case of the electron, by means of the simple substitutions W , p → W −eϕ, p − ec A, one finds a magnetic moment equal to + 12 μ0 , instead of −μ0 .

2 Quoted

by Oppenheimer, J.R.: Phys. Rev. 41, 763 (1932).

Relativistic Theory of Particles with Arbitrary Intrinsic Angular Momentum

97

Thus, if we want to specialize our theory to a theory for the electron and maintain as far as possible good agreement with the experimental data, we have to modify the magnetic moment by introducing additional terms. However, the electron theory obtained in this way is a useless copy of Dirac’s theory, the latter remaining completely preferable thanks to its simplicity and to the wide support from experiment. On the other hand, the advantage of the present theory lies in its applicability to particles with angular momentum different from 1/2. The equations, including both the external field and the additional terms which modify the intrinsic magnetic moment, have the following form: 

!  W e e " − ϕ + a, p − A − βmc + λ(a  , H ) + λ(b , E) ψ = 0, c c c

(22)

where a  stands for (ax , ay , az ) and b for (bx , by , bz ), while E and H represent the electric and magnetic field. The matrix ax can be deduced from ax of Eq. (9) by means of the rule 

   1   j, max j  , m = $! "! " j, m|ax |j , m j + 12 j  + 12

(23)

and similarly for ay , az , by , bz . For particles having intrinsic angular momentum s = 1/2 the choice λ = 2c μ should be made if μ is the magnetic moment which one wants to add to the one which naturally arises from the introduction of the electromagnetic potentials into the wave equation. As seen before, the latter magnetic moment has in this case, the value − 12 4πehmc . For particles having no intrinsic magnetic moment it is natural to choose λ = 0. Regarding the practical solutions of the wave equations, we recall that for slow movements they are finite and that those ψj,m which satisfy to the Schrödinger equations are only those which have j equal to the intrinsic angular momentum in units h/2π . For instance, for particles having no intrinsic momentum, one is left with one component only, namely ψ0,0 , while ψ1,m are of order v/c, v being the speed of the particle, ψ2,m are of order v 2 /c2 , and so on. In this way one succeeds in eliminating, by successive approximations, the small components and in particular one arrives at very simple expansions for the calculation of the first relativistic corrections. I particularly thank Prof. E. Fermi for discussions of the present theory.

98

N. CABIBBO

Comment on: “Relativistic theory of particles with arbitrary intrinsic angular momentum”. The central problem in constructing a purely wave-mechanical relativistic generalization of Schrödinger’s equation is the emergence of negative-energy solutions. Even if we assume that only positive-energy states are physically meaningful, the resulting theory would be unstable with respect to transitions to negative-energy states. Dirac tried to avoid this problem by writing a first order-wave equation i

! " dψ  + βm ψ, = −i α · ∇ dt

or :

(iγ μ

∂ − m)ψ = 0. ∂xμ

(24)

If, following Dirac we assume anticommutation relations, {γ μ , γ ν } = 2g μν ,l this equation describes a particle of mass m, but admits both positive and negative energy solutions. Dirac’s solution for this problem was to assume that all negative-energy states are occupied, so that thanks to the Pauli esclusion principle a positiveenergy particle cannot jump into a negative-energy state. In this version Dirac’s theory goes far beyond wave mechanics, and is essentially equivalent to the modern field-theoretical formulation, admirably presented in Majorana’s “Teoria simmetrica dell’elettrone e del positrone”. In his paper on particles of arbitrary spin, Majorana tried to construct a fully relativistic wave mechanics that completely avoids the negative-energy states. He was able to show that a necessary condition for the absence of negative-energy states is that the operator β has only positive eigenvalues, and in turn this implies that φ = β 1/2 ψ transforms according to a unitary representation of the Lorentz group. In this paper Majorana displays a complete mastery of the theory of groups. E. Amaldi3 recalls Majorana’s admiration for the work of H. Weyl and E. Wigner on the application of group theory to quantum mechanics. In this paper we can find traces of Weyl’s discussion of the Lie algebra of the rotation group, extended here to the case of the Lorentz group, in particular when he introduces the commutation rules of the group generators as “integrability conditions”. Starting from the commutation relations, Majorana builds the “simplest” infinite-dimensional unitary representations of the Lorentz group, one for integer angular momentum and one for semi-integer angular momentum, and constructs explicitly the corresponding infinite-dimensional representations of the α and β matrices that appear in Dirac’s equation. The Majorana versions of Eq. (24) have solutions that respectively describe particles of arbitrary integer or semi-integer spin angular momentum j , with mass Mj =

3 E.

Amaldi, op. cit.

m j+

1 2

.

(25)

Comment on: Relativistic theory of particles with arbitrary intrinsic angular. . .

99

The avoidance of negative-energy solutions has a very high price: the existence of an infinite sequence of states with increasing spin and decreasing mass. Majorana could not accept this conclusion and leaves open the possibility that his equation could describe a single particle of arbitrary spin j0 by declaring that all solutions with j = j0 are to be considered as “unphysical”. He mentions in particular the possibility of using the integer-spin version of the equation to describe a spinless particle. He, however, realizes that in the presence of interactions it would be difficult to ensure the absence of transitions to states of different spin. In a different context this is the same problem that plagued the wave mechanics interpretation of Dirac’s equation. The disease is made worse here by the presence, briefly mentioned in Majorana’s paper, of what we now call “tachyonic” # solutions, that would correspond to states with imaginary mass ik, with E = ± c2 p2 − k 2 c4 , that exist for |p|  > k and for any value of k. Majorana’s paper was written in the early summer of 1932, just before Anderson announced the discovery of the positive electron, thus sealing the triumph of Dirac’s electron theory. It received little attention, in spite of the brilliant and original results on the unitary representations of the Lorentz group that were rediscovered years later by Wigner. Extensive references to the successive developments in this field can be found in the review of Majorana’s paper by D. M. Fradkin.4 If Majorana’s paper on particles with arbitrary spin had not been totally forgotten, it could be considered a precursor of some of the most actively pursued recent developments in theoretical physics. In the 1960s the study of Regge-poles and the discovery of high-spin hadrons briefly rekindled the interest for theories, based on fields that transform according to infinite-dimensional representations of the Lorentz group, that describe a sequence of particles with increasing spin and increasing mass. These attempts had their highest expression in the dual models of hadrons but, as in the case of Majorana’s theory, were plagued by the existence of tachyonic states that could only be avoided in a space-time with a large number of dimensions. In view of these difficulties and of the emergence of Quantum Chromo Dynamics the application of these ideas to hadron physics was abandoned, but they resurfaced later in modern string theories. Nicola Cabibbo† Università di Roma “La Sapienza”, Rome, Italy

4 Fradkin,

D.M.: Am. J. Phys. 34, 314 (1966)

On Nuclear Theory Ettore Majorana

Summary: We discuss a new interpretation of HEISENBERG’s nuclear theory which leads to a slightly different Hamiltonian function. Accordingly we treat the nuclei statistically. The discovery of the neutron, a heavy and uncharged elementary particle, made it possible to develop a nuclear theory using ideas of quantum mechanics without, however, removing the fundamental difficulties that are connected with β-decay. According to Heisenberg1 , we can think of nuclei for many purposes as consisting h of protons and neutrons, i.e. of particles of almost the same mass, with spin 12 2π and obeying Fermi statistics. The problem is thus reduced to finding a suitable Hamiltonian which holds for this system of particles, and we need a non-relativistic approximation since the speed of the particles is presumably rather small compared c with the speed of light (v ∼ 10 ). In order to find a suitable interaction between the components of the nuclei Heisenberg was guided by an obvious analogy. He treats the neutron as a combination of a proton and an electron, i.e. like a hydrogen atom bound by a process not fully understood by present theories, in such a way that it changes its statistical properties and its spin. He further assumes that there are exchange forces between protons and neutrons similar to those responsible for the molecular binding of H and H+ . In addition to this interaction between protons and neutrons considered essential for the stability of the nucleus there are Coulomb repulsion between protons, van der Waals attraction between neutrons and some sort of electrostatic interaction between protons and neutrons.2

Translated from Über die Kerntheorie “Zeitschrift für Physik”, Bd. 82, 1933, pp. 137–145, by Verena Wehrli-Brink. Reprinted with permission of Elsevier from “Nuclear Forces”, edited by D. M. Brink, Pergamon Press, 1965. (Courtesy of D. M. Brink.) 1 Heisenberg, W.: Z Physik. 77, 1 (1932); 78, 156 (1933) 2 Heisenberg, W.: Z Physik. 80, 587 (1933) © Springer Nature Switzerland AG 2020 L. Cifarelli (ed.), Scientific Papers of Ettore Majorana, https://doi.org/10.1007/978-3-030-23509-3_10

101

102

E. Majorana

One may doubt the validity of this analogy as the theory does not explain the inner structure of the neutron, and the interaction between neutron and proton seems rather big compared with the mass-defect of the neutron as determined by Chadwick. We think, therefore, that it may be quite interesting to find a Hamiltonian very similar to Heisenberg’s which represents in the simplest way the most general and most obvious properties of the nucleus. We shall use a statistical method which should be permissable for determining orders of magnitude. We should also like to point out the exchange forces must have the opposite sign to Heisenberg’s forces because of the criterium we fixed for the Hamiltonian. Therefore, the symmetry characteristics of the eigenfunctions belonging to the normal state and the whole statistical treatment are different from Heisenberg’s. 1. The numerous sources of information we have on nuclear structure, i.e. radioactive decay, artificial decay, anomalous scattering of α-particles, mass-defect measurements etc. seem to indicate that nuclei, unlike atoms, are not uniformly organized. On the contrary, it looks as though nuclei consist of rather independent components which react only on immediate contact, i.e. of some sort of matter with the same properties of size and impenetrability as macroscopic matter. Light and heavy nuclei are built up of this matter and the difference between them depends mainly on their different content of “nuclear matter”. This theory can only be correct if the Coulomb repulsion between the positive components of nuclei is not very important. This certainly holds for rather light nuclei, whereas we have to have a slight correction for heavy nuclei. If we assume that nuclei consist of protons and neutrons we have to formulate the simplest law of interaction between them which will lead if the electrostatic repulsion is negligible, to a constant density for nuclear matter. We have to find three laws of interaction: One between protons, one between protons and neutrons and one between neutrons. We shall assume, however, that only Coulomb’s force acts between each pair of protons. This can be justified to a certain extent by the fact that the classical radius of protons is much smaller than the average distance between the particles in the nucleus. Also, the Coulomb force is not very important for light nuclei and, as they contain almost the same number of neutrons and protons, it seems reasonable to think that a special interaction between protons and neutrons is the main cause of nuclear stability. We assume that there is no noticeable interaction between the neutrons for there is no proof of the contrary.We now have to find a suitable interaction between protons and neutrons. Nuclear structure and the structure of solids and liquids seem to be somewhat similar and it mightbe possible to have an interaction of the same type as between atoms and molecules, i.e. attraction for large distances and strong repulsion for small distances so that the particles do not penetrate each other (see Fig. 1). We would also have to assume repulsive forces between neutrons with small separations in order to obtain the desired ratio between the number of particles and the nuclear volume. Such a solution would be aesthetically unsatisfactory, however, since we would have not only attractive forces of unknown origin between the particles, but also, for short

On Nuclear Theory

103

Fig. 1 Potential energy between two atoms

Fig. 2 Curve of the resonance forces

distances, repulsive forces of enormous magnitude corresponding to a potential of several million volts. We shall, therefore, try to find another solution and introduce as few arbitrary elements as possible. The main problem is this: How can we obtain a density independent of the nuclear mass without obstructing the free movement of the particles by an artificial impenetrability? We must try to find an interaction whose average energy per particle never exceeds a certain limit however great the density. This might occur through a sort of saturation phenomenon more or less analogous to valence saturation. Such an interaction is given, as we shall prove, by 

  Q , q  |J |Q , q  = −δ q  − Q δ q  − Q J (r),

(1)

where r = |q  − Q | and Q and q are the coordinates of a neutron and proton, respectively. The function J (r) is positive, and a possible form of it is shown in Fig. 2. Expression (1) implies that there is an attraction or a repulsion respectively between the neutron and the proton depending on whether the wave-function is approximately symmetrical or anti-symmetrical in the two particles. In order to account for the special stability of the α-particle we shall assume that Q and q in Eq. (1) are only the position coordinates without spin. Thus we find that both neutrons act on each proton in the α-particle instead of only one and vice versa, since we assume a symmetrical function in the position coordinates of all protons and neutrons (which is true only if we neglect the Coulomb energy of the protons). In the α-particle all four particles are in the same state so that it is a closed shell. If we proceed from an α-particle to heavier nuclei we can have no more particles in the same state because of the Pauli principle. Also, the exchange energy (1) is usually large only if a proton and a neutron are in the same state and we may expect, which agrees with experiments, that in heavy nuclei the mass defect per particle is not noticeably bigger than in the α-particle. Let us now compare expression (1) for the interaction between a proton and a neutron with the interaction deduced from the resonance term of Heisenberg’s

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E. Majorana

Hamiltonian by distinguishing between neutrons and protons and by eliminating the troublesome ρ-spin-coordinate. We find an expression similar to (1) which is, however, fundamentally different in two respects: Firstly, in Heisenberg’s expression Q and q stand for all coordinates including the spin. Secondly, Heisenberg assumes the opposite sign for the resonance forces. Statistically this is most important as there is no saturation because of the symmetry character of Heisenberg’s eigenfunctions and repulsive interactions at short distances are necessary.3 We shall now investigate the saturation that leads to the uniform density of the nuclear components found experimentally. 2. In a first approximation we take the eigenfunction of the nucleus as a product of two functions which depend on the coordinates of the n1 neutrons and n2 protons respectively:   ψ = ψN Q1 , 1 , . . . , Qn1 , n1 ψP q1 , σ1 , . . . , qn2 , σn2

(2)

and we assume that ψN and ψP can be obtained by anti-symmetrizing products of individual orthogonal single-particle eigenfunctions: ⎧ ⎪ ⎨ψN = ⎪ ⎩ψP =

√1 n1 ! √1 n2 !



R

R

  ±RψN Q1 , 1 · · · ψNn1 Qn1 , n1 ,  ±RψP (q1 , σ1 ) · · · ψPn2 qn2 , σn2 .

(3)

For many particles the individual-particle wave-function ψ may be identified with free-particle wave-packets. We can show that each proton is subject on the average to the interaction of a small number (one or two) of neutrons and vice versa, and the assumption of free-particle wave-functions introduces a slight error because of polarization effects. This method is, suitable for order-of-magnitude calculations. We have to calculate the mean value of the total energy using the wavefunction (2) and find its minimum. This energy consists of three parts: W = T + E + A,

(4)

where T is the kinetic energy, E the electrostatic energy of the protons and A the exchange energy. We assume that all individual particle states are either free or occupied twice with opposite spin direction. Then, n1 and n2 are even. We also introduce Dirac’s density matrices:

3 I would like to thank Professor Heisenberg very much for being able to see his paper before it was

published. (See 2 )

On Nuclear Theory

105

⎧ n1 2     ⎪ ⎪  ⎪ q = |ρ |q ψNi q  , σi ψ¯ Ni q  , σi , N ⎪ ⎪ ⎨ σ =1 i=1 i

n2 2 ⎪ ⎪     ⎪  ⎪ q |ρ |q ψPi q  , σi ψ¯ Pi q  , σi . = ⎪ P ⎩

(5)

σi =1 i=1

and have 2 ρN = 2ρN ,

ρP2 = 2ρP ,

(6)

where the factor 2 comes from the spin. The eigenvalues of the density matrices are ss 2 ρN =sKsKK , 0

ss 2 ρP =sKsKK . 0

(7)

If the mass M of each particle is approximately the same for neutrons and protons we obtain ) * 1 Tr ρN + ρP p2 , 2M    1 e2   q |ρP |q  dq  dq  + · · · . E= q |ρP |q    2 |q − q | T =

(8) (9)

In Eq. (9) we have left out a term which is essentially the Coulomb exchange energy of the protons. This term has been calculated by Dirac4 and is not very important when there are many particles. Finally we obtain:  A=−

    q |ρN |q  J q  − q   q  |ρP |q  dq  dq  .

(10)

If the number of particles is large ρN and ρP are almost diagonal matrices and even classical functions of p and q (See 4 ). The best relation between the matrices and the classical functions is given by ⎧! v v" ⎪ = ⎪ q − |ρN |q + ⎨ 2 2 ! " ⎪ ⎪ ⎩ q − v |ρP |q + v = 2 2

4 Dirac,

1 h3 1 h3



ρN (p, q)e−



2π i h (p,v)

dp, (11)

ρP (p, q)e

P.A.M.: Proc. Camb. Philos. Soc. 26, 376 (1930)

− 2πh i (p,v)

dq

106

E. Majorana

and by an inversion of the Fourier integrals. If we put Eq. (11) in the above expression we obtain T =

1 2M



ρN (p, q) + ρP (p, q) 2 p dp dq, h3

 e2 1 ρN (p, q)ρP (p , q  ) dp dq dp dq  , E= 6 2 |q − q  | h   ρN (p, q)VN (p, q) ρP (p, q)VP (p, q) A= dp dq = dp dq, h3 h3

(12) (13) (14)

where VN (p, q) and VP (p, q) are the classical functions corresponding to the matrices ⎧    ⎨ q  |VN |q  = − q  |ρP |q  J q  − q  , (15) ⎩q  |V |q  = −q  |ρ |q  J q  − q  . P N We now assume that near a point q the states of low energy are occupied by neutrons as well as by protons. There will be a maximum value of the momentum PN (q) for the neutrons and the protons, and from Eq. (7) it follows that 2, ss ρN (p, q) =sKsKK 0, 2, ss ρP (p, q) =sKsKK 0,

if

p < PN (q),

if

p > PN (q),

(16)

if p < PP (q), (17) if p > PP (q).

We first investigate a limiting case, i.e. a case of very high density when h/pN and h/pP , which are of the order of magnitude of the mutual distance between the particles in the nucleus, are small compared with the range of the resonance forces. We also assume that PN > PP , i.e. that the density of the neutrons is larger than the density of the protons. We note that in the second equation of (15) ρN is almost diagonal and J |q  − q  | can be substituted by J (0) if J (0) is finite. The equation then reads    q |VP |q  = −J (0) q  |ρN |q  , and from this follows VP (p, q) = −J (0)ρN (p, q).

(18)

On Nuclear Theory

107

Fig. 3 Kinetic and potential energy per particle

We put this in Eq. (14) and note that ρN = 2 if ρP (p, q) > 0 and obtain  A = −2J (0)

ρP (p, q) dp dq = −2J (0)n2 . h3

(19)

This means that the binding energy per proton due to the exchange forces is only −2J (0) if the particle density is high and the density of neutrons larger than that of protons. We neglect for the time being the Coulomb repulsion of the protons (which is approximately true for light nuclei) and fix the ratio n1 /n2 , but not the density. Then the potential energy per particle is a certain function of the total density a = a(μ),

μ=

8π  3 P + PP3 . 3h3 N

(20)

2 This vanishes for μ = 0 and approaches a constant value − n12n +n2 J (0) for μ → ∞. This limiting value will reach the minimum −J (0) if n1 = n2 . For intermediate densities the general expression of a(μ) follows from Eqs. (10) and (11) and is

1 a= μ(q)



  ρN (p, q)ρP p , q G p, p dp dp , 6 h

(21)

where G(p, p ) is a function of |p − p | which depends on J (r) in the following way  G p, p =



e−

2π i  h (p−p ,v)

The kinetic energy per particle is t = κμ2/3

J |v|dv.

(22)

108

E. Majorana

and the total energy a + t reaches a minimum for a certain value dependent only on the ratio n1 /n2 (Fig. 3). We obtain a constant density independent of the nuclear mass and thus a nuclear volume and binding energy proportional only to the number of particles, as is found by experiment. We can try to determine the function J (r) in a way that best represents the experimental results. The expression J (r) = λ

e2 , r

for instance, with an arbitrary constant is suitable even though it becomes infinite if r = 0. For great distances, however, it must be modified as it gives an infinite cross section for the collision between protons and neutrons. Also, it seems to provide too small a ratio for the mass defects of the α-particle and the hydrogen isotope. Thus, we have to use an expression with at least two constants, e.g. an exponential function, J (r) = Ae−βr . We shall not follow this up since it has been shown that the first statistical approximation can lead to considerable errors however large the number of particles. For heavy nuclei Coulomb’s force is very important which means that the nuclear extension increases slightly and the density of neurons and protons is no longer constant locally. The exchange binding energy not only depends on the ratio n1 /n2 , it is even slightly smaller than for light nuclei because of the smaller density caused by Coulomb forces. I would like to thank Professor HEISENBERG very much for his advice and numerous discussions. My thanks are due to Professor EHRENFEST for many valuable discussions and also to the Consiglio Nazionale delle Ricerche for making my stay in Leipzig possible.

Comment on: “On nuclear theory”.

109

Comment on: “On nuclear theory”. This is Majorana’s first and only article on nuclear physics5 ; it contains a version of exchange forces between neutrons (n) and protons (p) more elegant and deeper than the one suggested by Heisenberg in July 1932 (Z. Phys., 77, 1 (1932)). Majorana arrived in Leipzig on January 20, 1933, with a research grant from the Italian “Consiglio Nazionale delle Ricerche”. In the application for the grant he had stated that his plan was to study “under the guidance of Professor W. Heisenberg the structure of nuclei and the relativistic formulation of the new quantum theory”. Surely Heisenberg was at that time the best and most original guide to the mysterious world of nuclear structure. Indeed less than four months after Chadwick’s discovery of neutrons (February 1932) Heisenberg had sent to “Zeitschrift für Physik” an article “On the Structure of Atomic Nuclei” which marked the beginning of nuclear theory. Majorana was one of the first physicists to quickly realize its importance. In that paper Heisenberg suggested that “atomic nuclei are composed of protons and neutrons but do not contain electrons. If this is correct it means a very considerable simplification of nuclear theory”. According to Heisenberg a nucleus with mass number A and charge number Z is made up of Z protons and A − Z neutrons. Since neutrons have a mass of the same order as that of the protons (actually a little larger) their motions would be non-relativistic as that of the protons. Hence quantum mechanics would hold for all the components of the nucleus without the need of any electrons. As Heisenberg wrote to Bohr on June 20, 1932 “the basic idea [of my theory] is to shove all fundamental difficulties onto the neutron and to do quantum mechanics in the nucleus”. Obviously this was much to Heisenberg’s satisfaction. Heisenberg’s point of view represented a profound departure from the traditional nuclear model which was supposed to contain A protons and A − Z electrons. Protons and electrons were, before the discovery of the neutron, the only known elementary particles. Heisenberg’s model took some time to be accepted. Indeed more than six months after the experimental discovery of neutrons, Fermi, at the International Conference on Electricity in Paris (1932), was still discussing in details the difficulties posed by the existence of electrons inside the nuclei. As he explained in his talk, the uncertainty principle implied that electrons confined within a sphere of radius 10−13 cm would have an average kinetic energy of more than 60 MeV, much higher than that of any known β-ray. Moreover, according to the proton-electron model the nucleus 14 N was supposed to contain 14 protons and 7 electrons, i.e. an odd number of fermions; therefore, as proved by Ehrenfest and Oppenheimer, its spin should have been half-integer and the nucleus should have followed Fermi’s statistics. Experiment instead showed that the spin was integer and the statistics 5 Majorana’s

unpublished doctoral dissertation (discussed in July 1929), in spite of its title “Sulla meccanica dei Nuclei Radioattivi” (“On the mechanics of the radioactive nuclei”) does not really concern nuclear structure. It is an interesting study of the quantum potential barrier.

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was that of Bose. Fermi concluded: “Ceci semble indiquer encore une fois que les conceptions de la mécanique quantique ordinaire, ne sont pas applicable à l’étude de la dinamique des électrons du noyau atomique” (Rapport 22, p. 798, at the Paris Conference, 1932. The report was certainly written before the conference, but in the published text there was no correction). Rather than discarding the proton-electron model and accepting the existence of a new neutral elementary particle, Fermi seems to have been ready to give up quantum mechanics whose validity, he emphasized, had only been tested at the atomic scale but might well fail at the nuclear scale which is 100,000 times smaller. Relativity and quantum mechanics had taught physicists that the laws of physics may change when the scale of phenomena changes: nuclear physics was going to teach them that “there are more things in heaven and earth, Horatio, than are dreamt of in your philosophy” (W. Shakespeare, Hamlet, I, 5). Heisenberg’s charge exchange force was the only purely quantum-mechanical model of the nucleus when Majorana arrived in Leipzig on January 20, 1933. Did he really go there to study the structure of nuclei under the guidance of Professor Heisenberg, as he had stated in the application for the research grant, or did he arrive there with his own version of exchange forces in mind and, rather than guidance, did he only want to ask the famous Professor his opinion on ideas that he, Majorana, had already developed? There is no record in Majorana’s archives of any work on nuclear structure that he might have done in Rome inspired by Heisenberg’s 1932 paper. However, the speed with which Majorana completed his paper seems to me to favour the second alternative. In a letter of February 18, 1933, to his father, Majorana states that he had already written “an article on the structure of nuclei that Heisenberg liked very much, even though it contains some corrections to his [Heisenberg’s] theory”. Heisenberg’s approval of the paper is mentioned again in a letter to his mother of February 22: “Heisenberg has spoken [in the weekly colloquium] of nuclear theory: he did a lot of advertising for a paper I have written here. We are now on friendly terms, thanks to many scientific discussions and some chess games”. What are the corrections that Majorana made to Heisenberg’s model? Both agree that nuclei are composed exclusively of neutrons and protons. They assume that there are no n-n forces or at most they are negligible; p-p forces are only due to Coulomb interactions and can be neglected in a first approximation at least for light nuclei. Hence the only forces responsible for nuclear stability are those between n and p. To determine these forces, Heisenberg started from the analogy of the (n, p) pair with the molecular ion H+ 2 made up of a hydrogen atom H and a ionized hydrogen H+ = p. As Heitler and London had proved, H+ 2 is held together by the exchange of an electron (H, H+ ) ⇔ (H+ , H). The analogy does not mean that Heisenberg thought that the neutron is a bound state of a proton and an electron. He simply meant that the pair (n, p) is held together by the exchange of “charge” which he defines as the eigenvalue of an operator ρ ζ representing the electric charge, where ρ ζ takes the value +1 on n and −1 on p.

Comment on: “On nuclear theory”.

111

ρ ζ is the third component of a spin-like operator ρ that he calls ρ-spin, which we now call isospin and denote by τ . Neutrons and protons are thus to be thought of as two different states of one single particle that we now call nucleon. As Heisenberg says “each particle in a nucleus is characterised by 5 quantities: the three position coordinates (x, y, z), the spin σz along the z-axis and a fifth number ρ ζ which can be ±1”. In his rapporteur talk at the Solvay Conference on Nuclear Physics in Brussels (October 1933) Heisenberg represented graphically the charge exchange operator Pc between n and p with the following diagram x2 x1 x2 Pc x1 ← • ◦ → ⇔ ← ◦ • →, where • represents a proton at x1 and ◦ a neutron at x2 ; the arrows represent the orientation of the spins. The operator Pc reverses the sign of the eigenvalue of ρ ζ but not that of the spin operator. Majorana instead considered the analogy of the pair (n, p) with H+ 2 of dubious significance probably because he thought it implied necessarily that the neutron is a composite particle, n = (p, e). Moreover, he disliked Heisenberg’s abstract conception of charge as the eigenvalue of ρ ζ which he termed “troublesome” (German: umbequem; Italian: incomodo) and insisted in treating n and p as different particles. He suggested that the n-p force is due to the exchange of the space coordinates x1 , x2 , represented by an operator Px : x1 x2 Px x1 x2 ← • ◦ → ⇔ ◦ → ← •. Owing to some “saturation phenomenon”, this exchange force leads to the independence of nuclear density of the total mass. One easily verifies that Majorana and Heisenberg’s exchange are related by Px = Ps Pc , where Ps is the operator that reverses the sign of the ordinary spin. Thus the exchange of space coordinates is equivalent to exchanging spin and ρ-spin coordinates. Majorana’s exchange prevents the collapse of the nucleus without the need of any repulsive force at short distance provided one inverts the sign of Heisenberg’s potential function J (r). Majorana motivates his own model of exchange forces in preference to Heisenberg’s one by noticing that it is compatible with the constancy of the nuclear density and that it ensures the “particular stability of the α-particle” arising from a “saturation phenomenon more or less analogous to valence saturation”. Indeed in the α-particle “both neutrons act on each proton instead of only one [as in Heisenberg’s model] and vice-versa since we can assume a symmetrical function in the position

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coordinates of all protons and neutrons (which is true if we neglect the Coulomb energy of the protons). In the α-particle all four particles are in the same state so that it is a closed shell” (German: “so das es eine abgeschlossene Schale in höhere Sinne as das Heliumatom ist”; abgeschlossene Schale means in the language of spin and isospin that the shell is closed when both spin and isospin functions are anti-symmetrical). It was this saturation of nuclear forces at A = 4 that immediately convinced Heisenberg that Majorana’s model was preferable to his own. Heisenberg’s talk at the Solvay Conference on nuclei in October 1933 was the best advertisement for the work of his younger colleague. In their papers Heisenberg and Majorana were only concerned with nuclear stability, i.e. with what we now call strong interactions. Therefore they did not solve the mystery of the origin of β-rays, i.e. of the origin of the electron emitted by radioactive nuclei. The problem was finally solved by Fermi in his celebrated 1934 article (Z. Phys. 88, 161 (1934)) which marks the beginning of the theory of weak interactions. After 1933 the study of nuclear forces advanced very rapidly. The n-n forces were soon found to be equal to the n-p forces and to the p-p forces, apart from the Coulomb interaction. The charge independence of the strong interactions was thus established, and Heisenberg’s isospin formalism was recognised (pace to Majorana) to be the correct way to describe it. In 1937 Wigner noticed that a combination of Majorana forces with his own forces (that depend only upon the space distance) are not only spin and isospin independent but are invariant under a larger symmetry group, SU (4). This is the so-called super-multiplets theory which leads to a degeneracy of nuclear levels in very rough agreement with empirical evidence for light nuclei. After his 1933 paper Majorana seems to have lost interest in nuclear physics or in physics all together, perhaps owing to poor health. I wish to thank Nadia ROBOTTI and Francesco GUERRA for many enlightening discussions and for letting me read (and make ample use of) the preliminary version of a historical paper on the early developments of nuclear physics in Rome.

L. A. RADICATI DI BROZOLO† Scuola Normale Superiore, Pisa, Italy

A Symmetric Theory of Electrons and Positrons Ettore Majorana

Summary: It is shown that it is possible to achieve complete formal symmetrization in the electron and proton quantum theory by means of a new quantization process. The meaning of DIRAC equations is somewhat modified and there is no longer any reason to speak of negative-energy states nor to assume, for any other types of particles, especially neutral ones, the existence of antiparticles, corresponding to the “holes” of negative energy. The interpretation of the so-called “negative energy states” proposed by Dirac1 leads, as is well known, to a substantially symmetric description of electrons and positrons. The substantial symmetry of the formalism consists precisely in that the theory itself gives completely symmetric results, whenever it is possible to apply it while overcoming divergence problems. The prescriptions needed to cast the theory into a symmetric form, in conformity with its content, are however not entirely satisfactory, because one always starts from an asymmetric form or because symmetric results are obtained only after one applies appropriate procedures, such as the cancellation of divergent constants, that one should possibly avoid. For these reasons, we have attempted a new approach, which leads more directly to the desired result. In the case of electrons and positrons, we may anticipate only a formal progress; but we consider it important, for possible extensions by analogy, that the very notion of negative energy states can be avoided. We shall see, in fact, that it is perfectly,

Translated from Teoria simmetrica dell’elettrone e del positrone, “Il Nuovo Cimento”, vol. 14, 1937, pp. 171–184, by Luciano Maiani. Originally published in, and reprinted with permission from, Soryushiron Kenkyu, vol. 63, issue 3, 1981, pp. 149–162. (Courtesy of L. Maiani). The present translation has been revised by the Editor with the addition of the summary which was missing in “Soryushiron Kenkyu”. 1 Dirac, P.A.M.: Proc. Camb. Philos. Soc. 30, 150 (1924). See also Heinsenberg, W.: Z Physik. 90, 209 (1934) © Springer Nature Switzerland AG 2020 L. Cifarelli (ed.), Scientific Papers of Ettore Majorana, https://doi.org/10.1007/978-3-030-23509-3_11

113

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E. Majorana

and most naturally, possible to formulate a theory of elementary neutral particles which do not have negative (energy) states. 1. It is well known that quantum electrodynamics can be deduced by quantizing a system of equations which include the Dirac wave equations for the electron and the Maxwell equations. In the latter, the charge density and current are represented by certain expressions containing the electron wave function. The form given to these expressions adds, in reality, something new because it allows to derive the asymmetry with respect to the sign of the electric charge, an asymmetry which is not present in the Dirac equations. These expressions can be derived directly from a variational principle, which yields the Maxwell and the Dirac equations at the same time. Therefore, our first problem will be to examine the foundation of the variational principle itself, and the possibility of replacing it with a more appropriate one. The Maxwell-Dirac equations contain quantities of two different types. On the one side, we have the electromagnetic potentials, which can be given a classical interpretation, within the limits posed by the correspondence principle. On the other side, there are the matter waves, which represent particles obeying the Fermi Statistic, and which have only a quantum interpretation. In this respect, it seems little satisfactory that the equations as well as the whole quantization procedure have to be derived from a variational principle which can be given only a classical interpretation. It seems more natural to search for a generalization of the variational method, such that the variables appearing in the Lagrange function assume, from the very beginning, their final significance, and, therefore, represent not necessarily commuting quantities. This is the approach we shall follow. This approach is most important for fields obeying the Fermi statistics; reasons of simplicity may indicate, on the other hand, that nothing has to be added to the old method in the case of the electromagnetic field. In fact, we shall not perform a systematic study of all the logical possibilities offered by the new point of view we are adopting. Rather, we limit ourselves to the description of a quantization procedure for the matterwaves, which is the only important case for applications, at present; this method appears as a natural generalization of the Jordan-Wigner method,2 and it allows not only to cast the electron-positron theory into a symmetric form, but also to construct an essentially new theory for particles not endowed with an electric charge (neutrons and the hypothetical neutrinos). Even though it is perhaps not yet possible to ask experiments to decide between the new theory and a simple extension of the Dirac equations to neutral particles, one should keep in mind that the new theory introduces a smaller number of hypothetical entities, in this yet unexplored field. Leaving to the reader the obvious extension of the formulae to the continuous systems, which we shall consider later on, we illustrate in the following the quantization procedure for discrete systems. Let a physical system be described by the real variables q1 q2 , . . . , qn (symmetric, Hermitian matrices). We define a Lagrange function: 2 Jordan,

P., Wigner, E.: Z Physik. 47, 631 (1928)

A Symmetric Theory of Electrons and Positrons

L=i

115

 Ars qr q˙s + Brs qr qs ,

(1)

r,s

and set:  Ldt = 0,

δ

(2)

we understand that Ars and Brs are ordinary real numbers, constant the former and, eventually, time-dependent the latter, which obey the relations: Ars = Asr ;

Brs = −Bsr ,

(3)

and, furthermore, with detArs  = 0. If the q’s were ordinary, commuting, variables, the variational principle (2) would have no meaning because it would be identically satisfied. In the case of noncommuting variables, Eq. (2) implies the vanishing, at any time, of the Hermitian matrix:      i Ars q˙s + Brs qs δqr = 0, Ars q˙s + Brs qs − δqr r

s

s

+ for arbitrary variations δqr . This is only possible if the expression s (Ars q˙s + Brs qs ) are multiple of the unit matrix so that, after some appropriate modification of the variational principle (2) (e.g. by requiring the sum of the diagonal terms in the above expressions to vanish3 ) we may consider the following equations of motion:  Ars q˙s + Brs qs = 0

r = 1, 2, . . . , n.

(4)

s

We now show that these equations can be derived, following the usual procedure: q˙r = −

2π i  qr H − H qr h

from the Hamiltonian: H = −i



Brs qr qs ,

(5)

r,s

(whose exact form will be better justified in the following) provided we assume suitable anticommutation relations for the qr . Substituting in Eq. (4) the successive

3 The

physical application which will be illustrated later on suggests the more rigorous restriction that, in any linear combination of qr and q˙r to any given eigenvalue there corresponds another one, equal in absolute value and opposite in sign.

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E. Majorana

equations, one finds:

Brs qs =

s

 2π Ars Blm qs ql qm − ql qm qs = h s,l,m

)  * 2π = Ars Blm qs ql + ql qs qm − ql qs qm + qm qs = h s,l,m    ,     2π = Blm qm Ars qs ql + ql qs + Ars qs ql + ql qs qm , h s s lm

so that it suffices to set: s

 h δrl , Ars qs ql + ql qs = 4π

(6)

for Eqs. (4) to be satisfied. Denoting by A−1 rs  the inverse matrix of Ars , Eq. (6) can be written as: qr qs + qs qr =

h −1 A . 4π rs

(6 )

In the special case where A is reduced to the diagonal form: Ars = ar δrs , we have therefore: qr qs + qs qr =

h δrs . 4π ar

(7)

We shall now apply the present scheme to the Dirac equations. 2. It is well known that one can eliminate the imaginary unit from the Dirac equations with no external field: 

 W + (α, p) + βmc ψ = 0, c

(8)

with an appropriate choice of the operators α and β (and this can be done in a relativistically invariant fashion). We shall, in fact, refer to a system of intrinsic coordinates such as to make Eqs. (8) real, keeping explicitly in mind that the formulae we shall derive are not valid, without suitable modification, in a more general coordinate system. Denoting, as usual, with σx , σy , σz and ρ1 , ρ2 , ρ3 two independent sets of Pauli matrices, we set:

A Symmetric Theory of Electrons and Positrons

αx = ρ1 σx ;

αy = ρ3 ;

117

αz = ρ1 σz ;

Dividing Eqs. (9) by − 2πh i and defining β  = −iβ, μ = equations: 

β = −ρ1 σy . 2π mc h ,

(9)

we obtain the real

 1 ∂ − (α, grad) + β  μ ψ = 0. c ∂t

(8 )

As a consequence, Eqs. (8) separate into two independent sets of equations, one for the real and one for the imaginary part of ψ. We set ψ = U + iV and consider the real equations (8 ) as acting on U : 

 1 ∂ − (α, grad) + β  μ U = 0. c ∂t

(10)

The latter equations, by themselves,4 i.e. without the similar equations involving V , can be derived from the variational principle previously illustrated and quantized, as indicated above. Nothing similar could be done with elementary methods. Equation (10) can be obtained from the variational principle:  δ

i

  hc ∗ 1 ∂ U − (α, grad) + β  μ U dqdt = 0. 2π c ∂t

(11)

It is easy to verify that the conditions (3), in their natural extension to a continuous system, are obeyed. Following Eqs. (7) the anticommutation relations hold:    1 Ui (q)Uk q  + Uk q  Ui (q) = δik δ q − q  , 2

(12)

while the energy, according to (5) is:  H =

) * U ∗ − c(α, p) − βmc2 U dq.

(13)

The relativistic invariance of (12) and (13) does not require a separate demonstration. If one adds to these equations the analogous ones involving V , as well as the anticommutation relations: Ur (q)Vs (q  ) + Vs (q  )Ur (q) = 0, one reobtains the usual Jordan-Wigner scheme, applied to the Dirac equations without external field.

4 The

behaviour of U under space reflection can be conveniently defined taking into account that a simultaneous change of sign of Ur has no physical significance, as already implied by other reasons. In our scheme: U  (q) = RU (−q) with R = iρ1 σy and R 2 = −1. Similarly, for a time reflection: U  (q, t) = iρ2 U (q, −t).

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It is remarkable, however, that the part of the formalism which refers to U (or V ) can be considered, in itself, as the theoretical descriptions of some material system, in conformity with the general methods of quantum mechanics. The fact that the reduced formalism cannot be applied to the description of positive and negative electrons may well be attributed to the presence of the electric charge, and it does not invalidate the statement that, at the present level of knowledge, Eqs. (12) and (13) constitute the simplest theoretical representation of neutral particles. The advantage, with respect to the elementary interpretation of the Dirac equation, is that there is now no need to assume the existence of antineutrons or antineutrinos (as we shall see shortly). The latter particles are indeed introduced in the theory of positive β-ray emission5 ; the theory, however, can be obviously modified so that the β-emission, both positive and negative, is always accompanied by the emission of a neutrino. Considering the interest that the above-mentioned hypothesis gives to Eqs. (12) and (13), it seems useful to examine their meaning more closely. To this aim, we developed U , inside a cube of side L, over the system of periodical functions: fγ (q) =  γ = γ x , γy , γz ;

γx =

1 2π i(γ ,q) e , L3/2 n1 , L

γy =

(14) n2 , L

γz =

n3 ; L

n1 , n2 , n3 = 0, ±1, ±2, . . . setting: Ur (q) =



ar (γ )fγ (q).

(15)

γ

As a consequence of the reality of U , we have: ar (γ ) = a¯ r (−γ ).

(16)

In the general case, γ = 0, it follows from (12) that: ⎧ 1 ⎪ ⎪ ⎪ ⎨ar (γ )a¯ s (γ ) + a¯ s (γ )ar (γ ) = 2 δrs , ar (γ )as (γ ) + as (γ )ar (γ ) = 0, ⎪ ⎪ ⎪ ⎩a¯ (γ )a¯ (γ ) + a¯ (γ )a¯ (γ ) = 0. r

s

s

(17)

r

Furthermore, these quantities anticommute with a(γ  ) and a(γ ¯  ), when γ  differs both from γ and from −γ .

5 Wick,

G.: Rend. Accad. Lincei. 21, 170 (1935)

A Symmetric Theory of Electrons and Positrons

119

The expression of the energy resulting from (13) is: H =

4 )

* − hc(γ , α rs ) − mc2 β rs a¯ r (γ )as (γ ).

(18)

γ r,s=1

The x component of the linear momentum corresponds to the unit translation h along x, up to the factor 2π i, as usual:  Mx =

U ∗ px U dq =

γ

hγx a¯ r (γ )ak (γ ),

(19)

r=1

and similarly for My and Mz . For any value of γ we have in (18) a Hermitian form which has, notoriously, # two positive and two negative eigenvalues, all equal in absolute value to c m2 c 2 + h2 γ 2 . We can thus replace (18) by: H=

. % c m2 c2 + h2 γ 2 b¯1 (γ )b1 (γ ) + b¯2 (γ )b2 (γ ) − b¯3 (γ )b3 (γ ) − b¯4 (γ )b4 (γ ) γ

(18 )

br being appropriate linear combinations of the ar , obtained by a unitary transformation. Furthermore, it follows from (16) that br (γ ) are linearly related to b¯r (−γ ). The Hermitian form (18), for a given value of γ , remains invariant under the exchange of γ with γ , as a consequence of (16) and (17). From this, and keeping again (17) into account, it follows that we can set: b3 (γ ) = b¯1 (−γ );

b4 (γ ) = b¯2 (−γ ).

(20)

We introduce, for simplicity, the new variables: B1 (γ ) =

√

B2 (γ ) =

2b1 (γ );

√ 2b2 (γ ),

(21)

and we obtain:  2  % 1 2 2 2 2 nr (γ ) − , H = c m c +h γ 2 γ

(22)

r=1

Mx =

γ

hγx

2  r=1

 1 nr (γ ) − , 2

(23)

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E. Majorana

where we have set: 0 rr nr (γ ) = B¯ r (γ )Br (γ ) =rLrLL L 1 considering, furthermore, that the following relations hold: ⎧ ⎪ ¯  ¯  ⎪ ⎨Br (γ )Bs (γ ) + Bs (γ )Br (γ ) = δγ γ  δrs , Br (γ )Bs (γ  ) + Bs (γ  )Br (γ ) = 0, ⎪ ⎪ ⎩B¯ (γ )B¯ (γ  ) + B¯ (γ  )B¯ (γ ) = 0, r s s r

(24)

as it would follow formally, in the Jordan-Wigner scheme, for the coefficients in the development of a two-component matter-wave. The preceding formulae are entirely analogous to those obtained in the quantization of the Maxwell equations, except for the different statistic. In the place of massless quanta, we have particles with a finite mass and also for them we have two available polarization states. In the present case, as in the case of the electromagnetic radiation, the half-quanta of rest energy and momentum are present, except that they appear with the opposite sign, in apparent connection with the different statistic. They do not constitute a specific difficulty, and they must be considered simply as additive constants, with no physical significance. Similarly to the case of light quanta, it is not possible to describe with eigenfunctions the states of such particles. In the present case, however, the presence of a rest mass allows one to consider the non-relativistic approximation, where all the notions of elementary quantum mechanics apply, obviously. The non-relativistic approximation may be useful primarily in the case of the heavy particles (neutrons). The simplest way to go to the configuration space representation is to associate the following plane wave to each oscillator: 1 2π i(γ ,q) e δσ σr , L3/2

(r = 1, 2),

corresponding to the same value of the momentum, and with two possible polarization states, to keep into account the multiplicity of oscillators. We can go further, and describe not a simple particle, but a system with an indefinite number of particles with the two-valued, complex eigenfunction  = (1 , 2 ), according to the Jordan-Wigner method. It is sufficient to set: ⎧ 1 ⎪ 1 (q) = e2π i(γ ,q) B1 (γ ), ⎪ ⎪ 3/2 ⎨ L γ 1 ⎪ ⎪ e2π i(γ ,q) B2 (γ ). ⎪ ⎩2 (q) = L3/2 γ

(25)

A Symmetric Theory of Electrons and Positrons

121

 In the non-relativistic approximation (|γ |  mc h ) the constants br (γ ) in (18 ) are linear combinations of ar (γ ), with γ -independent coefficients. The latter coefficients depend only upon the elements of γ and, according to (9), we have:

b1 (γ ) =

a3 (γ ) − ia2 (γ ) ; √ 2

b3 (γ ) =

a3 (γ ) + ia2 (γ ) , √ 2

b2 (γ ) =

a4 (γ ) + ia1 (γ ) ; √ 2

b4 (γ ) =

a4 (γ ) − ia1 (γ ) , √ 2

which satisfy also Eq. (20), as a consequence of (16). From Eqs. (15) and (25) we have, in the non-relativistic approximation:  1 (q) = U3 (q) − iU2 (q), 2 (q) = U4 (q) + iU1 (q).

(26)

√ On the purely formal side, we note that  = (1 , 2 ) coincides, up to a 2 factor, with the pair of large eigenfunctions of Eqs. (10), when interpreted in the usual way, that is with no reality restriction. 1−ρ σ To prove this, it is enough to verify that the transformation ψ = √2 y U allows 2 one to go from the scheme (9) to the usual Dirac scheme (α = ρ1 σ ; β = ρ3 ), so that, effectively: 1 ψ3 = √ 1 , 2

1 ψ4 = √ 2 ; 2

notoriously, in the latter scheme, ψ3 and ψ4 are the large components. This relation clarifies the transformation law of  with respect to space rotations, but it has no meaning, obviously, with respect to general Lorentz transformations. The existence of simple formulae such as (26) could lead one to suspect that, to a certain extent, the passage through plane waves is superfluous. As a matter of fact, such a passage is conceptually needed to obtain the cancellation of the rest-energy half-quanta. In fact, after such cancellation, the method of the energy is naturally given by:  H =

1 2 2 ˜ p dq,  mc + 2m

(27)

to first approximation, and it differs in an essential way from (13). 3. As we have already said, the scheme (12) is not sufficient to describe charged particles; but, upon the introduction of a further quadruplet of real quantities Vr , analogous to the Ur , one re-obtains the usual electrodynamics, in a form symmetric with respect to the electron and positron. We consider, therefore, two sets of

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E. Majorana

real quantities, representing the matter particles and the electromagnetic field, respectively. Quantities of the first kind are to be interpreted according to the scheme described in Sect. 1. Quantities of the second kind, i.e. the electromagnetic potentials ϕ and A = (Ax , Ay , Az ), can be interpreted as classical quantities, and have to be quantized according to the Heisenberg rule, based on the correspondence principle. The Maxwell and Dirac equations (with the above-mentioned restriction for the latter) can be obtained from a variational principle:  δ

Ldqdt = 0

L being the sum of three terms: L = L + L + L . The first term refers to the matter wave:  /  hc  ∗ 1 ∂  L =i (28) − (α, grad) + β μ U + U 2π c ∂t   0 ∗ 1 ∂  − (α, grad) + β μ V , +V c ∂t while the second describes the radiation field, which we suppose to be quantized according to the method of Fermi6 : 1 1  2 E − H2 − L = 8π 8π 



2 1 ϕ˙ + div A . c

(29)

We must therefore impose the auxiliary condition 1 ϕ˙ + div A = 0. c

(30)

The expression given in (29) differs from the one used by Fermi, but for integrable terms only. It leads to a definition of the momentum P0 , conjugate to ϕ, such as to allow one to eliminate immediately one of the two longitudinal waves, without having to go through the plane-wave development; in this respect, it is completely immaterial whether the second term in the expression (29) for L is multiplied by an arbitrary, non-vanishing constant. As for L it must be so chosen that ψ = U + iV obeys the Dirac equation (8) completed with the external field, i.e. the equation:

6 Fermi,

E.: Rend. Accad. Lincei. 9, 881 (1929)

A Symmetric Theory of Electrons and Positrons



123

 ! W e e " + ϕ + α, p + A + βmc ψ = 0. c c c

In practice, this requirement leads to: L = ieU ∗ [ϕ + (α, A)]V − ieV ∗ [ϕ + (α, A)]U.

(31)

Upon variation of the electromagnetic potentials, we obtain the following expressions for the charge and current densities: ⎧ ˜ − ψ ∗ ψ¯ ψψ ⎪ ⎪ , ⎨ρ = −ie(U ∗ V − V ∗ U ) = −e 2 ∗ ¯ ⎪ ˜ ⎪ ⎩I = ie(U ∗ αV − V ∗ αU ) = e ψαψ − ψ α ψ . 2

(32)

These expressions differ from the usual ones for infinite constants only. The cancellation of such infinite constants is required by the symmetry of the theory, which is already implicit in the form chosen for the variational principle; in fact, the exchange of Ur and Vr , which appear symmetrically in L , is equivalent to changing sign to the electric charge. U and V obey the anticommutation relations: 1 δ(q − q  )δrs , 2 1 Vr (q)Vs (q  ) + Vs (q  )Vr (q) = δ(q − q  )δrs , 2

Ur (q)Us (q  ) + Us (q  )Ur (q) =

Ur (q)Vs (q  ) + Vs (q  )Ur (q) = 0, which are equivalent to the usual Jordan-Wigner scheme, if we set ψ = U + iV . The electromagnetic potentials ϕ, Ax , Ay , Az , on the other side, obey to the usual commutation relations with their conjugate momenta, e.g. P0 (q)ϕ(q  ) − ϕ(q  )P0 (q) = 2πh i δ(q − q  ), with:

⎧ 1 1 ⎪ ⎪ ⎨P0 = − 4π c c ϕ˙ + div A , ⎪ 1 ⎪ ⎩P = − 1 E ; Ey ; Py = − x x 4π c 4π c

(33) 1 Ez . Pz = − 4π c

The energy is made up of three terms: H = H  + H  + H  is derived from L , according to the rules already illustrated. The second term is obtained from 1 the classical rules: H  = [P0 ϕ˙ + (P , A) − L ]dq, where P = (Px , Py , Pz ). As for H 1, it can be obtained from L , following either methods (in our case H  = − L dq) as it must be, since L is a function of both the matter and

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the electromagnetic field variables. This, by the way, proves the necessity of the ansatz (5). The continuity equation (30), is obeyed at any time, if it holds initially together with the divergence equation div E = 4πρ. It follows from (33) that the kinematics defined by the exchange rules has to be reduced by the use of the equations: ⎧ ⎪ ⎨P0 (q) = 0, 1 ⎪ ⎩div P + ρ = 0, c

(34)

and therefore by assigning fixed values to two field quantities, with the corresponding indeterminacy in the conjugate variables. The first of (34) implies therefore, the elimination of P0 and ϕ from the expression of H . The elimination is easily obtained by making use of (33), and one arrives, in this way, at the expression: 0  / 2 * ) 1  rot A dq. ψ˜ − c(α, p) − βmc2 ψ − (A, I ) + 2π cP 2 + 8π (35) As for relativistic invariance, we note that ψ = U +iV obeys the Dirac equations, and that the Maxwell equations also hold, with charge and current densities which obey the relativistic transformation law. These two facts guarantee that the complete proof of the invariance of the theory is already implicit in the results of Heisenberg and Pauli.7 We turn now to the interpretation of the formalism. H =

4. Upon developing the U in the basis of the periodical functions considered before, and similarly for the V , we find as the obvious extension of (22), and after cancellation of the rest-energy half-quanta: 2 % ) * 2 2 2 2 H = B¯ r (γ )Br (γ ) + B¯ r (γ )Br (γ ) , c m c +h γ 

γ

(36)

r=1

where Br and Br refer to the development of U and V , respectively; Br and Br and their conjugate variables obey the usual anticommutation relations. If, for each value of γ , we introduce four appropriate spin functions ξs (γ ) (s = 1, 2, 3, 4) assuming four complex values and forming a unitary system, we can set: ⎧ 3 1 2 ⎪ U=√ B1 (γ )ξ1 (γ )+B2 (γ )ξ2 (γ )+B¯ 1 (−γ )ξ3 (γ )+B¯ 2 (−γ )ξ4 (γ ) fγ (q), ⎪ ⎪ ⎨ 2 γ 2 3 1 ⎪ ⎪ B1 (γ )ξ1 (γ )+B2 (γ )ξ2 (γ )+B¯ 1 (−γ )ξ3 (γ )+B¯ 2 (−γ )ξ4 (γ ) fγ (q), ⎪ ⎩V = √ 2 γ (37) 7 Heisenberg,

W., Pauli, W.: Z Physik. 56, 1 (1929); 59, 168 (1930)

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the following relations being, furthermore, satisfied:  ξ3 (γ ) = ξ¯1 (−γ ), ξ4 (γ ) = ξ¯2 (−γ ).

(38)

It follows from the expression (32) for the electric charge density that the total charge is given by: Q=− =−

ie 2



* ) ∗ U (q)V (q) − V ∗ (q)U (q) dq =

(39)

2 * ie ) Br (γ )B¯ r (γ ) + B¯ r (γ )Br (γ ) − B¯ r (γ )Br (γ ) − Br (γ )B¯ r (γ ) . 2 γ r=1

If we set: Crel =

Br + iBr ; √ 2

pos

Cr

=

Br − iBr √ 2

(40)

we can transform the expressions (36) and (39) for the energy and charge into the form: H =

2 ! " % pos pos C¯ rel Crel + C¯ r Cr c m2 c 2 + h2 γ 2 γ

(41)

r=1

 2  1 1 pos pos el el ¯ ¯ − C r Cr − + Cr Cr − = Q=e 2 2 γ

(42)

r=1

=e

2 ! γ

" pos pos − C¯ rel Crel + C¯ r Cr .

r=1

The elimination of the half-quanta of electricity is, therefore, automatic, provided we perform the internal sum first. Equations (41) ans (42) represent a set of oscillators which are equivalent to a double system of particles obeying the Fermi pos statistic, with rest mass m and charge ±e; the variables Cr refer to positrons and el the Cr to electrons. The elimination of the longitudinal electric field by the second equation in (34) is somewhat different in a symmetric theory because it is not possible to cast ρ, as it results from (32), in a diagonal form. The result of the elimination is well known in ordinary electrodynamics (though partially illusory because of convergence ˜ but it is equally known if one starts from ρ = eψ ∗ ψ¯ difficulties) where ρ = −eψψ; because the latter position is fully equivalent to exchange the role of electron and positron, considering the latter as a real particle and the former as a positron “hole”.

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It seems plausible that those matrix elements which maintain the same form in the two opposite theories remain the same in the symmetric theory. We thus assume to have already eliminated the irrotational part of A and P . The expression (35) for H is modified in two ways: first by assuming that A and P in this expression represent only the divergence free part of such vectors; secondly by adding a term which represents the electrostatic energy. The latter term takes a different form in the ordinary theory (electron-electron hole) and in the opposite theory. Keeping the interaction of each particle with itself, one has in the first theory: Hels

e2 = 2



  1 ˜ ψ(q)ψ(q) ψ˜ q  ψ q  dqdq  , |q − q  |

while in the second theory: Hels =

e2 2



  1 ∗  ¯ ¯ ψ ∗ (q)ψ(q)ψ q ψ q  dqdq  .  |q − q |

Using (37) and (40) one can express the electrostatic energy as a function of the C’s. The only terms which have given rise to physical applications are identical in the two theories: they are those which can be interpreted, from the particle viewpoint, as repulsion or attraction between distinct particles of the same or of the opposite type. For what concerns the interaction with the radiation field, the only difference between the symmetric and the ordinary theory lies in the cancellation of undetermined constants, relative to the single oscillators, in the expression for the current density; again the formulae of interest for the applications remain unchanged.

Comment on: A symmetric theory of electrons and positrons

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Comment on: “A symmetric theory of electrons and positrons”. Written in 1937, 1 year before his tragic disappearence, in a concise and elegant Italian language, this article probably represents the best long-lasting contribution of Ettore Majorana to particle physics. The article tackles the problem of formulating the Dirac theory without the cumbersome sea of negative-energy states. In the usual formulation one would start from a highly asymmetric situation, to discover only at the end that there is a perfect symmetry between electrons and positrons. The symmetry is so little evident that Dirac himself tried at first to identify the positively charged particles, the holes, with protons! Referring to the usual formulation, M. notes that: the prescriptions needed to cast the theory into a symmetric form, in conformity to its content, are however not entirely satisfactory because one always starts from an asymmetric form or because symmetric results are obtained after one applies appropriate procedures, such as the cancellation of divergent constants, that one should possibly avoid. For these reasons, we have attempted a new approach, which leads more directly to the desired result.

From these premises, M. formulates a field theory based on anticommuting variables, hence without classical interpretation, and derives the Dirac equation from a variational principle. The electron is represented by a complex field, which can be divided into Hermitian and anti-Hermitian components. However, in the representation where the Dirac matrices are all imaginary (henceforth called the M. representation) each component “can be considered, in itself, as the theoretical description of some material system, in conformity with the general methods of quantum mechanics.” M. promptly recognizes, of course, that we cannot avoid introducing both components for the electron, which admits a conserved charge. But the simplicity of the scheme leads him to speculate that his theory can find application to the case of electrically neutral particles. The advantage. . . is that there is no reason now to infer the existence of antineutrons or antineutrinos. The latter particles are introduced in the theory of positive β-ray emission; the theory, however, can be obviously modified so that the β-emission, both positive and negative, is always accompanied by the emission of a neutrino.

M. refers here to the theory of positive β-rays formulated two years before, in Rome, by Giancarlo Wick. Unexpectedly, from a reformulation of the Dirac theory, a novel physical possibility emerges, which has since been the object of theoretical and experimental scrutiny. We have not succeeded, yet, to find a definitive answer to M.’s proposal. The M. neutrino has met with alternating fortunes, somehow superimposing to the “two-component neutrino” theory, formulated by Herman Weyl a few years before, in 1929. It is a fact that the neutral particles emitted in negative or positive β-decays behave differently: in the interaction with atomic nuclei, the former particles produce invariably positrons, the latter electrons. However, with a V −A interaction,

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we can associate the different behaviour to the different helicity of the emitted neutral particle. Since helicity is strictly conserved for massless particles, deviations from this pattern are due to terms in the amplitude of the order of the ratio: (neutrino mass)/(neutrino energy), which is unobservably small in all neutrino-induced reactions. The Majorana nature of the neutrino can be tested in the so-called neutrinoless double β-decays. These are second-order processes in the Fermi theory, whereby a virtual neutrino of positive elicity is emitted, together with an electron, and reabsorbed as if it were a neutrino (negative elicity), with emission of a second electron. The overall process: N ∗ → N + 2e, violates lepton number conservation and is proportional to the Majorana mass of the neutrino. The observation of neutrinoless double β-decay would be an evident proof that: “there is no reason to infer. . . the existence of. . . antineutrinos.” Long considered as an exotic possibility, the M. neutrino has emerged, in our times, as the most natural explanation for the surprisingly small value of neutrino masses. In addition, the non-conservation of a lepton number, L, leads to speculate that the decay of supermassive M. neutrinos in the primordial Universe may have given rise to an asymmetry in L, transformed in the presently observed baryon number asymmetry by virtue of B − L conservation. Several laboratories around the world host experiments to detect neutrinoless double β-decay, thus far with no success. A new-generation experiment, CUORE, is in preparation in the INFN laboratory below the Gran Sasso Mountain in Central Italy. With dimensions never reached before, CUORE should put very stringent limits to the process. . . or maybe observe it. I like to think that the answer to the question posed by Majorana more than half a century ago may be found precisely in our country giving, at the same time, a possible explanation to the dominance of matter over antimatter in our Universe, on which our very same existence depends. LUCIANO MAIANI Università di Roma “La Sapienza”, Rome, Italy

The Value of Statistical Laws in Physics and Social Sciences∗ Ettore Majorana

Summary: The deterministic conception of nature implies in itself a real cause of weakness in the irremediable contradiction that it faces with the most certain data of our consciousness. G. SOREL attempted to compose this disagreement with the distinction between artificial nature and natural nature (this last acausal), but in this way he denied the unity of science. On the other hand, the formal analogy between the statistical laws of physics and the ones of social sciences credited the opinion that human facts also undergo a rigid determinism. It is therefore important that quantum mechanics principles have brought to recognize the statistical character of basic laws of elementary processes, in addition to a certain absence of objectiveness in the description of phenomena. This conclusion has made essential the analogy between physics and social sciences, between which it turned out an identity of value and method.

∗ This

article of Ettore Majorana—the great theoretical physicist of Naples University who went missing on 25 March 1938—was originally written for a sociology journal. It was not published perhaps due to the reticence that the author had in interacting with others. Reticence that convinced him to put important papers inside a drawer too often. This article has been conserved by the dedicated care of his brother and it is presented here not only for the intrinsic interest of the topic but above all because it shows us one aspect of the rich personality of Majorana which so much impressed people who knew him, a thinker with a sharp realistic sense and with an extremely critical but not skeptical mind. He takes here a clear position concerning the debated problem of the statistical value of the basic physics law. This aspect was considered by several scholars as a defect similar to a charge of indeterminism in the evolution of nature; it is indeed for Majorana a reason to claim the intrinsic importance of the statistical method. Up to now this method has been applied only to social sciences and in the new interpretation of physics laws it fully recovers its original meaning. Giovanni Gentile jr, 1942. Translated from Il valore delle leggi statistiche nella fisica e nelle scienze sociali, “Scientia”, vol. 36, 1942, pp. 58–66, by R. N. Mantegna in “Quantitative Finance” 5 (2005) 133–140. Reproduced by permission of Taylor and Francis Ltd. (http://www.tandf.co.uk/journals). © Springer Nature Switzerland AG 2020 L. Cifarelli (ed.), Scientific Papers of Ettore Majorana, https://doi.org/10.1007/978-3-030-23509-3_12

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1 The Concept of Nature According to Classical Physics The study of the true or hypothetical relations between physics and other sciences has always been of notable interest due to the special influence that physics has played on the general course of scientific thought in modern times. It is known that the laws of mechanics have been seen for a long time as the ultimate kind of human knowledge about nature. Many scholars have also believed that the imperfect notions of other sciences should eventually be related back to the kind of notions observed in mechanics. The above concept justifies the study we consider here. The exceptional credit of physics evidently comes from the discovery of the so-called exact laws. These laws consist of relatively simple formulas, originally “excogitated” starting from fragmentary and approximate empirical indications, which turn out to be of universal validity both when these laws are applied to new orders of phenomena and when the progressive improvement of the art of experiments allows one to verify them in a more and more rigorous way. It is known to everybody that according to the fundamental concept of classical mechanics the motion of a physical body is completely determined by the initial conditions (position and velocity) of the body and by the forces that are applied to it. On the nature and size of forces that may be present in material systems, the general laws of mechanics state only some condition, or limitation, that always must be verified. Such a characteristic, for example, has the principle of action and reaction. To this principle one has added, more recently, other general rules such as the ones concerning constrained systems (principle of virtual work) or elastic reactions and, even more recently, the mechanical interpretation of heat and also the energy conservation principle, which is seen as a general principle of mechanics. Apart from these general indications, it is however a special task of physics to discover, case by case, all that is needed to effectively apply the principles of dynamics, which is the knowledge of all forces acting in the system being investigated. In one case, however, it has been possible to find the general expression for forces that are present between material bodies. This occurs in the case where material bodies are isolated one from the other and therefore the forces are reciprocally acting at distance only. In this last case, if we do not consider electromagnetic forces, which were discovered in the nineteenth century and which manifest themselves only under specific conditions, the only force acting is the force of gravitation, whose notion was suggested to Newton from the mathematical analysis of the Keplero laws. The Newton law is typically applicable to the study of the motion of celestial bodies which being separated by immense empty spaces, can indeed influence each other only through action at a distance As is known, this law is indeed sufficient to predict in any aspect and with a beautiful accuracy the complex dynamics of our planetary system. Only one minute exception, the secular displacement that undergoes Mercury’s perihelion, constitutes one of the major experimental proofs of the recent theory of general relativity. The sensational success of mechanics applied to astronomy has encouraged the assumption that more complicated phenomena of common experience must

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also be in the end reconducted to a similar mechanism, albeit more general than the gravitational law. According to this point of view, which has produced the mechanistic conception of nature, the entire material universe evolves obeying an inflexible law, where the state of the universe at a given instant of time is completely determined from its state at the previous instant of time. This is a sign of the fact that the future is implicit in the present. In other words, the future can be predicted with absolute certainty provided that the actual state of the universe is completely known. This fully deterministic conception of nature has had numerous confirmations since its introduction. Further developments of physics, from the discovery of the electromagnetism laws to the ones for the theory of relativity, have suggested a progressive enlargement of the principles of classical mechanics. On the other hand, they have vigorously confirmed an essential point, namely, the complete causality in physics. It is not disputable that determinism has the principal and almost exclusive merit of having made possible the magnificent modern development of science, and also in fields very far removed from physics. Determinism, which does not leave any rule to human freedom and forces one to consider all the phenomena of life as illusory, implicates a real cause of weakness. This is the irremediable and immediate contradiction with the most certain data of our conscience. Indeed, how the effective and, most probably, definitive overtaking of determinism has occurred in the physics of recent years will be discussed later. Indeed our final aim will be to illustrate the renovation that the traditional concept of statistical laws must undertake as a consequence of the new direction followed by contemporary physics. At the present stage we still wish to keep the classical conception of physics. This is done not only for its enormous historical interest but also because classical physics is still the only physics largely known except to specialists. Before ending this introductory part, we wish to point out that criticism of determinism has been raised in recent times. The philosophical reaction, when appropriate, did not extend beyond the philosophical field and essentially it has left the scientific problem untouched. An attempt devoted to solve this specific scientific problem can be found in the work of G. Sorel.1 He is an author representing the pragmatism or philosophical current in pluralism. According to the followers of this movement, an effective heterogeneity of natural phenomena excludes that a unitary knowledge of them might exist. Each scientific principle should be applied to a delimited ambit of phenomena without the possibility of achieving universal validity. G. Sorel develops the criticism of determinism by stating that this concept would apply only to phenomena which he calls artificial nature. These phenomena are characterized by the fact that they do not occur in the presence of an appreciable degradation of energy (in the sense of the second principle of thermodynamics). These phenomena sometimes occur spontaneously, especially in astronomy, where they constitute phenomena of simple observation. However, more often these phenomena are investigated in laboratories by experimenters. They devote special care to eliminating all possible passive resistances. The other phenomena, which

1 Sorel,

G.: De l’utilité du pragmatisme, Cap. IV. Marcel Rivière, Paris (1921).

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belong to the common experience or to natural nature and occur in the presence of passive resistances, would not be controlled by precise laws but they should be affected by chance to a various degree. Sorel explicitly uses a metaphysical principle of G. B. Vico. We do not want to discuss here the arbitrary accentuation given to a specific aspect of science as it has been represented in an epoch which is no longer ours. Here we have to note that the pragmatist principle of judging the scientific doctrines on the basis of their effective usefulness does not justify the attempt of condemning the ideal of the unity of science. This idea has acted as a powerful stimulus to the progress of science many times.

2 The Classical Meaning of Statistical Laws and Social Statistics To fully understand the meaning of statistical laws according to mechanics, one needs to recall a hypothesis about the structure of matter, which was already familiar to the ancients and entered the domain of science due to Dalton at the beginning of the nineteenth century. He first recognized in the atomic hypothesis the natural explanation for the general laws of chemistry, which had been recently discovered. According to modern atomic theory, which has been definitively confirmed with specific methods of physics, there exists an amount of species of indivisible elementary particles, or atoms, of the same number of simple chemical elements. The union of two or more atoms of the same or different species forms the molecules. Molecules are the last particles in which one can divide a definite chemical substance, which are capable of independent existence. Single molecules (and sometimes also atoms within molecules) are not located in a fixed position but rather they undergo a very fast movement of translation and rotation around themselves. The molecular structure of gases is very simple. Indeed single molecules of gases can be considered as rather independent in common conditions. The relative distances between molecules are very large with respect to their extremely limited dimensions. By applying the inertial principle one concludes that their motion is rectilinear and uniform most of the time. The motion undergoes abrupt changes of direction and speed only when impacts occur. Supposing we exactly know the laws governing the mutual influence of molecules we should expect, in terms of general principles of mechanics, that it is enough to know the position of all molecules and their translational and rotational velocities in addition in order to predict in principle the exact state of the system after a certain time interval (although these calculations could be too complex to be effectively realized in practice). The use of the deterministic scheme, which is specific to mechanics, is however subject to a real limitation of principle when we take into account the fact that the usual methods of observation are not able to provide us with the exact instantaneous conditions of the system. They provide us only with a certain number of global observables. For example, by considering the physical system given by

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a certain amount of gas, it is sufficient to know the gas pressure and density to determine other variables, such as the temperature, the viscosity coefficient, etc., that could be the object of specific measurements. In other words, in the present example the values of pressure and density are sufficient to fully determine the state of the system from a macroscopic point of view, although they are evidently not sufficient to establish the exact internal structure of the gas at each time, i.e. the distribution of position and velocity of all molecules. To discuss with clarity and conciseness and without any mathematical apparatus, the nature of the relationship between a macroscopic state (A) and a real state (a) of a system, we need to relax precision to a certain degree, although we avoid altering the true nature of the facts in an essential way. We therefore need to understand that the observed macroscopic state (A) corresponds to a large number of possibilities a, a  , a  . . . Our observations do not allow us to distinguish among them. The number N of these internal possibilities would be infinite within the framework of classical theory, but quantum theory has introduced an essential discontinuity in the description of natural phenomena, so that the number (N ) of these possibilities in the structure of a system is indeed finite although huge. The value of N gives a measure of the degree of hidden indeterminacy of the system. It is however practically preferable to consider a quantity proportional to its logarithm S = K log N K being the Boltzmann universal constant, which has been determined by imposing that S coincides with the entropy, which is a known quantity of thermodynamics. Indeed entropy is a physical quantity with the same importance as weight, energy, etc. This is mainly because entropy is an additive quantity in the same manner as the others. In other words, the entropy of a system composed of several independent parts is equal to the sum of the entropy of each single part. To prove this, it is enough to observe that the number of potential possibilities of a composed system is evidently equal to the product of analogous numbers describing the constituent parts together with the known elementary rule establishing the correspondence between the product of two or more numbers and the sum of their logarithms. In general, there are no difficulties in how to determine the ensemble of internal configurations a, a  , a  . . . corresponding to a macroscopic state A. However one can discuss if all the distinct possibilities a, a  , a  . . . should be considered as equally probable. According to the ergodic or quasi-ergodic hypothesis, which is widely believed to be verified, whether a system persists in a state A indefinitely, then one can state that it spends an equal fraction of its time in each of the configurations a, a  , a  . . . Therefore one considers all possible internal determinations as equally probable. This is indeed a new hypothesis because the universe, which is far from being in the same state indefinitely, is subjected to continuous transformations. We will therefore admit as an extremely plausible working hypothesis, whose far reaching consequences could sometime not be verified, that all the internal states of a system are a priori equally probable for specific physical conditions. Under this

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hypothesis, the statistical ensemble associated with each macroscopic state A turns out to be completely defined. The general problem of statistical mechanics can be summarized as follows: suppose the initial state A has been statistically defined; which predictions, therefore, are possible about its state at time t? At first sight, it may seem that this definition is too limited because other static problems can be considered in addition to the dynamic problem. For example, what is the temperature of a gas whose pressure and density are known? Similarly this applies in all cases when one wishes to determine a quantity of interest from certain known characteristics of the system, which are sufficient to define its state. The distinction can be formally ignored because by incorporating appropriate measuring instruments in the system one can always go back to the previous case. Let us suppose that the initial state of the considered system is described by a statistical ensemble A = (a, a  , a  . . .) of possible cases which are, as stated before, equally probable. Each of these specific determinations changes during time according to a law that we have to consider strictly causal in agreement with the general principles of mechanics. Therefore the system moves from the series a, a  , a  . . . to another specific series β, β  , β  . . . after a certain time. The statistical ensemble (β, β  , β  . . .); which is also constituted of N equally probable elements as the original ensemble A (Liouville theorem), defines all the possible predictions for the system evolution. Due to reasons that only a complex mathematical analysis could make precise, in general it turns out that all simple cases belonging to the series, β, β  , β  . . . except a negligible number of exceptions, wholly or in part constitute a new statistical ensemble B defined from a well-determined macroscopic state as in A. We can therefore state the statistical law according to which there is the practical certainty that the system should move from A to B. Due to the above discussion, the statistical ensemble B is at least as large as A. It contains a number of elements not less than N . It therefore follows that the entropy of B is equal to or larger than that for A. In the presence of any transformation occurring in agreement with statistical laws one therefore has a constant or increasing entropy, never a decrease. This is the statistical foundation of the famous second principle of thermodynamics. It is worth noting that the transition from A to B can be considered certain from a practical point of view. This explains why historically the statistical laws have been originally considered as accurate as the laws of mechanics and only because of the progress of theoretical investigation one has subsequently recognized their true character. The statistical laws include a large amount of physics. Among the most widespread applications, we cite: the gas state equation, the diffusion theory, the theory of thermal conductance, of viscosity, of osmotic pressure and several other similar ones. A specific mention is deserved for the statistical theory of irradiation, which introduces the discontinuum symbolized by the Planck constant for the first time in physics. Moreover there is an entire area of physics, thermodynamics, whose principles, although directly based on experience, can be related to the general notions of statistical mechanics. On the basis of what has been done before, one can summarize the meaning of statistical laws within classical physics in the following

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way: (1) natural phenomena obey a complete determinism; (2) the customary observation of a system does not allow one to identify the internal state of the system but only the ensemble of very large possibilities which are macroscopically indistinguishable; (3) by establishing a plausible hypothesis for the probability of different possibilities and by assuming valid the laws of mechanics, the probability calculus allows the probabilistic prediction of future phenomena. We are now ready to examine the relation present between the laws established by classical mechanics and the empirical regularities, which are known by the same name especially in social sciences. First of all, one should realize that the formal analogy could not be more stringent. For example, when one states the statistical law: “In a modern European society the annual marriage rate is about 8 for 1000 inhabitants”. It is clear enough that the investigated system is defined only with respect to certain global characters by deliberately renouncing the investigation of additional information, such as, for example, the biography of all individuals composing the society under investigation. This knowledge would certainly be useful in predicting the phenomenon with a precision and an accuracy higher than in the case for a generic statistical law. This is not different from when one defines the state of a gas by simply using pressure and volume and by deliberately renouncing investigation of the initial conditions for all single molecules. A substantial difference could indeed be detected in the definite mathematical character of the physics statistical laws, which has to be compared with the empirical character of social statistical laws. It is however plausible to attribute the empirism of social statistics (with the term empirism we precisely mean the lack of reproducibility of their results in addition to the random part) to the complexity of the considered phenomena. This last aspect implies that it is not possible to precisely define the conditions or the content of the law. On the other hand, physics also has empirical laws when it is studying phenomena of applied interest. Examples are the laws of friction among solid bodies or the magnetic properties of several types of iron and other similar materials. Lastly, one could express the special importance of the difference in the measurement methods, which are global in physics (it is sufficient to read a measurement instrument to know the pressure of a gas in spite of the fact that pressure arises from the sum of independent pulses that single molecules transmit to the walls) whereas in social statistics individual facts are recorded. This difference is however not an absolute antithesis as it is proved by the possibility of various indirect methods of detection. By admitting the arguments that suggest the existence of a real analogy between physics and statistical laws, we are induced to assume as plausible that the social statistical laws are the most direct proof that an absolute determinism also governs human facts in a way similar as physics statistical laws imply a rigid determinism. This is an argument that has had much better fortune because, as said before, one has detected the tendency to see the causality of classical physics as a model of universal value for independent reasons. Here it would be out of place to reopen old and never concluded discussions, but we wish to recall as a generally admitted fact that the absence of conciliation among contrasting intuitions of nature has for a long time played a role in modern thought and in moral values. It is therefore not just a

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scientific curiosity the announcement that physics has been forced to abandon its traditional course by rejecting the absolute determinism of classical mechanics in a definitive manner in recent years.

3 The New Concepts of Physics It is impossible to present with some completeness in a few lines the mathematical apparatus and the experimental content of quantum mechanics.2 We will therefore limit ourselves to some short description. There are experimental facts, known for a long time (interference phenomena), which undoubtedly support in favour of an undulatory theory of light. Conversely, other recently discovered facts (Compton effect) suggest, no less convincingly, the opposite corpuscular theory. All attempts of settling this contradiction within classical physics have been unsuccessful. This may not seem so relevant except that these inexplicable facts and others no less inexplicable and those of the most differing nature and lastly almost all phenomena known to physicists, and up to now insufficiently understood, have been explained with a unique and wonderful simple explanation. This is the one contained in the principles of quantum mechanics. This extraordinary theory is so solidly founded on experience in as much, perhaps, any other one has never been. The criticism that it has received and is receiving are not concerned at all with the legitimacy of its use for effective prediction of phenomena, but rather the widespread opinion that the new approach should be conserved and perhaps even grow in future developments of physics. The specific aspects of quantum mechanics as compared with classical mechanics are as follows. (a) There are no laws in nature which express a fatal succession of phenomena. Basic laws governing elementary phenomena (atomic systems) have a statistical character. They establish only the probability that a measure performed in a prepared system will give a certain result. This occurs in spite of the means by which we are disposed to determine the initial state of the system with the highest possible accuracy. These statistical laws indicate a real deficiency of determinism. They have nothing in common with the classical statistical laws where uncertainty of results derives from a voluntary renunciation for practical reasons to investigate the initial conditions of physical systems in the most minute aspects. Below we will see a well-known example of this new kind of natural law. (b) A certain lack of objectiveness in the description of phenomena. Any experiment performed on an atomic system exerts a finite perturbation on it that cannot be eliminated or reduced for principle reasons. The result of any measure seems, therefore, to be concerned with the state where the system is led during

2 The

reader desiring to study thoroughly the knowledge in this matter by avoiding where possible the mathematical difficulty can consult W. Heisenberg, Die Physikalischen Prinzipien der Quantentheorie, Lipsia, 1930.

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the same measurement rather than the undetectable state in which the system was before the perturbation. This aspect of quantum mechanics is without doubt more disquieting, i.e. farther from our customary intuitions, than the simple lack of determinism. Among the probabilistic laws concerning basic phenomena, the one governing radioactive processes has been known for a long time. Any atom of radioactive matter has a probability mdt in a time interval dt of transforming itself after the emission of an α particle (a helium nucleus) or, in other cases, of a β particle (an electron). The mortality rate m is constant, i.e. independent of the atom age. This gives a specific form to the survival curve, which is exponential. The mean lifetime is 1/m and one can estimate the probable lifetime, sometimes called the transformation period, in an elementary way. Both quantities are independent of the atom age. Indeed the atom does not manifest any sign of real aging when time elapses. Several methods of observation and automatic recording of the single transformation occurring within a radioactive matter exist. It has been therefore possible to verify through direct statistical measurements and applications of probability calculus that single radioactive atoms do not undergo any reciprocal influence or any external influence concerning the instant of transformation. Indeed the number of disintegrations occurring in a certain time interval is subject to random fluctuations, i.e. to the probabilistic character of the individual law of transformation. Quantum mechanics has taught us to see in the exponential law of radioactive transformations a basic law which is not reducible to a simpler causal mechanism. Of course the statistical laws concerning complex systems known in classical mechanics retain their validity according to quantum mechanics. Quantum mechanics modifies the rules of the determination for internal configurations in two different ways, depending on the nature of the physical systems, ending up with the statistical theories of Bose-Einstein and Fermi. However the introduction in physics of a new kind of statistical law or, better, simply a probabilistic law, which is hidden under the customary statistical laws, forces us to reconsider the basis of the analogy with the above-established statistical social laws. It is indisputable that the statistical character of social laws derives at least in part from the manner in which the conditions for phenomena are defined. It is a generic manner, i.e. strictly statistical, allowing a countless complex of different concrete possibilities. On the other hand, by remembering what has been said above on the mortality tables of radioactive atoms, we are induced to ask ourselves whether there also exists here a real analogy with social facts, which are described with a somewhat similar language. At first sight something seems to exclude this. The disintegration of an atom is a simple fact, which is unpredictable and which occurs abruptly and in isolation after a wait of thousands and even billions of years, whereas nothing similar occurs for facts which are recorded from social statistics. However, this is not an insurmountable objection.

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The disintegration of a radioactive atom can force an automatic counter to detect it with a mechanical effect, which is possible thanks to a suitable amplification. Common laboratory set-ups are therefore sufficient to prepare for whatever complex chain of rich phenomena which is produced from an accidental disintegration of a single radioactive atom. From a scientific point of view nothing prevents one from considering that an equally simple, invisible and unpredictable vital fact could be found at the origin of human events. If this is so, as we believe it is, the statistical laws of social sciences increase their function. Their function is not only of empirically establishing the resultant of a great number of unknown causes, but, above all, it is to provide an immediate and concrete evidence of reality. The interpretation of this evidence requires a special skill, which is an important support of the art of government.

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Comment on: “The value of statistical laws in physics and social sciences”. Majorana published nine articles before his disappearance and a 10th article, whose manuscript was found by Majorana’s brother among his files, was published in 1942, after his disappearance, in the international Italian journal Scientia, through the interest of his friend Giovanni Gentile jr. The article is a rather special article in several respects. In the original presentation for Scientia, Giovanni Gentile jr. wrote that the article was originally written for a sociology journal. This article was therefore intended to present the point of view of a physicist about the value of statistical laws in physics and social sciences to scholars of a broad spectrum of different disciplines such as sociology and economics. In his article, Majorana considers quantum mechanics as a fundamental and successful theory able to describe the basic processes involving particles and atoms. He explicitly considers quantum mechanics as an irreducible statistical theory because theory is not able to describe the time evolution of a single particle or atom in a precise environment at a deterministic level. As an example of the lack of determinism in the time evolution of a single system he discusses the case of the decay of a radioactive atom. 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 social sciences. In his article, he states that: “This conclusion has made essential the analogy between physics and social sciences, between which it turned out an identity of value and method”. According to the biographical and scientific note published in the book edited by Edoardo Amaldi,3 Majorana most probably wrote this article during the period from 1933 to 1937 after his 1933 travel to Lipsia and Copenhagen. According to Amaldi’s recollection, after his return to Rome during the fall of 1933 Majorana’s involvement in theoretical physics research declined until 1937. During this period Ettore Majorana was studying economics, politics, naval fleets of different countries and their relative strength and philosophical problems. There is a pioneering nature of this article both from the perspective of physics and of economics. From the physics point of view, Majorana stated that quantum mechanics forces scientists to use a statistical description down to events involving single entities. He took a clear position about the statistical nature of quantum mechanics by considering this property as irreducible in terms of an underlying deterministic theory. From the point of view of economics and social sciences, there is the observation that statistical laws are investigation tools to be used in economic and social modeling and are characterized by the same epistemological status of irreducible probabilistic laws as quantum mechanics. It should be noted that this position was not that of the majority of scholars working in the thirties of the twentieth century in both the disciplines considered. In fact, during the thirties of the last century the interaction between economics and physics was developed under the paradigm of celestial mechanics and therefore under a complete determinism (the only exception to this approach was the one pursued by Louis Bachelier with 3 E.

Amaldi, op. cit.

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the stochastic description of the time evolution of the price of a financial asset that, at that time, had no impact on the academy4 ). This interaction goes back to the development of the general equilibrium theory pursued by Walras, Pareto, Schlesinger and Wald. The emphasis of Majorana on the intrinsic statistical nature of quantum phenomena motivated him to support the idea that statistical laws should be incorporated into the scientific modeling of social phenomena. Physics might certainly benefit from a deeper understanding of the role, necessity and peculiarity of statistical laws in physics. Some of the statistical laws are eventually reinterpreted in terms of more fundamental and deterministic laws. However there are cases when a reduction seems to be impossible. One of these cases is indeed quantum mechanics and other more recent examples, in years when Majorana was not active any longer, concern topics of dynamical systems and critical-phenomena theory. In summary, the 10th article of Majorana raised the necessity of focusing the attention of several disciplines on the value and nature of statistical laws. From physics, to biology and to social sciences, several scientific disciplines present statistical laws and scholars need to reflect about their role and value within each discipline. Majorana took the view that quantum mechanics implies that a scientific description without statistical laws is impossible as far as the description of elementary processes is concerned. Today there is still the need to assess the status of statistical laws in different disciplines and to consider the validation procedures that are most appropriate to these sorts of laws and to their knowledge content. We wish to conclude this short note by saying that we believe Ettore Majorana today deserves a great tribute not only for his exceptional achievements in theoretical physics but also for his fresh and original views on the essential aspects, importance and role of statistical laws in physics and in other disciplines such as social sciences. R. N. MANTEGNA Università di Palermo, Palermo, Italy

4 Bachelier,

L.: Thèorie de la spèculation, Ph.D. thesis in mathematics. Annales Scientifiques de l’Ecole Normale Supèrieure. III-17, 21–86 (1900).

Viareggio pinewood, summer 1926. Ettore Majorana (third from the right), his mother and sisters (Rosina and Maria), his friend G. Piqué and his grandmother. (Courtesy of B. Piqué)

Karlsbad, autumn 1931. Ettore Majorana (second from the right) with his family, except his brother Luciano

In Venice with his family in summer 1930

Marino, winter 1934. Ettore Majorana (second from the right) with some friends

Are Neutrinos Completely Neutral Particles? Alessandro Bettini

In A symmetric theory of electrons and positrons [1] of 1937 Majorana published his theoretical discovery of the completely neutral spinors. Among the known elementary particles, neutrinos may be such, namely equal to their antiparticles. Other cases are hypothetically foreseen by Supersymmetry, such as the neutralino and the gluino. The Majorana solution of the Dirac equation, his representation of the Dirac γ matrices and the properties of his spinor are described in the first chapter of this book by A. Zichichi, to which I refer the reader (AZ in the following). Let me only recall that in the starting point of the original Dirac path to his equation there is no symmetry between electron and positron. It is, in the words of Majorana, the interpretation of the so-called “negative energy states” proposed by Dirac [2] that leads, as is well known, to a substantially symmetrical description of electrons and positrons. Not happy of this inelegance, Majorana attempted a novel approach, assuming symmetry from the very start. He succeeded, not only to achieve a complete particle-antiparticle symmetrisation and to show that there is no longer any reason to speak of negative-energy states, but even more importantly, to build a substantially novel theory for the particles deprived of electric charge. Even if, he adds, it is probably not yet possible to ask to the experience to decide between this new theory and the simple extension of the Dirac equations to the neutral particles. We are doing that now, but, after 80 years, we do not have an answer yet, as I shall describe. Completely neutral particle means that it has no “charge” of any type, including lepton and baryon numbers, which however are not, properly speaking, charges. We know completely neutral bosons, the higgs, the photon and the Z. Likewise, Majorana “particle” and “antiparticle” are the two helicity states of the same thing, similarly to a clockwise polarised photon and an anticlockwise polarised

A. Bettini () University and INFN, Padua, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2020 L. Cifarelli (ed.), Scientific Papers of Ettore Majorana, https://doi.org/10.1007/978-3-030-23509-3_13

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“antiphoton”. However (see AZ), while a completely neutral boson can be massive, like the Z, or massless, like the photon, a completely neutral spin-½ particle must be massive. In the past 20 years or so, we learned from experiments that, contrarily to what the Standard Model of particle physics assumes, neutrinos do have small, but non-zero masses, which (not yet exactly known) are in the 10–100 meV range. Majorana had not considered how to experimentally test his theory, but soon after his paper, G. Racah [3] noticed that Dirac neutrinos, if emitted in a β − decay can induce only an inverse β + process and vice versa. On the contrary, if they obey Majorana, any (electron) neutrino can produce both electrons and positrons. The argument of Racah is correct, but unfortunately it does not work for reasons unknown at his time. As we shall see, the V–A nature of the W-mediated weak interactions (charged currents) and the smallness of neutrino masses conspire in making the probability of the lepton number violating processes of Racah, not zero, but extremely small. In 1935 M. Goeppert-Mayer [4] considered an extremely rare phenomenon, the two-neutrino double-beta decay (2ν2β), of nuclei for which beta decay is energetically forbidden. In the process a nucleus of charge Z and atomic number A decays in a daughter of charge Z and atomic number A + 2, two neutrons disappear and two protons appear at the same time, with the creation and emission of two electrons and two antineutrinos, namely (in modern formalism) (A, Z) → (A, Z + 2) + 2e− + 2ν e .

(1)

She calculated, with the Fermi theory [5], that the half-lives had to be in excess of 1017 yr. This looked very long indeed, but, in fact, the half-lives are even much longer. Several of them have been measured and found typically on the order of 1021 yr. Recent results are, for example, for 136 Xe, in two different experiments, T1/2 = (2.165 ± 0.016 ± 0.059) × 1021 yr [6] and +0.14 T1/2 = (2.11 ± 0.04 ± 0.2) × 1021 yr [7], and for 76 Ge, T1/2 = 1.84−0.10 × 1021 yr [8]. In 1939 W. H. Furry [9] noticed that a Majorana neutrino could act as a propagator between the two decaying nucleons, originating double beta decay without neutrinos in the final state (0ν2β), namely (A, Z) → (A, Z + 2) + 2e− .

(2)

He observed that in the older theory, double-β disintegration involves emission of four particles, two electrons (or positrons) and two antineutrinos (or neutrinos), and the probability is extremely small. In the Majorana theory only two particles – the electrons or positrons – have to be emitted, and transition probability is much larger. We know now that the 0ν2β decay rate would have been indeed much larger than that of the 2ν2β, if neutrino masses were larger than about 10 eV, but, unfortunately once more, and for the same reasons as above, the opposite is true for sub-eV masses. Difficult as it is, however, the 0ν2β decay is the only way we have,

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at present, to investigate on the Majorana nature of neutrinos. I shall hint to other possible future ways in the closing paragraphs. The 0ν2β decay violates the lepton number (L) conservation by two units and is forbidden in the standard model. Notice, however, that lepton and baryon numbers are not “sacred” as the charges of the fundamental interactions are. Differently from the latter, they do not correspond to a gauge symmetry of the Hamiltonian. In addition, there is no experimental proof that lepton number is conserved in interactions with neutrinos. The reasons of the apparent conservation are the following. The V-A structure of the W-mediated weak interactions implies that only left, meaning negative chirality,1 neutrinos and antineutrinos are created and destroyed. In the standard model, neutrinos and antineutrinos, assumed to have zero mass, are eigenstates of both helicity and lepton number. In nature, however, neutrinos are massive, and are not helicity eigenstates. Both in the Dirac and in the Majorana case, the “particle” created by the neutrino field (negative chirality) is a superposition of  + − (dominant) negative and (small) positive helicity states, namely ∝ m + ν ν L, E  L − ν . where the upper index is the helicity, while the “antiparticle” is ∝ νL+ + m L E If they are Dirac particles, neutrinos and antineutrinos are lepton number eigenstates with opposite eigenvalues. When electron neutrinos or antineutrinos interact, both helicity components produce electrons or positrons respectively. Contrastingly, Majorana particles do not have a lepton number and the two states are distinguished by helicity only. A Majorana neutral particle produced with a positron has the same helicity components as in the Dirac case, but its negative helicity component produces electrons, while the (small) positive helicity one produces positrons. Vice versa for the particle produced with an electron. The factor (m/E)2 in the cross section of the processes proposed by Racah for energies of the order, say, of 10 MeV and masses of the order of 100 meV, is as small as 10−16 , making the detection in practice impossible. Under these conditions, lepton number is not, rigorously speaking, conserved, but its violation is so small to be undetectable. The discovery of the 0ν2β decay would be the discovery of the lepton number violation. From that, one would infer that the baryon number is violated too, and possibly understand why matter dominates in the Universe over antimatter, clearly an extremely important discovery. In addition, a measurement of the half-life of the decay would provide information on neutrino masses. Let us see in which limits. Neutrinos of different flavours, νe , νμ , and ντ , are linear combinations of the mass eigenstates ν1 , ν2 , and ν3 , of masses m1 , m2 and m3 . The flavour state νl , with l = e, μ, τ is given by

1 Unfortunately,

in the literature the words “left” and “right” are used both to mean negative and positive chirality respectively (both neutrinos and antineutrinos are left in this sense), and also negative and positive helicity (relativistic neutrinos would be called (almost) left, antineutrinos (almost) right). The confusion arose because for massless particles chirality eigenstates are also helicity eigenstates and the standard model assumes neutrinos to be massless.

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νl =

3

Uli νi ,

(3)

i=1

where U is the neutrino mixing matrix. It is usually expressed in terms of three mixing angles, similarly to the Euler angles in the rigid rotations in three dimensions, called θ 12 , θ 23 and θ 13 , and one phase factor δ, which, if different from 0 or π , induces CP violation (similar to quarks till here). If neutrinos are Majorana particles, two additional phases exist, α and β. These “Majorana phases” are important in the 0ν2β decay, but do not contribute to the oscillation probabilities, which are the quantities to which the experiments, different from those on that decay, are sensitive. Consequently, these phases are completely unknown. Global fits using all the experimental data are periodically performed by specialised theoretical groups. They find (see for example [10]) values of the squared sines of the three mixing angles to be known within a few per cent accuracy. The “Dirac phase” is not well known yet, but strong hints exist that it is large. Oscillations in a vacuum are sensitive to the absolute values of the square mass differences, while matter effects, in particular in the Sun, are sensitive to their sign too. We know, again at a few per cent accuracy, the smaller difference m22 –m21 in absolute value and sign    (which is positive) and the absolute value of the larger one, m23 − m21 + m22 /2. In other words, we do not know whether m3 (which the eigenstate for which the νe is the smallest) is larger or smaller than the other two masses. The two cases are called normal (NO) and inverted (IO) ordering (or also hierarchy) respectively. The mentioned global fit favours NO over IO by 3.1 σ. Another important unknown is the absolute neutrino mass scale. From the known larger mass splitting, we have the lower limit m3 > 50 meV for NO or m1 and m2 > 50 meV for IO. Cosmology provides, albeit indirect, strong+limits on the sum of the neutrino masses, dependent on the data sets one includes, mi 2 MeV of the 16 double-beta active ones are experimentally interesting. Are any favoured ones amongst these 11? Considering that the phase space available to the final electrons is a rapidly increasing function of Qββ , one might conclude that the larger it is the better. But the conclusion is naive, as one understands comparing, e. g., the limits on the 0ν2β half-life measured for 48 Ca, which is the most exothermic case (Qββ = 4272 keV), T1/2 > 0.58 × 1023 yr [20], with the 76 Ge, which has the smallest Q above 2 MeV (Qββ = 2039 keV), for which is T1/2 > 0.9 × 1026 yr [21], more than three orders of magnitude larger. There are theoretical, experimental and practical reasons for that. On the theoretical side, R. G. H. Robertson [22] has discussed the sensitivities to Mee achievable with a given sensitive mass of different isotopes and their uncertainties. Clearly, the higher is the decay rate per unit isotope mass the higher

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is the sensitivity, provided that the other experimental characteristics (background index, energy resolution, efficiency, etc.) are equal. Notice first that the phase space factor G in Eq. (5) has the dimensions of an activity per unit time per atom, while in the comparison one must consider the activity per unit time per unit mass, namely a “specific” phase space factor. In other words, in a kilogram of a light element there are more nuclei than in a kilogram of a heavier one. For the three isotopes used in the most sensitive experiments, for example, the specific phase space factor is about 3.5 larger for both 130 Te and 130 Xe than for 76 Ge [22]. Secondly, however, Robertson found that a strong uniform inverse correlation exists between phase space factor and NME, not unexpectedly on theoretical grounds. For the three above mentioned isotopes, considering, for example, the NME of the IBM model [16], and making the decay rate per unit isotope mass of 76 Ge equal to 1 for the sake of comparison, one finds 2.2 for 130 Te and 1.5 for 136 Xe. We see that the heavier isotopes are favoured, but only marginally, at the level of the uncertainties in the NME calculations. In conclusion, no isotope is theoretically either favoured or disfavoured; all have qualitatively the same decay rate per unit sensitive mass for any given value of the Majorana mass. The advantages of an isotope are to be credited to experimental and practical factors such as the cost of the isotope separation, the energy resolution that can be achieved, the volume to surface ratio, the scalability, etc. To fix the scale of the problem, consider that for the sensitivity of the present frontier experiments of T1/2 > 1026 yr, in a kmol detector mass (i. e. 76 kg of 76 Ge, 130 kg of 130 Te and 136 kg of 136 Xe), the expected signal is 4 counts per year (at 100% efficiency). Being Mee proportional to square root of the inverse half-life, in order to gain, say, one order of magnitude in neutrino mass, one needs to gain two orders in sensitivity on T1/2 . To increase the sensitivity the “exposure” must be increased, namely the product Mt of the source mass M times the exposure time t. Being the latter limited in practice at a few years, we must increase the isotope M and proportionally the costs. This is not enough, however. Indeed, the relevant factor is the expected number of background events in the ROI. We call background index b the number of counts per unit detector mass per unit exposure time and per unit energy interval. The lowest values reached so far are about b5 × 10−4 (keV kg yr)−1 . Defining the ROI as an energy interval around Qββ of one FWHM energy resolution E, the background rate per unit mass and unit time is b E. The total background for a live time t and a sensitive mass M will be (b E)(Mt). One can schematically consider two types of experiment. We call “zero background” conditions the cases in which the total background in Mt is quite smaller than one (it is never rigorously 0). In this case, the sensitivity on T1/2 increases linearly with Mt, and that on Mee as (Mt)1/2 . These conditions, up to the design exposure, have been reached by the GERDA experiment. If (b E)(Mt) is small, but non-zero, then the signal must be larger than the statistical background fluctuations and, consequently, the sensitivity to T1/2 increases only as (Mt)1/2 and that on Mee as (Mt)1/4 . Under these conditions, to gain one order of magnitude in Mee , at constant background in the ROI, one would need to increase Mt by four orders. Being

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Fig. 3 Limits on Mee from the searches on 136 Xe, 76 Ge and 130 Te and, indirect, from cosmology

this clearly impractical, an increase of detector mass by a certain factor must be accompanied by a decrease of b E by the same factor. Reviews of the experimental search for 0ν2β exist in the literature; for recent ones see [23, 24]. Here only the most sensitive experiments will be considered. The status is summarised in the cartoon of Fig. 3. The horizontal axis is the sum of neutrino masses mi , on which the already mentioned upper limits from cosmology exist. The two bands of dots show the possible values of Mee for IO (disfavoured by the oscillation data) and NO. Their widths are a consequence of being the Majorana phases unknown. The NO band extends, in principle, to even lower values, down almost to zero, but that can happen only for very few combinations of the phases. The dots have been obtained by Aldo Ianni uniformly populating the parameter space. The experiments give their limit on Mee in bands, from the most optimistic to the most pessimistic NME. In summary: Mee < 61–165 meV for 136 Xe, Mee < 95–205 meV for 76 Ge and Mee < 103–520 meV for 136 Te (for which the NME calculations span a wider range). The GERDA experiment works in the INFN underground LNGS lab, where the cosmic muon flux is attenuated by six orders of magnitude compared to the surface. The experiment employs hyper-pure Ge diodes enriched (at about 87%) in 76 Ge isotope, of two different detector geometry, called coaxial and broad energy germanium (BEGe), for a total of 35.6 kg. The detectors are deployed as bare crystals in liquid argon contained in a cryostat. The liquid argon serves three scopes: as the coolant for the diodes, as a high-purity shielding medium against external radiation and as a powerful active veto to discard internal background radiation which simultaneously deposits energy in a germanium detector and in the adjacent liquid argon. The cryostat is contained in a 10 m diameter tank full of ultra-pure water acting both as a further passive shield against the environmental radioactivity and as an active veto for cosmic muons. GERDA published results for a background free exposure of 82.4 kg yr [24]. The experiment adopted since the beginning a rigorous blind analysis strategy to ensure an unbiased definition of the cuts and of all the analysis procedures. The events with a reconstructed energy within the interval Qββ ± 25 keV are not accessible

Are Neutrinos Completely Neutral Particles? Table 1 Backgrounds in a FWHM per isotope ton per year

151 Experiment GERDA EXO-200 KamLAND-ZEN NEXT-100 CUORE a Design

b E (t−1 yr−1 ) 2.3 105 120 13a 400

figure

Fig. 4 The energy spectrum before and after the cuts. In the inserts the energy range in the vicinity of Qββ for coaxial and BEGe detectors

during the analysis. The energy resolution at Qββ = !2039.061 ± 0.007 " keV [25] is +3.4 −4 /(keV kg yr) 3–3.6 keV FWHM. The background index at Qββ is 5.6−2.4 × 10 ! " +4.1 for the BEGe detectors and of 5.7−2.6 × 10−4 /(keV kg yr) for the coaxial ones. The background index in the ROI per 76 Ge tons per year is in Table 1, together with the other experiments I shall mention. Under these conditions, one background event is expected in one FWHM for an exposure of about 500 kg yr. Figure 4 shows the measured spectrum before and after the cuts. The 2ν2β appears almost background free at the lowest energies, with two potassium lines superimposed. Qββ is well separated, due to the per mille level energy resolution. As can be seen in the inserts, no event was found passing the cuts in an energy 0ν > 9.1 × 1025 yr, to be window of Qββ ± 2σ . The derived limit (90% c.l.) is T1/2 0ν > 1.1 × 1026 yr. compared with a median sensitivity of T1/2 The MAJORANA-Demonstrator (MD) in SURF laboratory in the USA is the other experiment on 76 Ge, using different shielding techniques. The experiment has presented results at the Neutrino 2018 Conference of an exposure of 26 kg yr, 0ν > 2.7×1025 yr [26] (median sensitivity T 0ν > 4.8×1025 resulting in the limit T1/2 1/2 yr). The background index is not as small as that of GERDA, but being the exposure

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still limited, the experiment is still (almost) background free. Under these conditions the limits of the two experiments can be combined simply by addition, obtaining 0ν > 1.18 × 1026 yr. This corresponds to the limits on M mentioned above. T1/2 ee In the future, the two experiments will join forces in the LEGEND proposal [27]. Its first phase, with a 200 kg 76 Ge array in the existing GERDA cryostat and infrastructure, aims at reaching the sensitivity of 1027 yr (corresponding to Mee of about 30–60 meV) with a 1000 kg yr background free exposure. The background level should be reduced by about a factor 4 below that achieved by GERDA. If this phase will be successful, the next step would be a 10 t yr exposure, aiming at 1028 yr, in order to explore down to Mee of about 15 meV. At this level, the signal rate is of the order of 0.1 counts per ton per year. A background reduction by a further factor of 6 will be necessary. Three are the running experiments on 136 Xe, each using a different technique. EXO-200 in the WIPP facility in the USA employs a liquid Xe time projection chamber (TPC). The main asset of the technique is the self-shielding. By reconstructing the coordinates of the events one defines a clean fiducial volume in the centre of the chamber, using the external layer as an active absorber. There are no physical surfaces, which are always source of background, between signal space and shielding. Results published for an exposure of 177.6 kg yr [28] show a background index b = (1.5 ± 0.3) 10−3 (keV kg yr)−1 and an energy resolution (at Qββ = 2457.83 ± 0.37 keV) E = 70 keV, resulting in the background in the ROI of Table 1. Having not observed any signal, the limit on the half-life is 0ν > 1.8 × 1025 yr (median sensitivity T 0ν > 3.7 × 1025 yr). The future plans T1/2 1/2 are for the nEXO proposal for a 5000 kg enriched 136 Xe TPC in the SNOLab in Canada. KamLAND-Zen (KL-Zen) operates in the Kamioka Observatory in Japan. The basic idea, proposed by R. Raghavan back in 1993 [29], is to dissolve xenon, enriched in the 136 Xe isotope, in a liquid scintillator, considering that this liquid can be treated to extremely low radioactivity levels and that a limited Xe loading does not alter its transparency. KL-Zen used 13 t of ultra-pure liquid scintillator (LS) loaded with 300 kg of 136 Xe contained in a 3.08 diameter Inner Balloon (IB). This is in the centre of an Outer Balloon (OB) containing 1 kt of unloaded ultra-pure LS acting as an active shield. The photomultipliers (PM) are installed on the surface of a stainless-steel sphere containing IB and OB and a buffer oil outside OB, working as a shield against the PMs and steel radioactivity. The sphere fits inside a 17 m diameter tank of ultrapure water acting as a Cherenkov veto. KL-Zen published in 2016 results of a 534.5 kg yr exposure [30]. The largest background sources were found on the surface of the IB. However, the liquid in the IB is self-shielding and backgrounds can be substantially reduced by defining off-line a fiducial volume in the middle, taken by KL-Zen as a 1 m radius sphere. The final background index is b = 4.4 × 10−4 /(keV kg yr), the lowest so far. The energy resolution, on the other hand, is not very high. The FWHM at Qββ is E ≈ 270 keV, resulting in a background in the ROI of Table 1. The resulting limit is T1/2 > 1.07 × 1026 yr, corresponding to the quite lower median sensitivity of T1/2 > 5.6 × 1025 yr. The

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collaboration is presently working at reducing the background for the next phase that will use 750 kg of 136 Xe. The further step may be KamLAND2-Zen aiming at increasing the energy resolution with a larger coverage of light collection. The third experiment with Xe is NEXT, in the underground Canfranc laboratory in Spain. The technology is the high pressure (10 bar) gas TPC [31]. The readout is through electroluminescence, avoiding charge amplification, which would spoil energy resolution due to its gain fluctuations. While the experiment will use 100 kg of gas enriched in 136 Xe, a 10 kg R&D prototype is taking data. The FWHM energy resolution was measured with calibration sources, finding a linear behaviour with energy that extrapolates to a about 1% at Qββ , namely E = 25 keV, which is the design figure [32]. In this case, the main weapon against background is the topological reconstruction. The largest fraction of the background in the fiducial volume are the Compton electrons. Tracks are reconstructed and the specific ionisation measured along them, showing the Bragg peak at the stop point appearing as a “blob”. The signal is instead two electrons, appearing as a track (the decay point position is not distinguishable due to the multiple scattering) with blobs at both ends. According to a simulation [33], based on the background model of the experiment (including the measured activities of its components), the selection algorithms should lead to a background index as low as b = 5 × 10−4 /(keV kg yr). If achieved, this would be similar to that of KL-Zen, but in a much narrower ROI, resulting in the background in Table 1. The experiment is now running the mentioned prototype to validate the background model. Further progress, towards the 1028 yr sensitivity on T1/2 , requires controlling the background below 10−4 /(kg yr) in a FWHM. An interesting perspective for 136 Xe is the identification, atom by atom, or “tagging”, of the 136 Ba++ daughter. The single molecule fluorescent imaging technology, borrowed from biochemistry, where it had demonstrated single ion sensitivity, was proposed by B.J.P. Jones et al. in 2016 [34]. It employs a fluor that becomes fluorescent only when it traps a targeted ion. With this method, the NEXT collaboration, which has already the energy resolution needed to separate the two-neutrino decay, was able to detect and localise, within about 2 mm, the single barium dication (Ba++ ) [35]. On their side, the EXO collaboration develops a tagging scheme utilizing a cryogenic solid xenon probe to trap the barium atom and to extract it from the liquid xenon chamber. Recently, the group succeeded in fluorescence imagining of single barium atoms deposited in a solid xenon matrix [36]. These are examples of how much particle physics can learn from atomic and molecular physics. The CUORE experiment just successfully started its data taking phase, after many years of construction at the LNGS. The active isotope is 136 Te. Its natural abundance is quite high, namely 34%, and consequently natural tellurium can be used, sparing the costs of enrichment. The experiment is an array of 988 TeO2 bolometers operating at about 10 mK, containing a total of 203 kg of 130 Te. The innermost shield is made out of extremely radio-pure ancient lead taken from a wrecked Roman ship, under an agreement with the Cultural Heritage Department. First results have been published for an exposure of 86 kg yr, corresponding to 24 kg yr of 130 Te and a median sensitivity of T1/2 > 0.7 × 1025 yr [37]. The

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background index was b = (5.2 ± 0.7) × 10−2 (keV kgTe-130 yr)−1 and the energy resolution E = 7.7 keV, with still margins for improvement, corresponding to the background figure in the ROI of Table 1. The limit on the half-life, including the data from previous smaller experiments, is T1/2 > 1.5 × 1025 yr. The sensitivity in a 5-year run is extrapolated around 8 × 1025 yr. The bolometric technique provides a very good energy resolution, second only to that of Ge, but is limited by sizeable backgrounds. Its main sources are on the surfaces, both of the bolometers themselves and of their supports, which, being in a vacuum can shoot alphas on the detectors, even after the drastic reduction obtained be CUORE developing advanced techniques for ultra-cleaning the Cu surfaces. The CUPID project aims at suppressing these backgrounds using scintillating bolometers, in which a small fraction of the released energy is converted into scintillation light, which is detected. The discrimination is achieved because the scintillation induced by α-particles is characterised by a different amplitude and time development compared to those of electrons of the same energy. With such an array of scintillating Zn82 Se bolometers, CUPID-0 has already reached +1.9 a background index b=3.6−1.4 × 10−3 (keV kg yr)−1 and energy resolution E = (23.0 ± 0.6) keV [38]. SNO+ in the SNOLab in Canada is the second experiment on 130 Te, and is starting now. Natural tellurium will be loaded in 780 t of liquid scintillator [39]. While the isotope mass will be as large, already in the first phase, as 800 kg, the energy resolution will be poor, E ≈ 270 keV, making the signal appear as shoulder, rather than a peak. Under these conditions, the accurate knowledge of the background spectrum shape becomes extremely critical. Before closing, let me mention two possible other ways to understand if neutrinos are Majorana particles. As we have seen, in 0ν2β the effects are proportional to (m/E)2 , a factor that is extremely small for typical nuclear energies of several MeV. Is it possible to work at the eV or sub eV energy scales? This means working with atoms or molecules. However, at those energies, neutrino interaction cross sections become extremely small. One should then try to exploit coherent processes, where effects are proportional to the square of the number of atoms, and to induce transitions between atomic or molecular levels, by an external coherent source. The SPAN (Spectroscopy of Atomic Neutrino) [40] collaboration is working since several years on the Radiative Emission of Neutrino Pairs (RENP) process shown in Fig. 5. The difficulties are certainly present and have been discussed in an independent analysis [41]. The atoms of the gas under study are contained in a transparent cell that is irradiated by two counter-propagating trigger LASER beams tuned at one half of the energy difference between the excited state e and the ground state g, say Eeg /2. These states are chosen to have the e to g electric dipole (E1) forbidden transition. The trigger field induces macroscopic coherent polarisation of the atoms. The rate of the decay | e →| g + γ + νi + ν j is proportional to the number n of coherently active atoms squared. It occurs via E1-M1 coupling to the level p. Calculations show

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Fig. 5 The radiative emission of neutrino pairs, showing the transition from the excited state e to ground state g via the intermediate state p, with the emission of a photon and a neutrino-antineutrino pair (two mass eigenstates)

that detectable rates might be obtained with n of the order of one Avogadro number and volumes of only 100 cm3 . Six thresholds corresponding to six different possible neutrino pairs  mi + mj Eeg − Wij = 2 2Eeg

2

(7)

give redundant information on the absolute values of neutrino masses and the mixing matrix elements. This would provide a beautiful means for neutrino spectroscopy. Appreciating the difference between Majorana and Dirac is even more difficult, because it appears only in interference terms, and is consequently small, but might not be impossible. This would provide unique information on the Majorana phases [42]. But Nature offers us neutrinos of even much lower energies. They are around us in the neutrino cosmic background, analogous to the cosmic microwave (photon) background, in the incredible number of 3.36 × 108 per cubic metre, 50% neutrinos and 50% antineutrinos, but are extremely difficult to detect. Assume, however, it would be possible. In the Dirac case neutrinos have lepton number L = +1, antineutrinos L = –1. In the Majorana case L cannot be defined. In both cases, both particles are produced and absorbed with negative chirality (γ 5 = −1). In the evolution of the Universe, these particles were still relativistic when they froze out, namely their energy, say Ef , was Ef >> mi . Neutrinos had a dominant negative and a small positive helicity component (mi /Ef ); the opposite was true for antineutrinos. However, chirality is not a conserved quantity, as γ 5 does not commute even with the free Hamiltonian (for massive fermions), and during the expansion was not conserved. On the contrary, helicity was conserved because these particles did not interact any more. As the expansion went on, their energy decreased and decreased, and is now E0 ∼ 100 μeV. Namely neutrinos are now non-relativistic being mi >>E0 . The PTOLEMY project [43] aims at detecting relic neutrinos through the inverse beta decay (IBD) on tritium νe +3 H → 3 He + e− , which has no energy threshold. The extremely difficult experiment plans using 100 g of 3 H atoms attached to a 100 m2 graphene surface and measuring the final electron energy with the necessary sub-eV resolution. In the Dirac case, the IBD process is induced by only by those

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having L = +1, which are 50% of the total, and only by the fraction of these that have negative chirality. For a non-relativistic particle, negative chirality is a superposition of positive and negative helicity components in almost equal parts. But the relic neutrinos (L = +1) have still the original dominant negative and small (mi /Ef ) positive helicity components. Consequently, only about 50% of them are able to induce the IBD. If they are Majorana, what matters is the helicity, not the lepton number. Hence, in addition to the Dirac case, about 50% of the L = –1 ones induce the IBD. In conclusion, the expected rate for Majorana case is twice as large as for Dirac. Cocco, Mangano and Messina [44] calculated that PTOLEMY should observe about 7.5 events per year in the Dirac case, which, I say, becomes 15 if neutrinos are Majorana. If successful, in a few years the experiment should be able to choose (or, perhaps, to find surprises). It took exactly one century to discover gravitational waves since the theoretical prediction by Einstein. We can hope it will not take so long to discover that neutrinos are Majorana particles.

References 1. Majorana, E.: Nuovo Cim. 5, 171 (1937) 2. Dirac, P.A.M.: Proc. Camb. Soc. 30, 150 (1924) 3. Racah, G.: Nuovo Cim. 14, 322 (1937) 4. Goeppert-Mayer, M.: Phys. Rev. 48, 512 (1935) 5. Fermi, E.: La Ricerca Scientifica. 2(12), 1 (1933); Nuovo Cim. 11, 1 (1934) 6. Albert, J.B., et al.: Phys. Rev. C. 89, 015502 (2014) 7. Gando, A., et al.: Phys. Rev. C. 85, 045504 (2012) 8. Agostini, M., et al.: J. Phys. G. 40, 035110 (2013) 9. Furry, W.H.: Phys. Rev. 56, 1184 (1939) 10. Capozzi, F., Lisi, E., Marrone, A., Palazzo, A.: Astrophys. J. Suppl. Ser. 225, 8 (2018) 11. Lesgourgues, J., Verde, L.: Neutrinos in cosmology, in Tanabashi M., et al. (PDG), Phys. Rev. D 98, 030001 (2018) 12. Kotila, J., Iachello, F.: Phys. Rev. C. 85, 034316 (2012) 13. Šimkovic, F., Rodin, V., Faessler, A., Vogel, P.: Phys. Rev. C. 87, 045501 (2013) 14. Hyvärinen, J., Suhonen, J.: Phys. Rev. C. 91, 024613 (2015) 15. Menendez, J., Poves, A., Caurier, E., Nowacki, F.: Nucl. Phys. A. 818, 139 (2009) 16. Barea, J., Kotila, J., Iachello, F.: Phys. Rev. C. 91, 034304 (2015) 17. Vaquero, N.L., Rodríguez, T., Egido, J.L.: Phys. Rev. Lett. 111, 142501 (2013) 18. Yao, J.M., Song, L.S., Hagino, K., Ring, P., Meng, J.: Phys. Rev. C. 91, 024316 (2015) 19. Engel, J., Men’endez, J.: Rep. Prog. Phys. 80, 046301 (2017) 20. Umehara, S., et al.: Phys. Rev. C. 78, 058501 (2008) 21. Agostini, M., et al. (GERDA collab.): Sci. 365, 1445 (2019) 22. Robertson, R.G.H.: Mod. Phys. Lett. A. 28(8), 1350021 (2013) 23. Henning, R.: Rev. Phys. 1, 29 (2016) 24. Dell’Oro, S., Marocci, S., Viel, M., Vissanti, F.: Adv. High En. Phys. 2016, 2162659 (2016) 25. Mount, B.J., Redshaw, M., Meyers, E.G.: Phys. Rev. C. 81, 032501 (2010) 26. Aalseth, C.E., et al.: Phys. Rev. Lett. 120, 132502 (2018) 27. Abgrall, N., et al.: AIP Conf. Proc. 1894, 020027 (2017) 28. Albert, J.B., et al.: Phys. Rev. Lett. 120, 072701 (2018) 29. Raghavan, R.S.: Phys. Rev. Lett. 72, 1411 (1993)

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30. Gando, A., et al.: Phys. Rev. Lett. 117, 082503 (2016) 31. Martin-Albo, J., et al.: JHEP. 05, 159 (2016) 32. Renner, J., et al.: arXiv:180801804 33. Nebot-Guinot, M.: Nucl. Part. Phys. Proc. 273–275, 2612 (2016) 34. Jones, B.J.P., McDonald, A.D., Nygren, D.R.: J. Instrum. 11, P12011 (2016) 35. McDonald, A.D.: Phys. Rev. Lett. 120, 132504 (2018) 36. Chambers, C., et al.: arXive:1806.10694 (2018) 37. Alduino, C., et al.: Phys. Rev. Lett. 120, 132501 (2018) 38. Azzolini, O., et al.: Phys. Rev. Lett. 120, 232502 (2018) 39. Adringa, S., et al.: Adv. High En. Phys. 2016, 1–21 (2016) 40. Fukkumi, A., et al.: Progr. Theor. Exp. Phys. 04D002 (2012); Dinh, D. N., et al.: Phys. Lett. B 719, 154 (2013); Miyamoto, Y., et al.: Prog. Theor. Exp. Phys. 2015, 081C01, (2015) 41. Song, N., et al.: Phys. Rev. D. 93, 013020 (2016) 42. Tanaka, M., et al.: Phys. Rev. D. 96, 113005 (2017) 43. Betts, S., et al.: arXiv:1307.4738 44. Coppo, A. C., et al.: JCAP 0706, 015 (2007); J. Phys. Conf. Ser. 110, 082014 (2008)

Majorana Fermions in Condensed Matter Giorgio Benedek

Fermi’s opinion that Ettore Majorana ranked within the Olympus of geniuses finds its best demonstration in the ninth Majorana’s paper [1], the last being published before his misterious disappearance on the 26th March 1938 at the age of 31. The paper, perfectly translated from the original Italian by Luciano Maiani [2] and entitled “A symmetric theory of electrons and positron”, aimed at modifying Dirac equation through a new quantization process, so as to avoid the solutions of negative energy. While for charged fermions such as electrons and positrons the new symmetric theory necessarily implies complex fields, Majorana shows that for neutral fermions the theory leads to the identity of particles with their own antiparticles, thus arguing “that there is no reason now to infer the existence of antineutrons and antineutrinos.” [2]. The process consists in replacing Dirac’s anticommuting gamma matrices γ μ with a new set of fully imaginary Majorana’s anticommuting matrices γ˜ μ = − (γ˜ μ )∗ which fulfil the Clifford algebra. In this way the operator i γ˜ μ ∂μ is real, thus allowing for real eigenfunctions. The reality of field operators is a prerequisite for a particle being its own eigenfunction, as it implies the identity between particle and antiparticle creator operators, ψ* ≡ ψ. Among the numerous contributions made by Majorana in his short life, that of “Majorana fermions” is perhaps the most profound and far-reaching one. As Frank Wilczek

Just the day this note was completed I received the sad news that Shoucheng Zhang passed away on December 1st, 2018, at only 55. This note is dedicated to his memory in consideration of his fundamental theoretical and experimental contribution to the discovery of Majorana fermions in condensed matter. G. Benedek () University of Milano-Bicocca, Milan, Italy Donostia International Physics Center, Donostia/San Sebastian, Spain e-mail: [email protected] © Springer Nature Switzerland AG 2020 L. Cifarelli (ed.), Scientific Papers of Ettore Majorana, https://doi.org/10.1007/978-3-030-23509-3_14

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remarks in his fashinating 2009 Nature Physics perspective “Majorana returns” [3], “for 70 years his modified equation remained a rather obscure footnote in theoretical physics. Now suddenly . . . Majorana’s concept is ubiquitous, and his equation is central to recent work not only in neutrino physics, super symmetry and dark matter, but also on some exotic states of ordinary matter.” As nicely explained by Alessandro Bettini in his article [4], the race for Majorana fermions in particle physics is however mostly focused on the search of neutrinoless double-beta decay—a finding which would finally prove the conjecture that neutrinos are Majorana fermions [5]. Although the chase has been so far unsuccessful, the optimal conditions for observing this rare event are gradually approached, which leads Bettini to conclude: “It took exactly one century to discover gravitational waves since the theoretical prediction by Einstein. We can hope it will not take so long to discover that neutrinos are Majorana particles”. It took much less in condensed matter physics, where Majorana fermions manifested themselves in various ways and forms [6], although obviously not as elementary particles. Solids are made of charged electrons and ions, thus no Majorana fermion around of any sort. Electronic states are organized in energy bands with gaps produced by the periodic crystalline order. The band shapes around gaps, one above the Fermi level and therefore only populated by thermally excited electrons, the other below the Fermi level and only populated by thermally excited holes (missing electrons), are well described as functions of momentum by the Dirac energy for electrons and holes, with the sum of their effective masses equal to the energy gap and the speed of light replaced by the Fermi velocity. The analogy would be perfect, but the motion on a periodic potential rather than in a uniform space makes the hole effective mass slightly larger than that of the electron, thus breaking a particle physics dogma requiring the equality of particle and antiparticle masses. Still, in the limit of vanishing gap both effective masses vanish as well, thus providing pairs of crossing bands with massless fermions at the Fermi level. In solids where spin-orbit interaction breaks Kramer’s degeneracy, the crossing bands may host states of opposite spin. When represented in the reciprocal (momentum) space the energy bands may form hypersurfaces of complex topology. In solids with free carriers like metals, semimetals and semiconductors electrons, besides moving in a periodic potential, are subject to a variety of interactions like electron-electron, exchange-correlation, spin-orbit, spinspin and, last but not least, electron-phonon coupling. All these interactions concur in shaping the Fermi surface and band-structure topology, allowing for a zoo of quasi-particles and elementary excitations. Among these, Majorana fermions may appear as emergent excitations associated with peculiar topologies. An example is mentioned by Frank Wilczek in his Perspective [3], “there are certain types of superconductor in which Majorana-type excitations are predicted to emerge. . . . In particular, depending on the kind of superconductor and the electronic spectrum, the (Abrikosov) vortices may trap so-called (Majorana) zero modes, spin1/2 ‘excitons’ of very low (formally, zero) energy. . . . The existence of these modes is related to a profound result in mathematics, the Atiyah–(Patodi)-Singer (APS) index theorem [7], which connects the existence of special, symmetric solutions

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of differential equations to the topology of the parameters that appear in those equations” [8]. The bad news is that Majorana fermions in condensed matter are far from being elementary objects; the good news is that, at least in principle, the potential, the interactions and the corresponding topological parameters of the phase space can be realized in condensed matter by design, so as to artificially reproduce the desired quasi-particles and excitation spectra. In this way, profiting of the scale invariance enjoyed by current field equations, artificial condensed matter systems, eventually obtained by interfacing different functions at the nanoscale, can mimic hypothetical quantum fields, interactions and processes occurring, e.g., in high-energy physics, and eventually test models and theories presently beyond the reach of supercolliders or observational astrophysics. In other words the progress in topological materials science [9, 10], encompassing topological insulators and superconductors [11, 12], graphene [13] and related carbon schwarzites [14, 15], is providing an efficient playground for topological field theories [11, 16]. Whenever coherence in condensed phases or long-range correlations, like sp2 -conjugation in graphenes, play a major role, topology takes over, establishing a unifying paradigm and enabling productive interactions among apparently distant areas of physics and mathematics [17–19]. Triplet pairing, occurring in the superfluid phase A of 3 He as well as in strontium ruthenate superconductivity, allows for vortices of only ms = 1 or only ms = −1 Cooper pairs, leading to what is known as half-quantum vortices [20]. As remarked by Tony Leggett [20], in these conditions excitations with the character of Majorana fermions must exist in association with the vortices, as a consequence of the APS theorem. The existence of Majorana zero-energy modes bonded at the vortex cores of chiral superconductors has been thoroughly discussed by Grigori Volovik as early as 1999 [21]. It is however interesting that one of the earliest and far-reaching suggestions of Majorana fermions in condensed matter is the toy model proposed in 2001 by Alexei Yu. Kitaev in connection with quantum computing [22]. The model consists of a one-dimensional system (a quantum wire) of finite length with a bulk energy gap and localized zero-energy states at the two extremities. Kitaev argued that, under special topological conditions of the quantum wire electronic structure induced by the proximity of a p-wave superconductor, the two edge states are Majorana fermions, and suggested possible experimental configurations for their observation. Other possible heterostructures providing topological conditions suitable to the formation of Majorana edge states have been theoretically designed by Lutchin et al. [23] and by Oreg et al. [24]. The first proposal consists in a one-dimensional semiconductor wire with a strong spin-orbit interaction embedded into a superconducting quantum interference device; the second one considers a single-channel quantum wire hosting a helical electron liquid (e.g., a hcp-InAs wire) interfaced to a superconductor. The latter configuration, though with an InSb nanowire in contact with an s-wave superconducting niobium titanium nitride electrode, actually led in 2012 Mourik et al. [25] to the experimental observation, in presence of magnetic fields in the range of 0.1 T, of localized midgap states at zero bias voltage, which have been considered as a signature of Majorana fermions. An account on this finding, which had a large

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impact on the scientific community with more than 2500 citations, may be found in [26]. A few months later Das et al. [27] reported on the observation in an Al/InAs nanowire topological superconductor of a zero-bias conductance peak induced by a magnetic field, which was also considered as a possible signature of a Majorana state. In the years 2012–2014 other hybrid superconductor-semiconductor devices have been explored by Deng et al. [28], Finck et al. [29], He et al. [30, 31] and Lee et al. [32], in order to gather more convincing evidence of Majorana quasi-particles. In particular the last two works exploit the role played by Majorana fermions in Andreev reflection [33, 34]. In 2014 further evidence was achieved by Nadj-Perge et al. [35] with an entirely different 1D Fermi system. In their experiment quantum wires constituted by ferromagnetic chains of Fe atoms a few hundred Å long are grown on the (110) surface of lead. At the onset of superconductivity in the Pb substrate the spectroscopic mapping of the atomic chains performed with scanning tunnel microscopy (STM) reveals a zero-bias peak localized at the chain edges, which the authors assign on a solid ground to the Majorana fermions predicted by theory. It should be said that the association of Majorana fermions to zero-bias peaks requires some caution, as was remarked by Liu et al. [36] who showed that zero-bias peaks in the tunneling conductance of spin-orbit-coupled superconducting wires may occur with and without Majorana edge states. Low temperature and negligible disorder are necessary conditions to prevent a non-topological origin of the conductance zero-bias peaks. A confirmation came however from the other option—fishing for Majorana in the vortices of a 2D chiral topological superconductor [6, 21, 37–41] — which was also intensively pursued in the recent years. Xu et al. [42] were able to detect with STM/STS a Majorana mode in a heterostructure made of a layer topological insulator, Bi2 Te3 and a layer superconductor, NbSe2 . The choice of layer materials ensures an atomically perfect interface, thus preventing spurious defect-induced zero-bias peaks. The spin-selected Andreev reflection effect [32] was exploited by Sun et al. [43] in a similar 2D heterostructure where five quintuple layers of the topological insulator Bi2 Se3 are interfaced to a NbSe2 superconductor. Topological order and superconductivity were found to coexist in Bi2 Se3 [44], which enabled Sun et al. [43] to unveil Majorana zero modes by probing the vortex core states with spin-polarized scanning-tunneling microscopy/spectroscopy (SP-STM/STS). The assignment of zero-energy modes to Majorana fermions may be received as deceiving in some way, their existence as field excitations with known dispersion relations being a premise for transport and the envisaged applications in quantum computation. This crucial issue has been discussed by Chamon et al. [45], who show that Majorana’s quest for real wave functions from the reality of the operator i γ˜ μ ∂μ is not confined to zero-energy modes but involves the entire quantum field. In about the same years a series of ground-breaking papers by Shou-Cheng Zhang and his coworkers in Stanford and Beijing, published between 2010 and 2015 [46–50], gave exact indications about where and how to find chiral Majorana fermions in an unambiguous way. The key step is the discovery of the quantum anomalous Hall (QAH) effect in magnetic topological insulators [49]. A twodimensional hybrid device interfacing a quantum anomalous Hall insulator to

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Fig. 1 (a) The hybrid device, consisting of a quantum anomalous Hall insulator (QSHI) film sandwiched between a GaAs(111) substrate and a conventional Nb superconductor, used by He et al. [51] to detect Majorana fermions (adapted from [52]). (b) At 20 mK with the superconductivity on (gap Δ = 0), Majorana fermions are observed as pairs of half-integer quantum plateaus of the Hall conductance at 0.5 e2 /h, corresponding to the Chern numbers ±1, for both increasing and decreasing magnetic field. (c) The removal of superconductivity (Δ = 0) yields a regular quantum anomalous Hall effect, with plateaus at the integer conductance quanta ± e2 /h and 0 [51]. (Diagrams adapted from [51, 52])

a conventional superconductor permits to pinpoint the presence of a Majorana fermion as a quantum Hall plateau at e2 / 2h, one half a conductance quantum [51]. As explained by Shoucheng Zhang in his Stanford presentation [52], “since the Majorana fermion has no anti-particle counterpart, it is in some sense half of a conventional particle, therefore the additional half-quantized plateaus provide the “smoking gun” evidence of its existence as a particle propagating in space and time.” This should dispel doubts about other possible mechanisms like those hinted about the origin of zero-bias peaks [20, 36]. On the contrary previous observations with other methods receive a substantial support. The “smoking gun”, reported a year ago by He et al. [51], has been achieved with a magnetic topological insulator film of composition (Cr012 Bi0.26 Sb0.24 )2 Te3 and thickness of 6 nm, sandwiched between a GaAs substrate and a conventional Nb superconductor (Fig. 1a). At very low temperature (20 mK) the Nb bar is superconducting with about its largest gap Δ. In these conditions, the QAH conductance σ 12 , measured for both increasing and decreasing magnetic field (Fig. 1b), exhibits, besides the high-field values, which are however both positive and equal to the quantum e2 /h, also two small plateaus at half a conductance quantum e2 /2h corresponding to the Chern numbers C = ±1, on the positive field side for increasing field, and two similar plateaus on the negative field side for decreasing field. When superconductivity is switched off (Δ = 0) by removing the Nb bar, the QAH conductance behaves as expected (Fig. 1c), varying for increasing

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magnetic field between the integer quantum values – e2 /h and + e2 /h, with a plateau at zero conductance at positive fields, and vice versa for decreasing field, showing the peculiar large hysteresis of QAH effect [49]. These results provide undisputable evidence that Majorana fermions can exist in condensed matter, and are endowed with that topological protection against decoherence required for a practical use in quantum computing [52]. The application of chiral Majorana fermions to quantum computing is promising thanks to the topological protection of coherence at a distance. This implies a controlled transport, but Majorana fermions do not carry electric charge. They can carry heat, however, as was recently proved by the observation of fractional quantum thermal Hall conductance [53–55]. While ordinary Hall effect consists in an electric field along y generated by an electric current along x under a magnetic field along z, thermal Hall effects are observed when any of the electric components are replaced by thermal ones: given a magnetic field along z, an electric current along x generates also a temperature gradient along y (Ettinghausen effect), while a heat flow along x generates along y an electric field (Nernst effect) and also a temperature gradient (Righi-Leduc effect). In 2D systems the respective conductances are all subject to quantization [56]. It is in an experimental setting suitable to observe the quantum anomalous Righi-Leduc effect that Kasahara et al. [57] have detected fractional thermal conductance quanta, signatures of Majorana fermions in a heat transport experiment. The device consisted in a 2D quantum magnet, α-RuCl3 , realizing at low temperature a Kitaev quantum spin liquid [38, 57–59]. In these conditions the authors demonstrated the spin fractionalization into itinerant Majorana fermions. Similar observations of Majorana fermions involved in thermal transport have been reported in the same issue of Nature by Banerjee et al. [53], though in a device based on a semiconductor heterostructure operating in the mK range. Also fermionic quantum liquids, such as 3 He in the superfluid B phase [60] or ultracold atoms in an optical lattice either in an s-wave [61] or (px + ipy )-wave [62] superfluid phase, have been proposed as systems suitable to host Majorana fermion quasiparticles for quantum computation. Thermal transport, however, is also due to phonons, the quanta of atomic vibrations in condensed matter, which obey Bose-Einstein statistics and contribute at 0 K to the zero-point energy. Thus any practical application of Majorana fermion transport has to compete with phonons, as carefully considered in the recent studies mentioned above [52, 53]. Majorana fermion-phonon interaction is an issue in the same way the ordinary electron-phonon interaction is for many fundamental transport properties in solids. However the very recent report by Gül et al. [63] of ballistic transport of Majorana fermions in nanowire devices opens a new possible avenue towards the application of Majorana fermions in practical devices and quantum computation. A network of quantum nanowires hosting n pairs of Majorana fermions can encode a string of n q-bits, thus providing, as anticipated by Kitaev with his famous toy model [22], the basis for a quantum computer immune from decoherence thanks to topological protection. Any algorithm implemented in such a quantum

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computer is realized through a sequence of exchanges of Majorana fermions, through braiding procedures which keep them apart. Due to the non-Abelian statistics any different sequence of exchange corresponds to a different algorithm. A scheme for Majorana fermion braiding in a system of 2n half-quantum vortices in a p-wave superconductor has been formulated by Ivanov as early as 2000 [64], while a concrete realization of a Majorana fermion exchange process by moving around vortices in a T-shaped configuration have been recently proposed by Liang et al. [65] and Wu et al. [66]. An up-to-date introduction to the unitary braid group representations associated with Majorana fermions required for the realization of topological quantum computing is found in Ref. [67]. In conclusion braiding Majorana fermions in a device, although representing a major technological challenge, would definitely constitute a major and most promising step toward quantum computing. As argued by Kirill Shtengel in his presentation [54] of Majorana transport experiments, “the heat is on for Majorana fermions”! The almost 2000 papers published in last 5 years according to Web of Science under the label of Majorana fermions, with more than 22,860 cites, let us hope that soon the great intuition of Ettore Majorana will be honored by a major progress in fundamental knowledge as well as by applications beneficial to mankind progress.

References 1. Majorana, E.: Teoria simmetrica dell’elettrone e del positrone. Nuovo Cim. 14, 171–184 (1937) 2. The translation by Luciano Maiani appeared first in Soryushiron Kenkyu, 63, 149 (1981), then reproduced, with his Commentary and the inclusion of the original summary, in the bilingual centenary collection Ettore Majorana Scientific Papers, edited by G. F. Bassani and the Council of the Italian Physical Society. SIF & Springer (2006), pp. 201–233 3. Wilczek, F.: Majorana returns. Nat. Phys. 5, 614–619 (2009) 4. Bettini, A.: Are neutrinos completely neutral particles? this volume, p. 143 5. Avignone, T.F., Elliott, S.R., Engel, J.: Double beta decay, Majorana neutrinos, and neutrino mass. Rev. Mod. Phys. 80, 481–516 (2008) 6. Alicea, J.: New directions in the pursuit of Majorana fermions in solid state systems. Rep. Prog. Phys. 75, 076501 (2012). https://doi.org/10.1088/0034-4885/75/7/076501 7. Atiyah, M. F., Singer, I. M.: The index of elliptic operators on compact manifolds. Bull. Amer. Math. Soc. 69, 422–433 (1963); Atiyah, M. F., Patodi, V. K., Singer, I. M.: Spectral asymmetry and Riemannian geometry. Bull. London Math. Soc. 5, 229–234 (1973). The great Indian mathematician Vijay Patodi (1945–1976), whose name is associated to those of Atiyah and Singer for proving the index theorem for elliptic operators based on the heat equation (M. F. Atiyah, R. Bott, and V. K. Patodi, On the Heat Equation and the Index Theorem, Invent. Math. 19, 279 (1973)) passed away prematurely, having the same age as Majorana when he disappeared 8. Weinberg, E.J.: Phys. Rev. D. 24, 2669–2673 (1981) 9. Haldane, F.D.M.: Nobel Lecture:Topological quantum matter. Rev. Mod. Phys. 89, 040502 (2017) 10. Bianconi, A., Maeno, Y. (eds.): Majorana Fermions and Topological Materials Science. Superstripes Press, Rome (2018). This abstract book of the 75th Course, chaired by Antonio

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Bianconi and Yoshiteru Maeno, of the International School of Solid State Physics (ISSSP), held at the Ettore Majorana Foundation and Centre for Scientific Culture in Erice, Italy on 21–27 July 2018, offers an up-to-date overview of the subject. This course was a followup of the 59th Course of ISSSP on Majorana Physics in Condensed Matter, 12–18 July 2013 (web.nano.cnr.it/mpcm13/MPCM2013Booklet.pdf). The concepts exposed in the lecture delivered by Frank Wilczek at the 2013 course can be found in F. Wilczek, Majorana and Condensed Matter Physics, arXiv:1404.0637 [cond-mat.supr-con] reported in the book by S. Esposito, The Physics of Ettore Majorana – Theoretical, Mathematical, and Phenomenological. Cambridge University Press (2014), pp. 279–302 11. Qi, X.-L., Zhang, S.-C.: Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011) 12. Kitaev, A.: Periodic table for topological insulators and superconductors. AIP Conf. Proc. 1134, 22–31 (2009) 13. Castro Neto, A.H., Guinea, F., Peres, N.M.R., Novoselov, K.S., Geim, A.K.: The electronic properties of graphene. Rev. Mod. Phys. 81, 109–162 (2009) 14. Barborini, E., Piseri, P., Milani, P., Benedek, G., Ducati, C., Robertson, J.: Negatively curved spongy carbon. Appl. Phys. Lett. 81, 3359–3361 (2002); see also: Ed Gerstner, “Spongy carbon”, Nat. Mater. Update, 7 Nov 2002 15. Benedek, G., Bernasconi, M., Cinquanta, E., D’Alessio, L., De Corato, M.: Chapter 12: the topological background of schwarzite physics. In: Cataldo, F., Graovac, A., Ori, O. (eds.) Mathematics and topology of fullerenes, Springer series on carbon materials chemistry and physics, vol. 4. Springer, Heidelberg/Berlin (2011) 16. Benedek, G.: When topology matters: the Nobel prize in physics 2016. Revue Quest. Sci. 188 (2), 1–7 (2017); Benedek, G.: Graphene as a quantum playground, in modern aspects of the combined applications of heat-electricity-mechanics, Rigamonti, A., Varlamov, A. (eds.) Istituto Lombardo Accademia di Scienze e Lettere (2018) pp. 77–104. http://www.ilasl.org/ index.php/Incontri/issue/view/47 17. Volovik, G.E.: The Universe in a Helium Droplet. Oxford University Press, Oxford (2003) 18. See, e.g., Unruh, W. G., Schützhold R. (eds.): Quantum Analogues: From Phase Transitions to Black Holes and Cosmology, Springer Lecture Notes in Physics Vol. 718 (2007); in particular the chapter by G. E. Volovik, pp. 31–73 19. Another interesting example is: Hartnoll, S. A., Herzog, C. P., Horowitz, G. T.: Holographic superconductors. JHEP 12, 015 (2008) 20. Leggett, A. L.: Majorana fermions in condensed-matter physics. Int. J. Modern Phys. B. 30, 1630012 (11 pp.) (2016) 21. Volovik, G.E.: Fermion zero modes on vortices in chiral superconductor. JETP Lett. 70, 609– 614 (1999) 22. Yu Kitaev, A.: Unpaired Majorana fermions in quantum wires. Phys. Usp. 44, 131–136 (2001) 23. Lutchyn, R.M., Sau, J.D., Das Sarma, S.: Majorana fermions and a topological phase transition in semiconductor-superconductor heterostructures. Phys. Rev. Lett. 105, 077001 (2010) 24. Oreg, Y., Refael, G., von Oppen, F.: Helical liquids and Majorana bound states in quantum wires. Phys. Rev. Lett. 105, 177002 (2010) 25. Mourik, V., Zuo, K., Frolov, S.M., Plissard, S.R., Bakkers, E.P.A.M., Kouwenhoven, L.P.: Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices. Science. 336, 1003 (2012) 26. Wilson, R.M.: Phys. Today. 65(6), 14 (2012). https://doi.org/10.1063/PT.3.1587 27. Das, A., Ronen, Y., Most, Y., Oreg, Y., Heiblum, M., Shtrikman, H.: Zero-bias peaks and splitting in an Al-InAs nanowire topological superconductor as a signature of Majorana fermions. Nat. Phys. 8, 887–895 (2012) 28. Deng, M.T., Yu, C.L., Huang, G.Y., Larsson, M., Caroff, P., Xu, H.Q.: Anomalous zero-bias conductance peak in a Nb–InSb nanowire–Nb hybrid device. Nano Lett. 12, 6414 (2012) 29. Finck, A.D.K., Van Harlingen, D.J., Mohseni, P.K., Jung, K., Li, X.: Anomalous modulation of a zero-bias peak in a hybrid nanowire-superconductor device. Phys. Rev. Lett. 110, 126406 (2013)

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30. He, J.J., Ng, T.K., Lee, P.A., Law, K.T.: Selective equal-spin Andreev reflections induced by Majorana fermions. Phys. Rev. Lett. 112, 037001 (2014) 31. He, J.J., Wu, J., Choy, T.-P., Liu, X.-J., Tanaka, Y., Law, K.T.: Correlated spin currents generated by resonant-crossed Andreev reflections in topological superconductors. Nat. Commun. 5, 3232 (2014) 32. Lee, E.J.H., Jiang, X., Houzet, M., Aguado, R., Lieber, C.M., De Franceschi, S.: Spin-resolved Andreev levels and parity crossings in hybrid superconductor–semiconductor nanostructures. Nat. Nanotechnol. 9, 79 (2014) 33. Tanaka, Y., Yokoyama, T., Nagaosa, N.: Manipulation of the Majorana fermion, Andreev reflection, and Josephson current on topological insulators. Phys. Rev. Lett. 103, 107002 (2009) 34. Law, K.T., Lee, P.A., Ng, T.K.: Majorana fermion induced resonant Andreev reflection. Phys. Rev. Lett. 103, 237001 (2009) 35. Nadj-Perge, S., Drozdov, I.K., Li, J., Chen, H., Jeon, S., Seo, J., MacDonald, A., Bernevig, B.A., Yazdani, A.: Observation of Majorana fermions in ferro-magnetic atomic ahains on a superconductors. Science. 346, 602 (2014) 36. Liu, J., Potter, A.C., Law, K.T., Lee, P.A.: Zero-bias peaks in the tunneling conductance of spin-orbit-coupled superconducting wires with and without Majorana end-states. Phys. Rev. Lett. 109, 267002 (2012) 37. Read, N., Green, D.: Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect. Phys. Rev. B. 61, 10267 (2000) 38. Kitaev, A.: Anyons in an exactly solved model and beyond. Ann. Phys. 321, 2–111 (2006) 39. Fu, L., Kane, C.L.: Superconducting proximity effect and Majorana fermions at the surface of a topological insulator. Phys. Rev. Lett. 100, 096407 (2008) 40. Fu, L., Kane, C.L.: Probing neutral Majorana fermion edge modes with charge transport. Phys. Rev. Lett. 102, 216403 (2009) 41. Akhmerov, A.R., Nilsson, J., Beenakker, C.W.J.: Electrically detected interferometry of Majorana fermions in a topological insulator. Phys. Rev. Lett. 102, 216404 (2009) 42. Xu, J.-P., Wang, M.-X., Liu, Z.L., Ge, J.-F., Yang, X., Liu, C., Xu, Z.A., Guan, D., Gao, C.L., Qian, D., Liu, Y., Wang, Q.-H., Zhang, F.-C., Xue, Q.-K., Jia, J.-F.: Experimental detection of a Majorana mode in the core of a magnetic vortex inside a topological insulator-superconductor Bi2 Te3 /NbSe2 hetero-structure. Phys. Rev. Lett. 114, 017001 (2015) 43. Sun, H.-H., Zhang, K.-W., Hu, L.-H., Li, C., Wang, G.-Y., Ma, H.-Y., Xu, Z.-A., Gao, C.L., Guan, D.-D., Li, Y.-Y., Liu, C., Qian, D., Zhou, Y., Fu, L., Li, S.-C., Zhang, F.-C., Jia, J.-F.: Majorana zero mode detected with spin selective Andreev reflection in the vortex of a topological superconductor. Phys. Rev. Lett. 111, 257 (2016) 44. Wang, M.-X., Liu, C., Xu, J.-P., Yang, F., Miao, L., Yao, M.-Y., Gao, C.L., Shen, C., Ma, X., Chen, X., Xu, Z.-A., Liu, Y., Zhang, S.-C., Qian, D., Jia, J.-F., Xue, Q.-K.: The coexistence of superconductivity and topological order in the Bi2 Se3 thin films. Science. 336, 52 (2012) 45. Chamon, C., Jackiw, R., Nishida, Y., Pi, S.-Y., Santos, L.: Quantizing Majorana fermions in a superconductor. Phys. Rev. B. 81, 224515 (2010) 46. Qi, X.-L., Hughes, T.L., Zhang, S.-C.: Chiral topological superconductor from the quantum Hall state. Phys. Rev. B. 82, 184516 (2010) 47. Qi, X.-L., Zhang, S.-C.: Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057 (2011) 48. Chung, S.B., Qi, X.-L., Maciejko, J., Zhang, S.-C.: Conductance and noise signatures of Majorana backscattering. Phys. Rev. B. 83, 100512(R) (2011) 49. Chang, C.-Z., Zhang, J., Feng, X., Shen, J., Zhang, Z., Guo, M., Li, K., Ou, Y., Wei, P., Wang, L.-L., Ji, Z.-Q., Feng, Y., Ji, S., Chen, X., Jia, J., Dai, X., Fang, Z., Zhang, S.-C., He, K., Wang, Y., Lu, L., Ma, X.-C., Xue, Q.-K.: Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator. Science. 340, 167–170 (2013) 50. Wang, J., Zhou, Q., Lian, B., Zhang, S.-C.: Chiral topological superconductor and half-integer conductance plateau from quantum anomalous Hall plateau transition. Phys. Rev. B. 92, 064520 (2015)

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51. He, Q.L., Pan, L., Stern, A.L., Burks, E.C., Che, X., Yin, G., Wang, J., Lian, B., Zhou, Q., Choi, E.S., Murata, K., Kou, X., Chen, Z., Nie, T., Shao, Q., Fan, Y., Zhang, S.-C., Liu, K., Xia, J., Wang, K.L.: Chiral Majorana fermion modes in a quantum anomalous Hall insulator– superconductor structure. Science. 357, 294 (2017) 52. Zhang, S.-C.: Discovery of the chiral Majorana fermion, in https://sitp.stanford.edu/ sites/default/files/discovery_of_majorana_0.pdf and Stanford video-lecture in https:// www.youtube.com/watch?v=sbJSAOngiGI 53. Banerjee, M., Heiblum, M., Umansky, V., Feldman, D.E., Oreg, Y., Stern, A.: Observation of half-integer thermal Hall conductance. Nature. 559, 205–210 (2018) 54. Kasahara, Y., Ohnishi, T., Mizukami, Y., Tanaka, O., Ma, S., Sugii, K., Kurita, N., Tanaka, H., Nasu, J., Motome, Y., Shibauchi, T., Matsuda, Y.: Majorana quantization and half-integer thermal quantum Hall effect in a Kitaev spin liquid. Nature. 559, 227–231 (2018) 55. Shtengel, K.: The heat is on for Majorana fermions. Nature. 559, 189–190 (2018) 56. March, N.H., Paranjape, B.V., Robson, R.E.: Nernst, Ettingshausen and Righi-Leduc phenomena in relation to the quantum Hall effect in two-dimensional electron assemblies. J. Phys. Chem. Solids. 54, 745 (1993) 57. Jackeli, G., Khaliullin, G.: Mott insulators in the strong spin–orbit coupling limit: from Heisenberg to a quantum compass and Kitaev models. Phys. Rev. Lett. 102, 017205 (2009) 58. Kim, H.-S., Shankar, V.V., Catuneanu, A., Kee, H.-Y.: Kitaev magnetism in honeycomb RuCl3 with intermediate spin–orbit coupling. Phys. Rev. B. 91, 241110 (2015) 59. Banerjee, A. A., Bridges, C. A., Yan, J.-Q., Aczel, A. A., Li, L., Stone, M. B., Granroth, G. E., Lumsden, M. D., Yiu, Y., Knolle,J., Bhattacharjee, S., Kovrizhin, D. L., Moessner, R., Tennant, D. A., Mandrus, D. G., Nagler, S. E.: Proximate Kitaev quantum spin liquid behaviour in a honeycomb magnet. Nat. Mater. 15, 733–740 (2016) 60. Chung, S.B., Zhang, S.-C.: Detecting the Majorana fermion surface state of 3 He-B through spin relaxation. Phys. Rev. Lett. 103, 235301 (2009) 61. Sato, M., Takahashi, Y., Fujimoto, S.: Non-abelian topological order in s-wave superfluids of ultracold fermionic atoms. Phys. Rev. Lett. 103, 020401 (2009) 62. Tewari, S., das Sarma, S., Nayak, C., Zhang, C.W., Zoller, P.: Quantum computation using vortices and Majorana zero modes of a px + ipy superfluid of fermionic cold atoms. Phys. Rev. Lett. 98, 010506 (2007) 63. Gül, Ö., Zhang, H., Bommer, J. D. S., de Moor, M. W. A. K., Car, D., Plissard, S. R., Bakkers, E. P. A. M., Geresdi, A., Watanabe, K., Taniguchi, T., Kouwenhoven, L. P.: Ballistic Majorana nanowire devices. Nat. Nanotechnol. 13, 192–197 (2018) 64. Ivanov, D.A.: Non-abelian statistics of hall-quantum vortices in p-wave superconductors. Phys. Rev. Lett. 86, 268 (2001) 65. Liang, Q.F., Wang, Z., Hu, X.: Manipulation of Majorana fermions by point-like gate voltage in the vortex state of a topological superconductor. EPL 99, 50004 (2012) 66. Wu, L.-H., Liang, Q.-F., Hu, X.: New scheme for braiding Majorana fermions. Sci. Technol. Adv. Mater. 15, 064402 (2014) 67. Kauffman, L.H., Lomonaco, S.J.: Braiding, Majorana fermions, Fibonacci particles and topological quantum computing. Quant. Inform. Process. 17, 201 (2018)

Biographical Notes on Ettore Majorana Francesco Guerra and Nadia Robotti

1 The Years of Education Up to His Degree in Physics Ettore Majorana was born in Catania, Sicily, on 5 August 1906 at a quarter past eight in the evening, the son of Fabio Majorana (1875–1934) and Salvatrice Corso (1873–1965), in the family home on Via Etnea No. 251. His family enjoyed a comfortable economic and social position. In particular his father Fabio had graduated in Engineering and Physics and he had developed a brilliant career, first in Catania’s telephone company and then in the Ministero delle Comunicazioni (Communications Ministry) in Rome, also distinguishing himself for his refined work as an architect. In particular he designed and built Palazzo Rosa (1903–1906) in Via VI Aprile in Catania, an elegant example of Art Nouveau. His mother, Salvatrice, was the daughter of Luciano Corso, an affluent engineer. The forefather of the family on his father’s side, Ettore’s grandfather, was Salvatore Majorana Calatabiano (1825–1897), who came from Militello in Val di Catania, with a strong personality as lawyer, member of parliament, minister, senator, professor at Catania university, economist, legal expert, orator, writer, as his son Giuseppe described him in a brief biographical note. Salvatore, considered “the creator of an ambitious project for an intellectual dynasty”, marriedas his second

F. Guerra () Department of Physics, University of Rome “La Sapienza”, INFN, Rome Section, Rome, Italy Enrico Fermi Historical Museum of Physics and Study and Research Centre, Rome, Italy e-mail: [email protected] N. Robotti Department of Physics, University of Genoa, INFN, Genoa Section, Genoa, Italy Enrico Fermi Historical Museum of Physics and Study and Research Centre, Rome, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2020 L. Cifarelli (ed.), Scientific Papers of Ettore Majorana, https://doi.org/10.1007/978-3-030-23509-3_15

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wife Rosa Campisi (1834–1896), with whom he had no less than seven children in succession, Giuseppe (1863–1940), Angelo (1865–1910), Quirino (1871–1957), Dante (1874–1955), Fabio (1875–1934), Elvira (1877–1944), Emilia (1879–1967), all of whom achieved an excellent economic and social position. No less than three of the brothers, Giuseppe, Angelo, Dante, Ettore’s uncles, were appointed full professors in law and economics in the Law Faculty of the University of Catania. They also had a brilliant political career. Quirino became a famous experimental physicist, full professor at Bologna, president of the Società Italiana di Fisica (Italian Physical Society), director of the journal Nuovo Cimento, member of the Accademia dei Lincei, and winner of the prestigious Mussolini Prize in 1940. Ettore was the fourth child of Fabio and Salvatrice, preceded by Rosa (1901– 1972), Salvatore (1903–1971), Luciano (1905–1967), and then followed by Maria (1914–1997). Ettore’s early primary education took place at home, according to the custom of the times. He then transferred to Rome, from 1915, as a boarding pupil of the prestigious Istituto Massimiliano Massimo, run by the Jesuit Fathers, then situated in the Palazzo Massimo close to the Stazione Termini railway station, now the premises of the Museo Romano (Museum of Rome). He spent 7 years there, from the fourth year, the final year of primary education at the time, up to the second year of liceo, skipping the fourth year of the ginnasio. (The liceo covered the final 3 years of secondary education in Italy and the previous years took place in the ginnasio.) From 1920 he was a day pupil since his family had transferred from Catania to Rome. His school reports were good but not exceptional, even with a few resits in September, also because of low marks for bad behaviour. After the second year of liceo, Ettore left the Massimo, obtaining in the autumn of 1922, at the Reale Liceo “Nicola Spedalieri” in Catania, admission to the third year of liceo in the state school system. He then registered at the Reale Liceo “Torquato Tasso” in Rome, where he attended regularly for the third year and obtained the liceo final certificate without examinations, on the basis only of his final marks, in the summer of 1923. Following a clear family tradition in November 1923 Ettore registered for the preparatory 2 year course for engineering students at the Faculty of Mathematical, Physical and Natural Sciences of the University of Rome. After obtaining the relevant diploma, he passed to the School of Engineering, where he spent 3 years characterised by a bizarre curriculum. Indeed, in the first 2 years he followed the courses regularly and passed all the corresponding examinations, as can be seen from the records kept at the Student Office. With all due respect to those who have claimed the contrary even the examination in Hydraulics was passed with the respectable mark of 75/100. Instead, in the final year, which basically covered 1928, Ettore did not tackle the courses and examinations in Engineering and only sat one examination, for the course in Theoretical Physics taught by Enrico Fermi (1901– 1954) at the Faculty of Mathematical, Physical and Natural Sciences, on 5 July 1928 with a mark of 100/100 cum laude.

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It was only at the end of 1928 that Ettore asked to transfer to the Degree course in Physics, after careful scheduling that also included, as well as the examination in Theoretical Physics, attendance of the first year of the 2 year courses in Advanced Physics and Physics exercises, taught respectively by Antonino Lo Surdo (1880– 1949) and Franco Rasetti (1901–2001). It was precisely because of this curriculum that the Faculty Board, in the meeting of 19 November 1928, granted his registration for the fourth year of Physics. It was then easy for Ettore to pass in June–July 1929 the remaining examinations required in Mathematical Physics, Earth Physics, Advanced Physics and Physics Exercises, all with the mark of thirty cum laude except for Physics Exercises in which he had to settle for only the top mark, to then pass the Degree in Physics with 110/110 cum laude in the session of 6 July 1929 at the age of 23. The reasons for his disappointing mark in the examination in Physics Exercises, sat only 1 day before his degree examination, are certainly due to young Ettore’s excessively critical, sometimes even sarcastic, character with regard to the surrounding academic environment. In a significant document, housed in the Domus Galilaeana in Pisa and dated 13 February 1928, Ettore wrote a detailed report on an experiment, devoted to the tuning of an electrical instrument, an electrodynamometer, as part of the course of Physics Exercises taught by Franco Rasetti. He noted that the formula that had been proposed was not suitable for expressing with sufficient approximation the angle of deviation of the instrument relative to the electrical current passing through it. Moreover, from an operational point of view, he complained that the small movable mirror connected to the instrument did not reflect the light well so that the precision of the measurements was not very satisfactory. At the end of the report Ettore added the following comment in brackets “(The measurements were taken and interpreted according to rules imposed by the competent authority, the basis for which has remained obscure to us)”. It is natural to think that “the competent authority”, in other words the teacher of the course, was not indifferent to comments of this type, certainly repeated on other occasions. Here we can see in embryo the beginning of the underlying contrast between Ettore and the “Via Panisperna boys” and perhaps one of the first reasons for the nickname “Great Inquisitor” that would be given to him. So Ettore graduated in Physics aged only 23 but he stayed in the university for almost 6 years, from November 1923 to July 1929, earning the diploma of Degree in Physics which only required a 4 year course of study. His performance as a student may seem disappointing but that was not the case. In reality from early 1928 Ettore was involved in intense scientific research on highly advanced subjects in atomic physics and nuclear physics, closely connected to Enrico Fermi but essentially in circumstances of complete cultural independence and great originality.

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2 Ettore Majorana’s Scientific Research While Still a Student From the beginning of 1928 Ettore Majorana, while an Engineering student but with the intention of soon transferring officially to Physics, was involved in intense research activity on the applications, improvement and extension of the statistical model of atoms introduced by Enrico Fermi only a few months before. As is well known, Enrico Fermi was appointed professor of Theoretical Physics in Rome at the beginning of 1927, as part of a general strategy followed by the Director of the Physics Institute, Orso Mario Corbino (1876–1937), with the aim of developing in Rome research into the most advanced modern physics. Fermi’s scientific results and the stages of his career are well known. Amongst other things we recall Fermi’s statistics, the statistical model of the atom, the theory of weak interactions, the discovery of artificial radioactivity induced by neutrons, the effects of slow neutrons, the Nobel Prize for Physics in 1938, his later emigration to the United States, the nuclear pile, the Manhattan project, his involvement in the physics of elementary particles and of accelerators, and the first electronic computers. Majorana immediately mastered Fermi’s statistical model of the atom and provided an important application of it to the splitting of the energy levels of some atoms, as a consequence of the spin of the electron, in an article written in collaboration with Giovanni Gentile Jr. (1906–1942), presented by Orso Mario Corbino at a session of the Accademia dei Lincei on 24 July 1928 and then published in the Academy’s Proceedings. Remember that Majorana had passed Fermi’s Theoretical examination only a few days earlier. Giovanni Gentile Jr., son of the powerful philosopher and politician Senator Giovanni Gentile (1875–1944), after graduating in Pisa in 1927 was hired by Corbino for a temporary post as assistant in Rome. Here he developed his collaboration with Majorana. The work by Gentile and Majorana was developed completely within the setting of Fermi’s general framework and achieved considerable success, so much so that Fermi cited and used these results in his later work. Majorana then continued his research in complete autonomy and arrived at results that constituted a real improvement in Fermi’s general proposal. Majorana communicated these results at the session on 29 December of the Italian Physical Society to an audience made up of the most important Italian scientists, including amongst others Enrico Fermi, Orso Mario Corbino, Tullio Levi Civita (1873–1941), Quirino Majorana, Enrico Persico (1900–1969), Giovanni Polvani (1892–1970), Giancarlo Vallauri (1882–1957), Vito Volterra (1860–1940). The improvement developed by Majorana is based on very profound and brilliant physics ideas and contains a remarkable methodological content. Majorana’s scientific communication to the conference not only demonstrates the remarkable results that he had obtained in only a few months of activity but is also a proof of his courage and of his determination to make his results known immediately to the scientific community. Ettore was certainly not timid and he greatly desired his

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efforts to receive fair recognition. Moreover he showed an arrogant attitude, well aware of the value of his ideas. After his report to the Rome conference in December 1928 Majorana’s interests shifted towards nuclear physics, which then constituted the most advanced frontier of scientific research in physics, even if the structure of the nucleus was not yet completely known. Only a few months earlier George Gamow (1904–1968), a Soviet nuclear physicist with close ties to the West, had proposed a brilliant mechanism based on the quantum tunnelling effect in order to explain the natural radioactive decay that some heavy nuclei undergo spontaneously, with the emission of alpha particles identified as nuclei of helium with very high energy. Unfortunately his interpretation had been heavily criticised in scientific circles because of its insufficient rigour in the mathematical treatment of the quantum phenomenon. Majorana, in his usual style, stepped into the discussion with assurance and demonstrated rigorously, using Schrödinger’s equation for the alpha particle in the nucleus in an appropriate physical context, that there actually is a non zero probability that the particle can escape from the nuclear trap and be found far away from the nucleus with a very great velocity. These results, together with other developments, were at an international level and were presented lucidly in his degree thesis with the title “On the mechanics of radioactive nuclei”. His supervisor was Enrico Fermi. The possibility of successfully applying quantum mechanics to the nucleus, at least in the case of alpha decay, was an important prelude to the development of complete nuclear models which would only be possible after the discovery of the neutron and which would see Majorana as protagonist. So we have seen that Majorana, within a few months and while still a student, managed to obtain substantial, well documented results on the statistical model of the atom and on alpha nuclear decay that improved the original frameworks of Fermi and Gamow, respectively.

3 From His Degree to His Libera Docenza (Independent Lectureship Qualification) This period, that stretches from 1929 to 1932, shows very clearly a peculiar aspect of the young Majorana’s academic career. Despite the relevance of the scientific results obtained after his degree on cutting edge topics he was not offered any post, not even a temporary one, in the Physics Institute in flagrant contradiction of the strategy to develop advanced modern physics and with a treatment that was different to that reserved for other young people. So Majorana “frequented the institute freely, following the scientific flow”, as he himself wrote in his application for the libera docenza. In any case his scientific

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activity continued regularly. One of the topics of his research was molecular physics, in accordance with an international approach that aimed to explain chemical bonds using methods of quantum mechanics. His friend Giovanni Gentile Jr. was visiting Germany. An intense collaboration developed between them. Their correspondence shows Ettore’s strong influence on Giovanni. However the two preferred to publish their results separately. It must be said that at that time selection boards generally did not look favourably on papers with more than one name because it was difficult to separate the original contributions of each author. In particular, we can identify an accurate subdivision of research subjects between the two young men, an indication of another aspect of Ettore’s attention towards the procedures of academic conduct. At the end of 1930 Ettore had two articles ready, the first on the formation and stability of the positive molecular ion of helium, then published in Nuovo Cimento, the second on some interactions between hydrogen atoms, published in the Proceedings of the Accademia dei Lincei. The papers showed complete mastery of the methods of quantum mechanics affecting the explanation of chemical bonds and made Majorana a sort of pioneering precursor of Theoretical Chemistry, a sector that would only be fully developed in Italy 20 years later. Other papers followed on problems of spectroscopy. They focussed on the theoretical interpretation of some new lines discovered in the spectrum of Helium and of some anomalies present in the spectrum of Calcium. They are authoritative works, commonly appreciated as examples of elegant applications of mathematical group theory but they also contain new physics ideas. In particular Majorana was the first to recognise the spectroscopic consequences of the phenomenon of selfionisation in atoms. Later, in 1932, Majorana produced very interesting results on the behaviour of polarised molecular beams crossing through rapidly variable magnetic fields, a cutting edge topic at that time, studied experimentally in many laboratories. In a paper in Il Nuovo Cimento, with the title “Atomi orientati in campo magnetico variabile” (Oriented atoms in a variable magnetic field), Majorana not only expounded the whole quantitative theory of the reversal of the angular momenta of molecules due to quantum effects but also found that to enhance the phenomenon it is best to pass the molecular beams through zones close to points where the magnetic field cancels out. Given the importance of the proposal, he also took care to publish in advance a brief announcement of his results in a Letter to Ricerca Scientifica, the official Journal of the Consiglio Nazionale delle Ricerche (National Research Council), in order to ensure that he would have priority. Majorana’s proposal had an immediate impact. It was promptly adopted successfully by Otto Robert Frisch (1904–1979) and Emilio Segrè (1905–1989) who at that time were working at the laboratory of Otto Stern (1888–1969) in Hamburg. Majorana’s results survived over time. More than 80 years later his methods are still fruitful. 1932 also signifies for Majorana his entry into the sector of elementary particles and of relativistic quantum theory with his paper “Teoria relativistica di particelle con momento intrinseco arbitrario” (Relativistic theory of particles with arbitrary intrinsic angular momentum), published in Il Nuovo Cimento.

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His motivations were very clear and precise regarding his dissatisfaction with Dirac’s famous equation, both with regard to the presence of states with negative energy and for the limitation to particles with spin s = 1/ 2. He therefore launched himself into the effort of finding equations valid for any intrinsic angular momentum and without states of negative energy. From a mathematical point of view he arrived at a great discovery, never put forward by others. He constructed a large class of infinite-dimensional unitary representations of the group of Lorentz transformations of Einstein’s special relativity. In physical terms he identified the space of the quantum states of relativistic particles with arbitrary intrinsic angular momentum and positive energies. The subject was very advanced and perhaps was somewhat ahead of its time. The article passed almost unnoticed in the first few years but Majorana’s pioneering role was fully recognised in a famous article by Eugene Wigner (1902–1995) in 1939, in Annals of Mathematics, where all the representations are constructed with elegant group methods. Wigner observed: “The representations of the Lorentz group have been investigated repeatedly. The first investigation is due to Majorana who found all representations of the class to be dealt with in the present work, excepting two sets of representations”. Moreover Wigner himself, in an article immediately after the war, stated that the strategy based on general relativistic wave equations “was already followed in many ways and with important results for the first time in the fundamental work by Majorana, then followed by Dirac, Proca, . . . ” (“zuerst wohl in der grundlegen Arbeit von Majorana, später von Dirac, Proca, . . . ”). It must be said that the representations apparently excluded in the paper published in Nuovo Cimento are actually completely described in Majorana’s research notes kept in the Domus in Pisa. So Majorana preceded Wigner all down the line, even if he used different methods based on relativistic wave equations. Overall, the 3 year period 1930–1932 was a period of intense scientific activity for Majorana, characterised by results that were already of historic importance. But even more important results were appearing on the horizon. In May 1932 he applied for the libera docenza in Theoretical Physics, presenting six publications, a considerable number bearing in mind the fact that he had only graduated 3 years earlier. The scientific standard was still exceptional even though the paper on relativistic equations could not be presented because it had not yet been published. At that time obtaining the libera docenza, as the result of a national competition, offered the chance to teach an independent course at the university, without remuneration, for simple enthusiasts of the subject, while university assistants benefitted from a condition that they could not be dismissed or transferred against their will. The institution of libera docenza remained almost intact until 1971 when it was finally abolished. The Examining Board, made up of Enrico Fermi, Antonino Lo Surdo, Enrico Persico, in its meeting on 12 November 1932, greatly appreciated Majorana’s scientific publications which “are short but substantial notes, commendable for the interest of the topics and the elegance and acumen with which they are dealt”, and

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recognised that “in the lesson given in public on the theme “The second principle of thermodynamics”, the candidate showed original breadth of points of view and he appropriately demonstrated the logical links between the various questions dealt with, also with regard to their historical development”. Finally “the Board derives reasons for gratification in the truly remarkable activity and attitudes of this candidate and is glad to propose unanimously to Your Excellency [the Minister] that the libera docenza in Theoretical Physics be awarded to Dr Ettore Majorana”. The Ministry of National Education issued the order by decree on 21 January 1933.

4 His Visit to Leipzig in 1933 At the end of 1932 Ettore took a decision of great strategic significance. He asked the Consiglio Nazionale delle Ricerche, through Fermi, for a study grant to go abroad to work at the Theoretical Physics Institute in Leipzig, where Werner Heisenberg, with whom he immediately established excellent relations, was to be found. Werner Heisenberg was an extremely famous physicist, the main founder of modern quantum mechanics, for which he won the 1932 Nobel Prize for Physics announced in November 1933, and deeply involved in the forefront of scientific research. In the early months of 1932, after preliminary work by Walther Bothe (1891– 1957) at the University of Giessen, and by Frédéric Joliot (1900–1958) and Irène Curie (1897–1956) in Paris, James Chadwick (1891–1974), at the Cavendish Laboratory in Cambridge, discovered a new elementary particle, the neutron, with a mass similar to that of the proton, but with a zero electric charge, capable of interacting intensely with nuclear matter. Immediately after the announcement of the experimental discovery of the neutron by Chadwick, Heisenberg had the formidable intuition of developing a theory of the nucleus in the context of quantum mechanics, in which protons and neutrons are assumed as fundamental constituents interacting with each other, with a complex system of forces that included both the Coulomb repulsion between the protons and an exchange force between protons and neutrons, very similar to the one that governs chemical bonds, but of the repulsive type, as well as other forces between the protons and neutrons that had to ensure the stability of the nuclei. His results were expressed in a series of three papers published in the famous journal Zeitschrift für Physik in 1932–1933, under the common title “Über der Bau der Atomkerne” (“On the structure of the atomic nucleus”). In Leipzig Majorana had the chance to see Heisenberg’s third paper before it was published. He immediately realised that very significant changes were possible which greatly improved the model. All the forces were reduced to just the exchange force appropriately formulated, now of the attractive type, as well as, naturally, the Coulomb force. The changes Majorana made were the result of a profound

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conceptual methodological journey, in which the exchange force was introduced on a purely phenomenological basis, with the intent of providing the nuclei with substantial stability and constant nuclear density as the number of constituent protons and neutrons varied. Majorana’s nuclear model is very different from Heisenberg’s in its physical structure of the forces involved. Majorana published his results in the prestigious and widely read Zeitschrift für Physik in German in an article with the title “Über die Kerntheorie” (“On the theory of the nucleus”), which echoed the similar title of Heisenberg’s articles. He also published a brief announcement in Ricerca Scientifica and later an extended article in Italian in the same journal, to gratify those in the Consiglio Nazionale delle Ricerche who had provided him with financial support. It should be pointed out that Majorana’s publication strategy in this case proved to be particularly elegant and diplomatically effective. Despite its brevity of only 17 lines the announcement provided a clear and concise description of Majorana’s results in nuclear physics, and it was the opportunity to show everyone in Italy that he was obtaining results at the highest level in accordance with the planned programmes. When Heisenberg knew of Majorana’s results he immediately understood the advantages of the framework proposed by his young guest and began immediately to disseminate them during his lessons in Leipzig, but especially in his report to the important Solvay conference, held in Brussels in October 1933, that was already distributed to the major research centres that summer. It was young Ettore’s international scientific triumph. After Leipzig, Majorana also went to Copenhagen, on a visit to Niels Bohr (1885–1962) organised by his friend George Placzek (1905–1955), then he returned to Rome for a couple of weeks over the Easter vacation. Finally he went again to Leipzig, returning from there to Rome permanently in early August 1933, close to his twenty-seventh birthday. The year 1933 constituted a sort of crucial year of transition in the relationship between Ettore Majorana and the rest of the Physics Institute in Rome, in particular Enrico Fermi. First we transcribe a postcard from Emilio Segrè, dated 3 February 1933, and a letter from Enrico Fermi, dated 11 February1933, addressed to Majorana in Leipzig. These documents, kept in the Family archive, have been made available to us with commendable cultural awareness by Ettore Majorana Jr. Dear Ettorre, since you left we haven’t heard any news from you. I hope you are getting along and that you’ve already settled in with the local element. In any case get in touch and tell us a bit about you and the Leipzig physicists. There are no great changes here. Fermi and I are writing on hyperfine structures. Ado is making short waves or trying to, Rasetti Bi crystals and the others entsprechend. There’s a lot of flu going around and plenty in the institute have caught it but now they’ve recovered. There isn’t any sensational physics news around, unless you

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can provide us with some. Best wishes for a pleasant stay in Leipzig and affectionate greetings from all and in particular from me. Write to us. Emilio Segrè Rome 11 February 1933 Dear Majorana, thank you for your letter. I seem to understand that you have already settled in pretty well in the Leipzig environment. In a few days I will send you a copy of the manuscript on hyperfine structures that Segrè and I are laboriously refining these days. Nothing sensational is happening here. Not finding anything sensible to do in theory I have started working experimentally and together with Rasetti we are trying out various methods to make existing techniques for radioactive measurements worse; there is no denying that we have had some success in this direction. We are waiting for news from Peierls about the conference you have in these days. You should tell Bloch from me, if Peierls hasn’t already told him, that there is no problem with him coming here from the first of October because I am not usually here yet but the institute is open. I advise him not to come in September because it will still be pretty hot and he would find the institute deserted. Best wishes and greetings. Say hello from me also to Heisenberg, Debye, Hund, Bloch, etc . . . - signed E. Fermi – (Enrico Fermi) This correspondence bears witness above all to the cordial relations that existed at that time between Majorana on the one hand and Segrè and Fermi on the other, but it also provides important information on the scientific activity in Rome then. We can clearly see that research activity in nuclear physics was languishing. In reality in Rome the importance of the neutron had not yet been understood, and nuclear physics research was still programmed towards gamma nuclear spectroscopy to be studied by means of diffraction by bismuth crystals. A fanciful programme that would not lead to any outcome. After Majorana’s return to Rome there were no more direct contacts between Fermi and Majorana, in practice interrupted at the end of 1932, as Majorana himself declared in an official document presented for the competition for a university chair in 1937. Indeed Majorana’s relations with the whole Via Panisperna group had certainly deteriorated profoundly, for reasons that are not easy to analyse because of a lack of probative documentation. We cite a very significant episode. On 9 November 1933 the Swedish Science Academy announced that the 1932 Nobel Prize for Physics had been awarded to Werner Heisenberg. Immediately all exponents of world culture sent their congratulations, in the most varied forms. Heisenberg kept all the messages he received in a file in his personal archive. Amongst these a telegram transmitted by Deutsche Reichspost stands out, sent from Rome and dated 11-XI-33. The text in German, in a very formal and cold style, reads: “HERZLICHSTE GRATULATIONEN CORBINO FERMI RASETTI

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SEGRE AMALDI WICK” (“Most heartfelt congratulations . . . ”). We can note that the order of the signatures strictly follows the academic seniority at the Physics Institute in Rome. Ettore Majorana, who was in Rome at that time, is not included in the list, not even in final place. But Majorana, who had left Leipzig at the beginning of August three months before, on the same date also sent him his “Gratulationen” independently, following his own style naturally. His message, very intense, was written in touching Italian on the two sides of a very small personal business card, on which in the printed heading the title “Dr.” had been struck out with a pen stroke. This is the text: Professor, Allow me (if you have not forgotten me!) to express to you my ardent best wishes on the occasion of the new solemn acknowledgement of your prodigious work. With deep admiration Yours Ettore Majorana

5 Years of Silence 1933–1937 As we have seen, Ettore retuned to Rome at the beginning of August 1933 while he was at the peak of his personal scientific prestige, with full international recognition. There were therefore the best conditions for him to be integrated fully into the academic life of the Physics Institute in Rome. In any case, having obtained the libera docenza would have allowed him to give a prestigious course on cutting edge subjects, in particular on the new quantum theory of nuclei to which his contribution had been decisive. Moreover, one would have expected there to be a fruitful collaboration between Fermi and Majorana, especially on themes connected to the development of nuclear physics. In reality events developed very differently. Relations between Ettore and the Physics Institute in Rome faded away ever more progressively. Moreover we witness a remarkable and unexpected disappearance of Ettore Majorana’s name from all the scientific literature published over four long years from 1933, after the articles on nuclei, until 1937. No papers, no announcements, no reports to conferences are to be found in the scientific journals or the proceedings of conferences and meetings at that time. This is even more surprising since in that same period there was a sudden awakening of nuclear physics research in Rome due to Enrico Fermi. After a phase essentially of stagnation a cycle of impressive results opened up, including the formulation of a brilliant theory on beta nuclear decay, the discovery of neutron induced radioactivity, the recognition of the effects produced by slow neutrons, the study of the properties of diffusion, absorption and slowing down of neutrons in matter. This extraordinary cycle of research reached its crowning achievement when the 1938 Nobel Prize for Physics was awarded to Enrico Fermi.

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The success of this research is certainly due to Fermi’s profound physics intuition and to his courage and determination when tackling extremely difficult problems with the modest means available in Via Panisperna. In any case, Majorana’s contribution in launching this research cycle was decisive. We will try to give a brief summary. The Heisenberg-Majorana nuclear model assumes only protons and neutrons as elementary constituents of nuclei, held together by a system of forces that can be dealt with perfectly in the context of quantum mechanics. In Heisenberg’s formulation of the exchange forces the virtual existence of electrons in the nucleus, rapidly exchanged between protons and neutrons, still remains. But in the definitive formulation given by Majorana the electrons were completely eliminated from the nucleus, even as virtual particles. Indeed, the form of the exchange force introduced by Majorana did not presuppose that there were exchanged electrons. This was a fundamental observation. For Majorana there were no electrons in the nucleus. So let us follow the thread of Fermi’s reasoning on this theme down to its final consequences. At the Solvay conference in October 1933, which Fermi attended, Heisenberg’s report made it clear to everyone that the model of the nucleus, as constructed by Majorana, was the valid one. So there were no electrons in the nucleus, not even virtual ones. So there was the problem of understanding how an electron could be expelled from the nucleus during beta decay. The solution proposed by Fermi, in December 1933, was simple and brilliant. Let us start with the analogy of the emission and absorption of electromagnetic radiation in the form of photons (quanta of light) when the external electronic cloud of the atom is changed. These photons obviously do not pre-exist in the atom before the emission but are created at the moment of the emission, and analogously can be absorbed in the inverse process. This complex mechanism is described well in the context of quantum electrodynamics in which Fermi was a great expert. Fully analogously with the electromagnetic case, Fermi admitted that the electron of a beta decay did not pre-exist in the nucleus, in full agreement with what Majorana maintained, but was created in the moment when the phenomenon occurred, following the transformation of a nuclear neutron into a proton. To take account of the conservation of the angular momentum and of the energy distribution, Fermi assumed that in beta emission another particle with extremely small mass and zero charge, hypothesized a few years earlier by Wolfgang Pauli (1900–1958) and that Fermi called “neutrino”, was emitted at the same time. Fermi’s theory was put in a well defined form also from a quantitative point of view and turned out to be in agreement with experiment, successfully explaining perfectly all the physical properties of beta decay. It was one of the most important results in the history of physics. The whole description of weak interactions, one of the four fundamental interactions of Nature, responsible also for beta decay, is still based essentially on Fermi’s ideas. But this was only a first step. Later Fermi, guided by his theory of beta decay, discovered neutron induced radioactivity and the effects when the neutrons were slowed down.

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In this exciting season in nuclear physics research Majorana, although in a certain sense he had been its conceptual initiator, was completely absent. They were years of silence. And yet the documents show that in those years he led a completely normal life, also travelling a great deal between Rome, Catania in Sicily, Abbazia in the province of Fiume (now Opatija in Croatia), Merano (Meran) and Vipiteno (Sterzing) in the South Tyrol, Passopisciaro near Catania, Marino and Monte Porzio Catone in Lazio. He also collaborated willingly with his uncle Quirino’s research into a supposed new type of photoelectric effect and on a multiple interferometer. Certainly Majorana was actively engaged in cutting edge scientific research, as shown for example by a letter to his uncle Quirino in January 1936, in which he declared that he was involved in the study of quantum electrodynamics, a subject of very great interest. Here we can mention, to be thorough, a further conceptual consequence of Majorana’s theory of the atomic nucleus. As a matter of fact, a few years later in 1935, Hideki Yukawa was able to prove that Majorana’s exchange force can be interpreted through a quantum field where the exchanged virtual particles are new ones with zero spin and -1 intrinsic parity. It was only in the late Forties that Yukawa particles were discovered in cosmic rays (the π mesons). Yukawa was awarded the Nobel Prize in 1949 “for his prediction of the existence of mesons on the basis of theoretical work on nuclear forces”. Majorana’s persistent international fame in the nuclear sector is amply documented by a handwritten letter in English dated 21 April 1935, addressed to Signor Majorana, kept in the Family archive, in which he was invited to attend an international nuclear physics conference planned for the following 20–30 September at the Physical-Technical institute in Leningrad. The letter was signed by M. Bronstein (Matvei Petrovich Bronstein (1906–1938)), Secretary of the conference. Signor Majorana, by immediate return post, answered at once in French accepting the invitation: Je suis heureuse d’accepter e d’avoir l’occasion de connâıtre à la fois vos éminents physiciens et votre grand et beau pays (I am glad to accept and to have the opportunity at the same time to know both your eminent physicists and your great and beautiful country). Unfortunately, given the difficult times, the conference was only held 2 years later, in 1937 and in Moscow, without Majorana attending. It is significant that the other two Italians invited to the 1935 conference were Enrico Fermi and Bruno Rossi (1905–1993). So Fermi, Majorana and Rossi were the points of reference for Italian nuclear physics at that time. A fine acknowledgement for Majorana. Majorana’s interest in his academic career, even during these years of silence, is impressively borne witness to by his courageous and persistent attempt to carry out his libera docenza course. Every year he presented to the Consiglio di Facoltà (Faculty Board) different and very advanced programmes for the proposed course but he was unable to teach it. From the minutes of the Faculty one can deduce that the atmosphere was not favourable. His proposal for the academic year 1936– 1937 to hold a course of “Quantum Electrodynamics” was actually rejected by the Faculty with a resolution according to which the course should be considered “not

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equivalent” to the official courses. A decision that would discourage any student from following the course. The article “Il valore delle leggi statistiche nella fisica e nelle scienze sociali” (The value of statistical laws in physics and the social sciences), which was an invited contribution in 1936 to a volume, never actually published, celebrating his uncle Giuseppe’s appointment as emeritus professor also dates from this period. The article, that developed pioneering considerations on probabilistic modelling of phenomena of a social nature, was then published posthumously in the journal Scientia in 1942, edited by Giovanni Gentile Jr., in a context that completely distorted its significance. The editor even prefixed the text with a sort of summary that was absolutely not by Majorana, as can be seen from the original manuscript and the typescript sent for publication in the commemorative volume. Moreover it is regrettable that amongst the editor’s comments sentences appear such as: “This article by Ettore Majorana – the distinguished theoretical physicist from the university of Naples who disappeared without trace on 25 March 1938 – was originally written for a sociology journal. But it was not published, perhaps because of the author’s moody reluctance to open up to others that too often persuaded him to lock up even important works in a drawer”. The article certainly was not intended for a sociology journal. Such an aim would have been crazily fanciful and amateurish for a Majorana used to very different publication strategies and interested in a very different academic career. It was instead a dutiful homage to his uncle’s retirement from university teaching. The article was not published for reasons completely unrelated to Majorana’s wishes, and certainly not for a presumed and grotesque “moody reluctance”, completely made up as an element in the malevolent interpretative canon of Majorana’s personality. Also from this period was the paper “Teoria simmetrica dell’elettrone e del positrone” (Symmetric theory of the electron and positron), then published in 1937, that we will speak of later. A part of its contents appears in fact in the detailed programme of his free course on “Elettrodinamica quantistica” (Quantum electrodynamics), proposed for 1936–1937 but never given.

6 The Awakening in 1937 and the Competition for University Chair In the first months of 1937 there was a significant awakening. Ettore resumed his publishing activity, submitting to Nuovo Cimento what is perhaps the most important article of his life, devoted to the symmetric theory of the electron and positron, and also containing the hypothesis of Majorana’s neutrino, that is to say a very light particle that is identical to its antiparticle. We also note that the title of the paper echoes, mockingly, some phrases used by Dirac a few years earlier in the

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form “symmetric theory of the electron and positron”, in a preliminary attempt, then abandoned, to interpret the holes in the Dirac Sea as protons. In any case the relevance of these ideas of Majorana’s are not limited to the physics of elementary particles but they also extend to solid state physics, where special cases of Majorana’s equations for the neutrino can describe electronic states in superconducting materials, with remarkable physical properties, that might even be useful in the construction of quantum electronic computers. These electronic states, named after “Majorana”, are currently the object di intense research, both theoretical and experimental. Majorana’s article was immediately published in Nuovo Cimento in April 1937. His official curriculum was considerably strengthened by it. There were the best conditions to take part in the competition for a post of professor of Theoretical Physics at the university of Palermo, announced on 15 March, the first in this discipline since the one won by Fermi way back in 1926. There is evidence to believe that the essential part of the 1937 paper was developed during, or immediately after, his stay in Leipzig in 1933, immediately after the positron was conclusively confirmed. One could therefore suppose that the “reawakening” in his publishing activity was closely linked to his wish to take part in the imminent Palermo competition. This would be a further clue to Majorana’s absolute normality. Any candidate in the run up to an important competition tries to strengthen his or her curriculum as much as possible. Moreover it is not possible to ignore the following facts. At the moment when the competition was announced in the Gazzetta Ufficiale (the Italian government’s official journal) on 15 March 1937, Ettore Majorana could not be considered a credible applicant for the competition. Let us remember that competitions for university chairs were based solely on qualifications basically comprised of scientific publications. Majorana’s scientific output, although of the highest quality, had stopped the year before. The examining board could question this lack of scientific continuity, a circumstance that would place a candidate in a difficult position, especially at the moment of the votes to establish the list of three winners’ names. On the other hand, the actual announcement of the competition had been possible on the assumption that Majorana could not be a candidate. We know for example, from correspondence between them, that Emilio Segrè at Palermo had already been in contact with Gian Carlo Wick, the future candidate with greatest scientific strength, to ensure that he would go to Palermo, with the admirable intent of consolidating the university’s scientific and teaching activity. If in the months preceding the competition Majorana’s curriculum had been above reproach perhaps Segrè would not have pushed for the chair in Theoretical Physics at Palermo to be put out to competition because Majorana would have been the certain winner, the first on the list of three. Majorana published his article immediately after the announcement of the competition, officially in April 1937, well before the deadline of 15 June, in order to be able to present it for the competition. But in this way he automatically became classified first. One may suspect that Majorana delayed publication of the article,

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and displayed the abnormal behaviour spoken of by some authoritative biographers, precisely in order to facilitate the chair being put out to competition, in a situation where apparently he could not be a candidate. If there is even a partial basis for these suspicions then Majorana’s academic strategy can be considered surprisingly refined and effective. Majorana took care to send Fermi a printed copy of his paper with a handwritten dedication To H. E. Enrico Fermi with many cordial greetings Ettore Majorana. Another similar copy was sent to Gian Carlo Wick, the successful theoretical physicist in Rome, who would certainly have been a very strong candidate in the competition. The dedication was now To Gian Carlo Wick with many cordial greetings Ettore Majorana. Fermi was entitled to be called “His Excellency” (H. E.) as a member of the Accademia d’Italia. It almost seems as if Majorana wanted to inform the certain chairman of the examining board and the strongest candidate that he was also in the running. Another copy was also sent to Giovanni Gentile Jr., another certain candidate. The subsequent business of the competition was conducted according to the usual process, with some significant surprises. After the deadline for applications, set by the 15 June announcement, the ministry machinery went into action to create an examining board. In a totalitarian regime the procedure was very simple and did not require complicatedly coordinated ballots. The head of the Minister’s private office drew up a list of possible board members, chosen from amongst professors of the subject or of similar subjects. The Minister chose the five board members with a simple stroke of a red pencil. Amongst them were of course Enrico Fermi and Enrico Persico, the only two available professors of Theoretical Physics. They were flanked by Giovanni Polvani from the university of Milan, Antonio Carrelli (1900–1980), from Naples, and Orazio Lazzarino (1880– 1963), from Pisa. The board was of a high scientific level and well balanced with regard to subject expertise and to representation of the various universities. Certainly the Minister had efficient advisors and his authoritarian strokes in red pencil were well justified. The Minister’s concern, and that of the universities involved, lay in conducting the competition rapidly so that the three winners on the list could immediately be called to the various universities, ensuring teaching for the beginning of the academic year. So the Board was convened on 25 October, straight after the summer break and Fermi’s planned return from one of his usual trips to the United States. Officially the board members only knew the names of the candidates at the first meeting. It was immediately evident that there was a problem. The law established that there should be three winners, ranked in order, so that the university that had requested the competition and the other universities could make the appointments. Instead the list of candidates showed unequivocally that there were four who were fully qualified to win, on the basis of research experience acquired and the scientific results attained. According to the natural ranking of qualifications they were: Ettore

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Majorana, whose application was perhaps a surprise, Gian Carlo Wick, with whom Emilio Segrè had already made an informal agreement for him to be appointed at Palermo, Giulio Racah (1909–1965), heavily backed by Pisa, and Giovanni Gentile Jr., heavily backed by Milan. If the board decided to abide by the objective order of merit, unpleasant disasters would ensue. Majorana would come first, and Palermo would be forced to appoint a winner who perhaps was not wanted, instead of the expected winner. Moreover Giovanni Gentile Jr. would be left out of the list of three winners, much to the disappointment of Milan university. In any case Giovannino was an excellent theoretical physicist and his exclusion would have been an injustice. But it was not difficult to find a brilliant solution that would satisfy everyone. “After a thorough exchange of ideas, the Board is unanimous in recognising the absolutely exceptional scientific position of Prof. Majorana Ettore who is one of the competitors. Therefore the Board decides to send a letter and a report to H.E. the Minister to suggest to him the advisability of appointing Majorana professor of Theoretical Physics for his high and deserved repute at one of the Universities of the Kingdom, independently of the competition requested by the University of Palermo”. In the letter to the Minister the Board explained also that it “was unanimously aware, after a thorough exchange of ideas, that of all the applicants Prof. Majorana Ettore has such a resounding national and international scientific position that the Board is reluctant to apply to him the normal procedure for university competitions”. Naturally the Minister, who was certainly well prepared for a request of this kind, on 2 November 1937 immediately issued an official order in which he decreed: “From 16 November 1937-XVI Prof. Ettore Majorana, due to the high reputation for remarkable expertise which he has attained in the field of study of theoretical Physics, is nominated Full Professor of theoretical Physics at the Faculty of Mathematical, Physical and Natural Sciences of the Royal University of Naples”. On 4 November the Minister communicated the order to the President of Naples University, asking him to inform the Faculty and the Majorana himself. The procedure was perfectly compliant with the law. Guglielmo Marconi, Nobel Prize winner for Physics in 1909, had also been appointed a few years earlier as full professor in Rome with the same procedure. The two cases are profoundly different of course. Once Majorana had been nominated at Naples, the Board for the Palermo competition was reconvened for 8 November and was able to return to its work without being hampered by the presence of a very awkward candidate. The work proceeded rapidly with the foreseen outcome and it concluded on 10 November. Top of the list of three was Gian Carlo Wick who was appointed at Palermo, the second was Giulio Racah appointed at Pisa, and the third was Giovanni Gentile Jr. appointed at Milan.

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7 Majorana in Naples. His Disappearance According to an almost universally accepted version Majorana received the communication of his appointment directly at his Rome address only on 10 January 1938. He then went to Naples and started his course. In fact, according to the official yearbook, Ettore Majorana was performing his duties in Naples from 16 November 1937, as required by the ministry’s order. Moreover the copious correspondence, preserved in the Majorana family archive in Catania, confirms that Majorana was actually in Naples from the middle of November. His uncle Giuseppe, who was in Rome at that time, wrote on 10 November to his relatives in Catania telling them, amongst other things, that Ettore was going to Naples and that Antonio Carrelli, a member of the competition examining board and director of the Physics Institute in Naples, had come to the house to meet him and to “come to an agreement about some immediate absences of his” [Carrelli’s], asking Majorana to teach the lessons of Experimental Physics in Naples in his stead. So when the board concluded its work on 10 November Ettore already knew, from Carrelli’s visit, that he had been appointed at Naples because of his irrefutable reputation, he knew the outcome of the competition as far as the other three winners were concerned, and he certainly could not refuse Carrelli’s request to replace him in his lessons because of “his immediate absences”. Unfortunately, a complete analysis of Majorana’s activity in Naples in November 1937 would require the support of documentation that is not yet available. With regard to this period, up to the end of 1937, we give here the text of an important letter, written to his uncle Dante in Catania, where some acute considerations on his “method” are described. We recall that Dante was extremely interested in questions of method. His first paper, published in the Antologia Giuridica (Law Anthology), Catania 1892, bore the title Sulla funzione metafisica del metodo e sulla sua ricerca (On the metaphysical function of method and the search for it). At that time Dante was only 18 and was still a student at the university of Rome. The letter is a copy, distributed by Dante to his brothers Giuseppe and Quirino. Rome 27–12 – 1937 XVI Dear Uncle Dante, I thank you deeply for your greetings that I return warmly to you and your family. Thank you also for your comments on method. Allow me to add an impression of my own. I believe in the unity of science but precisely because I really believe in it I think that, as long as practically distinct sciences exist with different aims, no error is so pernicious as the confusion of methods. In particular the mathematical method cannot be of any substantial use in sciences that are currently alien to physics. In other words, if one day the mathematical face of the simplest facts of life or of conscience is discovered this will certainly not happen due to a natural evolution of biology or psychology but only because some further radical transformation of the general principles of physics will allow its dominion to be extended in fields that

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are still alien to it. The most significant example is given by chemistry which, after having lived for a long time with great glory as an independent science, has over the last few years been completely absorbed by physics. This was made possible by the emergence of quantum mechanics while no useful relationship could be established between chemistry and classical mechanics. So until physics accomplishes further miracles the enthusiasts of the other sciences should be advised to rely on the methods typical of each, and not to seek models or suggestions from the physics of today, much less from the physics of yesterday. That physics may one day to be able to say the final word on matters of biology or morals is something we do not yet have any inkling of. Affectionate greetings Ettore His considerations are extremely lucid on the relationship between chemistry and physics, which it had only been possible to establish after the arrival of quantum mechanics because classical mechanics does not admit the explanation of chemical bonds and of the nature of chemical reactions. It then remains to be understood whether recent developments in the physics of complex systems can constitute, at least in embryo, the “further miracles” necessary for physics methods to be extended to “matters of biology or morals”. So in January 1938 Majorana began his course in Theoretical Physics, held in a small lecture room in the Physics Institute in Via Tari. The course, preceded by an inaugural lesson given on 13 January, although the text does not survive, was followed by seven devoted students. Of these five were registered for the degree course in Physics, Nella Altieri, Laura Mercogliano, Nada Minghetti, Sebastiano Sciuti, Gilda Senatore, and one in engineering, Mario Cutolo. But there was also an external attendee, Don Savino Coronato, the assistant of the famous mathematician Renato Caccioppoli (1904–1959), who followed the course in order to report on it. The contents of the course can be reconstructed from notes kept at the Domus Galilaeana in Pisa, and from the testimony of his students such as Gilda Senatore and Sebastiano Sciuti. The course began with an introductory part where a phenomenological description of the main aspects of quantum mechanics was provided, in the context of Bohr-Sommerfeld’s original approach. Subsequently, the general physical and mathematical framework of the new quantum mechanics of Heisenberg, Born, Jordan, Dirac, Schrödinger was presented in full detail. It was dealt with in a lively and concise way which is still perfectly valid today. The course had certainly been prepared in the previous years, during his attempts to give the libera docenza lessons. Many of the themes that had earlier been proposed for the free courses now were accommodated in the official course that was actually taught. Examining carefully the contents in detail one can also spot Majorana’s intention of differentiating his course from the one taught by Fermi which he himself had followed earlier, while still an Engineering student. For example, Sommerfeld’s relativistic corrections to Bohr’s theory of quantum orbits were dealt with by Majorana from the very beginning in all the details of mathematical development,

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while in the text of Fermi’s lessons there is only a brief reference at the end. Ettore Majorana tried to spark interest in the relativistic extensions of quantum mechanics beginning with the most elementary cases. Overall it is a course of a remarkably high level and constituted a real challenge for the students that followed it since their education in physics and mathematics must be assumed to be fairly modest. At the end of March 1938 Ettore Majorana suddenly disappeared, in circumstances that are still not entirely known, without there having been any warning signs recognisable as the cause of this drastic decision. The manner of his disappearance has been told in various and contradictory ways by various sources. We state again that in accordance with our choices and our firm intentions, declared from the beginning of our research into the Majorana case and to which we have always strictly adhered, we refrain here too from any illegitimate intrusion into these delicate personal questions. We believe that his decisions, whatever they were, must be respected. Altogether a dense journalistic and media commotion developed around the figure of Majorana in which everything, and the contrary of everything, was alleged, with no concrete basis and with no respect for Majorana’s character and for his family’s reserve and pain. This fuss also tarnished the presentation of his scientific, academic and cultural personality, various aspects of which were interpreted not in themselves but in the light of a presumed particular interpretation of the reasons for his disappearance. We know that Police Headquarters launched a search on the morning of 31 March 1938 with a telegraphic circular to the regional police forces in which it was asked to “trace the missing person Ettore Majorana, without the interested party being aware of anything”. Later circulars continued to urge for further searches. One of the automatic effects of the circular was that Majorana was added to the Frontier List that contained the names of individuals to be identified if they tried to cross the border. Ettore Majorana’s name remained on the List for about a year, until 22 April 1939 when the order was given to cancel it. The Police were never able to clarify the reasons, means and outcome of his disappearance.