The Physics of Ettore Majorana: Theoretical, Mathematical, and Phenomenological 1107044022, 9781107044029

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T H E P H Y S I C S O F E T T O R E M A J O R A NA

Through just a handful of papers, Ettore Majorana left an indelible mark on the fields of physics, mathematics, computer science, and even economics before his mysterious disappearance in 1938. It is only now that the importance of Majorana’s work is being realized: Majorana fermions are intensely studied today, and his work on neutrino physics has provided possible explanations for the existence of dark matter. In this unique volume, Salvatore Esposito not only explores Majorana’s known papers but, even more interestingly, also unveils his unpublished works. These include powerful methods and results, ranging from the atomic two-center problem, the Thomas–Fermi model, and ferromagnetism to quasi-stationary states, n-component relativistic wave equations, and quantum scalar electrodynamics. Featuring biographical notes and contributions from leading experts Evgeny Akhmedov and Frank Wilczek, this fascinating book will captivate graduate students and researchers interested in both frontier science and the history of science. Salvatore Esposito is Professor of the History of Physics and Associate Professor of Theoretical Physics, and Associate Researcher at the Naples Unit of the Istituto Nazionale di Fisica Nucleare. His research interests range from neutrino physics to field theory, and he is considered to be the world expert on Majorana’s work.

THE PHYSICS O F ETTORE M A J O R A NA Phenomenological, Theoretical, and Mathematical S A LVAT O R E E S P O S I T O With contributions by E. Akhmedov and F. Wilczek

University Printing House, Cambridge CB2 8BS, United Kingdom Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107044029 © S. Esposito 2015 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2015 Printed in the United Kingdom by Clays, St Ives plc A catalog record for this publication is available from the British Library ISBN 978-1-107-04402-9 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

Acknowledgments

Part I Introducing the character

page xii

1

1

Life and myth 1.1 Fortunes and misfortunes of a genius 1.2 Family and university training 1.3 Lone physicist in the Fermi group 1.4 Leipzig–Rome–Naples: the later years

3 3 6 9 12

2

The visible side 2.1 Ten papers depicting the future 2.2 Introducing the Dirac equation into atomic spectroscopy 2.3 Spontaneous ionization 2.3.1 Anomalous terms in helium 2.3.2 Incomplete P triplets 2.3.3 Majorana–Fano–Feshbach resonances 2.4 Chemical bonding 2.4.1 Helium molecular ion 2.4.2 Majorana structures 2.5 Non-adiabatic spin-flip 2.5.1 Majorana sphere and a general theorem 2.5.2 Landau–Zener probability formula 2.5.3 Majorana’s holes 2.5.4 Majorana–Brossel effect 2.6 Nuclear forces 2.6.1 The Heisenberg model of nuclear interactions 2.6.2 Majorana’s exchange mechanism

17 17 18 19 19 21 23 24 24 26 27 28 30 31 32 33 33 35

vi

Contents

2.7

2.8

2.9

2.6.3 Thomas–Fermi formalism and Yukawa potential Infinite-component equation 2.7.1 A successful relativistic wave equation 2.7.2 Majorana equation 2.7.3 Infinite-dimensional representations of the Lorentz group 2.7.4 A difficult problem for Pauli and Fierz 2.7.5 Further elaborations Majorana neutrino theory 2.8.1 “Symmetric” Dirac equation 2.8.2 Neutrino–antineutrino identity 2.8.3 Racah and the neutrinoless double β-decay 2.8.4 Pontecorvo and the neutrino oscillations 2.8.5 Majorana fermions Complex systems in physics and economics 2.9.1 Genesis of paper N.10 2.9.2 Statistical laws in social sciences 2.9.3 A sensational success in econophysics

37 38 39 40 41 42 44 47 47 49 50 51 52 53 54 55 57

Part II Atomic physics

61

3

Two-electron problem 3.1 A long-lasting success for quantum mechanics 3.2 Known solutions to the helium atom problem 3.2.1 Perturbative calculations 3.2.2 Variational method I 3.2.3 Self-consistent field method 3.2.4 Slater’s refinement 3.2.5 Variational method II: Hylleraas variables 3.2.6 Helium-like ions 3.3 Majorana empirical relations 3.4 Helium wavefunctions and broad range estimates 3.5 Accurate numbers and a general theory 3.5.1 A simpler alternative to Hylleraas’s method 3.5.2 Majorana’s variant of the variational method 3.6 Conclusions

63 63 64 64 66 68 69 70 71 72 76 78 78 79 81

4

Thomas–Fermi model 4.1 Fermi universal potential 4.1.1 Thomas–Fermi equation

83 83 83

Contents

4.2

4.3

4.4

4.5

4.1.2 Numerical and approximate solutions 4.1.3 Mathematical properties Majorana solution of the Thomas–Fermi equation 4.2.1 Transformation into an Abel equation 4.2.2 Analytic series solution 4.2.3 Numerical tables Mathematical generalizations 4.3.1 Frobenius method 4.3.2 Scale-invariant differential equations Physical applications 4.4.1 Modified Fermi potential for heavy atoms 4.4.2 Second approximation for the atomic potential 4.4.3 Atomic polarizability 4.4.4 Applications to molecules Conclusions

vii

85 86 87 87 89 92 93 93 95 97 98 100 102 103 105

Part III Nuclear and statistical physics

107

5

Quasi-stationary nuclear states 5.1 Probing the atomic nucleus with α particles 5.2 Scattering of α particles on a radioactive nucleus 5.2.1 Quantum-mechanical theory 5.2.2 Thermodynamic approach 5.3 Transition probabilities of quasi-stationary states 5.3.1 Transitions from a discrete into a continuum state 5.3.2 Transitions into two continuous spectra 5.3.3 Transitions from a continuum state 5.4 Nuclear disintegration by α particles 5.4.1 Statement of the problem 5.4.2 The appropriate wavefunction 5.4.3 Cross section 5.5 Conclusions

109 109 111 111 115 116 116 118 118 119 119 121 122 124

6

Theory of ferromagnetism 6.1 Towards a statistical theory of ferromagnetism 6.1.1 Molecular fields 6.1.2 Heisenberg theory 6.1.3 Later refinements 6.2 Majorana statistical model 6.2.1 Distribution function

127 128 128 129 131 131 134

viii

Contents

6.3

6.4

6.5

Solution of the model in the continuum limit 6.3.1 Partition function at finite temperature 6.3.2 Mean magnetization Applications and further results 6.4.1 Particular ferromagnetic geometries 6.4.2 Critical temperature and dimensionality Conclusions

136 138 139 141 141 143 144

Part IV Relativistic fields and group theory

147

7

Groups and their applications to quantum mechanics 7.1 The “Gruppenpest” in quantum mechanics 7.2 Unitary transformations in two dimensions 7.2.1 Dj representation and group generators 7.3 Three-dimensional rotations 7.3.1 Group generators 7.4 Application: the anomalous Zeeman effect 7.5 Lorentz group and its representations 7.5.1 n-dimensional Dirac matrices 7.5.2 Special case: maximum allowed p for fixed n 7.5.3 Non-Hermitian operators 7.5.4 Infinite-dimensional unitary representations 7.6 Conclusions

149 150 153 154 156 157 160 164 164 167 169 170 173

8

Dirac equations and some alternatives 8.1 Searching for an equation 8.1.1 Massive photons and the DKP algebra 8.1.2 Dirac–Fierz–Pauli formalism 8.1.3 General equations for arbitrary spin 8.2 Majorana n-component spinor equations 8.2.1 The 16-component equation for a two-particle system 8.2.2 Equation for a six-component spinor 8.2.3 Five-component equation 8.3 Parallel lives (and findings) 8.4 Conclusions

175 175 178 180 182 184 185 187 189 189 191

Part V Quantum field theory

193

Scalar electrodynamics 9.1 Early quantum electrodynamics

195 196

9

Contents

9.2 9.3 9.4 9.5 10

9.1.1 Quantum field formalism 9.1.2 Particles and antiparticles 9.1.3 Pauli–Weisskopf theory Majorana theory of scalar electrodynamics I Majorana theory of scalar electrodynamics II Application to the nuclear structure Conclusions

ix

196 198 198 201 205 209 212

Photons and electrons 10.1 Photon wave mechanics 10.1.1 Majorana–Oppenheimer formulation of electrodynamics 10.1.2 Lorentz-invariant wave theory 10.1.3 Two-component theory 10.1.4 Field quantization 10.2 Dynamical theory of electrons and holes 10.3 Conclusions

215 217 219 220 221 224

Part VI

227

Fundamental theories and other topics

214 215

11

A “path integral” approach to quantum mechanics 11.1 Dirac and Feynman’s mathematical approach 11.2 Majorana’s physical approach 11.3 Conclusions

229 230 232 234

12

Fundamental lengths and times 12.1 Introducing elementary space-time lengths 12.2 Quasi-Coulombian scattering 12.3 Intrinsic time delay and retarded electromagnetic fields 12.4 Conclusions

236 236 239 242 244

13

Majorana’s multifaceted life 13.1 Majorana as a student 13.1.1 Melting point shift due to a magnetic field 13.1.2 Determination of a function from its moments 13.1.3 WKB method for differential equations 13.2 Majorana as a phenomenologist: spontaneous and induced ionization of a hydrogen atom 13.2.1 Hydrogen atom placed in a high potential region 13.2.2 Ionization of a hydrogen-like atom in an electric field 13.3 Majorana as a theoretician: a unifying model for the fundamental constants

246 246 246 248 251 254 255 260 263

x

14

15

Contents

13.4 Majorana as a mathematician 13.4.1 Improper operators 13.4.2 Cubic symmetry 13.5 Majorana as a teacher

264 264 266 270

Part VII Beyond Majorana

277

Majorana and condensed matter physics f. wilczek

279

14.1 Spin response and universal connection 14.2 Level crossing and generalized Laplace transform 14.3 Majorana fermions and Majorana mass: from neutrinos to electrons 14.3.1 Majorana’s equation 14.3.2 Analysis of Majorana neutrinos 14.3.3 Majorana mass 14.3.4 Majorana electrons 14.4 Majorinos 14.4.1 Kitaev chain 14.4.2 Junctions and the algebraic genesis of majorinos 14.4.3 Continuum majorinos

280 282

Majorana neutrinos and other Majorana particles: theory and experiment e. akhmedov 15.1 Weyl, Dirac, and Majorana fermions 15.1.1 Particle−antiparticle conjugation 15.1.2 Dirac dynamics and the Majorana condition 15.1.3 Fermion mass terms and U(1) symmetries 15.1.4 Feynman rules for Majorana particles 15.2 C, P, CP, and CPT properties of Majorana fermions 15.3 Mixing and oscillations of Majorana neutrinos 15.3.1 Neutrinos with a Majorana mass term 15.3.2 General case of Dirac + Majorana mass term 15.3.3 Dirac and pseudo-Dirac neutrino limits in the D + M case 15.4 Seesaw mechanism of neutrino mass generation 15.5 Electromagnetic properties of Majorana neutrinos 15.6 Majorana particles in SUSY theories

285 285 287 289 292 293 294 298 301 303 304 306 307 312 313 314 317 317 322 325 327 330 338

Contents

15.7 Experimental searches for Majorana neutrinos and other Majorana particles 15.7.1 Neutrinoless double β-decay and related processes 15.7.2 Other lepton-number-violating processes 15.8 Baryogenesis through leptogenesis and Majorana neutrinos 15.9 Miscellaneous 15.10 Summary and conclusions Appendix A.1 A.2 A.3

Molecular bonding in quantum mechanics On the meaning of quantum state Symmetry properties of a system in classical and quantum mechanics Resonance forces between states that cannot be symmetrized for small perturbations and spectroscopic consequences. Theory of homopolar valence according to the method of bonding electrons. Properties of the symmetrized states that are not obtained from non-symmetrized ones with a weak perturbation

References Author index Subject index

xi

339 339 344 346 350 352 354 354 356

358 364 377 379

Acknowledgments

My interest in the (unpublished) scientific work by Majorana was stimulated many years ago by Erasmo Recami. I take this opportunity to thank him warmly for his continuous encouragement and constant interest in my own historical and scientific work about the protagonist of the present book. I have also greatly appreciated several talks and discussions with a number of my colleagues about different topics treated here; in particular, I express my gratitude to A. De Gregorio, E. Di Grezia, A. Drago, A. Naddeo, and G. Salesi. I also thank my colleagues G. Mangano, G. Miele, and O. Pisanti for their patience, especially when I repeatedly evoked the phantom of Majorana (through his writings) in their rooms at the Department of Physics in Naples. Last but not least, I am particularly grateful to E. Akhmedov and F. Wilczek for their enthusiasm in joining this project, as well as for their excellent contributions, which have greatly increased the value of the book.

Part I Introducing the character

1 Life and myth

On Saturday March 26, 1938 the director of the Institute of Physics at the University of Naples in Italy, Antonio Carrelli, received a mysterious telegram. It had been sent the previous day from the Sicilian capital Palermo, some 300 km across the Tyrrhenian Sea, and read: “Don’t worry. A letter will follow. Majorana.” That same Saturday, Ettore Majorana – who had just been appointed as full professor of theoretical physics at the university at the age of 31 – had not turned up to give his three-weekly lecture on theoretical physics. By Sunday the promised letter had reached Carrelli. In it Majorana wrote that he had abandoned his suicidal intentions and would return to Naples, but it revealed no hint of where the illustrious physicist might be. The picture was quickly becoming clear: Majorana had disappeared. Worried by these circumstances, Carrelli called his friend Enrico Fermi in Rome, who immediately realized the seriousness of the situation. Fermi was working in his laboratory with the young physicist Giuseppe Cocconi at the time. In order to give him an idea of the seriousness of the loss to the community of physicists caused by Majorana’s disappearance, Fermi told Cocconi: You see, in the world there are various categories of scientists: there are people of a secondary or tertiary standing, who do their best but do not go very far. There are also those of high standing, who come to discoveries of great importance, fundamental for the development of science [...]. But then there are geniuses like Galileo and Newton. Well, Ettore was one of them. Majorana had what no-one else in the world had.1

1.1 Fortunes and misfortunes of a genius Physicists working in several areas of research know quite well the name of Ettore Majorana, since it is currently associated with fundamental concepts like Majorana neutrinos in particle physics and cosmology or Majorana fermions in condensed 1 Cocconi letter to Edoardo Amaldi, dated July 18, 1965. Reported in Ref. [1].

4

Life and myth

matter physics. For non-specialists, the name of Ettore Majorana is usually intimately related to the fact that he disappeared rather mysteriously in 1938 and was never seen again. However, Ettore Majorana’s fame rests mainly – until now – on testimonies by the scholars who had the chance to know him personally [1, 2], like, for example, Bruno Pontecorvo, a younger colleague of Majorana at the Institute of Physics in Rome directed by Fermi: Some time after joining the Fermi group, Majorana already had such an erudition and reached such a high level of comprehension of physics that he was able to discuss with Fermi about scientific problems. Fermi himself held him to be the greatest theoretical physicist of our time. He often was astounded [...]. I remember exactly these words that Fermi spoke: “Once a problem has already been posed, no one in the world is able of solving it better than Majorana.” [3, 4]

Indeed, Majorana earned great respect in Fermi’s Institute, where he was considered a prodigy by all its associates, including Emilio Segrè, Edoardo Amaldi, Gian Carlo Wick, and many other scientists visiting Rome from their native countries in Europe and America. Werner Heisenberg, for example, who knew him when he visited Leipzig in 1933 (see below), considered Majorana a “very good physicist. [...] He was a very brilliant man. [...] He did excellent work. [...] I tried always to induce him to write papers and so he did finally write a very good paper” [5]. And it is quite intriguing and surprising that, while lecturing on nuclear forces at the VII Solvay Congress in 1933, Heisenberg mentioned only very rarely, and in a very hidden way, his own contribution to the problem, while almost always used expressions such as “d’apres Majorana,” “en suivant l’exemple de Majorana,” “comme y a insisté Majorana,” “nous choisirons avec Majorana.” [6] A more complete acknowledgment of Majorana’s work, during his lifetime, comes again from Fermi; by borrowing his own words, given in 1937 in the occasion of a meeting of the board in charge of the competition for a new full professorship in theoretical physics in Italy: Without listing his works, all of which are highly notable both for their originality of the methods as well as for the importance of the results achieved, we limit ourselves to the following. In modern nuclear theories, the contribution made by this researcher to the introduction of the forces called “Majorana forces” is universally recognized as the one, among the most fundamental, that allows us to understand theoretically nuclear stability. The work of Majorana today serves as a basis for the most important research in this field. In atomic physics, the merit of having resolved some of the most intricate questions on the structure of spectra through simple and elegant considerations of symmetry is due to Majorana. Lastly, he devised a brilliant method that deals with positive and negative electrons in a symmetric way, eventually eliminating the necessity to rely on the extremely artificial and

1.1 Fortunes and misfortunes of a genius

5

unsatisfactory hypothesis of an infinitely large electric charge diffused in space, a question that had been tackled by many other scholars without success. [7, 8]

With this justification, the board, chaired by Fermi, proposed to apply (for only the second time) a special bill passed a few years earlier in order to give a chair to the Nobel laureate Guglielmo Marconi, and suggested to the Minister of National Education “to appoint Majorana as full professor of Theoretical Physics at some University of the Italian kingdom, for high and well-deserved repute, independently of the competition rule.” The Minister accepted the proposal: evidently, such a reputation was sufficiently established on the basis of just a few (nine) papers published by the Italian scientist. Unfortunately enough, however, the University of Naples hosted his talent for three months only, until the end of March 1938, when Majorana gave his last lesson. He was always extremely pessimistic about physics. [...] I would say that he was perhaps not pessimistic about physics especially but rather about life in general. He was that kind of difficult fellow. Well, sometimes I thought perhaps he had had very difficult experiences in his life with other people, perhaps his girls or so. I don’t know. Anyway, I couldn’t make out why he, being such a young man, and such a brilliant young man, could always be so pessimistic. He was a very attractive fellow, so I liked him in our Leipzig group. I tried to see him frequently, and we had him at our ping pong games. Then I would sit down with him and ask him about not only physics but his private things and so on. So I tried to keep in touch with him. He was a very attractive fellow but very nervous so that he would get in a state of some excitement if you talked to him. So he was a bit difficult. Then all of a sudden he disappeared. I was very sorry to hear of it. [5]

It is quite striking that Majorana left an indelible mark of his science on a number of people, like Heisenberg, almost entirely independent of his own will, due to his extremely shy nature. It was very difficult [to communicate with Majorana]. Of course, people tried to talk to him and he was always very kind and very polite and very shy. It’s very difficult to get something out from him. But still, one could see at once that he was a very good physicist. When he made a remark it was to the point. [9]

Indeed, any people who knew Majorana share the same opinion (starting, of course, with his friends and colleagues in Rome), and it is a matter of fact that several nice anecdotes developed about his genius. This only apparently contrasts with a rather poor list of papers (only nine) authored by Majorana, since his peculiar character prevented him from publishing the results he obtained during his own researches, as alluded to by Heisenberg in the preceding quotation and, especially, testified by his colleagues at the Institute of Physics in Rome. The direct result of such a business was that his published contributions to physics were not at all widespread for decades (with the notable exception of his work on nuclear forces,

6

Life and myth

sponsored by Heisenberg), while his unpublished contributions did not affect the development of physics. The fact remains, however, that a number of results he did obtain – and not publish – were re-discovered only later, and much later, by other scholars, while some others still wait for proper consideration. The relevance of such a gap can only be glimpsed – for the moment – from testimonies who personally met Majorana. It is, therefore, useful to recall a final quotation from Fermi, who was very well known to be not at all prone to rave about anyone: Able to develop audacious hypotheses and, at the same time, to criticize acutely his own work and that of others; highly skilled in calculations and a deep-routed mathematician who never loses the very essence of the physical problem behind the veil of numbers and algorithms, Ettore Majorana has at the highest level that rare combination of abilities which form a first-rank theoretical physicist. Indeed, in the few years during which his activity has been carried out, until now, he has been able to outclass the attention of scholars from all over the world, who recognized, in his works, the stamp of one of the greatest minds of our times and the promise of further conquests.2

These further conquests will be the subject of the following chapters.

1.2 Family and university training Ettore Majorana was born on August 5, 1906 in Catania, a town in Sicily, Italy, to Fabio and Salvatrice (Dorina) Corso. Engineer Fabio Majorana, the chief of the local telephone company, was the last of five sons of Salvatore Majorana Calatabiano, who was Minister of Agriculture, Industry, and Trade in the two cabinets chaired by Prime Minister Agostino Depretis at the end of the nineteenth century. Ettore’s uncles also played key roles in the Italian society of their time. Giuseppe, a professor of finance and economy, was Rector of the University of Catania and deputy in three legislatures of the Italian Parliament. Angelo, a professor of constitutional law and of sociology, was Rector of the same university and acted as under-secretary and as minister (twice) in the cabinet chaired by Giovanni Giolitti. Quirino Majorana was, instead, a talented and illustrious experimental physicist, professor at the University of Bologna after Augusto Righi; he directed the Italian Physical Society for many years. Finally Dante, a lawyer, was a deputy and Rector too. The two younger sisters of Fabio Majorana, Elvira and Emilia, married important political personalities. The relevance and importance of the Majorana family in Italian life is thus clear. Also, Ettore’s mother belonged to a rich family of farmers and, from her marriage to Fabio Majorana, three sons and two daughters were born: Rosina, Salvatore (a lawyer and a lover of philosophy), Luciano (a civil engineer who specialized in the making 2 Fermi letter to Prime Minister Benito Mussolini, dated July 27, 1938. Reported in Ref. [1]. See also Ref. [10].

1.2 Family and university training

7

of aeronautical constructions and instruments for optical astronomy), Ettore, and Maria. The pronounced attitudes of the child Ettore, as well as his interest in physical sciences and astronomy, were cultivated directly by his father Fabio, with whom a particularly affectionate feeling was established. Studies corresponding to the first school years were performed at home (under the direction of Ettore’s father), but when Ettore was eight years old his mother decided that the young genius should study at a prestigious boarding-school in Rome, namely at the Istituto parificato Massimiliano Massimo, managed by Jesuits. His mother Dorina moved with her sons and daughters to Rome in 1921, followed by Ettore’s father three years later, when the telephone company became a state company and he was transferred to Rome. Following a common procedure, while approaching his final high-school examinations, for the last years of his course of studies Ettore (and his brother Luciano) moved from the Istituto Massimo to the equally prestigious Liceo Statale Torquato Tasso, where Ettore received his degree of licenza liceale in the summer of 1923. Majorana joined the Faculty of Engineering at the University of Rome in November 1923, and remained there (where he excelled) until the start of the fifth year of his course. Among his teachers, we may recall the renowned mathematicians Guido Castelnuovo, Tullio Levi-Civita, Francesco Severi, and Francesco Tricomi, as well as the physicist Orso Mario Corbino, Director of the Institute of Physics; their high-level courses gave him a solid preparation in calculus, geometry, rational mechanics, and experimental physics. An amusing anecdote regarding Majorana, and narrated by a colleague of his at the Faculty of Engineering, Emilio Segrè, is the following: Once, not having sufficiently prepared a lecture, Severi started a proof of a theorem the wrong way. Majorana immediately whispered that he would soon be in trouble, so we all anticipated what was to come. After a minute or two, Severi’s face reddened, and it became obvious that he did not know how to proceed. Some voices then murmured: “Majorana predicted it.” Severi did not know who Majorana was, but said haughtily, “Then let Mr. Majorana come forward.” Ettore was pushed to the blackboard, where he erased what Severi had written and gave the correct proof. [11]

In addition to Emilio Segrè, Majorana could rely on several other friends also attending engineering. First, there were his brother Luciano and Gastone Piqué, a very dear friend of his since high school, while other classmates included Enrico Volterra (son of the well-known mathematician Vito Volterra), Giovanni Enriques (son of Federigo Enriques, another illustrious mathematician and epistemologist), and Giovanni Ferro-Luzzi. In his third year at university, Majorana – like his close friends – transferred to the School of Engineering (for the final three years of his course), finding the

8

Life and myth

courses much less interesting than in the previous preparatory biennium. Indeed, according to Edoardo Amaldi, another fellow student (still in the biennium): While he was at the School of Engineering, Majorana, together with some of his fellow students, grew very critical of the way in which some of the subjects were taught: he felt that too much time was spent on unnecessary detail and not enough on the general synthesis needed for serious and systematic scientific study. This deep-rooted conviction of his frequently gave rise to lively, and sometimes heated, discussions with some of the professors. [10]

Such a situation endured for almost two years, but then something unexpected happened. In June 1927, the Director of the Institute of Physics at the University of Rome, Orso Mario Corbino, launched a curious “appeal” to his students at the Faculty of Engineering. According to him, “in the present state of rapid change now prevailing in physics all over Europe, and with Fermi’s appointment at Rome, an exceptional period has opened up for young people who have already shown sufficient ability and are willing to make an exceptional effort in theoretical and experimental study” [11]. In practice, the intent was to entice some of the most brilliant young minds into studying physics, and then recruit appropriate members for the newborn group created by Fermi. Amaldi and, a few months later, Segrè rose to the challenge, and moved from engineering to physics, thus enlarging the group comprising only Fermi and the experimentalist Franco Rasetti, who had been hired earlier by Corbino as his assistant, in accordance with his plan of supporting a rapid development of physics in Italy [11]. In their new circle, Amaldi and Segrè obviously told Fermi and Rasetti of Majorana’s exceptional gifts: You could ask Majorana things that you couldn’t ask anybody else in the world. He was a prodigy. [...] You could get him to integrate, I don’t know, a very complicated integral and so on; he would look at it and tell you the answer without even writing it. He could do feats of this type. [12]

Urged repeatedly by his friends Amaldi and Segrè, in the fall of the same year, 1927, Majorana eventually agreed to meet Fermi. The two beautiful minds immediately started talking about the statistical model of atoms that Fermi was working on, later to be known as the Thomas–Fermi model. As is well known, such a model involves a complicated non-linear differential equation, whose analytic solution was unknown, but Fermi had managed to obtain a numerical table of approximate values for it. In that room of the Institute of Physics in Rome, Majorana carefully followed Fermi’s reasoning, asked a few questions, and left the Institute. The next day, towards the end of the morning, he again came into Fermi’s office and asked him without more ado to show him the table which he had seen for a few moments the day

1.3 Lone physicist in the Fermi group

9

before. Holding this table in his hand, he took from his pocket a piece of paper on which he had worked out a similar table at home in the last twenty-four hours. [...] He compared the two tables and, having noted that they agreed, said that Fermi’s table was correct: he then went out of the office and left the Institute. [10]

What seems nothing more than an amusing anecdote, as recalled by Rasetti, Segrè and Amaldi, has since been carefully tested on scientific grounds, as we will see in Chapter 4. As if satisfied that Fermi had passed his “examination,” Majorana decided to leave Engineering and join Fermi’s group: A few days later he switched over to Physics and began to attend the Institute regularly. [10]

1.3 Lone physicist in the Fermi group The final two years of Majorana’s university career then developed at the Institute of Physics and, among the last few (and new) courses that he attended we mention that of Higher Experimental Physics held by Antonino Lo Surdo (the discoverer, independently of Johannes Stark, of the homonymous physical effect induced by an electric field on light spectra), and that of Mathematical Physics delivered by Vito Volterra. However, in addition to the “institutional” courses, Majorana also participated in “private” lectures held by Fermi in his office, and addressed to his students and collaborators. Fermi taught us privately. [...] He gave a course in which he explained exactly what is contained in [his book] Introduzione alla fisica atomica, which one can get out of the library and see exactly what he taught. [...] Then, I would say three times a week, in the afternoon at five o’clock or something like this, he would give us a private lecture. To Amaldi, to me, to Rasetti, and Majorana, and occasionally Corbino would come. [...] Majorana very often would say, “Well, it’s beneath my dignity. Why should I learn these things? You are doing it in a childish way; it should be done this way.” [...] Fermi didn’t react to that. Except a year or two later he decided that we were not worthy to be present at the interviews [when] Majorana [was there], and then they would closet themselves together because they went very fast in very difficult theory and so on. [12]

Majorana’s personal contribution to the Fermi group was, then, substantial from the beginning, but it developed in different ways through the years. In July 1928, when still a student and well before his graduation in physics, Majorana completed his first published paper, “On the splitting of the Roentgen and optical terms caused by the electron rotation and on the intensity of the cesium lines,” in collaboration with his later close friend Giovanni Gentile Jr., where they derived the ionization energy of an electron in the 3d orbit of gadolinium and uranium, and, in addition, calculated the fine structure splitting of different (X-ray transition) spectroscopic terms in gadolinium, uranium, and caesium by applying first-order perturbation theory to the recently developed Dirac

10

Life and myth

equation. Gentile, the son of the renowned Italian philosopher Giovanni Gentile, graduated in physics at the University of Pisa in November 1927, but soon after he moved to the Institute of Physics in Rome, where he spent about six months before departing for a one-year stay in Berlin (working with Erwin Schrödinger) and, then, another one-year stay in Leipzig (working with Heisenberg). Fermi discussed the results obtained by Gentile and Majorana on spin-orbit couplings and intensity ratios at a restricted conference held in Leipzig that same year (1928) under the chairmanship of Peter Debye, where he was invited to report the results of the work performed in Rome [13]. This was, however, not at all the only contribution given by Majorana to the study of some applications of the Thomas–Fermi atomic model carried out in those years by the Fermi group. Apart from the semi-analytic solution of the Thomas–Fermi equation already mentioned, as we will see in Chapter 4, Majorana studied the problem of an atom in a weak external electric field, i.e. atomic polarizability, and obtained an expression for the electric dipole moment for a (neutral or arbitrarily ionized) atom. Furthermore, he also started to consider the application of the statistical method to molecules, rather than single atoms, studying the case of a diatomic molecule with identical nuclei. Finally, Majorana also considered the second approximation for the potential inside the atom, beyond the Thomas– Fermi approximation, with a generalization of the statistical model of neutral atoms to those ionized n times. Unfortunately, Majorana did not publish anything on these studies. The attitude of Majorana not to spread the results of a given research, until its perfect refinement (according to his hypercritical judgement) was completed or when they were considered premature, manifested here for the first time. However, Fermi succeeded in convincing him to present (some of) his results to the 1928 annual meeting of the Italian Physical Society, held at the Institute of Physics in Rome in December. Majorana effectively delivered a talk at the meeting, but refused to publish his own work in the form of a regular article: “the researches performed till now are still too much scarce to fully appreciate such results” [14]. We will see that this is typical of how the personality of Majorana developed. The first work on nuclear physics performed within the Fermi group is also associated with the name of Majorana: indeed, in July 1929 he graduated in physics by defending his master thesis on “the quantum theory of radioactive nuclei.” The Fermi group did not become effectively involved in nuclear physics until 1933, but it is not particularly strange that such a subject entered in a thesis: although not applicable to Majorana, Fermi did not like to assign subjects for a thesis since he did not easily find subjects simple enough for beginners (he usually thought of problems that interested him personally but were too difficult for students) [15]. Nuclear physics was among the subjects studied by Majorana for several years,

1.3 Lone physicist in the Fermi group

11

independently of the main research topics of the Fermi group, until his famous theory of nuclear exchange forces published in 1933 (see Section 1.4). After graduation, he continued to attend the Institute of Physics in Rome: He spent his time in the library, where he mainly studied the works of Dirac, Heisenberg, Pauli, Weyl and Wigner. The last two authors were perhaps the only ones for whom he expressed unqualified admiration. This was due, at least to a large extent, to his particularly lively, almost prophetic, interest in group theory and its application to physics. [10]

Indeed, particularly relevant for Majorana was his “discovery” of the book by Hermann Weyl on Gruppentheorie und Quantenmechanik [16], which profoundly influenced his entire scientific thought and work. The synthetic, clear, and general approach offered by group theory was well appreciated by Majorana and only a very few other scientists in the world, until its re-discovery in 1950s and 1960s. Although not canonically involved in the researches performed by the Fermi group, true to his style3 Majorana did get involved in the studies of his friends and colleagues (including Fermi). Some illuminating examples, among many others, are the following. In 1931–2 Segrè was in Hamburg, thanks to a fellowship that allowed him to work with Otto Robert Frisch in the group headed by Otto Stern, experimenting on atoms in a variable magnetic field (in practice, they were setting up an experiment to generalize the famous work by Stern and Gerlach on spatial quantization). However, the interpretation of their results required the solution of complicated theoretical problems, which they were not able to solve, so Segrè asked for help from his friend Majorana [12], who then produced the appropriate theory. In this case, Majorana did publish the theory in his paper N.6, which then became a seminal work for the treatment of non-adiabatic spin-flips, now included in quantum mechanics textbooks and recently re-discovered in atomic and molecular collision physics at low energy, as well as in nuclear and particle physics. At almost the same time, George Placzeck joined the Fermi group in Rome for one year, working with Amaldi on the rotational spectrum of ammonia molecules observed in the Raman effect [10]. Majorana was regularly informed of such studies, which evidently interested him so much that he autonomously solved the problem of the determination of ammonia oscillation frequencies, relating them to the tetrahedral structure of the molecule. This work was, however, not published, and we know about it only through his own research notes [17] [18]. Fermi benefitted from help given by Majorana on several occasions, as, for example, when he and Segrè were working on the hyperfine structures of atomic spectra. In the paper published by Fermi and Segrè in 1933, the authors explicitly 3 No papers published by Majorana have a co-author, with the exception of the already mentioned first paper,

in collaboration with Gentile.

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Life and myth

acknowledge “dr. Majorana for several discussions about the calculations” [19]. A trace of such calculations may be found, again, among his personal research notes [18]. Majorana’s attitude to getting involved in his friends studies, and thus helping them with technical problems, was experienced also by a number of visiting scientists in Rome (as well as by people he met abroad, as we will see). Just to quote an example, we recall what was mentioned by Rudolph Peierls: “I certainly got a lot of useful ideas of clarification from Fermi and from the other people, including Wick and Majorana” [20]. All this “offstage” work was, unfortunately, never published, but some of it is conserved in the personal notes already mentioned, and will be here the subject of dedicated chapters. By the end of 1931, Majorana had published just four more papers, two on the chemical bonding of molecules, and two on spectroscopy. However, things changed drastically in the following year, starting with a breakthrough discovery that occurred across the Alps, the news of which soon reached Rome. 1.4 Leipzig–Rome–Naples: the later years The period of time around the years 1931–2, as recalled by Peierls, was just the time when the artificial radioactivity had been discovered, and when there were the experiments beginning to come out which led to the discovery of the neutron. Fermi always had a slightly peculiar attitude to that. I think he felt that the Paris group, the Joliots, should really have seen the existence of the neutron from their experiments which were later pointed out by Chadwick. I had the impression that he knew what the experiments meant. [20]

This apparent “mystery” was unveiled by Amaldi years later: When the paper of Joliot arrived in Rome in January with the Comptes Rendus and the paper of Chadwick had not yet appeared, we all were very much interested. But Majorana said, “How stupid of Joliot! They have not understood that this is the neutron. [...] This is obvious. These gamma rays [of 50 MeV] make no sense in a nucleus. This should be the neutral particles.” [...] Nobody took seriously the suggestion of Majorana. [...] When the paper of Chadwick came out, we were all convinced. And we said, “Look how quick Ettore is – he has understood before Chadwick”. [21]

Majorana’s striking intuition was confined to the Fermi group in Rome, and the discovery of the neutron by James Chadwick (announced in February 1932) caused a sensation in the international physics community. Several scientists turned their attention to building a theory of the nuclear structure that was able to include, consistently, the neutron as a nuclear constituent along with the proton, in agreement with the experimental observations that were accumulating. Majorana himself revealed to his friends and colleagues in Rome [10] that he had built a theory of

1.4 Leipzig–Rome–Naples: the later years

13

light nuclei based on the quantum concept of exchange forces. Although encouraged by Fermi to go public with his results, Majorana’s hypercritical judgement prevented him from doing so, and, true to style, his work was not recognized until a few months later when it was independently elaborated (and published, in July) by Heisenberg. The appearance of Heisenberg’s paper took the entire Rome group by surprise, and Fermi then urged Majorana, now successfully, to visit Heisenberg in Leipzig, for a six-month period in 1933. In the original Heisenberg model, atomic nuclei were supposed to be composed of protons and neutrons only, without any need of electrons, as was previously largely accepted (before the discovery of the neutron, protons and electrons were the only known “elementary” particles). It was also assumed that the leading forces responsible for nuclear stability were exchange forces, in analogy to what had been proven by Walter Heitler and Fritz London for the H+ 2 molecular ion. According to Heisenberg, the underlying nuclear forces should be interpreted in terms of nucleons exchanging spinless electrons, implicitly assuming that the neutron was practically formed by a proton and an electron (such an ambiguity about the effective presence of electrons is a reflection of Heisenberg’s uncertain reasoning). Majorana immediately realized this defect of Heisenberg’s theory.4 Indeed, in Majorana’s view, the neutron was pictured as a “neutral proton” [10], as it effectively is, and in his model the forces between neutrons and protons were explained in terms of the exchange interaction arising from the quantum effect coming from interchanging the space coordinates of identical nucleons, rather than from interchanging their electric charge as in the Heisenberg model. The observed particular stability of the α-particle (and not the unobserved one of the deuterium nucleus, as instead predicted by the Heisenberg model) was thus explained by Majorana as a “saturation phenomenon more or less analogous to valence saturation.” It was especially this saturation of nuclear forces for the α-particle that quickly led Heisenberg himself and others to recognize the Majorana model as the most appropriate one. It is rather intriguing that, on several occasions, while discussing the “Heisenberg–Majorana” exchange forces, Heisenberg mentioned his own contribution only marginally, while emphasizing that of Majorana [6]. As a matter of fact, this model was certainly the most renowned contribution by Majorana recognized by the physics community of his time. Majorana started his six-month fellowship in January 1933 as a visiting scientist of the Institute of Physics in Leipzig, but he also visited (in April) the Institute directed by Niels Bohr in Copenhagen, although with far less benefit. Overall, during his stay abroad, he established “relationships with several illustrious people” 4 As we will see in Chapter 9, in a letter to Gentile in 1929, Majorana was already aware at that time of the

difficulties related to the possible existence of proton–electron “bound states” in nuclei.

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Life and myth

[2], including, among others, David R. Inglis, Eugene Feenberg, Felix Bloch, Paul Ehrenfest, Felix Hund, Peter Debye, Bartol L. van der Waerden, Gleb Wataghin, George Placzeck, Niels Bohr, Christian Møller, Victor Weisskopf, Wolfgang Pauli, Hans A. Bethe, Leon Rosenfeld, and Guido Beck. Particularly in Leipzig, Majorana immediately realized the different working conditions with respect to those he experienced in Rome, now more suitable (according to him) for his researches as a theoretical physicist. Indeed, where Fermi and his own friends had failed, now Heisenberg and others were able to convince Majorana to publish the results of his work on nuclear forces. Also, Majorana succeeded in seeing properly recognized a paper he published before his departure from Rome (and about which even Fermi missed the relevance), in which he generalized the relativistic Dirac equation to an infinite components equation, by inventing a mathematical technique for the representation of a group several years before Eugene Wigner. Remarkably, indeed, Majorana obtained the simplest infinite-dimensional unitary representations of the Lorentz group that were re-discovered by Wigner in his 1939 and 1948 works. The entire theory was later re-invented by Soviet mathematicians (in particular I.M. Gelfand and collaborators) in a series of articles starting from 1948 and finally applied by physicists, first to hadronic physics and then to modern string theories, years later. The relevance of Majorana’s article was realized in Leipzig by van der Waerden, one of the leading authorities in group theory: van der Waerden’s role in Leipzig was very important because he had a tremendous ability of understanding quickly what the people were talking about and then he knew all these things so well, so he would, by a few sentences of explanation, clarify at once a complicated situation at our seminar. [...] The understanding at Leipzig of the Dirac paper was very largely due to van der Waerden’s help. Well, we spoke about the Weyl spinor business. The Dirac spinor was the thing which everybody discussed, but then there was Weyl spinor business which van der Waerden knew. The others did not know about it, but then there was Majorana. [9]

Unfortunately, the paper by Majorana remained unnoticed for more than three decades, until David M. Fradkin, informed by Amaldi, re-examined that pioneering work in the light of later developments and clearly explained the relevance of Majorana’s approach and the results accomplished many years earlier [22]. In August 1933 Majorana went back to Rome, and a novel phase of his life started, known as his “gloomy years.” A number of hypotheses have been expressed about the reasons for his discomfort, including the worsening of his health (due to gastritis), his father’s death (in 1934), the return to less than optimal research conditions (which Majorana felt were not as good as those in Leipzig), and so on. As a matter of fact, as recalled by Amaldi,

1.4 Leipzig–Rome–Naples: the later years

15

he began to attend the Institute in the Via Panisperna only at intervals, and after some months no longer came at all: he tended more and more to spend his days at home immersed in study for a quite extraordinary number of hours. [...] A considerable number of attempts by Giovanni Gentile junior, Emilio Segrè and myself to bring him back to living a normal life met with no success. [...] None of us succeeded in finding out whether he was still doing theoretical physics research; I believe he was, but I have no proof. [10]

Such a conjecture has proved true only in recent years. First of all, Majorana constantly helped his uncle Quirino (a renowned experimental physicists) by elaborating the appropriate theoretical framework his uncle’s for experimental observations [23]. Of course, Quirino Majorana repeatedly urged his nephew to publish the results of his theory, but without any success, and was only able to include the corresponding relevant calculations by Ettore in the final paper that Quirino published about the observed phenomena. Here the subject of the collaboration was in no way at the frontier of research,5 but among Majorana’s personal research notes [18] we have evidence that he was also working on several theoretical problems including quantum electrodynamics and, very likely, the subject of his last paper (published years later, in 1937) on the famous “Majorana neutrino” (see Section 2.8). Also, and quite unexpectedly, Majorana revealed a genuine interest in advanced physics teaching starting from 1933, soon after he obtained at the end of 1932 the professorship degree of “libero docente” (analogous to the German privatdozent or English lecturer). In view of this position, he proposed some academic courses at the University of Rome, as testified by the programs of three courses he would have given between 1933 and 1937 [24]: Mathematical Methods of Quantum Mechanics in the academic year 1933–4, Mathematical Methods of Atomic Physics for 1935–6, and, finally, Quantum Electrodynamics for 1936–7. However, Majorana never gave lectures at the University of Rome, probably due to the lack of students, and he effectively lectured on theoretical physics only in 1938 when he obtained a position as a full professor at the University of Naples. Indeed, in 1937 Majorana took part in the national competition for a full professorship in theoretical physics at the University of Palermo (the first opening in Italy since Fermi became a professor), and for this occasion Fermi persuaded him finally to publish his earlier work on the symmetric theory of electrons and positrons, which would become famous for the appearance of the “Majorana neutrino” theory. As already mentioned, Majorana’s application apparently caused some trouble among the board of examiners, including Fermi, who resolved to suspend temporarily the competition for the position at the University of Palermo. Without going into further details [10, 1, 2, 8], as a matter of fact the board of 5 The observations performed by Quirino Majorana concerned a presumed novel photoelectric effect (in

addition to that discovered by Heinrich Hertz), not induced by the thermal action of light on a metal.

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Life and myth

examiners finally appointed Majorana as full professor of theoretical physics at the University of Naples “for high and well deserved repute, independently of the competition rules.” The last chapter of Majorana’s life had Naples as its center, where he arrived in January 1938. Here, Majorana effectively lectured on theoretical physics, presenting quantum mechanics in a very modern way [24] by skillfully sandwiching its mathematical formalism with practical applications. Unfortunately, however, this course was suddenly and unexpectedly interrupted by his mysterious disappearance just three months after his appointment at Naples. On Friday March 25, 1938 Majorana went to the Institute of Physics in Naples and handed over the lecture notes and some other papers to one of his students. After that, he returned to his hotel and, after writing farewell letters to his family and to the director of the Institute of Physics, Antonio Carrelli, he apparently embarked a ship to Palermo. He likely reached his destination on the following morning, where he lodged for a short time in the Grand Hotel Sole. Probably it was there that he wrote a telegram and a letter to Carrelli pointing out a change of mind about his decisions. On Saturday evening, Majorana apparently embarked a ship from Palermo to Naples. From here onwards, no other information about him is available. There have been several conjectures about the fate of Majorana, including suicide, a retreat in a monastery, and a flight to a foreign country [1, 2]. Understanding the root of such a dramatic decision is perhaps impossible; it could have been triggered by personal or familial reasons, such as Majorana’s peculiar relationship with his extremely possessive mother (especially after the death of his father in 1934), or more elaborate reasons reported in some nice literary tales. However, to quote Majorana himself on his approach to physics: “We cannot give to such hypothesis greater likelihood than to some other theoretical presumptions without a too much subjective appraisal.” A detailed, analytic examination of the disappearance of Majorana can be found in Ref. [25], but a useful caveat is that reported by Amaldi: Fermi observed that with his intelligence, once he had decided to disappear or to make his body disappear, Majorana would certainly have succeeded. [10]

This is, however, not the whole story.

2 The visible side

2.1 Ten papers depicting the future Majorana made fundamental contributions to several different areas of theoretical physics, in part as collaborator of the Fermi group in Rome, as seen in Chapter 1. However, he published only a few scientific articles, so his brilliant activity was not at all immediately recognized outside the Fermi group (with the notable exception of paper N.8, publicized by Heisenberg). The complete list of articles written by Majorana and published by him is as follows: N.1. Gentile Jr., G. and Majorana E., 1928. Sullo sdoppiamento dei termini Roentgen ottici a causa dell’elettrone rotante e sulla intensità delle righe del Cesio. Rend. Acc. Lincei, 8, 229. N.2. Majorana, E., 1931. Sulla formazione dello ione molecolare di He. Nuovo Cim., 8, 22. N.3. Majorana, E., 1931. I presunti termini anomali dell’Elio. Nuovo Cim., 8, 78. N.4. Majorana, E., 1931. Reazione pseudopolare fra atomi di Idrogeno. Rend. Acc. Lincei, 13, 58. N.5. Majorana, E., 1931. Teoria dei tripletti P incompleti. Nuovo Cim., 8, 107. N.6. Majorana, E., 1932. Atomi orientati in campo magnetico variabile. Nuovo Cim., 9, 43.

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The visible side

N.7. Majorana, E., 1932. Teoria relativistica di particelle con momento intrinseco arbitrario. Nuovo Cim., 9, 335. N.8a. Majorana, E., 1933. Über die Kerntheorie. Z. Phys., 82, 137. N.8b. Majorana, E., 1933. Sulla teoria dei nuclei. Ric. Scientifica, 4 (1), 559. N.9. Majorana E., 1937. Teoria simmetrica dell’elettrone e del positrone. Nuovo Cim., 14, 171. A further paper should be added to this list, published posthumously by Majorana’s friend Giovanni Gentile Jr. from an original manuscript he found among Majorana’s personal papers: N.10. Majorana, E., 1942. Il valore delle leggi statistiche nella fisica e nelle scienze sociali. Scientia, 36, 55. The largest part of Majorana’s work was, however, left unpublished. We are now left with his master thesis on “the quantum theory of radioactive nuclei,” 5 notebooks (“Volumetti”) [17], 18 booklets (“Quaderni”) [18], 12 folders with spare papers, and a set of the lecture notes on theoretical physics [26] prepared for a class at the University of Naples. In the following we give an account of what was effectively published by Majorana during his short scientific life, while, in the remaining part of this book, we will focus on some key contributions reported in his personal notes but left unpublished by Majorana. 2.2 Introducing the Dirac equation into atomic spectroscopy According to Amaldi [10], an in-depth examination of Majorana’s papers on atomic and molecular physics, the set of papers NN.1–5, leaves one struck by their quality. They indeed reveal both a deep knowledge of the experimental data, in the most minute detail, and an uncommon (and without equal at that time) ability in using the symmetry properties of the quantum states, resulting in a remarkable simplification of the problems and a brilliant choice of the most suitable method for their quantitative resolution. The corresponding work fits – more or less – into the general framework of the activity performed by the different members of the Fermi group during the years 1928–32. However, strictly speaking, only the first paper was a genuine product of a collaboration within that group, involved at that

2.3 Spontaneous ionization

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time in applying the Thomas–Fermi model of atoms, developed earlier by Fermi, to different problems of atomic physics. In paper N.1, “On the splitting of the Roentgen and optical terms caused by the spinning electron and on the intensity of the cesium lines,” written in collaboration with Gentile Jr., the authors applied the statistical model of atoms in order to obtain first-principle calculations of the spectra of some complex atoms. The ionization energy of an electron in the 3d orbit of gadolinium and uranium is derived (the “Roentgen terms” refer to X-ray transitions, according to modern notation), as well as the fine structure splitting of different (X-ray transition) spectroscopic terms in gadolinium, uranium, and cesium, obtained by applying the first-order perturbation theory to the Dirac equation. Note that the Dirac equation made its appearance just that same year (1928), so that such a paper represents one of the very first applications of relativistic quantum mechanics, and it is quite impressive that the authors succeeded to encompass substantial numerical computations, give a quantitative treatment of spin-orbit interaction, make a comparison with experimental data, and also introduce corrections to Fermi’s statistical potential to attain better agreement with fine structure data. The straightforward introduction of the statistical effective potential, deduced from the Thomas–Fermi model, into the Dirac equation produced, indeed, a somewhat good agreement with the experimental data on gadolinium and uranium, while a better fit with the data on spin-orbit splitting of cesium was obtained by replacing that potential with an effective one, deduced from the Thomas–Fermi potential and the central Coulomb potential, a method that became common in atomic physics. The particularly good agreement with the experimental values thus obtained allowed the authors to extend their spectroscopic analysis and then to evaluate the ratio of the transition probability for the optical transitions from the 6S ground state to the two upper 6P and 7P states of cesium. Remarkably, the value predicted agrees with present experiments within 5%, and only very recently (1997) has a better approximation for the effective potential produced an apparent better numerical accuracy [27] compared with that obtained by Gentile and Majorana.

2.3 Spontaneous ionization 2.3.1 Anomalous terms in helium The apparent discovery by P. Gerard Kruger in 1930 [28] of two new lines in the helium spectrum, and the ensuing discussion on their interpretation (were they true lines of the helium spectrum?), motivated the writing of paper N.3. Kruger proposed to interpret the new lines as due to transitions from a normal state to doubly excited levels, but Majorana was not convinced of this attribution and

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The visible side

considered the problem in detail in his paper “On the presumed anomalous terms of helium.” The point was that states with two excited electrons have energy above the ionization potential of the helium atom, so that transitions to states of the continuum spectrum formed by the ground state He+ ion and a free electron – namely, spontaneous ionization processes – could take place. According to Majorana, 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 lies 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.

That is, the energy levels suitable for a possible spontaneous ionization process are so highly unstable (and thus their energy is not at all well determined) that radiative transitions showing very narrow lines (like the ones observed by Kruger) are not expected. The remarkable character of Majorana’s analysis lies, however, in his unusual (for that time) approach based on studying the symmetry properties of the problem, as coded in the well-known (to him) book by Weyl [16]. Majorana considers all the unperturbed helium eigenfunctions corresponding to electronic levels generated by combining two hydrogen-like configurations with principal quantum number n = 2, and discusses in detail the symmetry properties of several relevant states with respect to spatial rotation, electron spin exchange (singlet and triplet states), and parity, following the approach employed earlier by Wigner [29]. The mutual repulsion between the electrons is, then, included as a perturbation in the adopted variational perturbative method, and the second-order calculations allowed him to confirm the Kruger interpretation for only one of the two lines, while ruling out the possibility for the other one to belong to the helium spectrum at all. Notably, there is some additional relevant work behind the results that Majorana presented in this paper; although far from being obvious, he decided not to publish it, but we will give an account of it in Chapter 3. Paper N.3 went, however, largely unnoticed, and the conclusions reported in it were independently re-discovered years later by Ta-You Wu1 [30], a collaborator of Samuel Goudsmit and others [31, 32, 33, 34], with no reference to Majorana’s paper. These authors, however, did not give evidence of knowledge of the appropriate LS-coupling requirements 1 The 1934 paper by Wu contained a major numerical mistake, and only after he corrected this mistake (one

year later) did his calculations definitely agree with Majorana’s. Ironically, Wu’s work was often quoted in subsequent years for reference to the Auger effect in helium, whereas Majorana’s was not.

2.3 Spontaneous ionization

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between electrons, as had been introduced by Majorana, so it is not surprising that some of the authors suggested spectroscopic classifications that Majorana had already shown to be physically unrealistic. Only in 1944 did Wu produce new calculations [35] with a statement of the appropriate symmetry-based conditions, and re-discovered what had been obtained by Majorana. The predictions made by Majorana in 1931 have only recently (1994) been confirmed to a high degree [36], within a few percent. 2.3.2 Incomplete P triplets Although it may seem strange, the most important article published in 1931 by Majorana, dealing again with spontaneous ionization, was his paper N.5 on the “Theory of the incomplete P triplets,” where the characterization of spectra of different atoms with two electrons in the outer shell was given. Here the problem was to give a suitable theoretical explanation to the fact that some lines in the predicted P triplet levels in the absorption spectra of Hg, Cd, and Zn had not been observed experimentally. It was already known that deviations (perturbations) in the position of the energy levels and in the line intensity of atomic spectra of atoms or ions with more than one electron in the outer shell, with respect to those predicted for simpler atoms, were due to a resonance process between two or more quasi-degenerate energy levels, the corresponding eigenstates being mixed up by the interactions between electrons. In his paper, Majorana follows exactly the same reasoning, but assumes a mixing between discrete and continuum levels, leading to a peculiar process – which he explicitly names spontaneous ionization – where the discrete levels mixed with the continuum may decay through a nonradiative transition, converting the excited atom into a free electron and a positive ion. In practice, Majorana introduced a process in the optical range completely equivalent to the Auger effect already known in X-ray emission, and now referred to as autoionization. In order to explain the Auger effect, in 1927 Gregor Wentzel had already considered [37] the simple case of the non-radiative decay of a discrete excited state of two electrons into final states in the continuum. He proposed that the Auger decay was due to the configuration interaction between the initial discrete state and the final continuum one, where one of the two electrons is emitted as a free electron, such a configuration interaction being driven by the Coulomb repulsion. With his theory, Wentzel was able to predict the ratio between the radiative and nonradiative recombination channel of the process, in agreement with the experimental results, thus proving the extension of the quantum resonance mechanism between two bound states to that between the discrete and continuum states of a manybody system to be meaningful. Majorana followed the reasoning of the Wentzel

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theory, but pointed out that, in order to control the probability of the non-radiative decay, it was necessary to include also the configuration interaction due to the spinorbit interaction between the atomic two-electron excited states, in addition to the Coulomb repulsion in the Wentzel theory. Indeed, the level mixing leading to autoionization can occur only if a discrete level above the ionization limit is resonant with a continuum having the same parity and angular momentum. According to Majorana, if Russell–Saunders coupling between electrons is assumed, the energy levels have definite parity, total spin, total orbital angular momentum, and total angular momentum: in absence of a radiative transition, the symmetry character of the given state cannot change, so that a doubly excited level can spontaneously ionize only into a continuum with the same symmetry character. In other words, configurations with two excited electrons above the ionization potential are stable (that is, no spontaneous ionization occurs) since, in the non-relativistic approximation, the corresponding energy levels have symmetry properties for which transitions to the continuum spectrum is strictly forbidden. This is just the case for the P levels of Hg, Cd, and Zn considered in paper N.5, where no continuum with the right parity is present. However, according to Majorana, by relaxing the non-relativistic approximation, the presence of the intrinsic magnetic moment of the electrons – mediating an electron coupling which is different from the LS Russell–Saunders case – may avoid such a stability that, for the J = 2 component of the anomalous triplets of Hg, Cd, and Zn. Majorana then assumes a more complex double interaction mixing, where a mixing between P and D discrete states, resulting from the inclusion of the spin-orbit interactions of the p electrons and leaving the atom in a mixed state with appropriate parity, is followed by a discrete–continuum mixing producing spontaneous ionization (in the D continuum). As recognized by Majorana, such a double mixing is necessary as long as a large enough autoionization is required in order to explain the disappearance of the mentioned lines, while, in general, a singlet–triplet admixture is not strictly a prerequisite to produce spontaneous ionization. The autoionization process in optical atomic spectra was introduced (with the name since adopted in the literature) almost simultaneously by Allen G. Shenstone [38], whose attention, however, focused on the more inspiring spectrum of Cu, for which spontaneous ionization is the rule rather than the exception. As the reader should now be accustomed, reference in the literature is usually made only to the work by Shenstone, not to that by Majorana, but, in the present case, this is probably due to an unclear understanding rather than ignorance. Indeed, the “bible” of atomic spectroscopy, published in 1935 by Edward U. Condon and George H. Shortley [39], recognized the simultaneous and independent contributions by the two authors, and a thorough summary of Majorana’s work is present there. However, Condon and Shortley were not fully convinced of the

2.3 Spontaneous ionization

23

spectroscopic assignments and the autoionization scheme proposed by Majorana, since a similar autoionization process should have been applied also to other P levels, the corresponding positive observations being unavailable at that time, and “Majorana’s theoretical discussion does not show clearly why this should [not] be so” [39]. As noted in Ref. [40] (to which the interested reader should refer for a comprehensive historical review of the birth of autoionization), it is quite surprising that no spectroscopist of that era correctly identified the mentioned levels, so that the puzzle of the missing P lines was eventually clarified on the experimental side in 1955 with the observations of W. Reginald S. Garton and Arthur Rajaratnam [41]. Even in this case, however, the authors just accepted Condon and Shortley’s summary of Majorana’s results, without recognition of his earlier identical assignments [40]. Only in 1970 did William C. Martin and Victor Kaufman reconsider the question [42], and proved explicitly the correctness of Majorana’s spectroscopic assignments: it is now finally recognized that any line from “incomplete P triplets” suffers from (large or small) perturbations induced by autoionization, following precisely the scheme predicted by Majorana, which he nevertheless perceived just from the low resolution spectra available at that time. 2.3.3 Majorana–Fano–Feshbach resonances Interestingly enough, Majorana did not publish all the results he obtained for autoionization, and we will give a thorough account of his unpublished work on quasi-stationary states, that is mixing between a discrete level and a continuum, in Chapter 5, including his applications to nuclear physics (in addition to atomic physics topics considered in the published papers). Here we only mention that Majorana derived many of the important theoretical results published later by Ugo Fano in 1935 [43],2 as well as an effect (the shift in the energy of the resonance due to interaction with the continuum) properly discussed by Fano only in 1961 [45]. As a matter of fact, the quantum interference between two transition amplitudes was pioneered only by these two scientists in the context of autoionization, and was later studied also in different areas. The inverse processes of the Fano theory (i.e. association processes; Fano’s autoionization is a dissociation process) were considered in 1958 [46], also in the framework of the general nuclear reaction theory for multi-channel reactions, and are usually known as Feshbach resonances. The theory of such nuclear scattering processes, arising from the configuration interaction between many different scattering channels (instead of only one), is, then, a further generalization of Majorana’s work, whose interest extends now also to the physics of high Tc superconductivity and ultra cold gases [47]. 2 Ref. [44] points out the possible, indirect role played by Majorana on the genesis of Fano’s paper, through

the mediation of Fermi.

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2.4 Chemical bonding The basis of the quantum-mechanical theory of the homopolar chemical bond is, as is well known, the concept of exchange forces, as opposed to that of polarization forces, which are not genuinely quantum in nature. Heitler and London introduced in 1927 [48] the idea of quantum exchange in order to explain the stability of the H2 molecule, and the success of their method relies mainly on the beautiful agreement with the experimental data and the remarkable accuracy with which the dependence of the total electronic energy on the internuclear distance R was predicted.

2.4.1 Helium molecular ion A more intriguing case study was that of the molecular ion He+ 2 , as the corresponding quantum states are more complicated both for the presence of a larger number of electrons and for the requirements imposed by the Pauli exclusion principle. Starting from a discussion of the still unclear experimental result on the band structure in the helium emission spectrum, which led to attributing the observed light bands just to the molecular ion He+ 2 , Majorana approached the problem of the possible formation of such an ion in his paper N.2, following the method proposed earlier by Heitler and London. The basic physical idea was to consider the He+ 2 system to be similar to that of + H2 rather than HeH, i.e. the chemical reaction He + He+ ↔ He+ 2 : the fact that only one of the two atoms bonded to form the molecular ion is ionized made the problem at variance with what was already known for the hydrogen molecule and, more generally, with the problems tackled by means of the homopolar valence theory. Majorana translated this idea in a suitable quantum-mechanical theory by constructing appropriate eigenfunctions of the system in accordance with its symmetry properties, analogously to the above-mentioned group-theoryinspired methods in his other papers. The formation of the He+ 2 molecular ion was considered, almost simultaneously (and independently), by Linus Pauling [49], who in 1931 described it qualitatively as being due to the combination of a neutral helium atom and a ionized one. However, Pauling arrived at about the same quantitative results obtained by Majorana only two years later [50], without employing a group theory-inspired method: it seems that, at least in 1935, Pauling became aware [51] of the Majorana paper (which was used in 1934 by Harrie S.W. Massey and Courtney B.O. Mohr in their important paper on transport phenomena in gases [52]). Like Heitler and London, Majorana also started from the asymptotic solution of the problem (for large internuclear distances R) – the wavefunctions of the system

2.4 Chemical bonding

25

are just those for a neutral helium atom and its ion – and, just as the former authors considered it to be very unlikely that both electrons resided on the same nucleus in the H2 molecule, Majorana also neglected the possibility that all three electrons in He+ 2 was located on the same nucleus. However, when the nuclei approach each other, their reciprocal interaction has to be taken into account; such an interaction mixes all the wavefunctions previously introduced, but Majorana was able to recognize the only two appropriate combinations satisfying general symmetry principles. Indeed, by emphasizing the relevance of inversion symmetry – the total electronic wavefunction must show a definite symmetry with respect to the midpoint of the internuclear line – he showed that two molecular states are possible for the He+ 2 molecular ion, only one of which corresponds to the bonding molecular orbital of the ion. Such a configuration reflects the fact that the ground state of the system is a resonance between the He:He· and He·He: configurations. In order to make explicit numerical predictions for the equilibrium distance, energy minimum, and oscillation frequency of the helium molecular ion, Majorana made recourse to variational calculations, for which explicit expressions for the helium wavefunctions were required. 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. Therefore we need to use rather simplified unperturbed eigenfunctions.

In his paper N.2, Majorana wrote the ground state of the helium atom simply as the product of two hydrogenoid wavefunctions, but introduced an effective nuclear charge (as a variational parameter) describing the screening effect of the nuclear charge by means of the atomic electrons. As we will see in Chapter 3, what is reported in this paper does not reflect faithfully the great amount of work performed by Majorana on the helium wavefunctions, always devoted as he was to obtaining generalizations of the simple approximations and to giving easy yet physically meaningful expressions. However, even within this simple approximation, he obtained good agreement with the experimental data on the equilibrium internuclear distance and not at all “accidental” (according to Majorana) perfect agreement with the experimentally determined value of the vibrational frequency of He+ 2 . Also, he estimated the dissociation energy of the molecular ion, obtaining a value Emin = −2.4 eV (including polarization forces effects) which Majorana could not compare with experiments, due to there being no available data, but which is now remarkably closer to the actual experimental determination of Emin = −2.4457 ± 0.0002 eV [53] than the recent theoretical prediction of Emin = −2.47 eV [54], which was obviously obtained with more refined mathematics than that used by Majorana.

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2.4.2 Majorana structures An even more intriguing paper on the chemical bond is Majorana’s paper N.4, regarding a “Pseudopolar reaction of hydrogen atoms.” Here, the original physical problem was an apparently unexplained phenomenon observed in the spectrum of the H2 molecule, namely the decay of the excited (2pσ )2 1 g (gerade) state into the (ungerade) 1sσ 2pσ 1 u state in the infrared spectral region, contrary to what happens in atomic systems, where the frequency corresponding to the transition 2p2p–1s2p involving two excited optical electrons was very close to that of the transition 1s2p–1s2s involving only one excited optical electron. What previously puzzled other authors [55] – the theoretical justification of the existence of the (2pσ )2 1 g term, along with an explanation of its abnormal energy level, compared to similar atomic systems – did not generically trouble Majorana, who promptly recognized the relevant difference between atomic and molecular systems: 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 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 [in atomic systems] where it is generally possible to neglect the interdependence of the electron motions (polarization) without losing sight of the essentials.

Majorana then generalized the Heitler–London theory of the hydrogen molecule, where only configurations corresponding to one electron in each atom of the molecule were considered, by including different configurations where both electrons or no electron belong to a given atom. In other words, while Heitler and London considered only the chemical reaction H + H ↔ H2 for the formation of the hydrogen molecule, Majorana introduced also the reaction H+ + H− ↔ H2 , where ionic structures are present. Of course, Majorana was aware of the fact that the apparent charge transfer via ionic structures has no proper physical interpretation in homopolar molecules, and, for this reason, he designated such a reaction between the two hydrogen atoms as “pseudopolar” rather than ionic (the same term we used above – ionic structure – is borrowed by more modern valence bond approaches). Nevertheless, he found that, while the normal (1sσ )2 1 g state predominantly refers to the Heitler–London H + H reaction, the term (2pσ )2 1 g [. . . ] 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.

2.5 Non-adiabatic spin-flip

27

He then succeeded to prove the existence of a stable molecular state with both electrons in excited 2p orbitals. The numerical results he obtained from his impressive calculations were amazingly close to the experimental observations, although Majorana concluded his paper with his usual aloof and humble tone: This result is even too favourable as, with the method we followed, we could have expected a value considerably smaller than the true one. [. . . ] 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.

It is interesting to note that, although a number of studies appeared on the ionic character of Hartree–Fock wavefunctions starting from 1949 [56, 57, 58], it was only after the Majorana centennial in 2006 (when Majorana’s paper N.4 was re-discovered) that it has been recognized that ionic structures in the homopolar molecules yield binding energy predictions close to the experimental values [59]. Since then, a novel designation has appeared in chemical computations, referring to ions that are not in the lowest ionic configuration as “Majorana structures” [60].

2.5 Non-adiabatic spin-flip An even more intriguing story is that of paper N.6 about “Oriented atoms in a variable magnetic field,” i.e. non-adiabatic spin-flip of atoms in a magnetic field. At the end of 1931, Segrè moved from the Fermi group in Rome to Hamburg for a Rockefeller fellowship, in order to learn about vacuum techniques and molecular beams under Otto Stern. Some years earlier, the famous Stern–Gerlach experiment established the quantization of angular momentum by means of measurements regarding the spin-flip of the atoms of a polarized vapor beam moving adiabatically in a static magnetic field gradient. Before Segrè’s arrival, Stern was still involved in a generalization of his earlier experiment to the non-adiabatic case with a rapidly varying magnetic field. A novel experimental device was prepared and tried out by Thomas E. Phipps and Stern, whereby a magnetic field with constant intensity rotating uniformly was employed, the required corresponding theoretical calculations (for the interpretation of the experimental data) being performed by Paul Güttinger. Such experiments did not prove very successful, and the device was handed over by Otto Frisch and Segrè when the latter arrived in Hamburg [61]. Segrè realized quite soon that, owing to technical difficulties, the original plan by Stern could not be fully carried out, and he instead proposed to induce nonadiabatic transitions by allowing the vapor beam pass near to a point where the magnetic field vanished. Stern approved my idea as soon as he heard it, got me the few extra parts I needed, and told me to rebuild the apparatus in the way I proposed. As to theory, I knew whom to look

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to for help. I wrote a letter submitting my problem to Ettore Majorana in Rome, and soon received the answer I needed. [11]

This is, then, the genesis of Majorana’s paper N.6, but its specific character does not provide a proper idea of its content. Indeed, Majorana discussed in this paper the dynamic behavior of a spin-1/2 particle in a time-dependent magnetic field that almost vanishes at a certain moment, as required by the Frisch and Segrè experiment. He calculated explicitly, “with extraordinary elegance and conciseness” [10], the corresponding transition probability in terms of the ratio between the Larmor precession frequency of the atomic dipole moment and the frequency of rotation of the magnetic field in the rest frame of the atom. The results – both specific and general – contained in the paper, however, largely exceeded the expectations. Paper N.6 may be schematically divided into two parts, the first dealing with the reduction of the general problem for a particle with arbitrary angular momentum to that for a spin-1/2, particle and the second discussing the dynamics of a two-state system in a time-dependent potential, as just addressed.3 2.5.1 Majorana sphere and a general theorem The “extraordinary elegance and conciseness” of Majorana’s solution of the general problem was achieved by means of his favorite group-theory methods, but the recognition of his results took place several years later – by Nobel laureate Isidor Isaac Rabi in 1937 and, in general, by Felix Bloch and Rabi in 1945 [62] – via a less difficult (for physicists) pictorial representation introduced by Majorana himself to represent graphically the state with arbitrary angular momentum J. In this representation, now known to mathematicians as the Majorana sphere (or even, but improperly,4 the extended Bloch sphere), an angular momentum state with J = N/2 is represented by N points on the unit sphere, since, as shown by Majorana, a J = N/2 state can be written as a symmetrized combination of N constituent spinors. The rotation of the angular momentum after interaction with the spatially varying magnetic field corresponds to a rigid rotation of the N representative points on the sphere. Due to its “conciseness,” such a representation has often been cited in quantum mechanics textbooks (for example in Ref. [64]). Although, as recognized by Bloch and Rabi, Majorana’s paper “has given some basic general results which are both very useful and greatly deepen our understanding of the process involved,” the language and the powerful methods of group theory were still particularly difficult to digest for physicists, such that Bloch and Rabi undertook a suitable “reduction”: 3 For further details, see Sections 14.1 and 14.2. 4 In Ref. [63], the so-called “Bloch sphere” referred just to spin-1/2 particles, represented by only one point on

the unit sphere.

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29

Majorana’s method, while remarkable in its elegance, has the disadvantage of somewhat obscuring the physical significance of the representative systems with spin 1/2. It is clear that a simple intuitive understanding of the procedure and of the essential formulae will be very useful to many. In this paper we shall arrive at such an understanding by the application of the familiar vector model. [62]

Instead, the very essence of what was obtained in general by Majorana was realized only slightly later by another Nobel laureate, Julian Schwinger, who, unfortunately, did not publish anything on the subject until 1977 [65]. At the heart of the attitude that I adopted toward angular momentum theory was the celebrated theorem and formula of Majorana. These established, qualitatively and quantitatively, how the behavior of an arbitrary magnetic moment in a time varying magnetic field is related to that of a spin 1/2 system. The original Majorana paper was baffling and it was obligatory to find a quantum mechanical derivation of the formula, in particular. My answer was only hinted at in a 1937 paper [given here as Ref. [66]] (“The evaluation of the matrix element (43) may be carried out, for an arbitrary J, by a method which will not be given here. The results are in complete agreement with those obtained from Majorana’s general theorem,” by which I meant the formula). That method was, of course, the explicit construction of an arbitrary angular momentum J as a superposition of 2J spin 1/2 systems. [. . . ] The subject remained quiescent until 1945 (the year for renewed attention to basic physics) when Bloch and Rabi [see Ref. [62]] remarked on the derivation of Majorana’s formula from the spin 1/2 representation, and I began to write a paper supporting the thesis that the expression of symmetry concepts in quantum mechanics does not require the injection of group theory as an independent mathematical discipline. A major piece of evidence was to be the derivation of Majorana’s theorem and formula, thereby finally making explicit the veiled reference of 1937. For some reason this paper was never completed. [. . . ] It may be argued that the mathematical methods of group theory are too general for the purposes of quantum mechanics, that quantum mechanics is adequate to describe, within its own framework, those symmetry operations that arise from physical considerations. It is the purpose of this note to bolster the last contention by deriving, with the aid of elementary quantal operator methods, a number of results that have thus far been considered striking examples of the power of group theoretic methods applied to quantum mechanics. These are: Majorana’s theorem; the matrices representing the finite rotation operators (in group theoretic language, the irreducible representations of the rotation group); the unitary transformation describing the composition of angular momenta; and the selection rules and magnetic quantum number dependence of tensor operator matrices. [65]

The specific terms Majorana’s formula and Majorana’s theorem, with the meaning quoted above, were only introduced by Schwinger in his 1977 paper [65], the relevance of which may be appreciated by noting its inclusion in the recent collection of “seminal papers” by Schwinger [67]. A more recent re-discovery of the general Majorana representation of a spin state has occurred through a popularized work by Roger Penrose [68], which points

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out in particular its geometric interpretation. Indeed, it has been found extremely useful in quantum information science, when dealing with multiparticle entanglement, where that representation leads to the classification of entanglement families of permutation symmetric qubits, based on the number and arrangement of spinors constituting the multiqubit state. In particular, the geometric representation obtained through the Majorana sphere gives a direct quantification of the entanglement of symmetric states, which is one of the central themes underlying quantum information theory, the Majorana representation leading to a simplified structure of the geometric measure of the entanglement [69]. Incidentally, it is also worth mentioning what the Majorana representation has brought (again, only in recent times) to a number of mathematical applications; we mention here only the connection between the group SU(∞) and the group SDiff (S2 ) of area-preserving diffeomorphism of the sphere [70], and the canonical representation of spherical functions (eigenfunction of the Laplacian on a sphere), where Maxwell’s multipoles are proved to be pairs of opposite vectors in Majorana’s sphere representation of quantum spins [71]. As previously stated, what we have discussed so far concerns only the first - the general – part of Majorana’s paper N.6; the second, specific, part relating directly to spin-flips has also been the progenitor of many key applications in different areas of physics.

2.5.2 Landau–Zener probability formula First of all, we have to note that, almost simultaneously, a problem similar to that studied by Majorana in his paper N.6 was carried out independently by Lev Landau [72, 73], Clarence Zener and Ernst C.G. Stückelberg [75], whose papers dealt explicitly with non-adiabatic transitions in atomic collisions for the avoided crossing situation. The formulae obtained by these authors for the probability of spin-flipping (sometimes known as the Landau–Zener probability of non-adiabatic level crossing) are identical to those reported by Majorana, but their mathematical treatment made recourse to higher transcendental functions (with an even more sophisticated treatment by Stückelberg), this explaining why only Majorana’s simpler approach is adopted in textbooks of quantum mechanics (almost without reference to him) [76]. The Landau–Zener probability formula is used in a number of atomic and molecular applications, as well as in the phenomenology of neutrino oscillation in matter, whenever a non-adiabatic level crossing takes place. Concerning, instead, further applications of the results obtained by Majorana and regarding the same physical problem, that is spinning particles in a variable magnetic field, we quote in the following only a couple of intriguing examples.

2.5 Non-adiabatic spin-flip

31

2.5.3 Majorana’s holes The first example is the previously mentioned work performed by Bloch and Rabi. This established the theoretical basis for the experimental method used to reverse the spin of neutrons by a radiofrequency field, a method that is still employed today in all polarized-neutron spectrometers used for studying magnetic structures, as well as in the manipulation of polarized beams in neutron physics. Majorana’s paper N.6 has also played a crucial role in the recent experimental realization of Bose–Einstein condensation in an atomic gas. Indeed, such a phenomenon can be observed only by cooling atoms down to microkelvin temperatures, made possible by using magnetic traps for atoms oriented and pre-cooled by laser radiation. When atoms move in a magnetic trap, they experience a varying magnetic field, and the slower they move around the magnetic field minimum the colder they are. The configuration is, then, completely similar to that studied by Majorana when atoms pass near to a region where the field vanishes, and his own quantitative analysis gives the probability that atoms undergo a spin-flip. Now, when atoms reverse their spin near the zero of the magnetic field, they are no longer confined, thus producing an undesired drawback in the cooling process; thus, the evaluation of the probability of such an effect is a crucial ingredient in achieving condensation. As recalled by the Nobel laureates Eric A. Cornell, Carl E. Wieman, and Wolfgang Ketterle, who shared the prize in 2001 “for the achievement of Bose– Einstein condensation in dilute gases of alkali atoms,” the key problem was, indeed, just to avoid Majorana’s holes: When constructing a trap for weak-field seeking atoms, with the aim of confining the atoms to a spatial size much smaller than the size of the magnets, one would like to use linear gradients. In that case, however, one is confronted with the problem of the minimum in the magnitude of the magnetic fields (and thus of the confining potential) occurring at a local zero in the magnetic field. This zero represents a “hole” in the trap, a site at which atoms can undergo Majorana transitions and thus escape from the trap. [. . . ] We knew that once the atoms became cold enough they would leak out the “hole” in the bottom of the trap, but the plan was to go ahead and get evaporation and worry about the hole later. [77]

The estimation of the Majorana flop rate was, nevertheless, performed, and the suppression of the Majorana spin-flip was realized by “plugging the hole” with a suitable magnetic field, thus enhancing significantly the atomic-trap lifetime and allowing the observation of Bose–Einstein condensation: We knew that this trap would ultimately be limited by Majorana flops in the center of the trap where the magnetic field is zero. Near zero magnetic field, the atomic spin doesn’t precess fast enough to follow the changing direction of the magnetic field – the result is a transition to another Zeeman sublevel which is untrapped, leading to trap loss. We estimated the Majorana flop rate. [. . . ] Why hadn’t we considered this trap earlier and avoided the detours with the quadrupole trap, Majorana flops, and plugging the hole? [78]

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The imagery of magnetic trapping and ultralow temperature atomic physics – as we can easily observe – is full of “Majorana” terms, all of them coming from his paper N.6 [79]. 2.5.4 Majorana–Brossel effect Finally, also worthy of mention here is the so-called Majorana–Brossel effect, employed in double-resonance spectroscopy. In the 1950s early, experiments were performed to investigate atomic structures in which atoms were doubly excited by electromagnetic radiation in the optical range by stimulating transitions between different electronic terms, and a radiofrequency magnetic field was applied to induce transitions between different magnetic sublevels. The basic experiment was that of Jean Brossel and Francis Bitter on the polarization of the fluorescence emitted from mercury, whose underlying idea was clearly summarized in the abstract of their paper [80]: Excitation of atoms in a vapor by polarized light produces unequal populations of the magnetic sublevels of the excited state. The emitted optical resonance radiation is therefore partly polarized. The application of radiofrequency or microwave fields at a magnetic resonance frequency will induce transitions between sublevels of the excited state. The degree of polarization of the emitted optical resonance radiation is altered when the magnetic resonance condition is fulfilled. As suggested by Brossel and Kastler, this effect may be used to reveal the structure of the energy level. [. . . ] The results obtained indicate that double resonance phenomena constitute a valuable new tool for investigating the structure of atomic energy levels.

Brossel and Bitter interpreted the effect they observed as induced by “Majorana transitions,” and the “Majorana formula” was used to explain correctly the experimental data. A review written by Bitter about ten years later is particularly illuminating [81]: During the decade of 1930, Majorana and Rabi discussed magnetic-resonance absorption theoretically, and Rabi and his collaborators detected nuclear magnetic resonance experimentally in atomic beams. [. . . ] The next step was the detection of magnetic resonance in solids and liquids, and also in gases. [. . . ] Finally, the optical methods of detecting magnetic resonance were developed after 1950. [. . . ] The first attempts to detect magnetic resonance optically were undertaken by Brossel, at the author’s [Bitter] suggestion [. . . ]; the effect was found by Brossel in the 3 P1 level of the even isotopes of mercury. [. . . ] The author [Bitter] was then able to show that the distinctive shape of these resonance curves was due to the multiple degeneracy of the magnetic-resonance transitions in this level with J = 1, and could be accounted for by a straightforward application of the Majorana formula for the magnetic resonance transition probabilities. [. . . ] As is apparent from the form of the Majorana expression for the transition probabilities, when more than two equally spaced levels are involved in a resonance, various multiples and submultiples of the normal Larmor frequency may be expected. The modulation of the resonance radiation emitted by

2.6 Nuclear forces

33

a precessing atom is formally equivalent to a “beat” phenomenon between two coherent radiations whose frequencies differ by the amount of the precessional frequency.

Non-trivial line shapes when large radiofrequency fields are applied, with particular reference to the splitting in the line shape as first observed by Brossel and Bitter, is now generally referred to as the Majorana–Brossel effect, and is still the object of experimental investigation on nonlinear magneto-optical rotation in the presence of radiofrequency fields [82]. Quite intriguing is the recent re-interpretation by the Nobel laureate Claude Cohen-Tannoudji (who founded the “atomes froids” group at the Laboratoire Kastler Brossel in Paris) of the double-resonance splitting in terms of a quantum interference effect between linear excitation (one-photon) and power broadening (three-photon) processes, and another characteristic higher-order three-photon process, which is responsible for the observed dip (between the two resonance peaks) in the line shape. Such a process is referred to by Cohen-Tannoudji as Majorana inversion [83]. 2.6 Nuclear forces The best-known contribution by Majorana during the 1930s is certainly his paper(s) N.8 “On nuclear theory,” mainly due to the peculiar advertising by Heisenberg, as explained in Chapter 1. 2.6.1 The Heisenberg model of nuclear interactions Before the discovery of the neutron in 1932, the accepted model for the nuclear structure assumed the nucleus to be composed of A protons and A − Z electrons (Z and A being the charge and mass number of the given atom, respectively). Very simply, this originated from the fact that only these two particles were known experimentally at that time, and – moreover – that in nuclear β-decay an electron was known to be emitted from the radioactive nucleus. Nevertheless, several problems were noted to arise from this naive model that prevented the development of a consistent nuclear theory according to the basic principles of the quantum theory. Two of them are particularly remarkable. The first one is that the nucleus 14 N was supposed to be composed of 14 protons and 7 electrons, an odd number of spin-1/2 particles, so that – as proved by Paul Ehrenfest and J. Robert Oppenheimer [84] – its spin should have been a half-integer number, and the nucleus should have followed Fermi statistics. Instead, spectroscopic experiments conducted on such a nucleus revealed that its spin was integer [85] with Bose statistics [86], as if the electrons in the nucleus were spinless. The second, as very clearly explained by Fermi in 1932 at the International Conference on Electricity in Paris, was that the uncertainty principle implied that electrons confined

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within a nucleus of size 10−13 cm would have an average kinetic energy of more than 60 million eV, a value exceedingly larger than that of any known beta ray, making it impossible to explain the confinement of the electron within an atomic nucleus. Chadwick’s discovery of the neutron allowed Heisenberg to suggest that, since neutrons have a mass of the same order as that of protons, their motion would be non-relativistic as well, thus overcoming the second difficulty. The first problem was instead solved by assuming that a nucleus with mass number A and charge number Z is made up of Z protons and A − Z neutrons. These were just the basic assumptions underlying the theory developed by Heisenberg in a series of three papers that appeared in June–December 1932 [87, 88, 89], apparently with no need for nuclear electrons. Starting from the analogy with the H+ 2 molecular ion, where H and H+ are held together by the exchange of an electron, as explained by Heitler and London, Heisenberg assumed that the nuclear forces holding together neutrons and protons in a nucleus are of the same “exchange” nature: neither the polarization forces among neutrons nor the Coulomb repulsion among protons were mainly responsible for the nuclear bonding. Indeed, forces between two neutrons were assumed to be negligible, while protons were supposed to interact only through Coulomb interactions, which (at least for light nuclei) could be neglected in a first approximation, so that the only forces responsible for nuclear stability were those between neutrons and protons. Heisenberg then assumed that a pair comprising a neutron and a proton is held together by the exchange of charge, defined as the eigenvalue of an abstract operator (taking the value +1 for a neutron and −1 for a proton) that could be regarded as the third component of a spin-like operator, which he called ρ-spin (and is now known as nuclear isospin). In a more transparent manner, as chemical binding between H and H+ is due to the exchange of an electron, likewise nuclear binding between a neutron and a proton is realized with the exchange of an “effective” spinless electron, as if a neutron were composed of a proton and an electron. It is intriguing that, in the entire series of his three 1932 papers, Heisenberg was constantly oscillating between holding the view that the neutron is an “elementary” particle and understanding it as a composite one, the latter position being strongly supported by the emission of electrons in nuclear β-decay [89]. Die Annahme, daß das Neutron in bezug auf Spin und Statistik einem Elementarteilchen, in bezug auf Polarisierbarkeit, Zerlegbarkeit usw. einem zusammengesetzten Gebilde gleicht, führt auf das Problem, zwei in der Quantenmechanik unvereinbare Eigenschaftstypen zu verschmelzen.5 5 “The assumption that the neutron appears as an elementary particle when considering spin and statistics,

while it resembles a composite structure when considering polarizability, decay, etc. leads to the problem of merging two incompatible types of properties in quantum mechanics.”

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Despite the striking improvements introduced by the neutrons + protons model with respect to the traditional protons + electrons one, this hesitation was typical of many physicists, and the Heisenberg model took some time to be accepted.

2.6.2 Majorana’s exchange mechanism The situation changed dramatically with the appearance of the Majorana contribution (paper N.8) in 1933 and, almost one year later, with the publication of “Fermi’s theory of beta decay” [90], which finally established that electrons emitted in such a decay did not pre-exist in the nucleus [6]. In order to understand why Majorana’s paper succeeded in changing the traditional view, we have only to follow Majorana’s reasoning and the results he obtained from that. The starting point was a very clear analysis of what had been introduced by Heisenberg:6 In the absence of other guiding criteria, Heisenberg was guided by an analogy presumably existing between the normal neutral hydrogen atom and the neutron, the last one being supposed – as commonly accepted – to be composed by a proton and an electron. [. . . ] The use of such an analogy is difficult to justify since, if the neutron would be effectively composed by a proton and an electron, their binding could not be described by the present theories, which would lead to associate the Bose–Einstein statistics and an integer multiple of h/2π for the mechanical moment to the neutron, contrary to fundamental assumptions. These come directly from empirical properties of the nuclei, and we cannot give them up. Given the present state of our knowledge, it is thus preferable to try to obtain the law of interaction between the elementary particles being guided by simplicity only, in order to predict the most general and characteristic properties of nuclei.

Simplicity is, indeed, the major feature of Majorana’s paper compared to the abstract formalism introduced by Heisenberg; even the assumption of focusing only on the neutron–proton interaction emerged there in a natural way from a welldefined problem, i.e. the quest for a description of an “impenetrable” nucleus or, in other words, for a justification of a constant-density nuclear matter: Our problem is, then, to find the most simple law of interaction between any elementary particle – protons and neutrons – leading to the definition of an impenetrable matter as long as Coulomb repulsion may be neglected. [. . . ] Electrostatic forces cannot be very important for the structure of light nuclei and, since they contain almost the same number of neutrons and protons, it seems reasonable to assume that a particular interaction between protons and neutrons is the main cause of nuclear stability, while we will as well assume that there is no noticeable interaction between the neutrons for there is no proof of the contrary. 6 Majorana published the same paper both in German and in Italian (after about three months) with almost

negligible differences between the two versions. The most relevant one (in my opinion) is that the author was a bit more explicit in his criticism of Heisenberg’s theory in the Italian version (I thank Prof. F. Panzera for useful discussions on this point). In what follows, we quote from that version.

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The basic assumptions were, then, the same as for Heisenberg, but now no hesitation exists about the nuclear elementary constituents, just protons and neutrons. This directly prevented some conclusions obtained by Heisenberg: It could seem natural to introduce an interaction [between protons and neutrons] as is generally done for any pair of atoms or molecules, i.e. long range attractive forces that are, nevertheless, strongly repulsive at a short range, for the particles to be “impenetrable.” In addition to this, however, we should also introduce short-ranged repulsive forces between neutrons, in order to get the desired proportionality between the number of constituent particles and the volume of nuclei. Such a solution to the problem is unsatisfactory, since the existence of attractive forces of unknown origin is required, along with the existence of short-ranged repulsive forces with exceptionally large strengths, again of unknown origin. We shall, therefore, try to find another solution able to explain how to predict a density independent of the nuclear mass, without obstructing the free movement of the particles by an artificial impenetrability.

It is now clear what Majorana meant by “simplicity,” and the stage is set for the introduction of an alternative, simpler assumption: We could try to focus on a given interaction whose average energy per particle never exceeds a certain limit however great the density, this occurring through a sort of saturation phenomenon more or less analogous to valence saturation.

Technically, this was accomplished by considering – in a mathematically simpler fashion – the same quantum-mechanical exchange mechanism envisaged by Heisenberg, but, exactly as in the Heitler–London theory, the “exchange” involves now the spatial coordinates of protons and neutrons, rather than their charge. In Heisenberg’s language, this corresponds to the simultaneous exchange of charge and spin, and is the most natural mechanism if one thinks of “elementary” protons and neutrons. Majorana’s exchange mechanism was then at variance with Heisenberg’s: There is no saturation because of the symmetry character of Heisenberg’s eigenfunctions, so that the introduction of repulsive interactions at short distances is required, contrary to our purpose.

Majorana’s exchange mechanism then prevented the collapse of the nucleus without the need for any repulsive force at short distance, but its main justification relied, according to Majorana himself, on the fact that it predicted the constancy of the nuclear density: 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 experiments.

Also, the saturation process predicted by Majorana led to thorough phenomenological predictions:

2.6 Nuclear forces

37

We find that both neutrons act on each proton in the α-particle instead of only one and viceversa, since we assume a symmetric 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 the constituent particles are in the same state so that it is a “closed shell” in a more strict sense than for the helium atom. 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, and since the exchange energy [See Ref. [87]] is usually large only if a proton and a neutron are in the same state, we may expect that in heavy nuclei the binding energy per particle is not noticeably bigger than in the α-particle, as shown by experiments.

It was just this “special stability of the α-particle” that immediately convinced Heisenberg that Majorana’s model was preferable to his own. As a matter of fact, it is indeed an empirical datum that Majorana’s exchange forces, along with the ordinary, non-exchange forces introduced by Eugene P. Wigner [91], are the most important ones for nuclear stability from the practical point of view [92]. 2.6.3 Thomas–Fermi formalism and Yukawa potential Mainly due to Heisenberg’s publicity (see Section 1.4), the scientific community quickly adopted Majorana’s exchange mechanism, with its description of nuclear density saturation (in modern terms, the density of nucleons is about the same for all nuclei) and binding energy saturation (the binding energy per nucleon is about the same for all nuclei). Credit is given not only in contemporary papers on nuclear physics, but also in subsequent reviews and books (see, for example, Ref. [92]), despite the fact that the forces commonly assumed today are very different from Majorana’s or Wigner’s forces acting only between unlike nucleons. However, it is intriguing that two more contributions, also contained in paper N.8 (though not emphasized), were later adopted by nuclear physicists, without reference to Majorana, probably due to the fact that not even Heisenberg realized their relevance. The first one is the formalism through which Majorana succeeded to obtain density saturation, i.e. that of the Thomas–Fermi model applied to nuclei, rather than atoms. Indeed, he defined the nuclear density just as in the atomic Thomas– Fermi model, by means of Fermi momenta of protons and neutrons, and then derived an asymptotic expression (for large densities) for the exchange energy per particle. From what he obtained for the total nuclear energy, Majorana then deduced that the competition between kinetic and potential energy resulted in an energy minimum (as a function of the intra-nuclear distance) corresponding to the stability condition for the nucleus, in full analogy with the molecular Heitler– London case, and that the density of neutrons and protons did not depend on the mass of the nucleus. This was just a manifestation of Majorana’s acquaintance with the statistical model of atoms (see Chapter 4), but it is remarkable that the application of the Thomas–Fermi model to nuclei was re-discovered only several

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years later (see, for example, the paper Ref. [93] and the more modern textbook Ref. [94]). The second interesting contribution concerns the explicit form of the interaction potential I(r) between protons and neutrons, which did not directly enter in to the calculations performed in the main body of either Majorana’s or Heisenberg’s paper. In the concluding remarks of his paper, however, Majorana commented on this issue: We can try to determine the function I(r) in a way that best represents the experimental results. The expression I(r) = λ

e2 , r

for instance, with an arbitrary constant, is suitable even though it becomes infinite for r = 0. For large values of r, however, it must be modified as it gives an infinite cross section for the scattering 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 should use an expression with at least two arbitrary constants, e.g. an exponential function I(r) = A e−sr .

It is apparent from this quotation that the well-known Yukawa potential −g2 e−λr /r, introduced two years later [95], follows directly.7 Majorana did not elaborate further on this point, whereas Yukawa explored the physical consequences of the assumption and introduced the so-called meson theory. The reason for this was due to purely practical considerations, and was explained by Majorana himself: “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.”

2.7 Infinite-component equation As pointed out in Ref. [96], the effective birth of theoretical particle physics – as we understand it today – may be traced back to Majorana’s paper N.7 (published in 1932) and, to some extent, to paper N.9 (published in 1937). The reason for such a claim lies mainly in the conception and use of quantum-relativistic field theory by Majorana in his papers, which was certainly ahead of his time; this will be clearly apparent from the following discussion. 7 Such a connection was pointed out to me by Prof. F. Panzera, to whom I owe my gratitude. There is no

definite proof that Hideki Yukawa knew about Majorana’s paper N.8 (whereas he mentioned Heisenberg’s papers), but it is certainly intriguing that he also stressed the introduction of two arbitrary constants, just as Majorana did: “Two constants g and λ appearing in the above equations should be determined by comparison with experiment” [95].

2.7 Infinite-component equation

39

2.7.1 A successful relativistic wave equation The Dirac equation was a favorite and long-lasting topic studied by Majorana [17, 18]. As is well known, the birth of quantum mechanics was driven by the basic principles of the theory of relativity, and Schrödinger himself wrote down first the relativistic wave equation, now known after Oskar Klein and Walter Gordon, and then his most famous non-relativistic equation. The original rejection by Schrödinger (and others) of the relativistic Klein–Gordon equation as the correct quantum equation describing an electron was based on the comparison of its prediction with the accurate experimental spectroscopic data on the hydrogen atom, and this discrepancy was correctly attributed to the missed consideration of the spin of the electron. However, when Dirac approached the problem of making a relativistic theory of the spinning electron, no reference to this discrepancy was made, but rather he focused on another theoretical problem, namely that of negative probabilities. Indeed, Dirac’s primary aim was to develop a relativistic formalism with positive probabilities, just as for the non-relativistic Schrödinger equation, different from what emerged from the relativistic Klein–Gordon equation. This was actually achieved by means of his famous first-order (in the time variable) equation [97, 98]. A second major problem, initially left unsolved by Dirac, concerned the appearance of negative energy states (in any relativistic field theory); a solution to it was later given with the interpretation of Dirac’s holes in terms of antiparticles with the same mass as electrons but with opposite charge [99]. With the exception of Pauli8 and a few other people, who gave some (justified) importance to the theoretical problem of the negative energy states, the vast majority of physicists granted the success of the Dirac equation even before its “final” confirmation with the discovery of the positron [101, 102]. The reason was due mainly to the successful predictions about the fine structure of hydrogen and the correct account for the magnetic moment of the electron (with the prediction of its gyromagnetic ratio g = 2), which were major troubles for the newly born quantum mechanics. People realized that such problems were intimately related to the spin of the electron, and the successful Dirac theory offered a reasonable and attractive way out, well founded on solid formal basis. Even better, quite a general consensus prevailed that the value of 1/2 for electron spin was a necessary consequence of the theory of relativity, i.e. of Lorentz invariance [103]. Just the contrary was instead demonstrated in 1932 by Majorana in his paper N.7, where he explicitly built a consistent relativistic quantum theory for particles with arbitrary spin, by starting from a lucid examination of the persisting problem of the negative energy states. 8 “I do not believe in your perception of ‘holes,’ even if the existence of the ‘antielectron’ is proved,” wrote

Pauli to Dirac on May 1, 1933 [100].

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2.7.2 Majorana equation Majorana adopted a relativistic wave equation of the same form as the Dirac equation, but nothing was assumed for the α, β “Dirac” matrices appearing in it. Instead, a thorough (mathematical) inspection of the occurrence of the negative energy states was presented. Equations of this kind present a difficulty in principle. Indeed, the operator β has to transform as the time component of a 4-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 [. . . ] 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 c4 + c2 p2 . As a matter of fact, the indeterminacy in the sign of the energy can be eliminated [. . . ] only if the wavefunction has infinitely many components that cannot be split into finite tensors or spinors.

Thus, Majorana did not require the ordinary energy-momentum relation W 2 = c p + m2 c4 to be satisfied for each component of the wavefunction, as Dirac did, in order to determine the expressions for the matrices α, β. Rather, he determined a priori a representation for the infinite matrices corresponding to the six infinitesimal Lorentz transformations, i.e. the transformation matrices for the components ψjm of the wavefunction forming a basis for that representation. The form of the α, β matrices was then deduced by imposing relativistic invariance on the action integral. The rest energy of the particles described by this theory is positive: 2 2

W0 =

mc2 , j + 12

(2.1)

but this reveals that the Majorana equation, contrary to the Dirac equation, is a multi-mass equation. “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.” Also, solutions with different energy eigenvalues are also present in the theory: “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 p2 − k2 c4 ; 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.” Tachyonic solutions thus enter for the first time in a relativistic wave equation. All these peculiarities were re-discovered and appreciated only years later, as we will see: the physical and mathematical content of the Majorana theory was, evidently, too much ahead of its time, during a period when people became

2.7 Infinite-component equation

41

convinced that spin-1/2 particles were enough for Nature, and that such a result directly followed from Lorentz invariance through the simple Dirac equation. It was only two years later [104] that Pauli and Weisskopf showed – by “secondquantizing” the Klein–Gordon theory – that the marriage between quantum mechanics and special relativity did not necessarily require spin 1/2 for the correct interpretation of the formalism, as erroneously believed. Thus, non-intrinsically wrong arguments were proved to exist in the Klein–Gordon theory that justified the derivation of the Dirac theory; such theories just applied to particles with different spin, as already deduced by Majorana himself (see also Chapter 9). 2.7.3 Infinite-dimensional representations of the Lorentz group The mathematical relevance of Majorana’s paper N.7 resides in the discovery of the (simplest) infinite-dimensional unitary representations of the Lorentz group, which were re-discovered by Wigner in his 1939 and 1948 works [105, 106]. We will consider in detail this topic in Section 7.5; here it is quite interesting to dwell on the extremely simple reasoning which led Majorana to consider the infinitedimensional representations, not deducible from his paper, but easily recognizable in his personal notebooks (see p. 446 of Ref. [17]): The representations of the Lorentz group are, except for the identity representation, essentially not unitary, i.e., they cannot be converted into unitary representations by some transformation. The reason for this is that the Lorentz group is an open group. However, in contrast to what happens for closed groups, open groups may have irreducible representations (even unitary) in infinite dimensions. In what follows, we shall give two classes of such representations for the Lorentz group, each of them composed of a continuous infinity of unitary representations.

Also interesting is the acknowledgment by Wigner of Majorana’s work, typical of a rigorous mathematician [105]: The representation of the Lorentz group has been investigated repeatedly. The first investigation is due to Majorana, who in fact found all representations of the class to be dealt with in the present work excepting two sets of representations. [. . . ] The difference between the present paper and that of Majorana [. . . ] lies – apart from the finding of new representations – mainly in its greater mathematical rigor. Majorana [. . . ] freely uses the notion of infinitesimal operators and a set of functions to all members of which every infinitesimal operator can be applied. This procedure cannot be mathematically justified at present, and no such assumption will be used in the present paper. Also the conditions of reducibility and irreducibility could be, in general, somewhat more complicated than assumed by Majorana.

The greater mathematical rigor by Wigner certainly justified the 56 pages of his paper, but the answer-at-a-distance of the physicist Majorana is illuminating: “in order to avoid exaggerated complications, we will give the transformation law

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only for infinitesimal Lorentz transformations, since any finite transformation can be obtained by integration of the former ones.” Such an approach is still used today by practically all physicists. 2.7.4 A difficult problem for Pauli and Fierz Despite such simplicity of reasoning, this Majorana theory was apparently not appreciated by many, given its very few mentions in the literature:9 apart from Wigner, it was quoted only in Refs. [107], [103], and [108] for the period of time considered. Nevertheless, this is only partially true, since much of the ideas underlying it and even some technical results obtained, but not published, by Majorana, would be re-discovered and re-considered later by other people in different times; such results concern mainly unpublished (preparatory) work, and we will discuss them at length in Chapters 7 and 8. Here, instead, we will focus on an intriguing occurrence that sheds some light on the direct influence of Majorana’s paper N.7 on leading scientists by looking at their careful studies of it, studies that were not published later in known publications. A key role in the understanding of Majorana’s paper N.7 was played by Pauli (probably informed by Heisenberg), who studied it at least from 1939 to 1947 in two distinct phases with his associate Marcus Fierz, around 1940 and in 1947. This is clearly testified by nine letters10 present in Pauli’s correspondence, now published in Ref. [109], from which we quote in the following. The reason for such a strong interest in that paper is declared in the first letter of this set, which Pauli wrote to Homi J. Bhabha on April 12, 1940: I believe in the existence of much more particles than known until now, particularly on particles with arbitrary values of the spin and of the charge. My considerations about the particles with higher spins came to some end now. I think that they exist really, but I can fancy that the complication of the theory comes from the assumptions, that one has to describe a set of particles with a finite number of spin values only. May be the matter becomes simpler, if one introduces a priori an infinite set of spin values (compare Majorana [. . . ]).

Such an indication (which evidently urged Bhabha to think about what he later published in 1945 [110]; see Chapter 8) clearly points out that Pauli only started 9 This is related also to the fact that it was published in a not internationally renowned journal. As already

noted by Fradkin [22], “Science Abstracts (Section A, Physics), the English language abstracting service, did not abstract from Nuovo Cimento until 1946. Majorana’s article was given a several line abstract by the contemporary German abstract service [Physikalische Berichte, 1933-I, p. 548] but the abstractor, whose major field was fluorescence of salts and crystal studies, failed to assess its significance or even mention the occurrence of the infinite-dimensional representations.” 10 Pauli to Bhabha, April 12, 1940; Pauli to Fierz, July, 3 1940; Pauli to Fierz, July, 17 1940; Pauli to Fierz, September, 3 1940; Pauli to Jauch, November, 1 1940; Pauli to Fierz, February, 12 1941; Pauli to Fierz, March, 29 1941; Fierz to Pauli, March, 17 1947; Pauli to Fierz, March, 30 1947.

2.7 Infinite-component equation

43

to study Majorana’s paper at that time, and further insights on it still had to follow. Indeed, a full understanding of the Majorana theory resulted to be not so easy to achieve: “Anyway, it seems to me that the case of the infinite-dimensional representations of the Lorentz group has not been studied sufficiently. [. . . ] Read Majorana!” (Pauli to Fierz, July 3, 1940). Or, in Pauli’s more characteristic style, when referring to Wigner’s paper [105]: “Wigner did not understand Majorana’s equations, as he admitted to me” (Pauli to Fierz, March, 29 1941). Pauli himself, however, had to read Majorana’s paper several times before coming to any definite conclusion. The first problem he envisaged was that of the solutions of the Majorana equation propagating with a superluminal speed (already noted by Majorana himself): The bad thing with his theory is that his plane waves correspond to an imaginary rest mass,  i.e. particles moving always with a velocity greater than that of light, that is E/c =

± p2 − p20 (p0 > 0 arbitrary, |p| > p0 ). [. . . ] The question would then be: does it exist an infinite number of eigenfunctions [. . . ] for those equations (or constraints) whose pathologic solutions with (v2 /c2 ) − k2 < 0 are excluded? This last condition plays a role similar to that of requiring a positive energy or charge density. (Pauli to Fierz, July 3, 1940)

It is remarkable that Pauli recognized such a problem as pathologic for the theory (and only this as a real problem: “I don’t see anything pathologic in the Majorana equation [. . . ]. The only thing which seems to be pathologic in Majorana’s theory are the solutions with imaginary mass” (Pauli to Fierz, July 17, 1940)), but nevertheless he continued to study it for a long time. Indeed, in a long letter to Fierz of September 3, 1940, Pauli re-derived (in an alternative way) the conclusions obtained by Majorana, even casting the original Majorana equation into a different, equivalent form and comparing them with the Dirac equation. The interesting conclusion about the “pathologic solutions” was that their exclusion should be related directly to the Dirac–Fierz–Pauli generalized equations for spin higher than 1/2 [111, 112] (see Chapter 8): “It seems to me that the exclusion of unphysical states with p20 − p2 < 0 should come essentially from our equations” (Pauli to Fierz, September 3, 1940). Such a feeling was likely the background for considering the Majorana equation as an interesting mathematical problem, but without interesting physical applications (Pauli to Fierz, September 3, 1940). Nevertheless, again, this did not prevent Pauli to look further inside the Majorana theory, even with the help of Valentine Bargmann,11 searching for possible alternative representations of the Majorana equations in terms of differential operators. This problem, however, was not so easy to solve, and several failures marked Pauli’s research: “Up to now we have not been able to obtain representations with 0 < J < 1 using differential 11 Bargmann published his final results on the unitary representations of the Lorentz group only years later

[113], but, evidently, Pauli was aware of some of his results earlier than this.

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operators. And we do not even know what it might mean the point J = 1 of the spectrum in group theory. [. . . ] So this is still an open problem” (Pauli to Fierz, February 12, 1941). However, Pauli did succeed in obtaining several mathematical results about the Majorana theory, and his first investigations of 1940–1 concluded with the important result that: The Majorana equation with an infinite-dimensional ζ -space corresponds, in the (ζ, p)space, to a reducible unitary representation of the inhomogeneous Lorentz group (including all the cases with p20 − p2 > 0, p20 − p2 = 0 and p20 − p2 < 0), besides the Dirac case and other cases. (Pauli to Fierz, March 29, 1941)

For a short period of time, Pauli again considered the problem in 1947, probably at the request of Fierz, who, in the meantime, went further into the mathematical inspection of it. Now, however, the trouble was with the mass spectrum predicted by Majorana, re-obtained in a different way by Fierz (see Eq. (2.1)), m =

M ,  + 12

(2.2)

by using an equivalent form of the Majorana equations. However, “in any case, the equations are really useless in this form because, for very large , masses are very small. Any coupling, for example with radiation fields, will result in transitions with any large ” (Fierz to Pauli, March 17, 1947). The attention then shifted to physics problems. Nevertheless, Pauli cut short any possible subsequent discussion, by recalling the result already obtained in 1941: It seems that the point of view of the unitary representations of the inhomogeneous Lorentz group for the classification of relativistic wave equations of particles in the absence of force is very reasonable, namely that to certain irreducible representations correspond definite values of the mass and the spin. From this point of view, for example, the equations of Majorana appear completely arbitrary, because they are reducible!” (Pauli to Fierz, March 30, 1947)

The initial confidence in the existence of “much more particles than known, with arbitrary values of the spin” finally changed: the appearance of the lucid analysis of Bhabha in 1945 [110] ended the game.

2.7.5 Further elaborations It seems that Majorana’s paper of 1932 remained very difficult to understand fully, even for first-order theoreticians like Wigner and Pauli, if it is true that it took about one year of intense study with Fierz to complete Pauli’s understanding of Majorana’s theory, including the recognition that Majorana’s equation corresponded to a reducible unitary representation of the Lorentz group (as in the Kemmer case [107],

2.7 Infinite-component equation

45

for example). This testifies to the difficulty of the problem at hand, and to the depth of Majorana’s reasoning and results, but it did not prevent several other scientists from “re-discovering” – about 20 years later – the Majorana equation and carefully studying its properties. The credit for this re-discovery certainly goes to Fradkin, who, informed by Amaldi, in 1966 re-examined the pioneering paper N.7 and clearly explained to a wide audience the relevance of Majorana’s original approach and results [22]. The first result was the appearance of an English translation of the whole of Majorana’s paper in 1968, made by Claudio A. Orzalesi and issued by the Department of Physics and Astronomy at the University of Maryland [114], this evidently stimulating a number of novel investigations about the Majorana infinite-component equation (see, for example, Refs. [115], [116], [117], [118], [119], [120], [121], etc.). The interest of physicists in Majorana’s paper N.7 at the end of 1960s was mainly driven by two reasons, both related to the emerging idea that hadrons were composite objects: the incoming use of dynamical groups on one hand, aimed at describing the interactions as generators of a group (in analogy to what happens for the electromagnetic interaction in the H atom), and, on the other hand, Murray Gell-Mann’s program of saturating the algebra of currents at infinite momentum transfer in terms of single-particle states [118]. It was proved, for example, that the Schrödinger equation for the hydrogen atom, when the electromagnetic interaction is included, can be rewritten in the form of a non-relativistic infinite-component wave equation of the Majorana type [116]. The major problem with the Majorana equation (and its possible generalizations formulated in the late 1960s) was, as first envisaged by Pauli (see above), its spacelike solutions, causing some problems with the CPT theorem (no negative energy solutions are present for the Majorana equation) and the spin-statistics theorem. Indeed, it became clear that Majorana’s paper N.7 showed that the CPT theorem can be violated in a local relativistic theory, so that relativity and locality are not the only basic ingredients to ensure the validity of the theorem, and a further hypothesis about the nature of the representations of the Lorentz group was necessary (the CPT theorem is valid for any finite-dimensional representation) [118]. The number of papers devoted to such a topic at the end of 1960s testify to a rather confusing situation, and, in this respect, the most remarkable contribution came in 1970, when E.C. George Sudarshan and Narasimhaiengar Mukunda succeeded [119] in formulating the quantum theory of the infinite-component field, thus providing a solid theoretical basis to the Majorana theory: In spite of renewed interest in infinite-component wave equations and their group-theoretic basis, there are several problems connected with the quantum theory of such a system. The first concerns the apparent possibility of restriction of the field to only one sign of the frequency and the consequent danger of violating basic features of conventional field

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theory like CPT invariance and the substitution law. The second difficulty concerns the existence of space-like solutions to these equations. It was also found that the usual proofs of the connection between spin and statistics were not valid in these cases. It is the purpose of this paper to resolve these difficulties and to present a quantum field theory of the infinite-component Majorana field. [. . . ] [Our results] enable us to interpret the second-quantized Majorana fields as assemblies of an infinite number of types of quantum-mechanical particles obeying Bose or Fermi statistics according as the spin is integral or is half-integral. This association extends also to the space-like particles, the relevant thing then being whether the representation of the little group is single or double valued. [119]

Many other aspects of the Majorana theory were studied, but we do not discuss them here.12 Instead, we focus on a curious re-evaluation of paper N.7 made by Dirac (in 1971) [120], who proposed a new relativistic wave equation for a onecomponent wavefunction but referring to “some inner degrees of freedom”: One of the surprising different physical consequences of the new theory is that it allows only positive energy values for the particle. A relativistic wave equation allowing only positive energies was proposed a long time ago by Majorana (1932). There is a connexion between Majorana’s equation and the present theory. [. . . ] It would just be Majorana’s, in a different representation. [. . . ] It was shown by Majorana that his equation [. . . ] allows an infinite succession of mass values, [. . . ] and the spin increases progressively to infinity as the mass goes to zero. A decreasing mass as the spin increases does not agree with what is observed experimentally, and for this reason Majorana’s equation did not arouse much interest among physicists. We shall here keep to the whole set of equations of part I and so have only one mass value. [120]

Dirac’s approach was, indeed, equivalent to extracting from the Majorana equation a single-mass spin-0 wave equation by adding to it a subsidiary condition, and considering appropriate “internal” dynamics for the given particle. The suggestion that, due to the presence of space-like solutions, an infinitecomponent equation may describe particles with a composite nature had been made already in 1968 [122],13 but the thorough study of the internal structure of those particles was pursued by Asim O. Barut and Ismail H. Duru, following Dirac: A wave equation with many mass and spin states can be interpreted as describing a composite system in a relativistically invariant manner. Such equations are known for systems like the H atom. [. . . ] Thus it is of interest to construct for the Majorana equation the equivalent composite system. [. . . ] We show in this paper that, given the infinite-component Majorana equation, one can introduce various internal coordinates such that the Majorana system looks like a two-dimensional oscillator. [. . . ] Because of its group structure one gets more naturally a two-dimensional dynamical structure. [121] 12 Note that, although just a couple of papers or so cited Majorana’s work from its appearance to 1960, about 70

papers appeared in the literature in the period 1966–1970, treating different problems of the Majorana theory. 13 The connection between composite systems and unitary representations of the Lorentz group had been

known since 1945 [123].

2.8 Majorana neutrino theory

47

Despite what was found by Sudarshan and Mukunda, the basic problem with the Majorana equation was the occurrence of space-like solutions and the related apparent causal problems. Remarkably, however, Barut and Duru showed that the space-like [light-like] solutions of the Majorana equation were the positive-energy [zero-energy] scattering solution of the corresponding two-dimensional internal oscillator, thus providing a direct physical interpretation of what originally discussed by Majorana. This later (1977) allowed Barut himself (and a collaborator) to conclude that “the existence of space-like solutions of infinite-component wave equations is not a ‘disease’ but a virtue,” since they “have a definite physical interpretation, they have experimentally demonstrable consequences and form an integral part of the theory” [124]. The “fortune” of Majorana’s paper N.7 has continued almost uninterrupted until today, mainly due to the related fortune of string theory, whose mass spectrum involves an infinite number of finite-dimensional representations: as noted recently by Augusto Sagnotti, that paper anticipated “to some extent, both Regge’s idea and its eventual realization in the Veneziano amplitude, and thus in String Theory altogether, by over thirty years!” [125]. 2.8 Majorana neutrino theory In 1937 Majorana published the paper N.9 containing a theory already elaborated some years earlier [18]. The main aim was not that of writing down a novel equation, but rather that of re-formulating the existing Dirac equation in order to achieve a complete “symmetry” between electron and positron components described by it. 2.8.1 “Symmetric” Dirac equation Heisenberg had already noted in 1933 the substantial symmetry, in the Dirac theory, between processes involving electrons and those involving positrons,14 and some time earlier Heisenberg himself elaborated an interesting application in which he considered the symmetry between holes and electrons in an occupied atomic level or in an occupied energy band of a crystal [126]. However, such a general idea of a particle–antiparticle symmetry was formally developed into a consistent theory only by Majorana in his most famous article on a “Symmetric theory of electrons and positrons.” The equation considered by Majorana was just the Dirac equation written in the form   mc 1∂ − α · ∇ − iβ ψ = 0, (2.3) c ∂t h¯ 14 See the letter by Heisenberg to Sommerfeld (June 17, 1933) reported in Ref. [100].

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but here he introduced a choice for the four independent α, β matrices different from the standard Dirac one, namely αx = σx ⊗ σx ,

αy = σz ⊗ 1,

αz = σx ⊗ σz ,

β = −σx ⊗ σy ,

(2.4)

where σ = (σx , σy , σz ) are Pauli matrices. This choice led to profound implications on the theory, since αx , αy , αz , and −iβ all have only real elements, so that Eq. (2.3) is an equation with real-valued coefficients. The bispinor field ψ can, of course, be decomposed again into a real and imaginary part, ψ = U + iV, as in the Dirac theory, but now the separate equations for the real functions U and V are completely identical. Thus, in the quantum description of charged particles, the present theory was completely symmetric with respect to particles and antiparticles. Majorana was aware of the fact that such an advantage was purely formal, since there was no distinction between the two theories in physical applications (but with the important result that the cancelations of infinite constants, relative to single field modes, is required by the symmetrization of the theory). In addition, however, Eqs. (2.3) and (2.4) had a different solution not present in the Dirac theory, that is a real solution ψ = U, without the introduction of the V field. In this case, the theory described a chargeless particle or, rather, according to Majorana’s own words, Eqs. (2.3) and (2.4) constituted “the simplest theoretical representation of neutral particles,” without the need for antiparticles: [Our approach] 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.

Majorana also provided the necessary formal developments aimed at giving a solid field-theoretic basis to Eqs. (2.3) and (2.4), which was derived from a variational principle by means of an appropriate Lagrangian function (containing only the U field). As showed earlier by Dirac and Pauli–Weisskopf, this was not a trivial task, but Majorana, guided by just mathematical elegance and symmetry, succeeded is making the idea that spin-1/2 particles could be their own antiparticles theoretically respectable, i.e. consistent with the general principles of relativity and quantum theory, already known for photons. This was acknowledged by a number of people, starting from the appearance of his paper, including Pauli,15 who appreciated both “the procedure of Majorana” [112] (that is, the field-theoretic derivation) and the “decomposition with respect to charge conjugate functions” 15 Among the first people who were impressed by Majorana’s theory and its consequences were Gïulio Racah

[127], Hans A. Kramers [128], Wendell Furry [129], Nicholas Kemmer [129], Wigner, and Frederik J. Belinfante [130], Pauli [131].

2.8 Majorana neutrino theory

49

[131]. At the end of 1937, Kramers obtained a generalization of the “symmetric theory” which was independent of the particular choice for the representation of the Dirac matrices [128], and, some time later, Furry was the first [129] to recognize “Majorana’s interesting new type of variation principle, concerning which we have nothing new to add,” and started to link indissolubly that theory to neutrinos (see below): Majorana has presented a derivation of a symmetric theory of the electron and positron from a new type of variation principle whose use depends essentially on the fact that the quantities involved are q-numbers. In spite of this novel approach, the positron theory he obtains is essentially just a subtraction theory of the simplest type; but Majorana also showed how his ideas can be applied in the theory of the neutral particle to obtain a formalism essentially different from that of the ordinary Dirac theory. [. . . ] For the neutrino, however, the Majorana theory is a priori just as acceptable as the ordinary Dirac theory. It is interesting to find that it is possible to accomplish all the purposes for which the neutrino theory was devised, including the discussion of both electron emission and positron emission, without the introduction of antineutrinos. [129]

From then on, it became customary to take for granted that the Majorana theory was “as acceptable as the ordinary Dirac theory” (see, for instance, the theoretical paper by Chen-Ning Yang and Jayme Tiomno [132]). We quote here only Shunichi Noma – a student of Yukawa – who introduced it into “the meson pair theory of nuclear forces” about ten years later [133], by following closely the original reasoning and formalism: “we assume that the neutral meson can be described by the Majorana’s abbreviated theory. [. . . ] In view of our interpretation that the three kinds of mesons are three different states of the same particle, this assumption will be natural.” 2.8.2 Neutrino–antineutrino identity The possibility that the antimatter partner of a given matter particle could be the particle itself was no doubt revolutionary, since it was in direct contradiction to what Dirac had successfully assumed in order to solve the problem of negative energy states in quantum field theory, i.e. the existence of the positron. The obvious question then arose: were there examples, physical realizations, of such a possibility? In the same paper N.9, with amazing farsightedness, Majorana suggested that the neutrino, which had just been postulated by Pauli and Fermi to explain the puzzling features of radioactive β-decay,16 could be such a particle: 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 16 Recall that the neutrino was discovered experimentally almost 20 years later, in 1956 [134].

50

The visible side

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.

The consequences of the Majorana theory for neutrons and, especially, neutrinos were further investigated just a few months after the publication of paper N.9 by a visitor to the Fermi group in Rome, Racah (who probably informed Pauli about that novel theory; see the acknowledgments in Ref. [127]). Indeed, while he showed that it cannot be applied to neutrons,17 nevertheless: The requirement that the neutrino wavefunction be real is not only a simpler mathematical method to represent neutrinos, but is also a logical consequence of the assumption that neutrinos and antineutrinos be identical. We thus recognize that, in addition to its formal value, the theory by E. Majorana leads to physical predictions essentially different from those coming from the Fermi theory [of β decay]. [. . . ] The theory by E. Majorana is equivalent to identifying particles with antiparticles. [127]

2.8.3 Racah and the neutrinoless double β-decay The explicit clarification made by Racah was not at all superfluous, since it paved the way to subsequent studies aimed at distinguishing phenomenologically a Dirac from a Majorana particle. Racah himself, according to B. Pontecorvo [135], analyzed this possibility by studying the processes of inverse beta decay originated by neutrinos. The best solution came, however, more than one year later, when Furry proposed [136] to look at a novel nuclear process, which could take place only if neutrinos are described by the Majorana theory, that is neutrinoless double β-decay: The phenomenon of double β-disintegration is one for which there is a marked difference between the results of Majorana’s symmetrical theory of the neutrino and those of the original Dirac-Fermi theory.

Soon after the appearance of the Fermi theory of β-decay in 1934, Wigner suggested that Maria Goeppert-Mayer considered a second-order nuclear process where the “simultaneous emission of two electrons (and two neutrinos)” occurred. Goeppert-Mayer evaluated the probability for this process in the framework of the Fermi theory, but found an exceedingly small value, even in the most favorable choice for the appropriate nuclear isotopes, inducing in physicists the unexpressed belief that a double β-disintegration would never be observed [136]. In his 1939 paper, Furry noted that “in the Majorana theory only two particles – electrons or 17 The real part of the solution of the wave equation for neutrons cannot be separated from the imaginary part

due to the presence of a magnetic moment term in the corresponding equation: in more modern terms, a Majorana particle has a vanishing magnetic moment, while the neutron doesn’t. Moreover, a real wavefunction for the neutron would lead to an equal probability for β − and β + decays, that is an equal probability to decay into protons and antiprotons, which is not supported by experiments.

2.8 Majorana neutrino theory

51

positrons – have to be emitted, and the transition probability is much larger” [136], thus providing the initial impetus to an experimental search that is still fully active. Indeed, just about ten years after the appearance of paper N.9, the first experiments on double β-decay were undertaken, and it is curious that the very first results were interpreted (incorrectly, as further studies showed) in terms of a neutrinoless process, given the apparently measured half-life [137]. Such an exciting search revealed to be extremely difficult, and while the two-neutrino mode was observed for the first time in a counter experiment only in 1987, the concluding evidence in favor of a neutrinoless decay, confirming the Majorana theory, is still lacking [138]. As the issue of the Dirac/Majorana nature of the neutrino is critical for elementary particle physics, the experimental search for the neutrinoless double β-decay continues actively in many laboratories around the world [139, 138]. 2.8.4 Pontecorvo and the neutrino oscillations Intriguing enough, the other hot topic of present-day neutrino physics – neutrino oscillations – made its first appearance through the indirect agency of the Majorana theory, again by means of another member of the former Fermi group in Rome. In 1957, Pontecorvo proposed [140] that the state of neutrinos produced in a given process is a superposition of “two Majorana neutrinos with different combined particles” (that is, two states with definite mass), in analogy with what was observed for the system of neutral K mesons: The question was raised as to whether there exist neutral particle mixtures, other than K 0 mesons, that is particles for which the transition particle → antiparticle is not strictly forbidden, although the particle at issue is an entity distinct from the corresponding antiparticle. It was noted that the neutrino may be such a particle mixture and consequently that there is a possibility of real transitions neutrino → antineutrino in vacuum. [141]

At that time, only one type of neutrino was known, and in Ref. [141] Pontecorvo formulated for the first time the hypothesis of neutrino oscillations, intended as neutrino–antineutrino transitions. Flavor oscillations were proposed only later [142], but, quite interestingly, the first phenomenological theory of them (i.e. a theory of neutrino mixing and oscillations), elaborated by Vladimir Gribov and Pontecorvo in 1969 [143], again assumed Majorana neutrinos. The idea that neutrinos were Majorana particles was so strong that only at the end of 1970s – after the final success of the quark mixing theory – was the possibility considered that neutrinos were particles described by the Dirac equation [144], analogous to the other leptons and quarks. The first conclusive evidence about neutrino (flavor) oscillations came only in 1998 [145], but experimental and theoretical research continues worldwide in order to assess the key features of such an elusive particle (see Chapter 15).

52

The visible side

Although the original Majorana proposal has not yet been given a definite answer, the observation of the phenomenon of neutrino oscillations has clearly posed the pressing question of describing theoretically the property of neutrinos to be massive (neutrino oscillations do not occur if neutrinos are massless), but “not so massive” as the other fundamental fermions such as electrons, quarks, etc. It is now a common belief that the answer to such a question inevitably involves the Majorana nature of neutrinos [146] (see Chapter 15). 2.8.5 Majorana fermions In the 1970s, Majorana particles invaded another sector of elementary particle physics, not associated with neutrino physics. Indeed, a novel mathematical formalism [147, 148] was developed that united particles of different spin into symmetry multiplets, leading to what became later known as supersymmetry, a topic that has dominated high-energy theoretical physics until the present day [149]. In a sense, supersymmetry is a quantum-mechanical enhancement of the properties and symmetries of space-time, predicting that any known elementary particle (matter fermions, such as quarks, electrons, etc., as well as bosonic force carriers, such as photons, gluons, etc.) has a “superpartner” particle with spin differing by a halfinteger amount. Now, several of the observed elementary bosons – the photon, for example – are charge self-conjugated particles (that is, antiparticles coincide with particles), so that the same is predicted for their fermionic superpartners: supersymmetric theories thus systematically require Majorana fermions as partners of spin-0 and spin-1 bosonic fields. Neutralinos are an example of such hypothetical particles. Also, the natural coordinate space of theories exhibiting supersymmetry is a “superspace” where, in addition to the ordinary space-time variables, other anticommuting fermionic parameters are present, which transform as Majorana spinors. Apart from neutrinos, the fortune of the Majorana theory of paper N.9 was therefore intimately related to the exceedingly large (theoretical) success of supersymmetry, testified to by tens of thousands of papers published over a few decades: terms like Majorana fermion or Majorana spinor owe their widespread popularity to this fact. The reason for such success lies mainly in the solutions it provides to several problems at the frontier of fundamental physics (such as, for example, the unification of the fundamental forces), as well as to the phenomenological problem of astronomical dark matter (supersymmetric Majorana particles are the favorite candidates to explain the observed mass of the Universe; see Section 15.6). Nevertheless, present day laboratory experiments have not yet confirmed the suggestive predictions of supersymmetric theories. Curiously enough, however, the

2.9 Complex systems in physics and economics

53

corresponding theoretical work has not been thrown away, rather several results previously obtained in the realm of formal quantum field theory have transmigrated into a completely different context, i.e. condensed matter physics. The first appearances of Majorana fermions in this novel area were related to ferromagnetic chains, when exploiting the known mapping of the Ising model to a field theory of Majorana particles [150, 151], or similar systems described by statistical models [152], including heavy fermions [153]. However, the true “revolution” began with the systematic consideration of Majorana zero-modes exhibited by solid state systems, that is emergent quasiparticles occurring at exactly zero energy and having the property of being their own antiparticles. Since the end of 1970s it has been known that fermion zero-modes are bound to topological defects [154] (for example, certain vortex excitations in given systems support Majorana fermions residing inside their cores), but the revolution started in 2000 with a thorough study of fractional quantum Hall liquids [155]. It then continued in the subsequent decade with other two-dimensional systems, including interacting quantum spin systems, spin-polarized p-wave superconductors, and interfaces between topological insulators or semiconductors and ordinary superconductors [156]. It would be impossible to list here all the recent applications of Majorana fermions in condensed matter physics, given their very large number. The reason for such a nearly incalculable number lies in the fact that Majorana zero-modes possess a remarkable property (nonAbelian exchange statistics) that allows them to be used as the building blocks of (forthcoming) topological quantum computers. They are practically immune to sources of decoherence [157], which plague other related devices containing quantum memories. We mention here only the current focus on one-dimensional (rather than twodimensional) structures exhibiting given topological properties, whose particular study originated in 2001 with the so-called Kitaev chain [158], describing spinless electrons hopping between the sites of a one-dimensional tight-binding chain and subject to p-wave superconducting pairing. The experimental race for detecting Majorana zero-modes in nanowires made of semiconductors with strong spin-orbit coupling and of topological insulators is at the frontier of the research on this topic [156] (see Chapter 14). 2.9 Complex systems in physics and economics In addition to the nine papers discussed so far, it is customary to include one more article in the list of papers published by Majorana, since it was published posthumously (i.e. after his disappearance) by Giovanni Gentile Jr. – one of Majorana’s closest friends – in 1942, when he was commissioned to write a biographical

54

The visible side

article for the Enciclopedia Italiana. This inclusion is fully justified by the fact that the manuscript was considered as “complete” and ready to publish by the author himself, who 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. As recalled by Gentile Jr. in his introduction to paper N.10, “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.” 2.9.1 Genesis of paper N.10 The topic chosen might seem somewhat unusual – and so it did until recent times – aided by the fact that Gentile Jr. assumed that the paper “was originally written for a sociology journal.” Actually, the story of the genesis of such a paper is somewhat different [159] and manifests a particular attitude of Majorana, closely related to family commitments, which he used to fulfill willingly. At the end of 1935, Giuseppe Majorana, full professor of general economics at the University of Catania, retired from teaching, and, due to his particularly brilliant academic achievements, the Faculty of Law decided to publish a volume to honor the illustrious scholar, with contributions from eminent personalities of that time. Giuseppe Majorana himself organized the structure of the volume, and in January 1936 he wrote a letter to his nephew Ettore to ask him for a personal contribution as a leading “mathematician”: Mathematics has many contact points – methodological, investigative, etc. – as well as dissonances with social disciplines. The Faculty [of Law] in Catania will publish a book in my honor for my retirement. Could you send me a paper or so related to your own fields of interest? One of our common interests is that of the “statistics” concerning infinitesimal corpuscles, upon which we previously talked about. Could you, e.g., outline the theoretical framework of such an investigation, even in comparison with the statistics as otherwise (or commonly) understood? [159]

Ettore accepted the invitation, as he did on several other occasions when a member of the extended Majorana family asked him for help (see, for example, the decisive theoretical support given by Ettore to his uncle Quirino, as recalled in Section 1.4). By the beginning of March 1936, the final version of the manuscript reached Giuseppe for later publication. Unfortunately, however, the scheduled volume was never published due to the economic restrictions imposed by the Fascist regime, and Ettore’s manuscript appeared in print only after his disappearance (and after Giuseppe passed away), due to the agency of Gentile Jr., who was not aware

2.9 Complex systems in physics and economics

55

of the story of the manuscript.18 The choice (made by Gentile Jr.) of the Italian journal Scientia was fully appropriate, due to previous contributions by renowned scholars as Vilfredo Pareto, as well as Emile Durkheim and Ernst Cassirer. Paper N.10 was, then, stimulated by Giuseppe Majorana, which explains the different nature of the topic considered with respect to those of the other published papers, but, as we will see, its content is far from obvious and was understood and accepted only more than half a century later. 2.9.2 Statistical laws in social sciences An appropriate introduction to what is expressed by Majorana in his paper N.10 can be deduced from a letter written by him to another uncle of his, Dante Majorana, on December 27, 1937: The mathematical method cannot be of any substantial utility to sciences that have nothing to do now with physics. In other words, if one day we will discover the mathematical description of the simplest facts regarding our life or consciousness, this most certainly will not happen for a natural evolution of biology or psychology, but only because some further radical renewal of the general principles of physics will extend its domain to fields that lie still outside it. The most relevant example is offered by chemistry which, after remaining an independent science for a long time and with great glory, in recent years it has been fully taken over by physics. This was made possible by the appearance of quantum mechanics, while no useful relationship had been established between chemistry and classical mechanics. While waiting for new miracles performed by physics, we should then recommend to scholars of other disciplines to rely on methods proper of its own discipline, and not to look for models or suggestions coming from present day physics, and even less than those coming from previous paradigms. And this, because we have still no feeling at all about the physics that, one day, may tell the ultimate word on biological or moral facts. [159]

Such a view was applied to social sciences in paper N.10. The starting point of Majorana’s analysis was the fact that, during the 1930s, the interaction between social sciences and natural scientists was developed mainly under the paradigm of celestial mechanics, with the resulting characteristic determinism: The sensational success of mechanics applied to astronomy has encouraged the assumption that more complicated phenomena of common experience must also be described, in the end, by a similar mechanism, albeit more general than the gravitational law. [. . . ] This point of view has produced the mechanistic conception of Nature.

Majorana noted, however, that such a deterministic scheme is “subject to a real limitation of principle when we take into account the fact that the usual methods of observations are not able to provide us with the exact instantaneous conditions 18 Note, however, that the abstract of the paper was probably not written by Majorana, but rather by Gentile Jr.

or some other editor [159].

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The visible side

of the system.” The best example is that of a gas in a given thermodynamic state, observations of which may provide only global information, such as the values of pressure and density, while the internal structure of the system remains undetermined, such “hidden indeterminacy” being measured by the Boltzmann entropy relation S = k log N. Statistical laws are, then, introduced into (classical) physics, their meaning being summarized by Majorana as follows: 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 the laws of mechanics to be valid, the probability calculus allows the probabilistic prediction of future phenomena.

This situation appeared to Majorana completely similar to that present in social sciences: 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 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.

Although it may now seem obvious, at the time this point of view was at variance with the general view of mathematical economists such as Léon Walras, Pareto, Karl Schlesinger and Abraham Wald, permeated as it was by a strict determinism. Such a view produced the development of general equilibrium theory [161]. This generic recognition of the role played by statistical laws in social sciences is, however, not the final result of Majorana’s analysis. Indeed, that envisaged above is only the classical view of statistical laws. In the 1930s a new paradigm had already developed in physics around the novel conception of quantum mechanics, and this necessarily led to further investigation: The statistical laws concerning complex systems known in classical mechanics retain their validity according to quantum mechanics. [. . . ] 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.

The basic point, according to Majorana, was that quantum mechanics is an irreducible statistical theory, since it is not able to describe the evolution of a single particle in a given environment at a deterministic level, contrary to what

2.9 Complex systems in physics and economics

57

happened in classical mechanics. It requires the use of a statistical description of events involving single entities. 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.

The intrinsic statistical nature of the underlying processes describing natural phenomena suggested to Majorana that statistical laws should be incorporated into a modeling approach to social and economic phenomena: those laws are proper investigation tools having an epistemological status similar to that of irreducible probabilistic laws in quantum mechanics. Majorana explained this point clearly by making recourse to the paradigmatic example of the decay of a radioactive atom, and then deducing his general conclusion for the corresponding social phenomena: 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 setups are therefore sufficient to prepare, for whatever complex chain of rich phenomena, which one 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.

Such a program of approaching society within the framework of statistical physics has transformed from a declaration of principles to a concrete research study only in recent times (see Section 2.9.3), the most famous achievement in finance being the Black and Scholes model of option pricing [162]. The fact remains that paper N.10 can be considered [163] to be the first article on complex systems, this same term being literally present in the paper. 2.9.3 A sensational success in econophysics Majorana’s paper N.10 went completely unnoticed (if we measure this information using the number of citations) from its publication to 1997, when a seminal article on “Physics investigation of financial markets” appeared [164], which opened with a quotation from paper N.10. That article reported the text of a lecture at a conference in Varenna, Lake Como, delivered by two founding fathers of econophysics – a term coined just two years before by H. Eugene Stanley – whose basic program is just that envisaged by Majorana more than 60 years earlier: During the last thirty years, physicists have achieved important results in the fields of phase transitions, statistical mechanics, nonlinear dynamics, disordered and self-organized

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The visible side

systems. New paradigms have been developed and a range of complex systems have been carefully investigated and described. [. . . ] Economic systems, strictly regulated and very frequently monitored, are ideal for a study performing tools and paradigms developed to describe physical systems. In this lecture, we have tried to give a concrete example on how and why physicists may consider economic systems (and in particular financial systems) as very interesting “complex systems”.

The name Majorana, however, began to be widely associated with such studies only two years later, when a bestseller from the same authors appeared, explaining the roots of econophysics: Recently, a growing number of physicists have attempted to analyze and model financial markets and, more generally, economic systems. The interest of this community in financial and economic systems has roots that date back to 1936, when Majorana wrote a pioneering paper on the essential analogy between statistical laws in physics and in the social sciences. This unorthodox point of view was considered of marginal interest until recently. Indeed, prior to the 1990s, very few professional physicists did any research associated with social or economic systems. The exceptions included Kadanoff, Montroll, and a group of physical scientists at the Santa Fe Institute. Since 1990, the physics research activity in this field has become less episodic and a research community has begun to emerge. [165]

This book (and other papers by the same authors) received (and still receives) several thousands of citations, so Majorana’s paper N.10 rapidly became well known to the interested community of physicists (and mathematical economists) who replaced the common notion of predictability usually associated with Newtonian linear systems by the less stringent one that is the property of nonlinear and complex systems, just as Majorana foresaw. Instrumental for such a change of perspective, occurring only decades after Majorana wrote his paper, has been the availability of new large databases and the appearance of brand new social phenomena related to the worldwide web, which prompted scientists to formulate statistical models to be analyzed quantitatively. It is quite intriguing that, after the success of econophysics, we find also several re-interpretations of the role played by Majorana, which do not hesitate to associate his name with the founding fathers of statistical mechanics, such as James Clerk Maxwell and Ludwig Boltzmann [166]: The discovery of quantitative laws in the collective properties of a large number of people, as revealed for example by birth and death rates or crime statistics, was one of the factors pushing for the development of statistics and led many scientists and philosophers to call for some quantitative understanding (in the sense of physics) on how such precise regularities arise out of the apparently erratic behavior of single individuals. [. . . ] This point of view was well known to Maxwell and Boltzmann and probably played a role when they abandoned the idea of describing the trajectory of single particles and introduced a statistical description for gases, laying the foundations of modern statistical physics.

2.9 Complex systems in physics and economics

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The value of statistical laws for social sciences has been foreseen also by Majorana in his famous tenth article. [166]

Even more intriguingly, Majorana’s paper N.10 has also stimulated epistemological and philosophical reflections in recent years, likely due to his critical discussion of determinism in physics – very simply put, indeed, Majorana did not consider determinism compatible with our personal experience of consciousness. We do not dwell here on such issues, but refer the interested reader to some appropriate references.19

19 See the general discussion in Ref. [161] (and references therein), as well as specific reflections in Ref. [167]

(“Ettore Majorana and the objectivist interpretation of social indeterminism”) and the most recent one in Ref. [168] (“Ettore Majorana’s transversal epistemology”).

Part II Atomic physics

3 Two-electron problem

Majorana was involved in studying helium on at least two occasions, as we have seen in Chapter 2, namely for his papers N.2 and N.3 published in 1931. However, in his unpublished personal study [17] and research [18] notes we find several additional interesting results and methods directly related to the basic two-electron problem, dating back to 1928–9, some of which are apparently preliminary studies for his published papers (especially paper N.3). The theoretical contributions and numerical calculations, including empirical relations, contained in those notes were largely deduced by making recourse to novel methods not yet in the literature (both of that time and in present day studies), and while part of those numerical results were (and are) inaccurate when compared with the experimental data, the novel methods are nevertheless quite useful in the frontier research related to atomic and nuclear physics [169]. 3.1 A long-lasting success for quantum mechanics Following the discovery of the atomic nucleus [170], in 1913 Bohr succeeded [171, 172, 173] in explaining the energy levels of the hydrogen atom in terms of quantization of the action for the classical Kepler orbits. Numerous attempts were then explored to explain the ground state of helium by quantizing different two-electron periodic orbits in a similar manner, but without success. For example, Bohr first discussed a simple model where both electrons in the helium atom move along the same circular orbit and are located at the opposite ends of a diameter [171, 172, 173]. This followed the 1904 proof by the Japanese physicist Hantaro Nagaoka [174, 175] (obviously in the framework of classical mechanics) that such motion is mechanically stable (for sufficiently large attractive forces) and realizes the lowest possible energy. In general, the attempts to quantize the helium atom in the Bohr–Sommerfeld theory [176, 177] were always based on the assumptions that the ground state is related to a single periodic orbit of the electron pair, and that

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the electrons move on symmetric orbits with equal radii at all times. Furthermore, it was assumed that orbits where the electrons hit the nucleus are excluded, and that the principal quantum number n in the quantization condition is an integer. Such an approach proved unsuccessful mainly because it failed to explain the diamagnetism of a two-electron atom in its lowest energy state and the value of the obtained energy was in rather poor agreement with experiments. It then became clear that “some radical modification in the ordinary conceptions of the quantum theory or of the electron may be necessary” [178]. The failure of the old quantum theory to describe successfully two-electron atoms, in fact, triggered (at least in part) the development of quantum mechanics in the 1920s [179], and, once the basic formalism had been established by Heisenberg and Schrödinger, early variational calculations (the problem couldn’t be solved analytically, as it had for the hydrogen atom) produced remarkably good results for the ground state of the helium atom, thus breaking the ground for the general acceptance of quantum mechanics. Within the Schrödinger framework of quantum mechanics, atoms with two electrons are described by a wavefunction ψ satisfying the eigenvalue equation (in electronic units):   Z Z 1 ψ = Wψ , (3.1) Hψ ≡ −∇ 2 ψ + 2 − − + r1 r2 r12 where r1 , r2 are the distances of the first and second electron from the nucleus, r12 is their mutual separation, and W is the energy of the system in Rydberg; ∇ 2 = ∇ 21 + ∇ 22 , with ∇ 21 , ∇ 22 the Laplacian operators in the space of the first and second electron, respectively. This differential equation is not separable, so that, unlike for the hydrogen atom, its solutions for the eigenfunctions and energy eigenvalues cannot be expressed in closed analytic form, and thus approximation methods must be used. As became clear, the key point at which different strategies could be tested (thus proving or disproving the underlying theory) was the prediction about the ground state energy – mainly for the neutral helium atom – so in the following we focus on this issue. Quite an exhaustive, though apparently outdated [180], review of the complete two-electron problem may be found in Ref. [181]. For future reference, it is useful to recall the experimental value of the ionization energy WI = −Z 2 − W (here it is sufficient to use the accuracy from late 1920s to the early 1930s): exp

WI

= 24.46 eV .

(3.2)

3.2 Known solutions to the helium atom problem 3.2.1 Perturbative calculations The simplest way of solving Eq. (3.1) is to find a soluble problem to serve as an unperturbed solution to which the theory of perturbations may be applied. This was

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carried out by Sommerfeld’s student Albrecht Unsöld in 1926 [182], who factorized the wavefunction in terms of spherical coordinates. The main idea and the basic calculations may be more easily understood by following a later derivation by Majorana, now reported in Ref. [18].1 To a first approximation, the interaction among the two electrons may be neglected, so that the eigenfunction of the unperturbed Hamiltonian,   Z Z , (3.3) + H0 = −∇ 2 ψ − 2 r1 r2 may be expressed as a product of two hydrogen 1s states, ψ = e−Zr1 e−Zr2 ,

(3.4)

with energy W = −2Z 2 . The inter-electron potential is then treated as a perturbation: by applying standard computational techniques, the first-order result for the ground state energy [ionization energy] is the following:   5 5 2 2 WI = Z − Z . (3.5) W = −2Z + Z 4 4 For the neutral helium and ionized lithium, this translates into the predictions WIHe = 20.3 eV and WILi = 71.0 eV, respectively [182].2 As noted by John C. Slater a few months after the appearance of the Unsöld results [183], “the solutions so far obtained, though qualitatively interesting, are quantitatively far from the truth” when compared (for example, for the helium) with the experimental result (3.2). This urged Slater to search for an alternative strategy to the problem, by changing the starting point (that is, the unperturbed system) of the perturbation theory. The motivation was, apparently, mathematical in nature: The method so far used in such cases has been to leave out, for the unperturbed problem, all those terms in the energy which prevent the separation of variables, regarding all these as perturbative terms, and solve the remaining problem by separation. Since these are, in general, the terms representing the interaction between electrons, it is obvious that the unperturbed problem will be a very poor approximation to the real case. We proceed in quite a different way, making no attempt to separate variables at all. [183]

The idea was to treat the variables of one of the two electrons as parameters in the Schrödinger equation of the other electron (that is, the equation for an electron in the field of the rest of the atom). From a more physical perspective, Slater imagined one electron in a fixed position and then obtained for the second electron a two-center problem, the eigenfunctions of which could be evaluated numerically as a function of the distance between the two centers. For the first electron he deduced a numerical equation in which the eigenvalue of the remaining 1 See, in particular, Sect. 3.1 in Ref. [18]. 2 Or, more precisely, according to Majorana, W He = 20.31 eV and W Li = 71.08 eV. I I

66

Two-electron problem

electron was introduced and which represented the motion of this electron in a central field. Then, both solutions were integrated into an approximate solution for the two-electron problem, and finally a perturbative calculation was performed. By using, initially, poorly accurate solutions to the unperturbed problem, Slater found a value WIHe = 24.35 eV, apparently very close to the experimental determination, but in a subsequent note added in proof [183] he corrected this prediction, by obtaining a numerical value of 5−6% greater than the experimental value (which was, however, a better result when compared with the Unsöld prediction). Almost a year later, Slater published a paper [184] in which he analyzed the reasons why his perturbative method became worse with increasing perturbative order, despite the fact that the outer electron in the helium atom was assumed to move not in a hydrogen-like field, as for Eq. (3.4), but rather in a certainly more accurate central field. The wavefunction replacing that in Eq. (3.4) (for the helium problem) for this purpose was the following: ψ = e−2(r1 +r2 )+r12 .

(3.6)

Slater found that his method indeed gave a good approximation to the energy levels and to the perturbed wavefunction, except at very small distances. More generally: Almost any method of setting up approximate wavefunctions will give fairly good results in the non-penetrating part of the orbit; and it is this part that is of almost complete importance in many practical applications that demand knowledge of the wavefunction. [...] The operator H, almost alone among the operators of physical interest, puts a severe strain on the most inaccurate part of the wavefunction, that connected with the penetrating part of the orbit, on account of the differentiation and the factors 1/r that it contains. For this reason, a wavefunction which is decidedly satisfactory for most physical applications may give a very inaccurate matrix of H. Yet it is just this matrix that is used in the perturbation method for solving the atomic structure problem. It seems necessary to conclude from this that the perturbation method is not a satisfactory one for actual calculations. [184]

A different method, then, appeared to be required for improving the predictions of quantum mechanics about two-electron atoms.3

3.2.2 Variational method I The variational Ritz method [185, 186] was introduced for the first time into quantum mechanics [187] by means of a student of Max von Laue in Berlin, Georg W. Kellner, who employed it in order to estimate the ground state energy of the helium atom. This is given by the minimum value of the Schrödinger variational integral, 3 Note, however, that Slater applied his conclusions to any atom.

3.2 Known solutions to the helium atom problem



W[ψ] = 

67

ψ ∗ Hψ dτ ∗

,

(3.7)

ψ ψ dτ with respect to any wavefunction ψ; however, when only a given set of functions ψ is considered, that minimum will correspond only to an approximate value of the ground state energy, the approximation improving more and more as the set is enlarged. When this set reduces to the unperturbed wavefunction in Eq. (3.4), the variational method gives the same result for the ground state energy seen above, so that a larger set of functions is required for a better approximation. This is achieved by choosing a trial wavefunction depending on some arbitrary parameters, the values of which are determined by requiring the energy functional in Eq. (3.7) to be the lowest possible. This minimum value then gives an upper limit for the ground state energy, the approximation of which is much better as the trial wavefunction is closer to the “true” eigenfunction of the problem, a suitable form for that being set up from given physical considerations. In principle, by including a sufficient number of arbitrary parameters, one can approximate the eigenfunction and eigenvalue of the problem as closely as desired, but it is clear that an appropriate choice of the form of the wavefunction will make the procedure converge rapidly. Kellner introduced the novel idea of an effective nuclear charge experienced by one of the two electrons in the field of the nucleus and of the other electron: this electron somewhat shields the nucleus, thus reducing the effective charge. This quantity was introduced as an arbitrary parameter in the trial wavefunction, replacing that in Eq. (3.4). By following, again, the simpler (and somewhat more general) later reworking by Majorana [18] the wavefunction thus takes the form ψ = e−k(r1 +r2 ) ,

(3.8)

and, on minimizing the energy functional in Eq. (3.7), the following value of the k parameter is obtained: 5 . (3.9) k=Z− 16 For the helium atom, then, the effective nuclear charge is less than two, kHe = 27/16, corresponding to an ionization energy WIHe = 217/128 (in Rydberg units) or WIHe = 22.95 eV, which is in better agreement with experiments when compared with the values obtained using perturbation methods. Majorana also found a general formula [18] for the ground state energy [ionization energy] by using the present method:     25 5 5 2 2 . (3.10) WI = Z − Z + W = −2 Z − 16 4 128

68

Two-electron problem

Returning to the helium atom, Kellner [187] re-iterated the method, by using more involved wavefunctions, where the exponential factor in Eq. (3.8) was multiplied by different combinations of Laguerre polynomials (in r1 and r2 ) and spherical harmonics (in the cosine of the angle between r1 and r2 ). After four approximation steps, the better result he obtained was WIHe = 23.75 eV, thus showing the Ritz variational method to be suitable for accurate quantum-mechanical treatment of the ground state of helium, dependent on making appropriate choices for the wavefunctions. 3.2.3 Self-consistent field method A different idea in managing, in general, many-electron atoms was introduced by Douglas R. Hartree at the end of 1927 [188]. He started by assuming the wavefunction of the given atom to be written in the form of a product of single-particle wavefunctions yi (ri ), one for each of the electrons, such as, for example, the one in Eq. (3.4) for the helium atom. The potential energy acting on the first electron is taken to be the Coulomb field of the nucleus corrected by the potential of the charge distribution |y2 (r2 )|2 of the second electron, averaged over all positions of the second electron:  Z |y2 (r2 )|2 dτ2 . (3.11) V1 (r1 ) = − + r1 r12 This corrected central potential was then substituted into the Schrödinger equation for the first electron and solved for the wavefunction y1 (r1 ). A similar procedure was carried out for the second electron and, from the solutions yi (ri ) for all electrons, Hartree calculated a charge distribution to be inserted again in Eq. (3.11) to obtain the field of the nucleus and a novel estimate of this same charge distribution. In general, the “final” field is not the same as the “initial” field, since the initial, trial wavefunctions yi (ri ) do not agree with the final ones yi (ri ). The whole procedure is repeated by using the final field of the first approximation as the initial field of the second one, and this is repeated over and over again until the initial and final wavefunctions agree to the desired accuracy. The final field is thus the same as the field produced by the charge distribution of the electrons, so that the field in which the electrons are assumed to move when calculating atomic quantities of interest (such as the ground state energy) is “self-consistent.” Here, Hartree’s basic assumption was that many-electron atoms could be described by wavefunctions given as the simple product of one-electron wavefunctions of individual electrons, rather than as one complete wavefunction (as, for example, that in Eq. (3.6)). Slater [190] later evaluated the error bars of Hartree’s results, by making a (general, not limited to helium) perturbative calculation of the energy levels using the wavefunction obtained with the Hartree method.

3.2 Known solutions to the helium atom problem

69

Hartree applied his method to Rb, Rb+ , Na+ , Cl− , and, first of all, to helium, obtaining a value for the ionization potential WIHe = 24.85 eV, very close to the experimental determination in Eq. (3.2).

3.2.4 Slater’s refinement An independent attempt was made by Slater [191] in order to find an accurate expression for the wavefunction of the ground state of (ortho-) helium, which can be considered as a further refinement of Eq. (3.6). The general idea was to use theoretical determined functions for the limiting cases of large and small rdistances and then interpolate between them. In particular, when one electron is at a considerable distance from the nucleus (large r2 ), while the other is close up to it (small r1 ), the wavefunction was assumed to be the product of an ionic function of the inner electron and a hydrogen-like function of the outer one:  −ar2  c −2r1 e (3.12) 1 + + ... ψ =e r2 r2b (the opposite case, with small r2 but large r1 , is obtained by interchanging r1 and r2 in this expression). Instead, when both electrons are close up, the wavefunction is assumed to be of the same form as in Eq. (3.6), but with a further correction term: ψ = e−2(r1 +r2 )+(1/2)r12 +d(r1 +r2 ) . 2

2

(3.13)

The novel term weighted by d was added by Slater in order to have a smooth interpolation between the two expressions in Eqs. (3.12) and (3.13), for intermediate values of r1 , r2 . Note that such wavefunctions were not intended for evaluating the energy eigenvalues, since the coefficients a, b, c, d were just deduced from the experimental spectroscopic data. Instead, they were introduced in order to compute several other atomic properties of helium, such as diamagnetic susceptibility, repulsive forces between two helium atoms, and so on. According to Slater [191]: The calculations were made before the writer saw Hartree’s paper, in which he obtains a charge density distribution for helium in quite a different way. The wavefunction found in the present paper is more complicated than Hartree’s in the matter of the way in which it takes the interaction energy between electrons into account. But the charge density can be computed equally well from either method, and this permits a comparison of the present results with Hartree’s. The discrepancies between the two are nowhere greater than one or two percent. This is highly satisfactory, both in that it verifies the present method and Hartree’s, and also that it justifies us in believing this density distribution to be correct within a narrow limit of error. The other numerical results are also gratifying.

In a sense, the trial stage of the novel theory of quantum mechanics was completed, and the subsequent need was to have a “standard” (and mathematically

70

Two-electron problem

reliable, for numerical purposes) procedure for evaluating atomic quantities to the desired accuracy. For many-electron atoms, this was found in the Hartree method, as modified by Vladimir Fock [192] to take exchange effects into account (fully symmetrized wavefunctions in the form of products). For helium, the story was somewhat different.

3.2.5 Variational method II: Hylleraas variables The variational approach to the ground state of the helium atom already exploited by Kellner was also adopted, in 1928, by a Norwegian student of Max Born in Gottingen [193], Egil A. Hylleraas, who pointed out that a “useful feature” of it was that “the eigenvalues obtained in successive orders decrease monotonically, and that therefore an exact eigenvalue can be given on the approach from one side” [194]. In the first of a series of papers devoted to the subject, Hylleraas soon noticed [194] that the wavefunction of the two-electron problem, which, in general, would depend on the six coordinates r1 , r2 , actually depended only on a subset of three, namely r1 , r2 , θ (θ being the angle between the two vectors r1 , r2 ), defining the shape of the electron–nucleus triangle. Its absolute orientation in space was, indeed, not of interest for the problem at hand. This introduced an enormous simplification into the variational problem: indeed, Hylleraas himself was able to perform computations to order 11 (instead of the order 4 calculations performed by Kellner), thus obtaining an excellent approximation for the ionization energy of helium, WIHe = 24.35 eV. The predictions by Kellner [187], Slater [183], and Hylleraas [194] agreed within the error bars for direct measurements (by electron collision), but they were not as accurate as the more precise spectroscopic measurements by Theodore Lyman [195], so that Hylleraas continued to search for possible improvements in subsequent years. In a second paper [196] on the topic, instead of adopting the r1 , r2 , θ coordinates, Hylleraas chose only “metric quantities appearing immediately in the expression for the potential energy,” that is r1 , r2 , r12 . In particular, he introduced the notations (later known as “Hylleraas variables”) s = r1 + r2 ,

t = r2 − r1 ,

u = r12 ,

(3.14)

so that the wavefunction had to be written in terms of these coordinates only. The variational method was then applied to the function ψ = ϕ(ks, kt, ku) – that is, with k as a variational parameter – with ϕ(s, t, u) = e−s/2 P(s, t, u),

(3.15)

3.2 Known solutions to the helium atom problem

71

P(s, t, u) being a polynomial in the variables s, t, u. Note that Kellner had already tried to force a faster convergence by multiplying the metric arguments in his wavefunctions by a free constant, but here the novelty was in the choice of the variables in Eq. (3.14). Very accurate results were obtained, depending – obviously – mainly on the number of terms included in P(s, t, u): increasing the number of accurately chosen polynomial terms resulted in better approximations. The method worked very well for parahelium (and a little less well for orthohelium) and, after several trials with different monomial terms s tm un (different values of , m, n), Hylleraas was able to obtain the value WIHe = 24.460 eV with order 6 calculations [196]. Soon after the appearance of Ref. [196], Bethe compared this result with that obtained using the Hartree method, showing that the overall charge distribution of the two electrons given by the Hartree self-consistent field method (with correlation effects not included) agreed very well with the predictions of the variational method [197].

3.2.6 Helium-like ions Later, in 1929, Hylleraas extended [198] his previous calculations [194, 196] to the helium-like ions Li+ and Be++ : A most exciting cooperation now started between [me and the famous spectroscopist Bengt Edlén]. No sooner had I found the energy of the lithium ion lying in between his limits of error than the doubly ionized beryllium ion was on the way, and so it continued with boron 3 plus and carbon 4 plus with me always behind. Suspecting there were no limits to Edlén’s power of producing ions I decided to overtake him with a good safety margin by introducing even an infinite nuclear charge. This was the origin of the energy formula E = −2Z 2 + 54 Z − 2 + 3 /Z − 4 /Z 2 + . . . as counted in Rydberg units. [193]

Hylleraas considered the term 1/Zr12 in the potential energy as a perturbation function and the factor 1/Z as a variational parameter. The energy eigenvalue E and eigenfunction ψ were then assumed to be of the forms E = E 0 Z 2 + E1 Z + E2 + E3

1 + ..., Z

(3.16)

1 , (3.17) Z and the following result for the ionization potential as a function of Z was obtained: ψ = ψ0 + ψ1

5 0.0147 . (3.18) WI = Z 2 − Z + 0.31455 − 4 Z It is interesting to compare the fully numerical evaluation leading to the last two terms in Eq. (3.18) with the corresponding analytic (and less accurate) evaluation in Eq. (3.10), accounting for a constant 25/128  0.1953 term. For the helium

72

Two-electron problem

ionization energy there is no particular improvement with respect to that in Ref. [196], but here Hylleraas pushed the calculations to order 10, with a prediction of WIHe = 24.469 eV, to be compared with the Lyman experimental result of 24.467 eV [195]. Given the success of the numerical calculations performed by Hylleraas, a more mathematical insight into the Ritz variational method, as modified by Kellner and, later, by Hylleraas with the introduction of screening constants, was in order. First of all, in 1930, Gregory Breit [199] showed that, due to the spherical symmetry of the field, the reduction of the general six-dimensional problem to a reduced three-dimensional one could always be performed for S-states. Then, a few months later, Carl Eckart [200] proved that the quantity E = ψ ∗ Hψ dτ , where H is the negative energy operator, was a lower limit to the term-value of the lowest level of a given spectral series, and that the best approximation to that term corresponded to the largest value of the integral evaluated over a set of various ψ functions. He also showed, however, that this approximation was not so good at large distances from the nucleus. Meanwhile, Hylleraas continued to improve his method, and to that anticipated in Ref. [198] he gave its final form in Ref. [201]. Now, the trial wavefunction was written as ψ = e−σ/2 P(σ, τ, ν),

(3.19)

with screened arguments: σ = 2Zs,

τ = 2Zt,

ν = 2Zu.

(3.20)

Equation (3.18) was then replaced by the more accurate equation: 5 0.01752 0.00548 , + WI = Z 2 − Z + 0.31488 − 4 Z Z2

(3.21)

with very small numerical coefficients for negative powers of Z. In particular, calculations of the helium ionization potential were made more precise by three orders with respect to the previous estimate, thus obtaining the value WIHe = 24.470 eV. 3.3 Majorana empirical relations The search for an optimum agreement between theory and experiments on twoelectron atoms, as outlined in Section 3.2, was considered to be very important for putting the quantum-mechanical approach to atomic spectra, based on the many-body Schrödinger equation, on a firm, quantitative ground. Majorana also participated in this exciting adventure, as transpired in his published papers N.2 and N.3 discussed in Chapter 2. Particularly interesting is his summary (reported

3.3 Majorana empirical relations

73

in paper N.2) of the wavefunctions for the helium atom being employed in calculations of observable quantities, depicting accurately the situation around 1930: 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 [see Refs. [191], [196]]. 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 criterion used for the evaluation; if we want to optimize (minimize) the average energy, we must then set Z = 2 − 5/16 = 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 theoretical calculations [see Ref. [191]], we must set Z = 1.60.

It was customary for Majorana to relate his theoretical calculations to experimental observations, and then to adopt appropriate approximations for the former in order to fit the accuracy of the latter. His summary, however, also explains his re-iterated use of hydrogenoid wavefunctions (as briefly mentioned in Section 3.2) and, in particular, his constant search for their possible generalizations giving easy but physically meaningful expressions, as we will see in the following. In paper N.3 Majorana also gave a further generalization of his (unpublished) formula reported in Eq. (3.10), the reasoning being as follows. In the zeroth approximation (with respect to the mutual interaction between the two electrons), the ground state energy (in Rydberg) is given by W = −2Z 2 , while in the first approximation it will be W = −2Z 2 + 4aZ since the interaction increases as Z. Now, the second approximation can be evaluated with the method of the variation of the unit of length [see Ref. [202]4 ] that is here equivalent to assuming hydrogen-like eigenfunctions with an effective Z ∗ .

The “more correct expression” he obtained with this method was the following: W = −2Z 2 + 4aZ − 2a2 ,

(3.22)

the screened charge being Z ∗ = Z −a, that is, just as in Eq. (3.10) but with arbitrary a.5 An explicit comparison of what Majorana obtained with this method with the available results was performed, which shows how good his estimate was: With this method we find that the value of the ground state is −W/Rh = 729/128 = 5.695, whereas the empirical value and the theoretical one obtained by Hylleraas is −W/Rh = 5.807, the difference being less than 2%. 4 Majorana refers simply to the Hylleraas method (see Section 3.2.5); the paper by Fock quoted here deals

with a general subject, i.e. the virial theorem in the framework of quantum mechanics, where the Hylleraas method is mentioned as an example. 5 Note that in paper N.3 Majorana applied this same formula to the ground state as well as to other levels (with different values of a).

74

Two-electron problem

Apparently, the first numerical studies on the two-electron problem performed by Majorana were aimed at finding empirical relations for the ground state energy.6 He was evidently aware of the difficulties in obtaining theoretical predictions,7 so that the first numerical efforts were likely devoted to finding suitable empirical expressions for W. According to Majorana himself, several of these relations lead to unsatisfactory results, but the reasoning behind them is nevertheless intriguing. Let us consider a two-electron atom with charge Z in its ground state. We denote by a = 1/r1 = 1/r2 the mean value of the inverse of the distance of each electron from the nucleus, and with b = 1/r12 the mean value of the inverse of the distance between the two electrons. Expressing the distances in electronic units and the energy in Ry, we have8 W = −2 a Z + b,

(3.23)

since the energy is equal to half the mean value of the potential energy. If we now consider an atom with atomic number Z + dZ, perturbation theory gives dW = − 4 a dZ,

(3.24)

and thus we have two equations in the three unknown Z functions E, a, b.

Interestingly enough, Majorana’s strategy for dealing with generic two-electron atoms with atomic number Z was to consider this quantity as a continuous variable, a small change of which induces a change in the ground state energy, the latter being evaluable with the aid of the perturbation theory. Now, the problem was to have two equations with three unknowns, so that the necessary additional relation was introduced as an empirical formula: We now add another empirical relation between a and b, which is presumably a good approximation: b = (2Z − 2a) (2a − Z).

(3.25)

This relation can be deduced from the following considerations. For sufficiently high values of Z, perturbation theory gives W = −2 Z 2 +

5 Z + ...; 4

(3.26)

but, on the other hand, b =

5 Z + ..., 8

(3.27)

6 See Sect. 3.15 in Ref. [17]. 7 Apart from the references in papers N.2 and N.3, we do not know precisely what papers were read and

studied by Majorana, but from the arguments, and especially from the symbols he used in his notes [17, 18], we can safely deduce that he was well aware of a large part of the literature quoted in Section 3.2. 8 See Sect. 3.15 in Ref. [17].

3.3 Majorana empirical relations

75

so that, from [Eq. (3.23)], a = Z −

5 + ..., 16

(3.28)

which, in first approximation, satisfies [Eq. (3.25)]. For very small values of Z we can consider that the first electron is next to the nucleus, while the other one is practically at an infinite distance; then we have a  Z/2, b  0, and [Eq. (3.25)] is again satisfied. We finally assume that it is also a good approximation for intermediate values of Z.

The next step was simply to substitute Eq. (3.25) into Eqs. (3.23) and (3.24), thus obtaining dZ = da, and “since we know the value of a for infinite Z, we deduce a = Z − 5/16.” The ground state energy is, accordingly, 5 25 Z − , (3.29) 4 64 to be compared with the perturbative result in Eq. (3.5) and subsequent improvements. Majorana noticed that this formula could be used only for Z ≥ 5/8; for Z = 5/8 the quantity b in Eq. (3.25) vanishes. Moreover, “the procedure used here is not very satisfactory, since for very small Z the quantity b would vanish faster than a first-order term and it would become negative.” A comparison with the data for the helium atom revealed that the predicted value for W was in excess of 1.13 eV with respect to the experimental value (25.59 eV instead of 24.46 eV), but here the intriguing thing is that Majorana applied the above formula also to hydrogen: “For the hydrogen atom (Z = 1) we find instead W = −1.141, from which the ionization potential would be 1.91 (electron affinity).”9 Given the poor agreement with the experimental values, Majorana also explored other relations for b as functions of Z, such as  5  2 k + Z2 − k , (3.30) b= 8 or  5  3 3 k + Z3 − k , (3.31) b= 8 or even 5 b = Z e−k/Z , (3.32) 8 where k is a parameter to be determined under suitable conditions. For example, from the last relation he obtained the following expression for a:    ∞ 5Z + 5k −k/Z a=Z 1 + e dZ , (3.33) 16 Z 3 0 W = −2 Z 2 +

9 Note that the predicted electron affinity of the hydrogen atom, i.e. the difference between the ground state

energies of the neutral atom and the once-ionized atom, is more than twice the actual value.

76

Two-electron problem

where it is evident the contribution from any Z that ranges from 0 to ∞. Since for small Z we must have a  Z/2, we can choose k such that  ∞ 5Z + 5k −k/Z 1 e dZ = , 3 2 16 Z 0

(3.34)

i.e., k = 5/4. However, in this way we obtain a bad approximation. Indeed, we would have [...] E = − Z 2 − Z 2 e−1.25/Z ,

(3.35)

and for helium (Z = 2) we would get E = −5.14, which is a value far from the experimental one.

Majorana was well aware that the introduction of such empirical relations was completely arbitrary (“there is no a priori reason to prefer one or the other”), so he soon abandoned these calculations. However, it remains the basic, novel assumption that Z could be treated – in an “effective” way – as a continuous variable, and that contributions to the ground state energy of a given atom (with given atomic number) come also from any other Z. 3.4 Helium wavefunctions and broad range estimates The conclusions achieved by Slater in 1928 [184] about the form of the unperturbed wavefunction and the role played by the operator H in the perturbative method (see Section 3.2), were analyzed repeatedly by Majorana,10 who also considered several different forms of the helium wavefunction as generalizations of Eq. (3.4), along the same lines as Slater (who introduced, in particular, the simple form in Eq. (3.6)). However, from the given wavefunctions, Majorana was able to deduce also (broad) ranges within which the value of the helium ground state energy would have lain, thus quantifying in a definite manner what had been generally deduced by Slater. Majorana realized very early what was later employed extensively by Hylleraas (but already latent in Slater’s calculations), namely that the helium wavefunction depends only on three metric quantities r1 , r2 , r12 . Probably inspired by Slater’s Eq. (3.13), Majorana first introduced the following general form: ψ = e−p , 1 2 2r1 + 2r2 − r12 + a(r12 + r22 ) + br1 r2 + cr12 + d(r1 + r2 )r12 2 , p= 1 + e(r1 + r2 ) + fr12

(3.36)

where a, b, . . . , f are numerical coefficients. The energy eigenvalue equation (3.1) was then written (in suitable units) as Lψ = λψ 10 See Sects. 3.2 and 3.3 in Ref. [18].

(3.37)

3.4 Helium wavefunctions and broad range estimates

77

with 4 2 4 + − + ∇2 r1 r2 r12 4 4 2 ∂2 ∂2 ∂2 2 ∂ 2 ∂ 4 ∂ = + − + 2 + 2 +2 2 + + + r1 r2 r12 ∂r1 ∂r2 ∂r12 r1 ∂r1 r2 ∂r2 r12 ∂r12 2 ∂ ∂2 + 2 cos α1 · + 2 cos α2 · , (3.38) ∂r1 ∂r12 ∂r2 ∂r12

L=

where α1 [α2 ] is the angle between r1 [r2 ] and r12 . However, he soon realized that calculations would become quite difficult with such a general form, so he shifted his attention to the simpler form 1

ψ0 = e−2r1 −2r2 + 2 r12 .

(3.39)

It should be noted, however, that this choice was not accidental (or dictated only by computational simplicity): indeed, he deduced the relations to hold among the a, . . . , f coefficients in order to obtain from Eq. (3.36) the same result as from Eq. (3.39).11 By substituting Eq. (3.39) into the eigenvalue equation (3.37), Majorana obtained simply: 17 (3.40) − 2 cos α1 − 2 cos α2 . 2 From this expression, by allowing the cosines to take any value in their domain, he deduced the range inside which the (negative) energy eigenvalue has to lie, namely 4.5 ≤ −W/Rh ≤ 8.5. When compared with the value of 5.807, this is indeed a very broad range, but, as we said above, it should give a quantitative idea of the results obtainable within the perturbative method, as induced by the particular form of the Hamiltonian operator on different unperturbed wavefunctions. Therefore, Majorana later considered an “approximated” wavefunction, that is   1 (3.41) ψ = 1 + r12 e−2r1 −2r2 . 2 λ=

Note that this expression approximates Eq. (3.39) (and then Eq. (3.36)) for small r12 , i.e. when the two electrons are close to each other; it can then be regarded as a correction to the simple one-electron case. By repeating the same calculations as above, Majorana obtained λ=8−

2 1 − (cos α1 + cos α2 ) , 1 1 + 2 r12 1 + 12 r12

11 For more technical details, see Sects. 3.2 and 3.3 in Ref. [18].

(3.42)

78

Two-electron problem

and thus 3 ≤ −W/Rh ≤ 8. The comparison of this range with that obtained previously already quantifies to a certain extent the result by Slater [184] addressed above, but Majorana did not limit himself to just these two forms of the possible wavefunctions. Without entering into computational details [18], we mention a different approximation of Eq. (3.36), namely 1 2 + cr1 r2 + d(r1 + r2 )r12 + . . . (3.43) ψ = 1 − 2r1 − 2r2 + r12 + a(r12 + r22 ) + br12 2 for small r1 , r2 , r12 ,12 and a generalization of Eq. (3.41),    −2r1 −(2−2α)r2  e−(2−2α)r1 −2r2 1 e + ψ = 1 + r12 (3.44) 2 1 + 2αr2 1 + 2αr1 (obviously symmetric under the exchange 1 ↔ 2), with the parameter α parameterizing an effective nuclear charge different for the two electrons, probably due to the different position of the two electrons with respect to the nucleus. 3.5 Accurate numbers and a general theory In addition to what we have already discussed, Majorana further considered the variational approach when studying the two-electron problem, mainly urged by his desire to obtain more accurate predictions for the ground state energy with respect to those coming from perturbative calculations, but with procedures and methods simpler and more transparent than those already employed in the literature. The results he obtained are particularly intriguing and, in some sense, astonishing. 3.5.1 A simpler alternative to Hylleraas’s method As recalled above, a sensible improvement in the predictions obtained with the variational method came with the recognition by Hylleraas that the wavefunction of the two-electron problem depends only on three coordinates (instead of six), namely the magnitude of the two vectors r1 , r2 and the angle θ between them or, rather, r1 , r2 , r12 . Likely before the appearance of the Hylleraas papers [194, 196] (or, in any case, independently of them), Majorana implemented such a result in one of his (unpublished) variational studies about the ground state energy of two-electron atoms,13 aimed at obtaining an improvement of his general result in Eq. (3.10). Instead of the wavefunction in Eq. (3.8), he considered the function ψ = e−kr1 e−kr2 er12

(3.45)

12 In particular, Majorana deduced the numerical values for the coefficients by requiring that ψ and its first

derivative have a node at the same position when the two-electron system collapses into a one-electron system, i.e. r1 = 0 (or r2 = 0) and r12 = 0. 13 See Sect. 3.1 in Ref. [18].

3.5 Accurate numbers and a general theory

79

with arbitrary k and  to be determined: “we will certainly obtain a better approximation.” The energy in Eq. (3.7) was then evaluated by noting that Z−k Z−k 2 − 4 ψ −2 ψ+ ψ, r1 r2 r12 (3.46) where a1 [a2 ] is the cosine of the angle between r1 [r2 ] and r12 . However, as subsequently carried out also by Hylleraas, the angular quantities were replaced by metric ones only by means of the following formulae: Hψ = −2k2 ψ − 22 ψ + 2kψa1 + 2kψa2 − 2

a1 =

2 r12 + r12 − r22 , 2r1 r12

a2 =

2 r22 + r12 − r12 . 2r2 r12

(3.47)

For some reason, Majorana apparently did not complete the full calculations [18], which would have resulted in the following expression for the energy to be minimized:

2(k − ) 8k3 + k2 (5 − 7 − 16Z) + 4k(Z +  − 1) − 2 ( − 1) . (3.48) W= 8k2 − 5k + 2 As expected, the minimization procedure gives a better result14 for the helium ground state energy with respect to the  = 0 case. However, despite the apparent complicated formula (3.48), the striking fact is the accuracy of the Majorana result, obtained just at first order, WIHe = 24.09 eV (corresponding to k = 1.8581,  = 0.2547), when compared with the analogous one produced by Kellner at fourth order, WIHe = 23.75 eV, and even with that obtained by Hylleraas with calculations at order 11, WIHe = 24.35 eV.

3.5.2 Majorana’s variant of the variational method Majorana did not limit himself to calculations within the known perturbative or variational methods, but also introduced novel general methods, which were obviously unknown to the physics community. Here we sketch two of them reported in his Quaderni.15 He considered an arbitrary function ϕ in terms of which the wavefunction of the ground state was written as ψ = a0 ϕ + a1 Hϕ,

(3.49)

H being the Hamiltonian operator. The two variational parameters a0 , a1 have to be determined (for a given ϕ) by minimizing the energy functional in Eq. (3.7), which 14 −W/Rh = 5.779 instead of −W/Rh = 729/128 = 5.695. 15 See Sect. 3.1 in Ref. [18].

80

Two-electron problem

assumes the form W= with

a20 A1 + 2a0 a1 A2 + a21 A3 , a20 + 2a0 a1 A1 + a21 A2  An = 

(3.50)

ϕ ∗ H n ϕ dτ ϕ ∗ ϕ dτ

(3.51)

(n = 1, 2, 3). By introducing the function f (a0 , a1 ) = a20 A1 + 2a0 a1 A2 + a21 A3 = Wa20 + 2Wa0 a1 A1 +Wa21 A2 , from the minimizing conditions ∂f /∂a0 = 0, ∂f /∂a1 = 0 the set of equations for deducing a0 , a1 is obtained: a0 (A1 − W) + a1 (A2 − WA1 ) = 0, a0 (A2 − WA1 ) + a1 (A3 − WA2 ) = 0.

(3.52)

Now, this is a homogeneous set of equations which admits non-trivial solutions for a0 , a1 only by requiring the matrix of coefficients to have a vanishing determinant:    A1 − W A2 − A1 W   (3.53)  A2 − A1 W A3 − A2 W  = 0. This last condition allows us to determine the energy W as the smallest root of the corresponding equation, that is  A3 − A1 A2 + (A3 − A1 A2 )2 − 4(A1 A3 − A22 )(A2 − A21 ) . (3.54) W= 2(A2 − A21 ) In his Quaderni, Majorana did not apply such a method to come up with numerical predictions about the ground state energy of helium, but it is nevertheless interesting to compare such predictions with already known results. For example, by choosing the form of ϕ as in Eq. (3.8), we have obtained WIHe = 21.62 eV (for k = 1.6711), which is certainly not a good result when compared with that obtained by Majorana himself using the standard variational method for the same trial wavefunction (see Section 3.2.2), i.e. WIHe = 22.95 eV. This is probably the reason why Majorana further generalized his method, by writing the wavefunction ψ = a0 ϕ + a1 Hϕ + a2 H 2 ϕ + . . . + an H n ϕ

(3.55)

for a given ϕ and a general index n, for which the energy to be minimized is (with the same notations as above) given by

3.6 Conclusions n 

W=

81

ai ak Ai+k+1

i,k=0 n 

.

(3.56)

ai ak Ai+k

i,k=0

Majorana states that “W will be the smallest root of the following equation:”     A1 − W A2 − A 1 W ... An − An−1 W     A2 − A1 W A3 − A2 W ... An+1 − An W    A3 − A2 W (3.57) A4 − A3 W ... An+2 − An+1 W  = 0.    ...    A −A W A A2n−1 − A2n−2 W  n n−1 n+1 − An W . . . Obviously, by increasing n, the calculations give better and better results, but they also become more and more difficult. However, Majorana focused on a different point, i.e. the appearance of divergencies: Often, this procedure does not converge, because, starting from a given value of n, quantity H n ϕ exhibits too many singularities, which forces us to consider only combinations of the form ψ = a0 ϕ + a1 Hϕ + . . . + an−1 H n−1 ϕ.

(3.58)

The inclusion of additional terms is not useful, since the corresponding a coefficients would necessarily vanish.

He referred to the calculations of integrals for r1 = 0, r2 = 0, or r12 = 0, whose degree of divergency increases with increasing n in H n , given the form of the Hamiltonian operator. The connection made by Majorana between the form (for a given ϕ) of ψ in Eq. (3.58), dictated by choosing a suitable n, and the divergency properties of the system, as described by its Hamiltonian is quite interesting.16

3.6 Conclusions The role played by Majorana’s contributions in the known literature on the twoelectron problem, as discussed in Section 2.3 and 2.4, is, in some sense, not as dominant as that in his published papers N.2 and N.3, due to the apparent paucity of those contributions. However, we have shown here that the known history of the solutions given to the two-electron problem should be supplemented by several, unpublished results contributed by Majorana, almost all of them not conveyed (even indirectly) in his two papers published in 1931. Just Majorana’s reproduction 16 Recall that this contribution dates from around 1928–9.

82

Two-electron problem

of standard perturbative and variational results, generalized to arbitrary Z, and summarized by the formulae in Eqs. (3.5) and (3.10), show their pedagogical relevance when compared to currently adopted deductions, given the simple reasoning and calculations behind them. Also, it is worth mentioning the simpler alternative to Hylleraas’s calculations, which led Majorana to obtain first-order numerical results comparable in accuracy with those at order 11 by Hylleraas (being certainly better when compared to fourth-order calculations by Kellner), just by choosing a suitable and quite “obvious” trial wavefunction. From a purely theoretical point of view, however, the most interesting contribution is a variant of the variational method, developed by Majorana in order to take directly into account, already in the trial wavefunction, the action of the full Hamiltonian operator of the given system. From the general wavefunction in Eq. (3.55) he was indeed able to obtain a determinantal equation (3.57) from which the ground state energy could be deduced. Though this method was originally devised for helium (or, more generally, for the two-electron problem), it is well suited for application to any variational problem, and, as such, it is worthy of being applied in several different fields of present day research. The same applies to the concept of an effective nuclear charge different for the two (or more) electrons, which generalizes known results. Finally, it is particularly intriguing to point out the application made by Majorana of the perturbative method, where the atomic number Z was treated effectively as a continuous variable, contributions to the ground state energy of an atom with given Z = Z0 coming also from any other Z = Z0 . Almost all of these novel ideas and methods wait for thorough applications in current physical problems.

4 Thomas–Fermi model

The first meeting between Majorana and Fermi, as recalled in Chapter 1, focused on the statistical model of atoms, known as the Thomas–Fermi model. Fermi developed the idea of applying his statistical method (the Fermi–Dirac statistics) to the completely degenerate state of electrons in an atom, in order to evaluate the effective potential acting on the electrons themselves. He was aware of the fact that, since the number of electrons involved is usually only moderately large, the results obtained would not present a very high accuracy, but the method turned out to be very simple, and gave an easy-to-use expression for the screening of the Coulomb potential accounted for by electrons as a whole. This was one of Fermi’s favorite issues, and the main activity of the Fermi group at the Institute of Physics in Rome from 1928 to 1932 was essentially devoted to this subject. As testified by a number of his notes [17] (and as announced, very briefly, at a conference [203]), Majorana repeatedly studied this same subject, contributing some very important results in both atomic and molecular physics [204, 205, 206]. In the following we shall see that the amusing anecdote recounted in Chapter 1 was only the tip of an iceberg that Amaldi, Segrè, and Rasetti were allowed to see (even without a full understanding of it), since – true to his style – Majorana did not publish anything on the subject. 4.1 Fermi universal potential The main idea of the statistical model of atoms is to consider the electrons around the atomic nucleus as a gas of particles at absolute zero temperature, all obeying the Pauli exclusion principle. 4.1.1 Thomas–Fermi equation The limiting case of the Fermi–Dirac statistics for strong degeneracy applies to such a gas, and Thomas [207] and Fermi [208] independently realized that its

84

Thomas–Fermi model

potential energy −eV satisfied a second-order inhomogeneous differential equation with a term proportional to V 3/2 :   4e 2 m 3/2 2 (e V)3/2 . (4.1) ∇ V= 3π h¯ 2 This equation allowed the evaluation of the potential inside an atom with atomic number Z, using boundary conditions such that for the radius r → 0 the potential becomes the Coulomb field of the nucleus, V(r) → Ze/r, while for r → ∞ it vanishes: V(r) → 0. In order to simplify the differential equation in Eq. (4.1), Fermi then introduced a suitable change of variables:   1 3π 2/3 h¯ 2 −1/3 Z , r = μx, μ= 2 4 me2 (4.2) Ze V(r) = ϕ(r), r in terms of which Eq. (4.1) becomes (for ϕ > 0): ϕ 3/2 ϕ  = √ x

(4.3)

(a prime denotes differentiation with respect to x), with the boundary conditions ϕ(0) = 1,

ϕ(∞) = 0.

(4.4)

The Fermi equation (4.3) was recognized to be a universal equation which did not depend on either the atomic number Z or the physical constants (h¯ , m, e). As noted by Fermi himself, its solution gave, from Eq. (4.2), a screened Coulomb potential which at any point is equal to that produced by an effective charge Ze ϕ (r/μ). Owing to the independence of Eq. (4.3) on Z, the method gave an effective potential which could be easily adapted to describe any atom with a suitable scaling factor. In order to evaluate ionization energies and similar quantities which were relevant for atomic physics observations, Fermi and Rasetti [13, 209, 210] proceeded to apply and enlarge the method to describe positive ions also. They considered the ion of nuclear charge Ze as a neutral atom of nuclear charge (Z − 1)e, to be treated with the statistical method, plus an extra proton in the nucleus. Since the electrostatic potential for the atom of atomic number Z − 1 is (Z − 1)e ϕ(r/μ)/r (with Z → Z − 1 for the parameter μ), the potential energy of one extra electron was shown to be, to first approximation,    r e2 1 + (Z − 1) ϕ . (4.5) −eV = − r μ

4.1 Fermi universal potential

85

With this formula, Fermi and his associates in Rome extended their calculations to many physical problems, obtaining quantities which fitted well the observations and the knowledge of the time. The method was further generalized for arbitrary positively charged ions by Edward B. Baker in 1930 [211] and corrected (for an interpretational misunderstanding) by Eugene Guth and Peierls six months later [212]. 4.1.2 Numerical and approximate solutions The problem of the theoretical calculation of observable atomic properties was solved, in the statistical model approximation, in terms of the function ϕ(x) satisfying the Fermi differential equation (4.3). However, it was believed that the solution of this equation satisfying the appropriate boundary conditions in Eq. (4.4) could not be expressed in closed form, so that some effort was made by Thomas [207], Fermi [13, 208], and others [211] in order to achieve the numerical integration of the differential equation. Thomas obtained a numerical table for some mathematical quantities from which the values of V as a function of the distance r from the nucleus could be deduced, but, as noted by Baker [211], Thomas’ numerical calculations of V near r = 0 were “slightly in error.” A similar effort was also pursued by Fermi, who built a numerical table for the values of the function ϕ(x), the numerical work being performed in approximately one week. The values obtained by Fermi were largely used not only by the members of the Rome group, but also by many other atomic physicists in the 1930s in their computations (see, for example, Refs. [211, 213]). Subsequent, more accurate, numerical evaluations of the solution of the Thomas– Fermi equation were performed by the mathematician Carlo Miranda in 1934 [214], who also gave a solid mathematical framework to the numerical integration of the Fermi equation. By using refined approximation procedures on a finite variation equation corresponding to the Fermi differential equation, he obtained numerical values for ϕ(x) which were accurate up to the fifth significant digit for small x (Fermi’s values were accurate “only” up to the third significant digit in the same x interval). By using standard but involved mathematical tools, Thomas also got an exact, “singular” solution of his differential equation satisfying only the condition V(r) → 0 for large r, such a solution being later (in 1930) considered by Sommerfeld [213] as an approximation of the function ϕ(x) for large x (and indeed known as the “Sommerfeld solution” of the Fermi equation), ϕ(x) =

144 . x3

(4.6)

86

Thomas–Fermi model

Sommerfeld also obtained corrections to this formula in order to improve the approximation ϕ(x) for large (but not extremely large) values of x, and the goodness of the resulting approximated expression was checked by him by comparing his results with the numerical values obtained by Fermi [176]. An approximate solution for ϕ(x) near x = 0 was, instead, first considered by Fermi [208], who obtained the following series expansion: 4 ϕ(x) = 1 − 1.58x + x3/2 + . . . . 3

(4.7)

This Fermi approximation of the function ϕ(x) with a polynomial with rational powers1 was later reconsidered by Baker [211], who obtained terms up to the power x9/2 , although the coefficient of the last term presented by author was wrong (see Section 4.3.1).

4.1.3 Mathematical properties In view of its relevance in atomic physics (and, later, also in nuclear physics), the nonlinear second-order differential equation (4.3) received quite a large amount of attention from mathematicians, who studied several formal properties of the solutions of the Thomas–Fermi equation. By neglecting discussions about the asymptotic expressions and theorems on numerical approximations (already mentioned above), from the mathematical point of view the first most important result for a nonlinear differential equation is the theorem establishing the existence and uniqueness of its solutions (the physically interesting solutions of Eq. (4.3) were those satisfying the boundary conditions Eqs. (4.4)). Such a theorem was clearly stated in 1929 by two Italian mathematicians, Antonio Mambriani [215] and Giuseppe Scorza-Dragoni [216, 217]. By looking at the analytical properties of the function appearing on the r.h.s. of Eq. (4.3) and using standard methods that hold for ordinary differential equations, they showed that an infinite number of integral curves of Eq. (4.3) pass through a given point (x0 , ϕ0 ) of the first quadrant of the x − ϕ plane, only one of which, ϕ ∗ (x), is always decreasing and approaching the x-axis. This solution corresponds to a “critical” value of the initial slope ϕ0 , given by −1.588 (found by Fermi). The other solutions with an initial slope greater than the critical value lie above ϕ ∗ (x) and diverge for diverging x, while those with an initial slope lower than the critical value lie below ϕ ∗ (x) and monotonically decrease for increasing x until they reach the x-axis. 1 The reason for the appearance of rational (rather than integer) powers is very simple. Due to the first of Eqs. (4.4), near the origin we have ϕ(x) ∼ 1, so that substitution into Eq. (4.3) results in ϕ  (x) ∼ x−1/2 ,

whose integration leads directly to a polynomial with rational powers.

4.2 Majorana solution of the Thomas–Fermi equation

87

Some other mathematical questions underlying the statistical model introduced by Thomas and Fermi and the solutions of the corresponding equation were only tackled later, in the 1940s [218]. In particular, the attention then shifted towards variational methods applied to a “Thomas–Fermi energy functional,” involving the screened potential ϕ(x), in order to obtain the ground state energy for the relevant atoms (and molecules). A renewed interest began later, in 1969, with the paper by Einar Hille [219], who studied analytically a number of mathematical aspects of the Thomas–Fermi equation for the atomic case. A generalization to the molecular case was subsequently analyzed by Elliot H. Lieb and Barry Simon in 1977 [220], by proving the existence and uniqueness of the corresponding Thomas– Fermi function.

4.2 Majorana solution of the Thomas–Fermi equation At the institute, [Majorana] found Fermi calculating the function central to the Thomas– Fermi statistical method for calculating atomic properties, cranking a small Brunsviga adding machine by hand. With its help, in about a week of work, he had obtained a numerical table of the function. Majorana informed himself in detail about the mathematical problem and went home without further comment. At home, he transformed Fermi’s nonlinear equation into a Riccati equation and solved it numerically using his brain as calculating machine. [11]

With these words, Segrè recalled the first meeting between Majorana and Fermi at the Institute of Physics in Rome, as discussed in Chapter 1. The whole work performed by Majorana on the solution of the Fermi equation was not published, but is contained in some spare sheets conserved at the Domus Galilaeana in Pisa, and largely reported by the author himself in his notebooks [17].

4.2.1 Transformation into an Abel equation The reduction of the Fermi equation to an Abel equation (rather than a Riccati one, as confused by Segrè) proceeded as follows.2 A change of variables, from (x, ϕ) to (t, u), was initially adopted, with the novel variables determined in such a way as to satisfy – if possible – automatically both the boundary conditions Eqs. (4.4). The function ϕ in Eq. (4.6) had the correct behavior for large x, but the wrong behavior near x = 0, such that the functional form of ϕ should be modified in order to take into account the first condition in Eqs. (4.4). An obvious modification was to write ϕ = (144/x3 )f (x), with f (x) a suitable function vanishing for x → 0 (for ϕ(x = 0) = 1 to hold). The simplest 2 See Sect. 2.7 in Ref. [17].

88

Thomas–Fermi model

choice for f (x) is a polynomial in the novel variable t, as later considered, in a similar way, by Sommerfeld [176]. The choice adopted by Majorana was ϕ(x) =

144 (1 − t)2 , x3

(4.8)

with t → 1 as x → 0, from which the first relation linking t to x, ϕ is obtained. The second relation, involving the dependent variable u, was chosen to be that typical of homogeneous differential equations (like the Fermi equation) for reducing the order of the equation, i.e. exponentiation with an integral of u(t). The transformation relations Majorana introduced were, therefore 1 3 x ϕ, 12 t ϕ = e 1 u(t) dt . t =1−

(4.9)

Substitution into Eq. (4.3) led to an Abel equation for u(t), du = α(t) + β(t) u + γ (t) u2 + δ(t) u3 , dt

(4.10)

16 , 3(1 − t) 1 β(t) = 8 + , 3(1 − t) 7 γ (t) = − 4t, 3 2 δ(t) = − t(1 − t). 3

(4.11)

with α(t) =

Note that both the boundary conditions in Eqs. (4.4) are automatically verified by the relations (4.9). We have reported the derivation of the Abel equation (4.10) mainly for historical reasons (to substantiate the Segrè testimony), since the precise numerical values for the Fermi function ϕ(x) were obtained by Majorana by solving a different firstorder differential equation, as we will see in the following. We do not know why Majorana transformed the original Thomas–Fermi equation into an Abel equation if he later switched to another type of differential equation;3 probably, his primary aim was to prove the existence and uniqueness of the solution he searched 3 It is, however, very unlikely that it was just a first unsuccessful “attempt” – later abandoned – to solve the

equation. Indeed, Majorana did not usually report in his notebooks, known as Volumetti [17], studies that he judged to be incomplete, such calculations appearing instead only in a number of spare sheets that he regularly used.

4.2 Majorana solution of the Thomas–Fermi equation

89

for, which followed directly from the corresponding theorem for the Abel differential equations. Note that, as recalled above, such a theorem for the Thomas– Fermi equation appeared in the literature only two years later, without following Majorana’s simple reasoning. 4.2.2 Analytic series solution Quite remarkably, none of Majorana’s colleagues and friends was aware of the actual method that Majorana used in order to solve the Thomas–Fermi equation. As appears in his notebooks,4 such a method did not start from Eq. (4.8), but rather ϕ(x) was chosen to be of the form 144 6 t , (4.12) x3 with the point x = 0 corresponding to t = 0. In order to obtain again a first-order differential equation for u(t), the transformation equation for the variable u involved ϕ and its first derivative. Majorana then introduced the following formulae: ϕ(x) =

t = 144−1/6 x1/2 ϕ 1/6 ,  1/3 16 ϕ −4/3 ϕ  . u=− 3

(4.13)

By taking the t-derivative of the second equation in (4.13) and inserting Eq. (4.3) into it, he obtained:  1/3   du 16 4 ϕ 2 ϕ 3/2 −4/3 =− + 1/2 . x˙ ϕ − (4.14) dt 3 3 ϕ x By using Eqs. (4.13) to eliminate x1/2 and ϕ 2 , the following equation resulted:  1/3 2 tu − 1 1/3 du 4 (4.15) = x˙ ϕ . dt 9 t Now the quantity x˙ ϕ 1/3 was expressed in terms of t and u by making use again of the first equation in (4.13) (and its t-derivative). After some algebra, the final result for the differential equation for u(t) was as follows: tu2 − 1 du =8 . (4.16) dt 1 − t2 u The obtained equation is again nonlinear but, unlike the original Fermi equation (4.3), it is first order in the novel variable t and the degree of nonlinearity is lower than that of Eq. (4.3). The boundary conditions for u(t) were easily taken 4 See Sect. 2.7 in Ref. [17].

90

Thomas–Fermi model

into account from the second equation in (4.13) and by requiring that for x → ∞ the Sommerfeld solution (Eq. (4.12) with t = 1) be recovered:  1/3 16 ϕ0 , u(1) = 1. (4.17) u(0) = − 3 Here we have denoted by ϕ0 = ϕ  (x = 0) the initial slope of the Thomas–Fermi function ϕ(x), which, for a neutral atom, is approximately equal to −1.588, as we have already seen. The branch of u(t) giving the Thomas–Fermi function (in parametric form) is that between t = 0 and t = 1. In this interval, the solution of Eq. (4.16) was achieved by Majorana in terms of a series expansion in powers of the variable τ = 1 − t: u = a0 + a1 τ + a2 τ 2 + a3 τ 3 + . . . .

(4.18)

From the condition u(1) = 1, the first coefficient immediately followed (a0 = 1), while the other ones were obtained from an iterative formula coming from the substitution of Eq. (4.18) into Eq. (4.16): ∞ ∞  

A(k, l) τ k+l = 0,

(4.19)

k=0 l=0

where A(k, l) = ak [(l + 1)al+1 − 2(l + 4)al + (l + 7)al−1 ] − (k + l + 1)δl0 ak+l+1 + 8δk0 δl0

(4.20)

(here a−1 = 0 is chosen). Equation (4.19) can also be cast in the following form (k + l = m, l = n):   m ∞   A(m − n, n) τ m = 0, (4.21) m=0

n=0

so that, for fixed m, the relation determining the series coefficients is as follows: m 

am−n [(n + 1)an+1 − 2(n + 4)an

n=0

+ (n + 7)(1 − δn0 )an−1 ] = (m + 1)am+1 − 8δm0 ,

(4.22)

with m = 0, 1, 2, 3, . . .. Equation (4.22) for m = 0, (a0 − 1)[a1 − 8(a0 + 1)] = 0, is identically satisfied due to the condition a0 = 1, while for m = 1 a second-degree algebraic equation for a1 was obtained, a21 − 18a1 + 8 = 0, of which the smallest √ root was chosen to be a1 = 9 − 73. The remaining coefficients were determined

4.2 Majorana solution of the Thomas–Fermi equation

91

by linear relations from the obtained values of a0 and a1 ; indeed, by excluding the cases with m = 0, 1 and after some algebra, Eq. (4.22) can be written as: m−2  1 am−n [(n + 1)an+1 am = 2(m + 8) − (m + 1)a1 n=1 − 2(n + 4)an + (n + 7)an−1 ] + am−1 [(m + 7)  − 2(m + 3)a1 ] + am−2 [(m + 6)a1 ]

.

(4.23)

The sum on the r.h.s. involves coefficient ai with indices i ≤ m − 1, so that the relation in Eq. (4.23) gave explicitly the value of am once the previous m − 1 coefficients am−1 , am−2 , . . . , a2 , a1 (and a0 ) were known. It is remarkable that the series expansion in Eq. (4.18) is uniformly convergent  in the interval [0, 1] for τ , since the series ∞ n=0 an of the coefficients converges to a finite value determined by the initial slope ϕ0 . Indeed, by setting τ = 1 (t = 0) in Eq. (4.18), from Eq. (4.17) it follows that:   ∞  16 ϕ0 . an = − (4.24) 3 n=0 Majorana was aware [17] of the fact that the an coefficients were positive definite and, more importantly, that the series in Eq. (4.18) exhibited geometrical convergence with an /an−1 ∼ 4/5 for n → ∞. Once the function u(t) was known, the final problem was to look for the Thomas–Fermi function ϕ(x). This was obtained in a parametric form x = x(t), ϕ = ϕ(t) in terms of the parameter t already introduced in Eqs. (4.13), and by writing ϕ(t) as ϕ(t) = e

t 0

w(t) dt

(4.25)

(with this choice, ϕ(t = 0) = 1 and the first condition in Eqs. (4.4) is automatically satisfied). Here w(t) was introduced as an auxiliary function, to be determined in terms of u(t) by substituting Eq. (4.25) into Eqs. (4.13); as a result, the parametric solution of Eq. (4.3), with boundary conditions (4.4), assumed the following form: x(t) = 1441/3 t2 e2I (t) , ϕ(t) = e−6 I (t) , with



t

I(t) = 0

ut dt. 1 − t2 u

(4.26)

(4.27)

92

Thomas–Fermi model 1.0

0.8

j(x)

0.6

0.4

0.2

0.0

0

1

2

3

4

5

x

Figure 4.1. Thomas–Fermi function ϕ(x) and the Majorana approximation of it. The dashed line refers to the exact (numerical) solution of Eq. (4.3) and the solid one corresponds to the parametric solution obtained from Eqs. (4.26) and (4.27).

Given the fact that, in more than 70 years, no one has been able to find an appropriate analytic solution to the Thomas–Fermi equation, it is quite remarkable that Majorana succeeded in doing it (just in one night!) in terms of only one quadrature.

4.2.3 Numerical tables Even with a very modest numerical program, one can test the accuracy of the solution obtained by Majorana. For example, by taking into account only ten terms in the series expansion for u(t), the numerical values obtained with the Majorana method approximate the values of the exact solution of the Thomas–Fermi equation with a relative error of the order of 0.1% [204], which is a very good result. Graphically, this is easily seen from the plot we present in Figure 4.1. It is probably more instructive to compare the numbers obtained by Fermi [13] and Majorana5 with their own methods (and, in addition, with the more accurate numbers obtained later by Miranda [214]); see Table 4.1. A quick look shows a satisfactory agreement between the Fermi numerical approach and the Majorana method, but it is even more remarkable that, for the first points, Majorana also obtained the values for the derivative of the function ϕ(x) (see Ref. [17]), not shown here. 5 See Sect. 2.7 in Ref. [17].

4.3 Mathematical generalizations

93

Table 4.1. Numerical values of the Thomas–Fermi function ϕ(x) x

ϕFer

ϕMaj

ϕMir

x

ϕFer

ϕMaj

ϕMir

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.2 1.4 2 3

1 0.882 0.793 0.721 0.660 0.607 0.562 0.521 0.485 0.453 0.425 0.375 0.333 0.244 0.157

1 0.882 0.793 0.721 0.660 0.607 0.561 0.521 0.486 0.453 0.424 0.374 0.333 0.243 0.157

1 0.88170 0.79307 0.72065 0.65955 0.607 0.56118 0.52081 0.48495 0.45288 0.42403 0.37427 0.33294 0.24306 0.15675

4 5 6 7 8 9 10 20 30 40 50 60 80 100

0.108 0.079 0.059 0.046 0.037 0.029 0.024 0.0056 0.0022 0.0011 0.00061 0.00039 0.00018 0.0001

0.108 0.079 0.059 0.046 0.036 0.029 0.024 0.0056 0.0022 0.0011 0.0006 0.0004 0.0002 0.0001

0.1086 0.0798 0.0599 0.0469 0.038 0.030 0.025 0.0063 0.0024 0.0012 0.00068 0.00042 0.00019 0.0001

After Fermi [13], Majorana [17], and Miranda [214].

4.3 Mathematical generalizations Although Majorana’s primary aim regarding the Thomas–Fermi model was essentially physical in nature, nevertheless the work he performed about the solution of the Fermi universal equation presents quite a mathematical interest as well, mainly driven – apart from what already discussed above – by explicit and implicit results he was able to get out.

4.3.1 Frobenius method As previously recalled, according to Rasetti, Segrè, and Amaldi, the paper by Fermi in Ref. [208] was shown to Majorana probably even before its publication. The same witnesses, however, did not know that Majorana proceeded to improve the degree of accuracy of the Fermi polynomial approximation in Eq. (4.7), by considering several other terms in the expansion up to the sixth power of x (including both integer and half-integer powers). The method Majorana followed was a generalization of the Frobenius method for differential equations [221], and goes as follows. The solution of the Fermi equation, recast by Majorana as x(ϕ  )2 − ϕ 3 = 0,

(4.28)

94

Thomas–Fermi model

was written in the form ϕ(x) =

∞  1 (a2n xn + a2n−1 xn+ 2 ).

(4.29)

n=0

The terms in Eq. (4.29) with integer powers of x accounted for the usual Taylor series expansion, while those with half-integer powers corresponded to the Frobenius expansion, the requirement for both terms being dictated by the Fermi approximation underlying Eq. (4.7), as already pointed out. The as yet unknown coefficients a2n and a2n−1 were then determined by substituting Eq. (4.29) into Eq. (4.28) and requiring that the coefficients of given powers of x, appearing on the l.h.s. of Eq. (4.28), vanish. In this way, Majorana sought to obtain two iterative formulae for the a2n and a2n−1 coefficients, but apparently he did not succeed in obtaining these general expressions (see the following), and stopped the series at the term x6 (thus considering a true polynomial), by evaluating only a few coefficients. The corresponding expression was as follows: 4 2p 1 3p2 7/2 2p 4 56 + 3p3 9/2 ϕ(x) = 1 − px + x3/2 − x5/2 + x3 + x − x + x 3 5 3 70 15 756 p2 5 −992p + 45p4 11/2 4(35 − 9p3 ) 6 + + (4.30) x + x x, 175 47520 14175 with p = 1.58, in according with the Fermi value. Note that, contrary to the similar (but more limited) calculation employed by Baker [211] (see Section 4.1.2), here all the numerical coefficients given by Majorana are correct. The reason why Majorana abandoned the complete series expansion in Eq. (4.29) is not known, but is likely related to the fact that such a series does not converge to the desired solution.6 Nevertheless, it is remarkable that Majorana stopped the series precisely at order 6. Indeed, it is easily shown that, by stopping the series in Eq. (4.29) at any power n ≤ 5, the obtained polynomial diverges towards +∞ for diverging x, while from n = 6 onward the corresponding polynomial diverges towards −∞, as can be seen in Figure 4.2. Neither behavior, of course, matches the correct condition ϕ(x → ∞) → 0, but the first one, ϕ(x → ∞) → +∞, is physically not acceptable (see, for example, the discussion in Ref. [212]). It is also evident from the same Figure 4.2 that the curve corresponding to n = 6 is the best possible one for not so small values of x: increasing the order in the series expansion results in a worsening of the approximation. 6 This is not surprising, since either differential equation (4.3) or (4.28) is nonlinear, so the series solution

method cannot, in general, be applied at all.

4.3 Mathematical generalizations

95

1.0

0.8 n=5 j(x)

0.6

0.4

0.2 n = 20 0.0 0.0

0.5

n = 15 1.0 x

n = 10

n=6

1.5

2.0

Figure 4.2. Thomas–Fermi function ϕ(x) (thick curve) and its (generalized) Frobenius polynomial approximations for small x. The labels n = 5, 6, 10, 15, 20 refer to the maximum power considered in the expansion in Eq. (4.29).

4.3.2 Scale-invariant differential equations The method developed by Majorana in order to obtain a parametric solution of the Thomas–Fermi equation (with appropriate boundary conditions) in terms of only one quadrature was based, as we have seen, on a particular double change of variables transforming the second-order Thomas–Fermi equation into a firstorder equation, whose solution was then obtained by series expansion. Remarkably, and (probably) contrary to naive expectations, the transformation method used by Majorana in this particular case may be applied to a large class of ordinary differential equations as well, for which a general theorem for reducing the order of those equations can be proved [205]. Such results were not obtained by Majorana (or, at least, there is apparently no written evidence that he was aware of them), but it is instructive to consider them briefly both for illustrating the great potential of Majorana’s original method for mathematical physics and to make clear the formal reasoning behind that method, which turns out to be not just a lucky attempt (for more technical mathematical details, see Ref. [205]). The class of ordinary differential equations to which the Majorana method can be applied is that of Majorana scale-invariant equations. A general differential equation of order n in the independent variable x and dependent variable y is an expression of the kind   F x, y, y , y , . . . , y(n) = 0,

(4.31)

96

Thomas–Fermi model

where a prime  denotes differentiation with respect to x. Such an equation is said to be Majorana scale invariant if it is invariant for x → α c x, y → αy for any α = 0 and some value c:     F α c x, αy, α 1−c y , α 1−2c y , . . . , α 1−nc y(n) = F x, y, y , y , . . . , y(n) . (4.32) Similar to what is already known for autonomous, homogeneous, or scale-invariant differential equations, it is always possible to reduce a Majorana scale-invariant equation of order n to a differential equation of order n − 1. This can be achieved, by generalizing Majorana’s original method, by seeking a parametric solution of Eq. (4.31) satisfying the condition (4.32) in the form ⎧ ⎨ x = z(t) yc (t), (4.33) ⎩ y = y(t), with z(t) an arbitrary but given function of the parameter t. Order reduction is achieved, as for homogeneous equations, by setting y(t) = e



u(t) dt

.

(4.34)

With some algebra it is then possible (once we have chosen the function z(t)) to translate the differential problem of searching for the solution y(x) into the corresponding one of searching for the solution u(t) of a suitable differential equation of order n − 1. Depending on the choice of the function z(t), it could happen that such a novel equation, though of order n − 1, is still much too complex to be solved, so that, in analogy to what was considered by Majorana, a further procedure aimed at reducing its degree of nonlinearity is desired. Following the same reasoning as in Section 4.2, this can be obtained by performing a second change of the dependent variable, i.e. u u −→ v = , (4.35) z˙ + cuz by means of which Eq. (4.34) is replaced by y(t) = e



v˙z 1−cvz

dt

,

with v(t) satisfying a suitable differential equation of order n − 1:   v˙ 2 F z, 1, v, (1 − cvz) + (1 − c)v , . . . = 0. z˙

(4.36)

(4.37)

To the non-dedicated scholar, this formalism may appear quite obscure, and a thorough example is certainly appropriate.

4.4 Physical applications

97

The most interesting one is probably that of the Emden–Fowler equation: y = xa yb

(4.38)

(where a, b are two real numbers), which is of particular relevance in mathematical physics [222] (the Thomas–Fermi equation, for example, is a particular case of the Emden–Fowler equation). It satisfies the condition in Eq. (4.32) provided that c = (1 − b)/(a + 2), so that, by following the procedure above (which, however, may apply only for a = −2), it is found to be equivalent to the following Abel equation of the first kind for u(t): du = α(t) + β(t) u + γ (t) u2 + δ(t) u3 , dt

(4.39)

where α(t) = za z˙2 , z¨ , z + 2c − 1 ,

β(t) = 3c za+1 z˙ + γ (t) = 3c2 za+2 δ(t) = c3

(4.40)

za+3 z + c(c − 1) . z˙ z˙

Note that it is usually not known by mathematicians that an Emden–Fowler equation may be transformed into an Abel equation. Alternatively, by adopting the transformation in Eq. (4.35), the corresponding equation for v(t) becomes

z˙ za − (1 − c)v2 dv = . (4.41) dt 1 − cvz Depending on the problem to be solved, the final equation for u(t) or v(t) can be further simplified with an appropriate choice for z(t).

4.4 Physical applications Majorana followed quite closely the activities of the Fermi group in Rome, when it was involved in studying several different applications of the Thomas–Fermi statistical model of atoms. Fermi himself [223] calculated the number of electrons in an atom with given values of the orbital angular momentum as a function of the atomic number Z, thus succeeding in giving an account of the appearance of the elements in the periodic table. Of course, it was readily realized that, since the potential V(r) was determined by means of statistical arguments, only average properties of the periodic system could be explained in terms of the Thomas– Fermi model, while peculiarities of the electronic structure, underlying particular

98

Thomas–Fermi model

properties of the elements, could not be accounted for by a statistical method. Nevertheless, as noted by Fermi, the agreement with the experimental observations was satisfactory, and this prompted him to proceed further with these studies. Indeed, Fermi [210] evaluated the S-levels of any (heavy) element, with particular reference to the calculation of the corresponding Rydberg correction by means of the knowledge of an approximate solution of the Schrödinger equation for an s-electron with zero energy, while Rasetti [224] considered the M-levels (corresponding to the angular momentum l = 3, m = 2) of the X-ray spectrum. Other applications, studied by the Fermi group during 1928, mainly focused on explaining the properties of the rare earths and the electron affinity of the halogens [13]. As previously mentioned, Majorana also participated in such a wide-ranging program, but his only published work was that contained in paper N.1 in collaboration with Gentile, discussed in section 2.2, where the doublet separation due to spin for optical and X-ray levels of some elements was calculated by applying the Dirac relativistic theory. The story, however, did not really end there, and a number of other unpublished results were obtained by Majorana (when he was still an undergraduate student), Fermi and coworkers being largely unaware of them. In the following, however, we will consider only the most intriguing ones, referring the interested readers directly to the original papers [17].

4.4.1 Modified Fermi potential for heavy atoms Although his first effort was to follow quite closely the Fermi approach to the statistical model of atom, quite soon7 Majorana developed his own approach to the problem of atoms with very large atomic number Z. To first order, the problem of the electron distribution in heavy atoms may be solved as follows: neglect the inversions of the periodic system and suppose all the electron orbits are circular. From Pauli’s principle we have two electrons in a circular orbit of quantum number 1, eight electrons in an orbit of quantum number 2, and in general 2n2 electrons in an orbit of quantum number n. If Z denotes the atomic number, then Z=

n 

2n2

(4.42)

1

and, for very large Z, Z= 7 See Sect. 2.8 in Ref. [17].

2 3 n . 3

(4.43)

4.4 Physical applications The p-th electron will be in an orbit having a quantum number  Q = 3 3p/2 ;

99

(4.44)

and, since it feels an effective charge Z − p, its distance from the nucleus will be r=

h¯ 2 (3p/2)2/3 . m e2 (Z − p)

(4.45)

In other words, Majorana used a rescaled screened potential, replacing Fermi’s Eqs. (4.2) with h¯ 2 (3/2)2/3 r 1 r π 2/3 x =  , , μ1 = x1 = μ1 2μ 2 1.480 m e2 Z 1/3 (4.46) Ze V1 = φ1 . r In such a way, since (Z − p)/Z = φ1 − xφ1 , by employing the same method that he used for seeking the (parametric) solution of the Thomas–Fermi equation but with t = φ1 − xφ1 , he found that the novel function φ1 (with the limiting condition φ1 (∞) = 0) was parametrically given by  x1 (1 − t)2/3 t , φ1 = − x1 dx . (4.47) x1 = 2 t ∞ x By performing the integration, Majorana was then able to obtain a closed parametric form for the modified Fermi potential:  2/3 (1 − t)2/3 2 , x=2 π t (4.48)

9 t 3 2/3 1 − (1 − t) − + , φ1 = 4t 2 4 from which he computed the novel numerical values to be compared with the corresponding ones for the Fermi function (see Table 4.2). It is interesting to make a comparison with Fermi’s ϕ [see Table 4.2]. From this, we conclude that our approximate method yields a value for the electron density near the nucleus that is roughly a sixth less than the actual value, and, for the potential generated by the negative charges in the vicinity of the nucleus, a value that is smaller than the actual one by roughly 4%. The approximate potential derived above can thus be used for the K and L series and, with a small error, also for the M series. It fails though to give correct results for the outermost regions of the atom. The reason for this is twofold: we have neglected the inversions of the periodic system and we have approximated the elliptical with circular orbits. The elliptical orbits are particularly relevant for strongly non-Coulombian fields, such as the ones that are present in the outermost regions, since, for any given total quantum number, they are closer to the nucleus than the circular orbits.8 8 See Sect. 2.8 in Ref. [17].

100

Thomas–Fermi model Table 4.2. Comparison between the Fermi ϕ and Majorana φ1 functions for several points x

φ

ϕ1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2

1 0.883 0.793 0.722 0.660 0.607 0.562 0.521 0.486 0.453 0.424 0.243

1 0.883 0.793 0.721 0.660 0.608 0.564 0.525 0.491 0.462 0.435 0.276

See Ref. [17].

4.4.2 Second approximation for the atomic potential Probably unsatisfied by the “rough” approximation employed in the basic Thomas– Fermi model, Majorana considered some refinements in the form of second approximation, including a generalization of the model to atoms ionized n times (with n ≥ 0). His starting point was the physical fact that the potential inside an atom is defined up to an additive constant C, as also clearly realized by Fermi in his 1927 paper [208]. This occurrence was exploited by Majorana in order to shift the attention from the local potential V0 at a given point inside the atom or the ion (that is, the potential generated by the nucleus and Z − n electrons), to the effective potential V acting on one electron at that point (that is, the potential generated by the nucleus and Z − n − 1 electrons). The two potentials are connected, approximately, by a simple scaling relation that Majorana wrote as ∇ 2V =

Z−n−1 2 ∇ V0 . Z−n

(4.49)

Since V0 satisfies the Poisson equation with a charge density ρ, ∇ 2 V0 = −4πρ, the effective potential V is that generated by a rescaled charge density ρ(Z − n − 1)/ (Z − n), taking into account the finite charge of the given electron on which V (rather than V0 ) acts. Note that, although in a completely different context, the procedure is rather similar to that now adopted in the renormalization of physical quantities in modern gauge theories.

4.4 Physical applications

The effective potential was then written as follows:   r Ze ϕ + C, V= r μ

101

(4.50)

where ϕ is the usual Thomas–Fermi screening function, while the dimensional parameter μ is as in Eq. (4.2) but rescaled by a factor [(Z − n)/(Z − n − 1)]2/3 . The constant C was interpreted as the value of the potential at the boundary of the atom (or the ion), which thus acquired a finite radius r0 (= μx0 ): C=

(n + 1)e . μx0

(4.51)

As a consequence, the maximum energy U of one bounded electron is different from zero and proportional to C, i.e. U = −Ce; from this expression, the Rydberg corrections to the energy levels of a given atom or ion can be easily deduced. It is worth mentioning that, in Majorana’s approach, the key role is played by the additive constant C, rather than by the finite radius r0 or x0 . Indeed, although the question of the effective atomic radius rests on solid physical grounds (an atom or an ion indeed has a finite extension), it seems that Majorana was aware of the fact that, in a sense, a finite radius r0 can always be obtained by requiring a finite range for the potential V, and this was simply achieved through the introduction of a suitable constant C, which could always be added to the potential. This fact was quite clear also to Fermi. In the papers dealing with neutral atoms (starting from Ref. [208] of 1927), though not expressly mentioned, the constant was chosen to be zero, or, in other words, it was imposed for simplicity that the atomic radius be infinite: at the accuracy level adopted there, such an approximation sufficed. Instead, in the articles of 1930–1 [225, 226], which discussed the energy spectra of ionized atoms, a finite radius r0 was explicitly assumed (as it should be), and it was introduced along the same lines followed by Majorana, that is by means of an additive constant. However, the subtleties envisaged by Majorana, in distinguishing the local potential from the effective one, were not present (again, they could be neglected at the degree of approximation considered), this being clearly apparent when comparing the above-mentioned relevant expressions obtained by Majorana with the corresponding ones employed by Fermi, where the former reduce to the latter on replacing n + 1 by n. Finally, in the final paper of the Fermi group on the applications of the Thomas–Fermi model, published in 1934 [227], the “choice” of the atomic potential was made by following completely9 the reasoning of Majorana. This, indeed, was required by the improvement in the degree of approximation pursued by the authors, with respect to previous 9 With the exception that Fermi and Amaldi replaced Eq. (4.49) with the direct one V = Z−n−1 V , that is the 0 Z−n

scaling relation was assumed to hold directly on the potentials.

102

Thomas–Fermi model

calculations, which now also included relativistic corrections. In fact, the “second approximation” for the effective potential depended on the ratio between the degree of ionization and the atomic number (this ratio was written, approximately, as n/Z in Refs. [225, 226] and, correctly, as (n + 1)/Z in Ref. [227], following Majorana), so that it had some (small) influence only for lighter elements, as considered in Ref. [227] only. Contrary to many other cases, here Fermi was aware of what had been obtained by Majorana [203] but used it only when explicitly required, according to his general attitude to avoiding unnecessary mathematical complications. Nevertheless, in December 1928 Fermi invited Majorana (who was still an undergraduate student) to present his work at the XXII General Meeting of the Italian Physical Society, “sandwiched” between two of his own talks; evidently, he was impressed with Majorana’s results. However, as happened in many other cases, the “second approximation” elaborated by Majorana has not been correctly credited to him, but rather has become known in the literature as the Fermi–Amaldi correction [228].

4.4.3 Atomic polarizability The information obtained about the finite radius of atoms or ions translated directly into the calculations that Majorana performed on the dipole polarizability of an atom in an external electric field:10 The potential inside an atom satisfies, to first or second approximation (as shown in the previous section), an equation of the kind ∇ 2 V = K (V − C)3/2 .

(4.52)

Let us now consider the atom in a weak field E. Because of the mutual dependence between the variations of the atomic quantities and the applied field, if the latter is weak, we deduce δV = − f (r) E r cos(r·E),

δC = 0.

(4.53)

When the atom is placed in a weak uniform electric field E, the potential V of the isolated atom considered in the preceding pages changes by a small value δV, and, according to Majorana (who assumed −E to lie along the x-axis), the total potential can be written as V1 = V + E x f (r). The function f (r) has to be determined by substitution into Eq. (4.52) and adopting a suitable linearization of the r.h.s. with respect to f (r), which Majorana obtained as follows: (V1 − C)3/2 = (V − C)3/2 + 10 See Sect. 2.16 in Ref. [17].

3 (V − C)1/2 E x f (r) + · · · . 2

(4.54)

4.4 Physical applications

103

The resulting equation was found to be f  (r) + 3

1  3 f (r) = K (V − C)1/2 f (r) , r 2

(4.55)

or, by changing the variable into y = r3/2 f (r) (and then V1 = V + x y E/r3/2 ),   √ 3 1 (4.56) y = K V − C + 2 y. 2 2r In order to find f (r) (or y(r)), and thus the atomic polarizability, the solution of the differential equation (4.56) was required to be finite as r → 0 and, in addition, to satisfy the condition that the potential inside the atom matches the external polarizing potential (which is the sum of a leading term given by the applied field and a second one given by the atomic dipole induced by the field) at the boundary of the atom or ion (at r = r0 ). In Majorana’s words: The condition that f (0) be finite allows us to obtain f or y up to a constant factor. This may then be determined by the requirement that the average value of −∂V/∂x on the surface of the ion be equal to −E, that is, to the external field. This requirement reads f (r0 ) +

1 r0 f  (r0 ) = 1. 3

(4.57)

Once the function f (r) is known, the atomic polarizability follows accordingly, and Majorana concluded: The electric moment of the ion is then M = E r03 (1 − f (r0 )) .

(4.58)

Interestingly, he then found that the value of the (static) dipole polarizability depends only on the radial part of the polarizing potential at r = r0 . No numerical calculations were apparently carried out by Majorana for particular atomic cases. Although the subject of atomic polarizabilities within the framework of the Thomas–Fermi model was considered (quite roughly) only in 1949 [228], we have to wait until 1972 [229] to see in the literature the same results obtained by Majorana. And it is quite intriguing that the Russian authors followed very closely the same physical and mathematical reasoning by Majorana, the only differences being in the notation used (u(r) replacing f (r), etc.)11 4.4.4 Applications to molecules In his notebooks, Majorana also started to consider the application of the Thomas– Fermi statistical method to molecules, rather than to single atoms, studying in 11 This is particularly evident when we compare the calculations given here with those discussed in Sect. 2 of

Ref. [230], a later reassessment by (almost) the same authors.

104

Thomas–Fermi model

particular the case of a diatomic molecule with identical nuclei.12 The effective potential of such a molecule was cast in the following form: V = V1 + V2 − α

2V1 V2 , V1 + V2

(4.59)

where V1 and V2 are the potentials generated by each of the two atoms, and α is a suitable function that he assumed depends only on the distance from the center of the molecule. Such a function must obviously obey the differential equation for V, ∇ 2 V = −KV 3/2 (there was no need for C = 0) with appropriate boundary conditions, but Majorana did not go much further in his calculations since, in his own words, “the hypothesis that α depends on the distance from the center of the molecule is, however, too far from reality.” Later on, Majorana returned to a possible description of molecules in the framework of the Thomas–Fermi model by devising a general method to determine V when the equipotential surfaces inside the molecule were approximately known.13 By writing the approximate expression for the equipotential surfaces as functions of a parameter p, f (x, y, z) = p, he was able to deduce a thorough equation, from which it was possible to determine V = V(p): y1 V  + y1 V  = − K V 3/2 y2 .

(4.60)

The quantities y1 (p), y2 (p) are known functions (in the mentioned approximation) given by    −1 ∂p ∂p dσ, (4.61) dσ, y2 (p) = y1 (p) = ∂n σ ∂n σ n being the outward normal to the equipotential surface σ . Once the boundary conditions were assigned, the equation (4.60) then made it possible to determine V(p). In order to illustrate his original method and evaluate the functions y1 (p) and y2 (p), Majorana again considered the particular case of a diatomic molecule with identical nuclei, by assuming that its equipotential surfaces were given by   1 −1 1 + . (4.62) p= r1 r2 By applying several theorems of differential geometry and using elliptic coordinates,14 Majorana did succeed in finding the following expressions for the desired functions: 12 See Sect. 2.12 in Ref. [17]. 13 See Sect. 3.12 in Ref. [17]. 14 See Sect. 3.12 in Ref. [17].

4.5 Conclusions

y1 (p) = 8π p2 ,

y2 (p) =

105

∂S , ∂p

(4.63)

where S is the volume enclosed by the equipotential surface labeled by p, which he evaluated as:     4p2 1 + 3t2 4p2 t2 4πp t2 2 2 − a − dt. (4.64) S= 3 a a (1 − t2 )2 (1 − t2 )2 (1 − t2 )3 t1  The upper limit of the integral was t2 = a/(p + p2 + a2 ), where 2a is the internuclear distance, and the lower limit is t1 = 0 for p ≥ a/2 and t1 = a2 − 2ap/a when p < a/2. Although the above integral can be evaluated exactly (and hence y2 (p) can be computed), Majorana did not continue his calculations, probably due to the resulting involved mathematical expressions. Nevertheless, the method he devised is extremely original and worthy of being fully exploited in computer-assisted molecular physics. 4.5 Conclusions Majorana was introduced by Fermi to the subject of the Thomas–Fermi statistical model of atoms as early as the end of 1927, during their first meeting at the Institute of Physics in Rome. As a consequence, in his personal notebooks [17], Majorana practically paralleled the first three papers by Fermi [208, 223, 210] on this subject, following his original intuition, and even just the reproduction of known findings resulted in quite interesting elaborations, as had been the case of the modified Fermi potential for heavy atoms. However, many of these studies converged into more intriguing generalizations of known results, both mathematical (such as, for example, the Frobenius expansion) and physical, with the elaboration of a “second approximation” for the atomic potential (which later found application in the Fermi–Amaldi correction [227]), as well as a general method aimed at describing molecules starting from the equipotential surfaces, and so on. Although, for mathematical physics, the method of order reduction (and parametric solution) envisaged by Majorana for a given general class of scale-invariant differential equations is of some relevance, his most important contribution was probably his solution (or, rather, methods of solutions) of the Thomas–Fermi equation. Its transformation into an Abel equation favored the direct (and simple) deduction of given mathematical properties, later recovered by leading mathematicians, but the analytic series solution that Majorana obtained in terms of only one quadrature allowed the rapid and accurate numerical determination of the effective electrostatic potential in atoms. This solution would have been most useful in a number of physical applications that for about a decade, had to be studied in

106

Thomas–Fermi model

the literature only by making recourse to Fermi’s famous table of approximate numerical values. Majorana’s solution of the Thomas–Fermi equation, however, remained unknown for more than 70 years, and a similar situation occurred concerning his results on the atomic polarizability of atoms in the framework of the Thomas– Fermi model, which were (independently) re-discovered only in the 1970s. It is impressive that such a large number of diverse contributions came from just one scholar over a very narrow period of time!

Part III Nuclear and statistical physics

5 Quasi-stationary nuclear states

At the end of 1920s, the activities of the Fermi group in Rome focused almost exclusively on atomic physics and related subjects, as did the other major research centers throughout Europe. After having obtained several significant results in atomic and molecular spectroscopy, around 1930 some people in the group realized that such a field could no longer offer any great prospects, and Fermi himself predicted that the interest would shift from the study of the external parts of the atom to its nucleus. However, no general consensus was reached initially inside the group, and the spectroscopic activities continued, along with the acquisition of theoretical knowledge and experimental technologies required by nuclear physics, until 1934, when the Fermi group discovered the radioactivity induced by neutrons and the important properties of slow neutrons [231]. Majorana also actively participated in scientific discussions within the Rome group [10, 11], but, as we have seen in Section 2.6, we are left with only one published paper of his on nuclear physics topics (that is, paper(s) N.8), dealing with the Heisenberg–Majorana exchange forces in nuclei, while nothing was published of his previous works on these matters. Remarkably, Majorana was the first in Rome to study nuclear physics; in 1929 (on July 6) he defended his degree thesis on “The quantum theory of radioactive nuclei,” and his studies on such topics continued for several years, independent of the main line of research carried out by the Fermi group. In this chapter we will focus on these (unpublished) studies [232], performed around 1929–30, devoted mainly to nuclear reactions induced by α particles. 5.1 Probing the atomic nucleus with α particles The first steps of nuclear physics were taken in the realm of radioactive phenomena, among which those involving α emission were readily recognized to induce a profound modification of the atomic nucleus by virtue of the large mass of the

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Quasi-stationary nuclear states

emitted particles. A great deal of information on nuclear structure and dynamics resulted, during the 1920s and 1930s, just from experiments on nuclei bombarded with α particles; later, there followed the discovery of the neutron anticipated by research on α irradiation on boron and beryllium. On the theoretical side, the first important achievement in nuclear physics was due to George Gamow, who provided the theory for the spontaneous α emission from a nucleus in the framework of quantum mechanics [233]. Intriguing results were obtained in 1929 from the artificial disintegration of nuclei by means of α particles. It was already known that, during bombardment of certain elements by these particles, protons were emitted as a result of the disintegration of the given nuclei. On the basis of Blackett’s photographs [234] of the disintegration of the nitrogen nucleus, it was assumed that in such a process the α particle is absorbed by the atomic nucleus, which is thereby transformed into a nucleus of the element of the following atomic number. According to this view, by applying energy-momentum conservation laws, simple relations between the energy of the incident α particle and that of the emitted proton should hold and, furthermore, the energy change from the original to the final nucleus should be a fixed amount. In fact, denoting by −Ep0 the energy level of a proton in a given nucleus and by −Eα0 that for an α particle, when such a particle with energy Eα is captured by the nucleus, the energy of the proton emitted in the disintegration is given by Ep = Eα + Eα0 − Ep0 (neglecting the small kinetic energy of the residual nucleus). Thus, in general, protons with definite energy are emitted during this process, their energy spectrum being discrete. However, Ernest Rutherford and Chadwick clearly showed in 1929 [235] that, at least in some cases (for example in the transition between aluminum and silicon), this is not true, and the change of energy was not always the same. This should lead, as a consequence, to the assumption that the energy levels of the protons and the α particles in the nucleus were not well defined, contrary to what emerged from the application of quantum mechanics to the nuclear structure. Such results were later confirmed by Chadwick, Constable, and Pollard [234], but, in the meantime, a novel idea for interpreting them emerged. Chadwick and Gamow [236, 237] indeed assumed that, in some cases, the disintegration of the nucleus might occur by the emission of a proton without α capture. In this case, the energy of the incident α particle is distributed between the emitted proton and the escaping α particle (neglecting the recoil of the nucleus), so that the protons emitted due to disintegration may have any energy between 0 and Eα − Ep0 , thus explaining the experimental results. As a consequence, if both types of disintegration (with and without α capture) take place, in the simplest case the energy spectrum of the emitted protons is composed of a continuous spectrum and – after its endpoint – a line (see Figure 5.1). The importance of the

5.2 Scattering of α particles

111

Np

0

Ea Ep Ea - Ep0

Ep0

0 0 Ea Ea - Ep

Figure 5.1. Energy spectrum of the proton emitted in α-induced nuclear disintegration as considered by Chadwick and Gamov (see the text).

identification of the continuous spectrum was readily recognized by Chadwick, Constable, and Pollard themselves: “It is most important to find the continuous spectrum of protons corresponding to disintegration without capture, for this gives immediately the level of the proton in the nucleus” [234]. The theoretical interpretation of the results from (α, p) reactions was mainly due to Chadwick and Gamow, but no dynamical theory for those processes, describing the superposition of a continuous spectrum and a discrete level, was ever published. This gap was filled – anonymously, without any publication – by Majorana, along with the achievement of further results on the scattering of α particles on nuclei. 5.2 Scattering of α particles on a radioactive nucleus In 1928, just after the appearance of the Gamow paper on the theory of nuclear α decay [233], Majorana studied the corresponding inverse process,1 calculating the probability of an α particle being captured by a nucleus that had undergone α decay, in such a way that a nucleus that had existed before decay might be reconstituted. He indeed succeeded in obtaining an expression for the absorption probability both within the framework of wave mechanics and, by considering the equilibrium between emitted and absorbed particles, within a thermodynamic approach. 5.2.1 Quantum-mechanical theory The motivation for the study (which formed part of Majorana’s master thesis) was declared at the beginning of his paper: 1 See Sect. 2.34 in Ref. [17]. In the following, any quotation refers solely to this reference.

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Quasi-stationary nuclear states

Let us consider the emission of an α particle by a radioactive nucleus and assume that such a particle is described by a quasi-stationary wave. As Gamow has shown, after some time this wave scatters at infinity. In other words, the particle spends some time near the nucleus but eventually ends up far from it. We now begin to study the features of such a quasi-stationary wave, and then address the inverse of the problem studied by Gamow. Namely, we want to determine the probability that an α particle, colliding with a nucleus that has just undergone an α radioactive transmutation, will be captured by the nucleus so as to reconstruct a nucleus of the element preceding the original one in the radioactive genealogy. This issue has somewhat been addressed by Kudar, although not deeply enough. It is directly related to our hypothesis according to which, under conditions rather different from the ones we are usually concerned with, a process can take place that reconstitutes the radioactive element.

The starting point was to assume, following Johann Kudar [238], spherical symmetry (and to neglect the recoil of the nucleus), so that the stationary states of the α particle are described by the relevant wavefunction ψ = χ /x (x is the radial variable), where χ satisfies the differential equation 2m d2 χ + 2 (E − U) χ = 0. 2 dx h¯

(5.1)

The potential energy U was assumed practically to vanish beyond a sufficiently large distance R, which was chosen to be of the order of the atomic dimensions. Let us now imagine that these exists a quasi-stationary state such that it is possible to construct a function u0 which vanishes for x > R, satisfies the constraint  R |u0 |2 dx = 1, (5.2) 0

and approximately obeys (for an approximately determined value of t, while being almost real) the differential [equation (5.1)] at the points where its value is large. This function u0 will be suited to represent the α particle at the initial time. It is possible to expand it in terms of the functions χ that are obtained by varying E within a limited range. Let us then set E = E0 + W.

(5.3)

The existence of such a quasi-stationary state is revealed by the fact that for x < R the functions χ [...] and their derivatives are small for small W.

As we have seen, the superposition of a discrete level with a continuum state would be considered (qualitatively) later by Chadwick and Gamow, but in a rather different context. Majorana instead solved the corresponding Schrödinger equation (5.1) for the functions χ , labeled by the continuum energy W and properly normalized.2 2 Majorana normalized the χ functions with respect to the variable γ = W/2π hv, v being the (average) ¯ W

speed of the emitted α particles.

5.2 Scattering of α particles

113

For small x values, the wavefunction representing the α particle at a generic time t was given by u = u0 eiE0 t/h¯ e−t/2T ,

(5.4)

where T denotes the half-life of the not strictly stationary state. By “neglecting what happens for values of x that are not too small, but still lower than R, and √ considering, instead, the case x > R,” Majorana obtained (v = 2E0 /m) ⎧ iE t/h −imv(x−R)/h¯ −t/2T (x−R)/(2vT) e e , ⎨ αe 0 ¯e u = (5.5) ⎩ 0, for vt − (x √ − R) > 0 and vt − (x − R) < 0, respectively. Here, α is a constant given by α = 1/ vT. Let us now assume that the nucleus has lost the α particle; this means that, initially, it is u0 = 0 near the nucleus. We now evaluate the probability that such a nucleus will re-absorb an α particle when bombarded with a parallel beam of particles. To characterize the beam we’ll have to give the intensity per unit area, the energy per particle, and the duration of the bombardment. The only particles with a high absorption probability are those having energy close to E0 , with an uncertainty of the order h/T. On the other hand, in order to make clear the interpretation of the results, the duration τ of the bombardment must be small compared to T. Then it follows that, from the uncertainty relations, the energy of the incident particles will be determined with an error greater than h/T. Thus, instead of fixing the intensity per unit area, it is more appropriate to give the intensity per unit area and unit energy close to E0 ; so, let N be the total number of particles incident on the nucleus during the entire duration of the bombardment, per unit area and unit energy.

Majorana then assumed that, initially, the incident flux is described by a plane wave confined between two parallel planes at distances ξ = d1 and ξ = d2 ≡ d1 + from the nucleus, and that d1 (and, consequently, d2 ) are arbitrarily large in such a way that  = d2 −d1 is small, “because the duration of the bombardment, which is of the order /v, must be negligible with respect to T.” For ξ < d1 or ξ > d2 it is u0 = 0; otherwise, by supposing d1 > R, he measured the inter-plane distance in terms of the wavelength λ of the emitted α particle,  = ρλ (ρ being an integer number), and expanded the relevant wavefunction in a Fourier series labeled by such integers ρ (neglecting negative ρ terms, representing – roughly – outgoing α particles). More precisely, Majorana split the wavefunction u0 of the incident particles into a term related to the principal energy E0 plus another term which he Fourier expanded in the energy E. Furthermore, he focused only on that part of the expansion, expressed in terms of the eigenfunctions associated with the nucleus central field, containing spherically symmetric eigenfunctions with energy eigenvalues very close to E0 , these being the only ones to be significantly different from zero near the nucleus.

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Within the given hypotheses,3 Majorana finally obtained the expression for the relevant wavefunction: ⎧ d ⎪ −imvd/h¯ − t−d/v ⎪ e 2T , for t > , ⎪ ⎨ qαe v (5.6) u = ⎪ ⎪ d ⎪ ⎩ 0, for t < , v √ where d = d1 − R and q = h3/2 N 1/2 /i m v1/2 4π . The meaning of these formulae is very clear: the α particle beam, which by assumption does not last for a long time, reaches the nucleus at the time t = d/v, and there is a probability |qα|2 that a particle is captured (obviously, q2 α 2  1). The effect of the beam then ceases and, if a particle has been absorbed, it is re-emitted on the time scale predicted by the laws of radioactive phenomena. If we set n = |qα|2 , then [...] we get n =

2π 2 h¯ 3 N, m2 v2 T

(5.7)

which tells us that the absorption probabilities are completely independent of any hypothesis on the form of the potential near the nucleus, and that they depend on the half-life T.

Majorana then found that the cross section4 for the absorption of an α particle with an energy close to an (unstable) nuclear energy level went as the inverse of the square of the particle velocity. However, as is clear from the preceding discussion, this result is not limited to the nuclear scattering of α particles, but applies to any massive particle within the stated approximations. Indeed, the same formula for the cross section appeared several times (and in different contexts and formalisms) in the literature [239, 240], mainly based on partial wave analysis of potential scattering. It was also explicitly re-derived in 1933 by Chadwick [241] (again assuming spherical symmetry) for neutron–proton scattering, the predicted 1/v2 behavior playing a key role in later investigations on neutron-induced nuclear reactions [242]. Majorana’s original approach, nevertheless, produced quite a general result that – as he pointed out – did not depend on the particular assumption about the form of the scattering potential (compare, for example, with the many particular cases considered in Ref. [240]). 3 In his notes, Majorana remarked that the starting assumption was that an additional “quasi-stationary state

different from the one we are considering does not exist.” 4 The corresponding expression for the cross section could be written as

σ =

π h¯ 2 . m2 v2

(5.8)

5.2 Scattering of α particles

115

5.2.2 Thermodynamic approach This result probably spurred Majorana to re-obtain it by using a thermodynamic approach, which was later adopted (in different forms) by others [242] in studying neutron-induced nuclear reactions. The common root for these studies was a series of earlier papers by Owen W. Richardson [243], who showed that electron absorption and emission could be treated as reversible phenomena subject to the laws of thermodynamics, his method being applicable to any kind of particle. Majorana was probably introduced to the subject by Fermi [242]. Let us consider one of our radioactive nuclei in a bath of α particles in thermal motion. To the degree of approximation we have treated the problem so far, we can consider the nucleus to be at rest. Due to the assumed spherical symmetry of the system, a particle in contact with the nucleus is in a quantum state with a simple statistical weight. Such a state, of energy E0 , is not strictly stationary, but has a finite half-life; this should be considered, as in all similar cases, as a second-order effect.

Denoting by D the number of α particles in the bath for unit volume and unit energy (close to E0 ), as the volume in the momentum space occupied by D dE particles is given by  4π p2 dp = 4π m2 2E0 /m dE = 4π m2 v dE , (5.9) the corresponding number of quantum states is given by m2 v dE . 2π 2 h¯ 3

(5.10)

Therefore, according to Majorana, we have 2π 2 h¯ 3 (5.11) m2 v particles in every quantum state with energy close to E0 . This is also the mean number of particles inside the nucleus, provided that [the expression (5.11)] is much smaller than 1, so that we can neglect the interactions between the particles. Since the half-life of the particles in the nucleus is T, then D

2π 2 h¯ 3 D (5.12) m2 vT particles will be emitted per unit time and, in order to maintain the equilibrium, the same number of particles will be absorbed. n =

Finally, Majorana related the quantity D to the number N (introduced earlier) of particles per unit area, unit energy, and unit time through the relation N = Dv, so that, by substitution into Eq. (5.12), Eq. (5.7) was recovered in just a few mathematical passages.

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Quasi-stationary nuclear states

As stated above, the present approach was quite general as well as being worthy of application to different situations, provided that – according to Majorana – interactions between the particles were small enough and that spherical symmetry approximately held. Later developments by Fermi and others [242], mainly concerning neutron radiative capture and deuteron photodissociation, existed outside this framework, but similar reasonings applied.

5.3 Transition probabilities of quasi-stationary states The quantum-mechanical theory of quasi-stationary states, as described in Section 5.2, was further considered – and generalized – by Majorana, who produced a consistent dynamical theory5 describing the interaction between the discrete level and the continuum states, later to be applied to the (α, p) reactions discussed by Gamow. 5.3.1 Transitions from a discrete into a continuum state As before, the focus was on a general physical system for which a discrete state ψ0 exists with energy E0 together with a continuum one ψW with energy E0 + W (the state ψW was normalized with respect to dW). The perturbation mixing the discrete and the continuum states was denoted by Majorana as  (5.13) IW = ψ¯ 0 Hp ψW dτ, where Hp is the generic perturbation potential and dτ is the volume element. The main goal is, then, to obtain the perturbed eigenfunctions ψW of the total Hamiltonian H: HψW = (E + W)ψW , with



(5.14)

I¯W ψW dW,

(5.15)

HψW = (E0 + W)ψW + IW ψ0 .

(5.16)

Hψ0 = E0 ψ0 +

The analytic solutions for the eigenfunctions were found to be as follows:    ψW  1   ¯ ψ0 − IW   ψW = dW + aψW , NW W −W 5 See Sect. 4.28 in Ref. [17]. In the following, any quotation refers solely to this reference.

(5.17)

5.3 Transition probabilities

117

 where NW = |a|2 + |b|2 is a normalization factor written in terms of the parameters6     −1 2 dW a = IW W + |IW |  , (5.18) W −W (5.19) b = πIW . The discrete state ψ0 was thus expanded in terms of the perturbed eigenfunctions ψW as  1  ψ0 = ψ dW. (5.20) NW W In order to get explicit results, Majorana assumed that terms higher than the second in IW could be neglected: Since the interesting values of W, that is, the values entering [Eq. (5.20)] in a relevant way, 2 , we can treat the quantities in the above formulae are of the same order as IW  dW  k (5.21) IW  I, |IW |2  W −W as constants.

In a sense, according also to Majorana, the meaning of the first of the approximations in Eq. (5.21) was to replace IW by its value I averagedover W. Furthermore,

by setting W = ε − k, he obtained a = ε/I, b = πI, N = the perturbed eigenfunctions are given by:   1 ψW   ψW   dW  + ψ0 − I  W −W ε2 /|I|2 + π 2 |I|2

ε2 /|I|2 + π 2 |I|2 , and

ε I

 ψW .

(5.22)

For the sake of definiteness, it was further assumed that, at time t = 0, the state of the system is ψ0 , and the time-dependent factor of ψW was taken to be e−iEt/h¯ = e−i(E0 −k)t/h¯ e−iεt/h¯ , so that the time evolution of the state of the system is given by     −iεt/h¯  ψW −i(E0 −k)t/h¯ −t/2T −t/2T ψ0 + I −e e dε . e ψ =e ε + iπ|I|2

(5.23)

(5.24)

Here, 2π 2 1 = (5.25) |I| T h¯ gives the transition probability per unit time between the initial (discrete) state ψ0 and the states ψW . 6 Here and in the following, the integrals were assumed to be evaluated as principal values.

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Quasi-stationary nuclear states

5.3.2 Transitions into two continuous spectra Let us now assume that for the unperturbed system there exist a discrete state ψ0 of energy E0 and two continuum spectra ψW and φW with energy E0 + W. Consider then a perturbation connecting the state ψ0 with both sets of states ψW and φW :   IW = ψ 0 Hp ψW dτ, LW = ψ 0 Hp φW dτ. (5.26)

In this case, by following similar steps to before, and within the same approximations, Majorana arrived at the following generalization of his Eq. (5.24): ψ = e−i(E0 −k)t/h¯ e−t/2T ψ0    ψW e−iEt/h¯ +I 1 − ei(W+k)t/h¯ e−t/2T dW 2 2 W + k + iπ(|I| + |L| )    φW e−iEt/h¯ 1 − ei(W+k)t/h¯ e−t/2T dW, +L 2 2 W + k + iπ(|I| + |L| ) with

 |IW  |2

dW  + W − W

 |LW  |2

dW  = k. W − W

(5.27)

(5.28)

The total transition probability is now given by  2π  2 1 = |I| + |L|2 . T h¯

(5.29)

The probability per unit time for the transition between the state ψ0 and the states ψW is then 2π |I|2 /h, ¯ while that for the transition between the state ψ0 and the states φW is 2π |L|2 /h, ¯ as expected.

5.3.3 Transitions from a continuum state We now consider another problem. Let us assume that the system is initially in the continuum state ψW and let us calculate the relative probability for the system to be at time t in the state ψ0 or in the states φW  or ψW  , with W  different from W. We may use the usual interpretation, that is, the probability for the system to be in the arbitrary state Y is unity if 2     Y ψ dτ  = 1,   so that |ψ|2 is the probability density in the configuration space τ . However, we prefer to use the concept of “number of systems” in a considered state rather than that of the relative probability for the system to be in that state. Now, although the continuum state ψW is not strictly stationary and represents an infinite number of systems, only a finite number of them has an energy that differs from E0 + W by a finite quantity. Thus we can expect that only transitions to states next to ψW and φW will increase in number with time (and we can presume this increase to be linear).

5.4 Nuclear disintegration by α particles

119

In this case, although the generic, t-dependent wavefunction describing the system could be expanded in terms of the unperturbed states ψ0 , ψW , φW (with the condition that ψ = ψW at time t = 0), Majorana preferred to express it (for t > 0) as the sum of two particular solutions, ψ = ψ1 + ψ2 , such that the state ψ1 substantially described the transition for sufficiently large values of time, while ψ2 was one of the discrete states of the form given in Eq. (5.27): I ψ1 = e−iEt/h¯ ψW + e−iEt/h¯ ψ0  + iπQ2   I IψW  + LφW  −iEt/h¯

i(E−E )t/h¯ e − 1 − e dE ,  + iπQ2  −  (5.30)



ψ2 =

I ei(E0 −k)t/h¯ e−t/2T ψ0  + iπQ2    IψW  + LφW  −iE t/h¯

 + e 1 − ei(E −E0 +k)t/h¯ e−t/2T dE .   + iπQ2

Here E = E0 + W, E =E0 + W  (and, consequently,  = E − E0 + k,   = E − E0 + k), while Q = |I|2 + |L|2 , and the total transition probability is given by 1/T = (2π /h¯ ) Q2 . The conclusion that Majorana drew was the following: The number of transitions per unit time from the state ψW into the states ψW  and φW  with energy close to E depends on the resonance denominator 1/(  − ) in the expression for ψ1 for sufficiently large values of the time. Denoting by A the number of transitions per unit time to states ψW  (W  = W) and by B the same number for the states φW  , we find A=

2π 2 |I|2 , |I| 2 h¯  + π 2 Q4

B=

|I|2 2π . |L|2 2 h¯  + π 2 Q4

(5.31)

5.4 Nuclear disintegration by α particles The general calculations reported in Section 5.3 were instrumental in studying an even more general case, suitable for the theoretical description of the artificial disintegration of nuclei by means of α particles, with and without α absorption.

5.4.1 Statement of the problem Majorana approached the problem by considering the simplest case with an unstable state (described by ψ0 ) of the system formed by the nucleus plus an α particle, which spontaneously decayed with the emission of an α particle or a proton.

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Quasi-stationary nuclear states

As in the other cases considered, he assumed that such a proton or α particle, coming from the disintegration of the nucleus described by ψ0 , is emitted as an s-wave and that the daughter nucleus was always in its ground state. The initial system formed by the parent nucleus plus the incoming α particle (in a “hyperbolic s-orbit”) is described by the states ψW , while the final one formed by the daughter nucleus and the free proton (in an s-orbit) is described by the states φW , both with energy E0 +W. The state ψ0 is coupled to both ψW and φW states by the perturbation potential Hp , as in Eqs. (5.26). Assuming ψW to be normalized with respect to dW (and neglecting the motion of the nucleus), it is simple to see that it represents a converging or diverging flux (number of particles per unit time) of α particles equal to 1/2π h¯ and, in the same way, φW represents an ingoing or outgoing flux of protons equal to 1/2π h. ¯

Majorana, however, also introduced some linear combinations of the states ψW and φW : ψW = u1W + u2W , where 1 i u1W = ψW − 2 2π u2W

1 i = ψW + 2 2π

v1W

1 i = φW − 2 2π

1 i v2W = φW + 2 2π

φW = v1W + v2W ,  

 

(5.32)

IW  ψW  dW  , IW (W  − W) IW  ψW  dW  , IW (W  − W)

(5.33)

φW  LW  dW  ,  LW (W − W) LW  φW  dW  .  LW (W − W)

(5.34)

These novel states have a definite physical meaning; indeed, the non-stationary states u1W and u2W [defined in Eqs. (5.33)] represent at large distances only outgoing or ingoing flux of α particles, respectively, whose intensity is again 1/2π h. ¯ Analogously, v1W and v2W [defined in Eqs. (5.34)] represent an outgoing or ingoing flux of protons.

The task here was to investigate the scattering of (a unitary flux per unit area of) α particles interacting with the parent nucleus and to determine how many of them were necessary to achieve disintegration of the nucleus. To this purpose, Majorana considered a stationary state, representing the incident plane wave plus a diverging spherical wave of α particles and a diverging spherical wave of protons, as obtained by a sum of particular solutions.

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121

The particular solutions corresponding to the parent nucleus plus α particles with azimuthal quantum number different from zero represent the usual scattering processes, which have the well-known form given by the theory of scattering from a Coulomb field. However, the considered state must be composed also of particular solutions representing incident α particles with  = 0 as well as of a diverging wave of α particles with  = 0.

Also, by virtue of the coupling between ψ0 and φW, the stationary state should be composed of the ψ0 state as well as of a diverging wave of protons. The main problem was then to search for an expression for such a particular solution, which Majorana called ZW .

5.4.2 The appropriate wavefunction The starting point was the introduction (for any value of W) of the two stationary 1 2 and ZW , defined by states ZW 1 1 HZW = (E0 + W)ZW , 2 2 = (E0 + W)ZW , HZW

chosen to be orthogonal and normalized in such a way that   IW LW 1 IW  ψW  1 ZW =  ψ0 + εW 2 ψW + εW 2 φW − dW  NW W − W QW QW   LW  φW   − dW , W − W LW ψW IW φW 2 = − , ZW QW QW where

 εW = W +

dW  |IW  |  + W −W 2

QW =  = NW

 

 |LW  |2

dW  = W + kW , W − W

(5.35)

(5.36)

(5.37)

|IW |2 + |LW |2 ,

(5.38)

2 εW + π Q2W . Q2W

(5.39)

The most general stationary state ZW corresponding to the energy E0 + W was 1 2 , ZW in Eqs. (5.36): ZW = then written as a linear combination of the states ZW 1 2 λZW + μZW . More generally, Majorana wrote such a state as ZW = c ψ0 + c1 u1W + c2 u2W + C1 v1W + C2 v2W ,

(5.40)

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Quasi-stationary nuclear states

with the coefficients given by λ  , NW IW =λ  NW IW =λ  NW LW =λ  NW LW =λ  NW

c= c1 c2 C1 C2



W Q2  W W Q2W  W Q2  W W Q2W



LW QW  LW + iπ + μ QW  IW − iπ − μ QW  IW + iπ − μ QW − iπ

+μ

, ,

(5.41)

, .

The state v2W – as we have seen already – describes an ingoing flux of protons that, in the particular problem considered here, is not present, so that Majorana set C2 = 0. From this condition, and from the mathematical relations coming from the orthonormality conditions of the states involved, he determined the coefficients λ, μ, and thus obtained the relevant expressions for the remaining coefficients c, c1 , c2 , C1 , except for a proportionality factor: c=

IW  NW QW

,

c1 =

1  NW QW

 1  εW + iπQ2W , c2 =  NW QW

 ! εW − iπ |IW |2 − |LW |2 , IW L¯ W C1 = −2π  . NW QW

(5.42)

5.4.3 Cross section In order to obtain predictions about the cross section of the α-induced nuclear disintegration, the coefficients in Eqs. (5.42) should be related to directly observable quantities. This was achieved by Majorana according to the following reasoning. The coefficient c2 was determined from the condition that the incoming flux of α particles results from the incident plane wave, this flux being equal to |c2 |2 /2π h¯ . On the other hand, the number of α particles (with  = 0) impinging per unit time on the nucleus can be set equal to the flux through a circular cross section, normal to the propagation direction of the wave, with radius λ/2π, λ being the wavelength of the α particles. “Since the incident wave represents a unit flux per unit area,” the following relations are obtained:  2 λ π h¯ 2 |c2 |2 =π = 2 2, (5.43) 2π h¯ 2π M v

5.4 Nuclear disintegration by α particles

123

where M and v are the mass and velocity of the particles, respectively. The expressions for the (moduli of the) other coefficients were deduced from Eqs. (5.42), once a “translation” from the normalization adopted in Eqs. (5.41) and that coming from Eqs. (5.43) was performed: |c1 |2 =

h¯ λ2  2 + π 2 (|IW |2 − |LW |2 )2 , 2  2 + π 2 Q4W

h¯ λ2 4π 2 |IW |2 |LW |2 , 2  2 + π 2 Q4W h¯ λ2 , λ = 2π h/Mv . |c2 |2 = ¯ 2

|C1 |2 =

(5.44)

We are interested in the moduli of c1 and C1 , since we only study the frequency of disintegration processes, while we disregard scattering anomalies that also depend on the phase of c1 .

The cross section7 S(ε) for the nuclear disintegration process considered here is given by the outgoing proton flux, |C1 |2 /2π h¯ , and, within the assumption (introduced in Section 5.3) according to which the terms IW = I and LW = L can be considered as constants, may be written as follows: S(ε) =

λ2 λ2 4π 2 |I|2 |L|2 = p(ε) . 4π ε2 + π 2 Q4 4π

(5.45)

The quantity p(ε), introduced here by Majorana, corresponded to the ratio |C1 |2 /|c2 |2 , whose meaning is straightforward: Since λ2 /4π gives the cross section for α particles with vanishing azimuthal quantum number, p() is the probability that one of such particles will induce a disintegration.

Majorana pointed out that this probability reached a maximum for “the most favorable value for the energy,” that is ε = 0, so he focused on the numerical values of p(0), which he wrote as p(0) =

4 4K , = (K + 1)2 (1 + K)(1 + 1/K)

(5.46)

where K = |I|2 /|L|2 . He noted that p(0) was symmetric under the exchange K → 1/K, and took the numerical values as in Table 5.1. In particular, it is interesting to note that p(0) can take the value 1 for k = 1. In other words, if the state ψ0 has the same probability to emit a proton or an α particle and the energy of the incident α particles takes its most favorable value, then all the incident particles with vanishing azimuthal quantum number will induce disintegration. 7 For simplicity, the subscript W was avoided in the notation for the energy, ε.

124

Quasi-stationary nuclear states Table 5.1. Maximum disintegration probability p(0) k

p(0)

1 2 or 1/2 3 or 1/3 6 or 1/6 10 or 1/10 100 or 1/100

1 0.889 0.750 0.490 0.331 0.039

Majorana’s analysis, however, was not limited to such “generic” predictions, though fully physically meaningful; he also calculated explicitly the directly observable integrated cross section: “it is often impossible to measure the cross section S() for particles with definite energy E0 + k + . In these cases, only the quantity S() d is measurable.” The integral to be evaluated is, simply,  2 2 4π 2 |I|2 |L|2 2 |I| |L| = 4π Q2 , (5.47) p() d = Q2 Q4 but Majorana rewrote it in order to introduce the disintegration probability: 2π 2 1 = Q. T h¯ The final result was that  λ2 π h¯ h¯ λ2 p(0) S() d = = p(0). 2T 4 4π 2T

(5.48)

(5.49)

5.5 Conclusions As a rule, in the late 1920s little attention was paid to nuclear physics experiments performed via the artificial disintegration of nuclei by bombardment with α particles, probably because of the lack of their comprehensive theoretical interpretation in the framework of quantum mechanics. In this respect, the original results from Rutherford and Chadwick in 1929 [235], on the peculiarities of proton emission in the artificial disintegration of some nuclei, were puzzling when it was assumed that the atomic nucleus captured the incident α particle. Only later, in 1930–1, with the fundamental contribution from Gamow, was it recognized that such disintegrations could occur even without the capture of α particles [236, 237], with the assumption that the protons and α particles contained in the nucleus belonged to definite energy levels. Thus, in general, the energy spectrum of the ejected protons turned out to be

5.5 Conclusions

125

the superposition of a continuous spectrum and a line, as observed experimentally. The next step, then, should have been the elaboration of a complete theory of such processes in the general framework of quantum mechanics, but this was not considered in the known literature. We have seen, however, that this issue was effectively studied by Majorana, although he did not publish anything about it. The first problem he tackled was that of the absorption of an α particle by a nucleus that had previously undergone α decay, a process originally considered by Kudar [238] (following Gamow). With very simple and clear reasoning, Majorana succeeded in deducing the absorption probability for such a process within quantum mechanics, along the lines developed earlier by Gamow [233]. He also devised a – mathematically more direct – thermodynamic approach to the problem (providing the same result), by studying the equilibrium between the different α particles emitted and absorbed by the given nucleus. Although such an approach was quite familiar within the Fermi group [242], it was only adopted by other scientists in the 1930s when studying different nuclear problems (mainly neutron-induced reactions). The main result obtained by Majorana, which he proved not to depend on the particular form of the scattering potential, was the cross section for the absorption of an α particle with energy close to an unstable nuclear energy level, that he found went as the inverse square of the particle velocity. This result (rather, the method used by Majorana for obtaining it) was quite general (within the adopted approximations) and applicable to the nuclear scattering of any massive particle; indeed, it was repeatedly re-derived in different contexts, mainly starting from a partial wave analysis of potential scattering, including a key derivation in 1933 by Chadwick for neutron–proton scattering [241]. The basic premise in Majorana’s studies of α-induced nuclear reactions was to follow the dynamics of a quantum state resulting from the superposition of a discrete state with a continuous one. The general quantum-mechanical theory that he elaborated, as described in Section 5.3, allowed him to obtain the transition probability in three different cases: namely, for the transition from a discrete level to one or two continuum states, as well as for transitions from a continuum state coupled with both a discrete level and another continuum state. These general calculations were probably an introduction to the far more interesting physical case regarding the artificial disintegration of nuclei induced by α particles, both with and without α absorption. The problem was approached by assuming the physical system to be formed by a nucleus plus an α particle, this system being unstable and thus likely to decay spontaneously with the emission of an α particle or a proton. As a result, Majorana succeeded in obtaining the explicit expression for the cross section of the nuclear process (and, in particular, for the integrated cross section, which was the direct measurable quantity of interest in the experiments), in terms of the disintegration probability. An interesting prediction

126

Quasi-stationary nuclear states

was that, provided the incident α particles had a particular energy value, if the unstable state had an equal probability of emitting a proton or an α particle, then all the incident particles induce disintegration. Majorana’s interest in true physical applications (rather than just mathematical-ones) is, thus, evident in these notes. The study of quasi-stationary states,8 which Majorana brought to a very satisfactory level, was later resumed by Fano, as discussed in Section 2.3. In the fundamental paper in Ref. [43], Fano investigated the stationary states with configuration mixing under conditions of autoionization, when interpreting the strange looking shapes of spectral absorption lines of atoms in the continuum. Intriguingly enough, Fano’s analysis was based on an expression of the perturbed eigenfunction of the system considered that was written in a manner surprisingly similar to that introduced by Majorana, although Fano then proceeded in his calculations in a completely different way. This was, possibly, another illuminating example of influence-at-a-distance on a physicist, who elaborated (independently) his theory just a short time after moving to Rome (in 1934–5) to work with Fermi.

8 The superposition of discrete and continuum states was also introduced by Oscar K. Rice – at about the same

time as Majorana – in a completely different context, that is the phenomenon of predissociation in molecules [244, 245], but Majorana was probably not aware of this.

6 Theory of ferromagnetism

At the October 1930 Solvay conference in Paris, Pauli made a comprehensive review of magnetism, including the intriguing results obtained by Heisenberg [246], Bloch [247, 248], and Slater [249] on the phenomenon of ferromagnetism. That meeting was also attended by Fermi, who, very likely, spread the novel results obtained by those scholars within his group in Rome, and Majorana, used as he was to spending time studying papers published by Heisenberg (and a few other authors), probably became (or was already) aware of his theory of ferromagnetism, in which exchange forces again play a key role in the understanding of the phenomenon. Fermi and his close collaborators never worked or published anything on this subject, but several people visiting the Institute of Physics in Rome in the early 1930s did, including, primarily, Bethe [250], who introduced the famous Bethe ansatz for the case of two interacting spin waves. Other minor (to some extent) contributions came from Inglis [251, 252, 253], who visited Rome in 1932, and from Majorana’s friend Gentile [254], who provided a thorough justification of Bloch’s results on elementary groups (Uebergangsgebiet) of aligned spins inside a ferromagnet [255]. As a matter of fact, a theory of ferromagnetism analogous to Heisenberg’s can be found among Majorana’s personal research notes:1 it was probably developed by Majorana at the end of the 1920s or early in the 1930s, but was never published by the author, so it remained unknown until recently [256]. As in Heisenberg’s theory, the framework of Majorana’s theory is that of the statistical Ising model [257], where the forces responsible for ferromagnetism are derived from the quantum-mechanical exchange interactions. However, Majorana’s theory differs substantially from Heisenberg’s in the methods employed and in the results obtained, though it describes successfully the main features of ferromagnetism. The key equation for the spontaneous mean magnetization and the expression for 1 See Sect. 5.3 in Ref. [18].

128

Theory of ferromagnetism

the Curie temperature deduced by Majorana are different from those deduced in the Heisenberg theory (and in the original phenomenological Weiss theory), and the novel theory presents also a peculiar prediction about the practical realization of ferromagnetism which avoids arbitrary assumptions based purely on experimental observations. It is, thus, of particular interest to follow Majorana’s argument, and in this chapter we will unveil the entire unknown theory with some original applications, adding further elaborations which shed some light on the relevance of Majorana’s findings. 6.1 Towards a statistical theory of ferromagnetism After the pioneering work by Paul Langevin [258], who successfully provided a theory of diamagnetism within the framework of the Lorentz theory of the electron, the paramagnetism of given materials was rightly related to the fact that atoms or molecules can have permanent dipole moments arising from the current loops generated by orbiting electrons (and proportional to the total angular momentum), thus acting as tiny magnetic shells. However, the correct interpretation of ferromagnetic phenomena did not follow immediately, and their understanding on an atomic basis was achieved only after the advent of quantum mechanics. The striking property to be explained was that ferromagnetic substances (like iron, cobalt, nickel, etc.) develop a spontaneous magnetization only when cooled below a certain temperature Tc , called the Curie point. Unlike an ordinary paramagnet, above this critical value the temperature variation of the magnetic susceptibility χ of a ferromagnet was found empirically to follow the Curie–Weiss law, χ = C/ (T − Tc ) (where C is a constant), whereas for a paramagnet the Curie law χ = C/T holds. 6.1.1 Molecular fields A first important step towards the understanding of the phenomenon was performed by Pierre Weiss in 1907 [259], when he postulated that atomic magnets tend to be brought into parallel orientation not only by an applied, external magnetic field He , but also by an inner field Hi – called by Weiss the “molecular field” – which is proportional to the magnetization M of the material: Hi = αM. By introducing such an idea into the Langevin theory of magnetism of classical dipoles, Weiss was able to account for most of the phenomenology of ferromagnets [260]. If a dipole is brought from outside and placed in the molecular field, it will be aligned in the direction of this field (i.e. in the direction of the magnetization itself), giving rise to a spontaneous ordering of the elementary magnets. Such an ordering effect competes with random thermal effects, with energies of the order of kT, trying to

6.1 Towards a statistical theory of ferromagnetism

129

flip the dipole away from the ordered state. In addition, Weiss showed that (in the simple case that dipoles can align only in two opposite directions), even in the absence of an external magnetic field (He = 0), a non-vanishing magnetization may arise, satisfying the following equation: M = M0 tanh

μαM , kT

(6.1)

where μ is the magnetic moment of the N atoms per unit volume of the body and M0 = Nμ.2 In fact, apart from the trivial solution M = 0, the implicit equation (6.1) admits also another non-zero solution for the magnetization M, provided that T < Tc = Nμ2 α/k. The magnetic susceptibility was calculated as well, leading to the experimentally observed Curie–Weiss law (with C = Nμ2 /k), thus finally showing the good agreement between the phenomenological Weiss theory and the known properties of ferromagnetism. Despite this success, the question concerning the origin of the local field remained unanswered. On one hand, it could not arise from magnetic interactions of the magnetic moments of atoms or molecules, these being extremely small, while, on the other hand, classical electrostatic forces leading to interactions of the right order of magnitude do not give a linear proportionality between the Weiss molecular field Hi and the magnetization M [261].

6.1.2 Heisenberg theory The solution of this puzzle came when Heisenberg [246] showed that the forces leading to ferromagnetism are purely quantum mechanical in nature, being due to electron exchange. The key idea was to show how strong non-magnetic forces between electrons that favor spin alignment (and, thus, the alignment of the elementary magnets) arise from quantum-mechanical exchange interactions, in the same way as they produce level splittings between electrons in singlet (para-) and triplet (ortho-) states of two-electron atoms. Exchange forces are electrostatic in nature, but, because of the constraints imposed by the Pauli principle, they are formally equivalent to a tremendously large coupling between spins, as seemingly required by the Weiss phenomenological theory. With a suitable assumption (a positive exchange integral), Heisenberg showed that the state in which spins are aligned in the same direction is energetically favored, helping in the ordering of the spins and thus proving his model to be well fitted to describe ferromagnetism. 2 Actually, in the original formulation, the Langevin function cosh x − 1/x appeared in Eq. (6.1) instead of the

function tanh x, which was considered only later by Heisenberg. However, as it is easily recognizable (as done by Heisenberg himself), the physical interpretation of the phenomena does not change, and no alteration in reasoning is required.

130

Theory of ferromagnetism

This result was, actually, achieved by following the statistical model constructed by Ernst Ising a few years before [257]. In the Ising model of a ferromagnetic body, N elementary magnets of moment μ are assumed to be arranged in a regular lattice, each of them having only two possible orientations, as in the Weiss theory. The magnets experience only short-range interactions between each other, the interaction energy (assumed to be a constant) being non-vanishing and negative (positive) only for each pair of neighboring magnets of the same (opposite) direction. This interaction tends, then, to make neighboring spins the same (an additional energy ±μHe is present for an applied magnetic field He ), so that in principle this model could be suitable in describing ferromagnetic phenomena. The problem was just to find an analytic expression for the partition function  E e− kT , (6.2) Z= config.

where E is the total energy of the system (and the sum is over any configuration of the system), from which all the thermodynamic quantities of interest (including the mean magnetization) can be derived. This was effectively achieved by Ising for magnets arranged into a linear chain, but he “succeeded in showing that the assumption of directed, sufficiently great forces between two neighboring atoms of a chain is not sufficient to explain ferromagnetism” [246], since non-zero magnetization is not predicted at any temperature. This occurs because the ordered state is unstable against random thermal fluctuations tending to destroy spin alignment, and Ising showed (incorrectly) that such negative results apply also in three dimensions. This problem was not properly solved by Heisenberg in 1928, although his different (non-constant) form of the interaction energy depends not only on the arrangement of the elementary magnets, but also on the speed with which they exchange their places. It was correctly addressed only in 1936 by Peierls [262] who, contrary to the commonly expressed opinion “that the solution of the threedimensional problem could be reduced to that of the linear model and would lead to similar results,” finally showed that the above-mentioned statistical model of ferromagnetism gives non-zero spontaneous magnetization at finite temperature in two (or more) dimensions. Returning to Heisenberg’s theory, his derivation of Weiss’s formula (6.1) proceeded through standard calculations on the partition function (6.2). The first step was to find the distribution of the energy levels of 2n localized electrons (one valence electron per atom) to first order in the coupling by evaluating the exchange integral between each electron and its nearest neighbors. Given the exponential decrease of the exchange integrals, the couplings among more distant electrons are, in fact, negligible. Formal group-theoretic methods were used to evaluate the

6.2 Majorana statistical model

131

energy levels and, in order to calculate the partition function, their fluctuations were neglected. This last assumption effectively led to the replacement of the energy levels of the spin multiplets of the microscopic magnets forming a given total spin S by the average energy corresponding to S. Subsequently, making a “somewhat arbitrary” assumption, Heisenberg introduced a Gaussian distribution of energies for S around its mean energy E0 with variance E02 ; as he himself wrote, “it is of course only to be expected that the temporary theory sketched here offers but a qualitative scheme into which ferromagnetic phenomena will perhaps be later incorporated” [246]. 6.1.3 Later refinements Subsequent improvements, both in the mathematical formalism and the comprehension of several physical effects, resulted between the end of the 1920s and the beginning of the 1930s, mainly due to Bloch, Slater, and Bethe. First, Bloch and Slater concentrated (independently) on understanding the physical basis of the Heisenberg model. While considering a free electron gas (instead of bound electrons contemplated by Heisenberg in his Heitler–London-like [48] approach) in order to study the role of conduction electrons in ferromagnets (to obtain its thermal properties), Bloch avoided Heisenberg’s Gaussian assumption and found a contribution “when two electrons are close together, in the neighborhood of the same atom” [247]. This was an important effect, since the exchange integral can become negative, thus decreasing the possibility of ferromagnetism, but it cannot be taken into account in the Heitler–London-like approach used by Heisenberg. Furthermore, Bloch replaced the difficult group-theoretic method employed by Heisenberg with the simpler determinantal method introduced shortly before by Slater in his theory of complex spectra [249]. The great advantage of writing a many-electron wavefunction in the form of a Slater determinant was that it took into account from the very beginning the correct antisymmetry properties imposed by the Pauli principle. This guaranteed that no two electrons in the system could be found in identical quantum states. Such a novel approach directly led Bloch to discover the so-called “spin waves” [248], i.e. the states of the system corresponding to a single or a few spin-flips in the fully aligned ground state, in the region of low temperatures, where both the Weiss phenomenological theory and the Heisenberg calculations could not be applied. 6.2 Majorana statistical model By following the same general scheme adopted by Heisenberg, i.e. the statistical Ising model, Majorana developed his own theory of ferromagnetism by taking into

132

Theory of ferromagnetism

account directly the constraints imposed by the Pauli principle, as also introduced by Bloch and Slater in their papers. Majorana focused on a set of n atoms, considered as magnetic dipoles (or, simply, spins), arranged in a given geometric array at locations q1 , q2 , . . . , qn . Among these n elements, he assumed that a number i of them have a spin parallel to a given direction (spin-up), while the remaining n − i elements are antiparallel to that direction (spin-down). Spin-up elements are characterized by the quantum numbers ↑ ↑ . . . ↑,

r 1 , r2 , . . . , ri

while those with spin-down are characterized by ↓↓ ... ↓ .

ri+1 , ri+2 , . . . , rn

The wavefunctions A(r1 . . . ri |ri+1 . . . rn ) describing such a system were assumed to have the form of a Slater determinant; in the following expression, the δ-functions (or, more appropriately, the Kronecker symbols) δ(sk ∓ 1) guarantee that spin-up/spin-down elements have s = ±1 spin (in some units), respectively:    ψ (q )δ(s − 1) . . . ψ (q )δ(s − 1)  1 r1 n n  r1 1   ...      . . . ψri (qn )δ(sn − 1)   ψri (q1 )δ(s1 − 1) A(r1 . . . ri |ri+1 . . . rn ) =   . (6.3)  ψri+1 (q1 )δ(s1 + 1) . . . ψri+1 (qn )δ(sn + 1)     ...     ψrn (q1 )δ(s1 + 1) . . . ψrn (qn )δ(sn + 1)  Since there are

 τ=

n i

 =

n! i!(n − i)!

ways to have i spin-up and n − i spin-down on a total of n elements (the order of r1 . . . ri or ri+1 . . . rn is not important), the wavefunctions describing the system are as follows: 1 1 , ri+2 , . . . , rn1 ), Ai (r11 , r21 , . . . , ri1 |ri+1

(6.4)

... τ Aτ (r1τ , . . . , r1τ |ri+1

, . . . , rnτ ).

Denoting by H the Hamiltonian operator describing the interaction acting on each element, the electrostatic interaction potential V0 is given by  (6.5) V0 = Hψ1 (q1 )ψ 1 (q1 )ψ2 (q2 )ψ 2 (q2 ) . . . ψn (qn )ψ n (qn ) dq1 . . . dqn .

6.2 Majorana statistical model

133

The exchange energy between states characterized by r and s quantum numbers (or, in Majorana’s language, the exchange energy between r and s orbits) is instead given by  e2 (6.6) ψr (q1 )ψ s (q1 )ψ r (q2 )ψs (q2 ) dq1 dq2 , Vrs = |q1 − q2 | with Vrs = Vsr . The energy eigenvalues are then Hmm = V0 −



Vrs +

r p of inner [energetic] levels, while any other level has a much greater energy. We deduce that the states of the system as a whole may be divided into two classes. The first one is composed of those configurations for which all the electrons belong to one of the inner states. However, the second one is formed by those configurations in which at least one electron belongs to a higher level not included in the n levels already mentioned. We will also assume that it is possible, with a sufficient degree of approximation, to neglect the interaction between the states of the two classes. In other words we will neglect the matrix elements of the energy corresponding to the coupling of different classes, so that we may consider the motion of the p particles in the n inner states, as if only these states exist. Then, our aim is to translate this problem into that of the motion of n − p particles in the same states, such new particles representing the holes, according to the Pauli principle. [321]

Majorana indeed succeeded (as did Heisenberg) in obtaining the appropriate expression for the energy of the system in terms of holes by adopting the standard formalism of quantum field theory. The Hamiltonian entering into the Schrödinger equation is split into two terms, H = H0 + V, the second highlighting “such a part depending on the mutual interaction of the electrons” and written as  V(P) = G(P, P ) ψ ∗ (P )ψ(P ) dτ, (10.19) where G(P , P) denotes the potential between two particles located in P and P and, according explicitly to Majorana, the field ψ “does not represent a single particle but, rather, a very large number of particles, in order to neglect the granular structure of matter.” By expanding ψ in terms of a set of orthogonal functions ϕi  (referring to the given ith particle), ψ = ai ϕi , the coefficients ai satisfy the following time evolution equation deduced by the Schrödinger equation:    i  0 ∗ a˙ i = − Hik ak + Oi,km a ak am , (10.20) h¯ k ,k,m where

 Hik0  Oi,km =

=

ϕi∗ H0 ϕk dτ,

G(P, P ) ϕi∗ (P) ϕ∗ (P ) ϕk (P) ϕm (P ) dτ dτ  .

10.2 Dynamical theory of electrons and holes

223

The resulting expression for the energy is given by W=



Hik0 a∗i ak +

i,k

1  Oi,km a∗i a∗ ak am . 2 i,,k,m

(10.21)

The quantization of the theory then proceeds in a standard way, but particularly interesting is the “justification” that Majorana gives for the adoption of anticommutators instead of commutators for fermions, for which he refers to the particular form of the Hamiltonian and to the corresponding equations of motion to be satisfied. Taking into account these equations, we will now give them a quantum meaning by setting i a˙ i = − (aW − Wa) , h¯

a˙ ∗i = −

 i ∗ a W − Wa∗ , h¯

(10.22)

the quantities a being now matrices. In order that the [equations (10.22)] be equivalent to [Eqs. (10.20)], we easily see that the quantities a should satisfy the exchange relations:

∗ ∗ ai , a∗k = δik , [ai , ak ] = 0, ai , ak = 0. This means quantizing [the theory] according to the classical Heisenberg rules since, indeed, the momenta conjugate to the variables a are classically the quantities a∗ multiplied by the factor ih. ¯ The Heisenberg exchange relations will lead us to consider particles obeying Bose statistics, while we are interested in the other case, namely that of particles obeying the Fermi statistics. As proved by Jordan and Wigner, to this end we have to change the signs in the Heisenberg relations: ! ! a∗i , a∗k = 0. ai , a∗k = δik , {ai , ak } = 0, This cannot be justified on general grounds, but only by the particular form of the Hamiltonian. In fact, we may verify that the equations of motion are best satisfied by these last exchange relations rather than by the Heisenberg ones. [321]

Quite intriguing is the “suitable solution,” which Majorana reports explicitly in his calculations, for the matrices ai appearing in Eqs. (10.22) and those in the extract. By setting         1 0 1 0 0 1 0 0  ∗ , Ik = , αk = , αk = , Ik = 0 1 0 −1 0 0 1 0 the matrices ai , a∗i are given by  × αi × Ii+1 × Ii+2 . . . , ai = I1 × I2 . . . Ii−1  × αi∗ × Ii+1 × Ii+2 . . . . a∗i = I1 × I2 . . . Ii−1

(10.23)

224

Photons and electrons

The normalization condition for ψ just expresses the total number of particles described by the theory:   αi∗ αi = n. (10.24) ψ ∗ ψ dτ = i

The introduction of hole operators bi is achieved by Majorana simply by setting bi = a∗i , b∗i = ai , and replacing – for example – the anticommutator a∗i ai + ai a∗i = 1 by a∗i ai + b∗i bi = 1, from which he deduces that the corresponding occupation numbers are a∗i ai = 1 and b∗i bi = 0, or a∗i ai = 0 and b∗i bi = 1. The energy W, now written as  1 (10.25) Hik bi b∗k + Oi,km bi b b∗k b∗m , 2 requires normal ordering, which Majorana introduces from the standard anticommutation relation, b∗i bk = δik − b∗k bi . The first term in the Hamiltonian leads simply   to Hii − H ik b∗i bk , while the four-particle operator is normal ordered through the expansion   bi b b∗k b∗m = bi δk − b∗k b b∗m = δk bi b∗m − bi b∗k b b∗m = . . . = b∗k b∗m bi b + δk bi b∗m + δim b b∗k − δik b b∗m − δm bi b∗k , leading to the non-trivial decomposition 1 1 Oi,km bi b b∗k b∗m = Oi,km b∗k b∗m bi b 2 2 1 1 + Oi,m bi b∗m + Oi,ki b b∗k + . . . . 2 2 10.3 Conclusions Despite his only published paper N.9 on quantum field theory, in its “symmetric” version with respect to particles and antiparticles, Majorana repeatedly studied different topics in quantum electrodynamics in his unpublished writings. Quite a large effort was devoted to developing a consistent (non-quantized) wave theory of photons by following an original idea by Oppenheimer [283] aimed at exploring the possibility of describing the electromagnetic field by means of the Dirac formalism. The advantage of such a formulation lies in considering the properties of the electromagnetic field as described by a wavefunction for the photon which is a directly observable quantity, a study which was well beyond the contemporary works of other authors. This peculiarity is, however, exceedingly more transparent in the mathematical treatment elaborated by Majorana (rather than by

10.3 Conclusions

225

Oppenheimer), who was able to realize (at the time) the basic phenomenological properties of photons upon which the theory has to be built. A first version of this, as envisaged by Oppenheimer, was plagued by the use of a non-explicit Lorentzinvariant formalism, but Majorana also developed a completely invariant theory (for a four-vector wavefunction) by starting from the known group properties. Also, a simple two-component theory for a wavefunction referring to the two different polarization states of the photon was considered, again focusing on directly observable properties. The Lorentz-invariant theory was cast in a form suitable for field quantization by following the standard Lagrangian formalism and imposing the usual commutation relations for the four fields introduced by Majorana. It is, thus, the first consistent theory alternative to quantum electrodynamics. The most striking quantum field calculations, however, concerned a dynamical theory of particles and antiparticles that Majorana developed by following an original paper by Heisenberg [126] and applying the formalism of field quantization to Dirac’s hole (rather than electron) quantities. Here, again, in addition to the relevant (and clear) mathematical results obtained, the physically pregnant position of the problem of describing the quantum dynamics of a system of particles and antiparticles is particularly worthy of mention. It concisely denotes a complete mastering of the quantum field theory formalism, which was recognizable only years later in the known literature.

Part VI Fundamental theories and other topics

11 A “path integral” approach to quantum mechanics

From January to March 1938 – as we have seen in Chapter 1 – Majorana delivered his only lectures on quantum mechanics at the University of Naples, where he obtained a position as a full professor of theoretical physics. He prepared a set of lecture notes [26] for his students (see Section 13.5), and, among the original manuscripts of these lectures, we find some additional spare papers that cannot be considered as notes for academic lectures, even for an advanced course such as that taught by Majorana. Instead, as suggested in Refs. [328, 329], they probably refer to a seminar or conference held at the University of Naples, whose main topic was likely the theoretical interpretation of the molecular bonding in the framework of quantum mechanics; the notes were prepared by Majorana for his own personal use. Scientific interest in this dissertation, however, focuses not on its main topic, nor on the calculations (which are practically absent from the manuscript), but rather on the interpretation given by Majorana to key concepts of the quantum theory, i.e. the concept of quantum state and the direct application of the quantum theory to the particular case of molecular bonding. The manuscript indeed discloses a peculiar cleverness on the part of Majorana in referring to this pivotal argument of quantum mechanics, and also reveals the concept was at least ten years ahead of its time. This link had been already noted earlier by Nicola Cabibbo [330], who saw in the Majorana manuscript a vague and approximate anticipation of the idea underlying the Feynman interpretation of quantum mechanics in terms of path integrals. A more analytic study has revealed, however, some intriguing surprises, upon which we will focus here. In addition to the historical interest of Majorana’s manuscript, we stress also the particularly powerful didactic method used by him on the given subject: his original presentation of the quantum-mechanical problems is exceedingly useful in teaching now those or related topics. For this reason, in the Appendix we will report the entire text (translated from the Italian) as prepared by Majorana.

230

A “path integral” approach to quantum mechanics

11.1 Dirac and Feynman’s mathematical approach According to a general postulate of quantum mechanics, the “state” of a certain physical system may be represented by a complex quantity ψ, considered as a (normalized) vector in a given Hilbert space of the corresponding physical system, where all the information on the system is contained. The time evolution of the state vector is ruled by the Schrödinger equation, which may be written in the general form ih¯

dψ = H ψ, dt

(11.1)

where H is the Hamiltonian operator of the considered system. The initial state at time t0 is specified by its choice among the possible eigenstates of a complete set of commuting operators (including, for example, the Hamiltonian), while the Hamiltonian H itself determines the state of the system at a subsequent time t through Eq. (11.1). The dynamical evolution of the system is thus completely determined if we evaluate the transition amplitude between the state at time t0 and that at time t. The usual quantum-mechanical description of a given system is, thus, strongly centered on the role played by the Hamiltonian H and, as a consequence, the time variable itself plays a key role in this description, such a dissymmetry between space and time variables being, obviously, not satisfactory in the light of the postulates of the theory of special relativity. This was first realized in 1932 by Dirac [331], who put forward the idea of reformulating the whole quantum mechanics in Lagrangian rather than Hamiltonian terms, although a formulation of the quantum theory of wave fields already existed in terms of a variational principle applied to a Lagrangian function (see, for example, the classical book by Heisenberg [332], considered also by Majorana for some of his own work). The starting point for Dirac was to exploit an analogy, which held at the quantum level, with the Hamilton principal function in classical mechanics [333]. From this, the transition amplitude from a state in the spatial configuration qa at time ta to a state in the spatial configuration qb at time tb is written as  qb |qa ∼ exp

   tb  i i L dt , S = exp h¯ h¯ ta

(11.2)

where L is the Lagrangian of the system and S[q] is the action functional defined on the paths from qa to qb . The previous relationship, however, cannot be considered as an equality as long as the time interval from ta to tb is finite, since it would lead to incorrect results, as Dirac himself realized. (In his paper he introduced several unjustified assumptions in order to overcome such a difficulty.) In fact, by splitting

11.1 Dirac and Feynman’s mathematical approach

231

the integration field in Eq. (11.2) into N intervals, ta = t0 < t1 < t2 < · · · < tN−1 < tN = tb , the transition amplitude can be written as a product of terms, qb |qa = qb |qN−1 qN−1 |qN−2 . . . q2 |q1 q1 |qa ,

(11.3)

while it is well known that, by using the completeness relations, the correct formula contains the integration over the intermediate regions:   qb |qa = qb | dqN−1 |qN−1 qN−1 | . . . dq1 |q1 q1 |qa  (11.4) = dq1 dq2 . . . dqN−1 qb |qN−1 . . . q2 |q1 q1 |qa . About ten years after the appearance of the Dirac paper, Feynman [334, 335] guessed that the relation in Eq. (11.2) should hold as an equality, up to a constant factor A, only for transitions between states spaced by an infinitesimal time interval. In such a case, by employing the correct formula in Eq. (11.4), the wellknown Feynman expression for the transition amplitude between two given states is obtained:   AN dq1 dq2 . . . dqN−1 eiS/h¯ ≡ Dq eiS/h¯ . (11.5) qb |qa = lim N→∞ tb − ta finite

The meaning of this formula can be understood as follows. In the integrations  present in it, which we have generically denoted with Dq, the limits ta and tb in the integration interval are kept fixed, while integration is performed over the space specified by the intermediate points. Since every spatial configuration qi of these intermediate points corresponds to a given dynamical trajectory joining the initial point ta with the final one tb , the integration over all these configurations is equivalent to the sum over all the possible paths from the initial point to the final one. In other words, the Feynman path integral’s formula points out that the transition amplitude between an initial and a final state can be expressed as a sum of the factor eiS[q]/h¯ over all the paths with fixed endpoints. In one way, this result is not surprising, since the transition amplitude for a given quantum process occurring through different ways is given by the sum of the partial amplitudes corresponding to all the possible ways through which the process happens. This is particularly evident, for example, in the classical experiment with an electron beam impinging on a screen through a double slit: the interference pattern observed on the screen, at a given distance from the slit, should suggest that a single electron has crossed through both slits. It can then be explained by assuming that the probability of one electron going from the source to the screen through the double slit is obtained by summing over all the possible paths covered by the electron. However, what

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A “path integral” approach to quantum mechanics

is crucial but unexpected in Feynman’s approach is that the sum is made over the phase factor eiS[q]/h¯ , which is generated by the classical action S[q]. One of the major merits of the Feynman approach to quantum mechanics is the possibility of obtaining, in a very clear manner, the classical limit when h¯ → 0 (the other merits being its versatile applicability to field quantization in Abelian or non-Abelian gauge theories, with or without spontaneous symmetry breaking). Indeed, for large values of S compared to h¯ , the phase factor in Eq. (11.5) undergoes large fluctuations and thus contributes with terms that average to zero. From a mathematical point of view, it is then clear that in the limit h¯ → 0 the dominant contribution to Eq. (11.5) comes out when the phase factor does not vary much or, in other words, when the action S is stationary. This result is precisely what emerges in classical mechanics, where the classical dynamical trajectories are obtained from the least action principle. This occurrence was first noted by Dirac [331], who realized the key role played by the action functional. The intuitive interpretation of the classical limit is very simple. Let us consider, in the q, t space, a given path far away from the classical trajectory qcl (t); since h¯ is very small, the phase S/h¯ along this path will be quite large. For each path there is another which is infinitely close to it, where the action S will change by only a small quantity, but since this is multiplied by a very large constant (1/h¯ ), the resulting phase will have a correspondingly large value: on average, these paths will give a vanishing contribution to the sum in Eq. (11.5). Instead, near the classical trajectory qcl (t), the action is stationary, so that, when passing from a path infinitely close to that classical trajectory to the classical trajectory, the action will not change at all; the corresponding contributions in Eq. (11.5) will sum coherently, and, as a result, the dominant term is obtained for h¯ → 0. In this approach, the classical trajectory is thus picked out in the limit h¯ → 0 not because of its dominant contribution to the dynamical evolution of the system, but rather because there are paths infinitely close to it that give contributions which sum coherently. The integration region is, actually, very narrow for classical systems, while it becomes wider for quantum ones. As a consequence, the concept of orbit itself, which is well defined in the classical case, loses its meaning in quantum systems, as for an electron orbiting around the atomic nucleus, for example. 11.2 Majorana’s physical approach It is quite clear from the argument in Section 11.1 that the mathematical aspect of the path integral formulation of quantum mechanics played a key role in the development of this formulation of quantum mechanics; it is also clear that this mathematical formalism comprised the basic foundation of the resulting impressive physical interpretation.

11.2 Majorana’s physical approach

233

It is intriguing to realize that, in the Majorana paper reported in the Appendix, there is practically nothing of the distinguishing mathematical aspect of that approach to quantum mechanics; instead, it reveals directly the presence of the physical foundations of the novel approach. Three items are particularly worth mentioning in this respect. The starting point upon which Majorana focuses is the search for a meaningful and clear formulation of the concept of quantum state, as opposed to an old quantum-theory-derived misconception: According to the Heisenberg theory, a quantum state corresponds not to a strangely privileged solution of the classical equations but rather to a set of solutions which differ for the initial conditions and even for the energy, i.e. what is meant as precisely defined energy for the quantum state corresponds to a sort of average over the infinite classical orbits belonging to that state. Thus the quantum states come to be the minimal statistical sets of classical motions, slightly different from each other, accessible to the observations. These minimal statistical sets cannot be further partitioned due to the uncertainty principle, introduced by Heisenberg himself, which forbids the precise simultaneous measurement of the position and the velocity of a particle, that is the determination of its orbit.

In Feynman’s language of ten years later, the “solutions which differ for the initial conditions” correspond just to the different integration paths, the different initial conditions being, indeed, always referred to the same initial time (ta ), while the determined quantum state corresponds to a fixed end time (tb ). The introduced remark about “slightly different” classical motion (emphasis as in the original manuscript), according to that specified by Heisenberg’s uncertainty principle, is instead related to that of the sufficiently wide integration region in Eq. (11.5) for quantum systems. In Majorana’s view, such a mathematical point is thus intimately related to a fundamental physical principle. The crucial point in the Feynman formulation of quantum mechanics, as recalled in Section 11.1, was to consider not only paths corresponding to classical trajectories, but also all the possible paths joining the initial points and the endpoints. This is also contained in the Majorana manuscript, although in a more cryptic way. Indeed, Majorana first points out the following general remark: Obviously the correspondence between quantum states and sets of classical solutions is only approximate, since the equations describing the quantum dynamics are in general independent of the corresponding classical equations, but denote a real modification of the mechanical laws, as well as a constraint on the feasibility of a given observation; however it is better founded than the representation of the quantum states in terms of quantized orbits, and can be usefully employed in qualitative studies.

This passage comes after an interesting discussion on the harmonic oscillator, in which its quantum energy levels are interpreted in terms of classical oscillations: “we can say that the ground state with energy E0 = hν/2 corresponds roughly to

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A “path integral” approach to quantum mechanics

all classical oscillations with energy between 0 and hν, the first excited state with energy E1 = 3h nu/2 corresponds to the classical solutions with energy between hν and 2hν, and so on.” Then, he actually writes more explicitly that the wavefunction “corresponds in quantum mechanics to any possible state of the electron,” whose interpretation just follows Feynman’s, according to the comprehensive discussion made by Majorana on the concept of quantum state. Finally, a key role is played in Majorana’s analysis by the symmetry properties of the physical system: Under given assumptions, that are verified in the very simple problems which we will consider, we can say that every quantum state possesses all the symmetry properties of the constraints of the system.

The relationship with the path integral formulation is made explicit as follows. In discussing a given atomic system, Majorana points out how from one quantum state S of the system, another one S can be obtained by means of a symmetry operation: Differently from what happens in classical mechanics for the single solutions of the dynamical equations, in general it is no longer true that S will be distinct from S. We can realize this easily by representing S with a set of classical solutions, as seen above; it then suffices that S includes, for any given solution, even the other one obtained from that solution by applying a symmetry property of the motions of the systems, in order that S results to be identical to S.

This passage is particularly intriguing if we observe that the issue of the redundant counting in the integration measure in gauge theories, leading to infinite expressions for the transition amplitudes [336, 337], was raised (and solved) only long after the appearance of Feynman’s papers. Several interesting applications to atomic and molecular systems are also present in Majorana’s paper, where known results are deduced or re-interpreted according to the novel point of view. We do not discuss them here, but certainly urge the interested reader to discover them by directly inspecting the text reported in the Appendix.

11.3 Conclusions Undoubtedly, no trace can be found of the mathematical formalism underlying Feynman’s path integral approach to quantum mechanics in the Majorana paper considered here, but, nevertheless, it is very intriguing that the main physical issue, the novel way of interpreting the quantum theory, was indeed realized well in advance by Majorana.

11.3 Conclusions

235

This is particularly impressive if we take into account that, within the known historical path, the physical interpretation of the formalism only followed the mathematical development of the formalism itself, this being its basic foundation. Unfortunately, probably due to the fact that the manuscript discussed in this chapter refers to a seminar for not a particularly high-level audience [7, 8, 2], no further details are present in Majorana’s notes that allow to investigate more deeply his own views on this subject. The reading of the text reported in the Appendix, however, discloses some interesting views on already well-known issues.

12 Fundamental lengths and times

Among a plethora of different topics considered by Majorana in his personal notes [17, 18], a preferred theme was certainly that of electrodynamics, which he studied extensively both at a classical and at a quantum level, as we have already seen in the preceding chapters. Many of these studies concerned topics and methods which were “commonly” discussed in the scientific workplaces of 1930s, although they were dealt with in a very original manner and, sometimes, with extraordinary results. A few, however, were referred to as apparently “unusual” topics, such as the introduction in physics of fundamental constants with the meaning of elementary lengths or times. Indeed, some unpublished contributions contained among Majorana’s personal notes focus solely on his own elaboration on such themes, the distinguishing feature with respect to other scholars being that they always concern specific physical examples rather than generic theoretical issues. In this chapter we discuss two of these studies [338] concerning the scattering of particles by a quasi-Coulombian potential, where a non-vanishing radius for the scatterer is considered, as well as the introduction of an intrinsic time delay in the propagation of a (classical) electromagnetic field, taking into account the possible effect of a fundamental (constant) length on the electromagnetic potentials. A short review of known results, related to the introduction of an elementary space length and of a fundamental time scale, which is exceedingly relevant in modern theoretical physics, will precede the discussion of Majorana’s own contributions, in order to appreciate them properly. 12.1 Introducing elementary space-time lengths The idea of elementary space or time intervals has recurred quite often in the scientific literature. The first appearance in physics of a fundamental length was suggested by Hendrik A. Lorentz, who introduced a classical electron radius

12.1 Introducing elementary space-time lengths

237

Rcl ≡ e2 /mc2  2.82 · 10−13 cm in his model of the electron, corresponding – roughly speaking – to the electron size required for obtaining its mass from the electrostatic potential energy (assuming that quantum-mechanical effects are irrelevant). Actually, it plays a non-negligible role in the dynamical behavior of charged particles at very short distances, the classical electron radius being considered as the length scale at which renormalization effects become relevant in quantum electrodynamics. Thus Rcl denotes the watershed between classical and quantum electrodynamics; it appears in the semiclassical theory for the non-relativistic Thomson scattering, as well as in the relativistic Klein–Nishina formula [339]. In early semiclassical models, the electron was substantially regarded as a charged sphere, but it was readily recognized that the electromagnetic self-energy of such a sphere diverges in the limit of pointlike charge. This led Max Abraham, Lorentz, and, later, Dirac to derive an equation of motion for the electron where a finite, non-vanishing size is fully taken into account [341, 342, 343], thus effectively introducing a fundamental length in order to describe properly the motion of charged particles. Extended, not pointlike, charges were then extensively studied by Sommerfeld [344], who obtained the following “retarded-like” firstorder finite difference equation (T ≡ 2R/c) for the motion of a sphere of radius R with a uniform surface charge: ma = e(E + v × B) +

2 e2 v(t − T) − v(t) . 3 Rc2 T

(12.1)

Here, the electromagnetic field of the given charge causes a self-force, which has a delayed effect on its motion, the “delay time” 2R/c, corresponding to the time the light takes to go across the sphere, being one of the earlier elementary times introduced in microphysics. Later, in the 1920s, J.J. Thomson [345] suggested that the electric force may act in a discontinuous way, producing finite increments of momentum separated by finite intervals of time. The relativistic version of Sommerfeld’s approach was developed by Piero Caldirola [346, 347] in the 1950s, his theory of the electron being one of the first and simplest theories that assumes a priori the existence of a minimum proper-time interval, the so-called chronon. In this theory, time flows continuously, but, when an external force acts on the electron, the reaction of the particle to the applied force is not continuous: the value of the electron velocity uμ is supposed to jump from uμ (τ − τ0 ) to uμ (τ ) only at certain positions sn along its world line. These “discrete positions” are such that the electron takes a time τ0 to travel from one position sn−1 to the next one sn . In principle, the electron is still considered as pointlike, but the Abraham–Lorentz–Dirac equation for the relativistic radiating electron is replaced by two equations. The first is a finite difference (retarded) equation in the velocity uμ (τ ).

238



Fundamental lengths and times

m μ uμ (τ ) uν (τ ) u (τ ) − uμ (τ − τ0 ) + [uν (τ ) − uν (τ − τ0 )] τ0 c2 e = F μν (τ ) uν (τ ) , c



(12.2)

which reduces to the Abraham–Lorentz–Dirac equation when τ0  τ . The second equation relates the discrete positions xμ (τ ) along the world line of the particle to each other:

τ0 (12.3) xμ (nτ0 ) − xμ ((n − 1)τ0 ) = uμ (nτ0 ) − uμ ((n − 1) τ0 ) ; 2 this is valid inside each discrete interval τ0 and describes the “internal” motion of the electron. In such equations the chronon associated with the electron is found to be 2 ke2 τ0  6.266 · 10−24 s ≡ θ0 = 2 3 mc3

(k ≡ 1/4πε0 ),

depending, therefore, on the particle properties (namely on its charge e and on its rest mass m). A different elementary length appearing in modern physics is the Compton wavelength, introduced in 1923 by Arthur H. Compton [348, 349] in his explanation of the scattering of photons by electrons. It is defined as λC ≡ h/mc, and it corresponds to the wavelength of a photon whose energy equals the particle rest mass; for the electron, the Compton wavelength is about 2.42·10−10 cm. It indicates the watershed between classical mechanics and quantum wave-mechanics, since, for spatial distances below h/mc, a microsystem behaves as a very quantum object, as realized by Schrödinger in 1930–1 [351] in his study of zitterbewegung emerging in the Dirac theory of the electron. Indeed, zitterbewegung induces the electron to undergo extremely rapid fluctuations on scales just of the order of the Compton wavelength, causing, for example, the electrons moving inside an atom to experience a smeared nuclear Coulomb potential. The appearance of quantum mechanics, with its discrete energy levels and the uncertainty principle, also encouraged physicists to speculate about space-time discreteness. Heisenberg, for example, in 1938 [353] noted that an elementary length scale is apparently required by the possible derivation of the mass of particles from fundamental constants as h and c. In his view, such a length scale would have been around 10−13 cm, thus corresponding to the classical electron radius. Later, Hartland S. Snyder [354] in 1947 introduced a minimum length in order to “quantize” the space-time, thus anticipating by more than half a century what is now known as “non-commutative geometry” models. Space-time coordinates were represented by quantum operators, and, in Snyder’s approach, these operators have a discrete spectrum, thus entailing a discrete interpretation of space-time.

12.2 Quasi-Coulombian scattering

239

Subsequently, Tsung-Dao Lee [356, 357] introduced an effective time discretization on the basis of the necessarily finite number of measurements performable in any finite interval of time. Finally, the concept of non-continuous space-time has recently come back into fashion in grand unified theories [358], string theories [359, 360], quantum gravity [361, 362], and other different theoretical approaches. Now, the mostimportant case of an elementary space scale is doubtless the “Planck length” P = h¯ G/c3  1.62 · 10−33 cm, which is related, for example, to the minimum length of a typical string in string theory. As a consequence, any space distance smaller than P is deprived of any physical meaning or possibility of being measured. The origin of the name traces back to Max Planck, who, early in 1899 [472], proposed a system of units to be used in physics (a “system of natural units”), where the fundamental units are the Planck length, the Planck time, and the Planck mass. Although Planck did not yet know about quantum mechanics and general relativity, it is interesting to observe that such scales mark out the ranges of validity of these theories as they apply reciprocally, such that a proper quantum gravity theory is required. At present, the Planck length is the minimum space-time scale appearing in physics at which a substantial “unification” of high-energy microphysics and early cosmology is expected. 12.2 Quasi-Coulombian scattering In his research notebooks,1 Majorana studied in detail the problem of the scattering of particles from a quasi-Coulombian potential of the form V(r) = √

k r2 + a2

,

(12.4)

where k is a positive constant (related to the electric charge of the potential source) and a is, according to Majorana, the “magnitude of the radius of the scatterer.” Although the motivations for such a study (probably for applications to atomic and nuclear physics problems) are not precisely known, as a matter of fact the original intention of Majorana was to extend the well-known Rutherford formula for the scattering of a beam of particles (with charge Z  e and mass m) from a given body (of charge Z e). This formula corresponds to a pure Coulomb scattering, and the cross section (number of scattering particles at an angle θ per unit time and solid angle) is given by the classical formula f (θ) = 1 See Sect. 6.6 in Ref. [18].

Z 2 Z 2 e4 Z 2 Z 2 e4 = , 4m2 v4 sin4 θ/2 16T 2 sin4 θ/2

(12.5)

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Fundamental lengths and times

where v is the velocity of the incident particles and T is their kinetic energy. Majorana deduced the above formula in his personal notebooks2 by using both classical-mechanics arguments and the quantum Born approximation method. The modification of the Coulomb scattering potential, considered by Majorana in Eq. (12.4), was aimed at improving the description of the physical phenomenon, with the introduction of a finite size for the scattering center (in the Rutherford formula it was assumed to be pointlike, a = 0), but the story does not end here. Indeed, the most simple approximation in this line of thinking is to consider the scattering center as a uniformly charged sphere of radius a, but, in such a case, the potential outside the sphere is strictly Coulombian (thanks to the Gauss law), and Eq. (12.4) would not apply. Of course, at the time when Majorana performed his calculations, it was known that the nucleus of a given atom is not uniformly charged, as it is formed by individual particles (see Section 9.4), but, again, such a situation is not described by the simple formula in Eq. (12.4). An example is the Yukawa potential [95], where the Coulomb potential acquires a screening factor with an exponential form ruled by one more parameter, related to the range of nuclear forces. In any case, it is striking that the only application of Eq. (12.4) reported in Majorana’s notebooks [18] was to the hydrogen atom, where just one proton forms its nucleus, thus being considered effectively as a uniformly charged particle. Therefore, Majorana’s reasoning behind Eq. (12.4) must have had a different starting point. From a strictly mathematical point of view, the introduction of a non-vanishing radius for the scattering center has the effect of regularizing the Coulomb potential, which, otherwise, would diverge for r → 0. By contrast to other similar cases, Majorana did not restore the full Coulomb potential by taking the limit a → 0 at the end of his calculations, but always maintained a finite value for a. Within the proposed problem, Majorana studied the deviations from pure Coulomb scattering by parameterizing it with the ratio i/iR of the effective scattering intensity under an angle θ (i.e. the flux of scattered particles per unit surface (normal to the incident direction) and per unit time) with respect to that deduced in the Rutherford approximation. Besides the radius a, this ratio depends also on the energy and momentum of the incident particles, but Majorana preferred to parameterize these by means of the scattering parameter (or, as he denoted himself, the “minimum approach distance”) b in the Coulomb limit, defined by k/b = T, and from the wavelength λ of the free particle. By setting α = a/(λ/2π), β = b/(λ/2π), he obtained i = f (α, β, θ) iR . 2 See Sects. 4.19 and 4.10 in Ref. [17].

(12.6)

12.2 Quasi-Coulombian scattering

241

Of course, in the pure Coulomb limit, the Rutherford formula has to be recovered, so that f (0, β, θ) = 1. The calculations then proceeded to evaluate the function f (α, β, θ) by keeping α fixed (that is, for fixed scatterer size) and considering the limit β → 0 (for scattering particles with increasing momentum, approaching closer and closer to the center). The wavefunction of the system is then evaluated perturbatively by expanding it in a series ruled by the experimental parameter β, at zeroth order (β = 0) using the Wentzel–Kramers–Brillouin (WKB) method [364, 365, 366]. In order to avoid convergence problems in the usage of the Green method, Majorana assumed that the scattering force acts only over distances less then a quantity R, and then he let R → ∞ at the end of his calculations. The poten  1 1 − . tial in Eq. (12.4) is thus replaced, during the calculations, by k √ r2 + a2 R After some passages, he then obtained the following analytic result within the approximation considered (ρ = r/(λ/2π)):    ρ θ θ ∞ dρ , (12.7) sin 2ρ sin f (α, β, θ) = 2 sin  2 0 2 ρ 2 + α2 which can also be expressed in terms of the Bessel K-function K (α sin θ/2) [443]. In this approximation, Majorana found that the actual scattering intensity may be substantially different from that of the Rutherford formula for backward scattering, provided that the radius a is appreciably different from zero (the ratio i/iR approaching zero for θ = π and increasing α). This remarkable result is not explicitly reported by Majorana, but he attempted to tabulate the radial wavefunction u , satisfying the equation   β ( + 1) ue = 0 , − (12.8) u + 1 −  r2 ρ 2 + α2 obtained numerically with the method of the particular solutions. Although numerical solutions are searched only for the particular case of  = 0 and β = 0.4 (and not β = 0), here the interesting point is that Majorana explicitly reports that “for the hydrogen atom we consider the values β = 0.4, 0.5, 0.6, 0.7 and α = 0, 0.2, 0.4, 0.6, 0.8, 1.” Unfortunately, no further discussion is given in his notebooks; the key point is, nevertheless, the fact that Majorana uses λ/2π as the length scale for both the lengths a and b. Now, it is very natural to measure the scattering parameter (or “the minimum approach distance”) in terms of the free particle wavelength or, equivalently, (the inverse of) the free particle momentum (λ = h/p), since it is obvious that, by increasing the momentum (or decreasing the probe wavelength), the incident particle approaches closer and closer to the scattering center, thus decreasing b. The same would not apply, however, to the

242

Fundamental lengths and times

radius of the scattering center, which should be independent of the incident particle properties, unless Majorana was thinking of an effective, momentum-dependent, size for the scattering center. In such a case, his attention was evidently shifted from the investigation of an actual, particular physical system (particles interacting with a scattering potential) to the study of more general properties of the background field or space (see Section 12.3). Interestingly enough, Majorana also considered another modification of the Coulomb potential, reminiscent of the Gamow potential for the description of α-decay processes, namely

V=

⎧ V, ⎪ ⎨ 0 ⎪ ⎩k , r

for r < R, (12.9) for r > R.

The role of the scattering center radius is now played by the range R, with α = R/(λ/2π). One more parameter is present here: the depth V0 of the potential, measured with respect to the kinetic energy of the incident particles, A = V0 /T. Although, in this case, the calculations were only sketched, the above-mentioned key points are also present. Now the correction function in Eq (12.6) is replaced by   V0 R b i =f , , ,θ (12.10) iR T λ/2π λ/2π where “for the hydrogen” Majorana considers the same values as before for the α, β parameters (except, obviously, α = 0, 0.2), while A = 2, 1.5, 1, 0.5, 0, −0.5, −1, −1.5, −2, −2.5, −3, ..., −8. Interestingly, besides negative values for V0 (as in the Gamow model), Majorana also considers a few positive values for the depth of the potential. As above, however, the notable point is that both the potential energy parameters V0 and R appear to be strictly related to the energy and momentum of the probe particles. For both cases studied, either in the presence of continuum states (Eq. (12.4)) or bound states (Eq. (12.9)), the reasoning was the same, and Majorana’s original motivations for the studies of quasi-Coulombian scattering were different from the standard ones related to atomic and nuclear physics, as we will see in Section 12.3.

12.3 Intrinsic time delay and retarded electromagnetic fields The modification of the standard Coulomb potential is apparently related to other calculations of Majorana, where he considered the possibility of introducing an intrinsic constant time delay (or, equivalently, an intrinsic space constant) in the

12.3 Intrinsic time delay and retarded electromagnetic fields

243

expression for the retarded electromagnetic fields.3 Here, however, the treatment was fully classical, and went as follows. The starting point is the wave equation satisfied by any component of the electromagnetic potentials, denoted generically by f (x, y, z, t), and then the evaluation of the D’Alembert operator for the standard retarded field denoted by

r ≡ f (x, y, z, t) . (12.11) ϕ(x, y, x, t) = f x, y, z, t − c The known result is, of course, the following: f = ∇ 2 ϕ +

2 2 ∂ 2ϕ ϕ˙ + , rc c ∂r∂t

(12.12)

where a dot indicates time differentiation. Majorana then introduced an intrinsic space constant ε, so that Eq. (12.11) is replaced by   √ r2 + ε2  = f (x, y, z, t) . (12.13) ϕ(x, y, z, t) = f x, y, z, t − c The D’Alembertian term entering into the wave equation is thus / = ∇ 2ϕ − f

2r2 + 3ε2 ε2 2r ∂ 2ϕ ϕ ¨ + , ϕ ˙ + √ c2 (r2 + ε2 ) c(r2 + ε2 )3/2 c r2 + ε2 ∂r∂t

(12.14)

which replaces Eq. (12.12). Majorana also introduced explicitly a time delay τ which, in his view, is a “universal constant” taking the value τ = 0 classically; in his calculations, however, he always uses the length ε = τc

(12.15)

(already introduced), so we continue to follow this line of reasoning here. The wave equation is apparently solved by using the Green method; denoting by S the generic source function (charge density ρ or current density J), the modified generic potential, , solution of the wave equation assumes now the following form:   √  R2 + ε 2 1    dx dy dz S x ,y ,z ,t − (12.16) = √ 2 2 c R +ε   (with R = r − r ). Majorana then ends his explicit calculations with the expansion of Eq. (12.16) up to second order in ε, for ε → 0 (thus approaching the classical limit): 3 See Sect. 2.14 in Ref. [18].

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Fundamental lengths and times

  1 R    S x ,y ,z ,t − dx dy dz = R c    R 1 2    dx dy dz −ε S x ,y ,z ,t − 2R3 c     R 1        S x ,y ,z ,t − dx dy dz + . . . . + 2R2 c c 

(12.17)

Some comments are in order. By taking the ordinary electric  monopole limit r − r   ∼ [368] into Eq. (12.16), with S(r, t − r/c) = ρ(r), = |r| = r, and  ρ(r )dx dy dz = Q being the electric charge, one easily obtains = √

Q r2

+ ε2

,

(12.18)

i.e. exactly the potential considered in Eq. (12.4). The “magnitude of the radius of the scatterer” in the scattering problem considered in Section 12.2 is thus interpreted as the “universal” space constant introduced here. Majorana’s original intention in the study of quasi-Coulombian scattering was, then, not that of exploring the properties of the scattering particle (an atom or its nucleus, or even other elementary particles), but rather that of investigating the underlying properties of the physical space. The intrinsic space constant plays the role of a fundamental length, like the one introduced by Planck in 1899 or that conjectured by Heisenberg in 1938. We do not know if Majorana effectively thought of a particular value, but, in any case, this is the first time that such a detailed study was effectively undertaken in the 1930s.

12.4 Conclusions The problems concerning the existence and effects of a fundamental length date back to the early 1900s. Although such a concept was introduced (and re-discovered) in physics in order to solve difficulties of a different kind (typically, e.g., in order to regularize some divergent results), it is quite notable that it led also to some deeper results. In Heisenberg’s view, for example, the existence of elementary space intervals allows the derivation of the mass of the particles, or even plays a key role in the physical interpretation of the spin variable. Strictly related to the introduction of a fundamental length is the assumption of the existence of a fundamental time, although such a hypothesis is less fashionable among physicists. In more recent times, indeed, the whole subject has been embedded in the more general framework of the quantization of space-time, where

12.4 Conclusions

245

attempts to provide a well-defined quantum theory of gravity manifest into different theoretical approaches, with the common denominator of a fundamental length scale. Intriguingly, as early as the beginning of the 1930s, Majorana not only conjectured (as already Planck and, later, others did) the existence of both an elementary length and a fundamental time scale, but also investigated in some detail the immediate physical consequences of such a hypothesis. Indeed, he considered the scattering from electrically charged particles when the underlying properties of space are such that an intrinsic length exists, resulting in the modification of the pure Coulomb potential, as in Eq. (12.4) or Eq. (12.18). Even just at the classical level, he obtained the same effect by introducing an intrinsic time delay, considered as a “universal constant” in the expression for the retarded electromagnetic fields. Once again, his contributions reveal a farsighted intuition, and are precious pieces of research both from the historical point of view and for present day theoretical investigations about the fundamental properties of physical space-time.

13 Majorana’s multifaceted life

The reader who has patiently followed Majorana’s calculations in the previous chapters is probably now well aware of what kind of problems he used to study, how he solved them, and what the relevance is of the results he obtained. In this chapter we will focus on a few other topics (among many others) considered in his personal notebooks, which add more details to particular aspects of Majorana’s multifaceted scientific personality. 13.1 Majorana as a student A large part of Majorana’s personal notebooks [17] is devoted to topics related, more or less directly, to studies that he performed when he was a student at the University of Rome, and, apparently, they do not represent his true scientific interest, since they concern issues typical of university courses (of the time). However, a close inspection shows that this is not the case, since the methods he followed in studying the different topics are almost always original and always extremely clear in their development, so that they present – at least – a true pedagogical interest. Moreover, in a few cases, the originality of the method leads to interesting results not previously considered. In the following we report just three examples of different natures (the last of which refers to Majorana as a self-taught student) that illustrate well this matter. 13.1.1 Melting point shift due to a magnetic field For this study,1 Majorana considered a system like that depicted in Figure 13.1, where a vessel containing a solid (S) substance in equilibrium with its liquid (L) is located between the polar expansions of a magnet, while connected to a similar second vessel outside the expansions. The reasoning went as follows. 1 See Sect. 1.28 in Ref. [17].

13.1 Majorana as a student

247

N

2

1 L p

S

h

L S

p + dp

S

Figure 13.1. Solid–liquid phase transition in the presence of a magnetic field.

In order to move (by any procedure) a unit volume of solid from vessel 2 to vessel 1 and place it in a thin layer at the boundary between the solid and the liquid, an amount of work W1 = h (γ1 − γ2 )

(13.1)

against gravity is required, where γ1 and γ2 are the specific weights of the solid and the liquid, respectively. As a first case, Majorana assumes that the solid is a magnetic material with magnetic permeability μ1 (the liquid is not), and then computes how much work the magnetic field B has to perform on the unit volume of solid during the corresponding process: W2 =

B2 μ 1 − 1 . 8π μ1

(13.2)

In order to exclude the possibility of perpetual motion, W1 = W2

(13.3)

must hold, wherefrom we obtain h=

B2 μ1 − 1 1 . 8π μ1 γ1 − γ2

(13.4)

The pressure difference between the boundary solid–liquid surfaces in the two vessels (see Figure 13.1) is determined solely by gravity (and is then hydrostatic), within the assumption that the liquid is non-magnetic, so that it is given by p = h γ2 =

B2 μ1 − 1 γ2 H 2 μ 1 − 1 V1 = , 8π μ1 γ1 − γ2 8π μ1 V2 − V1

(13.5)

where the specific volumes V1 , V2 have been introduced in the last equality. Denoting by T the melting temperature in the absence of a magnetic field, at

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Majorana’s multifaceted life

pressure p, the temperature change at the same pressure is given by the Clapeyron equation, so that:   TB2 μ1 − 1 V1 T , (13.6) (V2 − V1 ) p = T = ρ 8π μ1 ρ ρ being the density of the substance. The generalization to the case where the liquid has an arbitrary magnetic permeability μ2 is straightforward:   μ2 − 1 γ1 B2 μ1 − 1 γ2 + p = , (13.7) 8π μ1 γ1 − γ2 μ2 γ2 − γ1   μ2 − 1 V2 TB2 μ1 − 1 V1 . (13.8) − T = 8π μ1 ρ μ2 ρ Majorana then considers the special case when μ1 = μ2 = μ, and observes that, contrary to naive expectations, the boundary surface between solid and liquid is not at the same level in both vessels since, due to the magnetization of the liquid, the pressure distribution is not hydrostatic, so that p = 0. Finally, he concludes with a generalization of the obtained result: Similar relations hold if the magnetic field is replaced with an electric field or if the different phases are liquid-vapor or solid-vapor instead of solid–liquid.2

Incidentally, we note that the above study is not purely academic in nature, having been revived by recent puzzling observations (see Ref. [369] and references therein). 13.1.2 Determination of a function from its moments Since his student days at the University of Rome, Majorana had studied repeatedly the theory of probability, mainly related to practical examples. Here we refer to a more general issue (and some related applications) which corresponds to some physics applications, including quantum mechanics.3 The mathematical problem focuses on a function y = y(x) that vanishes for 2 x > a2 (for some a), where the integral  ∞ |y| dx (13.9) −∞

is finite. By defining the moments μ0 , μ1 , ..., μn of order 0, 1, 2, ..., n as  μn = xn y dx, 2 See Sect. 1.28 in Ref. [17]. 3 See Sect. 3.4 in Ref. [17].

(13.10)

13.1 Majorana as a student

249

the problem is to find an integral representation of the function y(x) that is completely determined by its moments μn . By setting  (13.11) z(t) = y eixt dx, so that y=

1 2π



∞

e−ixt z dt,

(13.12)

−∞

Majorana computes the successive derivative of z(t),   dz dn z ixt n ... = i xn y eixt dx, = i x y e dx, dt dtn from which he deduces z(0) = μ0 and, in general,  n  dz = in μ n . dtn t=0

(13.13)

Therefore, from the given assumptions, the function z(t) can be expanded in an absolutely convergent MacLaurin series: z=

∞  n=0

μn

(it)n . n!

Substituting into Eq. (13.12), we then have the desired representation  ∞ ∞  (it)n 1 −ixt e μn dt, y= 2π −∞ n! 0 which can also be written as  ∞ ∞  t2r 1 y= cos xt (−1)r μ2r dt π 0 (2r)! 0  ∞ ∞  t2r+1 1 sin xt (−1)r μ2r+1 dt. + π 0 (2r + 1)! 0

(13.14)

(13.15)

(13.16)

Majorana rightly points out that “obviously, the integral and the series cannot be inverted.” Several simple examples followed in order to illustrate the method, including the function defined by μn = 1/(n + 1) (corresponding to the constant function), μn = n! (decreasing exponential function), μ2r = (r − 1/2)! and μ2r+1 = 0 (Gaussian function), and μn = 1/(n + 1)2 (logarithmic function). Particularly relevant cases are those corresponding to given probability distributions, such as the following ones explicitly worked out by Majorana:

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Majorana’s multifaceted life

1. the probability that the distance between two points belonging to two concentric spherical surfaces, one with unit radius and the other with a radius a < 1, lies between r and r + dr, described by the moments μn =

(1 + a)n+2 − (1 − a)n+2 , 2(n + 2)a

(13.17)

and corresponding to the function ⎧ 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ r , y= ⎪ 2a ⎪ ⎪ ⎪ ⎪ ⎩ 0,

for r < 1 − a, for 1 − a < r < 1 + a,

(13.18)

for r > 1 + a;

2. the probability that two points belonging to two concentric spherical surfaces with radii a and b < a are at a distance r from each other, described by the moments μn =

(a + b)n+2 − (a − b)n+2 , 2ab(n + 2)

(13.19)

and corresponding to the function ⎧ 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ r , y= ⎪ 2ab ⎪ ⎪ ⎪ ⎪ ⎩ 0,

for r < a − b, for a − b < r < a + b,

(13.20)

for r > a + b;

3. the probability that two points on a segment of length  are at a distance r from each other, described by the moments μn =

2n , (n + 1) (n + 2)

and corresponding to the function ⎧ 2( − r) ⎪ ⎪ , ⎨ 2  y= ⎪ ⎪ ⎩ 0,

(13.21)

for 0 < r < , (13.22) otherwise.

13.1 Majorana as a student

251

13.1.3 WKB method for differential equations The method devised by Harold Jeffreys, Gregor Wentzel, Hendrik A. Kramers and Léon Brillouin as early as the 1920s is usually adopted to perform semiclassical calculations in quantum mechanics and to compute the stationary solution of the Schrödinger equation, recast as a suitable exponential function. However, it can also be used to find approximate solutions to any linear differential equation with non-constant coefficients, and Majorana applied4 the general idea of the WKB method to equations of the form y + P y = 0, with P = P(x). By means of the position    k dx , y = u exp i u2

(13.23)

(13.24)

where k is a suitable constant, the basic differential equation for u becomes u −

k2 + u P = 0. u3

(13.25)

Given the initial conditions y0 and y0 at x = x0 , we set u0 = |y0 |,

(13.26)

so that the arbitrary (real) additive constant of the integral [in Eq. (13.24)] is determined modulo 2π . For y0 = 0 [...] we then put   y0 k u0 + i . (13.27) y0 = |y0 | |y0 | We can suppose that both u0 and k are real. Then, if P is also real, the integration [of Eq. (13.23)] with a complex variable is equivalent to the integration [of Eq. (13.25)] with a real variable. Note that, if y0 /y0 is real, then k = 0 and [Eq. (13.25)] reduces to [(13.23)].

The general case was found by Majorana by setting u u1 = √ k

(13.28)

and noting that the resulting equation for u1 is just Eq. (13.25) with k = 1. He concluded that “we can always reduce the problem to the k = 1 case,” and the general solution can be written as:        dx dx + B exp −i , (13.29) y = u1 A exp i u21 u21 with the coefficients A, B determined from the initial conditions. 4 See Sect. 2.31 in Ref. [17].

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Majorana’s multifaceted life

Majorana then focused on the case with a slowly varying P function, that is when the condition   P     1 P is fulfilled. In such a case, as a first approximation he posed u = P−1/4 , and he found the general solution to first order:   ⎧  √  √ 1 ⎪ ⎪ A cos P dx + B sin P dx , P > 0, √ ⎪ ⎪ ⎨ 4P y=       ⎪ ⎪ √ √ 1 ⎪ ⎪ A exp −P dx + B exp − −P dx , P < 0. ⎩ √ 4 −P (13.30) The second-order approximation (and, thus, any successive approximation) is instead obtained by replacing u in Eq. (13.25) (with k = 1) with the value coming from the first-order approximation u = P−1/4 . Then, by writing the second-order function u as P−1/4 + u, the small correction u is determined by direct substitution into the relevant differential equation, obtaining   P P − (5/4) P2 −1/4 1+ . (13.31) u=P 16 P3 The general solution of Eq. (13.23) at second order is, thus,   ⎧ P P − (5/4) P2 1 ⎪ ⎪ ⎪ 1+ √ ⎪ 4 ⎪ 16 P3 P ⎪ ⎪      √  ⎪ ⎪ ⎪ P2 P ⎪ ⎪ dx P 1− × A cos − 3/2 + ⎪ ⎪ 8P 32P3   ⎪ ⎪    ⎪ √ ⎪ P2 P ⎪ ⎪ dx , + P 1 − + B sin − ⎪ ⎪ ⎨ 8P3/2 32P3 y=   ⎪ ⎪  2 ⎪ P P 1 − (5/4) P ⎪ ⎪ 1+ √ ⎪ ⎪ 4 ⎪ 16 P3 −P ⎪ ⎪       ⎪  ⎪ √ P2 P ⎪ ⎪ ⎪ + −P 1 − × A exp − dx ⎪ ⎪ 8(−P)3/2 32P3   ⎪    ⎪ ⎪ √ ⎪ P2 P ⎪ ⎩ − −P 1 − + B exp + dx , 8(−P)3/2 32P3

P > 0,

P < 0. (13.32)

Majorana concluded his analysis by giving a more general result:

13.1 Majorana as a student If y1 is a solution [of Eq. (13.23)], the general solution reads  dx y = A y1 + B y1 . y21 Indeed, on setting

 y2 = y1

253

(13.33)

dx , y21

we get [...] 0 = y2 −

y1 y2 = y2 + P y2 . y1

(13.34)

As an application of the WKB method, Majorana then applied5 it to the solution of the equation6 y = xy. This is a straightforward exercise, but, as he pointed out, the expressions thus obtained no longer hold when x approaches zero: Therefore the problem that arises is how to connect the asymptotic expressions for x > 0 (by a few units, at least) with those for x < 0. Since the equation is homogenous, we need only to know how to match two particular solutions, to be able to perform the matching for any solution. [17]

This is achieved by considering the following two particular solutions coming from the WKB method: M = 1+ N = x+

x6 x9 x3 + + + ..., 2·3 2·3·5·6 2·3·5·6·8·9 4

7

(13.35)

10

x x x + + + .... 3·4 3·4·6·7 3·4·6·7·9·10

For |x| > 4 the first and, even better, second-approximation asymptotic expressions are practically exact. It is then enough to compute, from [Eq. (13.35)], the values of M, N, M  , N  for x = ±4. [17]

Majorana’s numerical ability can be envisaged by looking at the values we report in Table 13.1, as well as at the plot of the M and N functions he obtained, which we present in Figure 13.2.7 5 See Sect. 2.5 in Ref. [17]. 6 This equation appeared when he considered the wave quantization of a particle attracted by a constant force

towards a perfectly elastic wall, that is one-dimensional motion along the direction perpendicular to an elastic surface. 7 Very probably, Majorana employed the formulae in Eqs. (13.35) by taking the expansions up to the non-vanishing tenth term, which means up to the x27 and x28 power terms for M and N, respectively.

254

Majorana’s multifaceted life Table 13.1. Matching values for the solutions of the equation y = xy x

M

M

N

N

−4 0 4

0.2199 1 68.1777

−1.2082 0 131.6581

0.5732 0 93.5172

1.3972 1 180.6092

1.0

0.5

0.0

-0.5

-1.0 -4

-3

-2

-1

0

Figure 13.2. The functions M (solid line) and N (dashed line) in the interval −4 < x < 0.

13.2 Majorana as a phenomenologist: spontaneous and induced ionization of a hydrogen atom Probably inspired by Gamow’s theory of the spontaneous α emission from a nucleus (see Chapter 5), in 1928 Majorana considered in detail – within the framework of quantum mechanics – the problem of spontaneous ionization of a hydrogen atom placed in a high potential region.8 As we will see, the reasoning behind this study follows quite closely the one he adopted in his calculations on the scattering of α particles from a radioactive nucleus (reported in Section 5.2); in a certain sense, the present study was “preliminary” to the nuclear problem he later considered for his Master’s thesis. As a matter of fact, however, his findings put Gamow’s calculations on a solid theoretical base. Furthermore, in addition to the particular results he obtained for the given problem of spontaneous ionization, Majorana was also able to draw a general conclusion about quasi-stationary states that largely goes beyond Gamow’s results. 8 See Sect. 2.33 in Ref. [17].

13.2 Majorana as a phenomenologist

255

Later in his studies, Majorana considered9 also the ionization of a hydrogen-like atom when placed in a constant electric field. This problem was first discussed in 1928 by Oppenheimer [370], who gave a sort of “golden rule” for the ionization rate, and was later (1930–1) studied by Cornelius Lanczos [371, 372] within the WKB approximation, while only in more recent times has the problem received more rigorous treatments [373]. The WKB method is traditionally adopted for two reasons: first, because ionization can usually be observed only for high values of the electric field, so that perturbative calculations do not apply (contrary to calculations for the Stark effect without ionization), and second because the method provides expressions for the relevant quantities (ionization probability) in terms of integrals that can be easily computed numerically. It is particularly intriguing that the lack of a simple analytic treatment, though approximate, has generated some confusion, and in quite recent times it has been recognized that “the field ionization of the hydrogen atom has exhibited a peculiar perverseness, with unsuspected pitfalls marring some of the earlier calculations” [374]. The reason for this is that “the barrier region dominates practical calculations, which are surprisingly sensitive to the accuracy of the wavefunction there” [374]. In his personal notebooks, Majorana considered just the same problem and, although he focused only on a very particular case (the ground state of the hydrogen-like atom, with no linear Stark effect, and for small fields), he was able to obtain in a simple way a perturbative second-order expression for the relevant wavefunction (and thus for its square, from which the ionization probability may be evaluated), which – to the best of our knowledge – is still lacking in the literature. The interest in such a derivation is, then, not only didactic. 13.2.1 Hydrogen atom placed in a high potential region The physical system10 is that of a hydrogen atom placed at the common center of two spheres of radii R and R + dR, upon which charges −Q /dR and Q /dR − e are located, respectively. It is assumed that Q = QR and that dR is an infinitesimal. By denoting the distance from the center by x, the potential experienced by the electron is as follows: ⎧ e ⎪ x < R, ⎨ V = − A, x (13.36) ⎪ ⎩ V = 0, x > R, with A a constant that can be fixed by requiring the continuity of the potential (A = e/R). The Schrödinger equation for χ = ψ/x, when adopting units for which 9 See Sect. 4.2 in Ref. [17]. 10 See Sect. 2.33 in Ref. [17].

256

Majorana’s multifaceted life

h¯ = e = m = 1 (e, m being the electron charge and mass, respectively) reduces to   ⎧ 1 ⎪  ⎪ χ +2 E −A+ χ = 0, x < R, ⎨ x (13.37) ⎪ ⎪ ⎩  x > R. χ + 2E χ = 0, Since the ground state energy of the hydrogen atom is equal to −1/2 in the units adopted, Majorana parameterized the energy of the system at hand in terms of the quantity α according to the relation 1 1 − α. (13.38) 2 2 For α = 0, the solution of the first Eq. (13.37) is just χ = x e−x , so that he casts the solution for α = 0 in the form E=A −

χ = x e−x + α y,

(13.39)

with the constraints y(0) = 0, y (0) = 0, where y satisfies the following equation:   2  −x y. (13.40) y = xe + 1 + α − x For large values of x, the asymptotic solution of this equation is given by y=

√ 1+α kα √ x1/ 1+α

ex

,

(13.41)

with kα a suitable constant. According to Majorana, since we have assumed that α is small, as an approximation we could set kα = k0 , and k0 will be evaluated from the asymptotic form for y with the constraints y(0) = 0, y (0) = 0. The differential equation for such y is   2 y, (13.42) y = x e−x + 1 − x and the asymptotic expression for the solution will be of the form y = k0

ex . x

(13.43)

The numerical value of the constant was found to be k0 = 1/8 by asymptotically expanding the exact analytic solution, obtained by summing the power series expansion of y: 1 3 1 x − x4 + . . . + an xn + . . . , 6 9 an−2 − 2an−1 n−2 + . an = − (−1)n n! (n − 1)n y=

(13.44) (13.45)

13.2 Majorana as a phenomenologist

257

The analytic result is the following: y = x e−x



x 0



 1 1 1 + x − x2 e−x , 4 2 2

(13.46)

 1 1 + + higher-order terms , 4x γ x2

(13.47)

e2x − 1 1 dx − ex + 2x 4

from which, for x → ∞, since 

x 0

e2x − 1 dx = e2x 2x



the asymptotic expression y=

1 ex 8 x

(13.48)

is immediately deduced. Majorana also gave the expression for the χ function, but focused on the case where R is large compared to atomic dimensions, and for small α. In this approximation,  √ 

√ R e 2B 1 2B + 2πγ (x − R) R e−R − 2πγ cos χ= A 4 R   √ 

√ 1 eR 2B −R 2B + 2πγ (x − R) , Re − 2πγ sin + − A 4 R 

(13.49) where A−1 1 + 4R2 e−2R , 2 A   4(A − 1) 2 −2R 1 1 1 + α γ =− R e √ 2π 2E A 2 B=A −

(13.50) (13.51)

(γ is the correction to the momentum of the system, in the units adopted). In a more compact form, we have " χ=

+ * √  2 2 −2R A + R−2 e2R 4π 2 γ 2 cos 2B + 2πγ (x − R) + z , R e A 8 (13.52)

258

Majorana’s multifaceted life

where z denotes a suitable quantity depending on γ . The normalized (with respect to dx) u wavefunction is, therefore, ⎧ " ⎪ A eR ⎪   ⎪ ⎪ √ ⎪ α ex ⎪ 2 R −x ⎪ eiBt e2π i 2Bγ t , for x < R, 2x e + " ⎪ ⎪ 4 x ⎪ π 2 A2 e4R 2 ⎨ 1+ γ u= 4 R4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  * √ + √ ⎪ ⎪ ⎩ 2 cos 2B + 2πγ (x − R) + z eiBt e2π i 2Bγ t , for x > R. (13.53) Majorana then assumed that, at t = 0, the electron is in its ground state, so that the eigenfunction ψ is spherically symmetric and can be written as ψ = U(x)/x. Within the same approximations as given before, he thus found ⎧ " 0 1  √ ⎪ ⎪ 8 R eiBt exp −i arcsin (2A − 1)/2A − i 2B(x − R) ⎪ ⎪ ⎪ A eR 0 1 ⎪ ⎪ ⎨ × exp 4R2 (x − R)/(Ae2R ) − 4R2 √2B t/(Ae2R ) , U= √ ⎪ for 2B t − (x − R) > 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ √ ⎩ 0, for 2B t − (x − R) < 0. (13.54) The results obtained by Majorana are – according also to his own recognition [17] – the theoretical basis for what was assumed by Gamow in his paper on nuclear α decay [233], where he postulated (rather than deduced) the characteristic exponential time dependence. At large distances from the nucleus, the eigenfunction of the α particle was found to be a spherical progressive wave, from which the lifetime T was determined (see Eq. (13.56) in the following). Majorana’s calculations, then, definitively answered also the criticism by Kudar [238] about alleged inconsistencies in Gamow’s findings arising from the space-dependent growth factor in U from Eq. (13.54). √ For 2Bt − (x − R) > 0, independently of the small time-dependent damping factor and of the space-dependent growth factor, [Eq. (13.54)] represents a progressive plane wave moving towards high values of x. For sufficiently small values of t and x − R, the electron flux per unit time is √ 8R2 2B F= . Ae2R

(13.55)

13.2 Majorana as a phenomenologist

259

On the other hand, the damping factor can be written as e−t/2T , where T is the timeconstant. It follows, as it is natural, that F = 1/T, T=

Ae2R √ . 8R2 2B

(13.56)

In order to check this result, Majorana considered the limit when A = 1/2, i.e. when A is equal to the ionization potential: in this case, B (≈ A − 1/2) approaches zero, √ and the lifetime becomes infinite. On the other hand, since T is proportional √ to A/ 2B ≈ (B + 1/2)/ 2B, the ionization probability per unit time (that is, the reciprocal of the lifetime) increases with increasing B until it reaches a maximum value for B = 1/2, after which it falls off to zero as B → ∞. The maximum corresponds to A = 1, i.e. twice the value of the ionization potential, and the minimum lifetime is given by T=

e2R . 8R2

(13.57)

The non-monotonic behavior is explained by Majorana as follows: Whenever there is a surface that sharply separates two regions with different potentials, it behaves as a reflecting surface not only for the particles coming from the region with lower potential energy, but also for the ones coming from the opposite side, provided that the absolute value of the kinetic energy (positive or negative) is small with respect to the abrupt potential energy jump.

The solution found in Eq. (13.54) is valid, for times close to t = 0, only near the nucleus, but with the √ passing of time it remains valid (under the approximations used) within a radius 2B t = vt, where v is the velocity of the emitted electron. According to the general principles of quantum mechanics, the finite lifetime of the quasi-stationary state induces an uncertainty in the emission velocity, and, intriguingly enough, Majorana was able to highlight such an uncertainty – that is, to determine the velocity curve of the emitted electron – independently of those principles; namely, by using statistical arguments only. By assuming that the energy (or the velocity) of the electron is determined only up to a given uncertainty, he focuses on the probability that it lies between E and E + dE (or, analogously, the probability that the velocity of the emitted electron falls between v and v + dv). In the approximations adopted, √ √ v = 2E  2B + 2π γ , dv  2π dγ , and, from Eq. (13.55) and subsequent calculations, Majorana deduced that the probability that the velocity of the emitted electron lies in the interval v, v + dv is

260

Majorana’s multifaceted life

(Ae2R /4πR2 ) dv (Ae2R /4πR2 ) dv  . √ 1 + (π 2 A2 e4R /4R4 )γ 2 1 + (A2 e4R /16R4 ) (v − 2B)2

(13.58)

A similar result holds for the energy, at first approximation; the probability per unit energy is given by √ 1/πK (Ae2R /4π 2BR2 ) = , (13.59) 2 4R 4 2 1 + (A e /32BR ) (E − B) 1 + (E − B)2 /K 2 with (obviously) the parameter K defining the probability amplitude. From Eq. (13.56), this quantity is related to the lifetime T by the relation K = 1/2T, or, going back to the usual units, K=

h¯ , 2T

(13.60)

which clearly expresses the time–energy uncertainty relation. The final general conclusion that Majorana drew from his calculations was, once more, astonishing, and goes well beyond Gamow’s conclusions [233]: If we deal with quasi-stationary states, we always find the same probability, independently of the form of the potential, provided it has spherical symmetry.

13.2.2 Ionization of a hydrogen-like atom in an electric field The problem studied here11 is that of a hydrogen-like atom (with atomic number Z) in an electric field of strength F along the x-direction, which is described by the equation (in units where e = h¯ = a0 = 1, a0 being the Bohr radius)   Z 2 (13.61) ∇ ψ + 2 E + − F x ψ = 0. r The Schrödinger equation separates by using parabolic coordinates, ξ = r + x,

η = r − x,

tan φ =

x , y

(13.62)

also adopted by Majorana, but the problem is apparently more complicated than a simple determination of bound state energy levels, and no exact solution has been undertaken. Indeed, at large distances from the proton, the potential energy becomes infinitely negative, and the electric field, however weak, eventually strips off the atomic electron: this was precisely the regime considered by Majorana in his notebooks. Usually [181], separation is achieved by means of two separation 11 See Sect. 4.2 in Ref. [17].

13.2 Majorana as a phenomenologist

261

constants Z1 , Z2 with the constraint Z1 + Z2 = Z, but here he introduced just one separation constant λ (thus effectively neglecting the linear Stark effect), the reasoning going as follows. The solution of Eq. (13.61) is written, in terms of parabolic coordinates, as ψ = P(ξ ) Q(η) eimφ , where m is the magnetic quantum number, and the functions P, Q satisfy (from Eq. (13.61)) the following equations:   1  m2 1 Z+λ F  − ξ P = 0, P + P + − 2 + E + ξ 2 2ξ 4   4ξ 2 (13.63) Z − λ F 1 m 1 Q + Q + − 2 + E + + η Q = 0. η 4η 2 2η 4 Majorana focused on the ground state of the atom, for which m = 0, and on the effect induced by small fields, F = 2 ( being a small quantity). The functions P, Q are written as √

P = u e−

−E/2 ξ

,

√

Q = v e−

−E/2 η

,

and the quantities u, v are the solutions of the following equations:   √   √  1 Z + λ − −2E   − −2E + u − ξ = 0, u +u ξ 2ξ 2   √   √  −2E 1 Z − λ − − −2E + v + η = 0. v + v η 2η 2

(13.64)

(13.65) (13.66)

For the solution of the first equation (similar considerations hold for the second one), we set u = 1 + a1 ξ + a2 ξ 2 + a3 ξ 3 + . . . ;

(13.67)

and, [from Eq. (13.65)], the coefficients will be determined from the relation an =

+ √ 1 *  an−3 . (2n − 1) −2E − (Z + λ) an−1 + 2 2n 2n2

(13.68)

However, we can no longer require that u be a finite polynomial; in the presence of an electric field, strictly stationary discrete states do not exist. Thus, we have to use a method of iterative approximations in which u is a finite polynomial except for terms that go to zero with  more rapidly than  n . This means that we neglect quantities that become appreciable at larger distances from the nucleus as long as  decreases. We then set (1) 2 (2) 3 (3) an = a(0) n +  an +  an +  an + . . . , (i)

and the order at which each series i for the constants an terminates, depends on n.

(13.69)

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A perturbative expansion is also applied to the separation constant λ and to the “energy,” λ =  λ1 +  2 λ2 +  3 λ3 + . . .

√ −2E = Z +  k1 +  2 k2 +  3 k3 + . . . , and, after some algebra, Majorana arrived at the following second-order expression for the complete eigenfunction:     9 2 ξ +η u(ξ ) v(η), F ψ = exp − Z + 4Z 5 r

(13.70)



u=1 − + v=1 + +

 1 1 2 F ξ + ξ 2Z 2 8Z   11 2 1 1 9 2 3 4 ξ + ξ + ξ + ξ , F 8Z 5 32Z 4 12Z 3 128Z 2   1 1 2 η η + F 2 2Z 8Z   11 2 1 1 9 2 3 4 η + η + η + η , F 8Z 5 32Z 4 12Z 3 128Z 2

corresponding to the energy E=−

9 2 Z2 F − 2 4Z 4

(13.71)

(λ = F/Z 2 + 0 F 2 = F/Z 2 ). Equation (13.71) is the well-known limiting expression for the second-order Stark energy for the case considered (see, for example, Ref. [181]) but, as noted above, apart for the mathematical method employed, the relevance of Majorana’s study lies in the analytic derivation of the secondorder wavefunction in Eq. (13.70). Majorana also provided the same expression in rectangular coordinates and, even more relevantly, its expansion in spherical functions; at second order in F he found    2   3r r r3 r4 r2 P1 (cos θ) ψ = e−Zr 1 + F 2 − F + + + 4Z 4 4Z 3 24Z 2 Z2 2Z  2   5r 5r3 r4 2 P2 (cos θ) , +F + + (13.72) 8Z 4 12Z 3 12Z 2 where P (cos θ) are Legendre polynomials. He concluded his analysis by calculating also the square of the above expression (at second order), obviously related to

13.3 Majorana as a theoretician: a unifying model for the

fundamental constants263

the ionization probability:      2 5r3 r4 r2 2r 2 −2Zr 2 11r + + + ψ =e P1 (cos θ) 1+F −F 6Z 4 6Z 3 6Z 2 Z2 Z    23r2 3r3 r4 P + + (cos θ) . (13.73) + F2 2 12Z 4 2Z 3 3Z 2 13.3 Majorana as a theoretician: a unifying model for the fundamental constants Majorana’s great interest in electrodynamics, in its classical or (especially) quantum form, has been illustrated in some detail in Chapters 9, 10, and 12. His line of thinking goes well beyond mere applications or generalizations of standard electrodynamics, and it is intriguing that once again Majorana tackled some fundamental questions when he was still a student, as early as 1928. Indeed, in his personal notebooks,12 Majorana attempted to find a relation between the fundamental constants e, h, and c, or, more explicitly, an expression for the elementary charge e appearing in the Coulomb law, describing, for example, the electrostatic force between two electrons. Majorana’s reasoning went as follows. In the region of space surrounding the two electrons, which are a distance  apart, an electromagnetic field is present that “in some sense” (Majorana’s words) is quantized. Here we should point out the first interesting thing since, in what follows it is evident that what is considered “quantized” by Majorana is undoubtedly the electromagnetic field acting between the two electrons. Nevertheless, differently from what appears elsewhere in his notes, Majorana does not refer explicitly to an electromagnetic field, but rather to the “aether” surrounding the two electrons, that is the physical space itself (in the improper language of that time). The two electrons are assumed to interact between themselves by means of a point particle (what we could term the quantum of the electromagnetic field) moving with a group velocity equal to the speed of light, c. This particle is further assumed to propagate freely from one electron to the other, whereas it reverses its motion when “colliding” with the electrons. The momentum of the quantum particle is deduced by a sort of Sommerfeld quantization rule, nh |p| = , 2 with n = 1. The “kick” received by the electron is therefore 2 |p| = h/, and, since the number of collisions per unit time is given by 1/T = c/2, the magnitude 12 See Sect. 2.30 in Ref. [17].

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of the force acting on each electron is then F = 2 |p| /T = hc/22 . By equating this expression to the Coulomb law, Majorana finally  deduces the expression he was searching for regarding the elementary charge e = hc/2, which, however, as pointed out explicitly by himself, gives a value “21 times greater than the real one.” Although his calculations led to erroneous results, the discussion is interesting for many reasons, one of which we have already mentioned. Indeed, first of all, the mechanical model considered here is the first to have been proposed as an attempt to deduce the value of the electric charge (the famous paper by Dirac in Ref. [99] was published a few years after). However, as should be evident the interpretation given by Majorana of the quantized electromagnetic field substantially coincides with that introduced more than a decade later by Feynman in quantum electrodynamics [334, 335], the point particles exchanged by electrons being assumed to be the photons. It is, then, remarkable that fundamental questions regarding the constants of Nature discussed formally (and also extensively) only in more recent times were addressed in pioneering works by Majorana as early as the end of the 1920s, when the key ideas of quantum field theory had just begun to appear.

13.4 Majorana as a mathematician We can find a number of mathematical derivations, formulae, etc., in Majorana’s personal notebooks [17, 18], ranging from numerical calculations to topics in advanced calculus, differential geometry, probability, group theory, and so on. Practically all of them were aimed to serve as useful tools in his elaboration of theoretical physics (or to be used in his undergraduate studies), and the originality of many of them lies only in their simple and clear derivation. Nonetheless, several of them also present interesting and original results, as we have seen throughout this book, especially concerning solutions of differential equations and in grouptheory topics. In the following, we will dwell on just two more issues, pointing out Majorana’s complete mastery of the subject, which are particularly useful in present-day research.

13.4.1 Improper operators Probably, the studies concerning relativistic wave equations stimulated Majorana to consider an integral representation of the square root of the Laplacian operator (which appears, however, also in different branches of physics). The method he adopted (and the result obtained) is quite general, and goes as follows.13 13 See Sect. 4.17 in Ref. [17].

13.4 Majorana as a mathematician

265

Let u(x, y, z) be an arbitrary function that can be expanded in harmonic components:  (13.74) u(x, y, z) = α(x, y, z) e2π i(γ1 x+γ2 y+γ3 z) dγ1 dγ2 dγ3 ,  (13.75) α(x, y, z) = u(x, y, z) e−2π i(γ1 x+γ2 y+γ3 z) dx dy dz, and define the operator F r transforming u into the function v(x, y, z) = F r u(x, y, z), whose Fourier integral expansion is given by  v(x, y, z) = λr α(x, y, z) e2π i(γ1 x+γ2 y+γ3 z) dγ1 dγ2 dγ3 ,

(13.76)

(13.77)

where λ =

1 1 =  γ γ12 + γ22 + γ32

(13.78)

is the wavelength of the (γ1 , γ2 , γ3 ) harmonic component. The operator F satisfies the following properties: F r F s = F s F r = F r+s ,

F 0 = 1.

The function v can be determined from u by setting14  v(x, y, z) = Kr (x, y, z; x , y , z ) u(x , y , z ) dx dy dz , where Kr is given by 





Kr (x, y, z; x , y , z ) =



λr e2π i(γ1 ξ +γ2 η+γ3 ζ ) dγ1 dγ2 dγ3 ,

(13.79)

(13.80)

(13.81)

where ξ = x − x , η = y − y , and ζ = z − z . The  integration in Eq. (13.81) may be performed on a sphere of radius D = 1/λ = γ12 + γ22 + γ32 , to obtain  (R = ξ 2 + η2 + ζ 2 ):  ∞  (2πR)r−1 ∞ sin t 2 sin 2πsR Kr (x, y, z; x , y , z ) = Kr (R) = ds = dt. R sr−1 πR2 tr−1 0 0 (13.82) 14 Majorana was aware that possible convergence issues can arise for given r; see the following text.

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Majorana’s multifaceted life

This formula can be used for 1 ≤ r < 3; the expression corresponding to the case r = 1 can be obtained from that corresponding to r = 1 +  by taking the limit  → 0 or, in an equivalent way, by taking the mean value of the integral with arbitrary upper limit. We, thus, find K1 =

1 , π R2

K2 =

π . R

(13.83)

These results allowed Majorana to consider the following integral representations:  1 1 u(x , y , z ) dx dy dz , (13.84) F u(x, y, z) = πR2  π F 2 u(x, y, z) = (13.85) u(x , y , z ) dx dy dz . R Moreover, on applying the Laplace operator on the two sides of Eq. (13.85), he got ∇ 2 F 2 = −4π 2 , and, since (from Eq. (13.77)) F 2 is invertible, ∇ 2 = − 4π 2 F −2 .

(13.86)

The square root of the Laplacian operator is, then, introduced as follows. We can define the operator

√

∇ 2 by setting  ∇ 2 = 2π i F −1 ,

which can be written, using Eqs. (13.86) and (13.79), as  1 1 2 F ∇ . ∇ 2 = 2π i F 1 F −2 = 2π i

(13.87)

(13.88)

√ From this and, from the reciprocal of Eq. (13.87), 1/ ∇ 2 = F 1 /2πi, Majorana √ finally obtained integral representations for ∇ 2 and its reciprocal:   1 ∇ 2 u(x, y, z) = (13.89) ∇ 2 u(x , y , z ) dx dy dz , 2π 2 R2 i  1 1 u(x , y , z ) dx dy dz . (13.90) √ u(x, y, z) = 2 R2 i 2 2π ∇ Note that, with the given representation, the square root of the Laplacian operator is intimately connected to the same Laplacian operator, through Eq. (13.89).

13.4.2 Cubic symmetry As we have seen in Chapter 7, Majorana extensively studied group theory in relation to several problems in the quantum theory, mainly following the text by Weyl [16]. However, he also considered a few other topics, apparently unrelated to his

13.4 Majorana as a mathematician

267

Table 13.2. Direction cosines for class II rotations Direction cosines of the rotation √ √ 0√ 1/ 2 1/√2 1/√2 0√ 1/ 2 1/ 2 1/√2 0√ 0√ 1/ 2 −1/√ 2 −1/√ 2 0√ 1/ 2 1/ 2 −1/ 2 0

Rotation angle (◦ )

Permutation

180 180 180 180 180 180

(14) (24) (34) (23) (31) (12)

physics studies (but still related to given physics problems). Particularly interesting is his treatment of cubic symmetry,15 with reference to rotational octahedral symmetry. The corresponding group is that of the 24 orientation-preserving symmetry of a cube or a regular octahedron, that is the 24 (proper) rotations that transform the x, y, z axes into themselves (except for the order and the direction). This group is holomorphic to the group S4 of permutations of four objects, and Majorana established this holomorphic correspondence by focusing on the corresponding conjugacy classes. These five classes describe isometries of the cube as follows: I – identity; II – six rotations about an axis from the center of an edge to the center of the opposite edge by an angle of 180o ; III – three rotations about an axis from the center of a face to the center of the opposite face by an angle of 180o ; IV – eight rotations about a body diagonal by an angle of 120o ; V – six rotations about an axis from the center of a face to the center of the opposite face by an angle of 90o . Majorana realized the correspondence by giving, for each transformation, the direction cosines of the rotation (with the corresponding rotation angle) and the permutations of the body diagonal (in brackets). For example, for classes II and III he explicitly gives the data reported in Tables 13.2 and 13.3. Given the equivalence, the considered group has five irreducible representations χs (s = 1, 2, 3, 4, 5), whose characters are just that of the group S4 (see Table 13.4), where the “Partitio [numerorum]” denotes the number of ways one can collect the four objects to permute. An irreducible representation Dj (with integer j) of the complete group of spatial rotations is also a (reducible) representation of the considered group. If ns is the mean value of χj ·χs∗ over the elements of this group, the above-mentioned representation can be reduced to ns representations χs . 15 See Sect. 4.14 in Ref. [17].

268

Majorana’s multifaceted life Table 13.3. Direction cosines for class III rotations Direction cosines of the rotation 1 0 0

0 1 0

0 0 1

Rotation angle (◦ )

Permutation

180 180 180

(14) (23) (24) (31) (34) (12)

Table 13.4.

ns

Partitio → Class ↓

4

3+ 1

2+ 2

2+ 1+ 1

1+ 6− 3+ 8+ 6−

(1) . . . (12) . . . (12)(34) (123) . . . (1234)

1 1 1 1 1

3 1 −1 0 −1

2 0 2 −1 0

3 −1 −1 0 1

1+ 1+ 1+ 1 1 −1 1 1 −1

Following Weyl [16], Majorana showed that the characters of Dj are given by sin(2j + 1)ω/ sin ω, where ω is half the rotation angle, so that the values of χj for the five classes of the group considered are 2 j + 1,

(−1)j ,

(−1)j ;

j 1 − rest of , 3

1 + rest of

j j − rest of , 2 4

respectively. By keeping the same order as above, the frequency ns of each irreducible representation is given by n1 = n2 = n3 = n4 = n5 =

      j j 1 j 1 j 1 rest of − rest of − rest of , +1− 12 2 2 3 3 4 4     j j 1 j 1 rest of + rest of , − 4 2 2 4 4     j j 1 j 1 rest of + rest of , − 6 2 2 3 3     j 1 j j + rest of − rest of , 4 2 4 4     1 j 1 j j − rest of + rest of . 12 3 3 4 4

13.4 Majorana as a mathematician

269

Since the degrees of the irreducible representations are 4

3+1

2+2

2+1+1

1+1+1+1

1

3

2

3

1

respectively, Majorana then deduced the following relation: n1 + 3 n2 + 2 n3 + 3 n4 + n5 = 2 j + 1.

(13.91)

We observe that, as in the normal representation, for large j the frequencies of appearance of the irreducible representations are proportional to their degrees. Moreover, if the values of ns for a given value of j are known, we can obtain the values corresponding to j + 12q from the following scheme: j

=

j + 12 q

n1 n2 n3 n4 n5

= = = = =

n1 n2 n3 n4 n5

+ + + + +

1·q 3·q 2·q 3·q 1·q

where the coefficients of q are exactly the degrees of the irreducible representations.

This allowed Majorana to evaluate the values of ns from j = 0 to j = 11 only, the others being deducible from these. The results he obtained are summarized in Table 13.5.

Table 13.5. Frequencies of the irreducible representations j

n1

n2

n3

n4

n5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 ...

1 0 0 0 1 0 1 0 1 1 1 0 1+1 0+1 ...

0 0 1 1 1 1 2 2 2 2 3 3 0+3 0+3 ...

0 0 1 0 1 1 1 1 2 1 2 2 0+2 0+2 ...

0 1 0 1 1 2 1 2 2 3 2 1 0+3 1+3 ...

0 0 0 1 0 0 1 1 0 1 1 1 0+1 0+1 ...

270

Majorana’s multifaceted life

13.5 Majorana as a teacher Majorana revealed a genuine interest in advanced physics teaching early in 1933, soon after he obtained (at the end of 1932; see Chapter 1) the professorship degree of libero docente. He did propose some academic courses to be delivered at the University of Rome [24], and the programs of three courses that he would have taught between 1933 and 1937 are reported in Tables 13.6–13.8. Although Majorana never gave these lectures, probably due to the lack of students, they are particularly interesting and informative due to a very careful choice of the topics he intended to cover. The first two programs were for courses in mathematical methods of quantum mechanics and atomic physics, respectively, and the second program contains references to more phenomenological topics. In presenting quantum mechanics, Majorana aimed to use group-theoretic methods, a very unusual approach for the time, as recalled in Section 7.1: at least in Italy, this approach would become standard only many years after World War II. The third program deals with quantum electrodynamics, which was again an unusual topic for Italian academic courses of that time, and one that had been fascinating Majorana for a long time. Majorana, instead, effectively lectured on theoretical physics only in 1938 when, as recounted in Chapter 1, he obtained a position as a full professor at the

Table 13.6. Program of Majorana’s first course Title: Mathematical methods of quantum mechanics Academic year: 1933–4 Topics to be covered: (1) Unitary geometry. Linear transformations. Hermitian operators. Unitary transformations. Eigenvalues and eigenvectors. (2) Phase space and the quantum of action. Modifications to classical kinematics. General framework of quantum mechanics. (3) Hamiltonians which are invariant under a transformation group. Transformations as complex quantities. Non-compatible systems. Representations of finite or continuous groups. (4) General elements on abstract groups. Representation theorems. The group of spatial rotations. Symmetric groups of permutations and other finite groups. (5) Properties of the systems endowed with spherical symmetry. Orbital and intrinsic momenta. Theory of the rigid rotator. (6) Systems with identical particles. Fermi and Bose–Einstein statistics. Symmetries of eigenfunctions in center-of-mass frames. (7) The Lorentz group and spinor calculus. Applications to the relativistic theory of elementary particles. Date of submission: May, 1933

13.5 Majorana as a teacher

271

Table 13.7. Program of Majorana’s second course Title: Mathematical methods of atomic physics Academic year: 1935–6 Topics to be covered: Matrix calculus. Phase space and the correspondence principle. Minimal statistical sets or elementary cells. Elements of quantum dynamics. Statistical theories. General definition of symmetry problems. Representations of groups. Complex atomic spectra. Kinematics of the rigid body. Diatomic and polyatomic molecules. Relativistic theory of the electron and the foundations of electrodynamics. Hyperfine structures and alternating bands. Elements of nuclear physics. Date of submission: April 30, 1935

Table 13.8. Program of Majorana’s third course Title: Quantum electrodynamics Academic year: 1936–7 Topics to be covered: Relativistic theory of the electron. Quantization procedures. Field quantities defined by commutability and anticommutability laws. Their kinematical equivalence with sets with an undetermined number of objects obeying Bose–Einstein or Fermi statistics, respectively. Dynamic equivalence. Quantization of the Maxwell–Dirac equations. Study of relativistic invariance. The positive electron and the symmetry of charges. Several applications of the theory. Radiation and scattering processes. Creation and annihilation of opposite charges. Collisions of fast electrons. Date of submission: April 28, 1936

University of Naples. He started lecturing on January 13 and continued to do so until March 24, one day before his disappearance. Even though he was at the university for only two months, his activity was intense, and his interest in teaching was extremely high. Indeed, he prepared careful notes for his lectures [26] for the benefit of his students, from which (see Table 13.9) we can easily infer how innovative was the design of his course. Majorana gave a broad outline of his course in the opening lecture, delivered on January 13, 1938. His goal was the study of quantum mechanics and its applications to atomic physics. He also stated his teaching method, which would consist of a combination of: the “mathematical method,” through which “the quantum formalism is presented in its most general and, therefore, clearest propositions from the very beginning and only afterwards are the criteria for applying it explained”;

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Table 13.9. Majorana’s lectures in Naples (see Ref. [26]) Title: Theoretical physics Academic year: 1937–8 Lecture N.1 Opening address to the course. Introduction: topics to be covered during the course. Notes for lectures N.2–N.5 were probably never written by Majorana. Lecture N.6 1. Fine structure formula. Its experimental validation and interpretative difficulties. 2. Finite nuclear mass correction. Lecture N.7 1. Space quantization and magnetic properties of atoms. 2. About atomic spectra of alkalis. The spinning electron hypothesis. Lecture N.8 1. The Pauli principle (or exclusion principle) and the interpretation of the atomic table of elements. 2. Sommerfeld conditions for the energetic levels of alkalis. Lecture N.9 1. Spectrum of atoms with two valence electrons. 2. Classical radiation theory. Lecture N.10 1. Integration of the Maxwell equations and application to the radiation of an oscillating system with an amplitude that is small with respect to the wavelength of the emitted wave. 2. Scattering of sunlight by atmosphere. Lecture N.11 1. The relativity principle in classical mechanics. 2. Michelson–Morley experiment. 3. Lorentz transformations. Lecture N.12 1. The relativity principle according to Einstein. 2. Transformation law for the electromagnetic potentials. Lecture N.13 1. Fresnel formula and the Fizeau experiment. 2. Relativistic invariance of the electric charge. 3. Minkowski space. 4. Equations of motion for an electron in an arbitrary electromagnetic field. Continued on next page

13.5 Majorana as a teacher

273

Table 13.9 – Continued from previous page Lecture N.14 1. Relativistic dynamics of the electron. 2. Photoelectric effect – relation between the electric potential and the wavelength. 3. Free electron scattering (Thomson formula). Lecture N.15 1. Compton effect. 2. Franck–Hertz experiment. Lecture N.16 Matrix calculus. 1. Vector spaces in n dimensions. 2. Matrices and linear operators. Lecture N.17 1. Unitary systems. 2. Hermitian operators. Hermitian forms. Lecture N.18 1. Diagonalization of commutable operators. 2. Infinite matrices. Lecture N.19 1. Fourier integrals. Wave mechanics. 2. de Broglie waves. Lecture N.20 1. Phase and group velocity. 2. Non-relativistic wave equation. Statistical interpretation of the wave packets. Lecture N.21 1. First extension of the statistical interpretation and uncertainty relations. Class start date: January 13, 1938 Class end date (effective): March 24, 1938

and the “historical method,” which “explains how the first idea of the formalism appeared.” Majorana regarded them as two opposite methods, but he aimed at a fruitful synergy between them.16 The first part of Majorana’s course on theoretical physics dealt with the phenomenology of the atomic physics and its interpretation in the framework of the 16 Majorana’s ideas might be compared with the preface to Dirac’s 1930 book on quantum mechanics [375],

where Dirac wrote that the mathematical presentation of quantum theory implied “a complete break from the historical line of development.”

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old quantum theory of Bohr–Sommerfeld. In particular, starting from the available spectroscopic data, he introduced the spin and Pauli’s exclusion principle, accounting for the periodic table of the elements and the features of the spectra of one- and two-electron atoms in the Sommerfeld theory. This part of the course was quite analogous to the course given by Fermi in Rome (1927–8) attended by Majorana as a student. The second part started with classical radiation theory, reporting explicit solutions of the Maxwell equations, scattering of the solar light, and some other applications. He then discussed the theory of special relativity, starting from simple phenomenology and later introducing the appropriate mathematical formalism. Lorentz transformations along with their immediate consequences are introduced in a simple and original way, and applications to the electromagnetic field are extensively considered. Particular emphasis is given to the relativistic dynamics of electrons, which Majorana obtained from a variational principle. Finally, a discussion of several effects, such as the photoelectric effect, Thomson scattering, the Compton effect, and the Franck–Hertz experiment were addressed: Majorana considered these phenomena as the true phenomenological bases for quantum mechanics, although the topics of the first part of his course relied upon spectroscopic results. The last part of the course was, instead, more mathematically oriented. After a detailed discussion of some relevant topics on matrix and operator theory and Fourier transforms, Majorana presented the (non-relativistic) Schrödinger wave equation along with its statistical interpretation based on the Heisenberg uncertainty principle. This part did not follow the Fermi approach, but rather referred to previous personal studies by Majorana, also following the original book by Weyl [16] on group theory and quantum mechanics, the use of group theory reflecting Majorana’s attention to conciseness and generality. A similar approach is present in the three proposed programs of 1933–6, but some comments are needed here. The 1933 course on “Mathematical methods of quantum mechanics” (see Table 13.6) is based entirely on group-theoretic methods. In the corresponding program, Majorana mentioned the elements of unitary geometry and transformations, and the invariance under groups of transformation. Then he went through the rotation and permutation groups, the applications (for example, angular momenta and systems with identical particles) remaining within the formalism of group theory. There, the standard presentation of quantum mechanics (in both the Schrödinger and the Heisenberg version) was completely ignored, and the theory of relativity was mentioned in the framework of the Lorentz group and of spinor calculus. Almost all the topics of the 1933–4 program were treated explicitly in Weyl’s book. For the 1935 course on “Mathematical methods of atomic physics” (see Table 13.7), a more phenomenological approach emerges, and the group-theoretic

13.5 Majorana as a teacher

275

methods appeared only marginally: they are subordinate to practical applications, such as complex atomic spectra and hyperfine structures.17 The 1936 course on “Quantum electrodynamics” was largely based on personal studies, and has, therefore, no relation to the other courses. From a historical and teaching viewpoint, the main feature of the 1938 course on “Theoretical physics” is, evidently, the judicious mixture of physical intuition and mathematical formalism (see Table 13.9), which both contribute to Majorana’s clear overview of the arguments discussed, without sacrificing completeness and generality. As an example, the discussion of some advanced topics of classical electromagnetism in lectures N.9 and N.10 is preparatory to understand fully the photoelectric effect and the Compton effect, the explanations of which also involve the theory of relativity expounded in the subsequent four lectures. Upon these effects, Majorana established, in lecture N.19, the foundations of Schrödinger’s quantum mechanics, through the agency of de Broglie waves, whose justification requires, in turn, the basic concepts of Einstein’s theory of relativity. Also, if it is true that, in lecture N.20, atomic physics applications drive the elaboration of the novel quantum mechanics, then the full understanding of this new theory comes from Majorana through the development of the appropriate mathematical formalism, i.e. (finally) Fourier integrals. As a result, Majorana’s legacy in teaching quantum mechanics consists of a fruitful mixture of an original approach – very similar to that of today’s courses on quantum mechanics – and of some consolidated lines of development, which he clearly inherited from Fermi’s courses that he followed in Rome a decade earlier [24]. Both the lecture notes for the Naples course and the programs of the three courses that Majorana submitted in Rome between 1933 and 1936 reveal his leading-edge approach and his search for new ways of dealing with quantum mechanics.

17 This decision likely reflects the research program pursued by Fermi’s group, which was engaged in

spectroscopic research before switching to nuclear physics [24].

Part VII Beyond Majorana

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14 Majorana and condensed matter physics F. Wilczek

Ettore Majorana contributed several ideas that have had a significant, lasting impact on condensed matter physics, broadly construed. In this chapter I will discuss, from a modern perspective, four important topics that have deep roots in Majorana’s work. 1. Spin response and universal connection: In paper N.6, Majorana considered the coupling of spins to magnetic fields. The paper is brief, but it contains two ingenious ideas whose importance extends well beyond the immediate problem he treated, and should be part of every physicist’s toolkit. The first of those ideas is that, having solved the problem for spin 1/2, one can deduce the solution for general spins by pure algebra. Majorana’s original construction uses a rather specialized mathematical apparatus. Bloch and Rabi, in a classic paper [62], brought it close to the form discussed in Section 14.1. Rephrased in modern terms, it is a realization – the first, I think, in physics – of the universality of non-Abelian charge transport (Wilson lines). 2. Level crossing and generalized Laplace transform: In the same paper, Majorana used an elegant mathematical technique to solve the hard part of the spin dynamics, which occurs near level crossings. This technique involves, at its center, a more general version of the Laplace transform than its usual use in constant-coefficient differential equations. Among other things, it gives an independent and transparent derivation of the celebrated Landau–Zener formula for non-adiabatic transitions. (Historically, Majorana’s work on the problem, and also that of Stückelberg [75], was essentially simultaneous with Landau’s [73] and Zener’s [74].) Majorana’s method is smooth and capable of considerable generalization. It continues to be relevant to contemporary problems. 3. Majorana fermions: from neutrinos to electrons: Majorana’s most famous paper N.9 concerns the possibility of formulating a purely real version of

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the Dirac equation. In a modern interpretation, this is the problem of formulating equations for the description of spin-1/2 particles that are their own antiparticles: Majorana fermions. Majorana’s original investigation was stimulated in part by issues of mathematical esthetics, and in part by the physical problem of describing the then hypothetical neutrino. As described in Chapter 15, the immediate issue it raised – i.e. whether neutrinos are their own antiparticles – has become a central issue in modern particle theory that still remains unresolved experimentally. Here I will show that Majorana’s basic idea is best viewed within a larger context, where we consider it as a tool for analyzing unusual, number-violating forms of mass. Within that larger conception we find, notably, that electrons within superconductors, when their energy is near to the Fermi surface, behave as Majorana fermions. 4. Majorinos and emergent symmetry: A very special and unusual kind of Majorana fermion – which, indeed, stretches the notion of “fermion” – has been the focus of much recent attention, both theoretical and experimental. These are zero-energy excitations that are localized around specific points, such as the ends of wires, or nodes within circuits of specially designed superconducting materials. Regarded as particles, they represent (0 + 1)-dimensional (0 space, 1 time) versions of the more familiar (3 + 1)-dimensional Majorana fermions, and in addition they have zero mass. Since they are so small, both in extent and in weight, I have proposed [376] to call them majorinos. Majorinos have remarkable properties, including unusual quantum statistics, and might afford an important platform for quantum information processing [158].

14.1 Spin response and universal connection The time-dependent Hamiltonian H(t) = γ B(t) · J

(14.1)

governs the interaction between a spin J and a time-dependent magnetic field B(t). The dynamical evolution implied by Eq. (14.1), i

dψ = Hψ, dt

(14.2)

for the spin wavefunction ψ, is the most basic building block for all the many applications of magnetic resonance physics. Its solution (see paper N.6) is foundational. The J matrices obey the commutation relations of angular momentum. Applied to a particle (which may be a nucleus, or an atom or molecule) of spin j, they obey the algebra

14.1 Spin response and universal connection

281

[Jk , Jl ] = i klm Jm , J2 = j(j + 1),

(14.3)

which is realized in a (2j + 1)-dimensional Hilbert space. The representation is essentially unique, up to an overall unitary transformation. In the simplest case, j = 1/2, we have a two-dimensional Hilbert space and we can take J = σ /2, with the Pauli matrices       0 1 0 −i 1 0 , σ2 = , σ3 = . (14.4) σ1 = 1 0 i 0 0 −1 The differential equation Eq. (14.2) becomes a system of two first-order equations that is reasonably tractable. (See the following.) For larger values of j we need to organize the calculation cunningly, lest it spin out of control. As Majorana recognized, group theory comes to our rescue. Here I will rephrase his basic insight in modern language. We can write the solution of Eqs. (14.1) and (14.2) as a time-ordered exponential, ⎧ ⎫ t ⎨ ⎬ ψ(t) = T exp −iγ dt B · J ψ(0) ≡ U(t)ψ(0). (14.5) ⎩ ⎭ 0 j

Now, if we write Bj → A0 , and view it as the scalar potential associated with an SU(2) symmetry, we recognize that the ordered exponential in Eq. (14.5) is an example of a Wilson line, which implements parallel transport according to the non-Abelian potential A. In the context of gauge theories, it is a familiar and basic result that if one knows how to parallel transport the basic representation (vectors), then one can construct the parallel transport of more complicated representations (tensors) uniquely, by algebraic manipulations, without solving any additional differential equations. This is a consequence – in some sense, it is the essence – of the universality of non-Abelian gauge couplings. In the present case, we can carry the procedure out quite explicitly. The most convenient way is to realize the spin-j object as a symmetric spinor [64]. Thus we write ψ = ψ α1 ...α2j ,

(14.6)

where the αk are two-valued indices and ψ is invariant under their permutation. 1 Then if U ( 2 ) = Vβα is the Wilson line for spin 1/2, the Wilson line for spin j will be α ...α

α

U (j) = (U (j) )β11 ...β2j2j = Vβα11 ...Vβ2j2j ≡ V ⊗ ... ⊗ V,

(14.7)

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Majorana and condensed matter physics

where the tensor product contains 2j copies. This follows from the underlying realization of the algebra of J(j) for spin j as 1

1

J(j) = J( 2 ) ⊗ 1 ⊗ ... ⊗ 1 + ... + 1 ⊗ ... ⊗ 1 ⊗ J( 2 ) ,

(14.8)

including 2j summands. From this point on, starting from Eq. (14.7), the algebra required to develop concrete formulae, ready for use, is straightforward. For Majorana to have exploited non-Abelian symmetry in such a sophisticated way, in 1932, was not only useful, but also visionary.

14.2 Level crossing and generalized Laplace transform Having emphasized the value of solutions of Eqs. (14.1) and (14.2) for spin 1/2, we turn to the task of obtaining them. The general problem can only be solved numerically. There is, however, a powerful approximate result available when the evolution is slow and smooth, namely the adiabatic theorem. It states, roughly speaking, that if the magnetic field varies slowly on the scale set by the (inverse) energy splittings |γ B|−1 , states initially occupying an energy eigenstate will remain within that energy eigenstate, as it evolves continuously – there are no “quantum jumps.” (One can refine this further to make statements about the phase of the wavefunctions. This leads into interesting issues connected with the geometric or Berry phase, but I will not pursue that line here.) The conditions for the adiabatic theorem will fail when energy levels approach one another, so it is important to have a bridging formula, to interpolate through two such extreme cases. The discussion that follows is basically an unpacking of Majorana’s extremely terse presentation in his paper N.6. As an interesting, experimentally relevant situation, which captures the essence of the problem, let us consider the Schrödinger equation describing a situation where the field-induced splitting between two nearby levels goes through a zero. We suppose that during the crucial time, when the total splitting is small, we can take the field to behave linearly. This leads us to analyze the Schrödinger equation  ψ≡

α β

iψ˙ = Hψ,

 ,

H = − t σ3 + ε σ1 .

(14.9)

In terms of components, we have iα˙ = − t α + εβ, iβ˙ = + t β + εα.

(14.10)

14.2 Level crossing and Laplace transform

283

By differentiating the first member of Eq. (14.10), using the second member to eliminate β˙ in terms of α, β, and finally using the first member again to eliminate β in favor of α, α˙ we derive a second-order equation for α alone: α¨ = i  α − ε2 α − ( t)2 α.

(14.11)

In order to use the generalized Laplace transform in a simple way, it is desirable to have t appear only linearly. We can achieve this by introducing γ ≡ e−i  t

2 /2

(14.12)

α.

In this way we find γ¨ + 2i  t γ˙ + ε2 γ = 0.

(14.13)

For later use, note that γ˙ = −i  e−i  t

2 /2

β,

so that we can obtain both β, and of course α, from γ . Now we express γ (t) in the form  γ (t) = ds es t f (s),

(14.14)

(14.15)

over an integration contour to be determined. Operating formally, we thereby re-express Eq. (14.13) as  (14.16) ds es t (−i  t s + s2 + ε2 ) f (s) = 0. Using d st d (e s f ) − es t (s f ), ds ds we cast Eq. (14.16) into the tractable form  

ds es t  s f  + ( + i s2 + iε2 ) f = 0, es t s t f (s) =

(14.17)

(14.18)

under the assumption that es t s f vanishes at the ends of the contour (if any). This leads to a simple differential equation for f (s), which is solved by f (s) = As−1−iε

2 /2

e−is

2 /2

(14.19)

for constant A, which we will choose to be A = 1. We will also assume  > 0. As τ → ±∞ the levels are highly split, and the adiabatic theorem comes into play, forbidding further transitions. So the most interesting problem, to compute the probability for transitions which violate the adiabatic theorem – “quantum

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Majorana and condensed matter physics

jumps” – can be investigated by studying the asymptotics as τ → ±∞. In those limits our contour integral contains a parametrically large exponential, and we can try to exploit the stationary phase approximation. The stationary phase points, in those limits, occur at s = −i  t.

(14.20)

A short calculation reveals that we should orient our contour with s − s0 ∝ (1 − i) near these points, to achieve steepest descent. This suggests that we use the contours s = p(1 − i) − i  t,

(14.21)

with p real. One verifies that es t sf vanishes rapidly for large |p|, so these contours satisfy our consistency condition. There is a subtlety here that proves crucial. In defining s−1−iε

2 /2

≡ eln s (−1−iε

2 /2)

we must choose a branch of the logarithm. Let us take its principal value, maintaining a cut along the negative real axis. The candidate contours of Eq. (14.21) cross the real axis at s = p = −i  τ . We see – noting  > 0 – that for negative t the contour avoids the cut, but for positive t we must deform the contour to avoid the cut. We do this by allowing it to run just above the cut from s = i  − i  τ to s = i  and then back below the cut from s = −i  to s = −i  − i  τ , where it rejoins its original trajectory. For τ < 0 (and |τ | large) the saddle point saturates the integration, and we find √ 2 2 2 (14.22) γ ≈ π · e−i t /2 · ( |t|)−1−iε /2 · e−iπ/2 eπ ε /4 . Let us review the origin of the various factors. The first arises from the final Gaussian integral over p. The second is the value of the exponential at the stationary phase point. The third and fourth come, respectively, from inserting the two terms of the logarithm π ln(−i  t) = ln(i  |t|) = ln( |t|) + i (14.23) 2 into s−1−iε

2 /2

= eln s (−1−iε

2 /2)

.

(14.24)

We see that γ vanishes for large −t while, according to Eq. (14.14), β has a finite limit. For τ > 0 we still have a saddle-point contribution, very similar to Eq. (14.22), 2 2 but with the exponential factor eπ ε /4 → e−π ε /4 . The difference arises from the

14.3 Majorana fermions and mass

285

change in sign of the imaginary part of the complex logarithm in Eq. (14.23). We will also have a contribution from the integral over the cut. It is not impossible to evaluate that integral, but we can get the most important result without it. We need only to observe that the contour integral is non-oscillatory, so it does not contribute to β. Thus in the limit τ → −∞ the wavefunction contains only a lower (β) 2 term, and is proportional to eπ ε /4 , while in the limit τ → −∞ the β term is 2 proportional to e−π ε /4 , with the other factors either in common or pure phases. We conclude that the probability for the state to evolve from negative to positive energy, making a quantum jump, is the square of the ratio, or e−π ε

2 /

(14.25)

– which is the classic “Landau–Zener” result. Majorana’s method is very well adapted to generalizations involving multiple level crossings. By keeping track of the phases, one can also find useful analogs of the geometric phase; and, by applying group theory, also analyze multiplet crossings systematically.1 It is interesting to contemplate an ambitious program in which one might infer the time dependence of states from the time-dependent energy levels, by patching together integral representations of the type just discussed, using these sorts of linear models to interpolate between adiabatic evolution.

14.3 Majorana fermions and Majorana mass: from neutrinos to electrons 14.3.1 Majorana’s equation In 1928, Dirac [97] proposed his relativistic wave equation for electrons, today of course known as the Dirac equation. This was a watershed event in theoretical physics, leading to a new understanding of spin, predicting the existence of antimatter, and impelling – for its adequate interpretation – the creation of quantum field theory. It also inaugurated a new method in theoretical physics, emphasizing mathematical esthetics as a source of inspiration. Indeed, when we venture into the depths of the quantum world, “physical intuition” derived from the common experience of humans interacting with the world of macroscopic objects is of dubious value, and it seems inevitable that other, more abstract, principles must take its place to guide us. Majorana’s most influential work, paper N.9, builds directly on Dirac’s, both in content and in method. By posing, and answering, a simple but profound question about Dirac’s equation, Majorana expanded our concept of what a particle might be, and, on deeper analysis, of what mass is. For 1 Shapere A. and Wilczek F. Transitions and phases in nonabelian crossing (paper in preparation at time of

writing).

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Majorana and condensed matter physics

many years Majorana’s ideas appeared to be ingenious but unfulfilled speculations; recently, however, they have come into their own. They now occupy a central place at several of the most vibrant frontiers of modern physics, including – perhaps surprisingly – condensed matter physics. To appreciate the continuing influence of Majorana’s famous “Majorana fermion” paper N.9 in contemporary condensed matter physics, it will be helpful first to review briefly its central idea within its original context of particle physics. I will do this in modern language, but with an unusual emphasis that leads naturally into the generalizations we will be considering. The companion Chapter 15 contains a much more extensive discussion of the particle physics around Majorana fermions. Dirac’s equation connects the four components of a field ψ. In modern (covariant) notation it reads (iγ μ ∂μ − m)ψ = 0.

(14.26)

The γ matrices must obey the rules of Clifford algebra, i.e. {γ μ γ ν } ≡ γ μ γ ν + γ ν γ μ = 2δ μν ,

(14.27)

where δ μν is the metric tensor of flat space. Spelling it out, we have (γ 0 )2 = −(γ 1 )2 = −(γ 2 )2 = −(γ 3 )2 = 1; γ j γ k = −γ k γ j ,

for i = j.

(14.28) (14.29)

(I have adopted units such that h¯ = c = 1.) Furthermore, we require that γ 0 be Hermitian, the others anti-Hermitian. These conditions insure that the equation properly describes the wavefunction of a spin-1/2 particle with mass m. Dirac found a suitable set of 4×4 γ matrices, whose entries contain both real and imaginary numbers. For the equation to make sense, then, ψ must be a complex field. Dirac, and most other physicists, regarded this as a good feature, because electrons are electrically charged, and the description of charged particles requires complex fields, even at the level of the Schrödinger equation. Another perspective comes from quantum field theory. In quantum field theory, if a given field φ creates ¯ then the complex conjugate φ ∗ will the particle A (and destroys its antiparticle A), create A¯ and destroy A. Particles that are their own antiparticles must be associated with fields obeying φ = φ ∗ , that is, real fields. The equations for particles with spin 0 (Klein–Gordon equation), spin 1 (Maxwell equations), and spin 2 (Einstein equations, derived from general relativity) readily accommodate real fields, since the equations are formulated using real numbers. Neutral π mesons π 0 , photons, and gravitons are their own antiparticles, with spins 0, 1, 2, respectively. But since electrons and positrons are distinct, the associated fields ψ and ψ ∗ must be distinct; and this feature appeared to be a natural consequence of Dirac’s equation.

14.3 Majorana fermions and mass

287

In his 1937 paper, Majorana posed, and answered, the question of whether equations for spin-1/2 fields must necessarily, like Dirac’s original equation, involve complex numbers. Considerations of mathematical elegance and symmetry both motivated and guided his investigation. Majorana discovered that, to the contrary, there is a simple, clever modification of Dirac’s equation that involves only real numbers. Indeed, once having posed the problem, it is not overly difficult to find a solution. The matrices γ˜ 0 = σ2 ⊗ σ1 ,

γ˜ 1 = iσ1 ⊗ 1,

or, in expanded form, ⎛ ⎞ 0 0 0 −i ⎜ 0 0 −i 0 ⎟ ⎟ γ˜ 0 = ⎜ ⎝ 0 i 0 0 ⎠, i 0 0 0 ⎛

i ⎜ 0 γ˜ 2 = ⎜ ⎝ 0 0

⎞

0 0 0 i 0 0 ⎟ ⎟, 0 −i 0 ⎠ 0 0 −i

γ˜ 2 = iσ3 ⊗ 1, ⎛

0 ⎜ 0 γ˜ 1 = ⎜ ⎝ i 0 ⎛

0 ⎜ 0 γ˜ 3 = ⎜ ⎝ 0 −i

γ˜ 3 = iσ2 ⊗ σ2 ,

0 0 0 i 0 0 i 0

i 0 0 0

(14.30)

⎞ 0 i ⎟ ⎟, 0 ⎠ 0 ⎞

(14.31)

0 −i i 0 ⎟ ⎟, 0 0 ⎠ 0 0

satisfy the same algebra Eq. (14.27) as Dirac’s, and are purely imaginary. Majorana’s version of the Dirac equation, (iγ˜ μ ∂μ − m)ψ˜ = 0,

(14.32)

therefore has the same desirable symmetry properties, including Lorentz invariance. But, since the γ˜ μ are pure imaginary, the iγ˜ μ are real, and so Majorana’s ˜ equation can govern a real field ψ. With this discovery, Majorana made the idea that spin-1/2 particles could be their own antiparticles theoretically respectable, that is, consistent with the general principles of relativity and quantum theory. In his honor, we call such hypothetical particles Majorana fermions.

14.3.2 Analysis of Majorana neutrinos Majorana speculated that his equation might apply to neutrinos. In 1937, neutrinos were themselves hypothetical, their properties unknown. The experimental study of neutrinos commenced with their discovery [134] in 1956, but their observed properties seemed to disfavor Majorana’s idea. Specifically, there seemed to be a strict distinction between neutrinos and antineutrinos. In recent years, however, Majorana’s question has come back to life.

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Majorana and condensed matter physics

The turning point came with the discovery of neutrino oscillations [145]. Neutrino oscillations provide evidence for mass terms. Indeed, it is non-diagonal mass terms, connecting neutrinos with different lepton numbers, that cause freely propagating neutrinos to mix. Mass terms, diagonal or not, are incompatible with chiral projections. Thus the familiar “left-handed neutrino,” which particle physicists worked with for decades, can only be an approximation to reality. The physical neutrino must have some admixture of right-handed chirality. Thereby a fundamental question arises: Are the right-handed components of neutrinos something entirely new – or could they involve the same degrees of freedom we met before, in antineutrinos? (Usually these questions are phrased in the historically appropriate but cryptic form: Are neutrinos Majorana particles?) At first hearing, that question might sound quite strange, since neutrinos and antineutrinos have quite different properties. How could there be a piece of the neutrino that acts like an antineutrino? But of course, if the size of the unexpected piece is small enough, it can be compatible with observations. Quantitatively: if the energy of our neutrinos is large compared to their mass, the admixture of opposite chirality will be proportional to m/E. To explain the phenomenology of neutrino oscillations, and taking into account cosmological constraints, we are led to masses m < eV, and so, in most practical experiments, m/E is a very small parameter. These considerations raise the possibility that neutrinos and antineutrinos are the same particles, just observed in different states of motion. The observed distinctions might just represent unusual spin-dependent (or, more properly, helicitydependent) interactions. To pose the issues mathematically, we must describe a massive spin-1/2 particle using just two – not four – degrees of freedom. We want the antiparticle to involve the same degrees of freedom as the particle. Concretely, we want to investigate how the hypothesis ψR = ψL∗ ?

(14.33)

might be compatible with non-zero mass. Equation (14.33) embodies, in precise mathematical form, the idea that antineutrinos are simply neutrinos in a different state of motion, i.e. with different helicity. If ψ is a real field, described by Majorana’s version of the Dirac equation, then     1 − γ5 ∗ 1 + γ5 ∗ (ψL ) ≡ ψ ≡ ψR , ψ= (14.34) 2 2 since γ5 ≡ iγ 0 γ 1 γ 2 γ 3 is pure imaginary. Conversely, if Eq. (14.33) holds, we can derive both ψL and ψR by projection from a single four-component real field, i.e. ψ ≡ ψL + ψR = ψL + ψL∗ .

(14.35)

This is the link between Majorana’s mathematics and modern neutrino physics.

14.3 Majorana fermions and mass

289

14.3.3 Majorana mass With that background and inspiration, we can distill the essential novelty in the Majorana equation, which is a bit more subtle than is commonly stated. What is distinctive is not merely the use of real fields. After all, a complex field can always be written in terms of two real ones, as ψ = Re ψ + i Im ψ, and so any system of equations involving the complex field ψ can be written as a larger system of equations involving only the real fields Re ψ and Im ψ. Rather, what is distinctive is the possibility of passing, in the description of a massive spin-1/2 particle, from a Dirac field with eight real degrees of freedom (four complex components) to a Majorana field with four real degrees of freedom. As we shall see, this requires an unusual, symmetry-breaking form of mass: Majorana mass. Broken symmetry aspect To highlight the innovation this requires, let us consider the chiral projection of the Majorana equation. In general, applying a chiral projection to the Dirac equation gives us iγ μ ∂μ ψL + MψR = 0,

(14.36)

where both ψL and ψR naturally contain two complex components. But in the Majorana specialization, ψR is not independent, for it satisfies Eq. (14.33). We have, therefore, iγ μ ∂μ ψL + MψL∗ = 0.

(14.37)

In this formulation, we see that the mass term is of an unusual form. It involves complex conjugating the field, and thus reads differently – by a minus sign – for its real and imaginary components. It is naturally associated with the breaking of the phase symmetry ψL → eiλ ψL ,

(14.38)

or, of course, the corresponding number symmetry. Formal aspect: Grassmann variables The appearance of Eq. (14.37) is unusual, and we may wonder, as Majorana did, how it could arise as a field equation, following the usual procedure of varying a Lagrangian density. That consideration led Majorana to another major insight. The unprojected mass term, ¯ = ψ † γ0 ψ, Lmass ∝ ψψ

(14.39)

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Majorana and condensed matter physics

becomes, if we write everything in terms of ψL (using Eq. (14.33)), Lmass ∝ ψ † γ0 ψ → (ψL )T γ0 ψL + (ψL∗ )T γ0 ψL∗ ,

(14.40)

where T denotes transpose. In verifying that these terms are non-trivial, whereas the remaining cross-terms vanish, it is important to note that γ5 is antisymmetric, i.e. that it changes sign under transpose. That is true because γ5 is both Hermitian and purely imaginary. Thus we have, for example,     1 − γ5 T 1 + γ5 (ψL )T γ0 ψR = (ψL )T γ0 ψR 2 2     1 + γ5 1 + γ5 T = (ψL ) γ0 ψR 2 2    1 + γ5 1 − γ5 ψR = (ψL )T γ0 2 2 = 0.

(14.41)

(If we do not adopt Majorana’s hypothesis Eq. (14.33), the mass term takes the form Lconventional ∝ (ψR∗ )T ψL + (ψL∗ )T ψR ,

(14.42)

which supports number symmetry ψL,R → eiλ ψL,R .) The survival of the remaining terms in Eq. (14.40) is, as Majorana noted, also non-trivial. In components, we have (ψL )T γ0 ψL = (γ0 )jk (ψL )j (ψL )k .

(14.43)

Now γ0 , like γ5 , is antisymmetric (for the same reasons). So, in order for this term to survive, we must assume that the fields ψj are anticommuting variables. Majorana’s bold invocation of such “Grassmann numbers,” which have become ubiquitous in the modern field theory of fermions, was ahead of its time. With this understanding, variation of the mass term, together with the conventional kinetic term, L ∝ (ψL∗ )T γ0 iγ μ ∂μ ψL + h.c.,

(14.44)

will give us Eq. (14.37). Majorana mass as symmetry-breaking perturbation Equation (14.40) affords an instructive perspective on the Majorana mass term, which, we have argued, is a central innovation of paper N.9. This perspective will be helpful in assessing the “shocking” result of Section 14.3.4.

14.3 Majorana fermions and mass

291

By stripping away all kinematic details, we can define a faithful, transparent analog of “Majoranization” through mass acquisition, for spin-0 bosons. Let φ and  be two complex boson fields, supporting a global U(1) symmetry (φ, ) → (eiα φ, e2iα ).

(14.45)

Our focus will be on how the properties of the φ quanta – in particular, their masses – change as  acquires a symmetry-breaking vacuum expectation value. The relevant Lagrangian is Lmass = Lconventional + LMajorana , ∗

Lconventional = −M φ φ, 2



∗ 2

∗ 2

LMajorana = −κ  φ + (φ )



(14.46)

(taking, for convenience, κ real). If the vacuum expectation value  = 0, then we have only the conventional mass term. The quanta of φ are a degenerate pair – particle and antiparticle – with opposite charge but common mass M. The states of definite charge are produced by φ and φ ∗ . If the vacuum expectation value  = v = 0 (assumed, for convenience, real), then the quadratic terms in φ read



 (14.47) Lmass = −M 2 (Re φ)2 + (Im φ)2 − 2κv (Re φ)2 − (Im φ)2 . In this case, the quanta of definite mass are associated with the real fields Re φ, Im φ, and have masses   mIm = M 2 − 2κ. (14.48) mRe = M 2 + 2κ, These particles with definite mass are “Majorana” particles, in the sense that they are their own antiparticles. On the other hand, if κ is small, the practical effect of the splitting might be quite limited. For example, let us suppose that other interactions (besides the mass) are more nearly diagonal in terms of φ and φ ∗ , not Re φ and Im φ, as will occur if the symmetry breaking is small. Then the typical φ quantum produced in an interaction will evolve (in its rest frame) as 1 |φ(t = 0) = √ (|Re φ + i|Im φ ) 2 ↓ (14.49)  1  −imRe t |Re φ + ie−imIm t |Im φ √ e 2   mRe − mIm ∗ mRe − mIm −i(mRe +mIm )t/2 =e |φ − sin |φ . cos 2 2

292

Majorana and condensed matter physics

in an evident notation. The oscillation time   mRe − mIm −1 M ≈ , 2 κ

for κ  M,

(14.50)

will be very long. For shorter times, and considering the dominant interactions, it will be a good approximation to work with the “non-eigenstates” |φ , |φ ∗ , wherein the underlying Majorana structure is hidden. These considerations, which of course have their parallel for fermions, show that the concept “Majorana particle” should not be regarded as a binary, yes-or-no predicate. For, as we have just seen, particles can be their own antiparticles and yet behave, for practical purposes (with arbitrary accuracy), as if they were not. Rather, the physically meaningful issues are the magnitude of Majorana mass terms, and the circumstances in which such mass terms induce significant physical effects.

14.3.4 Majorana electrons After these preliminaries we are prepared to discuss the concept of Majorana electrons, which might otherwise sound absurd. Electrons and antielectrons have opposite electric charge, and electric charge is most definitely an observable quantity, which might seem to preclude that electrons might be their own antiparticles. Inside a superconductor, however, we have a different situation: a condensate of Cooper pairs. Heuristically: a hole, in this environment, can be “dressed” by a Cooper pair, and come to look like a particle. More formally: inside a superconductor, the gauge symmetry associated to charge conservation is spontaneously broken, and electric charge is not a good quantum number, so one cannot invoke it to distinguish particles from holes. To connect superconductivity with Majorana mass, consider how the formation of an electron pair condensate affects the ambient electrons. From the electron– electron interaction term e¯ e¯ ee – suppressing spin indices and considering only simple s-wave ordering – we derive an effective interaction between electrons and the ambient condensate, Lelectron−condensate = ∗ ee + h.c. ← κ e¯ e¯ ee + h.c.,

(14.51)

that precisely mirrors the Majorana mass term! Famously, the interaction Eq. (14.51) of electrons (and holes) with the condensate both mixes electrons with holes and opens a gap in the electron spectrum at the Fermi surface. A close analogy between the opening of that gap and the generation of mass, by condensation, for relativistic fermions had already been noted in Nambu’s great work on spontaneously broken symmetry in relativistic particle physics [377, 378]. Indeed, that analogy largely inspired Nambu’s work.

14.4 Majorinos

293

We are emphasizing here that in superconductors the mass in question is actually a mass of Majorana’s unusual, number-violating form. The induced Majorana mass, according to Eq. (14.51), is 10−3 eV, which is minuscule on the scale of the electron’s intrinsic (normal) mass, and even small on the scale of ordinary Fermi energies. It will dominate only for quasiparticles within a narrow range of the nominal Fermi surface. Nevertheless, most of the phenomena of superconductivity follow from the existence of the gap and therefore, implicitly, from this Majorana mass. Very recently, Beenakker has proposed a more pointed experimental demonstration of the Majorana nature of electrons in superconductors [379]. While the details are complicated, the essential idea is that the Majorana mass term depends on the phase eiδ of , and the relative phase of the particle and hole components of the Majorana quasiparticle will reflect that phase. Thus if we bring together quasiparticles from two different superconductors, their overlap, and therefore their 2 . annihilation probability, will be proportional to cos2 δ1 −δ 2 14.4 Majorinos As we have just discussed, inside superconductors the near-Fermi surface quasiparticle excitations have a Majorana character. Another interesting theme in recent condensed matter physics is the importance of zero modes – roughly speaking, mid-gap states – in quasiparticle spectra. They are typically associated with topological features, such as domain walls, vortices, or boundaries. (Mathematical aside: the existence of these modes is connected to index theorems.) The conjunction of these two ideas leads us Majorana zero modes. In a particle interpretation, the quanta associated with these modes are zero-mass Majorana particles. Since they are localized on specific points, they can be considered as particles in 0 + 1 (0 space, 1 time) dimension. They are, in many respects, the most extreme Majorana (self-conjugate) particles, consistent with their origin as midgap states in superconductors, where the electron Majorana mass is most dominant. They are not quite fermions, as we shall see, but obey an interesting generalization of Fermi statistics. It will be convenient to have a name for such particles. In view of their smallness both in mass and in spatial extension, and their extreme Majorana character, the name majorino suggests itself, and I shall adopt it here. The existence of Majorana modes in condensed matter systems [380, 381, 382, 155, 383, 384] is intrinsically interesting, in that it embodies a qualitatively new and deeply quantum-mechanical phenomenon [157]. It is also possible that such modes might have useful applications, particularly in quantum information processing [385, 386]. One feature that makes Majorana modes useful is that they generate a doubled spectrum. When we have several Majorana modes, each

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produces an independent doubling. Such repeated doubling generates a huge Hilbert space of degenerate states. If we can control the dynamics of majorinos, we can navigate through that Hilbert space. That vision inspires research to enable quantum information processing using majorinos. This section is structured as follows. We first discuss the occurrence of majorinos in a simple, discrete one-dimensional model, following Kitaev’s simple but profound analysis [158]. We then generalize to a more realistic model wire (still following Kitaev), and then to circuits. In that context we identify a remarkable algebraic structure, which allows us to identify the majorinos, to show that their existence is robust against interactions, and to exhibit the doubling phenomenon, in a transparent fashion. Finally, we discuss a continuum field theory approach to majorinos.

14.4.1 Kitaev chain Schematic Hamiltonian Let us briefly recall the simplest, yet representative, model for such modes, Kitaev’s wire segment. We imagine that N ordered sites are available to our electrons, so we have creation and destruction operators a†j , ak , 1 ≤ j, k ≤ N, with {aj , ak } = {a†j , a†k } = 0 and {a†j , ak } = δjk . The same commutation relations can be expressed using the Hermitian and anti-Hermitian parts of the aj , leading to a Clifford algebra, as follows: γ2j−1 = aj + a†j ,

γ2j =

aj − a†j i

,

{γk , γl } = 2 δkl .

(14.52)

Now let us compare the Hamiltonians: H0 = −i

N 

γ2j−1 γ2j ,

(14.53)

γ2j γ2j+1 .

(14.54)

j=1

H1 = −i

N−1  j=1

Since −iγ2j−1 γ2j = 2a†j aj − 1, H0 simply measures the total occupancy. It is a normal, if unusually trivial, electron Hamiltonian. H1 strongly resembles H0 , but there are three major differences. 1. One difference emerges, if we re-express H1 in terms of the aj . We find that it is local in terms of those variables, in the sense that only neighboring sites

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295

are connected, but that, in addition to conventional hopping terms of the type aj a†j+1 , we have terms of the type aj aj+1 , and their Hermitian conjugates. The aa type, which we may call superconductive hopping, does not conserve electron number, and is characteristic of a superconducting (pairing) state. 2. A second difference grows out of a similarity: since the algebra Eq. (14.52) of the γj is uniform in j, we can interpret the products γ2j γ2j+1 that appear in H1 in the same fashion that we interpret the products γ2j−1 γ2j that appear in H0 , that is as occupancy numbers. The effective fermions that appear in these numbers, however, are not the original electrons, but mixtures of electrons and holes on neighboring sites. 3. The third and most profound difference is that the operators γ1 , γ2N do not appear at all in H1 . These are the Majorana mode operators. They commute with the Hamiltonian, square to the identity, and anticommute with each other. The action of γ1 and γ2N on the ground state implies a degeneracy of that state, and the corresponding modes have zero energy. Kitaev [158] shows that similar behavior occurs for a family of Hamiltonians allowing continuous variation of microscopic parameters, i.e. for a universality class. (We will reproduce the core of that analysis immediately, and extend it in a different direction subsequently.) Within that universality class one has Hermitian operators bL , bR on the two ends of the wire whose action is exponentially (in N) localized and commute with the Hamiltonian up to exponentially small corrections, which satisfy the characteristic relations b2L = b2R = 1. In principle, there is a correction Hamiltonian, Hc ∝ −ibL bR ,

(14.55)

that will encourage us to re-assemble bL , bR into an effective fermion creation– destruction pair, and to realize Hc as its occupation number, by inverting the construction of Eq. (14.52). But for a long wire and weak interactions we expect the coefficient of Hc to be very small, since the modes excited by bL , bR are spatially distant, and for most physical purposes it will be more appropriate to work with the local variables bL , bR , since these respond independently to spatially uncorrelated perturbations. Parametric Hamiltonian H1 involved a very particular choice of parameters. Here we show what can arise with a class of Hamiltonians of a physically plausible form, which contain continuous parameters, yet share its central qualitative feature – that is, the emergence of majorinos at the ends of the wire.

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Majorana and condensed matter physics

Indeed, consider H=

N 

−w



a†j aj+1

+

a†j+1 aj

j=1



  1 † + aj aj+1 + ∗ a†j+1 a†j , − μ aj aj − 2 (14.56)

with the understanding a†N+1 = aN+1 = 0. It describes a superconducting wire, with N sites. The first term, with coefficient w, is a conventional hopping term; the second term, proportional to μ, is a chemical potential; the remaining terms represent residual interactions with the superconducting gap parameter  ≡ eiθ ||, here regarded as a given external field. The crucial issue for us is the existence (or not) of zero-energy modes. So we look for solutions of the equation ⎤ ⎡ 2N  ⎣H, cj γj ⎦ = 0. (14.57) j=1

With the understanding that γk → 0 when k falls outside the allowed range 1 ≤ k ≤ 2N, we have, for j even,

(j even); (14.58) H, γj ∝ μγj−1 + (w + ||)γj+1 + (w − ||)γj−3 for j odd,

H, γj ∝ −μγj+1 − (w + ||)γj−1 − (w − ||)γj+3

(j odd).

(14.59)

( odd), ( even),

(14.60)

Gathering terms proportional to γl in Eq. (14.57), we have μc+1 + (w + ||) c−1 + (w − ||) c+3 = 0 −μc−1 − (w + ||) c+1 − (w − ||) c−3 = 0

with the understanding that cj → 0 unless 1 ≤ j ≤ 2N. Equation (14.60) is a three-term recurrence relation that connects coefficients whose subscripts differ by two units. At first neglecting boundary conditions, we construct candidate solutions in the form of power series which contain only even or only odd terms. Putting c = αx(+1)/2 , for all odd , we get a quadratic equation for x (from the second line of Eq. (14.60)):  μ2 + 4||2 − 4w2 −μ ± odd = . (14.61) x± 2(w + ||) This leads us to look for zero modes in the form  

j j α+ x+ + α− x− γ2j−1 . b1 =

(14.62)

j

The zero-mode condition Eq. (14.57) is then satisfied, by construction, in bulk.

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297

Now we must attend to the boundaries. In our formal manipulations we ignored the prescriptions cl , γl → 0 for l outside the physical range. To have a legitimate solution at the ends, we must also impose α+ x+N+1 + α− x−N+1 = 0

(14.63)

0 0 α+ x+ + α− x− = α+ + α− = 0.

(14.64)

and

We reach different conclusions depending upon whether our candidate solutions are both growing (i.e. |x+ |, |x− | > 1), both shrinking (|x+ |, |x− | < 1), or we have one with each behavior. If both are growing, then, after imposing Eq. (14.63) and N+1 = 1, we find normalizing the coefficient α+ x+ −N−1 −N−1 α+ + α− = x+ − x− → 0,

for N → ∞,

(14.65)

so that Eq. (14.64) is satisfied approximately, up to quantities that are exponentially small in N. For a half-infinite wire we can take the limit, and we have exact normalizable zero modes, concentrated on the (right-hand) boundary. For large but finite N we will need another small parameter in order to satisfy the second boundary condition. Such a parameter is available: the energy. So we will not get an exact zero-energy mode, but rather a normalizable mode with exponentially small energy, concentrated on the right-hand boundary. If both of our candidate solutions are shrinking, then a similar analysis reveals a normalizable near-zero-energy mode concentrated on the left-hand boundary. Note that our earlier H1 corresponds to x+ = x− = 0, so we have solutions concentrated on one site! If one candidate solution is growing while the other is shrinking, the construction fails, for upon imposing one boundary condition we are forced into gross violations of the other. That completes our analysis of the candidate zero modes associated with linear combinations of γ2l−1 , i.e. concentrated on odd sites. We can of course perform a parallel analysis of candidate zero modes concentrated on even sites. Due to the re-sequencing of terms appearing in the two lines of Eq. (14.60) the quadratic equation for the factor x is reversed, so that it is solved instead by x−1 . Thus candidate zero modes concentrated on even sites take the form  

−j −j (14.66) b2 = β+ x+ + β− x− γ2j . j

We find that a normalizable near-zero mode of this even type occurs in precisely the same circumstances where we have a near-zero mode of the odd type; and that when both solutions exist they are localized at opposite ends of the wire.

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Majorana and condensed matter physics

14.4.2 Junctions and the algebraic genesis of majorinos Now let us consider the situation where multiple Majorana modes come together to form a junction, as might occur in a network of superconducting wires of the sort analyzed above. Several experimental groups are developing physical embodiments of Majorana modes, for eventual use in such quantum circuits. (For a useful sampling of recent activity, see the collection of abstracts from the July 12–18 2013 Erice workshop [387] and references therein.) A fundamental issue, in analyzing such circuits, is the behavior at junctions. Pairs of Majorana modes can be organized into fermion creation and destruction operators and their allowed interaction terms which make that appropriate, but we can anticipate that general principles might be enough to give us one surviving Majorana mode at an odd junction. We will, in fact, identify a remarkably simple, explicit nonlinear operator  that can be considered the creator/destroyer of majorinos:  obeys simple algebraic relations with the Hamiltonian, and implements a general doubling of the spectrum. Its existence and properties are tightly connected to fermion number parity. The emergent algebraic structure, in its power and simplicity, seems encouraging for further analysis and development of Majorana wire circuits. The following considerations will appear more pointed if we explain their origin in the following little puzzle. Let us imagine we bring together the ends of three wires supporting Majorana modes b1 , b2 , b3 . Thus we have the algebra {bj , bk } = 2δjk .

(14.67)

The bj do not appear in their separate wire Hamiltonians, but we can expect to have interactions Hint = −i(α b1 b2 + β b2 b3 + γ b3 b1 ),

(14.68)

which plausibly arise from normal or superconductive inter-wire electron hopping. We assume here that the only important couplings among the wires involve the Majorana modes. This is appropriate if the remaining modes are gapped and the interaction is weak – for example, if we only include effects of resonant tunneling. We shall relax this assumption in due course. We might expect, heuristically, that the interactions cause two Majorana degrees of freedom to pair up to form a conventional fermion degree of freedom, leaving one Majorana mode behind. On the other hand, the algebra in Eq. (14.67) can be realized using Pauli σ matrices, in the form bj = σj . In that realization, wehave simply H = α σ3 + β σ1 + γ σ2 . But that Hamiltonian has eigenvalues ± |α|2 + |β|2 + |γ |2 , with neither degeneracy nor zero mode. In fact, a similar

14.4 Majorinos

299

problem arises even for “junctions” containing a single wire, since we could use bR = σ1 (and bL = σ2 ). The point is that the algebra of Eq. (14.67) is conceptually incomplete. It does not incorporate relevant implications of electron number parity, or, in other words, electron number modulo 2, which remains valid even in the presence of a pairing condensate. The operator P ≡ (−1)Ne

(14.69)

that implements electron number parity should obey P2 = 1,

(14.70)

[P, Heff. ] = 0,

(14.71)

{P, bj } = 0.

(14.72)

Equation (14.70) follows directly from the motivating definition; Eq. (14.71) reflects the fundamental constraint that electron number modulo 2 is conserved in the theories under consideration, and indeed under very broad – possibly universal – conditions; Eq. (14.72) reflects, in the context of Ref. [158], that the bj are linear functions of the ak , a†l , but is more general. Indeed, it will persist under any “dressing” of the bj operators induced by interactions that conserve P. We will see striking examples of this persistence in the following. The preceding puzzle can now be addressed. Including the algebra of electron parity operator, we take a concrete realization of operators as b1 = σ1 ⊗ I, b2 = σ3 ⊗ I, b3 = σ2 ⊗ σ1 , and P = σ2 ⊗ σ3 . This choice represents the algebra Eqs. (14.67) and (14.70)–(14.72). The Hamiltonian represented in this enlarged space contains doublets at each energy level. (Related algebraic structures are implicit in Ref. [388]. See also Refs. [389, 390, 391, 392] for additional constructions.) Returning now to the abstract analysis, consider the special operator  ≡ −ib1 b2 b3 .

(14.73)

 2 = 1,

, bj = 0,

(14.74)

[, Heff. ] = 0,

(14.76)

{, P} = 0.

(14.77)

It satisfies

(14.75)

Equations (14.74) and (14.75) follow directly from the definition, while Eq. (14.76) follows, given Eq. (14.75), from the requirement that Heff. should contain only terms of even degree in the bi s. That requirement, in turn, follows from the

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Majorana and condensed matter physics

restriction of the Hamiltonian to terms even under P. Finally, Eq. (14.77) is a direct consequence of Eq. (14.72) and the definition of . This emergent  has the characteristic properties of a Majorana mode operator: it is Hermitian, it squares to 1, and it has odd electron number parity. Most crucially, it commutes with the Hamiltonian, but is not a function of the Hamiltonian. We can bring the relevant structure into focus by going to a basis where H and P are both diagonal. Then from Eq. (14.77) we see that  takes states with P = ±1 into states of the same energy with P = ∓1. This doubling applies to all energy eigenstates, not only to the ground state. It is reminiscent of, but differs from, Kramers doubling. (No antiunitary operation appears, nor is T symmetry assumed.) One also has a linear operator w ≡ α b3 + β b1 + γ b2

(14.78)

that commutes with the Hamiltonian. But this w is not independent of , since we have w = H ,

(14.79)

and it is , not w, which generalizes smoothly. The same considerations apply to a junction supporting any odd number p of Majorana mode operators, with  ≡ i

p(p−1) 2

p 6

γj .

(14.80)

j=1

For even p, however, we get a commutator instead of an anticommutator in Eq. (14.77), and the doubling construction fails. For odd p ≥ 5 generally there is no linear operator, analogous to the w of Eq. (14.78), that commutes with H. (If the Hamiltonian is quadratic, the existence of a linear zero mode follows from simple linear algebra – namely, the existence of a zero eigenvalue of an odd-dimensional antisymmetric matrix, as discussed in many earlier analyses. But for more complex, realistic Hamiltonians, including nearby electron modes as envisaged here, that argument is insufficient, even for p = 3. The emergent operator , on the other hand, always commutes with the Hamiltonian (Eqs. (14.76)), even allowing for higher-order contributions such as quartic or higher polynomials in the bi s.) Now let us revisit the approximation of keeping only the interactions of the Majorana modes from the separate wires. We can in fact, without difficulty, include any finite number of “ordinary” creation–annihilation modes from each wire, thus including all degrees of freedom that overlap significantly with the junction under consideration. These can be expressed, as in Eq. (14.52), in terms of an even number of additional γ operators, to be included with the odd number of bj . But

14.4 Majorinos

301

then the product   of all these operators, including both types (and the appropriate power of i), retains the good properties Eq. (14.74) of the  operator we had before. If p ≥ 5, or even at p = 3 with nearby electron interaction effects included, the emergent zero mode is a highly nonlinear entangled state involving all the wires that participate at the junction. The robustness of these conclusions results from the identified algebraic properties of . If we have a circuit with several junctions j, the emergent j will obey the Clifford algebra {j , k } = 2δjk .

(14.81)

This applies also to junctions with p = 1, i.e. simple terminals; also, the circuit does not need to be connected. The algebraic structure defined by Eqs. (14.70)–(14.72) is fully non-perturbative. It may be taken as the definition of the universality class supporting Majorana modes. The construction of  (in its most general form) and its consequences Eqs. (14.74)–(14.77) reproduces that structure, allowing for additional interactions, with  playing the role of an emergent b. The definition of , the consequences Eqs. (14.74)–(14.77), and the deduction of doubling are likewise fully non-perturbative. It is noteworthy that our construction of emergent majorinos is at the opposite extreme from a single-particle operator: the mode it excites is associated with the product wavefunction over the modes associated with the bj , rather than a linear combination. In this sense we have extreme valencebond (Heitler–London) as opposed to linear (Mulliken) orbitals. The contrast is especially marked, of course, for large p. When there are several junctions k, each has its own k operator, and together they define a Clifford algebra {k , l } = 2δkl ,

(14.82)

as is easily demonstrated. Thus majorinos at different junctions anticommute, similar to fermions. But the square of each k is unity, not zero, so they do not obey the Pauli exclusion principle. They are neither bosons nor fermions, but a type of non-Abelian anyon, whose quantum statistics is effectively defined by Eq. (14.82). The simple, explicit construction of emergent majorinos in terms of the underlying microscopic variables might be helpful in the design of useful circuit operations, based on their response to the variation of the Hamiltonian parameters, which could be controlled by applying external fields.

14.4.3 Continuum majorinos We can also define a simple continuum construction, based directly on the Majorana equation, that supports majorinos. It is a modification of the classic

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Jackiw–Rebbi [154] model, where, in place of ordinary mass, the background field imparts Majorana mass to an electron, and is closely related to considerations in Ref. [393]. To describe it in a self-contained manner, let us consider a (1 + 1)-dimensional version of Majorana’s equation Eq. (14.37), allowing both for Majorana mass M and conventional mass m, iγ μ ∂μ ψ + M(x)ψ ∗ + mψ = 0,

(14.83)

where we allow the Majorana mass M(x) to depend on x. For our γ matrices, we take simply γ 0 = σ2 ,

γ 1 = iσ3 .

(14.84)

The equation for a zero-energy solution is then dψ = M(x)ψ ∗ + mψ, dx or, in terms of the real and imaginary parts ψ = φ + iη, σ3

σ3

(14.85)

 dφ  = m + M(x) φ, dx

(14.86)  dη  σ3 = m − M(x) η. dx Writing the spinors in terms of σ3 eigenspinors (i.e. up and down components) φ± , η± , we find the formal solutions: ⎫ ⎧ ⎨ x  ⎬ φ± (x) = φ± (0) exp ± dy m + M(y) , ⎭ ⎩ 0

⎫ ⎧ ⎨ x  ⎬ η± (x) = η± (0) exp ± dy m − M(y) . ⎭ ⎩

(14.87)

0

These solutions will be normalizable only if the integrand is negative for large positive y and positive for large negative y. It is not difficult to construct monotonic profiles for M(y) and values for m such that exactly one of the candidate solutions is normalizable. In that case one will have a single Majorana mode localized, after an appropriate shift, around x = 0.

15 Majorana neutrinos and other Majorana particles: theory and experiment E. Akhmedov

What are Majorana particles? They are massive fermions that are their own antiparticles. In this chapter we will concentrate on spin-1/2 Majorana particles, though fermions of higher spin can also be of Majorana nature. Obviously, Majorana particles must be genuinely neutral, i.e. they cannot possess any conserved charge-like quantum number that would allow one to discriminate between the particle and its antiparticle. In particular, they must be electrically neutral. Among the known spin-1/2 particles, only neutrinos can be of Majorana nature. Another known quasi-stable neutral fermion, the neutron, has non-zero magnetic moment, which disqualifies it from being a Majorana particle: the antineutron exists, and its magnetic moment is opposite to that of the neutron.1 Neutrinos are exactly massless in the original version of the Standard Model of electroweak interaction [314], and are massive Majorana particles in most of its extensions. Although massive Dirac neutrinos are also a possibility, most economical and natural models of neutrino mass lead to Majorana neutrinos. Since only massive neutrinos can oscillate, interest in the possibility of neutrinos being Majorana particles rose significantly after the first hints of neutrino oscillations obtained in solar and atmospheric neutrino experiments. This interest increased greatly after the oscillations were firmly established in the experiments with solar, atmospheric, accelerator, and reactor neutrinos [394, 395, 396]. In addition to being the simplest and most economical possibility, Majorana neutrinos have two important added bonuses: they can explain the smallness of the neutrino mass in a very natural way through the so-called seesaw mechanism, and they can account for the observed baryon asymmetry of the Universe through “baryogenesis via leptogenesis.” We shall discuss both here.

1 One could have also argued that the neutron and antineutron are distinguished by their baryon number

(+1 and −1, respectively), but conservation of baryon number is not an exact symmetry of Nature.

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Majorana neutrinos and other Majorana particles

In the limit of vanishingly small mass, the difference between Dirac and Majorana fermions disappears. Therefore the observed smallness of the neutrino mass makes it very difficult to discriminate between different types of massive neutrinos, and it is not currently known if neutrinos are Majorana or Dirac particles. The most promising means of finding this out is through the experiments on neutrinoless double β-decay. Such experiments are currently being conducted in a number of laboratories. In this chapter we review the properties of Majorana neutrinos and other Majorana particles. We start by discussing Weyl, Dirac, and Majorana fermions and comparing the Dirac and Majorana mass terms. We then proceed to discuss C, P, CP, and CPT properties of Majorana particles in Section 15.2. This is followed by a discussion of mixing and oscillations of neutrinos in the Majorana and general Dirac + Majorana cases in Section 15.3. In Section 15.4 we discuss the seesaw mechanism of neutrino mass generation, which is the leading candidate for the explanation of the smallness of the neutrino mass. In Section 15.5 we consider the electromagnetic properties of Majorana neutrinos, and Section 15.6 contains a brief discussion of the Majorana particles predicted by supersymmetric theories. In Section 15.7 we review theoretical foundations and the experimental status of the neutrinoless double β-decay as well as of other processes that could distinguish between Majorana and Dirac neutrinos. Our next topic is baryogenesis via leptogenesis due to lepton-number-violating processes caused by Majorana neutrinos (Section 15.8). Finally, in Section 15.9, we collect together a few assorted remarks on Majorana particles, and in Section 15.10 we summarize the main points of our discussion.

15.1 Weyl, Dirac, and Majorana fermions In his famous paper N.9, Majorana sought to cast the Dirac equation in a form that would be completely symmetric with respect to particles and antiparticles. He succeeded in doing that by finding a new form of the Dirac equation in which all coefficients were real. While it led to only a formal improvement for charged fermions, the Majorana form of the Dirac equation opened up a very important new possibility for neutral ones – they can be their own antiparticles. The Majorana particles are thus fermionic analogs of genuinely neutral bosons, such as the π 0 -meson or the photon. Recall that a free spin-1/2 fermion field in general satisfies the Dirac equation2 (iγ μ ∂μ − m)ψ(x) = 0 , 2 We use the natural units h = c = 1 and assume summation over repeated indices in this chapter. ¯

(15.1)

15.1 Weyl, Dirac, and Majorana fermions

305

where ∂μ ≡ ∂/∂xμ , ψ(x) is a four-component spinor field, m is the mass of the fermion, and γ μ (μ = 0, 1, 2, 3) are 4 × 4 matrices satisfying {γ μ , γ ν } = 2δ μν · 1 ,

γ 0 γ μ† γ 0 = γ μ ,

(15.2)

with δ μν = diag(1, −1, −1, −1) and 1 being the flat space-time metric tensor and the 4 × 4 unit matrix, respectively. Note that the Dirac equation (15.1) can be cast in the Schrödinger form i(∂/∂t)ψ(x) = HD ψ(x), where HD = −iγ 0 γ · ∇ + γ 0 m. The first equality in Eq. (15.2) follows from the requirement that the solutions of the Dirac equation obey the usual dispersion law of free relativistic particles E2 = p2 + m2 , while the second equality follows from the hermiticity of the Dirac Hamiltonian HD . In addition to the matrices γ μ , a very important role is played by the matrix γ5 ≡ iγ 0 γ 1 γ 2 γ 3 , which satisfies {γ5 , γ μ } = 0 ,

γ5† = γ5 ,

γ52 = 1 .

(15.3)

There are infinitely many unitarily equivalent representations of the Dirac matrices. In this chapter, unless otherwise specified, we will use the so-called chiral (or Weyl) representation       0 1 0 σi −1 0 0 i γ = , γ = , (15.4) , γ5 = 1 0 0 1 −σ i 0 where 1 and 0 are the unit and zero 2 × 2 matrices, and σ i (i = 1, 2, 3) are the standard Pauli matrices. The left-handed and right-handed chirality projector operators PL,R are defined as 1 − γ5 1 + γ5 , PR = . (15.5) PL = 2 2 They have the following properties: P2L = PL ,

P2R = PR ,

PL PR = PR PL = 0 ,

PL + PR = 1 .

(15.6)

Any spin-1/2 fermion field ψ can be decomposed into the sum of its left-handed and right-handed components according to ψ = ψL + ψR ,

where

ψL,R = PL,R ψ =

1 ∓ γ5 ψ. 2

(15.7)

Note that the chiral fields ψL,R are eigenstates of γ5 : γ5 ψL,R = ∓ψL,R . The terms “left-handed” and “right-handed” originate from the fact that for relativistic particles chirality almost coincides with helicity defined as the projection of the spin of the particle on its momentum. More precisely, in the relativistic limit, for positiveenergy solutions of the Dirac equation the left- and right-handed chirality fields

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Majorana neutrinos and other Majorana particles

approximately coincide with those of negative and positive helicity, respectively. The helicity projection operators are   σ ·p 1 1± . (15.8) P± = 2 |p| They satisfy relations similar to Eqs. (15.6). For a free fermion, helicity is conserved but chirality in general is not; it is only conserved in the limit m = 0, when it coincides with helicity. However, for relativistic particles chirality is nearly conserved, and the description in terms of chiral states is useful.

15.1.1 Particle−antiparticle conjugation ˆ For our discussion we will need the particle−antiparticle conjugation operator C. Its action on a fermion field ψ is defined as Cˆ : ψ → ψ c = C ψ¯ T ,

(15.9)

where ψ¯ ≡ ψ † γ 0 is the adjoint field, and the matrix C satisfies C −1 γ μ C = −γ μT ,

C −1 γ5 C = γ5T ,

C † = C −1 = −C ∗ .

(15.10)

Note that the second equality here follows from the first one and the definition of ˆ field ψ c (x) satisfies the same Dirac equation γ5 . For free particles, the C-conjugate as ψ(x). Some useful relations that follow from Eqs. (15.9) and (15.10) are ψ c = −ψ T C −1 ,

(ψ c )c = ψ , ψk ψic = ψi ψkc ,

ψk Aψi = ψic (CAT C −1 )ψkc ,

(15.11)

where ψ, ψi , ψk are anticommuting four-component fermion fields and A is an arbitrary 4 × 4 matrix. Note that the third Eq. (15.10) means that the matrix C is antisymmetric. In the representation Eqs. (15.4), as well as in a number of other representations of the Dirac matrices, one can choose e.g. C = iγ 2 γ 0 . In this case C is real, C −1 = −C, and ψ c = ψ T C. For future use, we give here the expressions CAT C −1 for several matrices A: Cγ μT C −1 = −γ μ ,

C(γ μ γ5 )T C −1 = γ μ γ5 ,

μν T −1

C(σ

(15.12) C(σ

) C

= −σ

μν

,

μν

T −1

γ5 ) C

= −σ

μν

γ5 ,

where σ μν ≡ 2i [γ μ , γ ν ]. Using the anticommutation properties of the Dirac γ -matrices it is easy to see that, acting on a chiral field, Cˆ flips its chirality: Cˆ : ψL → (ψL )c = (ψ c )R ,

ψR → (ψR )c = (ψ c )L ,

(15.13)

15.1 Weyl, Dirac, and Majorana fermions

307

i.e. the antiparticle of a left-handed fermion is right handed. This fact plays a very important role in the theory of Majorana particles. The particle−antiparticle conjugation operation Cˆ must not be confused with the charge conjugation operation C, which, by definition, flips all the charge-like quantum numbers of a field (electric charge, baryon number B, lepton number L, etc.) but leaves all the other quantum numbers (including chirality) intact. In particular, charge conjugation would take a left-handed neutrino into a left-handed antineutrino that does not exist, which is a consequence of maximal C-violation in ˆ weak interactions. At the same time, C-conjugation converts a left-handed neutrino into a right-handed antineutrino that does exist and is the antiparticle of the lefthanded neutrino. A little caveat should be added to this. Strictly speaking, a particle and its antiparticle are related by the CPT transformation, as only this combination of the charge conjugation C, space parity P, and time reversal T is exactly conserved in any “normal” theory (i.e. local Poincaré invariant Lagrangian quantum field theory with the usual relation between spin and statistics). However, the CP conjugation does essentially the same job as far as (typically very small) effects of CP violation can be neglected. The Cˆ conjugation introduced in Eq. (15.9) acts very similarly to the CP conjugation as it flips all the non-zero charges of the fermion as well as its chirality, which is odd under P transformation. We discuss these points in more detail in Section 15.2. It should be added that when we say that the charge conjugation C flips the baryon and lepton numbers of the particles we assume that these numbers are well defined, i.e. that small effects of B and L violation can be ignored.

15.1.2 Dirac dynamics and the Majorana condition Let us now return to the discussion of the Dirac equation. Adopting the Weyl representation of the Dirac γ -matrices Eqs. (15.4) and writing the four-component spinor field ψ(x) in terms of the two-component spinors φ(x) and ξ(x) as   φ , (15.14) ψ= ξ one can rewrite the Dirac equation Eq. (15.1) as a set of two coupled equations for φ and ξ : (i∂0 − iσ · ∇)φ − mξ = 0 , (i∂0 + iσ · ∇)ξ − mφ = 0 .

(15.15)

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Majorana neutrinos and other Majorana particles

From the expression for γ5 in Eq. (15.4) and Eq. (15.14), one obtains     φ 0 , ψR = , ψL = 0 ξ

(15.16)

i.e. the two-component spinor fields φ and ξ determine the left- and right-handed components of the four-component field ψ, respectively. Thus, the chiral fields are actually two-component rather than four-component objects. From Eq. (15.15) it follows that in the limit m = 0 the equations for φ and ξ decouple, i.e. the left-handed and right-handed components of ψ evolve independently. The resulting equations are called the Weyl equations, and the corresponding chiral solutions describe massless spin-1/2 particles called Weyl fermions. At the same time, as follows from Eq. (15.15), to describe a massive fermion one needs both left-handed and right-handed chiral fields. The latter statement can also be demonstrated as follows. The Dirac equation for a free spin-1/2 particle can be obtained as the Euler–Lagrange equation applied to the Dirac Lagrangian μ ¯ ∂μ − m)ψ . (15.17) L = ψ(iγ The mass term of this Lagrangian can be written as ¯ = m(ψL + ψR )(ψL + ψR ) = m(ψL ψR + ψR ψL ) , − Lm = mψψ

(15.18)

i.e. only the cross terms survive, while the ψL ψL and ψR ψR terms vanish identically. Thus, one needs both left-handed and right-handed chiral fields to construct the mass term of the Lagrangian, and a massive fermion field must be a sum of them: ψ = ψL + ψR .3 Now, there are essentially two possibilities. First, the right-handed component of a massive field can be completely independent of the left-handed one; in this case we have a Dirac field. The second, and most important for us, possibility is based on the discussed fact that the particle−antiparticle conjugate of a left-handed field is right handed. Therefore the right-handed component of a massive spin-1/2 field ˆ can be just the C-conjugate of its left-handed component: ψR = (ψL )c = (ψ c )R , or ψ = ψL + (ψL )c = ψL + (ψ c )R .

(15.19)

In this case we have a Majorana field; one can construct it with just one chiral field. ˆ From Eq. (15.19) it immediately follows that the C-conjugate field coincides with the original one: ψc = ψ .

(15.20)

3 Note that the kinetic term of the Lagrangian Eq. (15.17) is decomposed as ψiγ ¯ μ ∂μ ψ = ψ¯ L i γ μ ∂μ ψL + ψ¯ R iγ μ ∂μ ψR . In other words, for each chiral component the kinetic term can be written

separately, and therefore it does not require the existence of both components.

15.1 Weyl, Dirac, and Majorana fermions

309

This means that particles associated with Majorana fields are genuinely neutral, i.e. they are their own antiparticles. The condition in Eq. (15.20) is called the Majorana condition.

In his paper N.9 Majorana found a representation of the γ -matrices in which they were all purely imaginary, so that the Dirac equation Eq. (15.1) did not contain any complex coefficients. As a result, the equation admitted real solutions ψ∗ = ψ ,

(15.21)

which describe genuinely neutral particles. Equation (15.20) generalizes the Majorana condition Eq. (15.21) to the case of an arbitrary representation of the γ -matrices (see e.g. Ref. [397] for a formal proof). It is easy to see that the general self-conjugacy condition Eq. (15.20) indeed reduces to Eq. (15.21) in the Majorana basis. In the Majorana representation the γ -matrices satisfying Eq. (15.2) can be chosen as   0 σ2 0 , γM = σ2 0  1 γM =i

σ3 0

0 σ3



 2 , γM =

0 σ2

−σ 2 0

  1 σ 3 , γM = −i 0

0 σ1

 ,

where the subscript M stands for the Majorana basis. All the γ -matrices are purely imaginary, as required (note that this representation is not unique). The matrix γ5M is then given by   2 0 σ 0 . = γ5M 0 −σ 2 0 is antisymmetric, whereas γ i (i = 1, 2, 3) are symmetric; the particle– Note that γM M antiparticle conjugation matrix C satisfying Eq. (15.10) can therefore be chosen as   0 σ2 0 . (15.22) CM = −γM = − σ2 0

From Eq. (15.9) we then find c T 0T ∗ 0 0T ∗ ∗ ψM = CM ψ¯ M = C M γM ψM = −γM γM ψM = ψM ,

(15.23)

i.e. the condition that the particle is its own antiparticle ψ c = ψ reduces in the Majorana basis to the requirement that the field ψM be real.

As was discussed in the text following Eq. (15.18), to construct a massive Dirac field one needs two independent two-component chiral fields, ψL and ψR ; this yields four degrees of freedom. In contrast with this, a Majorana fermion has only two degrees of freedom, because its right-handed component is constructed

310

Majorana neutrinos and other Majorana particles

from the left-handed one. Thus, Majorana fermions are actually simpler and more economical constructions than the Dirac ones. While Majorana fields are essentially two-component objects, it is often useful to write them in the four-component notation, especially when considering processes in which Majorana particles participate along with Dirac ones. It is easy to see that in the chiral representation of the Dirac matrices the Majorana field can be written in the four-component form as   φ . (15.24) ψ= −iσ 2 φ ∗ Indeed, from C = iγ 2 γ 0 and Eq. (15.9) we have      φ∗ φ 0 iσ 2 c 2 ∗ =ψ. = ψ = iγ ψ = −iσ 2 0 −iσ 2 φ −iσ 2 φ ∗

(15.25)

To understand better the difference between the Dirac and Majorana particles it is instructive to look at the expansions of their quantum fields in terms of the plane-wave modes. Recall that for a Dirac field the expansion has the form  

d3 p ψ(x) = bs (p)us (p)e−ipx + ds† (p)vs (p)eipx , (15.26)  (2π)3 2Ep s where s = ±1/2is the projection of the particle’s spin on a fixed spatial direction, Ep = p0 = + p2 + m2 , us (p) and vs (p) are the positive- and negative-energy solutions of the Dirac equation in the momentum space, and bs (p) and ds† (p) are the annihilation operator for the particle and the creation operator for the antiparticle, respectively. The field ψ thus annihilates the particle and creates its antiparticle, whereas the Hermitian conjugate field annihilates the antiparticle and creates the particle. Because for Majorana fermions the particle and antiparticle coincide, for them one has to identify bs (p) and ds (p), i.e. the Fourier expansion of Majorana fields takes the form4  

d3 p ψ(x) = bs (p)us (p)e−ipx + b†s (p)vs (p)eipx . (15.27)  3 (2π) 2Ep s It is possible (and convenient) to choose the phases of the spinors us (p) and vs (p) in such a way that vs (p) = C u¯ Ts (p) ,

us (p) = C v¯ Ts (p) .

(15.28)

4 Expansions (15.26) and (15.27) are sometimes defined with a phase factor λ in front of the creation

operators. This factor, however, enters physical observables only together with other phase factors, discussed in Section 15.2, i.e. it is not separately observable. We therefore choose λ = 1 throughout this chapter.

15.1 Weyl, Dirac, and Majorana fermions

311

From these relations it immediately follows that the field Eq. (15.27) satisfies the Majorana self-conjugacy condition Eq. (15.20). The plane-wave decomposition of the Majorana fields Eq. (15.27) is reminiscent of the familiar Fourier expansion of the photon field Aμ (x), which also contains the creation and annihilation operators of only one kind, aλ (p) and a†λ (p), because the photon is its own antiparticle. The action of the charge conjugation operation C amounts to interchanging the particle with its antiparticle without changing its momentum or spin polarization state. For a Dirac fermion field Eq. (15.26) it can therefore be represented as C bs (p) C−1 = ds (p) ,

C ds† (p) C−1 = b†s (p) .

(15.29)

With the help of Eq. (15.28) one can readily make sure that applying to (15.26) the particle–antiparticle conjugation defined in Eq. (15.9) yields exactly the same result as the C conjugation Eqs. (15.29). How about the Majorana fields? For them, ds (p) = bs (p), so that the operation in Eq. (15.29) is just the trivial identity transformation, which has no effect on the fields. The Cˆ operation also leaves the Majorana fields unchanged – we have actually defined them through this condition, Eq. (15.20). Thus, we conclude that for free massive fermion fields, both of Dirac and Majorana nature, the C and Cˆ conjugations are equivalent. As already pointed out, the two operations are not equivalent when acting on chiral fields. Consider now the equations of motion for the left-handed and right-handed components of a Majorana field. Equation (15.24) tells us that in the four-component notation Eq. (15.14) the lower 2-spinor is given by ξ = −iσ 2 φ ∗ . Substituting this into Eq. (15.15) we find [398] (∂0 − σ · ∇) φ + mσ 2 φ ∗ = 0 ,

(15.30)

(∂0 + σ · ∇) σ 2 φ ∗ − mφ = 0 .

(15.31)

It is easy to see that the second of these equations is equivalent to the first one. Indeed, taking the complex conjugate of Eq. (15.31), multiplying on the left by σ 2 , and using the relation σ 2 σ ∗ σ 2 = −σ , we obtain Eq. (15.30). Next, let us exclude φ ∗ from Eqs. (15.30) and (15.31). By acting on Eq. (15.30) with (∂0 + σ · ∇) and making use of Eq. (15.31) we find that φ satisfies the Klein–Gordon equation (∂ 2 + m2 )φ = 0 .

(15.32)

This means that free Majorana particles obey the standard dispersion relation E2 = p2 + m2 . Thus, kinematically Dirac and Majorana fermions are indistinguishable. They can, however, in principle be told apart through their interactions, as we discuss in Section 15.5.

312

Majorana neutrinos and other Majorana particles

15.1.3 Fermion mass terms and U(1) symmetries Let us now turn to the Lagrangian of a free Majorana field. From Eqs. (15.18) and (15.19) we find that the mass term in the Lagrangian is + m* (ψL )c ψL + ψL (ψL )c Lm = − 2 + m

m * T −1 T (15.33) = ψL C ψL + ψL C −1 ψL = ψLT C −1 ψL + h.c. , 2 2 where we have used the second equality in Eq. (15.11), and the factor 1/2 was introduced because Lm is quadratic in ψL . Thus, the Majorana Lagrangian can be written as

m T −1 L = ψL iγ μ ∂μ ψL + ψL C ψL + h.c. . (15.34) 2 Note that it is expressed solely in terms of ψL . In particular, there is no kinetic term for the field ψR because the left-handed and right-handed components of the Majorana field are not independent. The Lagrangian in Eq. (15.34) can be cast in a more familiar form if we use the notation ψ = ψL + (ψL )c . Then, up to a total derivative term that does not contribute to the action, the Lagrangian Eq. (15.34) can be rewritten as5 1 m ¯ μ ∂μ ψ − ψψ ¯ . (15.35) L = ψiγ 2 2 It is also not difficult to write down the Majorana Lagrangian in the twocomponent notation. From ψL = (φ, 0)T and Eq. (15.34) we have 1 L = φ † i(∂0 − σ · ∇)φ − (φ T iσ 2 φ + h.c.) . 2

(15.36)

By comparing the mass term in this expression with Eq. (15.33) one can see that in the two-component formalism the role of the particle–antiparticle conjugation matrix C is played by iσ 2 . From Eq. (15.33) a very important difference between the Dirac and Majorana ¯ are invariant with respect to the mass terms follows. The Dirac mass terms ψψ U(1) transformations ψ → eiα ψ ,

ψ¯ → ψ¯ e−iα ,

(15.37)

i.e. they conserve the charges associated with the corresponding transformations (electric charge, lepton or baryon number, etc.). At the same time, the Majorana mass terms have the structure ψL ψL + h.c., and therefore they break all 5 Indeed, ψiγ ¯ μ ∂μ ψ = ψL iγ μ ∂μ ψL + (ψL )c iγ μ ∂μ (ψL )c , and using Eqs. (15.9) and (15.10) one can rewrite the last term as (ψL )c iγ μ ∂μ (ψL )c = −∂μ [ψ¯ L iγ μ ψL ] + ψ¯ L iγ μ ∂μ ψL . Thus, we have ψ¯ L iγ μ ∂μ ψL = (1/2)ψ¯ iγ μ ∂μ ψ + total derivative term.

15.1 Weyl, Dirac, and Majorana fermions

313

U(1)-charges by two units. Since the electric charge is exactly conserved, this in particular means that no charged particle can have Majorana mass. Another important point is that the Majorana mass term in Eqs. (15.33) and (15.34) do not vanish, even though the matrix C −1 is antisymmetric (because so is C). This follows from the fact that the fermionic quantum fields anticommute, and so the interchange of the two ψL in ψLT C −1 ψL yields an extra minus sign. A similar argument applies to the Majorana mass term in the two-component formalism in Eq. (15.36) (note that the matrix σ 2 is antisymmetric). Thus, the Majorana mass is of essentially quantum nature.6 In the massless limit the difference between Dirac and Majorana particles disappears as both actually become Weyl particles. In particular, vanishing Majorana mass means that the free Lagrangian now conserves a U(1) charge corresponding to the transformations Eqs. (15.37).

15.1.4 Feynman rules for Majorana particles Let us now briefly review the Feynman rules for Majorana particles [399, 400, 401, 402, 403, 404, 405]. Unlike for a Dirac fermion, whose quantum field ψ annihilates the particle and creates its antiparticle while ψ † annihilates the antiparticle and creates the particle, in the Majorana case the same field χ creates and annihilates the corresponding Majorana fermion. This leads to the existence of Wick contractions that are different from the standard ones. As a result, in addition to the usual Feynman propagator  i(p/ + m) −ip(x−x ) d4 p   (15.38) e SF (x − x ) ≡ 0|Tχ (x)χ¯ (x )|0 = 4 2 (2π) p − m2 + iε which coincides with the propagator of the Dirac fermion, there exist new types of propagators [399, 400], 0|Tχ (x)χ T (x )|0 = −SF (x − x )C

(15.39)

0|T χ¯ T (x)χ¯ (x )|0 = C −1 SF (x − x ) ,

(15.40)

and

where p/ ≡ γ μ pμ and we have used the second Eq. (15.11) and the Majorana condition χ c = χ . Recall that Dirac fermions carry a conserved additive charge which is generically called the fermion number. The flow of this number is usually indicated on Feynman diagrams by arrows on the fermion lines which correspond to the standard propagator Eq. (15.38). If a diagram contains a chain of fermion 6 Note, however, that formally one can also write the Majorana mass term at the classical level if one assumes

that ψ(x) is an anticommuting classical field, i.e. a field that takes Grassmann numbers as its values.

314

Majorana neutrinos and other Majorana particles

lines, the fermion number flow is continuous through this chain. As Majorana particles do not carry any conserved additive quantum number, there is no continuous flow of fermion number through Feynman diagrams in the Majorana case. This is reflected in the existence of the fermion-number-violating propagators Eq. (15.40), which can be graphically represented as lines with two arrows pointing in opposite directions (outwards for the first propagator in Eq. (15.40) and inwards for the second one). In addition, each term of the interaction Lagrangian that contains Majorana fields gives rise to several vertices, depending on the direction of the arrows on the incoming and outgoing lines of Majorana particles. Some of these vertices also contain the particle–antiparticle conjugation matrix C [399, 400]. Special care should be taken to get the correct relative signs between different diagrams contributing coherently to the same amplitude. The rules are completed by requiring that diagrams with Majorana fermion loops have an extra factor 1/2 due to the permutation symmetry of the Majorana particles. The resulting Majorana Feynman rules are rather complicated. They can, however, be simplified by noting that the matrices C (or C −1 ) that are present in some vertices are always either canceled by the corresponding matrices in the propagators in Eq. (15.40) or eliminated through the proper attribution of the spinors to the external fermionic legs of the diagram with the help of Eq. (15.28). This leads to much simpler Majorana Feynman rules, with propagators and vertices not containing explicitly the matrix C [401, 402, 403, 404]. In this case the Feynman rules include just the usual propagator Eq. (15.38) for Majorana fermions, and the number of vertices corresponding to each term of the interaction Lagrangian is at most two. The Majorana fermion propagators are depicted by lines with no arrows. Instead of the fermion number flow (which is not conserved) the notion of a fermion flow is introduced. To each diagram a certain (but arbitrary) direction of the fermion flow is attributed, which is used simply as a bookkeeping device; the analytic expressions for the amplitudes are independent of the chosen direction of this flow. Finally, in appendix B of Ref. [405] a very simple set of Majorana Feynman rules is suggested, based on the elimination of the adjoint Majorana fields χ¯ from the kinetic as well as the interaction terms of the Lagrangian through the relation χ¯ = χ c = −χ T C −1 . For more detailed discussions of Majorana Feynman rules, we refer the reader to Refs. [399, 400, 401, 402, 403, 404, 405]. 15.2 C, P, CP, and CPT properties of Majorana fermions Since Majorana fermions are their own antiparticles, they are expected to have special properties with respect to C, P, CP, and CPT transformations.

15.2 C, P, CP, and CPT properties of Majorana fermions

315

Consider first the charge conjugation C.7 As already discussed in Section 15.1.1, for massive spinor fields this operation coincides with the particle–antiparticle conjugation Cˆ defined in Eq. (15.9). The latter, however, without loss of generality, can be modified by introducing an arbitrary phase factor ηC∗ on the right-hand side. That is, instead of Eq. (15.9) we can define ψ c ≡ ηC∗ C ψ¯ T = ηC∗ iγ 2 ψ ∗ .

(15.41)

Indeed, this will not affect the evolution equation satisfied by ψ c (x) as well as the relation (ψ c )c = ψ. The charge conjugation transformation C Eq. (15.29) can be modified accordingly, so that the equivalence between the C and Cˆ conjugations is maintained: C bs (p) C−1 = ηC∗ ds (p) ,

C ds† (p) C−1 = ηC∗ b†s (p) .

(15.42)

Equations (15.41) and (15.42) apply to arbitrary spin-1/2 fermions; let us now discuss Majorana fields. Because, for them, C ψ¯ T = ψ, the new definition of Cˆ conjugation Eq. (15.41) implies that the Majorana condition Eq. (15.20) now takes the form ψ c (x) = ηC∗ ψ(x) .

(15.43)

Since for Majorana fields one has to identify ds (p) = bs (p), Eq. (15.42) becomes C bs (p) C−1 = ηC∗ bs (p) ,

C b†s (p) C−1 = ηC∗ b†s (p) .

(15.44)

Hermitian conjugation of the first of these two equalities yields C b†s (p) C−1 = ηC b†s (p). The consistency of this relation and the second one in Eq. (15.44) requires that ηC be real, i.e. ηC = ±1. Next, let us apply the second Eq. (15.44) to the vacuum state. Assuming the vacuum to be even under the charge conjugation, we find C|p, s = ηC |p, s ,

(15.45)

where |p, s is the one-particle Majorana state with momentum p and spin projection s. The Majorana condition Eq. (15.43) and Eq. (15.45) imply that the Majorana state is an eigenstate of charge conjugation C, and ηC is its charge parity. It should be stressed, however, that this is, strictly speaking, only valid when C is exactly conserved. The preceding description certainly applies to free Majorana fermions, since the corresponding action is charge conjugation invariant.8 However, the 7 Here we mostly follow Ref. [406], though some of our phase conventions are different. 8 This can be most easily seen if we rewrite the kinetic term of the Lagrangian L = ψiγ ¯ μ ∂μ ψ as k ¯ μ ∂μ ψ − ∂μ ψ¯ · iγ μ ψ] + total derivative term and apply the Cˆ conjugation (which for Lk = (1/2)[ψiγ

massive fermion fields is equivalent to charge conjugation C) to the full Lagrangian of free Majorana particles.

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Majorana neutrinos and other Majorana particles

charge parity ηC , apart from being real, is completely arbitrary and therefore unphysical in this case. A physical (i.e. interacting) Majorana particle is an eigenstate of C only when all its interactions are C invariant.9 The C parity of a Majorana particle is then constrained by the C-transformation properties of the other fields that enter its interaction Lagrangian. If C is only an approximate symmetry of the theory, the Majorana particle will be an approximate eigenstate of C, to the same extent to which charge conjugation invariance is satisfied. The situation is completely different for neutrinos, which are the prime candidates for being Majorana particles. The point is that their charged-current weak interactions are maximally C violating. Indeed, these interactions are left handed (i.e. of the V − A form), whereas charge conjugation would transform them into the right-handed (V + A) interactions which do not exist in the Standard Model based on the gauge group SU(2)L × U(1). Thus, for Majorana neutrinos, C parity does not bear any physical sense. However, CP is a good approximate symmetry of the leptonic sector of the Standard Model. Indeed, it is an exact symmetry of the gauge interactions, and in the minimally extended (to include non-zero neutrino mass) Standard Model it can only be violated by the neutrino mass generating sector. The corresponding CP-violation effects are very difficult to observe – in particular, they have not been unambiguously observed by the time of publication of this book. Therefore, in many situations CP violation in the leptonic sector can be ignored. In other words, in some regards Majorana neutrinos can be considered as CP eigenstates with certain CP parities. This, however, is not in general true when possible CP-violating effects play a major role. We discuss these effects in Sections 15.3, 15.7, and 15.8. The properties of Majorana particles with respect to CP (assuming that it is a good symmetry) and CPT can be studied similarly to their properties under C transformation [406]. The results are summarized in Table 15.1. In deriving the properties of Majorana neutrinos under the discrete symmetries one can make use of the following properties of the spinors us (p) and vs (p): γ 0 us (p) = us (−p), u∗s (p) = (−1)s+1/2 γ 1 γ 3 u−s (−p),

γ 0 vs (p) = −vs (−p), v∗s (p) = (−1)s+1/2 γ 1 γ 3 v−s (−p),

where the sign factors in the second line correspond to the phase convention γ5 us (p) = (−1)s−1/2 v−s (p). This choice of the phases is consistent with that in Eq. (15.28). It should also be kept in mind that, while C and P are unitary operators, T is antiunitary, and so is CPT.

9 An example of such a Majorana fermion is the photino – the supersymmetric partner of the photon

[407, 400, 408, 468].

15.3 Mixing and oscillations of Majorana neutrinos

317

Table 15.1. Effects of C, CP, and CPT operations on a Majorana field ψ(t, x) and on the corresponding one-particle Majorana state |p, s Symmetry operation

Effect on ψ(t, x)

Effect on |p, s

Restriction

C

ηC∗ iγ 2 ψ ∗ (t, x)

ηC |p, s

ηC = ±1

CP CPT

∗ iγ 0 γ 2 ψ ∗ (t, − x) ηCP ∗ ∗ −ηCPT γ5 ψ (−t, −x)

ηCP | − p, s s |p, −s ηCPT

ηCP = ±i ηCPT = ±i

s Here ηCPT ≡ (−1)s−1/2 ηCPT.

15.3 Mixing and oscillations of Majorana neutrinos In Section 15.1 we considered the mass term of a lone Majorana particle, Eq. (15.33). This expression is readily generalized to the case when there are n Majorana fermions which in general can mix with each other: + 1

1* (15.46) Lm = − (ψL )c m ψL + ψL m† (ψL )c = ψLT C −1 mψL + h.c. . 2 2 Here ψ = (ψ1 , . . . , ψn )T and m is an n × n matrix. Using the anticommutation property of the fermion fields and C T = −C, it is easy to show that the matrix m must be symmetric: mT = m. Equation (15.46) applies to any set of Majorana particles; in the rest of this section we shall specifically consider Majorana neutrinos and their oscillations. 15.3.1 Neutrinos with a Majorana mass term Consider first the case of n standard lepton generations, consisting each of an SU(2)L -doublet of left-handed neutrino and charged lepton fields lα = (ναL , eαL )T and an SU(2)L -singlet right-handed charged lepton field eαR (α = e, μ, τ, ...) [143, 409]. In the Standard Model extended to include the mass generation mechanism for Majorana neutrinos, the terms of the Lagrangian that are relevant to neutrino oscillations include the charged-current (CC) weak interaction term and the mass terms of the charged leptons and neutrinos: g 1 Lw+m = − √ (¯eL γ μ νL ) Wμ − e¯ L ml eR + νLT C −1 m νL + h.c. 2 2

(15.47)

Here g is the CC gauge coupling constant, Wμ is the W − boson field, and all leptons are assembled in vectors in the generation space: νL = (νeL , νμL , ντ L , ...)T , and similarly for eL and eR . Since the CC weak interaction Lagrangian in Eq. (15.47)

318

Majorana neutrinos and other Majorana particles

is diagonal in the chosen basis, the latter is called the weak-eigenstate basis. The matrices ml and m are, however, in general not diagonal in this basis. For n leptonic generations, the mass matrix of the charged leptons ml is a general complex n × n matrix, whereas the Majorana mass matrix of neutrinos m is a complex symmetric n × n matrix. Recall now that an arbitrary square matrix A can be diagonalized by a bi-unitary transformation according to Adiag = V1† AV2 , where Adiag is a diagonal matrix with non-negative diagonal elements. Similarly, a symmetric square matrix B is diagonalized by a transformation with a single unitary matrix: Bdiag = U T BU, where all the diagonal elements of Bdiag are non-negative. We therefore perform the basis transformations of the lepton fields according to eL = VL eL ,

eR = VR eR ,

νL = UL νL ,

(15.48)

with VL , VR , and UL being unitary matrices chosen such that they diagonalize the charged-lepton and neutrino mass matrices: VL† ml VR = ml ,

ULT m UL = m

(15.49)

(ml , m are diagonal mass matrices). Note that the kinetic Lagrangians of neutrinos and of the left- and right-handed charged leptons are invariant under these transformations. In the new (unprimed) basis Eq. (15.47) takes the form  g  e¯ αL γ μ Uαi νiL Wμ − mlα e¯ αL eαR Lw+m = − √ 2 α,i α 1 T −1 + mi νiL C νiL + h.c. (15.50) 2 i Here mlα (a = e, μ, τ, ...) and mi (i = 1, 2, 3, ...) are the diagonal elements of the mass matrices ml and m, respectively, i.e. they are the masses of the charged leptons and of the mass-eigenstate neutrinos. The matrix U ≡ VL† UL

(15.51)

is the leptonic mixing matrix, also called the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix [140, 141, 142]. The flavor-eigenstate neutrino fields are defined as ναL =

n 

Uαi νiL .

(15.52)

i=1

In terms of these fields the CC-interaction part of the Lagrangian Eq. (15.50) takes  the form Lw = − √g2 α e¯ αL γ μ ναL Wμ +h.c., i.e. the flavor eigenstates νe , νμ , ντ ,... are the neutrinos emitted or absorbed together with the charged leptons e, μ, τ ,. . . , respectively.

15.3 Mixing and oscillations of Majorana neutrinos

319

Let us introduce the four-component neutrino fields χi = νiL + (νiL )c ,

i = 1, . . . , n .

(15.53)

T −1 T −1 Using the relations νiL C νiL = −(νiL )c νiL , (νiL C νiL )† = −νiL (νiL )c which follow from Eq. (15.11), one can rewrite the neutrino mass term in Eq. (15.50) as

1 =− mi χ¯ i χi . 2 i=1 n

Lνm

(15.54)

Equations (15.52) and (15.53) mean that the neutrino flavor eigenstates ναL are linear superpositions of the left-handed components of the n mass eigenstates χi . From Eq. (15.53) it follows that χic = χi , i.e. the massive neutrino fields are Majorana fields in the case we consider. The Lagrangian Eq. (15.47) (or equivalently Eq. (15.50)) implies that massive neutrinos are in general mixed, and leads to the phenomenon of neutrino flavor oscillations [140, 141, 142]. The oscillation probability, i.e. the probability that a relativistic neutrino produced as a flavor eigenstate να will be in a flavor eigenstate νβ after having propagated a distance L in vacuum, is [410, 411] 2    m2  −i 2pij L ∗ Uβi e Uαi  , (15.55) P(να → νβ ; L) =    i

where p is the modulus of the mean momentum of the neutrino state, m2ij = m2i − m2j , and for the index j one can take any fixed value between 1 and n. Equation (15.55) can be written equivalently as    m2ij ∗ ∗ L Re(Uαj Uβj Uαi Uβi ) cos P(να → νβ ; L) = 2p i,j    m2ij ∗ ∗ + Im(Uαj Uβj Uαi Uβi ) sin L . (15.56) 2p i,j Let us now discuss the general properties of the leptonic mixing matrix U. Being an n × n unitary matrix, it depends on n2 independent parameters, of which n(n − 1)/2 are mixing angles and n(n + 1)/2 are complex phases. Not all of these phases are physical, though. As follows from Eq. (15.50), one can always remove n phases from U by rephasing the left-handed fields of charged leptons according to eαL → eiϕα eαL , which allows one to fix the phases of one column of the matrix U. If one also rephases the right-handed charged lepton fields in the same way, i.e. eαR → eiϕα eαR , the phase change of the fields eαL in the mass term of the charged leptons in Eq. (15.50) will be compensated, and therefore Eq. (15.50) will remain unchanged. It is easy to see that the kinetic terms of the Lagrangian of eL and eR are

320

Majorana neutrinos and other Majorana particles

also invariant under the field rephasing. Thus, the leptonic Lagrangian is invariant with respect to the above rephasing of the charged lepton fields, which means that n out of the n(n + 1)/2 phases in U are unphysical. How about rephasing the neutrino fields? Consider first the case when neutrinos are Dirac particles (which means that we should add n right-handed neutrino fields to our model). Then their mass term is similar to that of the charged leptons. In that case it is possible similarly to rephase the left-handed and right-handed neutrino fields without modifying their mass term, which could be used to fix the phases of the elements of one line of the matrix U. However, the number of the phases that can be removed from U in this way is n − 1 rather than n, because the phase of one element which is at the intersection of the selected line of U and the column whose phases have already been fixed by the rephasing of the charged lepton fields can no longer be modified. Thus, the total number of physical phases characterizing the leptonic matrix U in the case of Dirac neutrinos is D = Nph

n(n + 1) (n − 1)(n − 2) − n − (n − 1) = . 2 2

(15.57)

∗ ∗ Uαi Uβi that enter Eq. (15.56) for the oscillation Note that the quantities Uαj Uβj probabilities are invariant with respect to the rephasing of the charged-lepton and neutrino fields. Let us now return to Majorana neutrinos. In this case the neutrino fields cannot be rephased since the Majorana mass term is of the type νL νL + h.c. rather than ν¯ L νR + h.c., and therefore it is not rephasing invariant. As a result, the total number of physical phases characterizing the leptonic matrix U in the case of the Majorana neutrinos is n(n − 1) n(n + 1) M −n= . (15.58) = Nph 2 2

The extra n−1 physical phases that are present in the Majorana neutrino case can be collected in a diagonal matrix of phases which is factored out of U as a right-hand side multiplier. Indeed, from Eq. (15.50) it is seen that such a factorization isolates in a diagonal factor the phases that could have been absorbed into the rephasing of the νL fields if the neutrinos were Dirac particles. Thus, one can write ˜ . ˜ · diag(1, eiϕ1 , eiϕ2 , . . . , eiϕn−1 ) ≡ UK U=U

(15.59)

˜ contains only (n − 1)(n − 2)/2 phases that are relevant also in Here the matrix U the Dirac neutrino case, whereas the factor K contains the extra “Majorana-type” phases. The position of the unit element in K is irrelevant, as the overall phase of the matrix U is unobservable. From Eqs. (15.57) and (15.58) it follows that in the Dirac neutrino case the mixing matrix U contains physical complex phases only

15.3 Mixing and oscillations of Majorana neutrinos

321

for n ≥ 3 generations, whereas in the Majorana case the physical phases are in general there for n ≥ 2. The reason why we paid so much attention to the phases of the leptonic mixing matrix U is that they lead to CP-violating effects in the leptonic sector. CP conjugation transforms left-handed neutrinos into their antiparticles – right-handed antineutrinos. Since CP conjugation of a fermionic field includes complex conjugation (see Table 15.1), the oscillation probability P(¯νa → ν¯ b ; L) is described by the right-hand side of Eq. (15.55) with the matrix U substituted by U ∗ . Complex conjugation means that the signs of all the phases characterizing U must be flipped; for this reason these phases are called CP-violating phases. In particular, if U bears non-removable complex phases beyond those contained in the matrix K, neutrino oscillations violate CP invariance, i.e. να → ν¯ β ; L) = 0 . PCP αβ (L) ≡ P(να → νβ ; L) − P(¯

(15.60)

The quantity PCP αβ (L) coincides (up to the factor of two) with the CP-odd part of the να → νβ oscillation probability, which is given by the second term on the right-hand side of Eq. (15.56), whereas the first term in Eq. (15.56) corresponds to the CP-even part of the probability. Can one find out whether neutrinos are Dirac or Majorana particles by studying neutrino oscillations? Unfortunately, this is not possible. It turns out that for Dirac neutrinos the oscillation probabilities are given by exactly the same formulae, Eqs. (15.55) or (15.56), as those for Majorana neutrinos. Moreover, the extra Majorana-type phases that enter the leptonic mixing matrix U in the Majorana neutrino case are not observable in neutrino oscillations. Indeed, the matrix K in Eq. (15.59) and the matrix exp[−i(m2ij /2p)L] for fixed j that enters Eq. (15.55) are both diagonal and therefore commute, so that K drops out of the expression for the oscillation probability Eq. (15.55). A similar argument applies to neutrino oscillations in matter. The Majorana-type phases, however, enter some other physical observables and so are in general observable quantities. We discuss these observables in Sections 15.7 and 15.8. For future reference, we give here the leptonic mixing matrix U for the case of three leptonic generations with Majorana neutrinos in the so-called standard parameterization: ⎞ ⎛ c12 c13 s12 c13 s13 e−iδCP U = ⎝−s12 c23 − c12 s13 s23 eiδCP c12 c23 − s12 s13 s23 eiδCP c13 s23 ⎠ iδCP iδCP s12 s23 − c12 s13 c23 e −c12 s23 − s12 s13 c23 e c13 c23 ⎞ ⎛ 1 0 0 iϕ1 ⎝ (15.61) × 0 e 0 ⎠. iϕ2 0 0 e

322

Majorana neutrinos and other Majorana particles

Here cij ≡ cos θij , sij ≡ sin θij , where θij are the mixing angles, δCP is the Dirac-type CP-violating phase, and ϕ1,2 are the Majorana-type CP-violating phases.

15.3.2 General case of Dirac + Majorana mass term Consider now the case in which, in addition to n standard leptonic generations with left-handed neutrino fields ναL being parts of SU(2)L leptonic doublets, there exist k right-handed neutrino fields νσ R that are electroweak singlets, i.e. singlets with respect to both weak isospin group SU(2)L and hypercharge U(1) [412, 413, 414, 415, 416, 417]. Such neutrinos do not have electroweak gauge interactions (though they may have, e.g., Yukawa interactions with leptonic and Higgs doublets) and therefore are often called “sterile neutrinos.” In contrast to this, the usual SU(2)L doublet neutrinos νL are called “active neutrinos.” Since νR are electroweak singlets, they do not contribute to the so-called chiral gauge anomalies and so their number is not fixed by the requirement of the anomaly cancelation. In particular, their number need not coincide with the number of the leptonic generations n, i.e. in general k = n. Let us stress that these extra neutrinos are sterile not because they are right-handed – their Cˆ conjugates are left handed and yet also sterile – but because they are electroweak singlets.

In the considered case the most general neutrino mass term contains the Majorana masses mL and mR for the left-handed and right-handed neutrino fields, respectively, as well as the Dirac mass mD that couples the νL s with the νR s: 1 1 Lm = νL T C −1 mL νL − νR mD νL + νR T C −1 m∗R νR + h.c. 2 2

(15.62)

Here we have assembled the n left-handed and k right-handed neutrino fields into the vectors νL and νR . The quantities mL and mR are complex symmetric n × n and k × k matrices, respectively, mD is a complex k × n matrix, and we have introduced the right-handed Majorana mass matrix through its complex conjugate to simplify the further notation. Introducing the vector of n + k left-handed fields,       νL νL = , (15.63) nL = (νR )c ν cL we can rewrite Eq. (15.62) as Lm =

1 T −1 n C M nL + h.c. , 2 L

(15.64)

15.3 Mixing and oscillations of Majorana neutrinos

where the matrix M has the form



M=

mL mTD mD mR

323

 (15.65)

.

In deriving Eq. (15.64) we have used the relations (ψRT C −1 m∗ ψR )† = (ψ cL )T C −1 m ψ cL , (15.66) ψR m ψL = −(ψ cL )T C −1 m ψL = −ψLT C −1 mT ψ cL , which follow from Eqs. (15.28) and (15.10). The matrix M is complex symmetric, so it can be diagonalized with a single unitary matrix. We therefore write naL =

n+k 

Uai χiL ,

U T MU = Md ,

(15.67)

i=1

where Md is a diagonal (m + n) × (m + n) matrix with non-negative diagonal elements Mdi . In terms of the fields χiL the neutrino mass term Eq. (15.64) reads Lm =

n+k 1  Mdi χiLT C −1 χiL + h.c. 2 i=1

(15.68)

Introducing the four-component massive neutrino fields χi as χi = χiL + (χiL )c ,

i = 1, . . . , n + k ,

(15.69)

we can rewrite the neutrino mass term in the mass-eigenstate basis Eq. (15.68) in the standard form: n+k 1 Lm = − mi χ¯ i χi . (15.70) 2 i=1 In Eq. (15.67) the index a can take n + k values; we will denote collectively the first n of them by α or β and the last k by σ or ρ. Equation (15.67) yields ναL =

n+k  i=1

Uαi χiL ,

(νσ R ) = c

n+k 

Uσ i χiL .

(15.71)

i=1

This means that left-handed active neutrinos and left-handed sterile antineutrinos are linear combinations of the left-handed components of all n + k mass-eigenstate fields χi . From Eq. (15.69) it follows that χic = χi , i.e. the neutrino mass eigenstates are Majorana fermions in the case we currently consider, just as in the pure Majorana mass term case discussed in Section 15.3.1. This is a general feature of fermion mass models: if the fermions possess Majorana mass terms, then, independently of whether or not the Dirac mass terms are also present, the mass eigenstates are always Majorana particles.

324

Majorana neutrinos and other Majorana particles

This is actually easy to understand by counting the number of the field degrees of freedom. In the Majorana mass case studied in Section 15.3.1 one has n twocomponent neutrino fields, and the neutrino mass matrix has in general n distinct eigenvalues. Each massive neutrino field then has two degrees of freedom, i.e. it should be a Majorana field. In the pure Dirac case there are 2n two-component fields (n left handed and n right handed), and the mass matrix has n eigenvalues. This means that each mass eigenstate has four degrees of freedom, i.e. is a Dirac field. In the Dirac + Majorana mass case there are n + k two-component fields, n left handed and k right handed. The matrix M has n + k in general distinct eigenvalues, which means that each neutrino mass eigenstate is characterized by two degrees of freedom, i.e. is a Majorana field. If some of the mass eigenvalues coincide, the corresponding twocomponent Majorana fields can merge into four-component Dirac ones. We consider this phenomenon in Section 15.3.3.

Let us now discuss neutrino oscillations in the Dirac + Majorana (D + M) mass scheme that we are now considering. Unlike in the pure Dirac or pure Majorana mass cases, in the (D + M) scheme two types of neutrino oscillations become possible: active–sterile and sterile–sterile oscillations. The oscillations between the active neutrino species are described by the same expression as in Eq. (15.55) but with the matrix U replaced by U and the summation over i extending from 1 to n + k. The probability of oscillations between the active and sterile neutrinos is given by 2  n+k   m2 ij  ∗ P(ναL → ν cσ L ; L) =  Uσ i e−i 2p L Uαi (15.72)  .   i=1

If a mechanism by which sterile neutrinos can be produced and detected exists,10 one can in principle observe sterile–sterile neutrino oscillations, whose probability is given by 2  n+k   m2 ij   Uρi e−i 2p L Uσ∗i  . (15.73) P(ν cσ L → ν cρL ; L) =    i=1

Equation (15.73) describes the oscillations between the left-handed sterile neutrino states ν cLσ = (νRσ )c and ν cLρ = (νRρ )c ; the oscillations between the corresponding right-handed states νRσ and νRρ can be obtained from Eq. (15.73) by replacing U ↔ U ∗. 10 Recall that, even though sterile neutrinos do not have gauge interactions in the Standard Model, they may

possess other interactions, such as Yukawa ones, or extra gauge interactions in the extensions of the Standard Model (e.g. SU(2)R gauge interactions in left–right symmetric models).

15.3 Mixing and oscillations of Majorana neutrinos

325

If the sterile neutrinos are completely undetectable, one can only observe active– active and active–sterile oscillations, the latter manifesting themselves through the disappearance of the active neutrinos.

15.3.3 Dirac and pseudo-Dirac neutrino limits in the D + M case As we have pointed out, if Majorana mass terms are present in a fermion mass model, the mass-eigenstate fermions are always Majorana particles, even when the Dirac mass terms are present as well. One can expect, however, that in the limit when the Majorana mass terms are much smaller than the Dirac ones, the properties of the mass eigenstates would come close to those of Dirac fermions. To see how this happens, it is instructive to consider the simple one-generation neutrino case with Majorana and Dirac mass terms. The quantities mL , mR , and mD are then just numbers, and M is a 2 × 2 matrix. For simplicity we shall assume all the mass parameters to be real. This, in particular, means that CP is conserved in the neutrino mass sector, i.e. the free mass eigenstates are also eigenstates of CP. The matrix M can be diagonalized by the transformation OT MO = Md , where O is a real orthogonal 2 × 2 matrix and Md = diag(m1 , m2 ). We introduce the fields χL through nL = OχL , or      cos θ sin θ χ1L νL = . (15.74) nL = ν cL χ2L − sin θ cos θ Here χ1L and χ2L are the left-handed components of the neutrino mass eigenstates. The mixing angle θ is given by tan 2θ = and the neutrino mass eigenvalues are m1,2

mR + mL = ∓ 2

2mD , mR − mL

 

mR − mL 2

(15.75)

2 + m2D .

(15.76)

They are real but can be of either sign. The neutrino mass term can now be rewritten as 1 1 Lm = nTL C −1 M nL + h.c. = χLT C −1 Md χL + h.c. 2 2 1 T T C −1 χ1L + m2 χ2L C −1 χ2L ) + h.c. = (m1 χ1L 2 1 = ( |m1 | χ 1 χ1 + |m2 | χ 2 χ2 ) . (15.77) 2

326

Majorana neutrinos and other Majorana particles

Here we have defined χ1 = χ1L + η1 (χ1L )c ,

χ2 = χ2L + η2 (χ2L )c ,

(15.78)

with ηi = 1 or −1 for mi > 0 or < 0, respectively. It follows immediately from Eq. (15.78) that the mass-eigenstate fields χ1 and χ2 describe Majorana neutrinos. The relative signs of the mass eigenvalues (η1 and η2 ) determine the relative CP parities of χ1 and χ2 ; the physical masses |m1 | and |m2 | are positive, as they should be. Instead of using a real orthogonal matrix O to diagonalize M one could employ a unitary matrix U = OK with K being a diagonal matrix of phases, as in Eq. (15.59). Choosing for the positive mass eigenvalues of M the diagonal elements of K to be 1 and for the negative ones ±i, one can write U T MU = Md , where Md now does not have any negative diagonal elements. Correspondingly, Eq. (15.74) should be replaced by nL = UχL . In this way it is no longer necessary to introduce the factors η1,2 , i.e. instead of Eq. (15.78) we have χ1 = χ1L + (χ1L )c ,

χ2 = χ2L + (χ2L )c .

(15.79)

That the neutrino mass eigenstates corresponding to opposite signs of the mass parameters m1 and m2 defined in Eq. (15.76) have opposite CP parities is now a consequence of the fact that the matrix U has one complex column. Indeed, let m1 defined in Eq. (15.76) be negative and m2 positive. Then from nL = UχL = OKχL and Eq. (15.79) we have χ1 = ∓i{c[νL − (νL )c ] + s[νR − (νR )c ]},

χ2 = s[νL + (νL )c ] + c[νR + (νR )c ]. (15.80) Making use of the definition of the CP conjugation given in Section 15.2 and taking into account that it is described by a linear operator, one can readily make sure that the CP parities of χ1 and χ2 are opposite.

It is instructive to consider some limiting cases. In the limit of no Majorana masses (mL = mR = 0), the pure Dirac case has to be recovered. Let us see how this limit can be obtained from the general D + M formalism. For mL = mR = 0 the mass matrix Eq. (15.65) takes the form   0 m M= . (15.81) m 0 This matrix corresponds to a conserved lepton number LνL − Lν cL = LνL + LνR , which can be identified with the total lepton number L. Thus, the lepton number is conserved in this limiting case, as expected. Let us now check that the usual Dirac mass term is recovered.

15.4 Seesaw mechanism of neutrino mass generation

327

The matrix M in Eq. (15.81) is diagonalized by the rotation Eq. (15.74) with θ = 45◦ , and its eigenvalues are −m and m. This means that we have two Majorana neutrinos that have the same mass, opposite CP parities, and are maximally mixed. Let us demonstrate that this is equivalent to having one Dirac neutrino of mass and (15.78) it then follows that m. We have√η2 = −η1 = 1; from Eqs. √ (15.74) c c χ1 + χ2 = 2(νL + νR ), χ1 − χ2 = − 2(ν L + ν R ) = −(χ1 + χ2 )c . This gives 1 1 m (χ 1 χ1 + χ 2 χ2 ) = m [(χ1 + χ2 )(χ1 + χ2 ) + [(χ1 − χ2 )(χ1 − χ2 )] 2 4 (15.82) = m ν¯ D νD , where νD ≡ νL + νR .

(15.83)

The counting of the degrees of freedom also shows that we must have a Dirac neutrino in this case – there are four degrees of freedom and just one physical mass. Thus, two maximally mixed degenerate Majorana neutrinos of opposite CP parities merge to form a Dirac neutrino. In Section 15.7 we shall discuss neutrinoless double β (0νββ) decay and show that this process can only take place if neutrinos are Majorana particles. We shall demonstrate there that in the limit when two degenerate in mass Majorana neutrinos merge into a Dirac neutrino, their contributions to the amplitude of the 0νββ decay exactly cancel, as they should. If the Majorana mass parameters mL and mR do not vanish but are small compared to mD , the resulting pair of Majorana neutrinos will be quasi-degenerate with almost maximal mixing and opposite CP parities. The physical neutrino masses in this case are |m1,2 |  mD ∓ (mL + mR )/2  mD . Such a pair in many respects behaves as a Dirac neutrino and therefore it is sometimes called a “pseudo-Dirac” (or a “quasi-Dirac”) neutrino. In particular, its contribution to the 0νββ decay amplitude is proportional to the mass difference |m2 | − |m1 |  (mL + mR ), which is much smaller than the mass of each component of the pair.

15.4 Seesaw mechanism of neutrino mass generation The seesaw mechanism [418, 419, 420, 421, 422] provides a very simple and attractive explanation of the smallness of neutrino mass by relating it to the existence of a very large mass scale. In the simplest and most popular version of this mechanism, the requisite large mass scale is given by the masses of heavy electroweaksinglet Majorana neutrinos. Although the seesaw mechanism is most natural in the framework of the grand unified theories (such as SO(10)) or left–right symmetric models [423], it also operates in the Standard Model extended to include the righthanded sterile neutrinos νR . Indeed, as soon as the νR s are introduced, one can add

328

Majorana neutrinos and other Majorana particles

to the Lagrangian of the model the Majorana mass term (1/2) νRT C −1 mR νR + h.c., which is allowed because νR are electroweak singlets. The Yukawa couplings of the right-handed neutrinos with the lepton doublets and the Higgs boson are also allowed, and after the spontaneous breaking of the electroweak symmetry they give rise to the Dirac mass term connecting the active and sterile neutrinos, similar to those that are present for quarks and charged leptons. The resulting neutrino mass scheme is just the D+M one discussed in Sections 15.3.2 and 15.3.3; see Eqs. (15.62)–(15.65). Note that in the Standard Model there is no Majorana mass term for left-handed neutrinos, i.e. mL = 0; however, mL is different from zero in some extensions of the Standard Model, so we shall keep it for generality. Because the right-handed neutrinos νR are electroweak singlets, the scale of their Majorana mass term need not be related to the electroweak scale. In particular, mR can be very large, possibly even at the Planck scale MPl  1.2 × 1019 GeV, grand unification scale, or at some intermediate scale MI ∼ 1010 − 1012 GeV which may be relevant for the physics of parity breaking. The seesaw suppression of the masses of active neutrinos is realized just in this case of a very large mR . We therefore consider the limit in which the characteristic scales of the Dirac and Majorana neutrino masses satisfy mL , mD  mR .

(15.84)

First let us discuss the one-generation case studied in Section 15.3.3. The diagonalization of the mass matrix is then performed by the simple rotation Eq. (15.74). A straightforward calculation gives for the rotation angle θ and eigenvalues of the mass matrix M θ

mD  1, mR

m1  mL −

m2D , mR

m2  mR ,

(15.85)

while the mass eigenstates are given by χ1  νL + η1 (νL )c ,

χ2  (νR )c + η2 νR .

(15.86)

Thus, we have a very light Majorana mass eigenstate χ1 predominantly composed ˆ of the active neutrino νL and its C-conjugate (νL )c , and a heavy eigenstate χ2 , mainly composed of the sterile νR and (νR )c . The admixture of the sterile neutrino state νR in χ1 and that of the usual active neutrinos νL in χ2 are of the order of mD /mR  1. As follows from Eq. (15.85), for mL  m2D /mR it is the sterile neutrino being heavy that makes the usual active one light (which explains the name “seesaw mechanism”). Consider now the case of n standard generations of left-handed leptons and k sterile neutrinos νR . This is actually the case discussed in Section 15.3.2, but now we want to consider specifically the limit of very high νR mass scale. Let us first

15.4 Seesaw mechanism of neutrino mass generation

329

decouple the light and heavy neutrino degrees of freedom. To this end, we blockdiagonalize the matrix M in Eqs. (15.64) and (15.65) according to nL = VχL ,  V MV = V T

T

mL mTD mD MR



 V =

m ˜L 0 ˜R 0 M



(15.87) ,

˜ R are symmetric where V is a unitary (n + k) × (n + k) matrix, m ˜ L and M n × n and k × k matrices, respectively, and we have changed the notation mR → MR . Note that the fields χ  that block-diagonalize M are not the fields ˜ R are not in general of mass-eigenstate neutrinos, since the matrices m ˜ L and M diagonal. They can be diagonalized by further unitary transformations. Correspondingly, V is not the leptonic mixing matrix. We shall be looking for the matrix V in the form [424] ⎞ ⎛  † 1 − ρρ ρ ⎠, (15.88) V=⎝  † −ρ 1 − ρ †ρ where ρ is an n × k matrix. Note that V is unitary by construction. Treating ρ as a perturbation and performing block-diagonalization of M approximately, we find ρ ∗  mTD MR−1 ,

˜ R  MR , M

m ˜ L  mL − mTD MR−1 mD .

(15.89) (15.90)

These relations generalize those of Eq. (15.85) to the case of n active and k sterile neutrinos. The diagonalization of the effective mass matrix m ˜ L then yields n light Majorana neutrino fields which are predominantly composed of the fields ˆ (νL )c , with very small of the usual (active) neutrinos νL and their C-conjugates ˜ R produces (∼ mD /MR ) admixture of sterile neutrinos νR ; the diagonalization of M k heavy Majorana neutrino fields which are mainly composed of νR and (νR )c . This, in particular, means that the oscillations between the active and sterile neutrinos are suppressed in this case. It is important that the active neutrinos get Majorana masses m ˜ L even if they have no “direct” masses, i.e. mL = 0, as is the case in the Standard Model. The masses of the active neutrinos are then of the order of m2D /MR . The generation of the effective Majorana mass of light neutrinos is diagrammatically illustrated in Figure 15.1.

330

Majorana neutrinos and other Majorana particles H

vL

mD

H

vR

MR

vR

mD

vL

Figure 15.1. Seesaw mechanism of m ˜ L generation. The vacuum expectation values of the Higgs field are denoted by H .

In the case where MR has one or more zero eigenvalues, MR−1 does not exist, and the usual seesaw approximation fails. However, it can be readily modified to produce meaningful results. One can just go to the νR basis where MR is diagonal and include the lines and columns of MR that contain zero eigenvalues into a redefined matrix mL . This situation is called the “singular seesaw,” and it generally leads to the existence of pseudo-Dirac light neutrinos.

What can one say about the expected mass scale of the right-handed neutrinos MR ? Let us take the mass of the heaviest among the light neutrinos to be mν ∼ 5 × 10−2 eV, as required by the data of the atmospheric and accelerator neutrino oscillation experiments under the assumption of the hierarchical neutrino masses. Then, assuming that the largest eigenvalue of the Dirac neutrino mass matrix is of the order of the electroweak scale, mD ∼ 200 GeV, and that mL  m2D /MR , from Eq. (15.90) we find MR ∼ 1015 GeV. Interestingly, this is very close to the expected grand unification scale mGUT ∼ 1016 GeV. Thus, neutrino oscillations, together with the seesaw mechanism of neutrino mass generation, may be giving us an indication in favor of the grand unification of weak, electromagnetic, and strong interactions. The version of the seesaw mechanism discussed here, with heavy sterile neutrinos responsible for the small light neutrino masses, is sometimes called type I seesaw. There also exist other versions – those with heavy SU(2)L -triplet Higgs scalars (type II seesaw), heavy triplet fermions (type III seesaw), as well as other realizations of the seesaw mechanism; see Ref. [425] for a review. In all these cases, neutrinos of definite mass are generically Majorana particles. 15.5 Electromagnetic properties of Majorana neutrinos As neutrinos are electrically neutral, they have no direct coupling to the photon, and their electromagnetic interactions arise entirely through loop effects (see Ref. [426] for a review). It is interesting to compare the electromagnetic properties

15.5 Electromagnetic properties of Majorana neutrinos

331

of Majorana neutrinos with those of Dirac neutrinos, which we discuss first. The matrix elements of the electromagnetic current jμ (x) between the one-particle onshell states of a Dirac neutrino (or any other Dirac fermion) can be written as p , s |jμ (x)|p, s = eiqx p , s |jμ (0)|p, s = eiqx u¯ s (p )

μ

(q)us (p) ,

(15.91)

where q ≡ p − p, us (p) and us (p ) are the free-particle plane wave spinors, and μ

(q) = FQ (q2 ) γ μ + FM (q2 ) iσ μν qν + FE (q2 ) σ μν γ5 qν + FA (q2 ) (q2 δ μν − qμ qν ) γν γ5 .

(15.92)

Here FQ (q2 ) and FM (q2 ) are the electric charge and magnetic dipole form factors, and FE (q2 ) and FA (q2 ) are the electric dipole form factor and anapole form factor, respectively. Unlike the magnetic and electric dipole moments, the anapole moment, first proposed in Ref. [427], has no simple classical multipolar analog. It can be modeled by a torus-shaped solenoid and therefore is sometimes called the toroidal moment. For a discussion of the properties of the anapole moment and its experimental manifestations, see Refs. [296, 428] and references therein.

The form of μ (q) in Eq. (15.92) follows from the requirements of Lorentz covariance and electromagnetic current conservation.11 For interactions with real photons the anapole form factor does not contribute. The hermiticity of the Hamiltonian of the electromagnetic interaction Hint = jμ (x)Aμ (x) implies that all the four form factors in Eq. (15.92) are real. The vector-current form factors FQ (q2 ) and FM (q2 ) are parity conserving, while the axial-vector ones FE (q2 ) and FA (q2 ) violate parity. In addition, FE (q2 ) violates CP invariance. As electroweak interactions that induce the effective neutrino electromagnetic current do violate parity (and possibly also CP), in general the form factors FA (q2 ) and FE (q2 ) need not vanish. The charge form factor taken at zero squared 4-momentum transfer yields the electric charge of the Dirac particle, FQ (0) = Q, whereas FM (0), FE (0), and FA (0), give, respectively, its anomalous magnetic moment g − 2, the electric dipole moment, and the anapole moment. The term FQ (q2 )γ μ in Eq. (15.92) describes in the static limit (q2 → 0) not only the electric charge interaction of the Dirac 11 There are alternative (but equivalent) forms of

μ (q).We prefer the one in Eq. (15.92) because each term in it separately conserves the electromagnetic current. The same applies to Eq. (15.98).

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Majorana neutrinos and other Majorana particles

fermion, but also its normal magnetic moment. This can be seen from the Gordon identity  μ  p + pμ iσ μν qν (15.93) us ( p) . + u¯ s ( p )γ μ us ( p) = u¯ s ( p ) 2m 2m The first term in the square brackets here corresponds to the convective part of the current, while the second one describes its spin part, i.e. the normal magnetic moment. The total magnetic moment of the particle is the sum of the normal and the anomalous ones. Because neutrinos have no electric charge (i.e. for them FQ (0) = 0), they do not have normal magnetic moments either. Note that electric neutrality does not mean that the entire charge form factor FQ (q2 ) vanishes. At small q2 one can write 1 FQ (q2 ) = FQ (0) + FQ (0)q2 + . . . ≡ FQ (0) + r 2 q2 + · · · . 6

(15.94)

The quantity r2 characterizes the charge distribution within the particle and is called its charge radius. It is in general different from zero even for neutral particles. Non-triviality of their charge distributions is related to the fact that interactions that “dress” the particles produce clouds of virtual particles of opposite charges and in general different configurations. Let us now discuss the electromagnetic properties of Majorana neutrinos [429, 430, 431, 432, 433, 434, 406]. For a Majorana particle all the electromagnetic form factors but one vanish identically, and the matrix element of the electromagnetic current takes the form p | jμ (0)| p = u¯ ( p )[FA (q2 )(q2 δ μν − qμ qν )γν γ5 ]u( p) .

(15.95)

That is, while the electromagnetic properties of a Dirac fermion are in general described by four form factors, for a Majorana particle only the anapole form factor survives. The simplest way to see this is to note that each term in Eq. (15.92) can be viewed as emerging from the matrix element of the ¯ μ ψ, where ψ is the free field operator and corresponding effective operator ψ μ μ μν μν μ  = (γ , σ qν , σ γ5 qν , γ γ5 ), between the free one-particle states. For Majorana neutrinos from Eqs. (15.11) and (15.12) we find ψ¯ k γ μ ψi = −ψ¯ i γ μ ψk , ψ¯ k σ μν ψi = −ψ¯ i σ μν ψk ,

ψ¯ k γ μ γ5 ψi = ψ¯ i γ μ γ5 ψk , ψ¯ k σ μν γ5 ψi = −ψ¯ i σ μν γ5 ψk ,

(15.96) (15.97)

where we have used the Majorana condition Eq. (15.20). In the case when the fields ψi and ψk correspond to the same particle, i.e. k = i, from Eqs. (15.96) and (15.97) it follows that the only non-zero operator is the axial-vector one, ψ¯ i γ μ γ5 ψi , whereas all the other operators vanish identically. Electromagnetic

15.5 Electromagnetic properties of Majorana neutrinos

333

current conservation then implies that the axial-vector operator can enter the matrix element Eq. (15.91) only through the anapole interaction. This result has a simple interpretation. The charge radius and magnetic and electric dipole moments have opposite signs for neutrinos and antineutrinos, and thus could be used to distinguish between them. Therefore, they must vanish if neutrinos are Majorana fermions. At the same time, the anapole moment does not change its sign under particle–antiparticle conjugation [427, 428], and so is allowed. Note that the vanishing of the charge, magnetic dipole, and electric dipole form factors of Majorana neutrinos has a deep reason – it is related to CPT invariance. Expressions (15.91) and (15.92) describe the matrix elements of the electromagnetic current jμ (x) between the one-particle states of an individual neutrino. They are actually a special case of more general matrix elements of jμ (x) between the states of neutrinos of different mass. These matrix elements have a form similar to Eq. (15.91), except that the quantity μ (q) and the form factors are now matrices in the space of neutrino mass eigenstates. The requirements of Lorentz invariance and electromagnetic current conservation yield μ

(q)ki = [F1 (q2 )ki − γ5 FA (q2 )ki ](q2 δ μν − qμ qν )γν + [FM (q2 )ki − iγ5 FE (q2 )ki ] iσ μν qν .

(15.98)

Equation (15.92) corresponds to the diagonal elements of Eq. (15.98), with FQ (q2 ) = FQ (q2 )i ≡ F1 (q2 )ii q2 (note that for i = k the expression u¯ ks (p )γ μ qμ uis (p) vanishes identically since the spinors satisfy the Dirac equation). The offdiagonal matrix elements of the form factors in Eq. (15.98) are called the transition form factors. They describe transitions νi → νk caused by the interaction of neutrinos with real or virtual photons or external electromagnetic fields. We will briefly discuss some of these processes towards the end of this section. Let us also note that, unlike for the diagonal elements of the form factors, hermiticity of the electromagnetic interaction Hamiltonian Hint does not by itself mean that the transition form factors are real. However, hermiticity of Hint combined with the assumption of CP invariance would require the form factors to be relatively real, i.e. for given i, k all Fa (q2 )ki (a = 1, A, M, E) would be allowed to differ from their respective complex conjugates only by the same phase factor. The expression for μ (q)ki has the same form Eq. (15.98) for Dirac and Majorana neutrinos, though in the Majorana case the form factors must satisfy some additional constraints. We have found that for Majorana neutrinos the diagonal matrix elements of the electromagnetic current contain only one nonzero form factor, FA (q2 )ii . This is an immediate consequence of Eqs. (15.96) and (15.97). However, for transition form factors the constraints are less severe. Equations (15.96) and (15.97) then simply imply that the form factors F1 (q2 )ki , FM (q2 )ki , and FE (q2 )ki are antisymmetric with respect to the interchange of the

334

Majorana neutrinos and other Majorana particles

indices i and k, whereas FA (q2 )ki is symmetric. Actually, this is a consequence of CPT invariance; one can readily check this by making use of the transformation properties of Majorana states under CPT given in Table 15.1 and taking into account that the CPT transformation is antiunitary and that the electromagnetic current jμ (0) is CPT odd.12 Another important point is that the non-vanishing transition electric dipole form factor FE (q2 )ki would not necessarily signify leptonic CP violation. It is only in the case when both the transition magnetic dipole and electric dipole form factors FM (q2 )ki and FE (q2 )ki are simultaneously different from zero that one would have to conclude that CP is violated. In fact, if CP is conserved in the leptonic sector, the form factors F1 (q2 )ki and FM (q2 )ki vanish when νi and νk have the same CP parity (either i or − i), whereas FA (q2 )ki and FE (q2 )ki vanish when νi and νk have opposite CP parities [434]. This can be readily shown using the transformation properties of Majorana states under CP given in Table 15.1. If CP is violated, neutrinos do not possess definite CP parities, and the simultaneous existence of non-zero FM (q2 )ki and FE (q2 )ki (or F1 (q2 )ki and FA (q2 )ki ) with k = i is allowed. We have mentioned in Section 15.1 that in the limit of vanishing neutrino mass Dirac and Majorana neutrinos become indistinguishable (as both actually become Weyl neutrinos). It is therefore interesting to see how the electromagnetic properties of Dirac and Majorana neutrinos converge in the massless limit. Let us first note that the vector and axial-vector operators ψ¯ i γ μ ψk and ψ¯ i γ μ γ5 ψk are chirality preserving, while ψ¯ i σ μν ψk and ψ¯ i σ μν γ5 ψk are chirality flipping: ψ¯ i σ μν ψk = ψ¯ iL σ μν ψkR + ψ¯ iR σ μν ψkL ,

(15.99)

and similarly for ψ¯ i σ μν γ5 ψk . No left–right transitions can be induced by loop effects in the massless neutrino limit, and so the dipole form factors FM (q2 ) and FE (q2 ) must vanish in this limit identically. This result is quite general and is easy to understand. If there is a loop diagram giving a contribution to the chirality-flipping neutrino magnetic or electric dipole form factors, then the same diagram with the external photon line removed will give a contribution to the neutrino mass term, which is also chirality flipping. Thus, for massless neutrinos their magnetic and electric dipole form factors must vanish identically. (It is possible to devise symmetries that allow neutrino magnetic moments but forbid neutrino masses [435], but such symmetries must be broken in the real world.)

12 In general, the symmetry or antisymmetry relations F (q2 ) = ±F (q2 ) should include an extra phase a a ik ki factor ηki related to the CPT parities of the Majorana neutrinos νi and νk . These CPT parities, however, are

not physically observable (unlike the CP parities in the case when CP is conserved), and so one can set ηki = 1 without loss of generality.

15.5 Electromagnetic properties of Majorana neutrinos

335

Thus, we are left with only vector and axial-vector operators. Next, we note that, for massless neutrinos, u¯ (p )γ μ qμ u(p) = u¯ (p )γ μ γ5 qμ u(p) = 0. The quantity μ (q)ki can therefore be written as μ

(q)ki = F1 (q2 )ki q2 γ μ + FA (q2 )ki q2 γ μ γ5 .

(15.100)

The two terms here contain the neutrino charge form factor and the anapole form factor. For Majorana neutrinos, the former is antisymmetric and the latter is symmetric with respect to the indices i and k, while no such constraints exist in the Dirac case. As massless neutrinos are chiral and γ5 uL,R = ∓uL,R , the two terms in Eq. (15.100) are actually indistinguishable and merge into one, which is neither symmetric nor antisymmetric. Thus, the restrictions on the neutrino electromagnetic interactions that are specific to the Majorana case disappear in the limit mν → 0. The same conclusion can be achieved in a different way [432]. As follows from the Majorana condition Eq. (15.20) (or equivalently from Eq. (15.27)), for each loop diagram contributing to the electromagnetic vertex of a Dirac neutrino, in the Majorana case there is an additional diagram with all particles replaced by 13 ˆ If the original diagram is caused by left-handed currents, their C-conjugates. ˆ then the C-conjugate one is due to the right-handed interactions of the antiparticles (see Eq. (15.13)). The additional diagrams contribute to the electromagnetic vertices of massive Majorana neutrinos because Majorana fields contain both the ˆ left-handed and right-handed parts, the latter being C-conjugates of the former. Decoupling of the left-handed and right-handed neutrino states in the massless limit means that the contribution of these additional diagrams to the amplitude of a given electromagnetic transition becomes negligible as mν → 0. Vanishing contributions of the diagrams that are specific to the Majorana case means that the electromagnetic properties of Majorana neutrinos converge to those of Dirac neutrinos when mν → 0. In Section 15.3 we pointed out that a pair of mass-degenerate Majorana neutrinos with maximal mixing and opposite CP parities merges into a Dirac neutrino. The electromagnetic properties of such a pair should then be the same as those of a Dirac neutrino. This is indeed the case; in particular, the transition magnetic moment of such a Majorana pair becomes the usual magnetic moment of the Dirac neutrino. We refer the reader to Ref. [431] for details. It follows that the electromagnetic properties of massive Dirac and Majorana neutrinos are very different. Can this be used to find out whether neutrinos are Dirac or Majorana particles? To answer this question, we should first examine how 13 There will also be extra diagrams in the Majorana case if the sector of the model responsible for the

Majorana mass generation contains charged particles. However, in the limit of vanishing neutrino mass, which we consider now, such diagrams can be neglected.

336

Majorana neutrinos and other Majorana particles

the neutrino electromagnetic properties can manifest themselves. First, through the photon exchange diagrams, they can contribute to the cross sections of νf scattering, where f is a charged lepton or a quark. In principle, such contributions can probe the neutrino magnetic and electric dipole moments, as well as the charge radius and anapole moment. Unfortunately, up to now experiments and observations (most notably experiments on ν¯ e e scattering with reactor antineutrinos as well as astrophysical data) have failed to discover neutrino electromagnetic properties and have so far only produced an upper limit on them [426]. This is actually not surprising, as in the Standard Model and its simplest extensions the neutrino electromagnetic interactions are expected to be extremely weak. As an example, the Standard Model prediction for the neutrino charge radius is r2 ∼ 10−33 cm2 , whereas on adding right-handed neutrinos to the model, one finds for the diagonal magnetic moments of Dirac neutrinos [436]

m  3eGF i mi ≈ 3.2 × 10−19 (15.101) μB , μi ≈ √ 2 eV 8 2π where e is the absolute value of the electron charge, GF is the Fermi constant, mi is the mass of the ith neutrino mass eigenstate, and μB = e/2me is the electron Bohr magneton. Similar expressions can be obtained for transition magnetic moments [430, 433]. In addition, as previously discussed, the smallness of the neutrino mass makes it very difficult to tell Majorana neutrinos from Dirac ones through their electromagnetic properties. In particular, it is difficult to distinguish experimentally the neutrino charge radius (which is non-zero only for Dirac neutrinos) from the anapole moment, which is the only non-vanishing diagonal electromagnetic moment of Majorana neutrinos.14 As we have seen, Dirac neutrinos can in general have both diagonal and transition dipole moments, whereas for Majorana neutrinos only transition dipole moments are allowed. For neutrinos of both types, transition magnetic and electric dipole moments will cause radiative decays of heavier neutrinos into lighter ones, νi → νk + γ . Although the rates of the radiative decay of Dirac and Majorana neutrinos are in general different, because of large uncertainties in the involved neutrino parameters it is impossible to establish the nature of neutrinos by measuring their radiative decay widths. However, the circular polarizations of the produced photons are very different in the Dirac and Majorana cases, and this is completely independent of the neutrino unknowns [433]. In addition, for polarized parent neutrinos, the angular distributions of the emitted photons are different for neutrinos of the two types [437, 433]. Thus, at least in principle, one could distinguish between the Dirac and Majorana neutrinos by measuring the 14 It should also be noted that there are some difficulties in defining the neutrino charge radius in a

gauge-invariant and process-independent way; see the discussion in Sect. 3.3 of Ref. [426].

15.5 Electromagnetic properties of Majorana neutrinos

337

polarization or angular distribution of the photons produced in radiative neutrino decay. Unfortunately, it is rather unlikely that neutrino radiative decays will ever be observed, as they are doubly suppressed by the smallness of the neutrino magnetic moments (which implies small transition amplitude) and of neutrino mass (which means very small phase space volume of the decay). There is, however, some chance of observing radiative neutrino transitions if relatively heavy sterile neutrinos exist. Neutrino diagonal and transition dipole moments can in principle manifest themselves differently – through neutrino spin precession in strong electromagnetic fields. Such a process can be caused by the interaction of neutrino magnetic [436] or electric [438] dipole moments with external fields. Transition dipole moments can give rise to spin-flavor precession, in which neutrino spin and flavor are flipped simultaneously [429, 439]. This process can be resonantly enhanced when neutrinos propagate in matter [440, 441]. Let us compare spin and spin-flavor precessions of Dirac and Majorana neutrinos. It is convenient to introduce the matrix of neutrino electromagnetic moments [439], μ˜ = μ + i ,

(15.102)

where μ and  are the Hermitian matrices of neutrino magnetic dipole and electric dipole moments, respectively. In the flavor-eigenstate basis the dipole moment couplings of neutrinos to an external electromagnetic field are described by the effective operators

μ˜ βα 1 νβR σ μν ναL Fμν + h.c. (15.103) ν¯ β σ μν (μ − iγ5 )βα να Fμν + h.c. = 2 2 (for Dirac neutrinos) and

μ˜ βα 1 ν¯ β σ μν (μ − iγ5 )βα να Fμν + h.c. = (νβL )c σ μν ναL Fμν + h.c. (15.104) 2 2 (for Majorana neutrinos), where Fμν = ∂μ Aν − ∂ν Aμ is the electromagnetic field tensor. Note that there is no extra factor 1/2 in Eq. (15.104) because in the Majorana case the matrix μ˜ is antisymmetric. It is instructive to look at the right-hand sides of Eqs. (15.103) and (15.104), which reveal the nature of the involved neutrinos. Equation (15.103) means that in a transverse15 external magnetic field, e.g. a left-handed (active) electron neutrino νeL can be converted into a right-handed sterile neutrino of the same or different flavor. At the same time, in the Majorana case only flavor-off-diagonal transitions are 15 Magnetic and electric dipole interactions of relativistic neutrinos with longitudinal (i.e. collinear with the

neutrino momentum) magnetic fields are strongly suppressed [436, 429].

338

Majorana neutrinos and other Majorana particles

allowed. For instance, for α = e and β = μ the interaction in Eq. (15.104) describes the transformation of an active left-handed electron neutrino νeL to the active right-handed muon neutrino state ν¯ μR = (νμL )c , which is usually called the muon antineutrino. From this discussion it is clear that neutrino spin and/or spin-flavor precession leads to physically very different final states for Dirac and Majorana neutrinos, and therefore could in principle be used to discriminate between them. Although the neutrino dipole moments are expected to be very small, the neutrino spin precession and spin-flavor precession can still occur with sizeable probabilities in the extremely strong magnetic fields that may be present in astrophysical objects. In particular, in the Majorana neutrino case, strong magnetic fields present in supernovae during the explosion stage may cause the resonantly enhanced conversion νe → ν¯ μ . The resulting muon antineutrinos will then experience the usual flavor transitions on their way from the supernova to the Earth, converting them to electron antineutrinos. As a result, the overall neutrino transmutation chain νe → ν¯ μ → ν¯ e will transform electron neutrinos into electron antineutrinos. Such a conversion of supernova νe s would have a very clear signature in the terrestrial detectors [442, 443], provided that the supernova event occurs in our galaxy and that the transition magnetic moments of Majorana neutrinos μ  10−14 μB .16 While such relatively large values of μ are not easily achieved, they are predicted in some models and are not excluded by the current data and observations. At the same time, the νe → ν¯ e conversion (which is a L = 2 process) cannot occur if neutrinos are Dirac particles. Thus, future supernova neutrino experiments may shed some light onto the Dirac versus Majorana nature of neutrinos. Yet, the most practical means of disentangling these two neutrino types is probably neutrinoless double β-decay, which will be discussed in Section 15.7.

15.6 Majorana particles in SUSY theories In supersymmetric (SUSY) theories each boson has a supersymmetric partner which is a fermion and each fermion has a bosonic superpartner. Such theories predict the existence of a plentitude of Majorana fermions, which are supersymmetric partners of neutral bosons [407, 400, 408]. These include the photino, as well as the gluino, the zino, and neutral higgsinos (the SUSY partners of the photon, gluon, Z 0 -boson, and of the neutral Higgs scalars, respectively). More precisely, since these particles can mix, what actually makes the Majorana fermions are the so-called neutralinos – the linear superpositions of the above-mentioned 16 Here we are assuming that at the resonance of spin-flavor conversion the supernova transverse magnetic field B⊥r can be as large as ∼109 G [442]. The νe → ν¯ e conversion efficiency depends on the product μB⊥r .

15.7 Experimental searches for Majorana particles

339

particles that have definite masses.17 In addition, if the spontaneous breaking of global supersymmetry occurs, there should exist the goldstino – a massless neutral Goldstone fermion. In supergravity the goldstino is absorbed, through a supersymmetric analog of the Higgs mechanism, into the gravitino, which is a massive spin-3/2 Majorana fermion (the SUSY partner of the graviton). In SUSY versions of the models where the so-called strong CP problem is solved through the existence of a light neutral pseudoscalar particle – the axion – there is yet another Majorana fermion, the axino. In SUSY models with conserved R parity the lightest supersymmetric particle is stable. If it is neutral, it can be the so-called WIMP (weakly interacting massive particle) and play the role of the dark matter particle, i.e. account for the missing matter of the Universe [444, 445, 446]. The lightest supersymmetric particle is then the lightest neutralino, the gravitino, or the axino.18 Thus the dark matter problem, which is one of the most important problems of modern cosmology, may have its solution through the existence of a Majorana fermion.

15.7 Experimental searches for Majorana neutrinos and other Majorana particles 15.7.1 Neutrinoless double β-decay and related processes As we have seen, the most practical way of discriminating between Dirac and Majorana neutrinos seems to be by looking for neutrinoless double β-decay (see Refs. [448, 449, 450, 451] for reviews). The usual double β-decay is the process in which a nucleus A(Z, N) decays into an isobar with the electric charge differing by two units: A(Z, N) → A(Z ± 2, N ∓ 2) + 2e∓ + 2¯νe (2νe ) .

(15.105)

In such decays two neutrons of the nucleus are simultaneously converted into two protons, or vice versa. At the fundamental (quark) level, these are transitions of two d quarks into two u quarks or vice versa (see Figure 15.2(a)). Double β-decay is a process of the second order in weak interaction, and the corresponding decay rates are very low: typical lifetimes of the nuclei with respect to the 2β decay are T  1019 years. Processes (15.105) are called 2νββ decays. Two-neutrino double β-decays with the emission of two electrons (2β − ) were experimentally observed for a number of isotopes with half-lives in the range ∼1019 –1024 years [451]; there 17 We assume here that these Majorana particles are non-degenerate in mass. Otherwise two Majorana fermions

can merge into a Dirac one, as discussed in Section 15.3.3. 18 The role of a dark matter particle can also be played by a non-SUSY sterile Majorana neutrino; see Refs.

[445, 446, 447] and references therein.

340

Majorana neutrinos and other Majorana particles (a)

(b)

dL

uL

w

(c)

dL

uL

dL

w

eL

eL

veL

veL veL

veL

eL

w dL

uL

veL

veR

uL

eL

mD MR eR

eL

w dL

WL

veR

mL

uL

WR dR

uR

Figure 15.2. Some Feynman diagrams for the amplitudes of double β decay.

are few candidate nuclei for 2β + decay, and the experimental observation of this process is difficult because of the very small energy release (Q values). If neutrinos are Majorana particles, the lepton number is not conserved, and the neutrino emitted in one of the elementary β-decay processes forming the double β decay can be absorbed in another (Figure 15.2(b)), leading to the neutrinoless 0νββ decay [452]: A(Z, N) → A(Z ± 2, N ∓ 2) + 2e∓ .

(15.106)

Such processes would have a very clear experimental signature: since the recoil energy of a daughter nucleus is negligibly small, the sum of the energies of the two electrons or positrons in the final state should be equal to the total energy release, i.e. should be represented by a discrete energy line. Therefore 0νββ decays could serve as a sensitive probe of the lepton number violation and Majorana nature of neutrinos. In some extensions of the Standard Model, exotic modes of 0νββ decay are possible, e.g. decays with a majoron emission [448, 450]. In this case the sum of the energies of two electrons or positrons is not a discrete line, but the double β energy spectra (as well as the single β-particle spectra) are expected to be different from those in the case of 2νββ decay. Neutrinoless double β decays not only break the lepton number; since the absorbed νe or ν¯ e has a “wrong” chirality, 0νββ decays also break chirality conservation. Therefore, if 0νββ decay is mediated by the standard weak interactions and exchange of light neutrinos, the amplitude of the process must be proportional to the neutrino mass. More precisely, as follows from Figure 15.2(b), it is proportional to the ee-entry of the neutrino Majorana mass matrix, whose modulus is usually called mββ :       2 A(0νββ) ∝  Uei mi  ≡ mββ . (15.107)   i

15.7 Experimental searches for Majorana particles

341

Note that this expression contains Uei2 rather than |Uei |2 . If CP is conserved in the leptonic sector, the mixing matrix Uai can always be made real; however, in this case, the mass parameters mi in Eq. (15.107) (the eigenvalues of the neutrino mass matrix) can be of either sign, their relative signs being related to the relative CP parities of neutrinos. This means that, in general, significant cancelations between various contributions to the sum in Eq. (15.107) are possible. As we discussed in Section 15.3.3, a pair of Majorana neutrinos with equal physical masses |mi |, opposite CP parities, and maximal mixing is equivalent to a Dirac neutrino. It is easy to see that such a pair does not contribute to the amplitude in Eq. (15.107) – the contributions of the two components of the pair cancel exactly. Analogously, the contribution of a pseudo-Dirac neutrino to Eq. (15.107) would be strongly suppressed. Partial cancelations of contributions of different neutrino mass eigenstates to mββ are also possible, of course, when CP is violated, i.e. when the leptonic mixing matrix is complex (with the convention that all neutrino masses are non-negative). In the case of just three usual light neutrino species, Eqs. (15.107) and (15.61) yield   mββ = c213 c212 m1 + c213 s212 e2iϕ1 m2 + s213 e2i(ϕ2 −δCP ) m3 . (15.108) By now, the neutrino oscillation experiments measured rather accurately the leptonic mixing angles, θ12 , θ23 , θ13 , and the neutrino mass squared differences m221 and |m231 |. Global analyses of the data [453, 454, 455] yield m221  7.5 × 10−5 eV2 , θ12  33◦ ,

|m231 |  2.4 × 10−3 eV2 ,

θ23  40◦ or 50◦ ,

θ13  9◦ .

(15.109) (15.110)

At the same time, at present there is essentially no information on the CP-violating phases and the neutrino mass ordering (the sign of m231 ), while for the absolute scale of the neutrino masses only upper limits exist: direct neutrino mass measurements in nuclear β-decay experiments and cosmology yield mi  O(1) eV. With these data, it follows from Eq. (15.108) that sizeable cancelations between the contributions of the different neutrino mass eigenstates to mββ are possible only in the case of the so-called normal neutrino mass hierarchy, m1 , m2  m3 . If, in addition to the usual three light neutrino species, there exist heavy neutrinos Ni , the active flavor-eigenstate neutrinos are linear superpositions of the left-handed components of both light and heavy neutrino mass eigenstates (see Eq. (15.52)). Since the chirality-flipping part of the fermion propagator m/(p2 − m2 )  −1/m for m2  p2 , the contribution of the diagram in Figure 15.2(b) with exchanges of heavy Majorana neutrinos to the amplitude of 0νββ decay is proportional to   n    −1 2 −1  mN ≡  Uei mi  . (15.111)   i=4

342

Majorana neutrinos and other Majorana particles W

v-e

d

e-

u

u

W

d

e-

ve

Figure 15.3. Black box argument for Majorana neutrino mass [456].

Thus, one should distinguish between effects of light and heavy Majorana neutrino exchanges. The latter requires the existence of extra neutrino species and can be considered as one of the non-standard mechanisms of 0νββ decay. The effect of Majorana neutrino exchanges on lepton-number-violating processes is expected to be maximal when the mass of the exchanged neutrino is of the order of the characteristic energy of the process. This applies not only to 0νββ decay, but also to all L = 2 processes, including those considered in Section 15.7.2. In extensions of the Standard Model, such as the left–right symmetric, SUSY, or grand unification models, additional mechanisms of 0νββ decay are possible, in which the process is mediated by right-handed currents, SUSY particles, or leptoquarks (see, e.g., Ref. [450]). One of the diagrams contributing to the amplitude of 0νββ decay in left–right symmetric theories is shown in Figure 15.2(c). It may appear that no Majorana mass mL of νL is necessary in such models, i.e. 0νββ decay can occur even if mL = 0 and the neutrinos are Dirac particles, or even if they are massless. This is, however, incorrect: in all models in which 0νββ decay occurs, the Majorana masses of νL must be different from zero. An elegant “black box” proof of this statement was presented in Ref. [456]. In Figure 15.3 the black box represents an unspecified mechanism by which two d-quarks can be converted to two u-quarks and two electrons, with no accompanying neutrinos. Next, we make use of the crossing symmetry to transform the initial-state d-quarks to the final¯ state d-quarks, join the ud¯ lines to produce W bosons, and then attach the other ends of the W-boson lines to the electron lines to produce neutrinos. This yields the diagram corresponding to the effective operator ν¯ eL (νeL )c describing the ν¯ eR → νeL transition, i.e. the Majorana mass term for νe . Thus, no matter what mechanism causes 0νββ decay, observation of this process would constitute an unambiguous proof that neutrinos are Majorana particles. If neutrinos are of Majorana nature, then, in addition to the usual 0νββ decay Eq. (15.105), some related processes should occur, such as, e.g., eB + A(Z, N) → A(Z − 2, N + 2) + e+ (neutrinoless electron capture) or 2eB + A(Z, N) → A(Z − 2, N+2)∗ → A(Z−2, N+2)+X (neutrinoless double electron capture) [457]. Here eB stands for a bound atomic electron and A(Z − 2, N + 2)∗ denotes the excited state of the A(Z − 2, N + 2) atom with two holes in the atomic 1S orbit, which

15.7 Experimental searches for Majorana particles

343

then de-excites with the emission of atomic X-rays, Auger electrons, etc. These processes are always energetically allowed if the usual double β + decay of A(Z, N) is allowed, and in some cases may also be allowed even if A(Z, N) is stable. The neutrinoless double electron capture may be resonantly enhanced provided that the total energy release is very close to the excitation energy of the A(Z − 2, N + 2)∗ atomic state. The search for isotopes with suitable atomic mass differences and excitation energies of the daughter atoms is currently underway [458]. Other processes related to 0νββ decay have been discussed, such as muon conversion on nuclei, μ− + A(Z, N) → A(Z − 2, N + 2) + e+ , or μ− + A(Z, N) → A(Z − 2, N + 2) + μ+ . Just like the 0νββ decay, such conversion processes break the total lepton number L = Le + Lμ + Lτ by two units (note, however, that the (μ− , e+ ) conversion conserves Le − Lμ ). If these processes are mediated by exchanges of light or heavy Majorana neutrinos, their expected rates are too small to render them observable in the foreseeable future [459, 460]. Thus, the muon conversion processes may be of interest only if they are dominated by non-standard mechanisms. It is customary to discuss the available data of 0νββ decay experiments as well as expected sensitivities of the ongoing and future experiments in terms of the 0ν and interpret them in terms of limits on the half-lives of the parent nuclei T1/2 the effective mass parameter mββ defined in Eq. (15.107). It should, however, be remembered that such an interpretation makes sense only when the standard diagram of Figure 15.2(a) with exchange of light Majorana neutrinos is the sole 0ν ∝ (or dominant) contribution to the amplitude of the process, in which case T1/2 2 1/ mββ . Otherwise, there is no or little connection between these two quantities, and the masses of light neutrinos cannot be directly probed in experiments on 0νββ decay.19 If 0νββ decay is dominated by non-standard mechanisms, experiments can only give an upper limit on the parameter mββ and therefore on the Majorana masses of the neutrino mass eigenstates. However, as we have stressed, even if the Majorana neutrino mass gives a negligible contribution to 0νββ decay, observation of this process would constitute unambiguous proof of the Majorana nature of neutrinos. Experimentally, one can in principle distinguish between different mechanisms of 0νββ decay by studying the properties of the decay products, e.g. angular correlations of the two produced β-particles, or by looking for processes related to 0νββ decay (see, e.g., discussion in Sect. 6 of Ref. [450]).

19 A similar argument applies to the contributions of heavy Majorana neutrino exchanges to 0νββ decay rates, which depend on the quantity m−1 N defined in Eq. (15.111), and the possibility of probing the masses of

heavy neutrinos.

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Majorana neutrinos and other Majorana particles

Neutrinoless double β-decay has been actively searched for but, up to now, not been experimentally discovered.20 The available data allow to put upper bounds on the effective Majorana neutrino mass mββ . The best current limits come from the EXO-200 experiment on 0νββ decay of 136 Xe [464] and GERDA experiment with 76 Ge [465] (90% C.L.): mββ < 0.14–0.38 eV mββ < 0.2–0.4 eV

(EXO-200) , (GERDA) ,

(15.112)

where the ranges are due to the uncertainties in the values of the nuclear matrix elements. The most promising current and forthcoming experiments are expected to be sensitive to the values mββ  0.03–0.1 eV [451]; as follows from Eqs. (15.109) and (15.110), they will be able to explore the Majorana neutrino mass only if neutrinos are quasi-degenerate in mass (m1 ∼ m2 ∼ m3 ) or have the inverted mass hierarchy (m3  m1 , m2 ). In order to see how it works, assume as an example that from independent experiments (e.g. neutrino oscillations plus cosmology) we know that neutrino masses obey the inverted hierarchy, i.e. m3  m1 , m2 . From Eq. (15.108) it then  follows

that the quantity mββ cannot be smaller than mββ min ≈ c213 cos 2θ12 |m231 | (∼0.02 eV). If from a 0νββ experiment an upper limit on mββ is inferred which is smaller than mββ min , this would rule out the Majorana nature of neutrinos, barring destructive interference with some non-standard 0νββ decay mechanisms.

If the neutrino masses obey the normal mass hierarchy, probing the Majorana neutrino mass through 0νββ decay will be problematic and will in any case require multiton-scale detectors and very low backgrounds. It would be extremely difficult to uncover the Majorana versus Dirac nature of neutrinos in this case, unless an efficient non-standard mechanism of 0νββ decay is at play.

15.7.2 Other lepton-number-violating processes The Majorana nature of neutrinos can be revealed not only through neutrinoless double β-decay, but also through other lepton-number-violating processes. One of these processes – neutrino spin-flavor precession in strong external magnetic 20 There is one positive claim of observation of 0νββ decay of 76 Ge by part of the Heidelberg–Moscow

Collaboration [461, 462, 463]. However, this result has been subject to criticism (see, e.g., Ref. [451] and references therein) and is now strongly disfavored (at 99% C.L.) by non-observation of 0νββ decay by GERDA [465].

15.7 Experimental searches for Majorana particles (b) u

(a) u

d W+

+ 1

N d

W+

u

+ 2

+ 1

W+

d

345

+ 2

N W+ d u-

Figure 15.4. Representative Majorana neutrino exchange diagrams contributing to like-sign dilepton production. (a) W-boson fusion with t-channel N exchange. (b) s-channel W-exchange diagram with production and subsequent decay of N.

fields – was discussed in Section 15.5. Here we shall briefly discuss two of the other types of L = 2 processes: rare particle decays and like-sign dilepton production at accelerators. If neutrinos are Majorana particles, they should mediate rare L = 2 particle decays, such as K + → π − e+ e+ , K + → π − μ+ μ+ , and similar decays of charged B and D mesons. The typical Feynman diagrams for such processes are essentially the same as that in Figure 15.2(b), except that some other quarks may be involved. However, the number of parent particles in the case of rare meson decays is suppressed by a huge factor of the order of the Avogadro number NA as compared to those in double β-decay experiments. Therefore, rare meson decays cannot compete with neutrinoless double β-decay in unraveling the neutrino nature when the exchanged Majorana neutrinos are very light or very heavy. Still, rare decays can provide tighter limits on the Majorana neutrino masses in the region of the order of the energy release of the corresponding process (for rare kaon decays, a few hundred MeV). In accelerator-based experiments the Majorana nature of neutrinos can be tested in the processes with like-sign dilepton production as well as in related reactions. ± The basic L = 2 processes are in this case W ± W ± ↔ ± 1 2 , where 1,2 = e, μ, or τ . A typical reaction that can be studied at hadron colliders (first discussed in ± Ref. [466] in the context of right-handed currents) is pp → ± 1 2 X. Majorana neutrinos contribute to the amplitude of this process through the diagrams of Figure 15.4 and similar diagrams with other quarks in the final state. Figure 15.4(a) is similar to Figure 15.2(b), which gives the standard contribution to the amplitude of 0νββ decay, whereas Figure 15.4(b) corresponds to the production and subsequent decay of a virtual or real Majorana neutrino N. If real Majorana neutrinos are kinematically accessible, the dilepton production process can be resonantly enhanced.

346

Majorana neutrinos and other Majorana particles

Lepton-number-violating rare decays and like-sign dilepton production processes have been actively looked for experimentally at accelerators, but no signals have been found so far. This allows one to put important constraints on the properties of Majorana neutrinos. For a detailed and comprehensive discussion of these (and other) L = 2 processes, as well as of the effects of Majorana neutrinos on electroweak precision observables, we refer the reader to Ref. [467] (see also Ref. [450] and references therein). There have also been extensive searches for the SUSY Majorana particles both at accelerators and in dark matter detectors. For recent discussions of these experiments see, e.g., Refs. [468, 469, 470]. Unfortunately, up to now no unambiguous evidence for such particles has been obtained.

15.8 Baryogenesis through leptogenesis and Majorana neutrinos In addition to providing a simple and natural way of explaining the observed smallness of the neutrino mass, the seesaw mechanism of the neutrino mass generation brings with it a free bonus: it furnishes a very simple and attractive mechanism of producing the observed baryon asymmetry of the Universe (BAU). The term baryon asymmetry simply reflects the fact that the observed Universe is made predominantly of matter rather than of an equal amount of matter and antimatter (there are very stringent constraints on the antimatter abundance in the Universe [471]). The ratio of the net baryon number to photon number in the Universe is now measured very accurately [472]: η≡

NB − NB¯ = (6.04 ± 0.08) × 10−10 , Nγ

(15.113)

where NB , NB¯ , and Nγ are the number densities of baryons, antibaryons, and photons at the present epoch, respectively. The observed BAU could not have resulted from an initial state of the Universe with B = 0, as any such pre-existing asymmetry would have been diluted to an absolutely negligible level during the stage of the accelerated expansion of the Universe predicted by cosmic inflation [473, 474, 475, 476] (which is the standard paradigm now). Thus, the BAU should have been generated dynamically in the post-inflationary epoch. Under what conditions can such a dynamical generation of the BAU occur? These conditions were actually formulated by Sakharov in 1967 [477]: (i) baryon number violation; (ii) C and CP violation; and (iii) deviation from thermal equilibrium. The first two conditions are necessary for the baryon asymmetry to be produced in the first place; the last condition ensures that the BAU produced in some processes is not destroyed by the inverse processes.

15.8 Baryogenesis through leptogenesis and Majorana neutrinos

347

As an illustration, consider a process X → Y + b, where X denotes an initial state with zero baryon number, Y stands for a set of final-state particles with vanishing net baryon number and b represents the produced excess baryons. Then, if condition (i) is not met, the process X → Y + b simply does not take place. If either C or CP is conserved, the processes X → Y + b and X¯ → Y¯ + b¯ occur at the same rate, and no net baryon number is produced (provided that the initial state of the system contained ¯ If the system is in thermal equilibrium, the equal numbers of X and X¯ or that X = X). processes X → Y + b and Y + b → X occur at the same rate (which is also true, of ¯ and the baryon asymmetry produced in direct course, for X¯ → Y¯ + b¯ and Y¯ + b¯ → X), processes is washed out by the inverse ones.

All these conditions are actually satisfied in the Standard Model of particle physics, though the amount of CP violation in this model is insufficient, and, most importantly, the deviation from thermal equilibrium is far too small [478] to account for the measured value of the BAU Eq. (15.113). Sakharov’s conditions are fulfilled and the successful generation of the BAU is possible in many extensions of the Standard Model, such as grand unification theories and SUSY models [478, 479]. Here we concentrate on the so-called baryogenesis via leptogenesis [480], which is built into the seesaw mechanism of the neutrino mass generation and does not require any new physics besides the existence of heavy Majorana neutrinos. We just outline the mechanism here; for details and ramifications, we refer the reader to the comprehensive review, Ref. [481]. To produce the phenomenologically acceptable mass spectrum of the usual light neutrinos, the seesaw mechanism should include at least two heavy electroweaksinglet (i.e. sterile) Majorana neutrinos Ni . The same number of Ni s turns out to be sufficient for the generation of the BAU. The mechanism works as follows. First, a lepton number L0 is produced through the out-of-equilibrium L- and (B − L)-violating decays of the Ni s. The produced lepton number is then reprocessed into the baryon number by the so-called sphaleron processes (hence the name baryogenesis through leptogenesis). Let us consider this in more detail. The singlet neutrinos Ni are actually not completely sterile: they cannot have gauge interactions in the Standard Model, but they can have the usual Yukawa couplings h∗αi ¯α NiR H + h.c.21 with the lepton doublets α = (ναL , eαL )T and the Higgs field H = (H 0 , H − )T , which are allowed by the electroweak gauge symmetry. The Yukawa couplings result in decays of Ni into the usual leptons and the Higgs particles. Since the Ni s are Majorana particles, their decay proceeds in a lepton-number-violating way, i.e. they can decay both 21 Following tradition, for the right-handed components of the singlet neutrinos N we use the notation N i iR

rather than νiR that was used in Sections 15.3 and 15.4.

348

Majorana neutrinos and other Majorana particles

¯ and through the CP-conjugate channel Ni → ¯α H. If the Yukawa via Ni → α H couplings hαi are complex, CP is not conserved in the leptonic sector, and the rates ¯ = (Ni → of the above decay modes are in general different: (Ni → α H) ¯ α H). This leads to the production of a non-zero net lepton number. Note that CP violation manifests itself through the interference between the tree-level and oneloop Feynman diagrams describing the Ni decay; at the tree level the decay rates are proportional to |hαi |2 , and the complexity of the Yukawa constants does not reveal itself. The parameter that describes the generation of the lepton asymmetry in the decay of Ni is given by  (Ni → α H)  ¯ − (Ni → ¯α H) 1 = Im[(h† h)2ij ] g(xj ) , † ¯ ¯ 8π(h h) (N →  →  H) H) + (N ii j=i i α i α α (15.114) 2 2 where xj ≡ Mj /Mi and g(x) is model dependent. In the Standard Model,    √ 2−x 1+x . (15.115) − (1 + x) ln g(x) = x 1−x x i =

In Eq. (15.114) we, for simplicity, summed over the flavors of final-state leptons (note that flavor effects may actually be important, see the discussion in Sect. 9 of Ref. [481]). Equation (15.114) is valid only when the mass differences of the heavy singlet Majorana neutrinos are large compared with their decay widths, |Mj − Mi |  i + j ; the opposite case, which leads to resonant leptogenesis [482, 483], requires a special consideration. The deviation from thermal equilibrium is provided by the expansion of the Universe, the rate of which is given by the Hubble parameter √ (15.116) H(T) = 1.66 g∗ T 2 /MPl . Here T is the temperature of the Universe, MPl = 1.2 × 1019 GeV is the Planck mass, and g∗ is the number of relativistic degrees of freedom in the thermal bath (in the Standard Model g∗ = 106.75). If the processes that create and destroy some particles are fast compared with the Hubble expansion rate H(T), they equilibrate particle distributions; otherwise, the thermal equilibrium is not achieved. For the singlet neutrino Ni , the condition of deviation from thermal equilibrium requires † that, at the time of the Ni decay (T ∼ Mi ), the decay rate i = (h8πh)ii Mi must be smaller than the Hubble rate: (h† h)ii √ T2 Mi  1.66 g∗ |T∼Mi . 8π MPl Thus, the lightest singlet neutrino Ni is typically the last one to go out of equilibrium in the course of the expansion and cooling of the Universe. Therefore the lepton asymmetry produced in decays of heavier Majorana neutrinos is washed out

15.8 Baryogenesis through leptogenesis and Majorana neutrinos

349

by the processes involving N1 , and the net lepton asymmetry of the Universe is produced in the decays of N1 .22 The out-of-equilibrium condition for these decays can be rewritten as m ˜1 ≡

(h† h)11 v2 √ v2  8π · 1.66 g∗  1.1 × 10−3 eV , M1 MPl

(15.117)

where v = 174 GeV is the Higgs vacuum expectation value (VEV). In the opposite case, the produced lepton asymmetry is strongly washed out. It is interesting to note that, since the Dirac-type neutrino mass matrix mD = hT v, the left-hand side of Eq. (15.117) is roughly of the same order of magnitude as the masses of the light active neutrinos predicted by the seesaw mechanism,23 whereas the righthand side is close to the light neutrino mass scale that follows from the oscillation experiments assuming that neutrino masses are hierarchical. Typically, one expects the left-hand side of Eq. (15.117) to slightly exceed its right-hand side, leading to a moderate washout of the lepton asymmetry. How can the produced lepton number be converted into the baryon one? In the Standard Model the baryon and lepton numbers are conserved at the tree level but are violated at the one-loop level by the so-called chiral anomalies.The anomalies of the baryon and lepton number currents are the same, and so B − L is exactly conserved, but the sum B + L is not. Although the (B + L)-violating processes are strongly suppressed at zero temperature, the situation at high temperatures is different [484]. The Standard Model predicts the existence of topologically non-trivial configurations of the gauge and Higgs fields (called sphalerons) which violate B+L with the rate sph , which exceeds the Hubble rate for 100 GeV  T  1012 GeV. In this temperature interval the sphaleron processes can efficiently wash out B+L and thus reprocess the lepton asymmetry into the baryon one. In a somewhat simplified way, the reprocessing mechanism can be described as follows. Assume that at time t = 0 a net lepton number L0 is produced, while the initial baryon number B0 = 0. Noting that B and L can be represented as linear combinations of B − L and B + L and that B + L is exponentially suppressed with time by the sphaleron processes, for the values of L and B at time t ≥ 0 we have 1 1 L(t) = − (B − L)0 + (B + L)0 e−sph t , 2 2

(15.118)

1 1 B(t) = (B − L)0 + (B + L)0 e−sph t . 2 2 22 Under certain circumstances decays of heavier singlet neutrinos can also be important; see Sect. 10.1 of

Ref. [481] and references therein. 23 If matrix h were real, for hierarchical masses of N the left-hand side of Eq. (15.117) would have been i

approximately equal to the trace of the mass matrix of light neutrinos, i.e. to the sum of the light neutrino mass eigenvalues.

350

Majorana neutrinos and other Majorana particles

−1 Thus, at the times t  sph we have L  L0 /2, B  −L0 /2, i.e. we end up with a non-zero baryon number. A realistic calculation, which takes into account that only left-handed quarks and leptons are coupled to the W-boson field, yields (in the Standard Model) B = −(29/78)L0 rather than −L0 /2. In addition, one should carefully take into account the processes that wash out the lepton asymmetry, such as inverse N1 decays and 2 → 2 scattering processes. This is usually done by solving a system of Boltzmann equations or quantum kinetic equations. As a result, one finds that, for hierarchical masses of heavy singlet neutrinos, the observed value of the BAU can be generated provided that the mass of the lightest among the heavy Majorana neutrinos N1 satisfies M1  108 GeV. For quasi-degenerate in mass heavy neutrinos the viable baryon asymmetry can be achieved, through the resonant leptogenesis, even for Mi as small as ∼1 TeV [482, 483]. In the discussed baryogenesis mechanism all three Sakharov conditions are satisfied. The baryon number violation is provided by the combination of L (and B − L) violation in decays of heavy Majorana neutrinos and B + L violation by the sphaleron processes. Thus, C violation follows from the chiral nature of the Yukawa couplings, while CP violation is a consequence of the complexity of the corresponding coupling constants. Finally, the condition of deviation from thermal equilibrium is met because in certain ranges of the parameters the rates of decay and inverse decay of the heavy Majorana neutrinos (as well as the rates of other L-violating processes) do not exceed significantly the Hubble expansion rate. We have so far discussed baryogenesis via leptogenesis in type I seesaw. Similar mechanisms work in the case of type II and type III seesaw [481]. There also exists an alternative leptogenesis mechanism [485, 486, 487], in which the lepton asymmetry is generated in CP-violating oscillations of the heavy Majorana neutrinos Ni rather than in their decays. The produced asymmetry is then communicated from the Ni s to the usual leptons through their Yukawa couplings, and the reprocessing of the lepton number to the baryon number proceeds through the sphaleron processes in the usual way. In all the discussed versions of baryogenesis through leptogenesis, the Majorana nature of the singlet neutrinos plays a crucial role. There also exist leptogenesis scenarios with Dirac neutrinos (see Sect. 10.3 of Ref. [481] and references therein), but they are based on more complicated and less economical models.

15.9 Miscellaneous Here we collect together a few assorted remarks on Majorana particles. It is usually said that in, e.g., nuclear β − -decay an electron antineutrino ν¯ e is emitted, while positron production in β + -decay is accompanied by the emission of an electron neutrino νe , and we know that these are distinct particles. Does this

15.9 Miscellaneous

351

mean that we have already established that neutrinos are Dirac particles and that νe = (νe )c ? Not really. The point is that the charged-current weak interactions ˆ are chiral, so that only left-handed particles and their right-handed C-conjugates can be emitted or absorbed. In the Dirac case, this means that only the left-handed component of the Dirac field νe = νeL +νeR (as well as the right-handed component (νeL )c ≡ ν¯ e of (νe )c = (νeL )c + (νeR )c ) take part in the interactions, while νe and (νe )c are indeed different particles. In the Majorana case we have νe = νeL + (νeL )c , and both chiral components of the field participate in weak interactions. What we call νe and ν¯ e are in this case merely the left-handed and right-handed components of the same Majorana field of the electron neutrino. They are, to a very good accuracy, distinct because the neutrinos we deal with are always highly relativistic, and the transitions between their left-handed and right-handed components are suppressed by the factor (mν /E)2  1. Thus, the role of the lepton number, which is conserved in the Dirac case, is played for relativistic Majorana neutrinos by chirality, which is nearly conserved. This illustrates once again the point we have already made more than once – the smallness of the neutrino mass makes it very difficult to discriminate between Dirac and Majorana neutrinos. Can one still tell these two neutrino types apart by studying neutrino propagation under extreme conditions, such as, e.g., the very high densities and/or strong magnetic fields which are expected to be present in stellar environments? This question was studied in Ref. [488], and the answer unfortunately turns out to be essentially negative. It is a well known but not yet completely understood fact that electric charge is quantized. Possible explanations include the existence of the magnetic monopole and the grand unification of particles and forces. It turns out, however, that electric charge quantization can be understood even outside these frameworks if neutrinos are Majorana particles [489, 490]. In the minimal Standard Model with no righthanded (singlet) neutrinos νR , charge quantization is a consequence of the hypercharge assignment of the particles that follows from the requirement of the anomaly cancelation. The cancelation of anomalies, in turn, is necessary for internal consistency of the theory. However, the minimal Standard Model is not realistic in the sense that neutrinos are massless in it. It therefore in any case has to be amended by a neutrino mass generating mechanism. If one adds right-handed singlet neutrinos to the Standard Model, there are essentially two possibilities. First, one imposes a lepton number conservation which allows only Dirac mass terms for neutrinos. In this case the anomaly cancelation condition no longer leads to electric charge quantization. If no lepton number conservation is imposed, massive neutrinos turn out to be Majorana particles. In this case anomaly cancelation always results in electric charge quantization [489, 490]. The authors of Ref. [489, 490] have also studied a wide class of non-grand-unified extensions of the Standard Model which

352

Majorana neutrinos and other Majorana particles

allow massive neutrinos, and found that in virtually all cases the Majorana nature of neutrinos led to electric charge quantization, whereas for Dirac neutrinos no such quantization occurred. Thus, the observed quantization of electric charge in Nature may have its explanation through the existence of Majorana neutrinos. It is conceivable that our (3+1)-dimensional world is actually embedded in a space-time of higher dimensionality; in particular, higher-dimensional space-times appear in Kaluza–Klein, supergravity, and superstring models. From the point of view of applications to condensed matter physics, it may also be interesting to consider space-times of lower dimensionality. Can Majorana particles live in such unconventional space-times? The answer is yes, but not in all of them. For ddimensional space-times with d − 1 space-like and one time-like dimensions, massive Majorana fermions can exist only if d = 2, 3, and 4 mod 8 (see, e.g., Sect. 2 of Ref. [491] and references therein). Massless self-conjugate fermions can live in the space-times of the same dimensionality, and in addition in d = 8 and 9 mod 8 dimensions.24 These results can also be extended to the case of n > 1 time-like dimensions.

15.10 Summary and conclusions The possibility of the existence of fermions, which are their own antiparticles, is certainly the most famous and arguably the most important result obtained by Ettore Majorana. Extensions of the Standard Model typically predict neutrinos to be massive Majorana particles. There are some experimental hints in favor of the possible existence of extra neutrino species (on top of the already known νe , νμ , and ντ ); if they exist, they are very likely Majorana particles. The Majorana nature of neutrinos would imply lepton number violation – a very interesting phenomenon which is now being intensely searched for experimentally. The possible existence of heavy electroweak-singlet Majorana neutrinos provides us, through the seesaw mechanism, with a natural and elegant explanation of the smallness of the masses of the usual neutrinos. Heavy (or relatively heavy) Majorana neutrinos furnish very simple and attractive mechanisms for generating the observed baryon asymmetry of the Universe. Majorana particles are abundant in SUSY models. Majorana fermions can play the role of the dark matter particles, and thus they provide a solution of one of the most important problems of modern cosmology. Majorana neutrinos may hold a clue to the understanding of electric charge quantization observed in Nature. 24 Note that massless spin-1/2 fermions admit more freedom in the definition of the particle–antiparticle

conjugation operation: the matrix C that enters Eq. (15.9) may be defined either through the usual relation C −1 γ μ C = −γ μT or through C −1 γ μ C = +γ μT .

15.10 Summary and conclusions

353

If Majorana particles exist, they should have special properties with respect to C, CP, and CPT transformations and possess very peculiar electromagnetic properties. By studying them we may be able to learn a great deal about the fundamental properties of particles and their interactions. Particle-like excitations of Majorana nature have been found in condensed matter systems (see Chapter 14). However, very active direct and indirect searches for Majorana neutrinos and other fundamental Majorana particles in many laboratories in the world have up to now borne no fruit. This should not discourage us too much – just remember that it took us over 40 years to discover neutrino oscillations after their possibility had first been proposed! After all, the idea of Majorana fermions is so elegant and attractive that Nature just could not have missed the opportunity to create them.

Appendix Molecular bonding in quantum mechanics

In what follows, we report the translation from the Italian of the text of an original note [26] written by Majorana, probably to deliver as a seminar the University of Naples in 1938, as discussed in Chapter 11.

A.1 On the meaning of quantum state The internal energy of a closed system (atom, molecule, etc.) can take, according to quantum mechanics, discrete values belonging to a set E0 , E1 , E2 , . . . composed of the so-called energy “eigenvalues.” To each given value of the energy we can associate a “quantum state,” that is a state where the system may remain indefinitely without external perturbations. As an example of these perturbations, we can in general consider the coupling of the system with the radiation field, by means of which the system may lose energy in form of electromagnetic radiation, jumping from an energy level Ek to a lower one Ei < Ek . It is only when the internal energy takes the minimum value E0 , that it cannot be further decreased by means of radiation; in this case the system is said to be in its “ground state” from which it cannot be removed without sufficiently strong external influences, such as the scattering with fast particles or with light quanta of large frequency. What is the corresponding concept of quantum state in classical mechanics? An answer is primarily required to this question in order for quantum mechanics to provide a correct representation of the results obtained within our field without entering, however, into the complex computational methods adopted by this theory. In classical mechanics the motion of a system composed of N mass points is entirely determined when the coordinates q1 , . . . , q3N of all the points are known as function of time: qi = qi (t);

(A.1)

A.1 On the meaning of quantum state

355

Eqs. (A.1) give the dynamical equations where all the internal and external forces acting on the system are present, and they can always be chosen in such a way that at a given instant all the coordinates qi (0) and their time derivatives q˙ i (0) take arbitrarily fixed values. Thus the general solution of the equations of motion must depend on 2 · 3N arbitrary constants. For systems with atomic dimensions the classical representation no longer holds and two successive modifications have been proposed. The first one, due to Bohr and Sommerfeld and that has provided very useful results, has been completely abandoned with the emergence of the novel quantum mechanics, which has been the only theory able to give an extremely general formalism, fully confirmed by the experiments on the study of the elementary processes. According to the old theory of Bohr–Sommerfeld, classical mechanics still holds when describing the atom, so that the motion of an electron, for example, around the hydrogen nucleus is still described by a solution (A.1) of the equations of classical mechanics. However, if we consider periodic motions, such as the revolution of an electron around the nucleus, not all the solutions of the classical equations are realized in Nature, but only a discrete infinity of those satisfying the so-called Sommerfeld conditions, that is certain cabalistic-like integral relations. For example in every periodic motion in one dimension, the integral of the double of the kinetic energy over the period τ : 

τ

2T(t) dt = nh

0

must be an integer multiple of the Planck constant (h = 6.55 · 10−27 ). The combination of classical mechanics with a principle which is unrelated to it, such as that of quantized orbits, appears so hybrid that it should not be surprising that the complete failure of that theory occurred in the last decade, irrespective of several favorable experimental tests which were supposed to be conclusive. The novel quantum mechanics, primarily due to Heisenberg, is substantially farther from the classical conceptions than the old quantum theory. According to the Heisenberg theory, a quantum state corresponds not to a strangely privileged solution of the classical equations but rather to a set of solutions which differ for the initial conditions and even for the energy, i.e. what it is meant as precisely defined energy for the quantum state corresponds to a sort of average over the infinite classical orbits belonging to that state. Thus the quantum states come to be the minimal statistical sets of classical motions, slightly different from each other, accessible to the observations. These minimal statistical sets cannot be further partitioned due to the uncertainty principle, introduced by Heisenberg himself, which forbids the precise simultaneous measurement of the position and the velocity of a particle, that is the determination of its orbit.

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Molecular bonding in quantum mechanics

An harmonic oscillator with frequency ν can oscillate classically with arbitrary amplitude and phase, its energy being given by E = 2π 2 mν 2 A20 , where m is its mass and A0 the maximum elongation. According to quantum mech1 3 anics the possible values for E are, as well known, E0 = hν, E = hν, . . . , 2 2   1 hν, . . . ; in this case we can say that the ground state with energy En = n + 2 1 E0 = hν corresponds roughly to all the classical oscillations with energy between 2 3 0 and hν, the first excited state with energy E0 = hν corresponds to the classical 2 solutions with energy between hν and 2 · hν, and so on. Obviously the correspondence between quantum states and sets of classical solutions is only approximate, since the equations describing the quantum dynamics are in general independent of the corresponding classical equations, but denote a real modification of the mechanical laws, as well as a constraint on the feasibility of a given observation; however it is better founded than the representation of the quantum states in terms of quantized orbits, and can be usefully employed in qualitative studies.

A.2 Symmetry properties of a system in classical and quantum mechanics Systems showing some symmetry property deserve particular study. For these systems, due to symmetry considerations alone, from one particular solution of the classical equations of motion qi = qi (t) we can deduce, in general, some other different ones qi = qi (t). For example if the system contains two or more electrons or, in general, two or more identical particles, from one given solution we can obtain another solution, which in general will be different from the previous one, just by changing the coordinates of two particles. Analogously if we consider an electron moving in the field of two identical nuclei or atoms (denoted by A and B

A

O

Figure A.1.

B

A.2 Symmetry properties

357

in the figure [A.1]), starting from an allowed orbit qi = qi (t) described around A with a given law of motion, we can deduce another orbit qi = qi (t) described by the electron around the nucleus or atom B by a reflection with respect to the center O of the line AB. The exchange operations between two identical particles, reflection with respect to one point or other ones corresponding to any symmetry property, keep their meaning in quantum mechanics. Thus it is possible to deduce from a state S another one S , corresponding to the same known value of the energy, if in the two mentioned examples we exchange two identical particles between them and reflect the system with respect to the point O. However, differently from what happens in classical mechanics for the single solutions of the dynamical equations, in general it is no longer true that S will be distinct from S. We can easily realize this by representing S with a set of classical solutions, as seen above; it then suffices that S includes, for any given solution, even the other one obtained from that solution by applying a symmetry property of the motions of the systems, in order that S results to be identical to S. In several cases, if the system satisfies sufficiently complex symmetry properties, it is instead possible to obtain, by symmetry on a given quantum state, other different states but with the same energy. In this case the system is said to be degenerate, i.e. it has many states with the same energy, exactly due to its symmetry properties. The study of degenerate systems and of the conditions under which degeneration can take place will bring us too far and, in any case, it is difficult to make such a study in terms of only classical analogies. Then we will leave it completely aside and limit our attention to problems without degeneration. This condition is always satisfied if the symmetry of the mechanical system allows such a simple transformation that its square, that is the transformation applied twice, reduces to the identity transformation. For example, by a double reflection of a system of mass points with respect a plane, a line or a point, we necessarily recover the same initial arrangement; analogously, the system remains unaltered by changing twice two identical particles. In all these cases we have only simple quantum states, i.e. to every possible value of the energy is associated only one quantum state. It follows that all the quantum states of the system containing two identical particles are symmetric with respect to the two particles, remaining unaltered under their exchange. Thus the states of an electron orbiting around two identical nuclei A and B are symmetric with respect to the middle point O of AB, or remain unaltered by reflection in O, and analogously for other similar cases. Under given assumptions, which are verified in the very simple problems that we will consider, we can say that every quantum state possesses all the symmetry properties of the constraints of the system.

358

Molecular bonding in quantum mechanics

A.3 Resonance forces between states that cannot be symmetrized for small perturbations and spectroscopic consequences. Theory of homopolar valence according to the method of bonding electrons. Properties of the symmetrized states that are not obtained from non-symmetrized ones with a weak perturbation Let us consider an electron moving in the field of two hydrogen nuclei or protons. The system composed by the two protons and the electron has a net resulting charge of +e and constitutes the simplest possible molecule, that is the positively ionized hydrogen molecule. In such a system the protons are able to move as well as the electron, but due to the large mass difference between the first ones and the second one (mass ratio 1840:1) the mean velocity of the protons is much lower than that of the electron, and the motion of this can be studied with great accuracy by assuming that the protons are at rest at a given mutual distance. This distance is determined, for stability reasons, in such a way that the total energy of the molecule is at a minimum. This energy is given, at a first approximation, by the sum of the mutual potential energy of the two protons and the energy of the electron moving in the field of the first two [protons]; it is different for different electron quantum states. The mutual potential energy of the protons is given by e2 /r if r is the distance between them, while the binding energy of the electron in its ground state is a negative function E(r) of r that does not have a simple analytic expression, but it can be obtained from quantum mechanics with an arbitrary large accuracy. The equilibrium distance r0 is then determined by the condition that the total energy is at a minimum: W(r0 ) =

e2 + E(r0 ). r0

The curve W(r) has a behavior like that shown in the figure [A.2], if we assume that zero energy corresponds to the molecule that is dissociated into a neutral hydrogen atom and an ionized atom at an infinite distance. The equilibrium distance has been theoretically evaluated by Burrau1 finding r0 = 1.05 · 10−8 cm and, for the corresponding energy, W(r0 ) = −2.75 electron volt. Both these results have been fully confirmed by observations on the spectrum emitted by the neutral or ionized molecule that indirectly depends on them. What is the origin of the force F = +dW/dr that tends to bring together the two hydrogen nuclei when they are at a distance larger than r0 apart? The answer given by quantum mechanics to this question is surprising since it seems to show that, 1 Majorana refers here to the paper by Ø. Burrau, Berechnung des Energiewertes des Wasserstoffmolekel-Ions (H+ 2 ) im Normalzustand, Kgl. Danske Videnskab. Selskab, Mat. Fys. Medd. 7 (1927) 14.

A.3 Resonance forces

359

W

r0

r

Figure A.2.

besides certain polarization forces which can be foreseen by classical mechanics, a predominant role is played by a completely novel kind of force, the so-called resonance forces. Let us suppose the distance r to be large with respect to the radius of the neutral hydrogen atom (∼0.5 · 10−8 cm). Then the electron undergoes the action of one or the other of the two protons, and around each of them they [the electrons] can classically describe closed orbits. The system composed by the electron and the nucleus around which it orbits forms a neutral hydrogen atom, so that our molecule results in being essentially composed by one neutral atom and one proton at a certain distance from the atom. The neutral hydrogen atom in its ground state has a charge distribution with spherical symmetry, classically meaning that all the orientations of the electronic orbit are equally possible, and the negative charge density exponentially decreases with the distance in such a way that the atomic radius can practically be considered as finite. It follows that no electric field is generated outside a neutral hydrogen atom, and thus no action can be exerted on a proton at a distance r which is large compared to the atomic dimensions. However, the neutral atom can be polarized under the action of the external proton and acquire an electric moment along the proton-neutral atom direction, and from the interaction of this electric moment with the non-uniform field generated by the proton an attractive force comes out which tends to combine the atom and the ion in a molecular system. The polarization forces, which can be easily predicted using classical arguments, can yield molecular compounds, which, however, are characterized by a pronounced fleetingness. More stable compounds can only be obtained if other forces are considered in addition to the polarization ones. In the polar molecules, composed of two ions of different signs, such forces are essentially given by the

360

Molecular bonding in quantum mechanics

electrostatic attraction between the ions; for example the HCl molecule is kept together essentially by the mutual attraction between the H+ positive ion and the Cl− negative one. However, in a molecule composed of two neutral atoms, or of a neutral atom and an ionized one, as in the case of the molecular ion H+ 2, the chemical affinity is essentially driven by the phenomenon of the resonance, according to the meaning assumed by this word in the novel [quantum] mechanics, which has no parallel in classical mechanics. When we study, from the quantum mechanics point of view, the motion of the electron in the field of the two protons, assumed to be fixed at a very large mutual distance r, at a first approximation we can determine the energy levels by assuming that the electron should move around the proton A (or B) and neglecting the influence of the other proton in B (or A), which exerts a weak perturbative action due to its distance. For the lowest energy eigenvalue E0 we thus obtain a state S corresponding to the formation of a neutral atom in its ground state consisting of the electron and the nucleus A, and a state S corresponding to a neutral atom composed by the electron and the nucleus B. Now if we take into account the perturbation that in both cases is exerted on the neutral atom by the positive ion, we again find, as long as the perturbation is small, not two eigenvalues equal to E0 but rather two eigenvalues E1 and E2 which are slightly different from E0 and both close to this value. However, the quantum states corresponding to them, let them be T1 and T2 , are not separately close to S and S , since, due to the fact that the potential field where the electron moves is symmetric with respect to the middle point of AB, the same symmetry must be shown, as mentioned above, by the effective states T1 and T2 of the electron, while it is not separately shown by S and S . According to the model representation of the quantum states introduced above, S consists of a set of electronic orbits around A, and analogously S of a set of orbits around B, while the true quantum states of the system T1 and T2 each corresponds, at a first approximation for very large r, to the orbits in S for one half and to those in S for the other half. The computations prove that, for sufficiently large nuclear distances, the mean value of the perturbed eigenvalues E1 and E2 coincides closely to the single unperturbed value E0 , while their difference is not negligible and has a conclusive importance in the present as well as in infinite other analogous cases of the study of the chemical reactions. We can thus suppose that E1 < E0 but E2 > E0 , and then T1 will be the ground state of the electron, while T2 will correspond to the excited state with a slightly higher energy. The electron in the T1 state, as well as in the T2 state, spends half of its time around the nucleus A and the other half around the nucleus B. We can also estimate the mean frequency of the periodic transit of the electron from A to B and vice versa, or that of the neutral or ionized state exchange between the two atoms, thus finding

A.3 Resonance forces

ν=

361

E2 − E1 , h

where h is the Planck constant. For large values of r, E2 − E1 decreases according to an exponential-like curve and thus the exchange frequency rapidly tends to zero, this meaning that the electron which was initially placed around A remains here for an increasingly longer time, as expected from a classical point of view. If the electron is in the state T1 , that is in its ground state, its energy (E1 ) is lower than it would have been without the mentioned exchange effect between nuclei A and B. This occurrence provides the basis for novel kinds of attractive forces among the nuclei, in addition to the polarization forces considered above, that are precisely the dominant cause of the molecular bonding. The resonance forces, as we have said, have no analogy in classical mechanics. However, as long as the analogy leading to the correspondence between a quantum state and a statistical set of classical motions can hold, the two states T1 and T2 , where the resonance forces also have opposite sign, are each composed identically by half of each of the original unperturbed states S and S . This, however, is true only at a certain approximation, that is exactly at the approximation where we can neglect the resonance forces. For an exact computation taking into account the resonance forces we must necessarily use quantum mechanics, and thus find a qualitative difference in the structure of the two quantum states that manifests itself mainly in the intermediate region between A and B through which a periodic transit of the electron between one atom and the other takes place, according to a mechanism that cannot be described by classical mechanics. Such a qualitative difference is purely formal in nature and we can deal with it only by introducing the wavefunction ψ(x, y, z) that, as is, known, corresponds in quantum mechanics to any possible state of the electron. The modulus of the square of ψ, which can also be a complex quantity, gives the probability that the electron lies in the volume unit around a generic point x, y, z. The wavefunction ψ must then satisfy a linear differential equation and thus we can always multiply ψ in any point by a fixed real or complex number of modulus 1, this constraint being required by the normalization condition  |ψ 2 | dx dy dz = 1, which is necessary for the mentioned physical interpretation of |ψ 2 |. The multiplication of ψ by a constant of modulus 1 leaves unaltered the spatial distribution of the electronic charge, and has in general no physical meaning. Now we will formally define the reflection of a quantum state with respect to the middle point O between the two nuclei A and B directly on the wavefunction ψ, by setting

362

Molecular bonding in quantum mechanics

ψ(x, y, z) = ψ  (−x, −y, −z) in a coordinate frame with origin in O. If ψ should represent a symmetric quantum state, and is thus invariant by reflection in O, the reflected wavefunction ψ  must have the same physical meaning of ψ and therefore differ from ψ, as previously stated, by a real or complex constant factor of modulus 1. Moreover, such a constant factor has to be ±1, since its square must yield unity, due to the fact that by a further reflection of ψ  with respect to the point O we again obtain the initial wavefunction ψ. For all states of the system we then must have: ψ(x, y, z) = ± ψ(−x, −y, −z), where the + sign holds for some of them, and the − one hold for the others. The formal difference between the T1 and T2 states considered above consists precisely in the fact that, in the previous equation, the upper sign holds for T1 while the lower one holds for T2 . The symmetry with respect to one point and, in general, any symmetry property, determines a formal splitting of the state of the system into two or more sectors, an important property of this splitting being that no transition between different sectors can be induced by external perturbations respecting the symmetries shown by the constraints of the system. Thus in systems containing two electrons, we have two kinds of not combinable states which are determined by the fact that the wavefunction, which now depends on the coordinates of both the electrons, remains unaltered or changes its sign by exchanging the two identical particles. In the special case of the helium atom this gives rise to the well known spectroscopic appearance of two distinct elements: parahelium and orthohelium. The theory of the chemical affinity between the neutral hydrogen atom and the ionized one, which is what we have considered until now, can be extended to the study of the neutral hydrogen molecule and, more generally, to all molecules resulting from two identical neutral atoms. Instead of only one electron moving around two fixed protons, for the neutral hydrogen molecule we should consider two electrons moving in the same field, neglecting at a first approximation their mutual repulsion. The stability of the molecule can then be understood by assuming that each of the two electrons lies in the T1 state, corresponding to attractive resonance forces. According to F. Hund we can say that the hydrogen molecule is kept together by two “bonding” electrons. However, the interaction between the two electrons is so large that it allows leave only a qualitative explanation for the schematic theory by Hund, but in principle we could predict exactly all the properties of the hydrogen molecule by solving to sufficient precision the equations introduced by quantum mechanics. In this way, using appropriate mathematical methods, we can effectively determine the chemical affinity between two neutral

A.3 Resonance forces

363

hydrogen atoms using only theoretical considerations, and the theoretical value agrees with the experimental one, given the precision of the computation imposed by practical reasons. For molecules other than hydrogen, the theory of chemical affinity is considerably more complex, due both to the larger number of electrons to be considered and to the Pauli principle, forbidding the simultaneous presence of more than two electrons in the same state. However, the different theories of chemical affinity proposed in the last few years, each of which has a greater or lesser applicability range, are practically consistent with the search for approximated computation methods for a mathematical problem that is exactly determined in itself, and not in the enunciation of novel physical principles. Then it is possible to bring the theory of the valence saturations back to more general principles of physics. Quantum mechanics opens the door to the logical unification of all the sciences having the inorganic world as a common object of study.

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Author index

Abraham, M., 237 Amaldi, E., 4, 8, 9, 11, 12, 14, 16, 83, 101 Anderson, C., 177

Eckart, C., 72 Ehrenfest, P., 14, 33 Enriques, F., 7

Baker, E.B., 85, 94 Bargmann, V., 43, 183 Barut, A.O., 46 Beck, G., 14 Belinfante, F.J., 48 Bethe, H.A., 14, 71, 127, 131 Bhabha, H.J., 42, 182, 191 Bitter, F., 32 Blackett, P.M.S., 110, 177 Bloch, F., 14, 28, 127, 131, 145 Bohr, N., 14, 63, 195, 355 Boltzmann, L., 56, 58 Born, M., 70, 196 Breit, G., 72 Brillouin, L., 251 Brossel, J., 32 Burrau, Ø., 358

Fano, U., 23, 126 Feenberg, E., 14 Fermi, E., 3–6, 8–15, 33, 49, 83–85, 100, 101, 109, 116, 126, 127, 195, 197, 274 Feynman, R.P., 231, 264 Fierz, M., 42, 182 Fock, V., 70 Fradkin, D., 14, 45 Frisch, O., 27 Furry, W., 198

Cabibbo, N., 229 Caldirola, P., 237 Carrelli, A., 3, 16 Cassirer, E., 55 Castelnuovo, G., 7 Chadwick, J., 12, 34, 110, 114, 124, 210 Cohen-Tannoudj, C., 33 Compton, A.H., 238 Condon, E.U., 22 Constable, J.E.R., 110 Corbino, O.M., 8, 9, 195 Cornell, E.A., 31 de Broglie, L., 178, 185, 190 Debye, P., 14 Dirac, P.A.M., 11, 14, 46, 175, 196, 220, 221, 230, 264, 273 Duffin, R.J., 180, 188 Durkheim, E., 55

Gamow, G., 110, 111, 124, 210, 242, 254, 258 Garton, W.R.S., 23 Geheniau, J., 188 Gelfand, I.M., 14 Gentile Jr, G., 9, 11, 15, 19, 53 Goeppert-Mayer, M., 50 Gordon, W., 39, 176 Goudsmith, S., 20 Güttinger, P., 27 Guth, E., 85 Hartree, D.R., 68 Heisenberg, W., 4–6, 10, 11, 13, 14, 33–37, 47, 64, 127, 129, 144, 195, 196, 221, 225, 230, 244, 355 Heitler, W., 24, 34, 195, 210 Herzberg, G., 210 Hille, E., 87 Hund, F., 14, 362 Hylleraas, E.A., 70, 79 Inglis, D.R., 14, 127, 142 Ising, E., 130 Jeffreys, H., 251 Joliot-Curie, F. & I., 12 Jordan, P., 196, 197, 221, 223 Kaufman, V., 23 Kellner, G.W., 66, 70, 79

378

Author index

Kemmer, N., 179, 188, 190 Ketterle, W., 31 Klein, O., 39, 176, 197 Kock, V., 198 Kramers, H.A., 251 Kruger, P.G., 19 Kudar, J., 112, 125, 258

Rajaratnam, A., 23 Rasetti, F., 8, 9, 83, 84 Rice, O.K., 126 Richardson, O.W., 115 Righi, A., 6 Rosenfeld, L., 14 Rutherford, E., 110, 124

Lanczos, C., 255 Landau, L.D., 30 Langevin, P., 128 Lee, T.D., 239 Levi-Civita, T., 7 Lieb, H., 87 London, F., 24, 34 Lorentz, H.A., 236 Lyman, T., 70

Sagnotti, A., 47 Sakharov, A.D., 346 Schlesinger, K., 56 Schrödinger, E., 39, 64, 176, 238 Schwinger, J., 29 Scorza-Dragoni, G., 86 Segrè, E., 4, 7–9, 11, 15, 27, 83 Severi, F., 7 Shenstone, A.G., 22 Shortley, G.H., 22 Simon, B., 87 Slater, J.C., 65, 68, 70, 127, 131 Snyder, H.S., 238 Sommerfeld, A., 85, 355 Stanley, H.E., 57 Stern, O., 27 Stuckelberg, E.C.G., 30 Sudarshan, E.C.G., 45

Majorana, G., 54 Majorana, Q., 6, 15, 54 Mambriani, A., 86 Martin, W.C., 23 Massey, H.S.W., 24 Maxwell, J.C., 58 Miranda, C., 85 Mohr, C.B.O., 24 Mukunda, N., 45 Møller, C., 14 Nagaoka, H., 63 Nambu, Y., 292 Noma, S., 49 Occhialini, G.P.S., 177 Oppenheimer, J.R., 33, 178, 198, 215, 224, 255 Orzalesi, C.A., 45 Pareto, V., 55, 56 Pauli, W., 11, 14, 42, 48, 49, 127, 177, 182, 190, 191, 195, 197, 198 Pauling, L., 24 Peierls, R., 12, 85, 130, 144 Penrose, R., 29 Petiau, G., 178 Phipps, T.E., 27 Placzeck, G., 11, 14 Planck, M., 239, 244 Pollard, E.C., 110 Pontecorvo, B., 50 Proca, A., 178 Rabi, I.I., 28 Racah, G., 48, 50

Thomas, L.H., 83, 85 Thomson, J.J., 237 Tricomi, F., 7 Unsöld, A., 65 van der Waerden, B.L., 14, 149, 180 Volterra, V., 7, 9 von Laue, M., 66, 150 Wald, A., 56 Walras, L., 56 Wataghin, G., 14 Weiss, P., 128, 144 Weisskopf, V.F., 14, 41, 195, 198, 212 Wentzel, G., 21, 251 Weyl, H., 11, 14, 20, 149, 177, 266, 274 Wick, G.C., 4, 12, 195 Wieman, C.E., 31 Wigner, E.P., 11, 14, 20, 37, 41, 50, 149, 183, 197, 223 Wu, T.Y., 20 Yukawa, H., 38, 49 Zener, C., 30

Subject index

(α, p) reactions, 111, 119 α particles decay Gamow theory, 110, 242, 254 scattering off nuclei, 254 nuclear disintegration, 119 cross section, 122 total probability, 124 scattering off nuclei, 109 cross section, 114 quantum-mechanical approach, 111 thermodynamic approach, 115 stability, 37 Abraham–Lorentz–Dirac theory, 237 action classical, 232 anapole moment, 332 anomaly cancelation, 322, 351 anyon, 301 Auger effect, 20, 21 autoionization, 19, 21, 23 Bargmann–Wigner equation, 184 baryogenesis, 346 baryon asymmetry, 303, 346 Berry phase, 282 Bethe ansatz, 127 Bhabha equation, 182 Black and Scholes modeling of option pricing, 57 Bohr–Sommerfeld theory, 63, 274 Boltzmann entropy relation, 56 bonding electron, 362 Bose–Einstein condensation, 31 Caldirola equations, 237 charge conjugation, 307, 314 chemical affinity, 360 chemical bond, 24 chirality projection operators, 305 chronon, 237 Clapeyron equation, 248

classical electron radius, 236 complex systems, 57 composite particles, 46 Compton wavelength, 238 Coulomb scattering, 239 CP violation, 321 CPT theorem, 45 critical temperature, 142 cross section α absorption by nuclei, 114 α-induced nuclear disintegration, 124 Curie law, 128 Curie temperature, 128 Curie–Weiss law, 128 current algebra, 45 D’Alembert wave equation, 215, 243 dark matter problem, 339 determinism, 55, 58 in philosophy, 59 diamagnetism, 128 differential equations Abel, 88 Emden–Fowler, 97 Majorana scale-invariant, 95 Thomas–Fermi, 84 WKB method, 251 Dirac algebra, 176, 286 Dirac equation, 10, 14, 19, 39, 47, 98, 176, 304 group properties, 164 Lorentz invariance, 177 Dirac Hamiltonian, 305 Dirac Lagrangian, 308 Dirac matrices, 40, 177, 305 n-dimensional, 164 non-Hermitian operators, 169 Dirac–Fierz–Pauli theory, 43, 180, 183 Dirac–Majorana mass term, 322 double β-decay, 50 neutrinoless, 51, 304, 327, 339

380 double-resonance spectroscopy, 32 Duffin–Kemmer–Petiau algebra, 180 Geheniau decomposition, 188 dynamical groups, 45 econophysics, 57 effective nuclear charge, 25 electric charge quantization, 351 electromagnetic self-force, 237 electron affinity, 75 entanglement, 30 exchange forces chemical bond, 24 ferromagnetism, 129 Fermi universal potential, 83 Fermi–Dirac statistics, 83 fermion number, 313 ferromagnetism critical temperature, 142 dimensionality, 144 distribution function, 134, 138 Heisenberg theory, 129 Majorana theory, 127 mean magnetization, 139 partition function, 140 spontaneous magnetization, 128 statistical model, 131 Weiss theory, 128 Feshbach resonances, 23 Feynman rules for Majorana particles, 313 form factors, 331 Frobenius method, Majorana generalization, 93 fundamental constants, 263 fundamental length/time, 236 Glashow–Weinberg–Salam theory, 209, 303 neutrino mixing, 317 Gordon identity, 332 grand unified theories, 327 Grassmann variables, 290 Green method, 243 group Lorentz, 41, 164 inhomogeneous, 44 permutations, 173, 267 rotations, 156 unitary transformations, 153 group theory and quantum mechanics, 149 Gruppenpest, 150 H2 molecule, 26, 362 H+ 2 molecular ion, 362 Hamilton principal function, 230 harmonic components, 265 Hartree method, 68 He+ 2 molecular ion, 24 Heisenberg theory of ferromagnetism, 129 Heisenberg–Majorana exchange forces, 33, 109 Heitler–London theory, 24, 26, 34 helicity projection operators, 306

Subject index helium, 362 ground state energy, 65 empirical relations, 72 ionization energy, 64 wavefunctions, 65, 66, 69, 76 homopolar valence, 358 Hubble expansion rate, 348 hydrogen ionization induced by an electric field, 260 in the old quantum theory, 63 pseudopolar reaction, 26 spontaneous ionization, 254 lifetime, 258 hydrostatic pressure, 247 Hylleraas variables, 70, 78 index theorems, 293 intrinsic time delay, 237, 242 retarded potentials, 242 inversion, 25 isospin, 34 Kemmer equation, 180 Kitaev chain, 53, 294 Klein–Gordon equation, 39, 41, 176, 179, 199 Kramers doubling, 300 Landau–Zener probability, 30, 279, 285 Langevin function, 129 Laplace transform, 279 generalized, 282 Laplacian, square root, 264 Larmor frequency, 28, 162, 165 least action principle, 232 lepton-number-violating processes, 344 level crossing, 282 Lorentz group, 164, 218 infinite-dimensional representations, 41, 170 inhomogeneous, 44 irreducible representations, 44, 170 Lorentz theory of the electron, 237 magnetic moment interaction, 22 neutron, 50 magnetic monopole, 351 magnetic susceptibility, 128 Majorana condition, 309 Majorana electron, 292 Majorana fermion, 52, 280, 287 C, CP and CPT parity, 314 Feynman rules, 313 in supersymmetry, 52, 338 Majorana field Fourier expansion, 310 Majorana formula, 29 Majorana hole, 31 Majorana infinite-component equation, 40 and the Bhabha equation, 183 composite particles, 46 and the CPT theorem, 45

Subject index group properties, 171 and the H atom, 45 quantum theory, 45 Majorana inversion, 33 Majorana Lagrangian, 312 Majorana mass term, 289, 290, 317 and superconductivity, 292 Majorana matrices, 287 Majorana neutrino, 47, 303, 304 anapole moment, 332 and electric charge quantization, 352 electromagnetic properties, 330 oscillations, 317 transition dipole moments, 333 Majorana representation (of a spin state), 29 Majorana sphere, 28 Majorana spinor, 52 16-component, 185 5-component, 189 6-component, 187 Majorana structure, 27 Majorana theorem, 29 Majorana theory of ferromagnetism, 127 critical temperature, 142 dimensionality, 144 distribution function, 134, 138 mean magnetization, 139 partition function, 140 statistical model, 131 Majorana theory of scalar electrodynamics, 201, 205 Majorana transition, 31, 32 Majorana zero mode, 53 Majorana–Brossel effect, 32 Majorana–Oppenheimer electrodynamics field quantization, 221 Lorentz-invariant formulation, 217 matrix algebra, 216 two-component theory, 219 3-vector formulation, 215 majorinos, 280, 293, 301 mathematical method, 55 melting point shift, 246 mixing matrix, 318 molecular bonding, 358 molecular field (Weiss), 128 molecules chemical bond, 24 diatomic, 104 equipotential surfaces, 104 H2 , 26, 362 H+ 2 , 362 He+ 2 , 24 Majorana structures, 26 and the Thomas–Fermi model, 103 muon conversion on nuclei, 343 neutralino, 338 neutrino

anapole moment, 336 charge radius, 336 magnetic moment, 336 mixing, 51, 317 rephasing, 319 oscillations, 288, 317 active-sterile, 324 CP violation, 321 flavor, 319 probability, 319, 324 pseudo-Dirac, 327, 341 radiative decay, 336 spin precession, 337 spin-flavor precession, 337 non-Abelian charge transport, 279 non-adiabatic level crossing, 30 non-adiabatic spin-flip, 27 non-Hermitian operators, 169 nuclear electrons, 33 nuclear forces Heisenberg, 34 Majorana, 33 saturation, 36 Wigner, 37 nuclear structure, 33, 209 and spin-statistics connection, 210 O(3) group, 156 paramagnetism, 128 particle−antiparticle conjugation, 306 partition function, 140 Paschen–Back effect, 164 path integrals Dirac approach, 230 Feynman approach, 231 Pauli–Weisskopf theory, 41, 198 perpetual motion, 247 photon basic properties, 220 neutrino theory of, 178 wave equation, 178, 215 wavefunction, 179, 215, 217 Planck natural units, 239 polarization forces, 359 Pontecorvo–Maki–Nakagawa–Sakata matrix, 318 Poynting theorem, 217 Proca equation, 179 projection operators chirality, 305 helicity, 306 pseudo-Dirac neutrino, 327, 341 quantum chromodynamics, 209 quantum field theory bosonic commutators, 196, 197, 223 electrons and holes, 221 fermionic anticommutators, 197, 223 fermionic commutators, 197 formalism, 196

381

382

Subject index

normal ordering, 224 photons, 221 scalar electrodynamics, 195 quantum mechanics classical limit, 232 Feynman path integral approach, 230 Majorana “path integral” approach, 232 quantum state, 229, 233, 354 classical analog, 354 quasi-Coulombian scattering, 239 quasi-stationary states, 23 coupling between discrete and continuum states, 116 nuclear, 112 relative probability, 118 spontaneous ionization of hydrogen, 259 transition probability from a continuum state, 118 transition probability from a discrete into a continuum state, 116 transition probability from a discrete into two continuous spectra, 118 quaternions, 154 qubit, 30, 53 rare particle decays, 345 renormalization of physical quantities, 100 resonance, 25, 359 Ritz method, 66, 72 Russell–Saunders coupling, 22 Rutherford formula, 239 Sakharov conditions, 346 scalar electrodynamics Majorana theory, 201, 205 Pauli–Weisskopf theory, 41, 198 Schrödinger equation, 230 screening effect, 25 seesaw mechanism, 303, 327 singular, 330 self-consistent field method, 68 spectrum absorption Cs, Gd, U, 19 He, 19 Zn, Cd, Hg, 21 emission He, 24 sphaleron, 347, 349 spin precession, 337 spin wave, 131 spin-flavor precession, 337

spin-statistics theorem, 45 spontaneous magnetization, 128, 130 Stark effect, 255 statistical laws in social sciences, 55 sterile neutrino, 322 Stern–Gerlach experiment, 27 subsidiary conditions, 182 supersymmetric theories, 338 symmetry cubic, 173, 266 and group theory, 151 Thomas–Fermi equation, 84 Fermi polynomial solution, 86 Frobenius method, 93 Majorana solution, 87, 91 mathematical properties, 86, 88 numerical solutions, 85, 92, 99 Sommerfeld solution, 85 Thomas equation, 84 transformation into an Abel equation, 87 Thomas–Fermi model, 8, 10, 19, 83 applied to molecules, 103 applied to nuclei, 37 atomic polarizability, 102 charged ions, 84 Fermi universal equation, 84 Fermi–Amaldi correction, 102 finite atomic radius, 101, 103 modified Fermi potential, 98 second approximation, 100 two-electron problem, 63 perturbative calculations, 64 self-consistent field method, 68, 71 variational method, 66, 70, 78 U(2) group, 153 variational method, 66, 72 divergencies, 81 Majorana’s variant, 79 Weiss theory of ferromagnetism, 128 Weyl fermions, 308 Wilson line, 279, 281 WKB method, 241 for differential equations, 251 Yukawa potential, 38, 240 Zeeman effect, 151 anomalous, 160 transition to the Paschen–Back effect, 164 Zitterbewegung, 238