Ettore Majorana Centennial
and Neutrino Legacy
S. Esposito
Dipartimento di Scienze Fisiche, University of Naples “Federico II”
and I.N.F.N. Sezione di Napoli
“In the world there are various categories of scientists: people of 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. But then there are geniuses like Galileo
and Newton. Well, Ettore was one of them. Majorana had what no one else in the world
Enrico Fermi
The family background
E. Majorana was born on 5 August 1906 in Catania, Sicily, to Fabio Majorana and
Dorina Corso, fourth of five sons.
He had a rich scientific, technological and political heritage..Three of his uncles
were chancellors of the University of Catania and members of the Italian
Quirino Majorana was a renowned experimental
physicist who was president of the Italian Physical
Ettore’s father was an engineer who founded
the first telephone company in Sicily and who
went on to become chief inspector of the
Ministry of Communications.
Academic studies
In 1923 he joined the
Faculty of Engineering at the
University of Rome, where he excelled.
Giovanni Gentile jr, Emilio Segrè,
Enrico Volterra, Giovanni Enriques and others
were some of his friends and colleagues.
Fermi passed an examination...
In 1927 O.M.Corbino, the director of the Institute of Physics at Rome launched
a famous appeal to the students of the engineering faculty to entice the most
brilliant young minds into studying physics. Segrè and his friend Amaldi
rose to the challenge, joining Fermi and Rasetti’s group and telling them of
Ettore’s exceptional gifts.
After some encouragement from Segrè and Amaldi, Majorana eventually
decided to meet Fermi in the autumn of that year. The pair immediately
started talking about the statistical model of atoms that Fermi was working
on, later to be known as the Thomas–Fermi model.
The model involves a complicated non-linear differential equation. The
analytical solution of the equation was then unknown, but Fermi had
managed to obtain a Numerical table of approximate values for it. Majorana
carefully followed what Fermi said and, after asking a few questions, left the
institute. The following morning he returned to Fermi’s office and asked for
a closer look at the numerical table so that he could compare it with an
analogous table he had drawn up the previous evening. Once he had
established agreement between the two tables, Majorana noted that Fermi’s
table was correct and left the institute with no further comment.
What did Majorana in that night?
He first transformed the TF equation
into an Abel equation, with a very original
method that can be used for a large class
of differetial equation. To the Abel equation,
known theorems on the existence and
uniqueness of the solution may be
Then, he transformed again the TF equation
into another first-order differential equation,
whose series solution is explicitly given in
terms of only one quadrature. From this
solution, Majorana obtained a table of
numerical values as accurate as (at least) that
of Fermi.
First studies in Physics
As if satisfied that Fermi had passed his “examination”, Majorana
decided to leave Engineering and join the Fermi group of the
“Via Panisperna boys”.
Majorana made substantial theoretical contributions to the group’s
research, and in 1928 – while still an undergraduate – published
his first paper, in which he calculated the splitting of some
spectroscopic terma in Gd, U and Cs due to the spin of electrons.
It is one among the first successfull applications of the Dirac
(1) Sullo sdoppiamento dei termini Roentgen ottici a causa dell’elettrone rotante e sulla
intensità delle righe del Cesio, in collaboration with Giovanni Gentile jr.:
Rend. Acc. Lincei, 8 (1928) 229-233
At the end of the same year, Fermi invited Majorana to give a talk at the General Meeting
of the Italian Physical Society on some applications of the Thomas-Fermi model.
Then on 6 July 1929, Majorana graduated with a master degree in Physics; his dissertation
was titled The quantum theory of radioactive nuclei..
Other published papers in 1931-1932
In 1931 he published two articles (2), (4) on the chemical bond of molecules and two more
papers (30, (5) on spectroscopy. In (3) Majorana anticipated results later obtained by a
collaborator of Goudmsith in 1934 on the Auger effect in helium.
(2) Sulla formazione dello ione molecolare di He: Nuovo Cimento 8 (1931) 22-28
(3) I presunti termini anomali dell’Elio: Nuovo Cimento, 8 (1931) 78-83
(4) Reazione pseudopolare fra atomi di Idrogeno: Rend. Acc. Lincei, 13 (1931) 58-61
(5) Teoria dei tripletti P’ incompleti: Nuovo Cimento, 8 (1931) 107-113
In 1932, stimulated by Segrè, Majorana published an important
paper on the non-adiabatic spin-flip of atoms in a magnetic field,
which was extended by Nobel laureate Rabi in 1937 and by Bloch
and Rabi in 1945.
This paper contains an independent derivation of the well-known
Landau-Zener formula (1932).
It also introduces a mathematical tool for representing spherical
functions (Majorana sphere) rediscovered only in recent times.
(6) Atomi orientati in un campo magnetico variabile:
Nuovo Cimento, 9 (1932) 43-50
(7) Teoria relativistica di particelle con momento intrinseco arbitrario:
Nuovo Cimento,. 9 (1932) 335-344
But.the most important paper of 1932 is that concerning a relativistic field theory of
particles with arbitrary spin, where Majorana introduced for the first time the unitary
infinite-dimensional representation of the Lorentz group, anticipating works by Nobel
laureates Wigner (in 1938) and Dirac (in 1945).
‘‘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.’’
The most famous paper on the neutrino
In 1937, probably after being invited by Fermi to compete for a full professorship, Majorana
published (but the theory was elaborated some years before) what was to become
his most famous paper, in which he introduced the so-called Majorana neutrino hypothesis.
(9) Teoria simmetrica dell’elettrone e del positrone: Nuovo Cimento 14 (1937) 171-184
The problem:
The Dirac theory is symmetric with respect to the electron
and the positron, but the field quantization method (used in
order to cancel divergencies) doesn’t.
This problem may be solved with a generalization of the
Jordan-Wigner method.
“The cancellation of infinite constants is required by the
symmetrization of the theory, which is already implicit in
the adopted form of the variational principle.”
Majorana is conscious that, for charged particles, the
advantage is purely formal, but... the situation may be
different for neutral particles...
The Majorana neutrino hypothesis was
revolutionary because it argued that the
antimatter partner of a given matter particle could
be the particle itself.
This was in direct contradiction to what Dirac had
successfully assumed in order to solve the
problem of negative energy states in Quantum
Field Theory.
With unprecedented farsightedness, Majorana proposed that the neutrino, which had just
been postulated by Pauli and Fermi, could be such a particle
Unpublished researches
The largest part of the Majorana’s work was left unpublished...
Master thesis
5 Notebooks (Volumetti)
18 Booklets (Quaderni)
12 folders with spare papers
Lecture notes for the course on Theoretical Physics at the University of Naples
Just very few examples...
Anticipating Feynman Q.E.D. ...
In an attempt to find a relation between fundamental
constants, Majorana gave an interpretaion of the
electromagnetic interaction in terms of particle exchange:
the space around charged particles is quantized, and two
electrons interact between them by means of the
exchange of particles from one to another.
Generalization of the Thomas-Fermi model to ions and molecules and its applications...
Anticipating Fano quasi-stationary states ...
Majorana was the first to study Nuclear Physics in
Rome (see also master thesis).
In the study of (α,p) reactions on light nuclei, he
generalized the Gamov model with the introduction
of quasi-stationary states in order to describe energy
states composed of continuous and discrete terms.
Anticipating Feynman path integral approach to QM ...
In some notes (probably prepared for a seminar at the
University of Naples), Majorana gave a physical
interpretation of Quantum Mechanics which anticipated
of several years the Feynman approach in terms of path
integral, independently of the underlying mathematical
and probably more...
“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.”
redundant counting in the integration measure in gauge theories?
However, even in the case of “standard”
or well-known topics, they were never
faced off in an obvious way:
Group-theoretical description of
Quantum Mechanics in terms of
Radiation theory...
His writings are a goldmine of seminal
new physical and mathematical ideas
and suggestions, all still quite
stimulating and useful for present-day
“Able at the same time to develop audacious hypothesis and criticize acutely
his work and that of others; very skilled calculating man, a deep-routed
mathematician that 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 collection of abilities which form the theoretical physicist of
very first-rank. 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 mind of our times and the promise of further conquests.”
Enrico Fermi
Published articles
(1) Sullo sdoppiamento dei termini Roentgen ottici a causa dell’elettrone rotante e
sulla intensità delle righe del Cesio, in collaboration with Giovanni Gentile
jr.: Rendiconti Accademia Lincei, vol. 8, pp. 229-233 (1928).
(2) Sulla formazione dello ione molecolare di He: Nuovo Cimento, vol. 8,
pp. 22-28 (1931).
(3) I presunti termini anomali dell’Elio: Nuovo Cimento, vol. 8, pp. 78-83 (1931).
(4) Reazione pseudopolare fra atomi di Idrogeno: Rendiconti Accademia Lincei,
vol. 13, pp. 58-61 (1931).
(5) Teoria dei tripletti P’ incompleti: Nuovo Cimento, vol. 8, pp. 107-113 (1931).
(6) Atomi orientati in un campo magnetico variabile: Nuovo Cimento, vol. 9,
pp. 43-50 (1932).
(7) Teoria relativistica di particelle con momento intrinseco arbitrario: Nuovo
Cimento, vol. 9, pp. 335-344 (1932).
(8) Über die Kerntheorie: Zeitschrift für Physik, vol. 82, pp. 137-145 (1933);
Sulla teoria dei nuclei: La Ricerca Scientifica, vol. 4 (1), pp. 559-565 (1933).
(9) Teoria simmetrica dell’elettrone e del positrone: Nuovo Cimento, vol. 14,
pp. 171-184 (1937).
(10) Il valore delle leggi statistiche nella fisica e nelle scienze sociali,
(posthumous, edited by G. Gentile jr.): Scientia, vol. 36, pp. 55-66 (1942).
E. Recami, Il caso Majorana (Di Renzo, Rome, 2004)
S. Esposito, Fleeting genius, Physics World 19 (2006) 34
S. Esposito, E. Majorana jr, A. van der Merwe, E. Recami,
Ettore Majorana: Notes on Theoretical Physics (Kluwer-Springer, New York, 2003)
E. Di Grezia, S. Esposito, Fermi, Majorana and the statistical model of atoms, Found. Phys. 34
(2004) 1431
S. Esposito, Majorana solution of the Thomas-Fermi equation, Am. J. Phys. 70 (2002) 852
S. Esposito, Again on Majorana and the Thomas-Fermi model: a comment to physics/0511222,
S. Esposito, Majorana transformation for differential equations,
Int. J. Theor. Phys. 41 (2002) 2417
E. Majorana, Lezioni di Fisica teorica, edited by S. Esposito (Bibliopolis, Naples, 2006)
A. Drago, S. Esposito, Ettore Majorana’s course on Theoretical Physics: the Moreno Lecture
Notes, arXiv:physics/0503084, to be published in Physics in Perspective
A. De Gregorio, S. Esposito, Teaching Theoretical Physics: the cases of Enrico Fermi and Ettore
Majorana, arXiv:physics/0602146
A. Drago, S. Esposito, Following Weyl on Quantum Mechanics: the contribution of Ettore
Majorana, Found. Phys. 34 (2004) 871
S. Esposito, A peculiar lecture by Ettore Majorana, Eur. J. Phys. 27 (2006) 1147
S. Esposito, Majorana and the path-integral approach to Quantum Mechanics,
arXiv:physics/0603140; to be published in the Annales de la fondation Louis De Broglie
A. Drago, S. Esposito, A logical analysis of Majorana papers on Theoretical Physics,
Electron, J, Theor. Phys.3 (2006) 249
S. Esposito, Four variation on Theoretical Physics by Ettore Majorana,
Electron, J, Theor. Phys.3 (2006) 265
S. Esposito, Un manoscritto inedito in francese di Ettore Majorana, arXiiv:physics/0607099

ppt - INFN