Ettore Majorana Centennial and Neutrino Legacy

1
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 has...


Enrico Fermi
2
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 parliament

• Quirino Majorana was a renowned experimental
• physicist who was president of the Italian
  Physical
• Society.

Ettores 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.
3
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.

4
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
  Rasettis group and telling them of Ettores
  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
  ThomasFermi 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 Fermis 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 Fermis table was correct and
  left the institute with no further comment.

5

• 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 applied...
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.
6
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 groups 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 equation...
(1) Sullo sdoppiamento dei termini Roentgen
ottici a causa dellelettrone 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..
7
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 dellElio 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
8
(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, essentiall
y 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.
9
1933 a theory with the neutral proton
In March 1932, after that Chadwick announced the
discovery of the neutron, Majorana revealed to
his friends and colleagues in Rome that he had
built a theory of light nuclei based on the
concept of exchange forces. Although incouraged
by Fermi, he did not publish his theory, which
was independently elaborated by Heisenberg. But...
According to Heisenberg, the underlying nuclear
forces should be interpreted in terms of nucleons
exchanging spinless electrons (implicitly n p
e)...
Instead, in Majoranas view, the neutron was
pictured as a neutral proton, and the erroneous
experimental consequences of the Heisenberg model
were quickly recognized by Heisenberg himself and
others. In many several occasions, the Nobel
lauerate Heisenberg only marginally mentioned his
own contribution, while annoyingly pointing out
that of Majorana...
(8) Über die Kerntheorie Zeits. Phys. 82 (1933).
137-145 Sulla teoria dei nuclei Ricerca
Scientifica, 4 (1933) 559-565
10
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 dellelettrone 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) doesnt. 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.
11
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
12
Unpublished researches
The largest part of the Majoranas 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

13
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 (a,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.
14
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 formulation.
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?
15
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 symmetries... Re
lativity... 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 research.
16
Epilogue
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
17
Published articles

• (1) Sullo sdoppiamento dei termini Roentgen
  ottici a causa dellelettrone 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 dellElio 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 dellelettrone 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).

18
References

• 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,
  arXivphysics/0512259
• 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 Majoranas course
  on Theoretical Physics the Moreno Lecture Notes,
  arXivphysics/0503084, to be published in Physics
  in Perspective
• A. De Gregorio, S. Esposito, Teaching Theoretical
  Physics the cases of Enrico Fermi and Ettore
  Majorana, arXivphysics/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, arXivphysics/06031
  40 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