| Фаддеев on the queer scene : интеревью Тян-Шанскому |
[Aug. 31st, 2007|01:49 pm] |
Еще одно интервью акад. Фаддева, на этот раз в Вестнике Европейского Математического Общества.
Faddeev: I remember at one point, after we had
evacuated to Kazan, my father was greatly excited with
his latest results. This was the time when he made important
discoveries in homological algebra. I asked him how
many people in the world would be able to understand
his results and he replied that there might be fi ve or six of
them. I thought that this would not suit me.
the secretary said to me in a grave voice, “Look, they say you are
reading the writings of a certain Knuth?”
[Akad. O.A.] Ladyzhenskaja She was in charge of our education at the Chair of Mathematical
Physics. This was, in fact, the first time ever that
this Chair had enrolled students on its own account. Before
that the Chair of Mathematics at the Physics Department,
created by academician Smirnov in the 1930’s,
was mainly considered an auxiliary one. His idea, supported
by academician Fock, was to dispense a special
mathematical education for physicists, with emphasis on
the mathematical background of quantum theory.
[a maxim] You know, I always believe that when you feel you are in a position
to start writing serial papers, you should better change
the subject…
Doing Lagrangian QFT in this country in the 1960’s
was not an easy deal.
.. Landau came to the conclusion
that quantum fi eld theory is inconsistent and [most famous paper of Faddeev]
could neither be published in any of the Soviet physics
journals nor abroad (for such a publication, one needed
a positive conclusion of the Nuclear Physics Division of
the Academy of Sciences). Finally,...
(1995), the Max Planck Medal (1996), the Euler Medal
(2002) and the Henri Poincaré prize (2006). He was also
awarded the USSR State Prize (1971), the Order of the
Red Banner of Labour (1981), the Order of Lenin (1987),
the Order of Friendship of Peoples (1994), the Russian
State Prize in Science and Technology (1995), the Order
of Merit of the 4th and 3rd degree (1999, 2005) and the
Russian State Prize (2005). He has been elected an honorary
member of the American Academy of Science and
Interview
Both your father and mother were prominent mathematicians
but you have chosen the Physics Department
of the University.
Well, it was only natural since my father was at the time
the Dean of the Mathematics Department and I wanted
to make my own way. I believe that in general a reasonable
share of stubbornness and non-conformism proved
to be of much importance in my formation as a scholar.
As a schoolboy, I did not have any particular interest in
mathematics. I was a passionate reader. For instance, I
learned much of English medieval history from Shakespeare’s
chronicles. Of course, the general intellectual
atmosphere of my family had a great infl uence upon me.
One precious thing that I owe to it is my love for music.
But I cannot say that it gave me a particularly professional
orientation. I remember at one point, after we had
evacuated to Kazan, my father was greatly excited with
his latest results. This was the time when he made important
discoveries in homological algebra. I asked him how
many people in the world would be able to understand
his results and he replied that there might be fi ve or six of
them. I thought that this would not suit me.
You entered the University in the early 50’s. This was
still a rather diffi cult time in our country, was not it?
Yes, I should say that I had a narrow escape in 1952
when the university offi cials heard that I was very fond
of Hamsun’s novels. I was invited to the local Comsomol
(Communist Youth) Committee and the secretary said to
me in a grave voice, “Look, they say you are reading the
writings of a certain Knuth?” Fortunately for me, Stalin
died a few months later and this story did not have any
sequel. But my overall impression in university was that
of tremendous freedom, which contrasted to secondary
school. I remember that one of the things that struck me
in my fi rst year at university was the elementary course
in analytic geometry, with simple routine calculations replacing
the rather refi ned reasoning of school geometry.
I should say that ever since, I have preferred simple calculations
to tricky arguments!
Your main teacher at university was Professor Ladyzhenskaya?
She was in charge of our education at the Chair of Mathematical
Physics. This was, in fact, the fi rst time ever that
this Chair had enrolled students on its own account. Before
that the Chair of Mathematics at the Physics Department,
created by academician Smirnov in the 1930’s,
was mainly considered an auxiliary one. His idea, supported
by academician Fock, was to dispense a special
mathematical education for physicists, with emphasis on
the mathematical background of quantum theory. Now,
for the fi rst time, it was given independent status and was
allowed to have its own students, give special courses and
supervise diploma work. Thus I was really in the very fi rst
group of students who in the following year, 1956, defended
their university thesis in mathematical physics.
O. A. Ladyzhenskaya was reading us a large variety of
courses, starting with a basic complex variables course,
which was followed by partial differential equations and
operator theory and by special seminars.
And what about academician Fock? Did he lecture at
the physics department at the time?
He gave us just about ten lectures on general relativity.
But he has always been very attentive to me personally.
He was not lecturing on quantum mechanics anymore
but through my mother he passed me his personal copy of
his lectures on quantum mechanics, a great bibliographical
rarity at the time.
Much later when I was
defending my second
thesis at the Steklov
Institute in Moscow, he
appeared at the meeting
quite unexpectedly.
This was a queer scene,
as a matter of fact. The
meeting of the Steklov
Scientifi c Council
was presided over by
academician Vinogradov,
seated deep in his
armchair, his back to
the blackboard, supervising the entire academic Areopagus
placed in the back benches. I had already started
my talk when the door went open and academician Fock
appeared. He placed himself in the fi rst bench in front
of Vinogradov and after a while asked him loudly, in his
characteristic high voice, “Ivan Matveevich, are you not
interested in what Ludwig is telling? You cannot see the
blackboard this way, can you?”
Going back to your university years, I remember you
saying you had a very early interest in quantum fi eld
theory?
My fi rst textbook in QFT was K. Friedrichs’ book, “Mathematical
aspects of the quantum theory of fi elds”, which
we started to study on the advice of Ladyzhenskaya in
1954/55. I was the main speaker in the seminar, which
started invariably with O. A.’s question, “First of all, Ludwig,
please do remind us of the defi nition of annihilation
operators.” This book has shaped my interest in mathematical
problems of QFT and also encouraged a kind of
aversion to the computation of Feynman diagrams that
was so very popular among my fellows from the Chair
of Theoretical Physics. Making an independent advance
in QFT was of course too diffi cult a bid at that stage and
my fi rst research work was on a much easier subject: the
potential scattering and spectral decomposition for the
Schrödinger operators with continuous spectrum. As I
was working on the subject, with the aim of combining
the ideas of Friedrichs with concrete methods borrowed
from the book of Levitan on singular Sturm–Liouville
operators, an important paper of Povzner on the continuous
spectrum expansions was published and it remained
for me only to refi ne and to generalize his work. In the
course of this I wrote my fi rst university thesis.
Another subject that you were deeply involved in starting
in the 1950’s was quantum inverse scattering.
A comprehensive study of the quantum inverse problem
was part of my offi cial curriculum as a research student.
Professor Ladyzhenskaya remained my scientifi c
supervisor. When it became known that I had prepared
an exhaustive exposition of the inverse scattering problem
for the radial Schrödinger operator, I was invited by
N. N. Bogolybov to give a plenary talk on the subject at
the inaugural meeting of the Laboratory of Theoretical
Interview
EMS Newsletter June 2007 33
Physics in Dubna, in the presence of Gelfand, Levitan,
Krein, Marchenko, and other senior fi gures. This was a
rather exceptional honour for me at the time. A written
version of this talk was published the next year in
Uspekhi. Simultaneously, I wrote a research paper that
gave a complete solution of the inverse problem for the
Schrödinger operator on the line. Much later this paper
became important once again as it contained all the
background of the future inverse scattering method in
the theory of integrable systems.
Your main technical achievement at this time is related
to multi-particle scattering.
I believed that it was important to resolve a really diffi -
cult technical problem before launching myself into the
insecure waters of QFT. One such problem was the threedimensional
inverse scattering problem (without any a
priory assumption of the symmetry of the potential). This
problem proved to be very diffi cult and I only managed
to solve it a few years later. And of course the quantum
three-body problem was a real challenge too. The experience
gained in the work on the so-called Friedrichs model
was of great importance for me in this venture. Some key
ideas also came from the study of a QFT model that was
widely discussed at the time, the so-called Thirring model.
Later on, your work on the quantum three-body problem
triggered tremendous activity.
Well, that is correct. Some of my pupils were continuing
in this direction for another decade or more. As for me, I
decided that it was really time to attack QFT. You know,
I always believe that when you feel you are in a position
to start writing serial papers, you should better change
the subject…
Yes, I know it only too well. Unfortunately, it’s a maxim
that is rather diffi cult to follow. But prior to your 1967
paper on Yang–Mills theory there were also a couple of
papers on automorphic functions and Selberg’s theory.
My fi rst paper on automorphic functions was written on
Gelfand’s command. By that time he was working on the
last volume of his ‘Generalized Functions’ and wanted to
include a special chapter on Selberg theory. He expected
that the methods of scattering theory would fi t in with
the setting of automorphic functions and gave me just
two weeks to settle the problem. This time schedule was
of course rather tough but I managed to complete the
task. However, Gelfand found the proof was too long to
be included in his book and instead proposed for me to
prepare a separate memoir, which awaited publication
for another two years. A couple of years later, together
with my students, A. Venkov and V. Kalinin, I gave a
complete non-arithmetic proof of the Selberg trace formula.
Another, very romantic, idea was to reformulate
the Riemann hypothesis in terms of automorphic scattering.
This was our paper with B. Pavlov written a few
years later, based on Lax–Phillips theory. Unfortunately,
this approach does not seem to bring us any closer to
the proof of the Riemann hypothesis but it was a very
interesting venture. A. Venkov has continued working in
automorphic harmonic analysis ever since but he gradually
returned to arithmetic methods.
I. M. Gelfand was in the jury for your second thesis.
You were also rather closely associated with his famous
seminar at Moscow University.
As far as I remember, I gave my fi rst talk at Gelfand’s
seminar in the spring of 1958. I had been well warned
about the dangers awaiting me. Gelfand would often stop
the speaker and continue the report in his place or make
rather acute remarks on the speaker’s mistakes. I was reporting
on the rather old paper of Friedrichs on perturbation
theory of continuous spectrum and on the defi nition
of the S-matrix, with my own additions, and since I was
very well prepared professionally Gelfand was not able to
shake me. Only V. B. Lidsky came under fi re because he
had once said that Friedrichs’ work was not interesting.
In Gelfand’s seminar I made acquaintance with F. A.
Berezin and we became close scientifi c friends for a long
time. Together with Berezin, as well as Victor Maslov,
whom I met at the 1958 Odessa conference, and Bob
(Robert) Minlos, we formed a small company of young
experts in quantum mathematical physics. I believe our
contacts were of much importance for the four of us.
Coming back to your 1967 work on Yang–Mills theory…
My starting point was not Yang–Mills theory! I did not
know much on it at the time, as a matter of fact. Of course,
the real thing I was concerned with was quantum gravity.
At that time, many theorists were studying oversimpli-
fi ed models, like ..4 as part of the so called “axiomatic
approach”, which was designed to overcome the diffi culties
of the conventional theory. But it was always intuitively
clear to me that a truly good theory, one that has
the chance to “exist”, should have some geometric motivation
and beauty. One important hint was provided by
a 1963 paper of Feynman, who was also concerned with
quantum gravity. Feynman noticed that the diagram expansion
in general relativity is inconsistent and requires
a correction to maintain unitarity. He also noticed that a
similar phenomenon occurs in Yang–Mills theory, which
he brought into play as a useful toy model, at the advice
of M. Gell-Mann. Almost simultaneously, I bought
a Russian translation of A. Lichnerowicz’s book, Théorie
globale des connexions et des groupes d’holonomie.
A similarity with formulae coming from physics was
striking. Soon I realized that Yang-Mills theory was as
good a theory geometrically as general relativity. The
book of Lichnerowicz was my fi rst textbook in modern
differential geometry. This gave me motivation to study
the Yang–Mills case on its own account. Its tremendous
importance in the theory of weak and strong interactions
was realized only a few years later.
Doing Lagrangian QFT in this country in the 1960’s
was not an easy deal.
It was even worse than that! Ever since the discovery of
the so-called “zero charge paradox” in quantum electrodynamics
in the 1950’s Landau came to the conclusion
that quantum fi eld theory is inconsistent and that the
Interview
34 EMS Newsletter June 2007
Hamiltonian method should be considered as completely
dead and may now be buried (“with all the honours it
deserves”). This was the verdict he made in his last published
paper, “Fundamental Problems”, written shortly
before the tragic car accident that put an end to his scientifi
c career. Landau’s pupils considered this maxim
as their teacher’s testament. So when, together with my
younger colleague, V. N. Popov, we prepared a paper on
the quantization of Yang–Mills theory, its complete text
could neither be published in any of the Soviet physics
journals nor abroad (for such a publication, one needed
a positive conclusion of the Nuclear Physics Division of
the Academy of Sciences). Finally, the full text was published
only as a preprint of the Kiev Institute of Theoretical
Physics, with hand-written formulas and a very
limited circulation. We were able to publish only a short
announcement in “Physics Letters”. The complete version
was translated into English by Fermilab only in 1973
when it fi nally received a wide circulation.
Victor Nikolaevich was one of the very rare experts in
functional integration methods at the time.
Yes, it really was a rarity in those days. Curiously enough,
R. Feynman, who was at the origin of the functional integration
method back in the 1940s, has never ever used
it in problems of QFT, although it is certainly the most
straightforward way to deduce the Feynman diagrams
expansion. In Yang–Mills theory, it also allows you to
resolve the problem of unitarity raised by Feynman in
a very concise form. In a way, we managed to outplay
Feynman in his own fi eld.
An exposition of your new method opened the very fi rst
issue of “Theoretical and Mathematical Physics” in
1969, which I remember to be the favourite reading of
young theorists (including myself) at that time.
The creation of the journal was of course very much in the
spirit of the time; the necessity to establish much closer
ties between fundamental physics and mathematics was
really pressing. My 1969 article, which treated the quantization
of Yang–Mills fi eld via the Hamiltonian functional
integral, emerged from my attempt to understand more
deeply the geometric aspects of our results (derived
at fi rst by a kind of trick, the so called Faddeev–Popov
trick). This brought me to the study of Dirac formalism
for constrained Hamiltonian systems and of symplectic
reduction (on the functional integral level). Working on
this problem, I had a very fruitful interaction with V. Arnold.
This gave me a big boost in my study of differential
geometry. In my turn, I tried to explain to him (probably
less successfully) the meaning of quantization.
Simultaneously you were also continuing your work on
the inverse scattering problem.
The 1960s were exceptionally productive years. I still regard
my work on three-dimensional inverse scattering as
probably my best analytic result, although, for various
reasons, it is much less known than many others. The key
point was to tackle the problem of over-determinacy of
the scattering data. The breakthrough came in 1966 when
I managed to fi nd a substitute for the “Volterra Green’s
functions” of the one-dimensional Sturm–Liouville equations
that could be applied in the multi-dimensional setting.
It took several years until all the technical details
were fi lled in. At fi rst I published just a short note, as I
was still fully in the Yang–Mills business, and the complete
exposition had to wait until 1976.
But before that there were many other exciting things
to happen.
Well, that’s true. At the very beginning of 1971 I went to
Novosibirsk to attend an international conference on the
inverse scattering problem, where I was going to report
on my results on the three-dimensional inverse problem.
It is there that I heard for the fi rst time (from the talk of
A. B. Shabat) about the exciting new development in the
theory of non-linear PDEs. The one-dimensional inverse
scattering problem has become a magic tool to solve the
KdV and the non-linear Schrödinger equations. Formulae
from my old PhD thesis on the inverse problem for a onedimensional
Schrödinger operator on the line were used
in an entirely new context. This was sort of a bolt from the
blue. During the conference we discussed the new method
with V. Zakharov and soon came to its new interpretation
in the spirit of the Liouville theorem in the theory of integrable
Hamiltonian systems. Our joint paper, published in
December 1971, started a new period in my scientifi c life.
By that time you were already heading a research group
at the Steklov Institute in Leningrad.
Already in the 1960s I was lecturing at the Leningrad
university (both in its Mathematics and Physics Departments)
and quite naturally my seminar at the Steklov
Institute became a point of attraction for many brilliant
young students. Gradually, this handful of youngsters became
a true research group I am really proud of. I should
say that the new research subject, the classical inverse
scattering method, which was launched in the 1970’s, was
perfectly adapted for a collective effort. In the university
Interview
EMS Newsletter June 2007 35
all my students had already received a very solid background
in scattering theory, so now they were completely
ready to join the race.
I remember that when you introduced our research
group to Peter Lax during his visit in 1976 you said you
reserved for yourself the role of a playing coach.
Well, you certainly agree with me that it was a really good
time and a truly creative atmosphere, which prevailed in
our weekly seminar and during the non-stop informal
discussions that followed.
The point of view on the classical inverse scattering
method that you advocated at the time was probably
different from that in other research groups we competed
with.
From the very beginning I was heading towards the possible
applications in QFT. Of course, the KdV equation
is a rather silly system to be quantized. But very quickly,
together with my student Leon Takhtajan we came upon
a truly exciting example: the Sine Gordon equation. Its
solitons and breathers have a clear interpretation of new
“relativistic particles”, which emerge as kind of “collective
excitations” of the original fi eld. What’s more, when
the coupling constant in the original Lagrangian is small,
the new particles appear to be strongly interacting, as the
coupling constant passes to the denominator. When this
mechanism was discovered I was of course excited and
we wrote a special notice for Physics Letters. Curiously
enough, this letter was lost somewhere on its way to the
journal (I had passed it to R. Haag during my visit to
Hamburg in 1972) so it has never been published. In the
mid 70s we started a systematic work on the quantization
problem for integrable models, mainly for the Sine–Gordon
and for the non-linear Schrödinger equation. The
fi rst step was the study of the semi-classical expansion
and of perturbation theory, which confi rmed, despite
the heavy opposition of the conservatives from the old
Landau school, that both integrability and the non-trivial
particle content of the model survive quantization. The
contribution of I. Arefyeva and V. Korepin was particularly
important at this stage.
Already at that time you were convinced that the role
of solitons in quantum physics goes beyond the two-dimensional
models.
Solitons gave the fi rst example where the old paradigm
of QFT “one particle – one fi eld” had been broken. Of
course I felt it would be great to make a similar mechanism
work in realistic space-time dimension. The idea to
associate stable solutions of nonlinear equations with a
new kind of particle was not at all generally acceptable
at the time; some of the physicists were mocking me and
speaking of the revival of ‘Einstein’s dream’ (which had a
strongly pejorative meaning). In the mid-70s we frequently
discussed the matter with young Moscow theorists, mainly
with Polyakov, and I believe these discussions have contributed
quite considerably to the discovery of the famous
quasiparticle solution, the ‘t Hooft–Polyakov monopole
and the instanton solution in Yang–Mills theory.
The semiclassical treatment of quantum integrable
models was then followed by an exact solution.
This was a truly important development of the late 1970’s.
The new method, which was baptized the quantum inverse
scattering method, brought together several ideas
that were completely disjoint at the time, ranging from
the old classical work of H. Bethe on lattice spin models
and the more recent results of R. Baxter on quantum statistical
mechanics to the inverse scattering method with
its emphasis on the role of “Lax operators”. The beautiful
algebra behind the new method was soon crystallized
in the notion of the quantum R-matrix and the quantum
Yang–Baxter equation. From the very beginning, my
young pupils played a key role in this development. I’d
like to mention, in particular, the work of E. Sklyanin
and L. Takhtajan. A couple of years later, Kulish and
Reshetikhin constructed the fi rst example of a “quantum
group”, which proved to be one of the most important byproducts
of the new method (the term itself was coined
a few years later by V. Drinfeld who was largely inspired
by our work on QISM in his search for the new algebraic
structures that underlie quantum integrability). For their
part, Izergin and Korepin developed powerful technical
tools to compute physically interesting quantities associated
with integrable models like correlation functions, etc.
Finally, Smirnov made a key contribution to the quantum
inverse scattering problem proper, i.e. to the reconstruction
of ‘local’ observables from spectral data. It’s at this
time that our research group won overall international
recognition. The very landscape of mathematical physics
was profoundly changed by these developments.
There were also various implications for the old classical
inverse scattering method itself.
I agree, and you know it perfectly well, as you were one
of the actors. The notion of the classical r-matrix (the
semi-classical counterpart of the quantum R-matrix
that was fi rst introduced by Sklyanin) has played here
the key role. The miracle of the Hamiltonian structures
that arise in the study of integrable systems has been
fully explained: the r-matrix method linked them to the
Riemann–Hilbert problems, which represent a technical
equivalent of the Gelfand–Levitan equations in inverse
scattering. Via this notion, the classical inverse scattering
method was also linked to the famous orbits method in
Lie group theory, making manifest the “hidden symmetry”
aspect of integrable systems. Finally, the quadratic
“Sklyanin Poisson brackets”, which naturally emerge in
the theory, started a truly new chapter in Poisson geometry,
which has developed very actively up to the present
day. In the 1980’s I wrote, together with L. Takhtajan, a
special book on the Hamiltonian methods in the theory
of integrable systems, which focused on the notion of
the classical r-matrix. We hoped this fi rst book would be
followed by another one treating the quantum inverse
scattering method but the big changes of the 1990s that
followed did not allow us to continue.
During all these crucial years, our laboratory at the
Steklov Mathematical Institute in Leningrad was a
point of attraction for the entire soliton community…
Interview
36 EMS Newsletter June 2007
That’s correct. I should mention in particular the so called
Quantum Soliton Meetings that we organized. The fi rst
one in the autumn of 1979 shortly followed the breakthrough
in the study of quantum integrable systems. This
was a truly exciting meeting that brought together the
leading lights of the previous stage of theoretical physics
like academician Migdal and V.N. Gribov and the best
experts in classical integrability like V. Zakharov and S.P.
Novikov, as well as key people of the younger generation
like Sasha Polyakov, Sasha Belavin, Sasha Zamolodchikov,
Volodia Drinfeld…
The name of Drinfeld inevitably brings us to the notion of
quantum groups, which you have already mentioned ...
Drinfeld took very seriously the discovery of the quantum
inverse scattering method; he was a very frequent
and most welcome guest of our laboratory in those days.
A few years later he managed to give a very nice and appealing
form to the ‘algebraic half’ of the QISM. In 1986,
during the Mathematical Congress in Berkeley, I brought
Drinfeld’s address with me and asked Cartier to present
it to the congress. In those late years of the Soviet Union,
Drinfeld still could not get permission to go the States
himself. This address immediately triggered the ‘Quantum
Groups Revolution’ of the late 1980’s, with important
implications for representation theory, knot theory,
geometry of low-dimensional manifolds, and more. In
our research group, I’d like to single out the contribution
of N. Reshetikhin to all these matters. For my part,
I always preferred the original R-matrix approach to
the high-brow algebraic language of Hopf algebras and
tensor categories. So I was very pleased to fi nd a simple
R-matrix equivalent for the q-deformed universal enveloping
algebras discovered by Drinfeld and Jimbo. This
was our joint work with Reshetikhin and Takhtajan on
the quantization of Lie groups and Lie algebras. I should
mention that, contrary to the popular belief, the QISM is
still considerably more rich than quantum group theory:
its key part is certainly the algebraic Bethe Ansatz and its
more refi ned modern versions, which still resist explanation
in the context of quantum group theory (or at least
any such explanation is much less elementary than our
original method). Of course, much work has been done in
connection to this over the past few years; the quantum
separation of variables method developed by Sklyanin is
probably the most important contribution I can mention.
You have mentioned q-deformations, which brings us to
the realm of deformations in general…
This was a very important idea for me, which I learned
in the old days, notably from the work of I. Segal. Quite
a few important new physical theories, like relativity or
quantum mechanics, are associated with nontrivial deformations
of the underlying algebraic structures. In the late
1970’s, deformation quantization formalism was systematically
developed by Flato, Lichnerowicz and their colleagues.
Quantum groups and the QISM provide another
striking example. One more project I launched in the late
1980’s was associated with the study of anomalies in QFT.
Together with my student S. Shatashvili we discovered
a nontrivial deformation of the three-dimensional nonabelian
gauge group. This deformation, which plays some
role in the Yang–Mills theory with chiral fermions, can be
traced down to the characteristic classes and cohomology
groups of the gauge groups. Curiously, this work has
brought me very close to the discoveries my father made
in homological algebra forty years earlier that I mentioned
at the beginning…
The late 1980’s were probably the best of times for our
old research group at the Steklov Institute. In the next
decade, things have changed…
By 1990 we had an incredible concentration of excellent
people in almost every part of mathematics in our laboratory
and in other fellow laboratories at the Leningrad
Branch of the Steklov Institute. The economic crisis of
the 1990’s, in combination with the new freedom, has
changed the scene profoundly. Many of my pupils were
offered prestigious positions abroad. Those who decided
to stay are travelling quite extensively as well and remain
in Petersburg for only a fraction of the year. For my part,
I am also travelling a lot. We also had some painful losses:
V. N. Popov and A. Izergin passed away at a very early
age. This new situation has prompted me to return to the
rather solitary working style of my younger years, which
contrasts so much with the team work of the 1970’s and
1980’s. Of course, now the isolation is only relative, as it
is moderated by the new Internet capabilities and also by
the freedom to travel. I have also established some new
collaborations, notably with Anti Niemi from Helsinki
University.
What were your main research topics during these
years?
Rather unexpectedly for me, I have returned to Yang–
Mills theory. Contacts with Niemi have revived my interest
in the modifi ed Skyrme model that I proposed in the
late 1970’s. Typical solitons that were studied at that time
are associated with certain “topological charges” and may
be described as localized solutions that resemble isolated
particles. In my version of the Skyrme model (which is
based on the Hopf invariant) excitations are more like
strings or pipelines and are classifi ed by various knots in
R3. Unfortunately, it is practically hopeless to fi nd explicit
solutions of this model in closed analytic form. In
Interview
36 EMS Newsletter June 2007
That’s correct. I should mention in particular the so called
Quantum Soliton Meetings that we organized. The fi rst
one in the autumn of 1979 shortly followed the breakthrough
in the study of quantum integrable systems. This
was a truly exciting meeting that brought together the
leading lights of the previous stage of theoretical physics
like academician Migdal and V.N. Gribov and the best
experts in classical integrability like V. Zakharov and S.P.
Novikov, as well as key people of the younger generation
like Sasha Polyakov, Sasha Belavin, Sasha Zamolodchikov,
Volodia Drinfeld…
The name of Drinfeld inevitably brings us to the notion of
quantum groups, which you have already mentioned ...
Drinfeld took very seriously the discovery of the quantum
inverse scattering method; he was a very frequent
and most welcome guest of our laboratory in those days.
A few years later he managed to give a very nice and appealing
form to the ‘algebraic half’ of the QISM. In 1986,
during the Mathematical Congress in Berkeley, I brought
Drinfeld’s address with me and asked Cartier to present
it to the congress. In those late years of the Soviet Union,
Drinfeld still could not get permission to go the States
himself. This address immediately triggered the ‘Quantum
Groups Revolution’ of the late 1980’s, with important
implications for representation theory, knot theory,
geometry of low-dimensional manifolds, and more. In
our research group, I’d like to single out the contribution
of N. Reshetikhin to all these matters. For my part,
I always preferred the original R-matrix approach to
the high-brow algebraic language of Hopf algebras and
tensor categories. So I was very pleased to fi nd a simple
R-matrix equivalent for the q-deformed universal enveloping
algebras discovered by Drinfeld and Jimbo. This
was our joint work with Reshetikhin and Takhtajan on
the quantization of Lie groups and Lie algebras. I should
mention that, contrary to the popular belief, the QISM is
still considerably more rich than quantum group theory:
its key part is certainly the algebraic Bethe Ansatz and its
more refi ned modern versions, which still resist explanation
in the context of quantum group theory (or at least
any such explanation is much less elementary than our
original method). Of course, much work has been done in
connection to this over the past few years; the quantum
separation of variables method developed by Sklyanin is
probably the most important contribution I can mention.
You have mentioned q-deformations, which brings us to
the realm of deformations in general…
This was a very important idea for me, which I learned
in the old days, notably from the work of I. Segal. Quite
a few important new physical theories, like relativity or
quantum mechanics, are associated with nontrivial deformations
of the underlying algebraic structures. In the late
1970’s, deformation quantization formalism was systematically
developed by Flato, Lichnerowicz and their colleagues.
Quantum groups and the QISM provide another
striking example. One more project I launched in the late
1980’s was associated with the study of anomalies in QFT.
Together with my student S. Shatashvili we discovered
a nontrivial deformation of the three-dimensional nonabelian
gauge group. This deformation, which plays some
role in the Yang–Mills theory with chiral fermions, can be
traced down to the characteristic classes and cohomology
groups of the gauge groups. Curiously, this work has
brought me very close to the discoveries my father made
in homological algebra forty years earlier that I mentioned
at the beginning…
The late 1980’s were probably the best of times for our
old research group at the Steklov Institute. In the next
decade, things have changed…
By 1990 we had an incredible concentration of excellent
people in almost every part of mathematics in our laboratory
and in other fellow laboratories at the Leningrad
Branch of the Steklov Institute. The economic crisis of
the 1990’s, in combination with the new freedom, has
changed the scene profoundly. Many of my pupils were
offered prestigious positions abroad. Those who decided
to stay are travelling quite extensively as well and remain
in Petersburg for only a fraction of the year. For my part,
I am also travelling a lot. We also had some painful losses:
V. N. Popov and A. Izergin passed away at a very early
age. This new situation has prompted me to return to the
rather solitary working style of my younger years, which
contrasts so much with the team work of the 1970’s and
1980’s. Of course, now the isolation is only relative, as it
is moderated by the new Internet capabilities and also by
the freedom to travel. I have also established some new
collaborations, notably with Anti Niemi from Helsinki
University.
What were your main research topics during these
years?
Rather unexpectedly for me, I have returned to Yang–
Mills theory. Contacts with Niemi have revived my interest
in the modifi ed Skyrme model that I proposed in the
late 1970’s. Typical solitons that were studied at that time
are associated with certain “topological charges” and may
be described as localized solutions that resemble isolated
particles. In my version of the Skyrme model (which is
based on the Hopf invariant) excitations are more like
strings or pipelines and are classifi ed by various knots in
R3. Unfortunately, it is practically hopeless to fi nd explicit
solutions of this model in closed analytic form. In
Interview
EMS Newsletter June 2007 37
the 1970’s, when I started to study this model, there were
no suffi ciently good computers to deal with it numerically.
In the 1990’s Niemi, who mastered modern numerical
methods, could already use the capacities of the supercomputer
of Helsinki University. His results confi rmed
my expectations for the simplest confi gurations (the unknot);
our joint publication in “Nature” attracted the attention
of the true professionals in numerical methods,
Sutcliff and Hietarinta, who have carried out still more
refi ned computations. Their results convincingly show the
existence of nontrivial knot-like excitations, in particular,
associated with the trefoil. You remember, of course, that
Escher’s etching of the trefoil knot has been hanging in
our seminar room since the early 1970’s.
Certainly, I do! And what was the next step?
Well, we started to look around in search of interesting
applications. As you know, the plausible scenario of the
quark confi nement in QCD is based on the following picture:
the gluon fi eld is supposed to be confi ned to tube-like
regions, the gluon strings, which tie up quarks and assure
the rapidly increasing force that keeps them together. It
is natural to expect that in pure Yang–Mills theory there
exist closed gluon strings (possibly knotted) that describe
“glueballs”. In order to trace them down, one has to look
for an appropriate “change of variables” in the Yang–Mills
Lagrangian, which allows you to fi nd an effective Lagrangian
describing the “collective gluonic excitations”. This
Lagrangian appears to be close to the Skyrme Lagrangian
with the Hopf term. This program has indeed been carried
through; the latest results have just been published. So far,
they have not attracted too much attention ...
In the last years, quantum fi eld theory is once again out
of fashion…
Well, maybe not quite. As you know, QFT is now particularly
beloved by topologists who have discovered for
themselves the importance of Feynman integrals, diagram
expansions, etc. But in what concerns fundamental
physics, QFT is now passing indeed through a diffi cult
period, due to the pressure of the largely speculative
ideas inspired by string theory. I should confess that my
attitude to fundamental physics is more conservative…
And what is your general opinion on the relation between
physics and mathematics?
I am an expert in mathematical physics. This term has
been given different meanings. Sometimes it is reduced
merely to the set of traditional mathematical methods associated
with the study of basic PDEs that arise in physical
problems. On the other hand, mathematical physics
was also considered as a synonym of theoretical physics
(as may be seen from the name of the Lucassian Chair in
Mathematics that was once occupied by Dirac and earlier
by Newton). In my own opinion, both mathematical
and theoretical physics are dealing with the same objects
and the same ultimate reality, although they are based on
different (and sometimes complementary) kinds of intuition.
My own type of intuition is rather mathematical and
I must confess that the simplifi ed arguments based on the
so-called “physical sense” are not too appealing to me.
On the other hand, really deep physical ideas can sometimes
provide a deep insight into purely mathematical
problems. Over my lifetime, my science has undergone
profound and unprecedented changes and I feel happy
to participate in its development, which brings us ever
closer to the understanding of the laws of nature.
Thank you for your story. Perhaps the readers would be
interested to learn more details ...
At several occasions I have already had an opportunity
to express my personal views on mathematical physics
and to outline the story of my own research engagement.
So I shall probably end this interview with a couple of
references. One can add to this list the short story of our
seminar in Steklov Institute, which you wrote a few years
ago.
Thank you for this suggestion. I am very glad you liked
this story and feel honoured to add this reference.
References
[1] Faddeev L.D. Modern mathematical physics: what it should be.
Mathematical physics 2000, 1–8, Imp. Coll. Press, London, 2000.
[2] Faddeev L.D. What is modern mathematical physics? Proc. Steklov
Inst. Math. 1999, no. 3 (226), 1–4.
[3] Faddeev, L.D. A mathematician’s view of the development of physics.
Les relations entre les mathématiques et la physique théorique,
73–79, Inst. Hautes Etudes Sci., Bures-sur-Yvette, 1998.
[4] Faddeev L.D. 40 years in mathematical physics. World Scientifi c
Series in 20th Century Mathematics, 2. World Scientifi c Publishing
Co., Inc., River Edge, NJ, 1995. x+471 pp.
[5] Faddeev, L. Instructive history of the quantum inverse scattering
method. KdV 95 (Amsterdam, 1995). Acta Appl. Math. 39 (1995),
no. 1–3, 69–84.
[6] M. Semenov-Tian-Shansky. Some personal historic notes on our
seminar. In: L.D. Faddeev’s Seminar on Mathematical Physics.
Advances in Math. Sciences, vol. 49 (M. Semenov-Tian-Shansky,
editor), American Mathematical Society Translations, ser. 2, v. 201,
AMS, Providence, R.I., 2000, pp. 1–8.
Michael Semenov-Tian-Shansky
[semenov@u-bourgogne.fr] graduated
from Leningrad (St. Petersburg)
University in 1972 and received his
PhD (1975) and Habilitation (1985)
from the Steklov Mathematical Institute.
After 1972 he worked at the
Leningrad (St. Petersburg) Branch
of the Steklov Mathematical Institute.
In 1992 he was appointed Professor
at the Université de Bourgogne, Dijon, France. His
research interests include representation theory of semisimple
Lie groups, integrable systems, quantum groups
and Poisson geometry. His interests outside of mathematics
include the history of science, music and hiking.
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