# Ludwig Faddeev wins 2008 Shaw Prize

In 2008 Vladimir Arnold and Ludwig Faddeev were awarded, in equal shares, the Shaw Prize, "for their widespread and influential contributions to Mathematical Physics." We present below the Biographical Note for Ludwig Faddeev and Ludwig Faddeev's autobiography written as Shaw Prize winner. We also give the Press announcement and the essay about Ludwig Faddeev's contributions.

**Biographical Note for Ludwig Faddeev**

Ludwig Faddeev, born 1934 in Leningrad (now St Petersburg), Russia, is a Director of the Euler International Mathematical Institute, Petersburg Department of Steklov Institute of Mathematics. He graduated from Leningrad State University in 1956 and became a Doctor of Physical and Mathematical Sciences in 1963. He has been a Professor of Leningrad State University since 1967 and was the President of the International Mathematical Union (1986 - 1990). Professor Faddeev is a member of the Russian Academy of Sciences, the US National Academy of Sciences and the French Academy of Sciences.

**Ludwig Faddeev's autobiography**

I was born on March 10, 1934 in Leningrad (now St Petersburg), where I have resided for most of my life. The exception was during the war from the middle of 1941 till the beginning of 1945, when I was evacuated from Leningrad and lived in several places in the East, mostly in Kazan.

Both my parents were mathematicians. My father's interests were very wide, but he considered himself an algebraist. It is acknowledged now, that he was an independent creator of the homological algebra. My mother worked on applied problems; her most known contributions are to the computational methods of linear algebra.

In high school I had many different interests including photography, radio modelling and cross-country skiing. I was good at mathematics in class, but was not an "Olympiad boy". I decided to get higher education at the Department of Physics of Leningrad University to be independent of my father, who was Professor at the Department of Mathematics.

However, mathematics caught me there. Due to the influence of academicians V A Fock and V I Smirnov the mathematical education of students of the Department of Physics was excellently organized. My tutor from the third year of undergraduate studies was Professor O A Ladyzhenskaya, a renowned specialist in partial differential equations. She did not push me into this field of classical mathematical physics. Rather, she proposed me as an additional reading papers on the quantum scattering problem. The first was a paper of N Levinson on the uniqueness in the reconstruction of potential in the radial Schrödinger operator from the phase shift. Also, I was to read and relate on the special seminar the book of K O Friedrichs "Mathematical Aspects of Quantum Field Theory". Of course I came through all traditional courses in Theoretical Physics. So I was very lucky to get an excellent education both in Theoretical Physics and Mathematics which defined my future career as a mathematical physicist with prime interest in quantum theory.

I finished my undergraduate studies in 1956 and graduated with the degree of Candidate of Sciences in 1959. During this time I was happy to marry Anna Veselova. We have two daughters and four grandchildren, already quite adult.

My first scientific paper was published in 1956, so I have been involved in active scientific work for more than 50 years. I began by treating the mathematical questions of the quantum scattering theory, both direct and inverse problems. The treatment of the quantum scattering theory for the system of three particles, based on the integral equations, now bearing my name, brought me my first success. The work was highly appreciated by the specialists in nuclear physics. The attention of mathematicians came later and now the theory of many body quantum scattering is an active subject of modern mathematical physics. However, personally I estimate higher my solution of the overdetermined many dimensional inverse problem for the Schrödinger operator with local potential. Recently, I have heard that this work gets practical applications in tomography.

The first success and defence of the Doctor of Science dissertation in 1963 allowed me to turn to Quantum Field Theory - my dream of younger years. At that time the QFT was practically forbidden in the Soviet Union because of the (pure scientific) censorship of Landau. Fortunately, living in Leningrad I was outside the scope of Moscow influence and was free to do what I wanted. After reading the Polish lecture of R Feynman and the book by A Lichnerowitz on the theory of connections, I decided to work on the problem of quantization of the Yang-Mills field. In the fall of 1966, in collaboration with a bright young colleague, Victor Popov, I came to the proper formulation of this theory in terms of the functional integral. We calculated the formal measure, on the manifold of the gauge equivalent classes of connections. Later it was said that we overplayed Feynman in his field. Our short paper, published in 1967, became popular only several years later, when the Yang-Mills field was incorporated into the unified theory of Electromagnetic and Weak Interactions by S Weinberg, A Salam.

In 1970 I was introduced by V Zakharov to the inverse scattering method of solving the nonlinear evolution equation on two dimensional space-time. Our first joint result - the Hamiltonian interpretation and complete integrability of the Korteveg - de Vries equation - defined my activity for 20 years. The main achievements here, made together with a large group of excellent students (now called "Leningrad School"), are the unravelling of the algebraic structure of quantum integrable models (the Yang-Baxter equation) and formulation of the Algebraic Bethe Ansatz. This development eventually became a base of construction of quantum groups by V Drinfeld.

It is quite invigorating for me to watch how this formalism was resurrected in the modern treatment the Yang-Mills theory.

In later years I also returned to the Yang-Mills theory, however without connection with integrability. Together with my colleague, A Niemi, I try to find an adequate picture for the particle-live excitations in this theory. We envision a possible knot-like soliton structure for them. However, the work is in its preliminary stage.

At present I live quite comfortably in St Petersburg. I enjoy contacts with my former students, living in Europe and the USA, and hope to recruit new ones from the generation more than 50 years younger than me.

9 September 2008, Hong Kong

**Ludwig Faddeev - Press Release**

Ludwig Faddeev has made many important contributions to quantum physics. Together with Victor Popov he showed the right way to quantize the famous Yang-Mills equations which underlie all contemporary work on sub-atomic physics. This led in particular to the work of 't Hooft and Veltman which was recognized by the Nobel Prize for Physics of 1999.

Faddeev also developed the quantum version of the beautiful theory of integrable systems in two dimensions which has important applications in solid state physics as well as in recent models of string theory.

In another application of the scattering theory of differential operators, Faddeev (jointly with Boris Pavlov) discovered a surprising link with number theory and the famous Riemann Hypothesis.

Mathematical Sciences Selection Committee

The Shaw Prize

10 June 2008, Hong Kong

**Ludwig Faddeev - The essay**

Mathematics and Physics have, over the centuries, had a long and close relationship. The modern era was ushered in by Galileo who said that the laws of nature were written in the language of mathematics. This was taken a giant step forward by Isaac Newton who developed and applied calculus to the study of dynamics. From that time on the whole theoretical framework of physics has been formulated in terms of differential equations.

Arnold's contributions are mainly in classical mechanics, emphasizing the geometrical aspects as developed over the centuries by Newton, Riemann and Poincare. Faddeev has been attracted more by the challenges of quantum theory and the algebraic formalism that is related to it.

Ludwig Faddeev has made many important contributions to quantum physics and to the differential equations that underpin it.

He is best known for his work with Victor Popov showing the right way to quantize the non-Abelian gauge theories which underlie all contemporary work on sub-atomic physics. Gauge symmetry is well understood in electromagnetic theory as formulated by Clerk Maxwell: it amounts to the ambiguity of the vector potential. The challenge was how to extend this simple linear situation to the non-linear case of the Yang-Mills equations relevant to particle physics. The answer lay in the introduction of what are now called "Faddeev - Popov ghosts". These have totally transformed the theory in an elegant and conceptual way, leading in due course to the work of 't Hooft and Veltman which was recognized by the Nobel Prize for Physics of 1999.

One of the surprising results of the past fifty years has been the discovery that a number of interesting non-linear partial differential equations that arise, in certain simplified physical situations, are "integrable". This means that they can be solved explicitly as a consequence of a beautiful and somewhat mysterious mathematical structure. The quantization of some of these theories is physically meaningful and Faddeev, in collaboration with many of his students, developed a quantum version of integrability, which led to the notion of quantum groups. It has had important applications in solid state physics as well as in recent work on string theory.

An important area of study in mathematical physics is that of "Scattering Theory". Here one envisages some kind of obstacle which diverts or reflects a flow of incoming waves, for example light waves, the scattering being the way incoming waves are related to outgoing waves. In practice one is frequently interested in the inverse problem, how to read information about the obstacle from the scattering data.

Peter Lax and Ralph Phillips developed this theory (in 1964), in the context of the spectral theory of linear differential operators. This was then brilliantly applied, by Faddeev and Boris Pavlov, to the geometry of the upper-half plane and the action of the modular group. They found a most surprising connection between this scattering theory and the famous (and still unsolved) Riemann hypothesis of number theory on the zeroes of the zeta function. This link between number theory and physics through subtle spectral analysis is illustrative of Faddeev's breadth of interest and insight.

While the detailed contributions of Arnold and Faddeev do not overlap, together they cover an enormous range of topics in mathematical physics. Rooted in the past, but with the incorporation of new and exciting ideas of our time, their work shows the continued vitality of mathematical physics in ways that would have gratified both Galileo and Newton. Arnold and Faddeev are worthy recipients of the Shaw Prize.

Mathematical Sciences Selection Committee

The Shaw Prize

9 September 2008, Hong Kong

Last Updated November 2019