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The Evil Spirit Equation

Aleksandr Mekhanik

Academician Ludwig Faddeyev believes mathematical rigor today is more important than physical intuition, and that mathematics will eventually help us build a “Universal Theory of Everything”

Picture by Konstantin Batynkov

The long-time discussion of whether mathematical rigor or physical meaning – a correctly resolved equation or intuitive understanding of a natural phenomenon – is more important in scientific research lasted all through the twentieth century. At a certain stage, it seemed, however, that physicists were gaining the upper hand in the battle: Albert Einstein as the creator of the special and general theories of relativity is more famous among the general public than Henri Poincare or David Gilbert; Erwin Schrödinger is more popular than André Weil; Lev Landau is more famous than Nikolai Bogoliubov. However, the situation has been changing over the past decades: it turned out that successful mathematical devices are not simple technical solutions, but that they have deep physical meaning. Mathematical intuition may become more important in resolving increasingly complicated physical problems. The situation has palpably irritated many a physicist. At the same time, in the second half of the twentieth century, many scientists appeared who could not be called either pure physicists or pure mathematicians. Ludwig Faddeyev is one of them. Having graduated from the physics department of the Leningrad University, he received world recognition as the person who, in collaboration with his pupil Viktor Popov, resolved most of the difficult mathematical problems of the Young-Mills theory, which later on became the basis of the “superstrings” theory. The effects they revealed in the course of their research entered all modern theoretical physics textbooks as the “Faddeyev-Popov ghosts”. Faddeyev is convinced that mathematics will make it possible to create a “universal theory of everything” and “close” physics, the same way physics had resolved all chemistry’s theoretical problems, thus “closing” chemistry. It is a radical view, but rather authoritative too.

Faddeyev’s works have received international acclaim. Faddeyev, a member of the Russian Academy of Sciences since 1976, leaves his colleagues, the Academy of Sciences Presidium members, far behind, from the citation index standpoint. Author of over two hundred research papers and five monographs, he has concentrated his main scientific interests on quantum mechanics and quantum field theory. Faddeyev has been awarded several USSR and Russian National Prizes, orders and decorations and the Max Planck medal; he is a foreign member of the US, French, and Swedish Academies. Additionally, Faddeyev is the founder and long-time director of the Euler International Mathematical Institute in St Petersburg, and the Secretary Academician of the Russian Academy of Sciences’ Mathematical Division.

Expert Magazine: The well-known American physicist Eugene Wigner called one of his papers “The Incredible Effectiveness of Mathematics in Natural Sciences”. Don’t you have a feeling that mathematics nowadays is generating “monsters” that humankind is incapable of comprehending? As Wigner wrote, quantum theory equations can be compared to an oracle’s prophecies; nevertheless, human mind cannot comprehend oracular predictions.

Ludwig Faddeyev: I am familiar with the paper, and believe its title reflects a sort of physicist’s irritation over the necessity to use mathematics.

EM: You call it irritation?

LF: The thing is that the mathematicians and the physicists’ communities have long been separated, starting in the twentieth century. Quantum mechanics was invented in the twenties. Mathematicians knew the mathematical aspects of the theory, but physicists were not familiar with them. Heisenberg was a genius, but he did not know the theory of matrices. Dirac, the English genius, created mathematics of his own; he even used his own symbols in it. However, there were also physicists who could understand quantum mechanics from the standpoint of familiar mathematics. Herman Weil was one of them. He explained to Schrödinger how to resolve the Schrödinger equation. However, Weil’s role has not been reflected in textbooks on the history of physics until recently. The fact demonstrates a kind of mutual hostility between physicists and mathematicians.

Gilbert once said that physics was too difficult for physicists, so physicists naturally got irritated. Landau used to say, “Who do mathematicians think they are? They are just athletes.” When Nikolai Bogoliubov, the great mathematician, developed his theory of super fluidity (now the Bose-gas theory), Landau did not at first trust its mathematical foundation. A textbook even said the theory was a mistake. It turned out the theory was right, but the offense had already been taken. The contradictions between physicists and mathematicians leveled out during my generation’s time, and nowadays they are nonexistent. In my view, the more physics uses mathematics, the more fundamental it becomes. Mathematics is the only adequate language for the quantum theory; you cannot formulate the theory without mathematics. We are offered the picture of the hydrogen atom as an electron rotating round a nucleus, but that is naïve. At the same time, the Schrödinger External Field Equation is the only consistent theory for the electron, a single electron. That is, pure mathematics.

One may say mathematics is the sixth sense. An electron has neither color nor smell. You can only sense it with the help of formulas. Then what is the difference between a mathematician and a physicist? The difference lies in the degree of intuition. Physical meaning plays the most important role for physicists. With Landau, it used to be very definite. Niels Bohr thought so too. Nevertheless, I predict that mathematical elegance, strictness – not the kind of strictness you use when formally proving a theorem, but logical rigor – will constitute the future of fundamental science. 

EM: Then how do you harmonize physical intuition and mathematical rigor? Some physicists explain their distrust of mathematical complexities by the latter’s being beyond physical intuition.

LF: Our physical intuition should be expanded, and new mathematical considerations ought to be included into it; otherwise, we won’t be able to move onto the next stage of physics. Nowadays, both physicists and mathematicians recognize my works. When I was younger, I used to change my language depending on who I was talking to. I talked to physicists the language they wanted to hear. However, I have decided I am on my own and I will be using the language I consider appropriate.

EM: That is, the language of mathematics?

LF: Exactly. I can explain my view using microscopic physics as an example. In the nineteenth century, atoms and molecules were thought to be the elementary components of matter. In the early twentieth century, scientists realized that atoms had nuclei, which, in their turn, consisted of nucleons. The picture is supported by experiments. However, we now believe nucleons consist of quarks, which cannot be seen as free particles. It is a purely mathematical model. So in describing it, you cannot use a language other than the language of mathematics.

EM: We read on a students’ forum: “The Maxwell’s Demon is a loser; the Faddeyev-Popov Ghost rules!” What are these ghosts all about?

LF: I like this one. Physicists are fond of giving unusual names to new phenomena. There are “charmed” quarks, “strange” quarks and their charges are termed “colors”. When Viktor Popov and I were developing the Young-Mills field theory, we introduced new variables that had not existed before, and we called them “ghosts”. The term survived. Our mathematical trick allowed further theoretical calculations. A funny story happened in the eighties. Young invited me to China then. (Although he worked full time in the US at that time, the Chinese, unlike Soviets, did not consider their citizens traitors if they worked in foreign countries. On the contrary, they treated them as their ambassadors.) The word “ghost” is more like “spirit” in Chinese, and, naturally, there are good spirits and evil spirits. So in one of their newspapers they wrote “Faddeyev concluded a contract with evil spirits and thus became famous”; a doctor Faustus of sorts.

EM: Do the superstring and supersymmetry theories continue your own theoretical studies?

LF: They develop the ideas of many scientists, but I am not actively involved in it. I occupy a conservative position here. I remember my teacher, academician Fock, saying that you cannot extrapolate a theory onto the whole world. I think string and supersymmetry were invented a priori. The developers of the theory have experienced many high and low points over the past twenty years. They are now experiencing difficult times. There are even books and TV interviews where they are being criticized for the absence of experimental prediction. I am skeptical about the criticism too, because I see the theory as an interesting option, although there are other ways of development too. In the US, where Bolshevism is stronger than in this country, people involved in the development of the superstring theory used to be so strong at one time that they prevented young scientists from exploring anything else. The supporters of the theory promise to unite cosmology and the quantum theory. However, what are cosmologists researching nowadays? Dark matter and dark energy. As if we are all inside a soup of some energy which does not interact with us. It is all quite naïve, and they are not using any superstrings. They’d like to, but… I have a pupil, Irina Arefyeva; she allegedly explains the dark matter with the help of strings. I don’t know; everything seems to me so speculative. I imagine a future unified theory will explain the universe in a different way. Nevertheless, no one so far has constructed a real theory and connected it with the gravitation theory by what we call the Einstein theory of quantum gravitation.

The generalization I count on will not stem from the Einstein line, because he was skeptical about quantum mechanics. Einstein created what we call the classical theory of gravitation or the general theory of relativity, although the theory’s main equations were written for the first time not by Einstein but by Gilbert, the great mathematician. Despite the difficulties in creating a unified theory, I am optimistic about creating a fundamental theory that will explain all micro and macro phenomena. You know, Lenin once wrote that the electron is as inexhaustible as the atom is, meaning that cognition is endless. I believe there is an end. As they explained it to me recently, such a view is called reductionism, and it is criticized severely. Let them think I am a reductionist. My verdict is: the basics of physics will finally be understood sometime.

EM: How do you assess Russia’s place in modern fundamental science?

LF: If we are to talk about Russian science in the world – not about its place inside the country – it is extremely influential even now. However, a great number of our compatriots work abroad. I will give you an example. Once every four years, they hold a world mathematics congress. Usually it consists of twenty invited reports and about a hundred reports within sections. Our share in Soviet times, although they used to subdue us somewhat, was three plenary session reports and twenty to twenty-five reports delivered at section meetings, which was one sixth to one seventh of the whole congress, This year, the congress was held in Madrid, and the situation was exactly the same; there were even four Russian plenary session reports scheduled. And that is how it would have been, had Perelman not refused. And twenty-two sectional reports. However, only two speakers actually came from Russia; the other Russian reporters came from outside the country.

We can notice some recovery nowadays. Financing is a little better than it used to be in 1995, when everybody fled. “Fled” is what I call it. When freedom came, they started fleeing. I am for freedom, but the market is saturated in Europe. It is much more difficult to find a job there, even for those who have settled down: they have problems with their career growth. On the other hand, we can see some sort of perspective here. We hope Russian authorities will finally realize that we are needed here.

EM: There is much talk nowadays that the Russian school traditionally used to give a fundamental education, but that it is unnecessary, excessive, and we are only wasting money and effort on something that will not be used.

LF: I have read something like that in a Russian newspaper. I can even quote. “The postindustrial society produces new requirements for people and their skills; they have to be flexible and creative. Many people do not need fundamental knowledge; they need technical skills.” First of all, let me ask you, what is fundamental and what is non-fundamental? Give me a definition. Secondly, in my view, there is no such thing as excessive knowledge, exactly because no one knows what a person will need in life – if not today, then tomorrow. A person who is capable of switching from one technology to another will be needed not only in science, but in production too. To achieve that, one should develop his or her scope and set of mind. This is only achievable with the help of knowledge. Knowledge helps develop a mind on the basis of an example. Later on, if there is a need, you can learn from other examples by yourself. The human brain is known to develop if you study a lot. It doesn’t matter much what you study – even  English history, if you wish. Only a knowledgeable person can be free; instead, they just suggest that we turn people into slaves. The article is full of contradictions, too. In the beginning, the author writes: “The Ministry of Science and Education officials admit that Russia is still lacking in advancement in education and cannot be compared to the leaders: the USA, the UK, etc.” And later on, literally in the same paragraph: “Russian students enjoy popularity in the West: they are quick, smart, and erudite”. Then let me ask you, why are they quick, smart and erudite? Oh, but because they received fundamental education. This is why it is extremely important to preserve fundamental education in its full capacity. If you need technicians, then train technicians. Let there be technical secondary schools. What do we need three thousand universities for? That is also an incorrect solution. We used to have technical schools: they were called institutes. Why turn all technical institutes into universities?

There are practically no technical institutes in Europe, while there are plenty of universities, just like in Russia, because it is their social policy. In France, they send all youngsters to universities because of the high unemployment rate among the younger generation. All of my pupils – and there are about fifteen full-time professors among them working abroad – mourn that the students are idiots who don’t want to do anything. They keep them because they have to keep them busy for five years somehow. At the same time, interest in science always existed in Russia. Competition to enroll in mathematics departments is on the rise once again, as it used to be. Nevertheless, when our graduates join corporations, they receive three thousand dollars a month, and they won’t receive that much in an Academy institute. The situation will certainly not last long. I keep shouting everywhere, “What’s going on, guys? Why don’t you consider Germany’s example? Their fundamental scientific progress had a break of twelve years, from 1933 to 1945, but they haven’t recovered until now. Traditions have been destroyed.” Many a colleague of mine is convinced that Europe’s problem lies in the destruction of the centuries-proven tradition of the so-called Gumboldtian or exploratory-type education.  

EM: What do you think of the reform of the Russian Academy of Sciences?

LF: I have a distinct feeling it is just another way of redistributing property. However, the unanimous vote at the Academy’s general assembly demonstrated both our solidarity and the difficulties awaiting the reformers. Khrushchev also wanted to reform the Academy, but finally gave it up, saying that it is like shearing a pig: too little wool and too much squeak. I hope everything will end up in a similar way. In the meantime, we are seeing outrageous sabotage in all governmental circles. The Academy sends them a document, and it lies there without being touched. Our Presidium recently discussed the situation with land taxes which have to be paid by the federal government on behalf of the Academy’s institutes. They just don’t do that. They keep returning our documents under different pretexts as if to say, you have voted against our reforms; now take that! They have already calculated penalty fees for our institute and will go to court soon. When did they invent this strange system of taking taxes first and returning the money afterwards?

EM: Is the well-known Grigori Perelman ( a Russian mathematician who in August 2006  was awarded the Fields Medal for "his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow". However, he declined to accept the award.) still working?

LF: Mathematicians are difficult people. You look at me and probably say: here we have a regular person. However, I graduated from the physics department. My father used to be a mathematics department professor. That is why I did not go there. Perelman is a difficult person. They say in the press that we bullied and fired him. That is a disgusting lie. He refused to defend his doctoral thesis. He was the only principal employee with the degree of a candidate of science. He did not yield any results for seven years. We were patient because we knew he was a remarkable scientist. At one moment, he decided to quit and said he would not go in for mathematics any longer, and left the institute. It was a kind of a nervous breakdown. When he was awarded the Fields medal, the highest decoration, awarded by the International Mathematical Union, I arranged a meeting for him with the IMU president, so as to persuade him receive the medal. It was against the rules: no laureate should know about the award beforehand. Perelman refused categorically.

EM: Not long ago you said Faraday and Maxwell had brought fundamental science to the breakeven level once and for all.

LF: I said it when they invited me to the Davos economic forum in 1996. I addressed the oligarchs and said, “Listen, guys, science costs nothing because Faraday and Maxwell had brought fundamental science to the breakeven level. You are using its fruit for free.” It certainly was a figure of speech, but there are two important circumstances in it. On the one hand, society ought to support fundamental science. You never know what will be of use afterwards. How long has electricity been in existence? Everyone thinks it has always been there. In fact, only several decades passed since Faraday was turning his frame inside the magnetic field before Siemens produced the dynamo machine. On the other hand, people engaged in fundamental science must be very frank in relation to society. That is, not to let money, allocated to fundamental research, be wasted. We certainly know of a great number of charlatans who call themselves fundamental scientists. Control should definitely be there. But then there is the question: who should be exercising control? The answer is, unfortunately, only one: only we, the scientists. Nobody else can; otherwise there would be much more corruption. We, the scientists, should rebuff dishonest people from our community.

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