# Paradise Lost

**Talk at Mitchell Feigenbaum Conference**

The initiation into Landau School of Thought started with famous Theoretical Minimum Exam. This was a sequence of increasing tests of that rare mix of knowledge, durability and passion, which was necessary to be accepted and survive the School.

I was lucky to be initiated by Landau himself, just one year before the tragic automobile accident, after which Landau never recovered.My friend Sasha Polyakov and I came to Landau in spring of 1961, with zero combined life experience and infinite combined self-confidence, after some success in Mathematical Olympics where we shared the First Prize.

Landau had that childish expression at his face — a mixture of surprise, impatience and arrogance — as if he was saying: “Let us see if you are as dumb as I think you are.”

He placed us in two different offices in “Kapichnik” (Kapitza Institute for Physical Problems), gave us some math problems and started pacing between these offices, looking over our shoulders with curiosity and making nasty remarks at every line we wrote.

I felt as if my Lord came to Earth and crucified me with His own hands. One of my problems was to integrate rational function—textbooks recommend the method of variation of constants — i.e. to represent

,

with a being roots of , and obtain coefficients c as residues at x=a and similarly for coefficients of polynomial as residues at .

But I have never read those textbooks — in fact I was very bad at reading textbooks all my life — I preferred to do it myself and then ask my educated friends whether my solution was right.

I had just a few minutes to invent my own method before Dau would have run out of his short patience. I do not remember now how I solved that problem — but I remember very well that I did, because I still hear his voice in my head — “Well, you finally got a solution — invented a wheel — should have known such triviality in a first place”.

Sashka had similar experience in his cell, after which we were pulled together in Dau’s office and he looked into our eyes without smiling. “Well, kids — you passed, but you got to really work on your mathematical skills — theoretician cannot live with the skills like that. You got to know Complex Analysis, PDE, Group Theory, all nice and easy stuff — and everything else they will teach you in the University you just flow around (“obtekaite”).”

I came back home totally humiliated. My father Arkady, former student, friend and rival of Landau, was waiting for me. He was initiated in the thirties without even being tested by TheorMinimum.

“Congratulations—you passed the first Landau Exam” — he hugged me, but I pushed back. “What are you talking about? Landau was cursing us, he said that a theoretician cannot live with the technique like that, he was mocking the way we solved problems…”

“He showed his respect to you idiots by treating 15-year old boys as grown-up theoreticians! Do you really think the grown-up theoretician could live with your mathematical technique? Go and learn Group Theory like he suggested, and always take good advice regardless how insulting it may sound! … And by the way, he called me up right after you two left his office and he was very excited. He said that you were so smart, you reminded him of himself when he was young.”

Landau did more than that to help us. He wrote a letter, recommending us to be accepted in the University, which later actually helped that miracle to happen. There were two miracles, in fact. One, young kids were allowed to attend the Physical Technical Institute entrance exams without graduating from high school. Second, Jews were actually accepted into PhysTech, which was next to impossible.

The Landau letter was so good, that it was never shown to us, until just a few years ago. We lived in tough country at tough times, when kids were not supposed to be spoiled by being praised. I guess now this letter cannot spoil us anymore.

After Landau’s death Isaak Khalatnikov and few other Apostles created the Landau Institute of Theoretical Physics. It was late sixties, when the Stalin’s Fear was starting to fade, and Iron Curtain was starting to rust a little. KGB still ruled the country, but it was already playing more sophisticated games, not necessarily the deadly ones.

Isaak Khalatnikov is truly a remarkable man, whose immense contribution to Theoretical Physics is underestimated in my view. He was involved in seminal work with Landau and Abrikosov, where they first discovered the famous “zero charge” problem, and laid the ground for a modern quest for consistent filed theory. But his life achievement is creation and leadership of Landau Institute, which played such an important role in History of Physics of the 20th Century.

Khalat was a genius of political intrigue. Being married into Inner Circle of the Soviet System (his wife Valya is the daughter of a legendary Revolution hero), he used all his connections and all the means to achieve his secret goal — assemble the best brains and let them Think Freely.

On the surface, his pitch to the Party went as follows.

“The West is attacking us for anti-Semitism. The best way to counter this slander is to create an Institute, where Jews are accepted, allowed to travel abroad and generally look happy. This can be a very small Institute, by standards of Atomic Project, it will have no secret military research, it will cost you very little, but it will help “Rasryadka” (Détente). These Jews will be so happy, they will tell all their Jewish friends in the West how well they live. And if they won’t –it is after all, us who decide which one goes abroad and which one stays home. They are smart kids, they will figure out which side of the toast is buttered.”

As I put it, Khalat sold half of his soul to Devil and used the money to save another half. I truly respect him for that, now once I learned what it takes to create a startup and try to protect it against hostile world.

As many crazy plans before it, this plan really worked. Best brains were assembled in Landau Institute, they were given a chance to happily solve problems without being forced to eat political shit like the whole country and – yes, they sometimes traveled abroad and made friends in the West.

In a way the plan worked too well — we became so worldly and so free that we could no longer be controlled. And, needless to say, our friends in the West became closer to us that our curators in KGB.

To some extent, KGB played similar games in other areas of Soviet Culture as well. Soviet Union improved its image in the eyes of the West, but at the same time the whole country, starting with KGB, got spoiled and seduced by Western influence. The Cold War was lost by USSR not just because of economical burden of arms race, but also because of loss of fighting spirit as a result of careless flirting with the West.

Political games and Cold War was the least of our worries in the 60s and 70s. I joined Landau Institute in 1969, a year after its inception, after defending the PHD on the scale invariant Reggeon Field Theory. When I look back at that time, it turns out to be the time of great discoveries in Solid State Physics (main specialty of Landau Institute) as well as in the Elementary Particle Physics.

Both streams of discoveries were fed by realization of remarkable analogy between these two fields, allowing for cross-fertilization. This analogy was foreseen by great Julian Schwinger, who noted that inverse temperature in statistical mechanics is equivalent to imaginary time in quantum theory, as it follows from the comparison of partition function

in statistics and

in quantum theory.

At the time I entered the field in the sixties, the analogy was barely noted, but its full implications were not known. I learned about this analogy from remarkable Physicist Tolya Larkin, who pioneered it in USSR with Valya Vaks. The whole discipline of Euclidean Field Theory, which makes no distinction between quantum and statistical applications and uses notions from both — was yet to be born.

I guess the missing link was a realization that this “statistical” imaginary time is the same as Minkovski imaginary time of special relativity. This is the same observation that drove Steven Hawking to interpret the imaginary time of the black hole as inverse temperature. Good ideas are so scarce they come back again and again in different disguise.

Moreover, the Field Theory was pronounced dead by Heisenberg and Landau. As Landau put it “The Lagrangean Field Theory is dead and should be buried, with all the proper honors of course”. The Landau’s motivation was the “zero charge”, which, indeed, indicated inconsistency of all known field theories except one, which was in its infancy and could not yet stand and defend itself. That one was, of course, the Yang-Mills Theory, born in the fifties, and never taken seriously until the seventies, when it was finally quantized and shown to be free of “zero charge” problem. Now it is The Theory of all elementary forces except Gravity.

Heisenberg’s motivation was even more ambitious. This is an example how one great leader can block the way to the whole army by falling down at a narrow pass. He dared to go one step further from his celebrated uncertainty principle and declare that Physics must only study observable quantities. His own approach was to study so called S-Matrix — collection of transition amplitudes between various observable incoming states, such as anti-proton flying towards the hydrogen atom and the observable out-coming states, such as beams of electrons, positrons, photons and mesons.

Pretty much like medieval Scholastic Magisters were extremely inventive in defending the Church Dogmas and blocking the way to experimental science, some great minds in the sixties developed the S-Matrix dogma with great perfection and skill before it was buried down in the seventies after discovery of quarks and asymptotic freedom.

It turned out — quite unfortunately for Physics — that one could deduce a lot about S-Matrix on purely phenomenological grounds without ever asking heretical questions “what is inside”. One could not, of course, even attempt to compute proton mass or its magnetic moment or explain anything about properties of so-called resonances — short living subatomic particles. Scattering amplitudes were the only numbers to be considered and computed.

In a way, this reactionary idea was a truly revolutionary one. For the first time since Galileo the quest for the structure of matter was stopped on philosophical grounds. There is nothing inside — total nuclear democracy! Everything consists of everything else — do not ask whether there was the rabbit inside the hat — you are only allowed to compute how far it will jump and in what direction.

It is a bitter irony of History, that such a restriction on a free thought was imposed by a German Scientist and so widely accepted in Russia in the second half of 20th century. My Physics teachers Gribov and Okun were respected as liberals and free thinkers, followers of great Landau, but still they would not even talk to me about Yang-Mills Theory because it was “unobservable”.

For the whole two years 1964 to 1966 JETP refused to publish our work with Sasha Polyakov “Spontaneous Symmetry Breaking of Strong Interaction and Absence of Massless Particles” where we (correctly!) argued that vector mesons of the Yang-Mills Theory must acquire mass by absorbing zero mass Goldstone particles. We were stomped to the ground at every seminar we tried to present this work at. The most disturbing thing was that nobody would even argue with us on the subject — the mere mention of “Spontaneous Symmetry Breaking” caused healthy laughter, which ended the conversation. Independently this effect was discovered and published by Higgs and rightfully is called Higgs Phenomenon.

This was the first and very useful lesson of the danger of independent thinking. There were much more, every one of them proving the same point: “ne visovivaisya” — do not stick out.

I eventually realized that I was born dissident and maverick, never to be part of any crowd or achieve any social success. I cannot be satisfied when I reach some harmony with the world — I have to abandon everything and go further.

As for the Lagrangean Field Theory, so respectfully buried by Heisenberg and Landau, my good friend Sasha Zamolodchikov (another Sasha from Landau Institute) summarized it like that: “They buried the Lagrangean Field Theory, but forgot to drive the stake through the heart”.

Another example — the early 70s when we were developing Conformal Field Theory. We were very excited by grandiose perspectives we saw in that theory (it eventually became one of the basic ingredients of the modern Mathematical Physics), so we kept trying to discuss it with our colleagues.

The rust in the Iron Curtain already made a bunch of holes, so that we had the privilege of publishing our work in the West, and discussing with Western colleagues: Leo Kadanoff, Ken Wilson, David Gross and Mitchell Feigenbaum.

These discussions helped formulate and develop the important concept of anomalous dimensions and renormalization group, which became the basis of the modern Euclidean Field Theory.

The bold idea that critical indexes may be transcendental numbers determined by self-consistency conditions of the Field theory equations was first expressed in my work with Volodya Gribov in 1966. Our argument was that in virtue of scale invariance these equations were homogeneous, so that generically they had only zero solution, unless some dimensionless parameter such as an Eigen value was tuned to allow for a family of solutions with arbitrary scales.

Critical index was left as the only free dimensionless parameter which led us to our conjecture. Later, when Conformal Symmetry was discovered, this conjecture got some mathematical ground, when the shape of 3-point functions was fixed by symmetry so that there were literally no more parameters left and equations reduced to the set of transcendental equations for critical index.

This idea was brilliantly developed by Ken Wilson, who uncovered global meaning of Renormalization Group and made it a quantitative theory by means of an ε-expansion.

I remember long arguments with Leo Kadanoff, who made a bet with me that anomalous dimensions were rational numbers (he turned out to be right only in two dimensions, as it was later proven by Sasha Zamolodchikov). In three dimensions, which we were arguing about, he lost his bet and presented me with the bottle of Whiskey. In return I presented him with ”irrational” bottle, reshaped in a professional kiln by my father (he was a very good sculptor in addition to his other talents) and filled by his famous home made alcohol flavored by Georgian herbs.

About the same time Mitchell Feigenbaum discovered his universal irrational index in the chaos theory, using renormalization group approach. David Gross and his student Frank Wilczek discovered that renormalization group leads to asymptotic freedom in the Yang Mills Theory of Strong Interaction.

What a heroic time it was!

This free collaboration was partly result of clever plot by Khalat who organized series of Soviet-American Symposia in spirit of Détente. The First Symposium, in Moscow in 1968 was when we presented our Phase Transition Theory, based on analogy with Relativistic Quantum Field Theory, and at the Second Symposium, in Leningrad 1972 Ken Wilson presented his famous ε-expansion, providing practical way to solve this theory.

Next Symposium was held in Aspen Co, in 1976. It was such a feast! In addition to the joy of endless theoretical discussions of emerging Filed Theory of Matter we hiked in Aspen mountains, danced at the Fourth of July celebration of 200 years of USA, smoked pot with friendly hippy crowds in the streets of Aspen (you bet I inhaled and enjoyed every drag!).

This was my last International Symposium, — afterwards I was approached by KGB and was too naïve in showing my disgust when I rejected their offer – as I was told later by wise people, the right thing to do was to completely lose the face in front of them and play a coward – then they would forgive the refusal. But like I said, I was and still is a maverick and never learn to play by the rules.

Conformal Field Theory was the next step of development of the idea of anomalous dimensions, based on remarkable observation that one-dimensional scale invariance in local Euclidean Field Theory necessarily leads to a wider symmetry, with 15 parameters in our four dimensions (including translations and rotations). The generators of conformal symmetry are related to various conserved currents

for dilatations

for special conformal transformations, etc.

Here the conserved symmetric stress-energy tensor is in addition traceless in case there is scale invariance (no massive fields), which leads to conservation of both of these currents simultaneously.

This invariance leads to very specific predictions such as explicit form of 3-point correlation functions and vanishing two-point correlations between fields with different dimensions.

It happened so, that there was an International Conference in Dubna, the main topic of which was so-called scale symmetry, promoted with great fanfare by the Bogoliubov School of Thought. This scale symmetry was mostly a political slogan, good for dissertations and career moves but not for any practical applications in a world of Physics.

After the Plenary Session devoted to the Scale Symmetry, one of the Western Physicists asked the speaker: “What is the difference between Scale Symmetry and Conformal Symmetry?” Apparently, the rumors about new symmetry were already spread, so this was what KGB used to call “provocative question”.

The speaker hesitated, but the Chairman of the Session, great mathematician N.N. Bogoliubov took the microphone and said literally the following: “There is no mathematical difference, but when some young people want to use a fancy word they call it Conformal Symmetry”.

Obviously, his ignorant lieutenants misinformed him, and he did not bother to look up for himself what was the Conformal Symmetry.

I could not stand it any longer – I raised the hand to give everybody brief introduction to Conformal Symmetry (naturally, nobody invited any of us, suckers, to speak at such an important International Conference, but I was allowed into the audience). Vigilant Organizers of the Conference ignored my raised hand, the break was quickly announced, so that my indignant cry: “15 parameters!” went apparently unnoticed. (By the way, somebody told me recently that he heard that cry and wondered for years what could that mean, until he learned conformal symmetry).

Here is an interesting part – I came home and said to my father: ”Look, what a fool N.N is really is ” – then I told him the story of 15 parameters. My father laughed with me – surely he knew what Conformal Symmetry was about – then he said something remarkable: “You know, Sasha, there are two kinds of intellect. The first kind helps you to say smart things. But the second kind helps you to do smart things. N.N used to have great intellect of the first kind, but later he switched to the second kind. Do you think he cares about parameters of Conformal Group? He is involved in Big Science, where political truth is more important than scientific truth. You would do yourself some good by borrowing the second kind of intellect from N.N.”

I wish I would know how to follow this wise advice! Fortunately, in my new life there are lawyers around to zip my mouth when the smart thing would be to keep it shut.

I feel obliged to Mitchell to say something about the atmosphere in Landau Institute: how we worked, how we discussed things, how we had fun. This is what he originally asked me to talk about, so I have to say at least something.

It was more like a Gentlemen’s Club than like any other Research Institute I ever knew. We worked at home the whole week, mostly alone, sometimes meeting at someone’s apartment or talking by phone. My students Volodya Kazakov and Ivan Kostov practically lived at my small apartment when we were developing our Matrix Models of Quantum Gravity.

These models were another example how original thinking gets you in trouble. They were just too simple as a solution for Quantum Gravity Problem, which at that time was attacked without much success by sophisticated methods of String Theory. Everybody was hypnotized by celebrated words of Ed Witten: “String Theory cannot be solved by methods of 20th Century Mathematics”.

Matrix Models were plain vulgar: they started by postulating that fluctuating curved space of Quantum Gravity can be made from small identical equilateral simplexes, dynamically reconnecting with each other, and computing the partition function (sum over all these re-connections) by good old combinatorial methods.

Naturally, nobody believed our dumb solutions for almost a decade, until Knizhnik, Polyakov and Zamolodchikov heroically reproduced some of these formulas, using all the heavy weaponry of String Theory and Conformal Field Theory. We got our share of ostracism before that happened.

As I bitterly commented it: “Ed was right as always — 2D String Theory was solved by methods of the 19th Century Mathematics”.

Every Friday we were obliged to go to Chernogolovka, and sit at the General Seminar. This was the time to exchange all the news and gossips, to get your paycheck, to fill out some paperwork for trips abroad (if you were entitled for those). Also there were informal seminars in one of a few small rooms owned by Landau Institute. This was where the new Physics was born and discussed, at those shabby blackboards with rough chalk, disintegrating in your hands and leaving almost no marks on the board.

I must also say something about the style of Landau Seminar, which was quite unique. The best analogy would be the dog hunt, with the speaker as a wolf and the rest of us as dogs pack. The role of the Hunter was initially reserved for Landau, but after his death it was vacant which added to the chaos of the hunt.

The Seminar lasted forever — we had nothing better to do than to hunt each other in Quest for Truth. No limits, nor any mercy, not to mention good manners or political correctness was allowed. The resulting stress and high emotions were relieved by a good drink where it all ended.

My naïve attempt to revive this Landau style of discussions (without the drinking part of course) at my undergrad class in Princeton ended with a disaster. When I said to them at the opening Lecture:” I will try to make you think, which is quite painful. But, as we say in Russia, -‘suffering purifies’ ” — half of the class dropped with indignant Child Abuse complaints to the Dean of the Faculty.

The remaining half stayed and suffered all the way. One of them — a nice Chinese student told me later at the exams: “I forgot the shortcut you showed us at your lecture so I solved this problem hard way — but this suffering purified me, Professor!”

“Oh yes it did,” — thought I with inner smile — “Long live Landau!”

The large facade at this picture is misleading: in fact, this is the place which was given to us by Fridays for Seminars – the rest of the Friday as well as the rest of the week Institute occupied only 8-10 small rooms at a first floor of annex not visible at this picture. We had a lot of fun in Landau Institute, at least in the 70-ties when it was one family, where everyone knew all secrets of everyone else.

We did not really suffer that much from our isolation – the pre-prints came regularly by mail, everyone who traveled abroad, dumped all the latest news on the rest of us after coming back.

In the 80-ties it was all starting to get sour – nothing lasts forever in this world. Some of us turned out to be more equal than others, and our Western friends, regardless how hard they tried to help us in isolation, could do nothing with the laws of a free market – if you are not present to explain and defend your ideas they will get stolen or simply ignored and reinvented.

As Giorgio Parisi – another hero of the Statistical Field Theory put it “All the credit goes to the last one to make an important discovery”. He made quite a lot of those discoveries himself, the most important being in my view the Stochastic Quantization, which explained how the Bohm’s idea of hidden variables can be implemented by introducing extra ‘stochastic’ time and adding two forces: Quantum Friction and Gaussian Random Force to quantize the Newton’s equations:

My own feeling is that this may be the way to go beyond Quantum Theory with its postulates of linear superposition of amplitudes of histories and resulting infinities coming from the quantum sea of zero point fluctuations. What if, in fact, there actually is an extra dimension, where some natural forces are present at Planck Scale, imitating stochastic noise at observable scales? Maybe God does not play dice after all? Besides, nothing is precisely linear in our world, why the superposition principle should be an exception? Here it is an approximation, coming about after averaging over hidden variables .

We had some wild parties in Chernogolovka, with rivers of vodka, lots of dances, flirtation, sometimes ending with fistfights. We were young, talented, brazen, careless and free. I have never been so free in my life since then. When I came into the Big Real World where I happily live now I realized that one cannot live without responsibilities, but we had almost no responsibilities back then, in the golden 70-ties. Money meant very little, nobody had any money by modern standards, but all the good things of life were free back then, or so it seemed to us.

We traveled to Siberia, we hiked in the mountains, we carelessly risked our lives and thought nothing of it. I remember Volodya Zakharov, Sasha Polyakov, Tolya Larkin, Yasha and Lena Sinai and myself hiking for two or three weeks in the Fan Mountains in what is now Tajikistan. It was less dangerous back then, but still pretty wild. We talked and laughed a lot in our tents after walking whole day under hot sun in those high mountains.

I remember one episode, when Volodya Zakharov, Sasha Polyakov and me left the rest of the team and went over the pass to get some food supplies from aul (the mountain village). We made it over the pass by late night, and camped at the safe distance to the village (those aul dogs would easily tear you apart if you enter uninvited). In the morning we had to make a cautious contact with locals and ask them to give us some food in exchange for alcohol and medicine (like I said, money did not count at those times).

I was awaken by screams of Sasha Polyakov who was the first to discover that we were all covered by huge hairy earwigs (“ukhovertka”). Each measuring an inch or more, they looked quite scary, and they were biting us. After defeating this army, we started laughing and immediately wrote a poem describing the feat.

It was joint creation of Zakharov and myself, and it sounds wildly funny in Russian. Unfortunately, I do not know how to translate it in English. Not to mention the rhymes, the whole mindset was totally Russian, to be more precise it was the mindset of the Declining Soviet Empire.

Am I missing these crazy, happy, funny, exciting times? Sure I am, but they are gone to never come back. I am quite happy with the world I live in now. It also had its share of craze and excitement in the 90s during Internet revolution. I feel that there is more craze and excitement ahead of us in the near future. Alas, it hardly will be Physics again!