*Interview with Alexander Zamolodchikov, the Breakthrough Prize winner.*

*On April 13, the Breakthrough Prize in Fundamental Physics 2024 was awarded in Los Angeles. The prize winners were *Alexander Zamolodchikov* (USA, Russia) and *John Cardy* (USA, UK). The award is given for “profound contributions to statistical physics and quantum field theory, with diverse and far-reaching applications in different branches of physics and mathematics.” T-invariant talked to *Alexander Zamolodchikov* about the quantum field theory and how it feels to be involved in a domain that is completely impossible to imagine and understand. We also discussed emigration and how, in the midst of Perestroika, he realized that things would not be good in Russia.*

**T-invariant: ***The things you do are beyond comprehension for most people, even those who are curious about science…*

**Alexander Zamolodchikov: **This is a normal situation. When I began to learn quantum field theory, I realized that I didn’t understand it at all by myself: not just physics, but also its mathematical machinery. It seemed to me shaky. I have decided that first I will understand the very basics, and for this purpose I will study the simplest possible systems where quantum field theory’s basic properties show up. These are two-dimensional systems, which can be thought of as reducing the three-dimensional space to a single line along which all particles move, leaving us with two dimensions: one-dimensional space and time.

Particles collide, leading to various intriguing phenomena. While this, of course, differs from what occurs in three-dimensional space, there are examples of such systems found in condensed matter physics — in certain polymers and layered crystals.

However, my interest in them wasn’t driven by these practical applications, but rather to delve into even simpler systems. In essence, I believed that by grasping these systems, I could gain a clearer understanding of the capabilities of quantum field theory: which phenomena it allows to describe, and which phenomena are inherently beyond its scope.

**T-i: ***What path did this motivate you towards?*

**AZ:** I started by seeking exact solutions of particular quantum-field problems. It became apparent that there exists an entire class of these problems that share a distinct similarity, and in such two-dimensional systems, they allow for exact solutions.

From that point onward, I became deeply intrigued by the nature of quantum field theories and how to classify them. When presented with a mathematical object, a mathematician typically seeks to classify all such objects. Similarly, my interest led me to seek the classification of quantum field theories. IGiven the complexity of categorizing real theories in three-dimensional space, I opted to develop a classification for the one-dimensional case, aiming to create a comprehensive map encompassing all conceivable quantum field theories.

**T-i: ***So, is it enough to have one-dimensional versions of these theories in order to organize them in some way?*

**AZ:** Not quite. My intention was to specifically map out the one-dimensional theories first, and then, with this map in hand, contemplate how the landscape of actual three-dimensional field theories is structured. Given their expected similarities, a map of one-dimensional field theories would prove highly beneficial in this endeavor.

**T-i: ***How does such a *map* even look like?*

**AZ:** There are “cities” on this map — these represent what are now referred to as conformal quantum field theories that look the same at all scales. Metaphorically, it’s like nothing changes when you turn the wheel of a microscope. There are roads in between them – that’s where the scale actually changes as you move along them. Interconnecting them are “roads” – here, the scale actually shifts as you traverse them. Traveling along a road signifies a shift in scale from finer to broader, where you cease to perceive minute details and start to see the large-scale structure of the theory. In this context, all “roads” are unidirectional. So, by changing the scale, on the map we “drive” from one conformal theory to another. But this is, of course, only an image.

I began working on this in the late 1970s. Fortunately, around the same time, Alexander Polyakov was advancing his work on this subject, particularly in relation to string theories. He is essentially the founder of the modern approach to strings. I immediately recognized that this was precisely what I needed. Through collaborative efforts with him and Sasha Belavin, conformal field theory — an apparatus widely employed in various realms of theoretical physics—came into fruition.

**T-i: ***With Polyakov and Belavin—this is indeed your primary work, with nearly 7 thousand citations since 1983.*

**AZ:** John Cardy, who shared the prize with me, was able to use this theory to bring many specific problems to a complete solution. One such example is the percolation problem. Imagine you have a plate made of insulating material, and you randomly drop small copper droplets onto it. Eventually, due to these droplets, conductivity emerges [across the entire plate]. Similarly, if you gradually fill a porous material with liquid, at a certain “critical” level of filling, the liquid starts to flow through the material. At the onset of flow, certain universal characteristics common to all such systems manifest. John adeptly employed conformal theories to obtain the exact solutions to such problems.

Why God devised such a link — perhaps, nobody knows.

AZ: In general, fields are dynamic systems distributed throughout space, so that a small part of them exists at each point in space, and they interact in a coordinated manner. We are all familiar with electromagnetic fields, gravitational fields.

However, there are situations where the overall configuration of the field is unknown. Instead there is a probability distribution that it will end up in one configuration or another. This is referred to as a random field. The probability distribution could be quantum or thermal. Mathematically, these concepts are closely related, and the intuition gained from thermal problems is highly beneficial in quantum problems, and vice versa.

The thermal probability is expressed through a distribution function. In quantum mechanics, a similar role is played by the wave function, which evolves in time according to the time evolution operator. If you put imaginary time (expressed by an imaginary number containing the square root of –1) in the time evolution operator, this operator is expressed by the same formula as the distribution function for the equilibrium classical field.

**T-i:*** So, a random field is a field whose configuration is determined by some probability. However, quantum probability is expressed using complex numbers, which consist of a real part (a regular number) and an imaginary part — a number multiplied by the square root of –1. To eliminate the imaginary part, time is required, which is also expressed by imaginary quantity. It’s completely baffling…*

**AZ:** Do not ask me what imaginary time is — I do not know. It’s a formal mathematical concept. But if you replace time in the formula with this imaginary time, the operator becomes real. The imaginary numbers (containing roots of –1) disappear. Why God devised such a link — perhaps nobody knows.

Here it is relevant to recall Landau’s famous statement about quantum mechanics. He said that the most remarkable aspect of it is its ability to describe phenomena that are beyond our imagination.

**T-i:*** What is it like to devote your entire life to things beyond imagination?*

AZ: When Newton discovered his law of universal gravitation, well, not discovered, but assumed its existence, he tried to understand how the planets move around the Sun. At first he was based on physical intuition, which told him that the planets would necessarily fall into the Sun. But they don’t. Newton had to invent a mathematical analysis because he had to understand a physical phenomenon that from the point of view of intuitive physics was not at all obvious.

Over time, one comes to understand that there are physical connections, and then there are, essentially, physico-mathematical ones. It’s when you start to see not just the physical object itself, but rather the equations that depict it.

In school, I naturally visualized physics: something pushes something else, a body speeds up, and so forth. In university, as we delved into analytical dynamics, it became clear that we were dealing with complex equations. That’s when I realized that everyday physical intuition could sometimes assist, and sometimes hinder, understanding.

In the late 1980s, I had a clear sense that things were going to be bad.

**T-i:*** You began your journey into physics alongside your late **twin brother, **Alexei. Could you please share who influenced whom? Did you compete with each other, or was it more collaborative? It’s quite rare for twins to both achieve eminence in the same field of science.*

**AZ:** This topic is both sad and very emotional for me, because we were really close since childhood. We always played together, and then we went to Fiztech together. Of course, nobody specially trained us, although my father was a chief engineer in the laboratory at JINR in Dubna. He understood physics well in many aspects and explained some things to us. Anyway, I think he helped us see that physics is worth devoting our life to it.

At some point, I decided to pursue one-dimensional field theory. Alyosha immediately agreed that it was what he wanted too. “Yes,” he said, “this is probably the most interesting thing right now.” And so, we started studying together.

It was an absolutely incredible time because we had such a deep understanding of each other. It felt like this purely magical situation where there was no need for lengthy explanations. I’d just listen to what Alyosha was saying, and I’d instantly grasp his meaning. And vice versa — he understood me flawlessly. It was like this perfect resonance between us.

No, of course, we weren’t thinking with one brain. I often find myself recalling a moment when Alyosha mentioned he knew how to solve a particular problem. Back then, I didn’t bother to ask him because I was preoccupied with something else. And now, it’s too late. I’m attempting to tackle this problem, but I’m not having much luck with it.

Alyosha and I had a well-known paper around 1976. It was about a whole class of solvable models for factorized S-matrix theory.

While we were in Moscow, we met almost daily, if not every day, and worked together. Then, as we dispersed across the Americas and Paris, our meetings became less frequent. However, we still saw each other regularly. He would come to visit me, and I would visit him. And we kept working together, producing more papers in collaboration. One of them is particularly significant — a paper on Liouville theory. I lost him in 2007, and of course, I miss him dearly.

**T-i: ***In the past two years, many scientists have left Russia during the wave of emigration. Your circumstances, however, were somewhat different. How did you come to the decision to leave, and how did you end up in the United States? Did you consider going somewhere together with your brother, or was it apparent from the start that you would pursue opportunities at different institutions, in different countries?*

**AZ:** Starting at some point in the late 1980s, I had a clear sense that things were going to be bad. You see, on one hand, I too hoped for democratization and believed it would lead to something positive. But I realized very soon that it would not. There were certain events that made this clear to me.

**T-i: ***What were those events?*

**AZ:** On one hand, everyone was saying, “There must be legality; since we are transitioning to something new, everything must adhere to the law.” On the other hand, there was an attempt to elect [the physicist and dissident Andrei] Sakharov to the Supreme Soviet [of USSR], but the Academy [of Sciences of USSR] refused to nominate him. Consequently, according to the law, he couldn’t become a deputy. Nonetheless, Sakharov’s supporters argued, “Yes, the law is the law, but this is an exceptional situation where the law is not applicable. We will try to approach things differently.”

When someone claims that the law is inapplicable in a certain exceptional situation, it sets a precedent for future exceptions. This was one of several such moments that prompted me to realize many things.

**T-i: ***This is a thought-provoking observation, because many are currently trying to understand what went wrong in Russia, at what point. And you offer a rather nuanced perspective. After all, the election of Sakharov to the Supreme Soviet seems like a positive event…*

**AZ:** Certainly, I fully supported Sakharov’s election to the Supreme Soviet. However, I felt that if exceptions to the laws were made right from the start, merely to advance someone somewhere, it would lead to a dead end.

**T-i: ***That was when you decided to emigrate?*

**AZ: **Everything was settled by a very tempting job offer I received from America. At that time, I didn’t make a decision about leaving for good or even for a long time. I thought: well, I would go for a year, two, or three — until it became clear what was happening in Russia. And then I would see whether to return or not.

Actually, I wanted to return, of course. To come back and live in my own country. But, you know, life happens, children go to school, and all those sorts of things. And even when the 1990s ended, returning wasn’t even a consideration. Alyosha went to France in much the same way.

**T-i: ***Is it true that your brother, at some point, actually decided to return to Russia?*

**АЗ**: He returned temporarily. He worked at the National Center for Scientific Research (CNRS — France’s leading state scientific institution — Ed.). In the 2000s, they established a department in Moscow. Various French researchers came there to interact with Russians and give lectures. At some point, Alyosha agreed to this arrangement. I believe he had a one-year contract for 2007.

I also secured a sabbatical for 2007 — in American (and not only) universities, there’s an opportunity to pause teaching for a year once every seven years to travel elsewhere — and I went to Russia too.

I was really looking forward to working together for the whole year. But it turned out that Alyosha started in September, and by November, he was gone. Anyway, I spent the whole sabbatical in Moscow, but I was working on my own.

**T-i:*** What were your impressions of Russian science upon your return after a 15-year absence? How had it evolved?*

**AZ:** There was nothing particularly surprising for me, as I was in nearly constant communication with my colleagues who stayed in Russia. I had a general idea of what was happening at the Landau Institute and in Dubna. Yes, there were significant funding challenges there, with a funding system quite different from that in America. However, that didn’t affect me much anymore. So even before taking the sabbatical, I had a rough understanding of how it all worked.

**T-i:*** What are your thoughts on the future trajectory of Russian science and what is happening to it now amidst the backdrop of war?*

**AZ:** Right now, I can’t make any predictions about the future at all. I’m even refraining from making them for myself, given how rapidly everything is evolving! Here’s how I see it: the world is like a cart that has careening off a mountain and is now hurtling forward, its battered wheels rattling as it gains speed. It’s unclear where it will all end up. I’m not particularly optimistic about either Russia or the United States. For a while, I thought things would be fine in Israel. But, as you can see, even there, there’s war.

**T-i:*** Now a great many scientists have left Russia and Ukraine. Have you met any of them in American universities?*

**AZ:** No, you know, I don’t want to say that there are none, but at the State University of New York, I didn’t meet any new Russian scientists. There were people who came to study before the war. And they are now flying to Russia a very long way via Istanbul. In general, communications between countries have become very complicated.

Now we are going to have a conference devoted to the 40th anniversary of our work with Belavin and Polyakov on conformal field theory. Absolutely, I will be there, and Polyakov too — he’s also in America. It is not clear how to bring Belavin, because he is in Russia, based in Chernogolovka.

**T-I: ***This is a great illustration because Belavin is among those who, in February 2022, signed an open letter of scientists denouncing Russia’s invasion of Ukraine. Despite this, he chose to stay in Russia. Even so, it’s surprising that someone like him—who not only has made significant contributions to science but also showed integrity and non-conformity in this situation—can’t come to America.*

**AZ:** Yeah, Americans are just not letting anyone with Russian passports in these days. Somehow, our students manage to get in. We were hoping to bring Belavin through a third country, but no luck. He’ll be joining remotely, unfortunately.

If you ask, “What’s the physical picture?” — lots of theoretical physicists simply won’t grasp what that means.

**T-i:*** What are you currently researching?*

**AZ:** I recently published a paper that got a lot of attention and means a lot to me, at least. It’s about that map of one-dimensional theories again, remember: cities, roads between them. And my work showed that it’s very natural to see this map as part of a much larger one, where the dominant part isn’t quantum field theories in the usual sense, but some other mathematical constructs.

You know, on old maps, unexplored territories were often marked with “Here be dragons.” That’s how I stumbled upon a region inhabited by dragons. What kind of dragons they are, I have no idea. But to me, it’s a clear sign that there’s a vast area of science closely tied to quantum field theory, yet distinct from it. In a way, it’s an expansion, a broader view.

**T-i:*** Are there any physical objects behind these mathematical structures?*

**AZ:** Well, I just don’t know what kind of physical objects these are. It’s always the same: as we delve into more abstract realms, the link to straightforward physical understanding becomes increasingly elusive. I still somewhat delude myself into believing that I maintain a grasp on the physical aspects. But now there’s a whole generation of theoretical physicists who work exclusively in these abstract realms, and they wouldn’t even comprehend the question if you asked them, “What’s the physical picture here?”

**T-i:** *The question isn’t even about what kind of objects they are, but whether they exist at all. You’ve found mathematical objects there. But are there any physical ones behind them?*

**AZ:** I hope they are. Well, either we’ll discover whether they’re there or not at some point in time.

**T-i: ***You mentioned that you found physical objects for some of your one-dimensional space systems.*

**AZ:** Yeah. It’s always satisfying when your findings turn out to have connections to actual experiments. It turned out that my research predicted a spectrum of excitations in a certain crystal. And this spectrum happened to be linked to one of the most complex groups in mathematics, the so-called E-8 group. It started as a purely mathematical concept, but it ended up directly describing the excitations in this system.

**T-i:*** And finally, a more tangible question. This prize is quite substantial financially. Do you have any plans for how you’ll use it?*

**AZ:** Firstly, half of it will be taken away by taxes. And secondly, despite living here for a long time, I still haven’t paid off my house. So, I think I’ll use most of it to pay off the mortgage on my condo in Long Island.

*Interviewer:* **Nikita Aronov**

Alexander Zamolodchikov, Nikita Aronov 16.04.2024