Albert Schwarz

Zierler:

OK, this is David Zierler, Oral Historian for the American Institute of Physics. It is June 2, 2021. It is my great pleasure to be here with Professor Albert Schwarz. Albert, it's great to see you. Thank you for joining me.

Schwarz:

Thank you for inviting me. I'm very pleased to have the opportunity to talk about my work and my life.

Zierler:

To start, would you please tell me your title and institutional affiliation?

Schwarz:

The title is Distinguished Professor of Mathematics, University of California at Davis. Or more completely, Distinguished Professor Emeritus.

Zierler:

What year did you retire?

Schwarz:

I retired when I was eighty-five in 2019.

Zierler:

What are you working on right now? What is interesting to you in the field?

Schwarz:

I'm working on several things. But all of them are related to the notion of inclusive scattering matrix and what I call the geometric approach to quantum theory. I think that I managed to give a very general approach to quantum theory, more general than the standard approach. This is related to my very first physics paper, which was called A New Formulation of the Quantum Theory that was published in '67, more than fifty years ago. In more detail, in this very, very old paper, I suggested a formulation of quantum theory that avoids using Hilbert space. The states are represented by functionals that I call L-functionals. This is related to algebraic approach to quantum theory. In some sense, this is an algebraic approach applied to Weyl algebras.

So, I worked in this direction with my students in this time, Tyupkin and Fateev. But at some point later, I switched to a very new and very exciting field. At first, I applied topology to physics, and after that, I applied physics to topology. The L-functionals were almost forgotten. Not quite because later this approach was rediscovered by Umezawa, but in a completely different form. I believe that Umezawa’s form is less transparent. Umezawa gave the name thermo-field dynamics to his approach, he and other people applied it to many problems of equilibrium and non-equilibrium statistical physics. It is closely related to what is called Keldysh formalism in statistical physics. Generalized Green functions appear in all of these approaches.

Tyupkin analyzed the scattering matrix in my approach and found that it is related to inclusive cross sections. What did I do quite recently? I defined the notion of inclusive scattering matrix. One can say that this is the scattering matrix in the formalism of L functionals. I proved that inclusive scattering matrix can be calculated in terms of generalized Green functions on shell; this is an analog of the famous LSZ formula. I am studying the inclusive scattering matrix. Now, about geometric approach. This is an approach where you have as a starting point, a set of states, not only pure states, but also mixed states. Together the states form a convex set. I have found that in this approach, based on space of states, you can do more or less everything that you can do in quantum mechanics. In particular, you can derive the probabilities from first principles without any additional axioms. This approach is more general than the usual quantum mechanics. You can construct models with any prescribed group of symmetries. But still, I do not have really interesting new models. I hope that they will come.

Zierler:

I would like to ask you a question about names. How do you understand the difference between mathematical physics and physical mathematics?

Schwarz:

I do not know what mathematical physics is. Probably, nobody knows. Because everything understands this in his own way. But for physical mathematics, I'm using the definition that was suggested in the journal, Nuclear Physics: physical mathematics is mathematics that is interesting to physicists. I say that I'm working in physical mathematics, not in mathematical physics.

Zierler:

Let's go all the way back to the beginning. Tell me about your parents.

Schwarz:

My father was a professor at the medical school in Kazan University. He was the department chair there. And my mother was an assistant professor at the same medical school.

Zierler:

Where are your parents from?

Schwarz:

My father was born in a small town in Lithuania, then a part of the Russian empire, and my mother was born in Russia, in Pinsk. Later, Pinsk became a part of Poland. After that, it was a part of the Soviet Union, now it is a part of Belarus. My mother was always very upset when she was forced to write that she was born in Poland. And she never visited Poland.

Zierler:

What was your family's experience like during the darkest times of Stalin?

Schwarz:

This was a very bad experience. My father supported the October Revolution. He joined the Communist Party in 1925. I should say that this was not surprising, really, because the October Revolution had very noble slogans. Particularly, decolonization, the right of nations for self-determination, the equality of all ethnicities, and gender equality. Education was emphasized, science was supported, abortions were allowed, homosexuality was decriminalized. But later, it became clear that these slogans were only slogans. They contradicted reality. The equality of all nations led to deportations of entire nations and government antisemitism. But this was later.

My father was expelled from the Party in 1936. He was arrested in the beginning of '37. And he was sentenced to ten years of prison without the right to write letters. Later, it was discovered that that was a euphemism for the death penalty. In reality, we found out several decades later that he was executed the same day he was sentenced in '37. And we did not know anything about his fate for a long time. My mother was also arrested. When she asked the interrogator what had happened with her husband, she got an answer “He does not confess. He is stubborn as a bull.” That's about my father. My mother was sentenced to eight years of prison. She spent a couple of years in the prison camp for wives of enemies of the people. My father was considered an enemy of the people.

So, she was definitely guilty. Her guilt was completely proven by the marriage certificate. It wasn't necessary to have anything else. At the time when my mother and father were exonerated, my mother had seen her dossier on the desk of the guy who talked to her, and it was very, very slim, a couple of pages. But she was in this prison camp only for a couple of years. Later, she was sent to Kolyma with the other wives of enemies of the people. Then she learned that her sentence was commuted. The People's Commissar [Minister] of Internal Affairs, Ezhov, was dismissed and executed. The new minister, Beria, commuted some sentences. My mother was very lucky, her prison sentence was replaced with exile to Kazakhstan for the same eight years. ten years after her arrest she was able to leave Kazakhstan with all of us.

We settled in Ivanovo, a city three hundred kilometers from Moscow. She chose this city because one of her friends from Kazanwas working there. This was probably ’48. But in ’53, many doctors that treated the Kremlin elite were arrested, they were accused of killing and trying to kill several party and government officials. One of the arrested was Professor Vovsi. A newspaper in Ivanovo published an article asserting that this friend of my mother that invited her to Ivanovo was a friend of Professor Vovsi, he said to my mother that he would be happy to be Vovsi’s friend, but he was not. This meant that my mother’s friend should await arrest. My mother was sure that after that, she would be arrested and exiled again. She did not have, at that moment, any winter coat. She purchased an ugly, but warm winter coat because she knew it was quite necessary to have one on the way to exile and in exile. But when Stalin died, the physicians were released. After that, the immediate fear disappeared.

Zierler:

With all of the tumult in your life, from Stalin to the war, when did you start to receive a formal education?

Schwarz:

Both my mother and father were arrested, and my grandmother was also arrested when my mother was arrested. Her guilt was that she was the mother-in-law of an enemy of the people. These guys decided that it wasn't a severe crime, and after a couple of months of prison, she was free. She found me in an orphanage. I was placed in an orphanage under a different name. The orphanage wasn't in Kazan, but it was nearby, so it was possible to find me under the name Alec Solomonov. After that, at first, we lived in Kazan in the same apartment. When my mother was exiled to Kazakhstan, we visited her for several months, then we came back to Kazan. When the war started, my grandmother decided it was better to go to Kazakhstan, and we left Kazan for Kazakhstan. This was '41. After that, the three of us lived together. I can't thank Stalin for a happy childhood, but I can thank him for a childhood where I wasn't hungry. Because in some sense, my mother was privileged. First of all, she was a doctor, and doctors are needed everywhere. Her medical profession saved her in prison camp. I don't think she was very well-adjusted to prison life. But doctors are needed also in prison camp.

Let me tell you a story about my mother. When she came to Kolyma, she met there her friend from Kazan time, Yevgenia Ginzburg. And twenty years later, Yevgenia Ginzburg wrote a book about her life in the gulag. It's really a good book, one of the best books about this time. It is translated in English under the name Journey into the Whirlwind. And in this book, Ginzburg tells about her meeting with my mother. She also says that due to her medical profession, my mother was in a privileged position, so much richer than Ginzburg. My mother, impressed by the extreme poverty of Ginzburg, gave her a woolen jacket that was almost immediately confiscated by the leader of the team. But still, this gift saved Ginzburg from hard labor. She was sent to do very light labor. So that saved her at least for several months.

The medical profession was really, really helpful in my mother’s life. She was a very good doctor of internal medicine and hematology. She was the deputy director of the main hospital in the city in which we lived called Kzyl-Orda. In some sense, I lived a privileged childhood. In some sense. I would not recommend such a privilege, but yes, it was a normal childhood. No hunger.

Zierler:

When did it first become obvious, either to you, your mother, or your teachers, that you had special mathematical ability?

Schwarz:

I don't know. The only thing I can say is, probably age ten or eleven, I became nearsighted. So, I got glasses. And with the glasses, I got the nickname Professor. So, I was obliged to become a professor, having this nickname (laughter). But about these magical abilities, this, I do not know. But I do remember when this became clear for me, and I decided to be a mathematician, I thought that my mother would be very upset by this decision because both of my parents were doctors. I thought that, although my father was dead already, my mother would be happy if I chose that profession. But when I told my mother that I would to be a mathematician, she was very happy and told me, "It's very, very good. You should not become a doctor."

Zierler:

Growing up, were you interested in physics at all?

Schwarz:

In high school, I already knew that mathematics was something I was really interested in, in particular because of the influence of Professor Efremovich, my future advisor. I'll talk about him later because he had a very important role in my life. In particular, he organized what was called mathematical circles, mathematical olympiads in Ivanovo. I participated in these circles and olympiads quite successfully. And I started to study calculus, non-Euclidean geometry. So, at the end of high school, I knew something which wasn't included in the school program.

Zierler:

Tell me about your education at the Ivanovo Pedagogical Institute.

Schwarz:

It was fine. I was able to skip lectures that were not necessary because I knew the material. I had good relations with other students. By the way, the profession of teacher was mostly a female specialty in the Soviet Union. The students were almost all female. But the high school was separated. There were high schools that were all-male and all-female. And I naturally studied in an all-male school. It was later that this was abolished, and schools were integrated. But this happened precisely after I graduated. I had the opportunity to work with Professor Efremovich. And this was really very good for me. I learned general topology for a while.

The first paper that I wrote was devoted to this subject. Later, Efremovich suggested to me more geometric problems, and one of these problems was really very good. It led to a paper that is now well-known and is considered the first paper in a new field called geometric group theory. This paper was not noticed for a long time. But after that, thirteen years later, a famous mathematician, John Milnor, wrote a paper which contained the same results as my paper. His work was completely independent. He didn't know of my paper. But his next paper cited my paper. The main result of this paper is called now the Schwarz-Milnor lemma. I think this is the first serious work that I wrote under the influence of Professor Efremovich.

Zierler:

Tell me about your decision to go to Moscow University for graduate school.

Schwarz:

Well, it was not a decision. This was an opportunity. I tried to get to Moscow University after high school. I was unsuccessful. I do not know the reason, but I know there were at least two very serious reasons. First of all, I was a son of an enemy of the people. And second, I was Jewish. So, I think either reason was sufficient. But this was '51. And in '55, when I graduated from the Pedagogical Institute, things changed. Stalin passed away in '53. My father was already exonerated. And I cannot say that government antisemitism disappeared, but it was at the lowest point starting '44, probably.

Of course, antisemitism didn't disappear after October revolution, but state antisemitism disappeared completely. I would say, there was state philo-semitism, not antisemitism. But about ’44 the state antisemitism reappeared; the maximal point was '53, the killer doctors arrest. After Stalin’s death, in '55, I suppose all was fine. I was very happy to be graduated precisely at the right time. My future wife was also very happy, she graduated from high school precisely at this time. And so, both of us were admitted, she became a student of Moscow University, I was admitted to graduate school. But this wasn't really my achievement. This was history that allowed me to become a graduate student at Moscow University.

Zierler:

Did you recognize in graduate school the value of your work for physics?

Schwarz:

I do not think that at that moment, any of my works were relevant for physics. In graduate school, I was quite productive. This was more or less pure mathematics. The paper that later became my PhD dissertation, candidate of science in Russian terminology, was devoted to homology of the space of closed curves. Later, it was found that these homologies are relevant for string theory. Sullivan gave the name “string homology” to this object. But at that moment, string theory did not exist. And so, my paper definitely was not relevant for physics at that moment. Another of my papers written in my graduate school was a part of my doctor of science dissertation. In Russia, you have two dissertations. One of them corresponds to a PhD, and another doesn't have an American analogue.

This paper had a very interesting fate that I discovered, really, a couple of days ago when I prepared for this interview. I looked at my papers on Google Scholar, and I found that this paper that was written sixty years ago, in the first forty years, collected, I believe, four or five citations. In the next twenty years, it collected more than one hundred and seventy citations. In some sense, this is also not my achievement. At some point, another famous mathematician, Steve Smale, rediscovered the main notion of my dissertation, the notion of genus of fiber space, and used it to estimate what he called the topological complexity of algorithms. The people that knew about my paper immediately found that, really, this is the same notion, first of all, and second, that my methods really give much stronger estimates for topological complexity than Smale gave.

My results give an estimate from below that almost coincides with an obvious estimate from above. And this was shown by a Russian mathematician, Vasiliev. Later, another Russian mathematician, Farber applied the same notion to what he calls topological robotics. He estimates the topological complexity of robots using my very old results. All these citations come either from a continuation of Smale’s paper or from a continuation of Farber’s paper. Farber wrote a book about topological robotics, in which my old results are widely used. So that's more or less all about my graduate schoolwork that I wanted to mention. I wrote many other papers in graduate school. I had about fifteen papers published, but I don't think I should talk about them.

Zierler:

Did you work on the genus of fiber space after graduate school?

Schwarz:

Yeah, I did. But this was not the main direction of my work. After a couple of years, I switched to other directions. Some of them also were related to what is called topological questions of calculus of variation. I’ll mention one of my papers that studied topology of the space of Fredholm maps. I can tell a funny story about my nonexistent paper that still has some citations. This was a paper about the degree of nonlinear Fredholm maps. I have found a generalization of the notion of the degree of the map to infinite dimensional Fredholm maps; I generalized not only the degree, but also other homotopy invariants of a map.

I gave a talk about this in the Congress of Mathematicians in Leningrad. I planned to write a paper about this, but I did not see that I had a complete picture and my colleague from Voronezh University, Borisovich, told me that an old paper of Cacciopoli had some results that are similar to my results. I looked at this paper and decided, before I have a complete picture, I will not publish my results. But Steve Smale, the mathematician that I have mentioned talking about the genus of fiber space, independently found the notion of the degree of Fredholm map and derived more or less the same results that I discussed in my talk in Leningrad. He did not know about results of Cacciopoli and he published his results. Later, Borisovich also worked in this direction. In his work, he cites my Leningrad talk, my nonexistent paper.

Zierler:

Can you tell me what it was like to get your first faculty position after graduate school? What might be unique in the Soviet system in terms of finding and applying to available jobs?

Schwarz:

I think there’s mostly nothing unique, except one thing. The first two jobs that I applied for were in the institutions where you needed the permission from the KGB. I was invited to apply to the Joint Institute of Nuclear Research in Dubna. And I was invited to apply to the Institute of Applied Mathematics in Moscow. In both cases, the KGB did not give the permission to hire me. I don't know why. At this time, it was still possible for a Jew to work in such an institute. Later, it was much harder. But at this time, this was still possible. But I had one more disadvantage. My uncle, my mother's brother, immigrated to Palestine around 1925. He died in a car accident shortly after the declaration of independence of Israel.

He was pretty active as the organizer of the air defense for the newly created state. But he died about '49, maybe '50. So, I had a problem. There was a question, "Do you have relatives abroad?" And I knew for sure that this was a very dangerous question for me. But I did not know what to write. One place, I wrote that I had an uncle in Israel, but he died in a car accident. In another place, I had written, "I do not have any relatives abroad." The result was the same. But in Voronezh University, this problem did not exist. The permission of KGB wasn't necessary. So, I was hired as an assistant professor.

Zierler:

Tell me how you got involved in this first opportunity in the duality of functors in autonomous categories?

Schwarz:

Oh, in Moscow, I had organized a seminar together with two professors, Botyansky and Postnikov. We organized a seminar called the Seminar in Geometric Topology. This seminar had, I believe, quite good influence on several of its participants. The famous mathematician Sergei Novikov was one of the participants of this seminar. He wrote a paper in topology that got him a Fields Medal, later he worked in mathematical physics very successfully. There was another very good student called Dmitry Fuchs, who became my friend. Later, he worked at the same university as me in the United States. We are still friends here. He worked with me, being a student of Moscow University. Later, when I went to Voronezh to teach, he was admitted to graduate school in Moscow University. He asked me to be his informal advisor. I agreed and he was my advisee informally. But he really invented the topic of his dissertation by himself. This was the duality of functors in topology. Later I worked with and without him also in the direction of the duality of functors, not only in topology, but also in general categories and in the category of Banach spaces together with Mityagin. So, I came to this topic at Dmitry's suggestion.

Zierler:

What were some of your contributions in quantum field theory with regard to scattering matrices?

Schwarz:

Well, I've already talked about my work I'm doing right now. But really, this is a continuation of my old work. I worked together with Likhachev and Tyupkin, my students in Moscow Engineering Physics Institute, on adiabatic definition of scattering matrix. I worked together with Fateev, also my student there, on the axiomatic approach to a scattering matrix. We proved that locality is irrelevant in the construction of scattering matrix, that we need only asymptotic commutativity. Later, I continued this work because I realized that the message of this work could be applied to string field theory. I wrote a paper with the title, Space and Time from Translation Symmetry. In this paper, I tried to explain how space and time appear in string field theory, which is, in some sense, two-dimensional, how ten dimensions or twenty-six dimensions appear in string field theory. This was another direction that's also closely related to scattering. Later, I worked a little bit together with M. Movshev in scattering theory in the framework of Nima Arkani-Hamed and his collaborators. I'm also working now in the same direction. The work is not finished, but it seems that one can generalize the approach, where not fields, but particles are basic objects, from four dimensions to any dimension, and from supersymmetric theories to any theories. But this is still a work in progress. I hope I will finish it together with J. Bourjaili and C. Langer. But this is still unfinished work. I'm not sure that I should talk about it in this interview. I have already told you about the geometric approach to quantum theory and about inclusive scattering matrix.

Zierler:

A more broad question, among all of the fantastic advances in theoretical particle physics in the 1970s, what was most useful for your research?

Schwarz:

I can say for sure that for me, the papers by Polyakov and 't Hooft about magnetic monopoles and gauge fields. I realized more or less immediately that these magnetic monopoles have topological origin, and this led to a paper by me, Fateev, and Tyupkin with an explanation how a very simple topological consideration leads to a proof of existence of magnetic monopoles. Practically, in all theories with simple Lie group as a gauge group. This was really more or less a result of the application of the standard tools of topology. It was obtained at the same moment in an independent paper by Monastyrski and Perelomov. It follows that in all grand unification theories should exist magnetic monopoles. This statement was, really, very important for cosmology because magnetic monopoles were not observed. And they are predicted. You should explain why this happens. This and a couple of other problems led to the creation of what is called inflation theory in cosmology, which was invented by Guth, and later, developed in papers by Andrei Linde that are very popular now. The inflation theory is not yet confirmed experimentally, but this is a very interesting and very popular development.

So Polyakov's and 't Hooft's papers were crucial for me in moving in this direction. As I have said, our first paper on heavy particles in gauge theories is very simple from a topological viewpoint. Probably not so simple from the viewpoint of physicists at this time. However now, it's all very simple, also, from the viewpoint of physicists because the elements of the homotopy theory are now common knowledge among physicists. So, I worked in this direction, starting with Polyakov's paper. I first heard of it from Polyakov, not from 't Hooft. We, I and my students V. Fateev, Yu. Tyupkin, I. Frolov, V. Romanov, obtained many interesting results in this direction. Would you like me to talk about these results?

Zierler:

I would like to ask first how you got involved in research on instantons.

Schwarz:

There was a conference, maybe a school, maybe a workshop in Taganrog, I don't remember. The important part was that Polyakov attended this conference. We met in the train on the way to Taganrog and started the discussions about his ideas. There was very hot at this time. So, some of these discussions were in the Sea of Azov, the only place it was not so hot. I should say that the very idea that you should consider extrema of Euclidian action was Polyakov's idea. We had talked about what happens in gauge theories, how to find these extrema. On the way back, I was on the train with my student, Yuri Tyupkin and we continued these discussions. Later, already in Moscow, Sasha Belavin joined our team. So, this was how this paper was written.

Zierler:

When did you first come across string theory? When did you first hear about the work of Schwarz and Green? Or even earlier, Veneziano and Nambu?

Schwarz:

I think John Schwarz came to Russia, and I attended his talks. I learned about string theory from these talks. But I started work on string theory later. This was '84 or '85. This was when superstring drew attention of many people.

Zierler:

Did you recognize immediately how your research would be valuable for string theory?

Schwarz:

My previous research didn't become important for string theory for a while. I'd worked on string theory with Baranov, Frolov, Rosly, and Voronov solving problems that were not fully related to my previous work. Polyakov used my methods of calculation infinite-dimensional determinants in in his famous paper where he introduced what is called now Polyakov action. Instantons appear everywhere now, including string theory. But at this moment, this wasn't clear. So, I don't think I can answer your question.

Zierler:

How did you become involved in supergravity?

Schwarz:

At that moment, reading the work of Ogievetsky and Sokatchev I realized that in some sense in their work, the coordinates and the fields were on equal footing. I thought that this was really an important remark. In my paper I used the name, Space-Field Democracy. The name for this phenomenon was suggested by Édouard Brézin. I think that this is really a reasonable idea. It was used for the theory of D-branes by Townsend and other people. However, I do not think this idea is popular now. Maybe it will become popular, as happened with my second dissertation. But at this moment, it isn't.

But also, there were other ways that led me to supergravity. I found that one can understand the geometry of supergravity in terms of complex geometry and general theory of G-structures.Together with my students A. Rosly, H. Khudaverdyan, V. Romanov, and M.Baranov I wrote several papers in this direction. Using the same methods A. Rosly invented the notion of isotwistor space and related it to supergravity. A little bit later, Ogievetsky and his collaborators also invented this notion under the name harmonic superspace and applied it very successfully. Later I worked with Rosly in the direction he started. The harmonic superspace was a very good invention. But I did not work in this direction very much.

Zierler:

What was so exciting about the second superstring revolution in the mid-1980s?

Schwarz:

Are you using this terminology of [Edward] Witten or the terminology of other people? I believe that this is terminology of Witten, right?

Zierler:

And others. Maybe it's the third revolution. But all of the excitement after the research from Schwarz and Green in 1984. What was exciting about it, or what new opportunities in research did it provide for you?

Schwarz:

Well, for me, this was really very exciting because this was an idea that really there is some very clear suggestion for the theory of everything. A small circle of people, thought that string theory leads to the theory of everything before 1984, but at this moment, many people thought that we were very close to the end, that we'd have the theory of everything in several years. I never thought we were very close to the final solution. However, this was definitely a very promising way. Not only me, but many other people thought this way, and, probably, this opinion is correct. In some sense. But not in the sense that many people thought in the eighties.

Zierler:

Before we move to 1989 and your decision to come to the United States, I'd like to ask, generally, as an academician in the Soviet Union, in what ways were the restrictions difficult for your research, and in what ways was your research not hampered by the broader restrictions in Soviet society?

Schwarz:

First of all, in theoretical physics, it is very difficult to have some restrictions. The restrictions were formal. For the first part of my life in science, I wasn't allowed to publish abroad at all. Later, I got this ability. But I could publish abroad only one paper per year. Therefore, at this time, I wrote longer papers than usual if I would like to publish some of my results abroad. Also, the contacts with foreign scholars were really quite difficult. I wasn't allowed to go abroad for a long time. Formally, I had an obligation to inform the KGB about my meetings with foreigners, but I neglected this requirement. Anyway, we had a very strong community of physicists and mathematicians who were interested in physics. So, in some sense, I wasn't alone. I think that maybe many American physicists were more isolated than I was.

Zierler:

In 1989, did you have a sense that the Soviet Union would fall apart soon? And was that part of your decision to come to the United States?

Schwarz:

It wasn't clear that it will be 15 states instead of one. However, the Soviet Union was falling apart since the communist rule was falling apart. This was really quite a dangerous situation. In Yugoslavia, they had wars. In the Soviet Union, it was mostly peaceful. It could [have been] worse. But the economy was falling apart. Nationalistic movements were very dangerous.

Zierler:

Was it difficult leaving the Soviet Union when you wanted to?

Schwarz:

Before it was '80, I wasn't allowed to go abroad at all. Sometime in the beginning of 80’s I visited Bulgaria and then Czechoslovakia. In '88, I visited Poland. That's it. But before '89, all of these countries were socialist countries, if you can name the regime in the Soviet Union, socialist regime. But '89 was the first time I was allowed to go to a capitalist country, Italy. At this time, it was much, much, much easier to leave the Soviet Union. I almost went to the UK for the Congress of Mathematical Physics. My trip was approved by all the organizations that had to approve. I was told at one point, "There are sixty out there." When I got this information, I was very surprised. I said, "It is impossible. All sixty of these organizations cannot be relevant.” The answer was, "No, all of them are important by a simple reason. If one of them does not reject any candidate, then this organization is not necessary. Therefore, it should disapprove of at least some candidates." All sixty organizations approved my trip. But later, I was excluded from the delegation to this congress because somebody wanted to include somebody else. Nothing personal. Simply, my place was needed for somebody else.

Zierler:

Why the Institute for Advanced Study in Princeton? What was your contact there?

Schwarz:

My first visit, really, was Trieste. In '89, I went to Trieste. I was invited for a month and a half together with my student, Baranov, to ICTP. And when I was there, I was immediately invited to come to the super membrane conference that was scheduled two months after my visit. I said "Nobody will allow to me to go formally to ICTP in two months. But maybe, you could give me a personal invitation, and I will come. Maybe with my wife. This could be a personal visit from the viewpoint of the Soviet Union, but it will be a formal visit to the conference from your side." And yeah, Sezgin, who had, at that moment, some formal position at ICTP, wrote a personal invitation for both my wife and I. I managed to get permission to go on this private visit to Italy to visit Sezgin. When I came. I told the people I knew in America that I would like to immigrate. There were several people that I could ask. One of them was Ed Witten. I received an invitation to Princeton, to Harvard, and to MIT. The Institute for Advanced Study was the first place. It wasn't my decision.

Zierler:

Did you know Ed Witten from correspondence? Had you communicated with him before you got to Princeton?

Schwarz:

I think I first met him in Russia, when he came to the Soviet Union. Moreover, Sasha Migdal told me that Ed attended my talk for Gelfand’s seminar, and Sasha Migdal translated my talk for him. But it seems that Ed knew already at this time the paper that I discussed in the talk, as I understand from his interview [with you]. He knew about my paper from Coleman. I met Ed in Trieste at ICTP during my visit. My second visit, probably. My friend Renata Kallosh talked to him about my wish to immigrate to America. He was very supportive. I did not communicate with him in letters, but I read his papers, and he read my papers.

Zierler:

How was your English when you got to the United States?

Schwarz:

Not very good, but still sufficient to give talks. I wrote papers in English in Russia, so that's why my written English was fine. At this time, spell checker did not exist, but it seems that I didn't need it. But oral English was a more difficult problem. I was able to give talks, I was able to give lectures. This wasn't a really serious problem for me. The more serious problem was the understanding of conversations. That was more difficult.

Zierler:

Did you enjoy your time at the Institute? Would you have stayed longer?

Schwarz:

Oh, I enjoyed my stay at the Institute, I enjoyed my stay at MIT, I enjoyed my stay at Harvard. They're all very, very nice places. All the people there were very supportive. Princeton organized a trip for me. The goal of this trip was to find a permanent job for me. So it ended with a job offer from Davis.

Zierler:

And how did that opportunity at UC Davis come about? Did someone recruit you from there?

Schwarz:

I think that Bruno Zumino recommended people from here to invite me. I discovered later that Davis is a very nice place. Now, it's much, much stronger than it was at the time. But it was a nice place at the time I arrived.

Zierler:

Did you enjoy the new experience of being a professor in America, teaching classes, taking on graduate students?

Schwarz:

Yeah, but I enjoyed my time in Russia also (laughter). I really enjoyed teaching. At Davis, I had the pleasure to teach mostly graduate classes. I also had the opportunity to recruit some good graduate students from Russia. So yeah, it was nice.

Zierler:

Tell me about your work on the geometry of the Batalin-Vilkovisky formalism.

Schwarz:

I think that the Batalin-Vilkovisky formalism is a part of homological algebra as well as general BRST formalism. In mathematics, there is a very general idea that if you have a complicated object, module, algebra, you can replace it by simple object. At least, a more simple object. However, this simple object should carry an action of a nilpotent operator, called differential. This idea is the basis of homological algebra. For example, we can start with a module over an algebra, physicists say that we are staring with a representation of an algebra, you can take a free resolution of this object, a simple object that carries a differential. You can use this idea to define some natural constructions- functors. The resolution is simpler, but larger, very often its construction uses anticommuting [Grassmann] variables. The general mathematical idea of replacing something by its free resolution, more generally projective resolution, works in many, many situations. It works in physics. This is a reason of appearance of ghosts and BRST formalism in physics. I wrote a small paper called Lefschetz Trace Formula and BRST. It is a very simple explanation of the BRST formalism. This formalism is very convenient to analyze the theories with constraints and the theories with degenerate action functionals, but it can be used in other situations also. BV formalism is closely related to BRST formalism, it is even more convenient for these goals. In mathematics differential modules appear as resolutions of usual modules.

But they are interesting by themselves, and this is really an important ide that you should develop a theory of differential modules. Similarly, in physics you can start with BRST or BV theories that do not come from theories with degenerate action functionals. This idea led to what is now called AKSZ model. It's a model that was invented in my work with Alexandrov, Kontsevich, and Zaboronsky. Our construction was used in many, many situations.

Zierler:

When did you realize the value of non-commutative geometry to string theory, and then later on to M-theory?

Schwarz:

I would say that it was in the opposite direction. It started with what is called the matrix model for M- theory (BFSS model). It was invented by Banks, Fischler, Shenker, and Susskind. This BFSS model can be constructed as a reduction of ten- dimensional super Yang-Mills theory to one dimension. There is another model called IKKT model, which can be constructed as zero-dimension reduction of the same super Yang-Mills theory. I concluded that there are some constructions in these models that lead to some objects that appear in non-commutative geometry. These objects are called non-commutative tori.

I realized this during my visit to IHES. And Alain Connes suggested to look at his papers about generalizations of connections to non-commutative geometry, in particular, he studied connections on non-commutative tori. This was a very good suggestion. So, I gave a talk about this stuff at the conference in École normale supérieure. At this moment, this was M-theory. But Michael Douglas, who attended my talk realized how this can be connected to string theory. This was the origin of our paper, Connes, Douglas, and I, about all this stuff.

Zierler:

What new opportunities were there to apply arithmetic geometry to physics?

Schwarz:

First of all, I should say a couple of thoughts about non-commutative geometry. I should say that my paper with Connes and Douglas was a beginning of series of my papers with Konechny, with Nekrasov, with Pioline, with Rieffel. I should mention the relation of Morita equivalence with T-duality in string theory. I believe that this relation is a precursor of more general theory explaining the dualities in M-theory on the basis of Morita equivalence and Koszul duality. There was a very important paper with Nekrasov. Or maybe better to say a paper that Nekrasov brought together with me, because this was his initiative. Later Nekrasov used the results of this paper to do explicit calculations in conventional gauge theories on the basis of localization theorem. I found out from Witten’s interview that this was the paper with Nekrasov that drew his attention to our work. Together with Seiberg, Witten wrote a paper String theory and Noncommutative Geometry where they answered many questions that came from our papers, in particular, they clarified the relation between gauge fields in commutative and non-commutative geometry. So, this was really a very important paper. Long paper. In arXiv version of this paper there is a comment “one hundred pages, sorry”. But several years later, Witten and Kapustin wrote another important paper, two hundred and twenty-five pages long with no “sorry.” So it seems that the papers are longer now.

Let us now come back to the question about arithmetic geometry. I think that also, this is an interesting direction. I would like to cite a joke from my paper with Shapiro, which is called Twisted Cohomology and "Physics Over a Ring," or something like this. The joke is as follows. Definition of physics: “Physics is a science about calculations of integrals where the integrand is a function multiplied by the exponent of some other function. The difference between different branches of physics is only in the name of these functions.”

And then, after this joke, we are trying to explain that the integral of this type can be defined not only for conventional functions, but also in the case when instead of functions we have elements of some ring. In particular, one can consider physics over p-adic numbers. It is possible that this physics is only auxiliary; it allows to answer some questions in conventional physics. Or maybe this can be fundamental. This is more or less the statements of the paper with Shapiro. In the paper with Vologodsky and Kontsevich, and later, papers with Vologodsky I applied physics over p-adic numbers to get some statements about usual physics.

We had some success. I think that this direction was really interesting. It seems that people now, in different ways, not in the way we did, but also in other ways relate number theory to physics. I don't follow this direction now, but there is a journal devoted to number theory and physics, there are conferences.

Zierler:

When you came to the United States, did you give thought about whether you belonged most in a math department or a physics department?

Schwarz:

I decided that I should go to the math department because I thought that nobody could say that I was not a mathematician, but I was not sure that I was a physicist. It seems that I was a little bit wrong on both statements. One of them is that nobody can say that I am not a mathematician. When I applied for a grant to the mathematical division of NSF one of referees wrote that, "Everything is very interesting, but he should apply to the physics branch of NSF." So, somebody could say that I was not a mathematician. I think that still, many people can say that I am not a physicist, but the existence of this interview probably means that at least many physicists think that I have the right to say that I'm a physicist. Bruno Zumino told me this many years ago. Now, I should agree.

Zierler:

In what ways has the department of math at UC Davis grown over the decades?

Schwarz:

Now, it's a very good department. We really have many very good people. We had very good people from the very beginning, some of them between mathematics and physics. Craig Tracy, who is now very well-known in theory of random matrices, was originally from the physics department, his first work was the calculation of correlation functions in the Ising model. We had Motohico Mulase whose work was a related to what is called KP hierarchy. Now he works in what is called topological recursion, a field started by physicists. Later we hired many very good people that work on the interface between mathematics and physics, some of them came from physics departments: E. Gorsky, G.Kuperberg, B. Nachtergaele, A. Soshnikov, A. Waldron. UC Davis organized a Center for Quantum Mathematics and Physics (QMAP) with the goal to promote the interaction between physicists and mathematicians; Department of Mathematics hired T. Dimofte in the framework of QMAP.

Zierler:

Over the past decade, you've been involved so much in homology. What's been new in this field that has captured your attention over these past ten years?

Schwarz:

My work on homology is in the same field as the work on BRST. So, this is not a new direction, it's a continuation of old directions. Some of homological problems we solved are related to my work with M. Movshev on maximally supersymmetric gauge theories that is closely related to the work of non-commutative geometry. Maximally supersymmetric gauge theories can be obtained as dimensional reductions of ten-dimensional supersymmetric gauge theory, they include BFSS and IKKT matrix models. Other homological problems we solved come from pure spinor formulation of superstring theory. This formulation is related to topological quantum field theory. Usually string theory is formulated in terms of conformal field theory with critical central charge. You should add ghosts to get string theory. Adding ghosts to conformal field theory with critical central charge, you get a topological theory. As Witten has remarked, it is not necessary to have separately conformal field theory (matter theory), and ghosts, you can consider theories, where matter and ghosts are not separated. So, the target of critical string theory is really a topological quantum field theory. This remark was used by N. Berkovits in what is called pure spinor string theory, where you do not have standard ghosts, you have a kind of bosonic ghosts. Some homological problems solved in my papers with M. Movshev, Renjun Xu, and A. Mikhailov come from this theory.

Zierler:

And more recently, you touched on this earlier, in what ways does taking a geometric approach allow us to better understand quantum field theory?

Schwarz:

First of all, I think that this is really quite important because it makes clearer that probabilities should be explained from basic principles of quantum mechanics. As in usual quantum mechanics, you can get probabilities from the decoherence in the geometric approach. However, in quantum mechanics, you have other ways to explain these probabilities. In the geometric approach, it's a necessary step. And the second thing is, in the geometric approach, you have many more ways to construct physical theories. You can construct theories with any symmetry group that you want.

The only problem is, at this moment, I did not manage to construct examples that are really new and interesting for physics. And I do not know if I have enough time in my life to do this. But I hope that other people will manage to do it. One more thing that I can say about the geometric approach, that it is very clear in this approach that you can get quantum theory from classical theory if you restrict the set of observables and identify states that cannot be distinguished by means of observables from the restricted set. This can be explained also in the standard approach to quantum mechanics; in some sense this statement appears in nonlinear quantum mechanics by Weinberg. This remark is very simple. Let me explain it in more detail. In the usual approach to classical mechanics you should consider a symplectic manifold, you should consider pure states corresponding to points of this manifold, and you should consider mixed states that are probability distributions on the symplectic manifold. Classical observables are functions on the symplectic manifold. Every measuring device measures one of observables. Now, you can assume that you cannot measure all observables, only part of them. Then your devices cannot distinguish some probability distributions. You identify indistinguishable distributions, indistinguishable states. The statement is that in this way, you will get quantum mechanics from classical mechanics. To get the textbook quantum mechanics one should take the complex projected space as a symplectic manifold. However, you can take any symplectic manifold and construct quantum theory with any symmetry group.

I believe it is an important remark, that in some sense, quantum mechanics is not really very different from classical mechanics. This is one of messages of the geometric approach. However, the main thing is to get new examples, probably using Jordan algebras, which are very natural in the geometric approach because Jordan algebras are closely related to homogeneous convex cones.

Zierler:

Now that we have worked right up to the present, for the last part of our talk, I'd like to ask a few broadly retrospective questions about your career. The first is, if you look at all of the things that you have worked on, are any of them purely mathematical? In other words, have you done anything that has not had an impact on physics?

Schwarz:

Oh, definitely. But I would like to say that I also have some physics papers that did not have an impact on physics (laughter). That's definitely true. Yes, of course, I had a lot of papers that are purely mathematical. For example, my first paper was about the relation of proximity spaces and uniform spaces. Probably never will be applied to physics. Nevertheless, I would like to say that in general, nobody knows. If you have a paper that is completely unrelated to physics at this moment, it is possible it will be related to physics in fifty years. There are many examples of this.

Zierler:

How do you see your work influencing some of the most fundamental questions? Like resolving quantum mechanics and general relativity, for example.

Schwarz:

First of all, I hope that the geometric approach is so general that it can contain the union of quantum mechanics and general relativity. It should contain string field theory, hence it should contain both quantum mechanics and general relativity. The second thing is that I, as many other people, believe general relativity should not be quantized. It should become a part of quantum theory, as it can be considered part of string theory. But maybe it separately shouldn't be quantized.

Zierler:

The hardships from your childhood relative to the success you have achieved in your career, do you see your origin story and what you are able to accomplish as connected somehow? In other words, did that provide you with certain life lessons to succeed?

Schwarz:

Yeah, I think so. First of all, as I have said to you, my mother was exiled to a place to which other people fled during the war. Hardships were later, as I have explained. The hardships that I've had are some reason for hard work. Several decades ago American Jews were discriminated against, now the states-sanctioned discrimination disappeared and they should work as hard as anybody else to succeed. In the Soviet Union it was necessary to work much harder. My children decided to immigrate because they did not want to have their options limited due to their ethnicity. They wanted to immigrate because they were afraid of being repressed and mistreated due to their Jewish roots.

And I immigrated to the United States because my children wanted to immigrate. I loved them dearly, I did not want to be in Russia, if they are in America. I told them, "No, I will immigrate first, and you after me" (laughter). My children were successful in America. My son got PhD in economics from Stanford. Now he is the Chief Economist at Microsoft. My daughter is quite successful too. She has PhD in economics from Princeton and works very productively in economics of the internet. Her papers are well cited. Still, my children's careers reflect a still persisting bias: regretfully, it remains far easier to be a male than a female in STEM fields.

In the Soviet Union my children even did not apply to Moscow University, the best university in the country, because they knew that the fate of the application will be determined not by their talents, but by their ethnicity. For them this was a very hard experience.

Zierler:

Between your childhood and your education, do you see your approach to math and physics in Russian or even Soviet terms?

Schwarz:

I think that the Russian approach to mathematics, in general, to science, is a little bit different than in America. I think that Russian scholars are broader than American ones. Not all of them. Not all Russian scholars are broad, not all American scholars are narrow. You know there are many examples of American scholars that are very, very broad. Now, this is much more common than earlier. But in Russia, the interaction between mathematics and physics that we see now started earlier than in America. At some point, Dyson wrote a paper about divorce between mathematics and physics, with some examples. But Dyson also came from mathematics. Russian mathematicians became interested early in physics. I believe this divorce between mathematics and physics ended in Russia earlier than in America. Now, the situation is completely different. Right now, here, the relationship between mathematics and physics is very tight. In this sense, the difference between Russian and American style, I wouldn't say it disappeared, but it's less pronounced.

Zierler:

As I'm sure you know, many in the physics community criticize string theory because it has not yet been proven to demonstrate the real world. A defense among string theorists has been perhaps, "Give us more time," but also, in the meantime, string theory has given us a wonderful mathematical toolbox. I wonder if you can react to that, both in terms of the utility of string theory and its ability to demonstrate what actually happens in the real world.

Schwarz:

So let me start from the very beginning. When you asked me about the eighties when I first came to string theory, at this time, I was quite pessimistic compared to other people working in string theory. I thought that string theory was very far from reality. Later, I did not change my opinion, but I became an optimist with the same opinion. Because people became pessimists. Later, there was another wave of optimism in the second or third string revolution. My opinion still did not change. So, to formalize my opinion today, as I explain to everybody, I am not sure that string theory is correct. I would say that I do not believe that string theory is a correct theory of everything.

But I believe that the right physics will come from string theory. String theory cannot be considered as physics today, there are no string theorists that are saying that today, string theory describes reality. There are no such people. There are people who hope that in the future, it will describe reality. But nobody can say that string theory is not a good science. It definitely is a very good science. You can call it mathematics. You can call it something else. But this is definitely good science. And it has a good chance to become physics. Maybe the good physics will come a different way. Maybe it will come from my geometric approach (laughter). But this doesn't matter. It's a good science. Another thing that matters, that there were already many physics papers that were mathematical for a pretty long time.

One example is gauge theory. Gauge theory was invented in '53, I believe. And it became physics probably about fifteen years later. I would say that general relativity, for a very, very long time, was not confirmed in experiments, there were about five experiments that confirmed it. And all of these experiments could be explained in many different ways. Everybody believed in general relativity for a simple reason. Because it was beautiful. But all experiments could be explained in different ways. So therefore, yes, there is a chance that string theory will become real physics. Yes, there is a chance that there will be another theory, not string theory. But string theory will not disappear.

Right now, the main problem, I believe, is that when the coupling constant is large, in most cases we do not have any rigorous ways to say anything. We have quantum electrodynamics, but we do not know what quantum electrodynamics is doing for large coupling constants. The same is true for standard model. You can't say anything. Hopefully we’ll find the answers, analytically or numerically.

Zierler:

In all of the problems that you've worked on in both math and physics, what stands out in your memory as most intractable, where no matter what you do, you always seem to hit a wall?

Schwarz:

I did already mention the problem of strong coupling. Basically, right now, we are always working with small couplings. Sometimes one can find a way to consider a theory with strong coupling as a theory with small coupling. Sometimes supersymmetry or topology allow us to prove statements that are correct for all coupling constants. This is the way physics works right now. Probably, the most intractable problem is the problem of intermediate couplings. Somebody, I don't know who, said that the Laplace method or stationary phase method in the calculation of integrals work very well in the case when it is necessary. For example, if you would like to calculate N! this is very easy when N is small and for large N you can get a very good approximation for example using Laplace method. However, for intermediate N you need a computer. No other way. So, I think that in physics, the really difficult problem is intermediate coupling constant. Or large coupling constant if you do not see a way to reduce it to small coupling constant.

Zierler:

My last question was on problems, but let's end our discussion by talking about solutions, looking to the future. Among all the things that you're currently working on right now, what are you most optimistic will be part of a solution to some major breakthrough in the future?

Schwarz:

You know, the life expectancy tables give me five years . When I thanked people that congratulated me for being eighty-five years old, I wrote on Facebook that if I've had no knee pain and cataracts, I would say that I am eighty-five years young. Now, I don't have knee pain, and I had cataract surgery. So, in less than a month from now, I will say that I am eighty-seven years young (laughter). I hope that I still have some time. But I understand very clearly that I have not very much. I'm working on two related things: inclusive scattering matrix and the geometric approach to quantum theory. I hope that I will be able to construct new interesting models in this geometric approach. Maybe to solve some of these models. I am not sure that I have time to solve them, but maybe I have enough time to construct. Maybe I'll have some help. I've had many very good collaborators. I should make a list of the people I've collaborated with. There are many people I did not collaborate with, but people who continued my work in some sense. Even more fantastic people.

Zierler:

I am so glad that after Ed Witten explained to me what a formative influence you were in his thinking, that he subsequently encouraged me to reach out to you directly, and I'm so glad he did. I'd like to thank you so much for doing this.

Schwarz:

Thank you very much.