«The world is a cart careening off a mountain and hurtling with acceleration»

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

Andrei Sakharov, Breakthrough Prize, Perestroyka, physics, quantum physics, russian scientists