# Category Archives: Physics

## Valery Rubakov (1955-2022)

On the morning of October 19, 2022, I woke up to a terrible news: Valery Rubakov has passed away.

Valery was my PhD thesis advisor. Every time I introduce myself to Russian physicists (or physicists from the former Soviet Union), after saying “I studied in Moscow,” I would add “I was a student of Valery Rubakov.” No words that I could use to describe myself would elevate my status in front of Russian physicists more immediately than that simple sentence “I was a student of Valery Rubakov.”

It is clear why it is so. Valery was a giant in particle physics; he has done so many groundbreaking works: the Rubakov-Callan effect (for the lay audience: how a hypothetical particle called the “magnetic monopole” can turn matter into energy), electroweak baryogenesis (an explanation of why the Universe contains only matter but no antimatter), an early suggestion of the braneworld scenario (that we are living in a world with more than 3 spatial dimensions, and that the extra dimensions do not have to be small) and many other things. Valery was also famous in Russia as a science popularizer and a staunch defender of scientific institutions against political interference.

I already heard legends about Valery as an undergrad at Moscow University, so deciding to do PhD under him was the easiest decision. I entered the PhD program at the Institute for Nuclear Research (INR) in September 1991. Valery was then the Vice Director of the Institute, responsible for the Institute’s scientific work. Our Theory Division (“Teorotdel”) is housed in a three-story building. Valery’s office was on the second floor, while I was in a shared office on the third floor. Normally Valery would work his day job as a Vice Director until some time in the afternoon, and come up to our office around 4pm or so to discuss physics with us over tea and cookies (pryaniki). I learned a lot from these conversations, but more importantly, they made doing physics an enjoyable, human endeavor.

I was a junior member in a small group at INR trying to understand whether very high-energy collisions lead to violation of baryon number (i.e., create more matter than antimatter). At the end, the problem remains unsolved, but there are many hints that the effect remains too small to be observed even at very high energy. At the beginning there were only three people in our group: Valery, Peter Tinyakov, and myself. Later a few younger people joined the group (Dima Semikoz, Maxim Libanov and Sergey Troitsky). The group felt like a tight-knit family: we were all members of the Rubakov school.

The possibility to have his own school was perhaps what kept Valery in Russia. He could have moved to the West in the 1990s, but he once told me that the mobility of scientists in the West, generally a good thing, also makes the existence of “schools” (in the Soviet sense, like in “Landau school” or “Bogoliubov school”) impossible. Valery enjoyed working with and cultivating younger scientists, and the structure of a Soviet physics school fits his style the best.

The INR was an oasis of sanity in the crazy world of 1990s Russia. The euphoria (if there was any) of August 1991 quickly subsided, replaced by a general tiredness and despair of a society undergoing a chaotic transition to Wild West capitalism. But Rubakov managed to get us enough money to not to have to worry too much about having enough to eat, so we could just concentrate on physics. Many friends of my age had to leave physics to go to business. Without Rubakov, I am not sure where I would end up now.

There was at least one occasion when the craziness of the external world seeped into our oasis. Once a visitor came to see Valery. Normally Valery received his guests in his office, but for some reason, he received that visitor in our third-floor shared office. The visitor came in and, seeing me sitting at my desk, asked Valery in a somewhat surprised voice “Your institute employs these?” From his voice and gesture it was obvious he thought that a Vietnamese should not be in a Russian institute. I tried to appear undisturbed. Valery was irritated, he said “Shon (that’s how my name sounds in Russian) is a PhD student (aspirant) in our theoretical division; he is a member of our collective.” Then the visitor started talking to Valery in a low voice. He talked about some issues in science politics, I wasn’t sure what these were. But at some point the visitor said “We should do something, otherwise the Jews are going to ruin us.” Valery immediately pointed to the door and told the visitor to get out.

For me, Valery set my standard of what constitutes a solution and what it means to understand. Whenever I think that I have solved a problem, I would step back and ask myself: do I understand the issue at the level that I would feel comfortable explaining it to Rubakov? When I write something, or make a presentation, sometimes I would ask myself: does my explanation meet Rubakov’s standard of clarity?

The last time I met Valery was in March 2019, at the informal conference RU-10000002 on the occasion of Rubakov’s 64th birthday (1000000=64 in binary) at the INR. I remember at the cocktail party before the conference dinner, somehow the conversation drifted to politics. I said “When I was in Moscow (i.e., until 1995), the things that are happening between Russia and Ukraine would not be imaginable.” Valery said: “This is also unfathomable to me. To be honest I don’t think I understand why the Ukrainians want to be in Europe so much. The European Union has tons of problems. But Ukraine is an independent country and the Ukrainians have every right to do whatever they want.”

After the war broke out, Valery was one of the first scientists to sign an open letter opposing the war. That was a strong letter which explicitly named Russia as the party solely responsible for starting the war. I was worried about Valery’s safety, so I sent an email to him. I wasn’t sure if the letter would be read by a third person, so I only tried to tell him what I thought of the current events using an expression from his open letter: “Shag v nikuda” (a step to nowhere). Rubakov wrote back to me saying “I agree with your assessment” and said that many people are frustrated, so moral support from people outside the country is valuable.

I was waiting for the war to end to visit Valery again. Alas, that is now impossible. But Valery will live forever in my memory.

(photo: Valery Rubakov and his wife Elvira at an anti-war protest in Moscow, March 2014)

## Origin of the term “ghost” used in quantum field theory

For a long time I had been wondering who was the first to use the term “ghost” into quantum field theory. My first encounter with the term was in the context of Faddeev-Popov’s approach to quantization of non-Abelian gauge theories. But Faddeev and Popov, in their first articles, did not use “ghost”; instead, they used a more innocuous term, something like “a fictitious scalar field.” From what I could find out, almost immediately after Faddeev and Popov that strange field, scalar but with fermionic statistics, was renamed “ghost.” The post-Faddeev-Popov literature, however, does not contain any indication on who came up with this term. It also seems that the Russian equivalent “духи” was imported from English, rather than the other way around.

Who was that person who has managed to introduce into the vocabulary of particle physics, the science of the 20th century, a word that has origin in the superstitious beliefs at the dawn of human history?

Several days ago I ran onto David Derbes, a retired physics teacher from the University of Chicago Laboratory High School, who has helped the publication of a number of historical documents, including Dyson’s and Coleman’s lectures in quantum field theory. He told me about his latest object of study—the first preprint of Faddeev and Popov, in Russian and never published. I asked him if he knew the origin of the term “ghost.” David said did not know, but he told me he would try to find out.

With David’s help, now I think I know who has introduced the term “ghost” and when.

It turned out that the term was introduced by Wolfgang Pauli, in a different context. Pauli, a giant in physics, was also responsible for the introduction of the particle now called “neutrino” with a famous letter which started with the words „Liebe radioaktive Damen und Herren“, “Dear radioactive Ladies and Gentlemen.” And as with the neutrino, Pauli introduced the term “ghost” in a letter. In fact, in two letters. The first letter, sent to Källén on December 9, 1954, contains Pauli’s announcement of his intention to send a letter to TD Lee, where he would propose the word “ghost.” (The letter to Källén is letter [1942] in the book edited by Karl von Meyenn, see the end of this posting). Pauli predicted that once the term is proposed, its use will spread epidemically in the literature. But Pauli wrote that he did not think “ghosts” are physical, citing a quote, allegedly by Lichtenberg, “There are more things in the compendiums of physics, than are dreamt of in heaven and Earth.”

The letter to TD Lee was sent five days later on December 14, 1954, written in English, and copied to Dyson. Here are the beginning and the end of that letter. Note the way Pauli signed the letter. (Text taken from the Karl von Mayenn’s book; this is letter [1946] in that volume.)

“Dear Lee!

It is already some time that I started to study your paper seriously. Days became weeks, weeks two months and my file „Lee-model“ is still increasing – a proof of its importance. It is true that in my way of looking at it, most of this importance is concentrated in your footnote 4, p. 1331 and the rest of the paper seems to me, at least in first approximation, negligible in comparison to this small printed note.

(… a lot of technical discussions follow …)

The essential occurrence of negative probabilities in your example makes it, of course, extremely unphysical. But this is not my whole story, and in some other respects, I may have good consolation for you, too. Until now I only told you results which are proved. In the following concluding part of this letter I shall formulate guesses or conjectures of a more general kind, which should be merely considered as the outline of a program of further mathematical investigations.

Let us call a new energy-state with negative probability (negative in comparison with the other states of normal behaviour for small coupling constant), whose energy tends to ±∞ for (renormalized) coupling constant g going to 0, a ‚ghost‘. It is my opinion that the occurrence of ‚ghosts‘ will soon turn out to be a general feature of coupling constant renormalization. This feature has been revealed first by your example, the importance of which should therefore not be underrated. If the unrenormalized theory diverges logarithmically, the energy of the ghosts will behave for small g as $\exp(\textrm{const}/g^2)$. This seems to offer an explanation for the fact that in the examples, which could be really investigated until now, Dyson’s power series have the convergence radius zero. I suggest that this result (Thirring and others), which is presumably general, be brought in connection with the fact that the ghost energy is of the mentioned essentially singular type at the point g = 0. If my conjecture is right, the renormalized field-theory should have a mathematically rigorous solution, which, however, is unphysical because of the occurrence of negative probabilities in it. The ghosts have no physical reality whatsoever, they are the formal reaction of mathematics to the tricks played on her by the method of renormalization.

If my conjecture is right, this should also hold for quantum electrodynamics, the only case where we are certain that renormalization has anything to do with nature. The ghosts would then be situated very roughly, at the extreme high energy of e137 times electron mass (factors like l/π in the exponent not excluded). These ghost-states will then be only very seldom excited and one can understand the possibility that quantum-electrodynamics including renormalization, can give good approximations. But, in principle, it would have this defect too.

The situation is too new for us to think about the therapy now, we have first to think about good mathematical methods, to check the diagnosis and make the bacillus in the renormalization method generally visible. There are great difficulties in this problem. If, for instance, a ‚ghost‘ would be discovered in a Tamm-Dancoff approximation, how could we be sure that it is not only a result of the insufficiency of this approximation?

Nevertheless I think that this mathematical problem can be attacked and I suggested to Källén, who is at present in Copenhagen, that he resumes his old work of 1952 in the light of this new aspect. Our results will certainly be published in due time in one form or another but I think they have first to be put on a broader basis.

Meanwhile I hope that the end of this long letter will be the beginning of something else and I conclude with all good wishes for Xmas and for a really new year in physics to yourself and to all friends at Columbia University.

Sincerely yours,

W.Pauli
The society of ghost hunters
The president

TD Lee’s paper mentioned by Pauli is Phys. Rev. 95, 1329 (1954). Pauli would continue to refer to particles with negative norm as “ghosts.” For example, in a letter sent to to Heisenberg on May 18, 1955, Pauli expressed doubt that modes with negative norm in Heisenberg’s model can be quarantined from the rest of the Hilbert space. He asked: „Wieso bleibt der Geist in der Flasche?“ “Why does the ghost stay in the bottle?” (it appears that in German the word Geist, meaning ghost, is also used for a genie in a bottle).

Pauli’s ghost is, in the modern language, related to the Landau pole. With the invention of asymptotic freedom, Landau pole is no longer inevitable in quantum field theory. Pauli’s prediction that the term “ghost” would be widely used remains correct. “Ghost” lives on, most notably as the colorful name for Faddeev and Popov’s “fictitious scalar field.”

Reference:

W. Pauli, Wissenschaftlicher Briefwechsel mit Bohr, Einstein, Heisenberg u.a. Band IV, Teil II: 1953-1954 (Scientific Correspondence with Bohr, Einstein, Heisenberg, a.o. Volume IV, Part II: 1953-1954), edited by Karl von Meyenn, Springer, 1999.

Addendum (July 6, 2019): Although the English word “ghost” was first proposed by Pauli in his December 9, 1954 letter to Källén, Pauli conceived the use of the German word “Geist” for the same purpose a few days earlier, in his December 6, 1954 letter to Fierz.

## Công thức Hardy-Ramanujan qua mô hình chất rắn Debye

Bạn nào đã xem cuốn phim The Man Who Knew Infinity chắc sẽ nhớ một công thức đóng vai trò rất quan trọng trong phim: công thức về hàm phân hoặch số nguyên, được Hardy và Ramanujan tìm ra năm 1918. Hàm phân hoạch p(n) định nghĩa rất đơn giản. Lấy ví dụ số 4; số này có thể biểu diễn bằng 5 cách khác nhau thành tổng các số nguyên:

4 = 4
4 = 3 + 1
4 = 2 + 2
4 = 2 + 1 + 1
4 = 1 + 1 + 1 + 1

Như vậy p(4) = 5. Tương tự p(5) = 7, p(10) = 42. Nhưng khi n tăng cao thì p(n) tăng lên rất nhanh, ví dụ, p(200) = 3.972.999.029.388. Trong phim MacMahon tính con số này bằng tay, không rõ bằng phương pháp nào. Hardy và Ramanujan tìm ra công thức cho tiệm cận của p(n) với n lớn,

$p(n) \approx \displaystyle{\frac1{4n\sqrt3}}\exp\left(\pi\sqrt{\frac{2n}3}\right)$

Nếu ta thay n = 200 vào công thức này thì ta sẽ tìm được p(200) = 4,10 × 1012, sai số 3.2% so với kế quả chính xác. n càng lớn thì sai số này càng bé.

Một cảnh trong phim

Có vẻ phương pháp mà Hardy và Ramanujan dùng để tìm được công thức này khá phức tạp. Trong bài này chúng ta sẽ dùng vật lý để tiếp cận công thức Hardy-Ramanujan. Tìm được toàn bộ tiệm cận của p(n) thì hơi khó, ta sẽ chỉ nhắm vào phần quan trọng nhất, phần exp thôi. Nói cách khác, chúng ta sẽ chứng minh:

$\ln p(n) \approx \pi\sqrt{\displaystyle{\frac{2n}3}}$

Để tìm được công thức này, chúng ta sẽ dùng một cách tiếp cận không chính quy. Ta sẽ dùng mô hình Debye của nhiệt dung của chất rắn. Công trình này của Debye được viết năm 1912, vài năm trước khi Hardy và Ramanujan công bố công thức cho p(n). Có lẽ Hardy và Ramanujan không biết về công trình của Debye.

Trước Debye người ta đã biết định luật Petit-Dulon, theo đó nhiện dung của một khối chất rắn là một hằng số không phụ thuộc vào nhiệt độ. Tuy nhiên thí nghiệm cho thấy định luật Petit-Dulon chỉ đúng ở nhiệt độ đủ cao, định luật này bị vi phạm ở nhiệt độ thấp. Einstein là người đầu tiên chỉ ra mối liên hệ giữa sự vi phạm định luật Petit-Dulon với cơ học lượng tử. Trong mô hình của Einstein, nhiệt dung là hằng số nếu nhiệt độ cao nhưng tiến tới 0 khi nhiệt độ giảm tới 0. Tuy nhiên trong mô hình của Einstein nhiệt dung tiến tới 0 nhanh hơn so với đo được trong thực nghiệm. Năm 1912 Debye đưa ra mô hình giải thích được sự biến thiên của nhiệt dung của chất rắn. Cách tiếp cận của Debye hết sức mới mẻ. Debye không nhìn chất rắn như một tập hợp các nguyên tử, ông nhìn chất rắn là một khí tạo ra bởi các hạt phonon – lượng tử của sóng âm thanh. Trong mô hình Debye, các nguyên tử chỉ là cái nền cho các hạt phonon lan truyển.

Để liên hệ với công thức Hardy-Ramanujan ta chỉ cần xem xét một chất rắn 1 chiều. Để dễ tưởng tượng, ta sẽ xét một chiếc dây đàn, căng giữa hai điểm A và B. Ta chọn trục x của hệ toạ độ chạy theo đường thẳng nối hai điểm A và B. Nếu độ dài dây đàn là L thì tại A ta chọn x = 0, tại B x = L.

Khi ta gẩy đàn sẽ có sóng lan truyền trên dây đàn. Sóng này coi như là âm thanh trong môi trường một chiều. Để cho đơn giản ta giả sử dây đàn chỉ dao động theo chiều y. Trạng thái của dây đàn tại một thời điểm nào đó được mô tả bới hàm y = y(x). Giả sử vận tốc lan truyền của sóng là v. Do hai đầu dây đàn bị đóng cứng, sóng trên dây đàn phải là sóng đứng, và biên độ của sóng biến thiên theo toạ độ và thời gian theo công thức

$y = \sum\limits_{k=1}^\infty A_k\cos(k\omega_1 t + \alpha_k) \sin\left( \displaystyle{k \frac{\pi x}L}\right)$

Trong công thức trên $\omega_1$ là tần số cơ bản của dao động của dây đàn,

$\omega_1 = \displaystyle{\frac {\pi v}L}$

và các hoạ ba (harmonic) cao hơn có tần số $\omega_k=k\omega_1$ với $k=2,3,\ldots$.

Bây giờ ta lượng tử hoá cái dây đàn. Mỗi tần số $\omega_k$ nay tương đương với một dao động tử điều hoà, và dây đàn là một tổ hợp các dao động tử điểu hoà với tần số $\omega_1$, $\omega_2$, v.v. Các mức năng lượng của dao động tử điều hoà với tần số $\omega$$\hbar\omega(n+\frac12)$. Như vậy để mô tả trạng thái lượng tử của dây đàn, ta cần một số vô hạn các số lượng tử $n_1, n_2,\ldots n_k,\ldots$ trong đó $n_k$ là số lượng tử của dao động tử với tần số $\omega_k=k\omega_1$. Như vậy

$E = E_0 + \sum\limits_{k=1}^\infty \hbar k \omega_1 n_k$

trong đó $E_0$ là năng lượng của trạng thái cơ bản. Để đơn giản từ nay ta sẽ đo năng lượng của dây đàn từ $E_0$, tức là cho $E_0=0$.

Bây giờ có thể nhận ra một điều như sau:

Có p(n) trạng thái lượng tử của dây đàn với năng lượng $n\hbar\omega_1$

Đây chính là điểm liên hệ giữa vật lý và công thức Hardy-Ramanujan. Nghĩ một lúc các bạn sẽ thấy điều này hầu như là hiển nhiên. Ví dụ ở mức năng lượng $4\hbar\omega_1$ có năm trạng thái:

$n_4=1; n_k=0, k\neq4$
$n_3=n_1=1; n_k=0, k\neq 1,3$
$n_2=2; n_k=0, k\neq 2$
$n_2=1, n_1=2; n_k=0, k\neq 1,2$
$n_1=4; n_k=0, k\neq 1$.

Khi đã biết số trạng thái có năng lượng $n\hbar\omega_1$ bằng p(n), ta kết luận $\ln p(n)$ chính là entropy khi năng lượng bằng $n\hbar\omega_1$, theo định nghĩa của entropy qua tập thống kê vi chính tắc (microcanonical ensenble).

Nhưng trong vật lý thống kê, ta có thể dùng tập thống kê chính tắc (canonical ensemble) để tính entropy của dây đàn, thay vì dùng vi chính tắc. Bình thường tính toán dùng tập thống kê chính tắc bao giờ cũng đơn giản hơn là dụng tập vi chính tắc.

Một điểm làm đơn giản bài toán là khi nhiệt độ lớn hơn tần số dao động cơ bản, ta có thể bỏ qua hiệu ứng bề mặt của hai đầu dây đàn. Bây giờ dây đàn có thể coi là một chất khí phonon một chiều. Bài toán như vậy được đưa về dạng một chiều của bài toán mà Debye đã giải quyết năm 1912 khi ông tính được nhiệt dung của chất rắn ở nhiệt độ thấp.

Bạn có thể làm tiếp những tính toán còn lại nếu bạn nào đã học vật lý thống kê; coi như đây là bài tập cho bạn. Bạn có thể tính entropy trực tiếp, hoặc tính nhiệt dung rồi lấy tích phân để tìm entropy. Kết quả là mô hình Debye của chất rắn cho ta phần exponent của công thức Hardy-Ramanujan.

$\ln p(n) \approx \pi\sqrt{\displaystyle{\frac{2n}3}}$

Cách tiếp cận vật lý cho công thức Hardy-Ramanujan trên đây có trong cuốn B. Zwiebach, A First Course in String Theory.