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3 January, 2014
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View mobile website按：1，原题为“INTERVIEW OF C. N. YANG FOR THE C. N. YANG ARCHIVE，THE CHINESE UNIVERSITY OF HONG KONG”，是黄克孙教授（以下称 Huang）于日在香港中文大学对杨振宁教授（以下称 Yang）进行的访谈，主要内容是关于统计物理的基础问题. 由于原文较长（26页，故我放弃翻译，好在现在大部分人英语也很好），做了删节，分为上下两部分转载. 原文为PDF版本，公式只好用文本方式显示.2，河北某大学，用了我本人为自己的统计物理课程制的图（见封面），也应该致一个谢？You are welcome!Huang: Chen-Ning, do you think ergodic theory gives us useful insight into the
foundation of statistical mechanics?Yang: I don’t think so. It is true that in the early days of statistical mechanics,
there was much discussion about ergodic theory. In particular, it was very useful for mathematicians to laun but in the 20th century, statistical mechanics mostly developed in the area of equilibrium statistical mechanics, and it has taken off really without much reference to ergodic theory. This does not mean that ergodic theory should not be pursued in the future, but I doubt that in the 21st century ergodic theory would have much influence on the development of statistical mechanics.Huang: As you know, there’s this Boltzmann factor e^{-βH}, where β is one over Boltzmann’s constant times the temperature. It turns into the quantum-mechanical time-translation operator e^{-itH}, if we make the temperature pure imaginary by setting β = it. Do you think this is just a mathematical coincidence?Yang: I think it is a coincidence. I myself do not see a deeper physical origin
of this “coincidence,” as you call it. But this is of course a deep subject. The path integral formalism of Feynman, which I regard as one of the great contributions to physics, indicates that the integral of the phase factor - that is, with the i - underlies the true meaning of quantum mechanics. It was not the way quantum mechanics was developed, but I think to understand what quantum mechanics is all about, the deepest way is through the path integral formalism, which deeply entangles this exponential phase factor. How that happens to be, in a formalistic way, related to the partition function expression. I do not see either a physical or mathematical significance, or deep meaning. It is of course very important, once one recognizes this relationship, to borrow from each other. It’s a very deep subject.Huang: Do you think we will some day have a theory of so-called “complexity”
comparable to thermodynamics?Yang: “Complexity” is a word which has been used and abused in recent
years. There were people who thought that complexity will overwhelm the whole of physics. I do not think that is likely to happen. Now, complexity, as relating to such subjects like dynamics, is likely to be a very important subject, both for physicists and mathematicians. But I doubt it will have any importance comparable to thermodynamics.On another level, complexity can have additional meaning, for example, in information theory and the structure of the human brain. These are mysterious subjects for future development. How that will play out in the 21st century I cannot guess.If you call that complexity, if you call the understanding of the structure of the
human brain a part of complexity, then I would say that would likely be the one
dominant subject of 21st century science.【我删去了以下一段关于统计物理对理解人类群体现象的部分，接下来关于精确求解Ising模型的部分很值得一读】Huang: I believe one of your earliest publications was on lattice models in
statistical mechanics. Can you tell us what attracted you to the subject?
Yang: When I was a graduate student in China in the wartime, 1942-44, I had
studied with Wang Zhuxi （注：王竹溪）on phase transitions, specifically, on such things as beta brass, which was a popular subject in the mid-thirties. So I was very familiar with lattice problems. Then, one day in 1944, I remember distinctly Professor Wang Zhuxi very excitedly told me that the two-dimensional Ising model had been solved exactly by a physicist named Onsager. He and I both studied Onsager’s paper, but we did not get anywhere. However, the existence of that model, and the fact that it could be exactly solved, made a deep impression on me. That was the reason why I later got involved in statistical mechanics, in particular lattice models.The involvement was not exactly smooth sailing at first, because, as I said a
few minutes ago, in China in the wartime, I didn’t understand what Onsager did. Later on, when I became a graduate student in Chicago, although Joseph and Maria Mayer were both there, they were not interested in phase transitions at that time. So I studied what Onsager did, on the side, by myself, and I still did not make much headway.All that I learned from Onsager’s paper was, he would commute A with B, and
commute the commutator with B. And he was commuting everything under the
sun. (Chuckle) I checked the computations, and they wer but he didn’t reveal the strategy, so you were led around by the nose, and suddenly the result pops out. That’s not understanding.But then, in 1949, during a van ride from Palmer Square in Princeton to the
Institute for Advanced Study - a 15 minute ride - Quinn Luttinger, who was a fellow post doc at the Institute, told me about Kaufman and Onsager’s simplification of Onsager’s calculations3. That led me to an understanding. After that, I like to tell graduate students that, if they launch into a field and get frustrated because they didn’t understand it, that is not necessarily time wasted. In this particular example - my launching into the Onsager solution - the reason I
could understand the key idea during the 15 minute ride with Luttinger was that I had “prelearned” the whole subject very well, though without true understanding. When the important point was revealed to me, I was able to immediately appreciate the whole strategy. That subject became one of my fascinations in physics, and that was truly the answer to your question.Huang: You calculated the spontaneous magnetization of the Ising model, and obtained a very simple formula after some intricate mathematics, in what Dyson called “an intricate Baroque music of elliptic functions.” What motivated you to study this problem? Was it the belief the answer would be simple?Yang: No, it was not because I felt the answer would be simple. The answer
was amazingly simple, much simpler than what I had expected, and much simpler than what the intermediate steps had led me to expect. What happened was the following.Onsager and Kaufman, and earlier Onsager, had calculated the partition function, and the specific heat of course. The mathematical method Onsager used was very powerful. It not only gave the largest eigenvalue of the transfer matrix. It gave, in fact, all eigenvalues at the same time. Now, in calculating the partition function and the specific heat, Onsager needed only the largest eigenvalue. So I thought, why waste all this knowledge. Couldn’t all these additional eigenvalues be useful? I immediately realized that to calculate the spontaneous magnetization, you need only the wave functions with the two largest eigenvalues - not just the largest, but the next largest one too. So I said, this seems to be a good problem, and I began to calculate that.Actually, there’s a bit of story before that. I told you earlier that Luttinger told me the main idea of the Kaufman-Onsager thing. When I got to the Institute, I spent a couple of hours, putting aside what I was doing in elementary particle
physics, and pushed through that idea, and understood thoroughly the Kaufman Onsager method. Then I said, well, why don’t I collaborate with Luttinger to do a similar calculation with, let’s say, a triangular model. That afternoon I talked to Quinn, but he was at that time doing some renormalization calculations, and he said no, he did not want to do it. So I thought about it a little longer, and decided it was a copycat thing, not challenging enough. So I put it aside.More then a year later, after I had a rest from what I was doing in elementary
particle physics, I came back to this model. As I said, I realized that to calculate spontaneous magnetization, all you need are the property of the largest two
eigenvalues and the wave functions. So I got launched into that.Huang: Was your method closer to Onsager’s original method, or Kaufman’s
method?Yang: Kaufman’s. Onsager’s original method was opaque. Very few people later refer to Onsager’s original method. Anyway, - I remember this was in 1951, around January or something - it was a tortuous mathematical problem, both frustrating and enticing. Frustrating, because every other day you would find the problem but after you think about if for a couple of days, you found you can turn the corner, and launch into a new direction. This repeated itself many times. Of course, the frustration sometimes led me to believe it was a totally useless exercise, and I would give up. But after a couple of days, something always happened to make me feel I could strike out on a different route. This went off and on, off and on, for about half a year.During that process, there were a lot of elliptic functions. I had learned that
elliptic functions was a beautiful subject, when I was a graduate student in China. I never thought it would be useful in a physics calculation. I was of course delighted that elliptic functions entered into the calculation, but the thing that really surprised me was that all the elliptic functions dropped out in the end. There were a lot of elliptic functions, and they were both in the denominator and numerator.I calculated the numerator - lots of elliptic functions. And I calculated the denominator - lots of elliptic functions. And when I put the two together, they all
dropped out, and I got an algebraic expression of great simplicity.As a matter of fact, the algebraic expression was so simple that it led to a further development. A year later, T.D. Lee and I got onto the subject of phase transitions. We began to guess at other algebraic expressions that might solve the Ising model in an imaginary magnetic field, and we guessed correctly. How do we know we guessed correctly? Because after our guess, we calculated the series expansion, and found that the results agree with the series expansion to - I forget - ten terms or something like that. So we asserted that this is the right expression. That guess was finally proven some 10 years later by T.T. Wu and McCoy.To answer your question directly, I did not expect the results to be simple. And in the process of that happy calculation, I did not believe it would come out with a simple answer.【以下我做了较多的删节，涉及他和李政道合作证明的著名的单位圆定理，讨论了相变的发生，然后是谈同爱因斯坦的见面，在此又一次谈及相变】Huang: Kirkwood conjectured sometime ago that a classical hard-sphere gas
has a phase transition. Computer simulations seem to indicate there is a first-order phase transition, arising from jamming at a smaller density than close packing. Do you think such a transition exists?Yang: Yes, I do. I was not so convinced many years ago, but I looked at some
of the computer calculations, and I’m convinced that the Kirkwood conjecture was correct. However, it might be very difficult to prove that statement. If you look at the calculations with bigger and bigger samples, I think it is very easy to be convinced that there should be a phase transition at a density close to, but not as large as, close packing.【以下是关于开普勒猜想的证明，做了删节，然后黄谈到著名的非对角长程序-ODLRO】Huang: You originated the concept of off-diagonal long-range order (ODLRO).
Is that the same as broken symmetry?Yang: (Long pause) Mmm - I wouldn’t say it’s the same, though it’s related.
In the first place, the ODLRO was already implicit in a paper by Penrose and Onsager. That’s not the famous Penrose. I forgot his initial. It’s a different Penrose.Huang: Oliver Penrose, his brother.Yang: Yes, his brother. They only applied it to bosons. When I was working
on flux quantization with Nina Byers, we dealt with fermions, of course. After the Byers-Yang paper, I began to look into this matter, and realized that the boson idea of Penrose and Onsager, which is quite simple, can be generalized to fermions, and that is a much more complicated, and in some sense a deeper idea. That was the origin of the ODLRO paper17. It is a little bit like the statement “Some atoms, under the right circumstances, would arrange themselves into a crystal.”In fact, I called that diagonal long-range order in my ODLRO paper. In quantum mechanics, there can exist diagonal long-range order, and there can exist off-diagonal long-range order. Aspects of ODLRO are strange to human perception. That’s why “super” phenomena are so fascinating and so strange.Huang: You seem to be saying broken symmetry is a more general concept.Yang: Yes, I would say so.Huang: I recall you once said, broken symmetry is useful only when it leads
to a renormalizable field theory. However, renormalization is not an issue in nonrelativistic problems, such as ferromagnetism and Bose-Einstein condensation. Do you think the idea of broken symmetry is still useful?Yang: I think the statement you recalled was made in the context of the use of
broken symmetry in field theory.Huang: The standard model?Yang: Yes. In general, I would say that broken symmetry, which is a very
general concept, is not related to renormalization.As for broken symmetry and elementary particle physics, I think the story is
not finished. Yes, we have a standard model, and there is broken symmetry, and that leads to a renormalizable field theory. These are very important developments, and amazingly they are in agreement with experiments. But I for one, and I think many people share my opinion, that it’s not the final story. I have no idea what fundamental future developments are needed, but I’m convinced that’s not the end.Huang: So perhaps you feel the so-called Higgs field is not a fundamental field?Yang: There are many proposals on what to do, but I did not find any proposal particularly beautiful and natural. So I think that is one of the open subjects.Huang: In your work with Byers on superconductivity, you explained flux quantization in terms of the supercurrent necessary to minimize the free energy. Is this a general mechanism to drive the system to off-diagonal long-ranged order?Yang: I wouldn’t put it that way. Let’s recall a little of what happened before
1961. Way before 1961, Onsager and London separately had conjectured flux
quantization in superconducting rings. Now, I had written that that showed great physical insight. Nevertheless, their argument was wrong. If you read London’s books Superfluids, in one of them he discussed flux quantization before it was discovered. It’s amazing that he and O but if you read it, you realize his reasoning was totally wrong. As Bloch had later pointed out, if London’s ideas were right, then all matter at any temperature would show flux quantization, because his argument had nothing to do with superconductivity.The important contribution Nina Byers and I made was that it is only because of pairing, which is characteristic of superconductors, that there is flux quantization. In fact, there is great confusion in the literature. Most people have not understood this point. I believe this point was not understood by Sakurai in his textbook.上半部分结束.SubtleTheLord(Subtle-is-the-Lord)　
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