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第 42 期 · 核心追问 Ⅳ·07「研究是怎样做成的？」
The Human Story Lurking behind the Hofstadter Butterfly
节目发布 2018-02-12章节 · 点击跳转视频
0:01
致敬西格巴恩与父亲：讲座缘起
2:59
少年数论：η序列与INT函数
13:50
INT的自相似结构与间断性
18:28
伯克利的抽象之痛与转投粒子物理
20:58
预印本洪流：信噪比困境
23:27
156个粒子与「我不干了」
26:04
三句话打破偏见：晶体也是真空
28:24
Wannier的问题：晶体+磁场
38:33
从Bloch能带到Harper方程
47:07
转移矩阵、胖x轴与能带计数
54:43
11月革命：看见蝴蝶的自相似
58:34
「数字命理学」：说服导师之战
62:27
康托集与抹平：调和数学与物理
67:06
退出物理与蝴蝶的后世影响
74:18
问答：运气、准备与父亲的身教
本期小问 · 档案清单
54:43
研究是怎样做成的？
74:18
伟大的事业从哪来？
62:27
数学是发现，还是发明？
2:59
知识从哪里来？
本期讲者
道格拉斯·霍夫施塔特美国认知科学家、印第安纳大学教授，1975年在俄勒冈大学取得物理学博士，博士论文提出「霍夫施塔特蝴蝶」能谱；著有《哥德尔、艾舍尔、巴赫》，获1980年普利策奖。中文世界通行译名「侯世达」。
01致敬西格巴恩与父亲：讲座缘起
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[Applause] [Music]
[掌声] [音乐]
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thank you very much all I want to say before plunging into the topic itself that I'm very fond of Sweden I lived here for six months quite a long time ago actually fifty-two years ago and founded the of the culture the people the language everything and so it's a great pleasure to be here and it's it's an honor to speak in this wonderful room and I don't know if other people this year have mentioned it but this is 2018 is the hundredth birth year it's not a hundredth year the anniversary at the anniversary of the birth year of kai siegbahn over there this works right there and that's Manas even I assume that's his father right the one right there you know and and so it's it's as I recall kai siegbahn was born on April 20th of 1918 and I saw it's a it's an appropriate year for to be in a Sieben Hall and I know that it's interesting to me I met kai siegbahn on a number of occasions and the last time was about 22 years ago and I noted that some of his work that he was well known for was in
非常感谢。在进入正题之前，我想说的是，我非常喜欢瑞典。我在这里住过六个月，其实是很久以前了，五十二年前，我爱上了这里的文化、这里的人，语言，一切的一切。所以能来到这里我非常高兴，能在这个美妙的大厅演讲也是一种荣幸。我不知道今年是否有其他人提到过，2018年是诞辰一百周年——不是一百年的周年，而是凯·西格巴恩（Kai Siegbahn）诞辰一百周年，就在那边，就是那位，那就是曼内（Manne），我猜那是他父亲，对吧，就在那边的那位。所以，就我记忆所及，凯·西格巴恩出生于1918年4月20日。我觉得在这一年身处西格巴恩大厅是很合适的。我觉得有意思的是，我曾在很多场合见过凯·西格巴恩，最后一次大约是22年前。我注意到，他为人所熟知的部分工作是测量康普顿效应的某些方面，
1:37
measuring aspects of the Compton effect which was actually something I talked about in my earlier lecture on Einstein and that's a very interesting thing because it binds him in a certain sense with my father who was a physicist and who worked also in his early days on measuring the timing of the Compton effect so that's something they had in common and I know they knew each other they met on many occasions so I want to mention also that today happens to be my father's birthday by coincidence and so I thought I would make this lecture in his memory and he had an enormous effect on me and in fact in my thesis which this lecture is really mostly about I had many words of thanks and among the words of thanks the last closing words of my words of thanks were to my father and I said this finally I would like to say that my eternal faith in the beauty and simplicity of nature comes straight from my father Robert Hofstadter and has here acted and will always act as the main guiding principle in my view of the
而这正是我在之前那场关于爱因斯坦的讲座中谈到的内容。这非常有意思，因为从某种意义上说，这把他和我父亲联系在了一起。我父亲是一位物理学家，他早年也从事康普顿效应的计时测量工作。所以这是他们的共同点，而且我知道他们彼此认识，他们在很多场合见过面。我还想提一句，今天恰好是我父亲的生日，所以我想把这场讲座献给对他的纪念。他对我有极其深远的影响，事实上在我的论文中——这场讲座主要讲的是我准备了很多感谢的话，而在这些感谢的话里，最后收尾的那几句是献给我父亲的，我说，最后我想说的是，我对自然之美与简洁性的永恒信念，直接来自我的父亲罗伯特·霍夫施塔特（Robert Hofstadter），它一直是、也将永远是我看待宇宙的
02少年数论：η序列与INT函数
2:59
universe and as certainly true still today and so that's what I wanted to say about my dad he'll come up a couple of times more in the lecture now this is an overview I'm not going to read this to you but basically this what what this lecture does is it tells about how I got into physics by a very circuitous and bumpy route how I was very happy to be in physics for a little while and then got very unhappy the unhappiness is starting around here and and then how I got happy again and then how I got unhappy again and then how I got happy again it's a it's a very bouncy story and it often appeals to graduate students and other young people who are struggling of course no one story appalling the one person will apply to other people but at least it gives a kind of a frame of reference with which you can compare your own travails and debacle --is as you struggle through your hurt your career so let me begin with the happy part which is when I was quite young I was fascinated by mathematics I particularly loved number
主要指导原则。这句话到今天依然千真万确。这就是我想说的关于我父亲的事，他在这场演讲里还会再出现几次。这是一个概要，我不打算念给你们听，但基本上这场演讲讲的是我如何通过一条非常曲折颠簸的路走进物理学，我有一阵子在物理学里非常快乐，后来又变得非常不快乐——不快乐大概是从这里开始的——然后我又变得快乐起来，接着又不快乐，然后又快乐起来。这是一个起起伏伏的故事，它常常能打动那些正在挣扎的研究生和其他年轻人。当然，没有哪一个人的故事能完全套用到别人身上，但至少它提供了一种参照系，让你可以拿来比照自己在职业道路上摸爬滚打时经历的辛劳和挫败。那么让我从快乐的部分讲起吧。在我很小的时候，我就着迷于数学，我特别喜欢数论。大约十六岁那年，我开始
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theory and when I was about 16 I started passionately exploring certain kinds of integer sequences now this is very relevant as it - what comes subsequently so bear with me for a moment although it will seem like it's just an excursion in elementary number theory it is in fact very central to the talk so oh well before I come to the element you think I have to say that I did a lot of the work that I did with friends and one of my best friends from elementary school was Robert burning her Robert and I were both fascinated by mathematics and physics and Robert I always wondered why number theory did not play a role in physics he wrote a letter in 1969 this is considerably after the number theory work was done but he wrote the letter asking this question and the letter got published in a book the book was this book the quotable Fineman edited by fineman's daughter Michelle she went through a lot of his correspondence and his writings and found some of the most interesting things he said and put them
热情地探索某些类型的整数序列。这跟后面要讲的内容非常相关，所以请稍微忍耐一下，虽然它看起来只是初等数论里的一段插曲，但它其实是这场演讲的核心。哦对了，在讲这些内容之前，我得说，我做的很多工作都是和朋友们一起做的，其中一位我小学时最好的朋友之一是罗伯特·伯宁格，罗伯特和我都痴迷于数学和物理，罗伯特一直很纳闷，为什么数论在物理里没有用武之地，他在1969年写了一封信，这已经是我做完那些数论工作很久以后了，但他写了那封信提出这个问题，而这封信后来被收进了一本书里，就是这本《费曼语录》，由费曼的女儿米歇尔编辑，她翻阅了大量他的书信和文稿，找出他说过的一些最有意思的话，编成了这本书，而我碰巧看到了，我
5:29
into this book and I happened to see I happen to come across Roberts question which I didn't know he had written a letter to Fineman and indeed he had and Fineman answered it and here is the quote from the book I don't know why number theory does not find application in physics we seem to need the mathematics of functions of continuous variables complex numbers in abstract algebra but he didn't believe and very few physicists ever thought that number theory would have any connection with physics so they thought so the the work that I did in at Stanford in the early 1960s was largely experimental or empirical if you wish in other words I did a lot of calculation and I used this old calculator fro that my dad did his income taxes on so that was but then I used a Burroughs 220 computer later on at Stanford and then I used other computers as as Stanford got more and more sophisticated in the world of computing so what kind of thing did I explore I explored what I called a de sequences but you can call them sidewalk
碰巧读到了罗伯特的那个问题，我原先并不知道他给费曼写过信，结果他真的写了，而且费曼还回了信，下面是书里的引文：我不知道为什么数论在物理中找不到应用，我们似乎需要的是连续变量函数的数学、复数、抽象代数——但他并不相信，而且极少有物理学家认为数论会和物理有什么联系，他们就是这么想的，所以说，我在斯坦福、在上世纪60年代初做的那些工作，基本上是实验性的，或者你愿意说是经验性的，换句话说，我做了大量的计算，我用的是我爸算个人所得税用的那台老式计算器，就是这样，不过后来我在斯坦福用上了一台 Burroughs 220 计算机，再后来随着斯坦福在计算领域越来越先进，我又用了别的计算机，那我到底探索了些什么呢？我探索的是我称之为 η 序列（eta sequences）的东西，不过你也可以叫它们“人行道
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sequences so what I'm going to is just a very simple idea imagine that there's a sidewalk whose squares are of side one exactly one we're talking precise precision now and I'm going to take steps on this sidewalk that have length square root of two and two every digit you know all the digits of the square root of two so when I walk I'm walking so perfectly and I'm crossing several a couple of lines one or two lines of the sidewalk because the sidewalk squares are a little bit smaller than my steps so what happens here is alpha is the step size and you those red dots are showing you where I land of course it would be infinitely precise and as my steps go on I cross in the first place here one line then two lines then one then two then one and one and two and so forth and so I get a sequence of ones and twos of course those are the two integers that surround the square root of two which is one point four one four and the closer one is one and the second closest one is two the twos occur separately that is isolated Lee the ones
序列”，我要讲的其实是个很简单的想法：想象有一条人行道，每块方砖的边长正好是1，精确地等于1，我们现在讲的是绝对精确，然后我要在这条人行道上迈步，每一步的长度是根号2，精确到每一位数字，你知道的，根号2的所有位数，所以当我走路时，我走得完美无缺，每一步会跨过好几条、或者说一两条人行道的砖缝，因为方砖比我的步长稍微小一点，所以这里发生的情况是：α 就是步长，那些红点显示的是我落脚的位置，当然它应该是无限精确的，随着我一步步走下去，第一步跨过一条线，然后两条线，然后一条，然后两条，然后一条、一条、两条，如此等等，于是我得到一个由1和2组成的序列，当然，这正是夹住根号2的那两个整数，根号2是1.414……，更近的那个是1，第二近的那个是2，那些2是孤立出现的，也就是彼此分开的，而那些1则是成小簇出现的，你可以
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however occur in little clumps you might say there are this clumps are small but sometimes a size one and sometimes of size two but anyway that's what they look like here we have a phenomenon that has involves two competing periodic phenomena one is the periodicity of the sidewalk near there's a periodicity of my steps and so we get some something where two periodicity ZAR sort of fighting with each other now I call the second closest integer there one that is farther away from the square root of two the set because it separates the groups and I call the other integer the closer one I call it the count and you can think of count as being the thing I count I'm going to count those things and you'll see in a moment but my just so that it's easy to remember town begins with a letter C and it also is the same C as in closest integer and set for separator begins with the letter S and it's the same letter as second closest so those are a little mnemonic for what a SEP an account are oh okay so went the wrong direction so like what
说这些簇很小，有时候是1个，有时候是2个，但总之它们看起来就是这样。这里我们看到的现象涉及两个相互竞争的周期现象：一个是人行道的周期性，另一个是我步伐的周期性，于是我们得到了两种周期性彼此较劲的局面。现在，我把第二近的那个整数、也就是离根号2更远的那个，称为 sep，因为它把各组分隔开，而另一个整数、更近的那个，我称之为 cot，你可以把 cot 想成我要去数的东西，我要去数它们，等一下你就明白了，为了便于记忆，cot 以字母 C 开头，跟 closest（最近的）整数的 C 是同一个字母；而 sep 是 separator（分隔符）的缩写，以字母 S 开头，跟 second closest（第二近的）是同一个字母，这就是记住 sep 和 cot 的小口诀。哦，好吧，翻错方向了。那么我接下来要做的，就是我
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I'm gonna do is I'm going to do what I call taking the derivative of the sequence which means I count the number of pounds between steps okay so that's one two one two one now I'm gonna show you a more general case because this turns out to be a very special case I'm gonna show you a more general case of a number where alpha is no longer the score to two office some number between six and seven so here here is the Aidid sequences it belongs to that number it has couns of seven that was the second closest integer and I said I said comes I mean it's EPS the steps are the sevens and accountants are the sixes and there they are occurring four top for them and for them then three of them then for them and then for them and so I count and taking the derivative here and what I get is another sequence consisting of two integers which looks suspiciously like another eight a sequence belonging however to a different number not belonging to this number alpha but to some other number which we could call
所说的“对序列求导”，意思是我数一数相邻两个 sep 之间有多少个 cot，好，于是就是1、2、1、2、1。现在我给你们看一个更一般的例子，因为刚才这个其实是非常特殊的情形，我要给你们看一个更一般的例子，这里的 α 不再是根号2，而是6和7之间的某个数，这就是属于那个数的 η 序列，它的 sep 是7，也就是第二近的整数，我说 cot，我是说 sep——那些7是 sep，那些6是 cot，它们在这里是这样出现的：4个、4个，然后3个，然后4个，然后又4个，于是我数一数，在这里求导，我得到的是另一个由两个整数组成的序列，它看起来很可疑地像是另一个 η 序列，只不过属于另外一个数，不属于这个 α，而是属于另外某个数，我们可以叫它 α′。它是不是某个 α 的 η 序列呢？答案是肯定的。这
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alpha prime is it an alpha and a two sequence and the answer is yes so this is what I called the fundamental theorem of Ada sequences which is when you take the derivative of a native sequence by counting the count between the steps you get another 80 sequence alpha prime of 8 a prime of alpha that's the derivative is a de belonging to another number alpha prime where you can get the Alpha prime by this simple formula which is the SEPA minus alpha over alpha minus the count and that means basically the distance to the second closest integer over the distance to the closest integer so it's another real number over greater than one so what you do then is you can take alpha and get alpha Prime from it and then of course you get off a double Prime and alpha triple prime and you can keep on going you can go on forever if alpha is irrational and we'll think about well temporarily just assume alpha is irrational so that this will go on forever now these these real numbers will be accompanied by the counts and
就是我所说的 η 序列基本定理：当你对一个 η 序列求导，也就是数相邻 sep 之间 cot 的个数时，你会得到另一个 η 序列，即 η(α) 的导数是属于另一个数 α′ 的 η 序列，而这个 α′ 可以由这个简单的公式得到，就是(sep − α) / (α − cot)，这基本上意味着：到第二近整数的距离，除以到最近整数的距离，所以它是另一个大于1的实数。那么接下来你可以从 α 得到 α′，当然接着又能得到 α″、α‴，你可以一直这样做下去，如果 α 是无理数，你就能永远做下去。我们暂且就假设 α是无理数，这样这个过程就会无限进行下去。现在，这些实数都会伴随着与之对应的 cot 和
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steps that are attached to them the closest and the second closest to integers to them so there they are these this is a column of integers and this is a column of integers and these are the second-closest and these are the closest for any real number you can do this as long as it's irrational you can do it for rationals as well but it will terminate so that is what that's what these columns are at real numbers around and integers now let's take an example with if alpha is the square root of 2 it turns out that the derivative sequence is the exact same sequence so in other words alpha prime equals alpha and that means that alpha double prime of course equals alpha as well and so forth so you just have the square root of 2 going down infinitely many times the cons are always 1 and the steps are always 2 that's a very simple example of this phenomenon another very famous constant that is located between 1 & 2 is the golden ratio 1 plus the square root of 5 over 2 and it turns out that it has the
sep，也就是离它们最近和第二近的整数，它们就在这儿：这是一列整数，这也是一列整数，这些是第二近的，这些是最近的。对任何实数你都可以这么做，只要它是无理数；有理数也可以做，只不过会终止。所以这些列就是这么回事：一列实数和一些整数。现在我们举个例子，如果 α 是根号2，结果发现求导后的序列就是原来那个序列，换句话说 α′ = α，这也就意味着 α″ 当然也等于 α，如此等等，所以你就是根号2一路无限地重复下去，cot永远是1，sep 永远是2，这是这个现象一个非常简单的例子。另一个非常著名的常数，同样位于1和2之间，就是黄金比例 (1+√5)/2，结果发现它的两列
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opposite columns that is where it's closer to 2 than it is to 1 and so it's cones are always 2 and B and it's its own derivative so the steps are always 1 so there's a curious complementarity between the square root of 2 and the golden ratio in this case and it might open up a question in more general terms of what happens if you swap the two columns what a number do you get that if alpha is given and you get the column that counts and steps and you swap them what number beta will have that as it comes and ceps so that's the the question I called that function int in this case the square root of 2 is into 5 fires into the square root of 2 but in more general terms you would say okay alpha gives these towns and these steps flip them cross them now what number beta is such that it has those and alpha equals into beta beta equals int of alpha and for interchange interchange the two columns that was the dysfunction I invented now I you know I did lots and lots of things in these several years
正好相反，也就是说它离2比离1更近，所以它的 cot 永远是2，而且它是它自己的导数，所以 sep 永远是1。所以在这个例子里，根号2和黄金比例之间存在一种奇妙的互补性。这可能会引出一个更一般的问题：如果把这两列互换会怎么样？给定 α，你得到 cot 那一列和 sep 那一列，把它们互换，那么哪个数 β 会以它们作为自己的 cot 和 sep？这就是那个问题，我把这个函数叫做 INT。在这个例子里，根号2 = INT(φ)，而 φ = INT(根号2)。但更一般地说，你会说：好，α 给出这些 cot 和这些 sep，把它们翻过来、交叉过来，那么哪个数 β 拥有这样的两列？于是 α = INT(β)，β = INT(α)。INT 取自 interchange（互换），互换这两列——这就是我发明的那个函数。要知道，在我疯狂探索数论的那几年里，我做了非常非常多的事情，这只是我所做工作的很小一部分，
03INT的自相似结构与间断性
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that I was exploring number theory like crazy this is only a very small part of what I did but it turns out to that it's the most relevant part and so it's important for me to to tell you about this now what does this function look like I when I was first calculating it I did it all by hand and then I did it with the help of the little calculator and then I did with a local computer I didn't have computer graphics I had to graph it all by hand this is of course a much later graph done by a computer and it shows you the solution which I did not see at the beginning I only saw what I saw was something that looked like this but it was much blurrier and it sort of looked like a little bumpy thing that went along and then kind of took a big leap here and then kind of bumpy here I couldn't tell what was going on later on I kind of realized I understand to understand what was going on and I'll tell you about that momentarily so I just want you to take a look at this because this shape this thing that it
但事实证明它是最相关的一部分，所以对我来说，把这个讲给你们听很重要。那么这个函数长什么样呢？我最初计算它的时候完全是手算，后来我借助那台小计算器算，再后来我用本地的计算机算。我没有计算机图形，只能全部手工绘图。这当然是很久以后由计算机画出来的图，它展示出了我当初没看出来的答案。我当时看到的只是类似这样的东西，但要模糊得多，看上去有点像一条坑坑洼洼的曲线往前走，然后在这里猛地跳一下，接着又是坑坑洼洼的，我完全看不出到底怎么回事。后来我才慢慢明白发生了什么，这个我待会儿再讲。我现在只想让你们好好看看这张图，因为这个形状，这些
14:53
where it gets these ribs we could call this a rib and another rib in another rib as they go toward the corner they get smaller and smaller and also this is a little curved and this gets a little less curved and a little less curved so they get a little bit less make it a little bit straighter you might say and they get shorter and they shrink as they go toward the corner that's a telltale phenomenon that occurs later in this talk so what I wanted to say is that that this is between 0 & 1 the same graph repeats itself here here between one and two and then between two and three and so forth so this is really the the main phenomenon and if you look at it you will see that the ribs themselves are composed of sub ribs and of course you can't see but the sub ribs are composed of sub sub ribs and so forth so there it turns out to be a structure that is made of copies of itself albeit distorted copies because this is straight that's a straight diagonal whereas this is curved and so is this each one of these is curved but
“肋”——我们可以把这叫做一条肋，这是另一条，这又是一条——随着它们越靠近这个角，就变得越来越小，而且这一条稍微有点弯曲，这一条弯曲得少一点，再一条又更少一点，所以它们越来越平直，可以说是越来越接近直线，同时它们越来越短，越靠近角就越收缩，这是一个很有代表性的现象，在这个讲座后面还会出现。我想说的是，这一段是在0和1之间，同样的图形在这里、在1和2之间又重复一遍，然后在2和3之间再重复，如此等等，所以这才是主要的现象。如果你仔细看，就会发现这些肋本身又是由次级的肋组成的，当然你看不见，但这些次级肋又由次次级肋组成，如此等等，所以结果是一个由自身的拷贝构成的结构，尽管是变形了的拷贝，因为这条是直的、是一条笔直的对角线，而这条是弯的，这条也是，每一条都是弯的，只不过越往下、越靠近末端就弯得越少。
16:01
they're curved less as they get down toward the end so what we can say is that int consists of copies of itself but that are bent and I might explain that the flipping of the cons and SEPs is responsible the top-level flipping of the top count and Sep is responsible for this anti-diagonal the second-level flipping of counting steps is responsible for these ribs the third level is responsible for the sub ribs etc so each level that you go down gives you another nesting and so forth so ant is composed of infinitely many smaller and distorted copies of itself that was exciting to me discovery I made at some point and I proved these results I was very very diligent about trying to prove everything that I discovered and it was very exciting and really intoxicating I would say for a young person to be discovering these things and I felt that I was going to go on and be a number theorist and that was my great hope and so I went on to graduate school Oh last thing about int before I quit here int it takes a jump discontinuity let's
所以我们可以说，INT 是由它自身的拷贝组成的，只不过这些拷贝被弯曲了。我还可以解释一下，cot 和 sep 的互换正是原因所在：最顶层 cot 和 sep 的互换，造就了这条反对角线；第二层 cot 和 sep 的互换，造就了这些肋；第三层造就了那些次级肋，以此类推。所以你每往下一层，就多一层嵌套，如此等等。所以 INT 是由无穷多个更小的、变形了的自身拷贝构成的。这是我在某个时刻做出的、让我非常兴奋的发现。而且我证明了这些结果，我极其勤勉地努力去证明我发现的每一件事，这非常令人兴奋，对一个年轻人来说，能发现这些东西真的可以说是令人陶醉的。我当时觉得我会继续走下去，成为一名数论学家，那是我极大的期望，于是我上了研究生院。哦，在结束之前，关于 INT 还有最后一件事：INT 有跳跃间断点。我们回到这儿，也许看这里，嗯，它在这里有一个跳跃，一个很大的跳跃间断，
17:17
go back here maybe see here um it takes a jump very big jump discontinuity here at 1/2 this is exactly 1/2 and also at 1/3 and also 2/4 and also 2/5 and also at 2/3 and at 3/4 and at 4/5 etc so it takes jump jumps at rationals it takes also at 2/5 at 3/7 and all sorts of place so it takes a jump discontinuity at every rational but it you could say it takes a jump discontinuity that is related to how big the denominator is and as the denominator gets bigger the jump gets smaller so for an irrational number if you think of an irrational number as being sort of a limiting process of getting bigger and bigger numerators and denominators then the irrational number is sort of an infinite integer of and another infinite integer so the jump discontinuity turns out to be of size zero meaning it is a continuous function that all Irrational's that's an interesting quality of int discontinuous that all rationals continues all Irrational's now I went to graduate school at Berkeley which was a very good school however I
就在 1/2 处，这正好是 1/2；在 1/3 处也有，1/4 处也有，1/5 处也有，还有 2/3、3/4、4/5 等等，所以它在有理数处发生跳跃，在 2/5、3/7 以及各种各样的地方也都有，所以它在每一个有理数处都有跳跃间断。但你可以说，这个跳跃间断的大小跟分母的大小有关，分母越大，跳跃就越小。所以对于一个无理数，如果你把无理数看作是分子分母越来越大的某种极限过程，那么无理数就相当于一个无穷大的整数比上另一个无穷大的整数，于是跳跃间断的大小就变成了零，也就是说，它在所有无理数处是连续的。这是 INT 一个有趣的性质：在所有有理数处不连续，在所有无理数处连续。那么，我去了伯克利读研究生，那是一所非常好的学校，但是我被要求修的课程
04伯克利的抽象之痛与转投粒子物理
18:28
was required to take courses that were very abstract and to my great surprise I found in them too abstract and and couldn't relate to them I couldn't see images I couldn't understand what I was being asked to absorb I had to do this as a fledgling graduate student and it was extremely traumatic I will have you I will tell you it was very very upsetting to me number theory I took a course in number three and they never mentioned the integers I mean you might say they mentioned it atures they mentioned numbers of some sort but there were very high abstractions they were not my familiar integers that I love and I was I got more and more upset and finally I realized I just had to get out mathematics was not going to be my future as I had hoped it would be and that was a very very difficult time in my life but I made the decision to make a leap away from mathematics and also away from Berkeley which at the time in the mid 60s was a a hotbed of extreme political activity which was just a little bit too intense for me
极其抽象，让我大为吃惊的是，我觉得它们太抽象了，我跟它们产生不了联系，我看不到图像，我没法理解人家要我吸收的东西。作为一个刚起步的研究生，我必须硬着头皮学，这对我造成了极大的创伤，我得告诉你们，这让我非常非常难受。数论——我修了一门数论课，他们从来不提整数。我是说，你也许可以说他们提到了整数、提到了某种意义上的数，但那是非常高度的抽象，它们不是我熟悉的、我热爱的那些整数，我越来越沮丧，最后我意识到我必须离开，数学不会是我的未来了，尽管我曾经那么希望它是。那是我一生中非常非常艰难的一段时期，但我做了决定，从数学纵身一跃跳开，同时也离开伯克利——那时候是60年代中期，伯克利是极端政治活动的温床，对我来说实在有点太激烈了。于是我在1968年跳了这一跃，从地理上说这一跃是
19:40
and so I took this leap and 1968 and this is the leap I made geographically from Berkeley to Eugene Oregon and I took the other leap of going from from mathematics to physics or if you wish in particular from number theory to particle physics that's a Fineman diagram to symbolize particles and I I was making this this leap but I as I said I didn't just make it from math to physics I made it deliberately into particle physics because I was suffering from a prejudice a prejudice that I later found out was definitely a prejudice which was the particle physics that physics is only about particles all the rest of physics is just engineering or not serious stuff the serious stuff is particles or maybe general relativity but other than that that was what physics was other than right there's nothing but that so now in this very auditorium a couple of months ago Shelley Glasgow gave a very fun talk about his days which were somewhat earlier than mine but nonetheless he said and I quote particle physics was
从伯克利到俄勒冈州的尤金；同时我还跳了另一跃，从数学跳到物理，或者更具体地说，从数论跳到粒子物理——那是一张代表粒子的费曼图。我当时就在完成这一跃，但正如我说的，我不只是从数学跳到物理，我是刻意跳进粒子物理，因为我抱有一种偏见，一种我后来才发现确实是偏见的偏见，那就是：物理就只关乎粒子，物理学的其余部分不过是工程学，或者不是什么正经东西，正经的东西是粒子，或者也许还有广义相对论，除此之外——那就是物理的全部，除此之外别无他物。就在几个月前，就在这个礼堂里，谢尔顿·格拉肖做了一个非常有趣的报告，讲他那个年代的事，比我要早一些，但不管怎样，他说——我引用一下——那个年代粒子物理很容易，因为
05预印本洪流：信噪比困境
20:58
easy in those days because there was so much low-hanging fruit well maybe so when you have an advisor like Schwinger who hands you a perfect thing and says work on this problem however it wasn't so easy for me and I'm going to tell you a little bit about it in my position so here we are lost up in Eugene we had a preprint library in the it stood for theoretical science and this would be about how much came in every week and it was very very difficult to figure out what to read and what not to read let I call it the vast reverberating din here are some of the particles for example that were in the air at that time quarks and gluons and Pomeranz and pree quarks and Parton's and instant ons and accion's and gravitons and solitons and tachyons and gravity nose and Goldstone bosons and Higgs bosons and squarks and foti knows quite a number of things for a graduate student to try to understand and make sense not to mention Reggie poles and magnetic monopoles and Veneziano model and my Rana neutrinos
到处都是唾手可得的果实。嗯，也许吧，当你有一位像施温格那样的导师，把一个完美的题目递给你说“去做这个问题”的时候。然而对我来说可没那么容易，我要跟你们讲讲我当时的处境。我们当时孤零零地待在尤金，我们有一个预印本图书馆，在 ITS，也就是理论科学研究所，每周大概会来这么多预印本，要弄清楚该读什么、不该读什么非常非常困难，我把它叫做“浩大的嗡嗡回响”。比如说，当时空气中飘着的那些粒子有：夸克、胶子、坡密子、前夸克、部分子……还有瞬子、轴子、引力子、孤立子、快子、引力微子、Goldstone 玻色子、Higgs 玻色子、标量夸克、光微子对一个研究生来说，要去理解、去弄懂的东西可真不少，更别提 Regge 极点、磁单极子还有 Veneziano 模型、Majorana 中微子——我发誓我不会全都念完——但这些只是其中的一部分
22:14
and I'm not going to read it all I swear to you but these are some of the things that we're coming in in that pile of preprints every week and so were these things coming in quantum flavor dynamics and spontaneous symmetry breaking and neutrino oscillations and asymptotic freedom and the kabhi bow angle and the renormalization group and bare Cain scaling Anna Weinberg angle and Susie guts and plunk lengthen PCAC and su3 cross su3 and dispersion relations and quark confinement in SU 12 and QCD and lattice gauge Theory etc not to mention strangeness changing neutral weak currents unitarity violating deep inelastic processes cabbie bo kobayashi maskawa matrix idler bellyache even amélie relativistic superconductors running coupling constants and so forth ad nauseam I found myself at my wit's end and I changed from well well okay signal-to-noise problem how can you recognize the great gems in the gigantic garbage dump how can you know what to follow and what to ignore and it was unbelievably difficult I jumped from one
每周涌进那一大堆预印本里的东西，而且这些也在不断涌来：量子味动力学自发对称性破缺、中微子振荡、渐近自由、Cabibbo 角重整化群、Bjorken 标度律、Weinberg 角、超对称大统一、Planck 长度、PCAC、SU(3)×SU(3)、色散关系夸克禁闭、SU(12)、QCD、格点规范理论等等，更别提改变奇异数的中性弱流破坏幺正性的深度非弹性过程、Cabibbo–Kobayashi–Maskawa 矩阵、Adler–Bell–Jackiw 反常、相对论性超导体跑动耦合常数等等等等，没完没了。我发现自己已经黔驴技穷，于是我就变了——嗯，好吧这是个信噪比问题：你怎么才能在一个巨大的垃圾堆里认出真正的宝石？你怎么知道该追随什么、该忽略什么？这难得令人难以置信。我从一个导师换到另一个，又换到另一个，再换到另一个
06156个粒子与「我不干了」
23:27
advisor to another to another to another and found nothing to give me clarity the big day was December 14th 1973 when I had been asked to give a talk on a paper that was about something engaged theory and I did my best to read it and to summarize it but I hated the paper because in this paper they introduced 156 new particles in one fell swoop and I was very familiar with the fact that in 1930 Paulie had had introduced the neutrino he called it a neutron because the neutron what we call today the neutron had not yet been discovered and so he was inventing the a neutral particle and he called it the neutron later on it was renamed a neutrino by Fermi but in any case he was that in order to save three fundamental conservation laws of physics conservation of energy conservation of momentum and conservation of angular momentum to save all three of them he could introduce one particle that would carry away momentum energy and angular momentum from a beta decay and and he was so frightened of this wild crazy
却始终找不到能让我看清方向的东西。转折的大日子是 1973 年 12 月 14 日，那天我被安排去讲一篇论文那是一篇关于某种规范理论的文章，我尽力去读它、总结它，但我讨厌这篇论文，因为在这篇论文里他们一口气引入了156 个新粒子。而我非常清楚这样一个事实：1930 年 Pauli 引入了中微子——他当时管它叫 neutron（中子），因为我们今天所说的中子那时还没有被发现所以他是在发明一种中性粒子，他把它叫做中子，后来被 Fermi 改名为中微子。总之他这么做是为了挽救物理学的三条基本守恒定律：能量守恒、动量守恒和角动量守恒。为了把这三条都保住，他可以引入一个粒子，让它从 β 衰变中带走动量、能量和角动量。而他对这个疯狂大胆的
24:45
hypothesis that he didn't dare write a paper about it but instead he he wrote a letter to a friend about it and the friend read the letter out loud in front of a conference and that's how paulie's idea became known so my point is the Pauli to introduce one particle and he was one of the most brash physicists imaginable was very frightened and here are these physicists inventing 156 new particles for almost no reason and I said to the people that I was I was giving this presentation in front of 10 people or something like that and I when I came to this point I said these people have no sense of shame and I threw the paper on the ground and I walked out and I said I quit and at that point I was lost and this blank slide represents that I had to invoke and invested six years at that point of my life into particle physics and I had nothing and I thought I'm never going to get a PhD what am I gonna do what can I do it was a very very frightening moment so it happened that my dad had at Princeton
假说害怕到不敢为此写一篇论文，而是给一位朋友写了一封信那位朋友在一个会议上当众把信念了出来，Pauli 的想法就是这样为人所知的。所以我想说的是Pauli 只是引入一个粒子——而他是你能想象到的最莽撞的物理学家之一——就已经非常害怕了；可如今这些物理学家几乎毫无理由地就发明了 156 个新粒子。于是我对在场的人说——我当时是在大概十个人面前做这个报告——讲到这一点时，我说：这些人一点羞耻心都没有。然后我把论文扔在地上，走了出去，我说：我不干了。而在那一刻我彻底迷失了。这张空白的幻灯片就代表着这一点：那时我已经把生命中的六年投进了粒子物理，却一无所获，我心想：我永远拿不到博士学位了，我该怎么办？我还能做什么？那是一个非常非常可怕的时刻。事情是这样的我父亲当年在普林斯顿读研究生，那里有一位瑞士物理学家，一位非常杰出的
07三句话打破偏见：晶体也是真空
26:04
gone to a graduate school there where a Swiss physicist a very distinguished Swiss citizen was a solid-state theorist was was there as a postdoc and my dad happened to Gregory wanye was at the University of Oregon so when I first went to the University of Oregon as a gung-ho particle person I introduced myself to wanye he was always extremely polite to me very nice and invited me to his house for dinner several times so I got to know one yay but I never dreamed that I would ever think of having him be my adviser because to me going to into solid state was slumming it was going into the slums of physics how could I even envision imagine such a thing this was my prejudice this is my very deep prejudice now I got out of this prejudice because of three people my first was my great friend Pete rim B who was in graduate school with me and he just simply said Greg long Gregory won yeah he's one of the deepest alive today well that was a pretty serious statement I took very seriously because I admired
瑞士人，是搞固体理论的，当时在那儿做博士后。而我父亲……Gregory Wannier 当时在俄勒冈大学。所以当我最初以一个热血的粒子物理人的身份来到俄勒冈大学时，我去向 Wannier 做了自我介绍。他对我一直极为客气、非常和善，还好几次请我去他家吃饭，所以我算是认识了Wannier，但我做梦也没想过会让他来当我的导师，因为在我看来，转去做固体物理就是自甘堕落，是走进物理学的贫民窟。我怎么可能设想、想象这样的事？这就是我的偏见，是我根深蒂固的偏见。而我之所以摆脱了这个偏见，是因为三个人。第一个是我的好朋友 Pete Rimbey，他和我一起读研究生，他就只是说了一句：Gregory Wannier 是当今在世最有深度的人之一这可是一句很郑重的话，我非常当真，因为我很敬佩 Pete。我的好朋友、来自智利的 Francisco Claro
27:15
Pete my great friend Francisco Claro from Chile said solid state is a wonderful subject because it builds a bridge between two worlds the microscopic and the macroscopic I had never thought of it that way and I was also very impressed by that and by the way Francisco did his work with Gregory 1yi uh and then finally the last person was an unknown Stanford physics graduate student who I was speaking with one day and this person made this amazing remark he says particle physics has only one vacuum the continuous vacuum of empty space but solid-state physics has many vacuums every single crystal is a different kind of vacuum and this was a poetic and beautiful thought because it really was saying that a crystal is like a medium which is a discrete medium as opposed to a continuous medium it has a periodicity it's not isotropic the way space is it is it has preferred axes and if all of a sudden I reminded me of number theory it reminded me of the distinction between the real line and the lattice the one-dimensional lattice
说：固体物理是个了不起的学科，因为它在两个世界之间架起了桥梁——微观世界和宏观世界。我从没那样想过，这也让我印象非常深刻。顺便说一句，Francisco 就是跟着 GregoryWannier 做的研究。然后最后一个人，是斯坦福一位我并不认识的物理系研究生，有一天我和他聊天这个人说了一句非常了不起的话，他说：粒子物理只有一个真空，也就是空荡空间那个连续的真空但固体物理有许多个真空，每一种晶体都是一种不同的真空。这是一个既有诗意又很美的想法，因为它其实是在说：晶体就像是一种介质，一种离散的介质，而不是连续的介质。它有周期性，它不像空间那样各向同性，它有偏好的轴向而这一下子让我想起了数论，让我想起了实数轴与格点之间的区别——整数构成的一维格点
08Wannier的问题：晶体+磁场
28:24
of integers so in some sense number theory is kind of like solid-state mathematics it's a one-dimensional crystal in superimposed on on the the vacuum the real vacuum which is the real line so that all of a sudden I started seeing to follow the state physics in a completely different way it wasn't slumming it there was something quite beautiful closely related to number theory well then could I go swimming with an Oregon duck well I say Oregon duck because that's the mascot of the University of Oregon so there's a picture of Juan EA and he said to me when I went to him to ask him if he might possibly be interested in having me as a graduate student he said I have three problems for you to consider I don't remember the first two that he told me about but I vividly remember the other one it had to do with block bands in a crystal mentioned block bands in discovered by Felix Bloch in 1928 that electrons in a crystal have a continuum of energy values forming a band this is a set of continuous band of energy levels that
所以在某种意义上，数论有点像是固体物理的数学版，它是一维晶体，叠加在真空之上——那个真正的真空，也就是实数轴。于是突然之间，我开始用一种完全不同的方式来看待固体物理。它并不是自甘堕落，那里面有某种相当美的东西，而且与数论密切相关。那么我是不是可以去跟一只俄勒冈鸭一起游泳呢？我说俄勒冈鸭，是因为那是俄勒冈大学的吉祥物。这是 Wannier 的照片。当我去找他、问他有没有可能愿意收我做研究生时，他对我说他说：我有三个问题可以给你考虑。他讲的前两个我记不得了，但我非常清楚地记得另外那一个，它和晶体中的 Bloch 能带有关。Bloch 能带是Felix Bloch 在 1928 年发现的：晶体中的电子拥有连续的一片能量值，形成一条能带，这是一组连续的、被允许的能级构成的带；而在它上方突然出现一个能隙，在它下方也有一个能隙
29:33
are allowed and then above it suddenly there's a gap and below it there's a gap so there's a band this is called a block band forbidden energies and allowed energies I might say that Felix Locke was at Stanford which is where I grew up and he was in the same department as my dad and in fact he was my dad's closest colleague so I had known Felix ever since I was a little boy and he was a very close friend of our families and here's a picture of my dad and Felix having just climbed the wonderful peak in the Sierras in 1953 my dad is on the right and Felix is on the left so these are two very dear people to me now the other thing that the problem that Gregory wanye was describing to me I had to do with Landau levels which is when you have electrons in not in a crystal but it's just in empty space and you ask what are the energy values of electrons in such a circumstance electrons in space in a in a magnetic field and the answer is that they are evenly spaced and then there's a gap between these
所以就有了一条带，这就叫 Bloch 能带——禁戒的能量和允许的能量。我要说一句，Felix Bloch当时在斯坦福，也就是我长大的地方，他和我父亲在同一个系，事实上他是我父亲最亲密的同事。所以我从小就认识 Felix，他也是我们家非常亲近的朋友这是我父亲和 Felix 的一张照片，1953 年他们刚爬完内华达山脉里一座很棒的山峰。我父亲在右边，Felix 在左边。这是两位对我来说非常珍贵的人。而另一件事，就是Gregory Wannier 向我描述的那个问题，它和朗道能级有关。那是说，电子不是在晶体里而只是在空荡的空间里，你要问在这种情形下电子的能量值是多少——处在空间中、处在磁场中的电子。答案是它们是等间距的，而这些
30:42
evenly spaced levels those are called Landau levels and so the natural question was to combine the two situations - in other words put the magnetic field onto this discreet vacuum and what happens now with electrons in a crystal in a magnetic field and its honor it was a very very fundamental problem basically you're taking of electrons in a vacuum I put it in quotes because it's a it's a crystal but it's it's a discrete vacuum and just applying a magnetic field to it what could be more natural question than that however it's it's not at all easy and it had been 40 years that the problem was first opposed and and it still didn't have an answer here is a picture we'd just imagine that if the crystal is only a two-dimensional crystal of a square lattice a on a side and a on the perpendicular side and so we look at the flux that passes through a lattice cell and that flux is the magnetic field times this the area of the cell of the lattice now it turns out that nature hands us a unit of flux that is natural
等间距的能级之间有间隔，这些就叫朗道能级。于是自然的问题就是把这两种情形结合起来——换句话说，把磁场加到这个离散的真空上，那么晶体中处在磁场里的电子会发生什么？这是一个非常非常基本的问题。基本上你是在把真空里的电子——我给真空加了引号因为它是晶体，但它是一种离散的真空——然后给它加上一个磁场。还有什么问题比这更自然呢？然而这一点都不容易，从这个问题最初被提出算起已经过了 40 年却依然没有答案。这里有一张图，我们不妨设想晶体只是一个二维的正方格子晶体，边长是 a，垂直方向也是 a，于是我们来看穿过一个格胞的磁通量这个通量就是磁场乘以格胞的面积。而结果是，大自然给了我们一个天然的通量单位，也就是所谓的磁通量子，磁通量子是 hc 乘以 1/e
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the so called flux quantum the flux quantum is a product of HC and 1 over E it's a natural amount of flux which occurs in physics and that it's a it's a natural measuring tool to measure other fluxes with so if you divide the flux that I already spoke of a squared H by this natural unit you get a pure number that measures the magnetic field in terms of the magnetic flux quantum ok this is a fundamental number this alpha number for electrons in a crystal a square lattice in this case in a magnetic field and so that number is this key number in this situation and Gregory said to me it turns out that rational and irrational values of alpha act differently now this sounded to me absolutely crazy and I couldn't believe it obviously how can how can nature think of take into account all the infinitely many decimals of a real number I mean no number is defined that well it made no sense but he said that this it turned out that this is a fact this is well known it's been it's been studied and people have studied this for many many
它是物理中出现的一个自然的通量大小，是一把天然的尺子，可以用来量度其他的通量。所以如果你把我刚才说的那个通量 a²H 除以这个自然单位，你就得到一个纯数它用磁通量子来度量磁场的大小。好，这是一个基本的数，这个 α对于处在磁场中的晶体（这里是正方格子）里的电子来说，这个数就是这个问题中的关键数。而Gregory 对我说：结果是 α 取有理数和取无理数时表现完全不同。这在我听来简直是疯了，我显然没法相信：大自然怎么可能去考虑一个实数无穷多位的小数呢？我是说，没有哪个数被定义得那么精确，这说不通。但他说，事实证明这就是事实，这是众所周知的，已经被研究过，人们研究这个已经很多很多年了。而我不信，我只是
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years and I didn't believe it I just thought surely you're joking mr. wani a rational and irrational and so that of course reminded me of my old work having to do with int and with many other things that I had explored in those number Theory days and I thought to myself go for it it's a good chance maybe some of your experience will come in handy I didn't know I was just taking taking a wild guess so I had to retool myself and become sort of familiar at least a little bit with solid-state physics and I had to learn about these kinds of things that that exist in solid-state physics so I'm gonna focus down on just two of them because they have to do with two crucial people blaah and wanye a block state is the most fundamental state of electron in a crystal it's a sort of a standing wave that is distributed over the entire crystal and it has a phase factor that is sort of e to the ikx where k is a wave number that kind of goes like this a helix as you go down the crystal and it didn't it didn't elude me that the
心想：Wannier 先生，您一定是在开玩笑吧——有理数和无理数？当然，这让我想起了自己早年的工作，那些和 INT 有关的东西，还有我在那段数论岁月里探索过的许多别的东西。我心想：干吧，这是个好机会也许你的一些经验会派上用场。我并不知道，我只是在瞎猜。所以我不得不重新武装自己，至少对固体物理稍微熟悉一点，我得去学这些固体物理里存在的东西。我只打算集中讲其中两个，因为它们和两位关键的人物有关：Bloch 和 Wannier。Bloch 态是晶体中电子最基本的状态，它是一种分布在整个晶体上的驻波，它带有一个相位因子，大致是 e^(ikx)，其中 k 是波数，它沿着晶体走下去的样子就像一条螺旋线（helix）。而我并没有忽略这样一件事：
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word helix for this phase factor rhymed with the name Felix and so I will show you that in a moment Gregory wanye took these lock states and he took the Fourier transform of them and that made them localized on a single atom or on a single nucleus these became known as wanye functions or wanye States and they're often used as an alternative to block states to expand wave functions so I decided I would write a limerick about Felix he was actually Heisenberg first PhD student a Heisenberg student named Felix thought crystals were swell psychedelics it boggled his mind when he happened to find that a block state repeats Helix then I wrote one about Gregory wanye the physicist gregory wanye left his land for the US of A there in solids with lots of knock waves but dots he transformed Bloc functions away so I had fun I wrote limericks about my fellow graduate students I wrote limericks about my professors and I was having fun while I was not doing limericks I was actually very involved on another
描述这个相位因子的词 helix 和 Felix 这个名字押韵，待会儿我会给你们看。Gregory Wannier 拿这些 Bloch 态做了傅里叶变换，这就使它们局域在单个原子或单个原子核上，这些后来被称为 Wannier 函数或 Wannier 态，人们常用它们代替 Bloch 态来展开波函数所以我决定给 Felix 写一首打油诗。他其实是 Heisenberg 的第一个博士生：有个 Heisenberg 的学生名叫 Felix，觉得晶体是绝妙的迷幻药；当他偶然发现 Bloch 态像螺旋线一样重复时，简直目瞪口呆。然后我又给 Gregory Wannier 写了一首：物理学家 Gregory Wannier，离开家乡来到美国，在固体中有一大堆波和结，他把 Bloch 函数变换掉了。我玩得很开心，我给我的同学们写打油诗，也给我的教授们写打油诗，我玩得不亦乐乎。而在不写打油诗的时候，我其实非常投入地在做另一个项目，它占用了我本该用来做正事的时间。我在写一本书，叫《哥德尔定理与人脑》
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project that was taking time away from Torah as they say I was working on a book called girdle serum and a human brain that wasn't the eventual title that it had but that was the title that I was using and it looks it it took a great deal of time it I really was working on it far too much this was it was it was I got very scared that I was not gonna finish my PhD if I kept on working on this book so recently I discovered in a box that I had not looked at in some 30 or 40 years this document I guess it's 40 years this is written up in 1974 at some point so it's actually 44 years ago I don't know if you can read it but it says Society for the sanity and survival of D H and that's me mmm Constitution and it consists of a whole bunch of handwritten rules including penalties that I would have if I did not work on my PhD thesis every day and I had all sorts of rules monetary fines that I would impose on myself and I put my manuscript of my book gödel serum in the human brain which of course became gödel Escher Bach but I
——那并不是它最后的书名，但当时我用的是这个名字。它花了我非常多的时间，我真的在它上面花得太多了。当时我非常害怕，如果继续写这本书，我就会拿不到博士学位。最近我在一个大概三四十年没打开过的箱子里翻到了这份文件——我想是 40 年吧，这是 1974 年某个时候写的，所以其实是 44 年前。我不知道你们能不能看清，上面写着《D.H. 神志清醒与生存协会》——D.H. 就是我——嗯，章程。里面是一大堆手写的规则，包括如果我没有每天做博士论文就要受的惩罚。我给自己定了各种各样的规矩还有要对自己处以的罚款。然后我把我那本《哥德尔定理与人脑》的手稿——它当然后来变成了《哥德尔、埃舍尔、巴赫》——把那份手稿放进了一个抽屉，而打开
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put that manuscript into a drawer and I it was very severe penalties for opening that drawer so that that's an interesting moment in my life I actually didn't touch the manuscript for well over a year didn't do it at all in 1974 Gregory wanye was going to Regensburg Germany and I asked if I could come along he worked out for me to be able to come and I there was a favela that did stir shefte an assistant chef kitchen assistant meaning teaching assistant in the layer stool Obermeyer Gustave obermeyer was the director of this particular part of the physics department and so here we have the four people that form this little group you may be able to recognize one of them and so that's me Alexander Rao was my best friend while I was in he was a young professor that's Gustavo virmire who ran the group and there's Gregory and we worked on this problem together now the thing about it is that the three of them the three on the right were all really master manipulators of differential equations and hyper geometric series and all sorts
那个抽屉的惩罚非常严厉。所以那是我人生中一个很有意思的时刻。我确实有一年多完全没有碰过那份手稿，一次都没碰。1974 年Gregory Wannier 要去德国雷根斯堡，我问我能不能一起去。他帮我安排好了，让我能去我在那儿有一个 Verwalter der Dienstgeschäfte 的职位——听起来像是助理厨师、厨房帮手，其实意思是助教，在 Obermair 教席里。Gustav Obermair 是物理系这一部分的负责人。所以这就是组成这个小组的四个人，你们也许能认出其中一位。那是我；Alexander Rauh 是我在那儿时最好的朋友他是一位年轻的教授；那是主持这个组的 Gustav Obermair；还有 Gregory。我们一起研究这个问题。而问题在于，他们三个——右边那三位——都是真正的高手，特别擅长摆弄微分方程、超几何级数之类的各种东西，而我不是。我
09从Bloch能带到Harper方程
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of things whereas I was not and I was always whenever we would meet in in somebody's office I would sit around feeling very inferior but I did have one advantage over them which I will come to in a moment but now we plunge into the physics of this problem for a while so you can understand what the situation is what is the Schrodinger equation for a physical situation that involves an electron in a crystal in a magnetic field so we need a Schrodinger equation so we need a Hamiltonian so we start with a block band which means that for every wave number K X K Y a two-dimensional crystal there is an energy that is the energy inside the band so given a wave number as the wave number varies the energy varies and covers the entire band a being the lattice spacing this is the simplest possible the simplest possible Hamiltonian for a block band not a Hamiltonian energy function for a block band we're going to use this energy function to create a Hamiltonian so in order to make an operator hamiltonian operator out of a out of a
每次我们在某个人的办公室碰头时，总是坐在那儿觉得自己很不如人。但我确实有一个胜过他们的优势，这个我待会儿再讲。现在我们先深入到这个问题的物理里去一会儿，好让你们明白情况是怎样的。对于一个涉及晶体中处在磁场里的电子的物理情形，薛定谔方程是什么？我们需要一个薛定谔方程，所以我们需要一个哈密顿量。于是我们从一条 Bloch 能带开始，这意味着对于二维晶体中每一个波数 kx、ky，都有一个能量，也就是带内的能量所以给定一个波数，随着波数变化，能量也随之变化，并覆盖整条带。a 是晶格间距。这是最简单的可能的——对一条 Bloch 能带来说最简单可能的哈密顿量，不，是能量函数。对于一条 Bloch 能带，我们要用这个能量函数来构造一个哈密顿量。为了从一个经典的数造出一个算符、一个哈密顿算符，我们用 Dirac 的配方
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classical number we use the Dirac recipe of substituting for a number a an operator a differential operator from a C number as Dirac called when to a queue number as he called them and we the the momentum becomes h-bar over I times del the differential operator and if there's an electromagnetic field then we have to add on this term this is the magnetic vector potential from which the electromagnetic field can be derived and and so this goes to that this operator goes to that operator so we want to ask well what what would be the vector potential when you have a uniform magnetic field and the answer is there are many answers there different gauges that work but we took the landau gauge the simplest way of getting a constant magnetic field the curl of this that a vector potential is a constant field in the Z direction so that was our vector potential now what we're gonna do is we're gonna perform the Dirac we're going to figure out how to perform the direct substitution on that on that energy
把一个数替换成一个算��、一个微分算符——用 Dirac 的说法，就是从 c 数变到 q 数数（他是这么叫它们的），然后动量就变成 h-bar 除以 i 再乘以 del，也就是那个微分算符；如果存在电磁场，那我们就得再加上这一项，这就是磁矢势，电磁场可以由它导出，所以这个就变成那个，这个算符就变成那个算符。那么我们要问，当你有一个均匀磁场时，矢势应该是什么？答案是有很多种答案，有不同的规范都可以用，但我们选了朗道规范，这是得到恒定磁场最简单的方式：这个矢势的旋度是一个沿 Z 方向的恒定场，所以那就是我们的矢势。现在我们要做的是，我们要做狄拉克那个——我们要弄清楚怎么对那个能量函数做狄拉克替换。所以我们从能量
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function so we begin with the energy function which is the block band and we have we what we first do is we simply multiply and divide by H or H bar and and so in the numerator we have H Bar K at a KX and H bar in the denominator but H Bar K has the dimensions of momentum and it acts it's called the crystal momentum it acts like a momentum or it's sometimes called a pseudo momentum as it says here and so it reminds people of momentum so what if we substitute what would it be just magic and do some magical thinking and replace H Bar K which looks like and feels like and acts like a momentum by that expression P minus e over C times C times a which is a momentum like quantity so we just do that we we would just go in here barge in and replace that by that which has no legal validity but it's just a magical act to do and now we do another substitution we do the door substitutional replace a/c number by a q number in other words now we put in differential operators we replace the P by D by DX times H bar over I there a
函数出发，也就是布洛赫能带。我们首先要做的，只是简单地乘以再除以 h 或者 h-bar，于是分子上我们有 h-bar k，在 kx 那儿，分母上有 h-bar。但 h-bar k 具有动量的量纲，而且它的行为——它被称为晶体动量，它表现得像动量，有时也叫赝动量，就像这里写的那样。所以它会让人联想到动量。那么如果我们做个替换会怎么样呢？就当是变魔术，做一点魔法式的思考，把 h-bar k——它看起来像、感觉像、表现得也像动量——替换成那个表达式：p减去 e 除以 c 乘以 A，这也是一个类似动量的量。所以我们就这么干，我们就直接闯进来把那个换成这个，这在逻辑上没有任何合法性，但这只是一个魔法般的操作。现在我们再做另一个替换，我们做狄拉克替换，把 c 数换成 q 数，换句话说，现在我们放进微分算符，我们把 p换成 d/dx 乘以 h-bar 除以 i。这里 A 的 x 分量是零，所以那一项就没了；A 的 y 分量不是零，就是这个小小的
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sub X is zero so what that goes away a sub y is not zero there it's this little quantity here and here we have another differential operator and there we have something quite interesting now we replace the cosine by sum of two Exponential's and we expand we just this cosine this factor two gets rid of the factor two that there would be down here and so this is this becomes e to the a times du over DX D by DX and so forth we have some phase factors here that come from right there but it looks like that now if you look at those two differential operators you will recognize that these you may recognize that these are translation operators along the x axis and they translate by an amount the distance a so this is really moving to the right by a this is moving to the left by a this is moving straight up by a this is moving straight down by a and these have the phase factors so this is our Hamiltonian that we've created that out of the block band we've created a Hamiltonian and now given a Hamiltonian we can make a
量，在这里；而这里我们又有一个微分算符，于是我们就得到了相当有意思的东西。现在我们把余弦换成两个指数的和，然后展开。这个余弦，这个因子 2 正好抵消掉本来会出现在下面的因子 2，于是这就变成 e 的 a 乘以 d/dx 次方，等等等等；这里我们还有一些相位因子，来自那边那一项。不过它看起来就是这样。现在如果你看那两个微分算符，你会认出来——你可能会认出来——这些是沿 x 轴的平移算符，它们平移的量正好是距离 a。所以这其实是向右移动 a，这是向左移动 a，这是向上移动 a，这是向下移动 a，而且这些还带着相位因子。所以这就是我们造出来的哈密顿量，我们从布洛赫能带里造出了一个哈密顿量。现在有了哈密顿量，我们就可以写出薛定谔方程。那么这个
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Schrodinger equation so what is the Schrodinger equation it relates five points in the in the crystal to each other it says the wave function here which is site XY is related to side eight X plus a times y that's that one and also that one and also that one and also that one it's related to those four neighbor points with these just as a sum of these with phase factors right here in here that's a pretty stunning equation now we can try to make it look more discrete oh well this is just the reminding you of the two different axes okay we can we can try to make bring out the discreteness by replacing X by MA because we're just talking about points on the ona lattice so integer times a integer times another integer times a we should use alpha to replace the magnetic field so we have alpha and then we'll create a over a zero is Epsilon that's a dimensionless energy variable so the equation that I just wrote down looks like this now written in this new form this is a integer subscripts to integer subscripts and
薛定谔方程是什么呢？它把晶体中的五个点联系了起来。它说这里的波函数，也就是 psi(x, y)，与 psi(x+a, y) 有关，就是那一个，还有那一个，还有那一个，还有那一个——它跟这四个近邻点有关，就是把它们带上相位因子加起来，相位因子就在这儿和这儿。这是个相当惊人的方程。现在我们可以试着让它看起来更离散一些。哦，这只是提醒你注意那两个不同的轴。好，我们可以试着把离散性凸显出来：把 x 换成 ma，因为我们讨论的只是格点上的点，所以是整数乘以 a、整数乘以 a、另一个整数乘以 a。我们应该用 alpha 来代替磁场，于是我们有了 alpha；然后我们定义 a 除以 a-零为 epsilon，这是一个无量纲的能量变量。所以我刚才写下的那个方程，用这个新形式写出来就是这样：整数下标对整数下标，
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this is still the Schrodinger equation now we're gonna make a guess that there can be a periodic behavior along one axis we're just going to assume that might be periodic purely periodic along one axis so that we could simplify it down to one dimension if we do that separation of variables we assume that this is the very behavior along one axis what does it do if we plug in and I won't do the math but here is the result that we get we get a one-dimensional equation in other words with one integer subscript now and it says the wave function at this point is is related to the wave functions at this at these two points by this very simple looking equation this is called harpers equation found in 55 also by harper and also found by wanye at that same time now this this thing here is analog an analogue to del squared in the normal Schrodinger equation it's it's basically that's a second derivative this is the second difference so it's a different difference equation instead of a differential equation but this is the
而这仍然是薛定谔方程。现在我们要做一个猜测：沿着某一个轴可能存在周期性的行为。我们就假设沿着某一个轴它可能是周期的、纯周期的，这样我们就能把它简化到一维。如果我们做这样的分离变量，我们假设沿一个轴是这样的行为，那会怎么样呢？我们代进去——数学过程我就不做了——但这里就是我们得到的结果：我们得到一个一维的方程，换句话说现在只有一个整数下标了。它说这一点上的波函数，与这两个点上的波函数由这个看起来非常简单的方程联系起来。这叫做哈珀方程，由哈珀在 55 年发现，同时期万尼尔也独立发现了。现在这里这个东西，是通常薛定谔方程里 del 平方的类比，它基本上就是——那个是二阶导数，这个是二阶差分，所以这是一个差分方程而不是微分方程，但这就是二阶导数的对应物。而这边我们有
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counterpart to the second different the second derivative and on here we have something that is acting like a potential energy and the cosine represents the crystal or if you wish that's kind of like the sidewalk and with its regular behavior and and the magnetic field is there alpha that has that it's not got that if it were if alpha were an integer then then there would it would be like that things are the to periodic phenomena are not competing but alpha is an irrational number in in many cases or certainly not an integer and so you have two competing phenomena to competing periodicity x' spatial periodicities one is the one having to do with the magnetic with the crystal and one having to do with the magnetic field and this is the energy eigenvalues so that is the that's the Schrodinger equation right there and we can see sort of traces of the crystal in a magnetic field and so these are the three points that are related to each other by that the wave functions are at those three points are related to each
一个表现得像势能的东西，那个余弦代表晶体，或者你愿意的话，那有点像人行道，带着它规则的行为。而磁场就在那儿，就是 alpha，如果——如果alpha 是个整数，那么就会是那样，两个周期性现象就不会相互竞争了。但 alpha 在很多情况下是个无理数，或者至少肯定不是整数，所以你就有两个相互竞争的现象、两个相互竞争的周期性、两个空间周期性：一个跟晶体有关，另一个跟磁场有关。而这个是能量本征值。所以那就是薛定谔方程，就在那儿。我们多少能看出磁场中晶体的一些痕迹。所以这就是那三个点，它们通过那个方程彼此关联起来——这三个点上的波函数由那个方程联系在一起。如果你
10转移矩阵、胖x轴与能带计数
47:07
other by that equation and if you rewrite the equation so that all the the fact the factors the two factors of G sub M are put together in a bracket then you get a recursion relation it says that this one is formed as a sum of these two where this one is modified by a product now again if alpha is an integer then this thing is just a constant so we just have a constant and and it really is almost like the Fibonacci sequence if alpha is not an integer then this number varies goes back and forth and back and forth but it it's it's a very simple recursion relation and so we can rewrite this in terms of little pairs of numbers if you take two of them at a time G sub n plus 1 and G sub M and relate it to the preceding one to the left then there's a little matrix a little two by two matrix mostly consisting of constants and ones and zeros but with this here and let me just remind you that epsilon is the energy which we'll assume is constant alpha which you may not be able to make out here but that's an alpha we have a
把方程重写一下，把两个 g_m 的因子放到一个括号里合并起来，你就得到一个递推关系。它说这一项是由这两项相加构成的，其中这一项还乘上了一个因子。再说一遍，如果 alpha 是整数，那这个东西就只是个常数，我们就只有一个常数，那它其实几乎就跟斐波那契数列一样。如果 alpha 不是整数，那这个数就会变化，来回来回地摆动，但它仍然是个非常简单的递推关系。所以我们可以用一对一对的数把它重写出来：如果你一次取两个，g_{n+1} 和 g_n，然后把它跟左边前一对联系起来，那就会出现一个小矩阵，一个二乘二的小矩阵，里面大部分是常数、1 和 0，只有这里有这个东西。让我提醒你一下，epsilon 是能量，我们假定它是常数；alpha——你在这儿可能看不太清，但那是个 alpha；我们还有一个 2π 就摆在那儿；然后 alpha 是磁场；
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two pi that's just sitting there and then alpha is the magnetic field and then M is an integer that's the one variable in this thing so here I'll just show you how it looks if we if we have the real line then those two if we multiply let me just go back from I said I'd call this matrix a sub M because the M is right there so this is a function of M it's it changes as M changes so if we multiply by a sub 1 that brings us to there and if we multiply a sub 2 that brings us there and we keep on multiplying each time we take a new matrix and we multiply and we get the wave function moving to the right we get the new wave function at these at these pair of discrete points well what for what values of the energy will this process yield a non divergence in other words as we go out very far away for certain values of the energy this will not blow up the wave function will not bow for other values the energy it'll blow up and that'll be physically unacceptable okay well I guess I have to take an example one third if alpha is
然后 m 是整数，那是这里面唯一的变量。所以这里我给你们看看它长什么样。如果我们有实数轴，那么这两个，如果我们相乘——让我先回去一下，我说过我要把这个矩阵叫做 A_m，因为 m 就在那儿。所以这是M 的函数，它会随着 M 变化。所以如果我们乘以 a₁，就把我们带到这里；如果乘以 a₂，就把我们带到那里我们就这样不断地乘下去，每次取一个新的矩阵相乘，波函数就往右移动我们就在这些离散的点对上得到新的波函数。那么，能量取什么值时这个过程才不会发散呢？换句话说，当我们走到很远的地方时，对某些能量值波函数不会爆掉；而对另一些能量值，它就会爆掉，那在物理上就是不可接受的好，我想我得举个例子，三分之一。如果 α 等于 1/3，也就是某个无量纲化表示的磁场
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1/3 that's a certain magnetic field in a dimensionless fashion then it turns out that that because of the periodicity of the cosine this matrix is identical to this matrix and this one is identical to this one and this one is identical to this one so we have these equations so in fact this product of three matrices is the same as this product and it's the same as this product and this product so in fact we're just putting a matrix to a high power a a we can call this a now and we're just putting a to a high power so if we want to go very far out we just put a to a very high power when will this slow up for large m and when will it stay bounded now it turns out as Gustavo Overmeyer showed that there is a simple condition on the matrix a which is a product of well I guess I should have said in the case of 1/3 it's a product of three matrices if it were 2/5 it would be a product of five matrices because for a rational value the then when you plug in that thing into the under the cosine it'll come back to
那么结果是，由于余弦的周期性，这个矩阵与这个矩阵完全相同，这一个和这一个相同，这一个和这一个也相同，于是我们就有了这些等式所以事实上，这三个矩阵的乘积和这个乘积是一样的，也和这个乘积、这个乘积一样所以实际上我们就是在把一个矩阵取很高的幂——我们现在可以把它叫做 A——我们就是在把 A 取很高的幂所以如果我们想走得很远，就只要把 A 取一个很高的幂。对于很大的 m，它什么时候会爆掉，什么时候会保持有界呢？结果正如 Gustav Obermair 所证明的，对矩阵 A 有一个很简单的条件，A是一个乘积——嗯，我应该先说，在 1/3 的情况下它是三个矩阵的乘积；如果是 2/5那就是五个矩阵的乘积。因为对于有理数值，当你把那个东西代进去放到余弦里面，它会回到自身，而当它回到自身时，这些矩阵就会重复；但它回到自身
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itself and so when it comes back to itself then the matrix these repeat but it comes back to itself only after five times that whatever the denominator of the rational number is so for rational numbers we'll have this matrix a which is a a product of Q different matrices where Q is the denominator if alpha equals P over Q a ratio of two integers then Q will tell you how many time how many matrices you need to multiply by and the condition on that matrix is that the trace should be less than or equal to four a very simple condition so and and that trace which involves Q matrices multiplied together each of which has epsilon linearly in it when you multiply Q of them together you get a terms that have the Q Tower of epsilon in them and so the trace will be a Q acute degree polynomial in Epson now here comes my piece of luck a very very major piece of luck that allowed me to get into the game because I was sitting in remember sitting in the office when wanyan and Rao and Obermeyer were flinging about
要等到五次之后，也就是那个有理数的分母是多少就要多少次。所以对于有理数，我们会得到这个矩阵 A，它是Q 个不同矩阵的乘积，其中 Q 是分母。如果 α 等于 P 比 Q，也就是两个整数之比，那么 Q就告诉你需要乘多少个矩阵。而对这个矩阵的条件是，它的迹应该小于或等于 4，一个非常简单的条件。而这个迹涉及 Q 个矩阵相乘，每一个里面都线性地含有 ε，当你把 Q 个相乘起来，你会得到含有ε 的 Q 次方的项，所以这个迹会是 ε 的 Q 次多项式。现在轮到我的一点运气了——非常非常重要的一点运气，它让我得以进入这个领域。因为我当时坐在，记得是坐在办公室里，看着 Wannier、Rauh 和 Obermair 在那儿摆弄他们的超几何函数、证明定理，而我坐在那里
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their hypergeometric functions and proving theorems and I was sitting there feeling inferior well it happened that in the leftover mile there was a computer a little computer sitting on a desktop an HP computer that I had actually used in Aspen Colorado five years earlier and I knew how to program it this is a picture of it this is the actual computer by the way this is they reviled it after 40 whatever it was years 43 42 years in Regensburg when I went there last October and they revived the actual machine and it says hello Doug its greeting me and so I got I programmed this computer to figure out when the trace was less than four for a lot of different rational uses a magnetic field and so let's take 1/3 1/3 we'll have a third degree polynomial and this is a cubic polynomial so it looks like that here's the x-axis going along here but the question is not where does it cross the x-axis the question is where is it less than or equal to 4 and absolute value and so I've widened the x-axis to become
感到很自卑。恰好在隔壁那间闲置的房间里有一台计算机，一台小小的、放在桌面上的惠普计算机，五年前我在科罗拉多州的阿斯彭其实用过它，我知道怎么给它编程。这是它的照片，这就是那台真正的计算机。顺便说一句，他们在四十几年——43 年还是 42 年之后，在雷根斯堡把它修复了。我去年十月去那里时，他们把这台真机复活了，它显示「你好，Doug」，在跟我打招呼。于是我给这台计算机编了程序，去算出对于很多不同的有理数磁场值，迹在什么时候小于 4。那么我们来看1/3。1/3 会得到一个三次多项式，这是一个三次多项式，所以它看起来像这样。这里是横轴，沿着这里走。但问题不在于它在哪里穿过横轴，问题在于它在哪里的绝对值小于或等于 4所以我把横轴加宽了，变成我所说的「胖横轴」，它现在是一条有厚度的线，而不是一条无限
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what I call a fat x-axis it's now a thick line rather than an infinitely thin line it's the fat x axis when is this cubic within this gray area and while it's in therefore it's in for this for this value set of values of epsilon right in there that's when it is crossing and staying within plus and minus 4 then it goes out and it comes back and it stays in for this length of time so to speak and then it goes out again and then it comes back in and and then goes away forever so there are three bands three bands for one-third what about 2/5 but 1/5 I'm sorry here's 1/5 but it's gonna be a five a fifth degree polynomial and this is what the Rumplestiltskin calculated for me it crosses then it goes way away then it comes back and it goes like this and so forth five different bands what about 2/5 2/5 looks like this it's the same it's a quintic polynomial again a fifth degree polynomial but the roots are the the fat roots you might say are distributed very differently to over here one over here and two over here and
细的线，这就是胖横轴。这条三次曲线什么时候落在这个灰色区域里？当它在里面时，对于这一组 ε 的值，就在那里，它就在里面。那时它穿过并保持在正负 4 之间，然后它跑出去，又回来，在这一段「时间」里一直待在里面，可以这么说，然后它又跑出去，然后又回来，然后就永远跑掉了。所以对于三分之一，有三条带，三条能带。那 2/5 呢？不过先看 1/5，抱歉，这里是1/5，它会是一个五次多项式。这就是那个「侏儒怪」给我算出来的结果，它穿过，然后跑得很远，然后又回来，然后像这样，等等——五条不同的能带。那 2/5 呢？2/5 看起来是这样，它也一样是个五次多项式，同样是五次的，但它的根——你可以说是「胖根」——分布得非常不同，那边两个、这边一个、这边两个。1/6 看起来是这样：1、2、3，它想要
1111月革命：看见蝴蝶的自相似
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1/6 looks like this 1 2 3 it tries to get out but it doesn't quite make it it goes tangent to the air to the to the the edge of the fat x-axis it can't quite get out but it's trying then it goes out and then it comes out again and then so forth the other a little teeny-weeny band here a little bit and then a wideband that touches this wideband then another one six bands for to kiss in the middle so in about September of 1974 I was graphing these things by hand this computer did not have a plotter attached to it and so I would get out numerical values that it would calculate for me overnight and and I would get out these numerical values that would come into the office and graph them on paper and this is what I was starting to see in and to begin in September of 1974 well a couple of months passed and this is what I started to see in November of 1974 I'll talk about the Nova the November revolution in a moment so what I would point out is that is what you may recognize from my integrand in other words you have
跑出去，但差一点没做到，它与那个胖横轴的边缘相切，就是跑不出去，但它在努力尝试。然后它跑出去，然后又出来一次，如此等等。另外还有一条小小的窄带在这里，一点点，然后是一条宽带，与这条宽带相接触，然后又一条——对 1/6 来说是六条带在中间相吻。所以大约在 1974 年 9 月，我开始手工画这些图，因为这台计算机没有接绘图仪，所以我会拿到它整夜为我算出来的数值，我拿到这些数值就到办公室里，把它们画在纸上。这就是我在 1974 年 9 月开始看到的东西。又过了几个月，这就是我在 1974 年 11 月开始看到的东西。我等一下会讲「11 月革命」。我要指出的是，这个东西你们可能认得，就是我那个被积函数——换句话说，你有个东西朝着一个角落往上走，越变越小，你多少
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something going up towards a corner getting smaller and smaller and you can sort of see that there are there's a sort of a periodicity here I can't explain it exactly but i intuited that there's something going on here moreover more importantly oh yeah we're reminded me of that I mean that's the comparison that I saw in my mind's eye but also this little area of the graph or this one or this one or this one looked to me suspiciously like a copy of the full graph where these two lines that kiss in the middle this is really two bands even though they touch right there are those two right there and these three these blue three are these three and these three here these blue three are these three and so forth and I thought my goodness it looks like this is a little copy over this this line is that line and this line here is that line so this is an entire copy of this graph distorted and that's what I started seeing and I call it my November revolution a posteriori because at that exact time in slack at Stan
能看出这里存在某种周期性。我没法确切解释它，但我直觉感到这里有某种规律。而且更重要的是——哦对，这让我想起了那个，我是说，那就是我在脑海里看到的类比。但另外，图上这一小块区域或者这一块、这一块、这一块，在我看来非常像是整幅图的一个复制品：这两条在中间相吻的线，其实是两条带，尽管它们在那里相接触，就是那边那两条；而这三条，这蓝色的三条，就是这三条；这里这三条，这蓝色的三条，就是这三条，如此等等。我心想天哪，看起来这是一个小小的复制品，这条线就是那条线，这里这条线就是那条线。所以这是整幅图的一个完整复制品，只是被扭曲了。这就是我当时开始看到的东西，我事后把它称为我的「11 月革命」，因为恰恰在那个时候，在斯坦福的 SLAC 和在 MIT——嗯，其实不是在 MIT，我想是那个叫什么来着，无所谓了
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and at MIT also well it wasn't at MIT I guess it was what's it called it doesn't matter anyway the East cousin on the west coast of the United States simultaneously two different groups were discovering the so called japes I particle which was a resonance that revealed that there was a fourth quark a charmed quark and this was considered to be one of the biggest moments in particle physics at that time and they later called it the November revolution because it was so exciting because they were discovering that there were not just up down and strange quarks but there there was a charm quark and later on they would find that there are a couple more but this was a very very big moment in physics and I remember talking to my dad about it over the telephone right at that time when I was in May at the park and he was at Stanford and so I was that's another copy by the way a little or copy and there was another copy right there and another copy there it's that right there all the way so I was starting to see something really
总之在美国的东岸和西岸，两个不同的小组同时发现了所谓的 J/ψ粒子，那是一个共振态，它揭示了存在第四种夸克，即粲夸克。这在当时被认为是粒子物理学最重大的时刻之一，后来人们把它称为「11 月革命」因为它太令人兴奋了，他们发现了不只有上夸克、下夸克和奇异夸克，而是还有有粲夸克（charm quark），后来人们又发现还有好几种，但这在物理学上是一个非常非常重大的时刻。我记得就在那个时候，我在电话里跟我爸爸聊起这件事，当时我在公园那边，他在斯坦福。所以我——顺便说一句，那儿又是一个副本，一个小一点的副本，那边还有另一个副本，就在那儿，还有一个在那儿，一直这样下去。所以我开始看到一些非常
12「数字命理学」：说服导师之战
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exciting here and it reminded me of that and I concluded that gee plot was made of infinitely many smaller distorted copies of itself and I was very very excited of course to find this and I went to Gregory Juan EA and I told him no chance favors have prepared I well yes that's the point I had a prepared I Gregory did not and what did he say well he said this is numerology you can do a library thesis and I said what is the library thesis he says it's where you write up the work of other people and and he said you you clearly can't get a PhD doing what you're doing and but if you really insist you can put your numerology work in an appendix and so I was pretty hurt but I was not deterred I continued working all by myself all alone now we went back from Regensburg oh one last thing yes my 30th birthday was coming up at that point and I gave myself an ultimatum I wanted because I was very very concerned with convincing Gregory obviously it was necessary to convince him he was my adviser I wanted to make a
激动人心的东西，它让我想起了之前那个情形，于是我得出结论：G 图是由无穷多个更小的、被扭曲了的自身副本构成的。发现这一点我当然非常非常兴奋，我就去找 Gregory Wannier，告诉他——（观众：机会偏爱有准备的人。）是啊，问题就在这儿，我有一个有准备的头脑，Gregory 没有。那他说了什么呢？他说这是数字命理学（numerology），你可以做一篇文献综述型的论文。我说什么是文献综述型论文，他说就是你把别人的工作整理写出来。他还说，照你现在这么搞，你显然拿不到博士学位；不过如果你真的坚持，可以把你那些数字命理学的东西放在附录里。我当时挺受伤的，但并没有被吓退，我继续完全靠自己一个人做下去。后来我们从雷根斯堡回来了——哦，还有最后一件事：当时我快满 30 岁了，我给自己下了最后通牒。因为我非常非常在意能不能说服 Gregory，显然我必须说服他，他是我的导师。我想给出一个数学证明，证明 G 图是由自身的副本构成的。我之前对 INT 函数
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mathematical proof that G plot consists of copies of itself and I had done that with int so I thought maybe I can do the same thing with G plot I was hoping that I could however I was unable to do it I worked extremely hard on doing this but I got nowhere then we went back to Eugene well another amazing coincidence took place in Eugene I was walking by the lab of Russ Donnelly who was a very good solid a very good low-temperature physicist at the University of Oregon and I saw a familiar HP desktop computer sitting there with a plotter this time and I asked him if I could use his computer and he said yes nobody's using it just like Gustav had said but those days physicists didn't use computers I don't know why they bought them because they didn't use them and and so he said sure so I used it and I put that computer to work calculating many many more values and plotting them for me very precisely using its plotter and this is what came out in May of 1975 and that was a very stunning graph and a
做到过这一点，所以我想，也许我可以对 G 图做同样的事。我一直希望能做到，但我没能做到。我为此拼了命地努力，可毫无进展。后来我们回到了尤金（Eugene）。在尤金又发生了一个惊人的巧合，我路过 Russ Donnelly 的实验室，他是俄勒冈大学一位非常出色的固体——非常出色的低温物理学家。我看到那儿摆着一台我熟悉的 HP 台式计算机，这次还配了绘图仪。我问他能不能用一下他的计算机，他说可以啊，没人在用——跟 Gustav 当年说的一模一样。可那些年头物理学家不用计算机，我不知道他们为什么要买，因为他们根本不用。总之他说没问题。于是我用上了它，让那台计算机去计算多得多的数值，并用绘图仪非常精确地把它们画出来，这就是 1975 年 5 月出来的结果。那是一张非常震撼的图，Gregory 很喜欢它，但他仍然
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Gregory liked it very much but he still thought what my ideas were nonsense he didn't believe my ideas whatsoever about the about the fact that this was a copy or that this was a copy I mean you can probably see these four gaps here right there those white gaps doesn't that look like that doesn't it look like that you can see it well he couldn't see it and he didn't believe it for a moment so what I did was I took a little teeny copy what I knew was a car down here that was located it was you know a funny angle and in some kind of funny trapezoid and I took this thing which inside it should contain a copy this is what does it say 1/7 and 1/6 so if I you know if I go back here it would be down around here I don't know where it is somewhere in there I took this little trapezoid and I calculated all the values in there and then I did what I called what rectangular ization I turned this thing into a rectangle I undistorted it and here's what I got that was that's inside that little structure when I showed that to Gregory
认为我的想法是胡说八道。他一点也不相信我说的——不相信这是一个副本，或者那是一个副本。我是说，你们大概能看到这里这四条缝隙，就在那儿，那些白色的缝隙，那看起来不就像那个吗？难道不像吗？你们能看出来吧。可他就是看不出来，他一秒钟都不相信。所以我做的是：我取了一个非常小的副本，我知道下面这儿有一个，它的位置角度很奇怪，处在某种奇怪的梯形里。我取了这块东西，它内部应该包含一个副本，这上面写的是 1/7 和 1/6，所以如果我回到这张图上，它大概在这下面，我也不确定具体在哪儿，大概在那一带。我取了这个小梯形，把里面所有的数值都算了出来，然后做了我称之为“矩形化”的操作，我把这东西变成了一个矩形，也就是把扭曲还原。这就是我得到的结果，那就是那个小结构内部的样子。当我把这个给 Gregory 看时，
13康托集与抹平：调和数学与物理
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all of a sudden the scales fell from his eyes and he said by God and he came around August 1975 and I said to myself I'm gonna get a PhD that was very exciting moment for me so as it turns out we've looked at rational numbers but what about irrational numbers well it turns out that this spectrum in this case consists of if we had for a rational number P over Q we had q bands well for an irrational number we should have P over Q where P and Q are infinitely large integers so it should be an infinite an infinitely large number of infinitely small bands and that's a cantor set and if you don't know the term I'll show you what a cantor set it looks like you begin with an interval say 0 1 eliminate the middle third then do that to the two remaining pieces eliminate the middle third again and keep on eliminating middle thirds and keep on repeating this infinitely many times what happens at the very end anything yes you're left with all the end points that you didn't eliminate and that's an uncountable set so it's a very
他一下子茅塞顿开，说了句“我的天”。他在 1975 年 8 月转变了看法，我对自己说：我要拿到博士学位了。那对我来说是非常激动人心的时刻。那么，我们已经看了有理数，无理数呢？结果表明，这种情况下的谱是这样的：对于有理数 P/Q，我们有 Q 条能带；而对于无理数，我们可以把它看成 P/Q，其中 P 和 Q 是无穷大的整数，所以应该是无穷多条无限窄的能带，这就是一个康托集。如果你不熟悉这个术语，我来给你看看康托集长什么样：你从一个区间开始，比如 0 到 1，去掉中间三分之一；然后对剩下的两段做同样的事，再去掉中间三分之一，就这样不断去掉中间三分之一，无穷多次地重复下去。到最后会剩下什么呢？什么都不剩吗？不，你会剩下所有你没去掉的端点，而那是一个不可数集，所以它其实是一个非常
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big infinity as a matter of fact but it's got no intervals in it whatsoever so it's got a measure zero so in some sense it weighs nothing if this thing had weighed 1 kilo then the Cantor set weighs zero exactly 0 but in uncountably any points in it so at every irrational field we have a different Cantor set because you can make Cantor sets in different ways you don't have to eliminate the middle third you can eliminate all sorts of other things as long as you keep on eliminating things inside the remaining intervals you'll get a Cantor set of some sort and also at every year a tional field we have a different Cantor set and that's what I found and put into my thesis how to reconcile this weird mathematical object with with physical reality well the idea is we have a magnetic field that is not defined to infinitely many decimal places so if you took if you take G plot which is what I call that graph and you taken the union of all the magnetic the energy is associated with all the magnetic fields within a very small
大的无穷。但它里面完全不包含任何区间，所以它的测度为零，从某种意义上说它没有重量。如果原来那段东西重 1 公斤，那么康托集的重量正好是零，可里面却有不可数无穷多个点。所以在每一个无理数取值处，我们都有一个不同的康托集，因为康托集可以用不同的方式构造，你不必去掉中间三分之一，你可以去掉各种各样别的部分，只要你不断地在剩下的区间内部继续去掉东西，你就会得到某种康托集。而且在每一个无理数取值处，我们都有一个不同的康托集。这就是我发现并写进论文里的东西。那么，怎么把这个奇怪的数学对象和物理现实调和起来呢？想法是这样的：磁场并不是被定义到无穷多位小数的。所以如果你取 G 图——我就是这么称呼那张图的——把所有磁场对应的能量取并集，也就是在一个非常小的 ΔH 或 Δα 范围内（你想怎么叫都行），你只要给磁场加上一点点
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Delta H or Delta alpha whatever you want to call it you just associated a little uncertainty with the magnetic field then take all the magnet all the energy values that are associated with those magnetic fields and take the union of them then it turns out that the graph you get looks very very smooth I call that the smeared graph because I actually I went and visited Fineman of all people in his office in 1975 I wrote him a letter and he was very willing to meet me and when he saw this graph he was so excited he couldn't believe it he immediately said to his secretary make a copy of this and he he was very excited about it and I and I told him about the the a my idea of smearing the graph and he called it jiggling the graph and so what here's what happens if you jiggle it quite a bit it looks just like a strange black object I mean it doesn't have any fractal structure left to it it's completely continuous it's got little it's like a like a map with little lakes in it these little white
不确定性，然后把与这些磁场相对应的所有能量值都取出来，取它们的并集，结果你得到的图看起来就非常非常光滑。我把它叫做“抹平后的图”。因为——我在 1975 年真的去他办公室拜访过费曼，我给他写了封信，他很乐意见我。当他看到这张图时，他兴奋得不敢相信，立刻对秘书说，把这个复印一份。他对此非常兴奋。我还跟他讲了我关于把图抹平的想法，而他把这叫做“抖动”这张图。那么，如果你把它抖动得比较厉害，就会变成这样，看起来就像一个奇怪的黑色物体，我是说它一点分形结构都不剩了，完全是连续的，上面有小小的——就像一张地图，上面有一些小湖泊，这些白色的小块可以看作湖，就
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things could be thought of as lakes it's just a map of some some zone when there's lakes in it if you diggle it less you see more lakes and more detail that's a slightly less more precise crap if you could jiggle it less you get back the original thing that's what zero jiggling and now I put this graph on page 137 of my thesis for physicists they'll know this number the reciprocal or the fine-structure constant I couldn't resist this touch and I guess it shows that I was a numerologist after all in any case I wound up deciding that I was I should be getting out of physics even though I had such a lovely discovery I didn't really think it was gonna happen again I gave a copy to Felix law and when he saw it he said when he heard about it he said there are two natural frequencies here one is due to the magnetic field one called the cyclotron frequency and one due to the crystal alone to different frequencies and I had never thought of that before and I wrote them down and I saw that their ratio was
像某个区域的地图，里面有些湖。如果你抖动得少一点，你就会看到更多的湖、更多的细节，这就是稍微少一点抖动、更精确一点的图。如果你完全不抖动，你就又得回原来那张图，也就是零抖动。然后我把这张图放在了我论文的第 137 页——物理学家会认识这个数字，它是精细结构常数的倒数。我实在忍不住来了这么一手，我想这说明我终究还是个数字命理学家吧。总之，我最后决定我应该离开物理学，尽管我有了这么美妙的一个发现，我并不真的认为这种事还会再发生一次。我把一份图给了 Felix Bloch，他看到之后——他听说这件事之后说，这里有两个自然频率，一个来自磁场，叫回旋频率；另一个来自晶体本身。两个不同的频率，而我以前从来没有想到过这一点。我把它们写下来，看到它们的比值正好就是 α——两个相互竞争的周期。所以
14退出物理与蝴蝶的后世影响
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exactly alpha two competing periods so what Felix said to me reminded me of ADA sequences and I was back so really I wound up getting a PhD in number theory even though I had left Berkeley dropped out and now I was doing things that I love going out in style well gee plot showed that number theory is intimately linked to physics and with Cantor sets and other things it also showed that topology is intimately linked to physics things that people had not suspected I was given a referee's report when I submitted my article to Physical Review it said I find this paper delightfully written almost like a Mozart divertimento well of course that was if I can be forbidden can be permitted forgiven for using the phrase music to my ears and it turns out this is a famous physicist who who wrote this he 20 years later introduced himself to me when I gave a talk somewhere and said you remember somebody once wrote in an anonymous review that and I said I certainly do and he said well I wrote that you know his name if you I don't know if
Felix 对我说的话让我想起了 eta 序列，我又回来了。所以说，我最后其实是拿了一个数论的博士学位，尽管我曾经离开伯克利、退过学。而现在我做的是我热爱的事情，也算是以漂亮的方式收场。G 图表明，数论和物理学有着密切的联系；而通过康托集之类的东西，它还表明拓扑学也和物理学密切相关——这些都是人们此前没有料到的。我把文章投给《物理评论》时，收到了一份审稿意见，上面说：我觉得这篇论文写得妙趣横生，几乎像莫扎特的一首嬉游曲。当然，如果允许我这么说的话，这真是我听过最动听的话。而且事实证明，写这份评审意见的是一位著名物理学家。20 年后，我在某处做讲座时他做了自我介绍，说：你还记得当年有人在一份匿名评审里写过那句话吗？我说我当然记得。他说，那是我写的。你们大概知道他的名字，我不确定你们有没有听说过，他叫 Leo Falicov，是一位
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you would've heard of him but his name was Leo phallic off he was an Argentinian solid-state theorist very distinguished person so I was very flattered but I chose to drop out why well I had found this very wonderful thing but I didn't think it was gonna happen again and I thought I might as well go out on a blaze of glory now from then on G plot started becoming famous I never thought of G plot as going to become famous I just thought it was my path it was my pathway to getting a PhD it was what allowed me to get a PhD but it turns out the people started exploring it in the 1980s mathematicians jumped into the game and they changed some terminology they called the idea that that the spectrum for every irrational value of alpha was a Cantor set they didn't associate it with me even though I published it in my thesis in my Physical Review article and even in gödel Escher Bach I said it in all three places that it was a cantor set they totally ignored that and and and attributed the hypothesis that the
阿根廷裔的固体理论物理学家，非常杰出的人物。所以我很受宠若惊。但我还是选择了退出，为什么呢？因为我确实找到了一个非常美妙的东西，可我不认为这种事还会再发生，我想那还不如就在这一片辉煌中收场。从那以后，G 图开始变得有名。我从来没想过 G 图会变得有名，我只把它当作我的路，是我拿到博士学位的途径，是它让我能拿到博士学位。但结果是，人们从 1980 年代开始研究它，数学家们也加入了这场游戏，并且改了一些术语。他们把“对每一个无理数 α，谱都是一个康托集”这个想法——他们并没有把它归到我名下，尽管我在自己的博士论文里、在《物理评论》的文章里，甚至在《哥德尔、艾舍尔、巴赫》里都发表过这一点。我在这三处都说过它是一个康托集，他们完全无视了这一点，而把“对应于无理数的谱是康托集”这个猜想归给了 Mark Kac，
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spectrum belonging to an irrational number is is the Cantor set to mark Katz who made a speech about it at the meeting of the American Mathematical Society they didn't call it Harper's equation but the almost Mathieu equation and in 2009 these two mathematicians Avila and g2 Mischka proved that the Harper spectrum is a cantor set for all irrational alpha and in 2014 a Fields Medal was given to a Vela for in part for his work on this spectrum and it does show why maybe I wasn't able to do it as a graduate student it was a very very very difficult problem physicists all around the world went on a grand expedition exploring variations on a theme so for example for this was calculated by Claire oh my good friend Francisco and wanye I mean wah Nia didn't do the computer computer work but Francisco did the computer work but they put it in a paper butterfly for a hexagonal lattice butterfly for a triangular lattice butterfly for some kind of I don't know a model of a molybdenum molybdenum disulphide lattice
因为他在美国数学会的一次会议上就此做过一个演讲。他们也不叫它 Harper 方程，而叫“殆 Mathieu 方程”。2009 年，Avila 和 Jitomirskaya 这两位数学家证明了：对所有无理数 α，Harper 谱都是康托集。2014 年，Avila 获得了菲尔兹奖，部分原因就是他在这个谱上的工作。这也说明了为什么我当研究生时也许做不出来——那是一个非常非常非常困难的问题。全世界的物理学家展开了一场大规模的探索，研究这个主题的各种变体。比如说，这个是由我的好朋友 Francisco Claro 计算的，还有 Wannier——我是说 Wannier 并没有做计算机方面的工作，计算是 Francisco 做的，但他们把它写进了一篇论文。六角晶格的蝴蝶，三角晶格的蝴蝶，还有某种——我也说不清——二硫化钼晶格模型的蝴蝶。都是些惊人的结构，但它们都有
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amazing structures but they all have something very profoundly in common copy native copies of themselves in nested infinitely many times distorted copies butterfly for a EEB raid structure and I don't know what that is butterfly for a cog oh man lattice I don't know what that is either but I think Kagame means basket butterfly for an octagon and docked tiling and on and on it goes you can find all sorts of variations on the butterfly butterfly with both magnetic and electric fields then in 1980 the quantum Hall effect was discovered and a nonconductor then manifests quantized conductance lo this was something nobody expected and it turns out that the butterfly is intimately related that these gaps that I pointed out to you earlier I've turned it 90 degrees here but other than that it's the same butterfly just with the gaps colored in now these gaps can be labeled with integer values in a certain number theoretical way involving daya Fontaine equations which I'm not going to go into these gaps they're called churning
一个非常深刻的共同点：自身的副本，无穷多层嵌套的、被扭曲的副本。还有某种结构的蝴蝶，我也不知道那是什么；还有 kagome 晶格的蝴蝶，我也不知道那是什么，不过我想 kagome 在日语里是“篮子（编目）”的意思。还有八边形铺砌的蝴蝶，等等等等，你能找到各种各样的蝴蝶变体，比如同时有磁场和电场的蝴蝶。后来在 1980 年，量子霍尔效应被发现了：一个非导体竟然表现出量子化的电导。这是没有人预料到的。而结果表明，蝴蝶与之密切相关——就是我前面给你们指出的那些缝隙。这里我把图转了 90 度，但除此之外还是同一只蝴蝶，只是把缝隙涂上了颜色。这些缝隙可以用整数来标记，方式来自某种数论手段，涉及丢番图方程，我就不细讲了。这些缝隙对应的数叫做陈数（Chern
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numbers and these gaps can be labeled and they correspond precisely to the conductance levels in the Hall effect of quantum Hall effect and the integer quantum Hall effect so this gap corresponds to this level this gap corresponds to this level this gap corresponds to this level and so forth all amazing relations that the butterfly had that of course I would have never suspected now a woman named induce a teacher a physicist at George Mason University even declared that the butterfly is the home the natural home of the hall-effect she wrote this book and sent it to me when not as a published book but as a manuscript and asked me if I would write something for her about my memories of the discovery of G plot and so I wrote something and that became it says here contributions by the South Sider that became my prologue to her book oh and so in her book she talks about all the connections between G plot and all these different things that I was completely unaware of Apollonian gaskets berry phases bose-einstein
numbers）。这些缝隙可以被标记，而它们精确地对应于霍尔效应中的电导台阶，也就是量子霍尔效应、整数量子霍尔效应中的台阶。所以这条缝隙对应这个台阶，这条缝隙对应这个台阶，这条缝隙对应这个台阶，如此等等。这些都是蝴蝶所具有的惊人关联，当然是我当年绝对不会想到的。有一位叫 Indubala Satija 的女士，乔治梅森大学的物理学家，甚至宣称蝴蝶就是霍尔效应的家、是它天然的归宿。她写了这本书，寄给我——当时还不是出版的书，而是一份手稿——问我能不能为她写点东西，讲讲我发现 G 图的回忆。于是我写了点东西，那就成了——这里写着“南区人供稿”——那成了她这本书的序言。在她书里，她谈到了 G 图和各种各样我完全不知道的东西之间的联系：阿波罗尼奥斯垫（Apollonian gasket）、贝里相位、玻色-爱因斯坦凝聚、陈数、马约拉纳
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condensates turn numbers my Rana fermions quantum Hall effect quasi crystals renormalization group topological insulators abelian and non abelian antion's and so forth all of those things ah she also talks about how in the last several years attempts to empirically to confirm this structure that the actual physically observed structure is identical to what was predicted there all sorts of ways very ingenious and experimental techniques that have been developed to try to confirm it and pieces have been done not the whole thing but it's all way above my head unfortunately I got out of physics 40 some years ago but it's nice to know that it has done all of this her book contains all sorts of wonderful things and I would suggest that you take a look at it if you're interested in this and a lot of poems it includes those two lute limericks that I wrote about block and Hwanhee and to conclude all we wrote write of we'd a poem that I wrote about my experience in discovering G plot it's called G plots
费米子、量子霍尔效应、准晶、重整化群、拓扑绝缘体、阿贝尔与非阿贝尔任意子（anyon），等等等等，所有这些东西。啊，她还讲到近几年人们如何尝试用实验来确证这个结构，也就是确认实际在物理上观察到的结构与当年预言的完全一致，用了各种各样非常巧妙的实验技术来试图验证它。已经做出了一部分，还不是全部，但这些对我来说都太高深了。很遗憾，我四十多年前就离开物理学了，但知道它带来了这么多东西，还是很欣慰的。她的书里有各种各样精彩的内容，如果你对这些感兴趣，我建议你去看看。里面还有很多诗，包括我写的那两首关于 Bloch 和 Wannier 的打油诗。最后我要念一首我写的、关于我发现 G 图这段经历的诗，题目叫《G 图的
15问答：运气、准备与父亲的身教
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grace what happens if a crystal is laced with the lines of a magnetic field what spectrum will the world be graced with what energies will nature yield it turns out that the matters crux is determined by just what the flux is P over Q cubans baguettes non ratio so give Cantor sets on hearing this a physicist will declare it numerology but once shown G plot all agreed deep magics lurking in a crystal this gem I found by luck that's why there but for G plots grace go I thank you [Applause] oh thanks I see I went overtime good very good excellent yes well sanity in the yeah I I kind of I I know just what you mean I mean in some sense gödel Escher Bach which is what it became was was much more me than working in physics was if you look at it a posteriori you're right that in some sense I needed that more than I needed the other but I needed a what is the word I'm looking for sort of a password or I can't think of the right term I had to get into I had to get a PhD I needed a PI if I didn't get a PhD what would I
恩典》：如果一块晶体被磁场的力线所贯穿，会发生什么？世界将被怎样的谱所装点，大自然又会给出怎样的能量？结果表明，问题的关键完全取决于磁通量是多少：P比 Q 给出 Q 条能带，而无理的比值则给出康托集。听到这个，物理学家会宣称这是数字命理学；可一旦看到 G 图，所有人都同意，深奥的魔法潜藏在晶体之中。这颗宝石是我靠运气找到的，所以说，若不是 G 图的恩典，哪有今天的我。谢谢大家。[掌声]哦，谢谢。我看我超时了。好，很好，非常好。是的，嗯……是啊，我大概明白你的意思，我是说，从某种意义上讲，《哥德尔、艾舍尔、巴赫》——也就是它最后变成的那本书——比在物理学里工作要更像我自己。如果事后回头看，你说得对，从某种意义上说，我更需要那个，而不是另一个。但我需要一个——我想找的那个词是什么呢——某种通行证吧，我想不出准确的说法。我必须进入……我必须拿到一个博士学位，我需要一个身份。如果我没拿到博士学位，我会变成什么样呢？我不知道。我是很认真的，你提出了一个非常严肃的问题，每次我想到
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have become I don't know I'm very serious here you've raised a very serious question which every time I look back to that period in my life and I asked myself what if Gregory wanye had not been at the University of Oregon what would have would I have done I had I certainly wouldn't have gotten a PhD in particle physics I was totally fed up with particle physics and I don't think really to be honest I would have gotten would have been able to get a PhD in any branch of physics unless it had somehow magically tied in with number theory so I was just phenomenally lucky and lucky in many ways wanye being there the computer being in the lab stool over my the other computer being in rust on Tony's office and so and of course the fact that I had worked on number theory in the 60s all of those pieces of luck were amazing but I really and I think I if I hadn't gotten a PhD I I don't know what in the world I would have done because it was I absolutely goal in life unshakable to be a professor and I had to get a PhD so I
回到我人生中的那段时期，我问自己：如果格雷戈里·万耶当时不在俄勒冈大学，会怎么样？我会做什么呢？我肯定不会拿到粒子物理学的博士学位，我已经对粒子物理彻底厌倦了说实话，我觉得我也不可能在物理学的任何分支拿到博士学位除非它能以某种神奇的方式和数论扯上关系。所以我真是运气好得离谱，而且是在很多方面都走运：万耶当时在那儿那台计算机就在实验室里，就在我旁边，另一台计算机在托尼办公室的角落里，还有当然还有我在六十年代研究过数论这件事。所有这些运气凑到一起，简直不可思议，但我真的我想，如果我没拿到博士学位，我真不知道自己会去干什么，因为那是我人生中绝对不可动摇的目标——当一名教授，而我必须拿到博士学位。所以我真是非常幸运。于是我意识到，我拿到博士学位的关键就是
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was just very lucky and so I realized my own mic my key to getting the PhD is to work on this damn thing this damn problem I mean because at that time I wasn't thinking that I was working on a problem that was gonna be exciting I just knew it was gonna be a problem that I had to work on so I just said put that damn manuscript away and work on this damn problem it was very difficult but sanity in that case didn't mean in mental sanity and insanity in the sense of it actually said Society for the sanity and survival of d-h and it was for the sort of intellectual survival but you're right I should I mean and of course I did go back as soon as I got my PhD went out very good yes I can i my my I was always interested in integers I'll tell you something I remember discovering when I was a kid I don't know how old I was but pretty young I remember that you know I loved squares and powers in general and of course I memorized lots of powers and of course I was familiar with the simple powers like three squared is nine and
去搞这个鬼东西、这个该死的问题。我是说，因为那个时候我并没有想着自己在研究一个多么激动人心的问题，我只知道那是一个我必须去研究的问题。所以我就对自己说：把那份该死的手稿收起来，去搞这个该死的问题。这非常困难，但那种情况下的"理智"并不是指精神上的理智或疯狂它的意思是——实际上它的全称是"为了d-h的理智与存续协会"，指的是那种智识上的存续。不过你说得对，我是说，当然，我一拿到博士学位就回去继续做了。非常好，是的，我可以，我一直都对整数很着迷。我给你讲个我小时候发现的东西，我不记得自己当时几岁了，但相当小。我记得，你知道，我喜欢平方数，喜欢各种幂，当然我背下了很多幂次，当然我也熟悉那些简单的平方，比如三的平方是九
79:17
five squared is 25 and I remember one time I multiplied nine and twenty-five together and I got 225 and I recognized that that was a square also I was very excited and I looked at which square it was it was teens squared and I thought oh my god 15 is 3 times 5 this is amazing 3 squared times 5 squared is 3 times 5 squared I thought I was a pioneer I thought I thought I was you know I had found something magical so yeah I was interested in integers ever since I was a small kid and yeah it goes way way back to very very early thank you for having bringing that up well I was 15 years old yeah I actually yeah right I was a high school student correct no but I had a friend I had a friend who was exactly my age who taught me how to program and and then when I went to Stanford a year later when I was 16 I had a mentor in the mathematics department name Gordon Latta who was a professor of mathematics and a brilliant teacher a wonderful wonderful teacher and I would go into Gordon latuza's office on a regular basis and he would
五的平方是二十五。我记得有一次我把九和二十五乘起来，得到二百二十五，然后我认出那也是一个平方数而且我特别兴奋，我看了一下那是哪个数的平方，是15的平方，我心想天哪，15是3乘5，这太神奇了，3的平方乘5的平方就等于3乘5的平方，我以为我是先驱，我以为我发现了什么神奇的东西。所以是的，我从小就对整数感兴趣，这可以追溯到非常非常早的时候，谢谢你提到这个。当时我15岁，是的，对，我那时候是个高中生，没错。不过我有个朋友，跟我同岁，他教我编程，然后一年后我16岁去了斯坦福，我在数学系有一位导师，叫Gordon Latta，他是数学教授，是一位杰出的老师，一位非常非常好的老师。我经常去Gordon Latta的办公室，他
80:55
always welcome me and I would tell him about things that I had discovered and he was always so warm and and welcoming and it really made a huge difference in my life so I think what you're saying is that there was a teacher it wasn't my high school teacher all those on very friendly terms with him but it was my it was my favorite math teacher at Stanford who who played that role way up at the top there it is it is more than 40 years old I couldn't tell you where I bought it but yeah I course did that deliberately I
总是很欢迎我，我会告诉他我发现的那些东西，他总是那么热情、那么欢迎我，这对我的人生产生了巨大的影响。所以我想你的意思是，确实有这么一位老师，不是我的高中老师，虽然我跟他关系也很好，但真正扮演这个角色的，是我在斯坦福最喜欢的数学老师。一直挂在最上面，就是它，它有40多年历史了，我说不上是在哪儿买的，但是的，我当然是故意那么做的。
81:46
talk about what currents patterns yeah well I I am a believer in the beauty like I said at the very early first the second slide you know I believe in the beauty and simplicity of nature as my dad instilled in me not by telling me or beating it into my head but just by exhibiting a love for the beauty of nature in everything he did that it transmitted itself automatically to me and so yes and I think that's a good question to conclude with so thank you very much [Applause]
谈谈现在的模式吧。是的，我是一个信奉美的人，就像我很早的时候在第一张、第二张幻灯片上说的，我相信自然的美和简洁，这是我父亲灌输给我的，不是靠说教或者硬塞进我脑子里，而只是通过他在做每一件事时都展现出对自然之美的热爱，这自然而然地传递给了我。所以是的，我觉得这是个很好的收尾问题，非常感谢你。（掌声）
视频总结 · 一句话概括与核心要点
一句话概括
Douglas Hofstadter 讲述自己如何从少年时痴迷的数论（"人行道序列"与自相似函数 int）出发，历经数学与粒子物理两次崩溃，最终在固体物理中借助台式计算机发现 Hofstadter 蝴蝶（G-plot）——一个由无穷多扭曲自身副本嵌套而成的分形能谱，并阐明它日后与康托集、量子霍尔效应和菲尔兹奖工作的深刻联系。
核心要点
- 少年数论探索为日后发现埋下伏笔：16 岁左右他在 Stanford 用父亲报税的计算器和 Burroughs 220 研究"人行道序列"（Eta 序列）——以无理数 α（如 √2）为步长走过单位格子，记录每步跨越的格线数，得到由两个相邻整数（"count"与"sep"）组成的序列。这本质上是两种周期性（格子周期与步长周期）的竞争。
- Eta 序列的"基本定理"与自相似函数 int：对序列"求导"（数两个 sep 之间的 count 个数）得到另一个属于 α′ 的 Eta 序列，α′ = (sep−α)/(α−count)，可无限迭代。√2 与黄金分割 φ 互为"列互换"，由此定义函数 int(α)。int 由无穷多缩小且弯曲的自身副本组成，在每个有理数处跳跃（跳幅随分母增大而缩小），在无理数处连续。
- 两次职业危机：Berkeley 数学研究生课程过度抽象（"数论课从不提整数"），1968 年他转投 Oregon 大学粒子物理；但面对每周涌入的海量预印本（夸克、胶子、瞬子、Regge 极点……）完全找不到信号。1973 年 12 月 14 日，他被要求报告一篇一次引入 156 个新粒子的规范理论论文，对比 Pauli 1930 年仅提出一个中微子都惶恐不安，他怒斥"这些人毫无羞耻心"，摔下论文退出，六年投入归零。
- 三句话打破"固体物理是贫民窟"的偏见：朋友 Pete Rimbey 说 Gregory Wannier 是"当今最深刻的人之一"；智利友人 Francisco Claro 称固体物理"连接微观与宏观两个世界"；一位 Stanford 学生说"粒子物理只有一个真空，固体物理每个晶体都是不同的真空"。晶体作为离散、各向异性的介质，让他立刻联想到整数格点与实数线的关系——数论就是"一维固体数学"。
- Wannier 给出的问题：晶体 + 磁场：Bloch 能带（1928）与 Landau 能级（真空中磁场里电子能级等间距）的结合，问题已悬置 40 年。关键参数 α = 每个晶格胞的磁通 ÷ 磁通量子 (hc/e)。Wannier 声称 α 为有理数与无理数时行为不同——他起初觉得"自然怎么可能在意一个实数的无穷多位小数"，但这正与 int 的性质呼应。
- 从 Bloch 能带到 Harper 方程：以最简单的 2D 方格能带 E = cos(k_x a)+cos(k_y a)，用 Dirac 的 c 数→q 数替换、Landau 规范、Peierls 替换 ħk→p−eA/c，得到联系五个格点的差分薛定谔方程；沿一轴假设周期性后降为一维的 Harper 方程（1955，Harper 与 Wannier 分别发现）：二阶差分项对应动能，cos(2παm) 项如"人行道"般代表晶体与磁场的周期竞争。
- 有理 α 的判据与"胖 x 轴"：把递推写成 2×2 转移矩阵，α = p/q 时矩阵每 q 步重复，Obermeyer 证明波函数有界当且仅当 q 个矩阵乘积的迹满足 |Tr| ≤ 4。迹是 ε 的 q 次多项式，落在 ±4 灰带内的区间即能带：α=1/3 有 3 条带，1/5 与 2/5 各 5 条但分布不同，1/6 有 6 条且中间两条相切。
- 一台无人使用的 HP 台式机是决定性优势：在 Regensburg（1974，Obermeyer 组）他不擅长同事们的超几何函数与微分方程，但会编程。1974 年 9 月开始手工绘图，11 月（恰逢粒子物理的"十一月革命"——J/ψ 粒子与粲夸克发现）看出图中局部是整体的扭曲副本，与 int 的"肋骨"结构如出一辙。Wannier 斥之为"数字命理学"，建议他写"图书馆论文"并把结果放附录。
- 说服导师与论文完成：回到 Eugene 后借用 Russ Donnelly 实验室带绘图仪的 HP 机，1975 年 5 月画出精细 G-plot；他截取 α∈(1/7,1/6) 间的小梯形并"矩形化"还原，Wannier 看后惊呼"By God"，1975 年 8 月转变态度。他推断无理 α 的谱是无穷多无穷小带即康托集（不可数点、测度零），并以"抹平/抖动（Feynman 称 jiggling）"磁场不确定度解释其物理可观测性；此图放在论文第 137 页（精细结构常数倒数）。
- 后续影响远超预期：数学界将无理谱为康托集的猜想归于 Mark Kac 并改称"almost Mathieu 方程"（尽管他已在论文、Phys. Rev. 与 GEB 三处发表）；Avila 与 Jitomirskaya 2009 年证明该猜想，Avila 2014 年部分因此获菲尔兹奖。1980 年量子霍尔效应发现后，蝴蝶的能隙可用丢番图方程标记的 Chern 数一一对应量子化电导台阶；六角、三角、Kagome、准晶等各种晶格的蝴蝶均具嵌套自相似性。
结论与值得注意的细节
- Hofstadter 自评："我最终拿的其实是数论博士。" Bloch 指出体系中有两个自然频率（回旋频率与晶格频率），其比值恰为 α——与他少年的两周期竞争 Eta 序列完全同构。
- 他把讲座献给当天生日的父亲 Robert Hofstadter（与 Kai Siegbahn 同样研究过康普顿效应），论文致谢末句称"对自然之美与简洁的永恒信念直接来自我父亲"。
- 为逼自己完成博士，他 1974 年手写"DH 理智与生存协会章程"，规定每日不写论文的罚款，并把《哥德尔、埃舍尔、巴赫》手稿锁进抽屉一年多。
- Feynman 1975 年看到 G-plot 时立刻让秘书复印；Physical Review 匿名审稿人（20 年后自曝为 Leo Falicov）称论文"像一首莫扎特嬉游曲"。
- 尽管有此发现，他仍选择"光荣退出"物理，因为"不认为这种运气会再来一次"。问答环节他坦言：若 Wannier 不在 Oregon、两台闲置计算机不存在、少年时没做数论，他大概拿不到任何博士学位——整个故事是"机会偏爱有准备的头脑"的多重巧合。
- Indubala Satija 的书称蝴蝶是"量子霍尔效应的自然家园"，并联系到 Apollonian 垫圈、Berry 相、Majorana 费米子、拓扑绝缘体、任意子等，近年已有部分实验验证；Hofstadter 为该书作序。
核心句型 · 9
1. before plunging into X, I want to say …
“All I want to say before plunging into the topic itself that I'm very fond of Sweden”
演讲开场白结构：先做一段与主题无关的铺垫再进入正题。plunge into 表示「一头扎进」，比 start 更生动，适合正式演讲或报告开头。
2. bear with me for a moment although it will seem like …
“So bear with me for a moment although it will seem like it's just an excursion in elementary number theory it is in fact very central to the talk”
请求听众耐心的礼貌句式，常用于即将展开一段看似离题的铺垫时。可仿写：Bear with me — this may seem tangential, but it matters later.
3. it didn't elude me that …
“It didn't elude me that the word helix for this phase factor rhymed with the name Felix”
双重否定表达「我注意到了」，比 I noticed 更书面、带点俏皮。elude 原义「逃脱」，此处指某事「没有逃过我的注意」。
4. how can you recognize the great gems in the gigantic garbage dump
“How can you recognize the great gems in the gigantic garbage dump how can you know what to follow and what to ignore”
用强烈对比意象（gems vs garbage dump）加头韵（g-g-g）表达信噪比问题。修辞性反问适合强调困境，可仿写于描述信息过载。
5. I was suffering from a prejudice, a prejudice that I later found out was definitely a prejudice
“Because I was suffering from a prejudice a prejudice that I later found out was definitely a prejudice”
同一词三次重复的自嘲式强调，口语演讲中营造幽默与坦诚。suffer from 搭配抽象名词（prejudice/illusion）很地道。
6. X was not slumming it; there was something quite beautiful …
“It wasn't slumming it there was something quite beautiful closely related to number theory”
slum it 是俚语「自降身份」。「否定旧看法 + there was something + 形容词」是转述观念转变的常用结构。
7. all of a sudden the scales fell from his eyes
“All of a sudden the scales fell from his eyes and he said by God”
源自《圣经·使徒行传》的成语，指顿悟、看清真相。用于描述他人态度骤变，语气正式而有画面感。
8. I was hurt but I was not deterred
“So I was pretty hurt but I was not deterred I continued working all by myself”
「受挫但不退缩」的对仗句式，deter 是被动用法。写个人经历或推荐信时可直接套用。
9. There but for X's grace go I
“There but for G plots grace go I”
改写英语谚语 There but for the grace of God go I（若非上帝恩典，我也会如此）。替换 God 为任意事物，表示「多亏了X我才有今天」。
生词精讲 · 141 · 按出现顺序
plunging into
phr.
0:12
一头扎进，直接进入（正题）
fond of
phr.
0:12
喜爱，钟情于
binds him
/baɪndz/
v.
1:37
把他和……联系在一起
eternal
/ɪˈtɜːrnl/
adj.
1:37
永恒的
circuitous
/sərˈkjuːɪtəs/
adj.
2:59
迂回曲折的
bumpy
/ˈbʌmpi/
adj.
2:59
颠簸的，坎坷的
frame of reference
phr.
2:59
参照系，参考框架
travails
/trəˈveɪlz/
n.
2:59
艰辛，劳苦（书面语）
debacle
/dɪˈbɑːkl/
n.
2:59
惨败，溃败
integer sequences
n. phr.
4:20
整数序列
bear with me
phr.
4:20
请耐心听我说
excursion
/ɪkˈskɜːrʒn/
n.
4:20
离题的插叙；短途旅行
correspondence
/ˌkɔːrəˈspɑːndəns/
n.
4:20
书信，通信
come across
phr. v.
5:29
偶然发现，碰巧遇到
continuous variables
n. phr.
5:29
连续变量
empirical
/ɪmˈpɪrɪkl/
adj.
5:29
经验的，基于观察/计算的
clumps
/klʌmps/
n.
8:02
簇，团块
periodicity
/ˌpɪriəˈdɪsəti/
n.
8:02
周期性
mnemonic
/nɪˈmɑːnɪk/
n.
8:02
助记口诀
derivative
/dɪˈrɪvətɪv/
n.
9:22
导数；此处指讲者自定义的「序列求导」
suspiciously like
phr.
9:22
可疑地像，看上去极像
irrational
/ɪˈræʃənl/
adj.
10:25
（数学）无理的
terminate
/ˈtɜːrmɪneɪt/
v.
11:33
终止
golden ratio
n. phr.
11:33
黄金比例 φ≈1.618
complementarity
/ˌkɑːmplɪmenˈterəti/
n.
12:38
互补性
interchange
/ˈɪntərtʃeɪndʒ/
v./n.
12:38
互换，交换
blurrier
/ˈblɜːriər/
adj.
13:50
更模糊的
momentarily
/ˌmoʊmənˈterəli/
adv.
13:50
（美）马上，一会儿
ribs
/rɪbz/
n.
14:53
肋状条纹（此处指图形中的分支）
telltale
/ˈtelteɪl/
adj.
14:53
泄露端倪的，标志性的
albeit
/ɔːlˈbiːɪt/
conj.
14:53
尽管，虽然
nesting
/ˈnestɪŋ/
n.
16:01
嵌套
diligent
/ˈdɪlɪdʒənt/
adj.
16:01
勤勉的
intoxicating
/ɪnˈtɑːksɪkeɪtɪŋ/
adj.
16:01
令人陶醉的
jump discontinuity
n. phr.
16:01
跳跃间断（点）
denominator
/dɪˈnɑːmɪneɪtər/
n.
17:17
分母
limiting process
n. phr.
17:17
极限过程
fledgling
/ˈfledʒlɪŋ/
adj.
18:28
羽翼未丰的，初出茅庐的
traumatic
/trəˈmætɪk/
adj.
18:28
造成创伤的
hotbed
/ˈhɑːtbed/
n.
18:28
温床
prejudice
/ˈpredʒədɪs/
n.
19:40
偏见
low-hanging fruit
idiom
20:58
唾手可得的成果
preprint
/ˈpriːprɪnt/
n.
20:58
预印本（未正式发表的论文）
reverberating din
n. phr.
20:58
回响不绝的喧嚣
ad nauseam
/æd ˈnɔːziæm/
adv.
22:14
（拉丁）令人作呕地没完没了
at my wit's end
idiom
22:14
黔驴技穷，束手无策
signal-to-noise
adj.
22:14
信噪比的
in one fell swoop
idiom
23:27
一举，一下子
conservation laws
n. phr.
23:27
守恒定律
brash
/bræʃ/
adj.
24:45
莽撞自负的
sense of shame
n. phr.
24:45
羞耻心
blank slate
n. phr.
24:45
白板，一片空白
distinguished
/dɪˈstɪŋɡwɪʃt/
adj.
26:04
杰出的，卓越的
gung-ho
/ˌɡʌŋ ˈhoʊ/
adj.
26:04
热情过头的，狂热的
slumming
/ˈslʌmɪŋ/
v.
26:04
自降身份，去低档地方（slum it）
isotropic
/ˌaɪsəˈtrɑːpɪk/
adj.
27:15
各向同性的
lattice
/ˈlætɪs/
n.
27:15
格点，晶格
superimposed on
phr.
28:24
叠加在……之上
mascot
/ˈmæskɑːt/
n.
28:24
吉祥物
vividly
/ˈvɪvɪdli/
adv.
28:24
生动地，清晰地
forbidden energies
n. phr.
29:33
禁戒能量（能隙）
evenly spaced
phr.
30:42
等间距的
flux
/flʌks/
n.
30:42
通量（磁通量）
pure number
n. phr.
31:57
纯数（无量纲数）
come in handy
idiom
33:16
派上用场
retool
/ˌriːˈtuːl/
v.
33:16
重新装备，重新学习技能
standing wave
n. phr.
33:16
驻波
elude
/ɪˈluːd/
v.
33:16
逃过（注意）；didn't elude me=我没忽略
localized
/ˈloʊkəlaɪzd/
adj.
34:34
局域化的
limerick
/ˈlɪmərɪk/
n.
34:34
五行打油诗
boggled his mind
idiom
34:34
令他目瞪口呆
sanity
/ˈsænəti/
n.
35:51
神志清醒，理智
monetary fines
n. phr.
35:51
罚款
master manipulators
n. phr.
37:11
操作高手（此处指擅长处理方程）
inferior
/ɪnˈfɪriər/
adj.
38:33
低人一等的
Hamiltonian
/ˌhæmɪlˈtoʊniən/
n.
38:33
哈密顿量（能量算符）
gauges
/ɡeɪdʒɪz/
n.
39:48
（物理）规范
curl
/kɜːrl/
n.
39:48
旋度
numerator
/ˈnuːməreɪtər/
n.
40:57
分子
barge in
phr. v.
40:57
闯入，硬闯
legal validity
n. phr.
40:57
合法性，正当性
translation operators
n. phr.
42:11
平移算符
stunning
/ˈstʌnɪŋ/
adj.
43:25
令人震惊的
dimensionless
/dɪˈmenʃnləs/
adj.
43:25
无量纲的
separation of variables
n. phr.
44:41
分离变量法
difference equation
n. phr.
44:41
差分方程
counterpart
/ˈkaʊntərpɑːrt/
n.
45:52
对应物
eigenvalues
/ˈaɪɡənˌvæljuːz/
n.
45:52
本征值
recursion relation
n. phr.
47:07
递推关系
divergence
/daɪˈvɜːrdʒəns/
n.
48:17
发散
blow up
phr. v.
48:17
（数值）爆掉，趋于无穷
bounded
/ˈbaʊndɪd/
adj.
49:39
有界的
trace
/treɪs/
n.
50:48
（矩阵的）迹
polynomial
/ˌpɑːliˈnoʊmiəl/
n.
50:48
多项式
flinging about
phr. v.
50:48
随手挥舞，信手摆弄
revived
/rɪˈvaɪvd/
v.
52:10
使复活，修复
cubic polynomial
n. phr.
52:10
三次多项式
quintic
/ˈkwɪntɪk/
adj.
53:26
五次的
tangent to
/ˈtændʒənt/
adj.
54:43
与……相切
teeny-weeny
/ˌtiːni ˈwiːni/
adj.
54:43
极小的（口语）
plotter
/ˈplɑːtər/
n.
54:43
绘图仪
intuited
/ɪnˈtuːɪtɪd/
v.
55:59
直觉感到
in my mind's eye
idiom
55:59
在我脑海里
a posteriori
/ˌeɪ pɒˌstɪriˈɔːraɪ/
adv.
55:59
（拉丁）事后地，后验地
resonance
/ˈrezənəns/
n.
57:18
共振（态）
numerology
/ˌnuːməˈrɑːlədʒi/
n.
58:34
数字命理学（贬义：无理论支撑的数字游戏）
deterred
/dɪˈtɜːrd/
v.
58:34
被吓退
ultimatum
/ˌʌltɪˈmeɪtəm/
n.
58:34
最后通牒
got nowhere
idiom
59:59
毫无进展
the scales fell from his eyes
idiom
62:27
茅塞顿开（源自《圣经》）
uncountable set
n. phr.
62:27
不可数集
measure zero
n. phr.
63:42
测度为零
reconcile
/ˈrekənsaɪl/
v.
63:42
调和，使一致
smeared
/smɪrd/
adj.
64:50
被抹平的，模糊化的
jiggling
/ˈdʒɪɡlɪŋ/
v.
64:50
轻轻抖动
reciprocal
/rɪˈsɪprəkl/
n.
65:55
倒数
couldn't resist
phr.
65:55
忍不住
cyclotron frequency
n. phr.
65:55
回旋频率
going out in style
idiom
67:06
漂亮收场
intimately linked
phr.
67:06
密切相关
divertimento
/dɪˌvɜːrtɪˈmentoʊ/
n.
67:06
嬉游曲（轻快的器乐组曲）
music to my ears
idiom
67:06
悦耳动听的话，正中下怀
flattered
/ˈflætərd/
adj.
68:15
受宠若惊的
blaze of glory
idiom
68:15
辉煌时刻（go out in a blaze of glory=在巅峰退场）
attributed
/əˈtrɪbjuːtɪd/
v.
68:15
归功于
expedition
/ˌekspəˈdɪʃn/
n.
69:30
探险，远征
variations on a theme
idiom
69:30
同一主题的变奏
profoundly
/prəˈfaʊndli/
adv.
70:44
深刻地
quantized conductance
n. phr.
70:44
量子化电导
manuscript
/ˈmænjuskrɪpt/
n.
71:56
手稿
prologue
/ˈproʊlɔːɡ/
n.
71:56
序言
ingenious
/ɪnˈdʒiːniəs/
adj.
73:07
精巧的，巧妙的
above my head
idiom
73:07
超出我的理解
laced with
phr.
74:18
穿插着，交织着
crux
/krʌks/
n.
74:18
关键，症结
lurking
/ˈlɜːrkɪŋ/
v.
74:18
潜伏，隐藏
fed up with
idiom
76:37
受够了
phenomenally
/fəˈnɑːmɪnəli/
adv.
76:37
惊人地，非凡地
unshakable
/ʌnˈʃeɪkəbl/
adj.
76:37
不可动摇的
on very friendly terms with
idiom
80:55
与……关系很好
instilled in
/ɪnˈstɪld/
v.
81:46
灌输给
理解自测 · 11 题 · 是真懂了，还是以为自己懂
1. 讲者少年时研究的「η序列」是如何定义的？它体现了什么核心概念？
η序列是这样定义的：想象一条方砖边长为1的人行道，以步长α（如√2）行走，记录每一步跨过的砖缝数，得到由两个相邻整数（如1和2）组成的序列。讲者在第6–7段解释，这个序列体现的核心概念是「两个相互竞争的周期性」——人行道的周期1与步长α不可通约。这一概念贯穿全篇：后来Harper方程中晶格周期与磁场回旋周期的竞争（第38段），以及Bloch指出的两个自然频率之比恰为α（第54段），都是同一结构的物理再现。
2. 讲者1973年12月14日发生了什么？他为什么用泡利的例子来对比？
那天讲者被安排讲解一篇规范理论论文，该论文一口气引入156个新粒子。他在约十人面前说「这些人一点羞耻心都没有」，把论文扔在地上，宣布「我不干了」（第20–21段）。他用泡利作对比：1930年泡利为挽救三条守恒定律只引入一个中微子，却害怕到不敢写论文，只写了一封信让朋友在会上宣读。泡利是最莽撞的物理学家之一尚且如此谨慎，而这篇论文毫无理由地发明156个粒子——讲者以此说明当时粒子物理在他眼中已失去理论纪律，这促使他彻底离开该领域。
3. 打破讲者对固体物理偏见的三个人分别说了什么？
第22–23段列出三人：(1) 同学Pete Rimbey说「Gregory Wannier是当今在世最有深度的人之一」；(2) 智利朋友Francisco Claro说「固体物理在微观与宏观两个世界之间架起桥梁」；(3) 一位不知名的斯坦福研究生说「粒子物理只有一个真空，固体物理有许多个真空，每种晶体都是一种不同的真空」。第三句对讲者最关键——它让他意识到晶体是离散、各向异性的介质，正如数论中整数格点相对于实数轴。这一类比（第24段「数论是固体物理的数学版」）是他思想转折的核心。
4. 为什么讲者说自己「最终其实拿了一个数论博士学位」？
这句自嘲出现在第55段。表面上他的博士论文属于固体物理（晶体电子在磁场中的能谱），但其核心发现完全依赖数论直觉：α为有理数p/q时有q条能带，无理数时为康托集；蝴蝶图由自身的变形副本无限嵌套而成——这与他16岁时研究的INT函数（第13–15段）在有理数处跳跃、无理数处连续、自相似嵌套的性质高度同构。Bloch指出两个频率之比恰为α，让他看到这正是少年时的η序列问题。他从伯克利数学系退学，却用数论思维完成了物理博士，因此说「回到了原点」。
5. 讲者相对于Wannier、Rauh、Obermair三位解析高手的「唯一优势」是什么？这个优势如何转化为发现？
他的优势是会编程（第42–43段）。三位合作者擅长微分方程和超几何级数的解析处理，讲者自认数学功底不如他们，坐在办公室里「感到自卑」。但隔壁恰好有一台闲置的HP台式计算机，他五年前在阿斯彭用过。他用它对大量有理数α计算转移矩阵的迹，画出「胖x轴」图，逐点手绘能带位置（第44–45段）。1974年11月，图形积累到足以显现内嵌的小蝴蝶时，他凭少年时对INT函数「肋越靠近角越小」的记忆，认出这是自相似结构。可以说，他把一个别人只能解析处理的问题变成了可视化问题，而视觉模式识别正是他的强项。
6. Wannier为什么起初把讲者的发现斥为「数字命理学」？讲者最终是如何说服他的？
第48–51段交代了这一过程。Wannier是传统解析物理学家，认为没有数学证明、仅凭图形观察得出的「G图由自身副本构成」的结论只是数字游戏，建议讲者做文献综述论文，把「命理学」放进附录。讲者尝试像证明INT那样给出数学证明，但失败了（这个问题直到2009年才被Avila和Jitomirskaya解决）。回到尤金后，他用Russ Donnelly实验室配有绘图仪的计算机画出精确蝴蝶图，Wannier仍不信。最后讲者取出图中一个倾斜梯形区域内的小副本，算出所有数值并「矩形化」拉直，展示它与整图一致，Wannier才「茅塞顿开」。这说明：在无法证明时，足够精确的可视化证据同样可以建立科学信念。
7. 「无理数α对应康托集」这一数学结论如何与物理现实调和？费曼对此有何反应？
第52–53段给出了答案。康托集测度为零却不可数，且每个无理α对应不同的康托集——这在物理上看似荒谬，因为没有磁场能被定义到无穷位小数。讲者的调和方法是引入不确定度Δα：取小邻域内所有α对应能谱的并集，得到的「抹平图」非常光滑，分形细节消失，像一张带湖泊的地图；抖动越少，细节越多，零抖动就回到原图。1975年他写信给费曼并到办公室拜访，费曼看到蝴蝶图兴奋得让秘书立刻复印，并把这一操作称为「抖动」（jiggling）图。这一段说明讲者并非纯粹沉迷数学美，而是认真考虑了测量精度对可观测结构的影响。
8. 讲者自称「运气好得离谱」，他列举了哪些运气？这与「机会偏爱有准备的头脑」是什么关系？
第62段讲者列举：Wannier恰好在俄勒冈大学；雷根斯堡隔壁恰有闲置的HP计算机；尤金Donnelly实验室恰有第二台配绘图仪的计算机；以及他在1960年代研究过数论。他坦言若无Wannier，他不可能在任何物理分支拿到博士，「除非它能神奇地与数论扯上关系」。第48段观众插话「机会偏爱有准备的头脑」（巴斯德名言），讲者回应「我有准备的头脑，Gregory没有」。两者的关系是互补的：运气提供了问题和工具，但看出蝴蝶图自相似的能力来自十年前对INT函数的深入研究——没有准备，同样的运气只会产生一堆数字。
9. 如果有人反驳说「蝴蝶图在实验上几十年无法验证，所以只是数学游戏」，讲者会如何回应？
讲者本人在第58–60段提供了回应材料。首先，1980年量子霍尔效应发现后，TKNN证明蝴蝶图每个能隙对应一个陈数，精确等于霍尔电导台阶——「蝴蝶是霍尔效应天然的家」（Satija语），拓扑学由此进入凝聚态物理。其次，2013年后石墨烯莫尔超晶格实验确实观测到了蝴蝶谱的部分结构。讲者可能还会引用自己的第53段：他早已用「抹平」处理回答了「无穷精度不物理」的质疑——真实测量看到的是光滑化后的谱，而分形结构是理想极限。他也会承认（第57段）证明康托集猜想极难，需要菲尔兹奖级别的数学，但「难以验证」不等于「没有物理内容」。
10. 讲者的经历对「研究是怎样做成的」这一问题有何启示？把它放到今天信息更过载的科研环境里，还成立吗？
讲者的路径揭示几点：(1) 领域切换时带着旧工具（数论直觉）进入新领域，能看到本领域专家看不到的模式；(2) 在解析方法主导的时代掌握计算与可视化，形成差异化优势；(3) 面对导师否定，用更精确的证据（矩形化）而非争辩来说服；(4) 自我约束机制（「理智与存续协会」的罚款章程）帮助他熬过低谷。放到今天，第19段描述的「预印本洪流」已被arXiv放大百倍，信噪比问题更严重，但讲者的解法仍适用：不是读完所有论文，而是找到一个足够基本、自己有独特工具切入的问题。不同之处在于，今天计算工具人人可得，差异化优势更可能来自跨领域的「有准备的头脑」而非工具本身。
11. 讲者说《哥德尔、艾舍尔、巴赫》「比在物理学里工作更像我自己」，却又说物理博士是「通行证」。这种手段与目的的关系，对年轻研究者的职业选择有什么可迁移的判断？
第61–63段讲者坦承：他真正想写的是GEB，博士学位是成为教授的「通行证」（他找不到更准确的词），因此把手稿锁进抽屉一年多去「搞这个该死的问题」。可迁移的判断有三层：第一，明确区分「我热爱的事」与「我必须完成的事」，两者可以并存但要排序，讲者为此制定了带罚款的章程；第二，被迫完成的「手段」可能意外成为最重要的成果——蝴蝶图被引用数千次、催生菲尔兹奖工作，而他当时只把它当作拿学位的路；第三，讲者在巅峰退出（第56段「blaze of glory」）说明他并未因意外成功而改变初衷。对年轻人的启示是：认真对待你不得不做的事，它可能比你计划的事走得更远，但不必因此放弃原本的方向。
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