内容来源：https://www.quantamagazine.org/ten-martini-proof-uses-number-theory-to-explain-quantum-fractals-20250825/

内容总结：

【科学前沿】数论破解量子分形之谜："十杯马提尼"证明终获圆满

在量子物理与纯数学的交叉领域，一项持续近半个世纪的重大猜想终于获得突破性解答。国际数学界知名的"十杯马提尼猜想"（Ten Martini Problem）近日被中美数学家团队通过创新性数论方法彻底解决，该成果不仅揭示了量子系统中电子能级的分形结构，更证实了抽象数学理论在解释物理现象中的深远意义。

故事始于1974年，当时还是俄勒冈大学物理学研究生的侯世达（Douglas Hofstadter）在德国访学期间，尝试用台式计算机求解磁场作用下晶体中电子能级的薛定谔方程。当磁场强度参数α取有理数时，他通过大量计算绘制出电子允许能级的分布图，意外发现其负空间图案呈现出类似蝴蝶翅膀的精细分形结构，后被学界称为"侯世达蝴蝶"。

侯世达敏锐地意识到，当α逼近无理数时，这些能级分布应该构成著名的康托尔集（Cantor set）。这一猜想在1981年被数学家马克·卡克（Mark Kac）和巴里·西蒙（Barry Simon）正式提出，并承诺以"十杯马提尼"作为解决猜想的奖励。

经过多代数学家的努力，2005年斯维特兰娜·吉托米尔斯卡娅（Svetlana Jitomirskaya）与时年24岁的阿图尔·阿维拉（Artur Avila）合作完成了大部分证明，阿维拉更因此项工作获得菲尔兹奖。但该证明仍存在局限：只能处理理想化模型，且需组合多种论证方法，被学界称为"拼凑式证明"。

转机出现在2019年。中国南开大学葛灵睿博士加入吉托米尔斯卡娅团队，与游建功、周麒等中国数学家合作，对阿维拉提出的"全局理论"进行革命性重构。通过将薛定谔方程转化为对偶方程，并构建新的几何解释框架，团队最终给出了适用于多种物理场景的统一证明。这项发表于《数学年刊》的成果，彻底消除了学界对量子分形是否真实存在的质疑。

尤为重要的是，2013年哥伦比亚大学实验团队在石墨烯磁场实验中首次观测到"侯世达蝴蝶"图案，使该理论预测得到实验验证。最新证明不仅确立了数学理论对物理现象的解释力，其创新的全局分析方法更为解决同类难题提供了强大工具。

正如葛灵睿博士所言："我们就像在黑暗海洋中发现了指引方向的灯塔。"这项跨越五十年的研究印证了基础数学与物理学的深刻联系，也为未来探索量子世界中的数学结构开辟了新路径。

中文翻译：

《十杯马提尼证明：数论揭开量子分形之谜》

1974年，在道格拉斯·霍夫施塔特凭借《哥德尔、埃舍尔、巴赫：集异璧之大成》斩获普利策奖的五年前，这位俄勒冈大学的物理学研究生随导师赴德国雷根斯堡学术休假。本想练习德语的霍夫施塔特，加入了一群正为量子理论难题绞尽脑汁的杰出理论物理学家团队——他们试图求解置于磁场中的晶格内电子能级。

作为团队里的异类，霍夫施塔特完全跟不上同行的思路。如今回望，他反而庆幸这种"脱节"："他们虽在证明定理，却未触及问题本质。"他决定另辟蹊径，用重达40磅的HP 9820A可编程台式计算器进行数值运算。

霍夫施塔特需要求解量子力学核心的薛定谔方程特定形式。该方程通过输入电子及其环境参数，可推演电子行为并解出其允许拥有的能量值。在他研究的案例中，方程包含关键变量α（磁场强度与晶格单胞面积的乘积），这个参数承载着作用于电子的力场信息。

德国数学家团队已知：当α为有理数（整数或分数）时，薛定谔方程虽求解艰难但仍可解；而当α为无理数时，他们全然无措。霍夫施塔特反其道而行，从有理数切入编程计算，将输出结果打印在纸卷上。每个通宵运算后的清晨，他都会用毡头笔在拼接的坐标纸上精心绘制能量值图谱——这就是后来因形似蝴蝶翅膀负空间图案而闻名的"霍夫施塔特蝴蝶图"。

同事们讥讽这种笨拙方法是"炼金术"，称计算器为"侏儒妖"。其导师更斥之为"数字命理学"并威胁停止资助。但逐渐显现的蝴蝶图谱令霍夫施塔特着迷：当输入分数时，允许能级被大段禁戒值隔断；分母位数越复杂，能级间隔越密集，最终形成视觉震撼的分形图案——局部与整体具有相似性的特殊结构。

他直觉这反映了深刻的数学真理："我显然抓住了关键要害。"这个"要害"正是数学家格奥尔格·康托尔1883年提出的康托尔集：将线段三等分后去除中段，无限重复此操作后形成的无限点集。虽然无法直接计算无理数α（需无穷位数无法编程），但当有理数α无限逼近无理数时，允许能级集越来越趋近康托尔集。他由此推测：无理数α对应的可能能级将构成真正的康托尔集。

数年后，巴里·西蒙和马克·卡克从几乎周期函数角度得出相同结论。1981年午餐会上，他们发现当α为无理数时，该薛定谔方程正属于其研究范畴。卡克当场悬赏："谁能证明，我请十杯马提尼！"于是这个难题被称为"十杯马提尼猜想"。

此后二十年，数学家们逐步推进证明，但始终未达终极目标。2003年，长期研究该问题的斯维特兰娜·季托米尔斯基娅已放弃证明猜想，却意外迎来24岁数学家阿图尔·阿维拉的合作邀约。他们2005年在线发表的证明最终刊于《数学年刊》，阿维拉后来凭此部分工作获得菲尔兹奖。两人以畅饮马提尼兑现了悬赏，但该证明存在缺陷：仅适用于特定无理数α值，且依赖于先前证明的拼凑组合。

更棘手的是，该证明基于理想化假设（原子规则排列、磁场恒定），而现实情况更为复杂。当调整薛定谔方程中α相关参数时，证明即告失效。这暗示霍夫施塔特蝴蝶可能只是数学奇观，而非物理现实。就连霍夫施塔特本人在著作中坦言："若实验真发现蝴蝶图，我将是全世界最惊讶的人。"

然而2013年哥伦比亚大学物理学家在实验室中捕捉到了蝴蝶图：他们将两层石墨烯置于磁场中测量电子能级，量子分形赫然显现。"它突然从数学幻想变成了实际存在，"季托米尔斯基娅表示，"这令人非常不安。"

转机出现在2019年。加入季托米尔斯基娅团队的葛灵睿受阿维拉"全局理论"启发，与中国南开大学游江弓和周麒合作，通过重新诠释阿维拉的几何对象并应用于对偶方程，构建出无需拼凑的统一证明。这项突破不仅证实霍夫施塔特蝴蝶是真实物理现象，更彰显数论在物理世界的强大解释力。

研究团队已运用该全局理论破解该领域另外两个关键难题。葛灵睿比喻："我们发现了全局理论背后的隐藏奥秘，它如同黑暗海洋中的灯塔，为我们指引正确方向。"

（更正说明：阿维拉与季托米尔斯基娅合作研究十杯马提尼问题时实为24岁，此前误作26岁。）

英文来源：

‘Ten Martini’ Proof Uses Number Theory to Explain Quantum Fractals
Introduction
In 1974, five years before he wrote his Pulitzer Prize–winning book Gödel, Escher, Bach: An Eternal Golden Braid, Douglas Hofstadter was a graduate student in physics at the University of Oregon. When his doctoral adviser went on sabbatical to Regensburg, Germany, Hofstadter tagged along, hoping to practice his German. The pair joined a group of brilliant theoretical physicists who were agonizing over a particular problem in quantum theory. They wanted to determine the energy levels of an electron in a crystal grid placed near a magnet.
Hofstadter was the odd one out, unable to follow the others’ line of thought. In retrospect, he’s glad. “Part of my luck was that I couldn’t keep up with them,” he said. “They were proving theorems, but they had nothing to do with the essence of the situation.”
Hofstadter instead decided to test out a more down-to-earth approach. Rather than proving theorems, he was going to crunch some numbers using an HP 9820A desk calculator — a computerlike machine that weighed nearly 40 pounds and could be programmed to perform complex computations.
Hofstadter needed it to solve a particular formulation of the Schrödinger equation, which lies at the core of quantum mechanics. When fed certain information about an electron and its environment as inputs, the Schrödinger equation describes how the electron will behave. In particular, its solutions tell you the amount of energy the electron can have.
In the case that Hofstadter was interested in, the Schrödinger equation includes a variable called alpha, the product of the magnetic field’s strength and the area of one grid square. Alpha captures information about the forces acting on the electron.
The team of mathematicians in Germany knew that when alpha was rational — that is, either a whole number or a fraction — solving the Schrödinger equation was arduous but possible (so long as you had a big enough calculator). But when alpha was irrational, meaning that it could not be written as a fraction, they had no idea how to solve it.
Instead of struggling with the irrational case like his colleagues, Hofstadter started with what he knew. He programmed his calculator to take a rational value of alpha as an input and print the output on a roll of paper. The result represented the electron’s permitted and forbidden energy levels.
Every evening, Hofstadter would leave his calculator whirring. The next morning, he’d return to a scroll of paper unfurling from the machine, listing the locations of the permitted energies for each rational value of alpha he’d set as an input. He taped together a few pieces of graph paper and, using a felt-tip pen, began meticulously graphing these energy values. That picture would come to be known as the Hofstadter butterfly, because of the resemblance of the graph’s negative space to the insect’s patterned wings.
Hofstadter’s colleagues couldn’t understand the point of his laborious approach. They joked that he was trying to spin straw into gold and took to calling his calculator “Rumpelstilzchen.”
Even his adviser dismissed the effort as “numerology” and threatened to drop his funding. “He was implying that I was being superstitious and talking nonsense,” Hofstadter said. “Finding meaning and patterns in numbers when they don’t exist.”
But the butterfly that began to emerge on his graph paper intrigued him. Hofstadter noticed that when he entered a fraction, the allowed energies would be broken up by long stretches of forbidden values. As the fraction got more complicated, with more digits in the denominator, the breaks between possible energies got more numerous. The energy values began forming a visually striking pattern — a fractal, meaning that smaller parts of it looked the same as the whole.
His gut told him that it reflected a deep mathematical truth. “It was very clear to me that I had a tiger by the tail,” he said. He recognized the tiger. It was the Cantor set.
The set is named for the mathematician Georg Cantor, who popularized it in 1883 by following a simple rule: Take a line segment, split it into three equal sections, then erase the middle third. This leaves you with two segments separated by a gap. Now erase the middle third of each of these, and so on. If you carry out this procedure infinitely many times, you get an infinite set of points, spread out like dust along the number line.
Hofstadter would never plug in an irrational value of alpha. Irrational numbers can’t be expressed as a fraction — it would require infinitely many digits in the numerator or denominator, something that was impossible to program the calculator to handle. But he noticed that as the rational values of alpha got closer and closer to an irrational number, the set of permitted energy values — the bands of ink in each row of his butterfly picture — looked more and more like a Cantor set. And so, he hypothesized, when alpha was irrational, the possible energies would form an actual Cantor set.
Several years later, two prominent mathematicians came to the same conclusion from a very different direction. Barry Simon and Mark Kac had been studying what they called almost-periodic functions. The outputs of a periodic function, like a sine wave, repeat over and over again. But an almost-periodic function traces out a path that comes very close to repeating, yet never does.
In 1981, Kac and Simon met for lunch and got to discussing the version of the Schrödinger equation that Hofstadter and his colleagues were trying to solve. When alpha was an irrational value, the equation turned into an almost-periodic function. It was exactly the kind of phenomenon they’d been studying. And based on what they knew about almost-periodic functions, Hofstadter was right: The allowed energy levels should form a Cantor set when alpha is irrational.
But Simon and Kac couldn’t prove it either. Kac said he’d buy 10 martinis for anyone who could. Simon began to publicize Kac’s offer, and the problem became known as the ten martini conjecture.
Over the years, mathematicians chipped away at it, proving the conjecture for certain irrational values of alpha (but not all). Simon announced one of these intermediate results in 1982. Kac offered him three martinis. When Kac died in 1984, the problem remained open. A proof worth all 10 martinis wouldn’t appear for another 20 years.
Just a Tad Dirty
In 2003, Svetlana Jitomirskaya, who had spent years studying the almost-periodic function embedded in the Schrödinger equation, had just given up on her career-long goal of proving the ten martini conjecture. A year earlier, a competitor named Joaquim Puig had proved it for all but a few classes of irrational alpha values. What’s more, he’d used techniques she’d published earlier to do it. “I was kicking myself,” she said. “All the hard work was in my proof, and then here he comes with this beautiful argument.”
So she was surprised when a 24-year-old mathematician named Artur Avila visited her and suggested they work on the remaining values of alpha. “I told him it would be very difficult, very time consuming, and no one would care,” she said.
People did. Their proof, which they posted online in 2005, was eventually published in the Annals of Mathematics, the field’s most prestigious journal. Avila later won a Fields Medal in part for his work on the problem.
They decided to honor the 10-martini contract themselves. “We had plenty of celebratory beverages, martinis included,” Jitomirskaya said.
But in some ways, the proof was a bit unsatisfying. Jitomirskaya and Avila had used a method that only applied to certain irrational values of alpha. By combining it with an intermediate proof that came before it, they could say the problem was solved. But this combined proof wasn’t elegant. It was a patchwork quilt, each square stitched out of distinct arguments.
Moreover, the proofs only settled the conjecture as it was originally stated, which involved making simplifying assumptions about the electron’s environment. More realistic situations are messier: Atoms in a solid are arranged in more complicated patterns, and magnetic fields aren’t quite constant. “You’ve verified it for this one model, but what does that have to do with reality?” said Simon Becker, a mathematician at the Swiss Federal Institute of Technology Zurich.
These more realistic situations require you to tweak the part of the Schrödinger equation where alpha appears. And when you do, the ten martini proof stops working. “This was always disturbing to me,” Jitomirskaya said.
The breakdown of the proof in these broader contexts also implied that the beautiful fractal patterns that had emerged — the Cantor sets, the Hofstadter butterfly — were nothing more than a mathematical curiosity, something that would disappear once the equation was made more realistic.
Avila and Jitomirskaya moved on to other problems. Even Hofstadter had doubts. If an experiment ever saw his butterfly, he’d written in Gödel, Escher, Bach, “I would be the most surprised person in the world.”
But in 2013, a group of physicists at Columbia University captured his butterfly in a lab. They placed two thin layers of graphene in a magnetic field, then measured the energy levels of the graphene’s electrons. The quantum fractal emerged in all its glory. “Suddenly it went from a figment of the mathematician’s imagination to something practical,” Jitomirskaya said. “It became very unsettling.”
She wanted to explain it with mathematics. And a new collaborator had an idea for how to do it.
Another Round, With a Twist
In 2019, Lingrui Ge joined Jitomirskaya’s group. He had been inspired by the work she and Avila had done on the ten martini problem, as well as by a direction of research that Avila had been trying to pursue ever since.
Avila had grown tired of the piecemeal approaches that mathematicians used to understand almost-periodic functions. He instead began to develop what he called a “global theory” — a way to uncover higher-level structure in all sorts of almost-periodic functions, which he could then use to solve entire classes of functions in one go.
To do this, he associated a geometric object to a given almost-periodic function and studied its properties. He realized that some of those geometric properties could help him solve the original function.
But it only worked for certain types of functions. It couldn’t handle the types of calculations that the ten martini problem required. It wasn’t clear that it ever could.
That’s because to prove the ten martini conjecture, mathematicians had to first transform the Schrödinger equation into a related equation called its dual, then solve that new equation. Avila’s theory couldn’t say anything about the higher-level structure of the dual.
Or so he thought. But Ge was intrigued by the geometric objects that Avila had described. He suspected that other properties of those objects hid even more information — information that might illuminate aspects of the dual equation. “I could see that it was a very beautiful and important theory,” Ge said.
He and Jitomirskaya — along with Jiangong You and Qi Zhou of Nankai University in China — figured out a new way to interpret Avila’s geometric object and apply it to the dual. This made the theory far more powerful. It also allowed Ge, Jitomirskaya and You to write a single proof that solved versions of the ten martini problem in lots of different settings. No patchwork quilt needed.
The result cements the Hofstadter butterfly as a real-life phenomenon. The abstract world of number theory holds power in the world of physics.
The mathematicians have since used their version of Avila’s global theory to solve two other key problems in the area. They predict that this is just the beginning of what they can do with the method they’ve uncovered. “We found this hidden mystery behind the global theory,” Ge said. “It was like a beacon on a dark sea that showed us the right direction.”
Correction: August 25, 2025
An earlier version of this article stated that Avila was 26 years old when he began to collaborate with Jitomirskaya on the ten martini problem. He was 24.