数学家证明了一个2D版本的量子重力工作

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亚历山大·莫塔科夫是普林斯顿大学的理论物理学家,1981年瞥了一眼量子理论的未来。一系列谜团,从弦乐队的夸克与夸克的结合到质子中,要求他可以剪影的新数学工具刚刚制作。

Polyakov的提案证明了很大。他的作品将物理学家带入了一个新的数学竞技场,一个是解锁称为弦的理论对象的行为并构建量子重力模型的一个必不可少的数学竞技场。

然而,过去七年来,一群数学家已经完成了许多研究人员认为不可能的。

亚历山大·莫塔科夫(Alexander Polyakov)是普林斯顿大学的理论物理学家,1981年瞥了一眼量子理论的未来。一系列谜团,从弦乐队的夸克与夸克的结合到质子中,要求他可以剪影的新数学工具刚刚制作。

“科学中存方法和公式,作为许多明显不同问题的掌握密钥,”他物理信件B中的一个现着名的四页信中写道,“目前我们必须开发一个艺术随机表面上处理总和。“

Polyakov的提案证明了很大。他的论文中,他勾勒出了一个粗略描述了如何计算如何计算狂野混乱类型的平均值的公式,“Liouville场”。他的作品将物理学家带入了一个新的数学竞技场,一个是解锁称为弦的理论对象的行为并构建量子重力模型的一个必不可少的数学竞技场。

多年的辛劳将领导Polyakov对物理学中其他理论的突破性解决方案,但他从未完全理解Liouville领域背后的数学。

然而,过去七年来,一群数学家已经完成了许多研究人员认为不可能的。一个有限的地标出版物中,他们使用完全严谨的数学语言重新译分Polyakov的公式,并证明了Liouville领域完美地模拟了现象Polyakov认为它会。

“数学中花费了40年来有四页,”法国国家科学研究中心的数学家,艾西克 - 马赛大学的Rémi罗得乐研究中心的法国科学研究和同招关的数学家Vincent Vargas说,Vincent Vargas说赫尔辛基,法国科学研究中心的François大卫,以及巴黎萨利大学的科林Guillarmou。

三篇论文数学世界和物理学的凌乱现实之间伪造了一座桥梁 - 他们通过概率理论的数学领域打破新的地面来这样做。这项工作还涉及关于基本物理领域中占据中心阶段的物体的哲学问题:量子领域。

“这是数学物理学中的杰作,”宾夕法尼亚大学的数学家辛孙说。

今天的物理学中,最成功的理论中的主要行为者是填补空间的田间对象,从一个地方到达不同的值。

例如,古典物理学中,单个字段会告诉您一切关于强制推动对象的一切。采取地球的磁场:指南针的抽搐揭示了该领域的影响(其力量和方向)地球上的各个点。

领域也是量子物理学的核心。然而,由于量子理论的深度随机性,这里的情况更加复杂。从量子的角度来看,地球不产生一个磁场,而是无限数量的不同。有些看起来几乎就像我们古典物理中观察的那种领域,但其他人都很疯狂地不同。

但物理学家仍然希望使预测预测是理想地匹配的预测,这种情况下,登山者指南针上读取的东西。将量子场的无限形式与单个预测同化,是“量子场理论”或QFT的强大任务。这是一个或多个量子字段的模型,每个量子域都具有其无限变化,行为和交互。

由巨大的实验支持驱动,QFT已成为粒子物理学的基本语言。所谓的标准模型是一种这样的QFT,描绘了像电子一样电子的基本颗粒,作为从电子领域的不确定出现的模糊凸块。它已经通过了迄今为止的每一个实验测试(尽管各种组可能找到第一孔的边缘上)。

物理学家玩很多不同的QFT。一些,如标准模型,渴望通过我们宇宙的四个维度(三个空间尺寸加上一维时间)来模拟真正的粒子。其他人描述了奇怪的宇宙中的异国情调颗粒,从二维平坦到六维超级世界。他们与现实的联系是偏远的,但物理学家希望他们能够获得他们可以回到我们自己的世界的见解。

Polyakov的Liouville Field理论是这样的一个例子。

基于来自法国数学家Joseph Liouville的1800年代的复杂分析的平等的延志领域描述了一个完全随机的二维表面 - 即表面,就像地壳一样,但其中的高度每一点都是随机选择的。这种行星将用无限高峰的山脉爆发,每个山脉都通过用无限脸部滚动模具来分配。

这样的对象似乎似乎不像物理学的信息模型,但随机性不是没有模式。例如,钟曲线告诉你,您将街上随机通过7英尺的篮球运动员。同样,球根云和斜斜线沿着随机模式,但仍然可以其大规模和小规模特征之间辨别一致的关系。

Liouville理论可用于识别所有可能随机锯齿状表面的无尽景观中的模式。 Polyakov意识到这种混乱的地形对于建模琴弦至关重要,它们移动时挖出曲面。该理论也已应用于二维世界中描述量子重力。爱因斯坦定义了重力作为时空曲率,但将他的描述转化为量子场理论的语言创造了一个无限数量的空间 - 随着地球产生无限的磁场集合。 Liouville理论将所有这些表面包装一起。它为物理学家提供了测量曲率的工具 - 而是随机2D表面上的每个位置处的引力。

“量子重力基本上意味着随机几何形状,因为量子意味着随机和重力装置几何形状,”太阳说。

Polyakov探索随机曲面世界的第一步是写下定义发现特定尖峰球的几率的表达,因为钟曲线定义了满足特定高度某人的可能性。但他的配方没有导致有用的数值预测。

为了解决量子场理论,能够使用该领域来预测观察。实践中,这意味着计算领域的“相关函数”,其通过描述一个点处的字段的测量值与另一个点的测量来捕获字段的行为。例如,计算光子字段中的相关函数可以为您提供量子电磁的教科书规律。

Polyakov更摘要之后:随机表面的本质,类似于使云或海岸线的云云或海岸线的统计关系类似。他需要Liouville领域的随意高度之间的相关性。几十年来,他尝试了两种不同的方式计算它们。他从一个名为Feynman路径积分的技术开始,并最终开发了一个围绕着称为引导的工作。两种方法以不同的方式出现,直到新工作背后的数学家以更精确的制定方式。

您可能想象一下,许多形式的核算量可以采取量子字段是不可能的。你会是对的。20世纪40年代Richard Feynman,一项量子物理先驱,开发了一个处理这种令人困惑的处方的处方,但该方法被证明是严重的限制。

再一次,地球的磁场。您的目标是使用量子场理论来预测当您特定位置读取罗盘读取时将观察到的。为此,Feynman建议将所有领域的表格汇总一起。他认为,您的阅读将代表所有领域可能表单的平均值。添加具有正确加权的这些无限字段配置的过程称为Feynman路径积分。

这是一个优雅的想法,仅对选择量子字段产生混凝土答案。没有已知的数学过程可以有意义地平均覆盖一般的无限空间的无限数量的物体。路径积分是比精确的数学配方的物理哲学更多。数学家质疑它的存作为有效的操作,并由物理学家依赖它的方式困扰。

德国波恩大学的数学家EveliinaPeltola说:“我被遗憾地被禁止为数学家。”

物理学家可以利用Feynman的路径积分来计算只有最乏味的字段字段计算确切的相关函数,这些字段不会与其他字段互动甚至是自己。否则,他们必须融合它,假装田地是免费的,并加入温和的相互作用或“扰动”。此过程称为扰动理论,使它们标准模型中大多数字段中的相关函数,因为大自然的力量发生得很弱。

但它不适用于Polyakov。虽然他初步推测,Liouville Field可能适用于添加轻度扰动的标准黑客,但他发现它自身互动太强烈。与自由领域相比,Liouville Field似乎数学上还有责备,其相关函数出现了无法实现的。

Polyakov很快开始寻找一个工作。 1984年,他与Alexander Belavin和Alexander Zamolodchikov合作,开发一种称为Bootstrap的技术 - 一个数学阶梯,逐渐导致Field的相关函数。

要开始攀爬梯子,您需要一个功能,它表达了Mere In The Mere三个点之间的测量之间的相关性。这种“三点相关函数”加上了有关能量的一些附加信息,可以采用粒子的粒子,形成引导梯的底部梯级。

从那里开始一次点一点:使用三点函数来构造四点函数,使用四点函数来构造五点函数,等等。但如果您第一个梯级中以错误的三点相关函数从错误的三点相关函数开始,则该过程会生成冲突的结果。

Polyakov,Belavin和Zamolodchikov使用自靴子成功地解决了各种简单的QFT理论,而是与Feynman路径积分一样,他们无法为Liouville领域工作。

然后20世纪90年代两对物理学家 - Harald Dorn和Hans-Jörg奥托和Zamolodchikov和他的兄弟Alexei-anged击中了三点相关函数,使得可以缩放梯子,完全解决Liouville领域(和其简单的量子重力描述)。他们的姓名缩写为冠军作为Dozz公式,让物理学家涉及刘维尔领域的任何预测。但即使是作者也知道他们已经偶然到达了它,而不是通过健全的数学来抵达。

“他们是这种猜测公式的那种天才,”Vargas说。

受过良好教育的猜测物理学中有用,但他们不满足数学家,后来想要知道Dozz配方来自哪里。解决了Liouville领域的等式应该来自该领域本身的一些描述,即使没有人有最新的想法如何获得它。

“它看着我就像科幻小说,”库基纳说。 “这永远不会被任何人证明。”

2010年代初,Vargas和Kupiainen与概率理论罗得罗德和物理学家François大卫相连。他们的目标是搭配刘维尔野外的数学松散目的 - 以形式化Polyakov被遗弃的Feynman路径积分,只可能会使Dozz公式揭示。

正如他们开始的那样,他们意识到一位名叫Jean-Pierre Kahane的法国数学家已经发现了几十年来,结果将成为Polyakov主理论的关键。

“某种意义上是完全疯狂的,刘维尔没有我们面前定义,”瓦尔加斯说。 “所有的成分都那里。”

洞察力2014年和2020年之间完成了三个里程碑纸张数学物理学。

他们首先抛光了Polyakov的失败的路径积分,因为Liouville Field本身非常互动,与Feynman的扰动工具不相容。所以相反,数学家使用了卡纳的想法,以重新狂野的Liouville领域作为一个较高的狂热的随机对象,称为高斯自由场。高斯自由领域的峰值不会Liouville领域的峰值波动到与Liouville领域的峰值相同,这使得数学家可以以明智的方式计算平均值和其他统计措施。

“以某种方式只是使用高斯自由领域,”Peltola说。 “从那时起,他们可以理论中构建一切。”

2014年,他们推出了它们的结果:1981年的路径积分多书的一个新的和改进版本,但可信高斯自由场方面完全定义。这是一个罕见的例子,其中Feynman的路径积分哲学已经发现了一个坚实的数学执行。

“可以存路径积分,确实存,”德国电子同步rotron的物理学家JörgTeschner说。

通过手中的严格定义的路径,研究人员然后试图看出它们是否可以使用它来从Liouville字段中获得答案并导出其相关函数。目标是神话般的dozz公式 - 但它之间的海湾和路径积分似乎很大。

“我们写我们的论文中,只是为了宣传原因,我们想要了解Dozz公式,”库圭金说。

该团队花了多年的概率散步,确认它真正拥有让自动启动工作所需的所有功能。正如他们所做的那样,他们建立早期的工作特质。最终,Vargas,Kupiainen和Rhodes成功地成功了一篇文章,2017年和20月20日10月的另一个纸上,有科林突齐德里。它们从路径积分派生DOZ和其他相关函数,并显示这些公式完全匹配的等式物理学家使用自举者。

“现我们已经完成了,”瓦尔加斯说。 “两个对象都是一样的。”

工作解释了Dozz公式的起源,并连接了引导程序 - 其中数学家被认为是粗略的数学对象。完全,它解决了Liouville领域的最终奥秘。

“这是一个以某种方式结束时代,”Peltola说。 “但我希望它也是一些新的有趣事情的开始。”

vargas和他的合作者现有一个独角兽,一个强烈地交互的qft,通过简短的数学公式以非触发方式完美地描述,这也使得数值预测。

现百万美元问题是:这些概率方法有多远?它们是否可以为所有QFT生成整洁的公式? vargas很快就会削弱这种希望,坚持认为他们的工具特定于刘维尔理论的二维环境。较高的尺寸中,即使是自由田过于不规则,所以他怀疑该集团的方法能够能够处理我们宇宙中的引力领域的量子行为。

但是,Polyakov的“Master Key”的新鲜薄荷将打开其他门。它的效果已经感受到概率理论,数学家现可以挥动以前的狡猾物理公式,不受惩罚。由Liouville Work,Sun和他的合作者已经从物理学进口方程来弥补,以解决有关随机曲线的两个问题。

物理学家也等待有形的福利,进一步走路上。 Liouville Field的严格建设可以激发数学家,以证明其他似乎棘手的QFTS的特点 - 不仅仅是玩具的重力,而且是直接现实最深刻的物理秘密上携带的真实颗粒和力的描述。

“[数学家]会做我们甚至无法想象的事情,”周边学院的理论物理学家Davide Gaiotto说。


英文译文:

Alexander Polyakov, a theoretical physicist now at Princeton University, caught a glimpse of the future of quantum theory in 1981. A range of mysteries, from the wiggling of strings to the binding of quarks into protons, demanded a new mathematical tool whose silhouette he could just make out.

“There are methods and formulae in science which serve as master keys to many apparently different problems,” he wrote in the introduction to a now famous four-page letter in Physics Letters B. “At the present time we have to develop an art of handling sums over random surfaces.”

Polyakov’s proposal proved powerful. In his paper he sketched out a formula that roughly described how to calculate averages of a wildly chaotic type of surface, the “Liouville field.” His work brought physicists into a new mathematical arena, one essential for unlocking the behavior of theoretical objects called strings and building a simplified model of quantum gravity.

Years of toil would lead Polyakov to breakthrough solutions for other theories in physics, but he never fully understood the mathematics behind the Liouville field.

Over the last seven years, however, a group of mathematicians has done what many researchers thought impossible. In a trilogy of landmark publications, they have recast Polyakov’s formula using fully rigorous mathematical language and proved that the Liouville field flawlessly models the phenomena Polyakov thought it would.

“It took us 40 years in math to make sense of four pages,” said Vincent Vargas, a mathematician at the French National Center for Scientific Research and coauthor of the research with Rémi Rhodes of Aix-Marseille University, Antti Kupiainen of the University of Helsinki, François David of the French National Center for Scientific Research, and Colin Guillarmou of Paris-Saclay University.

The three papers forge a bridge between the pristine world of mathematics and the messy reality of physics—and they do so by breaking new ground in the mathematical field of probability theory. The work also touches on philosophical questions regarding the objects that take center stage in the leading theories of fundamental physics: quantum fields.

“This is a masterpiece in mathematical physics,” said Xin Sun, a mathematician at the University of Pennsylvania.

In physics today, the main actors in the most successful theories are fields—objects that fill space, taking on different values from place to place.

In classical physics, for example, a single field tells you everything about how a force pushes objects around. Take Earth’s magnetic field: The twitches of a compass needle reveal the field’s influence (its strength and direction) at every point on the planet.

Fields are central to quantum physics, too. However, the situation here is more complicated due to the deep randomness of quantum theory. From the quantum perspective, Earth doesn’t generate one magnetic field, but rather an infinite number of different ones. Some look almost like the field we observe in classical physics, but others are wildly different.

But physicists still want to make predictions—predictions that ideally match, in this case, what a mountaineer reads on a compass. Assimilating the infinite forms of a quantum field into a single prediction is the formidable task of a “quantum field theory,” or QFT. This is a model of how one or more quantum fields, each with their infinite variations, act and interact.

Driven by immense experimental support, QFTs have become the basic language of particle physics. The so-called standard model is one such QFT, depicting fundamental particles like electrons as fuzzy bumps that emerge from an infinitude of electron fields. It has passed every experimental test to date (although various groups may be on the verge of finding the first holes).

Physicists play with many different QFTs. Some, like the standard model, aspire to model real particles moving through the four dimensions of our universe (three spatial dimensions plus one dimension of time). Others describe exotic particles in strange universes, from two-dimensional flatlands to six-dimensional uber-worlds. Their connection to reality is remote, but physicists study them in the hopes of gaining insights they can carry back into our own world.

Polyakov’s Liouville field theory is one such example.

The Liouville field, which is based on an equation from complex analysis developed in the 1800s by the French mathematician Joseph Liouville, describes a completely random two-dimensional surface—that is, a surface, like Earth’s crust, but one in which the height of every point is chosen randomly. Such a planet would erupt with mountain ranges of infinitely tall peaks, each assigned by rolling a die with infinite faces.

Such an object might not seem like an informative model for physics, but randomness is not devoid of patterns. The bell curve, for example, tells you how likely you are to randomly pass a 7-foot basketball player on the street. Similarly, bulbous clouds and crinkly coastlines follow random patterns, but it’s nevertheless possible to discern consistent relationships between their large-scale and small-scale features.

Liouville theory can be used to identify patterns in the endless landscape of all possible random, jagged surfaces. Polyakov realized this chaotic topography was essential for modeling strings, which trace out surfaces as they move. The theory has also been applied to describe quantum gravity in a two-dimensional world. Einstein defined gravity as space-time’s curvature, but translating his description into the language of quantum field theory creates an infinite number of space-times—much as the Earth produces an infinite collection of magnetic fields. Liouville theory packages all those surfaces together into one object. It gives physicists the tools to measure the curvature—and hence, gravitation—at every location on a random 2D surface.

“Quantum gravity basically means random geometry, because quantum means random and gravity means geometry,” said Sun.

Polyakov’s first step in exploring the world of random surfaces was to write down an expression defining the odds of finding a particular spiky planet, much as the bell curve defines the odds of meeting someone of a particular height. But his formula did not lead to useful numerical predictions.

To solve a quantum field theory is to be able to use the field to predict observations. In practice, this means calculating a field’s “correlation functions,” which capture the field’s behavior by describing the extent to which a measurement of the field at one point relates, or correlates, to a measurement at another point. Calculating correlation functions in the photon field, for instance, can give you the textbook laws of quantum electromagnetism.

Polyakov was after something more abstract: the essence of random surfaces, similar to the statistical relationships that make a cloud a cloud or a coastline a coastline. He needed the correlations between the haphazard heights of the Liouville field. Over the decades he tried two different ways of calculating them. He started with a technique called the Feynman path integral and ended up developing a work-around known as the bootstrap. Both methods came up short in different ways, until the mathematicians behind the new work united them in a more precise formulation.

You might imagine that accounting for the infinitely many forms a quantum field can take is next to impossible. And you would be right. In the 1940s Richard Feynman, a quantum physics pioneer, developed one prescription for dealing with this bewildering situation, but the method proved severely limited.

Take, again, Earth’s magnetic field. Your goal is to use quantum field theory to predict what you’ll observe when you take a compass reading at a particular location. To do this, Feynman proposed summing all the field’s forms together. He argued that your reading will represent some average of all the field’s possible forms. The procedure for adding up these infinite field configurations with the proper weighting is known as the Feynman path integral.

It’s an elegant idea that yields concrete answers only for select quantum fields. No known mathematical procedure can meaningfully average an infinite number of objects covering an infinite expanse of space in general. The path integral is more of a physics philosophy than an exact mathematical recipe. Mathematicians question its very existence as a valid operation and are bothered by the way physicists rely on it.

“I’m disturbed as a mathematician by something which is not defined,” said Eveliina Peltola, a mathematician at the University of Bonn in Germany.

Physicists can harness Feynman’s path integral to calculate exact correlation functions for only the most boring of fields—free fields, which do not interact with other fields or even with themselves. Otherwise, they have to fudge it, pretending the fields are free and adding in mild interactions, or “perturbations.” This procedure, known as perturbation theory, gets them correlation functions for most of the fields in the standard model, because nature’s forces happen to be quite feeble.

But it didn’t work for Polyakov. Although he initially speculated that the Liouville field might be amenable to the standard hack of adding mild perturbations, he found that it interacted with itself too strongly. Compared to a free field, the Liouville field seemed mathematically inscrutable, and its correlation functions appeared unattainable.

Polyakov soon began looking for a work-around. In 1984, he teamed up with Alexander Belavin and Alexander Zamolodchikov to develop a technique called the bootstrap—a mathematical ladder that gradually leads to a field’s correlation functions.

To start climbing the ladder, you need a function which expresses the correlations between measurements at a mere three points in the field. This “three-point correlation function,” plus some additional information about the energies a particle of the field can take, forms the bottom rung of the bootstrap ladder.

From there you climb one point at a time: Use the three-point function to construct the four-point function, use the four-point function to construct the five-point function, and so on. But the procedure generates conflicting results if you start with the wrong three-point correlation function in the first rung.

Polyakov, Belavin, and Zamolodchikov used the bootstrap to successfully solve a variety of simple QFT theories, but just as with the Feynman path integral, they couldn’t make it work for the Liouville field.

Then in the 1990s two pairs of physicists—Harald Dorn and Hans-Jörg Otto, and Zamolodchikov and his brother Alexei—managed to hit on the three-point correlation function that made it possible to scale the ladder, completely solving the Liouville field (and its simple description of quantum gravity). Their result, known by their initials as the DOZZ formula, let physicists make any prediction involving the Liouville field. But even the authors knew they had arrived at it partially by chance, not through sound mathematics.

“They were these kind of geniuses who guessed formulas,” said Vargas.

Educated guesses are useful in physics, but they don’t satisfy mathematicians, who afterward wanted to know where the DOZZ formula came from. The equation that solved the Liouville field should have come from some description of the field itself, even if no one had the faintest idea how to get it.

“It looked to me like science fiction,” said Kupiainen. “This is never going to be proven by anybody.”

In the early 2010s, Vargas and Kupiainen joined forces with the probability theorist Rémi Rhodes and the physicist François David. Their goal was to tie up the mathematical loose ends of the Liouville field—to formalize the Feynman path integral that Polyakov had abandoned and, just maybe, demystify the DOZZ formula.

As they began, they realized that a French mathematician named Jean-Pierre Kahane had discovered, decades earlier, what would turn out to be the key to Polyakov’s master theory.

“In some sense it’s completely crazy that Liouville was not defined before us,” Vargas said. “All the ingredients were there.”

The insight led to three milestone papers in mathematical physics completed between 2014 and 2020.

They first polished off the path integral, which had failed Polyakov because the Liouville field interacts strongly with itself, making it incompatible with Feynman’s perturbative tools. So instead, the mathematicians used Kahane’s ideas to recast the wild Liouville field as a somewhat milder random object known as the Gaussian free field. The peaks in the Gaussian free field don’t fluctuate to the same random extremes as the peaks in the Liouville field, making it possible for the mathematicians to calculate averages and other statistical measures in sensible ways.

“Somehow it’s all just using the Gaussian free field,” Peltola said. “From that they can construct everything in the theory.”

In 2014, they unveiled their result: a new and improved version of the path integral Polyakov had written down in 1981, but fully defined in terms of the trusted Gaussian free field. It’s a rare instance in which Feynman’s path integral philosophy has found a solid mathematical execution.

“Path integrals can exist, do exist,” said Jörg Teschner, a physicist at the German Electron Synchrotron.

With a rigorously defined path integral in hand, the researchers then tried to see if they could use it to get answers from the Liouville field and to derive its correlation functions. The target was the mythical DOZZ formula—but the gulf between it and the path integral seemed vast.

“We’d write in our papers, just for propaganda reasons, that we want to understand the DOZZ formula,” said Kupiainen.

The team spent years prodding their probabilistic path integral, confirming that it truly had all the features needed to make the bootstrap work. As they did so, they built on earlier work by Teschner. Eventually, Vargas, Kupiainen, and Rhodes succeeded with a paper posted in 2017 and another in October 2020, with Colin Guillarmou. They derived DOZZ and other correlation functions from the path integral and showed that these formulas perfectly matched the equations physicists had reached using the bootstrap.

“Now we’re done,” Vargas said. “Both objects are the same.”

The work explains the origins of the DOZZ formula and connects the bootstrap procedure—which mathematicians had considered sketchy—with verified mathematical objects. Altogether, it resolves the final mysteries of the Liouville field.

“It’s somehow the end of an era,” said Peltola. “But I hope it’s also the beginning of some new, interesting things.”

Vargas and his collaborators now have a unicorn on their hands, a strongly interacting QFT perfectly described in a nonperturbative way by a brief mathematical formula that also makes numerical predictions.

Now the literal million-dollar question is: How far can these probabilistic methods go? Can they generate tidy formulas for all QFTs? Vargas is quick to dash such hopes, insisting that their tools are specific to the two-dimensional environment of Liouville theory. In higher dimensions, even free fields are too irregular, so he doubts the group’s methods will ever be able to handle the quantum behavior of gravitational fields in our universe.

But the fresh minting of Polyakov’s “master key” will open other doors. Its effects are already being felt in probability theory, where mathematicians can now wield previously dodgy physics formulas with impunity. Emboldened by the Liouville work, Sun and his collaborators have already imported equations from physics to solve two problems regarding random curves.

Physicists await tangible benefits too, further down the road. The rigorous construction of the Liouville field could inspire mathematicians to try their hand at proving features of other seemingly intractable QFTs—not just toy theories of gravity but descriptions of real particles and forces that bear directly on the deepest physical secrets of reality.

“[Mathematicians] will do things that we can’t even imagine,” said Davide Gaiotto, a theoretical physicist at the Perimeter Institute.


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