From conformal loop ensembles to conformal field theory

有时，许多基本成分以简单易懂的方式相互作用，如金属中的电子，液体中的分子，或股票市场上的买卖双方，当大量存在时，会在大范围内产生意想不到的结果。这通常被称为突发行为，一般来说，很难预测这种行为是什么。物理学家感兴趣的许多系统是那些具有许多成分的系统，这些成分可以在不太远的邻居之间相互作用时波动（热或量子力学）。令人惊讶的是，虽然这种相互作用是局部的，但在某些情况下，这些成分会形成非常大的群体，由非常多的邻居组成的链，这些群体一起波动，就好像这些群体是一个新系统的新组成部分。这些都是突发行为。大群体倾向于形成的情况被称为临界，因为这时系统对外部干扰非常敏感：整个群体将对这种干扰做出反应，产生大而远距离的变化。自然，这些令人惊讶的突发集体行为导致了大量有趣的物理现象，比如近藤云的形成改变了含磁性杂质的金属的导电性能。将宏观经济学中个体行为与新兴行为联系起来也很诱人，而且在未来有望取得丰硕成果：一次小规模的次贷市场崩盘导致了一场全球性衰退！物理学家提出了一个非常强大的理论，基于物理原理，描述了关键系统中的紧急行为。这就是量子场论。事实上，20世纪理论物理学最伟大的成就之一就是认识到，在当今的实验中观察到的所有基本粒子都可以被理解为来自一个更简单、更对称的理论：这就是量子场论的标准模型。然后我们对这些突现行为有了理解，但这种理解还没有形成一个数学上连贯的整体，也不是对突现集体本身的完整理解。我们通过量子粒子和它们如何散射来理解突发行为，通过能量和它如何在局部变化，通过局部探针和它们如何对局部干扰做出反应。但我们常常不知道如何将这些不同的想法联系起来，以及如何将它们联系起来，并实际描述，组成成分的波动突现的集体。共形场论是量子场论的一系列模型，其中我们上面列举的理解的标准要素得到了更好的发展和相互联系。它们对应于量子场论中一个很小但很有启发性的角落。大约三年前，数学家们提出了一系列数学度量，这些度量被认为是，有时也被证明是，用来描述关键系统中远距离行为的某些方面，这些方面落入了共形场论所描述的角落。在一些作品中，我最近强调，这些度量实际上准确地描述了那个角落里所有出现的波动物体，至少对于一个广泛的模型家族来说是这样。我的研究包括使用这些数学方法，以便将涌现的集体与共形场论的强大结构完全联系起来。这将给我们提供一个全新的视角，让我们更细致地了解突现物体的行为方式，并将首次提供一条从潜在的多组分系统到量子场论的完整路径。

Sometimes, many basic constituents that are interacting amongst each other in simple and understood ways, such as electrons in a metal, molecules in a liquid, or buyers and sellers on the stock market, when present in large numbers, give rise to unexpected results on large scales. This is usually called emergent behaviours , and it is very hard to predict, in general, what such behaviours can be. Many systems of interest to physicists are those with very many constituents that can fluctuate (thermally or quantum mechanically) while interacting amongst not-too-far neighbours. Quite surprisingly, although the interaction is local , it happens in some situations that the constituents form very large groups, chains of very many neighbours, that fluctuate together, as if the groups were new constituents of a new system. These are emergent behaviours. Situations where big groups tend to form are called critical , because then the system is hyper-sensitive to external disturbances: whole groups will react to such disturbances, producing a big, large-distance change. Naturally, these quite surprising emergent collective behaviours are responsible for a wealth of interesting physical phenomena, like the formation of Kondo clouds that change conductive properties of metals with magnetic impurities. It is also tempting, and promise to be fruitful in the future, to make a connection with the emergent behaviours from individual agents in macroeconomics: a small sub-prime market crash gave us an international recession!Physicists came up with a very powerful theory, based on physical principles, that describes the emergent behaviours in critical systems. This is quantum field theory. In fact, one of the great achievements of theoretical physics of the twentieth century is the understanding that all fundamental particles that are observed in current-day experiments can be understood as emerging from a simpler, more symmetrical theory: this is the standard model of quantum field theory. We then have an understanding of such emergent behaviours, but this understanding does not form yet a mathematically coherent whole, neither is it a complete understanding of the emergent collectivities themselves. We understand emergent behaviours through quantum particles and how they scatter, through energy and how it varies locally, and through local probes and how they react to local disturbances. But we often don't know how to relate these various ideas, and how to connect them to, and actually describe, the fluctuating emergent collectivities of constituents.Conformal field theory is a family of models of quantum field theory where the standard elements of our understanding enumerated above are much better developed and connected to each other. They correspond to a small, but very instructive, corner of quantum field theory. About three years ago, mathematicians proposed a family of mathematical measures supposed, and sometimes proved, to describe certain aspects of large-distance behaviours in critical systems, aspects that fall into the corner described by conformal field theory. In some works, I recently emphasized that these measures in fact exactly describe all emergent fluctuating objects in that corner, at least for a wide family of models. My research consists in using these mathematical measures in order to fully connect the emergent collectivities with the powerful structure of conformal field theory. This will give us an entirely new insight into the more subtle way emergent objects behave, and will provide, for the first time, a complete path from underlying many-constituent systems, to quantum field theory.