从拉马努金到重整化：消除分歧和获得物理结果的艺术
From Ramanujan to renormalization: the art of doing away with divergences and arriving at physical results
摘要：
一个世纪前，Srinivasa Ramanujan --伟大的自学成才的印度数学天才--在从英国剑桥返回后不久去世，在那里他与Godfrey Hardy合作。拉马努金贡献了许多杰出的成果，不同的数学分支，如分析和数论，重点是特殊的功能和系列。在这里，我们指的是显然奇怪的价值观，他分配给两个简单的发散系列，$\sum_{n \geq 1} n$和$\sum_{n \geq 1} n^{3}$。然而，这些值作为解析延拓是合理的，它们对应于黎曼的$ζ$函数。此外，它们在物理学中也有应用：我们讨论了光子场的真空能量，从中可以推导出卡西米尔力，卡西米尔力已经被实验测量到。我们进一步讨论其解释，这仍然是有争议的。这是一个简单的方法来说明重整化的概念，这是至关重要的量子场论。
英文摘要：
A century ago, Srinivasa Ramanujan -- the great self-taught Indian genius of mathematics -- died, shortly after returning from Cambridge, UK, where he had collaborated with Godfrey Hardy. Ramanujan contributed numerous outstanding results to different branches of mathematics, like analysis and number theory, with a focus on special functions and series. Here we refer to apparently weird values which he assigned to two simple divergent series, $\sum_{n \geq 1} n$ and $\sum_{n \geq 1} n^{3}$. These values are sensible, however, as analytic continuations, which correspond to Riemann's $ζ$-function. Moreover, they have applications in physics: we discuss the vacuum energy of the photon field, from which one can derive the Casimir force, which has been experimentally measured. We further discuss its interpretation, which remains controversial. This is a simple way to illustrate the concept of renormalization, which is vital in quantum field theory.
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