数学 1918

不变变分问题

埃米·诺特

物理学中每一条守恒律的背后，都站着一个对称——反之亦然。

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In depth · the introduction

能量为何永远无法被创造、也无法被消灭，背后藏着一个理由——而找到它的人，是埃米·诺特。

核心想法

物理学家早就知道一份简短的「守恒律」清单：在一个封闭系统里，能量、动量、角动量的总量永不改变。诺特证明，这并不是三桩各自孤立的巧合。它们每一个，都是某个对称的后果——所谓对称，就是当你挪动某样东西时，自然法则依然分毫不变的那种「不变」。

正因为物理定律不在乎现在是几点，能量才守恒。正因为它们不在乎你身在何处，动量才守恒。正因为它们不在乎你面朝哪个方向，角动量才守恒。对称进去，守恒律出来——而她还证明了：反过来也成立。

它是如何诞生的

1915 年，两位在世的顶尖数学家——大卫·希尔伯特与费利克斯·克莱因——把诺特请到了哥廷根。他们正与爱因斯坦那套全新的引力理论搏斗，在那理论里，能量似乎会令人不安地「失常」——他们需要她对「不变量」无与伦比的掌握来理清头绪。诺特做的远不止理清头绪：她找到了潜伏在底下的那条普遍法则。

而她做成这一切时，连一个学者最寻常的权利都被剥夺着。身为女性，她无法担任带薪教授，一直以希尔伯特本人的名义讲课。当同僚反对给她一个职位时，据说希尔伯特顶了回去——大学评议会「又不是澡堂」。她直到 1919 年才正式获得任教的资格——正是这篇论文问世的次年。

它为何重要

在诺特之前，守恒律是靠实验发现的、被珍视的事实。在她之后，它们成了预言：说出自然的一个对称，她就能准确告诉你，什么必然守恒。这把物理学整个翻了过来——对称，成了构建新理论的首要工具，而粒子物理的标准模型，后来正是这样被建起来的。

一个可以想象的画面

想象一只完美的圆盘子。在桌上转动它，它看上去一模一样——这份「一样」，就是旋转对称。现在在它边缘压出一个凹痕：再转一下，你就看得出它动过了。圆盘子在旋转下「守住」了自己的模样；带凹痕的那只，则守不住。诺特定理说，自然也是这样运作的：凡是法则在某种改变之后看上去毫无二致，就总有某样东西保持精确不变——而你一旦在对称上压出凹痕，那样东西便开始漂移。

它的位置

诺特站在现代物理的枢纽上。组织起麦克斯韦电磁学、爱因斯坦相对论、以及希格斯场的那些对称——它们都在本馆的别处——正是通过她的定理，化作了守恒律。一个世纪之后，「是什么对称在守护这个量？」已是物理学家面对任何新想法时，最先发问的问题之一。

The original document

Original source text

Emmy Noether · Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Math.-phys. Klasse (1918): 235–257 · presented by F. Klein, 26 July 1918

The problem

Noether studies variational integrals I = ∫ L dx that are left unchanged by a continuous group of transformations (in the sense of Lie), and asks what follows for the associated Euler–Lagrange equations. Two theorems give the most general answer: one for finite groups (a fixed number of constant parameters), one for infinite groups whose transformations contain arbitrary functions.

Theorem I

If the integral I is invariant with respect to a G_ρ, then ρ linearly independent combinations of the Lagrange expressions become divergences — and from this, conversely, invariance of I with respect to a G_ρ will follow. The theorem holds good even in the limiting case of infinitely many parameters.

A divergence that vanishes is a conservation law: each of the ρ parameters of a finite symmetry group yields one conserved quantity — energy from invariance in time, momentum from invariance in space, angular momentum from invariance under rotation.

Theorem II

If the integral I is invariant with respect to a G_∞ρ in which the arbitrary functions occur up to the σ-th derivative, then there subsist ρ identity relationships between the Lagrange expressions and their derivatives up to the σ-th order. Here also the converse holds.

For symmetries that depend on arbitrary functions — the local, or “gauge,” symmetries, of which the general covariance of Einstein's gravitation is the great example — the field equations are no longer independent; they satisfy identities. The associated “conservation laws” become consequences of those identities: precisely the “improper” energy theorems Hilbert had noticed in general relativity.

[ … ]

The paper closes by noting that this is exactly the situation in Einstein's theory of gravitation, clarifying the assertions of Hilbert and Klein about the special status of the energy law there — the question that had brought Noether to the problem in the first place.

Göttingen · 1918