數學 1918

不變變分問題

埃米·諾特

物理學中每一條守恆律的背後，都站著一個對稱——反之亦然。

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In depth · the introduction

能量為何永遠無法被創造、也無法被消滅，背後藏著一個理由——而找到它的人，是埃米·諾特。

核心想法

物理學家早就知道一份簡短的「守恆律」清單：在一個封閉系統裡，能量、動量、角動量的總量永不改變。諾特證明，這並不是三樁各自孤立的巧合。它們每一個，都是某個對稱的後果——所謂對稱，就是當你挪動某樣東西時，自然法則依然分毫不變的那種「不變」。

正因為物理定律不在乎現在是幾點，能量才守恆。正因為它們不在乎你身在何處，動量才守恆。正因為它們不在乎你面朝哪個方向，角動量才守恆。對稱進去，守恆律出來——而她還證明了：反過來也成立。

它是如何誕生的

1915 年，兩位在世的頂尖數學家——大衛·希爾伯特與費利克斯·克萊因——把諾特請到了哥廷根。他們正與愛因斯坦那套全新的引力理論搏鬥，在那理論裡，能量似乎會令人不安地「失常」——他們需要她對「不變量」無與倫比的掌握來理清頭緒。諾特做的遠不止理清頭緒：她找到了潛伏在底下的那條普遍法則。

而她做成這一切時，連一個學者最尋常的權利都被剝奪著。身為女性，她無法擔任帶薪教授，一直以希爾伯特本人的名義講課。當同僚反對給她一個職位時，據說希爾伯特頂了回去——大學評議會「又不是澡堂」。她直到 1919 年才正式獲得任教的資格——正是這篇論文問世的次年。

它為何重要

在諾特之前，守恆律是靠實驗發現的、被珍視的事實。在她之後，它們成了預言：說出自然的一個對稱，她就能準確告訴你，什麼必然守恆。這把物理學整個翻了過來——對稱，成了構建新理論的首要工具，而粒子物理的標準模型，後來正是這樣被建起來的。

一個可以想像的畫面

想像一隻完美的圓盤子。在桌上轉動它，它看上去一模一樣——這份「一樣」，就是旋轉對稱。現在在它邊緣壓出一個凹痕：再轉一下，你就看得出它動過了。圓盤子在旋轉下「守住」了自己的模樣；帶凹痕的那隻，則守不住。諾特定理說，自然也是這樣運作的：凡是法則在某種改變之後看上去毫無二致，就總有某樣東西保持精確不變——而你一旦在對稱上壓出凹痕，那樣東西便開始漂移。

它的位置

諾特站在現代物理的樞紐上。組織起馬克士威電磁學、愛因斯坦相對論、以及希格斯場的那些對稱——它們都在本館的別處——正是通過她的定理，化作了守恆律。一個世紀之後，「是什麼對稱在守護這個量？」已是物理學家面對任何新想法時，最先發問的問題之一。

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