諾獎得主Wilczek：歐幾里德的幾何時空_風聞

撰文 | Frank Wilczek

翻譯 | 胡風、梁丁當

中文版

歷經兩千多年曆史，經歷了相對論和量子力學革命，古希臘的幾何學至今仍在發揮影響。

公元前約300年，古希臘數學家歐幾里德寫下了他著名的《幾何原本》 (Elements) 。這本書是思想和表達方式的不朽之傑作。它以幾條明確設定的、不言自明的公設或公理為前提，通過演繹與推理，得到了豐富的、強有力的、甚至驚人的結論。《幾何原本》不僅是空間幾何與測量科學的好教材，也廣泛用於培養與提高學生的邏輯思維能力。兩千多年來，儘管期間有發展非常緩慢的階段，科學已經有了非常大的進展，而歐幾里德幾何，儘管有些緩慢的拓展與變化，卻延續至今。

牛頓的經典力學與萬有引力以及麥克斯韋的電磁場理論，均建立在歐幾里德幾何的基礎上。這些理論引入了粒子、場和力的概念，但它們存在的空間卻是由歐幾里德幾何描述的空間。

在歐幾里得幾何的所有公設中，平行公設似乎不那麼令人信服。平行公設説 ：通過一條直線上不同點的兩條垂線永遠不會相交，但通過這兩點的所有其他線都會相交一次。19世紀的數學家發現，如果對平行公設進行修改，而保持其他幾個公設不變，就能漂亮地推理出關於曲面幾何的正確描述。

德國數學家黎曼則用了更激進的方法。為了描述曲面和高維的超曲面，他提出在小尺度上歐幾里德幾何總是適用的，因為此時曲率的影響可以忽略不計。而如果要描述大尺度上的空間，必須將局部的幾何描述編織在一起。比如，一名高山滑雪運動員在跌宕起伏的山上比賽時會盡力保持直滑下降，但她在整個過程中的滑雪軌跡卻是一條曲線。

1905年，愛因斯坦提出了狹義相對論。他的老師閔科夫斯基受到啓發，對歐氏幾何作了另一種推廣。1908年，閔科夫斯基在演講《空間與時間》 (Space and Time) 的結尾宣稱：“從現在起，孤立的空間與孤立的時間註定將黯然消失，只有兩者的某種統一才能保持獨立的存在。”但是閔可夫斯基時空仍然植根於歐氏幾何。雖然它對平行公設進行了簡單的推廣，但在固定的時間，其時空的“空間”部分是純粹的歐幾里得空間。直到1915年，類似黎曼對歐幾里得空間的修改，愛因斯坦將曲率引入閔可夫斯基時空，建立了廣義相對論。

基於彎曲時空的廣義相對論非常成功。它為眾多遠超古希臘人想象的科學預言提供了理論基礎，如宇宙膨脹、引力波、以及連接遙遠時空的蟲洞。然而，愛因斯坦的理論框架依然帶着鮮明的歐幾里德幾何的印記——通過對歐幾里得空間進行拓展與修改，納入了時間和大尺度的曲率。

量子效應似乎破壞了歐幾里得空間的核心基礎，即空間能夠被精細切分並被度規測量的可能性。一把真正的標尺是由原子組成的，而原子是由彌散在空間的電子波函數構成的。後來的數學發展還發現，除了平行公設，其它的歐幾里德公設也都並非不證自明。比如，他的連續體很難嚴格定義 ；如果你能一個一個地數物理空間的點，就像我們在（簡化的）數字圖像中所做的那樣，那就容易多了。

然而，在用來描述基本相互作用的標準模型中，我們依然看見了歐氏幾何的身影 ：相對論量子場仍然存在於歐幾里德的連續空間中，更準確地説，是存在於愛因斯坦的狹義相對論時空中。我想，這大概就是著名理論物理學家尤金 · 維格納 (Eugene Wigner) 所説的“數學在自然科學中不合理的有效性”吧。

英文版

The geometry of ancient Greece has stood for more than two millennia, even after relativity and quantum mechanics.

Euclid wrote his famous textbook of geometry, the “Elements,” around 300 B.C. It is a masterpiece of thought and exposition. The “Elements” deduces abundant, surprising, and powerful consequences from a few clearly stated, “self-evident” assumptions, or axioms. It served to train many generations of students not only in the science of space and measurement but in the art of clear thinking and logical deduction. A lot has happened in science since the book appeared more than two millennia ago—though there were some very slow stretches—but somehow Euclid endures.

ac Newton’s system of classical mechanics and gravity and James Clerk Maxwell’s system of electromagnetism were built upon the foundation of Euclidean geometry. They added particles, fields, and forces, but the space in which those things lived was Euclid’s.

One of Euclid’s axioms, the so-called parallel postulate, seemed to many later readers less compelling than the others. It says that perpendiculars drawn from two different points on a line never meet but that all other pairs of lines through those points intersect once. In the 19th century, mathematicians showed that by slightly modifying Euclid’s parallel postulate while keeping his other axioms, you can get a lovely—and correct—description of how geometry works on the surface of a sphere.

The German mathematician Bernhard Riemann took a more radical approach. Inspired by the prospect of describing surfaces and higher-dimensional hypersurfaces, he proposed that Euclid’s geometry becomes accurate at short distances (where the effect of curvature is negligible) but that to describe large-scale geometry one must weave together the local descriptions. Thus, for example, an Alpine skier racing down a bumpy mountain will keep doing her best to go straight down, but over the course she will trace a curve.

Albert Einstein’s 1905 special theory of relativity inspired one of his teachers, Hermann Minkowski, to propose another generalization of Euclidean geometry. At the end of his 1908 lecture “Space and Time,” he proclaimed, “Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.” Yet Minkowski’s space-time is still rooted in Euclid. It incorporates a simple generalization of the parallel postulate, and its “space” part, at any fixed time, is pure Euclid. It was left to Einstein, in his 1915 general theory of relativity, to do for Minkowski what Riemann had done for Euclid, that is, to bring in space-time curvature.

This framework has worked brilliantly. It supports applications that Euclid never dreamed of, such as the concepts of expanding universes, gravitational waves and (speculatively) wormholes that connect otherwise far-off places. Yet Einstein’s framework is still recognizably Euclidean, extended and adapted to bring in time and large-scale curvature.

Quantum phenomena, it might seem, undermine the most basic underpinnings of Euclid’s concept of space: the possibility to divide it finely and measure it with rulers and compasses. Real rulers are made of atoms, and atoms are cloudy assemblages of wavy electrons. Later developments in mathematics also rendered some Euclidean assumptions seem the opposite of “self-evident.” His continuum is quite challenging to define rigorously; it would be much easier if you could count the points of physical space, as in effect we do in our (simplified) digital images of it.

And yet our Standard Model of fundamental interactions still has Euclid under the hood. Its relativistic quantum fields still live in Euclid’s continuum—or more precisely, in Einstein’s update. To me, this is the most striking example of what Eugene Wigner called “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”.

Frank Wilczek

弗蘭克·維爾切克是麻省理工學院物理學教授、量子色動力學的奠基人之一。因發現了量子色動力學的漸近自由現象，他在2004年獲得了諾貝爾物理學獎。

本文經授權轉載自微信公眾號“蔻享學術”，編輯：王茹茹。

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