【诺奖得主Wilczek科普专栏】欧几里德的几何时空
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Frank Wilczek
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.
Isaac 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”.
编辑:王茹茹
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