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Frank Wilczek
Frank Wilczek is a professor of physics at the Massachusetts Institute of Technology and one of the founders of quantum chromodynamics. He was awarded the Nobel Prize in Physics in 2004 for discovering the asymptotic free phenomenon of quantum chromodynamics.
Author | Frank Wilczek
Translate | Hu Feng, Liang Dingdang
Chinese version
After more than two thousand years of history, the revolutions in relativity and quantum mechanics, the geometry of ancient Greece is still influential today.
Around 300 BC, the ancient Greek mathematician Euclid wrote his famous Elements. This book is an immortal masterpiece of thought and expression. It presupposes several clearly defined, self-explanatory postulates or axioms, and through deduction and reasoning, arrives at rich, powerful, and even astonishing conclusions. "Geometry" is not only a good textbook for spatial geometry and measurement science, but also widely used to cultivate and improve students' logical thinking ability. For more than two thousand years, science has made great progress despite a very slow period of development, and Euclid geometry, despite some slow expansion and change, continues to this day.
Newton's classical mechanics and gravity, as well as Maxwell's theory of electromagnetic fields, are based on Euclid geometry. These theories introduce concepts of particles, fields, and forces, but the spaces in which they exist are spaces described by Euclid geometry.
Of all the postulates of Euclidean geometry, parallel postulates seem less convincing. Parallel postulates say that two perpendicular lines passing through different points on a straight line will never intersect, but all other lines that pass through these two points will intersect once. Mathematicians in the 19th century discovered that if parallel postulates were modified while leaving several other postulates unchanged, they could beautifully deduce the correct description of surface geometry.
The German mathematician Riemann took a more radical approach. In order to describe surfaces and hypersurfaces of high dimension, he proposed that Euclidean geometry is always applicable on small scales, since the effect of curvature at this time is negligible. If you want to describe space on a large scale, you must weave together a local geometric description. For example, an alpine skier tries to keep the straight descent as she races in the undulating mountains, but her ski trajectory throughout the process is a curve.
In 1905, Einstein proposed the special theory of relativity. His teacher Minkowski was inspired to make another generalization of Euclidean geometry. In 1908, Minkowski declared at the end of his speech Space and Time: "From now on, isolated space and isolated time are destined to disappear, and only some unity of the two can maintain an independent existence." But Minkowski space-time is still rooted in Euclidean geometry. Although it is a simple generalization of parallel postulates, the "spatial" part of its space-time is pure Euclidean space at fixed times. It was not until 1915, similar to Riemann's modification of Euclidean space, that Einstein introduced curvature into Minkowski space-time and established the general theory of relativity.
General relativity based on curved space-time has been very successful. It provided the theoretical basis for many scientific predictions far beyond the imagination of the ancient Greeks, such as the expansion of the universe, gravitational waves, and wormholes connecting distant space-time. However, Einstein's theoretical framework still bears a distinct imprint on Euclid geometry – by expanding and modifying Euclidean space, incorporating time and large-scale curvature.
Quantum effects seem to have disrupted the core foundation of Euclidean space, the possibility that space can be finely sliced and measured by a degree gauge. A true ruler is made up of atoms, and atoms are made up of electron wave functions that diffuse in space. Later mathematical developments also found that, in addition to parallel postulates, other Euclid postulates were not self-evident. For example, his continuum is difficult to define rigorously; it's much easier if you can count the points of physical space one by one, as we do in (simplified) digital images.
However, in the Standard Model used to describe fundamental interactions, we still see Euclidean geometry: relativistic quantum fields still exist in Euclid's continuous space, or more precisely in Einstein's special relativity space-time. I think this is probably the famous theoretical physicist Eugene What Eugene Wigner called "the irrational validity of mathematics in the natural sciences."
English version
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”.
Editor: Wang Ruru
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