Mathematics might be more of an environmental science than we realize. Even though it is a search for eternal truths, many mathematical concepts trace their origins to everyday experience. Astrology and architecture inspired Egyptians and Babylonians to develop geometry. The study of mechanics during the scientific revolution of the 17th century brought us calculus.
Remarkably, ideas from quantum theory turn out to carry tremendous mathematical power as well, even though we have little daily experience dealing with elementary particles. The bizarre world of quantum theory — where things can seem to be in two places at the same time and are subject to the laws of probability — not only represents a more fundamental description of nature than what preceded it, it also provides a rich context for modern mathematics. Could the logical structure of quantum theory, once fully understood and absorbed, inspire a new realm of mathematics that might be called “quantum mathematics”?
A monthly column in which top researchers explore the process of discovery. This month’s columnist, Robbert Dijkgraaf, is the director of the Institute for Advanced Study in Princeton, New Jersey.
There is of course a long-standing and intimate relationship between mathematics and physics. Galileo famously wrote about a book of nature waiting to be decoded: “Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics.” From more modern times we can quote Richard Feynman, who was not known as a connoisseur of abstract mathematics: “To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature. … If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in.” (On the other hand, he also stated: “If all mathematics disappeared today, physics would be set back exactly one week,” to which a mathematician had the clever riposte: “This was the week that God created the world.”)
The mathematical physicist and Nobel laureate Eugene Wigner has written eloquently about the amazing ability of mathematics to describe reality, characterizing it as “the unreasonable effectiveness of mathematics in the natural sciences.” The same mathematical concepts turn up in a wide range of contexts. But these days we seem to be witnessing the reverse: the unreasonable effectiveness of quantum theory in modern mathematics. Ideas that originate in particle physics have an uncanny tendency to appear in the most diverse mathematical fields. This is especially true for string theory. Its stimulating influence in mathematics will have a lasting and rewarding impact, whatever its final role in fundamental physics turns out to be. The number of disciplines that it touches is dizzying: analysis, geometry, algebra, topology, representation theory, combinatorics, probability — the list goes on and on. One starts to feel sorry for the poor students who have to learn all this!
What could be the underlying reason for this unreasonable effectiveness of quantum theory? In my view, it is closely connected to the fact that in the quantum world everything that can happen does happen.
In a very schematic way, classical mechanics tries to compute how a particle travels from A to B. For example, the preferred path could be along a geodesic — a path of minimal length in a curved space. In quantum mechanics one considers instead the collection of all possible paths from A to B, however long and convoluted. This is Feynman’s famous “sum over histories” interpretation. The laws of physics will then assign to each path a certain weight that determines the probability that a particle will move along that particular trajectory. The classical solution that obeys Newton’s laws is simply the most likely one among many. So, in a natural way, quantum physics studies the set of all paths, as a weighted ensemble, allowing us to sum over all possibilities.
This holistic approach of considering everything at once is very much in the spirit of modern mathematics, where the study of “categories” of objects focuses much more on the mutual relations than on any specific individual example. It is this bird’s-eye view of quantum theory that brings out surprising new connections.
A striking example of the magic of quantum theory is mirror symmetry — a truly astonishing equivalence of spaces that has revolutionized geometry. The story starts in enumerative geometry, a well-established, but not very exciting branch of algebraic geometry that counts objects. For example, researchers might want to count the number of curves on Calabi-Yau spaces — six-dimensional solutions of Einstein’s equations of gravity that are of particular interest in string theory, where they are used to curl up extra space dimensions.
Just as you can wrap a rubber band around a cylinder multiple times, the curves on a Calabi-Yau space are classified by an integer, called the degree, that measures how often they wrap around. Finding the numbers of curves of a given degree is a famously hard problem, even for the simplest Calabi-Yau space, the so-called quintic. A classical result from the 19th century states that the number of lines — degree-one curves — is equal to 2,875. The number of degree-two curves was only computed around 1980 and turns out to be much larger: 609,250. But the number of curves of degree three required the help of string theorists.
Around 1990, a group of string theorists asked geometers to calculate this number. The geometers devised a complicated computer program and came back with an answer. But the string theorists suspected it was erroneous, which suggested a mistake in the code. Upon checking, the geometers confirmed there was, but how did the physicists know?
String theorists had already been working to translate this geometric problem into a physical one. In doing so, they had developed a way to calculate the number of curves of any degree all at once. It’s hard to overestimate the shock of this result in mathematical circles. It was a bit like devising a way to climb each and every mountain, no matter how high!
Within quantum theory it makes perfect sense to combine the numbers of curves of all degrees into a single elegant function. Assembled in this way, it has a straightforward physical interpretation. It can be seen as a probability amplitude for a string propagating in the Calabi–Yau space, where the sum-over-histories principle has been applied. A string can be thought to probe all possible curves of every possible degree at the same time and is thus a super-efficient “quantum calculator.”
But a second ingredient was necessary to find the actual solution: an equivalent formulation of the physics using a so-called “mirror” Calabi–Yau space. The term “mirror” is deceptively simple. In contrast to the way an ordinary mirror reflects an image, here the original space and its mirror are of very different shapes; they do not even have the same topology. But in the realm of quantum theory, they share many properties. In particular, the string propagation in both spaces turns out to be identical. The difficult computation on the original manifold translates into a much simpler expression on the mirror manifold, where it can be computed by a single integral. Et voilà!
Duality of Equals
Mirror symmetry illustrates a powerful property of quantum theory called duality: Two classical models can become equivalent when considered as quantum systems, as if a magic wand is waved and all the differences suddenly disappear. Dualities point to deep but often mysterious symmetries of the underlying quantum theory. In general, they are poorly understood and an indication that our understanding of quantum theory is incomplete at best.
The first and most famous example of such an equivalence is the well-known particle-wave duality that states that every quantum particle, such as an electron, can be considered both as a particle and as a wave. Both points of views have their advantages, offering different perspectives on the same physical phenomenon. The “correct” point of view — particle or wave — is determined solely by the nature of the question, not by the nature of the electron. The two sides of mirror symmetry offer dual and equally valid perspectives on “quantum geometry.”
Mathematics has the wonderful ability to connect different worlds. The most overlooked symbol in any equation is the humble equal sign. Ideas flow through it, as if the equal sign conducts the electric current that illuminates the “Aha!” lightbulb in our mind. And the double lines indicate that ideas can flow in both directions. Albert Einstein was an absolute master of finding equations that exemplify this property. Take E = mc2, without a doubt the most famous equation in history. In all its understated elegance, it connects the physical concepts of mass and energy that were seen as totally distinct before the advent of relativity. Through Einstein’s equation we learn that mass can be transformed into energy, and vice versa. The equation of Einstein’s general theory of relativity, although less catchy and well-known, links the worlds of geometry and matter in an equally surprising and beautiful manner. A succinct way to summarize that theory is that mass tells space how to curve, and space tells mass how to move.
Mirror symmetry is another perfect example of the power of the equal sign. It is capable of connecting two different mathematical worlds. One is the realm of symplectic geometry, the branch of mathematics that underlies much of mechanics. On the other side is the realm of algebraic geometry, the world of complex numbers. Quantum physics allows ideas to flow freely from one field to the other and provides an unexpected “grand unification” of these two mathematical disciplines.
It is comforting to see how mathematics has been able to absorb so much of the intuitive, often imprecise reasoning of quantum physics and string theory, and to transform many of these ideas into rigorous statements and proofs. Mathematicians are close to applying this exactitude to homological mirror symmetry, a program that vastly extends string theory’s original idea of mirror symmetry. In a sense, they’re writing a full dictionary of the objects that appear in the two separate mathematical worlds, including all the relations they satisfy. Remarkably, these proofs often do not follow the path that physical arguments had suggested. It is apparently not the role of mathematicians to clean up after physicists! On the contrary, in many cases completely new lines of thought had to be developed in order to find the proofs. This is further evidence of the deep and as yet undiscovered logic that underlies quantum theory and, ultimately, reality.
Niels Bohr was very fond of the notion of complementarity. The concept emerged from the fact that, as Werner Heisenberg proved with his uncertainty principle, in quantum mechanics one can measure either the momentum p of a particle or its position q, but not both at the same time. Wolfgang Pauli wittily summarized this duality in a letter to Heisenberg dated October 19, 1926, just a few weeks after the discovery: “One can see the world with the p-eye, and one can see it with the q-eye, but if one opens both eyes, then one becomes crazy.”
In his later years, Bohr tried to push this idea into a much broader philosophy. One of his favorite complementary pairs was truth and clarity. Perhaps the pair of mathematical rigor and physical intuition should be added as another example of two mutually exclusive qualities. You can look at the world with a mathematical eye or with a complementary physical eye, but don’t dare to open both.