To the editors:
In her essay, Natalie Paquette argues that the flow of ideas from mathematics to physics has been reversed in recent years due to string theory.1 While there is indeed some truth to such a sentiment, it does not fully capture the complex relationship between mathematics and physics. Not only was the contemporary division between mathematics and physics non-existent at the dawn of classical physics, but ideas from physics guided mathematics for centuries to come. How did mathematics and physics drift apart prior to their modern day reunion? And how did mathematicians gain an edge over physicists, if this was ever the case at all? In this letter, I would like to address these questions from a shamelessly revisionist point of view.2 I will argue that the deepest and most far-reaching ideas of physics are not the most elegant or beautiful, but the ideas that are confusing, not rigorous, improperly formulated, or, in fact, utterly incomprehensible to mathematicians.
The great Vladimir Arnold asserted that mathematics can be categorized into three disciplines: cryptography, hydrodynamics, and celestial mechanics.3 Cryptography, with its history dating back to the Old Kingdom of Egypt, is the realm of number theory, algebra, and combinatorics. Celestial mechanics and hydrodynamics, on the other hand, were the physics of earth and the heavens—or water, to be exact. They led to calculus, dynamical systems, and partial differential equations.
Isaac Newton did not have the mathematical tools at his disposal to describe mechanics. He invented new rules to deal with infinitesimal increments, guided largely by intuition and consistency rather than rigor, in a manner perhaps not all that different from modern day physicists. The lack of a firm logical foundation did not prevent Newton from deriving his universal law of gravitation, a spectacular result based on Kepler’s meticulous and religiously motivated observations of celestial bodies.4 Even more spectacular was the accuracy of Newton’s theory—for the next two hundred years, celestial mechanics was more closely associated with mathematics than physics.
Nonetheless, Newton had miscalculated, and so too had Leonhard Euler, Alexis Clairaut, and Jean le Rond d’Alembert.5 In studying the motion of the moon, an incorrect prediction about the difference between the sidereal month—the length of time taken by the moon to complete a single orbit of the earth with respect to the stars, 27.32166 days—and anomalistic month—the length of time taken by the moon to pass from perigee to perigee, 27.55455 days—put Newton’s theory in serious doubt. It was saved by Clairaut a few years later, when he discovered an error in an earlier calculation. The lunar theory was arguably one of the pinnacles of eighteenth and nineteenth century mathematical physics, as exemplified by Charles Delaunay’s Théorie du mouvement de la lune (1860), which comprised two volumes and nearly two thousand pages.6 A technical tour de force on this scale might generate a warm and fuzzy sense of camaraderie from the modern day mathematical physicist. Yet there is also the uneasy feeling that Delaunay’s work has been largely forgotten because it served to support, rather than dismantle Newton’s theory. At around the same time, Newton’s theory would be put in doubt again, when it failed to predict the correct rate of precession of the perihelion of Mercury.7 Its ultimate resolution required an overhaul of the very concept of gravitation.8
Compared to the orderly motion of celestial bodies, physics seemed a lot messier on earth, and took much longer to figure out without a grant and support from the church. The flow of water has baffled physicists and mathematicians since the beginning. It wasn’t until the first half of the nineteenth century that Claude-Louis Navier and George Stokes were able to formulate the equation that described the dynamics of water as an incompressible viscous fluid, except that it wasn’t clear then, and remains unclear today, that their equation can rigorously predict the motion of water.9 But it does seem to work, almost always. At times it can be really hard to make a prediction from the Navier-Stokes equation, so difficult, in fact, that it makes modern supercomputers sweat.10
Up until the early nineteenth century, it would be hard to draw the distinction between physicists and mathematicians. The forefront of physics always required cutting edge mathematics. Then along came some stubborn German fellows named Karl Weierstrass, Bernhard Riemann, and Georg Cantor, who were not satisfied with the intuitive and computational nature of mathematics at the time. Weierstrass reinvented calculus, putting it on logically sound foundation. Riemann reinvented geometry, looking at shapes from within rather than from the outside. Cantor taught us that we don’t actually know what we are talking about when we talk about mathematics, by inventing set theory. David Hilbert took it even further, establishing his important finiteness theorem on polynomial rings with a logically airtight non-constructive proof that offered no algorithm to compute the answer.11 This outrageous approach was criticized at the time by old-fashioned folks such as Paul Gordan, who described it as “theology, not mathematics”.12 Whatever it was, it became mathematics, and it remains that way today.
By the end of the nineteenth century, mathematics had undergone a revolution, acquiring entirely new levels of abstraction. Physics seemed rudimentary in comparison. Riemann was almost Einstein, but not quite. Riemann understood the intrinsic curvature of space long before Einstein learned about it, but Riemann did not think of time—a concept all too familiar to an observer of nature that had eluded the most brilliant mathematical minds. Riemannian geometry is rich, but general relativity—the Riemann geometry for space and time—is far more profound. The instincts of physicists have led them to spectacularly deep conjectures such as cosmic censorship, which asserts that while singularities may necessarily form under gravitational collapse, singularities must be shielded by black hole horizons.13 Physics would otherwise cease to make sense. But physics should always make sense. And with that belief, physicists could be one step ahead of mathematicians.
The focus of physics also shifted in the mid-nineteenth century. Thermodynamics, which began as a subject of chemists and engineers, was reformulated as a part of physics.14 Equipped with dry and uninteresting mathematical formulas, it talks about boring things like heat and work. Unlike Newton’s laws that were universal and absolute truth until they were not, the laws of thermodynamics always seemed like half-baked truths that arose from empirical observations rather than logical deduction. The attempt to make sense of thermodynamics, however, led to the development of theories of atoms and molecules. Theoretical physics was no longer concerned with accurate calculations of heavenly orders, but about approximations and statistics on one hand, and reductionism on the other.
The mechanism that made statistics work is chaos—the exponential growth of uncertainty, an unavoidable part of life. Even the mathematicians struggling with the three-body problem of moon-earth-sun would come to accept the failure of their predictive power and the statistical fate of our solar system.15 Thermodynamics would eventually reshape the objective of physics, as well as its foundation. Intuitions born from thermodynamics would lead to the black hole information paradox and other far-reaching conjectures, such as eigenstate thermalization hypothesis, that are still inspiring new mathematics and physics in the twenty-first century.16
The gap between mathematics and physics grew even wider in the early twentieth century. While David Hilbert and his contemporaries developed ever more sophisticated theories of infinite dimensional function spaces, physics was dominated by a brave new generation that possessed fantastic intuition, but minimal mathematical training. Physicists couldn’t be bothered with subtle differences between the bounded and the unbounded, the separable and the non-separable. New elementary particles were waiting to be discovered just around the corner.
It is not surprising that the intuition of even the best physicists would sometimes fail to guide their sloppy mathematics. Lev Landau thought that an integrable system—roughly speaking, a mechanical system that undergoes regular, quasi-periodic motions—would remain integrable when perturbed slightly, without realizing that in order to prove this he would need to sum up an infinite series that does not converge.17 Enrico Fermi thought about the same problem and came to the opposite conclusion, based on intuitions from thermodynamics, that a generic small perturbation ought to destroy integrability entirely. He was surprised to learn that this is not the case from the results of a numerical experiment.18 This debate about the so-called ergodic hypothesis was not settled until the mid-twentieth century and the landmark work of Andrey Kolmogorov, Vladimir Arnold, and Jürgen Moser.19 The truth lay somewhere between Landau’s sloppy calculation and Fermi’s incorrect intuition—make no mistake, both were titans of science—and, in this case, could only be revealed by rigorous mathematics.
The intuitive physicists were triumphant on many more occasions. Like Richard Feynman, who, from his wild imagination of virtual particles, wrote down outrageously ill-defined integral overall paths of virtual particles and fields through space and time.20 At the level of perturbation theory, that is, pretending that the quantum was infinitesimal, Feynman’s path integral is nothing more than a mathematical trick that helps with organizing calculations. In fairness, it was a damn fine trick. It helped predict the magnetic dipole moment of the electron to eleven digits, while also helping to solve difficult mathematical problems from the topological invariants of knots to the deformation quantization of Poisson manifolds.21
But perturbation theory was not enough. Freeman Dyson famously argued in 1952 that one cannot simply pretend that the quantum is infinitesimal, otherwise one could mathematically construct a world in which objects with like electric charges attract rather than repel one another, which ought to be catastrophic because the vacuum would be rendered unstable by pair-producing oppositely charged particles.22 Somehow or another, Feynman’s path integral knows better. It also made sense beyond perturbation theory, if only one approximates the continuous space and time with a discrete lattice.23 This lattice version of the path integral, pioneered by Ken Wilson, though mathematically well defined, is incalculable by the human hand, or, for that matter, the human brain.24 It was viewed as rather distasteful by posher theoretical physicists, who preferred to think about things like the 1/N expansion—infinitely-many-color limits of quarks and gluons.25 The lattice had to wait for the development of modern supercomputers before it could demonstrate its prowess by successfully calculating the correct mass of the proton from quantum chromodynamics.26
Ken Wilson, the master of dirty physics who thought long and hard about particles, magnets, and phase transitions, had another ground-shattering insight. He decided that the best way to understand all of this was to organize physics according to size and to focus on the phenomena that are self-replicating with ever-increasing sizes.27 With this idea, he turned the Gell-Mann-Low equation, previously a good trick for improving the accuracy of perturbation theory, into what is now known as the Wilsonian renormalization group (RG), and reshaped the way we think about quantum field theory.28 In the eyes of physicists, the Wilsonian RG was both a magic wand for conceptual reasoning and a slug monster for practical calculations. To mathematicians, the Wilsonian approach did not meet the classic standard of mathematical beauty. It was largely ignored until the work of Kevin Costello in 2011.29 He established a rigorous foundation of the perturbative aspect of Wilson’s RG theory using a Soviet machine named Batalin-Vilkovisky that still scares away most physicists.30
In 1994, in an effort to understand the confinement of quarks and gluons, Nathan Seiberg and Edward Witten applied Wilson’s RG to a theory of interacting gluons and their superpartners at short distances.31 They needed to employ another piece of dirty physics known as spontaneously symmetry breaking. In a quantum theory there can be more than one single vacuum state and the vacuum states need not respect the symmetries of the underlying dynamics. This was an idea born out of the study of superconductors and the theories of electroweak as well as strong interactions.32 It was deemed so ugly that it was rejected at birth by Chen-Ning Yang, the creator of Yang-Mills theory.33 The idea has since gained wide appreciation by physicists. Seiberg and Witten came to the surprising conclusion that in certain vacuum states of their theory, the particles that dominate the physics over long distances are magnetic monopoles rather than gluons. Coupling all of this with the belief that quantum field theories are supposed to make sense in curved spacetimes, the same Wilsonian RG paradigm would end up relating Donaldson’s topological invariant of smooth four-manifolds—having to do with the physics at short distances—to the solution space of the so-called monopole equations, now known to mathematicians as the Seiberg-Witten invariant—having to do with the physics at long distances.34 The spectacular mathematical breakthrough that followed was partly described in Paquette’s essay. A colleague once told me that, were it not for the insight of Seiberg and Witten, mathematicians might have stumbled upon the monopole equations and discovered S-W’s topological invariant, but no mathematician would have ever dreamed of its connection to Donaldson invariant.
How did quantum field theory (QFT) become a central pillar of modern theoretical physics? Some of the physicist’s confidence in QFT can be attributed to its success in explaining natural phenomena: the scattering of photons and electrons, or phase transitions of water and magnets alike. Nonetheless, the development of QFT has been tortuous. In its early days, QFT did not look like the correct theory of anything, let alone everything. It seemed to suffer from all kinds of blatant mathematical inconsistencies. These were eventually removed, one by one. Rivaling theories were proposed, such as source theory, S-matrix theory, and string theory. Some of these, like source theory, slid into obscurity.35 Others, such as S-matrix theory, became part of QFT.36 The singular case of string theory, however, grew into a greater framework that would eventually encompass QFT.
Two basic principles that arose from the intuition of physicists turned out to be far deeper than they seemed. Unitarity is the conservation of probability in the framework of quantum mechanics. Locality, which is just a more refined version of causality, states that signals cannot travel faster than the speed of light. When combined, unitarity and locality become unexpectedly powerful. From these principles, Sasha and Alyosha Zamolodchikov were able to bootstrap their way through the complexity of spin models of magnets and get nearly everything from nothing.37 Another application led to the construction of the conformal field theory that underlies monstrous moonshine, which is also discussed in Paquette’s essay.38 In this case, the essential ingredient is the so-called orbifold construction, which goes something like this: a symmetry in a two-dimensional world amounts to lassoing by a topological defect line. Locality requires that the defect line can end at a point. Unitarity implies that the end points are in fact states of a new quantum field theory, the orbifold theory.39
The last half century has seen the emergence of dramatically different approaches to QFT between physicists and mathematicians. Physicists have largely adopted the Wilsonian view. Mathematicians have attempted various axiomatic approaches, based on either the Hamiltonian formalism of quantum mechanics, the analytic properties of correlation functions, or the von Neumann algebra of local observables. Over the last decade, while the Wilsonian school of thinking has reached new heights, some physicists have recognized that the axiomatic approach may, in fact, lead to new practical and computational results.40 The conformal bootstrap program that began during the 1970s, based on the idea that locality, unitarity, and symmetry principles could be used to solve strongly coupled quantum field theories completely, was rebooted in 2008.41 Some spectacular successes were to follow.42 The analyticity of correlation functions and S-matrices are being subjected to a new level of scrutiny.43 The Hamiltonian approach has been revisited and injected with new ideas, such as matrix product states and tensor networks born out of the study of quantum entanglement in condensed matter physics.44 The algebraic approach is just starting to be taken seriously by physicists and is now being tied to quantum information theory and holography.45 It appears that the division between mathematics and physics has once again become blurred.
I have barely touched upon the role of string theory, whose connection to mathematics is too vast for me to elaborate in the limited space and time of this letter. Taking a step back, one could see some parallels between the early days of QFT and string theory. The initial effort to promote string theory from a classical mechanical model of mesons to a quantum theory suffered from the problem of tachyons. This problem was solved by introducing supersymmetry.46 Certain superstring theories were thought to be incompatible with quantum mechanics, but Michael Green and John Schwarz came along and fixed that as well.47 The theory of interacting strings that emerged during the 1980s came in different types, each one only making sense at the level of perturbation theory. Then came the golden years in which Witten unified different types of superstrings, Joseph Polchinski discovered D-branes, Andrew Strominger and Cumrun Vafa reconciled string theory with the thermodynamics of black holes, and Juan Maldacena discovered that string theory could, in fact, be QFT in disguise.48 Maldacena made a convincing case that a fully non-perturbative, consistent, quantum string theory exists. This dazzling development left disciplined mathematicians scratching their heads while struggling to understand the definitions of bits and pieces of this string theory.
As string theorists march on in their quest for the theory of everything, whilst also leaving a trail of mathematical gems along the way, some traditional physicists were outraged: “Is physics no longer rooted in observations of nature? Or is this theology?” I couldn’t help but notice a striking parallel with the way mathematics became detached from physics during the nineteenth century and, in particular, the outrage that accompanied Cantor’s transfinite set theory and Hilbert’s non-constructive proofs. Was the kind of mathematics that could never be exhibited with real objects actual mathematics, or was it theology? With the benefit of hindsight, we now know that the mathematics flourished like never before during the twentieth century. One can only hope the same thing happens with string theory in the decades to come.
As far the relationship between physics and mathematics is concerned, I was reminded that an experimental particle physicist once said that his job was to make the lives of theorists miserable. I believe that part of the job of a theoretical physicist is to make the lives of mathematicians miserable. There are, incidentally, few things I can think of that could make a mathematician more miserable than reading Leonard Susskind’s papers.49
