Introduction

“The single most profound result in all of physics”

Whether you’re a scientist, a student, or someone else with a general interest in the history of ideas, you certainly have some notion about what physics is and how it developed over the centuries. Whatever the origins of these notions, chances are that the name Emmy Noether doesn’t spring immediately to mind. Albert Einstein, Erwin Schrödinger, Paul Dirac, Niels Bohr, Werner Heisenberg… these are some of the names familiar to anyone who’s read books about the invention of modern physics—about the momentous discoveries of the twentieth century that changed our conceptions of time and space, and of the nature of reality itself. And these are the names that figure prominently in the textbooks used by physics students.

Who, then, was Emmy Noether?

In the pages ahead I hope to convince you that she deserves a place alongside the names mentioned above in the history of physics—and of science in general—and that this is due to the impact of discoveries that she published in 1918.

Her four closely connected results, collectively now called Noether’s theorem, supply the foundation of the present-day search for the holy grail of physics: a unified theory that will join quantum mechanics with gravity. Noether’s theorem has also supplied the methodology for constructing the most accurate theory in the history of physics: the standard model. This framework encompasses all the elementary particles and their interactions—it is our modern theory of matter. In addition, Noether’s discovery solved a persistent mystery in Einstein’s recently completed general theory of relativity: the conundrum of energy conservation, a problem whose solution had eluded Einstein and several of the greatest mathematicians in the world. Attacking this puzzle began the chain of reasoning that eventually led Noether to her great theorem. Along the way, she wound up tutoring Einstein in some of the math he needed to complete his work, thereby becoming one of several uncredited authors of what remains our current theory of gravity. I’ll have much more to say about unification, the standard model, general relativity, and their connections with Noether’s theorem throughout the book. For now, it’s enough to note that these subjects essentially define the principal concerns of what we think of as fundamental modern physics, and they are all intimately connected with, and often depend on, Noether’s theorem.

The theorem goes beyond providing the foundation of current physical theory and supplying the touchstone for the physics of the future. It provides the modern definition of the concept of energy and clarifies the importance of symmetry in nature. It brings order to the physics of the past, completing its theoretical structures and rendering it whole. The theorem’s revelation of the active role of symmetry throughout nature is so provocative that it is now being exploited in biology, computation, economics, and elsewhere.

In this book, I trace the surprising trajectory of the theorem: its origins and the events and lives that created the conditions for its birth. I explore how the unique genius of Emmy Noether allowed her to see something completely unexpected and arrive at insights that still astonish those who encounter them a century later. I’ll recount how the theorem became dormant for decades and was almost lost to the world, and how it was rediscovered by physicists creating a new theory of matter. Finally, we’ll see how Noether’s theorem has now gained a new life, guiding research in fields far removed from physics.

Emmy Noether’s story is that of a woman pursuing her passion for over three decades in a world offended by her simple intention to be a mathematician. But that is what she would be, however she could. Despite being denied opportunities, passed over, excluded, and expected to work without pay or position, she not only persisted but surpassed the men around her. Her story is also a story of solidarity and the loyalty of those few men who championed her cause.

This view of Noether’s importance is bolstered by the judgments of many of our leading physicists, including Nobel Prize winners such as Frank Wilczek. Wilczek and other top researchers, intimate with how their science works and its debt to Noether’s theorem, have occasionally stepped outside their offices and laboratories to write about science for the layperson, explaining the big ideas in physics, its history, and its possible future development. Along with me, they believe that Noether’s theorem is one of these big ideas, and they consider its creator a neglected figure in the history of science. According to Wilczek, Noether’s theorem is “the single most profound result in all of physics.”1

Renowned physicists Leon Lederman and Christopher Hill say that Noether’s theorem is “one of the most important mathematical theorems ever proved in guiding the development of modern physics, possibly on a par with the Pythagorean theorem” and that it “rules modern scientific methodology.”2 (Lederman is a Nobel Prize winner and coined the term God particle.)

Brian Greene, a theoretical physicist well known for explaining modern physics through books and television and a leader in current attempts at building a unified theory, believes that “Emmy Noether’s theorem is so vital to physics that she deserves to be as well known as Einstein. Yet, many have never even heard of her.”3

Emmy Noether was born in Germany in the late nineteenth century and died in the United States in the twentieth. She devoted her life to research and teaching in pure mathematics. She did so out of unquenchable passion for math, despite a cruel series of obstacles and injustices placed in her path for one reason only: that she was a woman.

She sought to study math in the university, but women were not allowed to enroll—so she audited, when permitted. After German society relented and allowed women to matriculate, she took her degree. She went to work at the premier university for mathematics in the world, by invitation of the greatest mathematician in the world—but without pay or position, because women were not allowed to be teachers. After about half a decade, that rule finally changed, too, and she was hired—but reluctantly, with minuscule salary, expressly excluded from any civil service benefits and warned that she had no authority.

She was one of the first victims of the Nazi purges of Germany’s faculty rolls, as she doubly offended by being both female and Jewish. Escaping to the United States, as her colleague Albert Einstein had already done, she was placed not alongside him and other refugee members of the intelligentsia at the new Institute for Advanced Study but in an obscure position, deliberately stowed at a distance from her male peers.

If the sound of Noether’s voice could somehow reach us one hundred years after she lived, we would hear no wail of complaint. Instead, we would hear the loud laughter so often described by her compatriots. She seems never to have voiced any objection to her lot or struggled to improve her position, although she was energetic in fighting for her friends. She simply proceeded to do whatever the circumstances permitted so that she could carry out her interests—her only interests—which were to discover and teach mathematics.

Denied the right to be a student, she audited; denied the right to a job, she worked without pay; forbidden by Hitler’s regime to teach at all, she taught in secret, in her apartment. Throughout it all, she enjoyed life, she raised the spirits of everyone around her, and, from joy, irony, and the delectation of absurdity, she laughed.

The full answer to the question of why Noether is so little known, given that she contributed as much to shape the history of physics over the past century as did any other person, is complex, sometimes subtle, and multifaceted. I will answer this question gradually throughout the book; in this introduction, I’ll give an outline of some of the reasons.

Noether’s relative obscurity is due, first of all, to the systematic suppression of her reputation and place in history by colleagues, officials, and scholars who minimized her contributions and importance. The evidence that this minimization was due to her sex is stark and abundant, sometimes taking the form of bald statements by the perpetrators themselves, who knew no reason to disguise their prejudice. In matters relating to her circumstances during her life, we can add to this the fact that she was a Jew in Germany in the 1930s.

For her part, Noether did nothing aimed at bolstering her own prominence or position. She was supremely generous, helping colleagues and students in the early stages of their careers by making gifts to them of mathematical results: theorems that she had proven and problems that she had solved but hadn’t bothered to publish. She encouraged these young colleagues to refine or extend her work and present it to the world as wholly their own. These were valuable gems that a normal academic type would have jealously guarded and polished until they were ripe for a career-boosting publication. But Noether, who was overflowing with ideas and results, was happy to pass many of them along for the benefit of her friends. Although aware of the importance of her own work in mathematics, she did not promote herself, and she rarely referred to her monumental result in physics, which was for her a side issue that she forgot about almost as soon as she had created it. This habit, combined with the casual attitude, even by her supporters, to issues of credit and priority, conspired to cause her role in her contributions to be overlooked, even after they had become part of the working machinery of physics.

In recent decades, we’ve seen a gradual rehabilitation of Noether’s reputation and a growing recognition of her theorem as a crucial component in the development of fundamental physics since 1918. As mentioned, this revision is largely carried out by scientist-writers who are aware that theory building rests on Noether’s theorem as a foundation, as an omnipresent guide and constraint. I intend this book to be a part of this revision, and I am confident that, before too long, the idea of a book chronicling the story of modern physics without Emmy Noether as a main character will be as unthinkable as one that forgot to mention Einstein.

How did I become aware of Emmy Noether and what she gave to physics?

During my college days, while browsing through an advanced textbook on classical mechanics, I came across something that stopped me in my tracks. A routine development of the subject had taken a minor detour to present some results that were surprising and that struck me as deeply profound—as a demonstration of how physics had the ability to capture the harmony and unity of nature.

What I had come across in the textbook were proofs that the conservation laws of classical physics, the familiar laws of the conservation of energy, momentum, and angular momentum, were each equivalent to a symmetry in time or space. Ideas like the conservation of energy were not just additions to mechanics that made it easier to solve problems involving, say, the trajectory of cannonballs; they were implied by the very structure of space and time. Such a connection among concepts that had previously seemed unrelated, demonstrated by unambiguous mathematics, not only was unexpected but also seemed to have implications that reached beyond physics and toward philosophy. At the very least, these beautiful connections suggested a myriad of questions and pushed me to look deeper into my subject.

These relationships stuck with me after college, all through graduate school. But I had gone all the way to a PhD without ever hearing about such things again, at least not directly. At the time, this omission didn’t seem strange. I assumed that the results were confined to classical mechanics, although they remained in the back of my mind throughout my studies.

Many years later, I convinced the editors of an online science magazine to let me write a popular article about the connections between conservation laws and symmetry. I still didn’t know much about the subject, but I knew it was intriguing and important, and not widely understood. I was sure that I could also make it interesting to nonspecialists, to show how physics could be provocative and fascinating apart from the wonders of the quantum world and relativity—to show how even classical mechanics could be beautiful.

In doing research for that article, I finally learned where these ideas had come from. They were simple, special cases of a deep theorem published in 1918 by someone I’d never heard of, someone whose name had not been mentioned once in my many years of physics training. Her name was Emmy Noether, and her result was called Noether’s theorem by the physicists who knew about it. In fact, those physicists seemed to belong to a secret club. They spoke of this theorem as one of the, if not the, single most important result in theoretical physics. They bemoaned the fact that its discoverer was not more widely known—was barely known at all, rarely mentioned either in classrooms or in popular histories of science. And these scientists were neither crackpots nor cultists. They were some of the biggest names in physics.

I kept reading about the theorem and the person behind it. I learned the amazing, inspiring, and tragic story of her life. I learned far more about some of the people whose lives had intersected hers, whose names I had encountered in my studies: David Hilbert, Felix Klein, Hermann Weyl, Einstein, and others. I went further, delving into archival materials that revealed the untold parts of the story about her brief time in the United States. I learned that I had, quite literally, walked on her grave without knowing it.

You don’t have to know any physics or advanced mathematics to read and understand this book. If you have some vague memory of what the Pythagorean theorem is, that should be plenty. I intend to describe the meaning and content of Noether’s theorem in a way that can be understood and appreciated by everyone. This is possible, even though the theorem requires some advanced mathematics to prove, because it possesses an intuitive core. Its essential meaning for physics and beyond is thoroughly explainable through prose. I will guide you through an honest understanding of the meaning of this result, so that you’ll be able to appreciate its more recent uses in other fields, outside of physics, such as in biology and economics. If you follow where I lead you, you’ll see how this one powerful idea unifies many realms of thought that seem superficially unrelated.

This is not a biography of Emmy Noether; rather, it’s the biography of an idea. She devoted her life to mathematics, and a faithful account of her life must stay close to the work that was important to her and its influence on the history of thought. To this end, I will take the time to dissect various topics in physics and mathematics as honestly as I can without resorting to equations, to relate them to the idea that will remain at the center of the story: Noether’s theorem. Much of this dissection is relegated to an appendix. There the interested reader will find a deeper layer of detail, where the history of, and relations between, ideas in physics and mathematics, and their connections to Noether’s thought, are explored. The somewhat more technical nature of the appendix will satisfy the more committed or mathematically inclined reader, while the sequestration of its details allows the other aspects of the story, namely, its intertwining, interpersonal threads, to proceed with more directness.

For those who want to delve deeper into anything that I mention, I supply many references to buttress my claims or to suggest further reading. Some of the references are to the technical literature, to serve the purpose of perhaps nonobvious sources for specialists; other references range from popular articles and books to videos and cartoons. In no case is a reference an endorsement. Many of the sources I cite contain errors (I’ve lost count of how many biographical notes about Noether state erroneously that she died of cancer), but aside from the mistakes, the sources contain interesting takes or information. You’ll notice many books, such as Constance Reid’s excellent biography of Hilbert, referred to repeatedly. Take this to stand in for the conventional suggestions for further reading.

I am not trained as a historian, yet to tell this story, I had to try to become one. I now have a new appreciation of the complexity and melancholy of the historian’s task. One wants to tell a story of the past, where each event leads naturally to a subsequent event, where people’s motivations are comprehensible, and where things hang together in a way that at least seems something other than utterly random and chaotic. Yet for each important event or turning point, there are contradictory accounts, lies, and fanciful imaginings in the sources that we dig through to try to construct a version of the past. Motivations are obscure. But somehow, one must make out of this stew of confounding details some kind of story, or else there would be nothing to call history. I find myself nodding knowingly at the words that Mark Twain put in Herodotus’s mouth: “Very few things happen at the right time, and the rest do not happen at all. The conscientious historian will correct this defect.”

There is also the need to constantly fight off the tendency to judge or understand people of the past as if they were our neighbors in odd outfits. The past is another world, cut off from our ideas no less than a geographically isolated region would be. We must aspire to be anthropologists of time.

From all this comes the complexity.

The melancholy arises from this: after a while we get close to the men and women of the past, with whom we have spent so much quality time. We care about them as if they were friends or relatives. And we are burdened with the bittersweet gift of hindsight. When we reconstruct the crossroads faced by our heroes, there are times when we know they are about to take the wrong path, which will lead them to misery. Or perhaps there is only one road to take, and as they march along it, we know that they are unprepared for the terror that awaits. In neither case can we warn them; there is nothing we can do to help. But perhaps if we can allow our small, irrational pangs of anguish at these times to animate our stories of the past, they will resonate more meaningfully for the reader. It is a consolation I allow myself, at any rate.

Finally, in getting to know these characters from history, I had a pleasant surprise. When we are young, we tend to look for heroes, in the present or in the past. But when we find them, and then look deeper into their lives, we’re almost always disappointed, or even horrified. They fail to live up to our standards. We become cynical. As I studied the main characters of my story, and as they seemed of reliably sterling character, therefore, I kept waiting for the other shoe to drop. Yet it never did. Our principal players, especially Emmy Noether and David Hilbert, did not disappoint. They were courageous, generous, and brilliant at every turn. I might even dare to say that they can be heroes for some of us.