WE PHYSICISTS have a habit of depicting our discipline as “beautiful” or “elegant”, where an outsider might be forgiven for seeing no more than an endless morass of equations. In an ideal world, those equations would be unnecessary; the ultimate goal of physics – and science generally – is to describe the world as simply as possible.
One hundred years ago, one person brought us a great step closer. In this centenary year of general relativity, Albert Einstein is getting the plaudits, and no one would gainsay him that. But that same year, 1915, the excitement surrounding relativity spawned another seminal piece of work. Even among physicists, though, it is not nearly as famous as it should be. Perhaps that is down to the complexity of its mathematics, but perhaps the author’s sex and sadly short life played their parts too.
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Yet there is no doubt that Amalie “Emmy” Noether transformed how we think about the universe. Despite the hairy mathematics, her great first theorem can be described conceptually in just a short sentence: Symmetries give rise to conservation laws.
This simplicity masks a penetrating insight. It provided a unifying perspective on the physics known at the time – and laid the groundwork for nearly every major fundamental discovery since.
Emmy Noether is a story unto herself. Despite wide recognition of her obvious brilliance, she was confounded by the prejudices of German academic tradition at the turn of the 20th century. Born into a prominent mathematical family in 1882 – her father, Max, was a professor at the University of Erlangen in the north of Bavaria – she was at first forbidden from enrolling at the university because of her gender.
Even though Noether was eventually able to gain both an undergraduate degree and a PhD, still no university would hire her for their faculty. Over the next decade, she became one of the world’s experts in the mathematics of symmetry – but without appointment, pay or formal title.
Symmetry may seem like a trifling subject at first blush. The mathematician Hermann Weyl, a contemporary of Noether’s who was greatly influenced by her work, once described a very simple way of thinking about the concept: “A thing is symmetrical if there is something you can do to it so that after you have finished doing it, it looks the same as before,” he wrote. A circle, for instance, can be rotated by any angle and looks the same.
“She became one of the world’s experts in the mathematics of symmetry – but without appointment, pay or formal title”
The idea that symmetries lie at the heart of physical laws is old. Aristotle and his contemporaries argued that the stars were pasted on celestial spheres, and that the globes moved in circular orbits. They were wrong, as it happens. As Johannes Kepler discovered through meticulous observation in the early 17th century, planets wander closer and further from the sun, in the geometric form of an ellipse. They travel faster when closer in, and slower when further out. An imaginary line connecting planets to the sun traces out equal areas in equal times: what we now know as conservation of angular momentum.
Beyond relativity
It wasn’t until later that century that Newton explained why this happens, with his universal law of gravitation. The source of this behaviour was indeed a symmetry – the symmetry of the invisible hand of gravity, which acts equally in all directions from a massive body such as the sun.
General relativity, Einstein’s much refined theory of gravity, was founded on a symmetry too, one known as the equivalence principle. This states that there is no practical difference between a body experiencing acceleration because of gravity and one experiencing an equivalent acceleration from a different source, such as the thrust of a rocket or the spin of a centrifuge. From the equivalence principle, Einstein developed his theory that yields everything from curved space-time and an expanding universe to black holes and the prediction (still unobserved) of gravitational waves rippling through space.
Einstein’s work revolutionised our view of the universe, but also spurred a great deal of interest in the role of symmetries in physical laws. Recognising Noether as an expert, in 1915 the eminent mathematicians David Hilbert and Felix Klein invited her to Göttingen, then the centre of the mathematical world – an offer, alas, which still didn’t extend to any financial remuneration. Hilbert did argue forcefully for an official appointment, but Noether wasn’t given even an honorary “extraordinary” professorship until 1922. In the interim, she was merely allowed to serve as a guest lecturer, unpaid, under Hilbert’s name.
Weyl, also at Göttingen in the 1920s, by contrast quickly achieved a prominent professorship, despite being Noether’s junior. “I was ashamed to occupy such a preferred position beside her whom I knew to be my superior as a mathematician in many respects,” he later remarked.
The indignities of Noether’s circumstances did not deter her work. Almost immediately on arrival, Noether developed her eponymous theorem. It formalised the idea, intrinsic but unstated in the examples of the two theories of gravity, that symmetries provide an express route to the heart of nature’s workings.
For another example, consider a puck placed on a very smooth, very large frozen lake. Wherever the puck slides, the lake is the same. Noether’s theorem provides a general way of turning that statement of symmetry into a conservation law.
Conservation laws are the bread and butter of physics. They are mathematical shortcuts that allow us to compute physical quantities once and then never again. Whatever you start with, that’s what you’ll end up with. That is incredibly useful: think how much trickier it would be to manage your time if the number of hours in the day changed constantly and were not conserved at 24; it’s bad enough twice in the year when the clocks go forward or back.
Most of the great laws of physics include some statement of conservation, implicitly or explicitly. Newton’s first law of motion crudely states that “objects in motion stay in motion, and objects at rest stay at rest”. That is nothing more than conservation of momentum, a consequence of the sort of spatial symmetry that governs the physics on top of our idealised frozen lake. Send a puck across the ice and, discounting friction, it will continue indefinitely. But the conservation law only holds as far as the symmetry does. A hole in the ice will disturb the symmetry, causing the puck to sink to the bottom of the lake and come to rest – violating Newton’s first law.
“The indignity of Noether’s circumstances did not deter her work”
It’s not always obvious what is conserved and what isn’t. For a long time, it was assumed that mass couldn’t be created or destroyed, but Einstein’s famous relation E=mc2 said otherwise. Matter can be created, if not out of thin air, then out of pure energy. In fact, although you are made of molecules that are made of protons and neutrons, those protons and neutrons are made of quarks. Quarks, as it happens, are so light that they make up only about 1 to 2 per cent of your body mass. The rest comes from the incredible energies with which these quarks interact.
Although matter can be created from energy, energy itself in all its myriad forms adds up to a constant and permanent total. Before Noether, energy was simply assumed to be conserved, an assumption so basic that it became known in the 19th century as the first law of thermodynamics. But do the mathematics associated with Noether’s theorem and it becomes plain that energy is conserved because of an even more basic symmetry: specifically, that the laws of physics aren’t changing with time. If they did, energy wouldn’t be conserved.
What Noether’s theorem adds up to is a practical prescription for making progress in physics: identify a symmetry in the world’s workings, and the associated conservation law will allow you to start meaningful calculation.
But it is also, in a sense, a statement about how the universe should be structured. When we look at the universe on the human scale, or even at the level of our solar system, space seems very different from a smooth lake: there are planet-sized bumps and wiggles. But take a broader picture – on the scale of hundreds of millions of light-years – and the universe appears much smoother. The assumption is that on the very largest scales, the universe is more or less the same.
As we lack the ability to travel billions of light years to beyond the observational horizon of our most powerful telescopes, this really is just an assumption, and it goes by the name of the cosmological principle. It tells us that what we call “down” on Earth is nothing more than a consequence of the relative position of us and the rock we’re standing on. The universe has no up or down, nor a centre for that matter. Its laws don’t seem to be in any way related to where we measure them, how our measuring devices are pointed, or even when we decide to make the measurements. Through Noether’s theorem, symmetries of space and time yield conservation of energy, momentum and angular momentum everywhere, all the time (see “Magic recipe” below).
“We assume the Universe has no up or down, nor a centre for that matter”
But there’s much more. Symmetries in space and time might be obvious to the naked eye, yet Noether’s theorem’s true strength comes from “internal symmetries”. To the uninitiated, the standard model of particle physics is nothing more than a list of fundamental forces and particles. But it is a model of internal symmetries writ large, and it was built on Noether’s theorem.
The most familiar force it deals with is electromagnetism, which describes the current running through our power cords, the behaviour of compasses and the shock of lightning. James Clerk Maxwell is generally credited for writing down a theory that unified electricity and magnetism into one working model in the 1860s. One of its assumptions is that electric charge is neither created nor destroyed, an idea that goes back even further to Benjamin Franklin in the 1740s.
Noether’s theorem shows that charge conservation, too, arises from a symmetry. Fundamental particles have a property called spin, and just as position doesn’t matter on a frozen lake, what’s known as the spin’s phase doesn’t change physical calculations. “Turn” every electron in the universe an extra degree, and neither energy nor anything else changes. What pops out, according to Noether’s mathematics, is charge conservation.
Weyl took this idea of phase symmetry a step further and supposed that every electron might be twisted by a different amount and still remain the same. Assume this and, almost by magic, all four of Maxwell’s equations emerge.
As the standard model has developed, so the symmetries of interest have become more subtle – but Noether’s theorem has been the gift that keeps on giving. It is hard to conceive, for example, that electrons, the particles that run through wires to power electronics, and neutrinos, which fly through us by the trillions every second without leaving a mark, are in some sense the same particle.
Neutrinos primarily interact through the weak force, which controls nuclear fusion in the sun. But the weak force is indifferent to whether a particle is an electron or a neutrino: switch them round, and weak interactions will be the same. This symmetry produces conservation of a quantity called weak isospin that, like electric charge, can be used to label particles and predict how they will behave (see “Hidden principles” below).
In the 1960s, researchers found that electromagnetism and the weak force could in fact be generated by a single symmetry, in what became known as the electroweak theory, a keystone of the standard model. “Breaking” that symmetry into two separate pieces produced a bunch of new interactions, along with the prediction of a new particle – what we now know as the Higgs boson. We waited a half-century for the confirmation of this prediction, which stemmed directly from the sort of considerations Noether’s theorem introduced into physics. It came, eventually, with the discovery of the Higgs at CERN’s Large Hadron Collider in 2012.
The other pillar of the standard model is the strong interaction, which holds individual protons and neutrons together. The quarks that make up these particles are labelled with one of three “colours”: red, green and blue. Shift all the colours by one, and all strong interactions will remain exactly the same.
Colour symmetry leads – in what might at first seem to be a tautology – to conservation of colour. Since that idea was first introduced, work on the nature of the strong force has found that all particles in nature exist in states without colour – “white”, effectively. Protons and neutrons are examples of particles called baryons that consist of three quarks, one red, one blue, one green. The universe as a whole seems to be colourless, just as it is electrically neutral, and the symmetry of the strong force is what makes particles like protons and neutrons possible in the first place.
The thrill of the chase
Physics is now at the point where new theories are built on the assumption of a fundamental symmetry, and an informed guess about what that symmetry might be. Unification is a holy grail of physics: the drive to develop theories that can describe everything in just a few, albeit possibly outstandingly difficult, equations. What sort of symmetry might unify the electroweak and strong forces we do not yet know, but the search for such a “grand unified theory” is an active area of physical endeavour. A good grand unified theory might predict where all of the protons and neutrons in the universe come from. The total number of these baryons seems to be conserved too. Experimentally, we’ve tried to see if protons, the lightest of the baryons, can decay into anything. If we ever observe this we’ll have some idea as to whether baryon number is really conserved, a key clue to grand unified theory.
Of particular interest as we look beyond the standard model is supersymmetry, a model at the heart of many fledgling grand unified theories. Supersymmetry is based on unifying the two major groups of fundamental particles: fermions (the particles that make up matter such as electrons and quarks), and bosons (including the photon, the Higgs and other particles governing forces). It supposes that ultimately every fermion has a partner boson and vice-versa: hypothetical exotics such as “selectrons” and “higgsinos”. At high enough energies, the supposition is that an electron and a selectron behave the same way, just as neutrinos and electrons behave identically under the weak force.
Supersymmetry neatly solves many problems of the standard model, as well as providing a motivation for why particles have the masses that they do. In principle, that is. The Large Hadron Collider is hard at work looking for signatures of supersymmetry, but the lack of any success so far suggests we’re barking up the wrong tree.
Even further away is the goal of folding gravity, that original object of symmetric study, and the forces covered by the standard model into a “theory of everything”. Indeed, physics is still far away from a final resolution. But in the thrill of the chase for better answers, it is studying symmetries that will guide us along the way – and it is Noether’s theorem that will magic useful physical insights from that.
Compared with this stellar legacy, the rest of Noether’s biography is kind of a downer. She left Germany to escape the Nazis in 1933 and came to Bryn Mawr College in Pennsylvania, dying of complications from cancer surgery two years later. As Einstein wrote after her death, “Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began”. Others might suggest the second part of that sentence is superfluous. Mathematicians do revere her, yet despite laying the groundwork for much of modern physics, physicists tend to gloss over her contributions. A century on, it’s time she had the recognition a genius of her stature deserves.
Read more: Unsung heroines: Six women denied scientific glory
This article appeared in print under the headline “The hidden law”
