Symmetry: A ‘Key to Nature’s Secrets’
When I first started doing research in the late 1950s, physics seemed to me to be in a dismal state. There had been a great success a decade earlier in quantum electrodynamics, the theory of electrons and light and their interactions. Physicists then had learned how to calculate things like the strength of the electron’s magnetic field with a precision unprecedented in all of science. But now we were confronted with newly discovered esoteric particles—muons and dozens of types of mesons and baryons—most existing nowhere in nature except in cosmic rays. And we had to deal with mysterious forces: strong nuclear forces that hold partiicles together inside atomic nuclei, and weak nuclear forces that can change the nature of these particles. We did not have a theory that would describe these particles and forces, and when we took a stab at a possible theory, we found that either we could not calculate its consequences, or when we could, we would come up with nonsensical results, like infinite energies or infinite probabilities. Nature, like an enemy, seemed intent on concealing from us its master plan.
At the same time, we did have a valuable key to nature’s secrets. The laws of nature evidently obeyed certain principles of symmetry, whose consequences we could work out and compare with observation, even without a detailed theory of particles and forces. There were symmetries that dictated that certain distinct processes all go at the same rate, and that also dictated the existence of families of distinct particles that all have the same mass. Once we observed such equalities of rates or of masses, we could infer the existence of a symmetry, and this we thought would give us a clearer idea of the further observations that should be made, and of the sort of underlying theories that might or might not be possible. It was like having a spy in the enemy’s high command.^{1}
1.
I had better pause to say something about what physicists mean by principles of symmetry. In conversations with friends who are not physicists or mathematicians, I find that they often take symmetry to mean the identity of the two sides of something symmetrical, like the human face or a butterfly. That is indeed a kind of symmetry, but it is only one simple example of a huge variety of possible symmetries.
The Oxford English Dictionary tells us that symmetry is “the quality of being made up of exactly similar parts.” A cube gives a good example. Every face, every edge, and every corner is just the same as every other face, edge, or corner. This is why cubes make good dice: if a cubical die is honestly made, when it is cast it has an equal chance of landing on any of its six faces.
The cube is one example of a small group of regular polyhedra—solid bodies with flat planes for faces, which satisfy the symmetry requirement that every face, every edge, and every corner should be precisely the same as every other face, edge, or corner. Thus the regular polyhedron called a triangular pyramid has four faces, each an equilateral triangle of the same size; six edges, at each of which two faces meet at the same angle; and four corners, at each of which three faces come together at the same angles. (See illustration on this page.)
These regular polyhedra fascinated Plato. He learned (probably from the mathematician Theaetetus) that regular polyhedra come in only five possible shapes, and he argued in Timaeus that these were the shapes of the bodies making up the elements: earth consists of little cubes, while fire, air, and water are made of polyhedra with four, eight, and twenty identical faces, respectively. The fifth regular polyhedron, with twelve identical faces, was supposed by Plato to symbolize the cosmos. Plato offered no evidence for all this—he wrote in Timaeus more as a poet than as a scientist, and the symmetries of these five bodies representing the elements evidently had a powerful hold on his poetic imagination.
The regular polyhedra in fact have nothing to do with the atoms that make up the material world, but they provide useful examples of a way of looking at symmetries, a way that is particularly congenial to physicists. A symmetry is a principle of invariance. That is, it tells us that something does not change its appearance when we make certain changes in our point of view—for instance, by rotating it or moving it. In addition to describing a cube by saying that it has six identical square faces, we can also say that its appearance does not change if we rotate it in certain ways—for instance by 90° around any direction parallel to the cube’s edges.
The set of all such transformations of point of view that will leave a particular object looking the same is called that object’s invariance group. This may seem like a fancy way of talking about things like cubes, but often in physics we make guesses about invariance groups, and test them experimentally, even when we know nothing else about the thing that is supposed to have the conjectured symmetry. There is a large and elegant branch of mathematics known as group theory, which catalogs and explores all possible invariance groups, and is described for general readers in two recently published books: Symmetry: A Journey into the Patterns of Nature by Marcus du Sautoy and Why Beauty Is Truth: A History of Symmetry by Ian Stewart.
2.
The symmetries that offered the way out of the problems of elementary particle physics in the 1950s were not the symmetries of objects, not even objects as important as atoms, but the symmetries of laws. A law of nature can be said to respect a certain symmetry if that law remains the same when we change the point of view from which we observe natural phenomena in certain definite ways. The particular set of ways that we can change our point of view without changing the law defines that symmetry.
Laws of nature, in the modern sense of mathematical equations that tell us precisely what will happen in various circumstances, first appeared as the laws of motion and gravitation that Newton developed as a basis for understanding Kepler’s description of the solar system. From the beginning, Newton’s laws incorporated symmetry: the laws that we observe to govern motion and gravitation do not change their form if we reset our clocks, or if we change the point from which distances are measured, or if we rotate our entire laboratory so it faces in a different direction.^{2}
There is another less obvious symmetry, known today as Galilean invariance, that had been anticipated in the fourteenth century by Jean Buridan and Nicole Oresme: the laws of nature that we discover do not change their form if we observe nature within a moving laboratory, traveling at constant velocity. The fact that the earth is speeding around the sun, for instance, does not affect the laws of motion of material objects that we observe on the earth’s surface.^{3}
Newton and his successors took these principles of invariance pretty much for granted, as an implicit basis for their theories, so it was quite a wrench when these principles themselves became a subject of serious physical investigation. The crux of Einstein’s 1905 Special Theory of Relativity was a modification of Galilean invariance. This was motivated in part by the persistent failure of physicists to find any effect of the earth’s motion on the measured speed of light, analogous to the effect of a boat’s motion on the observed speed of water waves.
It is still true in Special Relativity that making observations from a moving laboratory does not change the form of the observed laws of nature, but the effect of this motion on measured distances and times is different in Special Relativity from what Newton had thought. Motion causes lengths to shrink and clocks to slow down in such a way that the speed of light remains a constant, whatever the speed of the observer. This new symmetry, known as Lorentz invariance,^{4} required profound departures from Newtonian physics, including the convertibility of energy and mass.
The advent and success of Special Relativity alerted physicists in the twentieth century to the importance of symmetry principles. But by themselves, the symmetries of space and time that are incorporated in the Special Theory of Relativity could not take us very far. One can imagine a great variety of theories of particles and forces that would be consistent with these spacetime symmetries. Fortunately it was already clear in the 1950s that the laws of nature, whatever they are, also respect symmetries of other kinds, having nothing directly to do with space and time.
There are four forces that allow particles to interact with one another: the familiar gravity and electromagnetism, and the less wellknown weak nuclear force (which is responsible for certain types of radioactive decay) and strong nuclear force (which binds protons and neutrons in the nucleus of an atom). (I am writing of a time, during the 1950s, before the formulation of the modern Standard Model, in which the three known forces other than gravity are now united in a single theory.) It had been known since the 1930s that the unknown laws that govern the strong nuclear force respect a symmetry between protons and neutrons, the two particles that make up atomic nuclei.
Even though the equations governing the strong forces were not known, the observations of nuclear properties had revealed that whatever these equations are, they must not change if everywhere in these equations we replace the symbol representing protons with that representing neutrons, and vice versa. Not only that, but the equations are also unchanged if we replace the symbols representing protons and neutrons with algebraic combinations of these symbols that represent superpositions of protons and neutrons, superpositions that might for instance have a 40 percent chance of being a proton and a 60 percent chance of being a neutron. It is like replacing a photo of Alice or of Bob with a picture in which photos of both Alice and Bob are superimposed. One consequence of this symmetry is that the nuclear force between two protons is not only equal to the force between two neutrons—it is also related to the force between a proton and a neutron.
Then as more and more types of particles were discovered, it was found in the 1960s that this proton–neutron symmetry was part of a larger symmetry group: not only are the proton and neutron related by this symmetry to each other, they are also related to six other subatomic particles, known as hyperons. The symmetry among these eight particles came to be called “the eightfold way.” All the particles that feel the strong nuclear force fall into similar symmetrical families, with eight, ten, or more members.
But there was something puzzling about these internal symmetries: unlike the symmetries of space and time, these new symmetries were clearly neither universal nor exact. Electromagnetic phenomena did not respect these symmetries: protons and some hyperons are electrically charged; neutrons and other hyperons are not. Also, the masses of protons and neutrons differ by about 0.14 percent, and their masses differ from those of the lightest hyperon by 19 percent. If symmetry principles are an expression of the simplicity of nature at the deepest level, what are we to make of a symmetry that applies to only some forces, and even there is only approximate?
An even more puzzling discovery about symmetry was made in 1956–1957. The principle of mirror symmetry states that physical laws do not change if we observe nature in a mirror, which reverses distances perpendicular to the mirror (that is, something far behind your head looks in the mirror as if it is far behind your image, and hence far in front of you). This is not a rotation—there is no way of rotating your point of view that has the effect of reversing directions in and out of a mirror, but not sideways or vertically. It had generally been taken for granted that mirror symmetry, like the other symmetries of space and time, was exact and universal, but in 1957 experiments showed convincingly that, while the electromagnetic and strong nuclear forces do obey mirror symmetry, the weak nuclear force does not. Experiments showed, for example, that it was possible to distinguish a cobalt nucleus in the process of decaying—as a result of the weak nuclear force—from its mirror image, spinning in the opposite direction. (See illustration on this page.)
So we had a double mystery: What causes the observed violations of the eightfold way symmetry and of mirror symmetry? Theorists offered several possible answers, but as we will see, this was the wrong question.
The 1960s and 1970s witnessed a great expansion of our conception of the sort of symmetry that might be possible in physics. The approximate proton–neutron symmetry was originally understood to be rigid, in the sense that the equations governing the strong nuclear forces were supposed to be unchanged only if we changed protons and neutrons into mixtures of each other in the same way everywhere in space and time (physicists somewhat confusingly use the adjective “global” for what I am here calling rigid symmetries).
But what if the equations obeyed a more demanding symmetry, one that was local, in the sense that the equations would also be unchanged if we changed neutrons and protons into different mixtures of each other at different times and locations? In order to allow the different local mixtures to interact with one another without changing the equations, such a local symmetry would require some way for protons and neutrons to exert force on each other. Much as photons (the massless particles of light) are required to carry the electromagnetic force, a new massless particle, the gluon, would be needed to carry the force between protons and neutrons. It was hoped that this sort of theory of symmetrical forces might somehow explain the strong nuclear force that holds neutrons and protons together in atomic nuclei.
Conceptions of symmetry also expanded in a different direction. Theorists began in the 1960s to consider the possibility of symmetries that are “broken.” That is, the underlying equations of physics might respect symmetries that are nevertheless not apparent in the actual physical states observed. The physical states that are possible in nature are represented by solutions of the equations of physics. When we have a broken symmetry, the solutions of the equations do not respect the symmetries of the equations themselves.^{5}
The elliptical orbits of planets in the solar system provide a good example. The equations governing the gravitational field of the sun, and the motions of bodies in that field, respect rotational symmetry—there is nothing in these equations that distinguishes one direction in space from another. A circular planetary orbit of the sort imagined by Plato would also respect this symmetry, but the elliptical orbits actually encountered in the solar system do not: the long axis of an ellipse points in a definite direction in space.
At first it was widely thought that broken symmetry might have something to do with the small known violations of symmetries like mirror symmetry or the eightfold way. This was a false lead. A broken symmetry is nothing like an approximate symmetry, and is useless for putting particles into families like those of the eightfold way.
But broken symmetries do have consequences that can be checked empirically. Because of the spherical symmetry of the equations governing the sun’s gravitational field, the long axis of an elliptical planetary orbit can point in any direction in space. This makes these orbits acutely sensitive to any small perturbation that violates the symmetry, like the gravitational field of other planets. For instance, these perturbations cause the long axis of Mercury’s orbit to swing around 360° every 2,254 centuries.
In the 1960s theorists realized that the strong nuclear forces have a broken symmetry, known as chiral symmetry. Chiral symmetry is like the proton–neutron symmetry mentioned above, except that the symmetry transformations can be different for particles spinning clockwise or counterclockwise. The breaking of this symmetry requires the existence of the subatomic particles called pi mesons. The pi meson is in a sense the analog of the slow change in orientation of an elliptical planetary orbit; just as small perturbations can make large changes in an orbit’s orientation, pi mesons can be created in collisions of neutrons and protons with relatively low energy.
The path out of the dismal state of particle physics in the 1950s turned out to lead through local and broken symmetries. First, electromagnetic and weak nuclear forces were found to be governed by a broken local symmetry. (The experiments now underway at Fermilab in Illinois and the new accelerator at CERN in Switzerland have as their first aim to pin down just what it is that breaks this symmetry.) Then the strong nuclear forces were found to be described by a different local symmetry. The resulting theory of strong, weak, and electromagnetic forces is what is now known as the Standard Model, and does a good job of accounting for virtually all phenomena observed in our laboratories.
3.
It would take far more space than I have here to go into details about these symmetries and the Standard Model, or about other proposed symmetries that go beyond those of the Standard Model. Instead I want to take up one aspect of symmetry that as far as I know has not yet been described for general readers. When the Standard Model was put in its present form in the early 1970s, theorists to their delight encountered something quite unexpected. It turned out that the Standard Model obeys certain symmetries that are accidental, in the sense that, though they are not the exact local symmetries on which the Standard Model is based, they are automatic consequences of the Standard Model. These accidental symmetries accounted for a good deal of what had seemed so mysterious in earlier years, and raised interesting new possibilities.
The origin of accidental symmetries lies in the fact that acceptable theories of elementary particles tend to be of a particularly simple type. The reason has to do with avoidance of the nonsensical infinities I mentioned at the outset. In theories that are sufficiently simple these infinities can be canceled by a mathematical process called “renormalization.” In this process, certain physical constants, like masses and charges, are carefully redefined so that the infinite terms are canceled out, without affecting the results of the theory. In these simple theories, known as “renormalizable” theories, only a small number of particles can interact at any given location and time, and then the energy of interaction can depend in only a simple way on how the particles are moving and spinning.
For a long time many of us thought that to avoid intractable infinities, these renormalizable theories were the only ones physically possible. This posed a serious problem, because Einstein’s successful theory of gravitation, the General Theory of Relativity, is not a renormalizable theory; the fundamental symmetry of the theory, known as general covariance (which says that the equations have the same form whatever coordinates we use to describe events in space and time), does not allow any sufficiently simple interactions. In the 1970s it became clear that there are circumstances in which nonrenormalizable theories are allowed without incurring nonsensical infinities, but that the relatively complicated interactions that make these theories nonrenormalizable are expected, under normal circumstances, to be so weak that physicists can usually ignore them and still get reliable approximate results.
This is a good thing. It means that to a good approximation there are only a few kinds of renormalizable theories that we need to consider as possible descriptions of nature.
Now, it just so happens that under the constraints imposed by Lorentz invariance and the exact local symmetries of the Standard Model, the most general renormalizable theory of strong and electromagnetic forces simply can’t be complicated enough to violate mirror symmetry.^{6} Thus, the mirror symmetry of the electromagnetic and strong nuclear forces is an accident, having nothing to do with any symmetry built into nature at a fundamental level. The weak nuclear forces do not respect mirror symmetry because there was never any reason why they should. Instead of asking what breaks mirror symmetry, we should have been asking why there should be any mirror symmetry at all. And now we know. It is accidental.
The proton–neutron symmetry is explained in a similar way. The Standard Model does not actually refer to protons and neutrons, but to the particles of which they are composed, known as quarks and gluons.^{7} The proton consists of two quarks of a type called “up” and one of a type called “down”; the neutron consists of two down quarks and an up quark. It just so happens that in the most general renormalizable theory of quarks and gluons satisfying the symmetries of the Standard Model, the only things that can violate the proton–neutron symmetry are the masses of the quarks. The up and down quark masses are not at all equal—the down quark is nearly twice as heavy as the up quark—because there is no reason why they should be equal. But these masses are both very small—most of the masses of the protons and neutrons come from the strong nuclear force, not from the quark masses. To the extent that quark masses can be neglected, then, we have an accidental approximate symmetry between protons and neutrons. Chiral symmetry and the eightfold way arise in the same accidental way.
So mirror symmetry and the proton–neutron symmetry and its generalizations are not fundamental at all, but just accidents, approximate consequences of deeper principles. To the extent that these symmetries were our spies in the high command of nature, we were exaggerating their importance, as also often happens with real spies.
The recognition of accidental symmetry not only resolved the old puzzle about approximate symmetries; it also opened up exciting new possibilities. It turned out that there are certain symmetries that could not be violated in any theory that has the same particles and the same exact local symmetries as the Standard Model and that is simple enough to be renormalizable.^{8} If really valid, these symmetries, known as lepton and baryon conservation,^{9} would dictate that neutrinos (particles that feel only the weak and gravitational forces) have no mass, and that protons and many atomic nuclei are absolutely stable. Now, on experimental grounds these symmetries had been known long before the advent of the Standard Model, and had generally been thought to be exactly valid. But if they are actually accidental symmetries of the Standard Model, like the accidental proton–neutron symmetry of the strong forces, then they too might be only approximate. As I mentioned earlier, we now understand that interactions that make the theory nonrenormalizable are not impossible, though they are likely to be extremely weak. Once one admits such more complicated nonrenormalizable interactions, the neutrino no longer has to be strictly massless, and the proton no longer has to be absolutely stable.
There are in fact possible nonrenormalizable interactions that would give the neutrino a tiny mass, of the order of one hundred millionth of the electron mass, and give protons a finite average lifetime, though one so long that typical protons in matter today will last much longer than the universe already has. Experiments in recent years have revealed that neutrinos do indeed have such masses. Experiments are under way to detect the tiny fraction of protons that decay in a year or so, and I would bet that these decays will eventually be observed. If protons do decay, the universe will eventually contain only lighter particles like neutrinos and photons. Matter as we know it will be gone.
I said that I would be concerned here with the symmetries of laws, not of objects, but there is one thing that is so important that I need to say a bit about it. It is the universe. As far as we can see, when averaged over sufficiently large scales containing many galaxies, the universe seems to have no preferred position, and no preferred directions—it is symmetrical. But this too may be an accident.
There is an attractive theory, called chaotic inflation, according to which the universe began without any special spatial symmetries, in a completely chaotic state. Here and there by accident the fields pervading the universe were more or less uniform, and according to the gravitational field equations it is these patches of space that then underwent an exponentially rapid expansion, known as inflation, leading to something like our present universe, with all nonuniformities in these patches smoothed out by the expansion. In different patches of space the symmetries of the laws of nature would be broken in different ways. Much of the universe is still chaotic, and it is only in the patches that inflated sufficiently (and in which symmetries were broken in the right ways) that life could arise, so any beings who study the universe will find themselves in such patches.
This is all quite speculative. There is observational evidence for an exponential early expansion, which has left its traces in the microwave radiation filling the universe, but as yet no evidence for an earlier period of chaos. If it turns out that chaotic inflation is correct, then much of what we observe in nature will be due to the accident of our particular location, an accident that can never be explained, except by the fact that it is only in such locations that anyone could live.

1
This article is based in part on a talk given at a conference devoted to symmetry at the Technical University of Budapest in August 2009. ↩

2
For reasons that are difficult to explain without mathematics, these symmetries imply important conservation laws: the conservation of energy, momentum, and angular momentum (or spin). Some other symmetries imply the conservation of other quantities, such as electric charge. ↩

3
Strictly speaking, Galilean invariance applies only approximately to the motion of the earth, since the earth is not moving in a straight line at constant speed. It is true that the earth’s motion in its orbit does not affect the laws we observe, but this is because gravity balances the effects of the centrifugal force caused by the earth’s curved motion. This too is dictated by a symmetry, but the symmetry here is Einstein’s principle of general covariance, the basis of the general theory of relativity. ↩

4
Lorentz had tried to explain the constancy of the observed speed of light by studying the effect of motion on particles of matter. Einstein was instead explaining the same observation by a change in one of nature’s fundamental symmetries. ↩

5
Consider the equation x ^{3} equals x. This equation has a symmetry under the transformation that replaces x with– x; if we replace x with– x, we get the same equation. The equation has a solution x = 0, which respects the symmetry;–0 = 0. But it also has a solution in which x = 1. This does not respect the symmetry;–1 is not equal to 1. This is a broken symmetry. Of course, this equation is not much like the equations of physics. ↩

6
Honesty compels me to admit that here I am gliding over some technical complications. ↩

7
These particles are not observed experimentally, not because they are too heavy to be produced (gluons are massless, and some quarks are quite light), but because the strong nuclear forces bind them together in composite states like protons and neutrons. ↩

8
Again, I admit to passing over some technical complications. ↩

9
Lepton number is defined as the number of electrons and similar heavier charged particles plus the number of neutrinos, minus the number of their antiparticles. (This conservation law requires the neutrino to be massless because neutrinos and antineutrinos, respectively, spin only counterclockwise and clockwise around their directions of motion. If neutrinos have any mass then they travel at less than the speed of light, so it is possible to reverse their apparent direction of motion by travelling faster past them, hence converting the spin from counterclockwise to clockwise, and neutrinos to antineutrinos, which changes the lepton number.) Baryon number is proportional to the number of quarks minus the number of antiquarks. ↩