Symmetry: A ‘Key to Nature’s Secrets’
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.

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. ↩