# Section 8: Symmetries of Nature

Symmetries are a central tool in theoretical physics. They can play the role of an organizing principle in a new theory, or can allow for tremendous simplification of otherwise difficult problems. In particle physics, theorists speculate about new symmetry principles when they seek deeper explanations or theories of fundamental particles or forces. Condensed matter physicists use symmetries to characterize the molecular structure of different materials. Atomic physicists organize atomic states in terms of rotational symmetry. Without symmetries—even approximate symmetries—it is extremely difficult to characterize the properties of physical systems.

## Exact and approximate symmetries in particle physics

A system has a symmetry when changing the system in some way leaves it in an identical state. For example, a perfect circle, when rotated around the center, looks the same. We call rotating the circle—or any change of the system that leaves it looking the same—a symmetry transformation. The set of all symmetry transformations—all things that can be done to the system and leave it looking the same—form a group, a word with a precise mathematical definition. The transformations can be continuous, as a rotation by an arbitrary angle, or discrete, as a flip to a mirror image.

Figure 28: Rotations in physical space and "particle space."

Source: © David Kaplan. More info

Symmetries can apply not only to external properties, like physical rotations, but also to internal properties, like particle type. For example, a symmetry could exist where all physics experiments done with particle A would yield the same results with the same probabilities if they were done with particle B. This implies an exchange symmetry between A and B: You get the same result if you exchange particle A for particle B, and vice versa. In this case, the two particles have precisely the same properties.

More general and stranger symmetries can exist as well. For example, instead of simply exchanging particles A and B, one could replace particle A with a particle that is partially A and partially B. This occurs in the case of neutrinos, where one flavor is produced—for example, electron neutrinos in the sun—and another is measured far away—muon or tau neutrinos on earth. It is also true in the case of uncharged mesons made of quarks, like neutral Kaons or B-mesons. Thus, this kind of symmetry of A and B could be described as a "rotation" between particle types—it could be one or the other or a mixture of the two. It is very similar to physical direction, in the sense that one could be facing north or east or in some mixture of the two (e.g., east-north-east).

If one wanted to do a measurement to tell whether a particle is A or B, there would have to be something to distinguish the two—some difference. But a difference would mean the symmetry is not exact. One example is the three neutrinos of the Standard Model. They are almost, but not quite, the same. The distinction is that electron neutrinos interact in a special way with electrons, whereas muon and tau neutrinos interact in that same way with muon and tau particles, respectively. So here, the symmetry is approximate because the particles the neutrinos are associated with have very different masses. Exchanging a neutrino of one species with another changes how it interacts with the electron, for example. Also, when a neutrino scatters off of matter, the matter can 'pick out' the flavor of neutrino and change it to one type or another.

Figure 29: A combination particle that's a mixture of type A and type B can be turned into a pure type A or pure type B particle when it interacts with matter.

Source: © David Kaplan. More info

What if the three neutrinos of the Standard Model were exactly the same—how would we ever know that there are three? It turns out that we can determine the number of light neutrinos from experimental measurements of Z bosons. The decay rate of the Z boson depends on how many light particles are coupled to it and the size of their couplings. Since the couplings can be measured in other ways, and the decays of the Z that don’t involve neutrinos are visible, i.e., they leave energy in the detector, one can infer the number of neutrinos. The number measured in this way is ~2.984, in agreement (within errors) with the three neutrinos of the Standard Model.

## Spontaneous symmetry breaking

It might seem as though a system either has a symmetry or it doesn't: We can rotate a circle by any angle and it looks the same, but that doesn't work for a square. However, it is possible for a physical theory to have a symmetry that isn’t reflected in the current state of the system it describes. This can happen when a symmetry is spontaneously broken.

What does that mean? Consider, for example, a spinning top, whose point remains stationary on a table. The system (the top) is perfectly symmetric around its spinning axis—looking at the top from any side of the table, one sees the same image. Once the top has finished spinning, it lies on the table. The symmetry is gone, and the top no longer looks the same when viewed from any angle around the table. The top's handle now points in a specific direction, and we see different things from different vantage points. However, the top could have fallen in any direction—in fact, one could say that the top has equal probability of pointing in any direction. Thus, the symmetry is inherent in the theory of the top, while that state of the system breaks the symmetry because the top has fallen in a particular direction. The symmetry was spontaneously broken because the top just fell over naturally as its rotational speed decreased.

Figure 30: The wine-bottle potential that is characteristic of spontaneous symmetry breaking.

Source: © David Kaplan. More info

One can have a similar spontaneous breaking of an internal symmetry. Imagine two fields, A and B, whose potential energies depend on each other in the way illustrated in Figure 30. While typically in theories, the lowest energy value of a field is zero, here we see the minimum energy value lies along the circular ring at the bottom. While the potential energy shape is symmetric—it looks the same rotated around the center—the fields take a particular value along the minimum-energy ring at every point in space, thus breaking the symmetry.

In Section 2, we learned that particles are simply fluctuations of a field. Our fields A and B can fluctuate in a very special way because the potential energy minimum forms a ring. If the fields are valued such that they sit in that minimum, and they fluctuate only around the ring, the potential energy does not change as the field fluctuates. Because the field vibrations involve no potential energy, the waves of the field can be as long and as low-energy as one wishes. Thus, they correspond to one or more massless particles. The fact that spontaneously breaking a symmetry results in a massless particle is known as the Nambu-Goldstone theorem, after physicists Yoichiro Nambu and Jeffrey Goldstone.

Figure 31: A wave in field space corresponds to a physical particle.

Source: © David Kaplan. More info

Pions, originally thought to be carriers of the strong force, are a real-life example of Nambu-Goldstone bosons. They are associated with the breaking of a complicated symmetry that involves the change of left-handed quarks into each other, with a simultaneous opposite change to right-handed quarks. This symmetry is spontaneously broken by the dynamics of the strong force in, as of today's knowledge, some inexplicable way. Pions are light, rather than massless, because the symmetry is approximate rather than exact. Knowing that the pions are Nambu-Goldstone bosons allows physicists to determine some of the mysteries of how the strong force actually works.

## Recovering symmetry at high temperatures

In our initial example of the spinning top, the theory had an underlying symmetry that was broken when the top fell over. When the top had a lot of energy and was spinning quickly, the symmetry was obvious. It was only when the top lost enough energy that it fell over that the symmetry was broken. The high-energy top displays a symmetry that the low-energy top does not. Something similar happens in more complicated systems such as magnets, superconductors, and even the universe.

Let's take a magnet as an example. Make it a ball of iron to keep things nice and symmetric. The magnetization of iron comes about because the electrons, which are themselves tiny magnets, tend to want to line up their spine, and thus their magnetic fields, such that collectively, the entire material is magnetic. If one dumps energy in the form of heat into the magnet, the electrons effectively vibrate and twist more and more violently until at a critical temperature, 768 degrees Celsius for iron, the directions of their individual spins are totally randomized. At that point, the magnetic field from all of the electrons averages out to zero, and the iron ball is no longer magnetic.

When the iron is magnetized it is like the fallen top, having selected a particular direction as different from the rest. Once heated to the critical temperature, however, symmetry is restored and any direction within the magnet is equivalent to any other. Many symmetries that are spontaneously broken in the minimum energy state are restored at high temperatures. The Higgs mechanism we encountered in the previous section is a significant, if complicated, example. The restoration of symmetry at high temperatures has significant implications for the early universe, when the temperatures were extremely hot—it implies that at times very soon after the Big Bang, most or all of the spontaneously broken symmetries of the Standard Model (and its underlying theory) were intact.

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