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SYMMETRY PRINCIPLES

The natural world is a complicated place. Symmetries allow people—and scientists—to discern order in nature. In physics, it has long been understood that symmetries are closely connected to conservation laws. Three of the most familiar conservation laws are the conservation of energy, the conservation of momentum, and the conservation of angular momentum. Conservation of energy is a consequence of the fact that the laws of nature do not change with time. For example, in Newton's law of gravitation, one could imagine that GN, the gravitational constant, depended on time. In this case, energy would not be conserved. From experimental searches for violations of energy conservation, one can set strong limits on any such time variation (astronomical observations provide stronger constraints). This principle is quite broad and applies in quantum mechanics as well as classical mechanics. Physicists sometimes call this symmetry—that there is no special time—the "homogeneity of time." Similarly, conservation of momentum is a consequence of the fact that there is no special place. If one describes the world with Cartesian coordinates, the laws of nature don't care what one takes to be the origin. This symmetry is called "translation invariance," or the homogeneity of space. Finally, conservation of angular momentum is related to a familiar symmetry of daily life: the laws of nature are invariant under rotations. For example, not only does it not matter how we choose the origin or our coordinate system, but it doesn't matter how we choose to orient the axes.

The symmetries of time and space translation, and rotations, are called continuous symmetries because one can translate the coordinate axes by any arbitrary amount, and one can rotate through any angle. Another class of symmetries are called discrete symmetries. An example is the symmetry of reflection in a mirror, or "parity." Newton's laws possess this symmetry. Watch the motion of an object falling in a gravitational field, and then examine the same motion in a mirror. While the motion is different, each appears to obey Newton's laws. This is familiar to anyone who has ever stood in front of a clean, well-polished mirror and gotten confused as to what was the object and what the mirror image. Another way to describe this symmetry is as a symmetry between left and right. For example, three-dimensional Cartesian coordinates are usually written according to the "right hand rule," as in Figure 1. The positive direction along the z -axis lies in the direction in which your thumb points if you rotate your right hand about the z -axis, starting at the x -axis and moving to the y -axis. The unconventional coordinate system in Figure 2 is the opposite; here the z -axis points in the direction your left hand would point. The statement that Newton's laws are invariant is the statement that we can use either kind of coordinate system, and the laws of nature look the same. The symmetry of parity is usually denoted by the letter P .

Parity is not the only discrete symmetry of interest in science. Another is called time reversal. In Newtonian mechanics, one can imagine taking a video of an object falling under the influence of gravity. Now consider running the video backward. Both the motion "forward in time" and the motion "backward" will obey Newton's laws (the backward motion may describe a situation which is not very plausible, but

FIGURE 1

FIGURE 2

it will not violate the laws). Time reversal is usually denoted by the letter T .

A third discrete symmetry is called charge conjugation. For every known particle, (the electron, proton, etc.) there is an antiparticle. The antiparticle has exactly the same mass, but the opposite electric charge. The antiparticle of the electron is called the positron. The antiparticle of the proton is called the antiproton. Recently, antihydrogen has been produced and studied. Charge conjugation is a symmetry between particles and their antiparticles. Clearly particles and antiparticles are not the same. But the symmetry means that, for example, the behavior of an electron in an electric field is identical to that of a positron (antielectron) in the opposite field. Charge conjugation is denoted by the letter C .

These symmetries, however, are not exact symmetries of the laws of nature. In 1956, experiments showed, surprisingly, that in the type of radioactivity called beta decay, there is an asymmetry between left and right. The asymmetry was first studied in decays of atomic nuclei, but it is most easily described in the decay of the negatively charged π- meson, another strongly interacting particle. The π- meson decays either to a muon and its antineutrino or an electron and its antineutrino. But the decays to the electron are very rare. This is related (by an argument which uses special relativity) to the fact that the antineutrino always emerges with its spin parallel to its direction of motion. If nature were symmetric between left and right, one would find the neutrino half the time with its spin parallel and half with its spin antiparallel. This is because, in a mirror, the direction of motion does not change, but the spin or angular momentum flips. Related to this is the positively charged π+ meson, the antiparticle of the π-. This particle decays to an electron neutrino, with its spin parallel to its momentum. This difference between the behavior of the neutrino and its antiparticle is an example of the violation of charge conjugation invariance.

After these discoveries, the question was raised whether time reversal invariance, T , was violated. By general principles of quantum mechanics and relativity, violation of T is related to violation of C × P , the product of charge conjugation and parity. CP, if it is a good symmetry, would state that the decay of the π+e+ + ν should proceed at the same rate as π-e- + . In 1964, an example of a process which violates CP was discovered, involving another set of strongly interacting particles, called the K mesons. It turns out that these particles have very special properties which allow the very tiny violation of CP to be measured. Only in the year 2001 was CP violation persuasively measured in the decays of another set of particles, the B mesons.

These results clearly show that absence of symmetry is often as interesting as its presence. Indeed, shortly after the discovery of CP violation, Andrei Sakharov pointed out that violation of CP in the laws of nature is a necessary ingredient to understanding the predominance of matter over antimatter in the universe.

It is still believed that the combination CPT, charge conjugation times parity times time reversal, is conserved. This follows from rather general principles of relativity and quantum mechanics and is, to date, supported by experimental studies. If any violation of this symmetry should be discovered, it would have profound implications.

So far, the symmetries discussed are significant in that they lead to conservation laws, or relations between rates of reaction between particles. There is another class of symmetries which actually determine many of the forces between particles. These symmetries are known as local symmetries or gauge symmetries.

One such symmetry leads to the electromagnetic interactions. Another, in Einstein's theory, leads to gravitation (in the form of Einstein's theory of general relativity). Einstein asserted, in enunciating his principle of general relativity, that it should be possible to write the laws of nature not only so they are invariant, for example, under a rotation of coordinates all at once everywhere in space, but under any change of coordinates. The mathematics for describing this had been developed by Friedrich Riemann and others in the nineteenth century; Einstein in part adapted, and in part reinvented, it for his needs. It turns out that to write equations (laws) which obey the principle, it is necessary to introduce a field, similar in many ways to the electromagnetic field (except that it has spin two). This field couples in just the right way to give Newton's law of gravitation, for things which are not too massive or moving too fast or not too dense. For systems which are very fast (compared to the speed of light) or very dense, general relativity leads to a rich array of exotic phenomena such as black holes and gravitational waves. All of this follows from Einstein's rather innocuous sounding symmetry principle.

The symmetry which leads to electricity and magnetism is another example of a local symmetry. To introduce it requires a bit of mathematics. In quantum mechanics, the properties of the electron are described by a "wave function," ψ (x ). It is crucial to the way quantum mechanics works that ψ is a complex number, in other words, it is, in general, the sum of a real number and an imaginary number. A complex number can always be written as the product of a real number, ρ, and a phase, e. For example, In quantum mechanics, one can multiply the wave function by a constant phase, with no effect. But if one insists on something stronger, that the equations don't depend on the phase (more precisely, if there are many particles with different charges, as there are in nature, a particular combination of phases is not important), one must, as in general relativity, introduce another set of fields. These fields are the electromagnetic fields. Enforcing this symmetry principle (plus the symmetries of special relativity) requires that the electromagnetic field obey Maxwell's equations. Today, all of the interactions of the Standard Model are understood as arising from such local gauge symmetry principles. The existence of the W and Z bosons, as well as their masses, half-lives, and other detailed properties, were successfully predicted as consequences of these principles.

What might lie beyond? A number of other possible symmetry principles have been proposed, for a variety of reasons. One such hypothetical symmetry is known as supersymmetry. This symmetry has been suggested for two reasons. First, it might explain a longstanding puzzle: why are there very small dimensionless numbers in the laws of nature? For example, when Planck introduce his constant h , he realized that one could use this to write a quantity with the dimensions of mass starting with Newton's constant. This quantity is now known as the Planck mass. It is given by where c is the speed of light, Gn is Newton's constant, and mp is the proton mass. Related to the fact that gravity is a very weak force, Mp is a very large number. In fact, compared to the mass of the W and Z , Mp/Mz = 1017

The great quantum physicist Paul Dirac (who predicted the existence of antimatter) called this the "problem of the large numbers." It turns out that postulating that nature is supersymmetric can help with this problem. Supersymmetry also seems to be an integral part of understanding how the principles of general relativity can be reconciled with the principles of quantum mechanics.

What is supersymmetry? Supersymmetry, if it exists, relates fermions (particles with half integral spin, which obey the Pauli exclusion principle) to bosons (particles with integer spin, which obey what are known as Bose statistics—the statistics which gives rise to the behaviors of lasers and Bose condensates). At first sight, however, it seems silly to propose such a symmetry, since if it were manifest in nature, one would expect that for every fermion there would be a boson of exactly the same mass, and vice versa.

In other words, in addition to the familiar electron, there should be a particle called the selectron, which has no spin and does not obey the exclusion principle, but is otherwise the same in every way as the electron. Similarly, related to the photon there should be another particle with spin 1/2 (which obeys the exclusion principle, like the electron) with zero mass and properties in many ways similar to those of photons. No such particles have been seen. It turns out, however, that these facts can be reconciled, and this brings us to one final point about symmetries. Symmetries can be symmetries of the laws of nature but need not be manifest in the world around us. Space around us is not homogeneous. It is filled with all kinds of different stuff, sitting at particular (special) places. Yet from conservation of momentum, we know that the laws of nature are symmetric under translations. In these circumstances, the symmetries are "spontaneously broken." In particle physics, the term is used more narrowly: a symmetry is said to be spontaneously broken if the state of lowest energy is not symmetric. This phenomenon occurs in many instances in nature: in permanent magnets, where the alignment of the spins which gives rise to magnetism in the lowest energy state breaks rotational invariance; in the interactions of the π mesons, which violate a symmetry called chiral symmetry. Whether supersymmetry exists in such a broken state is now a subject of intense experimental investigation.

Bibliography

Feynman, R. P. Six Not-So-Easy Pieces: Lectures on Symmetry, Relativity and Space-Time (Addison-Wesley, Reading, MA,1997).

Icke, V. The Force of Symmetry (Cambridge University Press, Cambridge, UK, 1995).

Michael Dine