Fear of Physics (28 page)

Whether or not such lattice symmetries revolutionize electrical technology, they have already played a role in revolutionizing biology. In 1905, Sir William Bragg and his son Sir Lawrence Bragg were awarded a Nobel Prize for a remarkable discovery. If X-rays, whose wavelength is comparable to the distance between atoms in a regular crystal lattice, are shined at such materials, the scattered X-rays form a regular pattern on a detecting screen. The nature of the pattern can be traced directly to the symmetry of the lattice. In this way, X-ray crystallography, as it has become known, has provided a powerful tool to explore the spatial configuration of atoms in materials and, with this, the structure of large molecular systems that may contain tens of thousands of regularly arrayed atoms. The most well-known application of this technique is probably the X-ray crystallographic data interpreted by Watson and Crick and their colleagues, which led to the discovery of the double-helix pattern of DNA.

The physics of materials has not been confined to technological developments. It has provided perhaps the most intimate relationship known between symmetry and dynamics, via the modern understanding of phase transitions. I have already described how, near a certain critical value of some parameter such as temperature or magnetic fields, totally different materials can effectively act the same. This is because at the critical point, most detailed microphysics becomes irrelevant and symmetry takes over.

Water at its critical point and an iron magnet at its critical point behave the same because of two related reasons. First, I remind you that at this critical point, small fluctuations are occurring simultaneously on all scales so, for example, it is impossible to say
on any scale whether one is observing water or water vapor. Since the material looks the same at all scales, local microphysical properties such as the particular atomic configuration of a water molecule must become irrelevant. Second, because of this, all that is necessary to characterize the configuration of water is its density: Is the region under consideration overdense or underdense? Water can be completely characterized by the same two numbers, +1 or–1, which we can use to identify the configuration of microscopic magnets inside iron.

Both of these crucial features are integrally related to the idea of symmetry. Water, and our iron magnets at their critical point, have become in some sense like our chessboard. There are two degrees of freedom that can be mapped into each other—black into white, overdense into underdense, up into down. It didn’t have to be this way. The fundamental parameter describing the possible states of the system near its critical point could have, for example, had a larger set of possibilities, such as pointing anywhere around a circle, as microscopic magnets might in a material in which they were not constrained to point up and down. Such a material, at its critical point, might then look schematically like this:

You might imagine that the fundamental characteristics of such a material as it approached its critical point would be different than water or the idealized iron magnets that were compared to water. And you would be correct. But what is the key difference between this picture and that shown on page 168 for water at its critical point? It is the set of possible values of the parameter that describe the transition—density, magnetic field direction, and so on. And what characterizes this set of possible values? The underlying symmetry of this “order parameter,” describing the change in the “ordering” of the material. Can it take any value on a circle, a square, a line, a sphere?

Seen in this way, symmetry once again determines dynamics. The nature of a phase transition at the critical point is completely determined by the nature of the order parameter. But this order parameter is restricted by its symmetries. All materials with an order parameter having the same set of symmetries behave
identically
when they undergo a phase transition at the critical point. Once again, symmetry completely determines the physics.

This use of symmetry, in fact, allows us to make a powerful connection between the physics of materials and elementary-particle physics. For the above picture is nothing but an example of spontaneous symmetry breaking. The order parameter describing the direction of the local magnetic fields in the preceding picture can take any value on a circle. It possesses an intrinsic circular symmetry. Once it picks out some value, in some region, it “breaks” this symmetry by choosing one particular manifestation among all the possibilities. In the above example, at the critical point, this value continually fluctuates, no matter what scale you measure it over. However, away from the critical point the system will relax into one possible configuration on large enough scales, that is, liquid water, all magnets pointing up, all magnets pointing east, and so on. In elementary-particle physics, we describe the configuration
of the ground state of the universe, the “vacuum,” by the characteristics of any coherent configuration of elementary fields that have some fixed value in this state. The order parameters in this case are just the elementary fields themselves. If they relax to some value that is nonzero in otherwise empty space, then particles that interact with these fields will behave differently than particles that don’t. The preexisting symmetry that might have existed among various elementary particles has been broken.

As a result, we expect that the spontaneous symmetry breaking that characterizes nature as we observe it today dies away on sufficiently small scales—where the order parameters, that is, the background fields, both fluctuate a great deal and, in any case, cannot alter the properties of particle motion on such small scales. Moreover, we also think spontaneous symmetry breaking disappeared at a very early stage in the big bang expansion. At that time, the universe was very hot. The same kind of phase transition that characterizes the liquefaction of water as temperature changes near the critical point can, in fact, occur for the ground state of the universe itself. At sufficiently high temperature, symmetries may become manifest because the order parameters, the elementary fields in nature, cannot relax to their low temperature values. And just as symmetries guide water through its phase transition, so, too, the symmetries of nature can guide the universe through its transitions. We believe that for every symmetry that is spontaneously broken at elementary scales today, there was at some sufficiently early time a “cosmic” phase transition associated with its breaking. Much of cosmology today is devoted to exploring the implications of such transitions, again governed by symmetry.

 

 

Returning to Earth, symmetry plays an even more powerful role in the phase transitions that govern the behavior of ordinary materials.
We have seen that the symmetry of the order parameter of water, or magnets, or oatmeal, or whatever, can completely determine the behavior of these materials at their critical points. But perhaps the most powerful symmetry known in nature governs our very ability to describe these transitions. This symmetry, which has played a role from the beginning of this book, is scale invariance.

What is fundamental in being able to relate materials as diverse as magnets and water at their critical points is the fact that fluctuations at the critical point take place on all scales. The material becomes scale-invariant: It looks the same on all scales. This is a
very
special property, so special that it is not even shared by spherical cows! Recall that I was able to make my sweeping statements about the nature of biology by considering how spherical cows behave as you change their size. If the relevant physics remained scale-invariant, then cows of arbitrary magnitude would be allowed. But it doesn’t, because the material of which cows are made does not change in density as you increase the size of the cow. This would be essential for quantities such as the pressure on the surface of a cow’s belly to remain the same, and for the strength of a supercow’s neck to grow in fixed proportion to its size.

But materials at the critical point of a phase transition
are
scale-invariant. The schematic diagrams of water and magnets such as I drew momentarily completely characterize the system on all scales. If I had used a microscope with higher resolving power, I would have seen the same kind of distribution of fluctuations. Because of this, only a very, very special kind of model will properly describe such a system near its critical point. The interesting mathematics of such models has become a focus for large numbers of both mathematicians and physicists in recent years. If one could, for example, categorize all possible models that possess scale invariance, then one could categorize all possible
critical phenomena in nature. This, one of the most complex phenomena in nature, at least on a microscopic scale, could be completely predicted and thus, at least from a physics perspective, understood. Many of the people who are interested in scale invariance are, or were, particle physicists. This is because there is reason to believe, as I hope to make clear in the next chapter, that the ultimate Theory of Everything, if there is such a thing, may rest on scale invariance.

 

 

I want to return, at the end of this chapter, to a few additional remarks about where symmetry is taking us. As I alluded in my introduction to string theory, this is, in fact, one of the few areas where one can get a glimpse of the origins of scientific progress at the frontier, when paradigms shift and new realities appear. This is where I can talk not just about things that are under control but also about things that are not. For, as I described earlier in this chapter, the questions physicists ask about nature are often guided by symmetries we don’t fully understand. Let me give you a few more concrete examples of this.

I have, throughout this chapter, made certain tacit assumptions about nature that seem unimpeachable. That nature should not care where and when we choose to describe it, for example, is the source of the two most important constraints on the physical world: energy and momentum conservation. In addition, while I can usually tell my right hand from my left, nature does not seem to care which is which. Would the physics of a world viewed in a mirror be any different? The sensible answer may seem to be no. However, our picture of what is sensible changed dramatically in 1956. In order to explain a puzzling phenomenon having to do with nuclear decays, two young Chinese-American theorists proposed
the impossible: Perhaps nature herself can tell right from left! This proposal was quickly tested. The decay of a neutron, which produces an outgoing electron and a neutrino, inside a cobalt nucleus with its local magnetic field aligned in a certain direction, was observed. If left-right invariance were preserved, as many electrons should be emitted on average going off to the right as would go off to the left. Instead, the distribution was found to be asymmetrical. Parity, or left-right symmetry, was
not
a property of the weak interactions that governed this decay!

This came as a complete shock to the physics community. The two physicists, Chen Ning Yang, and Tsung Dao Lee, were awarded the Nobel Prize within a year of their prediction.
Parity violation,
as it became known, became an integral part of the theory of the weak interaction, and it is the reason that the neutrino, alone among particles in nature that feels only this interaction (as far as we know), has a very special property. Particles such as neutrinos, and also electrons, protons, and neutrons, behave as if they are “spinning,” in the sense that in their interactions they act like little tops or gyroscopes. In the case of electrons and protons, which are charged, this spinning causes them to act like little magnets, with a north and south pole. Now, for an electron that is moving along, the direction of its internal magnetic field is essentially arbitrary. A neutrino, which is neutral, may not have an internal magnetic field, but its spin still points in some direction. It is a property of the parity violation of the weak interactions, however, that only neutrinos whose spin points along the same direction as their motion are emitted or absorbed during processes mediated by this interaction. We call such neutrinos left-handed, for no good reason except that this property is intimately related to the “handedness” observed in the distribution of particles emitted during nuclear decay.

We have no idea whether “right-handed” neutrinos exist in nature. If so, they need not interact via the weak interaction, so we might not know about them. But this does not mean they cannot exist. One can show, in fact, that if neutrinos are not exactly massless, like photons, it is highly probable that right-handed neutrinos might exist. If, indeed, any neutrino was found to have a nonzero mass, this would be a direct signal that some new physics, beyond the Standard Model, is required. It is for this reason that there is so much interest in the experiments now being performed to detect neutrinos emitted from the core of the sun. It has been shown that if the deficit observed in these neutrinos is real, then one of the most likely possible sources of such a deficit would be the existence of a nonzero neutrino mass. If true, we will have opened a whole new window on the world. Parity violation, which shocked the world but has now become a central part of our model of it, may point us in the right direction to look for ever more fundamental laws of nature.