The identical twins don’t exist. Physicists didn’t blunder and miss a whole parallel world. Why, then, is there interest in such idle mathematical speculation—interest that has only intensified over the last thirty years? Physicists are always interested in possible mathematical symmetries, even if the only question is why they aren’t part of nature. But also both the real world and the physicist’s description of it are full of symmetry. Symmetry is one of the most far-reaching and powerful weapons in the arsenal of theoretical physics. It permeates every branch of modern physics, particularly those that have to do with quantum mechanics. In many cases all we really know about a physical system is its symmetries, but so powerful are these symmetries that they sometimes tell us almost everything we want to know. Symmetries are often at the core of what physicists find esthetically pleasing about their theories. But what are they?
Let’s begin with a snowflake. Every child knows that no two snowflakes are exactly alike, but they nevertheless all share a certain feature: a symmetry. Just looking at a snowflake, you can easily see its symmetry. Take the snowflake and rotate it by an angle. It will look different, tilted. But if you rotate it by exactly 60 degrees, it will look unchanged. A physicist would say that rotating a snowflake through 60 degrees is a symmetry.
Symmetries are concerned with the operations you can do to a system that don’t change the results of experiments. In the case of the snowflake, the operation is to rotate it by 60 degrees. Here is another example: suppose we build an experiment to determine the acceleration due to gravity at the earth’s surface. The simplest version would be to drop a rock from a known height and time its fall. The answer is approximately ten meters per second per second. Notice that I didn’t bother to tell you where I dropped the rock, whether it was in California or Calcutta. To a pretty good approximation, the answer is the same everywhere on the earth’s surface: the result of the experiment would not change if you moved the entire experiment from one point of the earth’s surface to another. In physics jargon moving or displacing something from one point to another is called a translation. So we speak about the symmetry of the earth’s gravitational field “under translation.” Of course some annoying irrelevant effect might contaminate the symmetry. For example, if we did the experiment just above a very large and heavy mineral deposit, we would get a slightly bigger result. In that case we would say that the symmetry is only approximate. Approximate symmetries are also called broken symmetries. The presence of the localized mineral deposit “broke the translation symmetry.”
Can the symmetry of the snowflake be broken? No doubt some snowflakes are not perfect. If the flake formed in less than ideal conditions, one side might be a little different from the others. You might still be able to detect the overall hexagon shape of the crystal, but it would have an imperfect, or broken, symmetry.
In outer space, far from any contaminating influences, we might measure the gravitational force between two masses and deduce Newton’s law of gravity. No matter where we did the experiment, we would get the same answer. So Newton’s law of gravity has translation invariance.
To measure the force law between two objects, you have to separate them along some direction in space. For example, you might measure the force by separating the masses along the x-axis or along the y-axis or any other direction for that matter. Could the force between two objects depend on the direction along which they are separated? In principle, yes, but only if the laws of nature were different from what they are. It’s a fact of nature that Newton’s law of gravity—force proportional to the product of masses and inversely proportional to the square of the distance—is the same in every direction. If the whole experiment were rotated in space to realign the separation, the results would be unchanged. This insensitivity to direction is called rotation symmetry. Translation and rotation symmetries are two of the most basic properties of the world we live in.
Look at yourself in the mirror as you get dressed in the morning. Your mirror image looks exactly the same as you do. The mirror image of your pants is exactly the same as your real pants. The mirror image of your left glove is the same as your left glove.
Wait—that’s wrong. Look again. The mirror image of your left glove is not at all like the real left glove. It’s identical to the real right glove! And the image of the right glove is actually a left glove.
Look a little more closely at your own mirror image. It’s not you. The freckle on your left cheek is on the right cheek of the mirror image. If you could open up your chest cavity, you would find something really strange about the image. Its heart would be on the right side. It’s like no real human being. Let’s call it a .
Let’s imagine that we had a futuristic technology that allowed us to build up any object we pleased by assembling atoms, one by one. Here is an interesting project. Let’s build a human whose mirror image is exactly like you—heart on the left, freckle on the left, and so on. Now the original would be a .
Does the work? Will it breathe? Will its heart work? If we feed it will it metabolize the sugar that we give it? Mostly the answers are yes. It will work as well as you do. But we will find a problem with its metabolism. It just won’t process ordinary sugar.
The reason is that sugar comes in two types just like gloves—left-handed and right-handed. Humans can metabolize only sugar. A can metabolize only . The ordinary sugar molecule comes in two varieties—the kind you eat and its mirror image, which you can’t. The technical names for sugar and are of course levulose (left-handed sugar molecules that we eat) and dextrose (right-handed sugar that we don’t metabolize). will work just as well as humans but only if you also replace everything in their environment, including their food, by reflected counterparts.
Replacing everything by its mirror image is called reflection symmetry or sometimes parity symmetry. The implication of this symmetry is probably clear, but let’s spell it out. If everything in the world is replaced by its mirror image, the world will behave in a completely unchanged way.
In actual fact reflection symmetry is not exact. It is an example of a broken symmetry. Something causes the mirror image of a neutrino to be many times heavier than the original. This infects other particles but only in the minutest way. It is as if nature’s mirror were a slightly distorted mirror—a funhouse mirror that gives a distorted image, but only very slightly. The distortion is so insignificant for ordinary matter that s would not notice at all. But high-energy elementary particles do notice the distortion and behave in ways that their mirror images don’t. However, let’s ignore the distortion for the time being and pretend that reflection symmetry is an exact symmetry of nature.
What do we mean when we say that “a symmetry relates particles”? Simply stated, it means that for each type of particle there is a partner or a twin with closely related properties. For reflection symmetry it means that if a left glove is possible, so is a right glove—if levulose can exist, so can dextrose. And if reflection symmetry were not broken, this would apply to every elementary particle. Every particle would have an identical but reflected twin. When reflecting humans to make , every elementary particle must be replaced by its twin.
Antimatter is another manifestation of symmetry called charge conjugation symmetry. Once again, the symmetry involves replacing everything by something else: in this case every particle is replaced by its antiparticle. That replaces a positive electric charge, such as the proton, by a negative charge—the antiproton. Likewise, a positron replaces every ordinary negative electron. Hydrogen atoms become antihydrogen atoms consisting of a positron and an antiproton. Such antiatoms have actually been made in the laboratory—just a handful, not enough to make even an antimolecule. But no one doubts that antimolecules are possible. Antihumans are possible but don’t forget to feed them antifood. In fact you’d best keep away from antipeople. When matter meets antimatter it explosively annihilates into a burst of photons. The explosion that would occur if you inadvertently shook hands with an antihuman would be simil
ar to a nuclear bomb explosion.
As it turns out charge conjugation symmetry is also a slightly broken symmetry. But as with reflections the effect would be completely insignificant except for very high-energy particles.
Now back to fermions and bosons. The original String Theory, the one that Nambu and I discovered, is called bosonic String Theory because all the particles that it describes are bosons. This was not a good thing for describing hadrons: after all, the proton is a fermion. Neither would it be good if the goal of String Theory were a theory of everything. Electrons, neutrinos, quarks—they are all fermions. But it didn’t take long before new versions of String Theory were discovered that contained fermions as well as bosons. And one of the remarkable mathematical properties of these so-called superString Theories is supersymmetry—the symmetry between bosons and fermions that requires particles to come in exactly matched pairs, a boson for every fermion and vice versa.
Supersymmetry is, again, an indispensable and very powerful mathematical tool for string theorists. Without it the mathematics is so hard that it is very difficult to be certain of the consistency of the theory. Almost all of the reliable work on the subject assumed the world was supersymmetric. But as I emphasized, supersymmetry is most definitely not an exact symmetry of nature. At best it is a fairly badly broken symmetry—the kind of distorted symmetry that a badly warped funhouse mirror would imply. In fact no superpartner has ever been discovered for any of the known elementary particles. If a boson existed with the same mass and charge as the electron, it would have been discovered long ago. Nevertheless, if you go to your Web browser and look up research papers in elementary-particle physics, you will find that since the mid-1970s the overwhelming majority have something to do with supersymmetry. Why is this? Why didn’t theoretical physicists toss out supersymmetry in disgust and, with it, superString Theory? The reasons vary.
The subject that was once called theoretical high-energy physics, or theoretical elementary-particle physics, long ago split into two separate disciplines called phenomenology and theory. If you type in the URL http://arXiv.org on your Web browser, you’ll find the site where physicists post their research papers. Separate subdisciplines are listed by subject: nuclear theory, condensed-matter physics, and so on. If you look for high-energy physics (hep), you will find four separate archives, only two of which are devoted to the theoretical aspects of the subject that concern us (the other two are for experimental physics and computer simulations). One archive (hep-ph) is for phenomenology (the meaning of which will become clear shortly). The other (hep-th) is for the more theoretical and mathematical papers. Open them both. You’ll find that the hep-ph papers are all about the problems of conventional-particle physics. They sometimes refer to experiments, either past or future, and the results of the papers are often numbers and graphs. By contrast, the hep-th papers are largely about String Theory and gravity. They are more mathematical, and for the most part, they have little to do with experiments. However, in the last few years, the lines between these sub-subdisciplines have been blurred a good deal, which I think is a good sign.
But in both archives you will find that many of the papers have something to do with supersymmetry. Each camp has its own reasons for this. For the hep-th people, the reasons are mathematical. Supersymmetry leads to amazing simplifications for problems that are much too hard otherwise. Remember that in chapter 2 I explained that the cosmological constant is guaranteed to be zero if particles come in superpartner pairs. This is one of many mathematical miracles that occur in supersymmetric theories. I won’t describe them, but the main point is that supersymmetry so simplifies the mathematics of quantum field theory and String Theory that it allows theorists to know things that would otherwise be far beyond calculation. The real world may not be supersymmetric, but there may be interesting phenomena that can be studied with the aid of supersymmetry: phenomena that ordinarily would be far too hard for us to comprehend. An example is black holes. Every theory that includes the force of gravity will have black holes among the objects it describes. Black holes have very mysterious and paradoxical properties that we will come upon later in this book. Speculations about these paradoxes have been much too complicated to check in an ordinary theory. But, as if by magic, the existence of superpartners makes black holes easy to study. For the string theorist this simplification is essential. The mathematics of the theory as it is now practiced relies almost wholly on supersymmetry. Even many old questions about the quantum mechanics of quarks and gluons become easy if we add superpartners. The supersymmetric world is not the real world (at least in our pocket universe), but it is close enough to hold many lessons about elementary particles and gravity.
Although the ultimate aims of hep-phers and hep-thers may be the same, phenomenologists have a different immediate agenda from string theorists. Their goal is to use the older methods and sometimes the new ideas of String Theory to describe Laws of Physics as they would have been understood for most of the twentieth century. They are usually not trying to build a theory from some first principles, a theory that would be mathematically complete, nor are they expecting to discover the ultimate theory. Their interest in supersymmetry is as a possible approximate or broken symmetry of nature, something that can be discovered in future laboratory experiments. For them the most important discoveries would be of a whole family of new particles—the missing superpartners.
Broken symmetries, remember, are not perfect. In a perfect mirror, an object and its mirror image are identical except that left and right are interchanged, but in a funhouse mirror the symmetry is imperfect. It may be good enough to recognize the image of an object, but image and object are distorted versions of each other. The image of a thin man might be a fat man who, if he were real, would have twice the weight of his thin twin.
In the funhouse that we call our universe, the mathematical supersymmetry “mirror” that reflects each particle into its superpartner is badly distorted, so badly distorted that the superpartners are like a very fat image. If they exist at all, they are many times heavier—more massive—than the known particles. No superpartner has ever been discovered: not the electron’s mate or the photon’s or the quarks’. Does that mean that they don’t exist and that supersymmetry is an irrelevant mathematical game? Perhaps so, but it could also mean the distortion is enough to make the superpartners so heavy that they are beyond the reach of current particle accelerators. If for some reason the superpartners were all a bit too massive—a couple of hundred times the proton mass—they could not be discovered until the next round of accelerators is built.
The superpartners all have names similar to their ordinary twins. It’s not too hard to remember them if you know the rule. If the ordinary particle is a boson like the photon or the Higgs boson, then the name of the fermion twin is obtained by adding ino. Thus the photino, the Higgsino, the Zino, and the gluino. If, on the other hand, the original particle is a fermion, then you just add the letter s at the beginning: selectron, smuon, sneutrino, squark, and so on. This last rule has generated some of the ugliest words in the physicist’s vocabulary.
There is always a strong tendency to hope that new discoveries are “just around the corner.” If the superpartners fail to show up at a hundred times the mass of the proton, then estimates will be revised, and an accelerator will have to be built in order to discover them at a thousand proton masses. Or at ten thousand proton masses. But is it all wishful thinking and no more? I don’t think so. There is a deep puzzle about the Higgs particle to which supersymmetry may hold the key. The problem is closely connected with the “mother of all physics problems” as well as with the surprising weakness of gravity.
The same quantum jitters that can create an enormous vacuum energy can also have an effect on the masses of elementary particles. Here’s the reason. Suppose a particle is placed into the jittery vacuum. That particle will interact with the quantum fluctuations and disturb the way the vacuum jitters in the immediate vicinity of the parti
cle. Some particles will damp the jittery behavior; others will increase it. The overall effect is to modify the energy due to the jitters. This change in the jitter energy, due to the presence of the particle, must be counted as part of its mass (remember E = mc2). A particularly violent example is the effect on the mass of the Higgs boson. Physicists know how to estimate the additional extra mass of the Higgs, and the result is almost as absurd as the estimate for the vacuum energy itself. The vacuum jitters in the neighborhood of the Higgs boson ought to add enough mass to the Higgs boson to make it as heavy as the Planck mass!
Why is this so problematic? Although theorists usually focus on the Higgs particle, the problem really infects all the elementary particles, with the exception of the photon and graviton. Any particle placed in a jittery vacuum will suffer an enormous increase in its mass. If all the particles had their masses increased, all matter would become heavier, and that would imply that the gravitational forces between objects would become stronger. It takes only a modest increase in the strength of gravity to render the world lifeless. This dilemma is conventionally called the Higgs mass problem and is another fine-tuning problem that belabors theorists’ attempts to understand the Laws of Physics. The two problems—the cosmological constant and the Higgs mass—are in many ways very similar. But what do they have to do with supersymmetry?
The Cosmic Landscape Page 26
