cphil_susy1
SUPERSYMMETRY
by Ray Shelton
INTRODUCTION
THE SEARCH FOR SUPERSYMMETRY
To put things in perspective, the quantum electrodynamic theory (QED) is an excellent theory, the electroweak theory is a very good theory, and so is the quantum chromodynamics theory (QCD), judging by the problems that they have so far solved and those that still need to be solved. The family resemblance between the three theories are perhaps the best guide that the theorists are really on the track of something more fundamental, which will unite all the forces of nature into one superforce. Electromagnetism is the simplest, and involves one charge. The weak field is characterized by a property which has two values, isospin, and relates doublets of quarks and doublets of leptons. Quarks come in triplets, and are described by a field one step more complicated. But the same common principles underlie the singletons of QED, the doubletons of the weak field, and the triplets of QCD, and that has enabled the first two to be combined into one successful unified theory. And the color of QCD is exactly analogous to the electric charge of QED, except that it comes in three varieties. Particles that do not carry charge cannot feel the electromagnetic field; particles that do not carry color, the leptons, cannot feel the field of QCD.
By pushing these ideas in the same direction, many theorists have attempted to construct Grand Unified Theories (GUT) that encapsulate the electroweak theory and QCD in one package. Most of these GUTs are members of the same family of theories, following a line of research pioneered by Glashow and a Harvard colleague, Howard Georgi, in the mid-1970s. Such theories each deal with particles in families of five — one such family, for example, consists of the three colors of the anti-down quark, plus the electron and its neutrino. Members of these families can be changed into one another by the same kind of transformation that converts protons into neutrons and one color of quark into another color, equivalent to rotating a pointer which has five positions on its scale. But now we have the possibility of turning leptons into quarks, and quarks into leptons. The GUTs describe a deeper symmetry than any of the simpler theories, but at a price.
The electroweak theory needs four bosons — the photon, two W‘s and the Z. The fivefold GUTs (known in mathematical shorthand as SU(5) theories) requires twenty-four bosons. Four of these are the four already needed by the electroweak theory; eight more are the gluons required by QCD. But that still leaves twelve “new” bosons, busy mediating new kinds of previously unsuspected interactions. Such hypotheical particles are collectively called X, for unknown quantity, or Y. They can change quarks into leptons, or vice versa, and carry charges of 1/3 or 4/3. But they are very massive — so massive that their lifetimes are extremely restricted in the universe today, and therefore they play a little part in the activity of particle world.
According to these theories, the three forces (electromagnetism, the weak interaction, and the strong force of QCD) would have been equal to each other at energies as great as 1015 GeV, that is, 1013 (10 million, million) times the energy at which the electromagnetic and weak forces were, or are, unified. That corresponds to a time when the universe was only 10-37 seconds old, at a temperature of 1029 Kelvin, and it means that the masses of the X particles themselves must be about 1015 GeV, a million, million times more than the greatest energy yet reached in a collision at CERN proton-antiproton collider. There is no prospect of creating such conditions artifically, and that is why physicists have to look to the Big Bang for evidence that X particles ever existed. Surprisingly, though, there is a possibility of detecting a side of effect of their existence here and now.
If a quark inside a proton could borrow enough energy from the uncertainty relation to create a virtual X boson and swap it with another quark, one of the quarks would become an electron (or positron). The two quarks left over will form a meson — a pion — and the proton will have decayed. Because the X boson is so massive, its virtual lifetime is so short that it could only cross from one quark to another if they were closer than 10-29 cm, and this is seventeen powers of ten smaller than the size of the proton itself (10-17 times the size of proton). Such very close encounters beween quarks must be rare indeed. But they will happen, from time to time, and the likelihood of such events can be calculated. It turns out that for an individual proton such an event will occur once in more than 1030 years — probably, depending on which theory you fancy, not for at least 1032 years. The universe is only some 1010 years old, so it is no surprise to find that protons are still around and seem pretty stable. But if the chance of one proton decaying in one year is one in 1030; if you have 1030 protons together, then there is a good chance that one of them (but you don’t know which one) will decay in each year that you are watching.
But all is not well with the GUTs. A line of research that started out with the simple idea of symmetry in guage theory has become ugly and complicated, with a proliferation of bosons and with the problem of what renormalization really implies still swept under the carpet, forming a bigger and harder-to-hide lump with every new force that is incorporated into the models. More quarks and leptons can be happily accommodated every time you want one, which indicates a certain lack of restraint on the part of the theories. But, embarrassingly, all of the GUTs predict the existence of magnetic monopoles, none of which have yet been found in the world we inhabit. And, indeed, since there are an infinite number of possible guage theories it is a mystery why these particular ones should be the ones that tell us anything about the real world at all. So what might happen if we cut loose from this step by step approach that builds a house of cards with one layer on top another, and get back to the roots?
That is what Julian Weiss, of the University of Karlsruhe, and Bruno Zumino, of the Berkeley campus of the University of California, did in 1974. GUTs surprise us by relating leptons to quarks, but they still leave bosons out on a limb as something different from material particles, merely the carriers of the forces. Weiss and Zumino said, in effect, if symmetry is a good idea, why not go the whole hog with supersymmetry, and relate the fermions to the bosons?
Stop and think about that for a minute. The distinction between fermions and bosons is the big one in quantum physics. Bosons do not obey the Pauli exclusion princple, but fermions do. The two seem far more unlike each other than the proverbial chalk and cheese. Can matter and force be really two faces of the same thing? Supersymmetry says yes; that every variety of fermion (every variety, not every individual particle) in the Universe should have a bosonic partner, and every kind of boson should have its own fermonic counterpart. What we see in our experiments, and feel the effects in everyday life, is only half of the Universe. Every type of quark, a fermion, ought to have a partner, a type of boson called squark; the photon, a boson, ought have a partner, a fermion called a photino; and so on. In the same vein, there ought to be winos, zinos, gluinos and sleptons. But there is no problem in explaining where the partners have gone; at this early stage of the game, the theorists can wave their mathematical magic wands and invoke some form of (unspecified) symmetry breaking that gave the unseen partners large masses and left them out in the cold when the Universe cooled.
Claiming that there is a symmetry between bosons and fermions sounds outrageous to anyone brought up to believe in the distinction between particles and forces. But is it so outrageous? Haven’t we come across something like it before? Quantum physics, after all, tells us that particles are waves and wave are particles. To a nineteenth century physicist such as Maxwell, electrons were particles and light was a wave; in the 1920s physicists learned that atoms are both particle and wave, while the photons are both wave and particle. And these are the archetypal members of the fermion and boson families. Is supersymmetry really doing anything more outrageous to our commonsense view of things than taking wave-particle duality to its logical limit, and saying that a particle-wave is the same as wave-particle? Is it only because we have got away from the roots of quantum physics, and, for convenience, described events in the subatomic world in terms of collisons and interactions between tiny hard particles, that supersymmetry strikes us as very odd at all. If only our minds were equipped to handle the same concepts in a more abstract form, in keeping with the quantum equations, so that we could properly understand the nature of quantum reality, where nothing is real unless it is observed, and there is no way of telling what “particles” are doing except at the moments when they are interacting with one another, then supersymmetry would seem much more natural. The flaw lies in our imaginations rather than in the theory. But even with our limited imaginations we can appreciate one feature of the new theory that makes it stand head and shoulders above most candidates for the title of “superforce”. The most dramatic thing about supersymmetry (SUSY for short) is that the mathematical tricks needed to change bosons into fermions, and vice versa, automatically, and inevitable, bring in the structure of spacetime, and gravity.
The symmetry operations involved in turning bosons into fermions are close mathematical relatives of the symmetry operations of general relativity, Einstein’s theory of gravity. If you apply the supersymmetry transformations to a fermion, you get its partner boson. A quark, say, becomes a squark. Apply the same transformation again, you get the original fermion back — but displaced slightly to one side. The supersymmetry transformations involve not only bosons and ferminons, but also spacetime itself. And general relativity tells us that gravity is simply a reflection of the geometry of spacetime.
But there is a peculiarity about the way physicists came up with the idea of supersymmetry. It all started in 1970, when Yoichiro Nambu, of the University of Chicago, came up with the idea of treating fundamental particles not as points, but as tiny one-dimensional entities, called strings. (Historically, the very first hints of string theory came in 1968, when two young researchers at CERN, Gabriel Veneziano and Mahiko Suzuki, were each looking for mathematical functions that could be used to describe the behavior of strongly interacting particles. They each, independently, noticed that a function written down in the nineteenth century by Leonhard Euler, and called the Euler beta function, might fit the bill. This turns out to be the mathematical underpinning string theory; but it was Nambu who turned mathematics into physics.) This was at about the time that the quark model was beginning to be taken seriously, and in the early 1970s Nambu’s idea was overshadowed by the rapid acceptance of the quark model — it was seen as a rival to the quark theory, not a complementary idea. The fundamental entities that Nambu was trying to model were not quarks, but hadrons (particles, such as the neutron and proton, which feel the strong force, and which we would now describe as being composed of quarks). The success of the quark model seemed to leave this string theory out in the cold; but a few mathematically inclined physicists played with it anyway. Nambu’s string theory involved spinning and vibrating length of strings only about 10-13 cm long. The properties of the particles he was trying to model in this way (their masses, electric charge, and so on) were thought of as corresponding to different states of vibration, like different notes played on a guitar string, or to be attached in some way to the whirling ends of the strings. And these vibrations also involved oscillations in more dimensions than in the three of space plus one of time that we are use to.
Embarrassingly, though, when the appropriate calculations were first carried through, they said the entities described by the strings would all have integer spin, in the usual quantum-mechanical sense. That is, they would all be bosons (force carriers, such as photons). And yet, the whole point of the model had been to describe hadrons, which are fermions and have half-integer spin! Then Pierre Ramond, of the University of Florida, found a way around the problem. He found a way to adapting Nambu’s equations to include strings with half-integer spin, describing fermions. But those fermionic strings were also allowed by the equations to join together in pairs, making strings with integer spin — bosons. John Schwarz, in Princeton, Joel Scherk, at Caltech, and the French physicist Andre Neveu developed this idea into a consistent mathematical theory of spinning strings which included both bosons and fermions, but required the strings to be vibrating in ten dimensions. It was Scherk, in particular, who established, by 1976, that fermions and bosons emerged from this string theory on an exactly equal footing, with every kind of boson having a fermionic partner, and every fermion having a bosonic partner. Supersymmmetry had been born.
There is a valuable way of looking at all of this, which is often emphasized by Ed Witten, one of the main players in the supersymmetry game of the 1990s. Bosons are entities whose properties can be described by ordinary commuting relationships, familar everyday rules such as A times B is equal to B times A, Fermions, though, have properties that do not always commute. (In fact, they do not commute in a special way; they are said to anticommute.) The appropriate mathematics that describes this behavior is quantum mechanics, not classical (Newtonian) mechanics. The concept of fermions is based entirely on the principles of quantum physics, while bosons are essentially classical in nature. Supersymmetry updates our understanding of spacetime to include fermions as well as bosons; it therefore updates the special theory relativity, Einstein’s first theory of space and time, by making it quantum mechanical.
This deep insignt was appreciated in 1976, and the next step was seen as being to seek to bring gravity into the fold, updating the General Theory of Relativity, Einstein’s second theory of space and time, the same sort of way. That might have speeded up the development of string theory by a decade. But it was not to be — even though gravity problem was in many people’s minds at the end of the 1970s, at that time they saw the next step in terms of an extension of supersymmetry to include gravity. in a theoretical package dubbed supergravity, without using the idea of strings at all.
Almost as soon as supersymmetry had burst upon the scene, the string theory that had given it birth had been forgotten. Never seen as more than a byway of physics by most researchers, it had by 1976 been totally eclipsed by the quark model. Once the idea of supersymmetry had been placed in the minds of physicists, it was easy to incorporate it into the standard model of the particle world. Indeed, that is the way generations of students after 1976 were introduced to supersymmetry, without any mention of strings at all. Physics moved on, and left string theory behind. Just about the only people who carried on working in the field were John Schwarz and, over in London, Michael Green (Scherk died young, and made no further contribution to the idea).
But while string theory languished, its offspring, supersymmetry flourished. A band of enthusiasts soon took up the ideas of SUSY, developing various lines of attack. One describes GUTs in terms of SUSY — the theories are known as SUSY GUTs. Another theory focuses upon gravity — supergravity, which itself comes in various forms with family resemblances but different detailed constructions. One great thing about all the supergravity models is that they each specify a different specific number of possible types of particles in the real world — so many leptons, so many photinos, so many quarks, and so on — instead of the endless proliferation of families allowed by the older GUTs. Nobody has yet succeeded in matching up the specific numbers allowed in any of these supergravity theories with the particles of the real world, but that is seen as a relatively minor problem compared with the previous one of the potentially infinite number of types of particles to worry about. A favored version of these theories is called “N=8” supergravity, and its enthusiasts claim that it could explain everything — forces, matter particles and the geometry of spacetime, in one package. But the best thing about N=8 supergravity is that it seems not merely to be renormalizable but in a sense to renomoralize itself — the infinities that have plagued field theory for half a century cancel out of N=8 theory all by themselves, without anyone having to lift a finger to encourage them. N=8 always comes up with finite answers to the questions physicists ask of it. “Superforce”, indeed!
But the one great puzzle about supergravity is that it requires eleven dimensions in which to operate. Where are they? All this success in the late 1970s and the early 1980s in finding potential ways to bring gravity and spacetime back into the fold of particle physics remineded physicist that way back in the 1920s there had already been attempts to explain all the forces of nature in terms of curved spacetime, the way gravity was explained by Einstein’s theory. And, from the outset, this approach had not involved higher dimensions (more than familiar four), but a neat trick for tucking them out of sight.
Early in 1919, Theodor Kalusa, a junior scholar at the University of Konigsberg, Germany, was setting at his desk in his study, working on the implications of the new General Theory of Relativity, which Einstein had first presented four years earlier and which was about to be confirmed, in spectacular fashion, by Arthur Eddington’s observations of light bending during the total eclipse of the Sun. As usual, Kalusa’ son, Theodor junior, age nine, was sitting quietly on the floor of the study, playing his own games. Suddenly, Kalusa senior stopped work. He sat still for several seconds, staring at his papers, covered with equations, that he was working on. Then he whistled softly, slapped both hands down hard on the table, and stood up. After another pause while he gazed at the work on the desk, he began to hum a faviourite aria, from Figaro, and started marching around the room, huming to himself all the while. This was not at all an usual behavior on the part of young Theodor’s father, and the image stuck in the boy’s mind, so that he was able to recall it vividly sixty-six years later, in an interview for BBC TV’s Horizon programme. The reason for his father’s unusual behavior was a discovery that is now, after decades in the wilderness, at the heart of research into the nature of the universe.
While tinkering with Einstein’s equations in which the gravitational force is explained in terms of the curvature of a four-dimensional continuum of spacetime, Kalusa had wondered, as mathematician do, how the equations would look if written down to represent five dimensions. He found that this five-dimensional version of General Relativity includes gravity, as before, but also another set of field equations, describing another force. The moment that struck in young Theodor’s mind so vividly was the moment that Kalusa senior wrote out the equations and saw that they were familiar — they were, indeed, Maxwell’s equations for electromagnetism. Kalusa had unified gravity and electromagnetism in one package, at the cost of adding in a fifth dimension to the universe. Electromagnetism seemed to be simply gravity operating in the fifth dimension.
Kalusa wrote up his idea and sent his paper to Einstein, as was then the procedure to publish was to first send the paper to an eminent authority, and getting his approval then send the paper on to a learned society to be published. In those days, a young researcher could not easily publish dramatic new discoveries out of the blue. Today, if you have a bright idea, you can write a paper and send it to a learned journal. The journal editors then send it out to an expert (or several experts) to assess before they decide whether or not to publish it. But in those days it was considered correct for the author to send the paper first to an eminent authority, who might then, if he approved, send the paper to a learned society with his recommendation that it be published. So Kaluza sent his results to Einstein.
Initially, Einstein was fascinated and enthusiatic. He wrote to Kalusa, in April of 1919, that his idea had never occurred to him, and said “at first glance, I like your idea enormously.” Although Einstein had no problem in “finding” four dimensions (three of space and one of time), there was no evidence that there really was a fifth dimension to the universe. Even so, Kaluza’s discovery was striking and looked important. But then Einstein began to pick at little points of detail. A perfectionist himself, Einstein urged Kalusa, in series of letters, to tidy up little details before publication. The correspondence, which now seems to be nit-picking, continued on into 1921, when all of sudden Einstein had a change of heart (nobody is quite sure why) and sent Kalusa a postcard telling him that he (Einstein) was going recommend publication. What Einstein recommended in 1921, no journal editor would argue with, and the article duly appeared in the proceedings of the Berlin Academy later that year, under the rather bland title of (in German) “On the Problem of Unification in Physics”.
The obvious defect with the theory presented in the paper was that it took no account of quantum theory — it was, like General Relativity itself, a “classical” theory, There was initially interest in Kalusa’s theory in 1922, but then nothing at all. Even Einstein, who himself spent the rest of his life seeking an unified field theory, seems to have ignored Kalusa’s idea from then on, in spite of the fact that in 1926 the Swedish physicist Oskar Klein found a way to incorporate Kalusa’s ideas in a quantum theory.
The behavior of an electron, or photon, or whatever, is described in quantum physics by a set of equations with four variables. A standard form of these equations is called Schrodinger’s equation, after the Austrian physicist who formulated it. Klein rewrote Schrodinger’s equation with five variables instead of four, and showed that solutions of this equation could now be represented in terms of particle-waves moving under the influence of both gravitational and electromagnetic fields. All theories of this kind, in which fields are represented geometrically in terms of more than four dimensions, now called Kaluza-Klein theories. (In fact, Gunnar Nordstrom, working at what was Helsinki University, had tried and failed to find a five-dimensional unification of gravity and electromagnetism in 1914, and in 1926, H. Mandel independently came up with the same basic idea as Kaluza, apparently in ignorance of Kaluza’s 1921 paper.) As early as 1926, they incorporated gravity and electromagnetism into one quantum theory.
One reason for overlooking, or neglecting, such theories as the Kaluza-Klein theory, was that there were now more forces to worry about, and so the Kaluza-Klein model seemed unrealistic. The “answer” was to invoke more dimensions, adding more variables to the equations to include the effects of all the new fields and their carriers, all described by the same geometrical effects of gravity. An electromagnetic wave (a photon) is a ripple in the fifth dimension; the Z, say, may be the ripple in the sixth; and so on. The more fields that there are, and the more force carriers, then the more dimensions that are needed. But the numbers are no worse than the numbers that come out of standard approaches to the unification of the four forces, such as supergravity.
Indeed, the numbers are exactly the same. The front runner among the supergravity candidates (indeed, the only good supergravity theory) is the N=8 theory. That theory describes a way to relate particles with different spins, under the operations of supersymmetry. The range of the spins available is +2 to -2, and spins come in half-integer quanta. (The hypothetical “particle” of gravity, the graviton, has a spin of 2, and theory suggests that this is greatest value possible.) So there are eight steps (eight SUSY transformations) involved in getting from one extreme to the other, hence the name. But there is another way of looking at all this. Just as Kaluza tinkered with Einstein’s equations to see how they would look in five dimensions, so modern mathematical physicists have tinkered with supergravity to see how they would look in different dimensions. It turns out that the simplest version of supergravity, the most beautiful and straightforward mathematical description, involves eleven dimensions — no more, no less. In eleven dimensions, there is a unique theory which just might be the sought-after superforce. If there are eleven dimensions to play with, all the complexity of the eight SUSY transformations disappears, and we are left with just one fundamental symmetry, an N=1 supergravity. And how many dimensions does the Kaluza-Klein theory need to accommodate all of the known forces of nature and their fields? Precisely eleven: the four familiar components of spacetime and the seven additional dimensions — no more, no less.
The implications of all of this have excited many physicists, and no less an authority than Abdus Salam described this geometrization of the world of particles and fields as “an incredible, miraculous idea.” [1] They are still a long way from producing a fully worked out theory of this kind, but the unification of Kaluza-Klein theories with supergravity was a key development in search for SUSY — although its true significance is only apparent in hindsight.