What about the nucleus, the positively charged “sun” at the center of the tiny atomic solar system? How might the proton and neutron have been deduced? The proton would not have been too difficult. Dalton in 1808 had made the first step. The mass of any atom is an integer multiple of a certain numerical value. That certainly suggests a discrete collection of basic constituents in the nucleus. Moreover, because the electric charge of a nucleus is generally smaller than the atomic number, the constituents cannot all have the same charge. The simplest possibility by far would have to be a single type of positively charged particle and a single neutral particle with practically identical masses. Smart theorists would have figured this out in no time.
Or would they? One thing might have led them astray, for how long I don’t know. There was a possibility even simpler than the neutron—a possibility that required no new particle. The nucleus might be understood as a number of protons stuck together with a smaller number of electrons. For example, a carbon nucleus with six protons and six neutrons might have been mistaken for six electrons stuck to twelve protons. In fact the mass of a neutron is close to the combined mass of a proton and an electron. Of course a new type of force would have to be introduced: the ordinary electrostatic force between electron and proton would not have been nearly strong enough to tightly bind the extra electrons to the protons—and with a new force, a new messenger particle. Perhaps in the end they would have decided the neutron was not such a bad idea.
Meanwhile, Einstein had developed his theory of gravity, and curious physicists were exploring its equations. Here again, we don’t need to guess. Karl Schwarzschild, even before Einstein had completed his theory, worked out the solution of Einstein’s equations that we now call the Schwarzschild black hole. Einstein himself derived the existence of gravitational waves that eventually led to the graviton idea. That most certainly required no experiment or observation. The consequences of the General Theory of Relativity were worked out without appeal to any empirical proof that the theory was correct. Even the modern theory of black holes, which we will encounter in the tenth chapter of this book, only involved the Schwarzschild solution combined with primitive ideas of quantum field theory.
Could theorists have guessed the full structure of the Standard Model? Protons and neutrons, perhaps, but quarks, neutrinos, muons, and all the rest? I don’t see any way that these things could have been guessed. But the basic underlying theoretical foundation—Yang Mills theory? Here I think I am on very firm ground. The experiment has been done, and the data are in. In 1953, with no other motivation than generalizing Kaluza’s theory of an extra dimension, one of history’s greatest theoretical physicists did invent the mathematical theory that today is called non-abelian gauge theory. Remember that Kaluza had added an extra dimension to the three dimensions of space and, in so doing, gave a unified description of gravity and electrodynamics. What Pauli did was to add one more dimension for a total of 5+1. The two extra dimensions he rolled up into a tiny 2-sphere. And what did he find? He found that the extra two dimensions gave rise to a new kind of theory, similar to electrodynamics but with a new twist. Instead of a single photon, the list of particles now had three photonlike particles. And, curiously, each photon carried charge; it could emit either of the other two. This was the first construction of a non-abelian, or Yang Mills, gauge theory.6 Today we recognize non-abelian gauge theory as the basis for the entire Standard Model. Gluons, photons, Z-particles, and W-particles are simple generalizations of Pauli’s three photonlike particles.
As I said, there was little or no chance that theorists would have been able to deduce the Standard Model with its quarks, neutrinos, muons, and Higgs bosons. And even if they had, it most likely would have been one of dozens of ideas. But I do think there is a possibility they could have found the basic theoretical ingredients.
Could they possibly have discovered String Theory? The discovery of String Theory is a good example of how the searching, probing minds of theorists often work. Again with absolutely no experimental basis, string theorists constructed a monumental mathematical edifice. The historical development of String Theory was somewhat accidental. But it easily could have arisen through other kinds of accidents. Stringlike objects play an important role in non-abelian gauge theories. Another plausible possibility is that it might have been developed through hydrodynamics, the theory of fluid flow. Think of the swirling vortex that forms when you let water drain from the sink. The actual center of the vortex forms a long, one-dimensional core that in many ways behaves like a string. Such vortices can form in air: tornadoes are an example. Smoke rings provide a more interesting example, vortex loops that resemble closed strings. Might fluid dynamics experts attempting to construct an idealized theory of vortices have invented String Theory? We will never know, but it doesn’t seem out of the question. Would physicists trying to explore the quantum theory of gravity have seized on it when the fluid people found closed strings that behaved like gravitons? I think they would have.
On the other hand, a skeptic could reasonably argue that for every good idea there would have been a hundred irrelevant, wrongheaded directions pursued. With no experiments to guide and discipline theorists, they would have gone off in every imaginable direction, with intellectual chaos ensuing. How would the good ideas ever be distinguished from the bad? Having every possible idea is just as bad as having no ideas.
The skeptics have a good point; they may be right. But it is also possible that good ideas have a kind of Darwinian survival value that bad ideas don’t. Good ideas tend to produce more good ideas—bad ones tend to lead nowhere. And mathematical consistency is a very unforgiving criterion. Perhaps it would have provided some of the discipline that would otherwise have come from experiment.
In a century without experiment, would physics have progressed the way I have suggested? Who knows? I don’t say it would have—only that it could have. In trying to gauge the limits of human ingenuity, I am certain that we are much more likely to underestimate where the limits lie than to overestimate.
In looking back I realize that in 1995 I was guilty of a very serious lack of imagination in speaking only of the ingenuity of theorists. In trying to console myself and the other physicists at the banquet about the poor prospects for future experimental data, I badly underestimated the ingenuity, imagination, and creativity of experimental physicists. Since that time they have gone on to create the revolutionary explosion of cosmological data that I described in chapter 5. In the last chapter of this book, I will discuss some other exciting experiments that will take place in the near future, but for now let’s return to String Theory and how it produces a huge Landscape of possibilities.
CHAPTER TEN
The Branes behind Rube Goldberg’s Greatest Machine
We come now to the heart of the matter. The unreasonable apparent design of the universe and the appeal to some form of Anthropic Principle is old stuff. What is really new, the earthquake that has caused enormous consternation and controversy among theoretical physicists and the reason that I wrote this book, is the recognition that the Landscape of String Theory has a stupendous number of diverse valleys. Earlier theories like QED (the theory of photons and electrons) and QCD (the theory of quarks and gluons) that had prevailed throughout the twentieth century had very boring Landscapes. The Standard Model, as complicated as it is, has only a single vacuum. No choices ever have to be made, or ever can be made, about which vacuum we actually live in.
The reason for the paucity of vacuums in older theories is not hard to understand. It was not that quantum field theories with rich Landscapes are mathematically impossible. By adding to the Standard Model a few hundred unobserved fields similar to the Higgs field, a huge Landscape can be generated. The reason that the vacuum of the Standard Model is unique is not any remarkable mathematical elegance of the kind that I explained in chapter 4. It has much more to do with the fact that it was constructed for the particular purpose of describing some limited facts
about our own world. They were built piecemeal, from experimental data, with the particular goal of describing (not explaining) our own vacuum. These theories admirably do the job that they were designed to do but no more. With this limited goal in mind, theorists had no reason to add loads of additional structure just to make a Landscape. In fact most physicists (with the exception of farsighted visionaries like Andrei Linde and Alex Vilenkin) throughout the twentieth century would have considered a diverse Landscape to be a blemish rather than an advantage.
Until recently string theorists were blinded by this old paradigm of a theory with a single vacuum. Despite the fact that at least a million different Calabi Yau manifolds could be utilized for compactifying (rolling up and hiding) the extra dimensions implied by String Theory, the leaders of the field continued to hope that some mathematical principle would be discovered that would eliminate all but a single possibility. But with all the effort that was spent on searching for such a vacuum selection principle, nothing ever turned up. They say that “hope springs eternal.” But by now most string theorists have realized that, although the theory may be correct, their aspirations were incorrect. The theory itself is demanding to be seen as a theory of diversity, not uniqueness.
What is it about String Theory that makes its Landscape so rich and diverse? The answer involves the enormous complexity of the tiny, rolled-up geometries that hide the extra six or seven dimensions of space. But before we get to this complexity, I want to explain a simpler and more familiar example of similar complexity. In fact this example was the original inspiration for the term Landscape.
The term Landscape did not originate with string theorists or cosmologists. When I first used it in 2003 to describe the large number of String Theory vacuums, I was borrowing it from a much older field of science: the physics and chemistry of large molecules. The possible configurations of a large molecule, made of hundreds or thousands of atoms, had long been described as landscapes or, sometimes, energy landscapes. The Landscape of String Theory has much less in common with the impoverished landscapes of quantum field theory than with the “configuration space” of large molecules. Let’s pursue this point before returning to the exploration of String Theory.
Begin with a single atom. Three numbers are required in order to specify the location of the atom: the coordinates of the atom along the x-, y-, and z-axes. If you don’t like x, y, and z, you may use longitude, latitude, and altitude instead. Thus, the possible configurations of a single atom are the points of ordinary three-dimensional space.
The next-simplest system made of atoms is a diatomic molecule—a molecule composed of two atoms. Specifying the position of two atoms requires six coordinates: three for each atom. It would be natural to call the six coordinates x1, y1, z1 and x2, y2, z2, the subscripts 1 and 2 referring to the two atoms. These six numbers describe two points of three-dimensional space, but we can also combine the six coordinates to form an abstract, six-dimensional space. That six-dimensional space is the landscape describing a diatomic molecule.
Now let’s jump to a molecule composed of one thousand atoms. For inorganic chemistry this would be a very large molecule, but for an organic biomolecule, it is fairly ordinary. How do we describe all the ways that the one thousand atoms can arrange themselves? This question is not entirely academic: biochemists and biophysicists who want to understand how protein molecules fold and unfold themselves think in terms of a molecular landscape.
Evidently, to specify the configuration of all one thousand atoms, we need to give three thousand numbers, which we can think of as the coordinates of a three-thousand-dimensional landscape: a landscape of possible molecular “designs.”
The collection of atoms has potential energy that varies as the atoms are moved around. For example, in the case of the diatomic molecule, if the two atoms are squeezed together, the potential energy becomes large. If the atoms move apart, they will eventually reach a point of minimum energy. Of course it is much more difficult to visualize the energy of one thousand atoms, but the principle is the same: the potential energy of the molecule varies as we move across the landscape. As in chapter 3, if we think of potential energy as altitude, the landscape will have a rich topography with mountains, valleys, ridges, and plains. It shouldn’t come as a surprise that the stable configurations of the molecule correspond to the bottoms of valleys.
The striking thing is that the number of these valleys is enormous: it grows exponentially with the number of atoms. For a large molecule the number of isolated valleys is way beyond millions or billions. The landscape of a molecule with one thousand atoms can easily have 10100 valleys. What does all of this have to do with the Landscape of vacuums and String Theory? The answer is that, like a molecule, a compactification of String Theory has a great many “moving parts.” Some of those parts we have already met. The compactification moduli were the quantities that determine the sizes and shapes of the various geometric features of Calabi Yau manifolds. In this chapter we are going to explore some additional moving parts and see why the Landscape is so complex and extraordinarily rich.
D-Branes
In chapter 8 I described how Ed Witten’s 1995 idea combined the multitude of String Theories into one big M(aster)-theory. But that theory had one serious problem: it needed new objects, objects that String Theory had not previously predicted. The theory would have to work something like this: each one of the String Theories must contain previously unsuspected objects deeply hidden in its mathematics. The fundamental strings of one version were not the same objects as the fundamental strings of another version. But as the moduli varied—as one moved through the Landscape—the new objects of version A would morph into the old objects of version B. One example that we have already seen is how the membranes of M-theory morph into the strings of Type IIa theory. Witten’s ideas were attractive—even compelling—but the nature of the new objects and their mathematical place in the theory was a complete mystery. That is, until Joe Polchinski discovered his branes.
Joe Polchinski has the good looks and sunny disposition of “the boy next door.” Commenting about food, Joe once said, “There are only two kinds of food—the kind you put chocolate sauce on and the kind you put ketchup on.” But the boyish exterior hides one of the deepest and most powerful minds to attack the problems of physics in the last half century. Even before Witten introduced his M-theory, Joe had been experimenting with a new idea in String Theory. More or less as a mathematical game, he postulated that there could be special places in space where strings could terminate. Picture a child holding the end of a jump rope and shaking it to make waves. The waves travel down to the far end of the rope, but what happens next depends on whether the far end is free to flop around or is attached to some anchor. Before Polchinski, open strings always had free ends—the floppy option—but Joe’s new idea was that there could be anchors in space that held string ends from flopping. The anchor could be a simple point in space: that would be more or less like a hand rigidly constraining the end from moving. But there are other possibilities. Suppose the end of the rope were attached to a ring that could slide up and down a pole. The end would be partly fixed but partly free to move. Although attached to the pole, the end would be free to slide along a line—the pole itself. What ropes with poles can do, so can strings, or so Polchinski reasoned. Why not have special lines in space to which string ends can attach? Like the rope and pole, the string end would be free to slide along the length of the line. The line might even be bent into a curve. But points and lines don’t exhaust the possibilities. The string end could be attached to a surface, a kind of membrane. Free to slide in any direction along the surface, the string end could not escape from the membrane.
These points, lines, and surfaces where strings could end needed a name. Joe called them Dirichlet-branes or just D-branes. Peter Dirichlet was a nineteenth-century German mathematician who had nothing whatever to do with String Theory. But 150 years earlier he had studied the mathematics of waves and how t
hey reflected off fixed objects. By all rights the new objects should be called Polchinski-branes, but the term P-branes was already in use by string theorists for another kind of object.
Joe is a good friend of mine. Over a period of twenty-five years we had worked closely together on a number of physics projects. The first I heard of D-branes was over coffee in Quackenbush’s Intergalactic Café and Espresso bar in Austin, Texas. I think the year was 1994. The idea seemed amusing but not the stuff of revolutions. I wasn’t alone in underestimating their importance. D-branes were not high on the to-do list of anyone at that time—maybe not even Joe’s list. It wasn’t until shortly after Witten’s 1995 lecture that D-branes exploded into the consciousness of theoretical physicists.