The God Equation

by Michio Kaku

  Physicist Edward Witten found that there was actually a hidden eleven-dimensional theory, called M-theory, that was based on membranes (like the surfaces of spheres and doughnuts) rather than just strings. He was able to explain why there were five different string theories, because there were five ways in which to collapse an eleven-dimensional membrane to a ten-dimensional string.

  In other words, all five versions of string theory were different mathematical representations of the same M-theory. (So string theory and M-theory are really the same theory, except that string theory is a reduction of eleven-dimensional M-theory to ten dimensions.) But how can a single eleven-dimensional theory give rise to five ten-dimensional theories?

  For example, think of a beach ball. If we let the air out, the ball collapses, gradually resembling a sausage. If we let even more air out, the sausage becomes a string. Hence, a string is actually a membrane in disguise, such that its air has been let out.

  If we start with a eleven-dimensional beach ball, you can show mathematically that there are five ways in which it can be collapsed to a ten-dimensional string.

  Or think of the tale of the blind men who encounter an elephant for the first time. One wise man, touching the ear of the elephant, declares the elephant is flat and two-dimensional like a fan. Another wise man touches the tail and assumes the elephant is like rope or a one-dimensional string. Another, touching a leg, concludes the elephant is a three-dimensional drum or a cylinder. But actually, if we step back and rise into the third dimension, we can see the elephant as a three-dimensional animal. In the same way, the five different string theories are like the ear, tail, and leg, but we still have yet to reveal the full elephant, M-theory.

  Holographic Universe

  As we mentioned, with time new layers have been uncovered in string theory. Soon after M-theory was proposed in 1995, another astonishing discovery was made by Juan Maldacena in 1997.

  He jolted the entire physics community by showing something that was once considered impossible: that a supersymmetric Yang-Mills theory, which describes the behavior of subatomic particles in four dimensions, was dual, or mathematically equivalent, to a certain string theory in ten dimensions. This sent the physics world into a tizzy. By 2015, there were ten thousand papers that referred to this paper, making it by far the most influential paper in high-energy physics. (Symmetry and duality are related but different. Symmetry arises when we rearrange the components of a single equation and it remains the same. Duality arises when we show that two entirely different theories are actually mathematically equivalent. Remarkably, string theory has both of these highly nontrivial features.)

  As we saw, Maxwell’s equations have a duality between electric and magnetic fields—that is, the equations remain the same if we reverse the two fields, turning electric fields into magnetic fields. (We can see this mathematically, because the EM equations often contain terms like E2 + B2, which remain the same when we rotate the two fields into each other, like in the Pythagorean theorem). Similarly, there are five distinct string theories in ten dimensions, which can be proven to be dual to each other, so they are really a single eleven-dimensional M-theory in disguise. So remarkably, duality shows that two different theories are actually two aspects of the same theory.

  Maldacena, however, showed that there was yet another duality between strings in ten dimensions and Yang-Mills theory in four dimensions. This was a totally unexpected development but one that has profound implications. It meant that there were deep, unexpected connections between the gravitational force and the nuclear force defined in totally different dimensions.

  Usually, dualities can be found between strings in the same dimension. By rearranging the terms describing those strings, for example, we can often change one string theory into another. This creates a web of dualities between different string theories, all defined in the same dimension. But a duality between two objects defined in different dimensions was unheard of.

  This is not an academic question, because it has far-reaching implications for understanding the nuclear force. For example, earlier we saw how gauge theory in four dimensions, as represented by the Yang-Mills field, gives us the best description of the nuclear force, but no one has ever been able to find an exact solution to the Yang-Mills field. But since gauge theory in four dimensions could be dual to string theory in ten dimensions, it meant that quantum gravity might hold the key to the nuclear force. This was an astonishing revelation, because it meant that basic features of the nuclear force (such as calculating the mass of the proton) might be best described by string theory.

  This created a bit of an identity crisis among physicists. Those who work exclusively on the nuclear force spend all their time studying three-dimensional objects, such as protons and neutrons, and often scoff at physicists theorizing in higher dimensions. But with this new duality between gravity and gauge theory, suddenly these physicists found themselves trying to learn all about ten-dimensional string theory, which might hold the key to understanding the nuclear force in four dimensions.

  Yet another unexpected development emerged from this bizarre duality, called the holographic principle. Holograms are two-dimensional flat sheets of plastic, containing the image of three-dimensional objects that have been specially encoded within them. By shining a laser beam at the flat screen, the three-dimensional image suddenly emerges. In other words, all the information needed to create a three-dimensional image has been encoded onto a flat two-dimensional screen using lasers, like the image of Princess Leia projected by R2-D2 or the haunted mansion at Disneyland where three-dimensional ghosts sail around us.

  This principle also works for black holes. As we saw earlier, if we throw an encyclopedia into a black hole, the information contained inside the books cannot disappear, according to quantum mechanics. So where does the information go? One theory posits that it is distributed onto the surface of the event horizon of the black hole. So the two-dimensional surface of a black hole contains all the information of all the three-dimensional objects that have been thrown into it.

  This also has implications for our conception of reality. We are convinced, of course, that we are three-dimensional objects that can move in space, defined by three numbers, length, width, and height. But perhaps this is an illusion. Perhaps we are living in a hologram.

  Perhaps the three-dimensional world we experience is just a shadow of the real world, which is actually ten- or eleven-dimensional. When we move in the three dimensions of space, we experience our real selves actually moving in ten or eleven dimensions. When we walk down the street, our shadow follows us and moves like us, except the shadow exists in two dimensions. Likewise, perhaps we are shadows moving in three dimensions, but our real selves are moving in ten or eleven dimensions.

  In summary, we see that with time, string theory reveals new, totally unexpected results. It means that we still do not really understand the basic fundamental principles behind it. Eventually, it may turn out that string theory is not really a theory about strings after all, since strings can be expressed as membranes when formulated in eleven dimensions.

  That is why it is premature to compare string theory with experiment. Once we have revealed the true principles behind string theory, we may find a way to test it, and maybe then we can say once and for all if it is a theory of everything or a theory of nothing.

  Testing the Theory

  But despite all the theoretical successes of string theory, it still has glaring weak spots. Any theory that makes claims as powerful as the ones made for string theory is naturally going to attract an army of detractors. One has to be continually reminded of the words of Carl Sagan, who said that “remarkable claims require remarkable proof.”

  (I am also reminded of the cynical words of Wolfgang Pauli, who was a master of the put-down. When listening to a talk, he might say, “What you said was so confused that one could not tel
l whether it was nonsense or not.” He would also say, “I do not mind if you think slowly, but I do object when you publish more quickly than you think.” If he were alive, he might apply these words to string theory.)

  The debate is so intense that the best minds in physics have split on this question. Not since the great sixth Solvay Conference of 1930, when Einstein and Bohr sparred with each other on the question of the quantum theory, has science witnessed such a grand schism.

  Nobel laureates have taken opposite positions on this question. Sheldon Glashow has written, “Years of intense effort by dozens of the best and the brightest have yielded not one verifiable prediction, nor should any soon be expected.” Gerard ’t Hooft went so far as to say that the interest surrounding string theory is comparable to “American television commercials”—that is, all hype and fanfare, but no substance.

  Others have praised the virtues of string theory. David Gross has written, “Einstein would have been pleased with this, at least with the goal, if not the realization….He would have liked the fact that there is an underlying geometrical principle—which, unfortunately, we don’t really yet understand.”

  Steven Weinberg has compared string theory to the historic effort to find the north pole. All ancient maps of the Earth had a huge, gaping hole, where the north pole should be, but no one had ever seen it. Anywhere on the Earth, all compass needles pointed to this mythical place. But all attempts to find the fabled north pole ended in failure. In their hearts, the ancient mariners knew that there must be a north pole, but no one could prove it. Some even doubted that it existed. However, after centuries of speculation, finally in 1909 Robert Peary actually set foot on the north pole.

  String theory critic Glashow has admitted that he is outnumbered in this debate. He once commented, “I find myself a dinosaur in a world of upstart mammals.”

  Criticisms of String Theory

  There are several main criticisms that have been leveled at string theory. The critics have claimed that the theory is all hype; that beauty by itself is an unreliable guide in physics; that it predicts too many universes; and, most important, that it is untestable.

  The great astronomer Kepler was once misled by the power of beauty. He was enamored of the fact that the solar system resembled a collection of regular polyhedrons stacked inside one another. Centuries earlier, the Greeks had enumerated five of these polyhedrons (e.g., the cube, pyramid, etc.). Kepler noticed that by sequentially putting these polyhedrons inside one another, like Russian dolls, one could reproduce some of the details of the solar system. It was a beautiful idea, but turned out to be totally wrong.

  Recently, some physicists have criticized string theory, stating that beauty is a misleading criterion for physics. Just because string theory has brilliant mathematical properties does not mean it holds a kernel of truth. They rightly point out that beautiful theories have sometimes been dead ends.

  But poets often quote the poem “Ode on a Grecian Urn” by John Keats:

  Beauty is truth, truth beauty,—that is all

  Ye know on earth, and all ye need to know.

  Paul Dirac was certainly a follower of this principle when he wrote, “The research worker, in his efforts to express the fundamental laws of Nature in mathematical form, should strive mainly for mathematical beauty.” In fact, he would write that he discovered his celebrated theory of the electron by fiddling with pure mathematical formulas rather than looking at the data.

  As powerful as beauty is in physics, certainly beauty can often lead you astray. As physicist Sabine Hossenfelder has written, “Beautiful theories have been ruled out in the hundreds, theories about unified forces and new particles and additional symmetries and other universes. All these theories were wrong, wrong, wrong. Relying on beauty is clearly not a successful strategy.”

  The critics claim string theory has beautiful mathematics, but this may have nothing to do with physical reality.

  There is some validity to this criticism, but one has to realize that aspects of string theory like supersymmetry are not useless and devoid of physical applications. Although evidence for supersymmetry has not yet been found, it has proven to be essential in eliminating many of the defects within the quantum theory. Supersymmetry, by canceling bosons against fermions, enables us to solve a long-standing problem, eliminating the divergences that plague quantum gravity.

  Not every beautiful theory has a physical application, but all fundamental physical theories found so far, without exception, have a type of beauty or symmetry built into them.

  Can It Be Tested?

  The foremost criticism of string theory is that it is untestable. The energy that gravitons possess is called the Planck energy, which is a quadrillion times greater than the energy produced by the LHC. Imagine trying to build a LHC that is a quadrillion times larger than the current one! One would probably need a particle accelerator the size of the galaxy for a direct test of the theory.

  Furthermore, each solution of string theory is an entire universe. And there seems to be an infinite number of solutions. For a direct test of the theory, one would need to create baby universes in the laboratory! In other words, only a god can truly test the theory directly, since the theory is based on universes, not just atoms or molecules.

  So at first, it seems that string theory fails the acid test for any theory, testability. But promoters of string theory are not fazed. As we have established, most science is done indirectly, by examining echoes from the sun, the Big Bang, etc.

  Similarly, we look for echoes from the tenth and eleventh dimension. Perhaps evidence for string theory is hidden all around us, but we have to listen for its echoes, rather than try to observe it directly.

  For example, one possible signal from hyperspace is the existence of dark matter. Until recently, it was widely believed that the universe is mainly made of atoms. Astronomers have been shocked to find that only 4.9 percent of the universe is made of atoms like hydrogen and helium. Actually, most of the universe is hidden from us, in the form of dark matter and dark energy. (We recall that dark matter and dark energy are two distinct things. Twenty-six point eight percent of the universe is made of dark matter, which is invisible matter that surrounds the galaxies and keep them from flying apart. And 68.3 percent of the universe is made of dark energy, which is even more mysterious, the energy of empty space that is driving the galaxies apart.) Perhaps evidence for the theory of everything lies hidden in this invisible universe.

  Search for Dark Matter

  Dark matter is strange, it is invisible, yet it holds the Milky Way galaxy together. But since it has weight and no charge, if you tried to hold dark matter in your hand it would sift through your fingers as if they weren’t there. It would fall right through the floor, through the core of the Earth, and then to the other side of the Earth, where gravity would eventually cause it to reverse course and fall back to your location. It would then oscillate between you and the other side of the planet, as if the Earth weren’t there.

  As strange as dark matter is, we know it must exist. If we analyze the spin of the Milky Way galaxy and use Newton’s laws, we find that there is not enough mass to counteract the centrifugal force. Given the amount of mass we see, the galaxies in the universe should be unstable and they should fly apart, but they have been stable for billions of years. So we have two choices: either Newton’s equations are incorrect when applied to galaxies, or else there is an unseen object that is keeping the galaxies intact. (We recall that the planet Neptune was found in the same way, by postulating a new planet that explained Uranus’s deviations from a perfect ellipse.)

  At present, one leading candidate for dark matter is called the weakly interacting massive particles (WIMPs). Among them, one likely possibility is the photino, the supersymmetric partner of the photon. The photino is stable, has mass, is invisible, and has no charge, which fits
precisely the characteristics of dark matter. Physicists believe the Earth moves in an invisible wind of dark matter that is probably passing through your body right now. If a photino collides with a proton, it may cause the proton to shatter into a shower of subatomic particles that can then be detected. In fact, even today there are huge swimming pool–sized detectors (with vast amounts of fluids containing xenon and argon) that may one day capture the spark created by a photino collision. There are about twenty active groups searching for dark matter, often deep inside mine shafts below the Earth’s surface, away from interfering cosmic ray interactions. So it is conceivable that the collision of dark matter may be captured by our instruments. Once dark matter collisions have been detected, then physicists will study the properties of dark matter particles and then compare them to the predicted properties of photinos. If the predictions of string theory match the experimental results on dark matter, this would go a long way toward convincing physicists that this is the correct path.

  Another possibility is that the photino may be produced by the next generation of particle accelerators being discussed.

  Beyond the LHC

  The Japanese are considering funding the International Linear Collider, which would shoot a beam of electrons down a straight tube, until it strikes a beam of anti-electrons. If approved, the device would be built in twelve years. The advantage of a collider like this is that it uses electrons rather than protons. Because protons consist of three quarks held together by gluons, the collision between protons is very messy, with an avalanche of extraneous particles being created. The electron, by contrast, is a single elementary particle, so the collision with an anti-electron is much cleaner and requires much less energy. As a result, at only 250 billion electron volts, it should be able to create Higgs bosons.