by Michio Kaku
Although research in general relativity stagnated for decades, the recent application of the quantum to relativity has opened up new unexpected vistas, especially as powerful new instruments go online. There has been a blossoming of new research.
But so far, we have only discussed applying quantum mechanics to the matter that moves within the gravity fields of Einstein’s theory. We have not discussed a much more difficult question: applying quantum mechanics to gravity itself in the form of gravitons.
And this is where we encounter the biggest question of all: finding a quantum theory of gravity, which has frustrated the world’s great physicists for decades. So let us review what we have learned so far. We recall that when we apply the quantum theory to light, we introduce the photon, a particle of light. As this photon moves, it is surrounded by electric and magnetic fields that oscillate and permeate space and obey Maxwell’s equations. This is the reason why light has both particle-like and wavelike properties. The power of Maxwell’s equations lies in their symmetries—that is, the ability to turn electric and magnetic fields into each other.
When the photon bumps into electrons, the equation that describes this interaction yields results that are infinite. However, using the bag of tricks devised by Feynman, Schwinger, Tomonaga, and many others, we are able to hide all the infinities. The resulting theory is called QED. Next, we applied this method to the nuclear force. We replaced the original Maxwell field with the Yang-Mills field, and replaced the electron with a series of quarks, neutrinos, etc. Then we introduced a new bag of tricks devised by ’t Hooft and his colleagues to eliminate all the infinities once again.
So three of the four forces of the universe could now be unified into a single theory, the Standard Model. The resulting theory was not very pretty, since it was created by cobbling together the symmetries of the strong, weak, and electromagnetic forces, but it worked. But when we apply this tried-and-true method to gravity, we have problems.
In theory, a particle of gravity should be called the graviton. Similar to the photon, it is a point particle, and as it moves at the speed of light, it is surrounded by waves of gravity that obey Einstein’s equations.
So far, so good. The problem occurs when the graviton bumps into other gravitons and also atoms. The resulting collision creates infinite answers. When one tries to apply the bag of tricks painfully formulated over the last seventy years, we find that they all fail. The greatest minds of the century have tried to solve this problem, but no one has been successful.
Clearly, an entirely new approach must be used, since all the easy ideas have been investigated and discarded. We need something truly fresh and original. And that leads us to perhaps the most controversial theory in physics, string theory, which might just be crazy enough to be the theory of everything.
6
RISE OF STRING THEORY: PROMISE AND PROBLEMS
We saw earlier that around 1900, there were two great pillars of physics: Newton’s law of gravity and Maxwell’s equations for light. Einstein realized that these two great pillars were in conflict with each other. One of them would have to collapse. The fall of Newtonian mechanics set into motion the great scientific revolutions of the twentieth century.
Today, history may be repeating itself. Once again we have two great pillars of physics. On one hand, we have the theory of the very big, Einstein’s theory of gravity, which gives us black holes, the Big Bang, and the expanding universe. On the other hand, we have the theory of the very small, the quantum theory, which explains the behavior of subatomic particles. The problem is that they stand in conflict with each other. They are based on two different principles, two different mathematics, and two different philosophies.
The next great revolution, we hope, will be to unify these two pillars into one.
String Theory
It all began in 1968, when two young physicists, Gabriele Veneziano and Mahiko Suzuki, were thumbing through math books and stumbled across a strange formula found by mathematician Leonhard Euler in the eighteenth century. This strange formula seemed to describe the scattering of two subatomic particles! How could an abstract formula from the eighteenth century describe the latest results from our atom smashers? Physics was not supposed to work this way.
Later, physicists, including Yoichiro Nambu, Holger Nielsen, and Leonard Susskind, realized that the properties of this formula represented the interaction of two strings. Very quickly, this formula was generalized to a whole army of equations, representing the scattering of multistrings. (This was, in fact, my Ph.D. thesis, calculating the complete set of interactions of an arbitrary number of strings.) Then researchers were able to introduce spinning particles into string theory.
String theory was like an oil well suddenly gushing forth a torrent of new equations. (Personally, I was not satisfied with this, because, ever since Faraday, physics had been represented by fields that concisely summarized vast amounts of information. String theory, by contrast, was a collection of disjointed equations. My colleague Keiji Kikkawa and I were then successful in writing all of string theory in the language of fields, creating what is called string field theory. All of string theory can be summarized by our equations in a field theory equation just one inch long.)
As a result of the torrent of equations, a new picture was beginning to emerge. Why were there so many particles? Like Pythagoras more than two thousand years ago, the theory said that each musical note—each vibration of a string—represented a particle. Electrons, quarks, and Yang-Mills particles were nothing but different notes on the same vibrating string.
What is so powerful and interesting about the theory is that gravity is necessarily included. Without any extra assumptions, the graviton emerges as one of the lowest vibrations of the string. In fact, even if Einstein had never been born, his entire theory of gravity might have been found simply by looking at the lowest vibration of the string.
As physicist Edward Witten once said, “String theory is extremely attractive because gravity is forced upon us. All known consistent string theories include gravity, so while gravity is impossible in quantum field theory as we have known it, it’s obligatory in string theory.”
Ten Dimensions
But as the theory began to evolve, more and more fantastic, totally unexpected features began to be revealed. For example, it was found that the theory can only exist in ten dimensions!
This shocked physicists, because no one had ever seen anything like it. Usually, any theory can be expressed in any dimension you like. We simply discard these other theories because we obviously live in a three-dimensional world. (We can only move forward, sideways, and up and down. If we add time, then it takes four dimensions to locate any event in the universe. If we want to meet someone in Manhattan, for example, we might say, Let’s meet at the corner of 5th Avenue and 42nd Street, on the tenth floor, at noon. However, moving in dimensions beyond four is impossible for us, no matter how we try. In fact, our brains cannot even visualize how to move in higher dimensions. Therefore all the research done in higher-dimensional string theory is done using pure mathematics.)
But in string theory, the dimensionality of space-time is fixed at ten dimensions. The theory breaks down mathematically in other dimensions.
I still remember the shock that physicists felt when string theory posited that we live in a universe of ten dimensions. Most physicists saw this as proof that the theory was wrong. When John Schwarz, one of the leading architects of string theory, was in the elevator at Caltech, Richard Feynman would prod him, asking, “Well, John, and how many dimensions are you in today?”
Yet over the years, physicists gradually began to show that all rival theories suffered from fatal flaws. For example, many could be ruled out because their quantum corrections were infinite or anomalous (that is, mathematically inconsistent).
So over time, physicists began to warm up to the ide
a that perhaps our universe might be ten-dimensional after all. Finally, in 1984, John Schwarz and Michael Green showed that string theory was free of all the problems that had doomed previous candidates for a unified field theory.
If string theory is correct, then the universe might have originally been ten-dimensional. But the universe was unstable and six of these dimensions somehow curled up and became too small to be observed. Hence, our universe might actually be ten-dimensional, but our atoms are too big to enter these tiny higher dimensions.
The Graviton
In spite of all the craziness of string theory, one thing that has kept it alive is that it successfully marries the two great theories of physics, general relativity and the quantum theory, giving us a finite theory of quantum gravity. That is what all the excitement is about.
Previously, we mentioned that if you add quantum corrections to QED, or the Yang-Mills particle, you get a flood of infinities that must be carefully and tediously removed.
But all this fails when we try to have a shotgun wedding between the two great theories of nature, relativity and the quantum theory. When we apply the quantum principle to gravity, we have to break it up into packets of energy, or quanta, called the graviton. Then we calculate the collision of these gravitons with other gravitons and with matter, like the electron. But when we do this, the entire bag of tricks found by Feynman and ’t Hooft fail miserably. The quantum corrections caused by gravitons interacting with other gravitons are infinite and defy all the methods found by previous generations of physicists.
This is where the next magic occurs. String theory can remove these troublesome infinites that have dogged physicists for almost a century. And this magic once again occurs through symmetry.
Supersymmetry
Historically, it was always considered nice to have our equations symmetrical, but it was a luxury that was not strictly necessary. But in the quantum theory, symmetry becomes the most important feature of the physics.
As we’ve established, when we calculate the quantum corrections to a theory, these quantum corrections are often divergent (that is, infinite), or anomalous (meaning that it violates the original symmetry of the theory). Physicists have realized only in the last few decades that symmetry, instead of being just a pleasing feature of a theory, is actually the central ingredient. Demanding a theory be symmetrical can often banish the divergences and anomalies that plague nonsymmetrical theories. Symmetry is the sword physicists use to vanquish the dragons unleashed by quantum corrections.
Figure 11. When calculating the collision of two gravitons (top), the answer is infinite and hence meaningless. But when two strings collide (bottom), we have two terms, one from the bosons and one from the fermions. In string theory, these two terms cancel exactly, helping to create a finite theory of quantum gravity.
As we mentioned earlier, Dirac found that his equation for the electron predicted that it had spin (which is a mathematical feature of the equations that resembles the familiar spin we see all around us). Later, physicists found that all subatomic particles have spin. But spin comes in two types.
In certain quantum units, the spin can be either integral (like 0, 1, or 2) or half-integral (like ½, 3/2). First, the particles that have integral spin describe the forces of the universe. They include the photon and Yang-Mills particle (with spin 1) and the particle of gravity, the graviton (with spin 2). These particles are named bosons (after the Indian physicist Satyendra Nath Bose). So the forces of nature are mediated by bosons.
Then there are particles that make up the matter in the universe. They have half-integral spin, such as electrons, neutrinos, and quarks (with spin ½). These particles are called fermions (after Enrico Fermi), out of which we can build up the other particles of the atom: protons and neutrons. So the atoms of our body are collections of fermions.
Two Types of Subatomic Particles
Fermions (matter) Bosons (forces)
electron, quark, photon, graviton,
neutrino, proton Yang-Mills
Bunji Sakita and Jean-Loup Gervais then demonstrated that string theory had a new type of symmetry, called supersymmetry. Since then, supersymmetry has been expanded so that it is now the largest symmetry ever found in physics. As we have emphasized, beauty to a physicist is symmetry, which allows us to find the link between different particles. All the particles of the universe could then be unified by supersymmetry. As we have emphasized, a symmetry rearranges the components of an object, leaving the original object the same. Here, one is rearranging the particles in our equations so that fermions are interchanged with bosons and vice versa. This becomes the central feature of string theory, so that the particles of the entire universe can be rearranged into one another.
This means that each particle has a super partner, called a sparticle, or super particle. For example, the super partner of the electron is called the selectron. The super partner of the quark is called the squark. The superpartner of the lepton (like the electron or neutrino) is called the slepton.
But in string theory, something remarkable happens. When calculating quantum corrections to string theory, you have two separate contributions. You have quantum corrections coming from fermions and also bosons. Miraculously, they are equal in size, but occur with the opposite sign. One term might have a positive sign, but there is another term that is negative. In fact, when they are added together, these terms cancel against each other, leaving a finite result.
The marriage between relativity and the quantum theory has dogged physicists for almost a century, but the symmetry between fermions and bosons, called supersymmetry, allows us to cancel many of these infinities against each other. Soon, physicists discovered other means of eliminating these infinities, leaving a finite result. So this is the origin of all the excitement surrounding string theory: it can unify gravity with the quantum theory. No other theory can make this claim. This may satisfy Dirac’s original objection. He hated renormalization theory because, in spite of its fantastic and undeniable successes, it involved adding and subtracting quantities that were infinite in size. Here, we see that string theory is finite all by itself, without renormalization.
This, in turn, may satisfy the picture originally proposed by Einstein himself. He once compared his theory of gravity to marble, which is smooth, elegant, polished. However, matter, by contrast, was more like wood. The trunk of a tree is gnarled, chaotic, rough, without a regular geometric pattern. His goal was to ultimately create a unified theory that combined the marble and the wood into a single form—that is, to create a theory entirely made of marble. That was Einstein’s dream.
String theory can complete this picture. Supersymmetry is a symmetry that can turn marble into wood and vice versa. They become two sides of the same coin. In this picture, marble is represented by bosons, and wood is represented by fermions. Although there is no experimental evidence for supersymmetry in nature, it is so elegant and beautiful that it has captured the imagination of the physics community.
As Steven Weinberg once said, “Although the symmetries are hidden from us, we can sense that they are latent in nature, governing everything about us. That’s the most exciting idea I know: that nature is much simpler than it looks. Nothing makes me more hopeful that our generation of human beings may actually hold the key to the universe in our hands—that perhaps in our lifetimes we may be able to tell why all of what we see in this immense universe of galaxies and particles is logically inevitable.”
In summary, we now see that symmetry may be the key to unifying all the laws of the universe, due to several remarkable achievements:
Symmetry creates order out of disorder. Out of the chaos of chemical elements and subatomic particles, the Mendeleyev periodic table and Standard Model can rearrange them in a tidy, symmetric fashion.
Symmetry helps fill in the gaps. Symmetry allows you to isolate gaps in these theories and hence predict the existence of new types of elements and subatomic particles.
Symmetry unifies totally unexpected and seemingly unrelated objects. Symmetry finds the link between space and time, matter and energy, electricity and magnetism, and fermions and bosons.
Symmetry reveals unexpected phenomena. Symmetry predicted the existence of new phenomena such as antimatter, spin, and quarks.
Symmetry eliminates unwanted consequences that can destroy the theory. Quantum corrections often have disastrous divergences and anomalies that can be eliminated by symmetry.
Symmetry alters the original classical theory. The quantum corrections to string theory are so stringent they actually alter the original theory, fixing the dimensionality of space-time.
Figure 12. At the beginning of time, it is believed there was a single superforce whose symmetry included all the particles of the universe. But it was unstable, and the symmetry began to break. The first to split off was gravity. Then the strong force and the weak force followed, leaving the electromagnetic force. So the universe today looks broken, with all the forces quite different from one another. It is the job of physicists to reassemble the pieces back together into a single force.
Superstring theory takes advantage of all these features. Its symmetry is supersymmetry (the symmetry that can interchange bosons and fermions). Supersymmetry, in turn, is the largest symmetry ever found in physics, capable of unifying all the known particles of the universe.
M-Theory
We have yet to complete the last step in string theory, finding its fundamental physical principles—that is, we still don’t understand how to derive the entire theory from a single equation. One shock wave came in 1995, when string theory underwent another metamorphosis and a new theory emerged, called M-theory. The problem with the original string theory was that there were five distinct versions of quantum gravity, each of them finite and well defined. These five string theories looked very similar, except their spins were arranged slightly differently. People began to ask: Why should there be five? Most physicists thought that the universe should be unique.