by Michio Kaku
Einstein once summarized his attitude toward unification by comparing marble and wood. Marble, thought Einstein, described the beautiful world of geometry, in which surfaces warped smoothly and continuously. The universe of stars and galaxies played out their cosmic game on the beautiful marble of space-time. Wood, on the other hand, represented the chaotic world of matter, with its jungle of subatomic particles, its nonsensical rules for the quantum. This wood, like gnarled vines, grows in unpredictable and random ways. New particles being discovered in the atom made the theory of matter quite ugly. Einstein saw the defect in his equations. The fatal flaw was that wood determined the structure of the marble. The amount of bending of space-time was determined by the amount of wood at any point.
Thus, to Einstein, his strategy was clear: to create a theory of pure marble, to eliminate the wood by reformulating it solely in terms of marble. If the wood itself could be shown to be made of marble, then he would have a theory that was purely geometric. For example, a point particle is infinitely tiny, having no extension in space. In field theory, a point particle is represented by a “singularity,” a point where the field strength goes to infinity. Einstein wanted to replace this singularity with a smooth deformation of space and time. Imagine, for example, a kink or knot in a rope. From a distance, the kink may look like a particle, but close-up the kink or knot is nothing but a wrinkle in the rope. Similarly, Einstein wanted to create a theory that was purely geometric and had no singularities whatsoever. Subatomic particles, like the electron, would emerge as kinks or as some kind of small wrinkle on the surface of space-time. The fundamental problem, however, was that he lacked a concrete symmetry and principle that could unify electromagnetism and gravity. As we saw earlier, the key to Einstein’s thinking was unification through symmetry. With special relativity, he had the picture that guided him constantly, running next to a light beam. This picture revealed the fundamental contradiction between Newtonian mechanics and Maxwell’s fields. From this, he was able to extract a principle, the constancy of the speed of light. Last, he was able to formulate the symmetry that unified space and time, the Lorentz transformations.
Similarly, with general relativity, he had a picture, that gravity was caused by the warping of space and time. This picture exposed the fundamental contradiction between Newton’s gravity (where gravity traveled instantaneously) and relativity (where nothing can go faster than light). From the picture, he extracted a principle, the equivalence principle, that accelerating and gravitating frames obeyed the same laws of physics. Last, he was able to formulate the generalized symmetry that described accelerations and gravity, which was general covariance.
The problem facing Einstein now was truly daunting, because he was working at least fifty years ahead of his time. In the 1920s, when he began work on the unified field theory, the only established forces were the gravitational and electromagnetic forces. The nucleus of the atom had only been discovered in 1911 by Ernest Rutherford, and the force that held it together was still shrouded in mystery. But without an understanding of the nuclear forces, Einstein lacked a crucial part of the puzzle. Furthermore, no experiment or observation exposed a contradiction between gravity and electromagnetism that would be the hook Einstein could grab onto.
Hermann Weyl, a mathematician who was inspired by Einstein’s search for a unified field theory, made the first serious attempt in 1918. At first, Einstein was very impressed. “It is a masterful symphony,” he wrote. Weyl expanded Einstein’s old theory of gravity by adding the Maxwell field directly into the equations. Then he demanded that the equations be covariant under even more symmetries than Einstein’s original theory, including scale transformations (i.e., transformations that expand or contract all distances). However, Einstein soon found some strange anomalies in the theory. For example, if you traveled in a circle and came back to your original point, you would find that you were shorter but had the same shape. In other words, lengths were not preserved. (In Einstein’s theory, lengths could also change, but they remained the same if you came back to where you started.) Time would also be shifted in a closed path, but this would violate our understanding of the physical world. For example, it meant that if vibrating atoms were moved around a complete circle, they would be vibrating at a different frequency when they came back. Although Weyl’s theory seemed ingenious, it had to be abandoned because it did not fit the data. (In hindsight, we can see that the Weyl theory had too much symmetry. Scale invariance is apparently a symmetry that nature does not use to describe our visible universe.)
In 1923, Arthur Eddington also caught the bug. Inspired by Weyl’s work, Eddington (and many others after him) tried his hand at a unified field theory. Like Einstein, he created a theory based on the Ricci curvature, but the concept of distance did not appear in the equations. In other words, it was impossible to define meters or seconds in his theory; the theory was “pre-geometrical.” Only in the last step would distance finally appear as a consequence of his equations. Electromagnetism was supposed to emerge as a piece of the Ricci curvature. The physicist Wolfgang Pauli did not like this theory at all, stating that it had “no significance for physics.” Einstein also panned it, thinking it had no physical content.
But what really rocked Einstein to the core was a paper that he saw in 1921, written by an obscure mathematician, Theodr Kaluza, from the University of Königsberg. Kaluza suggested that Einstein, who had pioneered the concept of the fourth dimension, add yet another dimension to his equations. Kaluza began by reformulating Einstein’s own general relativity in five dimensions (four dimensions of space and one dimension of time). This takes no work at all, since Einstein’s equations could easily be formulated in any dimension. Then, in a few lines, Kaluza showed that if the fifth dimension is separated from the other four, Einstein’s equations emerged, along with Maxwell’s equations! In other words, Maxwell’s equations, the horrible set of eight partial differential equations memorized by every engineer and physicist, can be reduced to waves traveling on the fifth dimension. To put it another way, Maxwell’s theory was already hidden inside Einstein’s theory if relativity were extended to five dimensions.
Einstein was surprised by the sheer audacity and beauty of Kaluza’s work. He wrote Kaluza, “The idea of achieving [unification] by means of a five-dimensional cylinder world never dawned on me…. At first glance, I liked your idea enormously.” A few weeks later, after studying the theory, he wrote, “The formal unity of your theory is startling.” In 1926, the mathematician Oskar Klein generalized Kaluza’s work and speculated that the fifth dimension was unobservable because it was small and possibly linked to the quantum theory. Kaluza and Klein were thus proposing an entirely different approach to unification. To them, electromagnetism was nothing but vibrations rippling along the surface of a small fifth dimension.
For example, if we think of fish living in a shallow pond, swimming just below the lily pads, the fish might conclude that their universe was two-dimensional. They can move forward and backward, left and right, but the concept of “up” into the third dimension would be alien to them. If their universe was two-dimensional, then how might they become aware of a mysterious third dimension? Imagine that it rains one day. Tiny ripples in the third dimension move along the surface of the pond, and they are clearly visible to the fish. As these ripples move along the surface, the fish might conclude that there was a mysterious force that could illuminate their universe. Similarly, in this picture, we are the fish. We conduct our affairs in three spatial dimensions, unaware that there could be higher dimensions existing just beyond our senses. The only direct contact that we might have with the unseen fifth dimension is light, now viewed as ripples traveling along the fifth dimension.
There was a reason why the Kaluza-Klein theory worked so well. Recall that unification through symmetry was one of Einstein’s great strategies that led to relativity. In the Kaluza-Klein theory, electromagnetism and gravity were united because of a new symmetry, five-dimensional gene
ral covariance. Although this picture was immediately appealing, unifying gravity and electromagnetism by introducing another dimension, there was still the nagging question, where was this fifth dimension? No experiment has ever, even to this day, picked up evidence of any higher dimension of space beyond length, width, and height. If these higher dimensions exist, then they must be extremely small, much smaller than an atom. For example, we know that if we release chlorine gas into a room, its atoms can slowly permeate all the nooks and crannies of any room without disappearing into some mysterious extra dimension. We know, therefore, that any hidden dimension must be smaller than any atom. In this new theory, if one makes the fifth dimension smaller than an atom, then it is consistent with all laboratory measurements, which have never detected the presence of the fifth dimension. Kaluza and Klein assumed that the fifth dimension was “curled up” into a small ball, too small to be experimentally observed.
Although the Kaluza-Klein theory was a fresh, intriguing approach to unifying electromagnetism with gravity, Einstein eventually had doubts. The thought that the fifth dimension might not exist, that it might be a mathematical fiction or mirage, bothered him. Also, he had problems finding subatomic particles in the Kaluza-Klein theory. His goal was to derive the electron from his gravitational field equations, and try as he could, he could find no such solution. (In hindsight, this was a tremendous missed opportunity for physics. If physicists had taken the Kaluza-Klein theory more seriously, they might have added more dimensions beyond five. As we increase the number of dimensions, Maxwell’s field increases in number into what are called “Yang-Mills fields.” Klein actually discovered the Yang-Mills fields in the late 1930s, but his work was forgotten because of the chaos of World War II. It would take almost two decades before they were rediscovered, in the mid-1950s. These Yang-Mills fields now form the foundation of the current theory of the nuclear force. Almost all of subatomic physics is formulated in terms of them. After another twenty years, the Kaluza-Klein theory would be resurrected in the form of a new theory, string theory, now considered the leading candidate for a unified field theory.)
Einstein hedged his bets. If the Kaluza-Klein theory failed, then he would have to explore a different avenue toward the unified field theory. His choice was to investigate geometries beyond Riemannian geometry. He consulted many mathematicians, and it became quickly obvious that this was a totally open field. In fact, at Einstein’s urging, many mathematicians began to look into “post-Riemannian” geometries, or the “theory of connections,” to help him explore new possible universes. New geometries involving “torsion” and “twisted spaces” were soon developed as a consequence. (These abstract spaces would have no application to physics for another seventy years, until the arrival of superstring theory.)
Working on post-Riemannian geometries was a nightmare, however. Einstein had no guiding physical principle to help him through the thicket of abstract equations. Previously, he used the equivalence principle and general covariance as compasses. Both were firmly rooted in experimental data. He had also relied on physical pictures to show him the way. With the unified field theory, however, he had no guiding physical principle or picture.
So curious was the world about Einstein’s work that a progress report he gave on the unified field theory to the Prussian Academy was reported to the New York Times, which even published parts of Einstein’s paper. Soon, there were hundreds of reporters swarming outside his home, hoping for a glimpse of him. Eddington wrote, “You may be amused to hear that one of our great department stores in London (Selfridges) has posted on its window your paper (the six pages pasted up side by side) so that passers-by can read it all through. Large crowds gather around to read it.” Einstein, however, would have traded all the adulation and praise in the world for a simple physical picture to guide his path.
Gradually, other physicists began to hint that Einstein was on the wrong track and that his physical intuition was failing him. One critic was his friend and colleague Wolfgang Pauli, one of the early pioneers of the quantum theory, who was famous in scientific circles for his unsparing wit. He once said of a misguided physics paper, “It is not even wrong.” To a colleague whose paper he had reviewed, he said, “I do not mind if you think slowly, but I do object when you publish more quickly than you think.” After he heard a confused, incoherent seminar, he would say, “What you said was so confusing that one cannot tell whether it was nonsense or not.” When fellow physicists complained that Pauli was too critical, he would reply, “Some people have very sensitive corns, and the only way to live with them is to step on these corns until they are used to it.” His impression of the unified field theory was reflected by his famous comment that what God has torn asunder, let no man put together. (Ironically, later Pauli would also catch the bug and propose his own version of the unified field theory.)
Pauli’s view would have been endorsed by many of his fellow physicists, who grew increasingly preoccupied with the quantum theory, the other great theory of the twentieth century. The quantum theory stands as one of the most successful physical theories of all time. It has had unparalleled success explaining the mysterious world of the atom, and by doing so has unleashed the power of lasers, modern electronics, computers, and nanotechnology. Ironically, however, the quantum theory is based on a foundation of sand. In the atomic world, electrons seemingly appear in two places at the same time, jump between orbits without warning, and disappear into the ghostly world between existence and nonexistence. As Einstein remarked as early as 1912, “The more success the quantum theory has, the sillier it looks.”
Some of the bizarre features of the quantum world were made apparent in 1924, when Einstein received a curious letter from an obscure Indian physicist, Satyendra Nath Bose, whose papers on statistical physics were so strange that they were flatly rejected for publication. Bose was proposing an extension of Einstein’s earlier work on statistical mechanics, searching for a fully quantum mechanical treatment of a gas, treating the atoms as quantum objects. Just as Einstein had extended Planck’s work to a theory of light, Bose was hinting that one could extend Einstein’s work into a fully quantum theory of atoms in a gas. Einstein, a master of the subject, found that though Bose had made a number of mistakes, making assumptions that could not be justified, his final answer appeared to be correct. Einstein was not only intrigued by the paper, he translated it into German and submitted it for publication.
He then extended Bose’s work and wrote a paper of his own, applying the result to extremely cold matter that hovers just above the temperature of absolute zero. Bose and Einstein found a curious fact about the quantum world: all atoms are indistinguishable; that is, you cannot put a label on each atom, as Boltzmann and Maxwell had thought. While rocks and trees and other ordinary matter can be labeled and given names, in the quantum world all atoms of hydrogen are identical in any experiment; there are no green or blue or yellow hydrogen atoms. Einstein then found that if a collection of atoms were supercooled to near absolute zero, where all atomic movement almost ceases, all the atoms would fall down to the lowest energy state, creating a single “superatom.” These atoms would condense into the same quantum state, behaving essentially like a single gigantic atom. He was proposing an entirely new state of matter, never seen before on Earth. However, before the atoms could tumble down to the lowest energy state, the temperatures would have to be fantastically small, much too small to be experimentally observed, about a millionth of a degree above absolute zero. (At these extremely low temperatures, the atoms vibrate in lockstep, and subtle quantum effects only seen at the level of individual atoms now become distributed throughout the entire condensate. Like the spectators at a football game who form “human waves” that sweep across the stands as they stand up and down in unison, the atoms in a “Bose-Einstein condensate” act as if everything is vibrating in unison.) But Einstein despaired of ever observing this Bose-Einstein condensation in his lifetime, since the technology of the 1920s did not permit ex
periments near temperatures of absolute zero. (In fact, Einstein was so ahead of his time that it would be about seventy years before that prediction could be tested.)
In addition to Bose-Einstein condensation, Einstein was interested in whether his principle of duality could be applied to matter as well as light. In his 1909 lecture, Einstein had showed that there was a dual nature to light, that it can simultaneously have both particle and wavelike properties. Although it was a heretical idea, it was supported fully by experimental results. Inspired by the duality program initiated by Einstein, a young graduate student, Prince Louis de Broglie, then speculated in 1923 that even matter itself can have both particle and wavelike properties. This was a bold, revolutionary concept, since it was a deep-seated prejudice that matter consisted of particles. Stimulated by Einstein’s work on duality, de Broglie could explain away some of the mysteries of the atom by introducing the concept that matter had wavelike properties.
Einstein liked the audacity of de Broglie’s “matter waves” and promoted his theory. (De Broglie would eventually be awarded the Nobel Prize for this seminal idea.) But if matter had wavelike properties, then what was the equation that the waves obeyed? Classical physicists had plenty of experience writing down the equations of ocean waves and sound waves, so an Austrian physicist, Erwin Schrödinger, was inspired to write down the equation of these matter waves. While Schrödinger, a well-known ladies’ man, was staying with one of his innumerable girlfriends in the Villa Herwig in Arosa during the Christmas holidays in 1925, he managed to divert himself long enough to formulate an equation that would soon be known as one of the most celebrated in all of quantum physics, the Schrödinger wave equation. Schrödinger’s biographer, Walter Moore, wrote, “Like the dark lady who inspired Shakespeare’s sonnets, the lady of Arosa may remain forever mysterious.”(Unfortunately, because Schrödinger had so many girlfriends and lovers in his life, as well as illegitimate children, it is impossible to determine precisely who served as the muse for this historic equation.) Over the next several months, in a remarkable series of papers, Schrödinger showed that the mysterious rules found by Niels Bohr for the hydrogen atom were simple consequences of his equation. For the first time, physicists had a detailed picture of the interior of the atom, by which one could, in principle, calculate the properties of more complex atoms, even molecules. Within months, the new quantum theory became a steamroller, obliterating many of the most puzzling questions about the atomic world, answering the greatest mysteries that had stumped scientists since the Greeks. The dance of electrons as they moved between orbits, releasing pulses of light or binding molecules together, suddenly became calculable, a matter of solving standard partial differential equations. One young brash quantum physicist, Paul Adrian Maurice Dirac, even boasted that all of chemistry could be explained as solutions of Schrödinger’s equation, reducing chemistry to applied physics.
