by Brian Greene
Kaluza revealed that in a universe with an additional dimension of space, gravity and electromagnetism can both be described in terms of spatial ripples. Gravity ripples through the familiar three spatial dimensions, while electromagnetism ripples through the fourth. An outstanding problem with Kaluza’s proposal was to explain why we don’t see this fourth spatial dimension. It was here that Klein made his mark by suggesting the resolution explained above: dimensions beyond those we directly experience can elude our senses and our equipment if they’re sufficiently small.
In 1919, after learning about the extra dimensional proposal for unification, Einstein vacillated. He was impressed by a framework that advanced his dream of unification but was hesitant about such an outlandish approach. After cogitating for a couple of years, in the process holding up publication of Kaluza’s paper, Einstein finally warmed to the idea and in time became one of the strongest champions of hidden spatial dimensions. In his own research toward a unified theory, he returned to this theme repeatedly.
Einstein’s blessing notwithstanding, subsequent research showed that the Kaluza-Klein program ran up against a number of hurdles, the most difficult being its inability to incorporate the detailed properties of matter particles, such as electrons, into its mathematical structure. Clever ways around this problem, as well as various generalizations and modifications of the original Kaluza-Klein proposal, were pursued on and off for a couple of decades, but as no pitfall-free framework emerged, by the mid-1940s the idea of unification through extra dimensions was largely dropped.
Thirty years later, along came string theory. Rather than allowing for a universe with more than three dimensions, the mathematics of string theory required it. And so string theory provided a new, ready-made setting for invoking the Kaluza-Klein program. In response to the question “If string theory is the long-sought unified theory, then why haven’t we seen the extra dimensions it needs?” Kaluza-Klein echoed across the decades, answering that the dimensions are all around us but are just too small to be seen. String theory resurrected the Kaluza-Klein program, and by the mid-1980s researchers worldwide were inspired to believe that it was only a matter of time—according to the most enthusiastic proponents, a short time—before string theory would provide a complete theory of all matter and all forces.
Great Expectations
During the early days of string theory, progress came at such a rapid clip that it was nearly impossible to keep up with all the developments. Many compared the atmosphere to that of the 1920s, when scientists stormed into the newly discovered realm of the quantum. With such excitement it’s understandable that some theoreticians spoke of a swift resolution to the major problems of fundamental physics: the merger of gravity and quantum mechanics; the unification of all of nature’s forces; an explanation of the properties of matter; a determination of the number of spatial dimensions; the elucidation of black hole singularities; and the unraveling of the origin of the universe. As more seasoned researchers anticipated, though, these expectations were premature. String theory is so rich, wide ranging, and mathematically difficult that research to date, nearly three decades after the initial euphoria, has taken us but partway down the road of exploration. And given that the realm of quantum gravity is some hundred billion billion times smaller than anything we can currently access experimentally, levelheaded assessments expect that the road will be long.
Where are we along it? In the rest of the chapter, I’ll survey the most advanced understanding in a number of key areas (saving those relevant to the theme of parallel universes for more detailed discussion in subsequent chapters), and I’ll appraise the achievements to date and the challenges still looming.
String Theory and the Properties of Particles
One of the deepest questions in all of physics is why nature’s particles have the properties they do. Why, for example, does the electron have its particular mass and the up-quark its particular electric charge? The question commands attention not only for its intrinsic interest but also because of a tantalizing fact we alluded to earlier. Had the particles’ properties been different—had, say, the electron been moderately heavier or lighter, or had the electric repulsion between electrons been stronger or weaker—the nuclear processes that power stars like our sun would have been disrupted. Without stars, the universe would be a very different place.11 Most pointedly, without the sun’s heat and light, the complex chain of events that led to life on earth would have failed to happen.
This leads to a grand challenge: using pen, paper, possibly a computer, and one’s best understanding of the laws of physics, calculate the particle properties and find results in agreement with the measured values. If we could meet this challenge, we’d take one of the most profound steps ever toward understanding why the universe is as it is.
In quantum field theory, the challenge is insurmountable. Permanently. Quantum field theory requires the measured particle properties as input—these features are part of the theory’s definition—and so can happily accommodate a broad range of values for their masses and charges.12 In an imaginary world where the electron’s mass or charge was larger or smaller than it is in ours, quantum field theory could cope without blinking an eye; it would simply be a matter of adjusting the value of a parameter within the theory’s equations.
Can string theory do better?
One of the most beautiful features of string theory (and the facet that most impressed me when I learned the subject) is that particle properties are determined by the size and shape of the extra dimensions. Because strings are so tiny, they don’t just vibrate within the three big dimensions of common experience; they also vibrate into the tiny, curled-up dimensions. And much as air streams flowing through a wind instrument have vibrational patterns dictated by the instrument’s geometrical form, the strings in string theory have vibrational patterns dictated by the geometrical form of the curled-up dimensions. Recalling that string vibrational patterns determine particle properties such as mass and electrical charge, we see that these properties are determined by the geometry of the extra dimensions.
So, if you knew exactly what the extra dimensions of string theory looked like, you’d be well on your way to predicting the detailed properties of vibrating strings, and hence the detailed properties of the elementary particles the strings vibrate into existence. The hurdle is, and has been for some time, that no one has been able to figure out the exact geometrical form of the extra dimensions. The equations of string theory place mathematical restrictions on the geometry of the extra dimensions, requiring them to belong to a particular class called Calabi-Yau shapes (or, in mathematical jargon, Calabi-Yau manifolds), named after the mathematicians Eugenio Calabi and Shing-Tung Yau, who investigated their properties well before their important role in string theory was discovered (Figure 4.6). The problem is that there’s not a single, unique Calabi-Yau shape. Instead, like musical instruments, the shapes come in a wide variety of sizes and contours. And just as different instruments generate different sounds, extra dimensions that differ in size and shape (as well as with respect to more detailed features we’ll come upon in the next chapter) generate different string vibrational patterns and hence different sets of particle properties. The lack of a unique specification of the extra dimensions is the main stumbling block preventing string theorists from making definitive predictions.
Figure 4.6 A close-up of the spatial fabric in string theory, showing an example of extra dimensions curled up into a Calabi-Yau shape. Like the pile and backing of a carpet, the Calabi-Yau shape would be attached to every point in the familiar three large spatial dimensions (represented by the two-dimensional grid), but for visual clarity the shapes are shown only on grid points.
When I started working on string theory, back in the mid-1980s, there were only a handful of known Calabi-Yau shapes, so one could imagine studying each, looking for a match to known physics. My doctoral dissertation was one of the earliest steps in this direction. A few years later, when I
was a postdoctoral fellow (working for the Yau of Calabi-Yau), the number of Calabi-Yau shapes had grown to a few thousand, which presented more of a challenge to exhaustive analysis—but that’s what graduate students are for. As time passed, however, the pages of the Calabi-Yau catalog continued to multiply; as we will see in Chapter 5, they have now grown more numerous than grains of sand on a beach. Every beach. Everywhere. By a long shot. To analyze mathematically each possibility for the extra dimensions is out of the question. String theorists have therefore continued the search for a mathematical directive from the theory that might single out a particular Calabi-Yau shape as “the one.” To date, no one has succeeded.
And so, when it comes to explaining the properties of the fundamental particles, string theory has yet to realize its promise. In this regard, it so far offers no improvement over quantum field theory.13
Bear in mind, however, that string theory’s claim to fame is its ability to resolve the central dilemma of twentieth-century theoretical physics: the raging hostility between general relativity and quantum mechanics. Within string theory, general relativity and quantum mechanics finally join together harmoniously. That’s where string theory provides a vital advance, taking us beyond a critical obstacle that confounded the standard methods of quantum field theory. Should a better understanding of the mathematics of string theory enable us to pick out a unique form for the extra dimensions, one that furthermore allows us to explain all observed particle properties, that would be a phenomenal triumph. But there’s no guarantee that string theory can rise to the challenge. There’s also no necessity for it to do so. Quantum field theory has been rightly lauded as hugely successful, and yet it can’t explain the fundamental particle properties. If string theory also can’t explain the particle properties but goes beyond quantum field theory in one key measure, by embracing gravity, that alone would be a monumental achievement.
Indeed, in Chapter 6 we’ll see that in a cosmos replete with parallel worlds—as suggested by one modern reading of string theory—it may be plainly wrongheaded to hope that mathematics would pick out a unique form for the extra dimensions. Instead, much as the many different forms for DNA provide for the abundant variety of life on earth, so the many different forms for the extra dimensions may provide for the abundant variety of universes populating a string-based multiverse.
String Theory and Experiment
If a typical string is as small as Figure 4.2 suggests, to probe its extended structure—the very characteristic that distinguishes it from a point—you’d need an accelerator some million billion times more powerful than even the Large Hadron Collider. Using known technology, such an accelerator would need to be about as large as the galaxy, and would consume enough energy each second to power the entire world for a millennium. Barring a spectacular technological breakthrough, this ensures that at the comparatively low energies our accelerators can reach, strings will appear as though they are point particles. This is the experimental version of the theoretical fact I emphasized earlier: at low energy, the mathematics of string theory transforms into the mathematics of quantum field theory. And so, even if string theory is the true fundamental theory, it will impersonate quantum field theory in a wide range of accessible experiments.
That’s a good thing. Although quantum field theory is not equipped to combine general relativity and quantum mechanics, nor to predict the fundamental properties of nature’s particles, it can explain a great many other experimental results. It does this by taking the measured properties of particles as input (input that dictates the choice of fields and energy curves in the quantum field theory) and then uses the mathematics of quantum field theory to predict how such particles will behave in other experiments, generally accelerator-based. The results are extremely accurate, which is why generations of particle physicists have made quantum field theory their primary approach.
The choice of fields and energy curves in quantum field theory is tantamount to the choice of the extra dimensional shape in string theory. The particular challenge facing string theory, though, is that the mathematics linking particle properties (such as their masses and charges) to the shape of the extra dimensions is extraordinarily intricate. This makes it difficult to work backwards—to use experimental data to guide the choice of the extra dimensions, much as such data guide the choices of fields and energy curves in quantum field theory. One day we may have the theoretical dexterity to use experimental data to fix the form of string theory’s extra dimensions, but not yet.
For the foreseeable future, then, the most promising avenue for linking string theory with data are predictions that, while open to explanations using more traditional methods, are far more naturally and convincingly explained using string theory. Just as you might theorize that I’m typing these words with my toes, a far more natural and convincing hypothesis—and one I can attest to as correct—is that I’m using my fingers. Analogous considerations applied to the experiments summarized in Table 4.1 have the capacity to build a circumstantial case for string theory.
The undertakings range from particle physics experiments at the Large Hadron Collider (searching for supersymmetric particles and for evidence of extra dimensions), to tabletop experiments (measuring the gravitational strength of attraction on scales of a millionth of a meter and smaller), to astronomical observations (looking for particular kinds of gravitational waves and fine temperature variations in the cosmic microwave background radiation). The table explains the individual approaches, but the overall assessment is readily summarized. A positive signature from any of these experiments could be explained without invoking string theory. For example, although the mathematical framework of supersymmetry (see the first entry in Table 4.1) was initially discovered through theoretical studies of string theory, it has since been incorporated into non-string theoretic approaches. Discovering supersymmetric particles would thus confirm a piece of string theory, but would not constitute a smoking gun. Similarly, although extra spatial dimensions have a natural home within string theory, we’ve seen that they too can be part of non-string theoretic proposals—Kaluza, as a case in point, was not thinking about string theory when he proposed the idea. The most favorable outcome from the approaches in Table 4.1, therefore, would be a series of positive results that would show pieces of the string theory puzzle falling into place. Like touting touch-typing toes, non-string explanations would become overwrought when faced with such a collection of positive results.
Table 4.1. Experiments and Observations with the Capacity to Link String Theory to Data
EXPERIMENT/OBSERVATION: Supersymmetry
EXPLANATION: The “super” in superstring theory refers to supersymmetry, a mathematical feature with a straightforward implication: for every known particle species there should be a partner species that has the same electrical and nuclear force properties. Theorists surmise that these particles have so far evaded detection because they are heavier than their known counterparts, and so lie beyond the reach of well-worn accelerators. The Large Hadron Collider may have enough energy to produce them, so there’s broad anticipation that we may be on the threshold of revealing nature’s supersymmetric quality.
EXPERIMENT/OBSERVATION: Extra Dimensions and Gravity
EXPLANATION: Because space is the medium for gravity, more dimensions supply a larger domain within which gravity can spread. And just as a drop of ink grows more diluted when it spreads in a vat of water, the strength of gravity would become diluted as it spreads through the additional dimensions—offering an explanation for why gravity appears weak (when you pick up a coffee cup, your muscles beat out the gravitational pull of the entire earth). If we could measure gravity’s strength over distances smaller than the size of the extra dimensions, we’d catch it before it’s fully spread and so we should find its strength to be stronger. To date, measurements on scales as short as a micron (10–6 meters) have found no deviation from expectations based on a world with three spatial dimensions. Should a deviation be f
ound as physicists push these experiments to ever-shorter distances, that would provide convincing evidence for additional dimensions.
EXPERIMENT/OBSERVATION: Extra Dimensions and Missing Energy
EXPLANATION: If the extra dimensions exist but are far smaller than a micron, they will be inaccessible to experiments that directly measure gravity’s strength. But the Large Hadron Collider provides another means of revealing their existence. Debris created by head-on collisions between fast-moving protons can be ejected from our familiar large dimensions and squeezed into the others (where, for reasons we’ll get to later, the debris would likely be particles of gravity, or gravitons). Were this to happen, the debris would carry away energy, and as a result our detectors would register a little less energy after the collision than was present before. Such missing energy signals could provide strong evidence for the existence of extra dimensions.
EXPERIMENT/OBSERVATION: Extra Dimensions and Mini Black Holes
EXPLANATION: Black holes are usually described as the remains of massive stars that have exhausted their nuclear fuel and collapsed under their own weight, but this is an unduly limited description. Anything would become a black hole if compressed sufficiently. Moreover, if there are extra dimensions that result in gravity being stronger when acting over short distances, it would be easier to form black holes, since a stronger gravitational force implies that it takes less compression to generate the same gravitational pull. Even just two protons, if slammed together at the velocities mustered by the Large Hadron Collider, may be able to cram enough energy into a sufficiently small volume to trigger the formation of a black hole. It would be a wisp of a black hole, but it would yield an unmistakable signature. Mathematical analysis, going back to the work of Stephen Hawking, shows that tiny black holes would quickly disintegrate into a spray of lighter particles whose tracks would be picked up by the collider’s detectors.