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THE FABRIC of the COSMOS, Brian Greene, 2004
```(annotated and with added bold highlights by Epsilon=One)
```(annotated and with added bold highlights by Epsilon=One)
Chapter 12 - The World on a String
Too Many Vibrations
Well, at first blush, string theory fails. For starters, there are an infinite number of different string vibrational patterns, with the first few of an endless series schematically illustrated in Figure 12.4. Yet Tables 12.1 and 12.2 contain only a finite list of particles, and so from the get-go we appear to have a vast mismatch between string theory and the real world. What's more, when we analyze mathematically the possible energies — and hence masses — of these vibrational patterns, we come upon another significant mismatch between theory and observation. The masses of the permissible string vibrational patterns bear no resemblance to the experimentally measured particle masses recorded in Tables 12.1 and 12.2. It's not hard to see why.
Since the early days of string theory, researchers have realized that the stiffness of a string is inversely proportional to its length (its length squared, to be more precise): while long strings are easy to bend, the shorter the string the more rigid it becomes. In 1974, when Schwarz and Scherk proposed decreasing the size of strings so that they'd embody a gravitational force of the right strength, they therefore also proposed increasing the tension of the strings — all the way, it turns out, to about a thousand trillion trillion trillion (10^39) tons,
times the tension on an average piano string. Now, if you imagine bending a tiny, extremely stiff string into one of the increasingly elaborate patterns in Figure 12.4, you'll realize that the more peaks and troughs there are, the more energy you'll have to exert. Conversely, once a string is vibrating in such an elaborate pattern, it embodies a huge amount of energy. Thus, all but the simplest string vibrational patterns are highly energetic and hence, via E = mc^2, correspond to particles with huge masses.
Figure 12.4: The first few examples of string theory vibrational patterns.
And by huge, I really mean huge. Calculations show that the masses of the sting vibrations follow a series analogous to musical harmonics: they are all multiples of a fundamental mass, the Planck mass, much as overtones are all multiples of a fundamental frequency or tone. By the standards of particle physics, the Planck mass is colossal — it is some 10 billion billion (10^19) times the mass of a proton, roughly the mass of a dust mote or a bacterium. Thus, the possible masses of string vibrations are 0 times the Planck mass, 1 times the Planck mass, 2 times the Planck mass, 3 times the Planck mass, and so on, showing that the masses of all but the 0-mass string vibrations are gargantuan. 15
As you can see, some of the particles in Tables 12.1 and 12.2 are indeed massless, but most aren't. And the nonzero masses in the tables are farther from the Planck mass than the Sultan of Brunei is from needing a loan. Thus, we see clearly that the known particle masses do not fit the pattern advanced by string theory. Does this mean that string theory is ruled out? You might think so, but it doesn't. Having an endless list of vibrational patterns whose masses become ever more remote from those of known particles is a challenge the theory must overcome. Years of research have revealed promising strategies for doing so.
As a start, note that experiments with the known particle species have taught us that heavy particles tend to be unstable; typically, heavy particles disintegrate quickly into a shower of lower-mass particles, ultimately generating the lightest and most familiar species in Tables 12.1 and 12.2. (For instance, the top-quark disintegrates in about 10-24 seconds.) We expect this lesson to hold true for the "superheavy" string vibrational patterns, and that would explain why, even if they were copiously produced in the hot, early universe, few if any would have survived until today. Even if string theory is right, our only chance to see the superheavy vibrational patterns would be to produce them through high-energy collisions in particle accelerators. However, as current accelerators can reach only energies equivalent to roughly 1,000 times the mass of a proton, they are far too feeble to excite any but string theory's most placid vibrational patterns. Thus, string theory's prediction of a tower of particles with masses starting some million billion times greater than that achievable with today's technology is not in conflict with observations.
This explanation also makes clear that contact between string theory and particle physics will involve only the lowest-energy — the massless — string vibrations, since the others are way beyond what we can reach with today's technology. But what of the fact that most of the particles in Tables 12.1 and 12.2 are not massless? It's an important issue, but less troubling than it might at first appear. Since the Planck mass is huge, even the most massive particle known, the top-quack, weighs in at only .0000000000000000116 (about 10^-17) times the Planck mass. As for the electron, it weighs in at .0000000000000000000000034 (about 10^-23) times the Planck mass. So, to a first approximation — valid to better than 1 part in 10^17 — all the particles in, Tables 12.1 and 12.2 do have masses equal to zero times the Planck mass (much as most earthlings' wealth, to a first approximation, is 0 times that of the Sultan of Brunei), just as "predicted" by string theory. Our goal is to better this approximation and show that string theory explains the tiny deviations from 0 times the Planck mass characteristic of the particles in Tables 12.1 and 12.2. But massless vibrational patterns are not as grossly at odds with the data as you might have initially thought.
This is encouraging, but detailed scrutiny reveals yet further challenges. Using the equations of superstring theory, physicists have listed every vibrational pattern. One entry is the spin-2 graviton, and that's the great success which launched the whole subject; it ensures that gravity is a part of quantum string theory. But the calculations also show that there are many more massless spin-1 vibrational patterns than there are particles in Table 12.2, and there are many more massless spin-1/2 vibrational patterns than there are particles in Table 12.1. Moreover, the list of spin-1/2 vibrational patterns shows no trace of any repetitive groupings like the family structure of Table 12.1. With a less cursory inspection, then, it seems increasingly difficult to see how string vibrations will align with the known particle species.
Thus, by the mid-1980s, while there were reasons to be excited about superstring theory, there were also reasons to be skeptical. Undeniably, superstring theory presented a bold step toward unification. By providing the first consistent approach for merging gravity and quantum mechanics, it did for physics what Roger Bannister did for the four-minute mile: it showed the seemingly impossible to be possible. Superstring theory established definitively that we could break through the seemingly impenetrable barrier separating the two pillars of twentieth-century physics.
Yet, in trying to go further and show that superstring theory could explain the detailed features of matter and nature's forces, physicists encountered difficulties. This led the skeptics to proclaim that superstring theory, despite all its potential for unification, was merely a mathematical structure with no direct relevance for the physical universe.
Even with the problems just discussed, at the top of the skeptics' list of superstring theory's shortcomings was a feature I've yet to introduce. Superstring theory does indeed provide a successful merger of gravity and quantum mechanics, one that is free of the mathematical inconsistencies that plagued all previous attempts. However, strange as it may sound, in the early years after its discovery, physicists found that the equations of superstring theory do not have these enviable properties if the universe has three spatial dimensions. Instead, the equations of superstring theory are mathematically consistent only if the universe has nine spatial dimensions, or, including the time dimension, they work only in a universe with ten spacetime dimensions!
In comparison to this bizarre-sounding claim, the difficulty in making a detailed alignment between string vibrational patterns and known particle species seems like a secondary issue. Superstring theory requires the existence of six dimensions of space that no one has ever seen. That's not a fine point — that's a problem.
Or is it?
Theoretical discoveries made during the early decades of the twentieth century, long before string theory came on the scene, suggested that
extra dimensions need not be a problem at all. And, with a late-twentieth-century updating, physicists showed that these extra dimensions have the capacity to bridge the gap between string theory's vibrational patterns and the elementary particles experimenters have discovered.
This is one of the theory's most gratifying developments; let's see how it works.
Since the early days of string theory, researchers have realized that the stiffness of a string is inversely proportional to its length (its length squared, to be more precise): while long strings are easy to bend, the shorter the string the more rigid it becomes. In 1974, when Schwarz and Scherk proposed decreasing the size of strings so that they'd embody a gravitational force of the right strength, they therefore also proposed increasing the tension of the strings — all the way, it turns out, to about a thousand trillion trillion trillion (10^39) tons,
about 100000000000000000000000000000000000000000 (10^41)
times the tension on an average piano string. Now, if you imagine bending a tiny, extremely stiff string into one of the increasingly elaborate patterns in Figure 12.4, you'll realize that the more peaks and troughs there are, the more energy you'll have to exert. Conversely, once a string is vibrating in such an elaborate pattern, it embodies a huge amount of energy. Thus, all but the simplest string vibrational patterns are highly energetic and hence, via E = mc^2, correspond to particles with huge masses.
Figure 12.4: The first few examples of string theory vibrational patterns.
And by huge, I really mean huge. Calculations show that the masses of the sting vibrations follow a series analogous to musical harmonics: they are all multiples of a fundamental mass, the Planck mass, much as overtones are all multiples of a fundamental frequency or tone. By the standards of particle physics, the Planck mass is colossal — it is some 10 billion billion (10^19) times the mass of a proton, roughly the mass of a dust mote or a bacterium. Thus, the possible masses of string vibrations are 0 times the Planck mass, 1 times the Planck mass, 2 times the Planck mass, 3 times the Planck mass, and so on, showing that the masses of all but the 0-mass string vibrations are gargantuan. 15
As you can see, some of the particles in Tables 12.1 and 12.2 are indeed massless, but most aren't. And the nonzero masses in the tables are farther from the Planck mass than the Sultan of Brunei is from needing a loan. Thus, we see clearly that the known particle masses do not fit the pattern advanced by string theory. Does this mean that string theory is ruled out? You might think so, but it doesn't. Having an endless list of vibrational patterns whose masses become ever more remote from those of known particles is a challenge the theory must overcome. Years of research have revealed promising strategies for doing so.
As a start, note that experiments with the known particle species have taught us that heavy particles tend to be unstable; typically, heavy particles disintegrate quickly into a shower of lower-mass particles, ultimately generating the lightest and most familiar species in Tables 12.1 and 12.2. (For instance, the top-quark disintegrates in about 10-24 seconds.) We expect this lesson to hold true for the "superheavy" string vibrational patterns, and that would explain why, even if they were copiously produced in the hot, early universe, few if any would have survived until today. Even if string theory is right, our only chance to see the superheavy vibrational patterns would be to produce them through high-energy collisions in particle accelerators. However, as current accelerators can reach only energies equivalent to roughly 1,000 times the mass of a proton, they are far too feeble to excite any but string theory's most placid vibrational patterns. Thus, string theory's prediction of a tower of particles with masses starting some million billion times greater than that achievable with today's technology is not in conflict with observations.
This explanation also makes clear that contact between string theory and particle physics will involve only the lowest-energy — the massless — string vibrations, since the others are way beyond what we can reach with today's technology. But what of the fact that most of the particles in Tables 12.1 and 12.2 are not massless? It's an important issue, but less troubling than it might at first appear. Since the Planck mass is huge, even the most massive particle known, the top-quack, weighs in at only .0000000000000000116 (about 10^-17) times the Planck mass. As for the electron, it weighs in at .0000000000000000000000034 (about 10^-23) times the Planck mass. So, to a first approximation — valid to better than 1 part in 10^17 — all the particles in, Tables 12.1 and 12.2 do have masses equal to zero times the Planck mass (much as most earthlings' wealth, to a first approximation, is 0 times that of the Sultan of Brunei), just as "predicted" by string theory. Our goal is to better this approximation and show that string theory explains the tiny deviations from 0 times the Planck mass characteristic of the particles in Tables 12.1 and 12.2. But massless vibrational patterns are not as grossly at odds with the data as you might have initially thought.
This is encouraging, but detailed scrutiny reveals yet further challenges. Using the equations of superstring theory, physicists have listed every vibrational pattern. One entry is the spin-2 graviton, and that's the great success which launched the whole subject; it ensures that gravity is a part of quantum string theory. But the calculations also show that there are many more massless spin-1 vibrational patterns than there are particles in Table 12.2, and there are many more massless spin-1/2 vibrational patterns than there are particles in Table 12.1. Moreover, the list of spin-1/2 vibrational patterns shows no trace of any repetitive groupings like the family structure of Table 12.1. With a less cursory inspection, then, it seems increasingly difficult to see how string vibrations will align with the known particle species.
Thus, by the mid-1980s, while there were reasons to be excited about superstring theory, there were also reasons to be skeptical. Undeniably, superstring theory presented a bold step toward unification. By providing the first consistent approach for merging gravity and quantum mechanics, it did for physics what Roger Bannister did for the four-minute mile: it showed the seemingly impossible to be possible. Superstring theory established definitively that we could break through the seemingly impenetrable barrier separating the two pillars of twentieth-century physics.
Yet, in trying to go further and show that superstring theory could explain the detailed features of matter and nature's forces, physicists encountered difficulties. This led the skeptics to proclaim that superstring theory, despite all its potential for unification, was merely a mathematical structure with no direct relevance for the physical universe.
Even with the problems just discussed, at the top of the skeptics' list of superstring theory's shortcomings was a feature I've yet to introduce. Superstring theory does indeed provide a successful merger of gravity and quantum mechanics, one that is free of the mathematical inconsistencies that plagued all previous attempts. However, strange as it may sound, in the early years after its discovery, physicists found that the equations of superstring theory do not have these enviable properties if the universe has three spatial dimensions. Instead, the equations of superstring theory are mathematically consistent only if the universe has nine spatial dimensions, or, including the time dimension, they work only in a universe with ten spacetime dimensions!
In comparison to this bizarre-sounding claim, the difficulty in making a detailed alignment between string vibrational patterns and known particle species seems like a secondary issue. Superstring theory requires the existence of six dimensions of space that no one has ever seen. That's not a fine point — that's a problem.
Or is it?
Theoretical discoveries made during the early decades of the twentieth century, long before string theory came on the scene, suggested that
extra dimensions need not be a problem at all. And, with a late-twentieth-century updating, physicists showed that these extra dimensions have the capacity to bridge the gap between string theory's vibrational patterns and the elementary particles experimenters have discovered.
This is one of the theory's most gratifying developments; let's see how it works.