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THE ELEGANT UNIVERSE, Brian Greene, 1999, 2003
```(annotated and with added bold highlights by Epsilon=One)
```(annotated and with added bold highlights by Epsilon=One)
Chapter 12 - Beyond Strings: In Search of M-Theory
The Power of Symmetry
Over the years, no one even attempted to study the properties of any of the five string theories for large, values of their string coupling constants because no one had any idea how to proceed without the perturbative framework. However, during the late 1980s and early 1990s, physicists made slow but steady progress in identifying certain special properties—including certain masses and force charges—that are part of the strong-coupling physics of a given string theory, and yet are still within our ability to calculate. The calculation of these properties, which necessarily transcends the perturhative framework, has played a central role in driving the progress of the second superstring revolution and is firmly rooted in the power of symmetry.
Symmetry principles provide insightful tools for understanding a great many things about the physical world. We have discussed, for instance, that the well-supported belief that the laws of physics do not treat any place in the universe or moment in time as special allows us to argue that the laws governing the here and now are the same ones at work everywhere and everywhen. This is a grandiose example, but symmetry principles can be equally important in less all-encompassing circumstances. For instance, if you witness a crime but were able to catch only a glimpse of the right side of the perpetrator's face, a police artist can nonetheless use your information to sketch the whole face. Symmetry is why. Although there are differences between the left and right sides of a person's face, most are symmetric enough that an image of one side can be flipped over to get a good approximation of the other.
In each of these widely different applications, the power of symmetry is its ability to nail down properties in an indirect manner—something that is often far easier than a more direct approach. We could learn about fundamental physics in the Andromeda galaxy by going there, finding a planet around some star, building accelerators, and performing the kinds of experiments carried out on earth. But the indirect approach of invoking symmetry under changes of locale is far easier. We could also learn about features on the left side of the perpetrator's face by tracking him down and examining it. But it is often far easier to invoke the left-right symmetry of faces. 7
Supersymmetry is a more abstract symmetry principle that relates physical properties of elementary constituents that carry different amounts of spin. At best there are only hints from experimental results that the microworld incorporates this symmetry, but, for reasons discussed earlier, there is a strong belief that it does. It is certainly an integral part of string theory. In the 1990s, led by the pioneering work of Nathan Seiberg of the Institute for Advanced Study, physicists have realized that supersymmetry provides a sharp and incisive tool that can answer some very difficult and important questions by indirect means.
Even without understanding intricate details of a theory, the fact that it has supersymmetry built in allows us to place significant constraints on the properties it can have. Using a linguistic analogy, imagine that we are told that a sequence of letters has been written on a slip of paper, that the sequence has exactly three occurrences, say, of the letter "y," and that the paper has been hidden within a sealed envelope. If we are given no further information, then there is no way that we can guess the sequence—for all we know it might be a random assortment of letters with three y's like mvcfojziyxidqfqzyycdi or any one of the infinitely many other possibilities. But imagine that we are subsequently given two further clues: The hidden sequence of letters spells out an English word and it has the minimum number of letters consistent with the first clue of having three y's. From the infinite number of letter sequences at the outset, these clues reduce the possibilities to one word—to the shortest English word containing three y's: syzygy.
Supersymmetry supplies similar constraining clues for those theories that incorporate its symmetry principles. To get a feel for this, imagine that we are presented with a physics puzzle analogous to the linguistic puzzle just described. Hidden inside a box there is something—its identity is left unspecified—that has a certain force charge. The charge might be electric, magnetic, or any of the other generalizations, but to be concrete let's say it has three units of electric charge. Without further information, the identity of the contents cannot be determined. It might be three particles of charge 1, like positrons or protons; it might be four particles of charge 1 and one particle of charge –1 (like the electron), as this combination still has a net charge of three; it might be nine particles of charge one-third (like the up-quark) or it might be the same nine particles accompanied by any number of chargeless particles (such as photons). As was the case with the hidden sequence of letters when we only had the clue about the three y's, the possibilities for the contents of the box are endless.
But let's now imagine that, as in the case of the linguistic puzzle, we are given two further clues: The theory describing the world—and hence the contents of the box—is supersymmetric, and the contents of the box has the minimum mass consistent with the first clue of having three units of charge. Based on the insights of E. Bogomol'nyi, Manoj Prasad, and Charles Sommerfield, physicists have shown that this specification of a tight organizational framework (the framework of supersymmetry, the analog of the English language) and a "minimality constraint" (minimum mass for a chosen amount of electric charge, the analog of a minimum word length for a chosen number of y's) implies that the identity of the hidden contents is nailed down uniquely. That is, merely by ensuring that the contents of the box is the lightest it could possibly be and still have the specified charge, physicists showed that its identity is fully established. Constituents of minimum mass for a chosen value of charge are known as [I]BPS states,[/B] in honor of their three discoverers. 8
The important thing about BPS states is that their properties are uniquely, easily, and exactly determined without resort to a perturbative calculation. This is true regardless of the value of the coupling constants. That is, even if the string coupling constant is large, implying that the perturbative approach is invalid, we are still able to deduce the exact properties of the BPS configurations. The properties are often called nonperturbative masses and charges since their values transcend the perturbative approximation scheme. For this reason, you can also think of BPS as standing for "beyond perturbative states."
The BPS properties exhaust only a small part of the full physics of a chosen string theory when its coupling constant is large, but they nonetheless give us a tangible grip on some of its strong coupling characteristics. As the coupling constant in a chosen string theory is increased beyond the realm accessible to perturbation theory, we anchor our limited understanding in the BPS states. Like a few choice words in a foreign tongue, we will find that they will take us quite far.
Symmetry principles provide insightful tools for understanding a great many things about the physical world. We have discussed, for instance, that the well-supported belief that the laws of physics do not treat any place in the universe or moment in time as special allows us to argue that the laws governing the here and now are the same ones at work everywhere and everywhen. This is a grandiose example, but symmetry principles can be equally important in less all-encompassing circumstances. For instance, if you witness a crime but were able to catch only a glimpse of the right side of the perpetrator's face, a police artist can nonetheless use your information to sketch the whole face. Symmetry is why. Although there are differences between the left and right sides of a person's face, most are symmetric enough that an image of one side can be flipped over to get a good approximation of the other.
In each of these widely different applications, the power of symmetry is its ability to nail down properties in an indirect manner—something that is often far easier than a more direct approach. We could learn about fundamental physics in the Andromeda galaxy by going there, finding a planet around some star, building accelerators, and performing the kinds of experiments carried out on earth. But the indirect approach of invoking symmetry under changes of locale is far easier. We could also learn about features on the left side of the perpetrator's face by tracking him down and examining it. But it is often far easier to invoke the left-right symmetry of faces. 7
Supersymmetry is a more abstract symmetry principle that relates physical properties of elementary constituents that carry different amounts of spin. At best there are only hints from experimental results that the microworld incorporates this symmetry, but, for reasons discussed earlier, there is a strong belief that it does. It is certainly an integral part of string theory. In the 1990s, led by the pioneering work of Nathan Seiberg of the Institute for Advanced Study, physicists have realized that supersymmetry provides a sharp and incisive tool that can answer some very difficult and important questions by indirect means.
Even without understanding intricate details of a theory, the fact that it has supersymmetry built in allows us to place significant constraints on the properties it can have. Using a linguistic analogy, imagine that we are told that a sequence of letters has been written on a slip of paper, that the sequence has exactly three occurrences, say, of the letter "y," and that the paper has been hidden within a sealed envelope. If we are given no further information, then there is no way that we can guess the sequence—for all we know it might be a random assortment of letters with three y's like mvcfojziyxidqfqzyycdi or any one of the infinitely many other possibilities. But imagine that we are subsequently given two further clues: The hidden sequence of letters spells out an English word and it has the minimum number of letters consistent with the first clue of having three y's. From the infinite number of letter sequences at the outset, these clues reduce the possibilities to one word—to the shortest English word containing three y's: syzygy.
Supersymmetry supplies similar constraining clues for those theories that incorporate its symmetry principles. To get a feel for this, imagine that we are presented with a physics puzzle analogous to the linguistic puzzle just described. Hidden inside a box there is something—its identity is left unspecified—that has a certain force charge. The charge might be electric, magnetic, or any of the other generalizations, but to be concrete let's say it has three units of electric charge. Without further information, the identity of the contents cannot be determined. It might be three particles of charge 1, like positrons or protons; it might be four particles of charge 1 and one particle of charge –1 (like the electron), as this combination still has a net charge of three; it might be nine particles of charge one-third (like the up-quark) or it might be the same nine particles accompanied by any number of chargeless particles (such as photons). As was the case with the hidden sequence of letters when we only had the clue about the three y's, the possibilities for the contents of the box are endless.
But let's now imagine that, as in the case of the linguistic puzzle, we are given two further clues: The theory describing the world—and hence the contents of the box—is supersymmetric, and the contents of the box has the minimum mass consistent with the first clue of having three units of charge. Based on the insights of E. Bogomol'nyi, Manoj Prasad, and Charles Sommerfield, physicists have shown that this specification of a tight organizational framework (the framework of supersymmetry, the analog of the English language) and a "minimality constraint" (minimum mass for a chosen amount of electric charge, the analog of a minimum word length for a chosen number of y's) implies that the identity of the hidden contents is nailed down uniquely. That is, merely by ensuring that the contents of the box is the lightest it could possibly be and still have the specified charge, physicists showed that its identity is fully established. Constituents of minimum mass for a chosen value of charge are known as [I]BPS states,[/B] in honor of their three discoverers. 8
The important thing about BPS states is that their properties are uniquely, easily, and exactly determined without resort to a perturbative calculation. This is true regardless of the value of the coupling constants. That is, even if the string coupling constant is large, implying that the perturbative approach is invalid, we are still able to deduce the exact properties of the BPS configurations. The properties are often called nonperturbative masses and charges since their values transcend the perturbative approximation scheme. For this reason, you can also think of BPS as standing for "beyond perturbative states."
The BPS properties exhaust only a small part of the full physics of a chosen string theory when its coupling constant is large, but they nonetheless give us a tangible grip on some of its strong coupling characteristics. As the coupling constant in a chosen string theory is increased beyond the realm accessible to perturbation theory, we anchor our limited understanding in the BPS states. Like a few choice words in a foreign tongue, we will find that they will take us quite far.