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The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory
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Part 3, Chapters 6-9Chapter Summaries & Analyses
Part 3: “The Cosmic Symphony”
Part 3, Chapter 6 Summary: “Nothing But Music: The Essentials of Superstring Theory”
Chapter 6 begins with music. According to Greene, “Music has long since provided the metaphors of choice for those puzzling over questions of cosmic concern” (135). In string theory, these metaphors become more literal because it argues that the universe is made of vibrating strings. Whereas the standard model of physics envisions the constituent particles of the universe as “pointlike ingredients with no internal structure” (135), string theory proposes that these ingredients are not particles but rather “one-dimensional filaments somewhat like infinitely thin rubber bands” (136) that vibrate. These strings are so small (about the size of the Planck length) that they appear pointlike when viewed via current technology.
Physicist Gabriele Veneziano first proposed string theory in 1968 when he found that certain purely abstract mathematical formulas by mathematician Leonhard Euler inexplicably worked for a series of strange observations in experimentation. Other scientists showed that if they shifted their thinking from pointlike particles to tiny vibrating strings, this math worked perfectly. In fact, this new math matched the hypothesized but unproven existence of the graviton. However, a series of failures led to dismissing string theory until the publication of a landmark paper in 1984 by physicists John Schwarz and Michael Green. They demonstrated that these mathematical failures could be resolved and the resultant theory could “encompass all of the four forces and all of matter as well” (138).
Greene was a graduate student in 1984, when this theory was proposed, and the excitement among physics graduate students was intense. Many new and established scientists began working on the theory, leading to a period from 1984 to 1986 called “the first superstring revolution” (139). However, lack of further progress stagnated the field, and many returned to their previous research, leading to a “dry spell.” Then, in 1995, physicist and pioneer of string theory Edward Witten gave a lecture at a physics conference that started the “second superstring revolution” (140).
String theory argues that the universe is made of tiny vibrating strings that are the “uncuttable constituents” first imagined with the atom. These strings cannot be cut into anything smaller. Additionally, “the different vibrational patterns of a fundamental string give rise to different masses and force charges” (143). Whereas the standard model envisions 12 fundamentally different matter particles, string theory proposes that all particles are a single identical string. The differences between these particles are determined by the way each string vibrates. String theorists believe that if they can discover the precise “vibrational patterns of fundamental strings—the ‘notes’ so to speak, that they play—[they] should be able to explain the observed properties” (146-47) of everything in the universe. Thus, string theory has potential as a unifying theory.
However, obstacles remain. Primarily, current technology does not allow examination at small and precise enough scales to directly observe or experiment on strings. Instead, various kinds of assumptions and approximations must be made to theoretically test the theory. Subsequently, physicists cannot yet determine the precise properties of string, such as its tension.
Theory suggests the tension of a string (called the Planck tension) would be colossally stiff in comparison with more familiar materials. This has three consequences. First, a huge tension would cause a string to contract to miniscule size (a size equal to the Planck length: 10-33 centimeters). Second, this enormous tension means that the vibrational energy of a string would likewise be enormous (consider how much harder you must pluck a string if it is extremely taut). Additionally, because energy and mass are intertwined, an enormous energy implies a much larger mass for a string than what is observable in subatomic particles like quarks—which does not make sense. However, according to string theory, energy cancellations occur to allow for lighter particles. Third, the high tension of the string indicates an infinite variety of vibrational patterns, which should also indicate an infinite variety of corresponding particles (rather than the 12 currently defined). Most of these would be enormously heavy (by subatomic standards) and could exist only in the high-energy moments right after the big bang.
One benefit of string theory is that it helps resolve the conflict between gravity and quantum mechanics by “smearing out” (152) the rough properties of space at miniscule scales. This is based on the idea that any physical measurement or description of an object depends on the size (and therefore precision) of the probe used to examine the object. Consider that a baseball bat dragged across a piece of paper will smooth over the fine textures in the paper (smearing them out), whereas a very tiny needle will feel those fine textures on the surface.
This concept applies at the subatomic level. If a string is shown as approximately the size of the Planck length, then it stands to reason that a string cannot be used to examine miniscule distances in spacetime smaller than the Planck length. The conflict between general relativity and quantum mechanics happens only at scales smaller than the Planck length. Therefore, “[i]f the elementary constituent of the universe cannot probe sub-Planck-scale distances, then neither it nor anything made from it can be affected by the supposedly disastrous short-distance quantum undulations” (156). Greene argues that these possible undulations are so sufficiently smeared out that, in essence, they do not even exist. The undulations proposed in quantum mechanics appear mathematically because scientists have inaccurately described particles as pointlike rather than stringlike, whereas when the math accounts for the size of a string, these undulations simply do not occur.
Part 3, Chapter 7 Summary: “The ‘Super’ in Superstrings”
The laws of physics rely heavily on symmetry. Greene explains that this is as much a matter of aesthetics as mathematical equations. Though “aesthetic judgments do not arbitrate scientific discourse” (166), physicists do sometimes make decisions based on a sense of “which theories have an elegance and beauty of structure on par with the world we experience” (167). And in physics, this sense of beauty is largely governed by principles of symmetry. Symmetry in physics has a specific meaning based on two properties: first, that the laws of physics do not change over time; and second, that the laws of physics remain the same no matter where in the universe one is located. These are symmetries of nature: Every moment in time and location in space is treated identically.
Einstein’s theories of special and general relativity extended this symmetry. The principle of relativity shows that all observers in constant-velocity relative motion must be treated identically—they are symmetrical. Likewise, general relativity demonstrates that the laws of physics remain identical for all observers even under complex accelerated motion. These two levels of symmetry are paramount to Einstein’s description of gravity.
Still other kinds of symmetry exist, such as the gauge symmetries viewed among the electromagnetic, weak, and strong forces. In addition, rotational symmetry shows that the laws of physics remain the same regardless of the angle from which one makes observations. In 1967, physicists Sydney Coleman and Jeffrey Mandula proved the existence of one more kind of symmetry, related to a concept called spin.
In the 1920s, physicists discovered an aspect of electrons called spin: a rotational motion that gives rise to certain magnetic properties. Spin is intrinsic to the electron: “[E]very electron in the universe, always and forever, spins at one fixed and never changing rate” (171). Every other matter particle in the three families also spins at the same rate as the electron (labeled spin-½). Moreover, the messenger particles associated with each nongravitational force (photons, weak gauge bosons, and gluons) also have an intrinsic spin valued at twice that of the matter particles (spin-1). In the context of string theory, this spin relates to the vibrational pattern of the string.
Coleman and Mandula demonstrated that one more kind of symmetry was mathematically possible when applied to spin, which they called supersymmetry. Although the exact nature of supersymmetry is abstract and subtle, Greene focuses on its primary consequence (should it prove true): Based on the supersymmetry of spin, physicists realized that “the particles of nature must come in pairs whose respective spins differ by half a unit” (173). This matches the existence of matter particles and messenger particles that differ by one half (spin-½ versus spin-1). However, an extension of the equations shows that many more partner particles should exist and thus have yet to be discovered. This proved too large a problem for supersymmetry theory, and it was rejected until it was eventually combined with string theory.
Initially, string theory did not incorporate supersymmetry (because it had not been discovered yet). This early version of string theory contained problems that indicated something was missing. However, with the inclusion of supersymmetry, these problems disappeared. Therefore, if string theory is right, supersymmetry must be as well.
However, this led to a new problem for proponents of string theory. By 1985, physicists realized that when combined with supersymmetry, string theory could be proposed in five ways, each similar in its basics but subtly different. This proved “quite an embarrassment” (183) because if multiple answers are all equally possible, none of them is likely correct. Fortunately, Witten’s 1995 lecture may have offered the path to fixing this problem: The five theories may merely be different aspects of a single overarching theory.
Part 3, Chapter 8 Summary: “More Dimensions Than Meets the Eye”
String theory so thoroughly upends modern physics that even the most basic concepts, like the accepted number of dimensions in the universe, are now in question. Greene explains that intuition is informed by experience, which “sets the frame within which we analyze and interpret” (184) the world. One such experience that everyone can agree on is the three spatial dimensions of the universe (generally described in terms of length, width, and height), combined with the fourth dimension of time. In 1919, mathematician Theodor Kaluza challenged this assumption by suggesting that the universe may contain more spatial dimensions. Mathematician Oskar Klein refined this idea in 1926, proposing that the “spatial fabric of our universe may have both extended and curled-up dimensions” (188).
Greene explains by using an analogy: Imagine a garden hose extended in a line. If a dimension is defined as the directions in which an observer can move (up-down, right-left), then the garden hose can be said to have two dimensions: the right-left dimension along the hose’s length, and the clockwise-counterclockwise dimension curled around the surface of the hose. The length dimension is extended, the clockwise dimension is curled-up. From a distance, the curled-up dimension is not visible, making the hose seem to be one dimension (as illustrated in Figures 8.1 (186), 8.2 (188), 8.5 (194), and 8.6 (195).
By extending this image, what Kaluza and Klein proposed becomes clear: The spatial universe has three extended dimensions and one curled-up dimension, or four spatial dimensions, plus the time dimension. They argued that this extra dimension exists “at every point in the extended dimensions, just as the circular girth of the hose exists at every point” (190) of its horizontal length. However, this dimension is so small that it cannot be detected. Specifically, Klein’s refinement suggests that the extra dimension is the size of the Planck length. The concept of extra tiny spatial dimensions is now called the Kaluza-Klein theory.
In his 1919 paper, Kaluza showed that adding one extra dimension to general relativity generated extra equations, which matched Maxwell’s equations for electromagnetism. By simply adding another dimension, Kaluza managed to unite gravity with Maxwell’s theory of light. Before this, theorists viewed these two forces as unrelated. However, Kaluza’s work suggested that electromagnetic waves travel through ripples in the curled-up dimension, just as gravity ripples through spacetime. Unfortunately, subsequent work showed that the initial Kaluza-Klein theory conflicted with other experimental data, and the idea was discarded until the 1970s, when physicists turned to uniting gravity with quantum mechanics.
Some physicists suggested that the initial failure of Kaluza’s idea was because he had not been able to include the strong and weak forces (which were unknown until the 1920s). They also believed he did not push the theory far enough. Physicists now believed that many more curled-up dimensions exist; string theory shows that 10 may exist.
Early versions of string theory accounting for vibrations in two dimensions (left-right, back-forth) often resulted in negative probabilities, which lay outside the allowed mathematical range. According to Greene, “the negative probabilities arose from a mismatch between what the theory required and what reality seemed to impose” (202). However, if strings are allowed to vibrate in nine spatial directions, these negative probabilities cancel each other out. String theory only works if it includes the Kaluza-Klein theory, requiring a universe that includes nine spatial dimensions (and six curled-up dimensions) plus time, for a total of 10.
These extra dimensions are so small they cannot be seen or interacted with, but they have significant indirect influence over the physical properties of the universe. Having more dimensions to vibrate in greatly influences the possible vibrational patterns of strings, and those vibrational patterns determine the properties of particles. Furthermore, string theorists surmise that the fundamental properties of the universe depend largely on the precise geometric shape and size of these extra dimensions, the exact description of which is the question in current research.
Theorists have made steps toward answering this question with Calabi-Yau shapes, named after two mathematicians, Eugenio Calabi and Shing-Tung Yau. Calabi-Yau shapes are a specific type of six-dimensional geometric shape that meet the requirements for use in string theory. These six-dimensional shapes are extraordinarily complex and abstract, but Figure 8.9 (207) offers an approximate representation via a knot of triangles and loops. One must then imagine these complex shapes existing at miniscule scales within every point of three-dimensional space. Greene explains that if one sweeps one’s hand through the air, the hand moves not only through the three spatial dimensions one knows but also through these knotted dimensions. However, this goes unnoticed because these dimensions are so tiny.
Part 3, Chapter 9 Summary: “The Smoking Gun: Experimental Signatures”
String theorists would be happiest if they could present experimental results that support the correctness of the theory. Unfortunately, this is not yet possible because current instruments do not allow for the precise experimentation required. Theorists hope that this will change in the future; in the meantime, they look to indirect clues that suggest string theory is right. Greene describes some of these clues in this chapter.
He first points to how physicists determined that all matter particles can be organized into three families with specific patterns. However, physicists continue to ask why matter particles follow this organizational structure (why three families exist and why these particles have these specific properties). String theory offers a proposal: A typical Calabi-Yau shape contains holes that exist in various arrangements in any of the six curled up dimensions. Theorists have found that the number and arrangement of these holes has mathematical implications that directly reflect the patterns of matter particles. Three holes in the Calabi-Yau shape leads to three families. Similarly, the way the dimensions within a Calabi-Yau shape interact and overlap appears to reflect the masses of each particle. Unfortunately, tens of thousands of possible Calabi-Yau shapes exist, and string theorists cannot yet determine which one corresponds to the observable universe.
One might ask why physicists cannot determine the right Calabi-Yau shape. Theorists largely blame this on the inadequate tools currently available. The mathematical framework required for string theory is so complex that for the moment physicists can “perform only approximate calculations through a formalism known as perturbation theory” (218). Nevertheless, theorists have found some examples of Calabi-Yau shapes that seem to work with the theory, which gives them hope.
Another clue toward string theory is the concept of supersymmetry. Mathematically, string theory and supersymmetry agree to an incredible level. In this way, supersymmetric string theory predicts that every known particle must have a “superpartner.” As yet, no such superpartner has been observed. However, theorists hope that this will change when the Large Hadron Collider in Geneva, Switzerland, is completed.
Additionally, five “possible experimental signatures” exist that, if found, would support string theory’s validity. First, string theory could provide a “compelling explanation of present and future neutrino data” (224) that the standard model has not yet explained. Second, experimenters could observe certain “hypothetical processes” such as the “disintegration of the proton” (224) whose existence string theory supports but which the standard model does not allow. Third, the discovery of new “tiny, long-range force fields” (225) could indicate a particular Calabi-Yau shape. Fourth, discovering certain properties of dark matter, a hypothetical substance for which string theory has several predictions, would support one or another version of string theory. Fifth, string theory could resolve the problems of the cosmological constant, a concept first proposed by Einstein, which the standard model has yet to address.
If any of these indications are found, they would represent compelling proof of string theory. However, any further progress may take generations, especially if science must wait for technological advancements. Current theorists risk wasting their lives on work that will not yield useful results, but this does not stop them from trying.
Part 3 Analysis
Having established the foundations of modern physics in Part 2, Greene now shifts his focus to the basics of string theory, using the building blocks he identified throughout Chapters 2-5. As before, each chapter is carefully organized to build on the previous one. Greene begins with basic concepts, condensing them to their essences, and explains them in a simplified way through analogies that illustrate the theory. He then expands on and complicates each basic concept in the subsequent chapter, allowing readers to slowly construct an understanding of abstract ideas piece by piece. One useful organizational element Greene uses is to explicitly number and list the components of each point he makes (such as “five questions” or “three conflicts”). He then discusses each in detail, clearly signposting each one to track them and conceptually tie them together. This helps readers keep track of the information and arguments Greene presents. By the end of Part 3, Greene effectively explains the complexities of string theory and makes a compelling argument for its potential as a theory to unify all of physics.
Although Greene uses several analogies throughout Part 3, the most important is the image of a garden hose. He establishes this analogy while discussing the concept of multiple spatial dimensions beyond the accepted three. He then returns to this analogy throughout Part 3, using a familiar and established image to further explain complications regarding the concept of Kaluza-Klein multidimensionality. This helps readers visualize the abstract hypothetical consequences of Calabi-Yau shapes. The garden hose analogy extends even into Part 4, becoming a useful way to explain string theory’s potential for examining bigger questions, like those concerning black holes and cosmology.
Furthermore, while Greene’s use of figures, diagrams, and graphs are useful throughout, they become especially helpful in Part 3. As the concepts become more abstract and esoteric, verbal descriptions become increasingly difficult. Visual representation of various concepts, such as the garden hose and Calabi-Yau shapes, are integral to fully absorbing and understanding Greene’s explanations. Greene notes, however, that these images are only approximations because they depict multidimensional concepts on the two-dimensional space of a printed piece of paper.
Another vital aspect of Part 3 is the introduction of narrative elements. While Parts 1 and 2 cover historical developments in physics, including a few quick biographical sketches of important figures like Newton and Einstein, the writing remains focused on scientific information. As scientific developments reach the current moment (of the 1990s) in Part 3, Greene includes more storytelling elements. He specifically tells the story of his own involvement as a graduate student and physics professor in the research and development of string theory. For instance, in Chapter 6, Greene discusses the excitement that he and other graduate students felt about the emergence of string theory. In addition, he writes about his experiences with Edward Witten at length. Greene returns to storytelling in Part 4, where he discusses his largest contributions to string theory research. These narrative elements likewise become prevalent when he discusses his direct predecessors and contemporaries like Feynman and Hawking.
As in the previous section, Part 3 thematically builds on The Human Need to Understand and the resultant search for a unifying theory, or The Unification of Physics. The way that humans use music as a metaphor for puzzling over “questions of cosmic concern” (135) embodies the drive to understand. Music continues as an effective metaphor in Greene’s descriptions of how theorists search for a unifying theory. Just as music is composed of many disparate elements (notes, rests, and so forth), so is string theory. Greene’s primary goal, like that of all string theorists, is to prove that string theory is the answer to this long search for a unifying theory. In this way, the book is part of a long history of humans striving to understand reality. However, as Greene explains the various components and developments of string theory, he also discusses the various obstacles to proving its validity. Chapter 9 focuses on these problems, which Greene expands on in the final parts of the book. Rather than discouraging string theorists, however, the challenges they face (primarily in technological capabilities) merely spur them on in their search for understanding and unification.
Although Greene does not discuss intuition as explicitly in Part 3 as in Part 2, he thematically touches on The Limitations of Intuition. He explains that humans have difficulty effectively imagining and understanding complex multidimensional shapes and spaces because intuition is informed by experience, which “sets the frame within which we analyze and interpret” (184) the world. Intuition is informed by the observable reality of the three-dimensional universe; therefore, it is incapable of making the necessary leaps to imagining a universe filled with many more dimensions. Intuition appears again in Part 4.
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