Farewell to Reality

by Jim Baggott

  Different versions of superstring theory have different symmetry properties. Type IIA is mirror-symmetric, and the physicists could be confident that this theory would exhibit no residual anomalies. But look in a mirror. The world we inhabit is not mirror-symmetric.

  In 1984, Witten and Spanish physicist Luis Alvarez-Gaumé pointed out that further anomalies would arise in superstring theory due to the gravitational field. But they went on to show that in a low-energy-approximation Type IIB theory, these anomalies did indeed cancel. This was encouraging, but still left a question mark over the Type I theory.

  In the summer of 1984, Green and Schwarz discovered that in a low-energy-approximation Type I superstring theory based on the symmetry group SO (32), a group of rotations in 32 dimensions,* the anomalies did indeed all cancel out. Witten picked up rumours of their breakthrough, and called to find out more. They sent him a draft manuscript by Federal Express.

  Things then happened very quickly.

  Green and Schwarz published their paper in September 1984. Witten submitted his first paper on superstrings to the same journal later that same month.

  At Princeton, David Gross, Jeffrey Harvey, Emil Martinec and Ryan Rohm (who would collectively come to be known as the Princeton String Quartet) found yet another version of the theory, called heterotic (or hybrid)* superstring theory, in which the anomalies also cancelled. It turned out that there were two types of heterotic superstring theory, one based on the symmetry group formed from the product E8 × E8, where E8 is an algebraic group with 248 dimensions, and the other based on the symmetry group SO(32). Both theories require ten spacetime dimensions. They submitted their paper for publication in November 1984.

  The first superstring revolution had begun.

  Hiding the extra dimensions: Calabi—Yau spaces

  Kaluza discovered that by solving Einstein’s field equations in a five-dimensional spacetime, Maxwell’s equations would emerge naturally. Klein deduced that by rolling up — compactifying — the extra spatial dimension into a cylinder with a radius of sub-nuclear scale, its role in the theory could be preserved without the embarrassment of having to experience it directly.

  Superstring theory couldn’t settle for just one extra spatial dimension. It demanded an extra six. The reason is not so difficult to understand, once you have accepted the assumption that unification can be achieved by expanding the number of dimensions. Kaluza—Klein theory uses five dimensions to join gravity and electromagnetism, to which we must now add further dimensions to accommodate the weak and strong nuclear forces. Alternatively, if we make the string assumption, then we need enough ‘degrees of freedom’ for the strings to vibrate in if these vibrational patterns are to represent all the elementary particles and all the forces between them.

  The point about superstring theory is that it demands precisely nine spatial dimensions, no more and no fewer. But what must we now do with this embarrassment of riches? A lifetime of experience tells me that the world is stubbornly three-dimensional and I have little doubt that I will not find another six dimensions no matter how hard I look.

  I feel another assumption coming on.

  The Compactification Assumption. The nine spatial dimensions demanded by superstring theory can be reconciled with our experience of a three-dimetisional world by assuming that six dimensions are compactifed into a manifold with a size of the order of the Planck length, about 1.6 billionths of a trillionth of a trillionth (1.6 × 10-33) centimetres.

  Klein had rolled up the extra dimension into a cylinder, but a further six dimensions demand a much more complex structure. In 1984, American theorist Andrew Strominger, then at the Institute for Advanced Study in Princeton, searched for an appropriate structure for this manifold in collaboration with British mathematical physicist Philip Candelas, then at the University of Texas. This was not a free choice. Supersymmetry places some fairly rigid constraints on precisely how these extra dimensions can be rolled up. Strominger’s search led to the library, and a recent paper by Chinese-born American mathematician Shing-Tung Yau. The paper contained a proof of something called the Calabi conjecture, named for Italian-American mathematician Eugenio Calabi.

  As Strominger explained: ‘I found Yau’s paper in the library and couldn’t make much sense of it, but from the little I did understand, I realized that these manifolds were just what the doctor ordered.’8

  Calabi’s conjecture concerns the geometric structures that are allowed by different topologies, the various shapes that ‘mathematical’ spaces can possess. Although this was a problem in abstract geometry, solutions to such problems have been of interest to physicists ever since Einstein established the connection between geometry and gravity in the general theory of relativity. One interpretation of the Calabi conjecture is that it suggests that in certain spaces, gravity is possible even in the absence of matter.

  Yau developed a proof of the conjecture, which he discussed with Calabi and Canadian-born American mathematician Louis Nirenberg on Christmas Day 1976.* The proof confirmed the existence of a series of shapes — now called Calabi—Yau spaces — that satisfy Einstein’s field equations in the absence of matter. It was indeed just what the doctor ordered.

  Strominger and Candelas got in touch with Gary Horowitz, at the University of California in Santa Barbara, a physicist who had worked with Yau as a postdoctoral associate. Strominger also visited Witten, to discover that the latter had independently arrived at the same conclusion. At Strominger and Witten’s request, on a flight from San Diego to Chicago Yau worked out the structure of a Calabi—Yau space that would generate three families or generations of matter particles.

  The four theorists collaborated on a paper which was published in 1985. Thus was born the idea of ‘hidden dimensions’. If I mark an infinitesimally small point on the desk in front of my keyboard, and could somehow zoom in on this point and magnify it so that a distance of a billionth of a trillionth of a trillionth of a centimetre becomes visible, then superstring theory says that I should perceive six further spatial dimensions, curled up into a Calabi—Yau shape (see Figure 8).

  On the one hand, this is a perfect example of how we make progress in science. We know that abstract point particles lead to problems. Abstract one-dimensional strings appear to offer better prospects. We reach for supersymmetry because we want a theory that describes fermions as well as bosons, and this eliminates some of the problems of the original string theory (such as tachyons) and yields theories free of anomalies. Superstrings demand a ten-dimensional spacetime, so we borrow concepts from mathematics and tuck the six extra spatial dimensions out of sight in a Calabi—Yau space. We can find a Calabi—Yau space that is consistent with three generations of elementary matter particles. This all seems perfectly logical and reasonable.

  Figure 8 The Calabi—Yau manifolds or Calabi—Yau shapes are complex, high-dimensional algebraic surfaces. They appear in superstring theory as manifolds containing the six additional spatial dimensions required for the strings to vibrate in. Because they have dimensions of the Planck length, these manifolds are far too small to be visible. Source: Wikimedia Commons.

  But then there is the other hand. On what basis do we choose strings, as opposed to any other kind of abstract construction? Because of a (possibly rather tenuous) relationship between the behaviour of strings and the beta function identified in the scattering of mesons by Veneziano. On what basis do we assume supersymmetry between fermions and bosons? Because this is the only way to get both kinds of particles into the same picture. What is the basis for assuming that six dimensions must be hidden in a Calabi—Yau space? Because it is our experience that the universe is four-dimensional.

  I think you get the point. Although this is all perfectly logical and reasonable, what we are actually doing is piling one grand assumption on top of another. I want to emphasize that there is no experimental or observational basis for these assumptions. This is a theory with little or no foundation in empirical reality. It is rather a loose
assemblage of assumptions, ideas or hypotheses that say rather more about how we would like the universe to be than how it really is.*

  I don’t think it’s uncharitable to suggest that this is looking increasingly like a house of cards.

  Of course, the Theory Principle tells us that this is still okay. Science is a very forgiving discipline in that it never really matters overmuch how you arrive at a theory. If it can be shown to work better than existing alternatives, then no matter how much speculation or luck was involved, you can sit back and wait patiently for the Nobel Prize committee to reach the right decision.

  Before we take a look at superstring theory’s predictions, we have to deal with the fact that there appear to be at least five different versions of the theory — Type I, Type IIA, Type IIB and two versions of heterotic superstring theory. This is a little embarrassing for a theory that has pretensions to be the theory of everything.

  Resolving this issue would require the biggest assumption of all, one that sparked the second superstring revolution.

  The M-theory conjecture and the second superstring revolution

  As theories proliferated and superstring theory lost any sense of uniqueness, interest began to wane. British superstring theorist Michael Duff explained it like this:

  Theorists love uniqueness; they like to think that the ultimate Theory of Everything will one day be singled out, not merely because all rival theories are in disagreement with experiment, but because they are mathematically inconsistent. In other words, that the universe is the way it is because it is the only possible universe.9

  There were also rumblings from some theorists that, based on calculations using supergravity, the right number of spacetime dimensions is actually eleven, not ten. It all seemed to be getting rather out of hand. Then, at a superstring theory conference at the University of Southern California in March 1995, Witten made a bold conjecture. Perhaps the five different ten-dimensional superstring theories are actually instances or approximations of a single, overarching eleven-dimensional structure. He called it M-theory. He was not specific on the meaning or significance of ‘M’.

  This was a conjecture, not a theory. Witten demonstrated the equivalence of a ten-dimensional superstring theory and eleven-dimensional supergravity but he could not formulate M-theory; he could only speculate that it must exist.

  The M-Theory Conjecture. We assume that the five variants of superstring theory can be subsumed into a single, eleven-dimensional framework. But nobody has yet been able to write this theory down on paper.

  This is a simple fact that many readers of popular presentations of superstring theory somehow tend to miss. M-theory is not a theory. Nobody knows what M-theory looks like, although many theorists have tinkered with structures that they believe it could or should possess. So, on top of a foundation built from a sequence of assumptions, we now erect the biggest assumption of all. We assume that a unique eleven-dimensional superstring theory is possible in principle, although we don’t yet know what this theory is.

  I think this is a truly remarkable state of affairs. Of course, this kind of speculative theorizing goes on all the time in physics, and there are plenty of examples from history. We might look again at the sequence of assumptions that have led us here and conclude that theorists who commit themselves to M-theory are either brave or foolhardy or both. We might express some concern for their future career development, but we might be ready and willing to acknowledge that the very idea of academic freedom means that there will always be some few committed to what seem to us to be mad or trivial pursuits.

  But M-theory is not the preserve of a few theorists who have become addicted to its beauty and the tightness of its structure, as Danish historian Helge Kragh explains:

  With M-theory and what followed, new recruits were attracted to the field. It is estimated that the string community amounts to some 1,500 scientists worldwide, a remarkably large number given the abstract and purely theoretical nature of string physics. Incidentally, this means that there are as many string theorists today as the total number of academic physicists in the world about 1900, all countries and fields of physics included.10

  Given that we can trace the first superstring revolution back nearly thirty years and the second revolution began seventeen years ago, we might well ask what all these theorists have been up to during this time.

  One of the big revelations of Witten’s M-theory conjecture concerns so-called dualities that prevail between different versions of superstring theory. For example, S-duality, or strong-weak duality, allows states with a coupling constant of a certain magnitude in one type of theory to be mapped to states with a coupling given by the reciprocal of this constant in its dual theory. This means that strong couplings which reflect the influence of strong forces in one theory would be weak couplings in the dual theory. So, the perturbation theory techniques used so successfully in QED, normally applicable only for systems of low energy and weak coupling, could now be applied to the dual theory to deal with problems involving high-energy, strong-coupling regimes.

  A lot of previously intractable calculations suddenly became tractable.

  Branes and braneworlds

  There was a lot more, however. Introducing the extra dimension in M-theory opened up an extraordinarily rich structure of mathematical objects. Superstring theory was no longer limited to one-dimensional strings. It could now accommodate higher-dimensional objects, called membranes, or ‘branes’, which quickly took over from strings as the primary focus of investigation, leading Duff to refer to it as ‘the theory formerly known as strings’.11

  Branes may have up to nine dimensions. A string is a ‘one-brane’. Two-dimensional membranes (sheets) are referred to as two-branes. Three-branes are actually three-dimensional spaces. Generically, these objects are referred to as p-branes, where p refers to the number of dimensions.

  Branes lie at the heart of superstring theory’s dualities. For example, particles in one version of the theory become branes in the dual theory. This kind of interrelationship lends credibility to the assumption that these are more than just mathematical curiosities — that branes reflect or represent physically real properties and behaviours of our universe. This is a perfectly logical inference, but we should be clear once again that there is no single piece of experimental or observational evidence to suggest that the fundamental elements of reality are multidimensional membranes.

  Branes are the principal objects of a theory which is conjectured to exist but which has yet to be written down. Exploring the mathematical relationships between and involving branes in various forms and connecting these with physically meaningful properties and behaviours requires another big assumption.

  The Brane Assumption. The higher-dimensional mathematical objects that arise in M-theory are assumed to have physical significance: they are assumed to describe aspects of empirical reality.

  A subcategory of p-branes, called ‘D-branes’, are particularly interesting because they represent locations in space where open strings can end.* Such objects not only represent different ways in which the various spatial dimensions can be occupied; they also possess shape and particlelike properties such as charge. Branes are dynamic objects; they move around and interact and are susceptible to forces.

  In fact, Witten and Czech theorist Petr Hořava constructed a model universe from two parallel ten-dimensional D-branes separated in the direction of the eleventh dimension. The space between branes is referred to as the ‘bulk’. They demonstrated that this structure is equivalent to a strong-coupling version of the original heterotic E8 × E8 superstring theory first devised by the Princeton String Quartet.

  The fascination with such ‘braneworlds’ derives from the fact that models can be created in which all material particles, which are composed of open strings, are permanently fastened to one or other of the D-branes, the spaces where the open strings end. Models can be set up so that these particles cannot detach from the brane and so cannot explore oth
er spatial dimensions that exist ‘at right angles’ to the dimensions of the D-brane. In other words, the particles cannot travel ‘at right angles to reality’.

  A ten-dimensional D-brane can be thought to consist of three ‘conventional’ spatial dimensions which extend off to infinity, six dimensions curled up and tucked away in a Calabi—Yau space, and time. The model demands that, once fixed to the D-brane, all the material particles of the standard model are then constrained to move in the three-dimensional space that we ourselves experience.

  But particles formed from closed strings, such as the graviton, are not so constrained. The gravitational force can therefore in principle explore all the spatial dimensions of the theory, including the eleventh dimension of the bulk. And because the standard model particles of familiar experience are stuck to one of the D-branes, there is no restriction in principle on the size of the eleventh dimension. It could be small, wrapped up into a tiny cylinder. Or it could be large.

  This was something of a revelation. It offers the possibility of using the difference between material particles and the standard model force-carriers (open strings) and the graviton (closed strings) to explain why gravity is so very different from the electromagnetic, weak and strong nuclear forces. The latter act on particles fixed to a D-brane. Gravity is free to roam.

  Could this be an answer to the hierarchy problem? It didn’t take superstring theorists too long to come up with some proposals. In 1998, theorists Nima Arkani-Hamed, Savas Dimopoulos and Gita Dvali (collectively referred to as ADD) devised a braneworld model in which the great gulf between the electro-weak mass-energy scale and the Planck mass-energy scale could be explained by the dilution of the force of gravity compared to other standard model forces.