The Universe Within

by Neil Turok

  Dirac ended his article by advocating the exploration of interesting mathematics as one way for us to discover new physical principles: “It seems to be one of the fundamental features of nature that fundamental physical laws are described in terms of a mathematical theory of great beauty and power, needing quite a high standard of mathematics for one to understand it. You may wonder: why is nature constructed along these lines? One can only answer that our present knowledge seems to show that nature is so constructed. We simply have to accept it. One could perhaps describe the situation by saying that God is a mathematician of a very high order and He used very advanced mathematics in constructing the universe. Our feeble attempts at mathematics enable us to understand a bit of the universe, and as we proceed to develop higher and higher mathematics we can hope to understand the universe better.”

  Dirac’s God was, I believe, the same one that Einstein or the ancient Greeks would have recognized: the God that is nature and the universe, and whose works epitomize the very best in rationality, order, and beauty. There is no higher compliment that Dirac can pay than to call God “a mathematician of a very high order.” Note, even here, Dirac’s understatement.

  Perhaps because of his shy, taciturn nature and his technical focus, Dirac is far less famous than other twentieth-­century physics icons. But his uniquely logical, mathematical mind allowed him to articulate quantum theory’s underlying principles more clearly than anyone else. After the 1930s, he initiated a number of research directions far ahead of his time. Above all, his uncompromising insistence on simplicity and absolute intellectual honesty continues to inspire attempts to improve on the formula he did so much to found.

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  AS BEAUTIFUL AS IT is, we know our magic formula isn’t a final description of nature. It includes neither dark matter nor the tiny masses of neutrinos, both of which we know to exist. However, it is easy to conceive of amendments to the formula that would correct these omissions. More experimental evidence is needed to tell us exactly which one of them to include.

  The second reason that the formula is unlikely to be the last word is an aesthetic one: as it stands it is only superficially “unified.” Buried in its compact notation are no less than nineteen adjustable parameters, fitted to experimental measurements.

  The formula also suffers from a profound logical flaw. Starting in the 1950s, it was realized that in theories like quantum electrodynamics or electroweak theory, vacuum fluctuations can alter the effective charges on matter particles at very short distances, in such a way as to make theories inconsistent. Technically, the problem is known as the “Landau ghost,” after the Russian physicist Lev D. Landau.

  The problem was circumvented by “grand unified” theories when they were introduced in the 1970s. The basic idea was to combine Glashow, Salam, and Weinberg’s electroweak force and Gross, Politzer, and Wilczek’s strong nuclear force into a single, grand unified force. At the same time, all the known matter particles would be combined into a single, grand unified particle. There would be new Higgs fields to separate out the strong and electroweak forces and distinguish the different matter particles from one another. These theories overcame Landau’s problem, and for a while they seemed to be mathematically consistent descriptions of all the known forces except gravity.

  Further encouragement came from calculations that extrapolated the strong force and the two electroweak forces to very short distances. All three seemed to unify nicely at a minuscule scale of around a ten-­thousand-trillionth the size of a proton, the atomic nucleus of hydrogen. For a while, from aesthetic and logical grounds as well as hints from the data, this idea of grand unification seemed very appealing. The devil is in the details, however. There turned out to be a great number of different possible grand unified theories, each involving different fields and symmetries. There are a large number of adjustable parameters that have to be fitted to the observed data. The early hints of unification at very tiny scales faded as measurements improved: unification could only be achieved by adding even more fields. Instead of making physics simpler and more beautiful, grand unified theories have, so far, turned out to make it more complex and arbitrary.

  A second reason to question grand unification is that its most striking predictions have not been confirmed. If at the most fundamental level there is only one type of particle, and if all of the differences between the particles we see are due to Higgs fields in the vacuum, then there should be physical processes allowing any one kind of particle to turn into any other kind of particle by burrowing quantum-mechanically through the grand unified Higgs field. One of the most dramatic such processes is proton decay, which would cause the proton, one of the basic constituents of atomic nuclei, to decay into lighter particles. If the prediction is correct, then all atoms will disappear, albeit at an extremely slow rate. For many years, researchers have searched for signals of this process in very large tanks of very clean water, observed with highly sensitive light detectors capable of detecting the process of nuclear decay, but so far without success.

  But the strongest reason to doubt grand unification is that it ignores the force of gravity. At a scale not too far below the grand unified scale — about a thousand times smaller — we reach the Planck scale, a ten-million-­trillionth the size of a proton, where the vacuum fluctuations start to wreak havoc with Einstein’s theory of gravity. As we go to shorter wavelengths, the quantum fluctuations become increasingly wild, causing space­time to become so curved and distorted that we cannot calculate anything. As beautiful as it is, we believe Einstein’s theory, as included in the formula, to be only a stand-in. We need new mathematical principles to understand how spacetime works at very short distances.

  At the far right of the formula, the Higgs potential energy, V, also poses a conundrum. Somehow, there is an extremely fine balance in the universe between the contribution from V and the contributions from vacuum fluctuations, a fine balance that results in a minuscule positive vacuum energy. We do not understand how this balance occurs. We can get the formula to agree with observations by adjusting V to 120 decimal places. It works, but it gives us no sense that we know what we are doing.

  To summarize: all the physics we know can be combined into a formula that, at a certain level, demonstrates how powerful and connected the basic principles are. The formula explains many things with exquisite precision. But in addition to its rather arbitrary-looking pattern of particles and forces, and its breakdown at extremely short distances due to quantum fluctuations, it has two glaring, overwhelming failures. So far, it fails to make sense of the universe’s singular beginning and its strange, vacuous future.

  In practice, physicists seldom use the complete formula. Most of physics is based on approximations, on knowing which parts of the formula to ignore and how to simplify the parts you keep. Nevertheless, many predictions based on the formula have been worked out and verified, sometimes with extreme precision. For example, an electron has spin, and this causes it to behave in some respects like a tiny bar magnet. The relevant parts of the formula allow you to calculate the strength of this little bar magnet to a precision of about one part in a trillion. And the calculations agree with experiment.

  For anything even slightly more complicated — like the structure of complex molecules, or the properties of glass or aluminum, or the flow of water — we are unable to work out all of the predictions because we are not good enough at doing the math, even though we believe the formula contains within it all the right answers. In the future, as I will describe in the next chapter, the development of quantum computers may completely transform our ability to calculate and to translate the magic formula directly into predictions for many processes far beyond the reach of computation today.

  HOW SHALL THE BASIC problems of the indescribable beginning and the puzzling future of the universe be resolved? The most popular candidate for replacing our formula for all known physics is a radically different framewo
rk known as string theory, as mentioned in the previous chapter. String theory was discovered more or less by accident in 1968, by a young Italian post-doctoral researcher named Gabriele Veneziano, working at the European Organization for Nuclear Research (CERN) in Geneva. Veneziano wasn’t looking for a unified theory; he was trying to fit experimental data on nuclear collisions. By chance, he came across a very interesting mathematical formula invented by the eighteenth-­century Swiss mathematician Leonhard Euler — the very same Euler whose mathematical discoveries are central to the formula for all known physics.

  Veneziano found he could use another formula of Euler’s, called “Euler’s beta function,” to describe the collisions of nuclear particles in an entirely new way. Veneziano’s calculations caused great excitement at the time, and even more so when it was realized that they were describing the particles as if they were little quantum pieces of string, an entirely different picture from that of quantum fields. Ultimately, the idea failed as a description of nuclear physics. It was superseded by the field theories of the strong and weak nuclear forces, and by the understanding that nuclear particles are complicated agglomerations of fields held together by vacuum fluctuations. But the mathematics of string theory turned out to be very rich and interesting, and during the early 1970s, it was developed rapidly.

  String is envisaged as a form of perfect elastic. It can exist as pieces with two ends or in the form of closed loops. Waves travel along it at the speed of light. And pieces of string can vibrate and spin in a myriad ways. One of string theory’s most attractive features is that just one entity — string — describes an infinite variety of objects. So string theory is a highly unified theory.

  In 1974, French physicist Joël Scherk and U.S. physicist John Schwarz realized that a closed loop of string, also spinning end over end, behaved like a graviton, the basic quantum of Einstein’s theory of gravity. And so it turned out that string theory automatically provided a theory of quantum gravity, a totally unexpected discovery. Even more surprising, string theory seems to be free of the infinities that plague more conventional approaches to quantum gravity. In the mid-1980s, just as hopes for a grand unified theory were fading, string theory came along as the next candidate for a theory of everything.

  As I discussed in Chapter Three, one of string theory’s features is that it requires the existence of extra dimensions of space. In addition to the familiar three dimensions of space — length, height, and breadth — the simplest string theories require six more space dimensions, and M-theory, which I also described, requires one more, bringing the number of extra dimensions to seven. The six extra dimensions of string theory can be curled up in a little ball, so small that we would not notice them in today’s universe. And the seventh dimension of M-theory is even more interesting. It takes the form of a gap between two three-dimensional worlds. This picture was the basis for the cyclic model of cosmology that I explained in the previous chapter.

  Although there were great expectations that string theory would solve the problem of the unification of forces, these hopes have also dimmed. The main problem is that, like grand unified theories, string theory is itself too arbitrary. For example, it turns out to be possible to curl up the six or seven extra dimensions in an almost infinite number of ways. Each one would lead to a three-dimensional world with a different pattern of particles and forces. Most of these models are hopelessly unrealistic. Still, many researchers hope that by scanning through this “landscape” of possible string theory universes, they may find the right one. Some even believe that every one of these landscapes of universes must be realized somewhere in the actual universe, although only one of them would be visible to us. This picture, called the “inflationary multiverse,” has to be one of the most extravagant proposals in the history of science.

  From my own point of view, none of these string theory universes is yet remotely realistic, because string theory has so far proven incapable of describing the initial singularity, the problem I outlined in the previous chapter. The string theory landscape, so far as it is currently understood, consists of a set of empty universe models. But there are serious grounds for doubt as to whether these empty models can actually be used to describe expanding universes full of matter and radiation, like ours.

  Rather than speculate about a “multiverse” of possible universes, I prefer to focus on the one we know exists, and try to understand the principles that might resolve its major puzzles: the singularity and the distant future. String theory is a powerful theoretical tool that has already provided completely new insights into quantum gravity. But there is some way to go before it is ready to convincingly describe our universe.

  THE SITUATION IN WHICH string theory finds itself is in many ways a reflection of how fundamental physics developed over the course of the twentieth century. In the early part of the century came the great ideas of quantum physics, spacetime, and general relativity. There was great philosophical richness in the debates over these matters, with much fewer publications and conferences than today, and a greater premium on originality. In the late 1920s, with the establishment of quantum theory and the quantum theory of fields, attention turned to more technical questions. Physicists focused on applications and became more like technicians. They extended the reach of physics to extremely small and large distances without having to add any revolutionary new ideas.

  Physics became a fertile source of new technologies — everything from nuclear power to radar and lasers, to transistors, LEDs, integrated circuits and other devices, to medical X-ray, PET, and NMR scans, and even superconducting trains. Particle accelerators probing very high energies made spectacular discoveries — of quarks, of the strong and electroweak forces, and most recently of the Higgs boson. Cosmology became a true observational science, and dedicated satellites mapped the whole universe with exquisite precision. Physics seemed to be steaming towards a final answer, towards a theory of everything.

  From the 1980s on, waves of enthusiasm swept the field only to die out nearly as quickly as they arose. Publications and citations soared and conferences multiplied, but genuinely new ideas were few and far between. The mainstreaming of grand unification and string theory, and the sheer pressure it created to force a realistic model out of incomplete theoretical frameworks so far has been dissatisfying.

  The development in physics is, I think, a kind of ultraviolet catastrophe, like the one Planck and Einstein discovered in classical physics at the start of the twentieth century. They are consequences of mechanical ways of thinking. I believe it is time for physics to step away from contrived models, whether artificial mathematical constructs or ad hoc fits to the data, and search for new unifying principles. We need to better appreciate the magic we have discovered, and all of its limitations, and find new ways to see into and beyond it.

  Every term in our formula required a giant leap of the imagination — from Einstein’s description of gravity, to Dirac’s description of the electron and other particles, to Feynman’s formulation of quantum mechanics as a sum over all possible histories. We need to foster opportunities for similar leaps to be made. We need to create a culture where the pursuit of deep questions is encouraged and enabled: where the philosophical richness and depth of an Einstein or Bohr combine with the technical brilliance of a Heisenberg or Dirac.

  As I have emphasized, some of the greatest contributions to physics were made by people from very ordinary backgrounds who, more or less through chance, came to work on fundamental problems. What they had in common was the boldness to follow logical ideas to their conclusion, to see connections everyone else had missed, to explore unknown territories, and to play with entirely new ideas. And this boldness produced leaps of understanding way beyond everyday experience, way beyond our circumstances and our history, leaps which we can all share.

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  WHEN CHILDREN GO TO school, we teach them algebra and geometry, physics according to Newton’s laws, and
so on, but as far as I know, nobody says anything about the fact that physics has discovered a blueprint of the universe. Although the formula takes many years of study to fully understand and appreciate, I believe it is inspirational to realize how far we have come towards combining the fundamental laws that govern the universe.

  In its harmonious and holistic nature, the formula is, I believe, a remarkable icon. All too often, our society today is driven by selfish behaviour and rigid agendas — on the one hand by people and groups pursuing their own short-term interests, and on the other by appeals to preconceived systems that are supposed to solve all our problems. But almost all of the traditional prescriptions have failed in the past, and they are all prone to being implemented in inhuman ways. It seems to me that as we enter a period of exploding human demand and increasingly limited resources, we need to look for more intelligent ways to behave.

  The formula suggests principles that might be more useful. In finding the right path for society, perhaps we need to consider all paths. Just as quantum theory explores all options and makes choices according to some measure of the “benefit,” we need to run our societies more creatively and responsively, based on a greater awareness of the whole. The world is not a machine that we can set in some perfect state or system and then forget about. Nor can we rely on selfish or dogmatic agendas as the drivers of progress. Instead, we need to take an informed view of the available options and be far-sighted enough to choose the best among them.

  It is all too easy to define ourselves by our language, nationality, religion, gender, politics, or culture. Certainly we should celebrate and draw strength from our diversity. But as our means of communications amplify, these differences can create confusion, misunderstanding, and tension. We need more sources of commonality, and our most basic understanding of the universe, the place we all share, serves as an example. It transcends all our differences and is by far the most reliable and cross-cultural description of the world we have. There is only one Dirac or Einstein or Maxwell equation, and each of these is so simple, accurate, and powerful that people from any and all backgrounds find it utterly compelling. Even the failures of our formula are something we can all agree about.