FOUR

THE WORLD IN AN EQUATION

“If you are receptive and humble, mathematics will lead you by the hand.
Again and again, when I have been at a loss how to proceed, I have just had
to wait until I have felt the mathematics lead me by the hand. It has led me along an unexpected path,
a path where new vistas open up, a path leading to new territory, where one ca
set up a base of operations, from which one can survey the surroundings and plan future progress.”
— Paul Dirac, 197575

USING THE MOST POWERFUL radio telescope on earth, astronomers have just detected an encrypted signal emanating from Vega, one of the brightest stars in the sky and about twenty-five light years away. The message contains instructions for building a machine to teleport five human beings across space to meet with the extraterrestrials. After an intense international search, world leaders select the five delegates. Among them is the brilliant young Nigerian physicist Dr. Abonnema Eda, who has just won the Nobel Prize for discovering the theory of superunification, combining all known physics into a single, unified picture.

The storyline comes from Carl Sagan’s 1985 novel Contact, later made into a movie starring Jodie Foster. Sagan was a renowned U.S. astronomer and, with his TV series Cosmos, one of science’s greatest popularizers. In casting Eda as a hero in his novel, Sagan was making two points: first, that discovering the basic laws of the universe is a global, cross-cultural field of research. People from every country are fascinated by the same questions about how the world works. Second, genius knows no national boundaries. Although Africa has so far been woefully under-represented in the history of physics, like other disadvantaged regions, in the future it may be a source of incredible talent. Science benefits greatly from a diversity of cultures, each bringing a new stimulus of energy and ideas.

OVER THE PAST DECADE I have led a dual existence. On the one hand, I have been trying to understand how to describe the beginning and the far future of the universe. On the other, I have been fascinated by the problem of how to enable young people to enter science, especially in the developing world.

My interest is rooted in my African origins. As I described earlier, I was born in South Africa, where my parents were imprisoned for resisting the apartheid regime. Upon their release, we left as refugees, first to East Africa and then the U.K. When I was seventeen, I returned to Africa to teach for a year in a village mission school in Lesotho, a tiny, landlocked country surrounded by South Africa. Lesotho is one of the poorest nations on Earth: 80 percent of the jobs available are migrant labour, mostly in the mines over the border. In the village of Makhakhe, where I worked, I met many wonderful people and great kids with loads of potential but zero opportunity. No matter how bright or talented they were, they would never have the chances I’d had. A clerical position in the mines was the height of their aspiration.

The kids in the mission school were eager, responsive, and bright. But by and large, the education they’d received consisted of rote learning: memorizing times tables, copying notes from the blackboard and reproducing them in exams. They had no real experience of figuring things out or learning to think for themselves. School was a dry exercise you had to submit to: its sole purpose was to get a certificate. The teachers had been taught that way themselves, and they perpetuated a cycle of harsh discipline and low expectations.

I tried to take the kids outside as often as possible, to try to connect what we did in class with the real world. One day I asked them to estimate the height of the school building. I expected them to put a ruler next to the wall, stand back and size it up with finger and thumb, and make an estimate of the wall’s height. But there was one boy, very small for his age and the son of one of the poorest families in the village, who was scribbling with chalk on the pavement. A bit annoyed, I said, “What are you doing? I want you to estimate the height of the building.” He said, “I measured the height of a brick. Then I counted the number of bricks and now I’m multiplying.” Well, needless to say, I hadn’t thought of that!

People often surprised me with their enthusiasms and interests. Watching a soccer match at the school one day, I sat next to a miner, at home on his annual leave. He told me, “There’s only one thing that I really loved at school: Shakespeare.” And he recited some lines to me. Many similar experiences convinced me of the vast potential for intellectual development which is sorely needed for the continent’s progress.

Evolution was not on the school curriculum, because the church objected, but we nevertheless had excellent classroom discussions about it. Most African children are unaware of modern scientific discoveries showing how Homo sapiens originated in Africa around two hundred thousand years ago and began to migrate out of Africa between seventy thousand and fifty thousand years ago. I believe they could draw motivation from learning how humankind, and mathematics, and music and art, arose in Africa. Instead, young Africans are all too often made to feel like helpless bystanders, with every advance happening elsewhere in the world.

With the end of apartheid in 1994, my parents were allowed to return to South Africa. Both won election as members of the new parliament for the African National Congress, alongside Nelson and Winnie Mandela. They kept saying to me, “Can’t you come back and help in some way?” At the time, I was too busy with my own scientific career. Eventually, in 2001, I took a leave from my position at Cambridge to visit the University of Cape Town, near where my parents lived. Most of the time, I was pursuing new cosmological theories, like the cyclic universe scenario. But I took time out from my research to meet with colleagues and discuss what we might do to help speed Africa’s scientific development.

In these conversations, it quickly emerged that Africa’s deficiency in maths is a serious problem. There is an acute scarcity of engineers, computer scientists, and statisticians, making it impossible for industry to innovate or, more generally, for governments to make well-informed decisions regarding health, education, industry, transport, or natural resources. African countries are highly dependent on the outside world; they export raw and unprocessed commodities, and import manufactured goods and packaged food. Cellphones have transformed the lives of many Africans, but none are yet made in Africa. If Africa is to become self-­sufficient, it urgently needs to develop its own community of skilled people and scientists to adapt and invent the technologies that will allow it to catch up to the rest of the world.

And so we came up with the idea of setting up a centre called the African Institute for Mathematical Sciences, or AIMS, which would serve the continent (click to see photo). The idea was very simple: to recruit the brightest students from across Africa and the best lecturers from around the world for a program designed to turn Africa’s top graduates into confident thinkers and problem solvers, skilled in a range of techniques like mathematical modelling, data analysis, and computing. We would provide exposure to many scientific fields of great current relevance to Africa, like epidemiology, resource and climate modelling, and communications, but we would also mix in foundational topics like basic physics and pure mathematics.

Above all, we wanted to create a centre with a culture of excellence and a commitment to Africa’s development. The goal of the institute would be to open doors; encourage students to explore and develop their interests; discover which fields excited them the most; and assist them in finding opportunities. AIMS would help them forge their path to becoming scientists, technologists, educators, advisors, and innovators contributing to their continent’s growth.

With my parents’ encouragement, my brothers and I used a small family inheritance to purchase a disused, derelict hotel. It’s an elegant eighty-room, 1920s art deco building in a seaside suburb of Cape Town. Then, with the support of my colleagues in Cambridge, we formed a partnership involving Cambridge and Oxford universities in the U.K., Orsay in France, and the three main universities in Cape Town. We recruited a South African nuclear physicist, Professor Fritz Hahne, as the institute’s first director. I persuaded many of my academic colleagues to teach for three weeks each, and we advertised the program across Africa by emailing academic contacts and by sending posters to universities. In 2003, we opened AIMS, receiving twenty-eight students from ten different countries.

AIMS was launched as an experiment. We started it out of belief and commitment but had no real idea how or whether it would work. Most of us working to develop the project were academics, with little experience of creating an institution from scratch. For everyone involved, it was a wonderful learning experience. What we discovered was that when you take students from across a continent as diverse as Africa and put them together with some of the best teachers in the world, sparks fly.

South Africa has a strong, largely white, scientific community. Many of the local academics said to us, “Are you sure you really want to do this? You’re going to be spending all your time on remedial teaching. These students won’t know a thing.” How quickly AIMS proved them wrong! What the students lack in preparation, they make up for with motivation. Many students have had to overcome incredible difficulties, whether poverty, war, or loss of family members. These experiences make them value life, and the opportunity AIMS provides them, even more. They work harder than any students I have ever seen, and they feel and behave as if the world is opening up in front of them (click to see photo).

Teaching at AIMS is an unforgettable experience. One feels a tremendous obligation to teach clearly and well, because the students really want to learn and need to learn fast. There is a profound shared sense at AIMS that we are participating in the scientific transformation of a continent. And we are filled with the belief that when young Africans are given the chance to contribute, they will astonish everyone.

Take Yves, from the heart of Cameroon, from a peasant family with nine children. His parents could afford to send only one of them to university. Yves was the lucky one, and he is determined to live up to the opportunity and to prove what he can do. After graduating from AIMS, he took a Ph.D. in pure mathematics. Soon after, he won the prize for the best Ph.D. student presentation at the annual South African Mathematical Society conference. What an achievement, and what a powerful symbol that someone who comes from a poor family in an African village can become a leading young scientist.

In just nine years of existence, AIMS has graduated nearly 450 students from thirty-one different African countries. Thirty percent of them are women. Most come from disadvantaged backgrounds, and almost all have gone on to successful careers in research, academia, enterprise, industry, and government. Their success sends a powerful message of hope which undermines prejudice and inspires countless others. What more cost-effective investment could one possibly make in the future of a continent?

Ever since we started AIMS in Cape Town, our dream was to create a network of centres, providing outstanding scientific education across the continent. Specifically, our plan was to open fifteen AIMS centres. Each should serve as a beacon, a jewel in the local scientific and educational firmament, helping to spark a transformation in young people’s aspirations and opportunities.

In 2008, I was invited by the TED organization to make “a wish to change the world” at their annual conference in California, attended by some of the most influential people in Silicon Valley. My wish was “that you help us unlock and nurture scientific talent across Africa, so that within our lifetimes we are celebrating an African Einstein.” Using Einstein’s name in this way is not something a theoretical physicist does lightly, and I must admit that I was nervous. Before I did, I sounded out some of my most critical physicist colleagues to see how they would respond. To my delight, they were unreservedly enthusiastic. Science needs more Einsteins, and it needs Africa’s participation.

The idea of using Einstein’s name in our slogan came from from another remarkable AIMS student. Esra comes from Darfur in western Sudan. Her family suffered from the genocide there, in which tens of thousands were murdered and millions displaced. Esra was doing physics at Khartoum University before she made her way to AIMS. In spite of her family and community’s desperate problems back home, she somehow manages to remain cheerful.

One evening at AIMS, I lectured on cosmology. As usual, there was a lot of very animated discussion. At one point, I showed the Einstein equation for the universe and, as an aside, said, “Of course, we hope that among you there will be another Einstein.” The next day, a potential donor was coming to visit, and we asked a number of the students, including Esra, to speak. She ended her short, moving speech with the words: “We want the next Einstein to be African.” So when TED called me, a few weeks later, and asked if I had a wish, I knew immediately what it would be.

The slogan is deliberately intended to reframe the goals of international aid and development. Instead of seeing Africa as a problem continent, beset by war, corruption, poverty, and disease, and deserving of our charity, let us see it for what it can and should be: one of the most beautiful places on Earth, filled with talented people. Africa is an enormous asset and opportunity for the world. For too long, Africa has been exploited for its diamonds, gold, and oil. But the future will be all about Africa’s people. We need to believe in them and what they are capable of.

Modern society is built upon science and scientific ways of thinking. These are our most precious possessions and the most valuable things we can share. The training of African scientists, mathematicians, engineers, doctors, technologists, teachers, and other skilled people should be given the highest priority. And this should be done not in a patronizing way but in a spirit of mutual respect and mutual benefit. We need to see Africa for what it is: the world’s greatest untapped pool of scientific talent.

In encouraging young Africans to aim for the heights of intellectual accomplishment, we will give them the courage and motivation to pursue advanced technical skills. Among them will be not only scientists, but also people entering government or creating new enterprises: the African Gateses, Brins, and Pages of the future.

Last year, we opened our second centre, AIMS-­Sénégal, in a beautiful coastal nature reserve just south of Dakar. This year, the third AIMS centre opened in Ghana, in another attractive seaside location. AIMS-­Ethiopia will be next.

AIMS now receives nearly five hundred applications per year, and our graduates are already having a big impact in many scientific fields, from biosciences to natural resources and materials science, engineering, information technologies, and mathematical finance, as well as many areas of pure maths and physics. They are blazing a trail for thousands more to follow. AIMS will, we hope, serve as the seed for building great science across Africa.

Very recently, South Africa won the international competition to host what will be the world’s largest radio telescope — the Square Kilometre Array (SKA). The array will span 5,000 kilometres and include countries from Namibia to Kenya and Madagascar. It will be one of the most advanced scientific facilities in the world, placing Africa at the leading edge of science and helping to inspire a new generation of young African scientists. Among them may well be an Abonnema Eda.

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OVER THE COURSE OF the twentieth century, in pursuit of superunification, physicists have produced a one-line formula summarizing all known physics: in other words, the world in an equation (click to see photo). Much of it is written in Greek, in homage to the ancients. The mathematics of the Pythagoreans, and most likely the ancient Sumerians and Egyptians before them, lies at its heart. Their beliefs in the power of mathematical reasoning and the fundamental simplicity of nature have been vindicated to an extent that would surely have delighted them.

This magic formula’s accuracy and its reach, from the tiniest subatomic scales to the entire visible universe, is without equal in all of science. It was deciphered through the combined insights and labours of many people from all over the world. The formula tells us that the world operates according to simple, powerful principles that we can understand. And in this, it tells us who we are: creators of explanatory knowledge. It is this ability that has brought us to where we are and will determine our future.

Every atom or molecule or quantum of light, right across the universe, follows the magic formula. The incredible reliability of physical laws is what allows us to build computers, smartphones, the internet, and all the rest of modern technology. But the universe is not like a machine or a digital computer. It operates on quantum laws whose full meaning and implications we are still discovering. According to these laws, we are not irrelevant bystanders. On the contrary, what we see depends upon what we decide to observe. Unlike classical physics, quantum physics allows for, but does not yet explain, an element of free will.

Let us start from the left of the formula with Schrödinger’s wavefunction, Ψ, the capital Greek letter pronounced psi. Every possible state of the world is represented by a number, which you get using Ψ. But it isn’t an ordinary number; it involves the mysterious number i, the square root of minus one, which we encountered in Chapter Two. Numbers like this are called “complex numbers.” They are unfamiliar, because we don’t use them for counting or measuring. But they are very useful in mathematics, and they are central to the inner workings of quantum theory.

A complex number has an ordinary number part and an imaginary number part telling you how much of i it contains. The Pythagorean theorem says that the square of the length of the long side of a right-angled triangle is the sum of the squares of the other two sides. In just the same way, the square of the length of a complex number is given by the the sum of the squares of its ordinary and imaginary parts. And this is how you get the probability from the complex number given by Ψ. It is a tribute to the earliest mathematicians, that the very first mathematical theorem we know of turns out to lie at the centre of quantum physics.

When we decide to measure some feature of a system, like the position of a ball or the spin of an electron, there is a certain set of possible outcomes. Quantum theory tells us how to convert the wavefunction Ψ into a probability for each outcome, using the Pythagorean theorem. And this is all quantum theory ever predicts. Often, when we are trying to predict the behaviour of large objects, the probabilities will hugely favour one outcome. For example, when you drop a ball, quantum theory predicts it will fall with near certainty. But if you let a tiny subatomic particle go, its position will soon become more and more uncertain. In quantum theory, it is in general only large collections of particles which together behave in highly predictable ways.

On the right of the equation, there are two funny symbols, which look like tall, thin, stretched-out S’s. They are called integral signs, and they tie everything together. The large one tells you to add up the contributions from every possible history of the world that ends at that particular state. For example, if we let our little particle go at one position and wanted to know how likely it was to turn up at some other position at some later time, we would consider all the possible ways it could have travelled between the two positions. It might go at a fixed speed and in a straight line. Or it could jump over to the moon and back. Each one of these possible paths contributes to the final wavefunction, Ψ. It is as if the world has this incredible ability to survey every possible route to every possible future, and all of them contribute to Ψ. The U.S. physicist Richard Feynman discovered this formulation of quantum theory, known as the “sum over histories,” and it is the language in which our formula for all known physics is phrased.

What is the contribution of any one history? That is given by everything to the right of the large integral sign, ∫. First, we see the number named e by the eighteenth-­century Swiss mathematician Leonhard Euler. Its value is 2.71828 . . . If you raise e to a power, it describes exponential growth, found in many real-life situations, from the multiplication of bacteria in a culture, to the growth of money according to compound interest, or the power of computers according to Moore’s law. It even describes the expansion of the universe driven by vacuum energy.

But the use of e in the formula is cleverer than that. Euler discovered what is sometimes called “the most remarkable formula in mathematics,” connecting algebra and analysis to geometry: if you raise e to a power that is imaginary — meaning it is an ordinary number times i — you get a complex number for which the sum of the squares of the ordinary and imaginary parts is one. In quantum theory, this fact ensures that the probabilities for all possible outcomes add up to one. Quantum theory therefore connects algebra, analysis, and geometry to probability, combining almost all of the major areas of mathematics into our most fundamental description of nature.

In the formula for all known physics, e is raised to a power that includes all the known laws of physics in a combination called the “action.” The action is the quantity starting with the small integral sign, ∫. That symbol means you have to add up all six terms to the right of it, over all space and all time, leading up to the moment for which you wish to know the Schrödinger wavefunction Ψ. The action is just an ordinary number, but one that is associated with any possible history of the world.

As we discussed in Chapter Two, the formulation of the laws of physics in terms of an action was developed early in the nineteenth century by the Irish mathematical physicist William Rowan Hamilton. This combination of the classical laws of physics (as represented in the action), the imaginary number i, Planck’s constant h, and Euler’s number e together represent the quantum world. The two stretched out S’s represent its exploratory, holistic character. If only we could see inside our formula and directly experience the weird and remote quantum world without having to reduce it to a set of outcomes, each assigned a probability, we might see a whole new universe inside it.

LET US NOW WALK through the six terms in the action, which together represent all the known physical laws. In sequence, they are: the law of gravity; the three forces of particle physics; all the matter particles; the mass term for matter particles; and finally, two terms for the Higgs field.

In the first term, gravity is represented by the curvature of spacetime, R, which is a central quantity in Einstein’s theory of gravity. Also appearing is G, Newton’s universal constant of gravitation. This is all that remains, in fundamental physics, of Newton’s original laws of motion and gravity.

In the second term, F stands for fields like those James Clerk Maxwell introduced to describe electric and magnetic forces. In our very compact notation, the term also represents the fields of the strong nuclear force, which holds atomic nuclei together, and the weak nuclear force, which governs radioactivity and the formation of the chemical elements in stars. Both are described using a generalization of Maxwell’s theory developed in the 1950s by Chinese physicist Chen-Ning Yang and U.S. physicist Robert Mills. In the 1960s, U.S. physicists Sheldon Lee Glashow and Steven Weinberg and Pakistani physicist Abdus Salam unified the weak nuclear force and electromagnetism into the “electroweak” theory. In the early 1970s, Dutch physicist Gerard ’t Hooft and his doctoral advisor, Martinus Veltman, demonstrated the mathematical consistency of quantum Yang-Mills theory, adding great impetus to these models. And soon after, U.S. physicists David J. Gross, H. David Politzer, and Frank Wilczek showed that the strong nuclear force could also be described by a version of Yang–Mills theory.

The third term was invented in 1928 by the English physicist Paul Dirac. In thinking about how to combine relativity with quantum mechanics, he discovered an equation that describes elementary particles like electrons. The equation turned out to also predict the existence of antimatter particles. Dirac noted that for every particle — like the electron, with a definite mass and electric charge — his equation predicted another particle, with exactly the same mass but the opposite electric charge. This stunning prediction was made in 1931; the following year, the U.S. physicist Carl D. Anderson detected the positron, the electron’s antimatter partner, with the exact predicted properties.

Dirac’s equation describes all the known matter particles, including electrons, muons, taons, and their neutrinos, and six different types of quarks. Each one has a corresponding antimatter particle. Both the particles and the antiparticles are quanta of a Dirac field, denoted by ψ, the lower case Greek letter psi. The Dirac term in the action also tells you how all these particles interact through the strong and electroweak forces and gravity.

The fourth term was introduced by the Japanese physicist Hideki Yukawa, and developed into its detailed, modern form by his compatriots Makoto Kobayashi and Toshihide Maskawa in 1973. This term connects Dirac’s field ψ to the Higgs field φ, which we shall discuss momentarily. The Yukawa–Kobayashi–Maskawa term describes how all the matter particles get their masses, and it also neatly explains why antimatter particles are not quite the perfect mirror images of their matter particle counterparts.

Finally, there are two terms describing Higgs field φ, the lower case Greek letter pronounced phi. The Higgs field is central to the electroweak theory.

One of the key ideas in particle physics is that the force-carrier fields and matter particles, all described by Maxwell–Yang–Mills theory or Dirac’s theory, come in several copies. In the early 1960s, a theoretical mechanism was discovered for creating differences between the copies, giving them different masses and charges. This is the famous Higgs mechanism. It was inspired by the theory of superconductivity, where the electromagnetic fields are squeezed out of superconductors. Philip Anderson, a famous U.S. condensed matter physicist, suggested that this mechanism might operate in the vacuum of empty space. The idea was subsequently combined with Einstein’s theory of relativity by several particle theorists, including the Belgian physicists Robert Brout and François Englert and the English physicist Peter Higgs. The idea was further developed by the U.S. physicists Gerald Guralink and Carl Hagen, working with the English physicist Tom Kibble, who I was fortunate to have as one of my mentors during my Ph.D.

The Higgs mechanism lies at the heart of Glashow, Salam, and Weinberg’s theory, in which the electroweak Higgs field φ is responsible for separating Maxwell’s electromagnetic force out from the weak nuclear force, and fixing the basic masses and charges of the matter particles.

The last term, the Higgs potential energy, V(φ), ensures that the Higgs field φ takes a fixed constant value in the vacuum, everywhere in space. It is this value that communicates a mass to the quanta of the force fields and to the matter particles. The Higgs field can also travel in waves — similar to electromagnetic waves in Maxwell’s theory — that carry energy quanta. These quanta are called “Higgs bosons.” (click to see photo) Unlike photons, they are fleetingly short-lived, decaying quickly into matter and antimatter particles. They have just been discovered at the Large Hadron Collider in the CERN laboratory in Geneva, confirming predictions made nearly half a century ago (click to see photo).

Finally, the value of the Higgs potential energy, V, in the vacuum also plays a role in fixing the energy of empty space — the vacuum energy — measured recently by cosmologists.

Taken together, these terms describe what is known as the “Standard Model of Particle Physics.” The quanta of force fields, like the photon and the Higgs boson, are called “force-carrier particles.” Including all the different spin states, there are thirty different force-carrier particles in total, including photons (quanta of the electromagnetic field), W and Z bosons (quanta of the weak nuclear force field), gluons (quanta of the strong force), gravitons (quanta of the gravitational field), and Higgs bosons (quanta of the Higgs field). The matter particles are all described by Dirac fields. Including all their spin and antiparticle states, there are a total of ninety different matter particles. So, in a sense, Dirac’s equation describes three-quarters of known physics.

When I was starting out as a graduate student I found the existence of this one-line formula, which summarizes everything we know about physics, hugely motivating. All you have to do is master the language and learn how to calculate, and in principle you understand at a basic level all of the laws governing every single physical process in the universe.

· · ·

YOU MAY WELL WONDER how it was that physics converged on this remarkably simple unified formula. One of the most important ideas guiding its development was that of “symmetry.” A symmetry of a physical system is a transformation under which the system does not change. For example, a watch ticks at exactly the same rate wherever you place it, because the laws governing the mechanism of the watch do not depend on where the watch is. We say the laws have a symmetry under moving the watch around in space. Similarly, the watch’s working is unchanged if we rotate the watch — we say the laws have a symmetry under rotations. And if the watch works just the same today, or tomorrow, or yesterday, or an hour from now, we say the laws that govern it have a symmetry under shifts in time.

The entry of these ideas of symmetry into physics traces back to a remarkable woman named Emmy Noether, who in 1915 discovered one of the most important results in mathematical physics. Noether showed mathematically that any system described by an action that is unchanged by shifts in time, as most familiar physical systems are, automatically has a conserved energy. Likewise, for many systems it makes no difference to the evolution of the system exactly where the system is located in space. What Noether showed in this case is that there are three conserved quantities — the three components of momentum. There is one of these for each independent direction in which you can move the system without changing it: east–west, north–south, up–down.

Ever since Newton, these quantities — energy and momentum — had been known, and found to be very useful in solving many practical problems. For example, energy can take a myriad forms: the heat energy stored in a boiling pan of water, the kinetic energy of a thrown ball, the potential energy of a ball sitting on a wall and waiting to fall, the radiation energy carried in sunlight, the chemical energy stored up in oil or gas, or the elastic energy stored in a stretched string. But as long as the system is isolated from the outside world, and as long as spacetime is not changing (which is an excellent approximation for any real experiment conducted on Earth), the total amount of energy will remain the same.

The total momentum of a system is another very useful conserved quantity, for example in describing the outcome of collisions. Similarly, Benjamin Franklin’s law of electric charge conservation, that you can move charge around but never change its total amount, is another consequence of Noether’s theorem.

Before Emmy Noether, no one had really understood why any of these quantities are conserved. What Noether realized was as simple as it was profound: the conservation laws are mathematical consequences of the symmetries of space and time and other basic ingredients in the laws of physics. Noether’s idea was critical to the development of the theories of the strong, weak, and electroweak forces. For example, in electroweak theory, there is an abstract symmetry under which an electron can be turned into a neutrino, and vice versa. The Higgs field differentiates between the particles and breaks the symmetry.

Noether was an extraordinary person. Born in Germany, she faced discrimination as both a Jew and a woman. Her father was a largely self-taught mathematician. The University of Erlangen, where he lectured, did not normally admit women. But Emmy was allowed to audit classes and was eventually given permission to graduate. After struggling to complete her Ph.D. thesis (which she later, with typical modesty, dismissed as “crap”) she taught for seven years at the university’s Mathematical Institute, without pay.

She attended seminars at Göttingen given by some of the most famous mathematicians of the time — David Hilbert, Felix Klein, Hermann Minkowski, and Hermann Weyl — and through these interactions her great potential became evident to them. As soon as Göttingen University’s restrictions on women lecturers were removed, Hilbert and Klein recruited Noether to teach there. Against great protests from other professors, she was eventually appointed — again, without pay. In 1915, shortly after her appointment, she discovered her famous theorem.

Noether’s theorem not only explains the basic conserved quantities in physics, like energy, momentum, and electric charge, it goes further. It explains how Einstein’s equations for general relativity are consistent even when space is expanding and energy is no longer conserved. For example, as we discussed in the previous chapter, it explains how the vacuum energy can drive the exponential expansion of the universe, creating more and more energy without violating any physical laws.

When Noether gave her explanation for conserved quantities and more general situations involving gravity, she did so in the context of classical physics and its formulation in terms of Hamilton’s action principle. Half a century later, it was realized — by the Irish physicist John Bell, along with U.S. physicists Steven Adler and Roman Jackiw — that quantum effects, included in Feynman’s sum over histories, could spoil the conservation laws that Noether’s argument predicted.

Nevertheless, it turns out that for the pattern of particles and forces seen in nature, there is a very delicate balance (known technically as “anomaly cancellation”) that allows Noether’s conservation laws to survive. This is another indication of the tremendous unity of fundamental physics: the whole works only because of all of the parts. If you tried, for example, to remove the electron, muon, and tanon and their neutrinos from physics and kept only the quarks, then Noether’s symmetries and conserved quantities would be ruined and the theory would be mathematically inconsistent. This idea, that Noether’s laws must be preserved within any consistent unified theory, has been a key guiding principle in the development of unified theories, including string theory, in the late twentieth century.

Noether’s dedicated mentorship of students was exemplary — she supervised a total of sixteen Ph.D. students through a very difficult time in Germany’s history. When Hitler came to power in 1933, Jews became targets. Noether was dismissed from Göttingen, as was her colleague Max Born. The great mathematical physicist Hermann Weyl, also working there, wrote later: “Emmy Noether — her courage, her frankness, her unconcern about her own fate, her conciliatory spirit — was, in the midst of all the hatred and meanness, despair and sorrow surrounding us, a moral solace.”76

Eventually, Noether fled to the United States, where she became a professor at Bryn Mawr College, a women’s college known particularly as a safe haven for Jewish women. Sadly, at the age of fifty-three she died of complications relating to an ovarian cyst.

In a letter to the New York Times, Albert Einstein wrote: “In the judgement of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began. In the realm of algebra, in which the most gifted mathematicians have been busy for centuries, she discovered methods that have proved of enormous importance in the development of the present-day younger generation of mathematicians.” 77 Emmy Noether was a pure soul whose mathematical discoveries opened many paths in physics and continue to exert great influence.

PAUL DIRAC WAS ANOTHER mathematical prodigy from a humble background. His discoveries laid the basis for the formula for all known physics. A master of quantum theory, he was largely responsible for its current formulation. When asked what his greatest discovery had been, he said he thought it was his “bra-ket” notation. This is a mathematical device which he introduced into quantum theory to represent the different possible states of a system. The initial state is called a “ket” and the final state a “bra.” It’s funny that someone who discovered the equation for three-quarters of all the known particles, who predicted antimatter, and who made countless other path-breaking discoveries would rate them all below a simple matter of notation. As with many other Dirac stories, one can’t help thinking: he can’t have been serious! But no one could tell.

A recent biography called Dirac “the strangest man.” He was born in Bristol, England, to a family of modest means. His Swiss father, Charles Dirac, was a French teacher and a strict disciplinarian. Paul led an unhappy, isolated childhood, although he was always his father’s favourite. He was fortunate to attend one of the best non-fee-paying schools for science and maths in England — Merchant Venturers’ Technical College in Bristol, where his father taught.

At school, it became clear that Paul had exceptional mathematical talent, and he went on to Cambridge to study engineering. In spite of graduating with a first-class degree, he could not find a job in the postwar economic climate. Engineering’s loss was physics’ gain: Dirac returned to Bristol University to take a second bachelor’s degree, this time in mathematics. And then, in 1923, at the ripe old age of twenty-one, he returned to St. John’s College in Cambridge to work towards a Ph.D. in general relativity and quantum theory.

Over the next few years, this shy, notoriously quiet young man — almost invisible, according to some — made a series of astonishing breakthroughs. His work combined deep sophistication with elegant simplicity and clarity. For his Ph.D., he developed a general theory of transformations that allowed him to present quantum theory in its most elegant form, still used today. At the age of twenty-six, he discovered the Dirac equation by combining relativity and quantum theory to describe the electron. The equation explained the electron’s spin and predicted the existence of the electron’s antiparticle, the positron. Positrons are now used every day in medical PET (positron emission tomography) scans, used to track the location of biological molecules introduced into the body.

When someone asked Dirac, “How did you find the Dirac equation?” he is said to have answered with: “I found it beautiful.” As often, he seemed to take pleasure in being deliberately literal and in using as few words as he could. His insistence on building physics on principled mathematical foundations was legendary. In spite of having initiated the theory of quantum electrodynamics, which was hugely successful, he was never satisfied with it. The theory has infinities, created by quantum fluctuations in the vacuum. Other physicists, including Richard Feynman, Julian Schwinger, and Sin-Itiro Tomonaga, found ways to control the infinities through a calculational technique known as “renormalization.” The technique produced many accurate predictions, but Dirac never trusted it because he felt that serious mathematical difficulties were being swept under the rug. He went so far as to say that all of the highly successful predictions of the theory were probably “flukes.”

Dirac also played a seminal role in anticipating the form of our formula for all known physics. For it was Dirac who saw the connection between Hamilton’s powerful action formalism for classical physics and the new quantum theory. He realized how to go from a classical theory to its quantum version, and how quantum physics extended the classical view of the world. In his famous textbook on quantum theory, written in 1930 and based upon this deep understanding, he outlined the relationship between the Schrödinger wavefunction, Hamilton’s action, and Planck’s action quantum. Nobody followed up this insightful remark until 1946, when Dirac’s comment inspired Feynman, who made the relation precise.

Dirac continued throughout his life to initiate surprising and original lines of research. He discussed the existence of magnetic monopoles and initiated the first serious attempt to quantize gravity. Although he was one of quantum theory’s founders, Dirac clearly loved the geometrical Einsteinian view of physics. In some ways, one can view Dirac as a brilliant technician, jumping off in directions that had been inspired by Einstein’s more philosophical work.

In his Scientific American article in May 1963, titled “The Evolution of the Physicists’ Picture of Nature,” he says, “Quantum theory has taught us that we have to take the process of observation into account, and observations usually require us to bring in the three-­dimensional sections of the four-dimensional picture of the universe.” What he meant by this was that in order to calculate and interpret the predictions of quantum theory, one often has to separate time from space. Dirac thought that Einstein’s spacetime picture and the split into space and time created by an observer were fundamental and unlikely to change. But he suspected that quantum theory and Heisenberg’s uncertainty relations would probably not survive in their current form. “Of course, there will not be a return to the determinism of classical physical theory. Evolution does not go backward,” he says. “There will have to be some new development that is quite unexpected, that we cannot make a guess about, which will take us still further from classical ideas.”

Many physicists regarded the unworldly Dirac with awe. Niels Bohr said, “Of all physicists, Dirac has the purest soul.” And “Dirac did not have a trivial bone in his body.” 78 The great U.S. physicist John Wheeler said, simply, “Dirac casts no penumbra.”79

I met Dirac twice, both times at summer schools for graduate students. At the first, in Italy, he gave a one-hour lecture on why physics would never make any progress until we understood how to predict the exact value of the electric charge carried by an electron. During the school, there was an evening event called “The Glorious Days of Physics,” to which many of the great physicists from earlier days had been invited. They did their best to inspire and encourage us students with stories of staying up all night poring over difficult problems. But Dirac, the most distinguished of them all, just stood up and said, “The 1920s really were the glorious days of physics, and they will never come again.” That was all he said — not exactly what we wanted to hear!

At the second summer school where I met him, in Edinburgh, another lecturer was excitedly explaining supersymmetry — a proposed symmetry between the forces and matter particles. He looked to Dirac for support, repeating Dirac’s well-known maxim that mathematical beauty was the single most important guiding principle in physics. But again Dirac rained on the parade, saying, “What people never quote is the second part of my statement, which is that if there is no experimental evidence for a beautiful idea after five years, you should abandon it.” I think he was, at least in part, just teasing us. In his Scientific American article80 he gave no such caveat. Writing about Schrödinger’s discovery of his wave equation, motivated far more by theoretical than experimental arguments, Dirac said, “I believe there is a moral to this story, namely that it is more important to have beauty in one’s equations than to have them fit experiment.”

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 framework 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.

And that, I think, will be the key to future scientific breakthroughs. If you look at the people who made the most important contributions in the past, many were among the first members of their societies to get involved in serious science. Many faced discrimination and prejudice. In overcoming these obstacles they had a point to prove, which encouraged them to question traditional thinking. As we saw in Chapter Two, many of the most prominent twentieth-century physicists were Jewish, yet until the mid-nineteenth century Jews had been deliberately excluded from science and technical subjects at many universities across Europe. When they finally gained access, they were hugely motivated to disprove their doubters, to show that Jews could do every bit as well as anyone else. Einstein, Bohr, Born, and Noether were part of an influx of new talent that completely revolutionized physics in the early twentieth century.

Which brings me back to the question of unification, both of people across the planet and of our understanding of the world. The search for a superunified theory is an extremely ambitious goal. A priori, it would seem to be hopeless: we are tiny, feeble creatures dwarfed by the universe around us. Our only tools are our minds and our ingenuity. But these have enabled us to come amazingly far. If we think of the world today, with seven billion minds, many in emerging economies and societies, it is clear that there is a potential gold mine of talent. What is needed is to open avenues for gifted young people to enter and contribute to science, no matter what their background. If opportunities are opened, we can anticipate waves of motivated, original young people capable of transformative discoveries.

Who are we, in the end? As far as we know, we represent something very rare in the universe — the organization of matter and energy into living, conscious beings. We have learned a great deal about our origins — about how the universe emerged from the singularity filled with a hot plasma; how the chemical elements were created in the big bang and stars and supernovae; how gravity and dark matter clumped molecules and atoms together into galaxies, stars, and planets; how Earth cooled and allowed lakes and oceans to condense, creating a primordial soup within which the first life arose. We do not know exactly how life started, but once the first self-regulating, self-replicating organisms formed, containing the DNA-protein machinery of life, reproduction, competition, and natural selection drove the evolution of more and more complex living organisms. We humans stand now on the threshold of a new phase of evolution, in which technology will play as much of a role as biology.

Great mysteries remain. Why did the universe emerge from the big bang with a set of physical laws that gave rise to heavy elements and allowed complex chemistry? Why did these laws allow for planets to form around stars, with water, organic molecules, an atmosphere, and the other requirements for life? Why did the DNA-protein machinery, developed and selected for in the evolution of primitive single-cell organisms, turn out to be able to code for complex creatures, like ourselves? How and why did consciousness emerge?

At every stage in the history of the universe, there was the potential for vastly more than what had been required to reach that stage. Today, this is more true than ever. Our understanding of the universe has grown faster than anyone could have imagined a century ago, way beyond anything that could be explained in terms of past evolutionary advantage. We cannot know what new technologies we will create, but if the past is any guide, they will be extraordinary. Commercial space travel is about to become a reality. Quantum computers are on the horizon, and they may completely transform our experience of the world. Are all these capabilities simply accidental? Or are we actually the door-openers to the future? Might we be the means for the universe to gain a consciousness of itself?