BERKELEY – The is the text of the J. Robert Oppenheimer Lecture in Physics, delivered March 13, 2007, by Stephen Hawking, the Lucasian professor of mathematics at Cambridge University. Hawking spoke at Zellerbach Hall on the campus of the University of California, Berkeley.
Can you hear me?
According to the Boshongo people of central Africa, in the beginning there was only darkness, water, and the great god Bumba. One day Bumba, in pain from a stomach ache, vomited up the sun. The sun dried up some of the water, leaving land. Still in pain, Bumba vomited up the moon, the stars, and then some animals. The leopard, the crocodile, the turtle, and, finally man.
This creation myth, like many others, tries to answer the questions we all ask. Why are we here? Where did we come from? The answer generally given, was that humans were of comparatively recent origin, because it must have been obvious, even at early times, that the human race was improving in knowledge and technology. So it can't have been around that long, or it would have progressed even more. For example, according to Bishop Usher, the Book of Genesis placed the creation of the world at 9 in the morning, on October the 27th, 4,004 BC. On the other hand, the physical surroundings, like mountains and rivers, change very little in a human life time. They were therefore thought to be a constant background, and either to have existed for ever as an empty landscape, or to have been created at the same time as the humans.
Not everyone however, was happy with the idea that the universe had a beginning. For example, Aristotle, the most famous of the Greek philosophers, believed the universe had existed for ever. Something eternal, is more perfect than something created. He suggested the reason we see progress, was that floods, or other natural disasters, had repeatedly set civilization back to the beginning. The motivation for believing in an eternal universe, was the desire to avoid invoking divine intervention, to create the universe, and set it going. Conversely, those who believed the universe had a beginning, used it as an argument for the existence of God, as the first cause, or prime mover of the universe.
If one believed that the universe had a beginning, the obvious question was, What happened before the beginning? What was God doing before He made the world? Was He preparing Hell for people who asked such questions? The problem of whether or not the universe had a beginning, was a great concern to the German philosopher, Immanuel Kant. He felt there were logical contradictions, or Antimonies, either way. If the universe had a beginning, why did it wait an infinite time before it began? He called that the thesis. On the other hand, if the universe had existed for ever, why did it take an infinite time to reach the present stage? He called that the anti thesis. Both the thesis, and the anti thesis, depended on Kant's assumption, along with almost everyone else, that time was Absolute. That is to say, it went from the infinite past, to the infinite future, independently of any universe that might or might not exist in this background.
This is still the picture in the mind of many scientists today. However in 1915, Einstein introduced his revolutionary General Theory of Relativity. In this, space and time were no longer Absolute, no longer a fixed background to events. Instead, they were dynamical quantities that were shaped by the matter and energy in the universe. They were defined only within the universe, so it made no sense to talk of a time before the universe began. It would be like asking for a point south of the South Pole. It is not defined.
If the universe was essentially unchanging in time, as was generally assumed before the 1920s, there would be no reason that time should not be defined arbitrarily far back. Any so-called beginning of the universe, would be artificial, in the sense that one could extend the history back to earlier times. Thus it might be that the universe was created last year, but with all the memories and physical evidence, to look like it was much older. This raises deep philosophical questions about the meaning of existence. I shall deal with these by adopting what is called, the positivist approach. In this, the idea is that we interpret the input from our senses in terms of a model we make of the world. One can not ask whether the model represents reality, only whether it works. A model is a good model, if first it interprets a wide range of observations, in terms of a simple and elegant model. And second, if the model makes definite predictions that can be tested, and possibly falsified, by observation.
In terms of the positivist approach, one can compare two models of the universe. One in which the universe was created last year, and one in which the universe existed much longer. The model in which the universe existed for longer than a year, can explain things like identical twins, that have a common cause more than a year ago. On the other hand, the model in which the universe was created last year, can not explain such events. So the first model is better. One can not ask whether the universe really existed before a year ago, or just appeared to. In the positivist approach, they are the same.
In an unchanging universe, there would be no natural starting point. The situation changed radically however, when Edwin Hubble began to make observations with the hundred inch telescope on Mount Wilson, in the 1920s.
Hubble found that stars are not uniformly distributed throughout space, but are gathered together in vast collections called galaxies.
By measuring the light from galaxies, Hubble could determine their velocities. He was expecting that as many galaxies would be moving towards us, as were moving away. This is what one would have in a universe that was unchanging with time. But to his surprise, Hubble found that nearly all the galaxies were moving away from us. Moreover, the further galaxies were from us, the faster they were moving away. The universe was not unchanging with time, as everyone had thought previously. It was expanding. The distance between distant galaxies, was increasing with time.
The expansion of the universe, was one of the most important intellectual discoveries of the 20th century, or of any century. It transformed the debate about whether the universe had a beginning. If galaxies are moving apart now, they must have been closer together in the past. If their speed had been constant, they would all have been on top of one another, about 15 billion years ago. Was this, the beginning of the universe.
Many scientists were still unhappy with the universe having a beginning, because it seemed to imply that physics broke down. One would have to invoke an outside agency, which for convenience, one can call God, to determine how the universe began. They therefore advanced theories in which the universe was expanding at the present time, but didn't have a beginning. One was the Steady State theory, proposed by Bondi, Gold, and Hoyle in 1948.
In the Steady State theory, as galaxies moved apart, the idea was that new galaxies would form from matter that was supposed to be continually being created throughout space. The universe would have existed for ever, and would have looked the same at all times. This last property had the great virtue, from a positivist point of view, of being a definite prediction, that could be tested by observation. The Cambridge radio astronomy group, under Martin Ryle, did a survey of weak radio sources in the early 1960s. These were distributed fairly uniformly across the sky, indicating that most of the sources, lay outside our galaxy. The weaker sources would be further away, on average.
The Steady State theory predicted the shape of the graph of the number of sources, against source Strength. But the observations showed more faint sources than predicted, indicating that the density sources was higher in the past. This was contrary to the basic assumption of the Steady State theory, that everything was constant in time. For this, and other reasons, the Steady State theory was abandoned.
Another attempt to avoid the universe having a beginning, was the suggestion that there was a previous contracting phase, but because of rotation and local irregularities, the matter would not all fall to the same point. Instead, different parts of the matter would miss each other, and the universe would expand again, with the density remaining finite. Two Russians, Lifshitz and Khalatnikov, actually claimed to have proved that a general contraction without exact symmetry, would always lead to a bounce, with the density remaining finite. This result was very convenient for Marxist Leninist dialectical materialism, because it avoided awkward questions about the creation of the universe. It therefore became an article of faith for Soviet scientists.
When Lifshitz and Khalatnikov published their claim, I was a 21-year-old research student, looking for something to complete my PhD thesis. I didn't believe their so-called proof, and set out with Roger Penrose to develop new mathematical techniques to study the question. We showed that the universe couldn't bounce. If Einstein's General Theory of Relativity is correct, there will be a singularity, a point of infinite density and space-time curvature, where time has a beginning.
Observational evidence to confirm the idea that the universe had a very dense beginning, came in October 1965, a few months after my first singularity result, with the discovery of a faint background of microwaves throughout space. These microwaves are the same as those in your microwave oven, but very much less powerful. They would heat your pizza only to minus 271 point 3 degrees centigrade, not much good for defrosting the pizza, let alone cooking it. You can actually observe these microwaves yourself. Set your television to an empty channel. A few percent of the snow you see on the screen, will be caused by this background of microwaves. The only reasonable interpretation of the background, is that it is radiation left over from an early very hot and dense state. As the universe expanded, the radiation would have cooled until it is just the faint remnant we observe today.
Although the singularity theorems of Penrose and myself, predicted that the universe had a beginning, they didn't say how it had begun. The equations of General Relativity would break down at the singularity. Thus Einstein's theory can not predict how the universe will begin, but only how it will evolve once it has begun. There are two attitudes one can take to the results of Penrose and myself. One is to that God chose how the universe began for reasons we could not understand. This was the view of Pope John Paul. At a conference on cosmology in the Vatican, the Pope told the delegates that it was OK to study the universe after it began. but they should not inquire into the beginning itself, because that was the moment of creation, and the work of God. I was glad he didn't realize I had presented a paper at the conference, suggesting how the universe began. I didn't fancy the thought of being handed over to the Inquisition, like Galileo.
The other interpretation of our results, which is favored by most scientists, is that it indicates that the General Theory of Relativity, breaks down in the very strong gravitational fields in the early universe. It has to be replaced by a more complete theory.. One would expect this anyway, because General Relativity does not take account of the small scale structure of matter, which is governed by quantum theory. This does not matter normally, because the scale of the universe, is enormous compared to the microscopic scales of quantum theory. But when the universe is the Planck size, a billion trillion trillionth of a centimeter, the two scales are the same, and quantum theory has to be taken into account.
In order to understand the Origin of the universe, we need to combine the General Theory of Relativity, with quantum theory. The best way of doing so, seems to be to use Feynman's idea of a sum over histories. Richard Feynman was a colorful character, who played the bongo drums in a strip joint in Pasadena, and was a brilliant physicist at the California Institute of Technology. He proposed that a system got from a state A, to a state B, by every possible path or history.
Each path or history, has a certain amplitude or intensity, and the probability of the system going from A- to B, is given by adding up the amplitudes for each path. There will be a history in which the moon is made of blue cheese, but the amplitude is low, which is bad news for mice.
The probability for a state of the universe at the present time, is given by adding up the amplitudes for all the histories that end with that state. But how did the histories start. This is the Origin question in another guise. Does it require a Creator to decree how the universe began. Or is the initial state of the universe, determined by a law of science.
In fact, this question would arise even if the histories of the universe went back to the infinite past. But it is more immediate if the universe began only 15 billion years ago. The problem of what happens at the beginning of time, is a bit like the question of what happened at the edge of the world, when people thought the world was flat. Is the world a flat plate, with the sea pouring over the edge. I have tested this experimentally. I have been round the world, and I have not fallen off.
As we all know, the problem of what happens at the edge of the world, was solved when people realized that the world was not a flat plate, but a curved surface. Time however, seemed to be different. It appeared to be separate from space, and to be like a model railway track. If it had a beginning, there would have to be someone to set the trains going.
Einstein's General Theory of Relativity, unified time and space as space-time, but time was still different from space, and was like a corridor, which either had a beginning and end, or went on for ever. However, when one combines General Relativity with Quantum Theory, Jim Hartle and I, realized that time can behave like another direction in space under extreme conditions. This means one can get rid of the problem of time having a beginning, in a similar way in which we got rid of the edge of the world. Suppose the beginning of the universe, was like the south pole of the Earth , with degrees of latitude, playing the role of time. The universe would start as a point at the South Pole. As one moves north, the circles of constant latitude, representing the size of the universe, would expand. To ask what happened before the beginning of the universe, would become a meaningless question, because there is nothing south of the South Pole.
Time, as measured in degrees of latitude, would have a beginning at the South Pole, but the South Pole is much like any other point, at least so I have been told. I have been to Antarctica, but not to the South Pole.
The same laws of Nature hold at the South Pole, as in other places. This would remove the age-old objection to the universe having a beginning, that it would be a place where the normal laws broke down. The beginning of the universe, would be governed by the laws of science.
The picture Jim Hartle and I developed, of the spontaneous quantum creation of the universe, would be a bit like the formation of bubbles of steam in boiling water.
The idea is that the most probable histories of the universe, would be like the surfaces of the bubbles. Many small bubbles would appear, and then disappear again. These would correspond to mini universes that would expand, but would collapse again while still of microscopic size. They are possible alternative universes, but they are not of much interest since they do not last long enough to develop galaxies and stars, let alone intelligent life. A few of the little bubbles, however, with grow to a certain size at which they are safe from recollapse. They will continue to expand at an ever increasing rate, and will form the bubbles we see. They will correspond to universes that would start off expanding at an ever increasing rate. This is called inflation, like the way prices go up every year.
The world record for inflation, was in Germany after the First World War. Prices rose by a factor of ten million in a period of 18 months. But that was nothing compared to inflation in the early universe. The universe expanded by a factor of million trillion trillion in a tiny fraction of a second. Unlike inflation in prices, inflation in the early universe was a very good thing. It produced a very large, and uniform universe, just as we observe. However, it would not be completely uniform. In the sum over histories, histories that are very slightly irregular, will have almost as high probabilities as the completely uniform and regular history.. The theory therefore predicts that the early universe is likely to be slightly non-uniform. These irregularities would produce small variations in the intensity of the microwave background from different directions. The microwave background has been observed by the Map satellite, and was found to have exactly the kind of variations predicted. So we know we are on the right lines.
The irregularities in the early universe, will mean that some regions will have slightly higher density than others. The gravitational attraction of the extra density, will slow the expansion of the region, and can eventually cause the region to collapse to form galaxies and stars. So look well at the map of the microwave sky. It is the blue print for all the structure in the universe. We are the product of quantum fluctuations in the very early universe. God really does play dice.
We have made tremendous progress in cosmology in the last hundred years. The General Theory of Relativity, and the discovery of the expansion of the universe, shattered the old picture of an ever existing, and ever lasting universe. Instead, general relativity predicted that the universe, and time itself, would begin in the big bang. It also predicted that time would come to an end in black holes. The discovery of the cosmic microwave background, and observations of black holes, support these conclusions. This is a profound change in our picture of the universe, and of reality itself.
Although the General Theory of Relativity, predicted that the universe must have come from a period of high curvature in the past, it could not predict how the universe would emerge from the big bang. Thus general relativity on its own, can not answer the central question in cosmology, Why is the universe, the way it is. However, if general relativity is combined with quantum theory, it may be possible to predict how the universe would start. It would initially expand at an ever increasing rate. During this so called inflationary period, the marriage of the two theories predicted that small fluctuations would develop, and lead to the formation of galaxies, stars, and all the other structure in the universe. This is confirmed by observations of small non uniformities in the cosmic microwave background, with exactly the predicted properties. So it seems we are on our way to understanding the origin of the universe, though much more work will be needed. A new window on the very early universe, will be opened when we can detect gravitational waves by accurately measuring the distances between space craft. Gravitational waves propagate freely to us from earliest times, unimpeded by any intervening material. By contrast, light is scattered many times by free electrons. The scattering goes on until the electrons freeze out, after 300,000 years.
Despite having had some great successes, not everything is solved. We do not yet have a good theoretical understanding, of the observations that the expansion of the universe, is accelerating again, after a long period of slowing down. Without such an understanding, we can not be sure of the future of the universe. Will it continue to expand forever? Is inflation a law of Nature? Or will the universe eventually collapse again? New observational results, and theoretical advances, are coming in rapidly. Cosmology is a very exciting and active subject. We are getting close to answering the age old questions. Why are we here? Where did we come from?
Thank you for listening to me.
Prime Numbers Get Hitched
Incubator by / March 27, 2006
In their search for patterns, mathematicians have uncovered unlikely connections between prime numbers and quantum physics. Will the subatomic world help reveal the elusive nature of the primes?
In 1972, the physicist Freeman Dyson wrote an article called “Missed Opportunities.” In it, he describes how relativity could have been discovered many years before Einstein announced his findings if mathematicians in places like Göttingen had spoken to physicists who were poring over Maxwell’s equations describing electromagnetism. The ingredients were there in 1865 to make the breakthrough—only announced by Einstein some 40 years later.
It is striking that Dyson should have written about scientific ships passing in the night. Shortly after he published the piece, he was responsible for an abrupt collision between physics and mathematics that produced one of the most remarkable scientific ideas of the last half century: that quantum physics and prime numbers are inextricably linked.
This unexpected connection with physics has given us a glimpse of the mathematics that might, ultimately, reveal the secret of these enigmatic numbers. At first the link seemed rather tenuous. But the important role played by the number 42 has recently persuaded even the deepest skeptics that the subatomic world might hold the key to one of the greatest unsolved problems in mathematics.
Prime numbers, such as 17 and 23, are those that can only be divided by themselves and one. They are the most important objects in mathematics because, as the ancient Greeks discovered, they are the building blocks of all numbers—any of which can be broken down into a product of primes. (For example, 105 = 3 x 5 x 7.) They are the hydrogen and oxygen of the world of mathematics, the atoms of arithmetic. They also represent one of the greatest challenges in mathematics.
As a mathematician, I’ve dedicated my life to trying to find patterns, structure and logic in the apparent chaos that surrounds me. Yet this science of patterns seems to be built from a set of numbers which have no logic to them at all. The primes look more like a set of lottery ticket numbers than a sequence generated by some simple formula or law.
For 2,000 years the problem of the pattern of the primes—or the lack thereof—has been like a magnet, drawing in perplexed mathematicians. Among them was Bernhard Riemann who, in 1859, the same year Darwin published his theory of evolution, put forward an equally-revolutionary thesis for the origin of the primes. Riemann was the mathematician in Göttingen responsible for creating the geometry that would become the foundation for Einstein’s great breakthrough. But it wasn’t only relativity that his theory would unlock.
Riemann discovered a geometric landscape, the contours of which held the secret to the way primes are distributed through the universe of numbers. He realized that he could use something called the zeta function to build a landscape where the peaks and troughs in a three-dimensional graph correspond to the outputs of the function. The zeta function provided a bridge between the primes and the world of geometry. As Riemann explored the significance of this new landscape, he realized that the places where the zeta function outputs zero (which correspond to the troughs, or places where the landscape dips to sea-level) hold crucial information about the nature of the primes. Mathematicians call these significant places the zeros.
Riemann’s discovery was as revolutionary as Einstein’s realization that E=mc2. Instead of matter turning into energy, Riemann’s equation transformed the primes into points at sea-level in the zeta landscape. But then Riemann noticed that it did something even more incredible. As he marked the locations of the first 10 zeros, a rather amazing pattern began to emerge. The zeros weren’t scattered all over; they seemed to be running in a straight line through the landscape. Riemann couldn’t believe this was just a coincidence. He proposed that all the zeros, infinitely many of them, would be sitting on this critical line—a conjecture that has become known as the Riemann Hypothesis.
But what did this amazing pattern mean for the primes? If Riemann’s discovery was right, it would imply that nature had distributed the primes as fairly as possible. It would mean that the primes behave rather like the random molecules of gas in a room: Although you might not know quite where each molecule is, you can be sure that there won’t be a vacuum at one corner and a concentration of molecules at the other.
For mathematicians, Riemann’s prediction about the distribution of primes has been very powerful. If true, it would imply the viability of thousands of other theorems, including several of my own, which have had to assume the validity of Riemann’s Hypothesis to make further progress. But despite nearly 150 years of effort, no one has been able to confirm that all the zeros really do line up as he predicted.
It was a chance meeting between physicist Freeman Dyson and number theorist Hugh Montgomery in 1972, over tea at Princeton’s Institute for Advanced Study, that revealed a stunning new connection in the story of the primes—one that might finally provide a clue about how to navigate Riemann’s landscape. They discovered that if you compare a strip of zeros from Riemann’s critical line to the experimentally recorded energy levels in the nucleus of a large atom like erbium, the 68th atom in the periodic table of elements, the two are uncannily similar.
It seemed the patterns Montgomery was predicting for the way zeros were distributed on Riemann’s critical line were the same as those predicted by quantum physicists for energy levels in the nucleus of heavy atoms. The implications of a connection were immense: If one could understand the mathematics describing the structure of the atomic nucleus in quantum physics, maybe the same math could solve the Riemann Hypothesis.
Mathematicians were skeptical. Though mathematics has often served physicists—Einstein, for instance—they wondered whether physics could really answer hard-core problems in number theory. So in 1996, Peter Sarnak at Princeton threw down the gauntlet and challenged physicists to tell the mathematicians something they didn’t know about primes. Recently, Jon Keating and Nina Snaith, of Bristol, duely obliged.
There is an important sequence of numbers called “the moments of the Riemann zeta function.” Although we know abstractly how to define it, mathematicians have had great difficulty explicitly calculating the numbers in the sequence. We have known since the 1920s that the first two numbers are 1 and 2, but it wasn’t until a few years ago that mathematicians conjectured that the third number in the sequence may be 42—a figure greatly significant to those well-versed in The Hitchhiker’s Guide to the Galaxy.
It would also prove to be significant in confirming the connection between primes and quantum physics. Using the connection, Keating and Snaith not only explained why the answer to life, the universe and the third moment of the Riemann zeta function should be 42, but also provided a formula to predict all the numbers in the sequence. Prior to this breakthrough, the evidence for a connection between quantum physics and the primes was based solely on interesting statistical comparisons. But mathematicians are very suspicious of statistics. We like things to be exact. Keating and Snaith had used physics to make a very precise prediction that left no room for the power of statistics to see patterns where there are none.
Mathematicians are now convinced. That chance meeting in the common room in Princeton resulted in one of the most exciting recent advances in the theory of prime numbers. Many of the great problems in mathematics, like Fermat’s Last Theorem, have only been cracked once connections were made to other parts of the mathematical world. For 150 years many have been too frightened to tackle the Riemann Hypothesis. The prospect that we might finally have the tools to understand the primes has persuaded many more mathematicians and physicists to take up the challenge. The feeling is in the air that we might be one step closer to a solution. Dyson might be right that the opportunity was missed to discover relativity 40 years earlier, but who knows how long we might still have had to wait for the discovery of connections between primes and quantum physics had mathematicians not enjoyed a good chat over tea. —Marcus du Sautoy is professor of mathematics at the University of Oxford, and is the author of The Music of the Primes (HarperCollins).
It is striking that Dyson should have written about scientific ships passing in the night. Shortly after he published the piece, he was responsible for an abrupt collision between physics and mathematics that produced one of the most remarkable scientific ideas of the last half century: that quantum physics and prime numbers are inextricably linked.
This unexpected connection with physics has given us a glimpse of the mathematics that might, ultimately, reveal the secret of these enigmatic numbers. At first the link seemed rather tenuous. But the important role played by the number 42 has recently persuaded even the deepest skeptics that the subatomic world might hold the key to one of the greatest unsolved problems in mathematics.
Prime numbers, such as 17 and 23, are those that can only be divided by themselves and one. They are the most important objects in mathematics because, as the ancient Greeks discovered, they are the building blocks of all numbers—any of which can be broken down into a product of primes. (For example, 105 = 3 x 5 x 7.) They are the hydrogen and oxygen of the world of mathematics, the atoms of arithmetic. They also represent one of the greatest challenges in mathematics.
As a mathematician, I’ve dedicated my life to trying to find patterns, structure and logic in the apparent chaos that surrounds me. Yet this science of patterns seems to be built from a set of numbers which have no logic to them at all. The primes look more like a set of lottery ticket numbers than a sequence generated by some simple formula or law.
For 2,000 years the problem of the pattern of the primes—or the lack thereof—has been like a magnet, drawing in perplexed mathematicians. Among them was Bernhard Riemann who, in 1859, the same year Darwin published his theory of evolution, put forward an equally-revolutionary thesis for the origin of the primes. Riemann was the mathematician in Göttingen responsible for creating the geometry that would become the foundation for Einstein’s great breakthrough. But it wasn’t only relativity that his theory would unlock.
Riemann discovered a geometric landscape, the contours of which held the secret to the way primes are distributed through the universe of numbers. He realized that he could use something called the zeta function to build a landscape where the peaks and troughs in a three-dimensional graph correspond to the outputs of the function. The zeta function provided a bridge between the primes and the world of geometry. As Riemann explored the significance of this new landscape, he realized that the places where the zeta function outputs zero (which correspond to the troughs, or places where the landscape dips to sea-level) hold crucial information about the nature of the primes. Mathematicians call these significant places the zeros.
Riemann’s discovery was as revolutionary as Einstein’s realization that E=mc2. Instead of matter turning into energy, Riemann’s equation transformed the primes into points at sea-level in the zeta landscape. But then Riemann noticed that it did something even more incredible. As he marked the locations of the first 10 zeros, a rather amazing pattern began to emerge. The zeros weren’t scattered all over; they seemed to be running in a straight line through the landscape. Riemann couldn’t believe this was just a coincidence. He proposed that all the zeros, infinitely many of them, would be sitting on this critical line—a conjecture that has become known as the Riemann Hypothesis.
But what did this amazing pattern mean for the primes? If Riemann’s discovery was right, it would imply that nature had distributed the primes as fairly as possible. It would mean that the primes behave rather like the random molecules of gas in a room: Although you might not know quite where each molecule is, you can be sure that there won’t be a vacuum at one corner and a concentration of molecules at the other.
For mathematicians, Riemann’s prediction about the distribution of primes has been very powerful. If true, it would imply the viability of thousands of other theorems, including several of my own, which have had to assume the validity of Riemann’s Hypothesis to make further progress. But despite nearly 150 years of effort, no one has been able to confirm that all the zeros really do line up as he predicted.
It was a chance meeting between physicist Freeman Dyson and number theorist Hugh Montgomery in 1972, over tea at Princeton’s Institute for Advanced Study, that revealed a stunning new connection in the story of the primes—one that might finally provide a clue about how to navigate Riemann’s landscape. They discovered that if you compare a strip of zeros from Riemann’s critical line to the experimentally recorded energy levels in the nucleus of a large atom like erbium, the 68th atom in the periodic table of elements, the two are uncannily similar.
It seemed the patterns Montgomery was predicting for the way zeros were distributed on Riemann’s critical line were the same as those predicted by quantum physicists for energy levels in the nucleus of heavy atoms. The implications of a connection were immense: If one could understand the mathematics describing the structure of the atomic nucleus in quantum physics, maybe the same math could solve the Riemann Hypothesis.
Mathematicians were skeptical. Though mathematics has often served physicists—Einstein, for instance—they wondered whether physics could really answer hard-core problems in number theory. So in 1996, Peter Sarnak at Princeton threw down the gauntlet and challenged physicists to tell the mathematicians something they didn’t know about primes. Recently, Jon Keating and Nina Snaith, of Bristol, duely obliged.
There is an important sequence of numbers called “the moments of the Riemann zeta function.” Although we know abstractly how to define it, mathematicians have had great difficulty explicitly calculating the numbers in the sequence. We have known since the 1920s that the first two numbers are 1 and 2, but it wasn’t until a few years ago that mathematicians conjectured that the third number in the sequence may be 42—a figure greatly significant to those well-versed in The Hitchhiker’s Guide to the Galaxy.
It would also prove to be significant in confirming the connection between primes and quantum physics. Using the connection, Keating and Snaith not only explained why the answer to life, the universe and the third moment of the Riemann zeta function should be 42, but also provided a formula to predict all the numbers in the sequence. Prior to this breakthrough, the evidence for a connection between quantum physics and the primes was based solely on interesting statistical comparisons. But mathematicians are very suspicious of statistics. We like things to be exact. Keating and Snaith had used physics to make a very precise prediction that left no room for the power of statistics to see patterns where there are none.
Mathematicians are now convinced. That chance meeting in the common room in Princeton resulted in one of the most exciting recent advances in the theory of prime numbers. Many of the great problems in mathematics, like Fermat’s Last Theorem, have only been cracked once connections were made to other parts of the mathematical world. For 150 years many have been too frightened to tackle the Riemann Hypothesis. The prospect that we might finally have the tools to understand the primes has persuaded many more mathematicians and physicists to take up the challenge. The feeling is in the air that we might be one step closer to a solution. Dyson might be right that the opportunity was missed to discover relativity 40 years earlier, but who knows how long we might still have had to wait for the discovery of connections between primes and quantum physics had mathematicians not enjoyed a good chat over tea. —Marcus du Sautoy is professor of mathematics at the University of Oxford, and is the author of The Music of the Primes (HarperCollins).
