The Universe Within (19 page)

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.

· · ·

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.