Q is for Quantum

INTRODUCTION

The quest for the quantum

This quick overview of a hundred years of scientific investigation of the microworld is intended to put the detail of the main section of this book in an historical perspective. All the technical terms are fully explained in the alphabetical section.

The quantum world is the world of the very small—the microworld. Although, as we shall see, quantum effects can be important for objects as large as molecules, the real quantum domain is in the subatomic world of particle physics. The first subatomic particle, the electron, was only identified, by J. J. Thomson, in 1897, exactly 100 years before this book, summing up our present understanding of the microworld, was completed. But it isn’t just the neatness of this anniversary that makes this a good time to take stock of the quantum world; particle physicists have now developed an understanding of what things are made of, and how those things interact with one another, that is more complete and satisfying than at any time since Thomson’s discovery changed the way people thought about the microworld. The standard model of particle physics, based upon the rules of quantum mechanics, tells us how the world is built up from the fundamental building blocks of quarks and leptons, held together by the exchange of particles called gluons and vector bosons.

But don’t imagine that even the physicists believe that the standard model is the last word. After all, it doesn’t include gravity. The structure of theoretical physics in the twentieth century was built on two great theories, the general theory of relativity (which describes gravity and the Universe at large) and quantum mechanics (which describes the microworld). Unifying those two great theories into one package, a theory of everything, is the Holy Grail that physicists seek as we enter the 21st century. Experiments that probe the accuracy of the standard model to greater and greater precision are being carried out using particle accelerators like those at CERN, in Geneva, and Fermilab, in Chicago. From time to time, hints that the standard theory is not the whole story emerge. This gives the opportunity for newspapers to run sensational headlines proclaiming that physics is in turmoil; in fact, these hints of something beyond the standard model are welcomed by the physicists, who are only too aware that their theory, beautiful though it is, is not the last word. Unfortunately, as yet none of those hints of what may lie beyond the standard model has stood up to further investigation. As of the spring of 1997, the standard model is still the best game in town.

But whatever lies beyond the standard model, it will still be based upon the rules of quantum physics. Just as the general theory of relativity includes the Newtonian version of gravity within itself as a special case, so that Newton’s theory is still a useful and accurate description of how things work in many applications (such as calculating the trajectory of a space probe being sent to Jupiter), so any improved theory of the microworld must include the quantum theory within itself. Apples didn’t start falling upwards when Albert Einstein came up with an improved theory of gravity; and no improved theory of physics will ever take away the weirdness of the quantum world.

By the standards of everyday common sense, the quantum world is very weird indeed. One of the key examples is the phenomenon of wave-particle duality. J. J. Thomson opened up the microworld to investigation when he found that the electron is a particle; three decades later, his son George proved that electrons are waves. Both of them were right (and they each won a Nobel Prize for their work). An electron is a particle, and it is a wave. Or rather, it is neither a particle nor a wave, but a quantum entity that will respond to one sort of experiment by behaving like a particle, and to another set of experiments by behaving like a wave. The same is true of light—it can behave either like a stream of particles (photons) or like a wave, depending on the circumstances. Indeed, it is, in principle, true of everything, although the duality does not show up with noticeable strength in the everyday world (which, of course, is why we do not regard the consequences of wave-particle duality as common sense).

All of this is related to the phenomenon of quantum uncertainty. A quantum entity, such as an electron or a photon, does not have a well-determined set of properties, in the way that a billiard ball rolling across the table has a precisely determined velocity and a precisely determined position at any instant. The photon and the electron (and other denizens of the microworld) do not know, and cannot know, both precisely where they are and precisely where they are going. It may seem an esoteric and bizarre idea, of no great practical consequence in the everyday world. But it is this quantum uncertainty that allows hydrogen nuclei to fuse together and generate heat inside the Sun, so without it we would not be here to wonder at such things (quantum uncertainty is also important in the process of radioactive decay, for substances such as uranium-235).

This highlights an important point about quantum physics. It is not just some exotic theory that academics in their ivory towers study as a kind of intellectual exercise, of no relevance to everyday life. You need quantum physics in order to calculate how to make an atom bomb, or a nuclear power station, that works properly—which is certainly relevant to the modern world. And you also need quantum physics in order to design much more domestic items of equipment, such as lasers. Not everybody immediately thinks of a laser as a piece of domestic equipment; but remember that a laser is at the heart of any CD player, reading the information stored on the disc itself; and the laser’s close cousin, the maser, is used in amplifying faint signals, including those from communications satellites that feed TV into your home.

Where does the quantum physics come in? Because lasers operate on a principle called stimulated emission, a purely quantum process, whose statistical principles were first spelled out by Albert Einstein as long ago as 1916. If an atom has absorbed energy in some way, so that it is in what is called an excited state, it can be triggered into releasing a pulse of electromagnetic energy (a photon) at a precisely determined wavelength (a wavelength that is determined by the quantum rules) by giving it a suitable nudge. A suitable nudge happens when a photon with exactly the right wavelength (the same wavelength as the photon that the excited atom is primed to emit) passes by. So, in a process rather like the chain reaction of atomic fission that goes on in a nuclear bomb, if a whole array of atoms has been excited in the right way, a single photon passing through the array (perhaps in a ruby crystal) can trigger all of them to emit electromagnetic radiation (light) in a pulse in which all of the waves are marching precisely in step with one another. Because all of the waves go up together and go down together, this produces a powerful beam of very pure electromagnetic radiation (that is, a very pure colour).

Quantum physics is also important in the design and operation of anything which contains a semiconductor, including computer chips—not just the computer chips in your home computer, but the ones in your TV, hi-fi, washing machine and car. Semiconductors are materials with conducting properties that are intermediate between those of insulators (in which the electrons are tightly bound to their respective atomic nuclei) and conductors (in which some electrons are able to roam more or less freely through the material). In a semiconductor, some electrons are only just attached to their atoms, and can be made to hop from one atom to the next under the right circumstances. The way the hopping takes place, and the behaviour of electrons in general, depends on a certain set of quantum rules known as Fermi-Dirac statistics (the behaviour of photons, in lasers and elsewhere, depends on another set of quantum rules, Bose-Einstein statistics).

After semiconductors, it is logical to mention superconductors—materials in which electricity flows without any resistance at all. Superconductors are beginning to have practical applications (including in computing), and once again the reason why they conduct electricity the way they do is explained in terms of quantum physics—in this case, because under the right circumstances in some materials electrons stop obeying Fermi-Dirac statistics, and start obeying Bose-Einstein statistics, behaving like photons.

Electrons, of course, are found in the outer parts of atoms, and form the interface between different atoms in molecules. The behaviour of electrons in atoms and molecules is entirely described by quantum physics; and since the interactions between atoms and molecules are the raw material of chemistry, this means that chemistry is described by quantum physics. And not just the kind of schoolboy chemistry used to make impressive smells and explosive interactions. Life itself is based upon complex chemical interactions, most notably involving the archetypal molecule of life, DNA. At the very heart of the process of life lies the ability of a DNA molecule, the famous double-stranded helix, to ‘unzip’ itself and make two copies of the original double helix by building up a new partner for each strand of the original molecules, using each unzipped single molecule as a template. The links that are used in this process to hold the strands together most of the time, but allow them to unzip in this way when it is appropriate, are a kind of chemical bond, known as the hydrogen bond. In a hydrogen bond, a single proton (the nucleus of a hydrogen atom) is shared between two atoms (or between two molecules), forming a link between them. The way fundamental life processes operate can only be explained if allowance is made for quantum processes at work in hydrogen-bonded systems.

As well as the importance of quantum physics in providing an understanding of the chemistry of life, an understanding of quantum chemistry is an integral part of the recent successes that have been achieved in the field of genetic engineering. In order to make progress in taking genes apart, adding bits of new genetic material and putting them back together again, you have to understand how and why atoms join together in certain sequences but not in others, why certain chemical bonds have a certain strength, and why those bonds hold atoms and molecules a certain distance apart from one another. You might make some progress by trial and error, without understanding the quantum physics involved; but it would take an awful long time before you got anywhere (evolution, of course, does operate by a process of trial and error, and has got somewhere because it has been going on for an awful long time).

In fact, although there are other forces which operate deep within the atom (and which form the subject of much of this book), if you understand the behaviour of electrons and the behaviour of photons (light) then you understand everything that matters in the everyday world, except gravity and nuclear power stations. Apart from gravity, everything that is important in the home (including the electricity generated in nuclear power stations) can be described in terms of the way electrons interact with one another, which determines the way that atoms interact with one another, and the way they interact with electromagnetic radiation, including light.

We don’t just mean that all of this can be described in general terms, in a qualitative, hand-waving fashion. It can be described quantitatively, to a staggering accuracy. The greatest triumph of theoretical quantum physics (indeed, of all physics) is the theory that describes light and matter in this way. It is called quantum electrodynamics (QED), and it was developed in its finished form in the 1940s, most notably by Richard Feynman. QED tells you about every possible interaction between light and matter (to a physicist, ‘light’ is used as shorthand for all electromagnetic radiation), and it does so to an accuracy of four parts in a hundred billion. It is the most accurate scientific theory ever developed, judged by the criterion of how closely the predictions of the theory agree with the results of experiments carried out in laboratories here on Earth.

Following the triumph of QED, it was used as the template for the construction of a similar theory of what goes on inside the protons and neutrons that make up the nuclei of atoms—a theory known as quantum chromodynamics, or QCD. Both QED and QCD are components of the standard model. J. J. Thomson could never have imagined what his discovery of the electron would lead to. But the first steps towards a complete theory of quantum physics, and the first hint of the existence of the entities known as quanta, appeared within three years of Thomson’s discovery, in 1900. That first step towards quantum physics came, though, not from the investigation of electrons, but from the investigation of the other key component of QED, photons.

At the end of the 19th century, nobody thought of light in terms of photons. Many observations—including the famous double-slit experiment carried out by Thomas Young—had shown that light is a form of wave. The equations of electromagnetism, discovered by James Clerk Maxwell, also described light as a wave. But Max Planck discovered that certain features of the way in which light is emitted and absorbed could be explained only if the radiation was being parcelled out in lumps of certain sizes, called quanta. Planck’s discovery was announced at a meeting of the Berlin Physical Society, in October 1900. But at that time nobody thought that what he had described implied that light only existed (or ever existed!) in the form of quanta; the assumption was that there was some property of atoms which meant that light could be emitted or absorbed only in lumps of a certain size, but that ‘really’ the light was a wave.

The first (and for a long time the only) person to take the idea of light quanta seriously was Einstein. But he was a junior patent office clerk at the time, with no formal academic connections, and hadn’t yet even finished his PhD. In 1905 he published a paper in which he used the idea of quanta to explain another puzzling feature of the way light is absorbed, the photoelectric effect. In order to explain this phenomenon (the way electrons are knocked out of a metal surface by light), Einstein used the idea that light actually travels as a stream of little particles, what we would now call photons. The idea was anathema to most physicists, and even Einstein was cautious about promoting the idea—it was not until 1909 that he made the first reference in print to light as being made up of ‘point-like quanta’. In spite of his caution, one physicist, Robert Millikan, was so annoyed by the suggestion that he spent the best part of ten years carrying out a series of superb experiments aimed at proving that Einstein’s idea was wrong. He succeeded only in proving—as he graciously acknowledged—that Einstein had been right. It was after Millikan’s experiments had established beyond doubt the reality of photons (which were not actually given that name until later) that Einstein received his Nobel Prize for this work (the 1921 prize, but actually awarded in 1922). Millikan received the Nobel Prize, partly for this work, in 1923.

While all this was going on, other physicists, led by Niels Bohr, had been making great strides by applying quantum ideas to an understanding of the structure of the atom. It was Bohr who came up with the image of an atom that is still basically the one we learn about when we first encounter the idea of atoms in school—a tiny central nucleus, around which electrons circle in a manner reminiscent of the way planets orbit around the Sun. Bohr’s model, in the form in which it was developed by 1913, had one spectacular success: it could explain the way in which atoms produce bright and dark lines at precisely defined wavelengths in the rainbow spectrum of light. The difference in energy between any two electron orbits was precisely defined by the model, and an electron jumping from one orbit to the other would emit or absorb light at a very precise wavelength, corresponding to that energy difference. But Bohr’s model introduced the bizarre idea that the electron did indeed ‘jump’, instantaneously, from one orbit to the other, without crossing the intervening space (this has become known as a ‘quantum leap’). First it was in one orbit, then it was in the other, without ever crossing the gap.

Bohr’s model of the atom also still used the idea of electrons as particles, like little billiard balls, and light as a wave. But by the time Einstein and Millikan received their Nobel Prizes, it was clear that there was more to light than this simple picture accounted for. As Einstein put it in 1924, ‘there are therefore now two theories of light, both indispensable... without any logical connection’. The next big step, which led to the first full quantum theory, came when Louis de Broglie pointed out that there was also more to electrons than the simple picture encapsulated in the Bohr model accounted for.

De Broglie made the leap of imagination (obvious with hindsight, but a breakthrough at the time) of suggesting that if something that had traditionally been regarded as a wave (light) could also be treated as a particle, then maybe something that had traditionally been regarded as a particle (the electron) could also be treated as a wave. Of course, he did more than just speculate along these lines. He took the same kind of quantum calculations that had been pioneered by Planck and Einstein in their description of light and turned the equations around, plugging in the numbers appropriate for electrons. And he suggested that what actually ‘travelled round’ an electron orbit in an atom was not a little particle, but a standing wave, like the wave corresponding to a pure note on a plucked violin string.

De Broglie’s idea was published in 1925. Although the idea of electrons behaving as waves was puzzling, this business of standing waves looked very attractive because it seemed to get rid of the strange quantum jumping. Now, it looked as if the transition of an electron from one energy level to another could be explained in terms of the vibration of the wave, changing from one harmonic (one note) to another. It was the way in which this idea seemed to restore a sense of normality to the quantum world that attracted Erwin Schrödinger, who worked out a complete mathematical description of the behaviour of electrons in atoms, based on the wave idea, by the end of 1926. He thought that his wave equation for the electron had done away with the need for what he called ‘damned quantum jumping’. But he was wrong.

Also by 1926, using a completely different approach based entirely on the idea of electrons as particles, Werner Heisenberg and his colleagues had found another way to describe the behaviour of electrons in atoms, and elsewhere—another complete mathematical quantum theory. And as if that weren’t enough, Paul Dirac had found yet another mathematical description of the quantum world. It soon turned out that all of these mathematical approaches were formally equivalent to one another, different views of the same quantum world (a bit like the choice between giving a date in Roman numerals or Arabic notation). It really didn’t matter which set of equations you used, since they all described the same thing and gave the same answers. To Schrödinger’s disgust, the ‘damned quantum jumping’ had not been eliminated after all; but, ironically, because most physicists are very familiar with how to manipulate wave equations, it was Schrödinger’s variation on the theme, based on his equation for the wave function of an electron, that soon became the conventional way to do calculations in quantum mechanics.

This tradition was reinforced by the mounting evidence (including the experiments carried out by George Thomson in 1927) that electrons did indeed behave like waves (the ultimate proof of this came when electrons were persuaded to participate in a version of the double-slit experiment, and produced the classic diffraction effects seen with light under the equivalent circumstances). But none of this stopped electrons behaving like particles in all the experiments where they had always behaved like particles.

By the end of the 1920s, physicists had a choice of different mathematical descriptions of the microworld, all of which worked perfectly and gave the right answers (in terms of predicting the outcome of experiments), but all of which included bizarre features such as quantum jumping, wave-particle duality and uncertainty. Niels Bohr developed a way of picturing what was going on that was taught as the standard version of quantum physics for half a century (and is still taught in far too many places), but which if anything made the situation even more confusing. This ‘Copenhagen interpretation’ says that entities such as electrons do not exist when they are not being observed or measured in some way, but spread out as a cloud of probability, with a definite probability of being found in one place, another probability of being detected somewhere else, and so on. When you decide to measure the position of the electron, there is a ‘collapse of the wave function’, and it chooses (at random, in accordance with the rules of probability, the same rules that operate in a casino) one position to be in. But as soon as you stop looking at it, it dissolves into a new cloud of probability, described by a wave function spreading out from the site where you last saw it.

It was their disgust with this image of the world that led Einstein and Schrödinger, in particular, to fight a rearguard battle against the Copenhagen interpretation over the next twenty years, each of them independently (but with moral support from each other) attempting to prove its logical absurdity with the aid of thought experiments, notably the famous example of Schrödinger’s hypothetical cat, a creature which, according to the strict rules of the Copenhagen interpretation, can be both dead and alive at the same time.

Although this debate (between Einstein and Schrödinger on one side, and Bohr on the other) was going on, most physicists ignored the weird philosophical implications of the Copenhagen interpretation, and just used the Schrödinger equation as a tool to do a job, working out how things like electrons behaved in the quantum world. Just as a car driver doesn’t need to understand what goes on beneath the bonnet of the car in order to get from A to B, as long as quantum mechanics worked, you didn’t have to understand it, even (as Linus Pauling showed) to get to grips with quantum chemistry.

The last thing most quantum physicists wanted was yet another mathematical description of the quantum world, and when Richard Feynman provided just that, in his PhD thesis in 1942, hardly anybody even noticed (most physicists at the time were, in any case, distracted by the Second World War). This has proved a great shame for subsequent generations of students, since Feynman’s approach, using path integrals, is actually simpler conceptually than any of the other approaches, and certainly no more difficult to handle mathematically. It also has the great merit of dealing with classical physics (the old ideas of Newton) and quantum physics in one package; it is literally true that if physics were taught Feynman’s way from the beginning, students would only ever have to learn the one approach to handle everything. As it is, although over the years the experts have come to accept that Feynman’s approach is the best one to use in tackling real problems at the research level, the way almost all students get to path integrals is by learning classical physics first (in school), then quantum physics the hard way (usually in the form of Schrödinger’s wave function, at undergraduate level) then, after completing at least one degree, being introduced to the simple way to do the job.

Don’t just take our word for this being the simplest way to tackle physics—John Wheeler, Feynman’s thesis supervisor, has said that the thesis marks the moment in the history of physics ‘when quantum theory became simpler than classical theory’. Feynman’s approach is not the standard way to teach quantum physics at undergraduate level (or classical physics in schools) for the same reason that the Betamax system is not the standard format for home video—because an inferior system got established in the market place first, and maintains its position as much through inertia as anything else.

Indeed, there is a deep flaw in the whole way in which science is taught, by recapitulating the work of the great scientists from Galileo to the present day, and it is no wonder that this approach bores the pants off kids in school. The right way to teach science is to start out with the exciting new ideas, things like quantum physics and black holes, building on the physical principles and not worrying too much too soon about the mathematical subtleties. Those children who don’t want a career in science will at least go away with some idea of what the excitement is all about, and those who do want a career in science will be strongly motivated to learn the maths when it becomes necessary. We speak from experience—one of us (JG) got turned on to science in just this way, by reading books that were allegedly too advanced for him and went way beyond the school curriculum, but which gave a feel for the mystery and excitement of quantum physics and cosmology even where the equations were at that time unintelligible to him.

In Feynman’s case, the path integral approach led him to quantum electrodynamics, and to the Feynman diagrams which have become an essential tool of all research in theoretical particle physics. But while these applications of quantum theory were providing the key to unlock an understanding of the microworld, even after the Second World War there were still a few theorists who worried about the fundamental philosophy of quantum mechanics, and what it was telling us about the nature of the Universe we live in.

For those who took the trouble to worry in this way, there was no getting away from the weirdness of the quantum world. Building from another thought experiment intended to prove the non-logical nature of quantum theory (the EPR experiment, dreamed up by Einstein and two of his colleagues), the work of David Bohm in the 1950s and John Bell in the 1960s led to the realization that it would actually be possible to carry out an experiment which would test the non-commonsensical aspects of quantum theory in a definitive manner.

What Einstein had correctly appreciated was that every version of quantum theory has built into it a breakdown of what is called ‘local reality’. ‘Local’, in this sense, means that no communication of any kind travels faster than light. ‘Reality’ means that the world exists when you are not looking at it, and that electrons, for example, do not dissolve into clouds of probability, wave functions waiting to collapse, when you stop looking at them. Quantum physics (any and every formulation of quantum physics) says that you can’t have both. It doesn’t say which one you have to do without, but one of them you must do without. What became known as the Bell test provided a way to see whether local reality applies in the (for want of a better word) real world—specifically, in the microworld.

The appropriate experiments were carried out by several teams in the 1980s, most definitively by Alain Aspect and his colleagues in Paris, using photons. They found that the predictions of quantum theory are indeed borne out by experiment—the quantum world is not both local and real.

So today you have no choice of options, if you want to think of the world as being made up of real entities which exist all the time, even when you are not looking at them; there is no escape from the conclusion that the world is non-local, meaning that there are communications between quantum entities that operate not just faster than light, but actually instantaneously. Einstein called this ‘spooky action at a distance’. The other option is to abandon both locality and reality, but most physicists prefer to cling on to one of the familiar features of the commonsense world, as long as that is allowed by the quantum rules.

Our own preference is for reality, even at the expense of locality; but that is just a personal preference, and you are quite free to choose the other option, the traditional Copenhagen interpretation involving both collapsing wave functions and spooky action at a distance, if that makes you happier. What you are not free to do, no matter how unhappy you are as a result, is to think that the microworld is both local and real.

The bottom line is that the microworld does not conform to the rules of common sense determined by our everyday experience. Why should it? We do not live in the microworld, and our everyday experience is severely limited to a middle range of scales (of both space and time) intermediate between the microworld and the cosmos. The important thing is not to worry about this. The greatest of all the quantum mechanics, Richard Feynman, gave a series of lectures at Cornell University on the theme The Character of Physical Law (published in book form by BBC Publications in 1965). In one of those lectures, he discussed the quantum mechanical view of nature, and in the introduction to that lecture he gave his audience a warning about the weirdness they were about to encounter. What he said then, many, applies with equal force today:

I think I can safely say that nobody understands quantum mechanics. So do not take the lecture too seriously, feeling that you really have to understand in terms of some model what I am going to describe, but just relax and enjoy it. I am going to tell you what nature behaves like. If you will simply admit that maybe she does behave like this, you will find her a delightful, entrancing thing. Do not keep saying to yourself, if you can possibly avoid it, ‘But how can it be like that?’ because you will go ‘down the drain’ into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that.

That is the spirit in which we offer you our guide to the quantum world; take the advice of the master—relax and enjoy it. Nobody knows how it can be like that.

NOTE TO THE READER

A few of the entries in the alphabetical section of this book overlap with entries in our earlier book, Companion to the Cosmos. In some cases, we have, in the spirit of quantum mechanics, tried to express the concepts in a complementary fashion on this occasion. But where we have found it impossible to improve on the form of words or analogy we used before, we have not made changes simply for the sake of making changes, although we have set the fragments of older material in their new context. See, for example, arrow of time.

A-Z DICTIONARY

A

Abelian group A group of mathematical transformations which can be carried out in any order and still give the same end result. Simple multiplication is Abelian: 3x2 is the same as 2 x 3, or, in more general terms, axb = bxa. This specific example, simple multiplication which forms an Abelian group, is said to follow a commutative law, or to commute. In the same way, in the everyday world addition is commutative, so that a + b = b + a.

But even in the everyday world, division does not obey the commutative law: a/b is not equal to b/a, and if, for example, you divide 4 by 2, the answer is 2, while if you divide 2 by 4, the answer is 0.5. Subtraction also does not commute; 4 - 2 is not the same as 2 - 4.

It is a key feature of the quantum world that many mathematical processes are non-Abelian, and in particular that multiplication is not necessarily commutative, which means that if a and b are quantum functions, not simple numbers, a x b may not be the same as bxa. Each component of a group (in this case, a and b) is called an element; if all the elements of the group commute, then the group is Abelian.

Named after the Norwegian mathematician Niels Henrik Abel (1802-1829).

absolute temperature See Kelvin scale.

absolute zero The lowest temperature that could ever be attained. At absolute zero, atoms and molecules would have the minimum amount of energy allowed by quantum theory. This is defined as 0 on the Kelvin scale of temperature; 0K is -273.15°C, and each unit on the Kelvin scale is the same size as one degree Celsius.

absorber theory See Wheeler-Feynman absorber theory.

accelerator A device which accelerates particles such as electrons and protons to very high speeds (close to the speed of light) using electric and magnetic fields. In modern accelerators, electrons can be accelerated to 99.999999986 per cent of the speed of light. The beams of fast-moving particles are then smashed into either stationary targets or beams of particles moving at similar speeds in the opposite direction. The way the particles bounce off the targets (‘scatter’) can be used to reveal details of the internal structure of the particles which make up the targets, rather as if those particles were being X-rayed. Also, when the fast-moving particles are brought to a halt, or dramatically slowed, in collisions, their energy of motion (kinetic energy) is converted into showers of new particles, in line with Albert Einstein’s famous equation E = mc² (in this case, the important point being that m = E/c²).

The showers of particles produced in this way can be studied to test theories of how the quantum world works. It is important to appreciate that in most cases there is no sense in which the particles in the showers were originally ‘inside’ the colliding particles and have been broken off, or knocked out, by the collisions. They have been created out of pure energy, and did not exist before the collision occurred. Indeed, the total mass of the particles in the shower can considerably exceed the rest mass of the particles involved in the collision.

In the most extreme experiments, beams of electrons collide with beams of positrons (antielectrons). Similarly, protons can be collided with antiprotons. When a particle meets its antiparticle counterpart, as well as the kinetic energy from each particle the mass-energy is available to make new particles, as the particle/antiparticle pair annihilate one another entirely.

Accelerator. Part of the particle accelerator ring at CERN.

aces See quarks.

action A mathematical quantity which depends upon the mass, velocity and distance travelled by a particle. Action is also associated with the way energy is carried from one place to another by a wave, but it can be understood most simply by imagining the trajectory of a ball tossed in a high arc from one person to another.

One of the most fundamental laws of science is the law of conservation of energy. Energy cannot be created or destroyed, only converted from one form to another. The ball leaves the thrower’s hand with a large kinetic energy, but as it climbs higher its speed slows down and the kinetic energy is reduced. But because the ball is higher above the ground (strictly speaking, because it is further from the centre of the Earth), it has gained gravitational potential energy. Leaving aside friction (which converts some of the energy of motion of the ball into heat energy as it passes through the air), the amount of gravitational energy it gains matches the amount of kinetic energy it has lost, for each point in its climb. At the top of its trajectory, the ball momentarily stops moving, so it has zero kinetic energy, but maximum gravitational energy for this particular trajectory. Then, as it falls towards the catcher, it gains kinetic energy at the expense of gravitational potential energy.

At any point along the trajectory, it is possible to calculate the kinetic energy and the potential energy of the ball. The total you get by adding the two is always the same. But if you subtract the potential energy from the kinetic energy, you get a different value of the difference at different points along the trajectory. If you add up this difference all along the trajectory, integrating the difference between the kinetic energy and the potential energy for the entire flight of the ball, the number you come up with is the action that corresponds to the flight of the ball. The action is a property not of a single point along the trajectory, but of the entire trajectory.

Action. The way light travels, to reach its destination by the quickest path possible, is an example of the principle of least action at work. The path which minimizes the time to go from A (in air) to B (in glass) is shown by the solid line; any other path takes longer.

There is a value of the action for each possible trajectory of the ball. In a similar way, there is a value of the action corresponding to each trajectory that might be taken by, say, an electron moving in a magnetic field. The way we have described it here, you would calculate the action using Newton’s laws of motion to describe the flight of the ball; but the process can be turned on its head, with the properties of the action used to determine the laws of motion. This works both for classical mechanics and for quantum mechanics, making the action one of the most important concepts in all of physics.

This is because objects following trajectories always follow the path of least action, in a way analogous to the way water runs downhill to the point of lowest energy available to it. There are many different curves that the ball could follow to get to the same end point, ranging from low, flat trajectories to highly curved flight paths in which it goes far above the destination before dropping on to it. Each curve is a parabola, one of the family of trajectories possible for a ball moving under the influence of the Earth’s gravity. But if you know how long the flight of the ball takes, from the moment it leaves the thrower’s hand to the moment it reaches its destination, that rules out all but one of the trajectories, specifying a unique path for the ball.

Given the time taken for the journey, the trajectory followed by the ball is always the one for which the difference, kinetic energy minus potential energy, added up all along the trajectory, is the least. This is the principle of least action, a property involving the whole path of the object.

Looking at the curved line on a blackboard representing the flight of the ball, you might think, for example, that you could make it take the same time for the journey by throwing it slightly more slowly, in a flatter arc, more nearly a straight line; or by throwing it faster along a longer trajectory, looping higher above the ground. But nature doesn’t work that way. There is only one possible path between two points for a given amount of time taken for the flight. Nature ‘chooses’ the path with the least action—and this applies not just to the flight of a ball, but to any kind of trajectory, at any scale.

It’s worth giving another example of the principle at work, this time in the guise of the principle of ‘least time’, because it is so important to science in general and to quantum physics in particular. This variation on the theme involves light. It happens that light travels slightly faster through air than it does through glass. Either in air or glass, light travels in straight lines—an example of the principle of least time because, since a straight line is the shortest distance between two points, that is the quickest way to get from A to B. But what if the journey from A to B starts out in air, and ends up inside a glass block? If the light still travelled in a single straight line, it would spend a relatively small amount of time moving swiftly through air, then a relatively long time moving slowly through glass. It turns out that there is a unique path which enables the light to take the least time on its journey, which involves travelling in a certain straight line up to the edge of the glass, then turning and travelling in a different straight line to its destination. The light seems to ‘know’ where it is going, apply the principle of least action, and ‘choose’ the optimum path for its journey.

In some ways, this is reminiscent of the way a quantum entity seems to ‘know’ about both holes in the double-slit experiment even though common sense says that it only goes through one hole; but remember that the principle of least action applies in the everyday world as well as in the quantum world. Richard Feynman used this to develop a version of mechanics, based on the principle of least action, which describes both classical and quantum mechanics in one package.

See also sum over histories.

WARNING! Unfortunately, physicists also use the word ‘action’ in a quite different way, as shorthand for the term ‘interaction’. See action at a distance. This has nothing to do with the action described here.

action at a distance The idea that interactions between objects, such as the gravitational interaction that holds the Earth in orbit around the Sun, operate without any intervening mechanism. The original version of the idea saw the interaction occurring instantaneously, regardless of the distance involved. Modern variations on the theme involve interactions that occur at a distance, but with a time delay related to the speed of light. See transactional interpretation, Wheeler-Feynman absorber theory.

Note that this use of the term ‘action’ as shorthand for ‘interaction’ has nothing to do with action.

adiabatic process A process which occurs without heat entering or leaving a system. This usually means that the temperature of the system changes. For example, if a gas expands adiabatically, pushing a piston out of a cylinder, the gas cools because it has to do work to make the piston move. Contrast this with an isothermal process.

In particle physics, the term ‘adiabatic’ is used to describe interactions in which there is no input of energy.

ADONE One of the first electron-positron colliders, built at Frascati, near Rome.

advanced wave A wave that travels backwards in time, from the future, to arrive at its ‘source’. See transactional interpretation, Wheeler-Feynman absorber theory.

Alhazen (Abu AM al-Hassan ibn al-Haytham) (about 965-1038) The greatest scientist of the Middle Ages, whose achievements were unsurpassed for more than 500 years, until the work of Galileo, Kepler and Newton. Usually referred to by the Europeanized version of his name.

Alhazen was born in Basra, now part of Iraq, in about 965. He later moved to Cairo, where he worked in the service of the Caliph al-Hakim, a reputedly mad tyrant. Having boasted that he could find a way to control the flooding of the Nile, Alhazen was sent south by the Caliph to make good his promise. When the expedition failed, in order to escape execution Alhazen himself had to pretend to be mad for several years, until the Caliph died in 1021. He then resumed his normal life.

Alhazen’s greatest scientific contribution was his work on optics, contained in a series of seven books (what we would now call scientific papers) written on either side of the year 1000. This work was translated into Latin, the scientific language of the day, at the end of the 12th century, and influenced the thinking of, among others, Roger Bacon. It was published in Basle in 1572, more than 500 years after Alhazen’s death, under the title Opticae thesaurus (Treasury of Optics).

The key insights in Alhazen’s work included a logical argument that sight does not (as earlier thinkers had taught) work by the eye sending out rays to probe the world outside, but is entirely due to light, produced in a flame or by the Sun, bouncing off objects and entering the eyes from outside. His key analogy was with the way images are formed in a ‘camera obscura’—a darkened room in which curtains are placed over the windows, with a pinhole in one curtain. Light from outside, passing through the pinhole, makes an image on the opposite wall of the world outside. This is indeed the way the eye (and a photographic camera) works. Alhazen realized that this means that light travels in straight lines, and thought of light as being made up of a stream of tiny particles that bounce off objects that they strike. This was the earliest introduction of the concept of what are now known as photons.

Alhazen measured both the reflection and the refraction of light, and tried to explain the occurrence of rainbows. He studied the Sun during an eclipse by using the camera obscura technique to cast its image on a wall, and he wrote scores of other ‘books’ on mathematics and scientific topics. He was the first scientific thinker to surpass the work of the ancient Greeks. He died in Cairo in 1038.

Alice matter See shadow matter.

alpha decay A process of radioactive decay in which the nucleus of an atom ejects an alpha particle. See tunnel effect.

alpha particle The nucleus of an atom of helium-4, made up of two protons and two neutrons held tightly together by the strong nuclear force. This nucleus is unusually stable, and is held together so tightly that alpha particles, produced in alpha decay, do indeed behave in many ways like single particles. Such particles, each carrying two units of positive charge, were the first probes used to investigate the structure of atoms, by Ernest Rutherford and his colleagues, at the beginning of the 20th century.

alpha radiation A stream of alpha particles, produced by alpha decay. The particles move at speeds of about 1,600 km per second, but can be stopped by a sheet of paper.

Alvarez, Luis Walter (1911-1988) American physicist who received the Nobel Prize in 1968 for his work in high-energy particle physics, including the development of the hydrogen bubble chamber technique.

Alvarez was born on 13 June 1911 in San Francisco. His father was a physician, who later became a medical journalist. Alvarez studied at the University of Chicago, switching from chemistry to physics after starting his degree. He graduated in 1932, and stayed there to complete an MSc in 1934 and his PhD in 1936. He then moved to the Berkeley campus of the University of California, where he worked at the Lawrence Radiation Laboratory for the rest of his career, apart from wartime work on radar and on the Manhattan Project.

Back at Berkeley, Alvarez became a full professor in 1945, and played a key role in the development of the first practical linear accelerator, which (in 1947) could accelerate protons to energies of 32MeV. In 1953, after a meeting with Donald Glaser (the inventor of the bubble chamber technique), Alvarez concentrated on developing a series of hydrogen bubble chambers, culminating in a large device, 72 inches in diameter, used to investigate the properties of particles at high energies. The data from these high-energy investigations provided key input for the theorists who developed the eightfold way concept and the idea of quarks.

Later, Alvarez became interested in less conventional ideas. He searched for hidden chambers in Egyptian pyramids using studies of cosmic rays to ‘X-ray’ the pyramids, and he gained worldwide fame in the 1980s for the suggestion, based on work carried out with his son Walter, that the ‘death of the dinosaurs’ was caused by the impact of a large meteorite with the Earth. He died in San Francisco on 1 September 1988.

amplitude For a wave, the amplitude is half the height from the peak of the wave to the trough. In quantum mechanics, the amplitude of a process is a number that is related to the probability of the process occurring. If there are several alternative processes (for example, if there are several different ways an electron can get from A to B), then each has its own amplitude. The probability of the electron ‘choosing’ a particular route (in this example) is equal to the square of the amplitude. But the probability that the electron will choose any of the routes (the probability that it goes from A to B by any means, rather than going off to C or D instead) is given not by adding up the probabilities, but by adding up the amplitudes and squaring the total amplitude (see sum over histories).

The numbers which measure quantum mechanical amplitudes are so-called complex numbers, which involve the square root of -1. This affects the way in which the squares (the probabilities) are calculated (see complex conjugation), and has important implications for the transactional interpretation of quantum mechanics.

See wave.

a.m.u. See atomic mass unit.

Anderson, Carl David (1905-1991) American physicist who received the Nobel Prize in 1936 for the discovery of the positron—the first proof of the existence of antimatter.

Anderson, the son of Swedish immigrants, was born in New York on 3 September 1905. He was brought up in Los Angeles, and studied at the California Institute of Technology, where he graduated in 1927. He stayed on as a graduate student, working with Robert Millikan, and completed his PhD in 1930. Still at Caltech, where he stayed for the rest of his career, at Millikan’s suggestion Anderson built a cloud chamber in order to study the tracks of electrons produced by cosmic rays. The detector soon (in 1932) revealed tracks produced by particles with the same mass as electrons but the opposite electric charge—positrons.

In the same year that he received the Nobel Prize, Anderson and his student Seth Neddermeyer announced the discovery of cosmic ray particles with mass larger than the mass of the electron but smaller than the mass of the proton. These particles became known as muons.

After wartime work on rockets, Anderson resumed his cosmic ray studies. He retired in 1976, and died on 11 January 1991.

Anderson, Philip Warren (1923-) American solid-state physicist, born in Indianapolis on 13 December 1923, who was awarded a share of the Nobel Prize in 1977 for his work on the behaviour of electrons in disordered (non-crystalline) solids. His work on semiconductors paved the way for the devices now used in computer memories.

Ångström, Anders Jonas (1814—1874) Swedish physicist who was one of the founders of the study of spectroscopy.

Born at Lödgö on 13 August 1814, Ångström was the son of a country chaplain. He studied at Uppsala, where he received his doctorate in 1839, and stayed to become first a lecturer and then (in 1858) professor of physics, a post he held until his death. Ångström’s studies of spectra led him to conclude that gases emit and absorb light at characteristic wavelengths, and in 1861, building on his own work and that of Gustav Kirchoff, Ångström began a study of the solar spectrum which showed that hydrogen is present in the Sun. Because he worked quietly and did not seek publicity (and because he published his results in Swedish), the value of Ångström’s work was only slowly recognized, but in 1870 he was elected a Fellow of the Royal Society, and after his death (at Uppsala on 21 June 1874) his name was given to a unit of wavelength used in spectroscopy.

angstrom An obsolete unit of wavelength defined as one hundred millionth (10-8) of a centimetre. Equal to one-tenth of a nanometre. Symbol Å.

angular momentum A property of rotating objects analogous to the momentum of an object moving in a straight line. The angular momentum of a spinning object depends on its mass, its size and the speed with which it is spinning. An object in orbit around another object (such as the Moon orbiting around the Earth) also has angular momentum, which depends on the mass of the object, the radius of its orbit and the speed with which it is moving. And the concept can be extended to any object moving in a curved trajectory.

In the quantum world, angular momentum is quantized, and can change only by amounts which are integer multiples of Planck’s constant ħ. One important implication of this is that if electrons are thought of as being in orbit around the central nucleus of an atom, they can only occupy distinct stable orbits which are separated by a whole number of angular momentum quanta. Similarly, the spin of an electron is quantized, and the electron can only exist with ‘spin up’ (+1/2) or ‘spin down’ (-1/2) measured against some external reference such as a magnetic field, not in an intermediate state.

annihilation See antimatter, pair production.

anode A positively charged electrode in, for example, a vacuum tube (electronic valve) or a battery (electric cell). Because electrons are negatively charged, they move towards an anode.

anomalous Zeeman effect See Zeeman effect.

anthropic principle The idea that the existence of life in the Universe (specifically, human life) can set constraints on the way the Universe is now, and how it got to be the way it is now.

The power of anthropic reasoning is best seen by an example. In order for us to exist, there has to be one star (the Sun), orbited at the appropriate distance by one planet (the Earth), made of the right mixture of chemical elements (particularly including carbon, nitrogen, oxygen and the primordial hydrogen left over from the Big Bang). At first sight, it may seem that the existence of the rest of the Universe, containing millions of galaxies scattered across billions of light years of space, is irrelevant to our existence.

But where did the elements of which we and the Earth are made come from? The Big Bang produced only hydrogen, helium and traces of a few light elements. Carbon and other heavy elements were manufactured inside stars (see nucleosynthesis) which had to run through their life cycles and explode, scattering the heavy elements across space to form clouds of material from which later generations of stars, including the Sun, and their attendant planets could form. This took billions of years. The evolution of life on a suitable planet to the point where intelligent beings could notice their surroundings and wonder about the size of the Universe took more billions of years. All the while, the Universe was expanding. After billions of years, it is inevitably billions of light years across. So the fact that we are here to ask questions about the size of the Universe means that the Universe must contain many stars, and must be billions of years old and billions of light years across.

This rests upon the assumption that there is nothing special about our place in the Universe, and nothing special about us—a proposition sometimes referred to as ‘the principle of terrestrial mediocrity’.

Further reading: John Barrow and Frank Tipler, The Anthropic Cosmological Principle; John Gribbin, Companion to the Cosmos.

antimatter A form of matter in which each particle has the opposite set of quantum properties (such as electric charge) to its counterpart in the everyday world. The classic example of a particle of antimatter (an antiparticle) is the antielectron, or positron, which has the same mass as an electron but a positive charge instead of a negative charge. The existence of antielectrons was predicted by Paul Dirac, at the end of the 1920s, when he found that the equation which represents a complete description of the electron in terms of both quantum mechanics and the special theory of relativity has two sets of solutions, one corresponding to negatively charged particles and one to positively charged particles. The exact meaning of this was not clear until 1932, when Carl Anderson discovered positrons from the traces they left in his cosmic ray detector.

Our visible Universe is almost entirely composed of matter, and very little antimatter has existed since the Big Bang in which the Universe was born. When an antiparticle meets its particle counterpart (for example, when a positron meets an electron), they annihilate, converting all of their rest mass into energy in line with Einstein’s equation E = mc². Antiparticles can be made out of energy in the reverse of this process, but only if a particle counterpart for every antiparticle is produced as well. This happens naturally in high-energy processes involving cosmic rays, and also in high-energy experiments in accelerators on Earth. Because the world is overwhelmingly made of matter, however, any antiparticle produced in this way soon meets up with a particle counterpart and annihilates.

It is now clear that there are antimatter counterparts for all the matter particles—antiprotons, antineutrons, antineutrinos and so on. All of these are produced routinely in accelerator experiments. In September 1995, a team of researchers at CERN succeeded for the first time in making complete antiatoms (of hydrogen, the simplest element) in which a negatively charged antiproton is associated with a positively charged antielectron. Just nine atoms of antihydrogen were manufactured in this first experiment, and they each survived for only about 40 billionths of a second before colliding with ordinary matter and annihilating. The researchers hope that they will soon be able to trap antiatoms for long enough to study them (for example, by probing them with laser beams) and find out if, as present theories predict, the laws of physics work in exactly the same way for antimatter as they do for matter.

Further reading: Yuval Ne’eman and Yoram Kirsch, The Particle Hunters.

antiparallel vectors Arrows that point in opposite directions. For example, the spin of an electron, measured against a background magnetic field, can point in only one of two directions, up or down. If two electrons have opposite spins, their spins can be said to be antiparallel. This allows pairs of electrons to share otherwise identical quantum ‘orbits’ in an atom.

antiparticles See antimatter.

Arago, (Dominique) François Jean (1786-1853) French physicist who made pioneering investigations into the nature of light, was instrumental in establishing the wave theory developed by Augustin Fresnel as a good description of how light travels, and paved the way for Leon Foucault’s determination of the speed of light in the laboratory.

Arago was born in Estagel, near Perpignan, on 26 February 1786. He studied at the Ecole Polytechnique in Paris, then worked for the Bureau de Longitude, and was a member of an expedition to Spain in 1806 to measure an arc of the meridian. He returned to Paris in 1809 and became a professor at the Ecole Polytechnique. Arago carried out experiments in many branches of physics, including electricity and magnetism, made astronomical observations, and was chairman of the