Laying hands on ordinary matter is no great problem; there’s a lot of it about. If we could also acquire (not by laying hands on) even a small amount of antimatter, we would have a compact source of almost unbounded energy. This potential has long been apparent to physicists and Star Trek writers. You just have to find or make antimatter, and store it in something where it won’t come into contact with ordinary matter, like a magnetic bottle. It works fine in Star Trek, but today’s technology falls woefully short of what will be available to starship captains in the 22nd century.20
In the current theories of particle physics, very well supported by experiment, every type of charged subatomic particle has an associated antiparticle, with the same mass but opposite electrical charge, and if the two ever meet ... bang! Now, Hoard is not about physics, but this particular bit of physics came about as an unintended side effect of a mathematical calculation. Sometimes a little bit of maths, taken seriously, can jump-start a scientific revolution.
In 1928, a young physicist named Paul Dirac was trying to reconcile the newfangled ideas of quantum mechanics with the slightly less newfangled ideas of relativity. He focused on the electron, one of the particles out of which atoms are made, and eventually wrote down an equation that both described the quantum properties of this particle and was also consistent with Einstein’s special theory of relativity. This, it must be added, was far from easy. The Dirac equation was a major event in physics, and it was one of the discoveries that led to his Nobel Prize in 1933. For all you equation-lovers out there: you’ll find it in the note on page 296.
Dirac started from the standard quantum-mechanical equation for the electron, which represents it as a wave; the difficulty was to tinker with this equation so that it respected the requirements of special relativity. To do so, he followed his celebrated nose for mathematical beauty, seeking an equation that treated energy and momentum on the same footing. One evening, sitting beside the fire in Cambridge and musing on this problem, he thought of a clever way to rewrite the ‘wave operator’ - a key feature of the traditional equation - as the square of something simpler. This step quickly led to some technical issues that were very familiar, and soon the desired equation was staring him in the face.
There was one snag, though. His reformulation introduced new solutions of his equation that did not solve the original version. This always happens when you square an equation; for instance, x = 2 becomes x2 = 4 when you square it, and now there is another solution, x = -2. Physically, one solution of Dirac’s equation has positive kinetic energy,21 while the other has negative kinetic energy. The first type of solution obeys all requirements for an electron - but what of the second type? On the face of it, negative kinetic energy makes no sense.
In classical (that is, non-quantum) relativity, this kind of thing also happens, but it can be evaded. A particle can never move from a state with positive energy to one with negative energy, because the system must change continuously. So the negative-energy states can be ruled out. But in quantum theory, particles can ‘jump’ discontinuously from one state to a completely different one. So the electron might, in principle, jump from a physically sensible positive-energy state to one of those baffling negative-energy states.
Dirac decided that he had to allow these puzzling solutions as well. But what were they?
The electron, like all subatomic particles, is characterised by various physical quantities, such as its mass, spin and electrical charge. The particle described by the Dirac equation has all the right properties for an electron; in particular, its spin is and its charge is -1, in suitable units. Working through the details, Dirac noticed that the puzzling solutions were just like electrons, with the same spin and the same mass, but their charge was +1, the exact opposite. Dirac had followed his mathematical nose and in effect had predicted a new particle.
Ironically, he stopped short of doing that, partly because he thought the ‘new’ particle was the familiar proton, which has positive charge. Now, a proton is 1,860 times as heavy as an electron, whereas the negative-energy solution of Dirac’s equation has to have the same mass as an electron. But Dirac thought that this discrepancy was caused by some asymmetry in electromagnetism, so he titled his paper ‘A Theory of Electrons and Protons’. It was a missed opportunity, because in 1932 Carl D. Anderson spotted a particle with the mass of the electron, but with opposite charge, in an experiment using a cloud chamber to detect cosmic rays. He named the newcomer the positron. When asked why he had not predicted the existence of this new particle, Dirac reportedly replied: ‘Pure cowardice!’
Not all the difficulties disappeared with the discovery of positrons. Individual positrons don’t have negative kinetic energy, so Dirac suggested that his equation really applies to a ‘sea’ of negative-energy electrons, which occupy almost all the available negative-energy states. ‘An unoccupied negative-energy state,’ he wrote, ‘will now appear as something with a positive energy, since to make it disappear . . . we should have to add to it an electron with negative energy.’ And he added that a quantum-mechanical vacuum provides just such a sea of particles. None of this is entirely satisfactory, even when reworked in terms of quantum field theory. But Dirac’s equation applies only to a single isolated particle, so it does not describe interactions, which is where the physical discrepancies arise. So physicists are happy to accept the Dirac equation provided that its interpretation is suitably restricted.
The consequences of these discoveries are enormous. Today, particle physicists see the existence of antimatter as a deep and beautiful symmetry in the fundamental laws of nature, called charge conjugation. To every particle there corresponds an antiparticle, differing mainly by having the opposite charge. An uncharged particle, such as the photon, can be its own antiparticle.22 If a particle and its antiparticle collide, they annihilate each other in a burst of photons.
The Big Bang ought to have created equal numbers of particles and antiparticles, so our universe ought to contain equal quantities of each type of matter - not counting photons. If the matter and antimatter were thoroughly mixed up, they would collide, so only photons would now exist. However, our universe isn’t like that; a lot of matter isn’t photons, and all of it seems to be ordinary matter. This is a big puzzle, called baryon asymmetry. No really satisfactory answer to this dilemma has been found. However, it turns out that charge conjugation symmetry is not quite exact, and it would have taken only a billion and one particles of matter for every billion particles of antimatter to lead to what we see today. Alternatively, there may be other regions of the universe where antimatter dominates, although that looks rather unlikely. Or maybe time travellers from the distant future may have stolen one particle of antimatter from every billion and one in the early universe, to power their time machines.
Antimatter certainly exists, however, because we can make it. Atoms of antihydrogen, made from one positron circling an antiproton, were first created in 1995 at the CERN particle accelerator facility in Geneva. No heavier antiatoms have yet been produced, although the nucleus of antideuterium (an atom that lacks its orbiting positron) has been made. The most common form of antimatter in laboratory experiments is the positron, which can be generated by certain radioactive atoms that undergo beta-plus decay. Here a proton turns into a neutron, a positron and a neutrino. These atoms include carbon- 11, potassium-40, nitrogen-13, and others.
The entire Furth poem can be found at:
www.cs.rice.edu/~ssiyer/minstrels/poems/795.html
For more on antimatter physics, see:
en.wikipedia.org/wiki/Antimatter
livefromcern.web.cern.ch/livefromcern/antimatter
For the Alcubierre drive and related topics, see:
en.wikipedia.org/wiki/Alcubierre_drive
hyperspace.wikia.com/wiki/Alcubierre_drive
How to See Inside Things
Antimatter isn’t just highbrow physics. Positrons have an important use in medical PET (positron emission tomography) scanners. These a
re often used in combination with CAT (computerised axial tomography) scanners, often now shortened to CT. Both are based on mathematical techniques invented long ago for no particular practical purpose. Those ideas have to be improved and tweaked, of course, to account for various practical issues - for example, keeping the patient’s exposure to X-rays as low as possible, which reduces the amount of data that can be collected.
No, not like that.
The technology goes back to the early days of X-rays; the mathematics goes back to Johann Radon, who was born in 1887 in Bohemia, which was then part of Austro-Hungary and is now in the Czech Republic. Among his discoveries was the Radon transform.
Johann Radon in 1920.
How to transform him.
The raw material for the Radon transform is a ‘function’ f defined on all points x of the plane. This means that f defines some rule, which, for any given choice of x, leads to a specific number f(x). Examples are things like ‘form the square of x’, in which case f(x) = x2, and so on. The transform turns f into a related function F defined on lines in the plane. The value F(L) of F at some line L can be thought of as the average of f(x) as x runs along the line.
That’s not terribly intuitive (except to professionals), so I’ll restate it in terms of something that, in this computer age, may be more familiar. Consider a ‘black and white’ picture such as the photo of Radon on the opposite page. We can can associate a number with each shade of grey in the picture. So, if 0 = white and 1 = black, then would be whatever grey you get by mixing equal amounts of black with white, and so on. These numbers determine a ‘grey scale’: the bigger the number, the darker the shade of grey. So points in Radon’s collar are at 0, most of his face is around 0.25 or so, his jacket is 0.5 or higher, and some of the shadows are close to 1.
We can associate a function f with the photo. To do so, let x be any point in the photo, and let f(x) be the number for the shade of grey at that point. For instance, f(point in collar) = 0, f(point in face) = 0.25, and so on. This function is defined at all points in the plane (within the edges of the photo). We can also reconstruct the photo from the function - in fact, that’s how the image is stored in a computer, give or take a few technicalities.
To define the Radon transform F, take any line in the plane - say the line marked L in the right-hand picture. Let F(L) be the average grey-scale value of the photo along line L. Here L cuts across Radon’s face, and the average is (say) 0.38. So F(L) = 0.38. The line M has a lot more dark grey along it, so maybe F(M) = 0.72. You have to do this for every possible line, not just these two: there’s a formula for the answer in terms of an integral.
Starting with a function and working out its Radon transform is straightforward, though a bit messy. It is less clear that, given the Radon transform, you can work out the function. Radon’s key discovery is that this is possible, and he gave another formula for that calculation. It implies that, if all we know is the average grey-scale value along every line across Radon’s photo, then we can work out what Radon looks like.
What does any of this have to do with CAT scans?
Suppose a doctor could take a ‘slice’ of your body, along a plane, and make a grey-scale image of the tissues that the slice cuts. Dense organs would show up as dark grey, less dense ones as light grey, and so on. It would be just like a plane slice through a sort of ‘three-dimensional X-ray’ image. And it would tell the doctor exactly where your bodily tissues are, relative to that slice.
Unfortunately, no X-ray machine exists that can take that sort of picture directly. But what you can do is pass an X-ray beam - essentially, a straight line - through the body, and measure how strong the radiation is when it comes out at the far side. This strength is related to the average density of tissue - the average grey-scale value of the hypothetical slice - observed along that line. The greater the average tissue density, the weaker the emerging rays are. So, if you shone such a beam along every possible line in the slice plane, you would be able to work out the Radon transform of the grey-scale function for that slice. Then Radon’s formula would tell you the grey-scale function itself, and that would be a direct representation of the image created by the plane slice. That is, what that slice of you looks like in real space. So it’s a way to see inside solid objects.
In practice you can’t measure the Radon transform along every line, but you can measure it along enough lines to reconstruct a useful approximation to the image. (Many of the tweaks are to do with this loss of precision.) And this, give or take a few million dollars’ worth of technicalities,23 is what a CAT scanner does. You lie inside a machine that takes X-ray images from a series of closely spaced angles in a plane that slices through your body. A computer uses tweaked versions of Radon’s formula, or related methods, to work out the corresponding cross-sectional image. The scanner does one more thing: it shifts you along a millimetre or so, and repeats the same process on a parallel slice. And then another, and then another ... building up a full three-dimensional image of your body.
Slices through a human head, made by a CAT scanner.
PET scans use similar technology, and are often performed using the same machines, but with positrons in place of X-rays. The patient is given a dose of a mildly radioactive version of a common body sugar, usually one called fluorodeoxyglucose. The sugar concentrates at different levels in different tissues. As the radioactive element decays, it emits positrons, and the more sugar there is in any location, the more positrons that region emits. The scanner picks up the positrons and measures how much activity there is along any given line. The rest is much as before.
If you ever need a medical scan, it could be worth bearing in mind that what makes it possible is some equations doodled by a mathematical physicist, and a formula discovered nearly a hundred years ago by a pure mathematician interested in a technical question about integral transforms.
Mathematicians Musing About Mathematics
Mathematics is written for mathematicians. Nicolaus Copernicus
Mathematics is the supreme arbiter. From its decisions there is no appeal. We cannot change the rules of the game; we cannot ascertain whether the game is fair. Tobias Dantzig
With me, everything turns into mathematics. René Descartes
Mathematics may be likened to a large rock whose interior composition we wish to examine. The older mathematicians appear as persevering stone-cutters slowly attempting to demolish the rock from the outside with hammer and chisel. The later mathematicians resemble expert miners who seek vulnerable veins, drill into these strategic places, and then blast the rock apart with well placed internal charges. Howard W. Eves
Nature’s great book is written in mathematical symbols. Galileo Galilei
Mathematics is the queen of the sciences. Carl Friedrich Gauss
Mathematics is a language. Josiah Willard Gibbs
Mathematics is an interesting intellectual sport but it should not be allowed to stand in the way of obtaining sensible information about physical processes. Richard W. Hamming
Pure mathematics is on the whole distinctly more useful than applied. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics. Godfrey Harold Hardy
One of the big misapprehensions about mathematics that we perpetrate in our classrooms is that the teacher always seems to know the answer to any problem that is discussed. This gives students the idea that there is a book somewhere with all the right answers to all the interesting questions, and that teachers know those answers. And if one could get hold of the book, one would have everything settled. That’s so unlike the true nature of mathematics. Leon Henkin
Mathematics is a game played according to certain simple rules with meaningless marks on paper. David Hilbert
Mathematics is the science of what is clear by itself. Carl Gustav Jacob Jacobi
Mathematics is the science which uses easy words for hard ideas. Edward Kasner and James Newman
The chief aim of all inve
stigations of the external world should be to discover the rational order and harmony which has been imposed on it by God and which He revealed to us in the language of mathematics. Johannes Kepler
In mathematics you don’t understand things. You just get used to them. John von Neumann
Mathematics is the science which draws necessary conclusions. Benjamin Peirce
Mathematics is the art of giving the same name to different things. Henri Poincaré
We often hear that mathematics consists mainly of ‘proving theorems’. Is a writer’s job mainly that of ‘writing sentences’? Gian-Carlo Rota
Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. Bertrand Russell
Mathematics is the science of significant form. Lynn Arthur Steen
Mathematics is not a book confined within a cover and bound between brazen clasps, whose contents it needs only patience to ransack; it is not a mine, whose treasures may take long to reduce into possession, but which fill only a limited number of veins and lodes; it is not a soil, whose fertility can be exhausted by the yield of successive harvests; it is not a continent or an ocean, whose area can be mapped out and its contour defined: it is limitless as that space which it finds too narrow for its aspirations; its possibilities are as infinite as the worlds which are forever crowding in and multiplying upon the astronomer’s gaze. James Joseph Sylvester
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