Professor Stewart's Hoard of Mathematical Treasures (17 page)

‘If only I’d paid more attention to my maths teacher at school,’ Roger sighed. ‘For then, by Beelzebub’s flaming breeches, I’d know how far X must be from the markers.’
¹⁹

What are the three distances?

[Hint: This is hard. You may find it helpful to know that if 7 divides a sum of two integer squares,
u
²
+
v
²
, then 7 divides each of u and v. Then again ... ]

 

Answer on page 294

Harold P. Furth was an Austrian-born American physicist who worked on nuclear fusion and related topics. In 2001, he wrote a short poem, ‘Perils of Modern Living’, which begins:

Edward Teller was a co-inventor of the hydrogen bomb, acquired huge political influence, and was the inspiration behind the Dr Strangelove character in the movie of the same name. The poem goes on to relate that one day a visitor from Earth turned up, and human and antihuman approached each other:

Anyone brought up on
Star Trek
is aware that antimatter is a kind of ‘mirror image’ of ordinary matter, and when the two are brought into contact they annihilate each other in a gigantic burst of photons (‘gamma rays’), particles of light. The combined
mass of the two types of matter is released as energy. Thanks to Einstein’s famous formula
E = mc
²
, a small mass
m
turns into a huge amount of energy E, because the speed of light
c
is very big, so
c
²
is even bigger.

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
x
²
= 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:

For more on antimatter physics, see:

For the Alcubierre drive and related topics, see:

Antimatter isn’t just highbrow physics. Positrons have an important use in medical PET (positron emission tomography) scanners. These are 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.