Farewell to Reality (11 page)

Dirac was a mathematician first, a physicist second. In devising his theoretical equation he had followed his mathematical instincts, and when these seemed to suggest some relatively unphysical conclusions,
he had nevertheless stood firm.
³
The theory produced twice as many solutions as he thought he had needed. Two of these solutions correspond to the spin-up and spin-down orientations of the electron, each carrying positive energy. But there were another two solutions of ‘negative' energy. Although these solutions were hard to understand, they couldn't be ignored.

In 1931, Dirac finally conceded that the two negative-energy solutions actually correspond to positive-energy spin-up and spin-down orientations of a
positively
charged electron. He had discovered the existence of antimatter, a previously unsuspected form of material substance. There had been no hints of antimatter in any experiment performed to that time and no good reason to suspect that it existed. It was completely ‘off the wall'.

Yet within a year, the positive electron had been discovered in experiments on cosmic rays. It was named the
positron.
Dirac's mathematical instincts were proved right.

Quantum fields and second quantization

Dirac's theory is actually a kind of
quantum field theory.
A field in physics is defined in terms of the magnitude of some physical property distributed over every point in space and time. Sprinkle iron filings on a sheet of paper held above a bar magnet. The iron filings organize themselves along the ‘lines of force' of the magnetic field, reflecting the strength of the field and its direction, stretching from north to south poles. The field exists in the ‘empty' space around the outside of the bar of magnetic material.

The nature of the field depends on the nature of the property being measured. The field may be
scalar,
meaning that it has magnitude but acts in no particular direction in space — it points but it doesn't push or pull. It may be
vector,
with both magnitude and direction, like a magnetic field or a Newtonian gravitational field. Finally, it may be
tensor,
a more complicated version of a vector field for situations in which the geometry of the field requires more parameters than would be required for three ‘Cartesian'
x, y
and
z
directions.

The field described by Dirac's theory is in fact none of the above. It is a
spinor field.
Spinors were discovered by mathematician Elié Cartan in 1913. They are vectors, but not in the sense of vectors in ‘ordinary'
space, and they cannot be constructed from ordinary vectors. They are, of course, spin vectors, related to the spin orientations of the electron. To make his theory conform to the requirements of special relativity, Dirac needed a spinor field with four components — two for the electron and two for the positron.

It might be best to think of the field used in Dirac's theory as an ‘electron field', the quantum field for which the corresponding quantum particle is the electron. The particle is, in essence, a fundamental field quantum, a basic fluctuation or disturbance of the field.

In contrast to conventional quantum theory, when we add or remove quanta in a quantum field theory, we're adding or removing particles to or from the field itself. We add energy to the field in the form of particles, rather than adding it to individual quantum particles. This is sometimes known as ‘second quantization'.

The strange theory of light and matter

Dirac's theory represented a major advance, but as a field theory it seemed rather obscure. It wasn't obvious from the structure of his equation that it met the demands of special relativity (though it did); neither was it clear how the theory should be related to much better understood classical field theories, such as Maxwell's theory of electromagnetism.

The attentions of some physicists were drawn instead to an alternative approach. Why not start with a well-understood classical wave field, and then find a way to impose quantum conditions on it? The obvious place to start was the field described by Maxwell's equations. If it could be done, the resulting quantum field theory would describe the interactions between an electron field (whose quanta are electrons) and the electromagnetic field (whose quanta are photons).

It began to dawn on physicists working on early versions of quantum field theory that they had figured out a very different way to understand how forces between particles actually work. Let's imagine that two electrons are ‘bounced' off each other (the technical term is ‘scattered'). We can suppose that as the two electrons approach each other in space and in time, they feel the mutually repulsive force generated by their negative charges.

But how? We might speculate that each moving electron generates an electromagnetic field and the mutual repulsion is felt in the space where these two fields overlap, much like we feel the repulsion between the north poles of two bar magnets in the space between them as we try to push them together. But in the quantum domain, fields are also associated with particles, and interacting fields with interacting particles. In 1932, German physicist Hans Bethe and Italian Enrico Fermi suggested that this experience of force is the result of the
exchange of a photon
between the two electrons.

As the two electrons come closer together, they reach some critical distance and exchange a photon. In this way the particles experience the electromagnetic force. The exchanged photon carries momentum from one electron to the other, thereby changing the momentum of both. The result is recoil, with both electrons changing speed and direction and moving apart.

The exchanged photon is a ‘virtual' photon, because it is transmitted directly between the two electrons and we don't actually see it pass from one to the other. In fact, there's no telling in which direction the photon actually goes. In diagrams drawn to represent the interaction, the passage of a virtual photon is simply illustrated using a squiggly line, with no direction indicated between the electrons.

Here was another revelation. The photon was no longer simply the quantum particle of light. It had become the ‘carrier' of the electromagnetic force.

Heisenberg and Pauli had tried to develop a formal quantum field theory of electromagnetism a few years before, in 1929. But this theory was plagued with problems. The worst of these was associated with the ‘self-energy' of the electron. When an electric charge moves through space, it generates an electromagnetic field. The ‘self-energy' of the electron results when an electron interacts with its own self-generated electromagnetic field. This interaction caused the equations to ‘blow up', producing physically unrealistic results. Some terms in the equation mushroomed to infinity.

A solution to this problem would be found only in 1947, when physicists were able to return to academic science after spending the war years working on the world's first atomic weapons.

Suppose the self-energy of the electron appears (using E = mc
²
) as an additional contribution to the electron's mass. The mass that we
observe in experiments would then include this additional contribution. It would be equal to an intrinsic or ‘bare' mass plus an ‘electromagnetic' mass arising from the electron's interaction with its own electromagnetic field.

The ‘bare' mass is a purely theoretical quantity. It is the mass that the electron would possess if it could ever be isolated from its own electromagnetic field. The mass that we have to deal with is the observed mass, so the theory has to be rewritten in terms of this. In other words, the theory has to be ‘renormalized'.

Suppose further that we want to apply quantum field theory to the calculation of the energies of the quantum states of an electron in a hydrogen atom. The theory predicts an infinite self-energy associated with the electron interacting with its own electromagnetic field. We identify this as an infinite contribution to the electron mass. But the equations for a freely moving electron also contain the same infinite mass contribution. So, what if we now subtract the expression for a free electron from the expression for the electron in a hydrogen atom? Would the two infinite terms cancel to give a finite answer?

It sounds as though subtracting infinity from infinity should lead only to nonsense, but it worked. A formal theory of quantum electrodynamics (QED), essentially a quantum version of Maxwell's theory, was eventually devised by rival American physicists Richard Feynman and Julian Schwinger, and independently by Japanese theorist Sin-Itiro Tomonaga. Although the approaches adopted by Schwinger and Tomonaga were similar, Feynman's was distinctly different, relying on pictorial representations of the different kinds of possible interactions which came to be called ‘Feynman diagrams'. English physicist Freeman Dyson subsequently demonstrated that their different approaches were entirely equivalent.

Not everyone was comfortable with the apparent sleight of hand involved in renormalization. Dyson asked Dirac: ‘Well, Professor Dirac, what do you think of these new developments in quantum electrodynamics?' Dirac, the mathematical purist, was not enamoured: ‘I might have thought that the new ideas were correct if they had not been so ugly.'
⁴

Ugly or not, there was no denying the power of the resulting theory. The g-factor for the electron, a physical constant governing the strength of the interaction of an electron with an external magnetic field, is
predicted by QED to have the value 2.00231930476. The comparable experimental value is 2.00231930482. ‘This accuracy', wrote Feynman, ‘is equivalent to measuring the distance from Los Angeles to New York, a distance of over 3,000 miles, to within the width of a human hair.'
⁵

The particle zoo

Dirac once speculated that it might one day be possible to describe all matter in terms of just one kind of elementary particle, some kind of ultimate quantum of ‘stuff'. Alas, as experimental physicists set to work with cosmic rays and early particle accelerators in the 1930s and 40s, Dirac's dream was shattered. Far from discovering an underlying simplicity in nature, they discovered instead a bewildering complexity.

American physicist Carl Anderson had discovered the positron in 1932. Four years later he discovered another particle, a heavier version of the electron, with a mass about 200 times that of an ordinary electron. This simply did not fit with any preconceptions of how the elementary building blocks of nature should be organized. Galician-born American physicist Isidor Rabi demanded to know: ‘Who ordered that?'
⁶
The new particle was given several names, but is today called the
muon.

In 1947, another new particle was discovered in cosmic rays by British physicist Cecil Powell. This was found to have a slightly larger mass than the muon; 273 times that of the electron. It came in positive, negative and, subsequently, neutral varieties. This was called the
pion.
As techniques for detecting particles became more sophisticated, the floodgates opened. The pion was quickly followed by the positive and negative kaon and the neutral lambda particle.

What was going on? One physicist expressed the prevailing sense of frustration: ‘… the finder of a new elementary particle used to be rewarded by a Nobel Prize, but such a discovery now ought to be punished by a $10,000 fine'.
⁷
Names proliferated. Responding to a question from one young physicist, Fermi remarked: ‘Young man, if I could remember the names of these particles, I would have been a botanist.'
⁸

In an attempt to give some sense of order to this ‘zoo' of particles, physicists introduced a new taxonomy. They defined two principal
classes:
hadrons
(from the Greek
hadros,
meaning thick or heavy) and
leptons
(from the Greek
leptos,
meaning small).

The class of hadrons includes a subclass of
baryons
(from the Greek
barys,
also meaning heavy). These are heavier particles which experience the strong nuclear force. The proton and neutron are baryons. It also includes the subclass of
mesons
(from the Greek
mésos,
meaning ‘middle'). These particles experience the strong force but are of intermediate mass. Examples include pions and kaons.

The class of leptons includes the electron, muon and neutrino. These are light particles which do not experience the strong nuclear force.

The baryons and the leptons are
matter particles.
They are also all fermions, characterized by half-integral spins, which obey Pauli's exclusion principle.

In addition to these there
are force particles.
These include the photon. These are
bosons,
characterized by integral spins. They are not subject to Pauli's exclusion principle. Bosons with zero spin are also possible, but these are not force particles. Mesons are examples.

The taxonomy was a little like organizing the chemical elements into a periodic table. It helped to establish the patterns among the different particle classes but gave no real clue to the underlying explanation.

The left hand of the electron

Things were about to get even more complicated. A wavefunction, such as a sine wave, moves up and down as it oscillates between peak and trough. Parts of the wavefunction have positive amplitude (as it rises to the peak and falls back) and parts have negative amplitude (as it dips below the axis heading for the trough and comes back up again).

The
parity
of the wavefunction is determined by its behaviour as we change the signs of the spatial co-ordinates in which the wave propagates. Think of this as changing left for right or up for down or front for back. Changing the signs of all three co-ordinates simultaneously is then a bit like reflecting the wavefunction in a special kind of mirror that also inverts the image and its perspective. The image is inverted left-to-right and up-to-down, and the front goes to the back as the back is brought forward to the front.