Magnetic Moments

It is only four years since we were treated to Abraham Pais’s authoritative study of Einstein, Subtle is the Lord, and now he presents an equally large and quite as impressive history of fundamental particle physics, in a style and at a price which calls for the warmest congratulations to him and to the Clarendon Press.

This is no book for the casual reader. The first chapters, to be sure, recount the rise of atomic and nuclear physics straightforwardly, presenting no problem for anyone with a modest appreciation of A-level physics. The years 1895-9 saw the discovery by Roentgen of X-rays, by Becquerel of radioactivity and by J.J. Thomson of the electron, while in 1900 Planck introduced the quantum. So long as this last remained separate from the others the elucidation of their properties and mutual relationships was a matter for imaginative experiment and descriptive interpretation, with little need for any but rudimentary mathematics. Almost contemporaries of the quantum, Pauli, Heisenberg, Fermi and Dirac were born into a world whose physics was soon to be dominated by Rutherford, Einstein and Bohr (Planck played little part after his initial idea). In a few years from 1925, they utterly transformed it by creating a mathematical structure that effected a merger (under entirely new management) of the concerns of Rutherford’s experiments on nuclei, Einstein’s relativity theory and Bohr’s quantum theory of the atom. From this point onwards the physics and the mathematics became inseparable.

The invisible sub-microscopic world of atoms and their nuclei rather suddenly ceased to be a visualisable world in which the particles could be imagined to move in ways little different from real-life billiard balls. Mathematics had indeed been required for exact calculation, but its primary function to a man like Rutherford was to confirm what his powerful intuition could guess by analogy. With quantum mechanics and its immediate offshoot of quantum field theory an entirely new outlook was demanded: electrons and protons ceased being real in the sense they were before. They must surely represent something which exists out there and which is manifested to the physicist in the form of flashes of light or instrument readings: but to pretend that they are actually independent little objects, bouncing against one another or caught up in orbits, gets in the way of describing the results correctly. Instead one must find a set of rules, expressed in mathematical form, whose application to a given problem will tell as much as one is permitted to know about the relation of one observation to another. Thus if one wishes to calculate the properties of a helium atom, naively pictured with two electrons attracted to the nucleus and repelling one another, only a very approximate solution can be obtained by treating them as separate particles, moving under their mutual influences. To reach an answer that agrees with experimental measurements demands putting both electrons together through the appropriate manipulations: it is as if they must be described as one particle moving in six dimensions rather than two moving in three dimensions. The theoretical physicist is no more able than anyone else to visualise six dimensions, but he knows the rules for writing down the correct equations, and if he (or his computer) works every hard, he may find the solution in the form of numbers to be checked against the experimenter’s – the wavelengths of spectral lines, for example. Bohr’s earlier quantum theory had not done this very well, but the new quantum mechanics succeeded brilliantly.

From now on the further development of the theory was in the hands of the mathematicians, among them in due course Pais himself. I make no pretence of understanding much of the second half of the book except at a superficial level – even for a practising physicist the going is tough. But I have every confidence in its essential correctness and fair apportioning of credit. At first sight there seems to be an overwhelming partiality for American work in the post-war period, but more critical reflection disposes of my insular prejudice. The transition from Europe, pre-1939, to the United States, post-1945, is truly astonishing. Nazi policies must take considerable responsibility for the Jewish influx and its vitalising of American theoretical physics, but there is more to it than that. There was already an embryonic indigenous school, led by Oppenheimer, one of Pais’s heroes; and young men like Feynman and Schwinger were growing up in New York, poised to astonish the world by their virtuosity and to stimulate a new generation’s sortie into further reaches of precise abstraction.

There is no reason to attribute all this to affluence, for the advances that took place immediately after the war were purely theoretical and must have been maturing during the war years in the corners of minds whose major tasks were radar and the atomic bomb. Later, of course, the lavish funding of US science by the Office of Naval Research, and then the National Science Foundation, encouraged the growth of strong research departments in dozens of universities. Academic mobility was much greater than in other countries, and the habit of frequent research conferences was soon built into the system. Nor must one overlook the strength of experimental particle physics in the United States, where the pre-war pioneer cyclotron-building of Lawrence had established a powerful tradition that paid off handsomely at Los Alamos. Within a few years from 1945 a new generation of machines was being built while the rest of the world was still wondering how it could afford even to restore its pre-war institutions.

The political and social influences, and the essential role of expensive and complex equipment, are two sides of the history with which Pais engages only lightly – and with good reason, since both are tasks for specialists. Modern accelerators, and the intricate arrangements for preparing the beams of particles, recording the details of their interactions with targets and their subsequent fates, and analysing the records of millions of encounters, wholly occupy the thoughts of thousands of professional physicists (it is not uncommon to see a four-page account of a new experimental result published under the names of eighty or more collaborators from ten universities). One cannot expect a mathematical physicist to add to his already formidable learning a balanced critical appreciation of the different kind of ingenuity that lies at the heart of an important experiment. All the same, I am sorry, because it may serve to consolidate a popular impression to which even physicists can succumb: that experimenters wait below stairs upon the needs of the theoreticians above. In fact, experiment and theory enjoy a symbiosis in which each depends crucially on the performance of the other. But who would guess unprompted, from bare statements of the mean lifetime (925 seconds) of a neutron before it disintegrates or of its magnetic moment and that of the electron (this last to about ten figures), that they record inventiveness of a high order as well as a superb skill in execution?

Why should anyone wish to know the magnetic moment of the electron to ten-figure accuracy? The answer is worth giving in some detail as an exemplary account of the research process. We must return to the early days of quantum mechanics and the task Dirac set himself in 1927 to make it consistent with relativity theory, as it was not when first formulated. In one of the masterpieces of physics he showed that the restrictions imposed by relativity, as well as others which he regarded as essential, allowed only one formulation. Applied to the electron, its solutions described, without any extra assumptions, a number of properties which the electron had been found experimentally to possess, but which had not been explained hitherto. Among them was the magnetic moment whose magnitude is conventionally expressed by a number g; in Dirac’s theory g turns out to be exactly 2, and the spectroscopists at the time were satisfied that this agreed with their extremely precise measurements.

There was, however, an exceedingly curious consequence of the theory: that it permitted electrons to have negative as well as positive energy, something that is inconceivable in classical mechanics. At first Dirac and others reluctantly concluded that one must ignore the negative energy solutions as an unfortunate artefact of the theory, but it soon became clear that this would lead to wrong answers to other problems. Perhaps, said Dirac, all states of negative energy are already occupied by electrons, so that all space is filled with them to an infinite density: paradoxically, if they are uniformly distributed inside and outside matter, their presence will not be perceived. All we observe are the few extra which can find no space in negative energy states and therefore take positive energies: not being pervasive, they do not escape our notice. Here we have the first germ of a concept that has proliferated through fundamental physical theory: that it is possible to live with infinities provided consistent procedures are adopted for cancelling them out. One must stress the word ‘consistent’ for, as Heisenberg said: ‘Just because a quantity is infinite is no reason for ignoring it.’ Juggling with infinities was (and still is) regarded as dangerously close to malpractice by physicists not engaged in the game, and was felt by Dirac and most of his contemporaries to be an interim palliative until the true theory appeared. In this they followed the pattern of their forbears, including Einstein, whose views on the quantum mechanics of the Twenties were much the same. Doubtless future generations will react similarly to new concepts, and one hopes they will, for the impetuosity of speculative theorists needs a rein.

We have not, however, finished with Dirac, who observed that a hard enough kick would knock an electron out of a state of negative energy into one of positive energy. There would then be one extra electron, and the hole left behind would seem to be a particle of the opposite charge. Could it be a proton? That was the only positive particle known at the time, but no one was very happy with that solution. In 1932, however, Carl Anderson discovered that high-energy cosmic rays could indeed create a pair of oppositely charged particles: an electron and something new, a positron. Moreover, as one would expect if the positron was the manifestation of an empty space in the negative energy states, an electron dropping into that space would itself disappear and take the positron with it – electron-positron annihilation such as is nowadays a commonplace in high-energy physics. So Dirac seemed to be vindicated, even if the vacuum was no longer void but was jam-packed with unobservable particles.

It soon became clear, however, that they have a residual influence: so that, among other things, the factor g, that was originally found to be exactly 2, must be slightly modified, to imply small changes to the predicted wavelength of spectral lines that the spectroscopists had already begun to suspect were needed. In the immediate post-war period techniques acquired from radar produced immense improvements in measurement, while simultaneously the theorists were applying new notions about coping with infinities to compute the modified value of g, with enough promise of agreement to generate the exhilaration to push theory and experiment to the limit. At present the theorists think g ought to be 2.0023193049, while the experimenters opt for 2.0023193044. There can be few more convincing confirmations of theory, let alone of theoretical advances into territory so treacherous with conceptual minefields.

When one reads the bare recitation, in a textbook, of how the tricks are done, it is hard to visualise the personalities of the pioneers, and we must be grateful to Pais that he has filled in the story with sketches of many of the principal characters: Rutherford the grownup schoolboy; the taciturn Dirac; the ebullient Feynman; Heisenberg and Pauli ceaselessly exchanging views, the one optimistic and overflowing with ideas, the other sardonic and critical. These are not bloodless thinking machines but real men, hopeful and despondent by turns, with personal pleasures and problems – above all, passionately devoted to exploring perhaps the most exciting range of intellectual challenges ever to present itself.

The developments of those days left their mark on the whole of science, indeed extended far beyond, for the discovery of the nature of the atom, and of how to calculate its behaviour in detail, provided a basis for all solid-state physics and chemistry and for the modern technologies which depend on them – computers, television and all the other innovations which are generally accepted as life-enhancing. Through molecular biology the new ideas have penetrated almost every branch of biology, including genetics and evolutionary theory, which along with advances in astronomy play a significant role in moulding the world-view of scientist and non-scientist alike. Without question, one may attach the label ‘important’ to such work, in the special sense that it captures the interest of many who are not personally engaged in the detailed investigation. By contrast, most research that achieves something beyond mere systematic application of accepted procedures should be described as, at best, ‘fascinating’, in that it arouses enthusiasm only in the actual participants. This is in no sense derogation, for fascinating research, even without any immediate outside relevance, makes for a lively mind that inspires in others the ambition to enjoy more fully the use of their own powers. There may also be accidental bonuses in the form of useful spin-offs, though the hard-headed businessman is probably right in not regarding investment in unrestricted research as a gold-mine.

Thoughts such as this come to mind when one attempts to judge whether fundamental physics is still important in the way it was once, and if not, as I believe, whether it deserves the lavish support it still receives. The transition from ‘importance’ to ‘fascination’ since, say, 1950 is closely related to the great expense of the game as it now stands. Hardly any of the experiments that led to the first important advances cost any significant sum, and the theorists were mercifully spared the basilisk gaze of computers. But to create new heavy particles demands, through Einstein’s E=mc², the bombardment of matter with enormously energetic protons or electrons; and to investigate the details of extremely close encounters between particles demands, through Heisenberg’s uncertainty principle, that the required precision of position measurement be accompanied by a correspondingly imprecise knowledge of momentum, only attainable if the momentum (and hence the energy) of the particle concerned is very large. No one has devised a process that will allow particles to be accelerated to the required energy except in a large machine. The present leading accelerators are a few kilometres across, but the latest contender for US funds (nicknamed the Desertron) will be 100 kilometres or more, if it ever gets built. The very fact that the new particles to be created are highly unstable, and that the energy needed to make them is a thousand times greater than that possessed by any particle in an atomic nucleus, means that one cannot expect new results in this field to have any application to anyone else’s interests, except for a few astrophysicists or cosmologists interested in the early history of the Universe immediately after the Big Bang.

In the light of this, it is easy to present a case for cutting off support for high-energy physics on the grounds of unjustifiable expense. Yet the expense is not really enormous: £1 a head of population per annum more than covers it, and is much less than what is spent on education, defence or even chickenfeed. It is small enough to make one wonder whether the withdrawing from what has been accepted as a great enterprise, a triumph of enlightened civilisation, might produce a backlash in the form of hopes disappointed among the intelligent young, that would cost the country far more than the rather trivial monetary saving. The problem is one, not merely of the wise use of money, but of human potential. Granted there are still intellectual paladins who find a challenge here, as nowhere else, and whom we should applaud with pride as they go to meet it, nevertheless behind them are all too many who are gifted indeed, but not gifted enough to rise from the ranks, whether they be calculators or machine-builders or instrument designers. Most of these could give more to the community than they do, if only they could be inspired to give it. It is the not-quite brilliant, the first-class brain of limited imagination, that is potentially available to work profitably and find satisfaction in almost any task that calls for skill and is rewarded by praise and recognition. Such as these have a place in every branch of science and technology, and many turn eventually to a field quite other than that which first captivated them. Having made just such a switch myself as a student, I can recommend without cynicism that one should hesitate to douse a beacon which has shone for so long and so brightly, even though one hopes it will guide travellers to a different land from any they had heard tell of.

Fermi once likened the big accelerators to the Pyramids, but perhaps their function is more that of heroic statuary, to keep us in mind of a past virtue. The virtue is more than pious legend – stripped of legendary accretions, the heroes of Pais’s narrative are heroes still. We cannot expect to be given an opportunity as golden as theirs, but at least, as Dirac remarked in a loquacious moment, one must try.