Because the procedure had partly tamed the behaviour of the equations he was able to obtain a rough estimate for the predicted size of the Lamb shift. Uncertain about a factor of 2 he had introduced into the calculation, on getting to the General Electric research laboratories he made a quick visit to the library and reassured himself that he had got it right. He had obtained a prediction for the Lamb shift which was just four per cent larger than the experimental value that Lamb had reported at the Shelter Island conference.
He was definitely on to something.
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A definitive, relativistic QED that could be renormalized in this way took a little while longer to develop. Schwinger described a version in a marathon, five-hour session delivered at a subsequent conference which took place in March 1948 at the Pocono Manor Inn in the Pocono Mountains near Scranton, Pennsylvania. His mathematics was largely impenetrable. Only Fermi and Bethe, it seemed, followed his derivation through to the end.
Schwinger’s New York rival Feynman had in the meantime developed a vastly different, much more intuitive approach to describing and keeping track of the perturbation corrections in QED. Neither understood the other’s approach, but when they compared notes at the end of Schwinger’s session, they found that their results were identical. ‘So I knew that I wasn’t crazy,’ Feynman said.2
This seemed to be the end of the matter, but Oppenheimer learned of yet another successful approach to QED from a letter he received from Japanese physicist Sin-Itiro Tomonaga shortly after returning from the Pocono conference. Tomonaga had used methods similar to Schwinger but his mathematics appeared a lot more straightforward. The situation was rather confusing. These very different approaches to relativistic QED all produced similar answers, but nobody quite understood why.
The challenge was taken up by a young English physicist called Freeman Dyson. On 2 September 1948, he boarded a bus from Berkeley, near San Francisco in California, bound for the East coast. ‘On the third day of the journey a remarkable thing happened,’ he wrote to his parents a few weeks later. ‘Going into a sort of semi-stupor as one does after 48 hours of bus-riding. I began to think very hard about physics, and particularly about the rival radiation theories of Schwinger and Feynman. Gradually my thoughts grew more coherent, and before I knew where I was, I had solved the problem that had been in the back of my mind all this year, which was to prove the equivalence of the two theories.’3
The result was a fully relativistic theory of QED that predicts the results of experiments to astonishing levels of accuracy and precision. The g-factor for the electron is predicted by QED to have the value 2.00231930476. The comparable experimental value is 2.00231930482.* ‘To give you a feeling for the accuracy of these numbers,’ Feynman later wrote, ‘it comes out something like this: If you were to measure the distance from Los Angeles and New York to this accuracy, it would be exact to the thickness of a human hair.’4
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The success of QED established some important precedents. It now seemed that the correct way to describe a fundamental particle and its interactions was in terms of a quantum field theory in which the force involved is carried by field particles. Like Maxwell’s theory of electromagnetism, QED is a U(1) gauge theory in which the local U(1) phase symmetry of the electron wavefunction is connected with the conservation of electric charge.
Attention now turned to a quantum field theory of the strong force between protons and neutrons inside the nucleus. But here was another puzzle. The connection between the conservation of electric charge and electromagnetism – classical or quantum – was intuitively obvious. If a quantum field theory of the strong force was to be discovered, it was first necessary to figure out what, precisely, was conserved in strong-force interactions and what continuous symmetry transformation this related to.
Chinese physicist Chen Ning Yang believed that the quantity conserved in nuclear interactions involving the strong force was isospin.
Yang was born in 1922, in Hefei, the capital city of Anhui province in eastern China. He studied in Kunming, at the National Southwestern Associated University which was formed from Tsinghua, Peking, and Nankai Universities following the invasion of Japanese forces in 1937. He graduated in 1942 and was awarded a master’s degree two years later. Armed with a scholarship known as a Boxer Indemnity,* in 1946 he headed for the University of Chicago.
In Chicago he studied nuclear physics under the supervision of Edward Teller. Inspired by reading the autobiography of the American inventor and politician Benjamin Franklin, he adopted a middle name ‘Franklin’ or just ‘Frank’. He obtained his doctorate in 1948 and worked for a further year as an assistant to Fermi. In 1949 he moved to the Institute for Advanced Study in Princeton.
It was in Princeton that he began to think about ways in which he could apply Noether’s theorem in search of a quantum field theory of the strong force.
The concept of isospin, or isotopic spin, grew out of the simple fact that the masses of the proton and neutron are very similar.* When the neutron was discovered in 1932, it was natural to assume that this was a composite particle consisting of a proton and an electron. It was well known that beta-radioactive decay involves the ejection of a high-speed electron directly from the nucleus, turning a neutron into a proton in the process. This seemed to imply that in beta-radioactivity, one of the composite neutrons was somehow shedding its ‘stuck-on’ electron.
Shortly after the discovery of the neutron, Heisenberg used the neutron-as-proton-plus-electron idea to develop an early theory of proton–neutron interactions in the nucleus. This was a model that was closely based on theories of chemical bonding.
Heisenberg hypothesized that the proton and neutron bind together in the nucleus by exchanging an electron between them, the proton turning into a neutron and the neutron turning into a proton in the process. The interaction between two neutrons would involve the exchange of two electrons, one in each ‘direction’.
This exchange suggests that in the nucleus, protons and neutrons tend to lose their identity, constantly flitting from one form to another. It suited Heisenberg’s purpose to imagine that the proton and neutron are simply different states of the same particle, distinguished by the different properties of these states. The different states possess different electrical charges, of course, one positive and one neutral. But to make his theory work he also needed to introduce a further property analogous to electron spin.
He therefore introduced the idea of isospin, not to be confused with electron spin, in which the proton is (arbitrarily) assigned a spin-up orientation and the neutron a spin-down orientation. These are orientations in an ‘isospin-space’ which has just two dimensions, up and down. Converting a neutron into a proton is then equivalent to ‘rotating’ the spin of the neutron in isospin space, from spin-down to spin-up.
This all sounds very mysterious, but in many ways isospin is like electrical charge. Our easy familiarity with electricity shouldn’t blind us to the fact that this, too, is a property which takes up ‘values’ (rather than ‘orientations’) in an abstract ‘charge-space’ with two dimensions – positive and negative.
Even as a simple analogy, Heisenberg’s theory was already a stretch. The strengths of chemical bonds formed by exchanging electrons are much weaker than the strength of the force binding protons and neutrons together inside the nucleus. But Heisenberg was able to use the theory to apply non-relativistic quantum mechanics to the nucleus itself. In a series of papers published in 1932 he accounted for many observations in nuclear physics, such as the relative stabilities of isotopes.
The weaknesses of the theory were exposed in experiments performed just a few years later. Because protons do not possess a ‘stuck-on’ electron, Heisenberg’s electron-exchange model did not allow for any kind of interaction between protons. In contrast, experiments showed that the strength of the interaction between protons is comparable to that between protons and neutrons.
Despite the shor
tcomings of the theory, Heisenberg’s electron-exchange model held at least a grain of truth. The exchange of electrons was abandoned, but the concept of isospin was retained. As far as the strong force is concerned, the proton and the neutron are essentially two states of the same particle, like the two spin orientations of the electron. The only difference between them is their isospin.
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The individual isospins of protons and neutrons can be added up to produce a total isospin, a concept first introduced by physicist Eugene Wigner in 1937. The literature on nuclear reactions seemed to support the idea that total isospin is conserved, just as electric charge is conserved in physical and chemical changes. Yang now identified isospin as a local gauge symmetry, like the phase symmetry of the electron wavefunction in QED, and began the search for a quantum field theory that would preserve it.
He quickly got bogged down, but became obsessed with the problem. ‘Occasionally an obsession does finally turn out to be something good,’ he later observed.5
In the summer of 1953 he took a short leave of absence from the Institute for Advanced Study and made a visit to Brookhaven National Laboratory on Long Island, New York. He found himself sharing an office with a young American physicist called Robert Mills.
Mills became absorbed by Yang’s obsession and together they worked on a quantum field theory of the strong nuclear force. ‘There was no other, more immediate motivation,’ Mills explained some years later. ‘He and I just asked ourselves “Here is something that occurs once. Why not again?”’6
In QED, changes in the phase of the electron wavefunction in space and time are compensated by corresponding changes in the electromagnetic field. The field ‘pushes back’, such that the phase symmetry is preserved. But a new quantum field theory of the strong force had to account for the fact that there are now two particles involved. If isospin symmetry is to be conserved, this means that the strong force sees no difference between the proton and the neutron. Changing the isospin symmetry by, for example, ‘rotating’ a neutron into a proton, therefore demands a field which ‘pushes back’ and so restores the symmetry. Yang and Mills therefore introduced a new field, which they called the ‘B’ field, designed to do just this.
The simple symmetry group U(1) is insufficient for this kind of complexity, and Yang and Mills reached for the symmetry group SU(2), the special unitary group of transformations of two complex variables. A larger symmetry group is needed simply because there are now two objects that can transform into each other.
The theory also needed three new field particles, responsible for carrying the strong force between the protons and neutrons inside the nucleus, analogues of the photon in QED. Two of the three field particles were required to carry electric charge, accounting for the change in charge resulting from proton–neutron and neutron–proton interactions. Yang and Mills referred to these particles as B+ and B−. The third particle was neutral, like the photon, and was meant to account for proton–proton and neutron–neutron interactions in which there is no change in charge. This was referred to as B0. It was found that these field particles interact not only with protons and neutrons, but also with each other.
By the end of the summer they had worked out a solution. But this was a solution with a whole new set of problems.
For one thing, the renormalization methods that had been used so successfully in QED could not be applied to the field theory that Yang and Mills had devised. Worse still, the zeroth-order term in the perturbation expansion indicated that the field particles should be massless, just like the photon. But this was self-contradictory. Heisenberg and Japanese physicist Hideki Yukawa had suggested in 1935 that the field particles of short-range forces like the strong force should be ‘heavy’, i.e. they should be large, massive particles. Massless field particles for the strong force made no sense whatsoever.
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Yang returned to Princeton. On 23 February 1954 he presented a seminar on the work he had done with Mills. Oppenheimer was in the audience, as was Pauli, who had moved to Princeton University in 1940.
It turned out that Pauli had earlier explored some of the same logic and had arrived at the same puzzling conclusions concerning the masses of the field particles. He had consequently abandoned the approach. As Yang drew his equations on the blackboard, Pauli piped up:
‘What is the mass of this B field?’ he asked, anticipating the answer.
‘I don’t know,’ Yang replied, somewhat feebly.
‘What is the mass of this B field?’ Pauli demanded.
‘We have investigated that question,’ Yang replied. ‘It is a very complex question, and we cannot answer it now.’
‘That is not a sufficient excuse,’ Pauli grumbled.7
Yang, taken aback, sat down to general embarrassment. ‘I think we should let Frank proceed,’ Oppenheimer suggested. Yang resumed his lecture. Pauli asked no more questions, but he was irked. The following day he left a note for Yang saying: ‘I regret that you made it almost impossible for me to talk with you after the seminar.’8
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It was a problem that simply would not go away. Without mass, the field particles of the Yang–Mills field theory did not fit with physical expectations. If they were massless, as the theory predicted, then they should be as ubiquitous as photons, yet no such particles had ever been observed. The accepted methods of renormalization wouldn’t work.
And yet, it was still a nice theory.
‘The idea was beautiful and should be published,’ Yang wrote, ‘But what is the mass of the gauge particle? We did not have firm conclusions, only frustrating experiences to show that [this] case is much more involved than electromagnetism. We tended to believe, on physical grounds, that the charged gauge particles cannot be massless.’9
Yang and Mills published a paper describing their results in October 1954. In it they wrote: ‘We next come to the question of the mass of the [B] quantum, to which we do not have a satisfactory answer.’10
They made no further progress, and turned their attentions elsewhere.
3
People Will Be Very Stupid About It
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In which Murray Gell-Mann discovers strangeness and the ‘Eightfold Way’, Sheldon Glashow applies Yang–Mills field theory to the weak nuclear force, and people are very stupid about it
Yang and Mills had tried to apply quantum field theory to the problem of strong-force interactions in the hope of repeating the success of QED. But they found that the theory could not be renormalized and yielded massless particles that should have been massive. Obviously, this could not be the solution to the strong force.
But what of the weak nuclear force?
The weak force was something of a mystery. Italian physicist Enrico Fermi had been obliged to invoke a new type of nuclear force in a detailed theory of beta-radioactivity in the early 1930s. He described this theory to his colleagues during a group skiing holiday in the Italian Alps at Christmas 1933. His colleague Emilio Segrè subsequently described the experience: ‘…we were all sitting on one bed in a hotelroom, and I could hardly keep still in that position, bruised as I was after several falls on icy snow. Fermi was fully aware of the importance of his accomplishment and said that he thought he would be remembered for this paper, his best so far.’1
Fermi drew parallels between the weak force and electromagnetism. From the resulting electromagnetic-like theory, he was able to deduce the range of energies (and hence speeds) of the emitted beta-electrons. His predictions were shown to be correct in experiments performed at Columbia University by Chinese–American physicist Chien-Shiung Wu in 1949. With some small adjustments, Fermi’s theory remains valid to this day.
Fermi deduced that the strength of the interactions between the particles involved in beta-radioactivity is some ten billion times weaker than electromagnetic interactions between charged particles. This is weak indeed, but the force has some profound consequences. Because of the weak force
, neutrons are inherently unstable. A neutron moving in free space will survive intact for an average of just 18 minutes. This is unusual behaviour for a particle that is meant to be fundamental or elementary.*
Of course, it was a bit much to have to invoke a novel force of nature just to explain a single type of interaction. But as experimentalists began to sift through the ‘zoo’ of new particles that were now being revealed in the debris of high-energy collisions, evidence began to emerge for other kinds of particles that were susceptible to the weak force.
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In the 1930s, if you wanted to study high-energy particle collisions then you needed to climb a mountain. Cosmic rays – streams of high-energy particles from outer space – wash constantly over the upper atmosphere. Some very highly energetic particles that constitute these rays can penetrate to lower levels of the atmosphere, levels that can be reached from the tops of mountains, where their collisions can be studied. Such studies rely on chance detection of the particles and, because of their randomness, no two events ever have quite the same conditions.
American physicist Carl Anderson had discovered Dirac’s positron in 1932. Four years later, he and fellow American Seth Neddermeyer loaded their particle detection apparatus onto a flat-bed truck and drove to the top of Pike’s Peak (now Pikes Peak) in the Rocky Mountains, about ten miles west of Colorado Springs.* In the tracks left by penetrating cosmic rays, the physicists identified another new particle. This particle behaved just like an electron, but was found to be deflected by a magnetic field to a much lesser extent.
Higgs:The invention and discovery of the 'God Particle' Page 5
