Born in 1885 in Elmshorn, a small town near Hamburg, Weyl had secured his doctorate under Hilbert’s supervision at Göttingen in 1908. He had then taken up a professorship at the Eidgenossische Technische Hochschule (ETH) in Zurich, where he met Albert Einstein and became fascinated by problems in mathematical physics.
In developing his general theory of relativity in 1915, Einstein had eliminated any sense of absolute space and time. Instead, he argued, physics should depend only on the distances between points and the curvature of space-time at each point. This is Einstein’s principle of general covariance, and the theory of gravitation that results is invariant to arbitrary changes of coordinate system. In other words, although there are natural physical laws, there is no ‘natural’ coordinate system of the universe. We invent coordinate systems to help describe the physics but the laws themselves should not (and do not) depend on these arbitrary choices.
There are two ways we can change the coordinate system. We can make a global change, applied uniformly at all points in space and time. An example of such a global symmetry transformation is a uniform shift in the lines of latitude and longitude used by cartographers to map the surface of the earth. So long as the change is uniform and applied consistently across the globe, this makes no difference to our ability to navigate from one place to another.
But changes can also be made locally, with different changes to the coordinates at different points in space-time. For example, in one particular part of space we could choose to rotate the axes of our coordinate system through a small angle, at the same time changing the scale. Provided this change is translated through to the measures of differences in position and differences in time, this makes no difference to the predictions of general relativity. General covariance is therefore an example of invariance to local symmetry transformations.
Weyl thought long and hard about Noether’s theorem and worked on the theory of groups of continuous symmetry transformations called Lie groups, named for the nineteenth century Norwegian mathematician Sophus Lie. In 1918 he concluded that the conservation laws are related to local symmetry transformations to which he gave the generic name gauge symmetry, an unfortunately rather obscure term. Guided by Einstein’s work, he was thinking of symmetry in relation to distances between points in space-time, as in the example of a train running on tracks with a fixed gauge.
He found that by generalizing the principle of general covariance to one of gauge invariance, he could use Einstein’s theory as a basis for the derivation of Maxwell’s equations for electromagnetism. What he had discovered appeared to be a theory that could unify the two forces then known to science – electromagnetism and gravity. The invariance identified with the conservation laws would then be related to arbitrary changes in the ‘gauge’ of the fields involved. In this way, Weyl hoped to demonstrate the conservation of energy, linear and angular momentum, and electric charge.
Weyl initially ascribed his gauge invariance to space itself. But, as Einstein quickly pointed out, this meant that the measured lengths of rods and the readings of clocks would come to depend on their recent history. A clock moved around a room would no longer keep time correctly. Einstein wrote to Weyl, complaining: ‘Apart from the agreement with reality, [your theory] is at any rate a grandiose achievement of the mind.’2
Weyl was disturbed by this criticism, but accepted that Einstein’s intuition in these matters was normally reliable. Weyl abandoned his theory.
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Austrian physicist Erwin Schrödinger joined the faculty at the University of Zurich three years later, in 1921. He was diagnosed with suspected pulmonary tuberculosis just a few months later. He was ordered to take a complete rest cure. He and his wife Anny retreated to a villa in the Alpine resort of Arosa, near the fashionable ski resort of Davos, where they stayed for nine months.
As Anny nursed him back to health, he pondered on the significance of Weyl’s gauge symmetry and, specifically, a periodic ‘gauge factor’ which appeared in Weyl’s theory. In 1913, Danish physicist Niels Bohr had published details of a theory of atomic structure in which electrons are required to orbit the nucleus at fixed energies characterized by their ‘quantum numbers’. These integral numbers govern the energies of the orbits, increasing in linear sequence (1, 2, 3,…) from the innermost to the outermost orbit. At the time, their origin was a complete mystery.
Schrödinger was struck by the possibility that there might be a connection between the periodicity implied by Weyl’s gauge factor and the periodicity implied by Bohr’s quantized atomic orbits. He examined a couple of possible forms for the gauge factor, including one containing a complex number, formed by multiplying a real number by the ‘imaginary’ number i, the square root of –1.* In a paper published in 1922 he suggested that this connection had a deep physical significance. But this was a vague intuition. The real significance of the connection would elude him until he studied the 1924 doctoral thesis of French physicist Louis de Broglie.
De Broglie had suggested that, just as electromagnetic waves can appear to behave like particles,† so perhaps particles like electrons could sometimes behave like waves. Whatever they were, these ‘matter waves’ could not be considered to be in any way like more familiar wave phenomena, such as sound waves or water waves. De Broglie concluded that the matter wave: ‘represents a spatial distribution of phase, that is to say, it is a “phase wave”.’3‡
Schrödinger was set to thinking: what would the electron look like if it was described mathematically as a wave? At Christmas 1925 he retreated once more to Arosa. His relationship with his wife was at an all-time low, so he chose to invite an old girlfriend from Vienna to join him. He also took with him his notes on de Broglie’s thesis. When he returned on 8 January 1926, he had discovered wave mechanics, a theory which describes the electron as a wave and the orbits of Bohr’s atomic theory in terms of electron ‘wavefunctions’.
It was now possible to make the connection. One example of a Lie group is the symmetry group U(1), referred to as the unitary group of transformations of one complex variable. This involves symmetry transformations that are, in many ways, entirely analogous to those involving continuous rotation in a circle. But whereas a circle is drawn in a two-dimensional plane formed from real dimensions, the transformations of the symmetry group U(1) involve rotations in a two-dimensional complex plane. This is formed from two ‘real’ dimensions, with one of them multiplied by i.
Another way of representing this symmetry group is in terms of continuous transformations of the phase angle of a sinusoidal wave (see Figure 7). Different phase angles correspond to different amplitudes of the wave in its peak–trough cycle. Weyl’s gauge symmetry is preserved if changes in the phase of the electron wavefunction are matched by changes in its accompanying electromagnetic field. The conservation of electric charge can be traced to the local phase symmetry of the electron wavefunction.
The connection between wave mechanics and Weyl’s gauge theory was made explicit in 1927 by young German theorist Fritz London and Soviet physicist Vladimir Fock. Weyl recast and extended his theory in the context of quantum mechanics in 1929.
FIGURE 7 The symmetry group U(1) is the unitary group of transformations of one complex variable. In a complex plane formed by one real axis and one imaginary axis, we can pinpoint any complex number on the circumference of the circle formed by rotating the line drawn from the origin to the point through the continuous angle, θ, that this line makes with the real axis. There is a deep connection between this continuous symmetry and simple wave motion, in which the angle θ is a phase angle.
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De Broglie’s wave–particle ‘duality’ implied that the electron was to be regarded as both wave and particle. But how could this be? Particles are localized bits of stuff, waves are delocalized disturbances in a medium (think of the ripples in a pond caused by the throw of a stone). Particles are ‘here’, waves are ‘there and everywhere’.
One of the ph
ysical consequences of wave–particle duality is that we cannot measure the simultaneous position and momentum (specifically the speed and direction) of a quantum particle precisely. Think about it. If we can measure the precise position of a wave-particle this must mean that it is localized in space and time. It is ‘here’. For a wave this is only possible if it is formed by combining a large number of wave forms of different frequencies, such that they add up to produce a wave which is large in one location in space and small everywhere else. This gives us the position, but at the cost of complete uncertainty in the wave frequency, because the wave must be composed of many waves with lots of different frequencies.
But in de Broglie’s hypothesis, the inverse frequency of the wave is directly related to the particle momentum.* Uncertainty in frequency therefore means uncertainty in momentum.
The converse is also true. If we want to be precise about the frequency of the wave, and hence the momentum of the particle, then we have to stick with a single wave with a single frequency. But then we can’t localize it. The wave-particle remains spread out in space and we can no longer measure a precise position.
This uncertainty in position and momentum is the basis for German physicist Werner Heisenberg’s famous uncertainty principle, discovered in 1927. It is a direct consequence of the duality of wave and particle behaviour in elementary quantum objects.
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Weyl returned to Göttingen in 1930, taking the professorship vacated by the retiring Hilbert. He joined Noether, who had remained in Göttingen but for a short period of study leave at Moscow State University during the winter of 1928–29.
In January 1933 Adolf Hitler became Chancellor of Germany. A few months later Hitler’s National Socialist government introduced the Law for the Reestablishment of the Career Civil Service, the first of four hundred such decrees. It provided a legal basis on which the Nazis could forbid Jews from holding positions in the Civil Service, including academic positions in German universities.
Weyl’s wife was Jewish, and he left Germany to join Einstein at the Institute for Advanced Study in Princeton, New Jersey. Noether was Jewish, and she lost her position at Göttingen. She had never been promoted to the status of full professor. She left for Bryn Mawr College, a liberal arts college in Pennsylvania. She died two years later, aged 53.
In an obituary that appeared in the New York Times shortly after her death, Einstein wrote:4
In the judgment of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began. In the realm of algebra, in which the most gifted mathematicians have been busy for centuries, she discovered methods which have proved of enormous importance in the development of the present-day younger generation of mathematicians. Pure mathematics is, in its way, the poetry of logical ideas. One seeks the most general ideas of operation which will bring together in a simple, logical, and unified form the largest possible circle of formal relationships. In this effort toward logical beauty spiritual formulas are discovered necessary for the deeper penetration into the laws of nature.
2
Not a Sufficient Excuse
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In which Chen Ning Yang and Robert Mills try to develop a quantum field theory of the strong nuclear force and annoy Wolfgang Pauli
When Dirac successfully combined quantum theory and Einstein’s special theory of relativity in 1927, the result was electron spin and anti-matter. Dirac’s equation was rightly regarded as an absolute wonder, but it was also quickly realized that this could not be the end of the story.
Physicists began to acknowledge that they needed a fully fledged relativistic theory of quantum electrodynamics, or QED. This would, in essence, be a quantum version of Maxwell’s equations that conformed to Einstein’s special theory of relativity. Such a theory would necessarily incorporate a quantum version of the electromagnetic field.
Some physicists believed that fields were more fundamental than particles. It was thought that a proper quantum field description should yield particles as the ‘quanta’ of the fields themselves, carrying the force from one interacting particle to another. It seemed clear that the photon was the field particle of the quantum electromagnetic field, created and destroyed when charged particles interact.
German physicist Werner Heisenberg and Austrian Wolfgang Pauli developed a version of just such a quantum field theory in 1929. But there was a big problem. The physicists found that they could not solve the field equations exactly. In other words, it was not possible to write down a solution to the field equations that took the form of a single, self-contained mathematical expression, applicable in all circumstances.
Heisenberg and Pauli had to resort to an alternative approach to solving the field equations based on a so-called perturbation expansion. In this approach, the equation is recast as the sum of a potentially infinite series of terms – x0 + x1 + x2 + x3 +…The series starts with a ‘zeroth-order’ (or zero-interaction) expression which can be solved exactly. To this is added additional (or perturbation) terms representing corrections to first-order (x1), second-order (x2), third-order (x3), etc. In principle, each term in the expansion provides a smaller and smaller correction to the zeroth-order result, gradually bringing the calculation closer and closer to the actual result. The accuracy of the final result then depends simply on the number of perturbation terms included in the calculation.
But instead of finding smaller and smaller corrections, they found that some terms in the perturbation expansion mushroomed to infinity. When applied to the quantum field theory of the electron, these terms were identified to result from the electron’s ‘self-energy’, a consequence of the electron interacting with its own electromagnetic field.
There was no obvious solution.
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There the matter rested. James Chadwick discovered the neutron in 1932. In the years following this discovery Italian physicist Enrico Fermi used high-energy neutrons to bombard atoms of different chemical elements in search of interesting new physics. Puzzled by some of Fermi’s results, German chemists Otto Hahn and Fritz Strassman studied the products from the neutron bombardment of uranium atoms. On Christmas Eve 1938, their even more puzzling results were discussed by Hahn’s long-time collaborator Lise Meitner and her physicist nephew Otto Frisch, by now both exiled from Nazi Germany. Their animated discussion led to the discovery of nuclear fission.
It was a portentous discovery, reported in January 1939, just nine months before the beginning of the Second World War. Transformed from ‘other-worldly eggheads’ into the most important military resources of nation-states, the physicists now worked to turn the discovery of nuclear fission into the world’s most dreadful weapon of war.
When the time finally came in 1947 to turn their attentions back to the problems that beset quantum electrodynamics, it was declared that theoretical physics had been in the doldrums for nearly two decades.
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But there quickly followed another great burst of creativity. In June 1947 a group of leading American physicists gathered for a small, invitation-only conference at the Ram’s Head Inn, a small clapboard hotel and inn on Shelter Island, at the eastern end of New York’s Long Island.
It was an illustrious group. Among them were J. Robert Oppenheimer, the ‘father’ of the atom bomb, Hans Bethe, who had led the Theoretical Division at Los Alamos, Victor Weisskopf, Isidor Rabi, Edward Teller, John Van Vleck, John von Neumann, Willis Lamb, and Hendrik Kramers. A new generation of physicists was represented by John Wheeler, Abraham Pais, Richard Feynman, Julian Schwinger, and former Oppenheimer students Robert Serber and David Bohm. Einstein had been invited to attend but declined for reasons of ill-health.
The physicists heard of some disturbing new experimental results. One of the quantum states of the hydrogen atom was found to be shifted in energy slightly in relation to another, a phenomenon that came to be called t
he Lamb shift, after its discoverer, Willis Lamb. Dirac’s theory predicted that both states should have precisely the same energy.
There was more. Rabi announced that a new measurement of the g-factor of the electron – a physical constant which reflects the strength of the interaction of an electron with a magnetic field – has a value of the order of 2.00244. Dirac’s theory predicted a g-factor of exactly 2.
These were results that simply could not be predicted without a fully fledged QED. It seemed that although the theory was beset with problems inherent in its mathematical structure, nature itself had no problems with infinities. The physicists had to find a way around this somehow.
The discussion continued long into the night. The physicists split into groups of two and three, the corridors echoing their arguments, as they regained their passion for physics. Schwinger later remarked: ‘It was the first time people who had all this physics pent up in them for five years could talk to each other without somebody peering over their shoulders and saying “Is this cleared?”’1
Then there came a glimmer of hope. Dutch physicist Kramers outlined a new approach to thinking about the mass of an electron in an electromagnetic field. He proposed to treat the self-energy of the electron as an additional contribution to its mass.
After the conference, Bethe returned to New York and took a train to Schenectady, where he was working as a part-time consultant to General Electric. As he sat on the train he played around with the equations of QED. The existing theories of QED predicted an infinite Lamb shift, a consequence of the electron’s self-interaction. Bethe now followed Kramers’ suggestion and identified the infinite term in the perturbation expansion as an electromagnetic mass effect. How could he now get rid of this?
He reasoned that he could just subtract it out. The perturbation expansion for an electron bound in a hydrogen atom includes an infinite mass term. But the expansion for a free electron also includes the same infinite mass term. Why not just subtract one perturbation series from the other, thereby eliminating the infinite terms? It sounds as though subtracting infinity from infinity should yield a nonsensical answer,* but Bethe now found that in a simple, non-relativistic version of QED this subtraction produced a result that, though it still had problems, behaved in a much more orderly manner. He figured that in a QED that fully complied with Einstein’s special theory of relativity, this ‘renormalization’ procedure would eliminate the problem completely and give a physically realistic answer.
Higgs:The invention and discovery of the 'God Particle' Page 4