by Colin Pask
11.1.1 Enter Paul Dirac
Graham Farmelo's recent biography of Paul Dirac (1902–1984) is called The Strangest Man, and perhaps it took such a man to create theories that lead to some of the strangest of all results in physics. Dirac began as an electrical engineer, but lacking a suitable job, he moved to Cambridge University to begin a career in physics. He was legendary for being concise and careful with words. If during question time at a lecture, someone said that they did not understand how the lecturer got to, say, equation six, expecting some words of clarification, they would be met with silence from Dirac who, when pushed by the chairman, would say something like, “oh, I thought that was just a statement of ignorance, not a question.”
It is no surprise then that Dirac's 1930 book The Principles of Quantum Mechanics is a model of precise writing and logical development of ideas. Recall that Einstein showed how to recast classical mechanics so that Newton's approach was made to accommodate his ideas in the special theory of relativity. Dirac did a similar thing for quantum mechanics, and the Schrödinger equation is replaced by the Dirac equation, which incorporates both quantum ideas and special relativity. Dirac's work on the basic principles of quantum mechanics is one of the pinnacles of theoretical physics. He was awarded the Nobel Prize in 1933 (shared with Erwin Schrödinger), and his acceptance speech (see bibliography) is a fine example of his economical writing style and mastery of physical concepts.
I will introduce three topics emanating from Dirac's ideas and his equation, leading finally to a truly amazing calculation (see section 11.1.4 below). First, the need to meet the requirements of relativity theory compelled Dirac to produce his quantum equation in which Schrödinger's wave function ψ (see section 10.3.2) is replaced by a mathematical object having four components. Clearly, more than the quantum state described by Schrödinger's ψ is involved, and the task was to interpret what the extra components could mean. It turns out that Dirac's equation naturally introduces the intrinsic “spin” of a particle. Already in atomic theory it was found that the individual particles must carry a unit of angular momentum. And since it is natural to think of a particle as some sort of spinning object, the idea of a particle spin came into quantum physics. Dirac showed that it was intrinsic to the full theory and not some added-on artifact.
Secondly, there were extra solutions in Dirac's equation that had strange properties and seemed to imply particles in negative energy states. Again, some interpretation is needed. It turns out that Dirac had found that a particle also has an “antiparticle” with the same mass as the original particle, but with some properties reversed. Thus there is a positively charged antiparticle, known as the positron, as well as the negatively charged electron. The electron and positron have the same mass and spin but opposite charges. The positron was first observed in cosmic rays by Carl D. Anderson in 1933. Dirac had increased the number of elementary particles, since, for example, there would now be a negatively charged antiproton.
Antiparticles are not directly part of our normal everyday matter, but they can be observed in cosmic rays and created in the accelerators used by particle physicists. There is one startling fact that will be important later: a particle and its antiparticle may come together to annihilate one another and change into gamma rays (photons); equally, under the right conditions, radiation may create a particle-antiparticle pair, and virtual pairs may be created and destroyed in the vacuum. Now perhaps you can see just what a strange world Dirac created! The process in which a positron and an electron convert into a pair of gamma rays is the basis for positron emission tomography, which was first explored in the 1950s. Today, it is a technique used widely in medical diagnosis and physiological research. I expect most readers will have seen those amazing maps of brain activity.
Thirdly, Dirac showed that his equation gave the natural way to describe how charged particles interact with radiation and how electrons and photons are coupled together. There were great technical difficulties with this theory (Victor Weisskopf gives a good history), but eventually it turned out to be the most accurate and all-encompassing theory in physics. In QED: The Strange Theory of Light and Matter, Richard Feynman gives a masterful account of the quantum electrodynamic theory and how it is used. In his introduction, Feynman writes that “the theory describes all the phenomena of the physical world except the gravitational effect…and radioactive phenomena, which involve nuclei shifting in their energy levels”1 (something I turn to later in this chapter). This is quite some claim, and certainly a pioneer like Dirac ought to be forgiven for his strangeness.
11.1.2 The Electron Magnetic Moment
A loop of wire carrying a current generates a magnetic field, and, in some way, the electron's spin property also produces a magnetic effect. Dirac found that the Hamiltonian, the theoretical construct describing how electrons and radiation interact, contains a term involving the electron spin and the magnetic field. He naturally came to the terms that had been introduced in a more ad hoc way by physicists trying to interpret experiments involving electrons in magnetic fields. The electron has a magnetic moment, and it is the product of that with the magnetic field that Dirac found in his new theory. I will write
electron magnetic moment = D × a constant.
Dirac's equation led naturally to the constant (in terms of the electron's charge and mass). The number D (the Dirac number, as Feynman calls it) is equal to 1 in Dirac's theory.
Developing the description of electron and radiation interactions forced physicists to introduce new elements into the theory and to create what Feynman refers to in the subtitle of his book as “the strange theory of light and matter.” In this theory, the number D is no longer 1, and the attempts to compare experimental measurements and calculations of D led to one of the most remarkably accurate, but surely unexpected, results in the whole of science. I give a few technical details in the next section and then present the results in section 11.1.4.
11.1.3 Picturing a Calculation
Among those who built on Dirac's ideas was Richard Feynman (1918–1988), who invented a new way to communicate what was involved in physical processes and how to keep track of calculations. We now know this as the Feynman-diagrams approach. In a space-time diagram, straight lines represent particles, and wavy lines are photons. Thus, when electrons scatter off one another, they do so by exchanging virtual photons as shown in figure 11.1 (a). The diagram is physically suggestive, and it is shorthand for a mathematical calculation that must be carried out to evaluate just what happens in the scattering process. There are rules for translating diagrams into detailed calculations that Feynman gave in his paper “Space-Time Approach to Quantum Electrodynamics.” Feynman, along with Julian Schwinger and Sin-itro Tomonaga, was awarded the 1965 Nobel Prize in physics for his work in quantum electrodynamics.
The interaction of charged particles using Coulomb forces, or by considering the electric fields they produce and how they move in an electric field, has been replaced by the quantum picture and the exchange of virtual photons—virtual because energy is not used to create real photons leaving the interaction region; virtual photons pop into existence and then pop out again.
But now we come to the fascinating (and maybe bewildering) part where we recognize that all sorts of other quantum processes may be part of the electron-electron interaction. There can be more than one virtual photon as in figure 11.1 (b). More disconcerting (for many) is the fact that virtual electron-positron pairs may pop in and out of existence as in figure 11.1 (c). A whole chain of increasingly complex processes occurs in the interaction region, and summing up all the contributions to the scattering process is needed to give the total description of what happens when two electrons scatter each other. Fortunately, there is a decrease in the importance of higher terms, and not everything must be calculated to give accurate scattering results.
The interaction of an electron with a magnetic field is also represented in terms of a photon as shown in figure 11.1 (d). Evaluating the implied cal
culation using Feynman's rules gives Dirac's result (so D = 1). But now we must remember that other quantum processes may occur, and some possibilities are shown in figures 11.1 (e)–(g). (Feynman's QED book, chapter 3, is a good place to start reading about this, and he shows lots of examples of possible processes.) These diagrams say that a series of calculations must be done which will give
In equation (11.1), α is the fine structure constant, which is very close to 1/137. The terms in equation (11.1) become increasingly complex (the Cn requires a more involved calculation) as they correspond to more complex Feynman diagrams, but they become of less importance because the αn multipliers rapidly decrease.
Figure 11.1. Feynman diagrams for particles (straight lines) and photons (wavy lines) in the space-time picture. Diagrams (a)–(c) show electron-electron scattering with a virtual electron-positron pair occurring in (c). Diagrams (d)–(g) indicate processes that must be considered when calculating the magnetic moment of the electron. Figure created by Annabelle Boag.
11.1.4 Results for the Electron Magnetic Moment
The calculation of D has been carried out with increasing accuracy over the years, especially as computer mathematical manipulations became available, and the latest (according to Gabrielse, see bibliography) involved the evaluation of almost 14,000 integrals. There has been a corresponding increase in the accuracy of the experimental results, and, according to Gabrielse's 2013 report, the comparison gives
D (calculated) = 1.001 159 652 181 78(77),
D (measured) = 1.001 159 652 180 73(28).
The figures in brackets give the estimated uncertainty in the final given figures.
The stunning agreement between theory and experiment leads me to take calculation 41, the electron magnetic moment as a worthy member of my list of important calculations. To emphasize the accuracy of calculation 41, Richard Feynman remarks that “if you were to measure the distance from Los Angeles to New York to this accuracy, it would be exact to the thickness of a human hair.”2 It is almost unbelievable that all those strange processes involving photons and virtual pairs of antiparticles can lead Feynman to write in his QED introduction:
The theory of quantum electrodynamics has now lasted for more than fifty years [Feynman is writing in 1985 but there has been no change], and has been tested more and more accurately over a wider and wider range of conditions. At the present time I can proudly say that there is no significant difference between experiment and theory!3
Yes, almost unbelievable, but true, and other cases are equally as impressive as the example I have chosen to illustrate the accomplishments of twentieth century physicists. Finally, for those of you struggling to make sense of the physical processes involved, let us go back to Dirac:
The only object of theoretical physics is to calculate results that can be compared with experiment…it is quite unnecessary that any satisfactory description of the whole course of the phenomena should be given.4
11.2 ONE MYSTERY AND TWO REVELATIONS
In the late nineteenth century, and for the first part of the twentieth century, scientists discovered radioactivity and its challenges to our picture of the particles making up matter. Here was something quite unexpected that required new experimental techniques and new ways of thinking for its investigation. Ernest (later Lord) Rutherford (1871–1937) made great advances with what was termed α-decay, the event in which a nucleus emits an object later identified as a helium nucleus. Rutherford's experiments on the scattering of α particles led to his theory of the atom.
For some nuclei, there was also β-decay, and by 1902, it was clear that the emitted “β rays,” or β particles, were in fact electrons. (There are also β-decay processes in which a positron is emitted.) The problem now was to understand the origin of those electrons and the way in which they were emitted by a nucleus.
(I will say more about nuclear physics in section 11.6. A good modern reference is the book by Dunlap. The historical development of the subject is extremely well covered by Pais. For the topic of this section, the article by Brown and the book by Sutton are recommended as easy reading.)
11.2.1 Energies in β-Decay
The α particles are emitted with particular velocities depending on the type of nucleus involved (see Dunlap's figure 8.1 for example), and it was expected that something similar should be the case for β-decay electrons. However, instead of a single “spectral line,” there tended to be a continuous spread of velocities. Sometimes several “lines” were observed, and the situation was confused. Furthermore, there is a process called internal conversion in which a gamma ray is emitted and also causes an electron to leave the atom with a particular fixed velocity. We must remember that this was a time when nuclear experiments were still quite new, and particle detection and measurement techniques were not well established.
By 1914, James Chadwick (whose discovery of the neutron I discussed in the previous chapter) was convinced that the β-decay electron energies fell on a continuous curve rather than a set of discrete values. There was then a lull during the First World War. In 1927, Charles D. Ellis and William A. Wooster performed what were taken to be definitive experiments leading them to write that “we may safely generalize this result for radium E to all β-ray bodies and the long controversy about the origin of the continuous spectrum of β-rays appears to be settled.”5
While it was true that it was accepted that a stream of electrons with variable energies was produced by β-decay, it was not clear how that process actually produced those electrons and which factors then determined their energies. Much of the problem was due to the fact that although the proton was an accepted particle, it was not clear which, if any, other particles helped to form the nucleus. For example, the so-called PE theory held that there were also combined proton-electron entities inside the nucleus. It was not until 1932 that Chadwick definitively established the existence of the neutron.
11.2.2 The Origin of the Continuous Energy Problem
Rather than follow the intricacies of the history of β-decay (see Pais and Brown), let me jump to the result: there are neutrons in the nucleus, and in the β-decay process, a neutron is converted into a proton and an electron is emitted. Note that the electric charge is conserved since the neutron's charge is neutral, and the electron (negative) and proton (positive) charges balance out. Clearly, the β-decay process only occurs for some neutrons in particular nuclei. However, the free neutron always decays with a half-life of 898 seconds.
To understand the electron energy problem, consider the decay of a stationary neutron as shown in figure 11.2. The electron has speed w, the proton has speed v, and they must travel in directly opposite directions to conserve momentum.
Figure 11.2. Decay of a neutron. Figure created by Annabelle Boag.
If the rest masses of the neutron, proton, and electron are Mn, Mp, and Me, the conservation of energy and momentum laws as expressed in Einstein's relativistic mechanics lead to
These equations may be solved to find the speed of the electron. And now comes the big shock: the calculation reveals that there is just one possible value for w; there is no continuous spectrum for electron energies as the experiments reveal. The single speed predicted is the maximum found for a β-decay electron, but experiments show a continuous set of values as in figure 11.3. Here was an example of a quite straightforward calculation that disagreed with experiment; resolving this difficulty led to a major advance in physics.
Figure 11.3. Schematic diagram showing electron energies observed in a β-decay process. Em is the maximum possible electron energy calculated assuming that in β-decay a neutron is converted into a proton and an electron. Figure created by Annabelle Boag.
11.2.3 A Technical Aside
For readers wishing to check details for themselves, it is easiest to make the approximation in which an energy mc2 is approximated as the rest mass energy Mc2 plus a classical kinetic energy ½Mv2. Then the conservation laws give
Notice that B is the energy released
because the rest mass of a neutron is greater than the sum of the rest masses of the proton and the electron. (We are of course using Einstein's famous E = mc2 equation.) These equations are easy to solve and
The calculation says this is the only speed that an electron should have after the β-decay of a neutron.
11.2.4 Revelation One: The Resolution of the Continuous Energy Problem
The phenomenon of β-decay had thrown up a profound problem: apparently the results were in direct conflict with the conservation laws that were absolutely fundamental in physics. It is a sign of the desperation felt at the time that Niels Bohr (of all people) actually contemplated the possibility that energy may not always be conserved. In his 1930 Faraday lecture, Bohr said:
At the present stage of atomic theory we have no argument, either empirical or theoretical, for upholding the energy principle in the case of β-ray disintegrations, and are even led to complications and difficulties in trying to do so.6
To give up such cherished principles would be a drastic thing to do, but the actual solution to the problem proved to be almost as dramatic.
Wolfgang Pauli (1900–1958) was one of the most colorful figures on the quantum-physics scene, and, in typical fashion, he suggested a solution to the β-decay problem in a letter he wrote to colleagues who were to meet in Tübingen in December 1930 (see Pais for full details). The letter is headed “Zürich 4 Dec 1930,” and begins, “Dear Radioactive Ladies and Gentlemen.”7 Pauli goes on to propose that there is a “neutron” inside the nucleus, the conservation of energy problem is solved, and
the continuous β spectrum would then be understandable, assuming that in the β decay together with the electron, in all cases, also a neutron is emitted, in such a way that the sum of the energy of the neutron and of the electron remains constant.
