A Theory of Electrons and Protons. Offprint from Proceedings of the Royal Society, Series A, Vol. 126, No. A801

First edition, extremely rare offprint, of Dirac's prediction of antimatter - although he did not accept it as such until a year later (see below) - and the advent of the dynamical view of the vacuum which was fundamental to the later development of quantum field theory. "The consensus among today's scientists is that Dirac's role in foreseeing the existence of the positron is one of the greatest achievements in science. In 2002, shortly after the centenary of Dirac's birth, the theoretical physicist Kurt Gottfried went further: 'Physics has produced other far-fetched predictions that have subsequently been confirmed by experiment. But Dirac's prediction of anti-matter stands alone in being motivated solely by faith in pure theory, without any hint from data, and yet revealing a deep and universal property of nature'" (Farmelo, p. 226). One troubling consequence of the famous Dirac relativistic wave equation was that it implied that electrons should exist in states of negative as well as positive energy. Therefore a Dirac electron with initially positive energy should fall indefinitely by spontaneous emission towards states of lower and lower energy. To avoid this Dirac postulated in 'A Theory of Electrons and Protons' that the states of negative energy were normally filled up - electrons could not then fall into them because the Pauli exclusion principle prevents two electrons from occupying the same state. Dirac observed that holes in this 'sea' would behave like positively-charged particles. Dirac initially believed that his holes could be identified with protons, but Hermann Weyl and others showed that they should have the same mass as the electron. The following year, in the introduction to another paper (on a different topic), Dirac conceded and called these holes 'anti-electrons'. The existence of this particle was demonstrated experimentally by Carl Anderson in 1932; he called it the 'positron'. In 1973, Heisenberg said that the proposal of the anti-electron was "perhaps the biggest jump of all big jumps in physics of our century." In Dirac's new picture 'empty space' had become much more complicated. It was no longer empty and in fact behaves in many ways like a semiconductor, where electrons can be excited out of a filled valence band while leaving holes (physicists now call it 'the vacuum'). Dirac shared the 1933 Nobel Prize in Physics with Schrödinger for "for the discovery of new productive forms of atomic theory." OCLC lists only the copy at the University of Florida, where Dirac spent his final years and where many of his papers are held. No copies in auction records.

In 1928, Dirac published his discovery of the 'Dirac equation', his relativistic wave equation for the electron, which "ranks among the highest achievements of twentieth-century science" (Pais, p. 290). However, "it was clear to Dirac and several of his colleagues that the relativistic electron theory of 1928 led to strange consequences. The problem, sometimes referred to as the '± difficulty,' had its basis in the Dirac equation, which formally included solutions with negative energy. Contrary to the situation in classical physics, these could not be dismissed as nonphysical but had to be taken seriously - that is, somehow to be related to some objects of nature" (Kragh (1999), p. 190).

"Dirac's equation consists of four components. That is, it contains four separate wave functions to describe electrons. Two components have an attractive and immediately successful interpretation, describing the two possible directions of an electron's spin. The extra doubling, by contrast, appeared at first to be quite problematic.

"In fact, the extra equations contain solutions with negative energy (and either direction of spin). In classical (non-quantum) physics the existence of extra solutions would be embarrassing, but not necessarily catastrophic. For in classical physics, you can simply choose not to use these solutions. This of course begs the question why nature chooses not to use them, but it is a logically consistent procedure. In quantum mechanics, even this option is not available. In quantum physics, generally 'that which is not forbidden is mandatory'. In the case at hand, we can be quite specific and precise about this. All solutions of the electron's wave equation represent possible behaviours of the electron that will arise in the right circumstances. Assuming Dirac's equation, if you start with an electron in one of the positive-energy solutions, you can calculate the rate for it to emit a photon and move into one of the negative-energy solutions. Energy must be conserved overall, but that is not a problem here - it just means that the emitted photon has higher energy than the electron that emitted it! Anyway, the rate turns out to be ridiculously fast, with the transition taking place in a small fraction of a second. So you can't ignore the negative-energy solutions for long. And since an electron has never been observed to do something so peculiar as radiating more energy than it began with, there was, on the face of it, a terrible problem with the quantum mechanics of Dirac's equation" (Wilczek, pp. 53-54).

In November 1929 Dirac believed he had found the solution to this problem. "His point of departure was the recognition, then generally accepted, that 'we cannot ignore the negative-energy states without giving rise to ambiguity in the interpretation of the theory.' But with which physical entities (or 'things,' as Dirac preferred it) should these states be associated? Formal reasons seemed to indicate that the negative-energy states could be represented by positive-energy electrons with positive charge. Dirac had no difficulty in showing that, according to the relativistic wave equation, 'an electron with negative energy moves in an external field as though it carries a positive charge.' However, he did not simply identify the negative-energy solutions with either protons or positively charged electrons. Particles with negative energy have no reality in physics, and Dirac emphasized that such ghost-entities as positively charged, negative-energy particles would produce physical absurdities. But he also added the straight empirical argument that 'no particles of this nature have ever been observed.'

"His way out of the dilemma was to introduce a world of negative-energy states uniformly occupied by an infinite number of electrons. As a crucial point, he supposed the distribution of negative-energy electrons to be governed by Pauli's exclusion principle. If the distribution of negative-energy electrons in the 'Dirac sea' is exactly uniform, they will be unobservable, merely serving to define a state of normal electrification. But if a few of the negative-energy states are unoccupied, these vacant states, or 'holes,' will appear as observable physical entities: 'Only the small departures from exact uniformity, brought about by some of the negative-energy states being unoccupied, can we hope to observe ... These holes will be things of positive energy and will therefore be in this respect like ordinary particles.'

"According to an interesting - but no doubt apocryphal - story, Dirac came upon his idea of holes unconsciously, while dreaming of a problem posed in a competition arranged by the Cambridge Students' Mathematical Union. More seriously, the main source of Dirac's daring theory seems to have been an argument by analogy, taken from the fields of X-ray theory and chemical atomic theory. For example, in the inert gases the electrons fill up closed shells, resulting in chemical inactivity (which means unobservability in the chemical sense); a hole in the outer shell yields a halogen atom with some distinct chemical properties. More than forty years later. Dirac recalled: 'It was not really so hard to get this idea [of holes] once one had the proper understanding of what one needed, because there was a very close analogy provided by the chemical theory of valency' ... Another source of inspiration may have been Dirac's theory of radiation from 1927. In this theory the absorption and emission of photons was pictured as taking place relative to an unobservable state of zero-energy photons; the emission of a photon was considered to be a jump from this zero-energy state to a state of positive energy.

"With the idea of holes, Dirac managed to account for the negative-energy solutions without introducing observable negative-energy particles. He then became confronted with the problem of supplying the holes with a physical identity. Two possibilities were worth considering, the proton and the positive electron. Dirac's first formal proposal was for the wrong candidate, the proton; he wrote, 'We are ... led to the assumption that the holes in the distribution of negative-energy electrons are the protons.' However, in his later recollections he claimed that his first inclination was actually toward what turned out to be the correct candidate, the positive electron, because of the appealing symmetry this choice would fulfill between the masses of the electron and the hole. In one interview he said, 'I really felt that it [the mass of the hole] should be the same [as the mass of the electron] but I didn't like to admit it to myself.' And at another occasion he remarked, 'as soon as I got this idea [of holes] it seemed to me that the negative-energy states would have to correspond to particles having a positive charge instead of the negative charge of the electron, and also having the same mass as the electron.' But regardless of this first leaning, he initially introduced protons as the holes, although he was quite aware of the inherent difficulties of this idea and, indeed, retrospectively thought of it as 'rather sick.'

"Why did Dirac consider the identification of the hole with the proton to be sick? First of all, the Dirac wave equation is symmetric with respect to negative and positive charges (electrons and anti-electrons), while nature shows no symmetry between the electron and the much heavier proton. Dirac, lacking a better solution, expressed the hope that it might be possible to account for the difference in mass by means of a future theory of the interactions of protons and electrons; but he admitted that he was at the time unable to work out such a theory.

"If Dirac so clearly recognized the difficulties of the proton theory, then why did he propose it in place of his original idea of positive electrons? At least two motives were operative, one being empirical and the other formal. In 1930, physicists almost universally believed that matter consisted of only two material particles, electrons and protons. Neither theory nor experiment indicated that there were other fundamental particles in nature (the neutron still existed only as a name for an electron-proton composite, and the neutrino, although recently proposed, was not taken seriously). Naturally, Dirac preferred to base his hole theory on known entities rather than to postulate a new elementary particle for which there was at the time no empirical evidence whatsoever. In 1930, the climate in the infant branch of particle physics was very conservative with respect to new elementary particles, and Dirac's fear of introducing an unobserved particle may have been the result of a sociological constraint, determined by the paradigms of the period. This conservatism of the physics community may have inhibited Dirac from making a proposal that he would have made under different sociological conditions. However, another motive also played an important role in Dirac's considerations, significantly catalyzing his suggestion of the proton theory. He considered the identification of protons with vacant negative-energy states to be a highly attractive idea because it promised a reduction of the known elementary particles to just one fundamental entity, the electron ... Dirac thought that the dream of philosophers was on its way to being realized, that the quantum theory had now intimated a positive answer to the age-old question of the unity of matter ...

"Having concluded that the proton was the most likely candidate for the hole, Dirac was faced with several difficulties. For example, one would expect that a positive-energy electron might occasionally make a quantum transition to fill a hole, under which circumstance the two particles would annihilate ... At the British Association for the Advancement of Science meeting in Bristol, he said:

'There appears to be no reason why such processes should not actually occur somewhere in the world. They would be consistent with all the general laws of Nature, in particular with the law of conservation of electric charge. But they would have to occur only very seldom under ordinary conditions, as they have never been observed in the laboratory' ...

"The lack of experimental evidence for proton-electron annihilation did not worry Dirac too much. He was content to observe that the predicted process was 'consistent with all the general laws of Nature.' The other main problem that faced the theory was a more serious one, namely, the difference in mass between the electron and its assumed anti-particle, the proton. If the difference in mass could not be explained, then 'the dream of philosophers' would remain a dream ...

"In the fall of 1930, Pauli reached the same conclusion as Heisenberg, that Dirac's unitary theory was inconsistent. [Igor] Tamm, who had met Pauli during a conference in Odessa, reported to Dirac: 'Pauli told us that he rigorously proved ... that on your theory of protons the interaction of electrons can't destroy the equality of the masses of an electron & a proton.' Another argument was advanced by [Hermann] Weyl in the second edition of his Gruppentheorie und Quantenmechanik. Weyl showed that according to Dirac's own theory of the electron the hole must necessarily have the same mass as an ordinary electron" (Kragh (1990), pp. 95-103).

A year later, Dirac bowed to the inevitable and accepted that a 'hole' is an anti-electron. This admission appeared almost incidentally in a paper that appeared in May 1931 ('Quantised singularities in the electromagnetic field', Proceedings of the Royal Society, Vol. 133, pp. 60-72). In this paper the anti-electron was introduced for the first time as "a new kind of particle, unknown to experimental physics, having the same mass and opposite charge to an electron". (Most of this paper was devoted to another hypothetical particle, the magnetic monopole, which has not been detected experimentally.)

"In 1931 the antielectron was a purely hypothetical particle, and most physicists declined to take Dirac's theory seriously ... The status of the antielectron ... changed during 1932-33. At the California Institute of Technology, Carl Anderson, a former student of Millikan, noted in cloud chamber photographs from the cosmic radiation some tracks that he thought might be due to protons. In a later paper in March 1933, he suggested that he had discovered a positively-charged electron, or a 'positron' as he called it" (Kragh (1999), p. 192).

As well as predicting the existence of antimatter, the present paper also transformed physicists' view of the nature of 'empty space.' "What Dirac proposed was a radically new conception of empty space. He proposed that what we consider 'empty' space is in reality chock-a-block with negative-energy electrons. In fact, according to Dirac, 'empty' space actually contains electrons obeying all the negative-energy solutions ...

'Since we are considering the idea that the ordinary state of 'empty' space is far from empty, it is helpful to have a different word for it. The one physicists like to use is 'vacuum.'

"In Dirac's proposal, the vacuum is full of negative-energy electrons. This makes the vacuum a medium, with dynamical properties of its own. For example, photons can interact with the vacuum. One thing that can happen is that if you shine a light on the vacuum, providing photons with enough energy, then a negative-energy electron can absorb one of these photons, and go into a positive-energy solution. The positive-energy solution would be observed as an ordinary electron, of course. But in the final state there is also a hole in the vacuum, because the solution originally occupied by the negative-energy electron is no longer occupied" (Wilczek, p. 55). This is the phenomenon of 'pair-production,' in which an electron-positron pair is created spontaneously.

This dynamical view of the vacuum is fundamental to quantum field theory, developed starting in the mid-1930s, which enables the positron to be treated as a 'real' particle rather than the absence of a particle, and makes the vacuum the state in which no particles exist instead of an infinite sea of particles.

The theory propounded in the present paper not only opened new vistas in particle physics and quantum field theory, it also inspired important developments in other areas of physics, some of which had practical applications. "The physical ideas of Dirac's hole theory fed back in a big way into solid-state physics. In solids one has a reference or ground configuration of electrons, with the lowest possible energy, in which electrons occupy all the available states up to a certain level. This ground configuration is the analogue of the vacuum in hole theory. There are also configurations of higher energy, wherein some of the low-energy states are not used by any electron. In these configurations there are vacancies, or 'holes - that's what they're called, technically - where an electron would ordinarily be. Such holes behave in many respects like positively charged particles. Solid-state diodes and transistors are based on clever manipulation of holes and electron densities at junctions between different materials. One also has the beautiful possibility to direct electrons and holes to a place where they can combine (annihilate). This allows you to design a source of photons that you can control quite precisely, and leads to such mainstays of modern technology as LEDs (light-emitting diodes) and solid-state lasers" (Wilczek, p. 57).

Farmelo, The Strangest Man: The Hidden Life of Paul Dirac, Quantum Genius, 2009; Kragh, Dirac: A scientific biography, 1990; Kragh, Quantum generations, 1999; Pais, Inward bound, 1988; Wilczek, 'The Dirac equation,' International Journal of Modern Physics A, Vol. 19 Supplement (2004), pp. 45-74.