## ‘The quantum theory of the electron,’ pp. 610-624 in Proceedings of the Royal Society of London, Series A, Vol. 117, No. 778.

London: for the Royal Society, 1928.

First edition, journal issue in the original printed wrappers, of the discovery of the ‘Dirac equation’. “The relativistic wave equation of the electron ranks among the highest achievements of twentieth-century science” (Pais, *Inward Bound*, p. 290). While the paper was in press, “Dirac wrote to Max Born in Göttingen, not mentioning his new equation except in a ten-line postscript, where he spelt out the reasoning that had led to it. Born showed these words to his colleagues, who regarded the equation as ‘an absolute wonder’. Jordan and Wigner, who were working on the problem that Dirac had solved, were flabbergasted. Jordan, seeing his rival walk off with the prize, sank into depression. When the equation appeared in print at the beginning of February, it was a sensation. Though most physicists struggled to understand the equation in all its mathematical complexities, the consensus was that Dirac had done something remarkable, the theorists equivalent of a hole in one. For the first time in his career, [Dirac] had shown that he was capable of tackling one of the toughest problems of the day and beating his competitors to the solution, hands down. The American theoretician John Van Vleck later likened Dirac’s explanation of electron spin to ‘a magician’s extraction of rabbits from a silk hat’. John Slater, soon to be a colleague of Van Vleck’s at Harvard, was even more effusive: ‘we can hardly conceive of anyone else having thought of [the equation]. It shows the peculiar power of the sort of intuitive genius which he has possessed more than perhaps any of the other scientists of the period’” (Farmelo, *The Strangest Man*, pp. 143-4). “Unlike many results in theoretical physics it was neither inspired by unexplained measurements nor by physical insight but only by considerations of mathematical ‘beauty’ or, in other words, simplicity. In the *Dirac equation *not only quantum mechanics and the special theory of relativity were married, but also the spin of the electron is contained in it without any ad hoc assumption. So far, so good. But the equation did not just beautifully describe known phenomena, it did more. It predicted the existence of electrons with negative energy. This was at first held to be a severe problem of the theory but was finally understood as great progress, because negative-energy electrons could be interpreted as hitherto unknown particles. Thus, the existence of new particles was predicted which had all properties of the electron except for the electric charge, which must be positive rather than negative. These particles were indeed found four years after the equation” (Brandt, *Harvest of a Century*, p. 183). “Dirac’s relativistic wave equation marked the end of the pioneering and heroic phase of quantum mechanics, and also marked the beginning of a new phase” (Kragh, *Quantum Generations*, p. 167). Dirac shared the 1933 Nobel Prize in Physics with Erwin Schrödinger ‘for the discovery of new productive forms of atomic theory.’

“Even with … the many successful applications of quantum mechanics to spectroscopy and other areas of physics, the theory was not without problems. There was, for example, the question of the relationship between relativity and quantum mechanics. If quantum mechanics was really a fundamental theory of the microcosmos, it ought to be consistent with the fundamental theory of macroscopic bodies, the (special) theory of relativity. Yet it was obvious from the very beginning that this was not the case. The Schrödinger equation is of the second order in the space derivative and of the first in the time derivative, in contradiction to the theory of relativity. It was not too difficult to construct a relativistic quantum wave equation, such as Schrödinger had already done privately and as Oskar Klein, Walter Gordon, and several other physicists did in 1926-27. Unfortunately, this equation, known as the Klein-Gordon equation, did not result in the correct fine structure of hydrogen and it proved impossible to combine it with the spin theory that Pauli had proposed in 1927” (Kragh, *Quantum Generations*, p. 167).

“Pauli developed a genuinely quantum-mechanical description of spin by constructing a vector with three components, which are themselves matrices in the sense of matrix mechanics. Using these *Pauli matrices*, which have two rows and two columns, he was able to fulfil all requirements posed by quantum mechanics on spin, including the mysterious two-valuedness. He described the electron by a *spinor*, in which both elements are ordinary wave functions, and in this way accounted for both spin and position. His *Pauli equations *are a system of two coupled Schrödinger equations.

“The Pauli equations, apart from being non-relativistic, had one essential drawback. An electron orbiting an atomic nucleus with a certain angular momentum behaves like a little magnet, it has a magnetic moment. So has the electron by virtue of its intrinsic angular momentum or spin. But the ratio of magnetic moment and angular momentum for spin is twice as large as for the orbital momentum. This factor of two in the *gyromagnetic ratio *of the spinning electron had to be introduced ‘by hand’ or ‘ad hoc’ in the Pauli equations …

“When Dirac began to work on the problem he, at first, did not consider spin but compared the non-relativistic and the relativistic Schrödinger equations. These are differential equations in space and time. If a differential has only a first derivative in a variable it is said to be of *first order *in that variable. If there is a second derivative, it is said to be of *second order*. The non-relativistic Schrödinger equation is of first order with respect to time but of second order with respect to the space coordinates. In special relativity, of course, time and space are treated on equal footing. The relativistic Schrödinger equation is of second order in time and in space. Following his sense of beauty or simplicity, Dirac searched for an equation that was only of first order in these variables. He made an *ansatz *of the form

(*iγ**μ**∂**μ **− m*)*Ψ* = 0.

It was clear that the equation made no sense if the quantities *γ**μ* were mere numbers.

“Dirac said about his work on the equation:

‘This was a problem for some months, and the solution came rather, I would say, out of the blue, one of my undeserved successes. It came from playing about with mathematics … It took me quite a while, studying over this dilemma, that there was no need to stick to quantities σ, which can be represented by matrices with just two rows and two columns. Why not go to four rows and four columns?’

“The equation was indeed fulfilled if the quantities *γ**μ* are taken to be four-by-four matrices with two-by-two submatrices, which are either the matrix 0 (containing zeroes only), or the unit matrix I, or the Pauli matrices. In his first paper on the subject [offered here], Dirac gives credit to Pauli for having introduced the σ matrices but, in reminiscences, half a century later, he said: ‘I believe I got these variables independently of Pauli, and possibly Pauli also got them independently of me.’

“The effect of the new matrices was that the wave function now had four components and not just two as in the case of the Pauli equations. They had to be interpreted as to correspond to two states of positive energy, each with another spin orientation, and two states of negative energy … In his first papers he chose to just ignore the negative energy states. Doing this, he could reap a triple harvest from his equation:

It yielded the correct *fine structure *of the hydrogen spectrum, which had been computed as a relativistic effect by Sommerfeld in the old quantum theory and which Schrödinger had been unable to reproduce.

It gave the correct results for the anomalous Zeeman effect.

It automatically gave the right gyromagnetic ratio g = 2 for the electron.

“In particular, the last point indicated that spin is a truly relativistic effect, since it was described correctly in an equation based on the special theory of relativity” (Brandt, pp. 184-7).

“In a certain, unhistorical sense, had spin not been discovered empirically, it would have turned up deductively from Dirac’s theory” (Kragh, *Quantum Generations*, p. 167).

“Paul Adrien Maurice Dirac was born on 8th August, 1902, at Bristol, England, his father being Swiss and his mother English. He was educated at the Merchant Venturer’s Secondary School, Bristol, then went on to Bristol University. Here, he studied electrical engineering, obtaining the B.Sc. (Engineering) degree in 1921. He then studied mathematics for two years at Bristol University, later going on to St. John’s College, Cambridge, as a research student in mathematics. He received his Ph.D. degree in 1926. The following year he became a Fellow of St.John’s College and, in 1932, Lucasian Professor of Mathematics at Cambridge.

“Dirac’s work has been concerned with the mathematical and theoretical aspects of quantum mechanics. He began work on the new quantum mechanics as soon as it was introduced by Heisenberg in 1925 – independently producing a mathematical equivalent which consisted essentially of a noncommutative algebra for calculating atomic properties – and wrote a series of papers on the subject, published mainly in the Proceedings of the Royal Society, leading up to his relativistic *theory of the electron* (1928) and the theory of holes (1930). This latter theory required the existence of a positive particle having the same mass and charge as the known (negative) electron. This, the positron was discovered experimentally at a later date (1932) by C. D. Anderson, while its existence was likewise proved by Blackett and Occhialini (1933 ) in the phenomena of ‘pair production’ and ‘annihilation’.

“The importance of Dirac’s work lies essentially in his famous wave equation, which introduced special relativity into Schrödinger’s equation. Taking into account the fact that, mathematically speaking, relativity theory and quantum theory are not only distinct from each other, but also oppose each other, Dirac’s work could be considered a fruitful reconciliation between the two theories.

“Dirac’s publications include the books *Quantum Theory of the Electron* (1928) [offered here] and *The Principles of Quantum Mechanics* (1930; 3rd ed. 1947).

“He was elected a Fellow of the Royal Society in 1930, being awarded the Society’s Royal Medal and the Copley Medal. He was elected a member of the Pontifical Academy of Sciences in 1961.

“Dirac has travelled extensively and studied at various foreign universities, including Copenhagen, Göttingen, Leyden, Wisconsin, Michigan, and Princeton (in 1934, as Visiting Professor). In 1929,after having spent five months in America, he went round the world, visiting Japan together with Heisenberg, and then returned across Siberia.

“In 1937 he married Margit Wigner, of Budapest [sister of Nobel Laureate Eugene Wigner]” (nobelprize.org/prizes/physics/1933/dirac/biographical/).

8vo (254 x 178 mm), pp. [4] 541-730, xxxvi, vi. Original printed wrappers (small tear in upper margin of front wrapper and another at foot of spine). An excellent copy.

Item #5817

**
Price:
$5,500.00
**