Simply Dirac

Helge Kragh

• Publisher: Simply Charly
• ISBN:
  9781943657001
• Topic: Technik & Maschinenbau
• Subtopic: Biographien der Naturwissenschaften & Technik

Paul Dirac (1902–1984) was a brilliant mathematician and a 1933 Nobel laureate whose work ranks alongside that of Albert Einstein and Sir Isaac Newton. Although not as well known as his famous contemporaries Werner Heisenberg and Richard Feynman, his influence on the course of physics was immense. His landmark book, The Principles of Quantum Mechanics, introduced that new science to the world and his "Dirac equation" was the first theory to reconcile special relativity and quantum mechanics.

Dirac held the Lucasian Chair of Mathematics at Cambridge University, a position also occupied by such luminaries as Isaac Newton and Stephen Hawking. Yet, during his 40-year career as a professor, he had only a few doctoral students due to his peculiar personality, which bordered on the bizarre. Taciturn and introverted, with virtually no social skills, he once turned down a knighthood because he didn't want to be addressed by his first name. Einstein described him as "balancing on the dizzying path between genius and madness."

In Simply Dirac, author Helge Kragh blends the scientific and the personal and invites the reader to get to know both Dirac the quantum genius and Dirac the social misfit. Featuring cameo appearances by some of the greatest scientists of the 20th century and highlighting the dramatic changes that occurred in the field of physics during Dirac's lifetime, this fascinating biography is an invaluable introduction to a truly singular man.

3

Anti-Worlds

With his fundamental papers published between 1925 and 1927, Dirac had proved himself a quantum wizard, a leading physicist of the quantum-mechanical revolution. However, although his contributions were invariably original and recognized to be highly significant, in almost all cases other European physicists obtained the same results—and in some cases, they even scooped him. Dirac felt that he still lived in Heisenberg’s shadow and had not yet produced a deep and really novel theory. The Nobel Committee in Stockholm even concluded that “Dirac is in the front rank of the group of researchers who have set themselves the task to realize Heisenberg’s bold thought,” but noted that “Dirac is a successor in relation to Heisenberg.”

The deep and really novel theory that he dreamed of came unexpectedly to him at the end of 1927. The eponymous Dirac equation is no less fundamental than the better-known Schrödinger equation, and its consequences even more amazing. The memorial stone for Dirac at Westminster Abbey reproduces the equation in a compact form familiar to modern physicists: iγ∂ψ = mψ. (Appropriately, the stone is placed near the tomb of Isaac Newton). To understand the origin and significance of this mysteriously-looking equation we need to go back to the spring of 1926.

When Schrödinger introduced his wave equation to the world of physics, he used it to calculate the energy levels of the hydrogen atom and derive the same spectral lines that Bohr had obtained from the old quantum theory in 1913. The hydrogen spectrum was a triumph of wave mechanics, but Schrödinger realized that the triumph was incomplete: the spectral lines were not sharp but consisted of narrowly spaced doublets, a phenomenon known as “fine structure.” The separation of the doublet lines, as given by the difference in frequency, is very small and can be expressed by a number known as the fine-structure constant. This number would play an important role in Dirac’s physics. Within the framework of the Bohr model, the fine structure could be accounted for by taking into regard the theory of relativity, but by 1926 Bohr’s orbital model was dead. Schrödinger was acutely aware that his equation did not satisfy the requirements of relativity and, probably, for this reason, was unable to explain hydrogen’s fine structure.

The Schrödinger wave equation contains terms that correspond to a particle’s energy E and its momentum p. According to relativity theory, energy is associated with time (t) while the momentum is associated with the space coordinates (x in one dimension). However, energy enters the quantum wave equation linearly (as E ~ t) and momentum enters as the square (as p² ~ x²), which means that the equation is not symmetrical in time and space. To Schrödinger’s despair, when he modified the equation according to the space-time symmetry requirement, it was still unable to reproduce the correct fine structure spectrum. There was something wrong, but he could not figure out what it was.

Nonetheless, in the summer of 1926 Schrödinger published the relativistic version of his wave equation and so did half a dozen other physicists. Pauli called it “the equation with the many fathers.” Because two of the “fathers” were Oskar Klein and Walter Gordon, a Swedish and German physicist respectively, it became known as the Klein-Gordon equation. It was a nice but apparently useless formula. The problem with relativity was not exclusive to wave mechanics for it also appeared in matrix mechanics and Dirac’s q-number algebra. After all, the three formulations were just different versions of the same theory, quantum mechanics.

There was another and possibly related problem. To characterize the behavior of an electron, it must be ascribed a “spin” quantum number that can attain only two values, + ½ or − ½. In the classical picture, it corresponds to a spherical charge that rotates around its axis in either of the two directions. Spin was discovered in the summer of 1925 and initially seemed foreign to the new quantum mechanics. It could be grafted upon the quantum equations but not derived from them. Physicists vaguely realized that spin and relativity were somewhat connected, but nobody could tell in what way or how they fit into the formalism of quantum mechanics. Attempts to make them fit were ad hoc and did not appeal to Dirac at all. He was slowly getting interested in the spin puzzle, and in December of 1926, when he stayed with Heisenberg in Copenhagen, the two physicists made a bet of when spin would be properly explained. Dirac thought three months, Heisenberg, at least, three years. They were both wrong, but Heisenberg more than Dirac. (The process took one year).

A little more than three months after the bet, Pauli came up with a quantum-mechanical theory of spin, if not a proper explanation. His idea was to extend Schrödinger’s wave function ψ from one to two components. An electron would then be characterized by (ψ₁, ψ₂) where the two functions represented the electron’s two spin states by means of new variables. The variables were 2 × 2 “Pauli matrices” with two rows and two columns. Pauli’s theory made sense of the spin within the framework of ordinary quantum mechanics, but since it was not relativistic, Dirac considered it to be merely a provisional solution. The easy way would be to integrate spin and the Klein-Gordon equation, but this turned out not to be possible. It was Dirac who solved the problem and perhaps won the bet with Heisenberg. However, the solution came in a roundabout way. “I was not interested in bringing the spin of the electron into the wave equation,” he recalled. “It was a great surprise to me when I later discovered that the simplest possible case did involve the spin.” Dirac’s discovery was an example of what is called serendipity, the almost accidental discovery of something the scientist is not looking for.

Having no faith in the Klein-Gordon equation, at the end of 1927 Dirac decided to find a better solution for a wave equation in accordance with the theory of relativity. Based on the relativity requirement and the general structure of quantum mechanics, he was convinced that the equation he looked for must be linear not only in energy, as in the ordinary Schrödinger equation, but also in momentum. This conviction brought him face to face with a purely mathematical problem, namely how to write the square root of a sum of four squares as a linear combination. Take the square root of (a² + b² + c² + d²) and try to write it as n₁a + n₂b + n₃c + n₄d, where the n’s are some coefficients. That’s not an easy problem, but Dirac needed to solve it. Guided by Pauli’s spin matrices and his own mathematical intuition, he realized that the trick could be done if the n-coefficients were 4 × 4 matrices, that is, quantities with four rows and columns comprising 16 numbers. With these “Dirac matrices,” he could straightforwardly write down the new wave equation for a free electron. As a consequence of the 4 × 4 matrices, the wave function had not only two components, as in Pauli’s theory, but four: ψ = (ψ₁, ψ_2, ψ₃, ψ₄).

The crucial step in Dirac’s derivation was the reduction of a physical problem to a mathematical one. The method was characteristic of his style of physics. “A great deal of my work is just playing with equations and seeing what they give,” Dirac said in a 1963 interview. “I think it’s a peculiarity of myself that I like to play about with equations, just looking for beautiful relations which maybe can’t have any physical meaning at all. Sometimes they do.” In this case, they did.

And what about the spin? Dirac had originally ignored the problem by considering, for reasons of simplicity, an electron without spin, knowing that such a particle did not exist. But when he extended his wave equation of the free electron to one where the electron interacted with an electromagnetic field, he discovered that the correct spin appeared, almost mysteriously. Without introducing the spin in advance, Dirac was able to deduce the electron’s spin from the first principles upon which his equation was built. In a certain if somewhat unhistorical sense, had spin not been discovered experimentally, it would have turned up deductively in Dirac’s theory. This was a great and unexpected triumph. It was less unexpected, but still very satisfactory that the equation could account in detail for the fine structure of the hydrogen spectrum.

Dirac submitted his paper on the new electron theory on the first day of 1928, and it appeared in print a month later. It came as a bombshell to the physics community. Dirac had worked alone, almost secretly, not for reasons of priority but because it was his habit. His senior colleague Charles Darwin, a professor at the University of Edinburgh and a grandson of the “real Darwin,” was one of the few who knew in advance what was going on. In a letter to Bohr of December 26, 1927, he wrote: “I was at Cambridge a few days ago and saw Dirac. He has now got a completely new system of equations for the electron, which does the spin right in all cases and seems to be ‘the thing.’ His equations are first order, not second, differential equations!” Dirac’s equation was indeed the thing. It was received with equal measures of surprise and enthusiasm—even by the physicists who had come close to solving the problem themselves. Among them was Jordan, who, in this case, lost to Dirac. A physicist who at the time worked in Göttingen recalled: “The general feeling was that Dirac had had more than he deserved! Doing physics in that way was not done! … [The Dirac equation] was immediately seen as the solution. It was regarded really as an absolute wonder.”

The new relativistic theory of the electron had a revolutionary effect on quantum physics, both pure and applied. It was as if it had a life of its own, full of surprises and subtleties undreamed of even by Dirac when he worked it out. For example, the 4 × 4 Dirac matrices attracted much interest among the pure mathematicians who eagerly studied the properties of the matrices and other mathematical objects related to them. This branch of mathematical physics eventually developed into a minor industry and is still an active field of research. Right after Dirac’s theory appeared, it seemed that although it had great explanatory power, there was no particular predictive power. It explained the electron’s spin and hydrogen’s fine structure most beautifully, but no predictions of novel phenomena followed from it. This situation soon changed. Within a few years, Dirac’s equation and the electron theory based on it proved successful over a wide range of physical areas, including high-energy scattering processes and the mysterious cosmic rays. Most dramatically, it led to the successful prediction of a new class of elementary particles, which Dirac called antiparticles.

The revolutionary concept of antimatter or antiparticles had its roots in conceptual problems of the Dirac equation noted at an early date. In the four-component wave function (ψ₁, ψ_2, ψ₃, ψ₄) two of the components, say ψ₁ and ψ₂, refer to the two spin states of an ordinary electron, just as in Pauli’s theory. What do ψ₃ and ψ₄ refer to? They enter because Dirac’s theory is a quantum translation of the energy-momentum relation in relativity theory, which involves the squares of energy and momentum. But Dirac’s equation, following the basic rules of quantum mechanics, was linear in energy. It involved E and not E². Now, when you take the square root of mc² you don’t get just the energy E but also the negative quantity − E. After all, (− E)² = E². Formally ψ₃ and ψ₄ refer to the two spin states of a negatively charged electron with negative energy. We can avoid negative-energy electrons, but in that case, the two wave functions need to be interpreted as belonging to electrons of positive charge. As Dirac pointed out in 1930, “an electron with negative energy moves in an external field as though it carries a positive charge.”

Both interpretations invited trouble. Take the hypothetical negative-energy (and negative-charge) electrons. Since energy is equivalent to mass by E = mc², the mass of a negative-energy electron at rest will also be negative. It can easily be shown that the energy of such a particle, when set in motion, will decrease the faster it moves; moreover, when a force is impressed on the particle, it will accelerate in the opposite direction of the force. Absurd! The Russian physicist George Gamow had first met Dirac in 1927, and the two established a lasting friendship. Gamow called the negative-energy electrons formally occurring in Dirac’s theory “donkey electrons”—the harder they were pushed, the slower they would move. From the standpoint of physics, by 1930 the positive-electron interpretation was no better, for the charge of electrons was known to be negative. Earlier speculations about positive electrons had turned out to be just that—speculations. The only known positive particle—or “positive electron”—was the proton, which is nearly 2,000 times heavier than an electron. According to the paradigm governing physics at the time, there were no other massive elementary particles than the electron and the proton (the photon has no mass). The dilemma was this: either one had to postulate electrons with negative energy that behaved absurdly or the existence of a particle for which there was not the slightest empirical evidence. As a third possibility, one might have discarded Dirac’s theory in toto, but this was an option no one seriously considered. There was no way back.

The intertwined problems of the energy and charge of Dirac’s electron were much debated in the physics community. Heisenberg, Jordan, and others were deeply worried about what they called the “± difficulty.” For example, in 1928 Jordan referred to “the murky problem of the asymmetry of the forms of electricity, that is, the inequality of mass for positive and negative electrons.” In relation to this problem, he judged that Dirac’s “theory is entangled in temporarily insoluble difficulties.” Heisenberg concurred: “The saddest chapter of modern physics is and remains the Dirac theory.” Dirac was worried too, but in the late autumn of 1929, he thought to have found an answer to the ± difficulty. His hypothesis was as original as it was audacious—vintage Dirac. He first reported it in letters to Bohr, through which it became known to other physicists, and publicly announced his “Theory of Electrons and Protons” in a paper written in early 1930.

Dirac realized that particles with negative energy could have no reality in nature, and he consequently focused on the ± di...