Paul Dirac: You'll Never Walk Alone

Dirac is a unique figure in the history of science. He has devoted himself to the field of theoretical physics for nearly 60 years with a unique research style: he combines intuition, imagination, linear logic and powerful mathematics. This is always confusing, how did he do it? He was a taciturn person. After the creative burst of results, he gradually became disappointed with quantum electrodynamics and moved away from the mainstream academic community, becoming a lonely thinker. But people still draw inspiration from his articles again and again, and move towards the route he has already marked. Dirac, a lonely man, never walks alone.

By Graham Farmelo

Translation | Snow

Paul Dirac is often called a "theorist's theorist." He was shy, silent, and seemingly lacking in empathy, a typical loner in the scientific community. In Dirac's later years, when a physicist called him out of the blue and asked if he would like to talk about the ideas in his article, he would firmly interrupt the other person and say, "I think people should study their own ideas," and then hang up the phone.

Dirac’s most famous contribution was the development of quantum mechanics, which had been begun by Werner Heisenberg and Erwin Schrödinger in 1925, when Dirac was only 23. In the early literature on the theory, Dirac’s papers stood out, as Freeman Dyson put it: “His great discoveries fell one after another from the sky like finely carved marble statues.”[1] Although Dirac was widely respected as a scientific magician, many physicists—particularly those in Berlin and Göttingen, Germany, where many of the foundational papers on quantum mechanics were produced—found his language to be incomprehensible, his arguments to be elusive, and his manner to be cold and distant. Einstein was among those who were puzzled: "I have trouble with Dirac. This balancing on the dizzying path between genius and madness is awful." Niels Bohr was impressed by Dirac, but was also puzzled by his indifference to the philosophical questions raised by the new theory, and called Dirac "the strangest man who ever visited my institute." [2]

Dirac's unique personality and attitude towards theoretical physics stemmed from his upbringing in Bristol, the largest city in southwest England. According to him, he had a miserable childhood without love and friends, but received a full education in science, mathematics and engineering. Eight weeks after his 21st birthday, Dirac arrived at Cambridge University to begin his doctoral studies - although his knowledge of modern physics was not comprehensive at the time, he already had two bachelor's degrees in electrical engineering and applied mathematics. He was an extremely unusual student, an outsider who was ready to leave a unique mark on science, and few people could have guessed that he was destined to become the most accomplished scholar in Britain in the 20th century.

Dirac later said that he had never had a childhood. His early memories of his family life were miserable—no one was immune, largely because of his overbearing father, a teacher, who insisted that the family have few visitors and that the children speak only to him in French. At mealtimes, the family would be separated: Dirac and his father in the front room, conversing only in French, while his mother and siblings were in the kitchen, speaking only English. A well-researched report from 1933 states that as a child, Dirac believed that men and women spoke different languages. His disciplinarian father would punish him for minor grammatical errors and even deny him access to the toilet. Dirac recalled that he believed silence was the best way to avoid punishment. This also explains his reluctance to speak unless there was a good reason.

Left: Dirac's mother Florence and three children, the photo was taken by his father on April 9, 1909 [Paul Dirac, less than 7 years old at the time, is on the left; his brother Felix is ​​on the right, and his sister Betty is in his mother's arms]. The right picture (taken in 1910) is Dirac's father Charles, who was born in Switzerland in 1866 and became a British citizen on October 22, 1919. Since then, his children have also become British citizens, while Paul Dirac's official nationality was Switzerland. Image source: Courtesy of Florida State University, Paul AM Dirac Collection.

Dirac was a good, but not exceptional, student in primary school (one of his classmates was Archie Leach, who later became the famous actor Cary Grant). Dirac began to shine when he entered high school. It was World War I, and many boys were going off to join the armed forces, leaving vacancies in the top classes that allowed bright students like Dirac to advance rapidly. The high school gave Dirac a first-rate practical education, allowing him to skip Latin, Greek, and other subjects that were unlikely to be useful for finding a job. He excelled in almost every subject, especially in mathematics, science, and technical drawing. By his teens, Dirac was far ahead of the rest of his class and began to ponder the nature of space and time, even though he knew nothing about relativity. His classmates thought he was eccentric and withdrawn; one described him as "a tall, thin boy in knickerbockers and curly hair who looked un-English." Dirac's math teacher, desperate for no assignments that would keep him focused, decided to invite Dirac to study Riemannian geometry, an invitation he gladly accepted.

When Dirac was 16, he was ready to go to university. Unsure of what he wanted to study, he decided to join his brother and go to Bristol University to study engineering. Dirac worked tirelessly on theoretical work, but he was hopelessly clumsy in the laboratory, spending most of his afternoons soldering circuits, operating a lathe, measuring beam loads, or busying himself with other skills necessary for a student engineer.

Flowing thoughts

Despite his busy schedule, Dirac needed a challenge. Sure enough, it came in late 1919, not long after the family renounced their Swiss citizenship and became British citizens, when, as Dirac put it, Einstein’s theory of general relativity “suddenly rocked the world.” New data from solar eclipse observations seemed to prove that Einstein’s theory was superior to Newton’s in describing the bending of light by the sun—and he and his classmates were thrilled by the sensational news. [See Daniel Kennefick in Physics Today (March 2009)] But for Dirac, it was hard to find the substance behind the big news. Details on the theory were scarce, and most of the pamphlets on Einstein’s work were thin on the ground, misleading, and often wrong.

Dirac’s desire for more detail was satisfied when he took a course on scientific thought taught by philosopher Charlie Broad, which focused on Einstein’s special and general theories of relativity. Broad, who had been trained in natural philosophy at Cambridge, had a gift for summarizing new ideas with precision and vividness. (He would read every sentence in his carefully prepared lecture notes twice, and jokes three times.) The idea of ​​expressing it in mathematical form, and thus being able to guess at the laws of nature, captured Dirac’s imagination, and at the age of 17 he was on his way to becoming a theoretical physicist.

In July 1921, Dirac received a first-class honors degree, but soon he also received an "unemployment certificate." At that time, the British economy was in recession and job opportunities were scarce. Dirac attended several interviews, but in the end, nothing came of it. One of his lecturers in the engineering department, David Robertson, took the initiative to arrange for him to take free university mathematics courses and skip the first year. During his studies in pure mathematics, Dirac attended courses by Peter Fraser. Fraser never wrote a research paper in his life, but was an extraordinary teacher - Dirac later said that this was the best teacher he had ever met. Fraser was keen on projective geometry - the study of geometric properties that are invariant under special transformations, a subject closely related to geometric drawing, which Dirac had been studying for nearly ten years. Although lectures on pure mathematics were Dirac's favorite, he spent most of his time on applied mathematics courses, solving many problems using Newtonian mechanics. He also attended several lectures on relativity, and he probably knew more than the speaker.

When Dirac arrived at Cambridge in October 1923 to pursue his doctorate, the university knew they had an unusual student. A report from a Bristol “intellectual scout” said that [Dirac] was “somewhat clumsy, preferred to sit and think, was a recluse, did not like to joke, and was very poor.” Dirac impressed the university with his excellent performance in the entrance examination and was eager to give him a place as a graduate student (he was not eligible for undergraduate courses because he had not studied Latin or Greek). Although he had great intellectual deficiencies and had never learned Maxwell’s equations, Dirac showed a natural talent for mathematics and had the technical skills and judgment of a trained engineer.

Dirac had wanted to begin his research career with relativity, so he was disappointed when he was told that his advisor was Ralph Fowler, an expert in statistical mechanics and quantum theory. However, Dirac soon realized that he had one of the best advisors in Cambridge - a well-connected, encouraging man with the ability to find manageable problems. Dirac quickly and imaginatively solved the problems raised by Fowler, thus establishing himself as a top-notch student. He also continued to study projective geometry in his spare time and looked for relativistic versions of various classical theories to satisfy his appetite for special relativity.

Dirac seemed content, as we can see from the extremely terse postcards he wrote home. But in the spring of 1925, Dirac was devastated by the news that his brother, by then estranged, had committed suicide by taking potassium cyanide. Dirac's initial reaction to the tragedy is not recorded; it remains a painful subject, and neither he nor his wife would discuss it. But he did talk to close friends, and he blamed their bullying father for his brother's death. For a while thereafter, Dirac's productivity dropped dramatically, and until he returned to Bristol that summer, he did not publish anything for several months. Towards the end of his vacation, he received a letter that changed his life.

The letter was from Fowler and contained a proof copy of an article now considered to be Heisenberg's first published work on quantum mechanics. [3] At first, Dirac thought it too complicated and set it aside. But about two weeks later, his attention was drawn to a few lines in the article in which Heisenberg pointed out an obvious flaw in his theory: the non-commutation of the two variables position and momentum, but he hinted that this problem was not insurmountable. Over the next few weeks, Dirac focused on this sentence and realized that it contained the key to quantum mechanics. Dirac constructed his own version of quantum mechanics by analogy with the Poisson bracket of classical mechanics, which plays an important role in determining the time evolution of dynamical systems. His first paper on the subject, "The Fundamental Equations of Quantum Mechanics," [4] made a great impression on Heisenberg, Max Born, and their colleagues in Göttingen. Forty years later, Heisenberg said in a BBC interview that none of them had heard of Dirac at the time, but guessed that he was a first-rate mathematician.

The photo was taken in 1927 when Dirac was 25 years old. The tree behind him is probably no coincidence: Dirac learned something from his Soviet friend Igor Tamm - Dirac liked to climb trees and often wore suits. Image credit: Courtesy of Florida State University, Paul AM Dirac Collection.

Dirac’s early papers on quantum mechanics are remarkable for their profound insights and elegance. Many of them still seem fresh and modern. In the mid-to-late 1920s, the book of nature seemed to open before him: he published one great paper after another, co-discovering quantum transformation theory and quantum field theory, dispersion theory, density matrices, and hole theory, and making several other groundbreaking contributions. Scholars were puzzled by Dirac’s many insights, but they did not get much information from him until the 1960s, when Dirac began to talk about his early work. In one comment, he opened up about his use of projective geometry in his earliest papers; he had not mentioned this mathematics in his papers in part because he thought other physicists were unfamiliar with it. When Roger Penrose asked Dirac to explain how he used geometry in these papers during a 1971 lecture at Boston University, Dirac gently shook his head and refused. However, he did explain his inspiration for the delta function in a 1963 interview, while recalling his studies in engineering:

When you think about... engineering structures, sometimes you have distributed loads, and sometimes you have concentrated loads at a point. Well, it's essentially the same thing... but you use different equations in each case. Basically, to unify these two cases, this leads to the delta function in a way.

Perhaps the highlight of Dirac’s creative explosion was his equation for the electron, published in 1928.[5] This equation made quantum mechanics and special relativity compatible, explaining both the spin and the magnetic moment of a particle. Three years later, in his seminal paper on magnetic monopoles, he used this equation in passing to foreshadow the existence of the antielectron.[6] At the end of a series of lectures at Princeton University in the fall of 1931, Dirac almost directly predicted the existence of the antielectron, although there is no evidence that he encouraged experimentalists to look for the new particle. In August 1932, Carl Anderson of Caltech published the first evidence for the existence of a particle with the same mass as the electron but opposite charge, but he did not mention Dirac’s work. It was not until several months later that the academic community realized that Anderson had discovered the antielectron predicted by Dirac. Thirty years later, with the Olympic detachment that became his trademark, Dirac said that his greatest satisfaction came not from the discovery of the antielectron but from the correctness of his equation.

This successful prediction impressed the Nobel Prize Committee, which had been reluctant to award a prize to quantum mechanics because it had not previously received sufficient experimental support. In November 1933, more than a year after Dirac became Lucasian Professor at Cambridge University, the Nobel Committee announced that Dirac and Schrödinger would share the Nobel Prize that year, each half, and retroactively awarded the 1932 Nobel Prize to Heisenberg. Dirac became the youngest Nobel Prize winner in physics at the time [Translator's note: for theory, for experiment it was 25-year-old William Lawrence Bragg], a record that was not broken until 1957 by Tsung-Dao Lee (only a few months later).

Opposition to QED

Weeks after Dirac won the Nobel Prize, he proposed the idea of ​​vacuum polarization, and his golden age came to an end. He was no longer fascinated by quantum electrodynamics (QED), and was troubled by the fact that many of the observables predicted by the theory were infinite, making calculations meaningless. In late 1936, he briefly turned his attention to cosmology and proposed the controversial large-numbers hypothesis, which posits that a few simple linear equations link the huge numbers on the scale of the universe (rather than coincidence).

A few years later, Dirac accepted an invitation from James Scott to give a lecture on his philosophy of physics. His acceptance was quite surprising, since Dirac openly disdained philosophy of science. In 1963, he described it as “just a way of talking about discoveries that have been made.” But Dirac did not disappoint his audience in his February 1939 lecture in Edinburgh on “The Relationship between Mathematics and Physics,” delivering profound insights in plain language without using a single abstract mathematical symbol. [7] Even his introduction was straightforward: “The mathematician plays a game for which he invents the rules; the physicist plays a game for which Nature gives the rules.”

He suggested that theoretical physicists should seek physical laws that embody the greatest possible mathematical beauty. However, he had no patience for the obvious question of what objectively constituted this aesthetic quality: “It is an indefinable quality, just as beauty in art cannot be defined, but it is not difficult for those who study mathematics to appreciate it.” Dirac later said that his belief in the so-called principles of mathematical beauty was “like a religion” to him and his friend Schrödinger.

At the same time as Dirac’s change of direction in his research, some important events were taking place in his personal life. In June 1936, Dirac’s father died, leaving Dirac under his control until the end of his life. After the funeral, Dirac breathed a sigh of relief: “I feel much freer now; I feel I am now my own master.” He wrote these words to his close friend Margit Balázs, the then divorced sister of his Hungarian friend and colleague Eugene Winger. Within six months, she married Dirac. It was an unlikely union, as she was in many ways Dirac’s opposite—talkative, gregarious, and opinionated. However, the marriage worked well, and they had two daughters and spent nearly 50 years together. Dirac considered himself a family man, keen on tending his garden and lawn, but he also worked on theoretical physics, but he became increasingly alienated from the mainstream academic world. During World War II, he served as an advisor to a secret British group working on nuclear weapons, and spent part of his time developing an idea he had proposed: separating isotopes with an instrument without moving parts. And he did not completely abandon theoretical physics. He was one of the few people who continued to work on QED during the war, and kept in touch with his fellow refugees Schrödinger and Wolfgang Pauli.

Margit Balázs, sister of theoretical physicist Eugene Wigner, was photographed in 1932, two years before she first met Dirac in a Princeton restaurant. They married in London in January 1937. Image: Courtesy of Florida State University, Paul AM Dirac Collection.

In the early 1950s, the next generation of theorists—notably Dyson, Richard Feynman, Julian Schwinger, and Shinichiro Tomonaga—developed a more complete theory of QED that systematically eliminated the troublesome infinities through renormalization, and the theory agreed closely with experiment. But Dirac was unimpressed. When Dyson asked him what he thought of the new theory, Dirac bluntly said, “If these new ideas were not so ugly, I might think they were right.”

Dirac thought it was foolish to try to push forward particle physics before the interaction between photons and electrons was better understood. As he all but ignored new results on the weak and strong interactions, he drifted away from academic circles and his productivity declined sharply. In the late 1950s and early 1960s, as he tried to develop a quantum theory of gravity, he did important work on the Hamiltonian form of general relativity and on the quantum theory of bound states. These were weighty contributions, but in the eyes of most of Dirac's colleagues he was simply floundering in his own scientific backwater—a man to be respected but not to be listened to. In 1969, two years after retiring from his Lucasian professorship at Cambridge, he joined the physics department at Florida State University in Tallahassee and traveled the world, lecturing mainly on his philosophical approach to physics; he took pains to point out what he saw as the fatal flaws of QED and to urge younger colleagues to develop a revolutionary theory to replace the one he had co-discovered.

In his 1980 lecture "Engineers and Physicists," Dirac laid out his adamant opposition to QED. His view was rooted in his training as an engineer, and renormalization required something that no self-respecting engineer would approve of: ignoring infinite terms in a series of approximations to real, measurable quantities. In Dirac's view, ignoring infinite quantities in equations was absurd.

Other engineers might have taken a more practical approach—does it work, does it agree with experiment, and thus accepted the theory. But Dirac could not accept that because he was an exceptional engineer—an engineer with the heart of an exceptional pure mathematician.

“The main problem for an engineer is deciding which approximations to make,” he said. A good engineer makes wise, often intuitive choices about what terms to ignore. “The ignored terms must be small and must not affect the result much. He must not ignore quantities that are not small.”

Max Born and his young colleagues in the back garden of his home in Göttingen in the spring of 1927. Dirac is reading a newspaper intently, and J. Robert Oppenheimer, a friend he met in Germany, is also here (fourth from left). Image source: Courtesy of Florida State University, Paul AM Dirac Collection.

Principled and eccentric

Like great poetry, Dirac’s papers are worth rereading. Again and again, researchers have found that the ideas and insights in Dirac’s papers had little impact when they were first published. A classic example is his 1939 paper on the relationship between mathematics and physics, which still circulates among theoretical physicists at the Institute for Advanced Study (IAS) in Princeton. One of them, Nathan Seiberg, told me, “If the date on the front of the text was 2009 instead of 1939, it would be just as impressive.”

In one particularly striking passage, Dirac speculates about the initial conditions of the universe (even as late as 1939, he had accepted the theory begun by his student Fred Hoyle, which later became known as the Big Bang). Dirac pointed out that if the universe simply obeyed a trivial set of equations of motion given a set of initial conditions, it could not explain the complexity displayed by the rich variety of life on Earth, or even by the universe itself. Quantum mechanics, he argued, could attribute this complexity to quantum jumps in the very early universe. Dirac seemed to know that he had stumbled upon an important insight, which he summed up very distinctively in italics:

“The quantum jumps now form the uncalcu-lable part of natural phenomena, to replace the initial conditions of the old mechanistic view.”

“It was an amazing insight,” Nima Arkani-Hamed, a colleague of Seiberg’s at the IAS, told me. “Even though Dirac didn’t know the details of how the universe evolved, like modern theories of inflation, he got the core concepts exactly right. So he’s a bit like Darwin, who was able to come up with a theory of evolution by natural selection even though he didn’t know anything about the underlying genetics.”

Arkani-Hamid also highlights the technical value of Dirac’s papers to modern physicists, including string theorists. In the early 1970s, a younger generation of physicists developing string theory realized that they were following in Dirac’s footsteps. Not only had he proposed an extended model of matter as elementary particles, but in his theory of the quantization of constrained mechanical systems, he had also developed the techniques that theorists needed to understand the quantum dynamics of relativistic string theory. In the mid-1970s, when physicists sought to understand the properties of magnetic monopoles, which naturally occur in many modern theories of elementary particles, they found that Dirac had once again set the course in his 1931 and 1948 papers.[8]

Dirac at the Institute for Advanced Study in Princeton, circa 1958. He enjoyed doing manual labor such as chopping trees and helping to clear paths in the woods near the institute. Image credit: Courtesy of Monica Dirac

Dirac seems to have paid little or no attention to the early string theory papers, or to the mainstream work of physicists in the 1970s, namely the Standard Model. Disillusioned with QED, he focused on connecting general relativity to his hypothesis of large numbers. And he knew that many physicists viewed him as a principled but eccentric man. Although Dirac was unfazed, he could sometimes become demoralized. No doubt this was noticed by Princeton physicist John Wheeler, who wrote him a particularly thoughtful note on Dirac’s 80th birthday:

I am writing to tell you, and I am not sure you have guessed it, that many scholars, young and old, look up to you as a hero of integrity and beauty, a role model for what is right.[9]

Dirac kept the letter in his desk. Less than two years later, on October 20, 1984, he died of heart failure at his home in Tallahassee, with his wife and nurse by his bedside. He worked until the end, and his contributions to physics did not end with his death. Like all truly great thinkers, he continues to provide the world with a steady stream of creative energy after his death.

About the Author

Graham Farmelo is a theoretical physicist, biographer, popular science writer, and senior researcher at the Natural History Museum in London. His book The Strangest Man: The Hidden Life of Paul Dirac, Mystic of the Atom won the 2010 Los Angeles Times Science Book Award and the 2009 Physics World Book of the Year Award.

References

[1] Unless otherwise stated, this article is based on G. Farmelo, The Strangest Man: The Hidden Life of Paul Dirac, Mystic of the Atom, Basic Books, New York (2009).

[2] K. Gottfried, http://arxiv.org/abs/quant-ph/0302041v1, p.9.3.

[3] W. Heisenberg, Z. Phys. 33, 879 (1925).

[4] PAM Dirac, Proc. R. Soc. London, Ser. A 109, 642 (1925).

[5] PAM Dirac, Proc. R. Soc. London, Ser. A 117, 610 (1928).

[6] PAM Dirac, Proc. R. Soc. London, Ser. A 133, 60 (1931).

[7] PAM Dirac, Proc. R. Soc. Edinburgh, Sect. A: Math. Phys. Sci. 59, 122(1938-39).

[8] PAM Dirac, Phys. Rev. 74,817 (1948).

[9] I. Wheeler to P AM Dirac, 8 August 1982, General Correspondence Paul AM Dirac Collection, Paul AM Dirac Library Florida State University, Tallahassee.

This article is translated and published in Fanpu with permission from the American Institute of Physics (AIP).

Graham Farmelo; Paul Dirac, a man apart. Physics Today 1 November 2009; 62 (11): 46–50. https://doi.org/10.1063/1.3265236.

Reproduced from [Graham Farmelo; Paul Dirac, a man apart. Physics Today 1 November 2009; 62 (11): 46–50. https://doi.org/10.1063/1.3265236], with the permission of the American Institute of Physics.

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