# Paul Dirac is the best physicist of all time

At Moscow University it is customary for internationally renowned physicists to leave a motto on the blackboard after their guest lecture. Niels Bohr (1885 to 1962, Nobel Prize 1922), the founder of the quantum theory of the atom, chose the motto of his principle of complementarity: "Contraria non contradictoria sed complementa sunt" (opposites do not contradict each other, but complement each other). Hideki Yukawa (1907 to 1981, Nobel Prize 1949), pioneer of the modern theory of strong nuclear power, noted: “In principle, nature is simple.” Paul Dirac (1902 to 1984, Nobel Prize 1933) decided on “A physical law must be mathematically beautiful be."

In the American edition of this magazine, Dirac wrote almost exactly 30 years ago: “God is a most brilliant mathematician. He built the universe according to profound and subtle mathematical laws ”(see“ The Evolution of the Physicist’s Picture of Nature ”, Scientific American, May 1963, page 45). Dirac was probably more concerned than any other modern physicist, inspired by his colleague Albert Einstein (1879 to 1955, Nobel Prize 1921) and the mathematician Hermann Weyl (1885 to 1955), with the concept of “mathematical beauty” as one inherent in nature Property and as a methodological aid for their scientific exploration. For him, “a mathematically beautiful theory was more correct than an ugly one that agrees with certain experimental results”.

His unusual focus on the aesthetics and logic of mathematical physics as well as his legendary silence and introversion made Dirac an eccentric among the great natural scientists of this century (Fig. 1 and box on pages 86 and 87). After an extraordinarily successful phase at the beginning of his scientific career, his extreme rationalism unfortunately led him to find himself on sterile side roads. While between the ages of 23 and 31 he not only developed an independent and extremely versatile formulation of quantum mechanics, but also a quantum theory of the emission and absorption of radiation by atoms (a first simple but important version of quantum electrodynamics), the relativistic wave equation of the Electrons, the concept of antiparticles, as well as a theory of magnetic monopoles, none of his later works had a similarly revolutionary character, and few of these were of lasting value.

**The ingenious loner**

Paul Adrien Maurice Dirac, born on August 8, 1902 in Bristol (England) as the second of three children, grew up in family relationships that today would be described as disturbed. His father was stressful.

Charles Adrien Ladislas Dirac had emigrated from Switzerland to England around 1890, where he married Florence Hannah Holten, the daughter of a captain. He worked as a French teacher at the Merchant Venturers Technical College in Bristol, where he was notorious for being obsessed with discipline. His family also suffered under his strict regime. By showing no feelings and equating parental love with discipline and order, he kept his children trapped in a climate of domestic tyranny and deprived them of all social and cultural life; he only spoke to his son when he addressed him in French.

Unwilling or unable to rebel, Paul sought protection in silent isolation and distanced himself from his father. Those unhappy years marked him for life. When his father died in 1936, Paul felt no grief. "I feel much freer now," he wrote to his future wife.

Fortunately, Paul had a rich spiritual world to retreat into. He showed a talent for mathematics at an early age. At the age of twelve he switched to the technical center where his father taught and which - unlike most secondary schools of the time - did not offer classical education with Latin and Greek, but courses in natural sciences, modern languages and practical subjects. This came in handy for Dirac because, as he himself said, "did not particularly appreciate the value of ancient cultures".

After graduation, he attended another school in the same building, Bristol University's Engineering School. There he took electrical engineering - not out of particular passion, but because he thought his father would like it.

The curriculum was little more than applied physics and mathematics. Nevertheless, Dirac was also enthusiastic about Einstein's new ideas of space, time and gravity, the special and general theory of relativity; he soon got used to her.

When Dirac graduated with honors in 1921, he was threatened with unemployment due to the economic depression of the post-war period. The salvation was a scholarship to study math in Bristol; and in the fall of 1923 he was able to transfer to St. John's College, Cambridge University, where he did applied mathematics and theoretical physics. At that time, such important scholars worked at the traditional university as the physicists Joseph Larmor (1857 to 1942) and Joseph John Thomson (1856 to 1940, Nobel Prize in Physics 1906), Ernest Rutherford (1871 to 1937, Nobel Prize in Chemistry 1908) and the astronomers Arthur Stanley Eddington (1882 to 1944) and James Jeans (1877 to 1946), but also the aspiring scientists James Chadwick (1891 to 1974, Nobel Prize in Physics 1935), Patrick Blackett (1897 to 1974, Nobel Prize in Physics 1948), the mathematician Ralph Fowler (1889 to 1944), the astrophysicist Edward A. Milne (1896 to 1950), the mathematician and theoretical physicist Douglas R. Hartree (1897 to 1956) and Pjotr Kapitza (1894 to 1984, Nobel Prize in Physics 1978).

Dirac was supervised by Fowler, who introduced him to the theory of atoms, which was new to him, and to statistical mechanics. He later wrote about these years: “I was completely focused on the scientific work. Only on Sundays did I relax and when the weather was nice I went for long, lonely walks in the country. "

Just six months after arriving in Cambridge, Dirac published his first scientific paper; in the following two years ten more publications were added. By the time he received his doctorate in May 1926, he had already developed his own formulation of quantum mechanics and had given the first lecture on this subject offered at a British university; and in 1933, just 31 years old, he and the Austrian Erwin Schrödinger (1887 to 1961) were awarded the Nobel Prize in Physics for his “discovery of new and fruitful forms of atomic theory ... and its applications”.

Dirac's eight most fertile years began on one day in August 1925 when Fowler gave him the proofs of an article by Werner Heisenberg (1901 to 1976, Nobel Prize 1932). In it, this 23-year-old theoretical physicist created the mathematical basis for a revolutionary theory of atomic phenomena, which soon became known as quantum mechanics (see "Werner Heisenberg and the principle of indeterminacy" by David C. Cassidy, Spectrum of Science, July 1992, page 92). Dirac understood at once that this work described the world of the very young in an entirely new way. During the following year, building on this, he came up with an independent formulation of quantum mechanics, the so-called q-number algebra (after English quantum numbers, the term Diracs for operators with which one can observe physical quantities - the observables - such as position, Represents momentum and energy).

Although Dirac's work quickly gained general recognition, many of his results were derived independently of him and almost simultaneously by a group of theoretical physicists working in Germany, including Heisenberg, Max Born (1882 to 1970, Nobel Prize 1954), Wolfgang Pauli (1900 to 1958 , Nobel Prize 1945) and Pascual Jordan (1902 to 1980). These were Dirac's immediate competitors.

Born, Heisenberg and Jordan developed Heisenberg's basic scheme using the matrix notation in the so-called matrix mechanics. In the spring of 1926, Schrödinger then presented another quantum theory, wave mechanics. This led to the same results as with the more abstract theories of Heisenberg and Dirac, but it simplified the calculations. Many physicists already suspected at that time that the three systems were different representations of a more general theory of quantum mechanics.

During a six-month stay at the Institute for Theoretical Physics in Copenhagen, headed by Bohr, Dirac finally found this general theory that so many scientists had hoped for - a framework that united all special cases and within which the rules for the transition from one form of representation to the others were set. This Diracian transformation theory formed - together with a similar theory developed by Jordan at about the same time - the basis for all later developments in quantum mechanics.

On December 26, 1927, the English physicist Charles G. Darwin (a grandson of the founder of the theory of evolution) wrote to Bohr: “A few days ago I was in Cambridge and met Dirac there. In the meantime he has developed a completely new system of equations for the electron that correctly reproduces the spin in all cases and appears to be 'the right thing'. His equations are first-order, not second-order differential equations! "

Dirac's equation for the electron was indeed “the right one”, since it simultaneously fulfilled the requirements of the special theory of relativity and took into account the experimentally observed spin, the intrinsic angular momentum of the electrons. This can take one of two values, +1/2 or -1/2 (which are also referred to as up and down). Schrödinger's original equation had failed here because it was non-relativistic; and even with its relativistic extension, the Klein-Gordon equation, the spin could not be described.

The formulation with first-order differential equations (i.e. only with first derivatives), which so impressed Darwin, was essential for two reasons: On the one hand, Dirac wanted to retain the formal structure of the Schrödinger equation, in which the first derivative occurs after time; on the other hand, he had to satisfy the theory of relativity, according to which space and time are equal. Dirac succeeded in fulfilling these two requirements in an elegant and functional way at the same time: if he applied the new equation to an electron moving in an electromagnetic field, the correct value for the electron spin resulted as if by itself.

This derivation of a physical property from fundamental principles impressed the specialist colleagues so much that they spoke of a "true miracle". During the investigation of the properties of Dirac's equation, spinor analysis (a versatile mathematical tool with which problems in almost all areas of physics can be investigated) and the relativistic wave equations for particles with a spin other than +1/2 or -1 emerged / 2 is. Another success was its application to the hydrogen atom, through which the spectral lines observed in the experiment could be reproduced exactly (box on page 89). Less than a year after its publication, the Dirac equation had become what it is today: a cornerstone of modern physics.

**From the hole theory to antimatter**

Dirac was not only an advocate of mathematical logic, he was also a master of inspiration. Nowhere were these seemingly contradicting abilities more evident than in the development of his so-called hole theory in the years 1929 to 1931, which revealed a previously neglected area of physics.

The starting point was Dirac's finding that his equation provided four solutions, two of which belonged to particles with positive energy and the spin +1/2 or -1/2 - namely the electrons - while the other two solutions were states of negative energy with this spin described. Such particles would have to have very strange properties. What's more, even for particles with positive energy, it shouldn't be unusual to fall into states of negative energy - which would mess up our world!

Towards the end of 1929, Dirac found an explanation for this oddity. He imagined the vacuum as a kind of uniform sea of states with negative energy, all of which were filled with electrons. Since - as Pauli had already recognized in 1925 - each quantum state can be occupied by only one single electron, the electrons of positive energy would then always have to be above this lake and thus form excited states, as it were. It should also be possible to transfer an electron of negative energy into an excited state by supplying a sufficiently large positive energy from the Dirac Sea. A hole would then remain in the lake, so to speak, into which another electron with negative energy could fall. Formally, such a hole could be interpreted as a particle of positive energy, but with a charge opposite to the electron: "These holes have positive energy and therefore behave like normal particles in this respect," wrote Dirac (Fig. 2).

But which particle should it be? At that time either the proton or a positively charged electron appeared possible. However, there were two serious problems with the proton. For one thing, one would expect that an electron might occasionally fall into a hole, with the two particles annihilating each other in a flash of gamma rays; however, such proton-electron annihilations had never been observed. On the other hand, the particle in question should be identical to the electron in all properties except the electrical charge - the proton, however, has about 2000 times the mass of the electron.

However, Dirac initially preferred the proton for the sake of simplicity. This and the ordinary electron were the only two known elementary particles in 1930, and Dirac did not feel comfortable introducing an additional one for which there was no experimental evidence. If he had also been able to interpret protons as negative energy states left by the electrons, the number of elementary parts would have been reduced to one - the electron. Such a simplification would be "the dream of the philosophers," as Dirac put it.

Because of the obvious discrepancies, however, he soon had to discard this interpretation. In May 1931 he reluctantly declared that the hole was an anti-electron, "a new, not yet experimentally observed particle that has the same mass as an electron but the opposite charge". Because of the complete symmetry between positive and negative charges in his theory, he consequently had to acknowledge the existence of an antiparticle for the proton. Dirac had thus doubled the number of elementary particles and given rise to speculation about the existence of parallel antimatter worlds.

Dirac also claimed the existence of another particle that, analogous to the electrical charge of an electron or proton, would have to carry an isolated magnetic charge - the magnetic monopole. A conclusive experimental proof of this is still pending today.

In September 1932 Dirac was elected to the Lucasian Chair in Mathematics at Cambridge University, which Isaac Newton once held for 30 years. (Dirac retained the professorship, which is currently held by Steven W. Hawking, for 37 years.) That same month, a young experimental physicist from the California Institute of Technology in Pasadena, Carl D. Anderson, submitted a paper to Science magazine, in which he described the discovery of a “positively charged particle with a mass comparable to that of an electron” in cosmic radiation. Although Anderson had taken his measurement for reasons other than checking Dirac's theory, the new particle - the positron - was generally equated with the anti-electron. On the occasion of receiving the Nobel Prize in December 1933, Dirac, then 31 years old, gave a lecture on the “theory of electrons and positrons”. Three years later, Anderson, also at the age of 31, received the Nobel Prize in turn for finding the hypothetical antiparticle of the electron (together with Anderson, the Austrian Victor Hess was awarded for the discovery of cosmic radiation in 1912).

**Search for an alternative quantum electrodynamics**

Quantum electrodynamics (QED) is a special quantum field theory that describes the electromagnetic interactions between elementary particles. By the mid-1930s all attempts to formulate a satisfactory relativistic form of this theory had reached a dead end; apparently fundamentally new physical ideas were necessary.

Dirac had already made groundbreaking contributions to QED in the late 1920s.He was aware of the formal inadequacies of the theoretical framework, which had been built mainly around a theory developed by Heisenberg and Pauli in 1929, which he considered illogical and simply "ugly". In addition, the calculations repeatedly generated diverging integrals ("infinities") to which no physical meaning could be assigned. Therefore Dirac developed an alternative theory in 1936, in which the law of conservation of energy was violated. Although this radical proposal was soon refuted experimentally, Dirac continued to criticize the Heisenberg-Pauli theory and searched almost obsessively for a more suitable solution. In 1979, looking back on his career, he wrote: "All my life I have been looking for better equations for quantum electrodynamics - so far without success, but I am still working on them."

A logical starting point for developing a better QED would have been to use an improved classical theory of the electron. Dirac followed this strategy and in 1938 put forward a classical, relativistic theory of the electron that was considerably better than that developed by the Dutch physicist Hendrik Antoon Lorentz (1853 to 1928, Nobel Prize 1902) shortly before the turn of the century. It provided an exact equation of motion for an electron assumed to be punctiform. Since neither infinities nor other undefined expressions appeared in this theory, it seemed possible to derive a divergence-free QED from it. However, the transformation into a satisfactory quantum mechanical version turned out to be considerably more difficult than expected. Dirac devoted itself to this problem for more than 20 years - in vain.

In 1947 and 1948 a new theory of quantum electrodynamics emerged that solved the problem of infinity in a practical sense. Its founders, Sin-itiro Tomonaga from Japan as well as Richard Feynman, Julian Schwinger and Freeman Dyson from the USA proposed a so-called renormalization, in which one essentially uses the infinite expressions following from the theory for the experimentally determinable quantities of electron mass and - charge subtracts infinite contributions in such a way that finite quantities remain. With this formal procedure one could make extremely accurate predictions. The numerous empirical successes of this theory mean that renormalization in QED has become the standard procedure.

Dirac, however, rejected this approach because it was just as "complicated and ugly" as the older approach by Heisenberg and Pauli. A theory that works with specially introduced mathematical tricks without a direct physical basis could not be good, he said, no matter how precisely it describes the experimental results - an objection that was largely ignored. Towards the end of his life he finally had to admit that not only was he largely isolated in the physicist community, but also that none of his numerous suggestions for improving QED had been successful.

Dirac's search for an alternative quantum field theory yielded some important side results. One was the important classical theory of the electron mentioned above. Another was a new notation that elegantly introduced the mathematics of vector spaces (also called Hilbert spaces) into quantum mechanics. This symbolism with so-called bra and ket vectors (after the English bracket) became generally known through the third edition of his influential textbook "The Principles of Quantum Mechanics", published in 1947, and has since become the preferred mathematical notation in quantum mechanics.

**The unsuccessful late years**

Dirac mostly worked on very specific questions in quantum theory. So it was rather unusual when he turned to cosmology in 1937 and developed a particular model of the universe. He was inspired to do this by two of his former teachers in Cambridge, Milne and Eddington, as well as by discussions with the talented young Indian astrophysicist Subrahmanyan Chandrasekhar (Nobel Prize 1983), whose doctoral thesis Dirac temporarily supervised.

Eddington had begun the attempt in the early thirties to derive the values of the natural constants by linking quantum theory with cosmology (see "Arthur Stanley Eddington" by Sir William McCrea, Spectrum of Science, December 1992, p. 82). This search for a really "fundamental theory", as Eddington called it, led him more and more to metaphysical speculation, which not only "paralyzed reason, but also poisoned judgment," as one critic wrote. Dirac was skeptical of Eddington's imaginative statements, but he was impressed by his philosophical approach to natural science - which was based on purely mathematical logic - and his belief in a fundamental connection between micro- and macrocosms.

In his first contribution to cosmology, Dirac concentrated on the very large, pure, i.e. dimensionless numbers that result when universal constants - for example the speed of light, Planck's quantum of action, gravitational constant and the charges and masses of electron and proton - are combined in such a way that their units cancel each other out. He was convinced that only these large numbers had a deep meaning in nature.

For example, Dirac noticed that the ratio of electrostatic to gravitational attraction between a proton and an electron - a value of about 1039 - corresponds to the age of the universe (assumed at the time), if it is expressed in suitable time units - e.g. as a multiple of time, that light needs to traverse the classic electron diameter.

Dirac was aware of some such relationships between large, pure numbers. He did not see them as simply coincidental coincidences, but held them to be the basis of an important new cosmological principle, which he called the hypothesis of large numbers: “There is a simple mathematical connection between every two of these very large, naturally occurring dimensionless numbers , in which the coefficients are of the order of magnitude. ”From this principle he concluded (with strong contradiction of his colleagues) that the gravitational constant g is not invariable, but inversely proportional to the age of the universe and therefore steadily decreasing.

By 1938 Dirac had derived several empirically verifiable conclusions from his hypothesis and sketched a model of the universe based on this principle. Most physicists and astronomers, increasingly angry with Dirac's rationalistic view of cosmology, rejected his ideas. It was not until the 1970s that Dirac dealt again with cosmology, essentially falling back on his original theory. He defended it and the assumption of a time-varying gravitational constant against objections based on observational results, and tried to incorporate new discoveries such as cosmic background radiation into his model. However, these efforts also went largely unnoticed, so that Dirac continued to be an outsider in both cosmology and QED.

**Away from the family**

Dirac was, as it were, married to his work and was long regarded by his colleagues as a die-hard bachelor. His marriage in 1937 to Margit Wigner, the sister of the Hungarian physicist Eugene Wigner, came as a surprise to many. Margit was widowed and brought a son and a daughter into the marriage; she and Paul had two other daughters.

Of course, Dirac had little sense in family life. "It is the irony of life that Paul suffered so badly from his father, who had the same difficulties with his family," Margit once wrote. “Paul wasn't a tyrannical father, but he kept too far a distance from his children. That history repeats itself is all too true with the Diracs. "

Dirac was never interested in art, music or literature, and he rarely went to the theater. His only hobbies were traveling and mountain tours. He was tireless when hiking; his persistence often surprised those who only knew him from conferences and receptions. His travels have taken him around the world three times and he conquered some of the highest mountains in Europe and America.

In September 1969, Dirac retired from the Lucasian Chair. The following year he and Margit decided to move to warm Florida, where he held the post of professor emeritus at Tallahassee State University from 1971. He continued to be active and attended many conferences until his health began to deteriorate. He died in Tallahassee on October 20, 1984.

From: Spektrum der Wissenschaft 7/1993, page 84

© Spektrum der Wissenschaft Verlagsgesellschaft mbH

#### This article is contained in Spectrum of Science 7/1993

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