# Paul Dirac

gigatos | February 21, 2022

## Summary

Paul Adrien Maurice Dirac (8 August 1902 (1902-08-08), Bristol – 20 October 1984, Tallahassee) was an English theoretical physicist and one of the creators of quantum mechanics. Winner of the 1933 Nobel Prize in Physics (together with Erwin Schrödinger).

Member of the Royal Society of London (1930), as well as a number of academies of sciences of the world, including member of the Pontifical Academy of Sciences (1961), foreign member of the USSR Academy of Sciences (1931), National Academy of Sciences of the USA (1949).

Dirac”s works are devoted to quantum physics, theory of elementary particles, general theory of relativity. He is the author of fundamental works on quantum mechanics (general theory of transformations), quantum electrodynamics (secondary quantization method and multitemporal formalism) and quantum field theory (quantization of coupled systems). His proposed relativistic equation of the electron allowed a natural explanation of spin and the introduction of the concept of antiparticles. Other well-known results of Dirac include the statistical distribution for fermions, the concept of the magnetic monopole, the hypothesis of large numbers, the Hamiltonian formulation of the theory of gravitation, etc.

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## Origins and Youth (1902-1923)

Paul Dirac was born on August 8, 1902, in Bristol into a teaching family. His father, Charles Adrien Ladislas Dirac (1866-1936), received a B.A. in literature from the University of Geneva and soon afterward moved to England. From 1896 he taught French at the Commercial School and Technical College of Bristol, which became part of Bristol University in the early twentieth century. Paul Dirac”s mother, Florence Hannah Holten (in addition to Paul is his older brother Reginald Felix (1900-1924, he committed suicide) and younger sister Beatrice (1906-1991). His father demanded that the family speak only French, which resulted in Paul”s character traits of reticence and a tendency to think alone. The father and children were registered as Swiss citizens and were not granted British citizenship until 1919.

At the age of 12, Paul Dirac became a high school student at the Technical College, whose curriculum had a practical and scientific orientation that fully matched Dirac”s aptitudes. In addition, his studies took place during the years of the First World War, which allowed him to get into high school faster than usual, from where many students went to war work.

In 1918 Dirac enrolled in engineering at Bristol University. Despite the fact that his favorite subject was mathematics, he repeatedly said that an engineering education gave him a lot:

I used to see sense only in exact equations. It seemed to me that if I used approximate methods, the work became unbearably ugly, whereas I was passionate about preserving mathematical beauty. The engineering education I received just taught me to come to terms with approximate methods, and I found that even in theories based on approximations one could see quite a lot of beauty… I found myself quite prepared to see all our equations as approximations, reflecting the existing level of knowledge, and to take them as a call to try to improve them. Had it not been for my engineering background, I probably would never have succeeded in my later work…

Dirac at this time was greatly influenced by his acquaintance with the theory of relativity, which was of great interest in society at that time. He attended lectures by Professor Brode, a professor of philosophy, from which he drew his initial knowledge of the field and which led him to pay close attention to geometrical ideas about the world. During his summer vacation, Dirac did an internship at a mechanical engineering plant in Rugby, but did not prove to be a good fit. So in 1921, after receiving a bachelor”s degree in electrical engineering, he failed to find a job. He also failed to continue his studies at Cambridge University: the scholarship was too small, and the Bristol authorities refused to support him financially, since Dirac had only recently taken English citizenship.

Dirac spent the next two years studying mathematics at Bristol University: members of the mathematics department invited him to attend classes unofficially. He was particularly influenced at this time by Professor Peter Fraser, through whom Dirac appreciated the importance of mathematical rigor and studied the methods of projective geometry, which proved a powerful tool in his later research. In 1923 Dirac passed his final examination with first-class honors.

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## Cambridge. Formalism of Quantum Mechanics (1923-1926)

After passing his examinations in mathematics, Dirac received a scholarship from Bristol University and a grant from the Bristol Education Department. He thus had the opportunity to enter graduate school at Cambridge University. He was soon admitted to St. John”s College. At Cambridge he attended lectures on a number of subjects which he had not studied at Bristol, such as Gibbs” statistical mechanics and classical electrodynamics, and he also studied Hamilton”s method of mechanics by reading Whittaker”s Analytic Dynamics.

He wanted to deal with the theory of relativity, but the famous theoretician Ralph Fowler, a specialist in statistical mechanics, was appointed his supervisor. It was to questions of statics and thermodynamics that Dirac”s first works were devoted; he also carried out calculations of the Compton effect, important for astrophysical applications. Fowler introduced Dirac to completely new ideas of atomic physics, which were put forward by Niels Bohr and developed by Arnold Sommerfeld and other scientists. Here is how Dirac himself recalled this episode in his biography:

I remember what a great impression Bohr”s theory made on me. I believe that the emergence of Bohr”s ideas was the most tremendous step in the history of quantum mechanics. The most unexpected, the most surprising thing was that such a radical departure from Newton”s laws yielded such remarkable results.

Dirac became involved in work on the theory of the atom, trying, like many other researchers, to extend Bohr”s ideas to multielectronic systems.

In the summer of 1925, Werner Heisenberg visited Cambridge and gave a talk on the anomalous Zeeman effect at the Kapitsa Club. At the end of his talk he mentioned some of his new ideas that formed the basis of matrix mechanics. However, Dirac did not pay attention to them at the time because of fatigue. At the end of the summer, while at his parents” house in Bristol, Dirac received by mail from Fowler a proof of Heisenberg”s article, but he could not immediately appreciate its main idea. It was not until a week or two later, returning to the article again, that he realized what was new in Heisenberg”s theory. Heisenberg”s dynamic variables did not describe a single Bohr orbit, but linked two atomic states and were expressed as matrices. The consequence of this was non-commutativity of variables, the meaning of which was not clear to Heisenberg himself. Dirac immediately understood the important role of this new property of the theory, which had to be interpreted correctly. The answer came in October 1925, already after his return to Cambridge, when Dirac during a walk came up with the idea of an analogy between the commutator and Poisson brackets. This connection made it possible to introduce the differentiation procedure into quantum theory (this result was stated in the article “The Fundamental Equations of Quantum Mechanics”, published at the end of 1925) and gave impetus to the construction of a coherent quantum-mechanical formalism based on the Hamiltonian approach. In the same direction Heisenberg, Max Born and Pasquale Jordan tried to develop the theory in Göttingen.

Subsequently, Dirac repeatedly noted Heisenberg”s crucial role in the construction of quantum mechanics. For example, prefacing one of the latter”s lectures, Dirac said:

I have the most compelling reason to be an admirer of Werner Heisenberg. We studied at the same time, were almost the same age, and worked on the same problem. Heisenberg succeeded where I failed. By that time, a huge amount of spectroscopic material had accumulated, and Heisenberg had found the right path in his labyrinth. By doing this, he ushered in a golden age of theoretical physics, and soon even a second-rate student was able to do first-rate work.

Dirac”s next step was to generalize the mathematical apparatus by constructing a quantum algebra for variables characterized by noncommutativity, which he called q-numbers. Examples of q-numbers are Heisenberg matrices. Working with such quantities, Dirac considered the problem of the hydrogen atom and obtained the Balmer formula. At the same time he tried to extend the algebra of q-numbers to encompass relativistic effects and peculiarities of multielectron systems, and he also continued his work on the theory of Compton scattering. His results were included in his Ph.D. thesis entitled “Quantum Mechanics,” which Dirac defended in May 1926.

By this time, the new theory developed by Erwin Schrödinger on the basis of ideas about wave properties of matter became known. Dirac”s attitude to this theory was at first not the most favorable, because, in his opinion, there already existed an approach that allowed obtaining correct results. However, it soon became clear that Heisenberg”s and Schrödinger”s theories are related and complement each other, so Dirac enthusiastically took up the study of the latter.

Dirac first applied it by considering the problem of a system of identical particles. He discovered that the type of statistics to which particles obey is determined by the symmetry properties of the wave function. Symmetric wave functions correspond to the statistics that was known by that time from the works of Chatyendranat Bose and Albert Einstein (Bose-Einstein statistics), while antisymmetric wave functions describe an entirely different situation and correspond to particles obeying the Pauli prohibition principle. Dirac studied the basic properties of these statistics and described them in his article “Toward a Theory of Quantum Mechanics” (August 1926). It soon turned out that this distribution had been introduced earlier by Enrico Fermi (for other reasons), and Dirac fully acknowledged its priority. Nevertheless, this type of quantum statistics is usually associated with the names of both scientists (Fermi – Dirac statistics).

In the same article “Toward a Theory of Quantum Mechanics” a time-dependent perturbation theory was developed (independently of Schrödinger) and applied to the atom in the radiation field. This allowed us to show the equality of Einstein”s coefficients for absorption and stimulated emission, but it was not possible to calculate the coefficients themselves.

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## Copenhagen and Göttingen. Theory of Transformations and Radiation Theory (1926-1927)

In September 1926, at Fowler”s suggestion, Dirac arrived in Copenhagen to spend some time at the Niels Bohr Institute. Here he became close friends with Paul Ehrenfest and Bohr himself, of whom he later recalled:

Bohr had a habit of thinking out loud… I was used to singling out from my reasoning those that could be written down in the form of equations, while Bohr”s reasoning had a much deeper meaning and went quite far from mathematics. I loved my relationship with Bohr, and … I can”t even estimate how much my work was influenced by hearing Bohr think aloud. <…> Ehrenfest always strove for absolute clarity in every detail of the discussion… In a lecture, at a colloquium, or at any event of this kind, Ehrenfest was the most helpful person.

While in Copenhagen, Dirac continued his work, trying to interpret his algebra of q-numbers. The result was a general theory of transformations that combined wave mechanics and matrix mechanics as special cases. This approach, analogous to the canonical transformations in classical Hamiltonian theory, made it possible to move between different sets of commuting variables. In order to be able to work with variables characterized by a continuous spectrum, Dirac introduced a new powerful mathematical tool, the so-called delta function, which now bears his name. The delta function was the first example of generalized functions, the theory of which was created in the works of Sergei Sobolev and Laurent Schwartz. The same article “Physical Interpretation of Quantum Dynamics,” presented in December 1926, introduced a number of notations that later became generally accepted in quantum mechanics. The theory of transformations constructed in the works of Dirac and Jordan allowed not to rely more on obscure considerations of the correspondence principle, but naturally introduced into the theory a statistical treatment of the formalism based on notions of probability amplitudes.

In Copenhagen Dirac began to deal with the theory of radiation. In “Quantum theory of emission and absorption of radiation” he showed its connection with Bose-Einstein statistics, and then, applying the quantization procedure to the wave function itself, he came to the method of secondary quantization for bosons. In this approach, the state of an ensemble of particles is given by their distribution over single-particle states defined by the so-called filling numbers, which change when the birth and annihilation operators act on the initial state. Dirac demonstrated the equivalence of two different approaches to considering the electromagnetic field, based on the notion of light quanta and on quantization of the field components. He also succeeded in obtaining expressions for Einstein”s coefficients as functions of the interaction potential and thus provided an interpretation of spontaneous radiation. In fact, this work introduced the notion of a new physical object – a quantum field, and the method of secondary quantization formed the basis for the construction of quantum electrodynamics and quantum field theory. A year later, Jordan and Eugene Wigner constructed a scheme of secondary quantization for fermions.

Dirac continued to study radiation theory (as well as dispersion and scattering theory) in Göttingen, where he arrived in February 1927 and spent the next few months. He attended Hermann Weil”s lectures on group theory and actively communicated with Born, Heisenberg and Robert Oppenheimer.

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## Relativistic quantum mechanics. Dirac equation (1927-1933)

By 1927, thanks to his pioneering work, Dirac had become widely known in scientific circles. This was evidenced by an invitation to the fifth Solvay Congress (“Electrons and Photons”), where he took part in the discussions. In the same year Dirac was elected to the board of St. John”s College, and in 1929 he was appointed senior lecturer in mathematical physics (however, he was not too burdened with teaching duties).

At that time Dirac was busy building an adequate relativistic theory of the electron. The existing approach based on the Klein-Gordon equation did not satisfy him: this equation includes the square of the time differential operator, so it could not be consistent with the usual probabilistic interpretation of the wave function and with the general theory of transformations developed by Dirac. His goal was an equation linear with respect to the differentiation operator and at the same time relativistically invariant. Several weeks of work led him to a suitable equation, for which he had to introduce 4×4 matrix operators. The wave function must also have four components. The resulting equation (Dirac equation) turned out to be quite successful, since it naturally includes the electron spin and its magnetic moment. The article “Quantum theory of the electron”, sent to the press in January 1928, also contained a calculation of the spectrum of the hydrogen atom based on the equation, which proved to be in perfect agreement with experimental data.

The same paper considered a new class of irreducible representations of the Lorentz group, for which Ehrenfest proposed the term “spinors. These objects interested “pure” mathematicians, and a year later Barthel van der Waarden published a paper on spinor analysis. It soon turned out that objects identical to spinors had been introduced by the mathematician Eli Kartan as early as 1913.

After the appearance of the Dirac equation it became clear that it contains one essential problem: in addition to the two states of the electron with different spin orientations, the four-component wave function contains two additional states characterized by negative energy. In experiments these states are not observed, but theory gives a finite probability of transition of the electron between states with positive and negative energies. Attempts to artificially exclude these transitions did not lead to anything. Finally, in 1930 Dirac took the next important step: he assumed that all states with negative energy are occupied (“Dirac”s sea”), which corresponds to a vacuum state with minimal energy. If a state with negative energy turns out to be free (“hole”), then a particle with positive energy is observed. When the electron moves to a state with negative energy, the “hole” disappears, i.e., annihilation occurs. It followed from general considerations that this hypothetical particle must be identical to the electron in everything, except for the opposite sign of the electric charge. At that time such a particle was not known, and Dirac did not dare to postulate its existence. Therefore, in “The Theory of Electrons and Protons” (1930) he suggested that such a particle is a proton, and its massiveness is due to the Coulomb interactions between electrons.

Weil soon showed that such a “hole” cannot be a proton, but must have the mass of an electron, for reasons of symmetry. Dirac agreed with these arguments and pointed out that then there must be not only a “positive electron,” or antielectron, but also a “negative proton” (antiproton). The antielectron was discovered a few years later. The first evidence of its existence in cosmic rays was obtained by Patrick Blackett, but while he was busy verifying the results, in August 1932 Carl Anderson independently discovered this particle, which was later called the positron.

In 1932, Dirac replaced Joseph Larmor as Lucas Professor of Mathematics (a post once held by Isaac Newton). In 1933, Dirac shared the Nobel Prize in Physics with Erwin Schrödinger “for the discovery of new forms of quantum theory. At first Dirac wanted to refuse, as he did not like to draw attention to himself, but Rutherford persuaded him, saying that with his refusal he would “make even more noise. On December 12, 1933, in Stockholm, Dirac gave a lecture on “The Theory of Electrons and Positrons,” in which he predicted the existence of antimatter. The prediction and discovery of the positron gave rise in the scientific community to the belief that the initial kinetic energy of some particles could be converted into resting energy of others, and led to a subsequent rapid increase in the number of known elementary particles.

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## Other works on quantum theory from the 1920s and 1930s

After trips to Copenhagen and Göttingen, Dirac developed a taste for traveling, visiting different countries and scientific centers. Since the late 1920s he gave lectures all over the world. For example, in 1929 he lectured at the University of Wisconsin and the University of Michigan in the United States, then crossed the Pacific Ocean with Heisenberg, and after lectures in Japan he returned to Europe by the Trans-Siberian Railway. This was not Dirac”s only visit to the Soviet Union. Thanks to his close scientific and friendly relations with Soviet physicists (Igor Tamm, Vladimir Fok, Pyotr Kapitsa and others) he visited the country many times (eight times in the pre-war period – in 1928-1930, 1932-1933, 1935-1937), and in 1936 he even took part in climbing Mount Elbrus. However, after 1937 he could not get a visa, so his next visits took place only after the war, in 1957, 1965 and 1973.

In addition to those discussed above, in the 1920s and 1930s Dirac published a number of papers containing significant results on various specific problems of quantum mechanics. He considered the density matrix introduced by John von Neumann (1929) and connected it with the Hartree-Fock method wave function (1931). In 1930 he analyzed the accounting of exchange effects for multi-electron atoms in the Thomas-Fermi approximation. In 1933, together with Kapitsa, Dirac examined the reflection of electrons from a standing light wave (the Kapitsa-Dirac effect), which was observed experimentally only many years later, after the appearance of laser technology. In his Lagrangian in Quantum Mechanics (1933), he proposed the idea of the path integral, which laid the foundation for the method of functional integration. This approach was the basis of the continuum integral formalism developed by Richard Feynman in the late 1940s, which proved to be extremely fruitful in solving problems of the theory of gauge fields.

In the 1930s Dirac wrote several fundamental works on quantum field theory. In 1932, in a joint article “Toward Quantum Electrodynamics” with Vladimir Fok and Boris Podolsky, he constructed the so-called “multitemporal formalism,” which made it possible to obtain relativistically invariant equations for a system of electrons in the electromagnetic field. This theory soon encountered a serious problem: divergences appeared in it. One of the reasons for this is the vacuum polarization effect predicted by Dirac in his 1933 Solvay paper and leading to a decrease of the observed charge of particles in comparison to their actual charges. Another reason for the appearance of divergences is the interaction of the electron with its own electromagnetic field (radiation friction, or self-excitation of the electron). Trying to solve this problem, Dirac considered the relativistic theory of the classical point electron and came close to the idea of renormalizations. The renormalization procedure was the basis of modern quantum electrodynamics, created in the second half of the 1940s in the works of Richard Feynman, Shinichiro Tomonagi, Julian Schwinger and Freeman Dyson.

An important contribution of Dirac to the dissemination of quantum ideas was the appearance of his famous monograph “Principles of Quantum Mechanics”, the first edition of which was published in 1930. This book was the first complete statement of quantum mechanics as a logically closed theory. The English physicist John Edward Lennard-Jones wrote on this subject (1931):

One famous European physicist who was fortunate enough to have a bound collection of Dr. Dirac”s original papers is said to have referred to it with reverence as his “bible. Those not so fortunate are now able to purchase an “authorized version” [i.e., a translation of the Bible approved by the church].

Subsequent editions (1935, 1947, 1958) contained significant additions and improvements to the presentation of the material. The 1976 edition differed from the fourth edition only with minor corrections.

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## Two unusual hypotheses: the magnetic monopole (1931) and the “big number hypothesis” (1937)

In 1931, in his article “Quantized singularities in the electromagnetic field,” Dirac introduced into physics the notion of a magnetic monopole, whose existence could explain the quantization of electric charge. Later, in 1948, he returned to this topic and developed a general theory of magnetic poles considered as ends of unobservable “strings” (singularity lines of vector potential). A number of attempts have been made to experimentally detect monopoles, but so far no definitive evidence of their existence has been obtained. Nevertheless, monopoles have become firmly embedded in modern theories of Great Unification and could serve as a source of important information on the structure and evolution of the Universe. Dirac monopoles were one of the first examples of using the ideas of topology to solve physical problems.

In 1937 Dirac formulated the so-called “hypothesis of large numbers”, according to which extremely large numbers (for example, the ratio of constants of electromagnetic and gravitational interactions of two particles) appearing in the theory must be related to the age of the Universe, also expressed by a huge number. This dependence must lead to the change of fundamental constants with time. Developing this hypothesis, Dirac put forward the idea of two time scales – atomic (enters into equations of quantum mechanics) and global (enters into equations of general relativity). These considerations may be reflected in the latest experimental results and theories of supergravity, introducing different dimensions of space for different types of interactions.

Dirac spent the academic year 1934-1935 at Princeton, where he met the sister of his close friend Eugene Wigner, Margit (Mancy), who came from Budapest. They married on January 2, 1937. Paul and Mancy had two daughters in 1940 and 1942. Mansi also had two children from her first marriage who took the surname Dirac.

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## Works on military subjects

After the outbreak of World War II, Dirac”s teaching load increased because of the shortage of staff. In addition, he had to take over the supervision of several graduate students. Before the war, Dirac tried to avoid such responsibility and generally preferred to work alone. It was not until 1930-1931 that he replaced Fowler as supervisor Subramanian Chandrasekar, and in 1935-1936 took on two graduate students, Max Born, who had left Cambridge and soon settled in Edinburgh. In all, Dirac supervised the work of no more than a dozen graduate students during his lifetime (mostly in the 1940s-50s). He relied on their independence, but when necessary he was ready to help with advice or answer questions. As his student S. Shanmugadhasan wrote,

Despite his “sink or swim” attitude toward students, I firmly believe that Dirac was the best supervisor one could wish for.

During the war, Dirac was involved in the development of methods of isotope separation, important from the point of view of atomic energy applications. Studies on the separation of isotopes in a gaseous mixture by centrifugation were conducted by Dirac together with Kapitsa as early as 1933, but these experiments stopped after a year, when Kapitsa was unable to return to England from the USSR. In 1941, Dirac began to collaborate with Francis Simon”s Oxford group, proposing several practical ideas for separation by statistical methods. He also gave a theoretical justification for the operation of the self-fractionation centrifuge invented by Harold Ury. The terminology proposed by Dirac in these studies is still in use today. In addition, he was an unofficial consultant to the Birmingham group, carrying out calculations of the critical mass of uranium, taking into account its shape.

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## Post-war activities. The last years

In the postwar period, Dirac resumed his activities, visiting various countries of the world. He gladly accepted invitations to work at such scientific institutions as the Princeton Institute for Advanced Study, the Institute for Basic Research in Bombay (where he contracted hepatitis in 1954), the National Research Council in Ottawa, and he lectured at various universities. However, sometimes unforeseen obstacles arose: for example, in 1954 Dirac was unable to obtain permission to come to the United States, which was apparently related to the Oppenheimer case and his pre-war visits to the Soviet Union. However, he spent most of his time in Cambridge, preferring to work at home and coming to his office mostly only for the purpose of communicating with students and university staff.

At this time Dirac continued to develop his own views on quantum electrodynamics, trying to rid it of divergences without resorting to such artificial tricks as renormalization. These attempts took several directions: one of them led to the concept of the “lambda-process,” another to a revision of the concept of the ether, etc. However, despite enormous efforts, Dirac never succeeded in achieving his goals and arriving at a satisfactory theory. After 1950, the most substantial concrete contribution to quantum field theory was the generalized Hamiltonian formalism for systems with bonds, developed in a number of papers. Later it allowed the quantization of Yang-Mills fields, which was of fundamental importance for the construction of the theory of gauge fields.

Another area of Dirac”s work was the general theory of relativity. He showed the validity of the equations of quantum mechanics in the transition to the space with GR metric (in particular, with de Sitter”s metric). In recent years he was engaged in the problem of quantization of the gravitational field, for which he extended the Hamiltonian approach to the problems of relativity theory.

In 1969, Dirac”s term as Lucas Professor ended. He soon accepted an invitation to take up a professorship at Florida State University in Tallahassee and moved to the United States. He also collaborated with the Center for Theoretical Studies in Miami, presenting the annual R. Oppenheimer Prize. His health weakened with each passing year, and in 1982 he underwent serious surgery. Dirac died on October 20, 1984, and was buried in a cemetery in Tallahassee.

Summarizing the life of Paul Dirac, it makes sense to quote Nobel laureate Abdus Salam:

Paul Adrien Maurice Dirac is without a doubt one of the greatest physicists of this and any other century. During three decisive years – 1925, 1926 and 1927 – with his three works he laid the foundations, first, of quantum physics in general, second, of quantum field theory and, third, of elementary particle theory… No other person, except Einstein, has had such a defining influence in such a short period of time on the development of physics in this century.

In evaluating Dirac”s work, an important place is occupied not only by the fundamental results obtained, but also by the very way in which they were obtained. In this sense, the notion of “mathematical beauty,” understood as the logical clarity and consistency of the theory, is of paramount importance. When Dirac was asked about his understanding of the philosophy of physics during a lecture at Moscow University in 1956, he wrote on the board:

Physical laws should have mathematical beauty. (Physical laws should have mathematical beauty).

This methodological attitude of Dirac was clearly and unambiguously expressed by him in the article devoted to the centennial of Einstein”s birth:

… one should be guided first and foremost by considerations of mathematical beauty, without giving much weight to discrepancies with experience. The discrepancies may well be caused by some secondary effects, which will become clear later. Although so far no discrepancy with Einstein”s theory of gravitation has been found, in the future such a discrepancy may appear. Then it will be explained not by falsity of initial assumptions, but by the necessity of further research and improvement of the theory.

For the same reasons Dirac could not put up with the way (renormalization procedure) by which it is customary to get rid of divergences in modern quantum field theory. The consequence of this was Dirac”s uncertainty even about the foundations of ordinary quantum mechanics. In one of his lectures, he said that all these difficulties

make me think that the foundations of quantum mechanics have not been established yet. Proceeding from modern foundations of quantum mechanics, people have spent enormous efforts to find by examples the rules of eliminating infinities in solving equations. But all these rules, in spite of the fact that results arising from them may agree with experience, are artificial and I cannot agree that modern foundations of quantum mechanics are correct.

Suggesting as a way out the trimming of integrals by replacing the infinite limits of integration by some sufficiently large finite value, he was ready to accept even the inevitable in this case relativistic invariance of the theory:

… quantum electrodynamics can be accommodated within a reasonable mathematical theory, but only at the cost of violating relativistic invariance. To me, however, this seems less evil than deviating from standard rules of mathematics and neglecting infinite quantities.

Dirac often spoke of his scientific work as a game with mathematical relations, considering the primary task to be the search for beautiful equations that could later be interpreted physically (he named the Dirac equation and the idea of the magnetic monopole as examples of the success of this approach).

Dirac paid much attention in his works to the choice of terms and notations, many of which turned out to be so successful that they have become a part of the arsenal of modern physics. As an example, we can mention key concepts of “observable” and “quantum state” in quantum mechanics. He introduced into quantum mechanics an idea of vectors in infinite-dimensional space and gave them familiar nowadays bracket designations (brackets and ket-vectors), introduced a word “commute” and designated commutator (quantum Poisson brackets) using square brackets, suggested terms “fermions” and “bosons” for two types of particles, named a unit of gravitational waves “graviton”, etc.

Even during his lifetime Dirac entered scientific folklore as a character of numerous anecdotal stories of varying degrees of authenticity. They allow to some extent to understand the peculiarities of his character: taciturnity, serious attitude to any topic of discussion, non-triviality of associations and thinking in general, desire for extremely clear expression of his thoughts, rational attitude to problems (even absolutely unrelated to the scientific search). He once gave a talk at a seminar; after finishing his presentation, Dirac turned to the audience, “Any questions?” – “I don”t understand how you got that expression,” said one of the attendees. “It”s a statement, not a question,” Dirac replied. – Any questions?”

He did not drink alcohol or smoke, was indifferent to food or comforts, and avoided attention to himself. Dirac was a long time non-believer, as reflected in Wolfgang Pauli”s famous joking phrase: “There is no God, and Dirac is his prophet. Over the years, his attitude toward religion softened (probably under the influence of his wife), and he even became a member of the Pontifical Academy of Sciences. In “The Evolution of Physicists” Views of the Picture of Nature,” Dirac concluded that:

Apparently, one of the fundamental properties of nature is that the basic physical laws are described by means of a mathematical theory that has so much finesse and power that it takes an extremely high level of mathematical thinking to understand it. You may ask: Why does nature work this way? You can only answer that our current knowledge shows that nature seems to be arranged in this way. We simply have to agree with it. In describing this situation, we can say that God is a very high-class mathematician and used very sophisticated mathematics in his construction of the universe.

“I have a problem with Dirac,” Einstein wrote to Paul Ehrenfest in August 1926. “This balancing on the dizzying edge between genius and madness is terrible.

Niels Bohr once said: “Of all physicists, Dirac has the purest soul.

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## Main articles

**Sources**