# Fok, Vladimir Aleksandrovich

# FOK, VLADIMIR ALEKSANDROVICH

(*b* St. Petersburg, Russia, 22 December 1898; *d*. Leningrad, U.S.S.R, 27 December 1974)

*physics*.

Fok (often spelled Fock) was the son of Aleksandr Aleksandrovich Fok, a forestry specialist whose works were well known at the beginning of the twentieth century, and of Nadezhda Alexeevna Fok. He graduated from the Practical School of the Reformation Churches in Petrograd with a gold medal in 1916 and then enrolled in the Faculty of Physics and Mathematics of Petrograd University. The following year he was called to military service and, after graduating from an accelerated course at the Artillery School, was sent to the front. He was back at the university in 1919, and that same year began work as a calculator at the State Optical Institute, which was directed by Dmitri Rozhdestvenskii. In 1922 Fok graduated from the university. In 1934 he received a doctor of science degree. The beginning of his scientific activity coincided with the dawn of Soviet physics and took place in poor conditions because there were almost no physics scholars in the country. Fok worked simultaneously in various scientific and educational institutions. In addition to his research in the department of theoretical physics of the State Optical Institute (1919–1923, 1928–1941), Fok was a researcher at the Leningrad Institute of Physics and Technology (1924–1936). He also taught at Leningrad University (1924–1974), becoming a professor there in 1932, and during the same period he conducted a seminar in experimental mathematics at the Faculty of Physics and Mechanics, Leningrad Polytechnical Institute. At this seminar students tried to solve actual problems of applied physics. Fok did research at the Lebedev Physics Institute (1944–1953) and at the Institute for Physical Problems (1954–1964), both part of the Soviet Academy of Sciences in Moscow.

Fok’s first paper, written while he was at the State Optical Institute(1924), concerned the theory of luminosity of surfaces of arbitrary shape. Fok was also involved in investigating the mathematical aspects of the process of optical-glass preparation. At the Institute of Physics and Technology, Fok worked out a rigorous theory of thermal breakdown of dielectrics and calculated the thermal resistance of a multicore cable. While serving as a consultant for various institutions, Fok solved the problem set by Joseph Lagrange (which proved to be too hard for G.F Bernhard Riemann): the problem of gas pressure within an artillery barrel before the ejection of the shell. Fok was also involved in theoretical aspects of geological surveys and electrical methods of prospecting for minerals, and he carried out investigations in the theory of elasticity and other problems of mathamatical physics. During World War II Fok performed ballistics calculations.

A number of Fok’s papers are purely mathematical (concerning integral equations, Bessel functions, and the theory of analytic functions and Airy functions. the latter tabulated by Fok in 1945). These papers are complemented by a series of investigations on the theory of diffraction and approximate solutions to some particular problems of radio-wave diffractions. He worked out how to determine the field in the region of half-shadow, or the so-called principle of locality. In his rigorous theory of diffraction of radio waves around the earth surface (1946), Fok took into account the nonhomogeneity of the atmosphere and the earth’s surface. The mathematical techniques developed by the Italian physicist Tullio Regge for the description of the scattering of elementary particles (the method of complex momenta) resemble those constructed by Fok for the solution of similar problems (spherical functions with a complex index used for the solution of diffraction problems).

Fok’s reputation rests mainly on his work on quantum mechanics, begun in the late 1920’s. In 1926 he generalized the SchrÖdinger wave equation to the relativistic case by replacing the momentum in the relativistic equation linking energy, momentum, and mass by the gradients. (The problem was solved independently by the Swedish physicist Oskar Klein, and the relativistic scalar equation describing particles with spin in an electromagnetic field was named the Klein-Fok equation.) Fok was also involved in quantum mechanics of multiparticle systems. Developing Douglas Hartree’s approach (1927) and using Wolfgang Pauli’s principle, Fok worked out the technique called the Hartree-Fok method (1930). The wave function of electrons in the atom is represented as a determinant of one-electron function, the latter being determined by a standard variational procedure. Theresulting self-consistent solution automatically takes into account correlations of the orbital electrons associated with their mutual exchange. Fok’s system of equations of a selfconsistent field with exchange is successfully used not only for calculations of multielectron atoms but also for all multiparticle problems of quantum mechanics, including the theory of superconductivity (for the latter, Fok’s equations have been generalized by Nikolai Bogolubov).

In 1931 Fok published his book *Fundamentals of Quantum Mechanics*, which played an important role in familiarizing Soviet physicists with a domain of physics already being explored in this country.

Between 1928 and 1934, Fok obtained important results in the quantum field theory. He completed the mathematical method of secondary quantization proposed by Paul A.M. Dirac and developed by Pascual Jordan and Eugene Wigner. It has been shown that this method does not go beyond the traditional framework of quantum mechanics (as Jordan argued) and that the two ways of description are completely equivalent. As early as 1934 Fok suggested describing a system with a variable number of particles (bosons) in the representations of the secondary quantization with the aid of a generating functional(“Fok’s functional”).In his papers Fok introduced a number of new notions. such as “the Fok space” (a Hilbert space in the representation of secondary quantization). In papers written with Dirac and Boris Podolsky (1932), Dirac’s results on interaction of charged particles by the exchange of virtual photons were extended. The one-dimensional problem solved by Dirac was generalized to the realistic three-dimensional case: the multitime formalism of Dirac, Podolsky, and Fok was introduced; and quantum electrodynamics was formulated in its modern form.

Fok also studied problems of the general theory of relativity. In 1939 he solved the problem of motion of a many-body system within the framework of this theory. Fok demonstrated that for the finite masses(not “point masses”) the general equations of the theory of gravitation (specifically, their compatibility conditions) make it possible to obtain the law of universal gravitation and Newton’s equations of motion without any additional considerations. Fok took into account the corrections to Newton’s equations of gravitation in the second approximation of Einstein’s theory. He summarized the results of his investigations in a special monograph(1955).

Fok paid a great deal of attention to the philosophical aspects of quantum mechanics (especially from the late 1940’s on) and of the theory of relativity. He was among those Soviet physicists (including A. Ioffe, Ya. Frenkel, L. Landau, I. Tamm, and S. Vavilov) who struggled against the vulgarization of both these great theories by some Soviet philosophers. In the postwar period Fok developed his own ideas on the interpretation of quantum mechanics and discussed them with Niels Bohr.

Fok loved literature and poetry, and wrote many facetious poems that became popular among physicists, including some dedicated to Bohr and Born that were written in Russian and in German. Fok wrote other lyrical poetry as well.

In 1932 Fok was elected as corresponding member, and in 1939 as full member, to the Soviet Academy of Sciences. In 1936 he received the Mendeleev Prize; in 1946, the State Prize; and in 1960, the Lenin Prize. In 1968 Fok was awarded the title Hero of Socialist Labor. He was a foreign member of the Norwegian and Danish Academies of Sciences and the Deutsche Akademies of Sciences and the Deutsche Akademie der Wissenschaften in Berlin, and held honorary degrees from the universities of New Delhi, Leipzig, and Michigan.

## BIBLIOGRAPHY

1. Original Works. “Zur Berechnung der Beleuchtigungsstärke,” in *Zeitschrift für Physik*, **28** , no. 2 (1924), 102–113; “Uber die invariant From der Wellen-und die Bewegungsgleichungen für einen geladenen Massenpunkt,” *ibid*., **39** , nos. 2–3 (1926), 226–232; “Zur Wärmetheorie des elektrischen Durchschlages,” in *Archiv für Elektrotechnik*, **19** , no. 1 (1927), 71–81; “Näherungsmethode Zur Lösung des quantenmechanischen Mehrkörperproblems,” in *Zeitschrift für Physik*, **61,** nos. 1–2 (1930), 126–148 “On Quantum Elektrodynamics,” in *Physikaliche Zeitschrift der Sowjetunion*, **2** , no. 6 (1932), 468–479, written with P. A. M. Dirac and Boris Podolsky; and “Zur Quantenelektrodynamik,” *ibid*., **6** , no. 5 (1934), 425–469.

*Nachala kvantovoi mekhaniki* (Moscow, 1931; 2nd ed., Moscow, 1976), trans. and rev. by Eugene Yankovsky as *Fundamentals of Quantum Mechanics* (Moscow, 1978;) *Teoriya Katorazha* (“Theory of [well] Logging”; Moscow and Leningrad, 1933); “Sur le mouvement des masses finies d’après la théories de gravitation Einsteinienne,” in *Journal of Physics* (Moscow), **1** , no. 2 (1939), 81–116; “Diffraction of Radio Waves Around the Earth’s Surface,” in *Journal of Physics* (Moscow), **9** , no. 4 (1945), 255–266; “New Methods in Diffraction Theory,” in *Philosophical Magazine, Vremeni i tyagoteniya* (Moscow, 1955), trans. by N. Kemmer as *The Theory of Space, Time and Gravitation* (London and New York, 1959; 2nd rev. ed., Oxford and New York, 1964); and “Quantum Physics and Philosophical Problems,” in *Foundations of Physics***1** , no. 4 (1971), 293–306.

II. Secondary Literature. “Sbornik statei posvyashchennikh 80-letiyu so dnya rozhdeniya V. A. Foka” (“Collection of Essays Dedicated to V. A. Fok on on the Occasion of the 80th Anniversary of His Birth”), in *Trudy GOI*, **43** no. 177 (1978), includes A bibliography of Fok’s work; M. G. Veselov, “Vladimir Aleksandrovich Fok,” in *Uspekhi fizicheskikh nauk*, **66** , no. 4 (1958), 695–699; M. G. Veselov, G. F. Drukarev, and Yu. V. Novozhilov, “Vladimir Aleksandrovich Fok,” in *Uspekhi fizicheskikh nauk*, **96** , no. 4 (1968) 741–743, trans. by E. Bergman in *Soviet Physics Uspekhi*, **11** , no. 6 (1969) 921–923; and M. G Veselov, P. L. Kapitsa, and M. A. Leontovich, “Pamyati Vladimira Aleksandrovicha Foka,” in *Uspekhi fizicheskikh nauk*, **117** , no. 2 (1975), 375–376, trans. as “In Memory of Vladimir Aleksandrovich Fok” by R. W. Bowers in *Soviet Physics Uspekhi*, **18** , no. 10 (1975), 840–841.

V. J. Frenkel

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