# Klein, Oskar Benjamin

# KLEIN, OSKAR BENJAMIN

(*b*. Mörby, Sweden, 15 September **1894;** *d.* Stockholm, Sweden, 5 February 1977)

*theoretical physics*.

Klein was the third and youngest child of Gottlieb Klein, Sweden’s first rabbi, who immigrated from Homonna, a small town in the southern Carpathians, via Eisenstadt (near Vienna) and Heidelberg, and of Toni Levy. From an early age his interest in the natural sciences was apparent; he began given to collecting small animals, observing the stars, and reading popular scientific books. Later he performed chemical experiments and studied such works as Darwin’s *On the Origin of Species*.

On the occasion of a peace conference held in the summer of 1910 at Stockholm, fifteen-year-old Oskar was introduced by his father to the chemist Svante Arrhenius, who subsequently invited the boy to work in his laboratory at the Nobel Institute on the solubility of salts with radioactive indicators. The results of this investigation were published the following year. Under the guidance of Arrhenius, Klein was introduced to more specialized scientific literature when he began his university studies. Klein considered Hendrik A. Lorentz’s *Leeboek der differentiaalen integraalrekening*, which was also available in German, as having been important to his scientific development.

In the spring of 1914, Klein arranged to stay in Germany and France. However, the outbreak of war in August 1914 forced him to return to Sweden. Where he fulfilled his military service. He continued his studies at the Stockholm Högskola and at the same time worked as scientific assistant at the Nobel Institute, where between 1917 and 1919 he published three papers on the dielectric properties of dipolar molecules and on electrolytes. In his doctoral dissertation, submitted in May 1921, he presented an important study of the statistical theory of suspensions and solutions based on Josiah Willard Gibbs’s statistical methods.

From 1918 on, Klein frequently visited Copenhagen, and finally remained there from September 1921 until September 1922. In June 1922 he accompanied Niels Bohr when he lectured at Göttingen. From then on, Klein and Hendrik Kramers were considered Bohr’s closest collaborators. Meanwhile, his interest had shifted from Physical chemistry to quantum theory. He and the Norwegian physicist Svein Rosseland, with whom he shared a grant from the Danish Rask-Ørsted Foundation (created to promote international scientific relations after the war), investigated the collisions of electrons with atoms, which were then of great interest for the theoretical development of quantum theory. In the course of these investigations, they introduced, as the counte)ts of normal collisions, “collisions of the second kind,” in which the colliding electron gains energy instead of losing it and the interacting atomic system undergoes a transition to a lower stationary state (1921). The successful application of this concept to various areas of atomic, molecular, and celestial physics contributed to Klein’s increasing reputation and aided the favorable reception of quantum theory in Sweden. But at the time, academic posts for theoretical physicists in Europe were few, and in 1922 Klein had to accept low-paid teaching positions at the universities of Stockholm and Lund.

In September 1923, Klein joined the spectroscopist Harrison M. Randall to help build up the physics department at the University of Michigan in Ann Arbor. In August 1923, before leaving for the United States, he married Gerda Koch: they had six children.

From 1923 to 1925, relatively isolated from European developments, Klein pursued his own attempts to formulate a unified relativistic theory embracing electromagnetism, gravitation, and quanta Inspired by the optical-mechanical analogy underlying William Rowan Hamilton’s mechanics, he achieved independently of Theodor Kaluza and others a five-dimensional generalization of Einstein“s general relativity theory that included electromagnetism.”^{1}In this work the wave-particle duality of matter was anticipated through the imposition of a quantum condition of periodicity on the wave function associated with the motion of a charged particle in five-dimensional space. Since these considerations were not published until April 1926, after the advent of wave mechanics, they had no influence on the early development of that area of physics.

There was a special interest in molecular spectra at Ann Arbor. Randall had been working for some years with Friedrich Paschen at Tubingen. Participating in these activities, Klein studied molecular interactions in terms of perturbation methods used by Bohr for the Stark effect.^{2} The outcome was a paper on the simultaneous action on a hydrogen atom of crossed homogeneous electric and magnetic fields (1924), which revealed a fundamental difficulty of the older quantum theory: that in this situation, transitions became possible from “allowed” to “prohibited” orbits. In his review of the state of quantum theory in 1925 (1926), Wolfgang Pauli concluded that this difficulty could be avoided only by a fundamental revision of the basic principles.

Klein returned to Europe in the summer of 1925 and resumed his post as docent in theoretical physics at the University of Lund. During a visit to Copenhagen in March 1926, he was informed of Erwin Schrüdinger’s wave mechanics, and after introducing some changes in his five-dimensional approach, he published his results, which included the relativistic wave equation (discovered independently by various authors) known as the Klein-Gordon equation (1926);

Attempts to relate the fifth dimension to known physical quantities remained unsuccessful, but some interest was aroused by Klein’s presentation at Leiden in June 1926. Under the influence of P. A. M. Dirac’s relativistic theory of the electron in 1928. Klein temporarily gave up his five-dimensional theory of quantum phenomena.

During 1926 Klein became deeply involved in Bohr’s work on correspondence and complementarity, which evolved into the Copenhagen interpretation of the quantum theory, offering a unified conception of the particle and wave character of microparticles. He obtained a relativistic extension of Schrödinger’s expressions for the electric charge and current density associated with the wave field (1927). The point of view of correspondence allowed him to establish a rule to determine the atomic transition probabilities before Dirac did so in a more satisfactory way by quantization of the electromagnetic field. However, Klein’s procedure remained accepted for many years.

Other important contributions to quantum field theory emerged in collaboration with Pascual Jordan, who visited the Bohr Institute in 1927. Klein had succeeded Kramers and Heisenberg as lecturer at the Copenhagen Institute of Theoretical Physics. He and Jordan became interested in the quantum theoretical treatment of the many-particle problem as a result of Dirac’s radiation theory. For the case of particles without spin (Bose particles), they found a method of obtaining the number of particles associated with the three-dimensional wave field by the introduction of commutation rules between the corresponding field operators (1927). This procedure, known as second quantization, is specially suited to describe processes involving particle creation and annihilation, and hence was important for the development of particle physics. Later the method was extended to fermions by Jordan and Eugene Wigner.

Dirac’s relativistic theory of the electron, published at the beginning of 1928, offered a new possibility for discussing the interaction between radiation and free electrons in a consistent manner. In a joint work with Yoshio Nishina, a Japanese guest at the Bohr Institute, Klein faced the laborious enterprise of calculating the angular intensity distribution of Compton Scattering according to the new theory (1929). The Klein-Nishina formula improved the earlier intensity formula obtained of electrodynamic potentials.

The experimental verification of the Klein-Nishina formula convinced many physicists of the soundness of Dirac’s relativistic equation, in spite of fundamental difficulties. One such difficulty is known as Klein’s paradox. By direct calculation Klein could show that, in appropriate potential fields, transitions to states with negative kinetic energy became possible. The interpretation of particles that appeared to accelerate in the opposite direction to an applied force was a major problem before the discovery of antiparticles.

In the last year of his stay at Copenhagen, Klein completed a paper containing an elegant method, still used in microwave spectroscopy, of determining rotational energy levels. Later he helped to develop a procedure, of great utility in modern molecular research, of calculating the interaction potentials directly from the energy levels of a diatomic molecule (1932).

In 1927 the chair of mechanics and mathematical physics at the Stockholm Högskola, occupied until then by the mathematician lvar Fredholm, became vacant. Klein applied for the post and won it, but because of the complicated Swedish nomination procedure, he was not appointed until 1930. Once in Stockholm, in addition to his scientific work and teaching obligations, Klein developed an active cultural and political life. Worthy of mention in the latter line is his help to refugees during World War II, and, in the former, his engagement in historical and philosophical studies, and in the diffusion of the new physical ideas.

In his first scientific paper at Stockholm (1931), Klein gave, in the tradition of Gibbs’s statistical mechanics, an explanation of the thermodynamic irreversibility paradoxically generated by mechanically reversible systems. For this purpose he used an inequality (later known as Klein’s lemma) for the statistical probability given by the density matrix introduced by Dirac. In 1933 he proposed an approximate method for solving the wave equation by a recursion procedure very similar to the Wentzel-Kramers-Brillouin method. In 1935 Klein used a generalized Dirac equation for an approximate description of nuclear system. In 1938 he presented some procedures for solving the Schrödinger equation in a periodic force field and also for describing electron interaction in a crystal. Also in 1938, at a conference on new theories in physics held in Warsaw, Klein presented a formulation of nonabelian vector field interactions that anticipated some aspects of the formalism of Chen Ning Yang and Robert Lawrence Mills. This work returned him to the fifth dimension.

The original five-dimensional theory of 1926 was an elegant attempt to unify the then known physical interactions, gravity and electromagnetism. In 1947 Klein tried to incorporate meson forces into this theory. In order to confine the charge of the mesons to integral positive and negative multiples of the elementary charge, the wave function must be a periodic function of the fifth coordinate, and this periodicity introduced a fundamental length scale into the theory. Klein was the first to recognize the similarity of the newly discovered muon decay to nuclear beta decay. This was the first indication of the universal Fermi interaction for weak interactions, and was discovered independently by E. Clementel and G. Puppi.

At about the same time, Klein also made important contributions to the theory of superconductivity. In order to obtain a quantum mechanical microscopic model of this phenomenon, he computed the diamagnetic properties of an electron gas (1944). Then, in 1945 in collaboration with Jens Lindhard, he pointed out that wave functions extending over large regions of space are favorable or large diamagnetism. In 1952, using the Bloch method, he reached a tentative description of superconductivity. The definitive solution was achieved in 1957 by John Bardeen, Leon N. Cooper, and John Robert Schrieffer.

In 1946, with the Swedish physicists Göran Beskov and Lars Treffenberg, Klein published a work on the distribution of the chemical elements. They intended to solve the basic problems connected with the socalled hypothesis of frozen equilibrium in accounting for the distribution of the chemical elements. This hypothesis assumes a special kind of chemical equilibrium among atomic species at high temperature and pressure, frozen by a rapid expansion. The simple equilibrium formula resulting from the hypothesis, however, does not agree with observation for large atomic masses. To correct this, Klein and his collaborators suggested that different atoms could be created at different places and at different times, in the interior of the primitive stars. Althoughingenious, their solution is not widely accepted. Also in relation to cosmology. Klein pursued the problem of finding exact solutions of Einstein’s field equations (1947). In 1954 he presented a solution that provided the relativistic analogue to the polytropic gas spheres.

Cosmology further stimulated Klein’s interest in statistical mechanics. In 1944 he derived the laws of thermodynamic equilibrium for a fluid in a static gravitational field. He showed that, in addition to the Tolman condition, one must introduce the condition α(g_{µv})½ = cte, where αis the chemical potential of the fluid and g_{µv} is the metric tensor. This relation constitutes the relativistic generalization of Gibb’s equilibrium condition.

Klein’s cosmology culminated in the mid 1950s, in a supergalactic system regarded as an ordinary, but very large, superstellar system, condensed from a thin, cold, hydrogen cloud. With some further assumptions. Klein deduced the Eddington relations and established the basis for the cosmological model that he later constructed with Hannes Alfvén. In 1956 and 1957 Klein proposed an extension of the five-dimensional theory of combined gravitational and electromagnetic fields including quantum fields, claiming that the inclusion of gravitational effects might eliminate the problem of the divergences of electron theory.

At the Solvay Conference on the Structure and Evolution of the Universe, held in 1958 at Brussels, Klein presented some speculative conjectures against the cosmological constant introduced by Einstein. In his report, “Some Considerations Regarding the Earlier Development of the System of Galaxies” (1959), he offered an alternative description of the evolution of a hypothetical gas cloud in the framework of general relativity.

As professor emeritus of the University of Stockholm in 1962, Klein pursued his studies of the classification schemes for the growing number of elementary particles. In 1959 he had presented an extension of the isobaric spin scheme to include K mesons and hyperons, suggesting also a scheme for weak interactions. In 1966 he proposed a classification of the baryon in terms of baryon and lepton numbers and strangeness, without using hypercharge or supercharge.

The experimental discovery of the antiproton by Owen Chamberlain and Emilio Segrè in 1955 confirmed the supposed matter-antimatter symmetry. In 1963, Klein and Alfvén presented a cosmological model based on a perfect symmetry between matter and antimatter on a cosmological scale. This model was inspired by Klein’s earlier speculations on the metagalactic structure of the universe, originating from a sphere of very dilute plasma containing equal quantities of matter and antimatter. When the density of the gravitationally contracting sphere reached a critical value, annihilation processes became dominant. Then the collapse stopped and reversed to the outward motion connected with the red shift of galaxies. The Klein-Alvén cosmology adopts the concept of a globally nonexpanding universe, considering the observed red shift only as a local phe nomenon. Difficulties with this cosmological model were soon recognized.

Also in connection with his cosmological model, Klein scrutinized the foundations of general relativity and concluded that many difficulties could be overcome by keeping close to the original ideas of Einstein. He claimed that the principle of equivalence between gravitation and inertial forces is incompatible with the idea proposed by Mach, and accepted by Einstein, to describe the universe by an analogy in three dimensions to the closed surface of a sphere. Along the same line, he derived an extension of Einstein’s principle of equivalence to include the case of particles described through the Dirac equation.

In 1970 Klein presented’ A Tentative Program for the Development of Quantum Field Theory as an Extension of the Equivalence Principle of General Relativity Theory, ’ which starts with his fivedimensional formulation of 1926:

Although the periodicity as an interpretation of the elementary quantum of electricity occurred to me almost in the beginning (in 1925) and later led me to the consideration of the corresponding group, from early times until recently

Ioscillated between this way and a formalism related to isospin, obtained by cutting down the higher harmonics of the Fourier expansion. This, however, led to two difficulties: the linearity in the momenta in the Lagrangian density could not be maintained unless the group was limited in analogy to the transformation group; and that the equivalence claim could not be satisfied. With the full periodicity these difficulties are absent. (p.253)

Klein’s last papers are devoted to this program.

In 1951 Klein received an honorary doctorate from the University of Oslo, and in 1965 from the University of Copenhagen. In 1954 he became a member of the Nobel Committee on Physics.

## NOTES

1. Since there is no documentary evidence of this early work, we must rely of Klein’s statements given in his autobiographical account on six interviews conducted in 1962 and 1963. See Kuhn *et al*., 103f.

2. See also interview 3 with Klein (Kuhn *et al*.) from 25 February 1963. p. 8f. of the transcript.

## BIBLIOGRAPHY

1. Original Works. A bibiography of Klein’s most important scientific articles is in Poggendorff, VI. pt. 2, 1330. and VIIb, pt. 4. 2492–2494. He also wrote many popular scientific essays and books on atomic and nuclear physics, on relativity, and on cosmology, most of them in Swedish: *Den Bohrsk a atomtheorien* (Stockholm, 1922);’ Vad vi veta om ljuset, ’ in *Natur och kultur*, **41** and **42** (1925); *Orsak och verkan i den nya atomteoriens belysning* (Stockholm, 1935);*Entretiens sur les idées fondamentales de la physique moderne*, L. Rosenfeld, trans (Paris, 1938); and *Les processus nucléaires dans les astres* (Liége, 1953).

Only a small portion of Klein’s scientific manuscripts and correspondence deposited at the Nield Bohr Institute. Copenhagen is listed in Thomas S. Kuhn, John L. Heilbron. Paul Forman, and Lini Allen, *Sources for History of Quantum Physics* (Philadelphia. 1967), 55. Some letters from the correspondence with Bohr and Kramers are in Niels Bohr’s *Collected Works*, Klaus Stolzenburger, ed., V (1984), 382–394, and IX, 592–596; and letters from the correspondence with Wolfang Pauli are in *Pauli’s Wissenschaftlicher Briefwechsel*, Karl von Meyenn, Armin Hermann, and Victor F. Weisskopf, eds., **I** , 488–492, 494–495, and **II** , 3–4, 7, 43–46, 51, 422–431, and 534.

“über die Löslichkeit von Zinkhydroxyd in Alkalien,” in *Zeitschrift für anorganische Chemie*. **74** (1912). 157– 169; “Zur statistischen Theorie der Suspensionen und Lösungen,” in *Arkiv för matematik, astronomi och fysik, 16* (1921), no.5, 1–51; über Zusammenstösse zwischen Atomen und freien Elektronen, ’ in *Zeitschrift für Physik*, **4** (1921). 46–51, with Svein Rosseland; “über die gleichzeitige Wirkung von gekreuzten homogenen elektrischen und magnetischen Feldern auf das Wasserstoffatom. **I** ’ *ibid*., **22** (1924), 109–118; “Quantentheorie und fünfdimensionale Relativitätstheorie,” *ibid*., **37** (1926), 895–906;“Elektrodynamik und Wellenmechnik vom Standpunkt des Korrespondenzprinzips,” *ibid*., **41** (1927), 407–442;’ Zur fünfdimensionalen Darstellung der Relativitätstheorie, ’ *ibid*., **46** (1927), 188–208; “Zum Mehrkörperproblem der Quantentheorie,” *ibid*., **45** (1927), 751– 765, with Pascual Jordan;’ uuml;ber die Streuung von Strahlung durch freie Elektronen nach der neuen relativistischen Quantendynamik von Dirac, ’ *ibid*., **52** (1929), 853–868 with Yoshio Nishina:’ Die Reflexion von Elektronenan einem Potentialsprung nach der relativistischen Dynamik von Dirac, ’ *ibid*., **53** (1929), 157–165; “Zur Frage der Quantelung des asymmetrischen Kreiseld,” *ibid*., **58** (1929), 730–734.

“Zur quantenmechanischen Begründung des zweiten Hauptsatzes der wärmelehre,” in *Zeitschrift für Physik*, **72** (1931), 767–775; “Zur Berechnung von Potentialkurven für zweiatomige Moleküle mit Hilfe von Spektraltermen,” *ibid*., **76** (1932), 226–235; Zur Frage der quasimechanischen Lösung der quantenmechanischen Wellengleichung, “in *Zeitschrift für Physik*, **80** (1933), 792–803;” Quelques remarques sur le traitement approximatif du probléme des électrons dans un réseau cristallin par la mécanique quantique, ’ in *Journal de physique et le radium* 7th ser., **9** (1938), 1–12; “Philosophy and Physics, ’ in *The oria* (Gotebrog-Lund) (1938). 59–61;” Sur la théorie des champs associés á des particules chargées, ’ in *Les nouvelles théories de la physique* (Paris, 1939), 81–98; “On the Magnetic Behaviour of Electrons in Crystals,” in *Arkiv fór matematik, astronomi och fysik*, **31A** (1944– 1945), no.12; “Some Remarks on the Quantum Theory of the Superconductive State,” in *Reviews of Modern Physics*, **17** (1945), 305–309, with J. Lindhard; “On the Origin of the Abundance Distribution of Chemical Elements,” in *Arkiv för matematik, astronomi och fysik*, 33B(1946–1947), no.1, with G.Beskov and L.Treffenberg; “Meson Fields and Nuclear Interaction,” *ibid*., **34A** 1947– 1948), no.1; “On a Case of Radiation Equilibrium in General Relativity Theory and Its Bearing on the Early Stage of Stellar Evolution,” *ibid*., no.19; and “On the Thermodynamical Equilibrium of Fluids in Gravitational Fields,” in *Reviews of Modern Physics*, **21** (1949), 531–533.

“Theory of Superconductivity,” in *Nature*, **169** (1952), 578–579; “On a Class of Spherically Symmetric Solutions of Einstein’s Gravitational Equations,” in Arkiv föf fysik, **7** (1954), 487–496; “Generalizations of Einstein” s Theory of Gravitation Considered from the Point of View of Quantum Field Theory, “in *Helvetica physica acta*, supp. **4** (1956), 58–71; “Some Remarks on General Relativity and the Divergence Problem of Quantum Field Theory, “in *Nuovo Cimento*, supp. **6** (1957), 344–348;” On the Systematics of Elementary Particles, ’ in Arkiv för fysik, **16** (1960), 191–196; “Matter–Antimatter Annihilation and Cosmology, : *ibid*., **23** (1963), 187–194, with Hannes Alfén;” Remark Concerning the Basic SU(3) Triplets, in *Physical Review Letters*, **16** (1966), 63–64; “A Tentative Program for the Development of Quantum Field Theory as an Extension of the Equivalence Principle of General Relativity Theory,” in *Nuclear Physics*, **B21** (1970), 253– 260; “Arguments Concerning Relativity and Cosmology,” in *Science*, **171** (1971), 339–345; “The Equivalence Principle and the Dirac Equation with Gravitation, in *Physica Norvegica*, 591–971), 145–147;” Ur mitt liv i fysiken.’ in Svensk naturvetenskap (1973), 159–172; “Generalization of Einstein”s Principle of Equivalence so as to Embrace the Field Equations of Gravitation, ’ in *Physica scripta*, **9** (1974), 69–72; “Einstein”s Principle of Equivalence Used for an Alternative Relativistic Cosmology Considering the System of Galaxies as Limited and not as the Universe, “in *Kongelige danske videnskabernes selskabs matematisk–fysiske meddelelser*, **39** (1974), 3–18; and’ Electromagnetic Theory Treated in Analogy to the Theory of Gravitation,” in *Nuclear Physics*, **B92** (1975), 541– 546.

II. SECONDARY LITERATURE. Obituaries are by Stanley Deser, in *Physics Today*, **30** (June1977), 67–68; Inga Fischer–Hjalmars and Bertel Laurent, in *Kosmos* (1978), 19–29; and /Christian Møller, in *Fysisk tidsskrift*, **75** (1977), 169–171.

Authoritative descriptions of Klein’s contributions to quantum theory are in Wolfgang Pauli, ’ Quantentheorie, ’ in *Handbuch der Physik*, XXIII (Berlin, 1926), 1–278, and’ Die allgemeinen Prinzipien der Wellenmechanik, ’ in *Handbuch der Physik*, XXIV, pt. 1 (Berlin, 1933), 83– 272.

Klein’s activity during his Copenhagen period is described in Max Dresden, Hendrik Anthony Kramers (New York, 1987); and in Peter Robertson, *The Early Years. The Niels Bohr Institute 1921–1930* (Copenhagen, 1979). A good survey of Klein’s contribution to the development of the relativistic wave equation is Helge Kragh, ’ Equation with Many Fathers. The Klein–Gordon Equations in 1926, ’ in *American Journal of Physics*, **52** (1984), 1024–1033, Klein’s cosmological speculations are summed up in Hannes Alfén, ’ Antimatter and Cosmology, ’ in *Scientific American* (April 1967), 106–113; and by Jagjit Singh, *Great Ideas and Theories of Modern Cosmology* (New York, 1961), 273–279. For the new formulations of the Kaluza–Klein theories, especially in the context of supergravity, see Venzo De Sabbata and Ernst Schmutzer, eds., *Unified Field Theories of More Than Four Dimensions* (Singapore, 1983).

Karl von Meyenn

Mariano Baig

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