{ "id": "ep2fnm10c8", "slug": "oskar-klein", "title": "Oskar Klein", "updated": "2026-10-25", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.scientists", "label": "Physical and mathematical scientists", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.scientists" }, { "id": "physical.scientists.physics-astronomy", "label": "Physicists and astronomers", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.scientists.physics-astronomy" }, { "id": "physical.scientists.physics-astronomy.phys-hep", "label": "Researchers in particle, nuclear, and high-energy theoretical physics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.scientists.physics-astronomy.phys-hep" }, { "id": "physical.scientists.physics-astronomy.phys-hep.quantum-field-theory-and-mathematical-physics", "label": "Quantum field theory and mathematical physics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.scientists.physics-astronomy.phys-hep.quantum-field-theory-and-mathematical-physics" } ], "geo": [ { "id": "geo.weu.t1946.physical.scientists.physics-astronomy.phys-hep.quantum-field-theory-and-mathematical-physics", "label": "Western Europe · 1946 to 2000: Quantum field theory and mathematical physics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.weu.t1946.physical.scientists.physics-astronomy.phys-hep.quantum-field-theory-and-mathematical-physics", "path": [ { "id": "geo.weu", "label": "Western Europe", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.weu" }, { "id": "geo.weu.t1946", "label": "Western Europe · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.weu.t1946" }, { "id": "geo.weu.t1946.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.weu.t1946.physical" }, { "id": "geo.weu.t1946.physical.scientists", "label": "Physical and mathematical scientists", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.weu.t1946.physical.scientists" }, { "id": "geo.weu.t1946.physical.scientists.physics-astronomy", "label": "Physicists and astronomers", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.weu.t1946.physical.scientists.physics-astronomy" }, { "id": "geo.weu.t1946.physical.scientists.physics-astronomy.phys-hep", "label": "Researchers in particle, nuclear, and high-energy theoretical physics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.weu.t1946.physical.scientists.physics-astronomy.phys-hep" }, { "id": "geo.weu.t1946.physical.scientists.physics-astronomy.phys-hep.quantum-field-theory-and-mathematical-physics", "label": "Quantum field theory and mathematical physics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.weu.t1946.physical.scientists.physics-astronomy.phys-hep.quantum-field-theory-and-mathematical-physics" } ] } ], "excerpt": "Oskar Benjamin Klein (1894–1977) was a Swedish theoretical physicist and longtime collaborator of Niels Bohr, known for the Klein–Gordon equation, Kaluza–Klein theory, and the Klein–Nishina formula.", "snippet": "Oskar Benjamin Klein (1894–1977) was a Swedish theoretical physicist and longtime collaborator of Niels Bohr, known for the Klein–Gordon equation, Kaluza–Klein theory, and the Klein–Nishina formula.", "node": "physical.scientists.physics-astronomy.phys-hep.quantum-field-theory-and-mathematical-physics", "markdown": "# Oskar Klein\n\n**Oskar Benjamin Klein** (15 September 1894 – 5 February 1977) was a Swedish theoretical physicist whose name is attached to several foundations of modern physics: the [Klein–Gordon equation](https://www.edgepedia.org/klein-gordon-equation), the first relativistic version of the Schrödinger wave equation; [Kaluza–Klein theory](https://www.edgepedia.org/kaluza-klein-theory), the five-dimensional unification of gravitation and electromagnetism; the Klein–Nishina formula for [Compton scattering](https://www.edgepedia.org/compton-scattering); the Klein paradox; and the Jordan–Klein method of second quantization.[1](https://www.su.se/english/divisions/oskar-klein-centre/about-the-okc/biography-of-oskar-klein) • [2](https://ar5iv.labs.arxiv.org/html/1309.4113) He worked for most of his career in Stockholm and was one of Niels Bohr's closest collaborators in the years when the Copenhagen interpretation of quantum mechanics took shape.[3](https://nba-old.nbi.dk/icos/klein.html)\n\n| Key fact | Detail |\n|---|---|\n| Born / died | 15 September 1894, Stockholm; 5 February 1977, Stockholm[1](https://www.su.se/english/divisions/oskar-klein-centre/about-the-okc/biography-of-oskar-klein) |\n| Klein–Gordon equation | Published April 1926, the first relativistic version of the Schrödinger wave equation; for spinless particles[1](https://www.su.se/english/divisions/oskar-klein-centre/about-the-okc/biography-of-oskar-klein) • [4](https://research.engineering.nyu.edu/~jbain/histlight/readings/81Kragh.pdf) |\n| Kaluza–Klein theory | 1926 five-dimensional unification with a compact extra dimension of order the Planck length, explaining charge quantization[2](https://ar5iv.labs.arxiv.org/html/1309.4113) |\n| Klein–Nishina formula | 1928 calculation of the intensity distribution in Compton scattering from the Dirac equation; its experimental confirmation became the main early support for Dirac's theory[5](https://pmc.ncbi.nlm.nih.gov/articles/PMC5709540/) |\n| Second quantization | 1927 Jordan–Klein commutation rules for field operators, suited to describing particle creation and annihilation[6](https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/klein-oskar-benjamin) |\n| Professorship | Professor of mathematical physics at Stockholms Högskola 1930–62[3](https://nba-old.nbi.dk/icos/klein.html) |\n\n## Life and career\n\nKlein was born in Stockholm, the son of Gottlieb Klein, a Hungarian rabbi who had moved to Stockholm the year before Oskar's birth.[1](https://www.su.se/english/divisions/oskar-klein-centre/about-the-okc/biography-of-oskar-klein) From the age of 15 he worked, on [Svante Arrhenius](https://www.edgepedia.org/svante-arrhenius)' invitation, at the Nobel Institute on the solubility of salts using radioactive indicators.[3](https://nba-old.nbi.dk/icos/klein.html) He completed his doctoral dissertation at Stockholms Högskola in 1921, having begun visiting Copenhagen in 1918; with Svein Rosseland he introduced the concept of collisions of the second kind.[3](https://nba-old.nbi.dk/icos/klein.html)\n\n**The Copenhagen years.** Klein became docent at [Lund University](https://www.edgepedia.org/lund-university) in 1926 and lecturer at the [Niels Bohr Institute](https://www.edgepedia.org/niels-bohr-institute) in 1927, where he was deeply involved in Bohr's work on the correspondence principle and complementarity.[3](https://nba-old.nbi.dk/icos/klein.html) The year 1926 was a productive one personally as well: he recovered from hepatitis, took up the Lund position, and produced his five-dimensional and wave-equation results.[7](https://mathshistory.st-andrews.ac.uk/Biographies/Klein_Oskar/)\n\n**Stockholm and later work.** Klein was professor of mathematical physics at Stockholms Högskola from 1930 to 1962.[3](https://nba-old.nbi.dk/icos/klein.html) His later research ranged over statistical mechanics, superconductivity (with Jens Lindhard, 1945), the distributions of chemical elements, and a cosmological model developed with [Hannes Alfvén](https://www.edgepedia.org/hannes-alfven) in 1963.[3](https://nba-old.nbi.dk/icos/klein.html) Work with Rydberg on molecular spectra produced the Rydberg–Klein–Rees method, still used to construct potential-energy curves for diatomic molecules.[1](https://www.su.se/english/divisions/oskar-klein-centre/about-the-okc/biography-of-oskar-klein) He retired in 1962 and died in Stockholm on 5 February 1977.[1](https://www.su.se/english/divisions/oskar-klein-centre/about-the-okc/biography-of-oskar-klein)\n\n## Scientific contributions\n\n**The Klein–Gordon equation.** In 1926 Klein published the first relativistic version of the Schrödinger wave equation, which [Paul Dirac](https://www.edgepedia.org/paul-dirac) later named the Klein–Gordon equation, to [Erwin Schrödinger](https://www.edgepedia.org/erwin-schrodinger)'s chagrin, since Schrödinger claimed to have found the same result but never published it.[1](https://www.su.se/english/divisions/oskar-klein-centre/about-the-okc/biography-of-oskar-klein) Walter Gordon derived the same equation the same year along similar lines, starting from the four-dimensional classical equation, which is why the name carries both.[2](https://ar5iv.labs.arxiv.org/html/1309.4113) The Klein–Gordon equation is used for spinless particles; Dirac's 1928 equation supplied the relativistic electron theory.[1](https://www.su.se/english/divisions/oskar-klein-centre/about-the-okc/biography-of-oskar-klein)\n\n**The Klein–Nishina formula.** Working with Yoshio Nishina, Klein calculated the intensity distribution of the scattered wave in Compton scattering using Dirac's relativistic electron theory; the formula was published in 1928.[5](https://pmc.ncbi.nlm.nih.gov/articles/PMC5709540/) • [1](https://www.su.se/english/divisions/oskar-klein-centre/about-the-okc/biography-of-oskar-klein) Bohr wrote to Nishina in 1934 that \"the striking confirmation which this formula has obtained became soon the main support for the correctness of Dirac's theory\".[5](https://pmc.ncbi.nlm.nih.gov/articles/PMC5709540/) The experiments run to confirm the formula also suggested the existence of then-unknown phenomena, the pair production and annihilation of positive and negative electrons.[5](https://pmc.ncbi.nlm.nih.gov/articles/PMC5709540/)\n\n**The Klein paradox.** By direct calculation Klein showed that, in appropriate potential fields, transitions to states with negative kinetic energy become possible.[6](https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/klein-oskar-benjamin) This was a major interpretive problem for the relativistic wave theories before the discovery of antiparticles, and it sits alongside the Klein–Nishina success in the record of how Dirac's equation was tested.[3](https://nba-old.nbi.dk/icos/klein.html)\n\n**Second quantization.** In 1927, working with [Pascual Jordan](https://www.edgepedia.org/pascual-jordan) at the Bohr Institute, Klein introduced commutation rules between field operators that determine particle number for spinless (Bose) particles, the procedure known as second quantization.[6](https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/klein-oskar-benjamin) It is specially suited to describing processes involving particle creation and annihilation, and it was important for the development of particle physics; Jordan and Wigner later extended the method to fermions.[6](https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/klein-oskar-benjamin) A CERN survey by the physicist-historian John Heilbron calls the 1927 Jordan–Klein paper \"the truly magisterial Jordan–Klein second quantization\", and notes that Heisenberg's 1955 summary expressed the fundamental importance of the work.[8](https://cds.cern.ch/record/277237/files/th-95-009.pdf)\n\n## Kaluza–Klein theory\n\n[Theodor Kaluza](https://www.edgepedia.org/theodor-kaluza) had proposed connecting Einstein's ten gravitational potentials and the four electromagnetic potentials with the coefficients of a line element in a Riemannian space containing a fifth dimension, so that the equations of motion of charged particles take the form of geodesics even in electromagnetic fields.[9](https://cdn.psiket.com/ds01/894afcdf4b5ffd365140c3cb4fea49df.pdf) Klein, working in relative isolation from 1923 to 1925, achieved a five-dimensional generalization of general relativity including electromagnetism independently of Kaluza, and learned of Schrödinger's wave mechanics during a March 1926 visit to Copenhagen before publishing.[6](https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/klein-oskar-benjamin) In 1926 he wrote two classic papers on the theory, the more elaborate one in *Zeitschrift für Physik*, connecting the quantum wave equation to five-dimensional null geodesics.[10](https://ar5iv.labs.arxiv.org/html/gr-qc/0009087) A short piece in *Nature* in the autumn of 1926 carried the quantum interpretation to a wider audience.[1](https://www.su.se/english/divisions/oskar-klein-centre/about-the-okc/biography-of-oskar-klein) • [7](https://mathshistory.st-andrews.ac.uk/Biographies/Klein_Oskar/)\n\n**Compactification.** Klein's decisive addition was to make the fifth dimension compact: a circle of radius R, with quantized momentum in the fifth direction, \\( p_{5} = n\\hbar/R \\). He found this intriguing because it would explain why particle charges come in quantized units of the electron's charge.[2](https://ar5iv.labs.arxiv.org/html/1309.4113) The fifth coordinate was a physical quantity conjugate to electrical charge, and the dimension's period was set by \\( l = \\hbar c\\sqrt{2k}/e \\), where \\( e \\) is the electron charge and \\( k \\) Einstein's gravitational constant, putting it on the order of the [Planck length](https://www.edgepedia.org/planck-length), \\( 10^{-33} \\) cm.[7](https://mathshistory.st-andrews.ac.uk/Biographies/Klein_Oskar/) Translations in the fifth direction, \\( x^{5} \\rightarrow x^{5} + \\Lambda(x) \\), correspond to gauge transformations of the wave function, with the gauge group of electrodynamics being U(1).[10](https://ar5iv.labs.arxiv.org/html/gr-qc/0009087)\n\n**Why Klein abandoned it.** His compactification calculation gave \\( \\alpha = 4(L_{P}/R)^{2} \\) with the Planck length \\( L_{P} = 1.62 \\times 10^{-33} \\) cm, implying \\( R = 23 L_{P} \\), far too small to detect. But the corresponding electron mass, \\( m = M_{P}/23 \\) with [Planck mass](https://www.edgepedia.org/planck-mass) \\( M_{P} = 1.22 \\times 10^{19} \\) GeV/c², is a factor \\( 10^{22} \\) larger than the physical electron mass, and Klein grew pessimistic within a year.[2](https://ar5iv.labs.arxiv.org/html/1309.4113) He recalled that \"Pauli and I drank some wine on the death of the fifth dimension in 1928\", and under the influence of Dirac's 1928 relativistic electron theory he temporarily gave up the five-dimensional approach.[1](https://www.su.se/english/divisions/oskar-klein-centre/about-the-okc/biography-of-oskar-klein) • [6](https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/klein-oskar-benjamin)\n\n## Role in the Copenhagen interpretation\n\nDuring 1926 Klein became deeply involved in Bohr's work on the correspondence principle and complementarity, which evolved into the [Copenhagen interpretation](https://www.edgepedia.org/copenhagen-interpretation) of quantum theory.[6](https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/klein-oskar-benjamin) The Bohr Archive record lists him as deeply involved in Bohr's work on correspondence and complementarity and on the Heisenberg uncertainty relations.[3](https://nba-old.nbi.dk/icos/klein.html) • [1](https://www.su.se/english/divisions/oskar-klein-centre/about-the-okc/biography-of-oskar-klein) In 1927 he obtained a relativistic extension of Schrödinger's expressions for electric charge and current density, and his correspondence-based rule for atomic transition probabilities remained accepted for many years before Dirac's quantization of the electromagnetic field.[6](https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/klein-oskar-benjamin) In 1929 he used the correspondence principle in his quantum treatment of rotationally symmetric molecules.[1](https://www.su.se/english/divisions/oskar-klein-centre/about-the-okc/biography-of-oskar-klein)\n\n## By the numbers\n\nThe compactification scale is the central quantity, and two values appear in the literature. One calculation chain gives \\( R = 23 L_{P} \\) with \\( L_{P} = 1.62 \\times 10^{-33} \\) cm, of order \\( 10^{-33} \\) cm.[2](https://ar5iv.labs.arxiv.org/html/1309.4113) • [7](https://mathshistory.st-andrews.ac.uk/Biographies/Klein_Oskar/) A review of Kaluza–Klein history instead gives the fifth dimension's scale parameter as \\( \\lambda_{5} = hc\\sqrt{2\\kappa}/e = 0.8 \\times 10^{-30} \\) cm, explaining the dimension's non-appearance in ordinary experiments by averaging over it.[10](https://ar5iv.labs.arxiv.org/html/gr-qc/0009087) These differ by more than an order of magnitude, and both trace to Klein's own papers; the discrepancy is unresolved in the literature. On either value the predicted electron mass fails badly: the \\( R = 23 L_{P} \\) route overshoots the physical electron mass by a factor \\( 10^{22} \\).[2](https://ar5iv.labs.arxiv.org/html/1309.4113)\n\n## Credit and contemporaries\n\n**Priority.** Helge Kragh, the historian of physics, states that priority in publication for the Klein–Gordon equations belongs to Klein, who proposed them in April 1926; several physicists worked the equations out that year.[4](https://research.engineering.nyu.edu/~jbain/histlight/readings/81Kragh.pdf) An American Journal of Physics study calls it \"the equation with the many fathers\", listing Klein, Vladimir Fock, Schrödinger, and de Broglie among those who announced it in 1926 as a candidate relativistic generalization of the [Schrödinger equation](https://www.edgepedia.org/schrodinger-equation).[11](https://pubs.aip.org/aapt/ajp/article/52/11/1024/1038484/Equation-with-the-many-fathers-The-Klein-Gordon) Fock's treatment of relativistic wave mechanics was probably the most detailed, calculating the relativistic Kepler motion.[4](https://research.engineering.nyu.edu/~jbain/histlight/readings/81Kragh.pdf) The same study notes that the five-dimensional attempt to embrace general relativity and electrodynamics had virtually nil impact on the mainstream development of quantum mechanics.[11](https://pubs.aip.org/aapt/ajp/article/52/11/1024/1038484/Equation-with-the-many-fathers-The-Klein-Gordon)\n\n**Klein's own view.** Late in life, not getting credit for discovering the Schrödinger equation was, in Heilbron's words, \"the one injustice I have ever heard him complain of\".[8](https://cds.cern.ch/record/277237/files/th-95-009.pdf) Among his honors were the Planck medal and the Lorentz professorship.[8](https://cds.cern.ch/record/277237/files/th-95-009.pdf)\n\n## References\n\n1. [Biography of Oskar Klein, Stockholm University, Oskar Klein Centre](https://www.su.se/english/divisions/oskar-klein-centre/about-the-okc/biography-of-oskar-klein)\n2. [On the history of the Klein–Gordon equation and Klein's five-dimensional theory, arXiv 1309.4113](https://ar5iv.labs.arxiv.org/html/1309.4113)\n3. [Klein, Oskar. 1894–1977, Oskar Klein Papers, Niels Bohr Archive](https://nba-old.nbi.dk/icos/klein.html)\n4. [Helge Kragh, The genesis of Dirac's relativistic theory of electrons](https://research.engineering.nyu.edu/~jbain/histlight/readings/81Kragh.pdf)\n5. [How the Klein–Nishina formula was derived, PMC](https://pmc.ncbi.nlm.nih.gov/articles/PMC5709540/)\n6. [Klein, Oskar Benjamin, Encyclopedia.com](https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/klein-oskar-benjamin)\n7. [Oskar Klein (1894–1977), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Klein_Oskar/)\n8. [Oskar Klein: from his life and physics, CERN preprint TH-95-009](https://cds.cern.ch/record/277237/files/th-95-009.pdf)\n9. [Oskar Klein, Quantum theory and five-dimensional relativity theory (1926, English translation)](https://cdn.psiket.com/ds01/894afcdf4b5ffd365140c3cb4fea49df.pdf)\n10. [Kaluza–Klein theory history, arXiv gr-qc/0009087](https://ar5iv.labs.arxiv.org/html/gr-qc/0009087)\n11. [Equation with the many fathers. The Klein–Gordon equation in 1926, American Journal of Physics](https://pubs.aip.org/aapt/ajp/article/52/11/1024/1038484/Equation-with-the-many-fathers-The-Klein-Gordon)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Physical and mathematical scientists › Physicists and astronomers › Researchers in particle, nuclear, and high-energy theoretical physics › Quantum field theory and mathematical physics*\n\n*Initially written Oct 10, 2026 · Edited: Oct 11, 2026; Oct 15, 2026; Oct 25, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [ "https://research.engineering.nyu.edu/~jbain/histlight/readings/81Kragh.pdf" ], "url": "https://www.edgepedia.org/oskar-klein", "markdown_url": "https://www.edgepedia.org/oskar-klein.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"Oskar Klein\", Edgepedia (EdgeChat), https://www.edgepedia.org/oskar-klein. Edgepedia Community License 1.0.", "credit_md": "\"[Oskar Klein](https://www.edgepedia.org/oskar-klein)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/oskar-klein](https://www.edgepedia.org/oskar-klein). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Oskar Klein\", Edgepedia (EdgeChat), https://www.edgepedia.org/oskar-klein. Edgepedia Community License 1.0.", "speakable": "Oskar Benjamin Klein was a Swedish theoretical physicist and longtime collaborator of Niels Bohr, known for the Klein–Gordon equation, Kaluza–Klein theory, and the Klein–Nishina formula." }