• Arnold Sommerfeld
Part of the Progress in Mathematical Physics book series (PMP, volume 35)


Arnold Johannes Wilhelm Sommerfeld, one of the great men of twentieth century physics, was born in Königsberg, Prussia in 1868. An intellectually gifted youth (“In school I was almost more interested in literature and history than in the exact sciences; I was equally good in all subjects, including ancient languages”1), he entered the University of Königsberg in 1886. German students at the time generally moved from place to place, but Sommerfeld remained at Königsberg for five years: “The excellence of the professorial chairs in mathematics (Lindemann2 as Ordinarius, Hurwitz3 as Extraordinarius, Hilbert4 as Privatdozent) prevented me from changing universities; this was in a way unfortunate, since I was at the same time a Burschenschaftler,5 and was thus diverted from consistent study.” His 1891 doctoral dissertation, Die willkürlichen Functionen in der mathematischen Physik, an analysis of series expansions in circular, cylindrical, and spherical functions, was “conceived and written down in a few weeks.” The dissertation contains the following vita:

I, Arnold Sommerfeld, son of the practicing physician Dr. F. Sommerfeld and his wife Cäcilie, born Mathiass, evangelical faith, was born on December 5, 1868 at Königsberg in Prussia. I attended the Altstädtische Gymnasium there, graduated with the Zeugnis der Reife 6 on Michaelmas7 1886, matriculated at the local Albertus-University, and devoted myself primarily to the study of mathematics. On July 28 of this year I passed the examen rigorosum.


Riemann Surface Diffraction Theory Diffraction Problem Shadow Boundary Luminous Line 
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  1. 1.
    Arnold Sommerfeld, Autobiographische Stizze, in Gesammelte Schriften (Friedr. Vieweg & Sohn, Braunschweig, 1968, Vol. 4, pp. 673–679). All subsequent unreferenced quotations in the introduction are from this source.Google Scholar
  2. 2.
    Carl Louis Ferdinand von Lindemann (1852-1939) is most famous for proving that the number π is transcendental (that is, not the root of an algebraic equation). In addition to Sommerfeld, he supervised more than sixty other doctoral students.Google Scholar
  3. 3.
    Adolf Hurwitz (1859-1919) worked in the fields of abstract algebra and complex function theory. He is known among engineers for the Routh-Hurwitz stability criterion, a method for determining the conditions under which all the roots of a polynomial lie in the left half of the complex plane.Google Scholar
  4. 4.
    David Hilbert (1862-1943) became one of the most important mathematicians of his time. His definition of twenty-three open problems, presented at the 1900 International Congress of Mathematicians in Paris, has influenced the development of mathematics for more than one hundred years.Google Scholar
  5. 5.
    Member of a German fraternal society.Google Scholar
  6. 6.
    Equivalent of the modern German Abitur or French baccalauréat.Google Scholar
  7. 7.
    September 29, the feast of St. Michael the Archangel.Google Scholar
  8. 8.
    Carl Chun (1852-1914), zoologist; Julius Sophus Felix Dahn (1834-1912), writer, poet and historian; Ludwig Hermann Alexander Elster (1856-1935), political economist; Wilhelm Clemens Lossen (1838-1906), chemist; Carl Johannes Wilhelm Theodor Pape (1836-1906), physicist; Johannes Rants (1854-1933), astronomer and statistician; Louis Saalschütz (1835-1913), mathematician; Günther Thiele (1841-1910), philosopher; Paul Oskar Eduard Volkmann (1856-1938), geophysicist; Julius Guido Wilhelm Hermann Walter (1841-1922), philosopher; Johann Emil Wiechert (1861-1928), physicist.Google Scholar
  9. 9.
    Theodor Liebisch (1852-1922) was a professor in Königsberg from 1884-1887 and was apparently an acquaintance of Sommerfeld’s family. In an 1894 letter to Sommerfeld’s mother Cäcilie, Liebisch’s wife Adelheid writes that she is concerned with Sommerfeld’s well-being; her husband is of the opinion “that Arnold will never be suitable as a mineralogist, that he is a pure mathematician through and through.”Google Scholar
  10. 10.
    Mathematiche Annalen, Vol. 47,1896, pp. 317–374. One of the editors of the Mathematische Annalen was Felix Klein, which may partly explain how the young Sommerfeld was able to publish a paper of fifty-seven pages. Klein and Sommerfeld later wrote the monumental Theorie des Kreisels, which was published in four volumes over the years 1897-1910.Google Scholar
  11. 11.
    On page 369 of his Mathematische Theorie der Diffraction, Sommerfeld defines geometric optics as “that entire way of thinking which decomposes the optical state into individual independently progressing rays.”Google Scholar
  12. 12.
    A facsimile of Physico-mathesis de lumine was published by the city of Bologna on the tercentenary of Grimaldi’s death (Arnaldo Forni Editore, Bologna, 1963).Google Scholar
  13. 13.
    The experiments of Grimaldi and Hooke are described by the English clergyman and chemist Joseph Priestley (1722-1804) in The History and Present State of Discoveries relating to Vision, Light and Colours (London, 1772). Priestley’s fascinating and delightful description is presented in Appendix I; the Rare Books and Manuscripts Department at the Boston Public Library is gratefully acknowledged for providing access to an original copy of Priestley’s book.Google Scholar
  14. l4.
    Werner Braunbek and Günther Laukien (“Einzelheiten zur Halbebenen-Beugung,” Optik, Vol. 9, 1952, pp. 174-179) have used Sommerfeld’s diffraction solution to compute the lines of average energy flow that result when a time-harmonic electromagnetic plane wave passes by a perfectly conducting screen. These lines of energy flow do indeed exhibit spatial oscillations near the edge of the screen. Michael Berry (“Exuberant interference: rainbows, tides, edges, (de)coherence …,” Philosophical Transactions of the Royal Society of London A, Vol. 360, 2002, pp. 1023-1037) has generously interpreted these oscillations as a confirmation of Newton’s conjecture.Google Scholar
  15. 15.
    The ether was imagined to be a fluid (and later a solid) which pervaded all space but which somehow did not interfere with the ordinary motion of bodies. The ether retained its place in physical theory throughout the nineteenth century. James Clerk Maxwell (1831-1879), the founder of the modern theory of the electromagnetic field, wrote an interesting article “Ether” for the 1878 Encyclopœdia Britannica. Sommerfeld “ventured for the first time onto the high seas of theoretical physics” with the 1892 paper “Mechanische Darstellung der elektromagnetischen Erscheinungen in ruhenden Körpern” (Annalen der Physik, Vol. 46, pp. 139-151), in which he developed a mechanical interpretation of Maxwell’s equations using a “gyro-static” model of the ether.Google Scholar
  16. 16.
    Newton’s description of the luminous line that appears when the edge of a sharp knife is viewed from the shadow region is given in his Opticks, Book Three, Part 1, Observation 5.Google Scholar
  17. 17.
    Young’s papers are reprinted in Miscellaneous Works of the late Thomas Young (George Peacock, ed., London, 1855, Vol. 1, articles VII, VIII, and IX).Google Scholar
  18. 18.
    According to George Peacock in his Life of Thomas Young, M.D., F.R.S., &c. (London, 1855, pp. 174-185), the anonymous review was probably written by Henry Peter, Lord Brougham (1778-1868), a sycophantic admirer of Newton who later became Lord Chancellor of England. Peacock gives several excerpts; the following is a disgustingly characteristic example: “It is difficult to deal with an author whose mind is filled with a medium of so fickle and vibratory a nature. Were we to take the trouble of refuting him, he might tell us ‘My opinion is changed and I have abandoned that hypothesis, but here is another for you.’ We demand, if the world of science which Newton once illuminated, is to be as changeable in its modes as the world of fashion, which is directed by the nod of a silly woman or a pampered fop? Has the Royal Society degraded its publications into bulletins of new and fashionable theories for the ladies of the Royal Institution? Proh pudor! Let the Professor continue to amuse his audience with an endless variety of such harmless trifles, but in the name of science, let them not find admittance into that venerable repository which contains the work of Newton, and Boyle, and Cavendish, and Maskelyne, and Herschel.”Google Scholar
  19. 19.
    Miscellaneous Works, Vol. 1, article X.Google Scholar
  20. 20.
    A series of cordial letters between Fresnel and Young is included in Young’s Miscellaneous Works, Vol. 1, article XVII.Google Scholar
  21. 21.
    References to the works of Fresnel and a more detailed account of the theories of Fresnel and Young are given by Charles F. Meyer in his very fine The Diffraction of Light, X-Rays, and Material Particles (University of Chicago Press, 1934, Chapter 1).Google Scholar
  22. 22.
    The original papers of Helmholtz and Kirchhoff are cited by B. B. Baker and E. T. Copson in The Mathematical Theory of Huygens’ Principle (Chelsea Publishing Co., New York, 1987, Chapter I, §4.2 and §5.1).Google Scholar
  23. 23.
    A modem version of the Kirchhoff diffraction theory is presented by the Nobel Laureate Max Born (1882-1970) and Emil Wolf in Principles of Optics (Macmillan Co., New York, 1964, Chapter VIII). Kirchhoff’s own theory, based on the elastic solid model of the ether, is discussed in translators’ note 49.Google Scholar
  24. 24.
    Poincaré’s work is cited and discussed by Sommerfeld in his paper and in translators’ note 48.Google Scholar
  25. 25.
    Sommerfeld had previously considered solutions of the two-dimensional heat conduction equation on a Riemann surface in “Zur analytischen Theorie der Wärmeleitung” (Mathematische Annalen, Vol. 45, 1894, pp. 263-277). He writes in his Autobiographische Skizze that he was motivated to consider such solutions by the problem of analyzing earth temperature data from an observation station in the Botanical Garden of Königsberg. “The ‘Physical-Economic Society,’ which my father attended as a member, had in 1891 established a prize for the processing of these data. The station lay near a declivity; thus there arose the problem of investigating heat conduction and integrating the corresponding differential equation in an angular space bounded by two arbitrary half rays inclined toward each other. At that time 1 had already reformulated the problem by the mirroring method onto a Riemann surface with a winding point, without, however, being able to carry out the integration of the heat conduction equation on such a surface. The work I submitted for the prize contained many unique and, as it seemed to me then, new methods, but was incorrect in an essential point concerning the fulfillment of the boundary conditions, and therefore had to be withdrawn. My work was not continued to numerical treatment, but radier remained mired symbolically in mathematical generalities.” Sommerfeld first published bis ideas on the use of Riemann surfaces for the solution of diffraction problems in “Zur mathematischen Theorie der Beugungserscheinungen” (Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, Nr. 4, 1894, pp. 338-342). A translation of this short paper is given in Appendix II. It is interesting to compare Sommerfeld’s initial announcement of his diffraction theory with the full paper published two years later.Google Scholar
  26. 26.
    H. Poincaré, “Sur la polarisation par diffraction (seconde partie)” (Acta mathematica, Vol. 20,1897, pp. 313–355). This second part of Poincaré’s diffraction analysis is discussed briefly at the end of translators’ note 48.MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    H. S. Carslaw, “Some Multiform Solutions of the Partial Differential Equations of Physical Mathematics and their Applications” (Proceedings of the London Mathematical Society, Vol. 30,1899, pp. 121–161). An additional alternative derivation of Sommerfeld’s solution was given by Horace Lamb (1849-1934) in his paper “On Sommerfeld’s Diffraction Problem; and on Reflection by a Parabolic Mirror” (Proceedings 8 of the London Mathematical Society, 2nd series, Vol. 4, 1906, pp. 190-203). Lamb writes that his paper may be “a matter of interest to many who do not feel at home in the refined mathematical theories on which Sommerfeld, in his well known paper, has drawn with such effect.” Sommerfeld gives a retrospective summary of bis diffraction theory in his Optics (Volume IV of the Lectures on Theoretical Physics, Academic Press, New York, 1954, pp. 247-272).MATHGoogle Scholar
  28. 28.
    A. Sommerfeld, “Asymptotische Darstellung von Formeln aus der Beugungstheorie des Lichtes” (Journal für die reine und angewandte Mathematik, Vol. 158, 1927/28, pp. 199–208).MATHGoogle Scholar
  29. 29.
    John William Struct, Lord Rayleigh (1842-1919), “Wave Theory of Light,” in the 1878 Encyclopœdia Britannica.Google Scholar
  30. 30.
    Sommerfeld’s life and work are recounted by Paul Forman and Armin Hermann in the Dictionary of Scientific Biography (C. G. Gillispie, ed., Charles Scribner’s Sons, New York, 1981, Vol. 12, pp. 525–532).Google Scholar

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