A beautiful sea: P. A. M. Dirac's epistemology and ontology of the vacuum

SUMMARY

This paper charts P.A.M. Dirac's development of his theory of the electron, and its radical picture of empty space as an almost-full plenum. Dirac's Quantum Electrodynamics famously accomplished more than the unification of special relativity and quantum mechanics. It also accounted for the ‘duplexity phenomena’ of spectral line splitting that we now attribute to electron spin. But the extra mathematical terms that allowed for spin were not alone, and this paper charts Dirac's struggle to ignore or account for them as a sea of strange, negative-energy, particles with positive ‘holes’. This work was not done in solitude, but rather in exchanges with Dirac's correspondence network. This social context for Dirac’s work contests his image as a lone genius, and documents a community wrestling with the ontological consequences of their work. Unification, consistency, causality, and community are common factors in explanations in the history of physics. This paper argues on the basis of materials in Dirac's archive that --- in addition --- mathematical beauty was an epistemological factor in the development of the electron and hole theory. In fact, if we believe that Dirac's beautiful mathematics captures something of the world, then there is both an epistemology and an ontology of mathematical beauty.

ORCID

Aaron Sidney Wright http://orcid.org/0000-0001-8428-890X

Notes

¹Lorentz transformations for special relativity, and more general transformations in general relativity.

²Olivier Darrigol, From c-Numbers to Q-Numbers: The Classical Analogy in the History of Quantum Theory (Berkeley: University of California Press, 1992).

³Cited in Abraham Pais, ‘Playing with equations, the Dirac way’, in Paul Dirac: the man and his work, ed. by Peter Goddard (Cambridge: Cambridge University Press, 1998), pp. 93–116 (pp. 96–97).

⁴P. A. M. Dirac, ‘The Physical Interpretation of the Quantum Dynamics’, Proceedings of the Royal Society of London. Series A, 113, no. 765 (1927), 621–41. doi:10.1098/rspa.1927.0012

⁵Helge Kragh, ‘The Genesis of Dirac's Relativistic Theory of Electrons’, Archive for History of Exact Sciences, 24, no. 1 (1981), 31–67 (p. 50). doi:10.1007/BF00327714

⁶Jagdish Mehra and Helmut Rechenberg, The Completion of Quantum Mechanics 1926–1941, vol. 6, The Historical Development of Quantum Theory 2 (New York: Springer, 2001); Helge Kragh, Quantum Generations: A History of Physics in the Twentieth Century (Princeton, NJ: Princeton University Press, 1999).

⁷Kragh, ‘The Genesis’ (note 5).

⁸Laurie M. Brown, ‘Some QED myths-in-the-making? : Silvan S. Schweber, QED and the Men Who Made It: Dyson, Feynman, Schwinger and Tomonaga (Princeton University Press, 1994)’, Studies In History and Philosophy of Science Part B: Studies In History and Philosophy of Modern Physics, 27, no. 1 (1996), 81–90. doi:10.1016/1355-2198(95)00023-2

⁹S. S. Schweber, QED and the men who made it: Dyson, Feynman, Schwinger, and Tomonaga (Princeton, NJ: Princeton University Press, 1994).

¹⁰Helge Kragh, Dirac: A Scientific Biography (Cambridge: Cambridge University Press, 1990).

¹¹Dalida Monti, ‘Dirac's Holes Model: From Proton to Positron’, Nuncius, 10, no. 1 (1995), 99–130.

¹²Darrigol, From c-Numbers to Q-Numbers (note 2), p. 302.

¹³Kragh, ‘The Genesis’ (note 5).

¹⁴Kragh, Dirac (note 10), pp. 87–117.

¹⁵P. A. M. Dirac, ‘The Quantum Theory of the Electron’, Proceedings of the Royal Society of London, Series A, 117, no. 778 (1928), 610–24 (p. 610). doi:10.1098/rspa.1928.0023

¹⁶To say that Dirac appealed to an aesthetics of simplicity here is not to equate simplicity with Dirac's concerns for beauty. They are related concepts, but he appealed to them in different contexts and explicitly rejected reducing beauty to simplicity. I thank an anonymous referee for prompting this clarification.

¹⁷P. A. M. Dirac, ‘Note on the Doppler Principle and Bohr's Frequency Condition’, Proceedings of the Cambridge Philosophical Society, 22 (1924), 432; Kragh, ‘The Genesis’ (note 5).

¹⁸Donald Franklin Moyer, ‘Origins of Dirac's Electron, 1925–1928′, American Journal of Physics, 49, no. 10 (1981), 944–9.

¹⁹Walter Gordon, ‘Der Comptoneffekt nach der Schrödingerschen Theorie’, Zeitschrift Für Physik, 40, no. 1/2 (November 1926), 117–77; Oskar Klein, ‘Quantentheorie und fünfdimensionale Relativitätstheorie’, Zeitschrift Für Physik, 37, no. 12 (1926), 895–906. doi:10.1007/BF01397481

²⁰Max Born, ‘On the quantum mechanics of collisions’, in Quantum Theory and Measurement, ed. and trans. by Wojciech Hubert Zurek and John Archibald Wheeler (Princeton, NJ: Princeton University Press, [1926] 1983), pp. 52–61.

²¹There are other senses of mathematical non-linearity, however this is the sense of linearity that was relevant to Dirac and his contemporaries. See C. G. Darwin, ‘The Wave Equations of the Electron’, Proceedings of the Royal Society of London. Series A, 118, no. 780 (1928), 654–80. doi:10.1098/rspa.1928.0076; G. Breit, ‘An Interpretation of Dirac's Theory of the Electron’, PNAS, 14, no. 7 (1928), 553–9.

²²Dirac, ‘The Quantum Theory of the Electron’ (note 15), p. 612.

²³Pais, ‘Playing with equations’ (note 3); Thomas Pashby, ‘Dirac's Prediction of the Positron: A Case Study for the Current Realism Debate’, Perspectives on Science, 20, no. 4 (2012), 440–75.

²⁴That is, new to physics.

²⁵Because the vectors in equation (2) have four components.

²⁶Dirac, ‘The Quantum Theory of the Electron’ (note 15), p. 615.

²⁷Paul Forman, ‘Alfred Landé and the Anomalous Zeeman Effect, 1919-1921’, Historical Studies in the Physical Sciences, 2 (1970), 153–261.

²⁸‘Since half the solutions must be rejected as referring to the charge on the electron, the correct number will be left to account for duplexity phenomena’. Dirac, ‘The Quantum Theory of the Electron’ (note 15), p. 618.

²⁹Interview of Oskar Klein by J. L. Heilbron and L. Rosenfeld on 1963 February 28, Niels Bohr Library & Archives, American Institute of Physics, College Park, MD USA, www.aip.org/history-programs/niels-bohr-library/oral-histories/4709-4 [accessed 31 May 2013].

³⁰N. Bohr to P. Dirac, 27 February 1928, Series II, Box 1, Folder 6, Paul A.M. Dirac Collection, Florida State University Libraries, Tallahassee, FL, 32306 USA (Hereafter ‘Dirac Papers’).

³¹Klein, ‘Quantentheorie und fünfdimensionale Relativitätstheorie’ (note 19); Koray Karaca, ‘Historical and Conceptual Foundations of the Higher Dimensional Unification Program in Physics’ (PhD thesis, Indiana University, 2010).

³²W. Heisenberg to P. Dirac, 13 February 1928, Dirac Papers, Series II, Box 1, Folder 6.

³³P. A. M. Dirac, ‘Über die Quantentheorie des Elektrons', Physikalische Zeitschrift, 29, no. 16 (August 1928), 561–63; P. A. M. Dirac, ‘Zur Quantentheorie Des Elektrons', in Leipziger Vorträge 1928: Quantentheorie Und Chemie, ed. by H. Falkenhagen, trans. by A. Eucken (Leipzig: S. Hirzel, 1928), pp. 85–94.

³⁴Dirac, ‘Über die Quantentheorie des Elektrons' (note 33), p. 563.

³⁵Ibid., p. 563.

³⁶Ibid., p. 563.

³⁷Ibid., p. 563.

³⁸Interview with P. A. M. Dirac by Thomas S. Kuhn and Eugene Paul Wigner, at Wigner's home, Princeton, New Jersey April 1, 1962, http://www.aip.org/history/ohilist/4575_1.html [accessed 21 October 2013].

³⁹Hermann Weyl, ‘Elektron Und Gravitation. I’, Zeitschrift Für Physik, 56, nos. 5-6 (1929), 330–52. doi:10.1007/BF01339504; Hermann Weyl, ‘Gravitation and the Electron’, PNAS, 15, no. 4 (1929), 323–34.

⁴⁰Cited in Pais, ‘Playing with equations' (note 3), p. 99.

⁴¹N. Bohr to P. Dirac, 24 November 1929, Dirac Papers, Series II, Box 1, Folder 9.

⁴²Kragh, Dirac (note 10), pp. 90–3.

⁴³Dirac papers, Series II, Box 14, Folder 2.

⁴⁴Niels Bohr, ‘The Quantum Postulate and the Recent Development of Atomic Theory,’ Nature, 121 (1928), 580–90.

⁴⁵Niels Bohr, ‘Space-Time-Continuity and Atomic Physics', in Niels Bohr Collected Works, ed. Jørgen Kalckar, vol. 6 Foundations of Quantum Physics I (Amsterdam: North-Holland, 1985), p. 369.

⁴⁶W. H. Furry and J. R. Oppenheimer, ‘On the Theory of the Electron and Positive’, Physical Review, 45, no. 4 (Feb 1934), 245–62. doi:10.1103/PhysRev.45.245

⁴⁷Dirac papers, Series II, Box 14, Folder 2.

⁴⁸P. A. M. Dirac, ‘A Theory of Electrons and Protons', Proceedings of the Royal Society of London. Series A, 126, no. 801 (1930), 360–5. doi:10.1098/rspa.1930.0013

⁴⁹Kragh, Dirac (note 10), p. 96.

⁵⁰Dirac's address took place Monday the 8th at 11am. P. A. M. Dirac, ‘The Proton’, Nature, 126, no. 3181 (October 1930), 605–6.

⁵¹The pencil manuscript is in Dirac Papers, Series II, Box 26, Folder 8. In the manuscript he also notes Oppenheimer’s proposal that each of electrons and protons should have their own infinite distribution of negative-energy states, cancelling the infinite charge density. J. R. Oppenheimer, ‘Note on the Theory of the Interaction of Field and Matter’, Physical Review, 35, no. 5 (Mar 1930), 461–77. doi:10.1103/PhysRev.35.461

⁵²Dirac Papers, Series II, Box 26, Folder 8.

⁵³Dirac Papers, Series II, Box 26, Folder 8.

⁵⁴Dirac Papers, Series II, Box 26, Folder 8.

⁵⁵Dirac Papers, Series II, Box 26, Folder 8.

⁵⁶Bruce J Hunt, The Maxwellians (Ithaca, NY: Cornell University Press, 1991), appendix.

⁵⁷W. Heisenberg to P. Dirac, 7 December 1929, Dirac Papers, Series II, Box 1, Folder 9, 1929/7–12.

⁵⁸W. Heisenberg and W. Pauli, ‘Zur Quantendynamik der Wellenfelder’, Zeitschrift Für Physik, 56, nos. 1-2 (1929), 1–61; W. Heisenberg and W. Pauli, ‘Zur Quantentheorie der Wellenfelder. II’, Zeitschrift Für Physik, 59, nos. 3--4 (1930), 168–90.

⁵⁹Karl Hall, ‘The Schooling of Lev Landau: The European Context of Postrevolutionary Soviet Theoretical Physics', Osiris, 23, no. 1 (2008), 230–59. doi:10.1086/591876

⁶⁰For example, Norwood Russell Hanson, ‘Discovering the Positron (I)’, The British Journal for the Philosophy of Science, 12, no. 47 (1961), 208; J. C. Polkinghorne, ‘At the feet of Dirac’, in Reminiscences About a Great Physicist: Paul Adrien Maurice Dirac, ed. by Behram N. Kursunoglu and Eugene Paul Wigner (Cambridge: Cambridge University Press, 1987), p. 221; Weyl's analysis is found in Hermann Weyl, The Theory of Groups and Quantum Mechanics, trans. by H. P. Robertson (London: Methuen & Co. Ltd, [1931] 1932).

⁶¹P. A. M. Dirac, ‘Quantised Singularities in the Electromagnetic Field’, Proceedings of the Royal Society of London. Series A, 133, no. 821 (1931), 60–72. doi:10.1098/rspa.1931.0130

⁶²Weyl's 1929 paper in PNAS was communicated 7 March 1929, his paper in Zeitschrift für Physik was communicated 9 May 1929.

⁶³Pashby, ‘Dirac's Prediction of the Positron’ (note 23), p. 453.

⁶⁴W. Heisenberg to P. Dirac, 16 January 1930, Dirac Papers, Series II, Box 2, Folder 1.

⁶⁵Dirac, ‘A Theory of Electrons and Protons' (note 48), p. 360.

⁶⁶W. Heisenberg to P.Dirac, 16 January 1930, Dirac Papers Series II, Box 2, Folder 1.

⁶⁷See the letter from Tamm of 13 September 1930, Dirac Papers Series II, Box 2, Folder 3.

⁶⁸Igor Tamm to P. Dirac, 3 March 1930, Dirac Papers, Series II, Box 2, Folder 1. See also the correspondence given in A B Kojevnikov, ed., Paul Dirac and Igor Tamm Correspondence; 1, 1928-1933, MPI-Ph-93-80 (Munich: Max-Planck Institut für Physik, 1993).

⁶⁹Igor Tamm to P. Dirac, 3 March 1930, Dirac Papers, Series II, Box 2, Folder 1.

⁷⁰See the published work, Igor Tamm, ‘Über die Wechselwirkung der freien Elektronen mit der Strahlung nach der Diracsehen Theorie des Elektrons und nach der Quantenelektrodynamik’, Zeitschrift Für Physik, 62, nos. 7-8 (1930), 545–68. doi:10.1007/BF01339679

⁷¹This may also be an example of Cambridge style mathematical physics problem solving, as derived from examination problems, spreading out beyond England. See Andrew Warwick, Masters of Theory: Cambridge and the Rise of Mathematical Physics (Chicago, IL: University of Chicago Press, 2003).

⁷²V. Fock to P. Dirac, 12 February 1930, Dirac Papers, Series II, Box 2, Folder 1.

⁷³Ibid.

⁷⁴I have been informed that the letter has not been kept at Fock's archive in Russia. Personal communication, Alexei Kojevnikov, July 2015.

⁷⁵V. Fock to P. Dirac, 1 March 1930, Dirac Papers, Series II, Box 2, Folder 1.

⁷⁶D. Iwanenko to P. Dirac, 16 May 1930, Dirac Papers, Series II, Box 2, Folder 1.

⁷⁷V. Fock and D. Iwanenko, ‘Zur Quantengeometrie’, Physikalische Zeitschrift, 30 (1929), 648; V. Fock and D. Iwanenko, ‘Quantum Geometry’, Nature, 123 (1929), 838.

⁷⁸V. Ambarzumian and D. Iwanenko, ‘Zur Frage nach Vermeidung der unendlichen Selbstrückwirkung des Elektrons', Zeitschrift Für Physik, 64, nos. 7-8 (1930), 563–7; Helge Kragh and Bruno Carazza, ‘From Time Atoms to Space-Time Quantization: The Idea of Discrete Time, ca 1925–1936’, Studies in History and Philosophy of Science Part A, 25, no. 3 (1994), 457–8. doi:http://dx.doi.org/10.1016/0039-3681(94)90061-2

⁷⁹Joan Bromberg, ‘The Impact of the Neutron: Bohr and Heisenberg’, Historical Studies in the Physical Sciences, 3 (1971), 324.

⁸⁰I thank an anonymous referee for directing me to this literature. W. Heisenberg, ‘The Self-Energy of the Electron’, in Early Quantum Electrodynamics: A Source Book, ed. by Arthur Miller (Cambridge: Cambridge University Press, [1930] 1994), pp. 121–8; B. Carazza and H. Kragh, ‘Heisenberg's lattice world: The 1930 theory sketch’, American Journal of Physics, 63 (July 1995), 595. doi:10.1119/1.17848

⁸¹W. Heisenberg to P. Dirac, 14 July 1930, Dirac Papers, Series II, Box 2, Folder 3.

⁸²Kragh, Quantum Generations (note 6), pp. 105–8.

⁸³W. Heisenberg to P. Dirac, 14 July 1930, Dirac Papers, Series II, Box 2, Folder 3.

⁸⁴Weyl, ‘Elektron Und Gravitation. I’ (note 39); Weyl, ‘Gravitation and the Electron’ (note 39).

⁸⁵Darwin, ‘The Wave Equations of the Electron’ (note 21), p. 680; Donald Franklin Moyer, ‘Evaluations of Dirac's Electron, 1928–1932’, American Journal of Physics, 49, no. 11 (1981), 1055–62.

⁸⁶In which Dirac cites Weyl's (1929) Zeitschrift für Physik paper. Dirac, 1930, page 361.

⁸⁷Tamm, ‘Über die Wechselwirkung der freien Elektronen mit der Strahlung nach der Diracsehen Theorie des Elektrons und nach der Quantenelektrodynamik’ (note 70); Oppenheimer, ‘Note on the Theory of the Interaction of Field and Matter’ (note 51).

⁸⁸P. A. M. Dirac, ‘The Origin of Quantum Field Theory’, in The Birth of Particle Physics, ed. by Laurie M. Brown and Lillian Hoddeson (Cambridge: Cambridge University Press, 1983), p. 52.

⁸⁹Joan Bromberg, ‘The Concept of Particle Creation Before and After Quantum Mechanics', Historical Studies in the Physical Sciences, 7 (1976), 186–7.

⁹⁰Dirac, ‘Quantised Singularities in the Electromagnetic Field’ (note 61).

⁹¹Kragh, Quantum Generations (note 6), pp. 174–7.

⁹²Dirac Papers, Series II, Box 26, Folder 15, ‘Lecture on Quantum Mechanics', 72.

⁹³Interview with Dr. P. A. M. Dirac by Thomas S. Kuhn at Cambridge, England, May 6, 1963, Niels Bohr Library & Archives, American Institute of Physics, College Park, MD USA, http://www.aip.org/history/ohilist/4575_2.html [accessed 21 October 2013].

⁹⁴Ibid.

⁹⁵Ibid.

⁹⁶Dirac Papers, Series II, Box 26, Folder 15, ‘Lecture on Quantum Mechanics', 102–3.

⁹⁷P. A. M. Dirac, ‘The Quantum Theory of the Emission and Absorption of Radiation’, Proceedings of the Royal Society of London. Series A, 114, no. 767 (1927), 260–1. doi:10.1098/rspa.1927.0039; Moyer, ‘Origins of Dirac's Electron, 1925–1928’ (note 18).

⁹⁸Dirac Papers, Series II, Box 26, Folder 15, ‘Lecture on Quantum Mechanics', 131–2.

⁹⁹Dirac Papers, Series II, Box 26, Folder 15, ‘Lecture on Quantum Mechanics', 132.

¹⁰⁰Dirac Papers, Series II, Box 26, Folder 15, ‘Lecture on Quantum Mechanics', 132.

¹⁰¹Ibid., 134.

¹⁰²Dirac, ‘Quantised Singularities in the Electromagnetic Field’ (note 61).

¹⁰³Laurie M. Brown and Lillian Hoddeson, ‘The Birth of Elementary Particle Physics: 1930–1950’, in The Birth of Particle Physics, ed. by Laurie M. Brown and Lillian Hoddeson (Cambridge: Cambridge University Press, 1983), pp. 3–36; Carl D. Anderson and Herbert L. Anderson, ‘Unraveling the Particle Content of Cosmic Rays', in The Birth of Particle Physics, ed. by Laurie M. Brown and Lillian Hoddeson (Cambridge: Cambridge University Press, 1983), pp. 131–54.

¹⁰⁴Carl D. Anderson, ‘The Apparent Existence of Easily Deflectable Positives', Science, 76 (9 September 1932), 238–39; Carl D. Anderson, ‘The Positive Electron’, Physical Review, 43, no. 6 (Mar 1933), 491–4. doi:10.1103/PhysRev.43.491; Patrick M.S. Blackett and G.P.S. Occhialini, ‘Some Photographs of the Tracks of Penetrating Radiation’, Proceedings of the Royal Society of London. Series A, 139, no. 839 (1933), 699–726.

¹⁰⁵Hanson, ‘Discovering the Positron (I)’ (note 60); Norwood Russell Hanson, ‘Discovering the Positron (II)’, The British Journal for the Philosophy of Science, 12, no. 48 (1962), 299–313.

¹⁰⁶Behram N. Kursunoglu and Eugene Paul Wigner, eds., Reminiscences About a Great Physicist: Paul Adrien Maurice Dirac (Cambridge: Cambridge University Press, 1987).

¹⁰⁷Xavier Roqué, ‘The Manufacture of the Positron’, Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 28, no. 1 (1997), 73–129.

¹⁰⁸Quoted in Pais, ‘Playing with equations' (note 3), pp. 101–2.

¹⁰⁹Dirac Papers, Series II, Box 26, Folder 17. The published version is P. A. M. Dirac, ‘Théorie du positron’, in Structure et propriétés des noyaux atomiques : rapports et discussions du septième Conseil de physique tenu à Bruxelles du 22 du 29 octobre 1933, sous les auspices de l'Institut international de physique Solvay (1934) Paris: Gauthier-Villars, 203--12.

¹¹⁰Ibid.

¹¹¹Here self-consistency refers to consistency within a given theory and inter-theoretical consistency refers to consistency between theories.

¹¹²Dirac Papers, Series II, Box 26, Folder 17.

¹¹³Olivier Darrigol, ‘The Electron Theories of Larmor and Lorentz: A Comparative Study’, Historical Studies in the Physical and Biological Sciences, 24, no. 2 (1994), 265–336, http://www.jstor.org/stable/27757725.

¹¹⁴Dirac papers, Series II, Box 26, Folder 17.

¹¹⁵Dirac papers, Series II, Box 26, Folder 17.

¹¹⁶P. A. M. Dirac, ‘Nobel Lecture: Theory of Electrons and Positrons', Nobel Media AB, [1933] 2013, http://www.nobelprize.org/nobel_prizes/physics/laureates/1933/dirac-lecture.html [accessed 21 October 2013].

¹¹⁷Dirac Papers, Series II, Box 36, Folder 22.

¹¹⁸See also Dirac's 6 May 1967 interview with T.S. Kuhn, available at http://www.aip.org/history/ohilist/4575_2.html [accessed 21 October 2013]; Peter Galison, ‘The Suppressed Drawing: Paul Dirac's Hidden Geometry’, Representations, no. 72 (2000), 145–66.

¹¹⁹Dirac Papers, Series II, Box 36, Folder 22.

¹²⁰P. A. M. Dirac, ‘Discussion of the Infinite Distribution of Electrons in the Theory of the Positron’, Mathematical Proceedings of the Cambridge Philosophical Society, 30, no. 2 (March 1934), 150–63. doi:10.1017/S030500410001656X; H. J. Bhabha, ‘The Scattering of Positrons by Electrons with Exchange on Dirac's Theory of the Positron’, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 154, no. 881 (1936), 195–206.

¹²¹P. A. M. Dirac, The Principles of Quantum Mechanics (Oxford: Clarendon Press, 1930); W. Heisenberg, ‘Bemerkungen Zur Diracschen Theorie Des Positrons', Zeitschrift Für Physik, 90, nos. 3--4 (1934), 209–31. doi:10.1007/BF01333516; W. Heisenberg and H. Euler, ‘Folgerungen aus der Diracschen Theorie des Positrons', Zeitschrift Für Physik, 98, nos. 11-12 (1936), 714–32; W. Heisenberg and H. Euler, ‘Consequences of Dirac Theory of the Positron’, trans. by W. Korolevski and H. Kleinert, ArXiv Physics E-Prints, May [1936] 2006.

¹²²P. A. M. Dirac, Lectures on Quantum Electrodynamics (Princeton, NJ: Institute for Advanced Study, 1935), Chapter IV.

¹²³V. Weisskopf, ‘Über Die Selbstenergie Des Elektrons', Zeitschrift Für Physik, 89, nos. 1-2 (1934), 27–39; V. Weisskopf, ‘Über die Elektrodynamik des Vakuums auf Grund der Quantentheorie des Elektrons', Det Kgl. Danske Videnskabernes Selskab. Mathematisk-Fysiske Meddelelser, XIV, no. 6 (1936), 1–39; V. F. Weisskopf, ‘On the Self-Energy and the Electromagnetic Field of the Electron’, Physical Review, 56 (July 1939), 72–85. Mehra and Rechenberg, The Completion of Quantum Mechanics 1926–1941 Part IV, 3; Moyer, ‘Evaluations of Dirac's Electron, 1928–1932’ (note 85).

¹²⁴Sophia Roosth and Susan Silbey, ‘Science and Technology Studies: From Controversies to Posthumanist Social Theory’, in The New Blackwell Companion to Social Theory, ed. Bryan S. Turner (Malden, MA: Blackwell Publishing Ltd., 2009), 451–73.

¹²⁵Schweber, QED and the men who made it (note 9), pp. 76–152.

¹²⁶Graham Farmelo, The Strangest Man: The Hidden Life of Paul Dirac, Mystic of the Atom (New York: Basic Books/Perseus Books, 2009).

¹²⁷Kragh, Dirac (note 10), Ch. 4.

¹²⁸Ludwik Fleck, Genesis and Development of a Scientific Fact (Chicago: The University of Chicago Press, [1935] 1981); K Knorr-Cetina, Epistemic Cultures: How the Sciences Make Knowledge (Cambridge, MA: Harvard University Press, 1999).

¹²⁹Darwin, ‘The Wave Equations of the Electron’ (note 21), p. 655.

¹³⁰Dirac, ‘The Quantum Theory of the Electron’ (note 15), p. 612.

¹³¹E. Uhlenbeck and S. Goudsmit, Naturwissenschaften 13, no. 47 (1925), 953; G. E. Uhlenbeck and S. Goudsmit, ‘Spinning Electrons and the Structure of Spectra’, Nature, 117 (February 1926), 264–5 and postscript by N. Bohr; S. A. Goudsmit, ‘The Discovery of the Electron Spin’, in Foundations of Modern EPR, ed. by G.R. Eaton, S.S. Eaton, and K.M. Salikhov, trans. by J. H. van der Waals (Singapore: World Scientific, [1971] 1998), [online].

¹³²Nadia Robotti, ‘Quantum Numbers and Electron Spin: The History of a Discovery’, Archives Internationale d’Histoire Des Sciences, 40, no. 125 (1990), 305–31.

¹³³Dirac, ‘The Quantum Theory of the Electron’ (note 15), p. 610.

¹³⁴Ibid., p. 610.

¹³⁵Dirac's understanding of Bohr's correspondence principle in terms of taking a large-quantum-number limit of a quantum equation dates from Fowler's lectures at Cambridge in the mid-1920s. See Dirac Papers, Series II, Box 32, Folder 4, ‘Fowler's Lectures of Quantum Theory. Cambridge 1924–1926’, [15].

¹³⁶‘Our problem is to obtain a wave equation of the [linear] form [ … ] which shall be invariant under a Lorentz transformation and shall be equivalent to [the classical case] in the limit of large quantum numbers'. Dirac, ‘The Quantum Theory of the Electron’ (note 15), p. 613.

¹³⁷Ibid., p. 610.

¹³⁸J.W. McAllister, ‘Dirac and the Aesthetic Evaluation of Theories', Methodology and Science, 23 (1990), 87–102.

¹³⁹Pais, ‘Playing with equations' (note 3), p. 98.

¹⁴⁰Ibid., p. 109.

¹⁴¹Dirac, ‘Quantised Singularities in the Electromagnetic Field’ (note 61).

¹⁴²P. A. M. Dirac, ‘The Relation between Mathematics and Physics', Proceedings of the Royal Society (Edinburgh), 59, no. Part II (1938–39), 122–9.

¹⁴³Quoted in Pais, ‘Playing with equations', p. 110.

¹⁴⁴Darrigol, From c-Numbers to Q-Numbers (note 2), p. 302.

¹⁴⁵Kragh, Dirac (note 10), pp. 275–92.

¹⁴⁶Ibid., p. 292.

¹⁴⁷But see Darrigol, From c-Numbers to Q-Numbers (note 2), p. 345.

¹⁴⁸Andrew Pickering, ‘Constraints on Controversy: The Case of the Magnetic Monopole’, Social Studies of Science, 11, no. 1 (February 1981), 63–93.

¹⁴⁹Aaron Sidney Wright, ‘The Advantages of Bringing Infinity to a Finite Place: Penrose diagrams as objects of intuition’, Historical Studies in the Natural Sciences, 44, no. 2 (2014), 99–139. doi:10.1525/hsns.2014.44.2.99

¹⁵⁰Dirac, ‘Quantised Singularities in the Electromagnetic Field’ (note 61), p. 60.

¹⁵¹David Hilbert, ’Mathematical problems', Bulletin of the American Mathematical Society, 8 (10) ([1900] 1902), 437–79.

¹⁵²Dirac, ‘Quantised Singularities in the Electromagnetic Field’ (note 61), p. 60.

¹⁵³‘A recent paper by the author may possibly be regarded as a small step according to this general scheme of advance’ (ibid., p. 61).

¹⁵⁴Dirac Papers, Series II, Box 26, Folder 15, ‘Lecture on Quantum Mechanics', p. 1.

¹⁵⁵Ibid., p. 60.

¹⁵⁶See also Darrigol, From c-Numbers to Q-Numbers (note 2), p. 308.

¹⁵⁷Helge Kragh, ‘Cosmo-Physics in the Thirties: Towards a History of Dirac Cosmology’, Historical Studies in the Physical Sciences, 13 (1982), 69–108.

¹⁵⁸I thank an anonymous referee for making this point. See Thomas Ryckman, The Reign of Relativity: Philosophy in Physics 1915-1925 (New York: Oxford University Press, 2005), p. 205.

¹⁵⁹Arthur Stanley Eddington, Space, Time and Gravitation: An Outline of the General Relativity Theory (Cambridge: Cambridge University Press, 1920); Arthur Stanley Eddington, The Mathematical Theory of Relativity (Cambridge: Cambridge University Press, 1923).

¹⁶⁰Darrigol, From c-Numbers to Q-Numbers (note 2), p. 300.

¹⁶¹C. W Kilmister, Eddington's Search for a Fundamental Theory: A Key to the Universe (Cambridge: Cambridge University Press, 1994), p. 61.

¹⁶²Darrigol, From c-Numbers to Q-Numbers (note 2), pp. 30102.

¹⁶³Ibid., p. 302.

¹⁶⁴Henry Frederick Baker's Cambridge tea parties, see ibid., pp. 295–6.

¹⁶⁵Mara Beller, Quantum Dialogue: The Making of a Revolution (Chicago, IL: University of Chicago Press, 1999), Ch. 4.

¹⁶⁶These notes are also available in the Archive for the History of Quantum Physics, American Philosophical Society, Microfilm 36, document 11.

¹⁶⁷Dirac Papers, Series II, Box 26, Folder 2, ‘Lectures on Modern Quantum Mechanics', 1.

¹⁶⁸Ibid.

¹⁶⁹Ibid.

¹⁷⁰Dirac Papers, Series II, Box 26, Folder 4, ‘Cambridge Adams Society Elements of Quantum Mechanics', 1.

¹⁷¹Ibid.

¹⁷²Mara Beller, ‘The Rhetoric of Antirealism and the Copenhagen Spirit’, Philosophy of Science, 63, no. 2 (Jun. 1996), 183–204.

¹⁷³Dirac, ‘The Relation between Mathematics and Physics' (note 142). Published as P.A.M. Dirac, ‘The Relation between Mathematics and Physics', Proceedings of the Royal Society (Edinburgh), 59(Part II) (1938-39), 122–9.

¹⁷⁴Dirac Papers, Series II, Box 36, Folder 27.

¹⁷⁵Dirac Papers, Series II, Box 29, Folder 2, Harvard ‘The Road that Led to Antimatter’, 1.

¹⁷⁶Dirac Papers, Series II, Box 29, Folder 18, ‘Lindau’ ‘Basic beliefs and Prejudices in Physics', 2.

¹⁷⁷Printed in P. A. M. Dirac, ‘The Mathematical Foundations of Quantum Theory’, in Mathematical Foundations of Quantum Theory, ed. by A. R. Marlow (New York: Academic Press, 1978), pp. 1–8.

¹⁷⁸Dirac Papers, Series II, Box 29, Folder 22, ‘New Orleans', ‘[Mathematical foundations of quantum theory]’, 1–2.

¹⁷⁹Dirac Papers, Series II, Box 29, Folder 42, ‘New Orleans' ‘Pretty Mathematics', 2.

¹⁸⁰Interview with P. A. M. Dirac by Thomas S. Kuhn and Eugene Paul Wigner, at Wigner's home, Princeton, New Jersey April 1, 1962, http://www.aip.org/history/ohilist/4575_1.html, [accessed 21 October 2013].

¹⁸¹Warwick, Masters of Theory (note 71), pp. 176–226.

¹⁸²And Dirac was not studying for the mathematical tripos exam that centres Warwick's analysis. Dirac arrived at Cambridge with a degree from Bristol as a research student studying for the PhD.

¹⁸³Aaron Sidney Wright, ‘The Advantages of Bringing Infinity to a Finite Place: Penrose diagrams as objects of intuition’, note 149.

¹⁸⁴Interview with P. A. M. Dirac by Thomas S. Kuhn and Eugene Paul Wigner, at Wigner's home, Princeton, New Jersey April 1, 1962, http://www.aip.org/history/ohilist/4575_1.html [accessed 21 October 2013].

¹⁸⁵Dirac Papers, Series II, Box 36, Folder 22, ‘Theory of the Positron’.

¹⁸⁶Galison, ‘The Suppressed Drawing’ (note 118).

¹⁸⁷Dirac Papers, Series II, Box 29 Folder 17 ‘[Mathematics will lead you by the hand]’.

¹⁸⁸Interview with Dr. P. A. M. Dirac, By Thomas S. Kuhn, At Cambridge, England, May 6, 1963, http://www.aip.org/history/ohilist/4575_2.html [accessed 27 October 2013].

¹⁸⁹P. A. M. Dirac, General Theory of Relativity (Princeton, NJ: Princeton University Press, [1975] 1996).

Additional information

Funding

The author gratefully acknowledges the support of the Social Sciences and Humanities Research Council of Canada [767-2011-2547, 756-2014-0128].