Pascual Jordan's resolution of the conundrum of the wave-particle duality of light

Anthony Duncan, Michel Janssen

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In 1909, Einstein derived a formula for the mean square energy fluctuation in black-body radiation. This formula is the sum of a wave term and a particle term. In a key contribution to the 1926 Dreimännerarbeit with Born and Heisenberg, Jordan showed that one recovers both terms in a simple model of quantized waves. So the two terms do not require separate mechanisms but arise from a unified dynamical framework. In this paper, we give a detailed reconstruction of Jordan's derivation of this result and discuss the curious story of its reception. Several authors have argued that various infinities invalidate Jordan's result. We defend it against such criticism. In particular, we note that the fluctuation in a narrow frequency range, which is what Jordan calculated, is perfectly finite. We also note, however, that Jordan's argument is incomplete. In modern terms, Jordan calculated the quantum uncertainty in the energy of a subsystem in an energy eigenstate of the system as a whole whereas the thermal fluctuation is the average of this quantity over an ensemble of such states. Still, our overall conclusion is that Jordan's argument is basically sound and that he deserves more credit than he received for having resolved a major conundrum in the development of quantum physics.

Original languageEnglish (US)
Pages (from-to)634-666
Number of pages33
JournalStudies in History and Philosophy of Science Part B - Studies in History and Philosophy of Modern Physics
Issue number3
StatePublished - Sep 2008

Bibliographical note

Funding Information:
A preliminary version of this paper was presented at HQ1, a conference on the history of quantum physics held at the Max-Planck-Institut für Wissenschaftsgeschichte, Berlin, July 2–6, 2007. We thank Clayton Gearhart, Don Howard, Joska Illy, Alexei Kojevnikov, Serge Rudaz, Rob Rynasiewicz, John Stachel, Jos Uffink, two anonymous referees, and, especially, Jürgen Ehlers for helpful comments, discussion, and references. We thank Michael Jordan for permission to quote extensively from his father's unpublished correspondence and from the interview with his father for the Archive for History of Quantum Physics (AHQP). We likewise thank Jochen Heisenberg for permission to quote from an unpublished letter from his father, which is available in microfilm as part of the AHQP. We also thank the staff at Walter Library at the University of Minnesota, where we consulted a copy of the AHQP. Finally, we thank the Einstein Archive at Hebrew University and Princeton University Press for permission to quote an unpublished postcard from Einstein to Jordan. The research of Anthony Duncan is supported in part by the National Science Foundation under Grant PHY-0554660.


  • Dreimännerarbeit
  • Fluctuations
  • Matrix mechanics
  • Pascual Jordan
  • Quantum field theory
  • Umdeutung
  • Wave-particle duality


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