Pascual Jordan’s legacy and the ongoing research in quantum field theory⋆

  • B. SchroerEmail author


Pascual Jordan’s path-breaking role as the protagonist of quantum field theory (QFT) is recalled and his friendly dispute with Dirac’s particle-based relativistic quantum theory is presented as the start of the field-particle conundrum which, though in modified form, persists up to this date. Jordan had an intuitive understanding that the existence of a causal propagation with finite propagation speed in a quantum theory led to radically different physical phenomena than those of QM. The conceptional-mathematical understanding for such an approach began to emerge only 30 years later. The strongest link between Jordan’s view of QFT and modern “local quantum physics” is the central role of causal locality as the defining principle of QFT as opposed to the Born localization in QM. The issue of causal localization is also the arena where misunderstandings led to a serious derailment of large part of particle theory e.g. the misinterpretation of an infinite component pointlike field resulting from the quantization of the Nambu-Goto Lagrangian as a spacetime quantum string. The new concept of modular localization, which replaces Jordan’s causal locality, is especially important to overcome the imperfections of gauge theories for which Jordan was the first to note nonlocal aspects of physical(not Lagrangian) charged fields. Two interesting subjects in which Jordan was far ahead of his contemporaries will be presented in two separate sections.


Gauge Theory Operator Algebra Vacuum Polarization Causal Localization Modular Localization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    D. Hoffmann, J. Ehlers, J. Renn, Pascual Jordan (1902–1980), Mainzer Symposium zum 100. Geburtstag. Max Planck Institute for History of Science, preprint 329 (2007)Google Scholar
  2. 2.
    Bert Schroer, Pascual Jordan, Biographical notes, his contributions to quantum mechanics and his role as a protagonist of quantum field theory, in Hoffmann et al. (2007), 47–68Google Scholar
  3. 3.
    Wolf D. Beiglböck, Ernst Pascual Jordan als Autor wissenschaftlicher und allgemeinbildender Schriften, pp. 145–206. in D. Hoffmann, J. Ehlers, J. Renn, Pascual Jordan (1902–1980), Mainzer Symposium zum 100. Geburtstag. Max Planck Institute for History of Science, preprint 329 (2007)Google Scholar
  4. 4.
    Max Born, Werner Heisenberg, Pascual Jordan, Zur Quantenmechanik II, Zeitschrift für Physik 35, 557–615 (1926)ADSGoogle Scholar
  5. 5.
    Albert Einstein, Zur Quantentheorie der Strahlung, Physikalische Zeitschrift 18, 121–128 (1917). English translation, in Van der Waerden (1968), 63–77Google Scholar
  6. 6.
    Anthony Duncan, Michel Janssen, Pascual Jordan’s resolution of the conundrum of the wave-particle duality of light, arXiv:0709.3812Google Scholar
  7. 7.
    Bert Schoer, The Einstein-Jordan conundrum and its relation to ongoing foundational research in local quantum physics, arXiv:1101.0569Google Scholar
  8. 8.
    Pascual Jordan, O. Klein, Zum Mehrkörperproblem in der Quantentheorie, Zeitschrift für Physik 45, 751–765 (1927)ADSCrossRefGoogle Scholar
  9. 9.
    Bert Schroer, Localization and the interface between quantum mechanics, quantum field theory and quantum gravity I (The two antagonistic localizations and their asymptotic compatibility), Studies in History and Philosophy of Modern Physics 41, 104–127 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bert Schroer, Localization and the interface between quantum mechanics, quantum field theory and quantum gravity II (The search of the interface between QFT and QG), Studies in History and Philosophy of Modern Physics 41, 293–308 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bert Schroer, Bondi-Metzner-Sachs symmetry, holography on null-surfaces and area proportionality of “light-slice” entropy, Foundations of Physics 41, 204–241 (2011)ADSzbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Pascual Jordan, Zur Theorie der Quantenstrahlung, Zeitschrift für Physik 30, 297–319 (1924)ADSCrossRefGoogle Scholar
  13. 13.
    Albert Einstein, Bemerkungen zu P. Jordans: Zur Theorie der Quantenstrahlung, Zeitschrift für Physik 31, 784–785 (1925)ADSCrossRefGoogle Scholar
  14. 14.
    W. Heitler, The quantum theory of radiation (Clarendon Press, Oxford, 1936)Google Scholar
  15. 15.
    Eugene Paul Wigner, On unitary representations of the inhomogeneous Lorentz group, Ann. Math. 40, 149–204 (1939)MathSciNetCrossRefGoogle Scholar
  16. 16.
    F. Coester, W.N. Polyzou, Relativistic quantum-mechanics of particles with direct interactions, Phys. Rev. D 26, 1348–1367 (1982)MathSciNetADSCrossRefGoogle Scholar
  17. 17.
    Steven Weinberg, What is Quantum Field Theory and what did we think it was? pp. 241–251, in Conceptual Foundations of Quantum Field Theory (ed. Tian Yu Cao, Cambridge University Press, 1999)Google Scholar
  18. 18.
    Pascual Jordan, Kausalitaet und Statistik in der Modernen Physik, Die Naturwissenschaften 15, 105–110 (1927)ADSCrossRefGoogle Scholar
  19. 19.
    H. Halvorson, Reeh-Schlieder defeats Newton-Wigner: on alternative localization schemes in quantum field theory, Philos. Sci. 68, 111–133 (2001)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Pascual Jordan, Wolfgang Pauli, Zur Quantenelektrodynamik ladungsfreier Felder, Zeitschrift für Physik 47, 151–173 (1928)ADSCrossRefGoogle Scholar
  21. 21.
    Werner Heisenberg, Über die mit der Entstehung von Materie aus Strahlung verknüpften Ladungsschwankungen, Verhandlungen der Sächsischen Akademie der Wissenschaften zu Leipzig 86, 317–322 (1934)Google Scholar
  22. 22.
    W.H. Furry, J. Robert Oppenheimer, On the theory of the electron and positive, Phys. Rev. 45, 245–262 (1934)ADSzbMATHCrossRefGoogle Scholar
  23. 23.
    Bert Schroer, A critical look at 50 years particle theory from the perspective of the crossing property, Found. Phys. 40, 1800–1857 (2010)MathSciNetADSCrossRefzbMATHGoogle Scholar
  24. 24.
    B. Bakamijan, L.H. Thomas, Relativistic particle dynamics II, Phys. Rev. 92, 1300–1310 (1953)ADSCrossRefGoogle Scholar
  25. 25.
    F. Coester, Scattering theory for relativistic particles, Helvetica Phys. Acta 38, 7–28 (1965)MathSciNetzbMATHGoogle Scholar
  26. 26.
    N.S. Sokolov, Interacting relativistic particles, Doklady Akad. Nauk USSR 233, 575–592 (1977)Google Scholar
  27. 27.
    W.N. Polyzou, Equivalent Hamiltonians. Phys. Rev. C. 82, 014002 (2010)ADSCrossRefGoogle Scholar
  28. 28.
    W.N. Polyzou, Cluster properties in relativistic quantum mechanics of N-particle systems, J. Math. Phys. 43, 6024–6038 (2002)MathSciNetADSzbMATHCrossRefGoogle Scholar
  29. 29.
    S. Weinberg, The Quantum Theory of Fields I (Cambridge University Press, 1995)Google Scholar
  30. 30.
    S.S. Schweber, QED and the men who made it (Dyson, Feynman, Schwinger and Tomonaga, Princeton University Press, 1994)Google Scholar
  31. 31.
    Rudolf Haag, Local Quantum Physics (Springer, 1996)Google Scholar
  32. 32.
    H.-J. Borchers, On revolutionizing quantum field theory with Tomita’s modular theory, J. Math. Phys. 41, 3604–3873 (2000)MathSciNetADSzbMATHCrossRefGoogle Scholar
  33. 33.
    W.G. Unruh, Notes on black hole evaporation, Phys. Rev. D 14, 870–892 (1976)ADSCrossRefGoogle Scholar
  34. 34.
    J.J. Bisognano, E.H. Wichmann, On the duality condition for quantum fields, J. Math. Phys. 17, 303–321 (1976)MathSciNetADSCrossRefGoogle Scholar
  35. 35.
    G.L. Sewell, PCT and gravitationally induced, Ann. Phys. 141, 201–224 (1982)MathSciNetADSCrossRefGoogle Scholar
  36. 36.
    Bert Schroer, Modular localization and the bootstrap-formfactor program, Nucl. Phys. B 499, 547–568 (1997)MathSciNetADSzbMATHCrossRefGoogle Scholar
  37. 37.
    R. Brunetti, D. Guido, R. Longo, Modular localization and Wigner particles, Rev. Math. Phys. 14, 759–785 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    L. Fassarella, Bert Schroer, Wigner particle theory and local quantum physics, J. Phys. A 35, 9123–9164 (2002)MathSciNetADSzbMATHCrossRefGoogle Scholar
  39. 39.
    Jens Mund, Bert Schroer, Jakob Yngvason, String-localized quantum fields and molecular localization, Commun. Math. Phys. 268, 621–672 (2006)MathSciNetADSzbMATHCrossRefGoogle Scholar
  40. 40.
    Jens Mund, Modular localization of massive particles with “any” spin in d = 2 + 1 dimensions, J. Math. Phys. 44, 2037–2057 (2003)MathSciNetADSzbMATHCrossRefGoogle Scholar
  41. 41.
    R. Longo, Simple proof of existence of modular automorphisms in approximately finite dimensional von Neumann algebras, Pacific J. Math. 75, 199–205 (1978)MathSciNetzbMATHGoogle Scholar
  42. 42.
    R.F. Streater, A.S. Wightman, PCT, Spin and Statistics and all that (New York, Benjamin, 1964)Google Scholar
  43. 43.
    Jens Mund, The Bisognano-Wichmann theorem for massive theories, Ann. Henri Poincare 2, 907–926 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Alain Connes, Caractérisation des espaces vectoriels ordonnées sous-jacents aux algèbres de von Neumann, Ann. Inst. Fourier 24, 121–155 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Bert Schroer, Unexplored regions in QFT and the conceptual foundations of the Standard Model, arXiv:1006.3543Google Scholar
  46. 46.
    Ozwaldo Zapata, On Facts in Superstring Theory, A Case Study: The AdS/CFT Correspondence, arXiv:0905.1439Google Scholar
  47. 47.
    Ozwaldo Zapata, Spinning the superweb, essays on the history of string theory,
  48. 48.
    K. Pohlmeyer, A group-theoretical approach to the quantization of the free relativistic closed string, Phys. Lett. B 119, 100–104 (1982)MathSciNetADSCrossRefGoogle Scholar
  49. 49.
    Y. Nambu, Lectures at the Copenhagen Symposium, 1970, unpublishedGoogle Scholar
  50. 50.
    Tetsuo Goto, Relativistic quantum mechanics of one-dimensional mechanical continuum and subsidiary condition of dual resonance model, Progr. Theor. Phys. 46, 1560–1569 (1971)MathSciNetADSzbMATHCrossRefGoogle Scholar
  51. 51.
    E. Martinec, The light-cone in string theory, Class. Quant. Grav. 10, 187–192 (1993)MathSciNetADSCrossRefGoogle Scholar
  52. 52.
    D.A. Lowe, Causal properties of free string field-theory, Phys. Lett. B 326, 223–230 (1994)ADSCrossRefGoogle Scholar
  53. 53.
    Paolo Di Vecchia, The birth of string theory, Lect. Not. Phys. 737, 59–118 (2008)MathSciNetADSCrossRefGoogle Scholar
  54. 54.
    E. Witten, Global aspects of current algebra, Nucl. Phys. B 223, 422–432 (1983)MathSciNetADSCrossRefGoogle Scholar
  55. 55.
    Y. Kawahigashi, R. Longo, U. Pennig, K.-H. Rehren, The classification of non-local chiral CFT with c < 1, Commun. Math. Phys. 271, 375–385 (2007)MathSciNetADSzbMATHCrossRefGoogle Scholar
  56. 56.
    Gandalf Lechner, On the Construction of Quantum Field Theories with Factorizing S-Matrices, PhD thesis, arXiv:math-ph/0611050Google Scholar
  57. 57.
    Gerhard Mack, D-dimensional Conformal Field Theories with anomalous dimensions as Dual Resonance Models, arXiv:0909.1024,Google Scholar
  58. 58.
    Gerhard Mack, D-independent representations of conformal field theories in D dimensions via transformations to auxiliary dual resonance models, Scalar amplitude, arXiv:0907:2407Google Scholar
  59. 59.
    N.N. Bogoliubov, A. Logunov, A.I. Oksak, I.T. Todorov, General principles of quantum field theory (Dordrecht Kluwer, 1990)Google Scholar
  60. 60.
    Detlev Buchholz, Klaus Fredenhagen, Locality and the structure of particle states, Commun. Math. Phys. 84, 1–54 (1982) MathSciNetADSzbMATHCrossRefGoogle Scholar
  61. 61.
    Pascual Jordan, The Present State of Quantum Electrodynamics, in Talks and Discussions of the Theoretical-Physical Conference in Kharkov (May 19.–25., 1929)Google Scholar
  62. 62.
    Pascual Jordan, The current position of quanta electro dynamics, Physik. Zeitschr. 30, 700–712 (1929)Google Scholar
  63. 63.
    Pascual Jordan, Zur Quantenelektrodynamik. III. Eichinvariante Quantelung und Diracsche Magnetpole, Zeitschrift für Physik 97, 535–537 (1935)ADSzbMATHCrossRefGoogle Scholar
  64. 64.
    Roman W. Jackiw, Dirac’s magnetic monopoles (again), Int. J. Mod. Phys. A 19S1, 137–143 (2004)MathSciNetCrossRefGoogle Scholar
  65. 65.
    Rudolf Haag, Discussion of the ‘axioms’ and the asymptotic properties of a local field theory with composite particles (historical document), Eur. Phys. J. H 35, 243–253 (2010)CrossRefMathSciNetGoogle Scholar
  66. 66.
    H. Epstein, V. Glaser, Role of locality in perturbation-theory, Ann. Inst. H. Poincaré A 19, 211–295 (1973)MathSciNetGoogle Scholar
  67. 67.
    S. Doplicher, J.E. Roberts, Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics, Commun. Math. Phys. 131, 51–107 (1990)MathSciNetADSzbMATHCrossRefGoogle Scholar
  68. 68.
    Bert Schroer, Modular Wedge Localization and the d = 1 + 1 Formfactor Program, Ann. Phys. 295, 190–223 (1999)MathSciNetADSGoogle Scholar
  69. 69.
    Pascual Jordan, Beiträge zur Neutrinotheorie des Lichts III, Zeitschrift für Physik 105, 229–231 (1937)ADSzbMATHCrossRefGoogle Scholar
  70. 70.
    Pascual Jordan, Beiträge zur Neutrinotheorie des Lichts II, Zeitschrift für Physik 105, 114–121 (1937)ADSzbMATHCrossRefGoogle Scholar
  71. 71.
    Pascual Jordan, Beiträge zur Neutrinotheorie des Lichts I, Zeitschrift für Physik 102, 243–252 (1937)ADSCrossRefGoogle Scholar
  72. 72.
    Pascual Jordan, Zur Neutrinotheorie des Lichtes, Zeitschrift für Physik 93, 464–472 (1935)ADSzbMATHCrossRefGoogle Scholar
  73. 73.
    B. Klaiber, The Thirring Model, edited by O.A. Barut, W.E. Brittin, Lectures in Theoretical Physics (Gordon and Breach, New York, 1968), volume 10 A, pp. 141–176Google Scholar
  74. 74.
    V. Fock, Inconsistency of the neutrino theory of light, Nature 136, N3502 (1936) 1011–1012ADSCrossRefGoogle Scholar
  75. 75.
    Julian Schwinger, Field theory commutators, Phys. Rev. Lett. 3, 296–297 (1959)ADSCrossRefGoogle Scholar
  76. 76.
    Detlev Buchholz, Gerhard Mack, Ivan Todorov, The current algebra on the circle as a germ of local field theories, Nucl. Phys. B (Proc. Suppl.) 5, 20–56 (1988) MathSciNetADSCrossRefGoogle Scholar
  77. 77.
    A. Pais, Inward Bound (Clarendon Press, Oxford University Press, 1986)Google Scholar
  78. 78.
    S. Doplicher, R. Longo, Standard and split inclusions of von Neumann algebras, Invent. Mat. 75, 493–536 (1984)MathSciNetADSzbMATHCrossRefGoogle Scholar
  79. 79.
    S. Hollands, R.E. Wald, Quantum Field Theory Is Not Merely Quantum Mechanics Applied to Low Energy Effective Degrees of Freedom, General Relativity and Gravitation 36, 2595–2603 (2004)MathSciNetADSzbMATHCrossRefGoogle Scholar
  80. 80.
    Pascual Jordan, Anschauliche Quantentheorie (Springer, Berlin, 1936)Google Scholar
  81. 81.
    Pascual Jordan, Eugene Paul Wigner, Über das Paulische Äquivalenzverbot, Zeitschrift für Physik 47, 631–651 (1928)ADSCrossRefGoogle Scholar
  82. 82.
    Julian Schwinger, Gauge invariance and mass, II. Phys. Rev. 128, 2425–2429 (1962)MathSciNetADSzbMATHGoogle Scholar
  83. 83.
    Julian Schwinger, Gauge theories of Vector particles. Theoretical Physics (Trieste Lectures, 1962) (I.A.E.A., Vienna 1963), p. 89Google Scholar
  84. 84.
    L.V. Belvedere, J.A. Swieca, K.D. Rothe, Bert Schroer, Generalized Twodimensional Abelian Gauge Theories and Confinement, Nucl. Phys. B 153, 112–140 (1979)MathSciNetADSCrossRefGoogle Scholar
  85. 85.
    Bert Schroer, Two dimensional models as testing ground for principles and concepts of local quantum physics, Ann. Phys. 321, 435–479 (2006)MathSciNetADSzbMATHCrossRefGoogle Scholar
  86. 86.
    Bert Schroer, Infrateilchen in des Quantenfeldtheorie, Fortschr. Phys. 11, 1–31 (1963), Bert Schroer, A note on infraparticles and unparticles, arXiv:0804.3563Google Scholar
  87. 87.
    Pascual Jordan, Zur Quantenelektrodynamik, I. Eichinvariante Operatoren, Zeitschrift für Physik 95, 202–209 (1935)zbMATHGoogle Scholar
  88. 88.
    O. Steinmann, A Jost-Schroer theorem for string fields, Commun. Math. Phys. 87, 259–264 (1982)MathSciNetADSzbMATHCrossRefGoogle Scholar
  89. 89.
    J. Langerholc, Bert Schroer. Can current – operators determine a complete theory? Commun. Math. Phys. 4, 123–136 (1967) MathSciNetADSCrossRefGoogle Scholar
  90. 90.
    S. Jacobs, Gauge bridges in classical field theory, DESY-THESIS-2009-009,
  91. 91.
    P. Leyland, J. Roberts, D. Testard, Duality for Quantum Free Fields, unpublished notes, CNRS Marseille, (1978)Google Scholar
  92. 92.
    Detlev Buchholz, Gauss law and the infraparticle problem, Phys. Lett. B 174, 331–334 (1986)MathSciNetADSCrossRefGoogle Scholar
  93. 93.
    Felix Bloch, A. Nordsiek, Note on the radiation field of the electron, Phys. Rev. 52, 54–59 (1937)ADSzbMATHCrossRefGoogle Scholar
  94. 94.
    Rudolf Haag, Bert Schroer, Postulates of quantum field theory, J. Math. Phys. 3, 248–256 (1962)MathSciNetADSzbMATHCrossRefGoogle Scholar
  95. 95.
    G.C. Hegerfeldt, Causality problems in Fermi’s two atom system, Phys. Rev. Lett. 72, 596–599 (1994)ADSzbMATHCrossRefGoogle Scholar
  96. 96.
    Detlev Buchholz, J. Yngvason, There are no causality problems for Fermi’s two atom system, Phys. Rev. Lett. 73, 613–616 (1994)MathSciNetADSzbMATHCrossRefGoogle Scholar
  97. 97.
    Matthew Norton Wise, Pascual Jordan: quantum mechanics, psychology, National Socialism, in: Science, Technology and National Socialism, edited by Monika Renneberg, Mark Walker (Cambridge, 1994), pp. 224–254Google Scholar
  98. 98.
    J. Cornwell, Hitler’s scientists, Science, War and the Devil’s Pact, Viking N4 (2000)Google Scholar
  99. 99.
    Richard H. Beyler, Targeting the organism, The scientific and cultural context of Pascual Jordan’s quantum biology, 1932–1947, Isis 87, 248–273 (1996)CrossRefGoogle Scholar
  100. 100.
    Olivier Darrigol, The origin of quantized matter waves, Hist. Stud. Phys. Sci. 16/2, 197–253 (1986)Google Scholar

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Authors and Affiliations

  1. 1.Institut für Theoretische Physik FU-BerlinBerlinGermany

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