2015/05/29 by John L. Friedman · 1 citation
Physics and Astronomy · Computer Science · Mathematics · #Quantum and electron transport phenomena #Quantum Mechanics and Applications #Quantum Information and Cryptography #Hopf fibration #Mathematical physics #Mathematics #Magnetic monopole #Fiber bundle #Equivalence (formal languages) #Gauge theory #Bundle #Physics #Pure mathematics #Quantum mechanics
paper · pdf · doi:10.1063/pt.3.2799
openalex publication_date 2015/05/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/05/19
C. N. Yang, in his clear review of Maxwell’s equations and gauge theory (Physics Today, November 2014, page 45), reports that his colleague mathematician James Simons exclaimed, “[Paul] Dirac had discovered trivial and nontrivial bundles before mathematicians.” Remarkably, however, in 1931, the same year that Dirac discovered his monopole, Heinz Hopf discovered its fiber-bundle equivalent, now known as the Hopf fibration of the 3-sphere.11. H. Hopf, Math. Ann. 104, 637 (1931). https://doi.org/10.1007/BF01457962Although Eli Lubkin pointed out the bundle structure of the Dirac monopole22. E. Lubkin, Ann. Phys. 23(2), 233 (1963). https://doi.org/10.1016/0003-4916(63)90194-5 in 1963 and Tai Tsun Wu and Yang provided a widely read description,33. T. T. Wu, C. N. Yang, Phys. Rev. D 12, 3845 (1975). https://doi.org/10.1103/PhysRevD.12.3845 Andrzej Trautman apparently first noted its identification with the Hopf fibration44. A. Trautman, Int. J. Theor. Phys. 16, 561 (1977). https://doi.org/10.1007/BF01811088 in 1977. Trautman’s 1967 lectures at King’s College London introduced some physicists to the mathematical equivalence of gauge theories and fiber-bundle theory, but not until 1970 were those lectures published.55. A. Trautman, Rep. Math. Phys. 1, 29 (1970). https://doi.org/10.1016/0034-4877(70)90003-0 Yang notes that the equivalence came as a shock to both physicists and mathematicians in the 1970s.REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. H. Hopf, Math. Ann. 104, 637 (1931). https://doi.org/10.1007/BF01457962, Google ScholarCrossref2. E. Lubkin, Ann. Phys. 23(2), 233 (1963). https://doi.org/10.1016/0003-4916(63)90194-5, Google ScholarCrossref3. T. T. Wu, C. N. Yang, Phys. Rev. D 12, 3845 (1975). https://doi.org/10.1103/PhysRevD.12.3845, Google ScholarCrossref, ISI4. A. Trautman, Int. J. Theor. Phys. 16, 561 (1977). https://doi.org/10.1007/BF01811088, Google ScholarCrossref5. A. Trautman, Rep. Math. Phys. 1, 29 (1970). https://doi.org/10.1016/0034-4877(70)90003-0, Google ScholarCrossref© 2015 American Institute of Physics.