Division Algebras and Quantum Theory


Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of axiomatic approaches. However, there are internal problems with real or quaternionic quantum theory. Here we argue that these problems can be resolved if we treat real, complex and quaternionic quantum theory as part of a unified structure. Dyson called this structure the ‘three-fold way’. It is perhaps easiest to see it in the study of irreducible unitary representations of groups on complex Hilbert spaces. These representations come in three kinds: those that are not isomorphic to their own dual (the truly ‘complex’ representations), those that are self-dual thanks to a symmetric bilinear pairing (which are ‘real’, in that they are the complexifications of representations on real Hilbert spaces), and those that are self-dual thanks to an antisymmetric bilinear pairing (which are ‘quaternionic’, in that they are the underlying complex representations of representations on quaternionic Hilbert spaces). This three-fold classification sheds light on the physics of time reversal symmetry, and it already plays an important role in particle physics. More generally, Hilbert spaces of any one of the three kinds—real, complex and quaternionic—can be seen as Hilbert spaces of the other kinds, equipped with extra structure.

This is a preview of subscription content, log in to check access.

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 199

This is the net price. Taxes to be calculated in checkout.


  1. 1.

    Abramsky, S.: Abstract scalars, loops, and free traced and strongly compact closed categories. In: Proceedings of CALCO 2005. Lecture Notes in Computer Science, vol. 3629, pp. 1–31. Springer, Berlin (2005). Also available at

  2. 2.

    Abramsky, S., Coecke, B.: A categorical semantics of quantum protocols. Available at arXiv:quant-ph/0402130

  3. 3.

    Adams, J.F.: Lectures on Lie Groups. Benjamin, New York (1969)

  4. 4.

    Adler, S.: Quaternionic Quantum Mechanics and Quantum Fields. Oxford University Press, Oxford (1995)

  5. 5.

    Amemiya, I., Araki, H.: A remark on Piron’s paper. Publ. Res. Inst. Math. Sci. 2, 423–427 (1966/67)

  6. 6.

    Anderson, F.W., Fuller, K.R.: Rings and Categories of Modules. Springer, Berlin (1998)

  7. 7.

    Arnold, V.I.: Symplectization, complexification and mathematical trinities. In: Bierstone, E., Khesin, B., Khovanskii, A., Marsden, J.E. (eds.) The Arnoldfest: Proceedings of a Conference in Honour of V.I. Arnold for His Sixtieth Birthday. AMS, Providence (1999)

  8. 8.

    Baez, J.: Higher-dimensional algebra II: 2-Hilbert spaces. Adv. Math. 127, 125–189 (1997). Also available as arXiv:q-alg/9609018

  9. 9.

    Baez, J.: The octonions. Bull. Am. Math. Soc. 39, 145–205 (2002). Errata in Bull. Am. Math. Soc. 42, 213 (2005). Also available as arXiv:math/0105155

  10. 10.

    Baez, J.: Quantum quandaries: a category-theoretic perspective. In: French, S., Rickles, D., Saatsi, J. (eds.) Structural Foundations of Quantum Gravity, pp. 240–265. Oxford University Press, Oxford (2006). Also available as arXiv:quant-ph/0404040

  11. 11.

    Baez, J., Huerta, J.: Division algebras and supersymmetry I. In: Doran, R., Friedman, G., Rosenberg, J. (eds.) Superstrings, Geometry, Topology, and C*-Algebras. Proc. Symp. Pure Math., vol. 81, pp. 65–80. AMS, Providence (2010). Also available as arXiv:0909.0551

  12. 12.

    Baez, J., Huerta, J.: Division algebras and supersymmetry II. Available as arXiv:1003.3436

  13. 13.

    Baez, J., Lauda, A.: A prehistory of n-categorical physics. In: Halvorson, H. (ed.) Deep Beauty: Mathematical Innovation and the Search for an Underlying Intelligibility of the Quantum World. Cambridge University Press, Cambridge (2011). Also available as arXiv:0908.2469

  14. 14.

    Baez, J., Stay, M.: Physics, topology, logic and computation: a Rosetta Stone. In: Coecke, B. (ed.) New Structures for Physics. Lecture Notes in Physics, vol. 813, pp. 95–174. Springer, Berlin (2000). Also available as arXiv:0903.0340

  15. 15.

    Barnum, H., Wilce, A.: Ordered linear spaces and categories as frameworks for information-processing characterizations of quantum and classical theory. Available as arXiv:0908.2354

  16. 16.

    Barnum, H., Duncan, R., Wilce, A.: Symmetry, compact closure and dagger compactness for categories of convex operational models. Available as arXiv:1004.2920

  17. 17.

    Barnum, H., Gaebler, C.P., Wilce, A.: Ensemble steering, weak self-duality, and the structure of probabilistic theories. Available as arXiv:0912.5532

  18. 18.

    Bartels, T.: Functional analysis with quaternions. Available at

  19. 19.

    Bourbaki, N.: Elements of Mathematics. Springer, Berlin (2008). Chapter VIII, Sect. 7, Prop. 12

  20. 20.

    Bourbaki, N.: Elements of Mathematics. Springer, Berlin (2008). Chapter IX, Appendix II, Prop. 4

  21. 21.

    Budinich, P., Trautman, A.: The Spinorial Chessboard. Springer, Berlin (1988)

  22. 22.

    Coecke, B.: New Structures for Physics. Lecture Notes in Physics, vol. 813. Springer, Berlin (2000)

  23. 23.

    Corrigan, E., Hollowood, T.J.: The exceptional Jordan algebra and the superstring. Commun. Math. Phys. 122, 393–410 (1989). Also available at Project Euclid

  24. 24.

    Dyson, F.: The threefold way: algebraic structure of symmetry groups and ensembles in quantum mechanics. J. Math. Phys. 3, 1199–1215 (1962)

  25. 25.

    Feynman, R.: The reason for antiparticles. In: Elementary Particles and the Laws of Physics: the 1986 Dirac Memorial Lectures, pp. 1–60. Cambridge University Press, Cambridge (1987)

  26. 26.

    Frobenius, F.G., Schur, I.: Über die reellen Darstellungen der endlichen Gruppen. Sitz. Akad. Preuss. Wiss. 186–208 (1906)

  27. 27.

    Hardy, L.: Quantum theory from five reasonable axioms. Available at arXiv:quant-ph/0101012

  28. 28.

    Holland, S.S. Jr.: Orthomodularity in infinite dimensions; a theorem of M. Solèr. Bull. Am. Math. Soc. 32, 205–234 (1995). Also available as arXiv:math/9504224

  29. 29.

    Hurwitz, A.: Über die Composition der quadratischen Formen von beliebig vielen Variabeln. Nachr. Ges. Wiss. Gött. 309–316 (1906)

  30. 30.

    Jordan, P.: Über eine Klasse nichtassociativer hyperkomplexer Algebren. Nachr. Ges. Wiss. Gött. 569–575 (1932)

  31. 31.

    Jordan, P., von Neumann, J., Wigner, E.: On an algebraic generalization of the quantum mechanical formalism. Ann. Math. 35, 29–64 (1934)

  32. 32.

    Koecher, M.: Positivitätsbereiche in ℝn. Am. J. Math. 79, 575–596 (1957)

  33. 33.

    Koecher, M.: In: Krieg, A., Walcher, S. (eds.) The Minnesota Notes on Jordan Algebras and Their Applications. Lecture Notes in Mathematics, vol. 1710. Springer, Berlin (1999)

  34. 34.

    McCrimmon, K.: A Taste of Jordan Algebras. Springer, Berlin (2004)

  35. 35.

    Ng, C.-K.: Quaternion functional analysis. Available as arXiv:math/0609160

  36. 36.

    Okubo, S.: Introduction to Octonion and Other Non-Associative Algebras in Physics. Cambridge University Press, Cambridge (1995)

  37. 37.

    Piron, C.: Foundations of Quantum Physics. Benjamin, New York (1976)

  38. 38.

    Piron, C.: “Axiomatique” quantique. Helv. Phys. Acta 37, 439–468 (1964)

  39. 39.

    Polchinski, J.: More on states and operators. In: String Theory, vol. I. Cambridge University Press, Cambridge (1998)

  40. 40.

    Selinger, P.: Dagger compact closed categories and completely positive maps. In: Proceedings of the 3rd International Workshop on Quantum Programming Languages (QPL 2005), pp. 139–163. Amsterdam, Elsevier (2007). Also available at

  41. 41.

    Solèr, M.P.: Characterization of Hilbert spaces by orthomodular spaces. Commun. Algebra 23, 219–243 (1995)

  42. 42.

    Urbanik, K., Wright, F.B.: Absolute-valued algebras. Proc. Am. Math. Soc. 11, 861–866 (1960). Freely available online from the AMS

  43. 43.

    Van Steirteghem, B., Stubbe, I.: Propositional systems, Hilbert lattices and generalized Hilbert spaces. In: Engesser, K., Gabbay, D.M., Lehmann, D. (eds.) Handbook of Quantum Logic and Quantum Structures: Quantum Structures. Elsevier, Amsterdam (2007)

  44. 44.

    Varadrajan, V.S.: Geometry of Quantum Theory. Springer, Berlin (1985)

  45. 45.

    Vicary, J.: Completeness of dagger-categories and the complex numbers. Available as arXiv:0807.2927

  46. 46.

    Vinberg, E.B.: Homogeneous cones. Sov. Math. Dokl. 1, 787–790 (1961)

  47. 47.

    Zelmanov, E.I.: On prime Jordan algebras. II. Sib. Mat. Zh. 24, 89–104 (1983)

Download references

Author information

Correspondence to John C. Baez.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Baez, J.C. Division Algebras and Quantum Theory. Found Phys 42, 819–855 (2012) doi:10.1007/s10701-011-9566-z

Download citation


  • Division algebra
  • Quantum theory
  • Jordan algebra
  • Quaternion
  • Octonion
  • Group representation
  • Convex cone
  • Duality