Foundations of Physics

, Volume 42, Issue 7, pp 819–855 | Cite as

Division Algebras and Quantum Theory

  • John C. BaezEmail author


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.


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


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  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 Google Scholar
  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) zbMATHGoogle Scholar
  4. 4.
    Adler, S.: Quaternionic Quantum Mechanics and Quantum Fields. Oxford University Press, Oxford (1995) zbMATHGoogle Scholar
  5. 5.
    Amemiya, I., Araki, H.: A remark on Piron’s paper. Publ. Res. Inst. Math. Sci. 2, 423–427 (1966/67) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Anderson, F.W., Fuller, K.R.: Rings and Categories of Modules. Springer, Berlin (1998) Google Scholar
  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) Google Scholar
  8. 8.
    Baez, J.: Higher-dimensional algebra II: 2-Hilbert spaces. Adv. Math. 127, 125–189 (1997). Also available as arXiv:q-alg/9609018 MathSciNetzbMATHCrossRefGoogle Scholar
  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 MathSciNetzbMATHCrossRefGoogle Scholar
  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 CrossRefGoogle Scholar
  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 Google Scholar
  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 Google Scholar
  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 CrossRefGoogle Scholar
  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 Google Scholar
  20. 20.
    Bourbaki, N.: Elements of Mathematics. Springer, Berlin (2008). Chapter IX, Appendix II, Prop. 4 Google Scholar
  21. 21.
    Budinich, P., Trautman, A.: The Spinorial Chessboard. Springer, Berlin (1988) CrossRefGoogle Scholar
  22. 22.
    Coecke, B.: New Structures for Physics. Lecture Notes in Physics, vol. 813. Springer, Berlin (2000) Google Scholar
  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 MathSciNetADSzbMATHCrossRefGoogle Scholar
  24. 24.
    Dyson, F.: The threefold way: algebraic structure of symmetry groups and ensembles in quantum mechanics. J. Math. Phys. 3, 1199–1215 (1962) MathSciNetADSzbMATHCrossRefGoogle Scholar
  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) Google Scholar
  26. 26.
    Frobenius, F.G., Schur, I.: Über die reellen Darstellungen der endlichen Gruppen. Sitz. Akad. Preuss. Wiss. 186–208 (1906) Google Scholar
  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 MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Hurwitz, A.: Über die Composition der quadratischen Formen von beliebig vielen Variabeln. Nachr. Ges. Wiss. Gött. 309–316 (1906) Google Scholar
  30. 30.
    Jordan, P.: Über eine Klasse nichtassociativer hyperkomplexer Algebren. Nachr. Ges. Wiss. Gött. 569–575 (1932) Google Scholar
  31. 31.
    Jordan, P., von Neumann, J., Wigner, E.: On an algebraic generalization of the quantum mechanical formalism. Ann. Math. 35, 29–64 (1934) CrossRefGoogle Scholar
  32. 32.
    Koecher, M.: Positivitätsbereiche in ℝn. Am. J. Math. 79, 575–596 (1957) MathSciNetzbMATHCrossRefGoogle Scholar
  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) Google Scholar
  34. 34.
    McCrimmon, K.: A Taste of Jordan Algebras. Springer, Berlin (2004) zbMATHGoogle Scholar
  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) zbMATHCrossRefGoogle Scholar
  37. 37.
    Piron, C.: Foundations of Quantum Physics. Benjamin, New York (1976) zbMATHGoogle Scholar
  38. 38.
    Piron, C.: “Axiomatique” quantique. Helv. Phys. Acta 37, 439–468 (1964) MathSciNetzbMATHGoogle Scholar
  39. 39.
    Polchinski, J.: More on states and operators. In: String Theory, vol. I. Cambridge University Press, Cambridge (1998) Google Scholar
  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 Google Scholar
  41. 41.
    Solèr, M.P.: Characterization of Hilbert spaces by orthomodular spaces. Commun. Algebra 23, 219–243 (1995) zbMATHCrossRefGoogle Scholar
  42. 42.
    Urbanik, K., Wright, F.B.: Absolute-valued algebras. Proc. Am. Math. Soc. 11, 861–866 (1960). Freely available online from the AMS MathSciNetCrossRefGoogle Scholar
  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) Google Scholar
  44. 44.
    Varadrajan, V.S.: Geometry of Quantum Theory. Springer, Berlin (1985) Google Scholar
  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) MathSciNetGoogle Scholar
  47. 47.
    Zelmanov, E.I.: On prime Jordan algebras. II. Sib. Mat. Zh. 24, 89–104 (1983) MathSciNetGoogle Scholar

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Centre for Quantum TechnologiesNational University of SingaporeSingaporeSingapore
  2. 2.Department of MathematicsUniversity of CaliforniaRiversideUSA

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