Foundations of Physics 42 (7):819-855 (2012)

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
Keywords Division algebra  Quantum theory  Jordan algebra  Quaternion  Octonion  Group representation  Convex cone  Duality
Categories (categorize this paper)
DOI 10.1007/s10701-011-9566-z
Edit this record
Mark as duplicate
Export citation
Find it on Scholar
Request removal from index
Revision history

Download options

PhilArchive copy

Upload a copy of this paper     Check publisher's policy     Papers currently archived: 70,307
External links

Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
Through your library

References found in this work BETA

Quantum Quandaries: A Category-Theoretic Perspective.J. C. Baez - 2006 - In Dean Rickles, Steven French & Juha T. Saatsi (eds.), The Structural Foundations of Quantum Gravity. Clarendon Press.

Add more references

Citations of this work BETA

Add more citations

Similar books and articles

Schwinger Algebra for Quaternionic Quantum Mechanics.L. P. Horwitz - 1997 - Foundations of Physics 27 (7):1011-1034.
Global Effects in Quaternionic Quantum Field Theory.S. P. Brumby & G. C. Joshi - 1996 - Foundations of Physics 26 (12):1591-1599.
Group-Theoretic Treatment of the Axioms of Quantum Mechanics.James Ax - 1976 - Foundations of Physics 6 (4):371-399.
Quantum MV Algebras.Roberto Giuntini - 1996 - Studia Logica 56 (3):393 - 417.
Second Quantized Quaternion Quantum Theory.James D. Edmonds - 1975 - Foundations of Physics 5 (4):643-648.
Hypercomplex Quantum Mechanics.L. P. Horwitz - 1996 - Foundations of Physics 26 (6):851-862.


Added to PP index

Total views
68 ( #168,928 of 2,507,700 )

Recent downloads (6 months)
1 ( #416,820 of 2,507,700 )

How can I increase my downloads?


My notes