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arXiv:2607.10833

Jordan Pair Quantum Theory and the Standard Model

John C. Baez, Endre Bokor, Latham Boyle

math-phhep-phhep-thquant-phSubmitted Jul 12, 2026• Announced Jul 14, 2026
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Abstract

Jordan pairs and hermitian Jordan triples were discovered by mathematicians studying Jordan algebras, which describe the possible algebras of observables in quantum mechanics. We point out a striking correspondence between the doubly exceptional hermitian Jordan triple (the so-called ``bi-Cayley'' triple) and the structure of the Standard Model of particle physics. We also point out how ordinary quantum mechanics may be reformulated, and generalized, using hermitian Jordan triples.

Comments
33 pages, 1 figure

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Jordan pairs and hermitian Jordan triples were discovered by mathematicians studying Jordan algebras, which describe the possible algebras of observables in quantum mechanics. We point out a striking correspondence between the doubly exceptional hermitian Jordan triple (the so-called ``bi-Cayley'' triple) and the structure of the Standard Model of particle physics. We also point out how ordinary quantum mechanics may be reformulated, and generalized, using hermitian Jordan triples.
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This is a transcription of a conference proceedings from 1985. It reviews the Jordan algebra formulation of quantum mechanics. A possible novelty is the discussion of time evolution; the associator takes over the role of iii times the commutator in the standard density matrix formulation, and for the Jordan algebra of complex Hermitian matrices this implies a Hamiltonian of the form H=i[x,y]+λIH= i[x,y] + λ\mathbb{I}H=i[x,y]+λI for traceless Hermitian x,yx,yx,y and real number λλλ. Other possibilities for time evolution in the Jordan formulation are briefly considered.
The basic structure of quantum mechanics was delineated in the early days of the theory and has not been modified since.One of the fundamental assumptions used in formulating the theory is that operators are represented by Hermitian matrices.In recent years it has been shown that quantum mechanics can be formulated consistently without making this assumption, using instead a combination of the parity (P) and time-reversal (T) operators and a number of other requirements related to P and T .Only the case of even T has been analyzed in the literature; here we generalize the principles to include odd time-reversal.We use this generalization to construct a non-Hermitian version of the Dirac equation, and in doing so discover a new type of particle not allowed within the (Hermitian) Standard Model.Finally we present a potential application of the ideas of non-Hermitian quantum mechanics to the unsolved problems of quantum magnetism and high temperature superconductivity.
We review attempts by Pascual Jordan and other researchers, most notably Lawrence Biedenharn to generalize quantum mechanics by passing from associative matrix or operator algebras to non-associative algebras. We start with Jordan's work from the early 1930ies leading to Jordan algebras and the first attempt to incorporate the alternative ring of octonions into physics. Jordan's work on the octonions from 1932 till 1952 will be covered, discussing aspects of the exceptional Jordan algebra and how to express probabilities when working with the octonions and the exceptional Jordan algebra. From the 1950ies onwards Jordan and others also considered one-sided distributive systems like near-fields, near-rings, quasi-fields and exceptional Segal systems (the last two examples being not necessarily associative). As the set of non-linear operators forms a near-ring and even a near-algebra, this may be of interest for attempts to pass from a linear to a non-linear setting in the study of quantum mechanics. Moreover, ideas introduced in the late 1960ies to use non-power-associative algebras to formulate a theory of a minimal length will be covered. Lawrence Biedenharn's and Jordan's ideas related to non-power-associative octonionic matrix algebras will be briefly mentioned, a long section is devoted to a summary of Horst Rühaak's PhD thesis from 1968 on matrix algebras over the octonions. Finally, recent attempts to use non-associative algebras in physics will be described.