The Exceptional Jordan Algebra in physics

[kodama]

TL;DR

The role of the Exceptional Jordan Algebra in physics

I found 3 papers on The Exceptional Jordan Algebra in physics

arXiv:2305.00668 (hep-ph)
[Submitted on 1 May 2023]
CKM matrix parameters from an algebra
Aditya Ankur Patel, Tejinder P. Singh
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We report a theoretical derivation of the Cabibbo-Kobayashi-Maskawa (CKM) matrix parameters and the accompanying mixing angles. These results are arrived at from the exceptional Jordan algebra applied to quark states, and from expressing flavor eigenstates (i.e. left-chiral states) as superposition of mass eigenstates (i.e. the right-chiral states) weighted by square-root of mass. Flavor mixing for quarks is mediated by the square-root mass eigenstates, and the mass ratios used have been derived in earlier work from a left-right symmetric extension of the standard model. This permits a construction of the CKM matrix from first principles. There exist only four normed division algebras, they can be listed as follows - the real numbers R, the complex numbers C, the quaternions H and the octonions O. The first three algebras are fairly well known; however, octonions as algebra are less studied. Recent research has pointed towards the importance of octonions in the study of high energy physics. Clifford algebras and the standard model are being studied closely. The main advantage of this approach is that the spinor representations of the fundamental fermions can be constructed easily here as the left ideals of the algebra. Also the action of various Spin Groups on these representations too can be studied easily. In this work, we build on some recent advances in the field and try to determine the CKM angles from an algebraic framework. We obtain the mixing angle values as θ12=11.093o,θ13=0.172o,θ23=4.054o. In comparison, the corresponding experimentally measured values for these angles are 13.04o±0.05o,0.201o±0.011o,2.38o±0.06o. The agreement of theory with experiment is likely to improve when running of quark masses is taken into account.

Comments: 35 pages, 8 tables, 4 figures
Subjects: High Energy Physics - Phenomenology (hep-ph)
Cite as: arXiv:2305.00668 [hep-ph]

arXiv:2304.01213 (physics)
[Submitted on 28 Mar 2023]
The exceptional Jordan algebra, and its implications for our understanding of gravitation and the weak force
Tejinder P. Singh
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The exceptional Jordan algebra is the algebra of 3×3 Hermitian matrices with octonionic entries. It is the only one from Jordan's algebraic formulation of quantum mechanics which is not equivalent to the conventional formulation of quantum theory. It has often been suggested that this exceptional algebra could explain physical phenomena not currently explained by the conventional approach, such as values of the fundamental constants of the standard model of particle physics, and their relation to gravitation. We show that this is indeed the case; and this also unravels the connection between general relativity and the weak force. The exceptional Jordan algebra also predicts a new U(1) gravitational interaction which modifies general relativity, and which provides a theoretical basis for understanding the Modified Newtonian Dynamics (MOND).

Comments: 12 pages
Subjects: General Physics (physics.gen-ph)
Cite as: arXiv:2304.01213 [physics.gen-ph]

arXiv:2006.16265 (hep-th)
[Submitted on 29 Jun 2020]
The Standard Model, The Exceptional Jordan Algebra, and Triality
Latham Boyle
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Jordan, Wigner and von Neumann classified the possible algebras of quantum mechanical observables, and found they fell into 4 "ordinary" families, plus one remarkable outlier: the exceptional Jordan algebra. We point out an intriguing relationship between the complexification of this algebra and the standard model of particle physics, its minimal left-right-symmetric SU(3)×SU(2)L×SU(2)R×U(1) extension, and Spin(10) unification. This suggests a geometric interpretation, where a single generation of standard model fermions is described by the tangent space (C⊗O)2 of the complex octonionic projective plane, and the existence of three generations is related to SO(8) triality.

Comments: 5 pages, 1 figure
Subjects: High Energy Physics - Theory (hep-th); High Energy Physics - Phenomenology (hep-ph); Quantum Physics (quant-ph)
Cite as: arXiv:2006.16265 [hep-th]

once again
the quaternions H and the octonions O Clifford algebras and the standard model are being studied closelycomments

 

[fresh_42]

TL;DR Summary: The role of the Exceptional Jordan Algebra in physics

comments

I understand that Lie algebras come into play via Noether's theorem. They are the simple consequence of invariants in differential equation systems. I had a bit of trouble understanding why it has to be semisimple ones, but as I asked about it here on PF years ago, someone said that it is the non-degenerate Killing form that is necessary to define a geometry, and in a way inverses.

Now, Jordan algebras are in a sense a counterpart to Lie algebras. However, I miss the corresponding Noether theorem. Such attempts via Jordan algebras, Lie superalgebras, Clifford algebras, Virasoro algebras, etc. always seem to me - and I might be all wrong, so please just take it as a comment, not as an insight - always seem to me what we call "fishing in the dark". Let's blow up the available algebras. If they are big enough, we can simulate everything with them.

In my opinion, an equivalent to Noether's theorem should be found for other algebras before we consider them as replacements. Maybe there are such theorems, e.g. for Lie superalgebras, but I haven't seen them - admittedly as a layman.

You asked for a comment. This was my 2 ct hoping to kickstart the discussion.

 

[phyzguy]

I think there are more important papers on this topic. I found this paper by Bhatt, et.al.
https://arxiv.org/abs/2108.05787
where they calculate the fermionic mass rations from the eigenvalues of the matrices of the Jordan Exceptional Algebra to be very interesting.

 

[arivero]

https://arxiv.org/abs/2303.11334 https://arxiv.org/abs/2206.07039 https://arxiv.org/abs/2006.16265 https://arxiv.org/abs/1806.09450 for a seed of recent preprints

 

[john baez]

In my opinion, an equivalent to Noether's theorem should be found for other algebras before we consider them as replacements.

You might like my paper about the role of Lie and Jordan algebras in physics, and how they interact:

• Getting to the bottom of Noether's theorem, in Philosophy and Physics of Noether'sTheorems: A Centenary Conference on the 1918 Work of Emmy Noether, eds. Bryan W. Roberts and Nicholas the, Cambridge U. Press, Cambridge, pp. 66-99.