The standard model, the Pati–Salam model, and ‘Jordan geometry’

Boyle, Latham; Farnsworth, Shane

doi:10.1088/1367-2630/ab9709

Cited by 24 publications

(25 citation statements)

References 35 publications

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“…There are but a few papers on the applications of Jordan algebras to quantum theory, [B93,B08,T85/16,GPR,DM,M,B12]. 2 Recently, a new paper, [BF19], was posted where an alternative approach, closer to Connes' real spectral triple, involving a different Jordan algebra, is been developed.…”

Section: Euclidean Jordan Algebrasmentioning

confidence: 99%

Jordan algebra approach to finite quantum geometry

Todorov¹

2020

Proceedings of Corfu Summer Institute 2019 "School and Workshops on Elementary Particle Physics and Gravity" — PoS(CORFU2019)

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The exceptional euclidean Jordan algebra J 8 3 , consisting of 3 × 3 hermitian octonionic matrices, appears to be tailor made for the internal space of the three generations of quarks and leptons. The maximal rank subgroup of the authomorphism group F 4 of J 8 3 that respects the lepton-quark splitting is (SU(3) c × SU(3) ew)/Z 3. Its restriction to the special Jordan subalgebra J 8 2 ⊂ J 8 3 , associated with a single generation of fundamental fermions, is precisely the symmetry group S(U(3) × U(2)) of the Standard Model. The Euclidean extension H 16 (C) ⊗ H 16 (C) of J 8 2 , the subalgebra of hermitian matrices of the complexification of the associative envelope of J 8 2 , involves 32 primitive idempotents giving the states of the first generation fermions. The triality relating left and right Spin(8) spinors to 8-vectors corresponds to the Yukawa coupling of the Higgs boson to quarks and leptons. The present study of J 8 3 originated in the paper arXiv:1604.01247v2 by Michel Dubois-Violette. It further develops ongoing work with him and with Svetla Drenska: arXiv:1806.09450; 1805.06739v2; 1808.08110.

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Section: Euclidean Jordan Algebrasmentioning

confidence: 99%

Jordan algebra approach to finite quantum geometry

Todorov¹

2020

Proceedings of Corfu Summer Institute 2019 "School and Workshops on Elementary Particle Physics and Gravity" — PoS(CORFU2019)

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“…The attempts to understand "the algebra of the Standard Model (SM) of particle physics" started with the Grand Unified Theories (GUT) (thus interpreted in the illuminating review [4]), was followed by a vigorous pursuit by Connes and collaborators of the noncommutative geometry approach to the SM (reviewed in [9,34]). The present work belongs to a more recent development, initiated by Dubois-Violette [11] and continued in [12,[38][39][40][41][42], that exploits the theory of euclidean Jordan algebras (see also [5,6]). We modify the superconnection associated with the Clifford algebra C 10 considered in [13].…”

Section: Introductionmentioning

confidence: 99%

Superselection of the weak hypercharge and the algebra of the Standard Model

Todorov

2021

J. High Energ. Phys.

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Restricting the ℤ2-graded tensor product of Clifford algebras $$ C{\mathrm{\ell}}_4\hat{\otimes}C{\mathrm{\ell}}_6 $$ C ℓ 4 ⊗ ̂ C ℓ 6 to the particle subspace allows a natural definition of the Higgs field Φ, the scalar part of Quillen’s superconnection, as an element of $$ C{\mathrm{\ell}}_4^1 $$ C ℓ 4 1 . We emphasize the role of the exactly conserved weak hypercharge Y, promoted here to a superselection rule for both observables and gauge transformations. This yields a change of the definition of the particle subspace adopted in recent work with Michel Dubois-Violette [13]; here we exclude the zero eigensubspace of Y consisting of the sterile (anti)neutrinos which are allowed to mix. One thus modifies the Lie superalgebra generated by the Higgs field. Equating the normalizations of Φ in the lepton and the quark subalgebras we obtain a relation between the masses of the W boson and the Higgs that fits the experimental values within one percent accuracy.

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“…There are some geometric interpretations of the Hopf map. However, we will show below the direct analytical construction of the Hopf map (15). Considering the four-dimensional real space R 4 with coordinates (u 1 , u 2 , v 1 , v 2 ), we can identify it with the two-dimensional complex space C 2 with coordinates (u, v) = (u 1 + iu 2 , v 1 + iv 2 ).…”

Section: Ii2 Hopf Mapsmentioning

confidence: 99%

“…As well known, the Standard model leaves so many unsolved questions for physicists to understand the nature of our world. In the scientific endeavor of solving these mysteries, two recent publications [14,15] showed the possibility of exploring and extending the Standard Model by using octonion algebra O.…”

Section: Introductionmentioning

confidence: 99%

Normed Division Algebras Application to the Monopole Physics

2021

Comm. in Phys.

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We review some normed division algebras (R, C, H, O) applications to the monopole physics and MICZ-Kepler problems. More specifically, we will briefly review some results in applying the normed division algebras to interpret the existence of Dirac, Yang, and SO(8) monopoles. These monopoles also appear during the examination of the duality between isotropic harmonic oscillators and the MICZ-Kepler problems. We also revisit some of our newest results in the nine-dimensional MICZ-Kepler problem using the generalized Hurwitz transformation.

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The standard model, the Pati–Salam model, and ‘Jordan geometry’

Cited by 24 publications

References 35 publications

Jordan algebra approach to finite quantum geometry

Jordan algebra approach to finite quantum geometry

Superselection of the weak hypercharge and the algebra of the Standard Model

Normed Division Algebras Application to the Monopole Physics

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