Standard model physics from an algebra (original) (raw)
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A simple geometric algebra is shown to contain automatically the leptons and quarks of a generation of the Standard Model, and the electroweak and color gauge symmetries. The algebra is just the Clifford algebra of a complex six-dimensional vector space endowed with a preferred Witt decomposition, and it is already implicitly present in the mathematical structure of the Standard Model. The minimal left ideals determined by the Witt decomposition correspond naturally pairs of leptons or quarks whose left chiral components interact weakly. The Dirac algebra is a distinguished subalgebra acting on the ideals representing leptons and quarks. The resulting representations on the ideals are invariant to the electromagnetic and color symmetries, which are generated by the bivectors of the algebra. The electroweak symmetry is also present, and it is already broken by the geometry of the algebra. The model predicts a bare Weinberg angle thetaW\theta_WthetaW given by sin2(thetaW)=0.25\sin^2(\theta_W)=0.25sin2(thetaW)=0.25.
Leptons, Quarks, and Gauge from the Complex Clifford Algebra \mathbb {C}\ell _6$$ C ℓ 6
Advances in Applied Clifford Algebras, 2018
A simple geometric algebra is shown to contain automatically the leptons and quarks of a family of the Standard Model, and the electroweak and color gauge symmetries, without predicting extra particles and symmetries. The algebra is already naturally present in the Standard Model, in two instances of the Clifford algebra Cℓ 6 , one being algebraically generated by the Dirac algebra and the weak symmetry generators, and the other by a complex three-dimensional representation of the color symmetry, which generates a Witt decomposition which leads to the decomposition of the algebra into ideals representing leptons and quarks. The two instances being isomorphic, the minimal approach is to identify them, resulting in the model proposed here. The Dirac and Lorentz algebras appear naturally as subalgebras acting on the ideals representing leptons and quarks. The resulting representations on the ideals are invariant to the electromagnetic and color symmetries, which are generated by the bivectors of the algebra. The electroweak symmetry is also present, and it is already broken by the geometry of the algebra. The model predicts a bare Weinberg angle θ W given by sin 2 θ W = 0.25. The model shares common ideas with previously known models, particularly with Chisholm
Three generations, two unbroken gauge symmetries, and one eight-dimensional algebra
Physics Letters B
A considerable amount of the standard model's three-generation structure can be realised from just the 8C-dimensional algebra of the complex octonions. Indeed, it is a little-known fact that the complex octonions can generate on their own a 64C-dimensional space. Here we identify an su(3) ⊕ u(1) action which splits this 64C-dimensional space into complexified generators of SU(3), together with 48 states. These 48 states exhibit the behaviour of exactly three generations of quarks and leptons under the standard model's two unbroken gauge symmetries. This article builds on a previous one, [1], by incorporating electric charge. Finally, we close this discussion by outlining a proposal for how the standard model's full set of states might be identified within the left action maps of R ⊗ C ⊗ H ⊗ O (the Clifford algebra Cl(8)). Our aim is to include not only the standard model's three generations of quarks and leptons, but also its gauge bosons.
New Algebraic Unified Theory of Leptons and Quarks
Progress of Theoretical Physics, 1987
A new algebraic theory is developed to describe the characteristic features of leptons and quarks as a whole. A pair of master fields with up and down 'weak-isospin is introduced and postulated to obey the generalized Dirac equations with coefficient matrices which belong to an algebra, a triplet algebra, consisting of triple-direct-products of Dirac's I-matrices. The triplet algebra is decomposed into three subalgebras, in a non-intersecting manner, which describe respectively the external Lorentz symmetry, the internal colour symmetry and the degrees of freedom for fourfold-family-replication of fundamental fermionic particle modes. The master fields belonging to a 64 dimensional multi-spinor space form non-irreducible representations of the Lorentz group and represent fourfold-replications of families of spin 1/2 particles, each one of which accomodates triply-degenerate quark modes and singlet leptonic modes. Canonical quantization of master fields leads naturally to the renormalizable unified field theories of fundamental fermions with universal gauge interactions of local symmetries having the route of descent from SUc(4) x SUL(2) x SUR(2) to SUc(3) x SUL(2) x Uy(l).
Lorentz and SU(3) Groups Derived from Cubic Quark Algebra
Acta Polytechnica, 2010
We show how Lorentz and SU(3) groups can be derived from the covariance principle conserving a Z3-graded three-form on a Z3-graded cubic algebra representing quarks endowed with non-standard commutation laws. This construction suggests that the geometry of space-time can be considered as a manifestation of symmetries of fundamental matter fields.
Unified theories for quarks and leptons based on Clifford algebras
Physics Letters B, 1980
The general standpoint is presented that unified theories arise from gauging of Clifford algebras describing the internal degrees of freedom (charge, color, generation, spin) of the fundamental fermions. The general formalism is presented and the ensuing theories for color and charge (with extension to N colors), and for generations, are discussed. The possibility of further including the spin is discussed, also in connection with generations.
the complexification of the exceptional Jordan algebra and applications to particle physics
2021
Recent papers of Todorov and Dubois-Violette[4] and Krasnov[7] contributed revitalizing the study of the exceptional Jordan algebra h3 (O) in its relations with the true Standard Model gauge group GSM. The absence of complex representations of F4 does not allow Aut (h3 (O)) to be a candidate for any Grand Unied Theory, but the group of automorphisms of the complexication of this algebra, i.e. h C 3 (O), is isomorphic to the compact form of E6. Following Boyle in [12], it is then easy to show that the gauge group of the minimal left-right symmetric extension of the Standard Model is isomorphic to a proper subgroup of Aut h C 3 (O). Recent papers of Todorov and Dubois-Violette[4] and Krasnov[7] characterized the Standard Model gauge group G SM as a subgroup of automorphisms of the exceptional Jordan algebra h 3 (O). These works, along with previous of Baez and Huerta[3], revitalized attention on the role octonions might have in the characterization of the Standard Model true gauge group G SM , i.e. [SU (3) × SU (2) × U (1)] /Z 6. One of the main issues with this approach to the Standard Model is that all groups related with Octonions, such as G 2 and F 4 , do not have complex representation. Recently, Boyle[12] proposed the analysis of the complexication of the Albert algebra, i.e. h C 3 (O), whose automorphisms are isomorphic to the compact form of E 6 and that evidentiated a relation with the gauge group of the minimal left-right symmetric extension of the Standard Model G LR , i.e. [SU (3) × SU (2) L × SU (2) R × U (1)] /Z 6 , along with the E 6 and Spin (10) unication. In section 2 we introduce the normed division algebras through the Cayley-Dickson construction. We then focus on the algebra of Octonions, their automorphisms and a generalization of complex analysis on Octonions developed by Gentili and Struppa in [8]. In section 3 we introduce Jordan algebras and focus on the exceptional Jordan algebra h 3 (O), while in section 4 we overview some of the possible applications to particle physics. Specically we show the canonical isomorphism between h 2 (K) and the Minkowski space-time for every normed division algebra K, we then proceed showing the relevance of the h 3 (O) algebra and its complexication for the gauge group of the Standard Model and its minimal left-right symmetric extension.
Lie Algebras in Particle Physics
American Journal of Physics, 1982
The Open Access version of this book, available at www.taylorfrancis.com, has been made available under a Creative Commons Attribution-Non Commercial 4.0 International.
Algebra of Current and the Nonrelativistic Quark Model
Progress of Theoretical Physics, 1968
Therefore, the results of the l10nrelativistic quark mudel are con~istent with those of the SU (6) theory.3) **'1:) These generators take the following form in the defining (()-climensional) representation:
Quarks, Leptons, and Hopf Algebra Propagators
2011
The weak quantum numbers of the elementary fermions arise as particular representations of the Lie symmetry SU(3)× SU(2)×U(1). Hopf algebras provide a generalization of Lie algebras with the advantage that they also naturally model the algebra of Feynman diagrams. The simplest Hopf algebras are the group algebras generated by finite groups such as the permutation group of three elements, P3. The simplest Feynman diagrams are those that define propagators. In this paper we examine the propagators of the Hopf algebra generated by the permutation group of three elements, C[P3]. The algebra consists of a 6-dimensional complex vector space, with basis given by the six elements of P3. Multiplication is defined by the group multiplication. We show that the propagators of this algebra naturally contain the quarks and leptons with their weak hypercharge, weak isospin, and baryon quantum numbers. We show that an extension of the algebra gives spin-1/2 and the generations.