The Unified Standard Model
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Advances in Applied Clifford Algebras
The Unified Standard Model Brage Gording∗
and Angnis Schmidt-May Communicated by Jayme Vaz
Abstract. The aim of this work is to find a simple mathematical framework for our established description of particle physics. We demonstrate that the particular gauge structure, group representations and charge assignments of the Standard Model particles are all captured by the algebra M(8, C) of complex 8 × 8 matrices. This algebra is well motivated by its close relation to the normed division algebra of octonions. (Anti)particle states are identified with basis elements of the vector space M(8, C). Gauge transformations are simply described by the algebra acting on itself. Our result shows that all particles and gauge structures of the Standard Model are contained in the tensor product of all four normed division algebras, with the quaternions providing the Lorentz representations. Interestingly, the space M(8, C) contains two additional elements independent of the Standard Model particles, hinting at a minimal amount of new physics. Mathematics Subject Classification. Primary 17A35, 81T99, Secondary 15A66. Keywords. Standard model, Unification, Division algebra, Gauge structure.
1. Introduction 1.1. Motivation The Standard Model of Particle Physics is our well-established theory for elementary particles and their fundamental interactions. It has been experimentally tested to a high precision and its last missing ingredient, the Higgs boson, was detected in 2012 [1,14]. While the establishment of this model has undoubtedly been one of the greatest achievements in fundamental science, the origin of its particular structure for particles and their interactions remains mysterious. ∗ Corresponding
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B. Gording and A. Schmidt-May
Adv. Appl. Clifford Algebras
The Standard Model is a quantum field theory invariant under the gauge group GSM = SU(3) × SU(2) × UY (1). In addition to the 12 gauge bosons, it contains 3 generations of fermions and the complex Higgs scalar which all transform in certain representations of the gauge group. More precisely, each generation of fermions appears in 1 (quark) triplet and 1 (lepton) singlet of SU(3). All fermions come in pairs where for left-handed fermions and the SU(3) singlet Higgs these pairs are in a doublet of SU(2). Additionally, fermions possess independent antiparticles. In total, this adds up to 12 + 2 · 2 · (3 · (3 + 1)) + 2 = 62 particle and anti-particle types in the Standard Model. Except for the gauge bosons, nearly all particles carry UY (1) hypercharges whose values are (almost) fixed by gauge anomaly cancellation [4,24, 37]. Apart from this consistency requirement, the particular gauge structure, particle content and charge assignments do not possess a fundamental motivation and their origin remains unexplained within the Standard Model. To many theoretical physicists such an ad-hoc choice made by Nature appears unsatisfactory and has inspired the search for unifying structures that could be unde
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