The underlying geometry of the CAM gauge model of the Standard Model of particle physics

Abstract

The Composition Algebra-based Methodology (CAM) [B. Wolk, Pap. Phys. 9, 090002 (2017); Phys. Scr. 94, 025301 (2019); Adv. Appl. Clifford Algebras 27, 3225 (2017); J. Appl. Math. Phys. 6, 1537 (2018); Phys. Scr. 94, 105301 (2019), Adv. Appl. Clifford Algebras 30, 4 (2020)], which provides a new model for generating the interactions of the Standard Model, is geometrically modeled for the electromagnetic and weak interactions on the parallelizable sphere operator fiber bundle [Formula: see text] consisting of base space, the tangent bundle [Formula: see text] of space–time [Formula: see text], projection operator [Formula: see text], the parallelizable spheres [Formula: see text] conceived as operator fibers [Formula: see text] attaching to and operating on [Formula: see text] [Formula: see text] as [Formula: see text] varies over [Formula: see text], and as structure group, the norm-preserving symmetry group [Formula: see text] for each of the division algebras which is simultaneously the isometry group of the associated unit sphere. The massless electroweak [Formula: see text] Lagrangian is shown to arise from [Formula: see text]’s generation of a local coupling operation on sections of Dirac spinor and Clifford algebra bundles over [Formula: see text]. Importantly, CAM is shown to be a new genre of gauge theory which subsumes Yang–Mills Standard Model gauge theory. Local gauge symmetry is shown to be at its core a geometric phenomenon inherent to CAM gauge theory. Lastly, the higher-dimensional, topological architecture which generates CAM from within a unified eleven [Formula: see text]-dimensional geometro-topological structure is introduced.

Quantum gravity through geometric algebra

“…The GA, though necessary to GA/QG, nevertheless is not sufficient to it, as GA does not entirely subsume the division algebras K = {R, C, H, O} [24,25,36,37]. The division algebras (and in particular the parallelizable spheres to which they are isomorphic) will be seen to form a necessary part of GA/QG, and will enter GA/QG through the SBM [5][6][7][8][9][10][11]. GA subsumes only the reals and the quaternions {R, H}-which are closed under the geometric product [24].…”

“…' This construction appears to be done by setting i = e 1 e 2 e 3 ≡ e 123 , the pseudoscalar in G 3with (e i ) 2 = 1 [18,25,26]. Baylis thus states 7 , 'The identification i ≡ e 123 endows the unit imaginary with geometrical significance and helps explain the widespread use of complex numbers in physics. '…”

“…6 Ablamowicz and Sobczyk [26], p 91 and chapter 4 generally. 7 Ablamowicz and Sobczyk [26], p 92. 8 See, e.g.…”

“…This model is designated GA/QG-Geometric Algebraic Quantum Gravity, to distinguish it from other models such as String/QG, Loop/QG, non-commutative geometry (NCG)/QG, and CDT/QG, to name a few [1][2][3][4]. The GA/QG model will be seen to incorporate as a necessary part of its theoretic structure the sphere fiber bundle gauge model (sphere bundle model (SBM)) [5][6][7][8][9][10][11].…”

Current survey of Clifford geometric algebra applications

“…The Composition Algebra‐based Methodology (CAM) 181 provides a new model for generating the interactions of the SM. It is geometrically modeled for electromagnetic and weak interactions on a parallelizable sphere operator fiber bundle consisting of base space, the tangent bundle of space‐time, a projection operator, parallelizable spheres conceived as operator fibers, and it has as structure group, the norm‐preserving symmetry group $$$ \mathrm{SO}\left(n+1\right) $$$ for each of the division algebras which is simultaneously the isometry group of the associated unit sphere.…”