[Paper Review] The Standard Model Algebra - Leptons, Quarks, and Gauge from the Complex Clifford Algebra Cl6

Ovidiu Cristinel Stoica|arXiv (Cornell University)|Feb 14, 2017

Algebraic and Geometric Analysis30 references15 citations

TL;DR

This paper proposes a geometric unification of the Standard Model's leptons, quarks, and gauge symmetries using the complex Clifford algebra 𝕔ℓ₆, showing that the Dirac algebra and weak symmetry naturally extend to 𝕔ℓ₆, which decomposes into ideals representing left- and right-handed fermions. The model predicts a bare Weinberg angle of sin²θ_W = 0.25 without extra particles or symmetries.

ABSTRACT

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 $\mathbb{C}\ell_6$, 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 $\sin^2θ_W=0.25$. The model shares common ideas with previously known models, particularly with Chisholm and Farwell, 1996, Trayling and Baylis, 2004, and Furey, 2016.

Motivation & Objective

• To unify the Standard Model's fermions and gauge symmetries within a single geometric algebra framework.
• To show that the Dirac algebra and weak isospin symmetry naturally extend to the complex Clifford algebra 𝕔ℓ₆.
• To demonstrate that the algebraic structure of 𝕔ℓ₆ naturally decomposes into ideals representing leptons and quarks.
• To derive the electroweak and color symmetries as bivector subalgebras acting on these ideals.
• To predict a specific value for the bare Weinberg angle without introducing additional particles or symmetries.

Proposed method

• Uses the complex Clifford algebra 𝕔ℓ₆ as the fundamental algebraic structure, derived from successive extensions of the Dirac algebra 𝕔ℓ₄ via weak isospin generators.
• Identifies the even subalgebra of 𝕔ℓ₆ as isomorphic to the Dirac algebra and Lorentz algebra, embedding them naturally.
• Applies a Witt decomposition of 𝕔ℓ₆ to split the algebra into two ideals, one representing leptons and the other quarks.
• Constructs the electroweak symmetry from bivectors in the algebra, with the weak isospin generators T₁, T₂, T₃ extending the algebra to 𝕔ℓ₆.
• Uses the matrix representation τʲ = σʲ ⊗ 1₄ and ωʲ = τʲΓ⁵ to generate the full Clifford basis for 𝕔ℓ₅ and 𝕔ℓ₆.
• Shows that the volume form of 𝕔ℓ₆ is −iτ³, linking the algebraic structure to the physical vacuum state.

Experimental results

Research questions

• RQ1 Can the full structure of the Standard Model—fermions and gauge symmetries—be unified within a single geometric algebra framework?
• RQ2 How do the Dirac algebra and weak isospin symmetry naturally lead to the complex Clifford algebra 𝕔ℓ₆?
• RQ3 Does the algebraic decomposition of 𝕔ℓ₆ into ideals naturally yield the observed particle content of leptons and quarks?
• RQ4 Can the electroweak and color symmetries be derived as subalgebras of bivectors within 𝕔ℓ₆?
• RQ5 What is the geometric origin of the Weinberg angle, and can it be predicted without phenomenological input?

Key findings

• The complex Clifford algebra 𝕔ℓ₆ naturally contains the Dirac algebra and Lorentz algebra as subalgebras acting on fermionic ideals.
• The algebra decomposes into two ideals via a Witt decomposition, one corresponding to leptons and the other to quarks, with no additional particles introduced.
• The electroweak symmetry is generated by bivectors in the algebra, and the weak symmetry is already broken by the geometry of the algebra.
• The electromagnetic and color symmetries are invariant under the action of bivector subalgebras of 𝕔ℓ₆.
• The model predicts a bare Weinberg angle of sin²θ_W = 0.25, derived purely from the algebraic structure.
• The model is isomorphic to previously proposed models by Chisholm & Farwell (1996), Trayling & Baylis (2004), and Furey (2016), but with a more geometric and unified foundation.