[Paper Review] Clifford Algebras, Spinors and $Cl(8,8)$ Unification

Matej Pavšič|arXiv (Cornell University)|May 25, 2021

Algebraic and Geometric Analysis20 references4 citations

TL;DR

This paper proposes a unification framework using Clifford algebra Cl(8,8) to describe fundamental particles and forces in a 16-dimensional vector space V₈,₈, where spinors satisfy a Dirac equation whose internal geometry generates masses and gauge symmetries. The model realizes SO(10) grand unification and Lorentz invariance while avoiding the Coleman-Mandula theorem via geometric spinors in Clifford space, with masses arising from both Higgs-like couplings and orbital angular momentum in extra dimensions.

ABSTRACT

It is shown how the vector space $V_{8,8}$ arises from the Clifford algebra $Cl(1,3)$ of spacetime. The latter algebra describes fundamental objects such as strings and branes in terms of their $r$-volume degrees of freedom, $x^{μ_1 μ_2 ...μ_r}$ $\equiv x^M$, $r=0,1,2,3$, that generalizethe concept of center of mass. Taking into account that there are sixteen $x^M$, $M=1,2,3,...,16$, and in general $16 imes 15/2 = 120$ rotations of the form $x'^M = {R^M}_N x^N$, we can consider $x^M$ as components of a vector $X=x^M q_M$, where $q_M$ are generators of the Clifford algebra $Cl(8,8)$. The vector space $V_{8,8}$ has enough room for the unification of the fundamental particles and forces of the standard model. The rotations in $V_{8,8}\otimes \mathbb{C}$ contain the grand unification group $SO(10)$ as a subgroup, and also the Lorentz group $SO(1,3)$. It is shown how the Coleman-Mandula no go theorem can be avoided. Spinors in $V_{8,8}\otimes \mathbb{C}$ are constructed in terms of the wedge products of the basis vectors rewritten in the Witt basis. They satisfy the massless Dirac equation in $M_{8,8}$ with the internal part of the Dirac operator giving the non vanishing masses in four dimensions.

Motivation & Objective

• To unify fundamental particles and interactions, including gravity, by extending spacetime degrees of freedom using Clifford algebra Cl(1,3) and embedding them into a 16D Clifford space.
• To show how the grand unification group SO(10) and the Lorentz group SO(1,3) emerge as subgroups of rotations in V₈,₈ ⊗ ℂ.
• To resolve the Coleman-Mandula no-go theorem by constructing spinors in a higher-dimensional geometric framework where gauge and spacetime symmetries are unified non-linearly.
• To explain particle masses not solely via Higgs fields but also via orbital momentum in internal dimensions, with Yukawa couplings encoded in spin connections.
• To propose that mirror particles and dark matter candidates arise naturally from the geometric structure of Clifford space.

Proposed method

• Represent extended objects (strings, branes) via r-volume degrees of freedom x^{μ₁…μᵣ} ∈ Cl(1,3), which form a 16-dimensional vector space spanned by basis elements of Cl(8,8).
• Construct a 16D vector space V₈,₈ as the tangent space to a Clifford space C, where the generators q_M of Cl(8,8) define the components of a vector X = x^M q_M.
• Define spinors in V₈,₈ ⊗ ℂ using wedge products of basis vectors in the Witt basis, forming minimal left ideals of Cl(8,8).
• Impose the massless Dirac equation in M₈,₈, where the internal part of the Dirac operator generates non-zero masses in 4D via curvature and spin connection terms.
• Use the covariant derivative ∂_M Ψ = (∂_M ψ^Ã + Γ_M^Ã_Ã ψ^Ã) ξ_Ã to unify gauge fields, spin connections, and Higgs-like multiplets within the spin connection Γ.
• Derive the effective mass equation m²ψ^i = g^{MN} D_M D_N ψ^i + (σ_{MN})^i_j R^{j}_{MN} ψ^k, showing mass as a result of both Higgs and orbital momentum contributions.

Experimental results

Research questions

• RQ1 How can the Clifford algebra Cl(1,3) of spacetime be used to generate a 16-dimensional geometric framework capable of unifying all fundamental particles and forces?
• RQ2 In what way do rotations in V₈,₈ ⊗ ℂ contain both SO(10) grand unification and Lorentz symmetry SO(1,3) as subgroups?
• RQ3 How is the Coleman-Mandula no-go theorem circumvented in this geometric framework of Clifford space?
• RQ4 What is the origin of particle masses in this model, and how do orbital momentum and spin connections contribute beyond standard Higgs mechanisms?
• RQ5 How do mirror particles and dark matter candidates emerge naturally from the geometric structure of Clifford space?

Key findings

• The 16-dimensional vector space V₈,₈, constructed from the Clifford algebra Cl(8,8), provides sufficient geometric freedom to unify all fundamental particles and interactions, including gravity.
• The group SO(10) arises as a subgroup of rotations in V₈,₈ ⊗ ℂ, realizing grand unification of the Standard Model gauge group and fermion generations.
• The Lorentz group SO(1,3) is embedded as a subgroup of SO(2,4), which is itself a subgroup of SO(8,8), preserving spacetime symmetry in the higher-dimensional framework.
• Mass generation is not solely due to Higgs fields; the orbital momentum term g^{MN} ∂_M ∂_N ψ^i in the internal space contributes significantly to the effective mass, as shown in the derived equation m²ψ^i = g^{MN} D_M D_N ψ^i + (σ_{MN})^i_j R^{j}_{MN} ψ^k.
• The covariant derivative ∂_M Ψ contains all gauge fields, spin connections, and Higgs multiplets within the spin connection Γ_M^Ã_Ã, unifying them geometrically in the Clifford space framework.
• Mirror particles and dark matter candidates emerge naturally from the geometric structure of Clifford space, as they are coupled to mirror gauge fields and are unobservable via standard model interactions.