[Paper Review] From the Geometry of Pure Spinors with their Division Algebras to Fermion's Physics

Paolo Budinich|arXiv (Cornell University)|Jul 19, 2001

Algebraic and Geometric Analysis9 references14 citations

TL;DR

This paper proposes a geometric construction of fermion physics from pure spinors and division algebras (complex numbers, quaternions, octonions), showing how equations of motion for fundamental particles—from neutrinos to the Standard Model—emerge naturally through a stepwise doubling of spinor dimensions from 2 to 32 components, culminating in a 10D Clifford algebra ${\mathbb{C}\ell}(1,9)$, where internal symmetries (U(1), SU(2), SU(3)) and fermion families arise from the algebraic structure of these division algebras without requiring extra spacetime dimensions beyond 4D Minkowski space.

ABSTRACT

The Cartan's equations definig simple spinors (renamed pure by C. Chevalley) are interpreted as equations of motion in momentum spaces, in a constructive approach in which at each step the dimesions of spinor space are doubled while those momentum space increased by two. The construction is possible only in the frame of geometry of simple or pure spinors, which imposes contraint equations on spinors with more than four components, and the momentum spaces result compact, isomorphic toinvariant-mass-spheres imbedded in each other, since the signatures appear to be unambiguously defined and result steadily lorentzian; up to dimension ten with Clifford algebra Cl(1,9), where the construction naturally ends. The equations of motion met in the construction are most of those traditionally postulated ad hoc for multicomponent fermions. The 3 division algebras: complex numbers, quaternions and octonions appear to be strictly correlated with this spinor geometry, from which they appear to gradually emerge in the construction, where they play a basic role for the physical interpretation. In fact they seem then to be at the origin of electroweak and strong charges, of the 3 families and of the groups of the standard model. In this approach there seems to be no need of higher dimensional (>4) space-time, here generated merely by Poincare translations, and dimensional reduction from Cl(1,9) to Cl(1,3) is equivalent to decoupling of the equations of motion.

Motivation & Objective

• To derive the equations of motion for fundamental fermions (neutrinos, electrons, quarks) from geometric constraints on pure spinors, avoiding ad hoc postulation.
• To explain the origin of internal symmetries (U(1), SU(2), SU(3)) and fermion families in the Standard Model through the algebraic structure of division algebras.
• To show that the 10-dimensional Clifford algebra ${\mathbb{C}\ell}(1,9)$ naturally encodes the full spectrum of known fermions and their interactions via spinor geometry.
• To demonstrate that dimensional reduction from 10D to 4D is not a compactification of extra dimensions but a decoupling of equations of motion in momentum space.
• To establish a constructive, step-by-step geometric framework where each stage corresponds to increasing energy scales and particle content, rooted in Cartan's equations for pure spinors.

Proposed method

• Starts from 2-component Weyl spinors in ${\mathbb{C}\ell}(2)$, corresponding to the Pauli algebra, and iteratively doubles the spinor space dimension while increasing momentum space by two dimensions at each step.
• Uses Cartan's equations for pure spinors to define null vectors in momentum space, which geometrically represent compact invariant-mass spheres with radii proportional to energy scales.
• Applies two key propositions to ensure the construction remains consistent with pure spinor geometry, which imposes constraints on higher-rank spinors.
• Introduces complex numbers at the third step to generate U(1) symmetry, quaternions at the fourth step to generate SU(2) symmetries (isospin and electroweak $SU(2)_L$), and octonions at the fifth step to generate SU(3) flavor and color symmetries.
• Defines complex octonions via projectors $u_{\pm} = \frac{1}{2}(1 \pm G_9)$ and $v^{(n)}_{\pm} = \frac{1}{2} G_{4+n}(1 \pm G_9)$, which close the octonion algebra and realize $SU(3)$ invariance.
• Relates the generators of the Clifford algebra ${\mathbb{C}\ell}(1,9)$ to physical momentum and internal symmetry generators, with $G_{\mu}$ and $G_9$ playing roles in defining the 4D and 10D momentum spaces respectively.

Experimental results

Research questions

• RQ1 How can the equations of motion for fundamental fermions (Weyl, Majorana, Dirac, electroweak, nucleon-pion) be derived from a single geometric principle without ad hoc assumptions?
• RQ2 What is the role of division algebras (complex numbers, quaternions, octonions) in generating the internal symmetries of the Standard Model?
• RQ3 Why does the construction naturally terminate at 32-component spinors in 10D, and what is the significance of the Clifford algebra ${\mathbb{C}\ell}(1,9)$ in this context?
• RQ4 How does the geometric approach explain the emergence of three fermion generations and the origin of electric and strong charges?
• RQ5 Can the Standard Model’s gauge groups and particle content be derived from pure spinor geometry without postulating extra spacetime dimensions?

Key findings

• The construction generates Weyl, Majorana, and Dirac equations as natural consequences of the pure spinor geometry, with Dirac’s equation emerging only as an approximation when interactions are neglected.
• The complex numbers at the third step generate a U(1) symmetry, which accounts for electric charge and the existence of charged-antiparticle pairs, including the proton-electron charge asymmetry.
• The quaternions at the fourth step generate two U(1) symmetries (one for electric charge, one for strong charge) and an SU(2) symmetry, which underlies isospin and the electroweak $SU(2)_L$ group.
• The octonions at the fifth step generate the SU(3) flavor and color groups, with the $G_2$ automorphism group of the octonions explaining the $SU(3)$ invariance of the strong interaction.
• The 32-component spinor space in ${\mathbb{C}\ell}(1,9)$ corresponds to the full fermion content of three generations, with the number of families equal to the number of quaternion units (three).
• Dimensional reduction from ${\mathbb{C}\ell}(1,9)$ to ${\mathbb{C}\ell}(1,3)$ is equivalent to decoupling the equations of motion, not to compactifying extra dimensions, preserving 4D Minkowski spacetime as fundamental.