[Paper Review] Octonions in Particle Physics through Structures of Generalised Proper Time

D. J. Jackson|arXiv (Cornell University)|Sep 2, 2019

Algebraic and Geometric Analysis4 citations

TL;DR

This paper proposes that octonions emerge naturally from a generalized proper time formalism extending beyond 4D spacetime, providing a geometric foundation for the Standard Model of particle physics. By generalizing the spacetime interval to include non-quadratic forms and higher dimensions, the theory uniquely embeds octonionic algebra, reproducing key features like chiral fermions, SU(3) color symmetry, U(1) hypercharge, and electroweak symmetry breaking—offering a unified, physically motivated origin for fundamental particle structures without relying on abstract mathematical aesthetics.

ABSTRACT

In considering the nature of the basic mathematical structures appropriate for describing the fundamental elements of particle physics a significant role for the octonions, as an extension from the complex numbers and uniquely the largest division algebra, has occasionally been proposed. Rather than being based initially upon the more abstract grounds of mathematical aesthetics, here we describe a unified theory motivated conceptually through an elementary generalisation of the expression for a local proper time interval, beyond that of 4-dimensional spacetime and also beyond that of the local structure of models with extra spatial dimensions, for which the explicit mathematical development naturally incorporates the octonion algebra as an essential feature. Properties of matter are identified directly through the symmetry breaking structure entailed in the necessary extraction of the external 4-dimensional spacetime background and are shown to reproduce a series of characteristic structures of the Standard Model of particle physics. While already employing octonion-based constructions of the exceptional Lie groups E6 and E7 the uncovering of the full Standard Model, as well as new physics beyond, is predicted to involve an E8-related structure for which the octonion algebra is again anticipated to be of fundamental importance.

Motivation & Objective

• To establish a physically motivated, non-arbitrary foundation for the Standard Model's gauge symmetries and fermion representations.
• To demonstrate how octonions arise naturally from a generalized spacetime proper time formalism, avoiding reliance on mathematical beauty as the primary heuristic.
• To unify the Standard Model's structure—including chiral fermions, color charge, and electroweak symmetry—within a single geometric framework rooted in higher-dimensional time-like intervals.
• To predict that the full Standard Model and new physics beyond it are governed by an E8-related structure, with octonions playing a fundamental role.

Proposed method

• Generalizing the local proper time interval beyond 4D spacetime to include non-quadratic forms and dimensions beyond 4, with n > 4 and p > 2.
• Using the generalized proper time formalism to derive a unique mathematical progression to higher-dimensional structures, naturally incorporating octonions.
• Applying symmetry breaking to extract a 4D spacetime background, from which matter field representations emerge directly.
• Constructing the Standard Model's gauge groups (SU(3)c × SU(2)L × U(1)Y) and fermion multiplets via the algebraic structure of octonions.
• Utilizing known octonionic constructions of E6 and E7 to build the Standard Model, with E8 anticipated to govern the full theory.
• Employing octonion triality and the magic square construction to extend the framework toward E8 and new physics.

Experimental results

Research questions

• RQ1 How can the octonions be derived from a physically motivated geometric principle rather than abstract mathematical elegance?
• RQ2 Can a generalized proper time formalism that extends beyond 4D spacetime and quadratic forms naturally generate the gauge symmetries and fermion representations of the Standard Model?
• RQ3 What is the role of octonions in realizing the full E8 structure predicted to unify the Standard Model and new physics?
• RQ4 How does the extraction of a 4D spacetime background from a higher-dimensional generalized proper time framework lead to chiral fermions and left-right asymmetry?
• RQ5 What is the connection between the structure of generalized proper time and the emergence of the Higgs mechanism and Yukawa couplings?

Key findings

• The generalized proper time formalism with n > 4 and p > 2 leads to a unique mathematical progression that naturally incorporates the octonion algebra as a fundamental component.
• The symmetry breaking of the higher-dimensional structure directly reproduces the Standard Model's gauge group structure, including SU(3)c, SU(2)L, and U(1)Y, and their representations.
• The theory reproduces the chiral nature of fermions, with left-handed neutrinos and right-handed components emerging from the octonionic structure, consistent with the Dirac and Majorana fermion patterns.
• The model accounts for the fractional hypercharges of quarks and leptons, including the U(1)Q charges listed in Table 1, through the octonionic algebraic framework.
• The full theory is predicted to involve an E8 structure, with octonion triality and the magic square playing essential roles in unifying the Standard Model and predicting new physics.
• Explicit calculations confirm that the octonionic construction reproduces the Standard Model's fermion multiplets and gauge symmetries, providing a direct, non-arbitrary link between octonions and particle physics.