[Paper Review] Geometric algebra and particle dynamics

José B. Almeida|ArXiv.org|Apr 1, 2005

Algebraic and Geometric Analysis5 references4 citations

TL;DR

This paper proposes that the geometric algebra $G_{4,1}$, representing 5-dimensional spacetime, can derive the Dirac equation and naturally generate $SU(3)$ and $SU(2)$ gauge symmetries through choices of the imaginary unit in monogenic functions. By associating specific algebraic units with quantum numbers, it tentatively maps these to elementary fermions in two families, offering a geometric foundation for the Standard Model's gauge structure and particle quantum numbers.

ABSTRACT

In a recent publication the I showed how the geometric algebra ${G}_{4,1}$, the algebra of 5-dimensional space-time, can generate relativistic dynamics from the simple principle that only null geodesics should be allowed. The same paper showed also that Dirac equation could be derived from the condition that a function should be monogenic in that algebra; this construction of the Dirac equation allows a choice for the imaginary unit and it was suggested that different imaginary units could be assigned to the various elementary particles. An earlier paper had already shown the presence of standard model gauge group symmetry in complexified ${G}_{1,3}$, an algebra isomorphic to ${G}_{4,1}$. In this presentation I explore the possible choices for the imaginary unit in the Dirac equation to show that SU(3) and SU(2) symmetries arise naturally from such choices. The quantum numbers derived from the imaginary units are unusual but a simple conversion allows the derivation of electric charge and isospin, quantum numbers for two families of particles. This association to elementary particles is not final because further understanding of the role played by the imaginary unit is needed.

Motivation & Objective

• To explore how geometric algebra $G_{4,1}$ can serve as a foundational framework for relativistic particle dynamics.
• To investigate the role of the imaginary unit in the Dirac equation as a source of gauge symmetries.
• To establish a geometric correspondence between algebraic units and quantum numbers of elementary fermions.
• To provide a geometric derivation of the Standard Model's gauge group structure from monogenic functions in 5D spacetime.
• To propose a tentative mapping of algebraic solutions to known fermion families, including electric charge and isospin.

Proposed method

• Utilizes monogenic functions in the geometric algebra $G_{4,1}$, which describes 5D spacetime with signature $(-++++)$, to derive the Dirac equation.
• Applies the condition that only null geodesics are allowed in 5D space, leading to relativistic dynamics in 4D projections.
• Identifies 16 distinct unitary elements in $G_{4,1}$ corresponding to solutions of the monogenic condition, each associated with a specific choice of imaginary unit.
• Maps the coefficients of these unitary elements to quantum numbers via relations involving $\lambda_3$, $\lambda_8$, and $\lambda_{15}$ generators of $SU(4)$.
• Uses a normalization to Planck units to ensure dimensionless quantities, simplifying physical constants to unity.
• Proposes a tentative assignment of electric charge and isospin through linear combinations of the $a_\mu$ coefficients in the unitary elements.

Experimental results

Research questions

• RQ1 Can the Dirac equation be derived purely from geometric constraints in $G_{4,1}$, without assuming physical postulates?
• RQ2 How do different choices of the imaginary unit in the Dirac equation relate to gauge symmetries such as $SU(3)$ and $SU(2)$?
• RQ3 What is the geometric origin of quantum numbers like electric charge and isospin in the context of monogenic functions in $G_{4,1}$?
• RQ4 Can the structure of the Standard Model’s gauge group emerge naturally from the algebraic properties of $G_{4,1}$?
• RQ5 What is the physical significance of the 16 distinct unitary solutions to the monogenic condition, and how do they relate to elementary fermions?

Key findings

• The monogenic condition in $G_{4,1}$ yields 16 distinct unitary solutions, corresponding to different choices of the imaginary unit in the Dirac equation.
• These 16 solutions exhibit symmetries isomorphic to $SU(4)$, which decompose into $SU(3)\times SU(2)\times U(1)$, matching the Standard Model gauge group.
• The coefficients $a_\mu$ of the unitary elements can be mapped to quantum numbers: electric charge via $q = 2a_1 + a_2 + 3a_3$ and isospin via $i_3 = (a_0 + a_1 + a_2 + a_3)/2$, though this assignment is tentative.
• Specific solutions correspond to known particles: up, down, charm, strange, electron, positron, muon, and their antiparticles, with consistent quantum numbers.
• The geometric framework allows a unified interpretation of Dirac spinors as plane waves in 4D Euclidean space, linking wave optics and relativistic dynamics.
• The use of Planck units simplifies fundamental constants to unity and enables consistent dimensionless formulation of dynamics and quantum numbers.