Fermions inside the geometric algebra of spacetime
Abstract
The (signed) basis versors of the geometric algebra of spacetime form the extraspecial groups of order 32. Ignoring signs and the unit element, there remain fifteen normalized basis versors. In a first main result, we show that each one of them can be written as, or factorizes into, a unit space bivector (of tridimensional space) and an isoversor, i.e., one of the three following elements: the trivector associated with space volume, the pseudoscalar, and the basis vector in the direction of the time axis. Our first result is to recognize that this central product structure, already known for extraspecial groups, also holds in the geometric algebra of spacetime and is crucial for exhibiting generators of continuous groups of the algebra. In a second main result, inspired by the 1982 paper of Hestenes ``Space-time structure of weak and electromagnetic interactions'', we unravel two pairs of special unitary groups of order two and three (in their fundamental and conjugate representations for the latter). A rich algebraic structure emerges, leading us to our third main result: The description inside the geometric algebra of spacetime of thirty-two flavorless particles of matter and antimatter. The sixteen matter particles, four leptons (left/right with up/down isospins) and four triplets of quarks (with three colors in addition to chirality and isospin), are gathered four by four into quadrispinors in two different ways: particles in a quadrispinor have either the same color or the same isospin and chirality. Thanks to this two-fold distribution of the sixteen matter particles inside four quadrispinors, the same continuous group based on isoversor produces the electroweak and strong gauge symmetry. However, chirality forbids the gauge mixing of left and right particles, which motivates the restriction to SU(2) × U(1) for electroweak interaction. Finally, as a fourth main result, we thoroughly study the dynamics of quadrispinors by generalizing the conventional treatment of bispinors in geometric algebra.
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BibTeX
@misc{verley2026fermions,
title = {{Fermions inside the geometric algebra of spacetime}},
author = {Gatien Verley},
howpublished = {HAL},
year = {2026},
month = nov,
url = {https://hal.science/hal-05852302v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Fermions inside the geometric algebra of spacetime
AU - Gatien Verley
PY - 2026
DA - 2026/11/04
PB - HAL
UR - https://hal.science/hal-05852302v1
AB - The (signed) basis versors of the geometric algebra of spacetime form the extraspecial groups of order 32. Ignoring signs and the unit element, there remain fifteen normalized basis versors. In a first main result, we show that each one of them can be written as, or factorizes into, a unit space bivector (of tridimensional space) and an isoversor, i.e., one of the three following elements: the trivector associated with space volume, the pseudoscalar, and the basis vector in the direction of the time axis. Our first result is to recognize that this central product structure, already known for extraspecial groups, also holds in the geometric algebra of spacetime and is crucial for exhibiting generators of continuous groups of the algebra. In a second main result, inspired by the 1982 paper of Hestenes ``Space-time structure of weak and electromagnetic interactions'', we unravel two pairs of special unitary groups of order two and three (in their fundamental and conjugate representations for the latter). A rich algebraic structure emerges, leading us to our third main result: The description inside the geometric algebra of spacetime of thirty-two flavorless particles of matter and antimatter. The sixteen matter particles, four leptons (left/right with up/down isospins) and four triplets of quarks (with three colors in addition to chirality and isospin), are gathered four by four into quadrispinors in two different ways: particles in a quadrispinor have either the same color or the same isospin and chirality. Thanks to this two-fold distribution of the sixteen matter particles inside four quadrispinors, the same continuous group based on isoversor produces the electroweak and strong gauge symmetry. However, chirality forbids the gauge mixing of left and right particles, which motivates the restriction to SU(2) × U(1) for electroweak interaction. Finally, as a fourth main result, we thoroughly study the dynamics of quadrispinors by generalizing the conventional treatment of bispinors in geometric algebra.
ER -