Star-Ordered Rotor Loops in Clifford Algebras
Abstract
We consider, in a real Clifford algebra, a family of unit bivectors Bi = C Si formed from k+1 pairwise anticommuting generators C, S1, …, Sk with square −1; we call this a \emph{star family} with \emph{central generator} C and \emph{leaf generators} Si. For the associated rotors Ui(θi) = cosθi/2 + Bi sinθi/2 we study two objects: the ordered chain Pk = U1 ⋯ Uk and the \emph{star-ordered rotor loop} Θk = Pk Qk with the \emph{same-order reversion product} Qk = Ũ1 ⋯ Ũk, as opposed to the trivially reversed order, for which U1 ⋯ Uk Ũk ⋯ Ũ1 = 1. The scalar part of Θk measures the \emph{order defect} of the family. We prove two theorems. First, the scalar part of the chain factorises, ⟨ Pk ⟩0 = ∏i=1k cosθi/2; this requires the star structure, and we give a counterexample. Second, we derive a closed expression for ⟨ Θk ⟩0, valid for every k, whose coefficients obey a \emph{period-four sign law} η(m) = (−1)⌊ m/2 ⌋ and which admits the compact form ⟨ Θk ⟩0 = Re [(1−i) ∏i (cos2θi/2 + isin2θi/2)], holding for every pairwise anticommuting unit family without further hypothesis. From the compact form we obtain a polar form, a complete zero criterion, a lacunary coefficient rule, and a proof of non-separability for k ≥ 2. Two supplementary theorems extend the result to arbitrary permutations of the index order and to families with mixed squares Bi2 ∈ {−1, +1}. All proofs use only anticommutation, reversion and the symmetry of the scalar form; no physical interpretation is assumed or claimed.
Versions
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BibTeX
@misc{jenzer2026starordered,
title = {{Star-Ordered Rotor Loops in Clifford Algebras}},
author = {Nik Jenzer},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21593},
eprint = {2610.21593},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21593},
}
RIS
TY - UNPB
TI - Star-Ordered Rotor Loops in Clifford Algebras
AU - Nik Jenzer
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21593
UR - https://arxiv.org/abs/2610.21593
AB - We consider, in a real Clifford algebra, a family of unit bivectors B_i = C S_i formed from k+1 pairwise anticommuting generators C, S_1, …, S_k with square −1; we call this a \emph{star family} with \emph{central generator} C and \emph{leaf generators} S_i. For the associated rotors U_i(θ_i) = cosθ_i/2 + B_i sinθ_i/2 we study two objects: the ordered chain P_k = U_1 ⋯ U_k and the \emph{star-ordered rotor loop} Θ_k = P_k Q_k with the \emph{same-order reversion product} Q_k = Ũ_1 ⋯ Ũ_k, as opposed to the trivially reversed order, for which U_1 ⋯ U_k Ũ_k ⋯ Ũ_1 = 1. The scalar part of Θ_k measures the \emph{order defect} of the family. We prove two theorems. First, the scalar part of the chain factorises, ⟨ P_k ⟩_0 = ∏_i=1^k cosθ_i/2; this requires the star structure, and we give a counterexample. Second, we derive a closed expression for ⟨ Θ_k ⟩_0, valid for every k, whose coefficients obey a \emph{period-four sign law} η(m) = (−1)^⌊ m/2 ⌋ and which admits the compact form ⟨ Θ_k ⟩_0 = Re [(1−i) ∏_i (cos^2θ_i/2 + isin^2θ_i/2)], holding for every pairwise anticommuting unit family without further hypothesis. From the compact form we obtain a polar form, a complete zero criterion, a lacunary coefficient rule, and a proof of non-separability for k ≥ 2. Two supplementary theorems extend the result to arbitrary permutations of the index order and to families with mixed squares B_i^2 ∈ {−1, +1}. All proofs use only anticommutation, reversion and the symmetry of the scalar form; no physical interpretation is assumed or claimed.
ER -