Star-Ordered Rotor Loops in Clifford Algebras

Source abstract

We consider, in a real Clifford algebra, a family of unit bivectors $B_i = C S_i$ formed from $k+1$ pairwise anticommuting generators $C, S_1, \ldots, S_k$ 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 $U_i(θ_i) = \cos\frac{θ_i}{2} + B_i \sin\frac{θ_i}{2}$ we study two objects: the ordered chain $P_k = U_1 \cdots U_k$ and the \emph{star-ordered rotor loop} $Θ_k = P_k \, Q_k$ with the \emph{same-order reversion product} $Q_k = \widetilde U_1 \cdots \widetilde U_k$, as opposed to the trivially reversed order, for which $U_1 \cdots U_k \widetilde U_k \cdots \widetilde U_1 = 1$. 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, $\langle P_k \rangle_0 = \prod_{i=1}^k \cos\frac{θ_i}{2}$; this requires the star structure, and we give a counterexample. Second, we derive a closed expression for $\langle Θ_k \rangle_0$, valid for every $k$, whose coefficients obey a \emph{period-four sign law} $η(m) = (-1)^{\lfloor m/2 \rfloor}$ and which admits the compact form $\langle Θ_k \rangle_0 = \operatorname{Re}\!\big[(1-\mathrm{i}) \prod_i (\cos^2\frac{θ_i}{2} + \mathrm{i}\sin^2\frac{θ_i}{2})\big]$, 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 $k \geq 2$. Two supplementary theorems extend the result to arbitrary permutations of the index order and to families with mixed squares $B_i^2 \in \{-1, +1\}$. All proofs use only anticommutation, reversion and the symmetry of the scalar form; no physical interpretation is assumed or claimed.