Star‑Ordered Rotor Loops in Clifford Algebras

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 *star family* with *central generator* $C$ and *leaf generators* $S_i$. The $B_i$ satisfy $B_i^2 = -1$ and anticommute pairwise. For the associated rotors $U_i(\theta_i) = \cos\frac{\theta_i}{2} + B_i \sin\frac{\theta_i}{2}$ we study two objects: the ordered chain $P_k = U_1 \cdots U_k$ and the *star-ordered rotor loop* $\Theta_k = P_k \, Q_k$ with the *same-order reversion product* $Q_k = \widetilde U_1 \cdots \widetilde U_k$. It is essential that $Q_k$ carries the same index order as $P_k$ and not the reversed one; for the reversed order one has trivially $U_1 \cdots U_k \widetilde U_k \cdots \widetilde U_1 = 1$. The scalar part of $\Theta_k$ therefore measures the *order defect* of the family. We prove two theorems. First, the scalar part of the chain factorises, $\langle P_k \rangle_0 = \prod_{i=1}^k \cos\frac{\theta_i}{2}$; this statement essentially requires the star structure, and we give a counterexample. Second, we derive a closed expression for $\langle \Theta_k \rangle_0$, valid for every $k$, whose coefficients obey a *period-four sign law* $\eta(m) = (-1)^{\lfloor m/2 \rfloor}$ and which admits the compact form $\langle \Theta_k \rangle_0 = \operatorname{Re}\!\big[(1-\mathrm{i}) \prod_i (\cos^2\frac{\theta_i}{2} + \mathrm{i}\sin^2\frac{\theta_i}{2})\big]$. Unlike the first, this second theorem holds for every pairwise anticommuting unit family without any further hypothesis. From the compact form we obtain a polar form, a complete zero criterion, a lacunary coefficient rule in the symmetric case, and a proof of non-separability for $k \geq 2$. Two supplementary theorems extend the result: to arbitrary permutations of the index order in the second factor, in which each inversion within a subset flips its sign contribution (under the star condition), and to families with mixed squares $B_i^2 \in \{-1, +1\}$, for which a uniform closed subset formula holds. All proofs use only anticommutation, reversion and the symmetry of the scalar form; no physical interpretation is assumed or claimed. > **Key Equations** > Compact form: $\langle \Theta_k \rangle_0 = \operatorname{Re}\!\big[(1-\mathrm{i}) \prod_{i=1}^k (\cos^2\frac{\theta_i}{2} + \mathrm{i}\sin^2\frac{\theta_i}{2})\big]$ (Chapter 6.2). > Polar form: $\langle \Theta_k \rangle_0 = \sqrt2 \big(\prod_{i=1}^k \rho_i\big) \cos\!\big(\sum_{i=1}^k \alpha_i - \tfrac{\pi}{4}\big)$ (Chapter 6.3).