Scope of the quaternion lift: the global sign ambiguity, closed loops, and the absence of a physical spin claim

The following qualifications are part of what the preceding items do and do not claim.

1. Two lifts and one rotation. Let $I\subseteq \mathbb{R}$ be a nondegenerate open interval and let $R:I\to SO(3)$ be of class $C^1$. By Existence, uniqueness and sign ambiguity of the quaternion lift of a $C^1$ rotation path, $R$ has exactly two $C^1$ quaternion lifts, namely $q$ and $-q$ for any one lift $q$, both solve $q^{\mathrm{\prime }}=\frac{1}{2}q\phi ({\omega }_R(t))$ for the same body angular velocity ${\omega }_R$, and $\rho (q)=\rho (-q)=R$. By the scaling recorded in The quaternion rotation map $\rho$: conjugation on the imaginary quaternions, the unit sphere $S^3$, and quaternion lifts, $\rho (\widehat{\lambda }q)=\rho (q)$ for every nonzero real $\lambda$, so a rotation is represented by the antipodal pair $\pm q$ of unit quaternions and by no other data of the quaternion itself.

2. No globally continuous sign choice. The sign of a lift is not determined by $R$, and the two signs cannot always be glued continuously along closed motion. The companion examples page exhibits a loop $R:[0,1]\to SO(3)$, continuous on $[0,1]$ and $C^1$ on $(0,1)$, with $R(0)=R(1)=I_3$, whose explicit lift from $1$ ends at $-1$. By uniqueness on the interior and continuity at the endpoints, this loop admits no closed lift through $1$ with that regularity; the explicit counterexample is linked in the Remarks below. In this sense the sign ambiguity is intrinsic and not an artifact of the formula $\rho (q)(x)=qx{q}^{-1}$.

3. No physical claim. Every statement on this page concerns curves into $S^3$ and into the matrix group $SO(3)$ and the differential equation $q^{\mathrm{\prime }}=\frac{1}{2}q\phi (\omega (t))$. No quantum-mechanical spin, relativistic, or half-integer-angular-momentum interpretation is asserted, no measurement is reported, and no empirical precision is claimed anywhere.

• What "double cover" means here. The map $q\mapsto \rho (q)$ from $S^3$ onto $SO(3)$ is two-to-one with fibres $\{q,-q\}$ and is a covering homomorphism; this is the content of the published canonical item ex-su-two-to-so-three-as-a-covering-homomorphism, restated in The quaternion rotation map $\rho$: conjugation on the imaginary quaternions, the unit sphere $S^3$, and quaternion lifts and linked from the page summary of this pair. Its countable-choice cost is part of that supplier's statement and is carried by every item that uses the kernel $\{1,-1\}$.

• What is done locally. The items of this page add the differential-equation content: for a continuous body angular velocity $\omega$ the lift equation has a unique solution in $S^3$ on the whole interval, the resulting rotation path has body angular velocity $\omega$, and conversely the body angular velocity of any $C^1$ rotation path is read from any lift by $\omega =2{\phi }^{-1}(q^{-1}{q}^{\mathrm{\prime }})$, where ${\phi }^{-1}$ is applied to the imaginary quaternion $q^{-1}{q}^{\mathrm{\prime }}$. None of these arguments re-proves the covering statement, which is taken from the published supplier.

• Half-angle parametrization. The factor $\frac{1}{2}$ in the lift equation is the algebraic half-angle relation of the quaternion parametrization: for a rotation of angle $\theta$ about a fixed axis the lift carries the angle $\theta \mathrm{/}2$, as the examples on the companion page verify on explicit curves. This is an identity about the map $\rho$, not a statement about any physical process.

• The explicit closed-loop counterexample of clause 2 is A closed rotation loop with no closed lift: the full turn about an axis lifts from $1$ to $-1$ ↗, on the companion examples page of this pair; it is a forward reference from this page and is declared as such in the frontmatter of this item.