Validity of the quaternion path-rotation rule for non-differentiable paths

Determine whether the proposed quaternion product rule for the rotation associated with a path—given by the ordered product of unit tangent-direction quaternions n1 · n2-bar · n3 · n4-bar · …—remains valid when the path is continuous but non-differentiable, specifically for Weierstrass-type curves that exhibit infinite discontinuities in slope.

Background

In the 3+1 dimensional extension of his path-integral ideas, Feynman proposes to encode the amplitude for a spin-1/2 particle’s path in terms of spatial rotations tracked by quaternions. For a path approximated by a sequence of unit tangent vectors, he asserts that the net rotation is captured by the ordered product n1 * n2-bar * n3 * n4-bar * …, arguing that in the limit of a continuous path the apparent factor-of-two issue cancels.

While this reasoning is intended to hold in the continuum limit for sufficiently smooth paths, Feynman explicitly questions whether it applies to continuous but non-differentiable trajectories (e.g., Weierstrass curves). Clarifying this would determine the mathematical domain of validity of the quaternion-based path amplitude rule and whether such paths contribute consistently to the proposed formulation.