Future- versus past-pointing step vectors in the biquaternion path integral

Ascertain whether, in the biquaternion-based one-step evolution equation (B) that integrates over unit timelike 4-vectors, the step vectors should be restricted to future-pointing directions, and justify the choice relative to including past-pointing vectors; clarify the implications of this choice for causality, locality, and convergence of the proposed formulation.

Background

Building on a Lorentz-covariant construction using complex quaternions (biquaternions), Feynman sketches a one-step evolution rule, Eq. (B), where the integration is taken over unit timelike 4-vectors. The commentary analyzes several technical and conceptual issues in this proposal.

One explicit unresolved point is the directional restriction of the step vectors in the integral to future-pointing timelike directions. Establishing whether this restriction is necessary or appropriate—and what changes if past-pointing directions are included—would clarify the physical and mathematical consistency of the approach.