Stochastic process with weak-momentum drift for moving post-selection

Identify whether a stochastic object exists beyond the phase-gradient and Wiseman interpretations whose drift equals the real part of the weak momentum for a moving, phase-carrying post-selection.

Background

The paper distinguishes the weak-momentum field’s real part, which is a phase gradient and, in the position-post-selected setting, a Wiseman current velocity, from the drift of a conditioned diffusion. For a free particle and for the explicit bridge constructions studied, the bridge drift does not generally coincide with the weak-momentum field; in particular, the moving two-endpoint bridge follows a classical connecting trajectory instead.

The unresolved issue is whether some other stochastic object—not necessarily the standard Doob or Bernstein bridge—could realize the weak-momentum field as its drift when the post-selected state moves and carries nontrivial phase information.

References

Identifying the stochastic object whose drift is $\Rea(p_w)$ for a moving, phase-carrying post-selection, if one exists beyond the phase-gradient and Wiseman readings, is left open.

— Nodal obstruction to conditioned-diffusion representations of the weak momentum  (2609.03242 - Kam et al., 3 Sep 2026) in Conclusion

What remains on the bridge side is to extend eq:etaexplicit to a moving, general-$k$ node. The two-endpoint process whose existence Corollary~\ref{cor:conditioning} invokes is confined to the moving cell by the drift repulsion of Proposition~\ref{prop:inaccessible}, but its closed form is left for future work.

— Nodal obstruction to conditioned-diffusion representations of the weak momentum  (2609.03242 - Kam et al., 3 Sep 2026) in Appendix, Section Reciprocal (Bernstein) process and positivity, subsection An explicit stationary node

It is very hard to tell whether this value can still be considered as a measurement result, let alone as a meter shift.

— Transition between weak and strong measurements in the presence of post-selection  (2609.04812 - Hanashiro et al., 4 Sep 2026) in Section 3, “Statistics of meter position,” paragraph following Eq. (approximation)