Hi Ken
Only the short path-memory experiments support the retrocausal picture (footnote 3 in paper). But on the other hand, I don't think any of the bouncing droplet experiments support Bohmian mechanics. The Qn is whether the pilot-wave is a phase wave (Bohmian Psi), or a standing wave that is distinct from the Psi wavefunction. In the bouncing droplet system, the pilot-wave is a standing wave.
If pilot-waves are standing-waves, then retrocausality is involved just because standing-waves have an advanced component. Of course, real bouncing droplets only simulate having an advanced field. But they do it well enough to show how things like double-slit experiments could work for electrons.
Everyday standing-waves (drum skins, guitar strings, skipping ropes, etc) arise because of reflections from boundaries. Retrocausality is irrelevant for these. But the standing wave surrounding a "deBrogle clock" exists only because the "ticks" of the clock propagate on both the forward and backward light-cones. The "inward" waves are not reflections from boundaries----they are "retro-sourced" directly by the particle.
For distinctions between de Broglie's theory of the Double Solution and Bohmian mechanics see, for example, de Broglie ref [25], Chapter 8. However, I'm pretty sure de Broglie never mentions standing waves. Likewise, his thesis is about phase waves, not standing waves (the reference for an English translation is given in [23]).
I imagine the wavefunction (a phase-wave) applying for an ensemble, with standing-waves generated by individual particles. The connection would be something like Huygens principle but I haven't given much thought to it yet. In fact, my main point for Section 2.1 was that if there is a URT, then I can ignore all the messy and difficult problems associated with interpreting the wavefunction until we get the dynamics of an individual electron right.
Re. your qns:
(1) I meant the future measurement process, not the measurement result.
I had in mind natural boundary conditions for the experiment. Eg. the electron hits a screen => the worldline terminates in the plane x(final)=0, leaving y(final) and z(final) as free boundary conditions for the variational problem. If the Lagrangian is 2nd order then the natural boundary conditions associated with the free data are non-trivial.
It seems proper that only those worldlines that satisfy whatever (if any) future boundary conditions that are imposed by the experiment should count towards working out probabilities. But I don't know what that implies, if anything, for the wave-function.
(2) Let A_ret and A_adv be the retarded and advanced Lienard-Wiechert fields for a point-charge current source in Maxwell theory. Then in free space (ie., no matter, no other currents, no boundaries) the potential due to the electron (in its point-like approximation) has the general form
A = (A_ret A_adv)/2 rho (A_ret - A_adv)/2 .
The 2nd term here is a free radiation field (everywhere non-singular and satisfying the source-free Maxwell eqns), and rho is an arbitrary constant.
If the electron is not in free space then one can expect A to be scattered and reflected, giving rise to an additional radiation field A_env, due to the response of the environment to the presence of the electron. If nonlinear materials are involved, then A_env will not even be linear in the charge q of the electron, nevertheless A_env --> 0 as q --> 0, so A_env should be regarded as part of the potential of the electron.
There are situations (eg., in a photonic crystal) where A_env needs to be taken into account because it radically changes how the electron radiates. These are special cases though. The best we can do if deriving an EOM for an electron in a given external potential A_ext, is to assume that A_env = 0.
When the approximation A_env = 0 is not good enough, then I think no EOM can be derived (though perhaps an iterative procedure may be applicable, starting with A_env=0). Possibly, one needs to solve the full Maxwell field equations with moving boundary conditions for a worldtube surrounding the electron (as proposed in Kijowski's formulation of electrodynamics -- GRG 26, 1994, 167-204).
Assuming that the approximation A_env = 0 is OK, the only freedom in A is our choice for the constant rho. (For example, the Lorentz-Dirac equation is derived with rho=1, which gives A = A_ret.)
If the electron has a persistent circular spin motion then it would emit infinite energy (to future null infinity) for rho>0 and absorb infinite energy (from past null infinity) for rho