The pilot-wave perspective on spin

Note that this expression for the field is simplified and idealized: A more precise expression would involve field components perpendicular to $ẑ$—which would allow the divergence of $B→$ to vanish, as it of course should—as well as a large constant field in the z-direction. As it is usually explained, this large constant field causes rapid precession about the z-axis: This makes any transverse forces (that would otherwise arise from the transverse magnetic field gradients) average to zero. It is much simpler, though, to simply look the other way in regard to the violation of Maxwell's equations, and thus avoid needing to talk about any of these subtleties, by simply pretending that Eq. (6) accurately describes the field to which the particles are subjected.

If the reflected wave term from Eq. (14) is retained, the detailed particle trajectories are a little more complicated. During the time period when the incident and reflected packets overlap, the particle velocity in that (y < 0) overlap region is slightly oscillatory, with an average velocity—in precisely, the sense explained and used in the earlier paper—that is slightly less than $ℏk/m$ by an amount proportional to $(κz)2$⁠. Thus, particles very near the trailing edge of the incident packet will in fact be overtaken by the trailing edge of the packet before reaching y = 0 and will hence be swept away, back out to the left, by the reflected packet. In short, particles that happen to begin in a small (parabolic) slice near the trailing edge of the incident packet will not behave as described in the main text, but will instead reflect. The behavior of the particles that do transmit through to the y > 0 region, however, is (in the applicable $κw→0$ limit) unaffected. So we will continue to suppress discussion at this level of detail in the main text.

Of course, the mere fact that the two sub-beams are spatially separated is not sufficient to prove that the empty packet is forevermore dynamically irrelevant to the motion of the particle. The “empty” spin-down sub-beam could be made to become dynamically relevant, for example, by arranging (by use, say, of appropriate magnetic fields) to deflect the spin-down beam back up toward the spin-up beam, at which point interference effects—including perhaps the re-constitution of a guiding wave proportional to $(10)$—could occur. But if, as suggested in Fig. 4, the spin-down sub-beam is blocked by a brick wall, then the usual process of decoherence will render it impossible, at least for all practical purposes, to arrange the appropriate overlapping of the two components of the wave function, and treating the wave as having “effectively collapsed” is entirely warranted.

It is perhaps worth noting explicitly that the non-locality cannot be used for signaling: if a large ensemble of pairs are prepared in the state (39), and Alice chooses at the last minute whether to subject all of the particles on her side to an $SGz$ measurement, or some other spin measurement, or no measurement, Bob will see the same 50/50 statistics on his side regardless of her choice.

In this respect it is somewhat telling that the worst offenses against the official orthodox view tend to occur in figures and illustrations. Almost all quantum mechanics textbooks, for example, include diagrams depicting the precession, induced by the presence of a magnetic field, of a particle's spin vector about some axis. Townsend's text (Ref. 31) avoids that particular misleading suggestion of a classical picture. But his front cover art—depicting a Stern-Gerlach spin measurement much like our Fig. 1—includes little circles with arrows (pointing, respectively, up and down) in the downstream sub-beams. The circles are even yellow, inviting us to recall Bell's warning, quoted earlier in Ref. 26 note.