Dear Bill,
Thank you for your comments. I regret that you found the logical foundation part of my essay hard going. I could not omit it, because without it the novel mathematical structures that are characterized by incompleteness would seem like non-sense, and I could not describe it more fully, partly because of the length restrictions and partly because I have not yet worked out all the details.
As for my argument for pseudo-nonlocality, perhaps it helps if I explain it in this way: The incomplete objects I defined imply that in between measurements there are no "beables": The only spacetime events associated with the incomplete spacetime vector are those labeled as x_i and x_f, and both of those are measurement events.
By definition, non-locality implies the superluminal transmission of some influence as well as a receiver. In my example, if there is no beable before Bob's measurement but after Alice's measurement, then there is nothing that can "receive" such an influence. So, in essence, the scenario altogether side-steps the usual arguments for non-locality (I actually think the arguments are correct!!) by giving a description that might be called radically "non-realist" precisely in the sense that the path integral and wave function are actualizable manifestations of intrinsically "incomplete objects" which are are ontologically distinct from complete (and therefore actual) objects that satisfy the criteria for being "beables".
What I am saying so far is really perfectly in line with the orthodox understanding of QM, which says that unless you measure a quantum object it has no definite properties, clothed in novel concepts and terms that are meant to take the mystery out of these phenomena and promote deeper understanding as well as the ability to make novel predictions.
Where the "deeper understanding" and "novel predictions" come in is in the explanation in how the correlations are enforced. It is here that this framework leads to novel and very unfamiliar ideas (but note, every truly new idea was highly unfamiliar in the beginning) that have implications both for geometry and topology. I believe that the key concept here is what I call an "incomplete embedding".
I believe that a space A which is incompletely embedded in another space B is not a subspace of the latter, in the sense that you cannot arrive uniquely at metric relations within A by projecting from the metric of B (the problem, for instance, in 3D is which 2D plane to project to, since in my example, the plane simply has no z-position). If I am right, this means that the metric intervals of A and B are independent, in the sense that two objects could be far apart in B but right next to each other in A (if A were a subspace of B, then this could not happen: Two objects would have to be either next to each other in both spaces, or else be in different subspaces, say object 1 would be in subspace A and 2 would be in subspace A', where the two subspaces are separated along a direction not contained in either but contained in B). This is the intuition my Euclidean analogy was meant to bring out, but the proportionality of the metric interval to proper time in Minkowski space complicates this somewhat: Instead of a plane, we need to consider the boundaries of successive lightcones.
If the metric intervals of the two spaces (In Minkoswki they are the 2+1D space that is the boundary of each one of the successive the light cones and the 3+1D space that is spacetime) are independent (in fact, I believe they are "orthogonal"), then the correlation could be enforced at a pre-emergence level (i.e. before the underlying incomplete object is completed) without requiring any sort of superluminal influence (recall that entangled particles are always either within or at most at the boundary of the lightcone of the entanglement event). Once the emergence occurs, the actual object that emerged out of the superposition of actualizabilies by which the incomplete object manifested itself will have the correlated property, even though in the higher dimensional space it is "far away" from the other.
This is the qualitative picture of the correlated measurement phenomena that I believe lies behind all these Bell phenomena. Of, course a quantitative description will be necessary before anyone other than me believes this, but I really hope I could at least convey the rough picture. It boils down to the idea that the correlations reflect metric relations which are not those that characterize ordinary (meaning actual or complete) spacetime objects.
On the paper by Chiribella et.al. : I just read it, and that particular paper does not make it obvious how they recover standard QM from their list of axioms. I did take a look at the original paper, and, as it seems quite technical, it would require an investment of time to be able to truly understand their ideas and thereby judge how similar their ideas are to mine.
But let me at least say this: While I do not a priori think of actualizability in terms of information, I suppose one could try to make such an identification, in which case their ideas do, at least to some extent, become similar to mine. In fact, I wrote an essay in the essay contest 2 years ago which, among other things, made this point (titled "It, bit; Object, Background").
http://fqxi.org/community/forum/topic/1919
BTW, this was before I had any concrete ideas of formalizing my default specification principle.
Two major differences that I see are that just saying that "quantum states are information", even if it reproduces standard QM does not seem to point any deeper insight (though, I could be perceiving this because I have a geometric bias) and that they do not, as far as I can tell, seem to make an ontological distinction between immediate pre-and post-measurement states. That would seem to imply that everything is information. I have serious difficulties with thinking of, say, myself as "made out of" pure information.
So, in short, there may be a limited analogy, but I do not see much of a point yet in trying to pursue it further.
I hope you found my answer understandable and useful.
Best wishes,
Armin
