mfb said:
It is just a mathematical tool to calculate known processes in quantum field theory. It is a new way to get the same results (probably with less effort). It is not related to Bell experiments in any way.
I think the questioner is asking whether a "hedron-based" theory could provide a hidden-variables theory. Note, not a
local hidden variables theory, because (we are being told) locality isn't fundamental here, it's emergent. So it's a logical question.
My general response would be to first remind everyone that the quantities you can calculate from this object are about the probability of having specified particles come in "from infinity" and having other specified particles leave "to infinity". And then to point out that in the new formalism, these probabilities come from calculating the volume of segments of the hedron - a calculation which is similar enough to evaluating an ordinary Feynman diagram, which will involve integration over a parameter space. It may even be a defensible proposition (I'm not sure yet) that the marvel of the amplituhedron can be reduced entirely to statements about integration domains for different path integrals - how they join up, and how they degenerate in singular limits.
One thing about the scattering picture (particles from infinity, to infinity) is that it doesn't say what happens in between. (Or it says that what happens in between is a superposition of all possibilities, if you take the path integral literally as a picture of reality.) Another thing is that it gets used in the real world, even though the scatterings only occur across finite space-time intervals, because of the "adiabatic hypothesis" that the particles (e.g. in colliders) start far enough apart that they can be regarded as "at infinity" - not yet interacting. And finally, that the scattering process in the hedron case is for a special sort of theory - first, it's conformal, and second, maximally supersymmetric.
It seems that these hedron calculations give us the scattering probabilities
without the path integral picture. That's certainly interesting, it is suggestive of quantum mechanics coming from something else. But one of the philosophical challenges of the S-matrix picture of the universe has always been, where do we fit in? We don't live at past infinity or future infinity, we live in real time, in the middle of the process somewhere.
I can think of basically two ways that relativistic quantum field theory reaches into this "real time". One is through real-time formalisms like Schwinger-Keldysh formalism. The other is via the focus on "resonances", transient objects which show up as poles in the S-matrix. In neither case is there a simple picture like nonrelativistic QM, where you just have a state that evolves in time. (Incidentally, the latter picture, of poles, is mentioned by Susskind in his "anthropic landscape" paper, where he ponders how to get a metastable de-Sitter universe from string theory.)
At this point, fans of the Copenhagen interpretation can still regard the amplituhedron the way they have also regarded state vectors in Hilbert space - just as mathematical constructs. The real world is the world of observables in a space-time, all these other objects are just mathematics. So that's one way to "interpret" it.
I am also expecting that some Everett fans will say that it is the shape of the multiverse. Whether that sort of claim can be turned into anything more than rhetoric remains to be seen.
One problem in extracting an interpretation or an ontological theory from the amplituhedron is that its volumes translate to probabilities. But I still think that the main problem is just the limitations of this first example. It doesn't even describe the whole of N=4 super-Yang-Mills, just the sector of Feynman diagrams without crossings. We will need to know how to get theories other than CFTs - though there is reason to hope there, in that generic QFTs can be understood as flows between CFTs, so it may be possible to stitch together amplituhedra from different CFTs, in order to describe a non-CFT. Here my main concern is whether we can move away from susy - I have no idea how integral susy is to the current construction, or indeed in what way it might be integral.
And there is also still the question of how to get something more than the from-infinity-to-infinity perspective; how to say something about what happens in the middle. It wouldn't surprise me if gravity was relevant here - if the way you get a gravitational theory is by stitching together "finite-time amplituhedra" for finite-time scattering events.
