While in the shower this morning I was thinking about an old idea I had concerning entanglement and multi-dimensional space. Many years ago, without thinking too much about it, I naïvely thought the locality problem in quantum mechanics could be solved by tossing in an extra dimension or two. But upon reflection, it occurred to me that if the basic principles of relativity are to be extended to (n+1) dimensions, any two spacelike separated events should remain spacelike separated regardless of the number of dimensions, at least in most geometries that I am aware of.
Now, since I tend to think that space and time are likely emergent and that, at the very least, our knowledge of the universe must be discrete, I’m not so convinced that there is a “smooth” geometry (whatever that might mean) that might allow for some kind of non-locality. My first question was, are there non-local geometries? A quick Google search turned up this by Mikhael Gromov and a slew of things related to turbulence. Gromov’s work is related to the homotopy principle (or h-principle) which he helped co-develop. So the next question then becomes, are there “discrete” geometries in the sense that all the points in such a geometry are non-adjacent to (i.e. not in the ε-neighborhood of) all other points? The answer this question turns out to be ‘yes.’ What I would call a “discrete” geometry is really known as a finite geometry. Euclidean geometry, for instance, is not finite because there are an infinite number of points on any given line. So a finite geometry has a finite number of points on it and thus could be construed, in a way, to be “non-adjacent” in the sense I mentioned above.
So here’s my proposal: can we construct a finite geometry, preferably in (3+1) dimensions, that is also non-local and that, via coarse-graining (or some other method), is locally curved? In other words, this geometry would be, in some limit, equivalent to the geometry of general relativity, but in some other limit, would allow for the non-locality of certain quantum states and, perhaps in the process, make entanglement less mysterious. Anyone have any thoughts on this?