## Entanglement and non-local, finite geometry

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?

November 8, 2012 at 12:30 pm

I vaguely remember that Yuval Ne’eman wrote something on the geometrical explanations of nonlocality (nonseparability). In example see “EPR non-separability and global aspects of QM” in Found. of Modern Physics (Joensuu 1986 Sump.), Lahti & Mittelstadt eds., World Scientific, 1985. And in Proc. Nat. Acad. Sci. 80, p. 7051 (1983). See also this paper http://link.springer.com/article/10.1007%2FBF01882693

November 8, 2012 at 11:12 pm

Interesting. Thanks for the links and references.

November 9, 2012 at 3:35 am

Hi Dr. Durham,

I’ve had similar ruminations in the past. Below are some papers I’ve previously found which appear to implement the idea in one way or another (at least qualitatively):

“An amusing analogy: modelling quantum-type behaviours with wormhole-based time travel”

Journal of Optics B: Quantum and Semiclassical Optics Volume 4 Number 4

Stéphane Durand 2002 J. Opt. B: Quantum Semiclass. Opt. 4 S351 doi:10.1088/1464-4266/4/4/319

http://iopscience.iop.org/1464-4266/4/4/319;jsessionid=6874FDED5DA17157FFF7635320CB919C.c1

“Might EPR particles communicate through a wormhole?”

E. Sergio Santin

EPL, 78 (2007) 30005

DOI: 10.1209/0295-5075/78/30005

http://epljournal.edpsciences.org/index.php?option=com_article&access=standard&Itemid=129&url=/articles/epl/abs/2007/09/epl10244/epl10244.html

The Logic of Quantum Mechanics Derived from Classical General Relativity

Mark J. Hadley

Journal reference: Found.Phys.Lett. 10 (1997) 43-60

http://arxiv.org/abs/quant-ph/9706018

I haven’t studied these papers in detail yet, so I’m not sure how helpful they are, but nevertheless.

Also, Peter Holland has discussed the possibility of explaining nonlocal correlations in QM via a non-trivial spacetime topology (i.e. microscopic wormholes connecting two spatially separated particles). See his monograph, The Quantum Theory of Motion: An Account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics, p. 481. He concludes however that the idea has many problems.

Cheers,

Maaneli

October 2, 2014 at 2:57 pm

Depends on what you mean? A point like discrete space that also is finite? I don’t think you need it, you need something explaining how ‘points’ come to be, you need something that connects those points, and you need it (the universe) to be observer dependent.

Any other suggestion reminds me of a universe as some thought up ‘sphere’. Introducing that you not only should need to define a ‘inside’ but also a ‘outside’, no matter if you turn the universe into a ‘möbius strip’ of some sort?

there is no experiments I know of(?) suggesting the universe to be the first way, but there is a lot of logic and experiments assuming the universe to be isotropic, homogeneous, infinite, with physics being the same wherever you can go to report back.

you can as easily go out from a discrete universe, a ‘point’, then defining what you think connect it to another. Introducing a large amount you now start to build your universe, and ‘dimensions’ (degrees of freedom). Doing it that way I think of it as there is no ‘outside’, although there definitely exist an ‘inside’ in which we exist, and measure.

October 2, 2014 at 3:10 pm

naively all points should be equivalent, meaning that they are one and the same, that’s where my brand of ‘locality’ comes from 🙂 The other type of locality in where we find action and reaction could then possibly be defined as a emergence of the dimensions, distances, we find. As if you had ‘opposites’, a equivalence locally measured with constants existing, the other ‘observer dependent’ as when measured/compared between ‘frames of reference’, relative ones local clock and ruler, them placed at some imaginary angle to each other. i know, weird ideas.

October 2, 2014 at 3:17 pm

Or maybe it could be described as a symmetry too?