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Stephen Wolfram in his book A New Kind of Science touches on a model of space itself based on automata theory. That it, he makes some suggestions about modelling not only the behaviour of matter through space, but the space itself in terms of state machines (a notion from computing). Here, the general topology of space arises from a small-scale connection lattice.

I wondered whether any theoretical work is being undertaken along these lines within the physics community.

The reason for my interest in this regards one of the mysteries of quantum mechanics, that of quantum entanglement and action at distance. I wondered whether, if space is imagined as having a topology that arises from a notion of neighbourhood at a fine level, then quantum entanglement might be a result of a 'short circuit' in the connection lattice. That is two points at a distance through 'normal' space might also still be neighbours at a fundamental level; there might be a short strand of connectivity in addition the all the long strands relating the two.

(I think Richard Feynman also alluded to this sort of model with his take on quantum electro-dynamics.)

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It should be said that we know very little about the true topology of spacetime, and that the topology of spacetime, barring some fundamental revelation from quantum gravity, is probably unknowable--we are constrained to only be able observe a limited subset of the whole universe--the visible universe, with the rest hidden behind a cosmological horizon. Most of the information about topology, however, involves going 'around' the universe, or investigating edge effects--thus, we will likely never find out about the true topology of the universe. –  Jerry Schirmer Feb 15 '11 at 0:29
    
@JerrySchirmer: recent work of Perelman on Poincare conjucture is the probable way out to find the shape of the universe. So, is it that now the above question stands valid ?? –  Rorschach Aug 28 '12 at 12:34
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Space_cadet mentioned already work about deriving spacetime as a smooth Lorentzian manifold from more "fundamental" concepts, there are a lot of others -like causal sets, but the motivation for the question was:

The reason for my interest in this regards one of the mysteries of quantum mechanics, that of quantum entanglement and action at distance. I wondered whether, if space is imagined as having a topology that arises from a notion of neighbourhood at a fine level, then quantum entanglement might be a result of a 'short circuit' in the connection lattice.

I'm not convinced that such an explanation is possible or warranted, the reason for this is the Reeh-Schlieder theorem from quantum field theory (I write "not convinced" because there is some subjectivity allowed, because the following paragraph describes an aspect of axiomatic quantum field theory which may become obsolete in the future with the development of a more complete theory):

It describes "action at a distance" in a mathematically precise way. According to the Reeh-Schlieder theorem there are correlations in the vacuum state between measurements at an arbitrary distance. The point is: The proof of the Reeh-Schlieder theorem is independent of any axiom describing causality, showing that quantum entanglement effects do not violate Einstein causality, and don't depend on the precise notion of causality. Therefore a change in spacetime topology in order to explain quantum entanglement effects won't work.

Discussions of the notion of quantum entanglement often conflate the notion of entanglement as "an action at a distance" and Einstein causality - these are two different things, and the first does not violate the second.

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I've heard you talk of the Reeh-Schlieder theorem before. I read its Wikipedia description and the mathematical definition (cyclic vector etc.) is impenetrable. I understand that it implies, as you mention correlations in the vacuum state between measurements at an arbitrary distance. It would be nice to have a more pedagogical description of this theorem. –  user346 Feb 13 '11 at 13:23
    
@space_cadet: this paper (referenced by wikipedia) is pretty good in explaining it. Also the ncatlab page is a good place to start (if a bit harder). But you should go ahead and ask this as a question; I'd also be interested to learn more. –  Marek Feb 13 '11 at 14:02
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In general, if you construct a manifold out of a combinatorial graph, or out of patches, then you're not finding anything "more fundamental". You're just describing the most straightforward and most superficial definition of the concept of topology.

I realize that this very modest sentence contradicts the whole philosophy of Stephen Wolfram's book, and the reason behind this contradiction is that Stephen Wolfram's book is misguided in this very basic respect.

If you want to find more fundamental laws of physics that produce topology, you always deal with phases of a physical system that have to be found by solving some equations; and/or with a geometric interpretation of individual terms in an expansion of a physical observable that doesn't have a topological interpretation.

To see the latter example, check e.g. the quantum foam paper

http://arxiv.org/abs/hep-th/0309208

by Okounkov (a Fields medalist), Reshetikhin, and (most importantly) Vafa. A partition sum of a topological string theory may be obtained as a sum over manifolds of many different topologies - but it may also be viewed as an expansion of a function that describes the propagation of (topological) strings in a flat background.

Both ways of writing the partition sum are also equivalent to the partition sum of a melting crystal. Their paper is a particular, quantitative realization of John Wheeler's concept of a "quantum foam".

In other contexts of physics, different topologies are simply allowed and they have to be summed over. That's the case of perturbative string theory where the scattering amplitudes are calculated as sums over world sheets of all topologies (Riemann surfaces).

This sum may be obtained in various other formalisms where it looks "derived". In the non-perturbative light cone gauge description of perturbative string theory, the so-called matrix string theory,

http://arxiv.org/abs/hep-th/9701025
http://arxiv.org/abs/hep-th/9703030

the very states containing $K$ strings are obtained from the same Hilbert space, by identifying various holonomies of the gauge field around a circular dimension of a Yang-Mills theory. The possible holonomies for which low-energy states exist may be labeled by permutations of $N$ eigenvalues, and the cycles from which these permutations are composed may literally be interpreted as strings of various lengths.

Now, 1 string is topologically different from $K$ strings for $K\neq 1$ but in matrix string theory, all these states are configurations of a single field theory in different limits. Analogously, the sum over histories will contain world sheets of all topologies which are generated as histories of switching in between the phases of the Yang-Mills theory.

In string theory, you get generically dual descriptions of a compactification in which the topology of the space is completely different. For example, mirror symmetry relates two very different topologies of a six-dimensional Calabi-Yau manifold. M-theory or string theory compactified on a four-dimensional K3 manifold is equivalent to heterotic strings on tori; the individual excitations of the heterotic string are mapped to nontrivial submanifolds of the K3 manifold, and so on.

So the summary is that there are many exciting ways in which topology of space may turn out to be "emergent", or at least "as fundamental as other, non-topological descriptions", but none of them is similar to your "discretized" template.

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This is essentially the philosophy which emerges from the spin-network picture of spacetime. A nice example of this is this paper by Smolin and Prescod-Weinstein which argues that the differing notions of locality arising due to the graph like nature of spacetime, might be relevant for understanding dark energy.

Also the notion of treating spacetime itself as a cellular automaton goes as far back as Konrad Zuse who designed and built some of the first digital computers. He has a nice monograph on this called "Rechrender Zaum" or "Rendering Space" which you can find along with other useful information on Jurgen Schmidhuber's page.

There are many people who have recently done work along these lines, such as Seth Lloyd, Livine, Terno, Girelli, and others (ref1, ref2). What precise implications such CA models have for topology is not clear. What is certain is that such models will naturally incorporate non-trivial topologies.

The model of Feynman that you refer to is known as the checkerboard model. This was his attempt to deduce the dynamics of an electron in a simplified setting from a stochastic model in which a priori the electron does not obey any predetermined equations of motion. One finds that the Dirac equation for the electron (in $1+1$ dimensions only) emerges from a path-integral calculation in this model.

Sidebar: Wolfram's book has received a great deal of criticism, some of it no doubt well-deserved. However it takes a great deal of courage to put forward a hypothesis such as that of the "computational universe" (my terminology) which has the potential to overturn 500 years of conventional wisdom.

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