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.)
Answer
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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