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My guess at a list of them would be: spin foams, casual sets, non-commutative geometry, Machian theories, twistor theory or strings and membranes existing in some higher-dimensional geometry...

Firstly, what have I missed?

Also, if there is a string theorist or similar on here that could briefly shed some light on what's involved with these approaches I'd be most grateful.

[Editted/expanded from here on]

OK so I'm not having much luck getting a response to this so I suppose I've not been clear enough. I'll try to expand a little:

In early 2010 I attended this inagural lecture by string theorist- Prof. Mavromatos entitled 'MAGIC strings'. In it he proposes that some string theory models may violate Lorentz symmetry at the Planck scale resulting in a kind-of foamy spacetime that could be observed by differing arrival times of photons of different energies reaching us from distant astronomical sources. See http://www.kcl.ac.uk/news/events_details.php?year=2010&event_id=2178 or here for one of the papers: http://iopscience.iop.org/1742-6596/174/1/012016

Furthermore, as I mentioned in the comment below, Prof. Sir Roger Penrose mentions that Wheeler and others have strongely argued that if we could examine spacetime at the Planck scale we would see a tubulent chaotic situation (from quantum-mechanical vacuum fluctuation I suppose) or perhaps a discrete granular one. Penrose goes on to list some other approaches that may suggest how this discete stucture may manifest itself. Hence my list above.

I have studied much QM, introductory QFT and the Standard Model as well as some basic GR but I have no formal experience of string theory. I therefore wondered if anyone here could jot down a sentace or two to explain what's involved with each of the above approaches and add to the list if I've overlooked some alternative popular(ish) approaches.

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pixie dust? j/k. Could you narrow your question down a bit? –  user346 Jan 31 '11 at 21:41
    
This is a question loosely based on a quote from Penrose's 'Cycles of Time'. I was trying to appreciate (at least in essense) the range of different theories that lead to a discretised spacetime, whether or not full GR has thus far been incorperated. –  qftme Feb 1 '11 at 14:42
    
The quote in question is on page 203 but if required I will transcribe it onto here.. –  qftme Feb 1 '11 at 14:43
    
Quantum Mechanics? –  user1708 Feb 1 '11 at 17:08
    
@kakemonsteret In the QM I've studied I don't recall ever coming across discretised space or time variables for free particles. I'm pretty sure they're continuous variables, or have I missed/forgotten something? –  qftme Feb 1 '11 at 17:45
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The main way in which space-time is reformed in string theory is in BFSS Matrix theory. In this idea, there are a collection of point particles (representing extremal charged black holes in a circle compactification of M-theory) and these point particles interact mutually with a special 1-dimensional matrix model. The BFSS Lagrangian is a bunch of points which carry a matrix position X, (with a huge amount of 1 dimensional supersymmetry) interacting with a potential energy proportional to the squared commutator of the matrices.

The least energy configurations have the position matrices commutative. When there are a large number of points, this description has a full 11 dimensional theory reemerging from a 1-dimensional collection of matrices.

This is an unbelievable realization of the program of space-time discretization, since the continuous 10 dimensional space is completely emergent from a large N limit of 1 dimensional oscillations. This type of miraculous emergence of a large space-time is the only well-established reasonably complete description of spacetime from a structure of much reduced complexity which is known to work for sure.

The Matrix theory is the only fully mathematically rigorous definition of M-theory known today. It is mathematically rigorous, because 1-dimensional supersymmetric systems can be simulated on a computer with known Nicolai map methods. This makes a large part of M-theory simulatable non-perturbatively on a computer, in the large N limit of the matrix theory.

Speculative proposals

Loop quantum gravity suggests that the holonomy variables of GR are the fundamental variables. In loop quantum gravity, the fundamental variables are a discrete collection of intertwined loops which carry an element of SO(3), a rotation to each link, like Penrose's spin network.

The hope of this type of description is that a continuum limit will reproduce ordinary space-time. So far, this hope is far from realization, because unlike string theory, there is no supersymmetry or analogous principle which demands a flat space-time emerge.

Causal networks are even more speculative, making the idea of past and future fundamental. This idea does not even have a precise model for reproducing a space-time, unlike loops, which have spin networks. But it is also a newer avenue, so it might just need time to mature.

I am biased in favor of the string approach. I believe that this description string theory gives is correct, and that any complementary approach will either turn out to be inconsistent with the holographic principle, or will be shown to emerge from string theory in a certain regime.

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