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It's comparatively easy (cum grano salis) to grasp the following concepts:

  • Euclidean space-time (continous space and continuous time)
  • classical mechanics (discretely distributed matter in continous space and continuous time)
  • Minkowskian space (continously intermingled space and time)
  • special relativistic mechanics (discretely distributed matter in continously intermingled space and time)
  • classical electrodynamics
  • classical quantum mechanics (discrete energies, continuously distributed matter in continous space and continuous time)
  • quantum electrodynamics
  • general relativity (continously intermingled space-time and matter)

Accordingly, there are lots of introductory texts and text-books.

It's also easy to grasp

  • numerical simulations (on artificially - and mostly unphysically - discretized spaces, times, and space-times)
  • cellular automata (on unphysical regular spatial grids)

It's definitely hard to grasp (for somehow graspable reasons)

  • quantum gravity

I do not know whether there are empirical evidences for a discrete space-time or only theoretical desiderata, anyhow I cannot figure a discrete space and/or time out.

Why is it so hard to introduce and explain the concept of a physical discrete space-time?

Why are there no easy to understand introductory texts or text-books on definitions, concepts, models, pros and cons of discrete space, time, and - finally - space-time?

Respectively: Where are they?

Are the reasons for this maybe related to the reasons why quantum gravity is so hard to grasp?

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This doesn't look like a question so much as a personal theory/prejudice rephrased as a grammatical question. –  Mark Eichenlaub Nov 23 '10 at 17:20
    
So, not taking the question seriously - as it was posed - means: understanding discrete space, time, and space-time is in principle as "easy" as any of the "easy" concepts listed above? So, please just help me to learn and give me one introductory text - as there are many for any of the other subjects. –  Hans Stricker Nov 23 '10 at 17:38
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I am not sure what you are asking for. If you want to learn about discrete quantum gravity theories (like LQG) then try to ask specifically about the concrete theory. Or ask about what approaches exist. Or perhaps do you want to learn about discretization in general? Because that is done all the time e.g. in condensed matter physics (think crystals) and lattice QFT (either space or time or both can be discretized). Although in lattice QFT there it's understood that discrete model is just an approximation to a continuous model. –  Marek Nov 23 '10 at 17:48
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@Hans: because "discrete geometry" is a much harder concept. What is usually meant by geometry is the space having a Lie (that is continuous) group acting on it. Or more generally, Riemannian geometry (of which the previous part is a homogenous case). But where do you get symmetries in discrete spaces? You lose almost all the rotations on the lattice (except for few that preserve it). So you lose Lorentz invariance and the theory is immediately shaky. To make the theory look at least a little like physics you have to introduce very nontrivial concepts. Does that answer your question? –  Marek Nov 23 '10 at 18:10
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Why are you assuming (or at least I think you are?) that quantum gravity has anything to do with discrete space and time? It probably doesn't. –  Matt Reece Nov 23 '10 at 21:39
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1 Answer

up vote 6 down vote accepted

I think you're looking for something like Regge Calculus (there are plenty of extra refs in this link, and many more in this one).

You can also check this article, Quantum Gravity and Regge Calculus.

Furthermore, this guy has a whole research area in this topic (discrete spaces, etc)!

So, there's plenty of stuff out there... you just have to look. ;-)

(Edited) PS: In fact, here's the name of the game: Discrete Differential Geometry. Google away...

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There's also en.wikipedia.org/wiki/Causal_dynamical_triangulation –  mtrencseni Nov 23 '10 at 22:08
    
@mtrencseni: Yup. But, i think you can understand CDTs as a Discrete Differential Geometry endowed with a Causal Structure. So, in the sense of "reducing to the basics", i think it'd be best to understand what it means to discretize geometrical objects (eg, Diff Forms), and then add causality to the mix. –  Daniel Nov 23 '10 at 22:25
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