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Is the universe finite, both in the sense of being a closed spacetime manifold, as viewed from the macro level, but also in the sense of being fully discrete and finite in all of its intricate quantum level construction?

The most popular current research, i.e. string theory, builds a machinery on top of a continous notion of spacetime. Would a more accurate model of the physical world have to build on top of a finite, discrete, network like, relational spacetime model?

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Related: – Qmechanic Mar 24 '12 at 16:58
@GrigoriStrassman, can you elaborate a bit on what you mean by "discrete and finite" in this context? – Terry Bollinger Mar 24 '12 at 19:12
If you mean: is the universe something like a cellular automaton, a number of smart people believe something like this, but I think that the possibility quantum computing rules this possibility out. As currently posed, this question is too vague to be answerable, and should be either edited or closed. – Peter Shor Mar 24 '12 at 20:51
@Terry, by "discrete and finite, I just mean the idea that anything in the universe can be viewed as a finite collection of seperate, yet related, objects, i.e. there are no entities which are either countably infinite, or uncountable. – Halfdan Faber Mar 25 '12 at 6:02
@Peter, I really only ask to consider a physical model where there are no objects or parameters that are continuous in the mathematical sense. I don't personally believe that the universe is constructed in such a simple manner as to in any way resemble cellular automatons, but some form of network or lattice structure seems to be most natural to this kind of thinking. Again, my interest here is really about using limits of the discrete for ease of mathematical manipulation versus truly believing that the underlying physical structures are continuous in a mathematical sense. – Halfdan Faber Mar 25 '12 at 6:15

Your question is good, but dangerously edgy even to try to answer. Alas, since I seem at times I'm prone to trying rather than doing nothing at all... :)

Let me suggest an intentionally different way of approaching your question: Only conservation is absolute. Both continuous and discrete behaviors are approximate and mutable expressions of the absolute conservation of certain quantities.

I'll point to the curious mix in quantum theory of continuous wave functions and discrete outcomes as a possible example. The most accurate way to represent a wave function mathematically is as precisely continuous, yet that same continuous perfection can only be accessed experimentally in terms of discrete result that sample many such nominally perfect wave functions. But the fully discrete particle view never fully wins either, since for example detecting an absolutely positioned particle is a physical impossibility in our universe. There is instead a sort of "bounce point" between the two views, one whose scale is captured by Planck's constant.

But what does always apply without exception in analyzing quantum problems, even across light years of separation in cases of entanglement,[1] is the absolute and unyielding conservation of a certain small set of properties that includes mass-energy, charge, momentum, spin, and a few more obscure quantities such as $T_3$. So why not just declare these conservation rules to be the real absolutes, with the variable interplay we observe between continuous and discrete views as more of an emergent perspective on how the conservation rules play out over time?

So: Since you asked a good but highly speculative question, I hope readers of this answer will have some mercy on me for giving an answer. While I don't think my answer is exactly radical -- few would debate the importance of the absolute conservation laws in physics, I think! -- I fully admit that it is highly speculative in terms of the priorities I am suggesting.

[1] Focusing on conservation first puts entanglement in a rather different light. It suggests that far from being an odd or minor side effect of QM, entanglement at the classical level reflects the unresolved remnants of deeper conservation laws that mostly work themselves out into something we call "locality of effect" when they are expanded out in a self-consistent fashion over that curious dimension we call time. By "time" in this context I mean the classical, entropic, macroscopic time we know on a daily basis. The quantum version of time, the wonderfully symmetric one, occurs when one or more of those absolute conservation laws insists on keeping its options open. That openness, expressed as the uncertainty principle, makes the irreversible time we know a lot less relevant at the quantum level.

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