Is space continuous or discrete?

[Nick1234]

Dear All,

I’ve been wondering about the “Is space continuous or discrete?”-debate recently.

My question is the following: as far as I know, Heisenberg’s uncertainty principle and quantum mechanics are the main reasons why we believe it is discrete. Are these the only theories which predict this?

The reason why I am asking is that there may be another way to conclude that space (or everything else) must be discrete. This is based on the following thought: continuous space implies the concept of “eternity” or "endlessness": a particle (or wave) – if moving from one position to another – passes through an endless number of positions. Now: if we succeeded in questioning eternity/endlessness fundamentally, would that also make space discrete?

Questioning eternity in general may appear difficult at first, but maybe it is not. After all, eternity is something which we have never observed it in practice and, even more importantly, never will (because: how should this be done?). Like in quantum physics, which includes the core thought that “if there is no conceivable experiment which proves it, why still assume reality” – can we also apply this thinking here and thereby reject eternity, making space non-continuous and discrete?

A quick feedback would be appreciated!

Thank you,

Nick

 

[Fredrik]

It doesn't make sense to "question eternity in general" since since both "infinity" and the field of real numbers are well-defined and well understood mathematical concepts. Quantum mechanics doesn't give us a reason to think space or time is discrete in any way. Neither does general relativity. There are however good reasons to think that the two of them combined would imply some sort of discreteness, but it would certainly be much more complicated than the discreteness of the pixels on your computer screen, so it's not the type of discreteness you have in mind.

 

[sokrates]

Fredrik,

I am just curious and would like to hear what you have in mind: What sort of discreteness you are referring to here? Position is routinely taken as a continuous variable, no?

 

[Fredrik]

* GR describes spacetime as a smooth manifold, and matter as classical. But matter isn't classical, so GR can't be correct. Since it's very accurate on large scales, and matter isn't classical on small scales, it's clear that GR can't be accurate at the smallest distance scales. A simple order of magnitude estimate suggests that the Planck scale is where things start going seriously wrong. This doesn't mean that spacetime is discrete, but it means that we don't know how to represent spacetime mathematically at small scales.

* I've read that loop quantum gravity defines area and volume operators, which have discrete spectra.

* QFT in curved spacetime says that the maximum amount of information that can be stored in a region of space is proportional to the area of a closed surface around that region. To me this suggests that something weird is going on on small scales.

* String theory also suggests weird things about short distances, but I don't know these things well enough to try to explain them.

 

[edpell]

Quantum mechanics doesn't give us a reason to think space or time is discrete in any way. Neither does general relativity. There are however good reasons to think that the two of them combined would imply some sort of discreteness...

Please explain? Thanks.

 

[Fredrik]

Didn't I just do that?

 

[msumm]

If you use seemingly reasonable assumptions and then apply statistics to determine the behavior of a system of many particles (statistical mechanics) you will need state space to consist of "cells" of size hbar^3 in order for the resulting equations to match experimental results. For a gas of electrons (or any fermion), no more than 1 electron can occupy such a cell, ...

 

[disregardthat]

Sorry to bring this up, but I didn't find a satisfactory answer googling. Is the debate between discrete and continuous space a serious debate? If so, what does it mean that space is continuous? Continuity is a mathematical concept, so what is its physical counterpart? I recognize the impossibility/difficulty? of demonstrating mathematical continuity of space by experiment, and I'd argue that it doesn't really makes sense to say that space is mathematically continuous. What are your opinions? Is there a context in which it makes sense to say that space is continuous?

 

[Fredrik]

Is there a context in which it makes sense to say that space is continuous?

It only makes sense to say that space (or spacetime) is continuous in a specific theory. That would mean that the mathematical structure that the theory uses to represent space (or space and time) mathematically is "continuous" in some sense. It's not at all obvious what that sense would be. I think we would all say that space is continuous in a theory that uses $\mathbb R^3$ (with the standard topology) to represent it mathematically, but why? I think an answer would have to be pretty technical. I expect that it would specify what sort of topology the set has. That's if it's a classical theory. If it's a quantum theory, I expect that it would be more complicated.