Is the space we live in continuous (as mathematically defined) or quantized?

Question

Let me start with stating my definitions in measuring the length of objects:

The way to measure an object properly is to measure it when it's static. So take the ultra resolution (which can describe the exact form) picture of the object. Now, we want to notate the length with some objective value. Fortunately, there is another object snapped which is shorter than the sample object. Now we can compare this short object with the sample and say n pieces of short objects can maximally fill the sample, and we can say that this is the length of the sample. In general,

Notice that u is a unit and the length coefficient can only be a natural number. For further generalization, there is a shorter unit so we can get more accurate length.

~(def1)

This is the concept what I'm thinking about length in real life (this demonstrates the difference as compared to mathematically defined concept). I strongly support this way and from now on I'll interpret the following assumptions with this view.

Assumption 1. The unit length of the space is 0.

Objection of 1. From the definition 1, let n≠k and L1=n[u(i=∞)]≠L2=k[u(i=∞)]. Since [u(i=∞)]=0, L1=L2 which leads to contradiction. For intuitive approach, the assumption implies that the unit does not local the space, nothing can be measured or comparable and is the contradiction that the objects have their independent localized space(length) and the distance from others exist.

Assumption 2. The unit length of the space is non-zero.

Interpreting of 2. From the definition 1, we want to find the smallest(or shortest) unit. If [u(i→∞)] where number of n is getting bigger which intuitively implies that the smallest unit is getting smaller with respect to time proceeds(take the picture when t=0 and t=1 or arbitrary non zero point), and this situation can be equivalent with mathematically defined continuous curve (or space). And [u(i=k)] where k is some number which implies that the space is statically quantized. And the last interpretation, when [u(i=f(t))] which means that the smallest unit changes with respect to time and also quantized.

I have thought hard about these final circumstances but I can't go any further. The three circumstances can stand independently in my opinion and this can't happen in reality. Thus my knowledge is not good enough to determine this problem. Any clue for determining the particular circumstance?

Answer 1

The idea of a quantized space sounds great for one dimensional spaces. However, it has strange side effects for two and three dimensional spaces.

• Because the square root of two is irrational, the diagonal and the side of a square are not multiples of a common smallest unit.
• Let's try to model three dimensional space by a lattice with a non-zero unit spacing. This model won't be rotational invariant, because some directions are distinguished. But there is no reason to expect that some directions of the space around us should be distinguished.

Quantum mechanics has to cope with worse side effects than that. How is it possible that light is both a wave and a particle? This seemingly contradictory state of affairs can be modeled mathematically by incorporating randomness into the model. I guess that modeling a quantized space will also require randomness in some form. Maybe the result of comparing the length of two objects is slightly random, if they differ only on the order of the Planck scale. Or we have some random lattice structure instead of a normal lattice. Perhaps there even exists (mathematically) "universal random lattice" structures, similar to the universal random graph? Can we define some Planck scale for such structures? The investigation of such structures might be interesting from a mathematical point of view, independent of whether these structures model any physical reality or not.

Answer 2

The question heading is understandable but I found the question body confusing.

First of all quantisation means many different things, but historically speaking it was the initial realisation that energy was not continuous but atomic by Planck, even though he didn't take it seriously as a physical possibility but as merely a helpful solution to the black-body problem.

Now, given matter was already understood atomically, one could view this as an extension of the atomic idea into a new area - energy. The question now remaining is whether space, or rather since Einstein, spacetime is also atomic. This is one of the reasons for lattice quantum theory where they model space as a lattice, and similarly for spin-foam models (which surprisingly has some broad similarity with Platos cosmological model); other approaches include causal nets and entropic gravity.

The supposition is that, if this is true, it will happen at the Planck-length. One supposes from this, that in a sense this is speculative physics, since it may be quite some time (decades or centuries) before we begin to probe that length.

One additional note of caution, although the notion of particle is used heavily in physics, it in fact combines continuous and discrete notions. Continuous because it is a field over space-time, and discrete in its interactions.

Answer 3

At the risk of trivializing a genuine question, I can't help but wonder whether the worry is just missing what it means to use the real numbers as a convenient modelling assumption.

Do you need to think that limits over arbitrary Cauchy Sequences need to be well defined in order to do physics? Well, no, of course not, because you're not going to need to appeal to every single Cauchy sequence that might possibly be defined in order to construct an effective physical theory in a finite (albeit expanding) universe. But the Cauchy sequences that aren't well defined at the limit but don't feature in our physical theories are, ipso facto, unlikely to make much of a difference to our physics. So long as the ones that we're using in our models of physics are well defined, it's okay for the Physicist to just rely on an assumption that their space forms a continuum.

Answer 4

First of all, one needs to separate mathematical model from reality. Does the "space we live in" correspond to the leading mathematical model of the space or to our perception of the reality?

Second, one needs to separate states from the observables. Even if all the observables you measure prove to be discrete that wouldn't imply that the states of the matter are discrete.

Third, nobody really knows what happens at very small length scales, such as Planck length. It's quite possible that the appropriate model at such scale is not even 3+1 dimensional.

Imagine that you are looking at the surface of the garden hose from very far away, so it seems 1-dimensional to you. So you ask "are possible measurements of the length of the garden hose discrete or continuous?" And, as you get closer, you notice that the surface hose is actually 2-dimensional, that besides the usual spacial dimension there is another "compact" one, across the hose. Moreover, you notice that when you try to measure the hose length you ruler wraps around it, and you have no control over how many times it would wrap around. That observation kind of invalidates you question about the length of the hose: first of all, it's not the appropriate measurement for the hose, and second, it's fundamentally impossible for you to measure anything smaller than the diameter of the hose.

Answer 5

Space is not quantified, space is. The human mind can not grasp or deal with this meaning due to two reasons: one: the brain is a finite organ and two, the frequency of the idea of space is not available in the everyday frequencies used by the mind. In order to understand space, you need to move to another frequency. This will enable you to understand space and infinity properly. Using an object to measure length is assuming the there is an object with a length (unitary) that has been already measured. This is a contradiction. Mathematics can provide representations of ideas when the ideas are clear, if the idea of space is not clear it is confusing to use mathematical terminology.

If you accept my proposal that space is, then let us symbolize it as [![S]]. Any portion of it will be [1[S]] or whatever denomination. In that feature it is clear that [1[S]] is included in [![S]] and its measurement can be done based on this. The idea that space is limited generates confusion about reality but it can be accepted and excused because of human brain limitations.

Answer 6

Assuming, that length is a measurable characteristica of an object, by counting the number of objects snapped to it, as supposed by you. This 'snapped' objects have to be smaller than the measured object itself, therefore assigning a length to the 'snapped' object, otherwise 'smaller than' would not be defined. This leads to the assumption, that the 'snapped' object has a length and is therefore measurable. This leads to the conclusion, that there is no measurable object with a minimum length.