# discrete space

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Hi everybody,

I have a question about the whole idea that space/spacetime is fundamentally discrete. It's been in the back of my mind since I first encountered it, because I still don't have any clear understanding of what it means. Keep in mind that I have absolutely no understanding of physics beyond some popular science books.

So my question is, what does it mean, if anything, for space to be fundamentally discrete? Does it mean that space is broken up into tiny parts? Physically what can it mean to have arrived at a smallest part? Or is it something more subtle than that?

Thanks for any input!

Peace,

Mike

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Discrete is just like digital.

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Does it mean that space is broken up into tiny parts?

In essence yes.

Classically the question is really one of topology. A discrete space is a set of point equipped with the discrete topology. Any such space has zero topological dimension. You should think of it as a collection of "isolated" points. There are models of quantum gravity that use such spaces: see causal set theory.

One may also be thinking about "quantum spaces" or noncommutative geometries. Instead of being described by a collection of points, it can only be cut up into fuzzy cells as the uncertainty principle applies. On such spaces there is a minimum volume that can be examined. This is also a kind of "discrete space" of sorts. But I must stress that such spaces (there is no "theory of NCG" as there is lots of work left) are not set theoretical objects. That is, not all the features of such spaces are described by points. In fact, the notion of a point can be totally lost. "Noncommutative geometry is pointless".

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Hi ajb, thanks for the response. You wrote,

Classically the question is really one of topology. A discrete space is a set of point equipped with the discrete topology. Any such space has zero topological dimension. You should think of it as a collection of "isolated" points.

And

This is also a kind of "discrete space" of sorts. But I must stress that such spaces (there is no "theory of NCG" as there is lots of work left) are not set theoretical objects. That is, not all the features of such spaces are described by points. In fact, the notion of a point can be totally lost. "Noncommutative geometry is pointless".

Like I said I am not educated in physics. Could you help me visualize this? What does it mean for a space to have zero topological dimension, classically speaking? If it has no dimension, what is it that we're describing?

As for quantum physics, what does it mean for spaces to not be 'set theoretical objects' and unable to be described by points?

I guess what I'm trying to wrap my brain around the idea that there is a smallest unit of space, because I immediately visualize some thing/object with dimensions. If it is a thing with dimensions then it cannot be fundamental, can it?

Mr. Skeptic calls it digital. If the idea here is that there is some limitation on how reality can operate (it must manifest itself wholly or not at all: quanta) this makes sense to me, though in a yet very fuzzy way. Forgive my metaphysics but it is my only substitute for the math.

Thanks for the input,

Mike

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What does it mean for a space to have zero topological dimension, classically speaking? If it has no dimension, what is it that we're describing?

A point by definition is zero dimensional. A zero dimensional space (in fact a manifold) is just a point or a collection of disjoint points. You think of these point as being isolated, so no "space in between them".

As for quantum physics, what does it mean for spaces to not be 'set theoretical objects' and unable to be described by points?

A topological space is a set of points and a topology which is really a way of defining a notion of points being close to each other.

Everything about a topological space is contained in the set of points and the topology.

We have a more general notion of a space than this, sometimes these have a topological space underlying them, sometimes not. Either way, not all the information about such spaces is encoded in the "set of points".

I guess what I'm trying to wrap my brain around the idea that there is a smallest unit of space, because I immediately visualize some thing/object with dimensions. If it is a thing with dimensions then it cannot be fundamental, can it?

It is a very non-intuitive notion. Especially as one likes to think of spaces as "collections of points".

Trying not to be too technical here, but a classical space is equivalent to the algebra of functions on it, which is a commutative algebra. So, a modern way of thinking is to think of classical spaces in terms of the functions on them. This way of thought does not require points and points can be thought of as a "derived" notion.

This way of thinking opens up a much more general ethos.

A "space" is an associative algebra.

Or maybe a slightly better way of thinking is that modern geometry is algebra in which one thinks along the lines of classical geometry.

Depending on exactly what you are doing, points can be defined but they are not really the same as the classical notion.

Mr. Skeptic calls it digital. If the idea here is that there is some limitation on how reality can operate (it must manifest itself wholly or not at all: quanta) this makes sense to me, though in a yet very fuzzy way. Forgive my metaphysics but it is my only substitute for the math.

Digital is not a bad way to think about it. Things come in "lumps". Think of your radio. You cannot single out an exact specific frequency, the best you can do is select a "lump" in which that frequency is included.

Thanks for the input,

Mike

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If space is really discreet, what's between each "packet" -- nothingness? A lack of space?

However...it could be all those (otherwise) empty areas would get filled in by random other discreet stuff, made 100% snug along their borders, so we'd never find a sliver of an empty place in the universe.

So basically in other words....space might not be continuous, yet the universe might be: existing as a seamless collage of discreet everythings. Who knows.

Unless of course, I'm entirely wrong about the (discreet space) concept's meaning or use in the first place.

Though it seems not exactly a foregone conclusion that space is discreet, according to what Martin had said a few years back.

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One would then expect a continuous smooth structure to emerge as a limit of the discrete or "fuzzy" space-time. If not then it would be difficult to understand the physics we see today.

Even the notion of topological dimension would be an emergent concept.

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One would then expect a continuous smooth structure to emerge as a limit of the discrete or "fuzzy" space-time.

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It is a very non-intuitive notion.

You can say that again. I find it very difficult to find meaning in the idea of an object (a point or a space) with zero dimension. I'm not saying there isn't any, just that I don't understand it. To my mind such spaces would be completely immaterial, and there would either be an infinite amount of space between each one or no space and therefore distinguishing between them would be meaningless. How could they ever 'add up' to the universe we see?

When we speak of these points and such, are we talking about anything close to a traditional 'thing' or 'object' that composes larger, macroscopic things, or is it something fundamentally more abstract and/or immaterial than that?

You've certainly given me a lot of ideas to look further into. On a philosophical level, I personally find the notion of a finite universe to be the more compelling choice. But I'm very much unable to visualize what it means to have arrived at a fundamental discrete 'part'. The question always arises, well what would that then be made of, or why can't you just make it smaller or break it apart? Of course this is assuming a very billiard-ball type object. Or any classical object. The only way, to my mind, is to find a way out of having to ask that question by a more subtle way of thinking about the problem - and I'm very sure modern physics has done quite a bit of that.

Thanks,

Mike

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On a fuzzy space, on large enough scales it would start to resemble a classical space. So even if space-time is made of "Planck-cells" at the scales we observe today we see a continuous structure. The fuzziness is smaller than our "microscope" allows us to resolve.

If one thinks about a lattice in say 3-d Euclidean space, then as the lattice spacing becomes smaller and smaller the lattice starts to look continuous. In the limit of the lattice space going to zero we are back in the continuous case. Presumably any discrete theory of gravity will in some correct limit be general relativity.

Merged post follows:

Consecutive posts merged
You can say that again. I find it very difficult to find meaning in the idea of an object (a point or a space) with zero dimension. I'm not saying there isn't any, just that I don't understand it. To my mind such spaces would be completely immaterial, and there would either be an infinite amount of space between each one or no space and therefore distinguishing between them would be meaningless. How could they ever 'add up' to the universe we see?

Topological dimension of a manifold, which is what we are talking about is intuitively easy. It is the number of numbers needed to specify a point on a manifold.

A manifold $M$ of dimension $n$ is a topological space (set of point + a topology) such it is locally homomorphic to $\mathbb{R}^{n}$.

So a one dimensional manifold can only be $\mathbb{R}$ or $S^{1}$ or the disjoint union of these.

What about a zero dimensional manifold? This can only be a point $p = \{ \}$ or a disjoint union of points.

You have some confusion here about "stuff in between points". This is because you are thinking about manifolds embedded into a higher Euclidean space. (You can always do this). You should think of manifolds as intrinsic objects in there own right.

For example, our zero dimensional manifolds as a collection of points. These are defined in terms of disjoint unions. There is nothing in between them. More than this, there is nothing to be defined between them.

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You have some confusion here about "stuff in between points". This is because you are thinking about manifolds embedded into a higher Euclidean space. (You can always do this). You should think of manifolds as intrinsic objects in there own right.

For example, our zero dimensional manifolds as a collection of points. These are defined in terms of disjoint unions. There is nothing in between them. More than this, there is nothing to be defined between them.

I'm interested in what this might mean philosophically and physically apart from mathematical formalisms on paper, because I realize that theoretically we can speak of zero-dimensional objects and point-particles. But it strikes me that these manifolds would exist immaterially as Plantonic- or as pseudo-objects, rather than as material/physical objects.

I'm also not sure how to take these points and manifolds as being 'isolated' or 'intrinsic objects', because if they were, in the senses that I think of 'isolated' and 'intrinsic', then there'd be no sense in speaking of them. They'd be their own reality all by themselves.

These are, of course, my own impressions rooted no doubt in the imprecise nature of words, and I do not project them onto the content of your responses.

Thanks,

Mike

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But it strikes me that these manifolds would exist immaterially as Plantonic- or as pseudo-objects, rather than as material/physical objects.

...

They'd be their own reality all by themselves.

I understand what you think of here. Theoretical physics is really a mathematical pursuit. All the constructions really live in the "mathematical world", whatever that is. The question one asks is "how does the mathematics relate to the physical world?". That is we require an association of part of the construction to a physical observable in the real world. I.e. something we can measure.

Not all of the construction will necessarily be observable in this sense.

So, I would not worry too much about the question "is the space and time around us really a manifold?" (and similar questions) The question is completely loaded! All we can really say is that "we model the physical universe, or part of it using the mathematical theory of manifolds". (for example).

A large part of theoretical physics is phenomenology which bridges the gap between theory and experiment. I see it, in a loose sense as a "map" between the mathematical world and the physical world.

Anyway that is my "philosophy". Others believe that the universe really many be mathematical. Max Tegmark has proposed the "mathematical universe hypothesis",Wikipedia has an article here and the original paper here.

All this is very interesting, but don't loose any sleep over "metaphysics" and "metamathematics". (Statements about not in physics and mathematics)

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I understand what you think of here. Theoretical physics is really a mathematical pursuit. All the constructions really live in the "mathematical world", whatever that is.

.....

So, I would not worry too much about the question "is the space and time around us really a manifold?" (and similar questions) The question is completely loaded! All we can really say is that "we model the physical universe, or part of it using the mathematical theory of manifolds". (for example).

A lot of things make sense now, ajb. Jeez, I was thinking of the constructions as real entities one can actually physically inspect given the right tools.

Question is, how can you tell it's only a mathematical construction when you encounter a scientific description or concept? Is there a list or database to check somewhere to see what's real (physical/observed) or mathematical?

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Question is, how can you tell it's only a mathematical construction when you encounter a scientific description or concept? Is there a list or database to check somewhere to see what's real (physical/observed) or mathematical?

Anything written down using mathematics is "mathematical" what you see in the lab is "real".

The game one is playing is to try to have a good agreement between the mathematical predictions and the actual observed phenomena. All "concepts" are really mathematical, they are our way of describing the world.

One can ask is momentum or energy "real"? I would say that they are mathematical ideas used to keep track of what we see. Such things may or may not have very clear physical interpretation.

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I understand what you think of here. Theoretical physics is really a mathematical pursuit. All the constructions really live in the "mathematical world", whatever that is. The question one asks is "how does the mathematics relate to the physical world?". That is we require an association of part of the construction to a physical observable in the real world. I.e. something we can measure.

Not all of the construction will necessarily be observable in this sense.

So, I would not worry too much about the question "is the space and time around us really a manifold?" (and similar questions) The question is completely loaded! All we can really say is that "we model the physical universe, or part of it using the mathematical theory of manifolds". (for example).

A large part of theoretical physics is phenomenology which bridges the gap between theory and experiment. I see it, in a loose sense as a "map" between the mathematical world and the physical world.

Anyway that is my "philosophy". Others believe that the universe really many be mathematical. Max Tegmark has proposed the "mathematical universe hypothesis",Wikipedia has an article here and the original paper here.

All this is very interesting, but don't loose any sleep over "metaphysics" and "metamathematics". (Statements about not in physics and mathematics)

Hi ajb,

I appreciate what you say here, and your taking the time to address these questions. Trying to find the simplest and therefore most general part or aspect of reality has been physic's main goal as I understand it, extending back to the atomism of the Greeks. But I wonder if, taken to the very extreme, such a search will ultimately prove to be meaningful. Not that we shouldn't try, since the limited results obtained are always insightful. But whatever set of axioms we glean from the universe, it seems logically that there will always be more, since any proposition seems to inescapably lead to more propositions and a synthesis of these propositions ad infinitum.

It seems to me that theoretical physics is ultimately trying to get us to the simplest conceivable principle or object. But how would we know it if we saw it? What would it really mean to find it? It seems that the simplest thing we can imagine, is nothing at all: zero dimensions, zero extension, etc. Such objects or whatever-you-call-its would be the ideal thing to work with. But do such objects explain nothing?

But really, what would it really mean to find an elementary particle or whatever? Would they fit the category of 'object' or would it be 'something' much different than that? In my understanding, a truly "fundamental" particle or philosophical atom, which many physicists seem to identify with what are commonly called the elementary particles, cannot fit the definition of a "thing" or "object" since they would have no structure or substructure. I question whether it makes any physical or even logical sense to posit something, using the category of "object" or "thing", that is not made of anything else, that is an irreducible or intrinsic object. It stretches the meaning of the word 'object' too far for me to find any meaning in it. And it is a contradiction in terms, I feel, that has been entertained for a long time.

Mike

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• 2 weeks later...

Hi all,

I was recommended Lee Smolin's book Three Roads to Quantum Gravity, and was wondering if I could get a few thoughts on a couple selections to clarify some issues.

"Perhaps it is hard to visualize space as something discrete. After all, why can something not be made to fit into half the volume of the smallest unit of space? The answer is that this is the wrong way to think, for to post this question is to presume that space has some absolute existence into which things can fit...we must put our minds completely into the relational way of thinking...the relationships define the space, not the other way around..." (p96)

"According to loop quantum gravity, space is made of discrete atoms each of which carries a very tiny unit of volume...a given region cannot have a volume which is arbitrarily big or small -- instead, the volume must be one of a finite set of numbers...as a result the volume of space is predicted to be quantized..."

"...If you tried to halve a region of this volume, the result would not be two regions each with half that volume. Instead, the process would create two new regions which together would have more volume than you started with."(p106)

My first question is, is Smolin's interpretation/definition of a discrete space pretty well agreed upon? What I mean to ask, is his definition of a discrete space essentially agreed upon and invariant, or are there other drastically opposed conceptions of a discrete space? In other words, when a physicist talks about 'discrete space', can it take on a fundamentally different picture than the one Smolin describes here?

My second question is simply if anyone can help clarify what Smolin is trying to say. It seems I can't help but fall into what he notes is the 'wrong way to think' about it.

Any help would very much be appreciated.

Thanks, and peace,

Mike

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I too believe in you and in my view not only space-time, whole thing in this universe is discrete.

for example:

consider our UNIVERSE when viewed as a whole it may seems continuous but actually it isn't continuous at all, all of its constituents are discrete like planets, moons, stars etc.

and in microscopic view also atoms are not also continuous all its constituents are discrete like electrons, protons and neutrons.

There is always a minimum limit beyond which one cannot have discreteness.

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A way to avoid conceptual problems is to consider the "discrete" concept as relative, not absolute. That is not the way established science works.

Although the "everything is relative" expression is widely used, when talking about dimensions, it is considered there exist a smallest absolute distance, the planck length. As far as i can understand (but many people here say I understand all wrong), there is no absolute smallest distance, but only relative smallest distance. So that the smallest "piece of space" must also be relative.

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• 3 weeks later...

I think that David Deutsch generally has the right approach to this question. In short, he thinks that “within each universe all observable quantities are discrete, but the multiverse as a whole is a continuum. When the equations of quantum theory describe a continuous but not-directly-observable transition between two values of a discrete quantity, what they are telling us is that the transition does not take place entirely within one universe. So perhaps the price of continuous motion is not an infinity of consecutive actions, but an infinity of concurrent actions taking place across the multiverse.”

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