Is space-time discrete or continuum?

[micromass]

What do we mean with "metric" here? A distance function that generates a metric space? Or a metric tensor?

 

[member 11137]

What would quantized geometry/gravity do to the topology of spacetime? Does QG destroy topology?

I have the temptation to ask the question in the converse direction: "Are all topologies" allowed by the GTR of A. Einstein?" If not, what are the acceptable solutions? Does the Cauchy problem (and its solutions) be the key ingredient for answering these questions? Does it not cost (quantized) energy to "go" from one allowed surface to the next one? Does the step between two successive allowed surfaces not be gouverned by the Heisenberg's uncertainty principle (energy x time) versus? Or with other words and, I am afraid at the boarder of science, does not this HUP tells us how the universe is growing and extending, looking its path in an unprobable future which nevertheless occurs?

Let me know if I am wrong; for me topos is related to the forms (thus the surfaces) and the metric is related to the science measuring the distances. We can measure distances, for example, on different surfaces.

 

[tom.stoer]

The question is whether we can answer the question "what it means that geometry is fundamentally discrete".

Suppose CDT is the correct description; it's a rather simple model using (discrete) triangulations of spacetime. Of course we can change the scale and look at spacetime at different resolutions, we can probe the triangulations, gravity and other interactions at different scales; we can calculate physical observables at different scales.

Now the following could happen: when changing scale and zooming to finer triangulations = to higher resolutions the physical answers we get become scale independent. That means that finer and finer triangulations do not have any effect on physical observables (below some "fundamental length").

So we make two observations
1) the theory allows for arbitrary small triangulations, i.e. it has a continuum limit
2) below some length scale physics doesn't change

1) means that the theory is not fundamentally discrete
2) means that it behaves as if it were fundamentally discrete

 

[friend]

The question is whether we can answer the question "what it means that geometry is fundamentally discrete".

Suppose CDT is the correct description; it's a rather simple model using (discrete) triangulations of spacetime. Of course we can change the scale and look at spacetime at different resolutions, we can probe the triangulations, gravity and other interactions at different scales; we can calculate physical observables at different scales.

Now the following could happen: when changing scale and zooming to finer triangulations = to higher resolutions the physical answers we get become scale independent. That means that finer and finer triangulations do not have any effect on physical observables (below some "fundamental length").

So we make two observations
1) the theory allows for arbitrary small triangulations, i.e. it has a continuum limit
2) below some length scale physics doesn't change

1) means that the theory is not fundamentally discrete
2) means that it behaves as if it were fundamentally discrete

I don't see how your description differs from a numerical computer method for solving a differential equation. Do numerical methods have anything to do with the underlying topology on which differential equations are formed?

 

[tom.stoer]

I don't see how your description differs from a numerical computer method for solving a differential equation. Do numerical methods have anything to do with the underlying topology on which differential equations are formed?

My description says that the model uses a discrete spacetime with length scale L, where
1) a continuum limit L→0 is possible, but where
2) a length scale L_min>0 exists below which no physical process can probe any smaller length scale L<L_min
My question to you is whether this means that spacetime is continuous b/c of (1) or whether it is discrete b/c of (2)

 

[friend]

My description says that the model uses a discrete spacetime with length scale L, where
1) a continuum limit L→0 is possible, but where
2) a length scale L_min>0 exists below which no physical process can probe any smaller length scale L<L_min
My question to you is whether this means that spacetime is continuous b/c of (1) or whether it is discrete b/c of (2)

I'd have to know more about what you mean by 2). What is meant by "no physical process can probe"?

 

[tom.stoer]

All I want to say is that it might happen that you have a description which uses continuous variables w/o minimal length, but from which a physical minimal length emerges, which is respected by all processes.

Another example: in LQG the fundamental variables are discrete, but still call for a continuum limit; the spectrum of the area operator is discrete, but this operator is not a physical observable; and as of today there is no proof that all physical observables probing length, area etc. have a discrete spectrum with some minimal length

So for me the relationship between quantization, discrete / continuous variables and discrete / continuous physical entities with / without minimal length is by no means obvious

 

[Natron]

i don't mean to be extremely simple here, but if Planck's constant is considered to discretely divide spacetime, then wouldn't there be a conundrum with gravitational forces over extreme distances? the obscure thing that comes to mind is imagine a complete vacuum of a universe that has 2 grains of sand (classical physical objects) but are placed 15 trillion light years apart, if spacetime is a continuum then these objects will affect each other, if spacetime is discrete, then either one of 2 things would happen, their gravitational forces against each other will be zero, or will be some sort of minimal constant that maintains regardless of the distance they are from each other.

 

[DimReg]

i don't mean to be extremely simple here, but if Planck's constant is considered to discretely divide spacetime, then wouldn't there be a conundrum with gravitational forces over extreme distances? the obscure thing that comes to mind is imagine a complete vacuum of a universe that has 2 grains of sand (classical physical objects) but are placed 15 trillion light years apart, if spacetime is a continuum then these objects will affect each other, if spacetime is discrete, then either one of 2 things would happen, their gravitational forces against each other will be zero, or will be some sort of minimal constant that maintains regardless of the distance they are from each other.

I imagine that since these would be quantum mechanical particles, they would always have some non-zero probability of propagating closer together, and this probability would increase as they get closer together. Further, I doubt that the "forbidden separations" would be positive potential regions, since two particles are falling down a potential gradient when they are gravitating together. That is, I don't think you could even think of the particles as "tunneling" to the next separation.

At this level, it would be difficult to apply your classical intuition.

 

[friend]

When you think of Hausdorff, you can think of "house" dorff. This is the property where each point in the topology can be enclosed in a neighborhood that does not include any other point you may choose. Even if you choose points very close together, you can always construct even smaller neighborhoods that exclude the other point you chose. No matter how close you choose the points, they each have their own little house to dwell in.

But if the metric is quantized and with it areas, then you cannot always construct a neighborhood that excludes a close point. The property of Hausdorff could not apply to such a space, and it would not be a manifold. Then since GR is constructed on manifolds, GR would not be applicable, right?

This all begs the question as to whether neighborhoods in topology are necessarily defined in terms of a metric. I mean the ususal description I've seen in textbooks is that neighborhoods are "balls" of radius r, and r is allowed to be any size. But the radius, r, is a measure of distance. So how do you escape the discrete area nature of a ball if the metric, which measures distance, is quantized? Can you define neighborhoods in topology without reference to a metric?

 

[micromass]

This all begs the question as to whether neighborhoods in topology are necessarily defined in terms of a metric.

They aren't. Please see a basic topology book such as Munkres.

I mean the ususal description I've seen in textbooks is that neighborhoods are "balls" of radius r, and r is allowed to be any size.

This is in a metric space.

Can you define neighborhoods in topology without reference to a metric?

Yes.

 

[Haelfix]

The question is whether we can answer the question "what it means that geometry is fundamentally discrete".

Suppose CDT is the correct description; it's a rather simple model using (discrete) triangulations of spacetime. Of course we can change the scale and look at spacetime at different resolutions, we can probe the triangulations, gravity and other interactions at different scales; we can calculate physical observables at different scales.

Now the following could happen: when changing scale and zooming to finer triangulations = to higher resolutions the physical answers we get become scale independent. That means that finer and finer triangulations do not have any effect on physical observables (below some "fundamental length").

So we make two observations
1) the theory allows for arbitrary small triangulations, i.e. it has a continuum limit
2) below some length scale physics doesn't change

1) means that the theory is not fundamentally discrete
2) means that it behaves as if it were fundamentally discrete

This is an excellent description.

 

[Natron]

my post was mainly based on a discrete minimal scale in which length, time, mass, and energy were based upon. if such a thing were to be considered, such as a minimal energy transfer on which every other transference of energy were a multiple of then we would have to reconsider a classical concept of gravity. let's say space/time was discrete, and that it progressed in small packets. and the advancement transference or adjustments or whatever could not be not on a smaller scale than these packets, then if the classical measurement of gravity were to stand, then at some distance, an object would have to fall below the threshold if minimal discreteness. for example if we determined that the smallest interval of anything were to be 1.0 m/s/kg/liters ^ 2 X 10^(-10000) [hypothetically speaking], then this could be transferred to a force measurement in which was claimed to be minimal. but since the classical definition of gravitational force between massive objects recedes based on distance, then mathematically it should be plausible that two objects can be placed at a distance in which the force enacted upon each other is below that discrete limit. so if spacetime is discrete, then the distance between massive objects should either 1- have a minimal gravitational force that extends throughout the cosmos regardless of distance equal to the minimal force within a discrete system, or 2 - jump to zero at some point and time. simply stating if there is a minimal number, our classical definition of gravity will eventually exceed that minimum regardless of what that number is, so we either have to accept that far flung objects are NOT affected by gravity (between themselves), or explain why they are despite their force being below the minimal barrier of a discrete system.

 

[friend]

They aren't. Please see a basic topology book such as Munkres.

Thank you, micromass. I think I will buy the paper back version of the book. But in the mean time, maybe you could give us a very brief definition of neighborhoods without use of the metric. For me it seems inescapable not to talk about some sort of size associated with neighborhoods, especially when considering concepts of continuity, where the neighborhood is allowed to strink in size to near zero, whatever that means without a metric.

As I recall, and it's been a while, a metric is an added structure to a topology. But once you define a metric on a topological space, it becomes impossible to talk about the size of neighborhoods of points without automatically saying something about their size in terms of the metric. So that if the metric were quantized, then so must be the neighborhoods, and one wonders what happens to the Hausdorff property of any manifold defined on that topology.

Perhaps this is worth a little more time since the thread is concerned with continuity, metrics, and the geometry used in physics.

 

[fanphysics]

The problem of quantisation of space and time is phylosophical problems and it will be solved when the basic principle is found which unifies the concepts of space and time.

 

[Fredrik]

maybe you could give us a very brief definition of neighborhoods without use of the metric.

There's a theorem about open sets in metric spaces that you may be familiar with. It says that if X is a metric space, the following statements are true:

(1) ∅ and X are open sets.
(2) Every union of open subsets of X is open.
(3) Every finite intersection of open subsets of X is open.

This theorem has inspired the following generalization. Let X be any set. A set $\tau$ whose elements are subsets of X is said to be a topology on X if the following statements are true:

(1) $\varnothing,X\in\tau$.
(2) Every union of elements of $\tau$ is an element of $\tau$.
(3) Every finite intersection of elements of $\tau$ is an element of $\tau$.

The pair $(X,\tau)$ is said to be a topological space if $\tau$ is a topology on X.

Suppose that $(X,\tau)$ is a topological space. A subset $E\subseteq X$ is said to be open if $E\in\tau$ and closed if $X-E\in\tau$.

Let $x\in X$ be arbitrary. There are at least two different definitions of "neighborhood of x" in the context of topological spaces:

1. A neighborhood of p is an open set that contains p.
2. A neighborhood of p is a set that contains an an open set that contains p.

 

[WannabeNewton]

As I recall, and it's been a while, a metric is an added structure to a topology. But once you define a metric on a topological space, it becomes impossible to talk about the size of neighborhoods of points without automatically saying something about their size in terms of the metric. So that if the metric were quantized, then so must be the neighborhoods, and one wonders what happens to the Hausdorff property of any manifold defined on that topology.

Topology doesn't care about size e.g. in topology one can show that the unit open ball $B^{n}\subseteq \mathbb{R}^{n}$ is homeomorphic to all of $\mathbb{R}^{n}$. That's the whole point of point-set topology: it removes the structure associated with metric spaces that gives us a notion of distance and size in the primitive geometric sense and instead just deals with neighborhoods in a more abstract sense. Also note that the metrics being spoken of in the context of space-times are pseudo-Riemannian metrics endowed on smooth manifolds, not metrics in the analysis sense. The two are completely different animals.

 

[tom.stoer]

... if the metric were quantized, then so must be the neighborhoods, and one wonders what happens to the Hausdorff property of any manifold defined on that topology.

As I tried to explain quantization (of the metric) does not necessarily lead to discretization (of the space, metric, ...); there are proposals with quantized but continuous gravitational field (in QM both x and p are quantized, i.e. they are operators, but nevertheless x is always continuous and p is only discrete for some eigenvalue problems; nevertheless the Hilberts space is a space of functions u(p) where p is a continuous variable); so again: quantizing the metric does not necessarily imply discretization.

But if there is discretization (either as a result of the quantization procedure or as a starting point put in by hand) then the usual topological properties will not survive. So what? Of course we expect that "quantum geometry" is different from classical one. Where's the problem?

 

[friend]

Topology doesn't care about size e.g. in topology one can show that the unit open ball $B^{n}\subseteq \mathbb{R}^{n}$ is homeomorphic to all of $\mathbb{R}^{n}$. That's the whole point of point-set topology: it removes the structure associated with metric spaces that gives us a notion of distance and size in the primitive geometric sense and instead just deals with neighborhoods in a more abstract sense. Also note that the metrics being spoken of in the context of space-times are pseudo-Riemannian metrics endowed on smooth manifolds, not metrics in the analysis sense. The two are completely different animals.

OK, so now we have two metrics to worry about and whether they are in any way connected to the size of neighborhoods. As I understand it, GR relies on the existence of an underlying manifold, and manifolds seem to rely on a continuous Euclidean metric, per wikipedia.org, which says,

A topological space X is called locally Euclidean if there is a non-negative integer n such that every point in X has a neighborhood which is homeomorphic to the Euclidean space Eⁿ (or, equivalently, to the real n-space Rⁿ, or to some connected open subset of either of two).[1]

A topological manifold is a locally Euclidean Hausdorff space.

A Euclidean space is a space with a Euclidean metric. And this Euclidean metric is continuous as indicated by the word "local". But it is not the pseudo-Riemannian metric of GR, since a pseudo-riemannian metric is not the Euclidean metric. All very confusing. What is the locally Euclidean metric on the manifolds associated with GR if not the pseudo-riemannian metric?

When you write,

in topology one can show that the unit open ball Bⁿ ⊆Rⁿ is homeomorphic to all of R

this only exacerbates the problem I have because it seems every time I read Rⁿ it always seems to be connected to the Euclidean metric. It would go a long way to clear things up for me if that distinction were made obvious with reliable sources.

 

[friend]

As I tried to explain quantization (of the metric) does not necessarily lead to discretization (of the space, metric, ...); there are proposals with quantized but continuous gravitational field...

I'm not aware of any metric quantization procedures that don't assume a result of a discrete spectrum. Maybe you could share some of these efforts with us.

... (in QM both x and p are quantized, i.e. they are operators, but nevertheless x is always continuous and p is only discrete for some eigenvalue problems; nevertheless the Hilberts space is a space of functions u(p) where p is a continuous variable); so again: quantizing the metric does not necessarily imply discretization.

I think we're talking about apples and oranges. There is quantizing fields on a background, and then there is quantizing the background itself. I'm concerned that trying to quantize the background will negate the validity of quantizing fields on the background.

Consider, a typical formulation in quantum mechanics is &lt; x|x&#039; &gt; \,\, = \,\,\,\delta (x - x&#039;). Can the Dirac delta function still be evaluated in a space with a quantized metric? I don't see it.

But if there is discretization (either as a result of the quantization procedure or as a starting point put in by hand) then the usual topological properties will not survive. So what? Of course we expect that "quantum geometry" is different from classical one. Where's the problem?

If you come up with a procedure that ultimately negates the premises, isn't that reducio ad absurdum?

 

[tom.stoer]

I'm not aware of any metric quantization procedures that don't assume a result of a discrete spectrum. Maybe you could share some of these efforts with us.

The Hawking at al. approach does not assume any discreteness; asymptotic safety approaches use a continuous metric; string theory and supergravity theories do use continuous fields for spacetime

 

[WannabeNewton]

OK, so now we have two metrics to worry about and whether they are in any way connected to the size of neighborhoods. As I understand it, GR relies on the existence of an underlying manifold, and manifolds seem to rely on a continuous Euclidean metric, per wikipedia.org, which says,
A Euclidean space is a space with a Euclidean metric. And this Euclidean metric is continuous as indicated by the word "local". But it is not the pseudo-Riemannian metric of GR, since a pseudo-riemannian metric is not the Euclidean metric. All very confusing. What is the locally Euclidean metric on the manifolds associated with GR if not the pseudo-riemannian metric?

Manifolds do not rely on any kind of metric. A smooth manifold is just a topological manifold with a smooth atlas; a topological manifold is a special kind of topological space. The locally euclidean property does not require a metric in any way; note that the property involves a homeomorphism which is an isomorphism in the category of topological spaces. Isomorphisms in the category of metric spaces are called isometries and these of course require a metric. It isn't your fault but you are confusing metrics from analysis with metrics from Riemannian geometry. The two are different. I don't blame you though because physicists tend to use the word metric when they really mean Riemannian metric. It's a horrible abuse of terminology but its rather ubiquitous. You must realize that the metrics from real analysis are in no way a priori related to Riemannian metrics. It is true that the topology of Euclidean space is usually defined as the one generated by the base of open balls of the Euclidean metric (the one induced by the 2-norm) but with regards to the locally Euclidean property we only care about the fact that topological manifolds are locally homeomorphic to Euclidean space i.e. we only want to check whether the topological space is locally topologically equivalent to Euclidean space. In this sense the metric space structure of Euclidean space is irrelevant to the locally Euclidean property, only its topological structure is of relevance.

If you want to properly learn the rudiments of point-set topology I would recommend Willard "General Topology". I'll post something a bit more detailed soon because I understand your confusion. Much of it arises from the hand-wavy mathematical language you see in most GR texts. If you want a rigorous and comprehensive account I would suggest Hawking and Ellis.

 

[friend]

Manifolds do not rely on any kind of metric.

I think I remember reading that somewhere. And that should be the end of the matter. But if I'm not mistaken, once you add the structure of a metric on the topological manifold, then you are also saying something about the manifold as well, that it has the characteristics of a metric space as well. Then it becomes impossible to talk about the open sets of the topology without also referring to the metric used to measure the size of those set, is this right?

If you have a metric on your manifold, then how can you have a quantized metric and a smooth point-set topology. Can a distance function only apply between some points in the topology but not between other points? A metric is not a metric if it does not give a distance to ANY pair of points in the topology. I thought manifolds were manifolds only because they are capable of having continuous coordinates imposed on them, which I think must mean that a continuous metric must be able to be defined on it. It would seem kind of arbitrary to have a metric that only applied between some points but not all points in the manifold.

I do appreciate the effort you're putting into this, wbn. I hope you understand that I'm really trying to get to the bottom of all this.

 

[tom.stoer]

If you have a metric on your manifold, then how can you have a quantized metric and a smooth point-set topology.

You are still confusing quantization and discreteness.

Suppose you have a topological manifold M with points P and some canonically conjugate functions A(P) and B(P). Then quantization means that you translate these functions A and B into operators, and that you have commutators

[A(P), B(P')] = δ_P,P'

with some delta-like functional on M.

This is the basic starting point for canonical quantization of fields A, B, ... on M.

Note that M does not vanish, nor does it become discrete.

The approaches I mentioned above use settings like this: canonical quantization or path integrals of fields on smooth manifolds.

 

[friend]

You are still confusing quantization and discreteness.

No, I accept that some efforts do not assume or calculate a discrete spectrum for the metric. I'm addressing those efforts that do.

And I accept that the manifold can be smooth but have different coordinate systems and different metrics defined on them. For example, in Special and General Relativity, different observers can calculate a different distance between the same two points of the underlying manifold.

What I don't understand is how you can have a discrete metric on a smooth manifold. I suppose you could have a distance function that gives the same answer for points that are near the two original points. E.g. suppose you have a distance function that gives the answer 4 length units for two points on a coordinate line that are at, say, (2,0,0) and (6,0,0) and also give 4 units for (2.1,0,0) and (5.8,0,0). That same distance function might also give 3 units between (2,0,0) and (5,0,0), and also give 3 units between (1.6,0,0) and (5.6,0,0). The question is: how do you assign a rule to decide where on the coordinate line you assign 3 units and where you start assigning 4 units.

 

[TrickyDicky]

What I don't understand is how you can have a discrete metric on a smooth manifold.

To understand this you just need to read any differential geometry text as adviced before.

But the important thing here is that such mathematical structure is not demanded in the standard current physical theories either quantized or not quantized , as Tom has said several times quantization doesn't imply a discrete topology.

 

[friend]

To understand this you just need to read any differential geometry text as adviced before.

Is this your idea of a reference? Could you kindly be more specific, please? What are the words or concepts I'd look up in a differential geometry book or on-line? I thought diff. geo. by definition involved continuous maps between coordinate patches so that you could define differentials to begin with. So how could I understand anything quantum mechanical in diff geo?

But the important thing here is that such mathematical structure is not demanded in the standard current physical theories either quantized or not quantized , as Tom has said several times quantization doesn't imply a discrete topology.

You misunderstand. I do not think that the underlying topology is discrete. I wonder how the metric could be discrete on a continuous topology. What exactly would that mean in those research programs that assume or calculate a discrete spectrum for the metric?

 

[WannabeNewton]

Correct me if I'm wrong but the thing being quantized is the space-time metric (in the sense that it is promoted to an operator field and quantized in the QFT sense) and not metrics in the analysis sense; a topological manifold can always be given some metric in the analysis sense but the proof of this is rather complicated-the proof for compact manifolds is simple if one uses Urysohn's lemma. Regardless, the existence theorem doesn't tell us explicitly with the metric actually is. There is no such thing as a "continuous" topology (note that discrete topology has a very strict definition: it is the finest topology that can be endowed on any set i.e. the power set of the original set).

 

[atyy]

Let me see if I can put tom.stoer's point in simple words.

Is a violin string discrete or continuous?

We'd probably say it is continuous.

However, when you strike a string, you get a definite note which consists of frequencies that are integer multiples of a fundamental frequency - so the frequencies are discrete even though the string is continuous.

In the same way, if you quantize spacetime, spacetime itself may be continuous, but the "notes" it gives off when struck may be discrete. In LQG, the "notes" would be the eigenvalues of the volume operator (or so it is hoped).

 

[friend]

Correct me if I'm wrong but the thing being quantized is the space-time metric (in the sense that it is promoted to an operator field and quantized in the QFT sense) and not metrics in the analysis sense; a topological manifold can always be given some metric in the analysis sense but the proof of this is rather complicated-the proof for compact manifolds is simple if one uses Urysohn's lemma. Regardless, the existence theorem doesn't tell us explicitly with the metric actually is. There is no such thing as a "continuous" topology (note that discrete topology has a very strict definition: it is the finest topology that can be endowed on any set i.e. the power set of the original set).

I suppose you could have more than one metric defined on the same manifold. That sounds like a somewhat arbitrary thing to do. As you say, there could be a metric assigned for analytic purposes and a spacetime metric. But they both result in a number given two points on the manifold. Is a metric still a metric if it results in the same number for various paris of points? Or does that contradict the definition of a metric?