Is space-time discrete or continuum?

[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?

 

[micromass]

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?

Sure, you can have various pairs of points which all lie in the same distance of each other. For example, we always have $d(p,p)=0$, so a point always lies within distances $0$ of itself. And this is true for any point.

It's true that a manifold admits many metrics. There is no metric that is the most natural. So yes, assigning a certain metric to the manifold is rather arbitrary.

If you're dealing with a Riemannian manifold however, then you can always find a metric that is most natural.

 

[WannabeNewton]

friend, a space-time metric is completely different from an analytic metric. A space-time metric allows you to compute inner products between vectors in each tangent space to the manifold at a given point. An analytic metric allows you to find the distance between different points on the manifold with respect to that metric. The input for a space-time metric is a specific point on the manifold and two vectors in the tangent space to the manifold at said point; the result is the inner product of these two vectors. The input for an analytic metric are two points on the manifold and the result is the distance between these two points as defined by that metric.

In the case of connected Riemannian manifolds, one can use the Riemannian metric to define a metric in the analysis sense: http://en.wikipedia.org/wiki/Riemannian_manifold#Riemannian_manifolds_as_metric_spaces_2 but note that this is for Riemannian and not pseudo-Riemannian manifolds.

 

[friend]

friend, a space-time metric is completely different from an analytic metric. A space-time metric allows you to compute inner products between vectors in each tangent space to the manifold at a given point. An analytic metric allows you to find the distance between different points on the manifold with respect to that metric. The input for a space-time metric is a specific point on the manifold and two vectors in the tangent space to the manifold at said point; the result is the inner product of these two vectors. The input for an analytic metric are two points on the manifold and the result is the distance between these two points as defined by that metric.

In the case of connected Riemannian manifolds, one can use the Riemannian metric to define a metric in the analysis sense: http://en.wikipedia.org/wiki/Riemannian_manifold#Riemannian_manifolds_as_metric_spaces_2 but note that this is for Riemannian and not pseudo-Riemannian manifolds.

Can you give examples of a degenerate metric, whether on the manifold or on the tangent space of it? What examples are there of a metric giving the same distance or the same inner product for basically an infinite number of nearby points or vectors, whatever close means with a degenerate metric?

 

[tom.stoer]

Suppose you have two light-like, orthogonal vectors, i.e. <x,x> = <y,y> = <x,y> = 0; then you also have <λx,λx> = <μy,μy> = <λx,μy> = 0 for arbitrary constants λ,μ. The inner product is defined in terms of coordinates <x,x> = x_ax^a = g_abx^ax^b

 

[WannabeNewton]

A metric tensor is non-degenerate by definition. The difference between a Riemannian metric and a pseudo-Riemannian one is that the latter is not positive definite. This leads to things like non-zero vectors having vanishing "norm" (such vectors are called null vectors/lightlike vectors).

 

[micromass]

A metric tensor is non-degenerate by definition. The difference between a Riemannian metric and a pseudo-Riemannian one is that the latter is not positive definite. This leads to things like non-zero vectors having vanishing "norm" (such vectors are called null vectors/lightlike vectors).

And although I risk to sound repetitive, I want to add that:

- A metric tensor (synonym: Riemannian metric) is defined on the tangent spaces. It is an inner product on the tangent space $T_pM$ that varies smoothly from tangent space to tangent space

- A distance function (also called a metric in analysis) is defined on the manifold itself. It is not an inner product. It simply is a function giving you the distance between two points.

The difference between these two concepts is crucial. Please try to understand it well. Not understanding this has lead to a lot of confusions in the past.

 

[friend]

Suppose you have two light-like, orthogonal vectors, i.e. <x,x> = <y,y> = <x,y> = 0; then you also have <λx,λx> = <μy,μy> = <λx,μy> = 0 for arbitrary constants λ,μ. The inner product is defined in terms of coordinates <x,x> = x_ax^a = g_abx^ax^b

OK, it seems all vectors on the light-cone have zero norm. This is not the same as a discrete spectrum, where zero in one possible values along with other possible values.

 

[tom.stoer]

There is absolutely no relation between the pseudo-norm ||x||² = <x,x> has absolutely nothing to do with eigenvalues.

 

[friend]

There is absolutely no relation between the pseudo-norm ||x||² = <x,x> has absolutely nothing to do with eigenvalues.

If the metric is quantized, then the inner product is quantized, right?

 

[tom.stoer]

You still confuse quantized and discrete

Quantized means that you are using quantum mechanics.

In QM:
- momentum is always quantized and sometimes it's discrete
- angular momentum is always quantized and discrete

In QG approaches
- the metric (or some other structure related to a manifold) is always quantized
- in LQG the manifold is replaced by a discrete structure during quantization
- in LQG length, area and volume will probably have discrete eigenvalues; this is not clear b/c the operators with discrete spectrum are no observables; other operators which are observables are not known afaik
- in other approaches with quantized metric but continuous manifold I do not know what happens to these eigenvalues; in these approaches the manifold is not replaced by a discrete structure
- in CDT the manifold is replaced by a discrete structure before quantization; length, area and volume are discrete b/c of discretization (trivially) w/ and w/o quantization
- whether the spectrum of physical observables remains discrete after the discretization of CDT is removed is unclear to me.

So even you are by precise regarding the question not all details are known; QG is work in progress.

 

[ryan albery]

If space-time has discreet values, or intervals- there's no way we have to measure those intervals. Every measurement we conduct (even weighing something depends on the relative passing of time) is based, fundamentally, upon time.

 

[tom.stoer]

If space-time has discreet values, or intervals- there's no way we have to measure those intervals. Every measurement we conduct (even weighing something depends on the relative passing of time) is based, fundamentally, upon time.

Perhaps I should add that given a quantum mechanical observable O does not imply that we know how to construct a measurement device for O. It simply means that O represents a quantity measurable in principle; how to measure it in practice cannot be derived from O.

 

[friend]

Perhaps I should add that given a quantum mechanical observable O does not imply that we know how to construct a measurement device for O. It simply means that O represents a quantity measurable in principle; how to measure it in practice cannot be derived from O.

Perhaps it would be instructive to tell us how in principle one would measure a discrete metric in quantum gravity. Or for that matter what it means and how we would measure, even in principle, ANY quantum nature of quantum gravity. I find myself not really understanding what that's supposed to mean.

 

[tom.stoer]

I do not see any direct way to measure space-time discreteness. But there are indirect methods, namely to measure effects induced by discreteness, especially violation or deformation of local Lorentz invariance, i.e. a corrections to E² = p² + m². For light propagation this means that speed of light propagation could become frequency-dependent. Experiments have ruled out these corrections up to a certain order.

 

[DimReg]

I do not see any direct way to measure space-time discreteness. But there are indirect methods, namely to measure effects induced by discreteness, especially violation or deformation of local Lorentz invariance, i.e. a corrections to E² = p² + m². For light propagation this means that speed of light propagation could become frequency-dependent. Experiments have ruled out these corrections up to a certain order.

I imagine you know this already, but it's worth pointing out for general audiences that not all theories with discreteness predict local lorentz invariance violation. As far as I know, LQG these days is believed to be local lorentz invariant.

 

[tom.stoer]

I imagine you know this already, but it's worth pointing out for general audiences that not all theories with discreteness predict local lorentz invariance violation. As far as I know, LQG these days is believed to be local lorentz invariant.

Yes, I agree, this is an important remark.

One must not confuse discreteness with a kind of fixed lattice structure or something like that. The main differences are that
1) spacetime discreteness may allow for dynamical creation and annihilation of "spacetime atoms"
2) spacetime becomes subject to "superpositions of spacetime states" in quantum gravity

This means that spacetime discreteness does not necessarily violate the quantum version of the continuous classical symmetries.

 

[friend]

Yes, I agree, this is an important remark.

One must not confuse discreteness with a kind of fixed lattice structure or something like that. The main differences are that
1) spacetime discreteness may allow for dynamical creation and annihilation of "spacetime atoms"
2) spacetime becomes subject to "superpositions of spacetime states" in quantum gravity

This means that spacetime discreteness does not necessarily violate the quantum version of the continuous classical symmetries.

Thank you Tom. Your efforts are appreciated.

If energy is quantized and mass of particles are quantized, then it stands to reason that curvature calculated from GR for that energy is quantized, at least in the rest frame of those particles.

 

[tom.stoer]

But energy is usually NOT quantized; action is quantized, and E=nhf is quantized in terms of number of photons n; but frequency is NOT quantized in general, only for specific systems and emission / absorption processes.

 

[ryan albery]

Interesting to contemplate the presence of big G in the equation of Newton's gravity (along with Einstein's) as being analogous to Planck's constant with the quantization of 'things'.

 

[friend]

How dependent is quantum gravity research on the assumption that Newton's constant, G, and Planck's constant h remains constant as the region of interest approaches the Planck scale? And what proof is there that these values don't change with very small scales?

 

[tom.stoer]

Regarding G (and Λ) expectation is that they become scale-dependent.

Asymptotic safety is an approach which tries to quantize gravity based on the assumption of smooth spacetime plus UV completeness using renormalization group theory http://www.percacci.it/roberto/physics/as/faq.html